Paper Title: Enhanced Interlocking in Granular Jamming Grippers through Hard and Soft Particle Mixtures.




Learning Ordinality in Semantic Segmentation
    Rafael Cristino^1 Ricardo P. M. Cruz^1,2* Jaime S. Cardoso^1,2*
^1 Faculty of Engineering, University of Porto
^2 INESC TEC, Porto
 rpcruz@fe.up.pt, jaime.cardoso@fe.up.pt
    
===============================================================================================================================================================================



§ ABSTRACT 


We investigate the influence of particle stiffness on the grasping performance of granular grippers, a class of soft robotic effectors that utilize granular jamming for object manipulation. Through experimental analyses and X-ray imaging, we show that grippers with soft particles exhibit improved wrapping of the object after jamming, in contrast to grippers with rigid particles. This results in significantly increased holding force through the interlocking. The addition of a small proportion of rigid particles into a predominantly soft particle mixture maintains the improved wrapping but also significantly increases the maximum holding force. These results suggest a tunable approach to optimizing the design of granular grippers for improved performance in soft robotics applications.







§ INTRODUCTION



Granular grippers are soft robotic effectors that exploit granular jamming to grip objects <cit.>. These grippers are highly adaptable and can easily manipulate objects of diverse geometry and surface properties without the need for reconfiguration between gripping cycles<cit.>. 
The typical design of a granular gripper consists of a granulate that is held in place by a flexible, hermetically sealed membrane.


In this state, the granular material easily flows when deformed, similar to a fluid. When pressed against an object, the gripper deforms and adapts to the object's contour. 
When the air is pumped out of the gripper, the membrane contracts and compresses the particles. This causes the granulate to jam and assume a mechanically robust, solid-like state characterized by a pronounced modulus of elasticity. The stiffness of the system increases significantly in the jammed state <cit.>. As the material jams, the gripper exerts force on the target object, that allows it to be gripped and held effectively. To release the object, the vacuum is broken, causing the membrane to relax and the granules to return to its deformable state.

The holding force that a granular gripper generates when engaging with an object is influenced by the mechanical characteristics of the target object. It emerges from the interplay of three distinct mechanisms<cit.>: static friction, suction, and geometrical interlocking. Friction arises from tangential stress at the contact between the object and the gripper membrane <cit.>. Suction is activated when the membrane seals the object's surface airtight <cit.>. When the gripper is retracted, the pressure within the sealed gap decreases, resulting in an attractive force on the object. Interlocking occurs when the membrane encloses the object (or any of its protrusions), creating geometric constraints with the object as the granulate hardens, effectively securing the grip <cit.>.

In granular grippers, the hoisting forces due to interlocking and suction are considerably larger than those produced through friction, with interlocking being the most efficient mechanism <cit.>. Thus, in cases lacking suction and interlocking, the grippers holding capability relies solely on friction, resulting in notably reduced performance. Strengthening the locking mechanism is, therefore, highly desirable. However, the contribution of interlock to the hoisting force depends significantly on the object's shape and the mechanical characteristics of the granulate. The granular material needs to flow and adapt to the object's contours, thereby establishing geometric constraints that improve the grip as jamming occurs.

To disengage interlocking, the gripper must flexibly relax, allowing the gripped object to be released smoothly. It was suggested that the peak hoisting force generated by interlocking is directly related to the strength of the granular material in its jammed state. This strength is specifically related to the maximum stress the jammed granular material can sustain under bending and stretching <cit.>. In this sense, it was shown <cit.> that the resistance to bending of a simply supported beam made of jammed granular matter encased in a membrane increases with the particle stiffness. Consequently, the use of rigid particles is anticipated to significantly enhance the interlocking capability of granular grippers. 

Brown et al. <cit.> measured the hoisting force of a gripper filled with glass beads when gripping a sphere. They found that the contribution of interlocking to the hoisting force increases with the extent to which the gripper envelops the object, quantified by the contact angle. In their experiments, geometrical interlocking was achieved by manually molding the gripper in the unjammed state around the object. Such manual manipulation of the gripper is impractical for real-world applications, like handling objects in factory automation.

In prior research <cit.>, we demonstrated that, unlike rigid particles, incorporating soft particles in granular grippers results in a distinct squeezing effect. Specifically, the gripper exerts significant compression on the object following the induction of jamming. This effect causes substantial normal and, consequently, frictional forces, contributing to large hoisting capability. However, the influence of particle elasticity on the interlocking mechanism remains unexplored.

In the present work, we show that soft particles, as opposed to rigid ones, significantly enhance the hoisting forces of granular grippers through interlocking. Utilizing X-ray computed tomography (CT), we reveal that grippers filled with soft particles undergo a notable reduction in volume compared to those with rigid particles. This reduction allows the gripper to more effectively envelop object protrusions or convex surfaces. Introducing a small proportion of rigid particles into a predominantly soft particle-filled gripper substantially enhances interlocking. X-ray CT scans reveal that the gripper's volume reduction is caused by soft particles, which, in turn, facilitates the displacement of the more rigid particles around the object's protrusions, largely increasing the force required to break the interlock.



§ MATERIALS AND METHODS




 §.§ Experimental setup and measurement procedure


Figure <ref> sketches the experimental setup: The granular gripper consists of a spherical elastic, air-tight bag with diameter (70.0 ± 0.5) mm, filled with granulate. Two types of beads are used for the experiments: glass beads of diameter (4.0 ± 0.3) mm and expanded polystyrene (EPS) beads of diameter (4.2 ± 0.5) mm. The gripper bag is filled with (200 ± 1) cm^3 of grains and is attached to a holder. A flange on the holder's top allows air pressure or vacuum to be applied within the gripper. Positioned beneath the gripper, the target object rests on a platform that can move vertically. The base's vertical position can be adjusted via two stepper motors driven by a microcontroller. A load cell, placed between the object and the platform, measures the force exerted on the target object.


We examine the gripping process involving a cylinder-symmetric object of a T-shaped cross-section, as depicted in Fig. <ref>. The specific shape of the object enables the gripper to envelop and interlock with it. The object consists of two cylindrical sections: a lower cylinder of diameter (14.0 ± 0.1) mm and height (44.0 ± 0.1) mm, and an upper flat cylinder of diameter (23.9 ± 0.1) mm and height (3.5 ± 0.1) mm. To inhibit the suction effect, the object's surface was intentionally roughened, thus, no air tight cavities could develop.

We studied the gripping performance using granulates of different compositions, namely, (a) glass beads, (b) EPS particles, and (c) different binary mixtures of both (fraction of glass beads: 

0.05, 0.1, 0,15, 0.20, 0.25, 0.5, and 0.75). 


Before each measurement, the gripper is pressurized to fluidize the enclosed granulate, effectively resetting any memory of previous gripping cycles. Each measurement comprises the following steps:

    
  (i)  Initially, the platform is raised, pressing the object against the gripper, which molds to the object's shape. Once a predetermined indentation depth is achieved, the upward motion of the platform halts.
    
  (ii) The granulate is then given a brief period to settle, during which the particles adjust and realign. 
    
  (iii) The air is evacuated from the gripper, creating a pressure differential between the gripper's interior and the external atmosphere, p_vac≈ 90 kPa. This pressure difference causes the membrane to tighten and compress the granulate, triggering jamming. The specific pressure value chosen corresponds to the maximum attainable by the vacuum pump. It was shown that large differential pressure corresponds to large values of the hoisting force of granular grippers <cit.>.
    
  (iv) Finally, the platform is lowered until the object is no longer in contact with the gripper. The force applied to the object is continuously recorded throughout the gripping process.  In particular, we consider the peak hoisting force attained by the gripper.
    



 §.§ X-ray computed tomography

The hoisting force generated by the gripper reflects the macroscopic response of the gripper due to the internal motion of the particles during the gripping process. To relate this macroscopic response to the rearrangement of the particles in the course of the gripping process, the granular packing post-jamming is examined using X-ray computed tomography (CT).

CT scans are conducted with a high-capacity laboratory X-ray scanner, accommodating the full experimental setup. The parameters chosen for the CT scans are detailed in Table <ref>. To mitigate beam hardening effects, a copper plate of 1.5 thickness is used as a filter <cit.>. The granular packing's three-dimensional (3D) structure is then reconstructed using the software XrayOffice (v2.0).




§ RESULTS AND DISCUSSIONS

We consider the impact of particle stiffness on the hoisting force of a granular gripper. Figure <ref> shows the peak hoisting forces of a gripper filled with either glass or EPS (Expanded Polystyrene) beads. The maximum holding force attained with EPS is considerably larger (∼3.5 times) than for glass beads, highlighting the enhanced role of interlocking when using EPS particles.

At first glance, the result appears counterintuitive. If geometrical constraints are formed between the gripper and the target object in both cases – glass beads and EPS beads – one would expect a stronger hoisting force for more rigid particles. This expectation follows from the logic that rigid particles confer greater stiffness to the gripper, which requires greater forces to deform the gripper sufficiently for object release <cit.>. 


To elucidate the enhanced hoisting force observed with soft particles, we examine the granular packing within the gripper at various stages of the gripping process. Tomograms are captured at two critical junctures: (1) after the gripper has molded around the object before evacuating the gripper and (2) after the evacuation and subsequent jamming. Figure <ref> shows vertical sections through the granular packing inside the gripper for both glass beads (Fig.<ref>(a,c)) and EPS beads (Fig.<ref>(b,d)), before and after the air is evacuated. For EPS particles, the volume of the packing reduces drastically (≈ 28 %), which enables the gripper to closely encase the object's protrusions, facilitating geometrical interlocking. In contrast, for glass beads, we see negligible volume change after evacuation (less than 1%). Consequently, in the process of jamming, the gripper hardly conforms around the object's protrusions, impeding interlocking. 

The hoisting force of a gripper filled with a mixture of hard gall and soft EPS beads can substantially exceed the values of pure glass and pure EPS beads. Figure <ref> shows the maximum holding forces achieved by a gripper filled with mixtures of glass and EPS beads at different fractions. 

The peak force, approximately 28 N, is attained at a 0.1 volume fraction of glass beads, highlighting the substantial role of interlocking to the hoisting force. Remarkably, the peak force for a mixture exceeds the values for pure granulate considerably, about six times for pure glass beads (Fig. <ref>) and two times for pure EPS beads (Fig. <ref>). 
For glass concentration ≥ 0.25 (Fig. <ref>), the hoisting force is ≈ 5 N, independently of the fraction of glass beads. From these results, we conclude that soft particles support the interlocking, while hard particles provide the final force.  

To explain the increase of the hoisting force, we use X-ray imaging. 

Figure <ref> shows tomogram slices through the gripper filled with a mixture of glass and EPS beads with a glass fraction  of 0.1 before and after the evacuation of the air. After evacuation, the gripper's volume reduces considerably, allowing it to embrace the object's protrusions, facilitating interlocking. Since glass beads are closer to the object, the gripper shrinkage causes the displacement of the more rigid particles around the object's edges. Therefore, geometrical interlocking is predominantly achieved by rigid particles, which significantly increases the force needed to break the interlock.






§ SUMMARY AND OUTLOOK


We studied the hoisting force of a granular gripper and find that it strongly depends on the composition of the granulate. For a mixture of soft and hard particles, the hoisting force exceeds the values obtained for pure granulates, hard or soft. The maximum force is achieved for a 10% volume fraction of hard beads. 

This increase is explained by the interlocking mechanism in granular grippers, which has previously been identified as the main contribution to the hoisting force <cit.>. While soft particles do not contribute much to the strength of the jammed granulate as a meta-material <cit.>, they enable the granulate to shrink in volume during the evacuation of the gripper. This shrinkage corresponds to a non-negligible motion of the particles, which in turn allows the granular gripper to better embrace the object. Both mechanisms, shrinkage and embracement by the dislocation of particles have been evidenced through X-ray CT tomography. 



Our findings can advance the development of robotic grippers by addressing their greatest weakness: the hoisting force, which has limited their practical use until now.




We gratefully acknowledge funding by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation)–Project Number 411517575. The authors thank Walter Pucheanu for technical support.








§ COMPLIANCE WITH ETHICAL STANDARDS


The authors declare that they have no conflict of interest.





unsrtnat

   



 Paper Title: Two-stage assembly of patchy ellipses: From bent-core particlesto liquid crystal analogs.

 





 
 



^1Department of Physics, Indian Institute of Technology Delhi, New Delhi 110016, India.
^2School of Chemistry, University of Birmingham, Birmingham B15 2TT, UK.
^3Institut für Theoretische Physik, TU Wien, Wiedner Hauptstraße 8-10, A-1040 Wien, Austria
^4CNR-ISC, Uos Sapienza, Piazzale A. Moro 2, 00185 Roma, Italy.





§ ABSTRACT 
We investigate the two-dimensional behavior of colloidal patchy ellipsoids specifically designed to follow a two-step assembly process from the monomer state to mesoscopic liquid-crystal phases, via the formation of so-called bent-core units at the intermediate stage. Our model comprises a binary mixture of ellipses interacting via the Gay-Berne potential and decorated by surface patches, with the binary components being mirror-image variants of each other - referred to as left-handed and right-handed ellipses according to the position of their patches. The surface patches are designed so as in the first stage of the assembly the monomers form bent-cores units, i.e. V-shaped dimers with a specific bent angle. The Gay-Berne interactions, which act between the ellipses, drive the dimers to subsequently form the characteristic phase observed in bent-core liquid crystals. We numerically investigate – by means of both Molecular Dynamics and Monte Carlo simulations – the described two-step process: we first optimize a target bent-core unit and we then fully characterize its state diagram in temperature and density, defining the regions where the different liquid crystalline phases dominate.


Two-stage assembly of patchy ellipses: From bent-core particles to liquid crystal analogs
    Varsha Banerjee^1
    August 5, 2024
=========================================================================================







§ INTRODUCTION



During the past years bent-core liquid crystals (BCLCs) have attracted a rapidly increasing interest both in experimental and theoretical investigations <cit.>. The constituent units of these systems are bent-core (BC) colloidal entities (typically ranging in size from nm to μm) that have a (sharp or smooth) bent shape. This characteristic shape can be realized in different ways, for instance by merging two ellipsoids (or sphero-cylinders) via their tips at a (possibly flexible) angle <cit.> or by creating bead-resolved, bent strings of spherical particles (often referred to as “banana-shaped” colloids) <cit.>. The reason for the considerable interest in such BC particles is the large variety of (ordered) phases that these entities can form: fascinating chiral nematic, smectic, splay, and twist-bend phases (despite the achiral nature of the BC constituents) are just a few examples. 


Here we propose a model system where the BC units result from the spontaneous assembly of suitably designed particles characterized by shape and bond anisotropy, namely we design anisotropic, patchy colloids such that they form V-shaped dimers. To introduce shape anisotropy, we choose – in our two-dimensional setup – elliptic particles, which interact via the well-established Gay-Berne (GB) potential <cit.>. The relevant parameters that characterize this interaction are the aspect ratio of the ellipses and their interaction range. The other ingredient to our model is the patchiness of the colloids <cit.>: patchy colloids have surfaces decorated with well defined regions (both in terms of position and of spatial extent) that support a different interaction behaviour with respect to the bare surface. Thus, depending on their mutual interaction properties these regions can either foster or suppress patch-patch directional bonding. Patchiness has become a widely used standard tool (both in experiments and in theoretical/simulation-based investigations) to trigger in bottom-up processes the self-assembly of decorated particles into highly complex bonding patterns, ranging from liquid-crystalline mesophases, over complex ordered structures and quasi-crystalline lattices to structural glasses <cit.>.

In the present contribution we describe in detail the process that starts from the formation of BCs via the self-assembly of elliptic patchy colloids and ends up in the self-organization of these entities into BCLCs. Our design of the elliptic patchy particles envisages a binary mixture of mirror-symmetric units: while both species have the same elliptic shape, the two types of particles are mirror symmetric with respect to their patch decoration (leading thus to left- and right-handed particles). In fact, the two patches (per particle) are located close to the tip of the colloid. The form of the patch-patch interaction is based on the work of Russo et al. <cit.>, while the choice of the position, interaction range and interaction strength of the patches influences the features of the BC units.
In a first step we demonstrate that our particles are indeed able to self-assemble via the related patch-patch bonding mechanism into stable BC entities; we demonstrate that our model is characterized by a high versatility, as the bonding angle can be tailored via the patch positions and/or the patch-patch interaction strength. The impact of all these parameters on the actual shape, stability and yield of these units is studied in detail. In a second step we consider an ensemble of such BCs and study their self-assembly into mesophases. Focusing for the moment on a particular set of model parameters we are able to trace out in a qualitative manner a diagram of states (in the temperature-density plane), which shows the emergence of isotropic and smectic phases.  All investigations have been carried out via extensive computer-simulations: we have applied both molecular dynamics (MD) and – in a complementary manner – Monte Carlo (MC) simulations, using different types of thermodynamic ensembles. As the proposed BC model can be easily extended to support BCs with different sets of features, it reveals itself as a powerful tool to gain a deeper insight into this fascinating class of functional materials with potential applications in optics, photonics, and sensing. The rest of our paper is organized as follows. Sec. <ref> provides the theoretical framework which includes the model and the methodology to characterize the bent-cores. The computational details of the MD and MC procedures are provided in Sec. <ref>. The results and discussion follow in Sec. <ref>. Finally, Sec. <ref> provides the summary and conclusion.  




§ THEORETICAL FRAMEWORK




 §.§ Model


We want to design a simple model in which colloidal monomers form a bent-core (BC) entity first, which then self-assemble into mesoscopic liquid crystalline phases. Such a targeted design process would allow for exploring a large number of phases which emerge by changing the shape as well as the surface chemistry of the colloidal monomers. 

To simulate such a system we use a simple Gay-Berne (GB) potential <cit.> for the elliptic colloidal monomers which are decorated on their surface by suitably placed patches (index `P'). The form of the patch potential is adopted from the work of Russo et al. <cit.>. Thus the interaction potential between a pair of particles i and j  is given by (where we have suppressed for the moment the arguments)


    V_ij = V_ GB + V_ P,


where V_ GB is the Gay-Berne potential acting between the centers of mass of the two particles, and V_ P is the interaction potential due to all pairs of patches located on the surfaces of particles i and j.

The GB potential V_ GB characterizes the shape-anisotropy of the elliptical particles and is defined as follows:

    V_GB(û_i, û_j, r⃗_ij) = 4ϵ(û_i, û_j,r̂_ij)[(σ_0/r_ij - σ(û_i, û_j, r̂_ij)+σ_0)^12. 
    . - (σ_0/r_ij - σ(û_i, û_j, r̂_ij)+σ_0)^6],


where σ_0 is the unit distance of the interactions.  r⃗_ij is the vector connecting the centers of mass of the particles and û_i and û_j are the unit vectors representing the orientations of the particles as shown in Fig. <ref>(a). ϵ = ϵ_0 ϵ_1^μϵ_2^ν is the orientation dependent interaction strength, where ϵ_0 represents the energy unit of the interactions; the parameters μ and ν modify the well depth of the potential; σ is the anisotropic contact distance. Further ϵ_1, ϵ_2 and σ are defined as:

    ϵ_1(û_i, û_j, r̂_ij)    = 1 - χ'/2{(r̂_ij.û_i + r̂_ij.û_j)^2/1+χ'û_i . û_j+(r̂_ij.û_i - r̂_ij.û_j)^2/1 - χ'û_i . û_j}, 
    ϵ_2(û_i, û_j)    = [1-χ^2(û_i . û_j)^2]^-1/2,
    σ(û_i, û_j, r̂_ij)    = σ_0[1-χ/2{(r̂_ij.û_i + r̂_ij.û_j)^2/1+χû_i . û_j+(r̂_ij.û_i - r̂_ij.û_j)^2/1 - χû_i . û_j}]^-1/2.

The energy anisotropy parameter χ' and the shape anisotropy parameter χ are defined as:

    χ'    = (κ'^1/μ-1) / (κ'^1/μ + 1), 
    χ   = (κ^2-1) / (κ^2 + 1),


where κ' = ϵ_s / ϵ_e is the energy anisotropy with ϵ_s and ϵ_e being the well depths of the side-to-side and end-to-end configurations. κ = σ_∥ / σ_⊥ is the aspect ratio of the elliptic particles with σ_∥ and σ_⊥ being the major and minor axes of the particles. The GB potential therefore contains four key parameters to describe interacting elliptic particles: κ, κ', μ, and ν. There exist a number of GB homologues in the context of nematic LCs depending on the values chosen for the four parameters  <cit.>. We will discuss the appropriate choice for obtaining the BCs and their LC-like self-assembly shortly.

The patch-patch interactions V_ P are described by a sum of inverted Gaussian potentials <cit.>:

    V_ P(d) = - ∑_k ∈ P_i∑_l ∈ P_jϵ_klexp[-1/2(d_kl/α)^2],


where ϵ_kl is the strength of the interaction between patches k and l, P_i and P_j are the set of patches on the particles i and j, respectively, and d_kl is the distance between the centers of patches k and l, which decorate the surface of the elliptical particles, as shown in the schematic of Fig. <ref>(a). The width of the inverted Gaussian, α, defines the range of the patch-patch interactions. In addition to being able to compute gradients required for molecular dynamics (MD) simulations, this choice of the patch potential also allows for oblique patch placements, i.e., at an angle from the tip of the ellipses. Such placements promote the assembly of stable BCs. 

Let us consider the schematic in Fig. <ref>(a) that shows ellipsoidal particles with two patches placed obliquely; the reason for a two-patch model will be clear soon. In our model there are two kinds of patch placements on the ellipsoids: one to the “left” (red particles) and the other to the “right” (green particles). The green particles have patches G_1 and G_2 at angles -θ_1 and -θ_2 from the major axis. Similarly, the red particles have patches R_1 and R_2 at angles θ_1 and θ_2 from the major axis. The interaction strengths ϵ_G_1 R_1 and  ϵ_G_2 R_2 are set equal to ϵ_p, while all other interactions are set to 0: these include notably ϵ_G_1G_1, ϵ_G_1G_2, ϵ_R_1R_1, ϵ_R_1R_2, ϵ_G_1 R_2 and ϵ_G_2 R_1. We consider a pair of left-handed and right-handed particles to be a BC unit if the patch interaction strength V_p is 
sufficiently large to prevent dissociation of this arrangement by thermal fluctuations. In our simulations, we find that a choice of V_p < -3k_ BT  (T being the temperature and k_ B being the Boltzmann constant) ensures that the BCs once formed remain bonded.  

Fig. <ref>(b) shows a schematic of a BC colloid corresponding to the above model description. This entity is characterized by the director n̂, the polarization p̂, and the bend angle γ (as indicated). In our investigations we have found that single patch particles are characterized by a broad distribution of the bend angles, as shown in Fig. <ref>(c). Reducing the width of the  angle distribution would require lowering the value of α, which, as we shall see later, negatively affects the kinetics of BC self-assembly. Further, multiple single-patch ellipses can adhere in clusters (or asters). 
Having two patches on the other hand results in smaller magnitudes of the bend angle and lesser flexibility.  
Thus the two-patch ellipsoids represent an optimal choice to obtain BCs at a high yield. 
The model parameters θ_i, ϵ_p, and α have a significant impact on the BC yield and phases. We will study their influence in the rest of the paper.




 §.§ Methods





  §.§.§ Bent-core characteristics


Let us refer to the schematic in Fig. <ref> that depicts a pair of left-handed and right-handed patchy ellipsoids. Let their center of mass, after the formation of the BC, be represented by r⃗_ cm^ l and r⃗_ cm^ r, respectively. The center of mass of the BC unit is given by the vector average:


    R⃗_ cm = r⃗_ cm^ l + r⃗_ cm^ r/2.


Further, the local director n̂ and the polarization p̂ can be expressed in terms of the orientations u⃗_l and u⃗_r as follows:


    n̂= u⃗_ l-u⃗_ r/|u⃗_ l-u⃗_ r|,   p̂ =u⃗_ l+u⃗_ r/|u⃗_ l+u⃗_ r|,


Similarly, the bend angle γ can also be defined in terms of the orientation vectors as follows:


    γ=cos^-1( u⃗_ l·u⃗_ r/|u⃗_ l||u⃗_ r|).


As discussed in the context of Fig. <ref>(b), the BC unit is characterized by n̂, p̂, and γ. We will use Eqs. (<ref>) and (<ref>) for the evaluation of these quantities in all our simulations. 



  §.§.§ Q-Tensor



To characterize the global orientational order of the mesoscopic phases formed by the BC units, we compute the Q-tensor as an appropriate order parameter for quantifying both the nematic and the polar order <cit.>. Its components are defined as follows:


    Q_αβ = 1/N_ b∑_i=1^N_ b1/2(3 n_i,αn_i,β - δ_αβ),


where N_ b is the number of BCs in the ensemble, α, β∈{x,y} represent the Cartesian components of n̂_i corresponding to the i^th BC. The largest eigenvalue of Q gives the nematic order parameter 𝒮 and the corresponding eigenvector represents the global director n. The nematic order parameter can also be interpreted as 𝒮=P_2(n̂_i·𝐧), where P_2(x) = (3x^2-1)/2 is the Legendre polynomial of degree two and the over-bar indicates an average over all the molecules. The isotropic phase corresponds to S = 0 and a fully aligned nematic phase has S = 1. 
The polar order can be computed as 𝒫 = |p̂_i·𝐩|, where 𝐩 is normal to 𝐧. A polarization 𝒫 = 1 implies that the polar vectors p̂_i of the BCs are all pointing in the same direction as 𝐩. In this case, aggregates of BCs exhibit nematic as well as polar order. 𝒫 = 0 indicates random or staggered (e.g., anti-ferroelectric) orientation of p̂_i of the BCs. Note that in this case, the aggregate could still exhibit nematic order.



  §.§.§ Radial distribution function



The radial distribution function g(r) (RDF) which measures correlations between pairs of particles can provide information about the presence of density modulations in the system and is a useful tool for identifying the structure of the aggregates or domains. This function is defined as <cit.>


    g(r) = 1/ρ_ b N_ b⟨∑_i,ji≠ j^N_ bδ(r-r_ij) ⟩,


where r_ij = |r⃗_ij| is the separation distance between the centers of mass of BCs i and j, N_ b is the number of BCs, and  ρ_ b = N_ b/L^2 is their number density. In practice, the δ-function is replaced by a counting function which is normalized appropriately by the area of the finite but small region in which the counting is done. The division by ρ_ b ensures that g(r) = 1 when the particles are completely uncorrelated, as this is, for instance realized in an ideal gas. Sharp peaks indicate the presence of structure and oscillatory behavior marks the presence of translational order in the system. We can also define a characteristic lengthscale (or correlation length) over which the particle positions are correlated. Typically, it is the distance r over which the peaks in g(r) decay to – say – 0.2 times the maximum value of the RDF <cit.>.

In order to identify layered phases such as the smectic one, we resolve the RDF along axes in the local frame of reference of the BC units and average the results. The RDF resolved parallel to the local director n̂_i of the BCs, g_∥(r_∥), is defined as follows <cit.>:


    g_∥(r_∥)=1/ρ_b N_b⟨∑_i,ji≠ j^N_bδ(r_∥ - r_ij, ∥)Θ(h/2 - r_ij,  ⊥)/h⟩,


where r_ij, ∥= |r⃗_⃗i⃗j⃗·n̂_i| and r_ij,⊥=|r⃗_⃗i⃗j⃗×n̂_i| are the parallel and perpendicular components of the separation r_ij with respect to n̂_i; further, h is the average height of the BCs (i.e., average extension of the BCs in the direction perpendicular to n̂_i, see Fig. <ref>b) and Θ (x) is the Heaviside step function, which ensures that only BCs within the same layer as the reference BC (with index i) are included in the computation. Similarly, g_⊥(r_⊥) which resolves RDF in the direction perpendicular to the local director is defined as <cit.>


    g_⊥(r_⊥)=1/ρ_ b N_ b⟨∑_i,ji≠ j^N_ bδ(r_⊥ - r_ij, ⊥)Θ(w/2 - r_ij,  ∥)/w⟩,


where w is the average width of the BCs (see Fig. <ref>b). 



§ COMPUTATIONAL DETAILS



Our starting point to obtain BC colloids and their phases is a collection of an equal number of right-handed and left-handed two-patch ellipsoids, and specified patch-patch interactions. We study the self-assembly using extensive molecular dynamics (MD) and Monte Carlo (MC) simulations. As discussed in detail in Sec. <ref>, the ellipsoids interact via the GB potential which is characterized by four essential parameters: κ, κ^', μ and ν. There are a large variety of GB homologues which differ from each other in terms of the values chosen for the four parameters. The primary focus in selecting their values has been (i) to obtain the nematic and smectic LC phases and (ii) to obtain convergence with experimental data <cit.>. The choice of κ=3.0 is elementary as for real LC systems, the length-to-breadth ratio of the constituent molecules is typically equal to or greater than 3:1. 
A phase diagram of the GB model in the κ^'-T indicates that the ordered phase as κ^'→ 1 is primarily nematic <cit.>. A value of κ^' > 1 introduces energy anisotropy, favoring side-to-side alignment of ellipses. These values promote nematic as well as smectic order characteristics of LCs. The parameters μ and ν modify the well depths of the potential, and hence their impact on the nematicity and smecticity is very subtle <cit.>. A popular choice of parameters is κ=3, κ^'=5, μ=1 and ν=3 due to Berardi et al. <cit.>. They ensure the formation of LC  phases and agreement with experimental data in many real systems <cit.>. In the present work, our focus is to first create BC units, and then study their assembly in ordered phases. Consequently, we choose κ^'=1 to promote patch-patch aggregation for the formation of BCs in the first stage, which then assemble to form ordered phases due to the GB interactions in the second stage. 


 


 §.§ Details of MD simulations

MD simulations are performed in NVT and NPT ensembles using the LAMMPS software <cit.>, employing the Nosé-Hoover thermostat and barostat for temperature and pressure coupling, as needed <cit.>. Periodic boundaries with minimum image convention have been used for the simulations. All calculations were done in reduced LJ units: T^*= k_BT/ϵ_0, Δ t^*= Δ t/ √(m σ_0^2/ ϵ_0), where k_B is the Boltzmann constant. The star is dropped in subsequent discussions. The equations of motion are integrated using the velocity-Verlet algorithm with simulation time step Δ t = 0.001 <cit.>. 

We initialize the system with N=4000 patchy ellipsoids in a square box of length L depending on the desired density ρ=(πσ_∥σ_⊥)N/L^2, where σ_∥  and σ_⊥ are the major and minor axes of the ellipsoids. For all studied densities, we initialize the particles on a regular lattice and allow the system to equilibrate at T=3 for 10^5 MD steps. The resulting homogeneous state is then quenched to the desired temperature. Simulations were performed up to 10^7 steps and relevant quantities measured at intervals of 5× 10^3 steps. All presented data have been averaged over 10 independent runs.



 §.§ Details of MC simulations



In a complementary fashion we have used Monte Carlo (MC) <cit.> simulations to investigate the structural properties of our BC entities. To be more specific we have used so-called “virtual-move” MC (VMMC) simulations <cit.>, an algorithm that proposes simultaneous translational and rotational moves of clusters of particles according to gradients of interaction energies of the particles. In short, a seed particle is subjected to a “virtual” trial move; based on the emerging bond energies before and after the trial move, neighbouring particles are added to a virtual cluster. This process is iterated for each particle in the cluster and the positions of all particles involved are updated simultaneously (for more details we refer to Refs. <cit.>). 

The VMMC simulations were performed in the NVT ensemble, applying standard periodic boundary conditions. Due to the fact that the degree of parallelization of the VMMC algorithm is strongly limited, we had to restrict ourselves to considerably smaller ensembles (as compared to the MD simulations): a typical ensemble size of N = 900 particles was used. Starting in a square box (whose size is imposed by the desired density ρ) half of the particles are left- and right-handed, respectively. At a given temperature the system is equilibrated during 4 × 10^7 VMMC steps, while the production part of the simulations extended over 2 × 10^7 VMMC steps. For each state point we considered ten independent MC runs, each of them starting from different initial conditions. For the highest densities we started from ordered arrays of parallel unbounded ellipses, while for the lowest densities we started from random configurations.  All quantities were averaged over 4 × 10^3 independent configurations per single run, adding to  4 × 10^4 per state point. 














§ RESULTS AND DISCUSSION





 §.§ First step assembly stage

Before studying the mesoscopic phases of the system, it is useful to understand the effect of the model parameters on the formation of BCs. The presented patchy colloidal analog of BCLCs has three model parameters and two system parameters to control the properties of the BCs: patch interaction strength ϵ_p (defined in units of ϵ_0), patch interaction range α, patch positions θ_i, density ρ, and temperature T. We perform simulations in the NVT ensemble with density ρ=0.4 and temperature T=1.2, corresponding to an equilibrium isotropic fluid state of the underlying Gay-Berne particles. Unless stated otherwise, the patch positions are fixed at θ_1 = 5 and θ_2 = 10. 

We start by demonstrating the effect of patch-patch interaction strength ϵ_p on the formation of BCs, keeping the patch range constant at α = 0.1. The important metrics to characterize the BCs are: (i) the fraction of particles f that have self-assembled into BCs, (ii) the average bend angle γ̅ of the BCs and (iii) the flexibility of the BCs. Figs. <ref>(a)-(c) show prototypical equilibrated snapshots at time t=10^4 for ϵ_p = 7.5, 15 and 26.25. The red and green patchy ellipsoids which yield a BC unit are coloured purple. The asters, formed when more than two ellipsoidal particles adhere, are coloured cyan. (See insets in Figs. <ref>(a)-(c).)  The BC yield f for different values of ϵ_p is shown in Fig. <ref>(d). The inset shows the patch-patch interaction V_p(d) (inverted Gaussian) for the different values of ϵ_p. The primary effect of varying ϵ_p is to vary the well-depth. Fig. <ref>(e) shows that the corresponding distribution of bend angles while Fig. <ref>(f) shows f and γ̅ as a function of ϵ_p.
For ϵ_p = 7.5, f remains quite low even after long simulation times. This is because V_p in this case is weaker than thermal energy k_BT and is not sufficient to keep the left-handed and right-handed particles bonded, see the morphology in Fig. <ref>(a). Fig. <ref>(e) shows that the corresponding distribution of bend angles is broad.
For an intermediate value of ϵ_p = 15, V_p is sufficiently strong with respect to k_BT, thus keeping the individual building blocks bonded, as shown in Fig. <ref>(b). As a result, we achieve a BC yield close to 1.
For ϵ_p=30, V_p is even stronger, making single bonds also stable against thermal fluctuations. In addition to the BCs, this leads to the creation of asters formed due to the patch at θ_1=5. Single bonds with more than one monomer of the opposite handedness are also stable in this case as seen in the morphology of Fig. <ref>(c). Consequently, the BC yield is also reduced, as observed in Fig. <ref>(d). The corresponding bend angle distribution shown in Fig. <ref>(e), which excludes the asters, is narrower than that for ϵ_p = 7.5, indicating the reduced flexibility of the BCs, as is expected with stronger patch-patch interactions. Interestingly, for the range of ϵ_p values over which the BC formation is efficient, we do not notice a significant variation in the flexibility of the bend angle as revealed by Fig. <ref>(f).



The other crucial parameter for controlling the self-assembly of BCs is the patch interaction range α. Graphically, it represents the well width of inverted Gaussian potential. 
Guided by the observations in Fig. <ref>, we set the patch strength ϵ_p=15 to explore the effect of α. Figs. <ref>(a)-(c) show the snapshots for α = 0.03, 0.1 and 0.15.
For the small value of 0.03, the effective region within which two complementary patches can make a stable bond is significantly reduced. This not only requires the bonding monomers to be close to each other but also that they are in near-perfect alignment in order to make a double bond. As a result, the probability of opposite-handed particles forming a BC is quite small. This negatively affects the kinetics of the self-assembly leading to a low rate of BC production, as seen in Fig. <ref>(d). The resultant BCs are also less flexible and prone to dissociate into monomers upon the smallest fluctuations. 
As α increases to 0.1 in Fig. <ref>(b), the monomers have a higher probability of making bonds and the rate of BC self-assembly increases. The flexibility of BCs also increases as the increased patch range allows for a larger window of angles over which the monomers can remain bonded, as shown in Fig <ref>(e) and (f). Both these factors contribute to a larger BC yield. For larger patch ranges α≥ 0.15 in Figs. <ref>(f), the rate of self-assembly of BCs is high. However, the yield is significantly reduced due to the formation of asters. They coexist with the BCs as shown in Fig <ref>(c), and the BCs display a large fluctuation in their bend angle for larger values of α. 

From Figs. <ref>(f) and <ref>(f), it can be seen that for fixed values of the patch angles, there is a sizable region in the (ϵ_p-α) parameter space which yields an efficient formation of BCs. The distribution of the bend angle γ is predominantly dictated by the patch range α and to a lesser extent by the patch strength ϵ_p. 

It is also educative to understand the role of the patch placements [θ_1, θ_2] on the BCs. Choosing the ϵ_p = 15 and α = 0.1, Fig. <ref> shows typical morphologies for four different choices of patch positions: (a) [0, 2]; (b) [0, 5]; (c) [3, 6]; (d) [5, 10]. The corresponding bend angle distributions P(γ) vs. γ and the BC yield f are also shown in Figs. <ref>(e) and (f). Their choice affects the bend angle. For example [0, 5] yields γ̅≃ 145, but shifted placements [5, 10] results in γ̅≃ 82. Note that this angle is dependent on the shape anisotropy κ as well. For instance, the mean bend angle would be larger for less eccentric ellipses, i.e., for κ < 3. BC yields are also affected as in cases such as [0, 2], there is a larger possibility of non-BC aggregations such as the asters. Further, we can also expect distinct phases and their textures as the patch positions are varied: [0, 2] nudges the arrangement in a nematic phase while [0, 5] introduces a splay in the texture. Our preferred choice has been [5, 10] as it yields BC formation ≃ 1. The interplay of α, θ_1 and θ_2 can be expected to yield a range of bend angles and novel phases. This is an exhaustive investigation that deserves attention. However, first it is essential to understand the methodology to create model BCs from patchy ellipsoids and subsequently study their self-assembly into states with long-range liquid crystalline order. We prioritize establishing these protocols in the present work and will address the α-θ_1-θ_2 state diagram in an ensuing study.



 §.§ Second step assembly stage

We now turn our attention to the ordering assemblies from the initial monomeric state. Figs. <ref>(a)-(d) show the time evolution of the morphology. The corresponding BC yields f are shown in Fig. <ref>(e).  
At time t=0, when the system is quenched from T=2 to 1.2, the system is in a homogeneous isotropic state where the monomers have not formed any BCs [Fig. <ref>(a)]. In the absence of any patches, this would remain an isotropic fluid, given our choice of the GB parameters. Due to the presence of attractive patches, opposite-handed monomers start adhering. The BC yield f rapidly approaches 1 as time increases. An unusual observation from Figs. <ref>(c) and (d) is that the evolving ordered state has anti-ferroelectric smectic order. 
We also evaluate the nematic and polar order parameters (𝒮 and 𝒫) for the evolving morphology in Fig. <ref>(f). Note that the polar direction p̂ of the neighbouring BCs in the ordered phase are antiparallel (or staggered). Consequently, we evaluate the polarization by considering only the magnitude of p̂ along the direction of polarization p⃗. In other words, we take the absolute value of (p̂_i·𝐩) before averaging over all BCs. As seen in Fig. <ref>(f), 𝒮 is close to zero at early times as the system is in an isotropic state. As time evolves, there is the emergence of BCs and subsequent liquid crystalline order due to the GB interactions between the BC units. The BCs start forming small locally aligned regions, which then coalesce into larger domains with liquid crystalline order [Fig. <ref>(c) and (d)]. Consequently, there is an increase in the nematic and polar order parameters, as seen in Fig. <ref>(f). These observations, along with Figs. <ref>(e), emphasize the two-stage self-assembly in our model: the BC formation followed by long-range liquid crystalline order in the smectic anti-ferroelectric phase.


The full set of BC states (phases) for a specific choice of model parameters (ϵ_p = 15, α = 0.1) in the optimal window is shown in the ρ - T state diagram of Fig. <ref>(a). The different symbols indicate the different states observed in our simulations: isotropic state (indigo circles), anti-ferroelectric smectic (magenta triangles), arrested or frozen state (green squares) and disordered state (blue diamonds). The background color represents the BC yield. We see that there is a sizeable region in the ρ-T space where the BC yield is greater than 0.9.
Panels (b) to (e) show typical morphologies obtained from MD (left) and MC (right) simulations for the four states obtained at indicated parameter values of the corresponding filled symbols in the state diagram. In (b),  there is formation of BCs, but the system is in an isotropic state as the Gay-Berne attraction between them is not sufficient to overcome the thermal fluctuations. 
The region between the dashed lines is where we obtain the smectic anti-ferroelectric phase shown in the snapshot (c). For these parameter values, the BCs are fluid enough to self-assemble into large ordered domains. At low temperatures (T < 0.9), there are locally aligned regions of BCs, but there is a lack of long-range order as these clusters are frozen. Such a kinetically arrested or frozen state due to low temperatures is shown in panel (d). At very high densities ρ > 0.6, the particles lack the free volume to move and reorient, leading to a dense disordered BC state, as shown in panel (e). Similar morphologies are also observed in the NPT ensemble with pressure-tuned to achieve corresponding densities, see Fig. <ref>.

In order to characterize the structure of the phases, we evaluate the directional radial distribution functions (RDF) resolved along axes in the local frame of reference of the BCs as described in section <ref>. This is schematically illustrated in Fig. <ref>(a) where for each BC i, we compute the RDF with other BCs along parallel or perpendicular directions to its local alignment n̂_i. Only those BCs that are within a finite region perpendicular to the direction of the correlation evaluation are included. The directional RDFs at ρ=0.4 and T=1.5, corresponding to the isotropic phase, are shown in Fig. <ref>(b). We observe only a few peaks in g_⊥(r_⊥) and hardly any in g_∥(r_∥), indicating that at most, only two or three BCs stack along their short axis. In the case of the smectic anti-ferroelectric phase (ρ=0.4 and T=1.2), we observe multiple equidistant peaks along either direction, indicating the arrangement of BCs into multiple layers, as shown in Fig. <ref>(c). Interestingly, the correlations decay much more slowly in the direction of the long axis of the BCs (g_⊥(r_⊥)). In the frozen state at ρ=0.4 and T=0.9 [Fig. <ref>(e)], we find equidistant peaks that decay rapidly, corresponding to short-range correlations along either direction. This is due to the presence of multiple small domains of locally aligned BCs. In the dense disordered state (ρ=0.7 and T=1.2), we only find short-range correlations along the short axis of the BCs, as seen from Fig. <ref>(f). In all cases, our MD and MC results are in agreement with each other. The variation in the correlation lengths along the two directions (l_n and l_p) with temperature T is shown in Fig. <ref>(d), for two densities (ρ = 0.4 and 0.7). At the higher density, the system is in a dense disordered state, and hence the correlation lengths do not vary with temperature. At the lower density, we see a gradual increase in the correlation lengths with a peak centered around T = 1.2, corresponding to the increased smectic anti-ferroelectric alignment, beyond which the correlations decay rapidly as the system enters the high-temperature isotropic phase. The plots also show that the size of the LC domains along the long axis of the BCs is larger than that in the orthogonal direction. The current investigation of parameters belonging to patch-patch interactions provides only isotropic and smectic anti-ferroelectric phases. This is because we have not fully explored the effect of patch positions on the formation of mesoscopic phases. Our preliminary explorations of the α-θ_1-θ_2 space reveal a variety of phases, e.g., nematic and splay-bends, through appropriate choice of the model parameters.







§ SUMMARY AND CONCLUSION



We have developed a patchy ellipsoid model capable of generating mesoscopic liquid crystalline phases in a hierarchical two-stage self-assembly process, via the formation of BC dimers in the first stage. This is achieved through a binary mixture of ellipses having a pair of complementary and obliquely placed patches with specific combination rules to guide the formation of BCs. The patches are modeled as points on the surface of the ellipses interacting via an inverted Gaussian potential. The second stage of the assembly into ordered liquid phases is governed by the Gay-Berne interactions acting between the ellipses themselves. Our model has three crucial parameters: (i) the patch interaction strength ϵ_p, (ii) the patch interaction range α and (iii) the patch positions θ_1 and θ_2, that control the formation of the BCs, their mean bend angle and flexibility. 

We perform extensive MD and MC simulations to elucidate the model and the effect of the model parameters. Our main observations are summarized below:

(a) For fixed patch positions θ_1 and θ_2 and patch range α, we observe a window of ϵ_p values for which the BC yield f is close to 1. For small values of ϵ_p, the particles do not make stable bonds, whereas for large values of ϵ_p, assemblies of more than two complementary monomers (asters) are formed due to increased stability of single bonds.

(b) Similarly, with all other parameters fixed, we obtain high BC yields for a wide range of values of α. Although smaller values of α also can produce BCs, we observe low yields in our simulations due to slow kinetics. Larger values of α promote the formation of asters as the monomers can make erroneous bonds due to increased range.

(c) For a choice of parameters in the optimal window, we study the evolution of the BC yield and the nematic and polar orders, which show that the assembly into ordered phases occurs in two-stages with the BC formation preceding the long-range ordering.

(d) We obtain the state diagram of the model in the ρ-T plane, which exhibits isotropic, smectic anti-ferroelectric and disordered arrested states (due to low temperatures or/and high densities) for the given choice of model parameters. We compute correlation functions and order parameters of these states and show that there is consistency between MD and MC simulations. Further, there is consistency in the morphologies between NVT and NPT ensembles as well, at appropriately chosen parameter values.

(e) The patch positions θ_1 and θ_2 can also be varied to generate BCs with different configurations leading to a myriad of interesting liquid crystalline phases, which we shall explore in subsequent works.

The assembly scenarios of these systems are very rich and, so far, the available models for BC entities from patches have not been able to predict all the observed BCLC phases. Our work sets the stage for these future investigations. In contrast to the molecular BCs often studied in experiments, our model provides a colloidal route to create BCLCs. Generating them at the colloidal scale in experiments is challenging but could have huge potential applications. We believe that our study offers one possible solution for the experimental realization of these interesting fluids.





§ ACKNOWLEDGEMENTS

AKS gratefully acknowledges the HPC computing facility of IIT Delhi. JT, EB and GK gratefully acknowledge the generous share of CPU time, offered by the Vienna Scientific Cluster (VSC) under project number fs71898. JT acknowledges financial support from the EU-funded doctorate college ENROL. EB acknowledges support from the Austrian Science Fund (FWF) under Proj. No. Y-1163-N27. VB acknowledges SERB (India) for research grants CRG/2022/004871 and CRG/2019/001836. AB acknowledges partial financial support from CRG/2019/001836 during a visit to    IIT Delhi that initiated this work.
















































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 Paper Title: Cybloids $-$ Creation and Control of Cybernetic Colloids.


	


debasish.saha@hhu.de
Condensed Matter Physics Laboratory, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D-40225 Düsseldorf, Germany



sonja.tarama@hhu.de
Institute for Theoretical Physics II: Soft Matter, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D-40225 Düsseldorf, Germany.
Present address: Laboratory of Biological Computation, College of Life Sciences, Department of Bioinformatics, Ritsumeikan University, 1 Chome-1-1 Nojihigashi, Kusatsu, Shiga 525-0058, Japan



Institute for Theoretical Physics II: Soft Matter, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D-40225 Düsseldorf, Germany.


Condensed Matter Physics Laboratory, Heinrich-Heine-Universität Düsseldorf, Universitätsstraße 1, D-40225 Düsseldorf, Germany





§ ABSTRACT 
Colloids play an important role in fundamental science as well as in nature and technology. They have had a strong impact on the fundamental understanding of statistical physics. For example, colloids have helped to obtain a better understanding of collective phenomena, ranging from phase transitions and glass formation to the swarming of active Brownian particles. 
Yet the success of colloidal systems hinges crucially on the specific physical and chemical properties of the colloidal particles, i.e. particles with the appropriate characteristics must be available. Here we present an idea to create particles with freely selectable properties. The properties might depend, for example, on the presence of other particles (hence mimicking specific pair or many-body interactions), previous configurations (hence introducing some memory or feedback), or a directional bias (hence changing the dynamics). Without directly interfering with the sample, each particle is fully controlled and can receive external commands through a predefined algorithm that can take into account any input parameters. This is realized with computer-controlled colloids, which we term cybloids —short for cybernetic colloids. The potential of cybloids is illustrated by programming a time-delayed external potential acting on a single colloid and interaction potentials for many colloids. Both an attractive harmonic potential and an annular potential are implemented. For a single particle, this programming can cause subdiffusive behavior or lend activity. For many colloids, the programmed interaction potential allows to select a crystal structure at wish.  Beyond these examples, we discuss further opportunities which cybloids offer.


Cybloids – Creation and Control of Cybernetic Colloids
    Stefan U. Egelhaaf
    August 5, 2024
======================================================

These authors contributed equally to this work




§ INTRODUCTION


The behaviour of a many-body system is determined by the interactions between its constituents, which can range from atoms, (bio)molecules and colloids to biological cells, animals and humans or even to planets, stars and galaxies. To establish a link between the interactions and the observed behaviour is one of the central tasks of statistical physics<cit.>. For a systematic and quantitative experimental test of theoretical predictions, it is crucial that the interactions can be tuned. In atomic systems, the available set of interactions is restricted by the number of atomic species. In contrast, in colloidal systems the interactions can in principle be tuned <cit.>. To change the colloid-colloid interactions, however, one usually needs to interfere with the sample, e.g., change the chemical composition or structure of the particles or add ingredients such as salts or polymers <cit.>. There are only few systems that allow for external control of the interactions. For example, the interactions of microgel particles and paramagnetic particles are susceptible to changes in temperature <cit.> and an external magnetic field <cit.>, respectively. However, the range of possible modifications remains quite limited. Irrespective of these limitations, colloids continue to play an important role in fundamental science and have had a particularly large impact on statistical physics. They have helped us to understand collective phenomena, ranging from glass formation
<cit.> to phase transitions <cit.> as well as the clustering of active Brownian particles <cit.>. Furthermore, this understanding has a large impact also on other areas of sciences, ranging from life science, e.g. the crowding in biological cells <cit.> and the swarming of bacteria <cit.>, to medicine, e.g. the mechanism leading to cataract <cit.>, and to material sciences, e.g. the rational design of materials such as the spontaneous assembly of hierarchical structures <cit.>. Since the success of colloidal systems originates from the specific properties (both chemical and physical) of the colloidal particles, the availability of particles with the appropriate characteristics is of prime importance.






Here we present a principle to create colloids with freely selectable properties. The key idea is to expose colloids to an externally created laser-optical potential which is programmed at wish. This allows to program the forces acting on each colloidal particle arbitrarily and thus gives rise to huge flexibility for generating novel colloidal properties. These properties might depend on several parameters, for example, previous configurations (hence introducing some memory or feedback), on the presence of other colloids (hence mimicking specific pair or many-body interactions) or a directional bias (hence changing the dynamics). Each individual particle is fully controlled and can receive commands through a predefined algorithm that can take into account any input parameter without directly interfering with the physical or chemical properties of the particles. We term these computer-controlled colloids
        cybloids as a portmanteau of  cybernetic colloids.
The desired properties are implemented by external programming which does neither require to change nor to directly interfere with the sample. This significantly extends the range of properties that can be realised, controlled, investigated and explored to, e.g., design and fabricate previously inaccessible states. Therefore, this method provides new possibilities ranging from novel model systems to new materials.


How can colloidal particles be programmed in detail? This is realised using a control circuit (Fig. <ref>) using time-delayed feedback <cit.>. (The details of the experimental setup are given in Section <ref>.) The particles are imaged using an optical microscope and their positions, r_i(t), are followed <cit.>. The actual and previous particle positions and possibly other parameters are used as input parameters to calculate the total potential, U_prog(r,t), that a particle at position r should experience at time t. The dependence of U_prog(r,t) on the input parameters can be freely chosen and determines the characteristics of the cybloids. The potential U_prog(r,t) is imposed on the particles by an appropriate light pattern I(r,t) <cit.>. This procedure exploits the susceptibility of colloids to external potentials. Here electromagnetic radiation is applied, as in optical tweezers <cit.>. 
Whereas tightly focused light beams are used in optical tweezers, extended light patterns are applied to control cybloids <cit.>. Modern optical methods allow us to create almost any time-varying light pattern I(r,t) <cit.> and hence to program a large variety of particle properties, including almost any particle-particle or particle-independent potential, cooperation rule or mode of activity.

In Section <ref>, we illustrate this approach by first implementing a feedback potential for a single colloidal particle. If the cybloid is coupled to a harmonic potential centered around its past position (“attractive” harmonic potential), this leads to a transient subdiffusive regime as the particle is retracted to its past position. However, if the harmonic potential is inverted, the particle is kicked away from its previous position resulting in an intermediate ballistic dynamical regime. In our experiment, we realize such a “repulsive” harmonic potential by an annular ring potential <cit.> centered around the previous particle position. In Section <ref>, we consider groups of cybloids and program their interaction potential as single or multiple attractive rings around them. By tuning the distance between different rings one can tune the crystalline lattice  and generate exotic crystallites such as ones dominated by a square structure in two dimensions which is relatively uncommon for single-component systems.
We compare our experimental results with theory and Brownian dynamics computer simulations and we find good agreement.




§ MATERIALS AND METHODS



 §.§ Sample preparation and experimental methods

We use colloidal suspensions of sulfate latex beads (IDC) with radius R = 1.4 μm in heavy water (D_2O). Due to the larger density of heavy water (1.11  g/ml), the particles (1.05 g/ml) are pushed to the top by gravity as well as by radiation pressure. The suspension is stirred together with ion exchange resin to reduce the salt content. Although no further effort is taken to keep the salt concentration low, this is sufficient to avoid particles sticking to the glass surface. Subsequently, the suspension is loaded into a sample cell. The cell is constructed from a microscope slide and a cover glass (number #1) separated by two cover glasses (number #0) used as spacers that are fixed using UV glue <cit.>. This results in a narrow capillary which is sealed with glue after filling.

The colloidal particles are observed using an inverted microscope (Eclipse TE2000-U, Nikon) equipped with a 60× objective. Images are recorded using a CMOS camera with USB interface (MAKO U-130, AVT) and the particle locations {r⃗_i(t)} are determined in real-time with a LabVIEW subroutine (IMAQ Particle Analysis Report VI) that is based on a standard particle tracking algorithm.


Light patterns I(r,t) are created with a holographic optical tweezers setup <cit.> that is combined with the inverted microscope. It is based on a SLM i.e. spatial light modulator (LCR-2500, Holoeye), 2-D galvanometer-mounted mirrors (Quantum Scan 30, Nutfield Technology Inc.) and a diode-pumped solid-state laser (wavelength λ = 532 nm, Ventus 532-1500, Laser Quantum). A control software based on LabVIEW and MATLAB handles the image acquisition, particle tracking and hologram generation. The holograms are calculated using an iterative algorithm, the Gerchberg-Saxton algorithm <cit.> and uploaded to the SLM. Finally, the light field is obtained by the speckle pattern created by the SLM. The particle tracking and light field generation naturally leads to a time delay τ between the measured particle positions {r⃗_i(t)} and the introduction of the corresponding light field I(r,t). The light field is kept the same while the next one is calculated, leading to discrete potential update times nτ (where ). Here, the delay time τ= 0.2s is much smaller than the Brownian time 
τ_B = R^2/D_0 ≈ 40s 
where D_0 ≈ 0.05 μm^2/s= is the diffusion coefficient without an external potential but in the presence of the confining glass plates. This implies that the feedback can be applied very quickly in an almost instantaneous manner.



 §.§ Langevin equation of motion and computer simulations

We model the time-dependent trajectory r(t)=(x(t),y(t)) of a single colloid in two spatial dimensions using Brownian dynamics. We describe the particle motion by the time-delayed over-damped Langevin equation<cit.>

    r/ t=1/γF_prog(r-r_p)+1/γf(t) ,

where γ denotes the friction coefficient and F_prog(r-r_p) is the systematic force the particle experiences at its position r(t) due to the feedback potential centered around its past position r_p.
f(t) is a Gaussian random force that describes the thermal motion of the particle and is characterized by its first two moments ⟨f(t)⟩=0 and , where ⟨…⟩ indicates an average over different realizations of the noise, D_0 is the diffusion coefficient of the particles and d=2 gives the spatial dimension.


The feedback force

    F_prog(r)=-∇ U_prog(r)

is derived from the programmed potential U_prog(r). We use two types of potentials: a harmonic attractive potential  


    U(r)=1/2κr^2


and a Gaussian ring potential

    V(r)   =V_0 ^-(|r|-)^2/2b^2 .




As in the experiment, we define times nτ with  at which the position of the particle is evaluated to update the past position r_p entering into the potential. 
The programmed potential U_prog(r) is then given by eq. (<ref>) and eq. (<ref>) respectively and centered around the past particle position

    r_p=r((k-1)τ)

for the time interval kτ≤ t < (k+1)τ, such that the last multiple of τ smaller than time t is kτ and the feedback potential introduced at this point corresponds to the particle positions a delay time τ prior to this which is (k-1)τ. The feedback potential is kept the same until the next potential update at time (k+1)τ. Consequently, the past position used in the feedback term has a time shift of between τ and 2τ to the present time t.



For the case of multiple particles, the programmed potential U_prog is composed of contributions of the form of eq. (<ref>) centered around all past particle positions r_p,j. Here, r_p,j=r_j((k-1)τ) indicates the past position of the jth particle. Further, we introduce an additional force to describe the direct repulsive steric interactions between the particles. This part is modelled via a smooth Weeks-Chandler-Anderson (WCA) pair-potential which takes the form

    V_WCA(r)=
    4ϵ[(σ/r)^12-(σ/r)^6]+ϵ ,    
    r≤ 2^1/6σ ,
    
    0    else ,

where σ is the particle diameter, r is the distance between the two particles and ϵ sets the interaction strength. Thus, the equation of motion in the multiple-particle case reads


    r_i/ t=1/γf_i(t)
       -1/γ∇_i∑_j=1
     j i^NV_WCA(|r_i-r_j|)
       +1/γ∑_j=1^NF_prog(r_i-r_p,j)


such that every particle i is experiencing a fluctuating Brownian force f_i, direct interaction via the WCA potential with all particles j but itself (j i) and feedback forces including the self-term j=i.





Brownian dynamics simulations are performed based on eqs. (<ref>) and (<ref>) with an explicit Euler forward integration scheme with a finite time step Δ t=10^-4s (all other parameters are given in the subsequent figure captions explicitly). For the simulations, the system is first equilibrated without the programmed potential (pure diffusive motion). Then, the feedback potential is introduced and the dynamics again equilibrated for ∼ 100s. 






§ FEEDBACK-DRIVEN DYNAMICS OF A SINGLE COLLOIDAL PARTICLE

First we investigate the effect of a feedback potential on the dynamics of a single colloidal particle. Here, we study simple forms of the programmed potential, given by a harmonic trap (eq. (<ref>)) and an annular ring potential (eq. (<ref>)). In general, any interaction potential can be implemented. We shall first present an analytic solution for a harmonic trap and then turn to experiment and simulation.



 §.§ Analytic solution for a harmonic feedback trap
 

In the following we derive an analytic expression for the mean squared displacement (MSD) of a single particle in a two-dimensional harmonic feedback potential. For the harmonic potential, the programming force is given by

    F(r-r_p)= -κ(r-r_p)

with the actual and past particle positions r=r(t) and  respectively. Here,  is determined by the relation kτ≤ t < (k+1)τ, i.e. kτ is the last update time of the feedback potential before time t.
For the harmonic potential, the x and y directions of motion decouple. In the following, we thus continue on to solve the one-dimensional equation of motion

    x/ t=-(x(t)-x_p)+1/γ f(t) ,

with x_p=x((k-1)τ). From this the two-dimensional MSD, follows as

    ⟨   ( r(+)-r())^2⟩
       =⟨( x(+)-x())^2⟩+⟨( y(+)-y())^2⟩
       =2⟨( x(+)-x())^2⟩ .

Here, we define the MSD as the mean squared distance the particle travels within a time  starting from a fixed reference time . It is thus given as the mean squared distance traveled between the fixed times  and +, while averaging over different realizations of the thermal noise with ⟨…⟩ denoting the thermal noise average.

We solve eq. (<ref>) on the intervals between feedback potential updates kτ≤ t < (k+1)τ. Within these time intervals x_p does not change, so we use the ansatz x(t)=x_0(t) ^- t to obtain the solution for the particle position

    x(t)=   ^- t∫_kτ^t t' f(t')/γ^ t' +x(kτ)^-(t-kτ)
       +x((k-1)τ)(1-^-(t-kτ))

where the integration constant is set by x(t=kτ)=x(kτ).
This solution still depends on the previous positions of the particle. For t→ (k+1)τ, we obtain the recursive formula for the particle position at the feedback potential update times

    x((k+1)τ)=   ^- (k+1)τ∫_kτ^(k+1)τ t' f(t')/γ^ t'+
    x(kτ)^-τ
       +x((k-1)τ)(1-^-τ) .




Now, from eq. (<ref>), we can see that the thermal noise within the last delay time (from kτ to (k+1)τ) enters with the term

    ^- (k+1)τ∫_kτ^(k+1)τ t' f(t')/γ^ t'

while previous positions (that depend on the noise integrals at earlier times) enter with additional constant prefactors. We thus use the ansatz 

    x(kτ)=^- k τ∑_n=0^k-1c_n∫_(k-n-1)τ^(k-n)τ t' f(t')/γ^ t'

for the particle position at the update times kτ, where c_n are (parameter-dependent) constants. Here, we cut the sum at k-1, i.e. the last interval entering into the sum is for t=0 to t=τ. This demands that the recursion according to eq. (<ref>) ends at

    x(τ)=   ^-τ∫_0^τ t' f(t')/γ^ t'+
    x(0)^-τ
       +x(-τ)(1-^-τ) .
 
We thus prescribe x(0)=0 and x(-τ)=0 at these two times, i.e., the particle starts from the position x(0)=0 and the first feedback potential is centered at this position.

Next, we plug eq. (<ref>) into both sides of eq. (<ref>). As the noise is δ-correlated, the integrals for different time intervals are completely independent of each other and should as such have matching prefactors on both sides of the equation. We thus compare the respective prefactors of the integrals to find

    c_0   =1
    
        c_1   =c_0
    
        c_n   =c_n-1+^2τ(1-^-τ)c_n-2 .

Finally, we solve this recurrence equation for c_n via the characteristic root technique to obtain

    c_n=^nτ1/2-^-τ(1-(^-τ-1)^n+1)

and arrive at the full solution for the particle position

    x(t)=   ^- t∫_kτ^t t'  f(t')/γ^ t'+∑_n=0^k-1c_n∫_(k-n-1)τ^(k-n)τ t' f(t')/γ^ t'^- t
       +∑_n=0^k-2c_n∫_(k-n-2)τ^(k-n-1)τ t' f(t')/γ^ t'^τ(^- kτ-^- t) .




The mean squared displacement (MSD) for τ≤ <(+1)τ and τ≤ t_0+< (+1)τ (and ≤), cf. Fig. <ref>, i.e. the mean squared distance the particle travels between two fixed times  and + while averaging over different realizations of the thermal noise, is then given by



    ⟨   ( x(+)-x())^2⟩κ/γ/D_0
       =2-^-2-^-2+(1-^-2τ)
    	[
    	A(-1,-1)^-2+A(-2,-1)^-τ^-(1-^-)
       
    	+A(-1,-2)^τ^-(1-^-)+A(-2,-2)(1-^-)^2
    	+A(-1,-1)^-2
       +A(-2,-1)^-τ^-(1-^-)+A(-1,-2)^τ^-(1-^-)
               +A(-2,-2)(1-^-)^2]
    
    
    
       +^-[ (1-^-2)δ_+A(-1,-1)(1-^-2τ)^-2+A(-2,-1)(1-^-2τ)^-^-τ(1-^-)
                +c_--1(1-^-2) ]
    
       +A(-1,-2)(1-^-2τ)^τ^-(^-(-)τ-^-( +))
       +A(-2,-2)(1-^-2τ)(1-^-) (^-(-)τ-^-(+))+c_--2 ^τ(1-^-2)(^-(-)τ^-^-) .






Here, c_n=0 for n<0 and =-τ, =+-τ are the time offsets to the respective last potential update  (, ∈ [0,τ)). Finally, the abbreviation A(i,j) is defined as

    A(i,j)=∑_n=0^ic_nc_n-i+j ^-2 nτ


which can be solved as a geometric series as


    A(i,j)=^(j-i)τ/(-2)^2{(-1)^2i+2-1/(-2)(-1)^j-i+2
       -(-1)^i+1-1/-2[1+(-1)^j-i](-1)+i+1} .

Eq. (<ref>) gives the mean squared distance that a particle travels between the times  and +.


For sufficiently large , i.e., large , the effect of the initial condition disappears. In this limit, the two terms (-1)^2i+2 and (-1)^i+1 in eq. (<ref>) vanish (as long as <2).
Note, however, that eq. (<ref>) not only depends on the time difference  as well as the number of potential renewals - during that time but also on the offsets =-τ and =+-τ that the times  and + have to the respective last renewal point of the potential. This is due to the discrete potential updates which cause the system to lose its invariance under translation in time (it is invariant only for multiples of the delay time τ). For measurements of the MSD, the resulting curve may thus vary depending on the protocol used. In particular, to obtain a full time average over t_0, the sampling of the particle position must be more frequent than the feedback update. Otherwise, i.e. for low sampling rates (= number of measurements of the particle position per delay time τ), the resulting MSD curve will not only be “rough” due to the small number of data points but actually differ from the full average over t_0, see Fig. <ref>





Further, we have not made any assumptions with respect to the sign of κ. The above expressions hold for both attractive (κ>0) and repulsive potentials (κ<0). However, for large negative values of κ, the particle continues to be accelerated away from its past position with increasing forces and the MSD diverges exponentially. An easy way to see this is from eq. (<ref>). Subtracting x(kτ) from this equation, we find that the distance the particle travels between two potential updates

    x((k+1)τ)-x(kτ)=   [ x(kτ)-x((k-1)τ)](^-τ-1)
       +^- (k+1)τ∫_kτ^(k+1)τdt' f(t')/γ^ t'

grows with a scaling factor of ^-τ-1 (on average, the additional noise term averages to zero). If this scaling factor is larger than 1 the distance increases exponentially for repeated iterations. This implies that ^-τ should be smaller than 2 to ensure convergence. 

We therefore constrain ourselves to potential strengths that fulfill . In this case, at long times the particle dynamics becomes diffusive with the long-time diffusion coefficient

    D_ℓ/D_0   =1/D_0lim_→∞⟨( x(+)-x())^2⟩/2
    =1-^-2τ/2τ(2-^-τ)^2

which only depends on the strength of the external potential κ and the delay time τ.


For an attractive potential (κ>0), the feedback potential leads to a decrease in the MSD compared to free diffusion as the particle is pulled back towards its past position, see Figs. <ref>,<ref>. Additionally, oscillations of period 2τ appear that are linked to the potential update: After a potential update at time t=0, the particle diffuses in a non-moving harmonic potential until the next potential update a delay time τ later. The new potential (at t=τ) is then centered around the position the particle had at the previous potential update (at t=0), i.e. the particle experiences a force drawing it back towards its past position, leading to a reduction in the MSD. At the next potential update (at t=2τ), the particle is then drawn towards its position at time t=τ, driving the particle away from the initial starting position and thus leading to an increase in the MSD (see also Fig.<ref>B). The repetition of this process leads to the observed oscillations while the additional spatial diffusion leads to overall diffusive behaviour at long times with diffusion coefficient D_ℓ<D_0. This long time diffusive motion can be understood as the diffusive motion of the feedback potential center.



For a weakly repulsive potential (κ<0, ^-τ<2), we find a ballistic (∝^2) regime at medium times between two diffusive regimes with diffusion constants D_0 and D_ℓ>D_0, see Fig. <ref>. At short times, the particle dynamics is dominated by diffusion. On intermediate time scales, the introduced feedback potential leads to an effective particle propulsion, as the particle is pushed away from its past position. Due to spatial diffusion, the angle between current and past position changes, causing the motion to become diffusive again on long time scales but now with a larger diffusion coefficient D_ℓ>D_0. 



Getting closer to the divergence point, the transition from propulsion to long-time diffusion shifts to longer times. This is accompanied by the divergence of the long-time diffusion coefficient as


    D_ℓ/D_0=   3/2ln(2)1/(p-2)^2+8ln(2)-3/4(ln 2)^21/(p-2)
       +6+8(ln 2)^2-13ln(2)/16(ln(2))^3+𝒪(p-2) ,

where p=^-τ.





It should also be noted that, as the memory of previous displacements decays with a factor ^-τ-1 over one decay time (see eq. (<ref>)), this decay becomes increasingly slow towards the divergence point and the dynamics becomes highly non-local in time. An increasingly long equilibration time is needed for the effect of the initial condition to disappear.





So for a system with long memory (^-τ≈ 2), what happens if, instead of an equilibrated feedback system, we start from t_0=0, i.e. the particle sitting in the centre of the feedback potential? 



In this case, the resulting MSD shows super-ballistic ∝^3 behaviour instead of ballistic motion, see Fig. <ref>.
From eq. (<ref>), we find that for the MSD in the limit τ→ -ln(2) the highest order in time is indeed given by

    lim_τ→ -ln(2)   ⟨( x(+)-x())^2⟩∝D_0τ/ln(2)^3 ∝D_0/ln(2)τ^2^3

where we used =(+t_0-)/τ and the limit values

    lim_τ→ 2c_n=-(1/2)^n(n+1)

and

    lim_τ→-ln(2) A(i,j)   =- 2^-(j-i+1)/3 (i^3-3i^2j-9ij-7i-6j-6) .

Mathematically, the ^3 appears from the A(i,j) terms, specifically 
A(-1,-1), A(-2,-1), A(-1,-2), A(-2,-2) all scale with ^3 as their highest order.

Physically, the ∝^3 behaviour can be understood in the following way:
Starting from the coarse-grained particle velocity , that follows from eq. (<ref>) as

    v̅_k   :=x(kτ)-x((k-1)τ)/τ
       =1/τ∑_n=0^k-1(^-τ-1)^n^- (k-n)τ∫_((k-n-1)τ^(k-n)τ t' f(t')/γ^ t'

we find that the velocity average ⟨v̅_k⟩=0 is zero, while its mean squared value changes as

    ⟨v̅_k^2⟩   =⟨[ x(kτ)-x((k-1)τ)] ^2 /τ^2⟩
       =1/γ^2τ^2∑_n=0^k-1∑_m=0^k-1(^-τ-1)^n+m^- (k-n)τ^- (k-m)τ
         ∫_(k-n-1)τ^(k-n)τ t'∫_(k-m-1)τ^(k-m)τ t”⟨f(t')f(t”)⟩_2D_0γ^2δ(t'-t”)^ (t'+t”)
       =2D_0/τ^2∑_n=0^k-1(^-τ-1)^2n^-2 (k-n)τ∫_(k-n-1)τ^(k-n)τ t' ^2 t'
       =γ D_0/κτ^2(1-^-2τ) ∑_n=0^k-1(^-τ-1)^2n .

For long memory, i.e., close to the divergence point ^-τ=2, the squared coarse-grained velocity is thus given by

    ⟨v̅_k^2⟩   ≈-3γ D_0/κτ^2∑_n=0^k-1(1)=-3γ D_0/κτ^2 k

and thus grows linearly in k (where k≈ t/τ), i.e. ∝ t, leading to a ⟨v̅^2⟩^2 ∝^3 behaviour in the MSD. The super-ballistic behaviour is thus caused by the growing variance of the particle velocity after the feedback potential is introduced which is similar to the ^3-scaling of an active particle in shear flow <cit.>. Further, for appropriate choices of  and t_0, the particle shows both ballistic and super-ballistic motion with the proportionality in the MSD changing as  - ^2 - ^3 - , see Fig. <ref>. The ^2 and ^3 regimes correspond to motion at constant velocity, given by the particle velocity at t_0 according to eq. (<ref>), and the increase in velocity due to the long memory in the system.





In our calculations, we have derived a full analytic solution for the trajectory a single particle in a harmonic feedback potential. We have shown that the feedback potential can lead to both reduced and increased long-time diffusion as well as particle propulsion at intermediate times. Furthermore, care must be taken when defining the mean squared displacement (MSD) as the system is not completely symmetric under shifts in time and strong memory effects can lead to the retaining of the initial condition for long times. In the following, we drop the tilde from  in the MSD, implying that the system is equilibrated long enough for the initial condition to be lost, and that the average over t_0 is executed properly such that only the relative time offset is relevant.





 §.§ Experimental realization and simulation results






In the experiments, two types of feedback potentials are realized: First, an attractive harmonic potential is implemented. Second, instead of a repulsive harmonic potential (which is technically hard to realize in the experiments) we use an attractive annular ring potential. This potential pulls the particle towards a bright ring centered around its past position, thus leading to an effective repulsion from its past position.








  §.§.§ Attractive harmonic potential

An attractive harmonic potential with stiffness κ>0 is imposed onto the particle centered around its previous position. While the actual light beam has a Gaussian shape, the particle stays close to the potential centre such that the potential can be approximated as harmonic (Fig.<ref>A). Due to the time delay τ of the feedback loop, the harmonic potential centre is located at the previous (rather than the actual) particle position and thus the particle tends to move back towards it (Fig.<ref>B).

The experimental setup was realized in three different ways: First, the spatial light modulator (SLM) was used as a mirror only and the Gaussian shape of the beam was exploited to create the desired light pattern. The beam location was controlled using galvanometer-mounted mirrors (GMM) (Fig.<ref>C). Second, the SLM was used to create a harmonic potential of different stiffnesses κ and the beam location was again controlled using the GMM (Fig.<ref>D). Third, the SLM was used to create a harmonic potential with different κ and to steer the beam (not shown in this figure, but discussed for other examples, see Figs. <ref>,<ref>). All three realizations yielded equivalent results and perfectly agree with our analytic calculations.







In particular, we measured the two-dimensional mean squared displacement (MSD) ⟨ (r(t+t_0)-r(t_0))^2⟩, as a function of time t (with the average taken over t_0) first for a static harmonic potential (no feedback) and then for the feedback case. The static harmonic potential case is used to calibrate the potential strength, i.e. determine κ. By fitting the MSD to the known result ⟨ (r(t+t_0)-r(t_0))^2⟩=4D_0/κ/γ(1-^-κ/γt) for a harmonic trap<cit.>, we obtain both the diffusion coefficient D_0 within the potential as well as the potential stiffness κ, see dotted and dashed lines in Figs.<ref>C,D. 
Next, the feedback loop is introduced, i.e. the same harmonic potential is applied with its centre at the position the particle was at a time τ = 0.2s earlier. As predicted by our analytic calculations, at short times, t < 0.1s, diffusive motion is observed; at intermediate times, 0.1s < t < 3s, the delayed feedback results in oscillations; at long times, t > 3s, diffusion is re-established with a significantly reduced diffusion coefficient. The long time diffusive behaviour then corresponds to the diffusive motion of the position of the potential centre. The experiments perfectly agree with our theoretical predictions (Fig.<ref>C,D).









  §.§.§ Attractive annular (=effectively repulsive) potential

Next, we applied an annular ring potential, see Fig. <ref>A,B. Due to the form of the potential, the particle is pulled towards the ring surrounding its past position and is thus effectively pushed away from its past position. We use a ring potential of radius 1.4R (where R is the particle radius). In this case, the circular minimum typically is not reached by the particle within the delay time τ before the potential is relocated (see Fig. <ref>A,B). The light pattern is created using the SLM.







Three regimes of the MSD are observed (Fig. <ref>C): At short times, the particle freely diffuses with diffusion coefficient D_0. 
These times are typically shorter than the delay time τ = 0.2 s and hence are not recorded in the experiments (Fig. <ref>C) but covered in the simulations (Fig. <ref>D). 
At intermediate times, 0.03s < t < 1s, the motion is superdiffusive, almost ballistic. This is also indicated by the dynamical exponent , which reaches a value of almost 2 in the experiments. 
At these times, the repeated relocation of the potential continuously pushes the particle away from the potential centre causing an effective propulsion. As the particle is typically off-centre in the refreshed potential, it tends to proceed in the same direction during successive steps. This propulsion direction is retained for long times (≈10s). Since the particle also undergoes spatial diffusion, the propulsion direction slowly changes, such that at very long times diffusion is re-established, with a diffusion coefficient higher than the short-time one. The long-time diffusive motion can be understood as the diffusive motion of the potential centre. Again, the experimental findings qualitatively agree with simulations, in particular the transition from ballistic ∝ t^2, to diffusive, ∝ t motion (Fig. <ref>C,D). The simulations show additional oscillations of period τ in μ that are caused by the discretized potential updates: The effective propulsion is caused by the feedback force whose strength varies with the distance of the particle 
from its past position. The force thus varies periodically with period τ.






§ FEEDBACK-DRIVEN DYNAMICS OF GROUPS OF COLLOIDAL PARTICLES

In addition to individual particles, we also studied an ensemble of particles to explore their cooperative behaviour. Here, we program a potential landscape made up of annular rings centered around all previous particle positions.








For multiple particles and no feedback potential in a dilute system, the particles are found in a disordered fluid state. By imposing an attractive interaction potential, however, ordered structures can be induced and the formation of “living” chains and small clusters (Fig. <ref>) was observed. Through collisions, these chains grow and only very rarely break. Similar to the individual particles, the chains undergo superdiffusion.
In very dilute samples, dimers exist for long times before forming trimers. They can be observed to not only perform translational motion but also rotate and slightly change their separation (Fig. <ref>). Similar chains and “rotators” were observed in computer simulations of the system. It is remarkable that the directed motion results from an instability-like situation: using an initial fluctuation the system is set into directed motion, while the subsequent iterations amplify the stochastic feedback systematically.






Further, we programmed an attractive annular interaction potential with radius 2R and width R/7 and found crystallization of the particles into a triangular lattice (Fig. <ref>A). The triangular arrangement optimises the particle separation (to the radius of the interaction potential) and maximizes the number of neighbours with this separation. During the assembly process, the particles are observed to rotate around each other (similar to the “rotators” in Fig. <ref>) thus keeping their ideal separation while more neighbours approach (similar to the chain formation in Fig. <ref>). This separation, i.e. the size of the crystal unit cell, can be tuned through the radius of the annular interaction potential (Fig. <ref>A-C).


An interaction potential with two concentric attractive annuli of ratio √(2) (e.g. radii 2R (as before) and 2√(2)R) and identical width R/7 (as before) leads to a square arrangement of the particles (Fig. <ref>D). The larger distance along the diagonal of the unit cell is favoured by the second annulus which overcompensates the effect of the reduced number of neighbours and hence stabilises the square lattice.
Again, an increase in the radii of the annuli leads to larger unit cells (Fig. <ref>E). The crystal formation was also found for computer simulations of the respective systems (Fig. <ref>F-J).

Note that these crystal structures are not the result of a fixed template that favors certain spatial positions, but is a result of the self-organization of the particles according to the programmed interaction potential. The formed crystals are therefore invariant under translation and rotation and hence retain all possible zero energy modes.










§ CONCLUSIONS

We have introduced cybloids, i.e. cybernetic colloids, whose interaction potential is computer-controlled through a programmed feedback potential. We have systematically and quantitatively investigated the characteristics of these cybloids, including their arrangement, dynamics and collective behaviour (Figs. <ref>, <ref>, <ref>, <ref> and <ref>).


Here, we have shown that the dynamics of the cybloids can be tuned by the programmed potential and delayed feedback. Through the programming, a cybloid particle is made to diffuse on short and long times with the diffusion coefficients given by the particle's and the potential's diffusion coefficients, respectively while for intermediate times transient sub- or superdiffusion appears. This demonstrates that a broad range of dynamics is accessible and can be comprehensively investigated using this approach.
Further, for groups of particle, we have shown the formation of “living” chains and small clusters as well as crystal lattices. 
Crystal formation can be induced and the crystal structure and the unit cell size controlled. This is achieved through an externally imposed interaction potential between the particles, not a stationary particle-independent potential, i.e. fixed template, for example, an array of tweezers. Programmed particles hence provide the opportunity to systematically and quantitatively investigate the relation between particle-particle interactions and particle arrangements, e.g. different crystal symmetries.
This example shows that programming can also be exploited to induce cooperative behaviour, similar to the recently reported implementation of specific cooperation rules <cit.>.

Beyond these examples, programming can be exploited to impose pair and multi-body interactions, to spur drifts, bias and swarming, to induce memory effects or to couple two otherwise unrelated samples and to “clone” samples. Many further dependencies, situations and couplings can be imagined. In any case, the properties of each particle can be prescribed and hence the sample is controlled on all relevant length scales, without the need to change or directly interfere with the sample nor the need that such a sample can actually be prepared. This ideally complements the already very popular possibility to observe and follow each particle <cit.> and hence characterize the system on all relevant length scales. Therefore, cybloids open a huge range of new opportunities which can be exploited to create previously inaccessible model systems, explore the rational design of materials or mimic natural and industrially relevant situations.





§ AUTHOR CONTRIBUTIONS

DS performed the experiments, ST did the theoretical calculations and performed the simulations. SUE and HL supervised the project. DS, ST, SUE and HL discussed the results and contributed to writing the paper. 



§ COMPETING INTEREST

The authors declare no competing interests.



§ ACKNOWLEDGEMENTS

We mourn the loss of our colleague Stefan Egelhaaf, who left us far too early. Many of the ideas in this paper originated from him. We are grateful for his creativity, his thoroughness, his persistence in pursuing questions and questioning data until they were consistent and understood, and for the many joint discussions that ultimately led to this publication. This work was financially supported by a joint project of SUE and HL of the German Research Foundation (DFG) with the project numbers EG 269/6 and LO 418/19. DS thanks Dennis Rohrschneider for his help with the experiments.







 Paper Title: Unsteady slip pulses under spatially-varying prestress.






eran.bouchbinder@weizmann.ac.il
Chemical and Biological Physics Department, Weizmann Institute of Science, Rehovot 7610001, Israel

§ ABSTRACT 
It has been recently established that self-healing slip pulses under uniform background/ambient stress (prestress) τ_ b are intrinsically unstable frictional rupture modes, i.e., they either slowly expand or decay with time t. Furthermore, their spatiotemporal dynamics have been shown to follow a reduced-dimensionality description corresponding to a special L(c) line in a plane defined by the pulse propagation velocity c(t) and size L(t). Yet, uniform prestress is rather the exception than the rule in natural faults. Here, we study the effects of a spatially-varying prestress τ_ b(x) (in the fault direction x) on 2D slip pulses, initially generated under a uniform τ_ b along a rate-and-state friction fault. We consider both periodic and constant-gradient prestress distributions τ_ b(x) around the reference uniform τ_ b. For a periodic τ_ b(x), pulses either sustain and form quasi-limit cycles in the L-c plane or decay predominantly monotonically along the L(c) line, depending on the instability index of the initial pulse and the properties of the periodic τ_ b(x). For a constant-gradient τ_ b(x), expanding and decaying pulses closely follow the L(c) line, with small systematic shifts determined by the sign and magnitude of the gradient. We also find that a spatially-varying τ_ b(x) can revert the expanding/decaying nature of the initial reference pulse. Finally, we show that a constant-gradient τ_ b(x), of sufficient magnitude and specific sign, can lead to the nucleation of a back-propagating rupture at the healing tail of the initial pulse, generating a bilateral crack-like rupture. This pulse-to-crack transition, along with the above-described effects, demonstrate that rather rich rupture dynamics can emerge from a simple, spatially-varying prestress. Furthermore, our results show that as long as pulses exist, their spatiotemporal dynamics are related to the special L(c) line, providing an effective, reduced-dimensionality description of unsteady slip pulses under spatially-varying prestress.

Unsteady slip pulses under spatially-varying prestress
    Eran Bouchbinder
    Received 2 August 2023 / Accepted 7 May 2024
======================================================





§ BACKGROUND AND MOTIVATION



Classical models of earthquakes, which consider a single frictional rupture that propagates along a homogeneous and planar fault separating unbounded linear elastic blocks in the presence of a uniform background/ambient stress (prestress) τ_ b, offer insight into the physics and mechanics of faulting <cit.>. Single frictional rupture is classified into crack-like modes and self-healing slip pulses <cit.>, where the latter feature a characteristic size L that implies a finite slip duration at points along the fault. Yet, our understanding of even these relatively simple models is incomplete.

Major recent progress has been made in understanding the spatiotemporal dynamics of self-healing slip pulses propagating along rate-and-state friction faults under a uniform prestress τ_ b <cit.>. It has been shown that slip pulses are intrinsically unstable objects that cannot propagate steadily, i.e., they either expand (L̇(t)>0) or decay (L̇(t)<0) with increasing time t. Yet, it has been shown that the unsteady pulses dynamics are slow and follow a special L(c) line in a plane defined by the pulse propagation velocity c(t) and size L(t). The special L(c) line corresponds to steady pulse solutions parameterized by τ_ b, which is composed of unstable points that nevertheless form a dynamic attractor at a given τ_ b. As such, special L(c) line should be regarded as an equation of motion for unsteady pulses under uniform prestress. This reduced-dimensionality description remains valid for other pairs of pulse quantities, e.g., the peak slip velocity v_ p(t) and L(t) <cit.>.

Real faults in the earth crust generally feature spatially-varying prestress <cit.>. Such nonuniform prestress distributions may emerge from previous inhomogeneous slip histories, or to effectively mimic either spatial variations in the frictional strength or geometrical fault irregularities. The latter include barriers <cit.>, bends <cit.>, double fault bends <cit.>, branched fault systems <cit.> and fault surface roughness <cit.>. Consequently, it has been recognized quite a long time ago that earthquake models should be extended to include spatially-varying prestress distributions <cit.>. It is particularly interesting to understand whether and to what extent a nonuniform prestress may contribute to the emergence of slip complexity, including phenomena such as non-self-healing slip pulses <cit.>, rupture arrest, back-propagating rupture <cit.> and pulse-to-crack transitions <cit.>.

Indeed, some works have studied the effects of nonuniform prestress on frictional rupture dynamics in various spatial dimensionalities, including in three dimensions (3D) <cit.>, in two dimensions (2D) <cit.> and in one dimensional (1D) space <cit.>, to list a few among others. The PhD thesis of <cit.> (see chapter 5 therein) focused on 2D pulse propagation in nonuniform prestress fields, arguing that the interaction of pulses with inhomogeneous prestress distributions is of particular interest as the finite size of pulses implies larger sensitivity to prestress spatial variations compared to crack-like rupture, which may tend to smooth out their effect.

Our goal in this paper is to study 2D unsteady pulse dynamics under spatially-varying prestress distributions τ_ b(x) of two classes. One is a periodic prestress and the other is a constant-gradient prestress. We will employ the above-mentioned reduced-dimensionality description of unsteady slip pulses, originally developed under uniform prestress <cit.>, to quantify pulse dynamics under nonuniform τ_ b(x) and to test the degree by which this theoretical framework extends to a broader class of problems. Our analysis also employs well defined initial pulse states, corresponding to controllably perturbed steady-state pulse solutions.



§ PROBLEM SETUP: INITIAL PULSES AND THE SPATIALLY-VARYING PRESTRESS Τ_ B(X)



We consider slip dynamics along a rate-and-state friction fault separating two infinite linear elastic blocks under anti-plane shear. The displacement field is u_z(x,y,t), where y=0 defines the fault and z is perpendicular to the xy plane. The frictional strength τ(v,ϕ) is a functional of the slip velocity field v(x,t)≡2∂_t u_z(x,y=0^+,t) and of the state field ϕ(x,t). The latter follows the “aging law” ∂_tϕ(x,t)=1-v(x,t) ϕ(x,t)/D, where D is a characteristic slip displacement <cit.>. τ(v,ϕ), whose explicit expression is given in the Supplementary Material (SM, appended below), features an N shape under steady-state sliding (for which ϕ=D/v), with a transition from rate-weakening to rate-strengthening friction occurring at a local minimum (v_ min,τ_ min) of the steady-state friction strength (see SM).

Spatiotemporal slip dynamics for t>0 are described by the boundary integral equation <cit.>

    τ[v(x,t),ϕ(x,t)] = τ_ b(x) -μ/2c_ sv(x,t) + s(x,t)  ,

balancing the frictional strength on the left-hand-side with the interfacial shear stress on the right-hand-side. The latter features three contributions; the first is the spatially-varying prestress τ_ b(x), the second is the so-called radiation-damping term (where μ is the shear modulus and c_ s is the shear wave-speed) and the third, s(x,t), is a spatiotemporal convolutional integral that accounts for the long-range interaction of different parts of the fault, mediated by bulk elastodynamic deformation. Equation (<ref>), along with the ϕ evolution equation, are solved in the spectral domain <cit.> using the open-source library cRacklet <cit.> in a periodic domain of size W, i.e., x∈[-12W, 12W] with periodic boundary conditions (see SM). The problem formulation is completed once the initial conditions and τ_ b(x) are specified.




In setting up the initial conditions, our goal is to employ well defined and controlled states that will allow us to address our main question regarding the interaction of slip pulses with nonuniform prestress distributions. Consequently, we are not interested in spontaneous frictional rupture nucleation under nonuniform prestress, but rather in generating propagating self-healing slip pulses with known properties. To this end, we invoke the steady-state pulse solutions of <cit.>, obtained under a uniform prestress, to be denoted as τ^_ b. Each such solution is characterized by time-independent slip velocity v^(x) and state ϕ^(x) fields, which propagate steadily at a velocity c^ and feature a characteristic size L^, see Fig. <ref>a. Such solutions were shown to correspond to “saddle configurations” separating expanding from decaying pulses. This implies that small positive perturbations of the steady-state pulse, which we quantify through an instability index Δϵ>0 (see SM), lead to a growing pulse, while small negative perturbations of the steady-state pulse with Δϵ<0 (see SM), lead to a decaying pulse, where |Δϵ| determines the initial rate of instability.

Yet another major advantage of using controllably perturbed steady-state pulses, obtained under uniform prestress τ^_ b, as initial conditions for Eq. (<ref>), is that they set a dynamical reference for the analysis under spatially-varying prestress τ_ b(x) to follow. Specifically, as already stated, it has been recently demonstrated that the unsteady pulse motion under uniform prestress τ^_ b, e.g., as quantified by the time evolution of L(t) and c(t), closely follows the special L(c) line in the L-c plane, whether the pulse grows/expands (L̇(t)>0 for Δϵ>0) or decays (L̇(t)<0 for Δϵ<0), see Fig. <ref>c. The special L(c) line, which corresponds to steady-state pulse solutions parameterized by τ^_ b <cit.>, has been shown to be a dynamic attractor (also for pulses nucleated away from it) and hence to provide a reduced-dimensionality description of unsteady slip pulses under uniform prestress <cit.>. The degree by which this reduced-dimensionality description extends to unsteady pulse dynamics under spatially-varying prestress is a central question addressed in this work.

To complete the problem formulation, we need to specify the spatially-varying prestress distributions τ_ b(x) in Eq. (<ref>). A first class of prestress distributions corresponds to a periodic (sinusoidal) variation of the form

    τ_ b(x) = τ^_ b[1+δsin(2π n x/W) ]  .

Here, the prestress distribution is characterized by two parameters: a dimensionless amplitude δ and a wavelength λ=W/n determined by n (expressed in terms of the domain size W). A second class of prestress distributions corresponds to a constant spatial gradient dτ_ b(x)/dx that is characterized by a tilt angle θ according to

    τ_ b(x) = τ^_ b[1+2tan(θ) x/W ]  ,

where θ>0 corresponds to a counterclockwise tilt. Examples of members of the two τ_ b(x) classes are shown in Fig. <ref>b.

The fault average of both distributions in Eqs. (<ref>)-(<ref>) satisfies ⟨τ_ b(x)⟩_x=τ^_ b, where the latter characterizes the uniform prestress for which the perturbed steady-state (serving as the t=0 initial condition) was obtained. This setup mimics physical situations in which an unsteady slip pulse obtained under a uniform prestress τ^_ b enters at t=0^+ a region of nonuniform prestress τ_ b(x) (with a fault average that equals τ^_ b), and its subsequent t>0 dynamics are tracked. Note also that we are interested in prestress distributions that vary rather slowly on the characteristic scale L^ (the size of the initial slip pulse). In the other limit, where the characteristic spatial scale of variation of τ_ b(x) is smaller than L^, we expect the pulse to smooth out the spatial variability of the prestress (see Sect. <ref> in the SM).






§ PULSE DYNAMICS UNDER PERIODIC PRESTRESS



We first consider the periodic prestress distributions in Eq. (<ref>). We set ⟨τ_ b(x)⟩_x/τ_ min=1.03 and study pulse dynamics in a system with W=480 m, for various Δϵ, δ and n. In Fig. <ref>a, we present the pulse dynamics in the L-c plane for Δϵ=-0.1, i.e., a pulse that decays under a uniform prestress. For δ=0.02 and n=3, it is observed that the pulse sustains, instead of decaying, and forms a limit cycle in the L-c plane (orange circles). The limit cycle, corresponding to a pulse whose L(t) and c(t) vary nearly periodically as it propagates (see Fig. <ref> in the SM), remains close to the special L(c) line (solid black line) and encircles the corresponding uniform prestress fixed point (square). In this case, the periodic prestress transforms a decaying pulse (obtained under a uniform prestress corresponding to the fault average of the periodic one) into a long-lived, sustained pulse. Changing the overall sign of the periodic prestress (i.e., setting δ=-0.02), leads to a decaying pulse, which closely follows the special L(c) line (green diamonds). These results indicate that the gradient of τ_ b(x) near x=0, in particular its sign, plays an important role in determining the fate of the pulse (i.e., whether it decays or sustains and follows a limit cycle in the L-c plane).

In Fig. <ref>b, we set Δϵ=-0.1 and δ=0.02, and vary the wavelength of the periodic prestress (by setting n=2,3,4, see legend). It is observed that sustained pulses, forming quasi-limit cycles in the L-c plane close to the special L(c) line, emerge. The quasi-limit cycles are pushed to larger L and c values with increasing wavelength (decreasing n). We use the term “quasi-limit cycle” as in some cases the limit cycle is not exact and since we did not always follow the dynamics over very long times (and system sizes W), due to computational constraints; consequently, we cannot always determine whether the limit cycle is exact or approximate (e.g., drifts with increasing time, see Fig. <ref> in the SM).

In Fig. <ref>c, we fix the periodic prestress parameters δ=0.01 and n=4, and vary the instability index Δϵ of the initial pulse. For Δϵ=0.1 (brown cycles), a quasi-limit cycle close to the special L(c) line emerges (the cycle in the L-c plane seems to be drifting upwards with time). Upon decreasing Δϵ, setting it to Δϵ=-0.1 (orange squares), a quasi-limit cycle also appears to emerge (recall that δ>0). Upon further decreasing Δϵ, setting it to Δϵ=-0.4 (green triangles), the pulse initially attempts to form a limit cycle in the L-c plane, but appears to be drifting to smaller L and c values, and to decay. Finally, in Fig. <ref>d, we fix Δϵ=-0.1 and n=4, and vary the amplitude δ>0 of the periodic prestress (see legend). It is observed that the emerging quasi-limit cycle broadens with increasing δ>0. Overall, the results presented in Fig. <ref> indicate that a periodic prestress distribution can reverse the fate of the original pulse (under uniform prestress), leading to either decaying pulses or to quasi-limit cycles in the L-c plane. In both cases, for the parameters considered, the unsteady pulse dynamics remain rather close to the special L(c) line, which corresponds to steady-state pulse solutions parameterized by a uniform pretress τ^_ b <cit.>.









§ PULSE DYNAMICS UNDER CONSTANT-GRADIENT PRESTRESS



We next consider the constant-gradient prestress distributions in Eq. (<ref>). We set ⟨τ_ b(x)⟩_x/τ_ min=1.05 and study pulse dynamics in a system with W=480 m, for various instability indices Δϵ and tilt angles θ. In Fig. <ref>, we present the pulse dynamics in the L-c plane for Δϵ=-0.295, i.e., a pulse that decays under a uniform prestress (corresponding to θ=0^∘ in the figure, see green squares), and various values of θ. Upon imposing a negative prestress gradient, corresponding to θ=-10^∘ (green circles), the pulse decays with a small down shift compared to the reference θ=0^∘ results, yet it still follows rather closely the special L(c) line (solid black line). Upon imposing a positive prestress gradient of θ=5^∘ (green diamonds), the pulse decays with a small up shift compared to the reference θ=0^∘ results.

The dynamics qualitatively change upon setting θ=6^∘ (yellow diamonds), where the pulse changes its character and becomes a growing/expanding pulse, which closely follows the special L(c) line. A similar growing/expanding pulse is observed for θ=10^∘ (brown circles), featuring a small up shift compared to the θ=6^∘ results. These results indicate that for a given instability index Δϵ<0, there exists a threshold tilt angle θ_ c>0 that can reverse the nature/fate of the pulse, from decaying to growing. For Δϵ=-0.295, θ_ c is between 5^∘ and 6^∘. In the inset of Fig. <ref>, we plot θ_ c against Δϵ<0, demonstrating that there is an interplay between the instability index of the initial pulse and the threshold tilt angle θ_ c>0 needed to induce a transition from decaying to growing/expanding pulses.

Our analysis above focused on initial self-healing slip pulses that remain pulses under various spatially-varying prestress conditions. Yet, as noted in Sect. <ref>, earthquakes can take the form of either pulse-like or crack-like rupture, with significant implications for the duration of slip, earthquake energy budget, rupture scaling laws and seismic radiation. The dynamic selection of these rupture modes/styles and the spontaneous transition between them are important topics under active investigation, e.g., see <cit.>. It is generally established <cit.> that crack-like rupture is more prevalent under larger average prestress levels. Consequently, it would be interesting to see whether for ⟨τ_ b(x)⟩_x/τ_ min=1.05 (a higher average prestress compared to the one considered in Sect. <ref>), a spatially-varying prestress distribution can spontaneously induce a pulse-to-crack rupture transition.

To address this question, we present in Fig. <ref> (top row) a series of equal-time snapshots of v(x,t) (Fig. <ref>a) and τ(x,t) (Fig. <ref>b) obtained for ⟨τ_ b(x)⟩_x/τ_ min=1.05 and Δϵ=0.005, under a constant-gradient prestress with θ=-9^∘. It is observed that the nonuniform prestress transforms a growing pulse (Δϵ>0) into a decaying pulse, which propagates a certain distance from left to right (in the positive x direction), until it arrests/dies off. Note that pulse arrest leaves behind it a heterogeneous stress distribution (brightest, rightmost snapshot in panel (b)). Interestingly, it is observed that near the healing tail of the initial pulse, around x/W≃-0.03 (also marked by an arrow), there seems to be a spontaneous attempt to nucleate secondary rupture. The latter is evident in τ(x,t) (Fig. <ref>b), but requires a logarithmic scale to be discerned in v(x,t), as done in the inset of Fig. <ref>a.

In Fig. <ref> (bottom row), we present the spatiotemporal dynamics for a slightly stronger negative gradient, corresponding to θ=-10^∘ (everything else being the same as in the top row). It is observed that this nonuniform prestress leads to a qualitative change in the dynamics. First, the attempted nucleation near the healing tail of the initial pulse, around x/W≃-0.03, becomes a successful nucleation event, giving rise to an asymmetric bilateral crack-like rupture. The rupture edge that propagates from right to left (in the negative x direction) corresponds to a back-propagating rupture. The rupture edge propagating in the forward (positive x) direction, catches up with the initial pulse that also propagates in the forward direction and merges with it. Second, upon merging of the two forward-propagating rupture fronts, a persistent asymmetric bilateral crack-like rupture emerges. Overall, it is observed that a spatially-varying prestress can induce a pulse-to-crack transition.




The crack-like nature of the two persistent, oppositely propagating rupture edges is evident in the inset of Fig. <ref>c (note the logarithmic y-axis), where a persistent finite slip velocity is left in the central part, while the two oppositely propagating rupture edges move further away. Similarly, the crack-like nature of the emerging rupture is also manifested in the residual stress observed in Fig. <ref>d. These results, which are presented again in a space-time plot in Fig. <ref>, demonstrate that rich spatiotemporal rupture dynamics may emerge from rather simple spatially-varying prestress distributions, revealing phenomena such as back-propagating rupture and pulse-to-crack transitions, which are of interest in various geophysical contexts, e.g., in relation to earthquake source models and seismological inversion analyses <cit.>. Finally, we note — as highlighted in <cit.> — that once pulse-like rupture transforms into a crack-like rupture, the dynamics of latter can no longer be meaningfully represented in the L-c plane and an intrinsic relation to the special L(c) line — which is essentially an equation of motion for self-healing slip pulses under uniform prestress — does not exist anymore.



§ SUMMARY AND CONCLUDING REMARKS



In this paper, we studied the interaction of well-defined initial self-healing slip pulses with two classes of spatially-varying prestress distributions τ_ b(x), one corresponding to a periodic variation and one to a constant gradient, see Eqs. (<ref>)-(<ref>). Our findings can be summarized along two major axes. The first concerns the richness (or to some extent the complexity) of rupture dynamics emerging from initial slip pulses interacting with a spatially-varying prestress (compared to their uniform prestress counterparts). The second concerns the theoretical description of unsteady slip pulse dynamics under spatially-varying prestress, particularly in relation to the recently established reduced-dimensionality description of pulse dynamics <cit.>.

We have demonstrated that simple nonuniform prestress distributions can give rise to rather rich rupture dynamics. In particular, we showed that a nonuniform prestress can reverse the nature/fate of a slip pulse, from decaying/shrinking to growing/expanding and vice versa, for a fixed fault-averaged prestress, see Figs. <ref>-<ref>. We also showed that pulses under periodic prestress can settle into propagating states in which their properties, e.g., pulse size L(t), peak slip velocity v_ p(t) and propagation velocity c(t), vary periodically or quasi-periodically as they propagate. Finally, we demonstrated that prestress spatial gradients can lead to the spontaneous nucleation of secondary rupture near the healing tail of a pulse, to back-propagating rupture and to pulse-to-crack transitions, see Figs. <ref>-<ref>.

The above mentioned rupture phenomena are geophysically documented phenomena of importance, which continue to pose challenges to our understanding. For example, it is known that while most seismic studies typically only estimate the average rupture velocity, seismic radiation is predominantly affected by the spatiotemporal variation of the rupture velocity <cit.>. Our finding, e.g., in relation to the emergence of quasi-periodically propagating slip pulses, can contribute in this context. Moreover, as noted in <cit.>, slip pulse propagation and arrest give rise to heterogeneous after-event stresses (cf., top row of Fig. <ref>) and hence may offer a mechanism for sustaining prestress heterogeneity over many earthquake cycles.

On the more theoretical side, our results demonstrate that the reduced-dimensionality description of unsteady pulses, originally established under uniform prestress conditions <cit.>, is still relevant under some nonuniform prestress conditions. Specifically, it has been shown that such pulse dynamics presented in a plane defined by a pair of pulse properties — e.g., its size L(t) and propagation velocity c(t) —, remain close to a special, previously derived line L(c). For example, periodically (or quasi-periodically) propagating slip pulses under nonuniform prestress correspond to limit cycles (or quasi-limit cycles) in the L-c plane that remain rather close to the special L(c) line at all times, see Figs. <ref>-<ref> and SM.

Consequently, while the special L(c) line is not a strict equation of motion for unsteady pulses under nonuniform prestress, in many cases it does provide a good approximation for the dynamics, in the sense that the knowledge of one pulse property, e.g., c(t), can be used to infer another property, e.g., L(t). Finally, we also demonstrated that similarly to the uniform prestress case <cit.>, the emerging theoretical picture remains valid upon using other pairs of pulse properties, e.g., L(t) and the peak slip velocity v_ p(t) (see SM).


Acknowledgements. We acknowledge support by the Israel Science Foundation (ISF grant no. 1085/20), the Minerva Foundation (with funding from the Federal German Ministry for Education and Research), the Ben May Center for Chemical Theory and Computation, and the Harold Perlman Family. We are grateful to Prof. Eric Dunham for posing to us the basic question, to Dr. Fabian Barras for his help with setting up the initial conditions in cRacklet and to Dr. Efim Brener for useful discussions.








              Supplementary material

























The goal of this document is to provide some technical details regarding the results presented in the manuscript and to offer some additional supporting data.





§ THE INTERFACIAL CONSTITUTIVE RELATION (FRICTION LAW)



The left-hand-side of Eq. (<ref>) corresponds to the frictional strength τ[v,ϕ], where v(x,t) is the slip velocity field and ϕ(x,t) is the internal state field. Exactly as in <cit.>, we have

    τ(v,ϕ)/σ = [1+blog(1+ϕ/ϕ^⋆)] [f_0/√(1+(v^⋆/v)^2)+α log(1+  v/v^⋆)] ,

where the frictional parameters appearing in the expression are given in the table in Fig. <ref>(left). This table also reports on the linear elastic bulk parameters of Eq. (<ref>) (the spatiotemporal convolutional integral s(x,t) therein features in addition to μ and c_ s, which appear in the radiation-damping term, also Poisson's ratio ν) and the value of the applied normal stress σ. As explained in the manuscript, the internal state field ϕ(x,t) evolves according the “aging law”, taking the form ∂_tϕ(x,t)=1-v(x,t) ϕ(x,t)√(1+v^*/v(x,t))/D, which includes the characteristic slip distance D, whose value is also given in the table in Fig. <ref>(left). Note that the regularization factor √(1+v^*/v(x,t)) was not included in the “aging law” expression in the manuscript, as it is relevant only at extremely small slip velocities (note the value of v^* in the table in Fig. <ref>). Under spatially homogeneous steady sliding, we have ∂_tϕ=0, which allows to express ϕ_ ss (the steady sliding value of the internal state field) in terms of v_ ss (the steady sliding value of the slip velocity field). Once done, one can obtain the steady sliding friction curve τ(v_ ss), plotted in Fig. <ref>(right). τ(v_ ss) follows a generic N shape, featuring a local minimum at (v_ min, τ_ min) <cit.> (marked on the figure). The existence of a minimum, though, does not play a crucial role in the problem we solved; yet, the characteristic slip velocity v_ min=0.005971 m/s and characteristic frictional strength τ_ min=0.3406679 MPa can be used to nondimensionalize various quantities, as done in the manuscript. We also note that using the previously-calculated effective fracture energy G_ c=0.65 J/m^2 of slip pulses <cit.> for our frictional parameters, one can construct a characteristic pulse length as L_ n=(c_ s/v_ min)^2 (G_ c/μ)=15.2 m. The latter is used to normalize the size of slip pulses L(t) in the manuscript.



§ PERTURBED STEADY-STATE PULSES AS INITIAL CONDITIONS



Our main goal in the manuscript is to systematically and controllably study the interaction of self-healing slip pulses with spatially-varying prestress distributions. To that aim, we need well-defined and controlled initial conditions that correspond to slip pulses. Here, we explain how this is achieved, providing additional details on top of the discussion offered in the manuscript.



 §.§ Steady-state self-healing pulses in finite periodic domains



Similarly to <cit.>, we employ the steady-state slip pulse solutions recently obtained under a uniform prestress in <cit.>. The latter solutions take the form v__ ss(ξ) and ϕ__ ss(ξ), with ξ=x-c^t being a co-moving space-time coordinate -∞<ξ<∞ and c^ is the steady pulse propagation velocity, see Fig. <ref>a, where the finite size of the steady-state pulse, L^, is also marked. Specifically, for a given spatially-varying prestress distribution τ_ b(x), we use a steady-state pulse solution (or a slightly perturbed version of it quantified by the instability index Δϵ, see Sect. <ref>) corresponding to a uniform prestress of magnitude ⟨τ_ b(x)⟩_x, i.e., the spatial fault average of τ_ b(x), at t=0. The latter is used as an initial condition for the t>0 spatiotemporal dynamics described by the boundary integral equation (Eq. (<ref>)), which is solved using the open-source library cRacklet <cit.>. cRacklet is a boundary element library based on a spectral formulation of linear elastodynamics <cit.>, and as such is applicable in finite domains of size W in the fault direction x (the domain is infinite in the perpendicular y direction, the fault is located at y=0), where periodic boundary conditions are employed.

The periodic domain of size W is clearly in conflict with steady-state pulses that propagated steadily at a velocity c^ indefinitely in the past, -∞<t<0. That is, generating slip history (past propagation) that extends beyond the domain boundary implies that the past spuriously affects future dynamics. The way to resolve this conflict has been discussed in detail in <cit.>, see in particular section S-3A of the Supplemental Materials file therein. In a nutshell, we require that the past propagation distance of the steady-state pulse, denoted by w, is much smaller than the spatial periodicity domain W and much larger than the steady-state pulse size L^, i.e., L^≪w≪W (see Fig. S1 in <cit.>). Effectively, this means that instead of indefinite past propagation (i.e., for -∞<t<0), we account for finite time past propagation over -t_0<t<0, where t_0>0 is determined by the spatial position in which the past propagation is initiated. Once a given slip history is generated, it is used to calculate the spatiotemporal convolutional integral and the resulting field s(x,t=0) is used in Eq. (<ref>) as an initial condition. The latter can be supplemented with a perturbation in the initial internal state field, quantified by the instability index Δϵ, as will be discussed in Sect. <ref>.





The above-described procedure for mimicking an initial condition corresponding to a propagating steady-state pulse in a finite, periodic system has one implication that has not been discussed in <cit.>. Specifically, the fact that the past indefinite propagation over -∞<t<0 is replaced by a finite past propagation over -t_0<t<0 gives rise to a frozen-in and localized artificial stress concentration due to the unaccounted for slip accumulated over -∞<t<-t_0. To demonstrate the point, we present in Fig. <ref>a the spatiotemporal evolution of the shear stress field τ(x,t), corresponding to one of the solutions discussed in <cit.> with a uniform prestress. The desired slip pulse initial stress distribution appears in the thick black curve. It is observed that the latter is indeed realized, except for a localized deviation near x/W≃-0.25. The latter persists as time progresses, upon which the pulse propagates to the right, further away from the artificial stress concentration. The corresponding dynamics of the slip velocity field v(x,t) are presented in Fig. <ref>b. In <cit.>, we made sure that the frozen-in, artificial stress concentration was located sufficiently away from the central part of the slip pulse such that the dynamics were not affected. The artifact can be readily removed by applying a static slip that exactly relaxes this artificial stress concentration at t=0, as done in Fig. <ref>c-d. Comparing the top and bottom rows of Fig. <ref>, we indeed observe that the frozen-in, artificial stress concentration does not affect the pulse dynamics. Nevertheless, in the present work, we eliminate it in generating the initial condition, at t=0, under uniform prestress. The spatially-varying prestress distribution τ_ b(x) is introduced at t=0^+, as explained in the manuscript.




The steady-state slip pulse solutions we employ as initial conditions have relatively long, small-amplitude tails at the healing front. An additional consequence of the computational domain periodicity in the presence of such initial conditions is that once the pulse propagates through the periodic boundary, it interacts with its own tail. Such interactions affect the dynamics of the pulse and are hence avoided, i.e., the simulations are stopped before this happens. Moreover, the initial pulse is placed such that its edge (e.g., defined by the location of its peak slip velocity) coincides with the center of the domain at x=0, where we also have τ_ b(x=0)=τ^_ b. Whenever a longer evolution time is required, the domain size W is increased (e.g., see results in Figs. <ref>-<ref>), but in general these computations are very demanding.

Finally, we note that for the constant-gradient prestress distributions τ_ b(x) there is a jump between τ_ b(x=-12W) and τ_ b(x=12W), which is formally inconsistent with the periodic boundary conditions. Yet, we verified that this jump does not affect the pulse evolution as long as its edge does not reach the domain boundary, which is indeed avoided, as explained above. Moreover, we verified that smoothly connecting τ_ b(x) across the domain boundary, to conform with the periodic boundary conditions, does not affect our results.



 §.§ Introducing initial pulse perturbations and the instability index



As shown in <cit.> and discussed in the manuscript, steady-state pulses under uniform prestress correspond to “saddle configurations” separating expanding from decaying pulses. That is, slightly perturbed steady-state pulses under uniform prestress either grow (expand) or decay (shrink), depending on the sign of the perturbation. Let us schematically denote the perturbation by ϵ, to be formally defined below. In an infinite system, steady-state pulses are expected to correspond to the marginal state, ϵ=0. However, fitting an infinite-system steady-state pulse into a finite, periodic domain of size W, as explained in Sect. <ref>, inevitably introduces perturbations (as also discussed in <cit.>). Consequently, initial pulses generated as explained in Sect. <ref> will either grow or decay under uniform prestress, i.e., they feature ϵ0, where the sign of ϵ determines whether the pulse grows or decays and its magnitude determines the initial growth/decay rate.

To quantify the latter, we adopt a similar perturbation scheme to that used in <cit.>. Let us denote by ϕ̃(x,t=0) the internal state field resulting from fitting an infinite-system steady-state pulse into a finite, periodic domain of size W, as discussed in Sect. <ref>. Next, we define ϵ through the following amplitude perturbation ϕ(x,t=0)=(1+ϵ) ϕ̃(x,t=0) and ask at what value of ϵ, the subsequent t>0 dynamics of ϕ(x,t=0) under uniform prestress (in the periodic, finite W domain) correspond to the marginal state (where decaying pulse become growing, or vice versa). We denote this value of ϵ by ϵ_ c, which obviously depends on W and on the level of uniform prestress τ^_ b. Specifically, for W=480 m, we find ϵ_ c=-0.4 for τ^_ b=1.03τ_ min and ϵ_ c=-0.295 for τ^_ b=1.05τ_ min.

We take advantage of the above results in order to generate initial conditions using well-controlled, perturbed initial steady-state pulses pulse by varying ϵ. For each uniform prestress τ^_ b, we define the instability index as Δϵ≡ϵ_ c-ϵ such that Δϵ>0 leads to an growing/expanding pulse, while Δϵ<0 leads to a decaying/shrinking pulse. Furthermore, |Δϵ| determines the initial rate of instability. For example, in Fig. <ref>, we set Δϵ=0.005>0 under uniform prestress and indeed the t>0 dynamics correspond to a very slowly growing/expanding pulse. In the manuscript, we used the above procedure to generate initial pulse states characterized by τ^_ b and Δϵ. These are used as initial conditions for the spatially-varying prestess problem, where a nonuniform τ_ b(x) is introduced at t=0^+ and the subsequent t>0^+ dynamics are studied, as discussed extensively in the manuscript.



§ ADDITIONAL SUPPORTING RESULTS


Here, we present a few additional results that further support the results discussed in the manuscript.



 §.§ Dynamics in the L-v_ p plane


In Figs. <ref>c, <ref>, and <ref>, we explored the reduced-dimensionality description of unsteady slip pulse dynamics in the L-c plane. Specifically, we tested the degree by which the pulse dynamics under various conditions follow the special L(c) line, corresponding to a family of steady-state pulse solutions parameterized by the value of the uniform prestress. In <cit.>, it was established that unsteady pulse dynamics under a given uniform prestress also follow the line corresponding to a family of steady-state pulse solutions when presented in a plane defined by other pairs of pulse properties/observables, e.g., the pulse peak slip velocity v_ p(t) and its size L(t).




In Fig. <ref>a, we plot the very same pulse dynamics under periodic prestress τ_ b(x) as in Fig. <ref>c, but in the L-v_ p plane (same color and symbol code as in Fig. <ref>c, see legend therein), where the solid line corresponds to the special L(v_ p) line. It is observed that the relations between the unsteady pulse dynamics and the special L(v_ p) line (and the proximity to it) are very similar to the corresponding relations to the special L(c) in Fig. <ref>c. Likewise, we present in Fig. <ref>b the same results for two cases presented in Fig. <ref> under constant-gradient prestress τ^_ b(x) (quantified by the tilt angle θ), but in the L-v_ p plane. Yet again, the results — in particular the relations to the special L(v_ p) line — are similar for the two reduced-dimensionality descriptions.


 

In Fig. <ref> and in Figs. <ref>c, <ref>, and <ref>, we focused on the reduced-dimensionality description of unsteady pulse dynamics. In Fig. <ref>, for completeness, we also present spatial snapshots of the pulse dynamics shown in Fig. <ref>a and in Fig. <ref>c. We use the very same parameters for the 3 pulses presented therein (see legend of Fig. <ref>c), but employ a system size twice as large, W=964 m. The latter allows us to probe pulse dynamics over larger propagation distances (and hence times), and to address the nature of the emerging limit cycles for growing pulses (e.g., whether the limit cycle is approximate or exact). Presenting pulse dynamics for the larger W in the reduced-dimensionality representation is, of course, possible but would be visually less clear due to the appearance of more overlapping cycles in the plane.

In Fig. <ref>a, we present equal-time snapshots corresponding to the pulse dynamics presented by the green triangles in Fig. <ref>a and in Fig. <ref>c. The spatial snapshots clearly demonstrate the decaying nature of this pulse, where we superimposed the envelope of the peak slip velocity (thicker line) for clarity. In Fig. <ref>b, we present equal-time snapshots corresponding to the pulse dynamics presented in the orange squares in Fig. <ref>a and in Fig. <ref>c. It is observed that the pulses seem to settle into a rather robust limit cycle, though there appears to be some increase in the pulse amplitude during the fourth cycle. Finally, we present in Fig. <ref>c equal-time snapshots corresponding to the pulse dynamics presented in the brown circles in Fig. <ref>a and in Fig. <ref>c. As is already clear from these two figures, the pulse amplitude shown in Fig. <ref>c indeed systematically increases with the cycle number.



 §.§ Pulse dynamics under prestress distributions featuring smaller scale spatial variations



Our analysis in this work focused on prestress distributions τ_ b(x) that vary over lengthscales that are larger than the size of the initial pulse L^. Here, we briefly discuss pulse dynamics when this scale separation does not hold. Similarly to Fig. <ref>, we consider larger systems with W=964 m. Since this value is different from the one used in the manuscript, to avoid any possible confusion we do not quantify the wavelength λ of the periodic prestress τ_ b(x) through n, according to n=W/λ, but rather use λ itself. That is, we rewrite Eq. (<ref>) as τ_ b(x)=τ^_ b[1+δsin(2π x/λ) ], making no reference to W. We set τ^_ b=1.03 τ_ min, for which L^=14.6 m. In Fig. <ref>a, we first consider λ=120 m, still featuring λ≫L^, with δ=0.01 and Δϵ=-0.1, which is identical to the case considered in Fig. <ref>b. As observed, and similarly to other examples presented in Fig. <ref>, in this case the periodic prestress reverses the fate of the pulse, from decaying under uniform prestress (Δϵ<0) to a sustained pulse, nearly forming a limit cycle in the L-c plane. Next, also in Fig. <ref>a, we reduced the periodic prestress wavelength to be λ=15 m, essentially identical to the initial pulse size L^=14.6 m. It is observed that the lack of scale separation between the prestress spatial variation and the initial pulse size leads to a qualitative change in the pulse dynamics in this case. That is, the pulse in this case follows a spiral decay trajectory close to the special L(c) line, see green circles in Fig. <ref>a.

In Fig. <ref>b, we again start with a reference case featuring λ≫L^, this time with with δ=0.01 and Δϵ=0.1>0. The latter is identical to the case considered in Fig. <ref>c, corresponding to a growing pulse, following upward drifting cycles in the L-c plane. Next, also in Fig. <ref>b, we reduced the periodic prestress wavelength to be λ=5 m, i.e., smaller than the initial pulse size L^=14.6 m. It is observed that under these conditions the pulse grows (while performing oscillations, which are not easily discerned in the figure), remaining close to the special L(c) line. This behavior, up to the short wavelength oscillations, is similar to the pulse dynamics under uniform prestress (recall that Δϵ=0.1>0). In fact, in the limit in which the prestress varies on a spatial scale much smaller than L^, we expect the pulse to smooth out the spatial variability of the prestress such that its dynamics are essentially the same for the average uniform prestress. To demonstrate this point, we set τ_ b(x)=τ^_ b[1+ζ(x) ], where ζ(x) is a random variable corresponding to a Gaussian distribution with zero mean and a standard deviation of 0.01. ζ(x) features no spatial correlations, i.e., its value at each numerical grid point x is independent of its values elsewhere. The results presented in Fig. <ref>c indeed demonstrate that the pulse dynamics, whether growing or decaying (see legend), smooth/average out the noise ζ(x) and identify with the corresponding uniform prestress results.















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 Paper Title: A dynamical mechanism rooted in the Lorenz system for giant negative and positive mobility of a memory-driven active particle.



APS/123-QED












]A dynamical mechanism rooted in the Lorenz system for giant negative and positive mobility of a memory-driven active particle

rahil.valani@physics.ox.ac.uk

^1Rudolf Peierls Centre for Theoretical Physics, Parks Road, University of Oxford, OX1 3PU, United Kingdom
^2Ecole Nationale Supérieure de Génie Mathématique et Modélisation (ENSGMM), Université Nationale des Sciences, Technologies, Ingénierie et Mathématiques, Abomey, Republique du Bénin



             

§ ABSTRACT 

On applying a small bias force, non-equilibrium systems may respond in paradoxical ways such as with giant negative mobility (GNM) – a large net drift opposite to the applied bias, or giant positive mobility (GPM) – an anomalously large drift in the same direction as the applied bias. 


We consider an idealized model of a memory-driven active particle inspired from experiments with walking/superwalking droplets that takes the form of the celebrated Lorenz system. By adding a small bias force to this Lorenz model, we uncover a mechanism for simultaneous emergence of GNM and GPM in the parameter space. Within the chaotic sea of the parameter space, a symmetric pair of coexisting asymmetric limit cycles separate and migrate under applied bias force, resulting in anomalous transport behaviors that are sensitive to the particle's memory. This provides a general dynamical mechanism for the emergence of anomalous transport behaviors for self-driven particles described by low-dimensional autonomous chaotic systems.



[
    Bruno S. Dandogbessi^2
    August 5, 2024
==========================



When a small bias force is applied to a system at equilibrium, it responds with a net drift in the direction of the applied bias. However, non-equilibrium systems can respond in counterintuitive ways to a small applied bias. A peculiar anomalous transport behavior that defies intuition is negative mobility, where a system responds with a net drift in the direction opposite to the applied bias force. Negative mobility has been observed in experiments of both classical as well as quantum systems, such as electrons in GaAs quantum wells <cit.>, transport through semiconductor superlattices <cit.> and particles moving in structured microfluidic channels <cit.>. Theoretically and numerically, negative mobility and related anomalous transport behaviors have been extensively investigated in various minimal models of a passive inertial particle, either deterministic or stochastic, with typical features of dissipation, spatially periodic potential and external time-periodic driving <cit.>. These studies have identified different mechanisms that give rise to negative mobility in these externally-driven non-autonomous systems, and the effect is attributed to at least one of the following ingredients: noise-induced fluctuations, chaotic attractors and periodic limit cycle attractors. Investigation of negative mobility in self-driven active systems has been relatively scarce but it is increasing with a recent surge of interest in active matter <cit.>. 

In this letter, we consider a simple model of a memory-driven active particle based on the experimental system of walking/superwalking droplets <cit.>, that takes the form of the celebrated Lorenz dynamical system <cit.>. We uncover a dynamical mechanism for simultaneous emergence of giant negative mobility (GNM) and giant positive mobility (GPM) rooted in the Lorenz system [Although existence of negative mobility has been reported by the first author for the Lorenz model in a previous work <cit.>, an underlying mechanism was not investigated]. 




 
A vertically vibrating liquid bath can support bouncing and walking droplets of the same liquid on its free surface <cit.>. Each impact in the periodic bouncing of the droplet generates a localized standing wave that decays in time. The droplet interacts with these self-generated waves on subsequent bounces to propel itself horizontally resulting in walking and superwalking droplets (see Fig. <ref>(a)). Such droplets have been shown to mimic hydrodynamic analogs of several quantum systems <cit.>. Three key features to note about this system - (i) the droplet and its underlying waves co-exist as a wave-particle entity; without the droplet, surface waves decay completely. (ii) There is memory in the system since at large amplitudes of bath vibrations, the droplet-generated waves decay very slowly in time. Hence the droplet motion is guided by the complex dynamical landscape sculpted by the history of waves along its trajectory. (iii) The droplet is self-driven or active <cit.> since the droplet locally extracts energy from the vibrating bath and converts it into persistent horizontal walking motion. A commonly used theoretical model that captures the key features of a droplet's walking dynamics is the stroboscopic model <cit.>. In this model, the vertical periodic bouncing motion of the droplet is averaged over and one gets a trajectory equation for the horizontal walking motion. As shown schematically in Fig. <ref>(b), we consider a simple setup where the droplet can move only in one horizontal dimension, resulting in the following trajectory equation of motion (in dimensionless units) <cit.>

    ẍ_d + ẋ_d = F_self + F_bias ,




where x_d is the position of the particle and an overdot denotes a time derivative. The left-hand-side of Eq. (<ref>) is composed of an inertial term ẍ_d and a dissipative force term proportional to particle velocity ẋ_d. The first term on the right-hand-side is the wave-memory force on the particle due to its self-generated waves. At each instant of time, the particle generates waves W(x) centered at the particle location and decaying exponentially in time. By integrating over the history of the particle's trajectory, one can calculate the overall wave field generated by a linear superposition of the individual waves. The wave-memory force is then proportional to the gradient of this overall wave field at the particle location and can be written as

    F_self=-R ∫_-∞^t W'( (x_d(t)-x_d(s)) ) e^-(t-s)/τ ds.

Here, the parameter R is a dimensionless wave-amplitude of particle-generated waves and the parameter τ is a dimensionless wave decay rate and hence determines the memory of the system (see <cit.> and <cit.> for more details on the parameters). The second term on the right hand side is the external bias force applied to the particle and we take it to be a constant force i.e. F_bias=F. In experiments, droplet-generated waves have features of both spatial oscillations and spatial decay (see Fig. <ref>(a)). However, if we only retain spatial oscillations and consider an idealized cosine waveform i.e. W(x)=cos(x), then this still captures the key qualitative features of walking dynamics <cit.> and significantly simplifies the system [In terms of the experimental system of walking/superwalking droplets, although in this letter we have only investigated an idealized wave form W(x)=cos(x) that does not include spatial decay, a modified wave form with weak spatial Gaussian decay also show a similar mechanism for GNM and GPM in a smaller region of the parameter space (see Supplemental Material).] by allowing us to map the integro-differential equation in Eq. (<ref>) onto the following system of three nonlinear ordinary differential equations <cit.> (see <cit.> for a derivation):

    Ẋ   =Y-X+F, 
    Ẏ   =-1/τY+XZ, 
    Ż   =R-1/τZ-XY.

Here, X=ẋ_d is the particle velocity whereas Y and Z are related to the memory forcing with Y=R ∫_-∞^tsin(x_d(t)-x_d(s)) e^-(t-s)/τ ds, as the wave-memory force on the particle, and Z=R∫_-∞^tcos(x_d(t)-x_d(s)) e^-(t-s)/τ ds, as the wave field height at the particle location. With no applied bias force, F=0, the system in Eq. (<ref>) is equivalent to the celebrated Lorenz system <cit.> (see Supplemental Material). Hence, the motion of our memory-driven active particle can be directly mapped onto the dynamics of the Lorenz system. We numerically solve the dynamical system of Eq. (<ref>) in MATLAB using the inbuilt ode45 solver.




With the above correspondence between the particle's trajectory equation and the Lorenz system, we can map the attractors and phase-space features of the Lorenz system onto the motion and trajectories of the active particle <cit.>. We proceed by reviewing different attractors realized in the Lorenz model <cit.> of Eq. (<ref>) as a function of the memory parameter τ and map them onto the different types of trajectories for the particle. For small values of τ, there is only one equilibrium point of the system at (X,Y,Z)=(0,0,0) which is stable. This corresponds to a (Type I) non-walking state for the particle since its velocity is ẋ_d=X=0 (see Fig. <ref>(a)). With increasing τ, the Lorenz system undergoes a pitchfork bifurcation where the non-walking state becomes unstable and a symmetric pair of stable equilibrium points emerge, (X,Y,Z)=(±√(R-1/τ^2),±√(R-1/τ^2),1/τ); this corresponds to steady walking motion (Type II) of the particle to right/left (see Fig. <ref>(b)). Further increase in τ leads to a complex set of bifurcations in the Lorenz system <cit.> and eventually chaos emerges at large τ with a strange attractor in the phase space. Phase-space dynamics on the Lorenz chaotic attractor corresponds to a  (Type III) chaotic walk <cit.> for the active particle where it unpredictably switches between left and right moving states as the phase space trajectory jump between left and right “wing" of the Lorenz attractor (see Fig. <ref>(c)). 

 


It is common for periodic windows to exist within a chaotic regime of parameter space. Since the Lorenz system obeys a symmetry i.e. the system is symmetric about a rotation of 180 degrees around the Z axis, it restricts the types of periodic limit cycles that can exist. Two types of limit cycles can be found in the Lorenz system: (i) a symmetric limit cycle about the Z axis which would correspond to a back-and-forth oscillation for the particle with no net drift, and (ii) a symmetric pair of asymmetric limit cycles, as shown in Fig. <ref>(d), that corresponds to the particle undergoing back-and-forth oscillations along with a net drift either towards the left or right (Type IV). A parameter space plot showing the time-averaged particle velocity ⟨ẋ_d ⟩ = ⟨ X ⟩ in the (R,τ) space is shown in Fig. <ref>(a) which helps identify these oscillating states that have a net drift. Region I corresponds to non-walking state (X=0), Region II is steady walking (X=±√(R-1/τ^2)), Region III is mostly filled with chaotic walks with no net drift (⟨ X ⟩ = 0). Of particular interest is Region IV corresponding to bands of net particle drift. The mixture of positive (blue) and negative (red) drift states along this band indicates the sensitivity of these states to small changes in parameter values. 

Now under the application of a small bias force of F=0.05, the corresponding parameter space plot in Fig. <ref>(a) transforms to Fig. <ref>(b). We clearly see that the band corresponding to Region IV splits into two in the parameter space, one corresponding to the asymmetric limit cycle with a net drift opposite to the applied bias (red) and the other corresponding to the asymmetric limit cycle with a net drift in the direction of the applied bias (blue). We identify the former as the GNM band while the latter the GPM band. Further, on these separated bands, all initial conditions lead to the same limit cycle attractor, unlike the F=0 band where the basin of attraction of left/right drifting states in the initial-condition space are intricately mixed. After complete separation, the two bands migrate away from each other with increasing small bias F such that the GNM band moves below the F=0 band (black dashed curve) while the GPM band migrates above the F=0 band. 

We note that this band separation mechanism of coexisting asymmetric limit cycles is closely related to the separation mechanism of phase-locked periodic attractors reported by <cit.> and <cit.> for negative mobility of a passive particle in a periodic potential with external time-periodic driving. However, in contrast, the negative and positive mobility bands completely separate and migrate in the parameter space of our Lorenz model, whereas the two bands only partially separate in the system of <cit.> along with the negative mobility band shrinking and positive mobility band growing in the parameter space. 

To demonstrate more clearly how this separation of bands manifests in GNM and GPM for our memory-driven active particle, let us consider the system at two different memory parameter τ values for a fixed R, one value below the region IV band with lower memory τ=6.9 and the other above the band τ=12.2 with higher memory. If we fixed the system parameter at these values, and investigate how the average velocity (both time and ensemble averaged) varies as a function of the bias force, we get the results shown in Fig. <ref>(a). For lower memory (red curve), the average velocity shows a strong dip of GNM near F=0.05. This is because, the GNM band after separation, migrates through the parameter space with increasing F and passes through this point in parameter space near F=0.05. Similarly, for the higher memory value (blue curve), the average velocity shows a strong spike near F=0.05 due to the passing of the GPM band. Figure <ref>(b) further shows the sensitivity of these behaviors as a function of the system memory τ. With no bias force (gray curve), there is no net mobility of the particle (when ensemble averaged) at all values of memory τ. However, with a small applied bias, the splitting of the region IV band results in GNM regions for lower memory and GPM regions for higher memory. This highlights the sensitivity of anomalous transport behaviors to memory of our system under small applied bias. 

In summary, we have shown a dynamical mechanism rooted in the celebrated Lorenz system model for the generation of giant negative mobility and giant positive mobility for a memory-driven active particle under a small applied bias. The degenerate pair of asymmetric limit cycles existing for the same parameter values completely separate and migrate in the parameter space upon small applied bias, resulting in the emergence of memory-dependent GNM and GPM for the active particle. We believe this dynamical mechanism to be general to many low-dimensional chaotic systems and it might be at play in generating anomalous transport behaviors for deterministic active particles. Further, such mechanisms may be used to tune the internal parameters of the system in order to control a particle's transport behaviors, or in sorting of active particles with different intrinsic properties.
Acknowledgments. This research was supported by the Leverhulme Trust [Grant No. LIP-2020-014]. 

Some of the numerical results were computed using the Hydra cluster in the Department of Physics at the University of Oxford.






 Paper Title: Fragmentation and aggregation of cyanobacterial colonies.








1]Yuri Z. Sinzato[y.z.sinzato@uva.nl, ORCID: 0000-0002-3324-5712]
2]Robert Uittenbogaard
3]Petra M. Visser
3]Jef Huisman
1]Maziyar Jalaal[m.jalaal@uva.nl, ORCID: 0000-0002-5654-8505]

[1]Van der Waals-Zeeman Institute, Institute of Physics, University of Amsterdam, Science Park 904, Amsterdam, 1098XH, The Netherlands
[2]Hydro-Key Ltd, Haelen 6081EA, The Netherlands
[3]Department of Freshwater and Marine Ecology, Institute for Biodiversity and Ecosystem Dynamics, University of Amsterdam, Science Park 904, Amsterdam, 1098XH, The Netherlands




Fragmentation and aggregation of cyanobacterial colonies
    [
    
========================================================





§ ABSTRACT 
Fluid flow has a major effect on the aggregation and fragmentation of bacterial colonies. Yet, a generic framework to understand and predict how hydrodynamics affects colony size remains elusive. This study investigates how fluid flow affects the formation and maintenance of large colonial structures in cyanobacteria. We performed experiments on laboratory cultures and lake samples of the cyanobacterium Microcystis, while their colony size distribution was measured simultaneously by direct microscopic imaging. 
We demonstrate that EPS-embedded cells formed by cell division exhibit significant mechanical resistance to shear forces. However, at elevated hydrodynamic stress levels (exceeding those typically generated by surface wind mixing) these colonies experience fragmentation through an erosion process. We also show that single cells can aggregate into small colonies due to fluid flow. However, the structural integrity of these flow-induced colonies is weaker than that of colonies formed by cell division. We provide a mathematical analysis to support the experiments and demonstrate that a population model with two categories of colonies describes the measured size distributions. Our results shed light on the specific conditions wherein flow-induced fragmentation and aggregation of cyanobacteria are decisive and indicate that colony formation under natural conditions is mainly driven by cell division, although flow-induced aggregation could play a role in dense bloom events. These findings can be used to improve prediction models and mitigation strategies for toxic cyanobacterial blooms and also offer potential applications in other areas 
such as algal biotechnology or medical settings where the dynamics of biological aggregates play a significant role.

Keywords: Algal blooms | Cyanobacteria | Hydrodynamics | Microcystis 





§ INTRODUCTION



Many environmental and pathogenic bacteria exhibit colonial forms of life, either as surface biofilms <cit.> or non-attached aggregates <cit.>. In most situations, these colonies are subjected to hydrodynamic stresses, which may lead to flow-induced fragmentation and aggregation <cit.>. These physical processes, combined with biological growth, determine the fate of aggregates and the microbial lifestyle. Nonetheless, despite their abundance and fundamental importance, a generic framework to study the effects of fluid flow on bacterial colonies, as well as their fragmentation and aggregation across scales, is still in its infancy <cit.>. Here, we focus on colonies of bloom-forming cyanobacteria and ask the question: how flow-induced aggregation and fragmentation regulate dynamic changes in colony size?


Toxic cyanobacterial blooms are a major nuisance for freshwater ecosystems worldwide <cit.>. These blooms often lead to increased turbidity, cause harm to other aquatic organisms, and deteriorate water quality. Recent evidence indicates that the severity and frequency of cyanobacterial blooms are increasing due to eutrophication and climate change <cit.>. Consequently, water management authorities show growing interest in techniques to prevent, monitor, and control these toxic blooms <cit.>. A particular effort is being taken to combat blooms of Microcystis, a non-motile colony-forming freshwater cyanobacterial genus that dominates many aquatic ecosystems worldwide <cit.>. Fluid dynamics is at the heart of Microcystis bloom formation, but also on some of the methods to control them. A large colony size and positive buoyancy give Microcystis a higher average light dose in comparison to non-buoyant algae <cit.>. Control methods such as deep artificial mixing can disrupt this competitive advantage by increasing the turbulence level in lakes <cit.>. Yet, the mechanisms of Microcystis colony formation, and in particular, the impact of fluid flow on the formation of colonies are poorly understood.

Microcystis cells are held together by a layer of extracellular polymeric substances (EPS), facilitating the formation of large colonies of up to a millimeter in diameter with diverse morphological arrangements <cit.>. Two main mechanisms for colony formation have been proposed <cit.>: (i) cell division, i.e., clonal expansion of cells which remain attached by their encapsulating EPS layer after division, and (ii) aggregation, i.e., initially independent cells/colonies collide and adhere to each other. On the one hand, sequencing of the genetic diversity within Microcystis  colonies supports the hypothesis that colony formation under natural conditions is primarily driven by cell division <cit.>. On the other hand, cell aggregation can occur on a shorter time scale and may offer improved protection against high grazing pressure <cit.>.
Aggregation becomes significant when the collision frequency is high enough for aggregation rates to overcome cell division rate <cit.>, and the stickiness of colonies is sufficiently large (controlled by the EPS composition, pH, and ionic strength of water or medium <cit.>). In addition to aggregation, fluid flow also influences colony formation through fragmentation, potentially serving as a mechanism for colony reproduction <cit.> to overcome the growth limitations of large colonies <cit.>. Beyond modulating colony size through aggregation and fragmentation, fluid flow has also been proposed as a driving factor in changes in colony morphology and growth rate <cit.>.

Flow-induced alterations in colony size and morphology impact the vertical migration and grazing resistance of colonies, thereby influencing the success of the Microcystis population. This bears significance for forecasting models which have often assumed a fixed colony size <cit.>. Information on colony size is also important for bloom control via artificial mixing systems, which rely on estimates of colony flotation velocity, determined by both colony shape and size  <cit.>. Moreover, it remains uncertain whether mixing systems, such as bubble plumes, can enhance their efficiency by fragmenting colonies or whether artificial mixing may inadvertently promote aggregation. Colony fragmentation has been observed in grid-stirred <cit.> and
propeller mixed tanks <cit.> at energy dissipation rates comparable to bubble plumes <cit.>. However, these studies reported averaged dissipation rates, while local values near the fast moving surfaces can be orders of magnitude higher <cit.>. Furthermore, the size distribution of the colony fragments has not been characterized.

In the present study, we investigate to what extent Microcystis colony fragmentation and/or aggregation occurs at levels of turbulence observed for naturally or artificially mixed lakes. To this end, colonies of a laboratory culture of Microcystis were subjected to various intensities of shear flow (Fig. <ref>). We identified the mechanism of colony fragmentation using direct measurements of dynamic changes in colony size distribution under controlled laboratory conditions and analyze the characteristics of colonies formed by aggregation and cell division. Furthermore, the fragmentation behavior of the laboratory culture was compared with field samples of Microcystis spp. We also developed a numerical model based on a two-category discrete population to simulate changes in the size of Microcystis colonies via aggregation and fragmentation. Finally, we provide a phase map based on flow characteristics and cyanobacterial abundance to highlight different regimes in which fluid flow plays a significant role in cyanobacterial colony formation.






§  FRAGMENTATION OF LARGE MICROCYSTIS COLONIES OCCURS THROUGH EROSION


A culture of Microcystis strain V163 was filtered to collect large colonies, and colonies on the filter were subsequently re-dispersed in fresh BG-11 medium up to the desired total biovolume fraction ϕ (Materials and Methods). The flow was generated by a cone-and-plate shear operating in an inertial regime, and it was characterized by the average energy dissipation rate ε̇, hereafter called dissipation rate (Materials and Methods). The size distribution of the colonies was measured over time by microscopic imaging of the suspension through the transparent lower surface (Fig. <ref>). The colony size is described here by its relative diameter l = d/d_1, where d is the Feret's diameter (maximum distance between two boundary points of the colony) and d_1 is the single-cell diameter. The number of cells in a colony, N_c, was calculated from its relative diameter via the relation N_c = l^f, where f = 2.09 is the average fractal dimension measured for colonies of Microcystis strain V163 (Materials and Methods). The size distribution was computed as the normalized biovolume distribution n(l), i.e., the number of cells in colonies in size bin { l; l+Δ l}, divided by the total number of cells in the suspension. 

The results show that the initial size distribution had a bimodal shape, with a subpopulation of small colonies peaking around l ∼ 1 - 2 (single cells, dimers, and trimers) and a subpopulation of large colonies, peaking around  l ∼ 20 - 40  (Fig. <ref>A). The cut-off size between the two subpopulations is defined here as the minimum of the bimodal distribution. Some small colonies (l ∼ 1 - 2), up to 20 % of the total biovolume, were always present in the initial distribution even after filtration, due to pre-stirring required to keep the suspension homogeneous. However, medium-sized colonies (l = 4 - 8) accounted for a negligible fraction of the distribution.

When subjected to an intense dissipation rate (ε̇ = 5.8 m^2/s^3), the large colonies of Microcystis strain V163 displayed considerable fragmentation, with their median diameter (50 % percentile of the biovolume distribution of the large colonies) decaying to about half of the initial value after a few hours (Fig. <ref>B). In contrast, the median diameter of small colonies showed little change in size and remained below l ≤ 2 during the entire experiment. The size distribution after 1 hour indicated that most colonies larger than l≳ 30 were fragmented, leading to a rise in the subpopulation of small colonies (Fig. <ref>C). Notably, the bimodal shape of the size distribution was preserved, with very few medium-sized colonies. This suggests that aggregation of single cells into larger colonies is negligible under this intense shear flow condition, and the main effect of the shear flow on the size distribution is the decrease in size of the large colonies concomitant with an increase in the fraction of small colonies. After a few hours of shear, about 80 % of the total biovolume was composed of small colonies (Fig. <ref>D). This time scale was much shorter than the cell division time, such that changes in the colony size distribution by division could be neglected within the experimental time <cit.>.

Direct measurement of the fragment distribution function g(l,l') (i.e., the frequency of fragments with size l arising from the fragmentation of a colony of size l') is an experimental challenge <cit.>. Individual fragmentation events could not be visualized as the residence time of colonies inside the field of view is too short. Instead, a suitable model can be identified from the rate of change in the biovolume distribution (Fig. <ref>E). A negative rate was observed for large colonies in the range l≳ 25, indicating a loss in that size range by fragmentation.  In contrast, the newly created fragments appeared as positive peaks, with a distinct peak close to the single-cell size. This result is compatible with an erosion mechanism, in which small fragments with one or a few cells are detached from the outer layer of the colony.  For an ideal binary erosion, a fragmentation event of a colony with i cells results in one cell being eroded from the surface of the colony with a complementary fragment of i-1 cells remaining.
Small deviations from an ideal binary erosion are observed in  Figure <ref>E, with most eroded fragments lying in the range 1 ≤ l ≤ 2. Nevertheless, ideal erosion provides a good approximation for the results. Accurate identification of complementary fragments from the rate of change in the biovolume distribution was more difficult, because the broad initial distribution of our suspension leaded to a broad distribution of complementary fragments. In this case, the rate of change in the biovolume distribution shifted from a net loss of eroded colonies at larger sizes ( l≳ 25) to a net gain of complementary fragments at intermediate sizes ( 5 ≲ l≲ 20).  

Assuming an ideal binary erosion mechanism, the fragmentation frequency κ(l) was computed from the temporal dependence of the biovolume distribution (Materials and Methods). Here, κ(l) is defined as the average number of fragmentation events (loss of a single cell) that one colony of size l suffers per unit of time. The fragmentation frequency of Microcystis strain V163 was calculated as a function of colony size for three values of dissipation rate (Fig. <ref>F). The results show that the fragmentation frequency increases with colony size. Furthermore, an increase in dissipation rate sharply intensifies the erosion of colonies, with a two order of magnitude increase in κ(l) when ε̇ varies from  1.1 to 9.6 m^2/s^3. 





§ AGGREGATION-FORMED COLONIES ARE LESS RESISTANT THAN DIVISION-FORMED COLONIES



In addition to inducing fragmentation, a turbulent flow may also increase the collision frequency of colonies. If the collisions are adhesive and the suspension is concentrated enough, then new colonies will be formed by aggregation. We subjected a single-cell suspension to the cone-and-plate shear flow to investigate whether aggregation can be a dominant colony-forming mechanism compared to cell division.  Cultures of the Microcystis strain V163 were filtered to obtain a single-cell suspension, and the total biovolume fraction was adjusted with fresh BG-11 medium (Materials and Methods). A typical initial bio-volume fraction distribution is presented in Figure <ref>A. In addition to single cells, some small colonies up to l = 3 were present in the filtered suspension due to imperfect filtering and/or aggregation during the pre-stirring. The size distribution was unimodal (single peak), in contrast with the bimodal distribution (double peak) in the suspension of division-formed colonies (Fig. <ref>A). After about 1 hour at a moderate dissipation rate (ε̇ = 0.019 m^2/s^3) and a total biovolume fraction of ϕ = 10^-4, the single-cell suspension showed aggregation of cells into small colonies, with a steady-state median diameter of l = 1.8 (Figs. <ref>B and <ref>C). Note that cell division can be neglected as the measurement time was much shorter than the division time. An increase in the collision rates due to a single-cell suspension led to an increase in the median diameter, although the increase was minor. At a high total biovolume fraction (ϕ = 5 · 10^-4), a steady-state median diameter of l= 2.4 was reached, with a negligible fraction of colonies above l ≳ 5 (SI Appendix, Fig. S4C). Therefore, the aggregation of single cells is likely limited by weak bonds between cells. At the steady-state regime of median diameter, the newly-formed aggregates with a large diameter would be quickly fragmented back into smaller parts.

A suspension of division-formed colonies under the same moderate dissipation rate (ε̇ = 0.019 m^2/s^3) did not suffer noticeable fragmentation (Fig.  <ref>D). The culture was grown in quiescent medium, without shaking, thus the colonies are assumed to have formed solely by cell division.
The erosion of division-formed colonies was negligible, as the fraction of small colonies remained below 20% (SI Appendix, Fig. S2B). Furthermore, the nearly constant median diameter of the large colonies (observed also for other total biovolume fraction values, SI Appendix, Fig. S2) shows that the division-formed colonies did not aggregate at this moderate dissipation rate.

The results above indicate that bonds formed during flow-induced aggregation (whether between two single cells or two division-formed colonies) are weak and cannot withstand intense dissipation rates. In contrast, the bonds between cells within a single division-formed colony are significantly more resistant. This difference in mechanical strength of the cellular binding in aggregation-formed versus division-formed colonies can be attributed to the structure of the EPS layer that surrounds the colonies. Aggregated colonies would have interstices between cells that are not filled with EPS. In contrast, cell division allows enough time for the gaps to be filled with secreted EPS, effectively increasing the bond strength between cells (Figs.  <ref>B and  <ref>D). However, we could not confirm this hypothesis visually, as the EPS layer in Microcystis strain V163 is too thin (< 1 μ m ) for a clear identification using optical microscopy.






§ DYNAMICAL CHANGES IN COLONY SIZE MODELED BY A TWO-CATEGORY DISTRIBUTION 


Our experimental results for colonies of Microcystis strain V163 (Fig. <ref>) suggest the presence of two categories of colonies: aggregated colonies (denoted as C_1) and 
division-formed colonies (C_2). The total biovolume distribution n(l) is given by the sum of both distributions n_1(l) + n_2(l), normalized such that ∫_0^∞ n(l) dl = 1. Within times shorter than the cell division time, the biovolume distribution of each colony category is described by a population model <cit.>:

    ∂ n_i∂ t(l,t) = A_i(l,t) + F_i(l,t) + T_i(l,t) ,

where A_i(l,t) and F_i(l,t) describe the aggregation and fragmentation rates as a function of colony size for each of the two categories, and T_i(l,t) is a transfer rate of biovolume between the two categories. The turbulent shear-induced collisions between colonies shift the distribution towards larger aggregates, and the aggregation rate is given by

    A_i(l,t)= 6ϕα_iπ[ 12∫_0^lβ(l',l_o) (ll' l_o)^2f -1  n_i(l',t)  n_i(l_o,t) dl'  . 
    .  -  n_i(l,t)∫_0^∞β(l,l') 1l'^f  n_i(l',t) dl' ]  ,

where l_o = (l^f - l'^f)^1/f. The collision frequency is described by the kernel β(l,l') and for a turbulent shear flow, we have β(l,l') = 0.163 (l + l')^3√(ε̇/ν) , where ν is kinematic viscosity of the fluid <cit.>. The ratio of aggregative collisions over total collisions is given by the stickiness parameter α_i. For the category C_1 (aggregated colonies), the stickiness is positive, α_1 > 0. In contrast, we neglect the aggregation of C_2 colonies (collisions between C_2-C_2 and C_1-C_2), since the bonds formed by aggregation would be weaker and quickly rupture again, such that α_2 = 0. This assumption is corroborated by the absence of aggregation between large colonies in our experiments (Fig. <ref>D).

The colony formation will be limited by fragmentation if the colony exceeds a critical size l^*_i = (ε̇/S_i)^-q_i/f, where S_i and q_i are constitutive parameters that depend on the aggregate structure and bond strength, with a lower bound at l^*_i≥1 <cit.>. 
We assume that aggregation-formed colonies have a small critical size, as the overlap between the EPS layer of each cell is narrow, leading to weak bonds. Meanwhile, division-formed colonies have an EPS layer that fills the pores between cells, thus increasing the resistance to hydrodynamic stress. The fragmentation rate is given by a first-order kinetic equation <cit.>:

    F_i(l,t) = - κ_i(l)  n_i(l,t) + ∫_l^∞g_i(l,l') κ_i(l')  n_i(l',t) dl' ,

where κ_i(l) is the fragmentation frequency, which is a function of the probability distribution function of the hydrodynamic stress. Assuming a Gaussian distribution, the fragmentation frequency is <cit.>:

    κ_i(l) = √(2 ε̇15 πν)exp(-M_i^2)erf(M_i) ,

where   M_i = √(15/2)(l/l^*_i)^-f/2q_i. Due to the scarcity of experimental data in the literature, simplified models are usually assumed for the fragment distribution function g_i(l,l'), such as erosion, equal fragments, or a uniform distribution <cit.>. Assuming a binary fragmentation (i.e., two fragments per fragmentation event), the fragment size distribution can be written in the form g_i(l,l') = δ(l - l_a) + δ(l - l_b), where l_a and l_b are the sizes of fragments. Based on the results from division-formed colonies under intense dissipation rate (Fig. <ref>E), an ideal erosion mechanism is adopted for category C_2 colonies, such that g_2(l,l') = δ(l - 1) + δ(l - √(l'^f -1)). In contrast, the experimental results for the small colonies (Fig. <ref>C) show a wider size distribution of eroded fragments which can not be captured by an ideal erosion mechanism, so an equal fragments mechanism is adopted for category C_1 colonies, i.e., g_2(l,l') = 2 δ(l - l'/√(2)). The size distributions of C_1 and C_2 colonies are coupled through a transfer rate of biovolume T_i(l,t). We propose that the cells eroded from C_2 colonies are transferred to category C_1, such that:

    T_1(l,t) = - T_2(l,t) = δ(l - 1) ∫_l^∞κ_2(l')  n_2(l',t) dl' ,

while the complementary fragments of C_2 colonies remain in that category.

We solved the balance equation (<ref>) using a zero-order discrete population balance, or sectional method, in which the size distribution is described by uniformly sized bins where the size distribution is evaluated at a representative size centered on the bin <cit.>. The simulations were initialized with the size distributions at t=0 obtained in our experiments. The two pairs of parameters for critical size ({q_1,S_1} and {q_2,S_2}) and the stickiness parameter (α_1) were used to fit the model to the experimental data (Materials and Methods). 

For an initial suspension composed mainly of large colonies at an intense dissipation rate (ε̇ = 5.8 m^/s^3), ), the model prediction of a decrease in the median size of C_2 colonies up to a steady-state diameter is in good agreement with the experimental findings (Fig. <ref>B). Moreover, the erosion mechanism accurately predicts the biovolume transfer from C_2 to C_1 colonies (Fig. <ref>D). The use of the erosion mechanism for the fragment distribution function is further corroborated by previous studies simulating bacterial aggregates <cit.>. For C_1 colonies at this intense dissipation rate, the model predicts that their median diameter remains near l^* = 1 (i.e., the lower bound for the critical size). However, the experimental results show a slightly wider size distribution of both the small and large colonies than predicted by an ideal erosion mechanism (Fig. <ref>C). Nevertheless, the global behavior of the size distribution is well described by the two-category model. Furthermore, a very good agreement is also observed between the measured fragmentation frequency and the model predictions given by Eq.(<ref>) (Fig. <ref>F).

To complement the mathematical model and to further investigate the mechanical origins of the resistance of division-formed colonies to intense shear, we performed rheology tests on the cyanobacterial colonies (Materials and Methods). Simple and oscillatory shear measurements revealed elasto-viscoplastic mechanics <cit.> of the concentrated aggregate with a yield stress of τ_y = 4.3 ± 0.3  Pa. The low volume fraction of cells in aggregate (∼ 12 %) indicates that the mechanical properties are mainly dictated by the EPS layer. Equating the average hydrodynamic shear stress to the yield stress results in a critical colony size of l^*_2 = 2.8 (Eq. <ref> in Materials and Methods).  The eroded fragments observed in the fragmentation experiments (Fig. <ref>E) are mostly smaller than this critical size. This suggests that large colonies will suffer erosion when the local hydrodynamic stress exceeds the yield stress of the EPS layer. Although the fracture behavior of colonies in suspension is expected to be different from a concentrated aggregate under shear rheology, the latter provides a reasonable and convenient estimate of the material properties of the colonies. 
Related to the critical size of the colony and their mechanical properties is the exponent q_2 (see equation <ref>). The best fit of the results in Fig. <ref> provided q_2 = 4.1. This value lies in the upper limit of previous results for colloidal aggregates <cit.>. Note that for colloids, q is related to the stress concentration around cracks in the fractal structure of the aggregate <cit.>, and low values of q are associated with brittle deformation, in which the stress concentration grows quickly with aggregate size. In contrast, cyanobacterial colonies have a soft plastic behavior that is less sensitive to large cracks, thus justifying the large value of q.

We further tested the model with the aggregation experiments. A suspension of C_1 colonies with an initial distribution close to the single-cell size (Fig. <ref>A) was simulated. The model agrees well with the experimental results for the median colony size, accurately predicting a steady-state value when there is a balance between collisions and fragmentation (Fig. <ref>B). The best fit of the results provided a small value for the stickiness parameter (α_1 = 0.023). The aggregation rate is linearly dependent on the stickiness and the total biovolume fraction (Eq. <ref>), thus an increase in these parameters (α_1 and ϕ) allows the suspension to reach the steady-state size faster. 
However, the steady-state size has low sensitivity to these parameters (α_1 and ϕ), as the fragmentation frequency has a higher-order dependence with colony size (Fig. <ref>F). The equal fragments hypothesis leads to a single-peaked size distribution, whose steady-state median increases with the total biovolume fraction. If an ideal erosion mechanism was assumed for the C_1 colonies, the peak would be fixed around the single-cell size (SI Appendix, Fig. S6). The experimental results for the C_1 colonies are in better agreement with the equal fragments hypothesis (Fig. <ref>C).

Note that the ideal erosion and equal fragments mechanisms are idealized scenarios, and the real distribution is expected to be more dispersed. Furthermore, a fractal description of small colonies is limited, as a continuum function between biovolume and diameter of colonies may not hold. Moreover, the turbulent model used for fragmentation (Eq. <ref>) poses two limitations: (i) the theory was developed for homogeneous Gaussian turbulence <cit.>, which differs considerably from the boundary-layer turbulence generated in a cone-and-plate shear <cit.>, and (ii) the colonies were larger than the Kolmogorov scale for the intense dissipation rate used in our fragmentation measurements (η = 20 μ m for a dissipation rate of ε̇ = 5.8 m^/s^3, SI Appendix), meaning that the colony size is comparable to the smallest eddies in the flow. Despite these limitations, the proposed two-category model provides a good approximation of the fragmentation process observed in our experiments. Simulations that take into account spatially inhomogeneous and non-Gaussian turbulence are demanding and lack an explicit analytic expression for κ(l) that can be applied to a population model <cit.>. 



§  FIELD SAMPLES OF MICROCYSTIS ARE MORE RESISTANT TO FRAGMENTATION DUE TO THICKER EPS LAYER





Samples of Microcystis spp. were collected from Lake Gaasperplas - the Netherlands (Materials and Methods). A major difference in fragmentation behavior is observed between the laboratory cultures of Microcystis strain V163 and the field samples of Microcystis spp. While the laboratory cultures suffered intense fragmentation at a dissipation rate of ε̇ = 5.8  m^2/s^3, the colonies collected from the field showed negligible fragmentation at this dissipation rate (Fig. <ref>A). The size distribution of field colonies also had a bimodal shape both initially and after 1 hour of flow (Figures <ref>C and E). The small colonies displayed an increase in size due to aggregation up to 2.5x times the initial median size (Fig. <ref>D), although the fraction of small colonies remained below 11% even after a few hours of shear (Fig. <ref>F). The large colonies displayed only a small increase in size due to aggregation and quickly reached a steady-state size. 

The higher resistance of field colonies to shear can be attributed to the thick EPS layer that surrounds the colony, which could be seen under brightfield microscopy using a negative staining (Fig. <ref>B, Materials and Methods). In contrast, this layer was barely visible in colonies of laboratory cultures of Microcystis strain V163. Nevertheless, the limited amount of aggregation observed for large field colonies indicates that aggregation-formed colonies have a smaller resistance to shear than division-formed colonies, similar to the results observed for laboratory colonies (Fig. <ref>). Differences in the morphology of colonies from field samples and laboratory cultures are frequently observed in the literature, with the latter often presenting only single cells or small colonies <cit.>. Colony formation may be influenced by various factors <cit.>, where especially high calcium and low nitrogen concentrations are identified as factors promoting high EPS production <cit.>. This is consistent with the calcium and nitrogen concentrations measured in Lake Gaasperplas (71 mg/L Ca2+ and 0.04 mg/L N) versus BG-11 medium (9.8 mg/L Ca2+ and 246 mg/L N) (Materials and Methods).




§  AGGREGATION, FRAGMENTATION, OR CELL DIVISION? A GUIDELINE


Under which conditions are aggregation, fragmentation or cell division the dominant processes in colony formation? In a quiescent fluid, cyanobacterial colonies are expected to grow solely due to cell division with a specific growth rate μ =N_c^-1d N_c / dt, which is a function of various environmental factors such as light intensity and nutrient availability <cit.>. Let l = ∫_0^∞l   n(l,t)dl be the average colony size of a suspension. Assuming that the cells remain attached after division, the rate of increase in the average colony size will be 

    1ldldt = μf .

The increase in colony size will progress until a maximum size l_m is reached. The upper limit in colony size is still a subject of intense discussion, with species variability, reduced division rate at large sizes, and dissolution of the EPS layer being important factors <cit.>. If the suspension is concentrated enough, collisions between colonies will also increase the average colony size due to aggregation. Let us consider initially a monodisperse suspension with a single size l. At short times, the rate of increase in the average colony size due to aggregation will be (SI Appendix)
  
    1ldldt = 2.49(√(2)-1) ϕ α l^ 4-2f√(ε̇ν) .

Assuming f = 2 for simplicity (which is similar to the measured value for Microcystis strain V163), we can use Eqs. (<ref>) and (<ref>) to define a dimensionless number Ag, quantifying the ratio between the rates of increase in the average colony size due to aggregation and cell division, given by

    Ag = 2.06 α ϕμ √(ε̇ν)

The dimensionless ratio Ag reveals the dominant mechanism for colony formation (cell division is dominant for Ag ≪ 1, while aggregation is dominant for Ag ≫ 1) and depends on the total biovolume fraction ϕ and dissipation rate ε̇, as shown by the phase diagram in Figure <ref>. For a lake with a total biovolume fraction corresponding to an early bloom (ϕ = 3 · 10^-7, according to WHO alert level 1 <cit.>), a low dissipation rate typical of wind-induced turbulence (ε̇∼ 10^-7 m^2/s^3) <cit.>, a specific growth rate of μ∼ 0.008 h^-1 <cit.>, a kinematic viscosity of ν = 10^-6m^2s^-1 and a stickiness of α∼ 0.02, we estimate a ratio Ag ∼ 10^-3. Under these conditions, the colony size increases due to cell division while the role of aggregation is negligible (region I in Fig. <ref>). In contrast, during a dense bloom event, the scum layer can reach a very high biovolume fraction of ϕ∼ 10^-2 <cit.>, which leads to Ag ∼ 10^2 (keeping all remaining parameters constant). Therefore, colony formation in dense scum layers is dominated by colony aggregation (region II in Fig. <ref>). The aggregation process can form larger colonies which will have an enhanced buoyancy, thus assisting in the persistence of the scum layer. 

Fragmentation limits the maximum colony size to a critical value l^*. Our experimental results (Fig. <ref>) indicate that aggregated colonies have a lower critical size than division-formed colonies for the same turbulence intensity. Conversely, a higher critical dissipation rate is needed for division-formed colonies ( ε̇_div) than for aggregated colonies (ε̇_agg) to achieve the same critical size l^* = l_m. This creates an interesting regime at high biovolume fractions and dissipation rates in the range ε̇_agg < ε̇< ε̇_div, in which only division-formed colonies can survive the turbulent flow, even though aggregation is faster than division (Ag ≫ 1) (region III in Fig. <ref>). Typical values of critical dissipation rate for our experiments are ε̇_agg = 2.1 · 10^-3 m^2/s^3 and ε̇_div = 3.7   m^2/s^3, assuming a maximum colony size of l_m = 50. Under natural turbulent conditions (e.g. surface wind-mixing), the dissipation rate is normally below ε̇∼ 10^-6 m^/s^3 <cit.>, which is insufficient to induce colony fragmentation. Artificial mixing systems can achieve levels of turbulence that are many orders of magnitude higher. In the core of bubble plumes, dissipation rates of ε̇∼ 10^-4 - 10^-1 m^2/s^3 are reached <cit.>. This value was sufficient to prevent aggregation of large colonies in our experiments but was not high enough to induce fragmentation of division-formed colonies. Moreover, 
colonies remain only a short time (∼ 1 min) in the bubble plume core, after which they are dispersed to weaker turbulent regions (dissipation rates of ε̇∼ 10^-7 - 10^-4 m^2/s^3 just a few meters away from the bubble plume core). The intense hydrodynamic stress required to fragment division-formed colonies (region IV in Fig. <ref>) can only be achieved by mixers with fast-moving surfaces, such as the cone-and-plate setup used in our experiments (ε̇∼ 10^-2 - 10^1 m^2/s^3) or propellers used in stirred tanks (ε̇∼ 10^0 - 10^2 m^2/s^3) <cit.>. These mixing devices can provide intense dissipation rates in small fluid volumes, but their implementation across entire lakes would be excessively energy-consuming.






§ CONCLUSIONS



Despite the widespread presence of bacterial colonies across species and environmental habitats, the influence of fluid flows on colony formation remains a fertile ground for continued exploration. A significant challenge is the lack of a generic framework in which the aggregation and fragmentation of colonies can be studied under various hydrodynamic stresses. We take a major step towards overcoming this challenge by presenting a combination of experiments and theory to elucidate the underlying mechanisms that determine the size distribution of cyanobacterial colonies over a wide range of flow intensities. Our experiments show that division-formed Microcystis colonies can be fragmented under intense dissipation rates ε̇∼ 3 - 10  m^2/s^3, following an erosion mechanism. Previous studies <cit.> have observed significant fragmentation of Microcystis colonies in field samples at dissipation rates as low as ε̇∼ 10^-4 m^2/s^3. However, such studies primarily report volume averages of the dissipation rate, overlooking the likelihood of fragmentation occurring near fast-moving solid surfaces (grid stirrer <cit.> and propeller <cit.>), where the local hydrodynamic stresses are orders of magnitude higher. Instead, our results indicate that fragmentation of division-formed colonies can not be achieved with wind mixing or aeration systems.  

We have demonstrated that flow-induced aggregated colonies of Microcystis form weaker structures than division-formed colonies, likely due to the weak EPS bonds between cells. Furthermore, we showed that colony aggregation is unlikely in lakes under early bloom conditions, as the collision frequency of colonies is too low. Our results, based on the mechanical properties of colonies, are corroborated by previous studies based on the sequencing of individual Microcystis colonies <cit.>. These studies have identified that the genetic diversity within Microcystis colonies is much smaller than the diversity between Microcystis colonies, thus supporting the hypothesis that colony formation is dominated by cell division rather than by aggregation of colonies. Nonetheless, our results demonstrate that during scum formation in very dense blooms or phytoplankton patches, flow-induced aggregation likely plays a role and may quickly increase colony size. 
A summary of these outcomes is presented in the phase map illustrated in Figure <ref>.

From a broader perspective, our findings have relevance beyond predicting and mitigating cyanobacterial blooms. Since fluid flow affects the colonies and aggregates of many species <cit.>, the experimental methods and theoretical framework presented here, along with the identified mechanisms, could serve as a foundation to better understand the relationship between mechanics and biological functions and traits of colonies under hydrodynamic stresses and their morphodynamics and lifestyle. Other examples include non-natural settings, such as algae or cyanobacteria cultured in bioreactors, where dense cell populations experience intense mixing <cit.>. The results presented here could serve as design guidelines for such applications.



§ MATERIALS AND METHODS




 §.§ Sample preparation


Microcystis strain V163 cultured under laboratory conditions, as well as samples collected at Lake Gaasperplas - The Netherlands containing Microcystis spp. colonies were used  for the experiments (Fig. <ref>). Microcystis strain V163 was originally isolated from Lake Volkerak - The Netherlands  <cit.>, grows in colonies and was obtained from the algal collection of the Institute for Biodiversity and Ecosystem Dynamics, University of Amsterdam. The Microcystis strain V163 was batch-cultured in Erlenmeyer flasks with standard BG-11 medium. The cultures were kept in an incubation chamber at a temperature of 20 ^oC. The light intensity was kept at 12 μmol photon m^-2s^-1 with a light:dark cycle of 14h:10h. The cultures were diluted with new BG-11 medium every week and kept at a maximum total biovolume fraction of ϕ = 5 · 10^-4. The cultures were not shaken except after the addition of medium and before filtration. The Microcystis strain V163 has a single-cell diameter of d_1 = 6.8 ± 1.4 μ m (SD, N = 189). We categorize partially divided cells also as single cells, as these also behave as single units in a fragmentation/aggregation process. When considering only spherical cells not undergoing division, a diameter of 5.1 ± 0.8 μ m (SD, N = 328) was obtained. Colonies of up to 500 μ m were present in the culture under the incubation conditions as described before. 

Samples of Microcystis spp. were collected from Lake Gaasperplas - the Netherlands (SI Appendix for water monitoring data and location of sampling point) on 14th August 2022. Water from the surface layer was filtered over a plankton net with a 100 μ m mesh size. Samples were kept refrigerated at 4 ^oC, and the experiments were conducted over the next three days. The plankton composition was dominated by Microcystis spp. (89 ± 7 % , SD, N=231), with  Microcystis aeruginosa being the main morphospecies, with an average single-cell diameter of d_1 = 7.0± 1.4  μ m (SD, N = 100). Colonies of Dolichospermum spp. were present in lower numbers. However, no differentiation of colony species was made during image processing, as it was not feasible to isolate them.

Negative staining was used to verify for the presence of an EPS layer surrounding the colonies of Microcystis strain V163 and of Microcystis spp. in field samples. Nigrosin was added to the surrounding medium, causing the EPS layer to appear bright under brightfield microscopy.

Upon appropriate filtering of the culture, we obtained a suspension with a specific size range. Filter paper with a 25 μ m pore size was used to obtain a suspension primarily consisting of single cells (colonies with 2 or 3 cells also present in small numbers). For a suspension mainly comprising large colonies, we filtered the culture with 53 μ m  nylon mesh. The colonies collected in the mesh were then redispersed in a new BG-11 medium. To estimate the total biovolume fraction of the suspension, 1 ml of the sample underwent a heat treatment to disaggregate colonies <cit.>. At least 1000 cells were counted in a Neubauer hemocytometer assisted by the image processing software ImageJ. The total biovolume fraction of the suspension was calculated as ϕ = cv_1, where c is the cell concentration and v_1 is the average single-cell volume. Prior to each experiment, the desired biovolume fraction was adjusted using new BG-11 medium for the Microcystis strain V163 culture and with filtered (0.8 μ m) water from Lake Gaasperplas for field samples of Microcystis spp..



 §.§ Cone-and-plate shear


A cone-and-plate shear flow was selected for our experiments (Fig. <ref>B and SI). This geometry is commonly used for standard rheological measurements because it provides a homogeneous shear rate under a laminar flow regime. Although less common in inertial regimes, it has been recommended by previous studies to investigate the effects of hydrodynamic stress in tissues <cit.>. Furthermore, the mixing provided by turbulence helps maintain a homogeneous concentration throughout the suspension. 

Our setup consists of an upper conical probe made from transparent acrylic, with an outer diameter of 42 mm and a cone complementary angle of θ = 5.44^o (Fig. SI). The conical probe is attached to a steel shaft, and it is driven by a rheometer head Anton Paar DSR 502. The lower surface consists of a 1mm thick glass slide. A cylindrical acrylic ring confines the fluid, preventing spillage of the sample due to strong centrifugal forces. The inner diameter of the ring is 46 mm, which ensures that the nominal shear rate between the upper probe's outer surface and the ring is the same as between the conical and lower surfaces. A clearance gap of 10  μm was set between the tip of the cone and the lower surface to avoid solid-solid friction. 

At low rotational speeds, the flow is laminar with a homogeneous shear rate γ̇_n = ω/θ, where ω is the angular velocity of the conical probe. The flow transitions to a turbulent regime at faster rotations, as the inertial forces become dominant. Since the shear rate is not uniform at the non-linear regime, the nominal shear rate γ̇_n is no longer an appropriate measure for shear intensity. Instead, the average intensity is given by the energy dissipation rate ε̇, which measures the rate of energy loss by viscous dissipation per unit fluid mass. Under steady-state conditions, this dissipation rate balances the power injected into the fluid, therefore:

    ε̇ = T ωρ   V ,

where ρ and V are, respectively, the density and volume of the fluid sample, and T is the torque on the conical probe, measured by the rheometer head. The flow regime can be identified from the energy dissipation rate curve against the angular velocity, with a ∼ω^2 dependence in the laminar regime and a  ∼ω^5/2 dependence in the turbulent regime (SI Appendix, Fig. S7B). All shear experiments conducted in this work were in the latter regime. The average hydrodynamic shear stress can be calculated from the dissipation rate by 

    τ_t = ρ √(ν ε̇).


The experimental setup provides a configuration where the dissipation rate can be controlled over a large range, while stress values can be measured with high precision. This, combined with optical access and simultaneous microscopy, presents a well-suited tool to study the response of biological samples under fluid flow.




 §.§ Fragmentation and aggregation experiments


The fragmentation and aggregation experiments involved subjecting a suspension of Microcystis colonies to the cone-and-plate shear and using brightfield microscopy to image the colonies under shear. The lower surface was positioned on top of an inverted microscope Nikon TS100 with a 4x objective lens. The center of the field of view was located 15 mm from the rotation axis and with the focal plane set 0.5 mm above the lower surface. Illumination was provided by a Schott KL 1500 HAL light source equipped with a cyan filter (low pass with a cut-off wavelength of 555 nm) to minimize the photosynthetic activity of the sample. Images were captured with a Prime BSI Express sCMOS camera with a resolution of 1.63  μ m per pixel. Prior to each experiment, a reference background image was created using distilled water, which was then discarded. A volume of 3.5 ml of the suspension of Microcystis colonies was pipetted in between the shear surfaces. The shear protocol was as follows: pre-shear at ε̇ = 1.1 m^2/s^3 for 1 minute to mix the suspension; 12 shear cycles of 5 minutes each followed by 12 cycles of 20 minutes each at the target dissipation rate (imaging cycles, at low dissipation rate of 0.019 m^2/s^3 for 1 min, were intercalated with each shear cycle). During the imaging cycles, 200 frames were captured at a rate of 10 fps, ensuring that each frame was sampled over a different set of colonies. Examples of the captured frames are shown in Figure <ref>. 



 §.§ Image analysis


The frames underwent pre-processing with background subtraction and thresholding (SI Appendix). Image segmentation was performed with the Scikit Image library for Python <cit.>. Colony size was determined by their Feret diameter d, which represents the maximum distance between two boundary points of the colony. The number of cells in a colony was estimated from the diameter using the formula N_c = (d/d_1)^f, where f is the average fractal dimension of the colonies. Since not all colonies were within the depth-of-focus, a correction function for the colony count was estimated using a suspension with a known size distribution. The image magnification was optimized for intermediate and large colonies Given the wide range of colonies (from single cells ≈ 7 μ m to large colonies ≈ 500 μ m). Colonies smaller than 25 μ m were measured with lower precision, necessitating a second correction function for accurate counting (SI Appendix, Eq. S1).  

To estimate the colony fractal dimension f, a reference sample of colonies had its size distribution characterized in a cylindrical counting chamber, followed by measurement of the total cell count in a hemocytometer. The fractal dimension was fitted such that the estimated cell count was equal to the measured count. For Microcystis strain V163, a value of f = 2.09 was obtained, while for the field samples of Microcystis spp. it was 1.64 .

For the direct computation of the fragmentation frequency, we assume a pure fragmentation with an ideal binary erosion, such that the time derivative of the colony size distribution is

    ddtN_p[i](t) = - κ[i]   N_p[i](t)l^f[i] - l^f[i-1] + κ[i+1]   N_p[i+1](t)l^f[i+1] - l^f[i] ,

where N_p[i](t), κ[i], and l[i] refer to the bin count, fragmentation frequency, and relative diameter at the center of bin i. This formula is only valid for the bins larger than the single cell, as the latter does not suffer fragmentation. The bin count is initialized with the experimental value and evolved in time using an implicit Radau scheme (SI Appendix). The fragmentation frequency of K_b of each bin was obtained by minimizing the squared error between the experimental and numerical bin counts. The script for image analysis is available https://github.com/FluidLab/CyanoB_Agg_Frag.githere. Detected features for all datasets and raw images for the dataset in Fig. <ref>A-E are available 10.5281/zenodo.11098426here.

 

 §.§ Yield stress of cyanobacterial colonies


The mechanical resistance of Microcystis colonies can be estimated by separating the colonies from the medium and measuring the yield stress of the concentrated aggregate. A suspension of Microcystis strain V163 was filtered through a 53 μ m mesh, and the large colonies were resuspended in fresh BG-11 medium. The sample was centrifuged for 30 min at 1000 g, and the pellet was separated. A dense gelatinous phase composed of cells and EPS was obtained, with cells accounting for 12 % of the total biovolume fraction. The yield stress of the concentrated aggregate was measured with an Anton Paar MCR 301 rheometer. The upper probe was a sand-blasted 50 mm cone with 1 ^o angle, while the lower surface was covered with sandpaper to avoid wall slip. Each sample was pre-sheared at 10 s^-1 for 2 minutes. Steady-shear and large amplitude oscillatory shear tests were performed (SI Appendix, Fig. S5). From a Herschel–Bulkley fit over the steady shear data, τ = τ_y + kγ̇^n, the dynamical yield stress τ_y = 4.3 ± 0.3  Pa (SD from best fit) was estimated. Due to the limited volume of samples available, only one repetition was possible for steady shear and oscillatory shear tests each. 



 §.§ Simulations


The balance equation (<ref>) for the total normalized biovolume distribution was solved numerically using a zero-order discrete population balance, or sectional method, in which the size distribution is described by uniformly sized bins, and the biovolume distribution is evaluated at a representative size centered on the bin<cit.>.  The integrals for the aggregation (Eq. <ref>), fragmentation (Eq. <ref>), and transfer (Eq. <ref>) rates were discretized with the approach by <cit.>. The biovolume distribution evolved in time with a Runge-Kutta 4-5 adaptive time step scheme written in Fortran. The simulation results were fitted with the experimental data by minimizing the absolute error of each category's 25th, 50th, and 75th percentiles of the biovolume distribution, using a Nelder-Mead algorithm <cit.>. The stickiness α_1 and for the pair of constitutive parameters  S_1 and q_1 for the critical size of small colonies (category C_1) were obtained by fitting the model to the dataset of a single-cell suspension under  ε̇ = 0.019 m^2/s^3 and ϕ = 0.01. The pair of constitutive parameters  S_2 and q_2 for the critical size of large colonies (category C_2) were obtained by fitting the model to the dataset of a large colony suspension under  ε̇ = 5.8 m^2/s^3 and ϕ = 0.01. These datasets were chosen as their aggregation/fragmentation times were similar to the measurement time, while for the other datasets, the variations were either too fast or slow to be captured in detail. The script for the simulation is available https://github.com/FluidLab/CyanoB_Agg_Frag.githere.



§ ACKNOWLEDGMENTS


We thank Maria van Herk, Pieter Slot, and Merijn Schuurmans for their support with laboratory cultures. We are grateful to Axel Gunderson for his assistance with fieldwork, and to Daan Giesen for his support with experimental setup design. We especially thank Johan Oosterbaan and Richard Steel for their contributions during the project's conception. We acknowledge funding from the Hoogheemraadschap van Rijnland. MJ acknowledges support from the ERC grant no. "2023-StG-101117025, FluMAB.



















§ SUPPORTING INFORMATION FOR


Title: Fragmentation and aggregation of cyanobacterial colonies


Authors: Yuri Z. Sinzato, Robert Uittenbogaard, Petra M. Visser, Jef Huisman and Maziyar Jalaal


Corresponding authors: Maziyar Jalaal (m.jalaal@uva.nl) / Yuri Z. Sinzato (y.z.sinzato@uva.nl)




§ SUPPORTING INFORMATION TEXT




 §.§ Field sample collection


Samples of Microcystis spp. were collected from Lake Gaasperplas - the Netherlands - at the sampling point 52°18'31.0"N 4°59'59.2"E on 14th August 2022, at 14:00. This is an artificial lake used for recreational purposes. No visible surface scums were present at the time of sampling. Water quality measurements of Lake Gaasperplas were obtained from the https://www.waterkwaliteitsportaal.nl/oppervlaktewaterkwaliteitDutch Water Authority (Rijkswaterstaat). Figure <ref> shows micrographic images of the phytoplankton samples. 



 §.§ Image analysis


This section describes the routine used to analyze raw images of the aggregation and fragmentation process of Microcystis colonies under a cone-and-plate shear setup. Each frame has dimensions of 2048x2048 pixels. Prior to each experiment, 3.5 ml of distilled water was pipette in the setup, and 200 frames were captured at ε̇ = 0.019 m^2/s^3 and 10 fps. A background image was generated by averaging over the frames and applying a Gaussian blur filter. After that, the distilled water was discarded, and 3.5 ml of cyanobacteria suspension was pipetted into the setup. Each time point in the data sets corresponds to a stack of 200 frames. Two treatments are made, one to detect small colonies(or single cells) and another to detect large colonies:

    
  * Small colonies (l<5): Image is cropped into a smaller section (500x500 pixels), to speed the processing. The background is subtracted using the reference image. The image intensity is renormalized and then thresholded. Small features (0.5<l<5) are segmented, measured, and counted.
    
  * Large colonies (l>5): A Gaussian blur is applied to filter small colonies. The background is subtracted using a reference image. The image intensity is renormalized and then thresholded. Small features (l>5) are segmented, measured, and counted.

For each time point, the counted features of the 200 frames were distributed in bins of width Δ l = 0.5. Since not all colonies are within the depth-of-focus, a correction function for the featured count was estimated using a suspension with a known size distribution, given by:

    N_c[i] =
        
             (λ_3 + λ_1  (λ_2 - λ_3))  N_p[i]    ,  if  l[i] ≤ 2 
    
            (λ_3 + λ_1  (λ_2 - λ_3)(4-l[i])/2)  N_p[i]     ,  if  2 < l[i] ≤ 4 
    λ_3  N_p[i]    ,  if  4 < l[i] ≤ 5 
    λ_4  N_p[i]    ,  if  l[i] > 5

Here, N_p[i] and N_c[i] are the number of features before and after correction in a bin centered at l[i]. To obtain the calibration parameters λ_i, a sample of filtered colonies of Microcystis strain V163 was subjected to the cone-and-plate shear setup for 2 min at ε̇ = 0.019 m^2/s^3, and then transferred to a counting chamber. The latter was used as a standard for the size distribution. The calculated parameters were λ_1 = 1, λ_2 = 73.13, λ_3 = 13.71 and λ_4 = 1.24. The small size range (l ≤ 4) has low precision due to the pixel resolution, therefore the parameter λ_1 was adjusted to ensure biovolume conservation. The value of λ=1 was used for t=0, and it was fitted in each subsequent time point so that the total cell count is constant through time. Finally, the normalized biovolume distribution for each time point is calculated.

For the direct computation of the fragmentation frequency, a pure fragmentation with an ideal binary erosion was assumed, such that the time derivative of the colony size distribution is

    ddtN_p[i](t) = - κ[i]   N_p[i](t)l^f[i] - l^f[i-1] + κ[i+1]   N_p[i+1](t)l^f[i+1] - l^f[i] ,

where N_p[i](t), κ[i], and l[i] refer to the bin count, fragmentation frequency, and relative diameter at the center of bin i. This formula is only valid for the bins larger than the single cell, as the latter does not suffer fragmentation. The bin count distribution N_p[i] is initialized with the experimental value, and an initial guess fragmentation frequency κ[i] is made for all bins larger than a single cell. The values of N_p[i] are evolved in time using the Python solver scipy.integrate.solve_ivp, with an implicit Radau method. For each bin, the cumulative squared error between the experimental and numerical bin counts is computed. The error is minimized using the values of κ[i] as the optimization parameter and the Python solver scipy.optimize.minimize. The uncertainty of κ[i] is propagated from (<ref>) talking into account the bin count uncertainty. The bin size for each dataset was chosen in order to minimize the bin count uncertainty. The script for image analysis is available https://github.com/FluidLab/CyanoB_Agg_Frag.githere.

 

 §.§ Fragmentation measurements with Microcystis strain V163


Additional measurements of fragmentation of large colonies of Microcystis strain V163 were performed. Colonies grown by cell division were filtered through a mesh of 53 μ m, and the collected colonies were redispersed in BG-11 medium. Under moderate dissipation rate of ε̇ = 0.019 m^/s^3 the fragmentation was negligible (Fig. <ref>), with the median size and fraction of large colonies remaining stable over 5 hours of flow. The behavior is independent of the total biovolume fraction, showing that aggregation rates are negligible for large colonies. In contrast, the steady median size for small colonies increases with the total biovolume fraction, although still an order of magnitude smaller than division-formed colonies. Significant fragmentation can be seen from a dissipation rate of ε̇ = 5.8 m^/s^3 (Fig. <ref>). For a large value of ε̇ = 9.6 m^/s^3, the small colonies become the largest fraction in below 5 min (the smallest time resolution in our setup). 

 

 §.§ Aggregation measurements with Microcystis strain V163


Additional measurements of aggregation of single cells of Microcystis strain V163 were performed. The suspension was filtered through a 25 μ m filter paper to select mostly single cells. Under moderate dissipation rate of ε̇ = 0.019 m^/s^3, the aggregation increased with the total biovolume fraction ϕ (Fig. <ref>). Under an intense dissipation rate of ε̇ = 5.8 m^/s^3, no aggregation was observed, and the median size of colonies remained equal to the initial value (near the single-cell size).




 §.§ Rate of increase in the average colony size due to pure aggregation


Consider a suspension of colonies subjected only to aggregation, neglecting cell division and fragmentation. In this case, the time derivative of the normalized biovolume distribution n(l,t) is given by <cit.>: 

    ∂∂ tn(l,t)= 6 ϕ απ[ 12∫_0^lβ(l',l_o) (ll' l_o)^2f -1  n(l',t)  n(l_o,t) dl'  . 
    .  -  n(l,t)∫_0^∞β(l,l') 1l'^f  n(l',t) dl' ]  ,

where l_o = (l^f - l'^f)^1/f, and the collision frequency kernel is β(l,l'). The average colony size is defined as l = ∫_0^∞l   n(l,t)dl. Applying this definition to Eq. (<ref>), we obtain that the time derivate of l(t) follows:

    ∂∂ tl(t)= 6 ϕ απ[ 12∫_0^∞∫_0^lβ(l',l_o)   l  (ll'   l_o)^2f -1  n(l',t)  n(l_o,t) dl'dl  . 
    .  -  ∫_0^∞∫_0^∞ n(l,t) β(l,l')   l 1l'^fn(l',t) dl'dl ]  .

If we assume that the size distribution remains monodisperse for short times, we can write that n(l,t) = δ(l - l(t)). Substituting this hypothesis into Eq. (<ref>) results that

    ∂∂ tl(t)= 6 ϕ απ[ 12∫_0^∞∫_0^lβ(l',l_o)   l  (ll' l_o)^2f -1 δ(l' - l) δ(l_o - l) dl'dl  . 
    .  -  ∫_0^∞∫_0^∞δ(l - l) β(l,l')   l  1l'^f δ(l' - l) dl'dl ]  .

Applying a variable change, the integral can be rewritten as:

    ∂∂ tl(t)= 6 ϕ απ[ 12∫_0^∞∫_0^∞β(l',l_o)   l  (ll' l_o)^2f -1  δ(l' - l) δ(l_o - l)  l_o^f-1  (l_o^f + l'^f)^1/f-1 dl_odl'  . 
    .  -  ∫_0^∞∫_0^∞δ(l - l)  β(l,l')   l  1l'^f δ(l' - l) dl'dl ]  .

Solving the integral and applying the model for the collision kernel in a turbulent shear flow <cit.>, β(l,l')=0.163 (l+l')^3 √(ε̇/ν), results that the rate of increase in the average colony size is:

    1ldldt = 2.49  (√(2)-1) ϕ α l^4-2f√(ε̇ν) .

In the limit of fractal dimension f = 2, the growth rate reduces to;

    1ldldt = 1.03 ϕ α √(ε̇ν) ,

which is independent of l.

 

 §.§ Rheological characterization of concentrated colonies of Microcystis strain V163

A steady-shear (Fig. <ref>A) and a large-amplitude-oscillatory-shear (Fig. <ref>B) tests were performed in a sample of concentrated colonies of Microcystis strain V163. Due to the limited volume of samples available after separation, only one repetition was possible for each test. Before each test, the sample was subject to a pre-shear at γ̇ = 10 s^-1 for 200s, followed by 100s of rest. For the steady-shear test, the minimum transient time was determined in a separate test (40s for γ̇ = 100 s^-1  to 200s for γ̇ = 0.001 s^-1). For the oscillatory shear, the minimum transient time was determined by the rheometer software. A Herschel–Bulkley fit was applied to the steady shear data, τ = τ_y + kγ̇^n, from which a dynamical yield stress τ_y = 4.3 ± 0.3  Pa (SD from best fit) was estimated.



 §.§ Comparison of hypothesis for the fragment distribution function


Assuming a binary fragmentation (i.e., two fragments per fragmentation event), the fragment size distribution can be written in the form g_i(l,l') = δ(l - l_a) + δ(l - l_b), where l_a and l_b are the size of fragments. Two ideal models for binary fragmentation are erosion and equal fragments, described by the following functions:

    g(l,l') = δ(l - 1) + δ(l - √(l'^f -1))    (Erosion),
    
            2 δ(l - l'/√(2))    (Equal fragments).

Due to a lack of experimental evidence of single colony fragmentation events, both models were tested for the category C_2 colonies.  The equal fragments hypothesis was chosen as it better captures the experimental results for the size distribution of Microcystis strain V163 colonies aggregated from single cells (Fig. <ref>).



 §.§ Cone-and-plate shear setup


The turbulent shear was generated by a cone-and-plate shear, with the dimensions depicted in Fig. <ref>A. The flow regime can be identified from the average energy dissipation rate ε̇ curve against the angular velocity ω, with a ∼ω^2 dependence in the laminar regime and a  ∼ω^5/2 dependence in the inertial region (Fig. <ref>B) <cit.>. The Kolmogorov scale η, which is an estimate of the size of the smallest eddies in a turbulent flow, can be calculated from the relation η = ν^3/4ε̇^-1/4. The Kolmogorov scale in our results range from η∼ 85  μm for the lowest dissipation rate (ε̇ = 0.019 m^2/s^3) to η∼ 18  μm for the highest dissipation rate (ε̇ = 9.6 m^2/s^3).
























 Paper Title: Line tension from dual-geometry sessile droplet measurements: Combining contact angle size dependence data for axisymmetric and cylindrical droplets to determine the line tension.




Line tension from dual-geometry sessile droplet measurements]Line tension from dual-geometry sessile droplet measurements: Combining contact angle size dependence data for axisymmetric and cylindrical droplets to determine the line tension


d.tatyanenko@spbu.ru



Department of Statistical Physics, Faculty of Physics, Saint Petersburg State University
  7–9 Universitetskaya nab., St. Petersburg, 199034, Russia



§ ABSTRACT 
We perform a thermodynamic analysis of various contributions to the size dependence of the contact angle cosine for axisymmetric and cylindrical sessile droplets. This shows that a widely used method to determine the line tension from the slope of the contact angle cosine dependence on the three-phase contact line curvature (for axisymmetric droplets) provides a certain combination of the line tension, adsorptions at the three interfaces, and the macroscopic contact angle. To extract the leading-order contact-line-related contribution and determine the line tension, we propose a simple technique using contact angle size dependences for axisymmetric and cylindrical droplets at the same conditions.




[
    Konstantin D. Apitsin
    August 5, 2024
=========================






§ INTRODUCTION


The contact angle of a large sessile droplet can be found from the well-known classical Young equation <cit.>

    cos =  -

with σ the surface tension of a corresponding interface,  the macroscopic contact angle; α, β, and γ mark the liquid, gaseous and solid phases, respectively, and double Greek superscripts mark the corresponding interfaces. The surface tensions here are thermodynamic (defined as the surface excesses of the grand potential per unit area of the interface). For small droplets, the thermodynamic line tension κ (defined as the line excess of the grand potential per unit length of the three-phase contact line <cit.>) affects the contact angle <cit.>:

    cos =  -  - κ/.

Here θ is the contact angle, r is the radius of the three-phase contact line, subscript “a” refers to axisymmetric droplet (see Fig. <ref>(a)); the line tension κ is implied to be constant. Eq. (<ref>) is usually called the modified Young equation.

Subtracting the modified Young equation (<ref>) from the classical  one (<ref>) and dividing the both parts of the resulting equation by , one usually arrives at <cit.>

    cos - cos = κ/.

This equation is widely used for contact-angle-based measurements of the line tension in laboratory experiments (see, e.g., <cit.>) and computer modeling (e.g., molecular dynamics simulations <cit.>). According to Eq. (<ref>), the slope of the graph cos vs 1/ must be equal to - κ/. The surface tension  is typically measured or calculated separately.

However, some authors have noted that Eq. (<ref>) is generally not correct <cit.>, since it assumes the corresponding surface tensions in Eqs. (<ref>) and  (<ref>) to be equal, whereas they correspond to different equilibrium sessile droplet sizes, which implies different values of the chemical potential(s). There are arguments that adsorption at the solid–liquid interface and its effect on the surface tension  affect cos in the same manner and with a similar magnitude as it is measured in experiments <cit.>, so the line tension itself is not necessary to explain the dependence of cos on 1/. Consideration with use of an interface displacement  model shows <cit.> that the contributions of the line tension and the adsorption at the solid–gas interface (and its effect on the surface tension ) to the  dependence of cos on 1/ can be comparable in the presence of a precursor film. Some authors <cit.> suppose only  to depend on the droplet size and Eq. (<ref>) to be correct, but the slope of the graph cos vs 1/ to give a quantity not equal to - κ/.  Sometimes the resulting “line tension” is called the apparent line tension <cit.>. General thermodynamic analysis shows <cit.> that the value of this slope has contributions both from the line tension and the adsorptions at all three interfaces (see also <cit.> for similar arguments).
             

How to “extract” the line-tension contribution and determine the value of the line tension from the measurements or calculations of cosθ vs 1/r? Thermodynamic analysis shows <cit.> that, for cylindrical droplets [long liquid “channels”, or “filaments”, or “ridges” attached to the substrate surface, see Fig. <ref>(b)], there is no κ/r term in the equation corresponding to (<ref>). It means that dependence of the contact angle  on the size of a cylindrical droplet (, its half-width  on the substrate) is governed by dependences of the surface tensions and the droplet size on the chemical potential(s). This suggests the idea to compare or combine data of size dependence of cosθ for axisymmetric and cylindrical droplets at the same conditions to exclude the surface-tension-related contributions and determine the line tension.

In section <ref>, we present a thermodynamic analysis of various contributions to droplet size dependence of the contact angle for both axisymmetric and cylindrical droplets in a system with a single-component fluid and an ideal smooth homogeneous solid substrate in absence of external fields. Based on it, in section <ref> we propose a simple technique that allows to determine the line tension from a combination of contact angle size dependences for axisymmetric and cylindrical droplets in a manner similar to use of Eq. (<ref>).

Cylindrical droplets are often used in modeling, e.g., in molecular dynamics (MD) simulations <cit.> due to convenient use of periodic boundary conditions along their longitudinal axes and due to expected independence of the contact angle of (or a weak dependence on) the droplet width. However, they are hardly available for experimental measurements due to Plateau–Rayleigh-type instability <cit.>. Thus, the technique we propose is mainly oriented to modeling, especially with methods not allowing for direct calculation of the line excess of the grand potential (e.g., MD simulations). The dimensions of such simulated axisymmetric and cylindrical droplets, i.e., R, r, and θ, can be “measured”, and these data can be used to determine the line tension.





§ THERMODYNAMICS AND CONTACT ANGLES





 §.§ Equilibrium conditions for sessile droplets

For thermodynamic description of a small sessile droplet on a partially wettable substrate, we consider a materially open system of a fixed volume including the droplet itself, the substrate, and the surrounding gas (vapor). The liquid and gaseous phases are supposed to be a single-component fluid at a given values of chemical potential μ and temperature T (that supposed to be constant).  The grand thermodynamic potential of the system decomposes into bulk, surface, and line parts:

    Ω =    - p^α - p^β + ω^γ V^γ
        +   +  +  + κ L

with V the volume, A the area of the interface, L the length of the three-phase contact line, p the pressure in the bulk phase, ω the bulk density of the grand thermodynamic potential (in fluid phases equals -p). Other quantities as well as notations of phases and interfaces has been explained above.

Let us consider two types of droplet geometry: (1) axisymmetric droplet [shaped as a spherical segment in our “macroscopic” thermodynamic description, see Fig. <ref>(a)] and (2) a fixed-length piece of a cylindrical droplet [shaped as a segment of a circular cylinder in our “macroscopic” thermodynamic description, see Fig. <ref>(b)]. In both geometries, the cross-section of the droplet in our “macroscopic” thermodynamic description is a circular segment, therefore,

    r = R sinθ.

Thus, the shape and size of the droplet can be described by two independent variables, e.g., (R, r), (R, θ) or (r, θ).

In both geometries, choosing the  dividing surface to be the surface of tension, one can obtain equilibrium conditions for sessile droplets by expressing all the volumes, areas and the length of the three-phase contact line in (<ref>), and solving one of the following sets of two equations


    ∂Ω(R,r,T,μ)/∂ R = 0     and     ∂Ω(R,r,T,μ)/∂ r = 0,
       ∂Ω(R,θ,T,μ)/∂ R = 0    and     ∂Ω(R,θ,T,μ)/∂θ = 0, 
       ∂Ω(r,θ,T,μ)/∂ r = 0    and     ∂Ω(r,θ,T,μ)/∂θ = 0,


and then taking into account the geometric relation (<ref>). Since T and μ are constant, the pressures and the surface tensions in (<ref>) will be constant at such variations (for , this also requires / R = 0 which is true namely for the surface of tension chosen as the  dividing surface).

The droplet-size-dependent surface tensions  and  for spherical and cylindrical interfaces can generally differ, though their difference is expected to be very small at the same value of μ (see subsection <ref>).


Let us chose R and θ as independent variables describing the sessile droplet. Introducing non-variable quantities for the total volume occupied by the fluid ≡ V^α + V^β and the total area of the substrate ≡ + and using relation (<ref>), one can express the volumes, areas, and the contact line length for an axisymmetric droplet as

    = π^3/3 (2 + cos)(1 - cos)^2,
       = 2πsin, 
     = 2π^2 (1 - cos),
       = π^2 sin^2 ,
     =  - ,
       =  - .


Decomposition (<ref>) of the grand potential then takes form

    (,) = - ( p^α - p^β) π^3/3 (2 + cos)(1 - cos)^2 
        + 2 π^2 (1 - cos) + ( - ) π^2 sin^2 
        + 2πsin - p^β - .

Application of the equilibrium conditions (<ref>) gives

    [t][8.5cm]
    - π^2 ( p^α - p^β) (2 + cos) (1 - cos)^2 
     + 4 π (1 - cos) + 2 π( - ) sin^2 
     + 2 πsin + 2 π∂/∂sin^2  = 0,
    
          - π^3  ( p^α - p^β) sin^3  + 2 π^2 sin + 2 πcos
        + 2 π^2 (  - ) sincos + 2 π^2 ∂/∂sincos = 0.

To exclude the pressure difference p^α - p^β, we multiply the first equation by (2 π)^-1sin^2   (1 - cos)^-2 and then subtract the second equation multiplied by (2 + cos)(2 π^2 sin)^-1 <cit.>. After some trigonometric simplifications and substitution of the result into one of the above equilibrium conditions, we arrive at the Laplace equation and the generalized Young equation as equilibrium conditions for an axisymmetric sessile droplet <cit.>

    p^α - p^β = 2/, 
    cos =  -  - / - /


For a cylindrical droplet,

    = ^2/4 (2  - sin 2),
     = ,
       = sin,
     =  - ,
       =  - ,

and the length  of the three-phase contact line does not depend on  and  [see Fig. <ref>(b)] and supposed to be constant. Contact angle  here is in radians. Decomposition (<ref>) of the grand potential then  takes form

    (,)    = - ( p^α - p^β) ^2/4  (2  - sin 2) + 
        + ( - ) sin +  - p^β - .

Application of the equilibrium conditions (<ref>) gives

    [t][8.1cm]
    - ( p^α - p^β)  ( - sincos) + 
     + ( - ) sin + ∂/∂sin = 0,
    
       [t][8.1cm]
    - ( p^α - p^β) ^2 sin^2  + 
      +  ( - ) cos + ∂/∂cos = 0.

Here we have employed trigonometric equalities  sin 2 = 2 sincos and 1 - cos 2 = 2 sin^2. To obtain a simpler equation for p^α - p^β, we multiply the first equation by cos and subtract the second equation multiplied by sin. After some simplifications and substitution of the result into one of the above equilibrium conditions, we arrive at the Laplace equation and the generalized Young equation as equilibrium conditions for an cylindrical sessile droplet, slightly different from Eqs. (<ref>) and (<ref>):

    p^α - p^β = /, 
    cos =  -  - /.


The absence of the coefficient 2 in the Laplace equation  (<ref>) for the cylindrical droplet can be explained by the fact that its  interface is curved only along one direction, whereas the spherical droplet has equal curvatures along two directions. The line-tension terms in the generalized Young equations (<ref>) and (<ref>) originate from variations of the line term κ L in (<ref>). For an axisymmetric droplet, = 2π and ()/ = 2π + 2π / giving the two terms / + / in Eq. (<ref>). For a cylindrical droplet,  does not depend on  and ()/ =  / giving the only term / in Eq. (<ref>).

For both geometries, we suppose the line tension to depend not only on T and μ, but also on the droplet size. This dependence is usually overlooked and, in this case, Eq. (<ref>) transforms into the modified Young equation (<ref>), whereas Eq. (<ref>) transforms into formally classical Young equation (<ref>) (with surface tensions, however, depending on the chemical potential, i.e. on the equilibrium droplet size, that makes the contact angle  different from , see subsection <ref>).

We suppose the contact line radius (half-width) r to be the only appropriate size variable the line tension can be considered to depend on. It is not clear for the derivation we used in this paper, and some authors suppose the line tension to depend also on the contact angle θ <cit.>, corresponding “notional” derivatives also appear in the work <cit.>. However, a variational derivation of Eqs. (<ref>), (<ref>), (<ref>), and (<ref>), which considers all three variables R, r, and θ to be independent in some sense, gives an argument in favor of r as the mathematically appropriate size variable for κ in both geometries <cit.>.

Simple dropping the size-dependent term κ/ r in the generalized Young equations (<ref>) and (<ref>) make them inexact (even if κ is supposed to depend on the droplet size). It can be seen in a model system if the line tension is calculated directly <cit.>. Also, the generalized Young equation is the simplest way to calculate κ/ r if θ and r are found and the surface and line tensions (the latter for axisymmetric droplet only) are calculated directly <cit.>.




 §.§ Size dependence of contact angle and relations between pressures, surface tensions, droplet size and chemical potential


Comparing the generalized Young equation (<ref>) or (<ref>) with the classical Young equation (<ref>), one should take into account that these two equations are written for droplets corresponding to different values of the chemical potential μ. The classical Young equation corresponds to a macroscopically large (strictly speaking, infinite) drop, i.e. liquid–gas bulk phase coexistence at the binodal. Let us mark the values of the quantities corresponding to the binodal with subscript “∞”. The surface tensions at the binodal then should be denoted as , , and , and the classical Young equation (<ref>) should be rewritten as

    cos =  - .


Subtracting the generalized Young equations (<ref>) and
(<ref>) from  Eq. (<ref>) we obtain, for an axisymmetric droplet,

    cos - cos = δ + / + / r
    and a similar equation for a cylindrical droplet:cos - cos = δ + /.

Here δ≡ - with ≡ - and ≡ -. To make these equations more similar to Eq. (<ref>), let us introduce δ≡ -. Then, Eqs. (<ref>) and (<ref>) can be rewritten as

    ( cos - cos ) = δ + δcos + / + /, 
        ( cos - cos) = δ + δcos + /.


Now we see, that, in the exact counterpart of Eq. (<ref>) for the difference cos - cosθ, we have 4 terms for an axisymmetric droplet (instead of 1) and 3 terms for a cylindrical droplet (instead of 0). How do they depend on the droplet size [the radius (half-width) of the droplet on the substrate]?

For the  liquid–gas interface, let us write down the isothermal Gibbs adsorption equations:

    = - μ,

where  is the adsorption (the surface excess of matter per unit surface area) at the  interface for corresponding geometry. Since the surface of tension is chosen as the dividing surface in both geometries, these equations do not contain (/ R)  R terms.

For each of the solid–fluid interfaces, similar equations can be written. Generally, they are more complicated (see, e.g., Sec. 6.3 in <cit.> and Sec. 3.4.3 in <cit.>), but in some simple cases (e.g., for non-deformable substrate and such a choice of the solid–fluid dividing surfaces that the amounts of matter of the immobile component of solid do not change in the solid phase at any change of μ) they take <cit.> the same simple form as (<ref>), so we can write down

    ≡(  - ) = (  - ) μ.

Integrating Eqs. (<ref>) and (<ref>) from the value  at the binodal to the current value of μ, we obtain, introducing δμ≡μ -,

    δ = - ∫_^μ(μ') μ' ≃ - δμ,  
    δ = ∫_^μ(  - ) μ' ≃(  - ) δμ.

Since () = () =, the asymptotic expression in (<ref>) is the same for spherical and cylindrical surfaces at the same value of μ, but the exact integral is generally not if (μ) ≠(μ). Thus,  and  generally differ, but at least in the second order in δμ:

    -  = δ - δ = ((δμ)^2).


The last asymptotic expressions in Eqs. (<ref>) and (<ref>) are valid for small values of δμ. How is this quantity related to the droplet size? This can be estimated using the Laplace equation (<ref>) or (<ref>) for the axisymmetric or cylindrical droplet, respectively. Applying the isothermal Gibbs–Duhem equations p = n μ to liquid and gaseous phases, respectively, we obtain p^α - p^β≃δμ with  the number density of molecules in α liquid phase. Here we have neglected the compressibility of the liquid (≃) and the number density of molecules  in the gas (≪). With use of the Laplace equations (<ref>) and (<ref>), and geometric relation (<ref>), we obtain

    2/ = 2sin/≃δμ, 
    / = sin/≃δμ.

We now see that δμ∝ 1/R = (1/r), i.e., small values of δμ correspond to large droplets, and δμ→ 0 expectedly corresponds to R →∞ and r →∞.

Thus, for the correction terms in Eqs. (<ref>) and (<ref>) we have

    δ≃2sin/(  - )  = (δμ) = (1/r), 
    δ≃ - 2sin/ = (δμ) = (1/r), 
    /≃/ = (δμ) = (1/r), where = at μ = (→∞) is the line tension of the macroscopic (infinite) droplet, andδ≃sin/(  - ) = (δμ) = (1/r), 
    δ≃ - sin/ = (δμ) = (1/r).


As for the terms κ/ r, it can be shown <cit.> that they are ( (δμ)^2 ) = ( 1/r^2 ). For sufficiently large droplets, sinθ can be approximated with sin, with a relative error (1/r), that makes Eqs. (<ref>), (<ref>), (<ref>), and (<ref>) valid with sin replaced by sin. The same applies to cos in the correction terms δcosθ in Eqs. (<ref>) and (<ref>).

Taking this into account, we can now write down the linear in 1/r approximations for Eq. (<ref>) and (<ref>):

    cos - cos≃/, 
    cos - cos≃/,




where

    ≡ + 2sin/(  -  - cos) 
    and   ≡sin/(  -  - cos)

are the quantities sometimes called the apparent line tensions <cit.> of axisymmetric and cylindrical droplets, respectively (see below).

Eqs. (<ref>) and (<ref>) show that the dependence of cos on 1/, even in the first order in 1/, is determined not only by the line tension, but also by the adsorptions at the three interfaces. In fact, Eq. (<ref>) should be treated as Eq. (<ref>) often is. This means that the slope of the graph cos vs 1/ at 1/→ 0 does not give the line tension itself but rather a combination of the line tension, the surface tension of the droplet free surface, adsorptions at the three interfaces, and the macroscopic contact angle. Some authors use the term “apparent line tension” meaning the value of the line tension calculated from the dependence of cosθ on 1/r (denoted further as cosθ (1/r)) at 1/r → 0 as if Eq. (<ref>) were the correct linear approximation. Then Eq. (<ref>) gives the value of the apparent line tension of axisymmetric droplets. More interesting, Eqs. (<ref>) and (<ref>) show that, for cylindrical droplets, cos also depends on 1/ in the first order in 1/. This has been found in simulations <cit.> and attributed to size dependence of the surface tension  expressed using the Tolman length <cit.> ≈/.  As it is seen from Eq. (<ref>), dependences of other surface tensions on the droplet size via the chemical potential could also made their contributions to the apparent line tension of cylindrical droplets.




§ LINE TENSION FROM DUAL-GEOMETRY CONTACT ANGLE DEPENDENCES ON DROPLET SIZE


One can easily notice that the apparent line tension (<ref>) of cylindrical droplets, proportional to the slope of the cos(1/)  graph at 1/→ 0, is exactly half of the second, adsorption-related, term in the apparent line tension (<ref>) of axisymmetric droplets, proportional to the slope of the cos(1/) graph at 1/→ 0. Thus, combining data for dependences cos(1/) and cos(1/), one can determine the line tension of a macroscopic droplet as

    =  - 2.


Using this relation, we have calculated the line tension  of a macroscopic droplet from the apparent line tensions  and  of sessile droplets simulated by Kanduč et al. <cit.> using data from Fig. 2(c) in <cit.>. As one can see in Fig. <ref>, the values of  calculated with Eq. (<ref>) are in good agreement with ones obtained in <cit.> by taking into account Tolman length <cit.> and “line tension stiffness” <cit.> corrections. It is clearly seen, that the values of  and  are comparable in these simulations that makes  visibly different from . In contrast, simulations of Isaiev et al. <cit.> show very weak dependence of cos(1/) comparing to cos(1/) (see Fig. 4 therein). This corresponds to ≪ and, thus, ≈.

             

Eqs. (<ref>)–(<ref>) suggest another idea. With dual-geometry sessile droplet measurements, it is not necessary to use the classical Young equation at all. Subtracting the generalized Young equation for axisymmetric droplets (<ref>)  from the one for cylindrical droplets (<ref>) we obtain

    cos - cos = / + / - /.

Taking into account Eqs. (<ref>), (<ref>) with subsequent comments and κ/ r = ( 1/^2 ), the first-order approximation to the combined equation (<ref>) will be

    cos - cos≃/.

Here dependences of  on μ and  and of  on μ are no longer taken into account since they give corrections ( 1/^2 ).

All the quantities in Eqs. (<ref>) and (<ref>) must be taken at the same values of the chemical potential and temperature. The chemical potential is often unknown in simulations or measurements. But, according to Eqs. (<ref>) and (<ref>) and taking into account Eq. (<ref>), the same value of δμ corresponds to the relation ≃/2 or, in the same order in δμ, to ≃/2. The former relation is expectedly more accurate, since it relies only on the Laplace equations (<ref>) and (<ref>), whereas the latter also uses approximation sin≃sin.

Thus, to determine the line tension from the measured (in the experiment or modeling) dependences of the cosθ on 1/r for both axisymmetric and cylindrical droplets at the same conditions, it is sufficient to calculate the difference

    cos|_ = /2 - cos|_  or  cos|_ = /2 - cos|_

and take a linear fit of it vs 1/ at sufficiently large values of . The slope of this graph will give /:

    cos|_ = /2 - cos|_≃/, 
    cos|_ = /2 - cos|_≃/.


An additional advantage of this technique is that the graph of the difference cos - cos vs 1/ must start from a known point (0 at 1/ = 0), thus the slope of the linear graph remains the only fitting parameter. When plotting cos(1/) and cos(1/) independently, each of them must be approximated by a straight line starting from an unknown point cos(0), which must be the same for axisymmetric and cylindrical droplets (and this, in principle, should be somehow taken into account to reduce the number of fitting parameters from 4 to 3). Unlike experimental measurements, droplet simulations require more computational resources as the droplet size increases, so approaching droplet sizes whose contact angles is macroscopic can be resource demanding. However, macroscopic contact angle , can be found, e.g., using the classical Young equation (<ref>) and calculating  and  in three simulations of  planar interfaces <cit.>.


Experiments with nanosized droplets suggest another possible advantage of using dual-geometry techniques to determine the line tension. For nanosized droplets, the graph cos vs 1/ can be fairly linear but this linear fit may work well only for small droplets and give a false value of cos at extrapolation down to 1/→ 0 (see Fig. 5 in <cit.>) and the slope that does not correspond to asymptotic Eq. (<ref>). In many simulations, the droplets are also nanosized, so the same problem can be with simulated droplets, both axisymmetric and cylindrical, giving false values of  and . If Eq. (<ref>) is then used, this would ultimately give a wrong value of . However, not only the slope would be wrong in such situation, but, most probably, the extrapolated values of cos would be different for cos vs 1/ and cos vs 1/ indicating an inconsistency. With use of the technique we proposed, such a situation would appear as a set of points of the quantity (<ref>) vs 1/ that do not fit a straight line starting from (0,0). Both such indicators would simply mean that the droplets are not large enough for the asymptotic Eqs. (<ref>), (<ref>), and (<ref>) to work and, thus, cannot be used to determine the contact angle  and the line tension  of macroscopic droplets.




§ OVERVIEW OF RESULTS AND DISCUSSION


Thermodynamic analysis we have performed shows that droplet size corrections to the contact angle cosine for axisymmetric sessile droplets attributed often to the line tension only, are determined generally by the line tension , its dependence on the droplet size (directly on the droplet radius  on the surface and via the chemical potential) and also by dependences of all three surface tensions on the droplet size (via the chemical potential). The same applies to corrections to the contact angle cosine for cylindrical sessile droplets, except for the line tension term / that is absent in the generalized Young equation (<ref>) for cylindrical droplets.

In the first order in 1/ (<ref>), this gives the slope of cos vs 1/ to be proportional not to the line tension  of macroscopic droplets [as it is widely believed according to Eq. (<ref>)] but rather to the so called “apparent line tension” , a sum of this line tension  and a combination of adsorptions at the three interfaces, the number density of molecules in the liquid phase and the macroscopic contact angle (<ref>).

For cylindrical droplets, the contact angle does also depend on the droplet half-width  on the surface already in the first order in 1/ (<ref>), giving the apparent line tension . It is determined by the same combination of the adsorptions, the number density of molecules in the liquid phase and the macroscopic contact angle (<ref>) but lacks the numerical coefficient 2 compared to the axisymmetric case (<ref>).

This gives a possibility to determine the line tension  from  and  found in dual-geometry measurements (e.g., for droplets simulated at the same conditions, i.e. at the same temperature with the same interaction potentials, etc.) as (<ref>). It was illustrated with calculations of  from  and  of MD-simulated droplets <cit.> (see Fig. <ref>).

We have also proposed a technique for  dual-geometry measurements based on calculation of the slope of the difference cos - cos at the same value of the chemical potential vs 1/. It can be implemented as a calculation of this difference (<ref>) at = /2 or = /2. The slope will equal / with  the surface tension of the planar liquid–gas interface. This technique has some advantages discussed in section <ref>.

The Laplace and the generalized Young equations in both geometries, as well as the Gibbs adsorption equation (<ref>) for the liquid–gas interface, were obtained and considered for the surface of tension as the liquid–gas dividing surface. At an arbitrary choice of this dividing surface, all these equations are more complicated <cit.>. Even though most researchers use these equations in their simple forms, the real choice of the dividing surface is often different or even not specified explicitly. In simulations, it can be done, e.g., by a boundary detection algorithm, and different algorithms can give <cit.> visibly different dependences of cosθ vs 1/r. This can be considered as an additional source of errors that need to be estimated.

As for the solid–liquid dividing surfaces, we have used the analogue of the Gibbs adsorption equation (<ref>) for the difference ≡ - written under some conditions that are not necessarily met at an arbitrary choice of these dividing surfaces. However, if the same choice is made in both geometries, these quantities will still cancel, both with use of Eq. (<ref>) and in the proposed technique where Eq. (<ref>) will remain valid. In contrast, the line tensions  and  (including their macroscopic value ) do generally depend on the choice of the solid–fluid dividing surfaces. For any specific choice, the dual-geometry approach should work, giving the value corresponding to chosen dividing surfaces.

Finally, let us discuss if the proposed technique can be easily adapted to determine the size-dependent line tension  and/or  (not only their limit  at r →∞). Eq. (<ref>) does not contain solid–fluid surface tensions, but it contains the surface tensions . They can be calculated separately for free spherical/cylindrical droplets of the same radius  of the liquid–gas surface as for the sessile droplet of the same type of geometry (thus, at the same value of chemical potential). However, since (μ,) and (μ,) are different functions, the expression / - / on the right hand side of Eq. (<ref>) is still ( (δμ)^2 ) = ( 1/^2 ) and, therefore, cannot be neglected even in the second order in 1/. Thus, Eq. (<ref>) can hardly be used to obtain more detailed information on the line tension from dual-geometry measurements of cosθ(1/r).



The authors thank  Volkov and  Polovinkin for valuable discussions and comments on the manuscript. The research has been carried out with financial support of the Russian Science Foundation, grant no. 22-13-00151, https://rscf.ru/en/project/22-13-00151/https://rscf.ru/en/project/22-13-00151/






 Paper Title: Critical surface adsorption of confined binary liquids with locally conserved mass and composition.





Department of Physics, University of Milan, via Celoria 16, Milano 20133, Italy


Freie Universität Berlin, Department of Mathematics and Computer Science, Arnimallee 6, 14195 Berlin, Germany
Zuse Institute Berlin, Takustr. 7, 14195 Berlin, Germany




§ ABSTRACT 
Close to a solid surface, the properties of a fluid deviate significantly from their bulk values.
In this context, we study the surface adsorption profiles of a symmetric binary liquid confined to a slit pore by means of molecular dynamics simulations; the latter naturally entails that mass and concentration are locally conserved.
Near a bulk consolute point, where the liquid exhibits a demixing transition with the local concentration as the order parameter, we determine the order parameter profiles and characterise the relevant critical scaling behaviour, in the regime of strong surface attraction, for a range of pore widths and temperatures.
The obtained order parameter profiles decay monotonically near the surfaces, also in the presence of a pronounced layering in the number density.
Overall, our results agree qualitatively with recent theoretical predictions from a mesoscopic field-theoretical approach for the canonical ensemble.





Critical surface adsorption of confined binary liquids with locally conserved mass and composition
    Felix Höfling
    August 5, 2024
==================================================================================================





§ INTRODUCTION









The investigation of fluids under nano-confinement has drawn substantial research attention in the past decades.
In this context, water doubtlessly belongs to the most prominent liquid substance whereas it is arguably also one of the most difficult one to understand.
To this end, molecular simulations have provided key insights into the properties of water, complementing experimental findings and theoretical analyses (see, e.g., Ref.  and references therein). This includes the calculation of phase diagrams, e.g., of liquid–vapour and fluid–solid equilibria <cit.>

and a putative liquid–liquid transition deep in the super-cooled state <cit.>, 
but also structural properties <cit.> and transport coefficients such as the dynamic shear viscosity <cit.>.

Confining a liquid in channels of only a few nanometers in diameter manifestly breaks the translational symmetry
and can significantly alter its physical properties relative to its behavior in bulk. For example, certain crystal structures can be suppressed and phase transition temperatures can be shifted <cit.>.
Confinement intricately modifies the relaxation dynamics of glass-forming <cit.> and polymeric <cit.> liquids, as well as water again <cit.>.
The presence of solid surfaces modifies the transversal density profiles showing, e.g., layering,
which can lead to surface-induced phase transitions such as
pre-freezing and pre-melting <cit.> and the rich scenario of wetting and drying transitions <cit.>; predictions that long-ranged dispersion forces lead to specific phase diagrams <cit.> were criticised recently <cit.>.
For complex liquids, consisting of anisotropic molecules, anchoring effects to the surface add another fascinating twist to the wetting problem <cit.>,
whereas orientational ordering can be observed for rod-like, stiff polymers <cit.>.

Going beyond single-component fluids, liquid mixtures exhibit phase separation, which in conjunction with confinement can lead to surface enrichment and domain growth of one component <cit.>.
The demixing transition of the bulk phase diagram is associated with a continuous phase transition and, thus, with the emergence of a macroscopic correlation length and relaxation time <cit.>.
Furthermore, the transversal pore dimensions delimit the growth of critical fluctuations so that the critical singularities near the consolute point appear rounded <cit.>.
It also gives rise to critical Casimir forces <cit.>
and to a splitting of the bulk universality class into few surface universality classes <cit.> (see below).
In particular, for the local excess concentration, playing the role of the order parameter (OP), one expects that its spatial dependence near a solid substrate exhibits critical scaling.
This surface-critical behaviour was corroborated within Monte Carlo simulations for
Ising-like lattice fluids <cit.> and for
single-component Lennard-Jones (LJ) liquids <cit.>, with the local density as the OP field.

Monte Carlo simulations on phase coexistence are typically carried out either in the grand canonical or in the Gibbs ensemble. Also the majority of theoretical studies on critical phenomena in confinement have been performed in the grand canonical ensemble.
Experimentally, however, the mass of each component of a mixture is locally conserved, the corresponding total masses are fixed, and thus the canonical ensemble appears as the proper description of confined liquids in thermal equilibrium.
Whereas the statistical ensembles are equivalent with respect to the mean values of thermodynamic observables, they are not with respect to their fluctuations <cit.>.
Further on the level of thermodynamic macrostates, the ensemble equivalence may not hold in the presence of long-ranged interactions (see, e.g., refs.  and  and references therein).
This issue becomes particularly relevant for confined fluids close to their critical points and, as a result, the choice of the ensemble can yield very different predictions for the behaviour of the fluid.
For example, the critical Casimir force emerging for a fluid confined in between two parallel walls with symmetric boundary conditions is repulsive for the canonical ensemble whereas it is attractive for the grand canonical ensemble <cit.>.



In the present study, we employ state-of-the-art molecular dynamics (MD) simulations to investigate the surface adsorption of a
confined binary liquid film at its bulk demixing (liquid–liquid) critical point.
The binary mixture considered is symmetric with respect to its components <cit.>, and both walls of the slit pore preferentially attract, with equal strength, the same component of the mixture.
Within the MD simulations, as well as in experiments, the partial masses of the two components are naturally conserved
so that the data obtained for equilibrium properties correspond to the canonical ensemble.
Specifically, we calculate spatial profiles of the excess adsorption and analyse their critical scaling.

Our findings are compared with existing mean-field predictions in the limit of infinite surface attraction strengths.

We note that computational studies of near-critical fluids are notoriously challenging due to critical slowing down <cit.>, a macroscopically large correlation length (entailing finite-size corrections <cit.>), and strong OP fluctuations (expressed by a high compressibility or generalised susceptibility); we have addressed these issues by performing a large number of independent simulations encompassing long time spans and huge particle numbers.




§ CRITICAL SURFACE ADSORPTION


Upon approaching the critical point of a continuous phase transition, the local OP near a boundary deviates in normal direction from its bulk value on the scale of the diverging bulk correlation length <cit.>.
The kind of thermodynamic singularities occurring in this surface layer depends on the boundary conditions for the OP such that each bulk universality class splits up into various surface universality classes.
Generically, fluids belong to the so-called normal surface universality class <cit.>, which is characterized by the presence of a symmetry breaking surface field inducing order at the surface even if the bulk is in the disordered phase.
Close to the critical point the bulk correlation length ξ_b diverges in the thermodynamic limit, thus becoming the most predominant length scale in the system.
In the following, we specialise to binary fluids near their bulk consolute point, given by the critical temperature T_c and the
critical composition (which plays a similar role as the critical density in case of a liquid–vapour transition).
In addition, we restrict the discussion to the mixed phase, i.e., to fluid temperatures T > T_c,
and we will often incorporate the temperature dependence via the bulk correlation length ξ_b = ξ_b(T).
For this choice of temperatures, wetting transitions are not expected to occur <cit.>.

The demixing OP of a binary liquid of species A and B in bulk is given by ϕ_b=x_A-x_B in terms of the bulk concentrations x_A and x_B of both species, defined as their corresponding mole fractions.
Upon confinement to a slit pore, these quantities depend on the perpendicular distance z from one of the confining walls,
so that the local OP reads Φ(z)=x_A(z)-x_B(z), where x_α(z) = ρ_α(z) / ρ(z) for α=A,B;
the partial number density ρ_α(z) counts the particles of species α in a narrow slab at position z, and
ρ(z) = ρ_A(z) + ρ_B(z) is the overall local number density.


For fluids in contact with a single wall, ξ_b competes with the range of the surface interaction of the fluid molecules;
in the case of a short ranged interaction, as studied here, this range is typically on the order of the molecular diameter σ.
A universal shape of the OP profile Φ(z), as function of the distance z to the surface, is anticipated except very close to the surface, i.e., universality holds for z ≫σ.
Depending on the distance z compared to the correlation length ξ_b, one distinguishes a standard regime with an exponential decay, Φ(z) ∼exp(-z/ξ_b) for z ≫ξ_b, and a critical regime exhibiting power-law behaviour, Φ(z) ∼ z^-β/ν for z ≪ξ_b.
The exponents β and ν denote the standard critical exponents of the three-dimensional Ising universality class; we use the reliable estimates <cit.> ν = 0.630 and β = 0.325.
In particular, they govern the critical singularities in the bulk behaviour of the correlation length and the OP, respectively <cit.>:
ξ_b(T) ≃ξ_0^+ ϵ^-ν in terms of ϵ = (T-T_c)/T_c ↓ 0, measuring the proximity to the bulk critical point, and ϕ_b(T) ≃ϕ_0 (-ϵ)^β for ϵ↑ 0,
where ξ_0^+ and ϕ_0 are non-universal amplitudes.

The OP profile exhibits critical scaling which, for a semi-infinite fluid, is condensed in a dimensionless scaling function P_1 with the rescaled distance ẑ := z/ξ_b as the only variable <cit.>:

    Φ(z;T>T_c) = φ_0 (ξ_b/ξ_0^+)^-β/ν P_1(z/ξ_b)  ,    z ≫σ .

The scaling function is universal, but depends on the boundary condition for the OP at the surface.
The scaling function obeys the asymptotics sketched above, namely
P_1(ẑ→∞) ≃exp(-ẑ) and
P_1(ẑ→ 0) ≃ c_0 ẑ^-β/ν with the universal surface amplitude c_0.
The excess mass of fluid adsorbed to the surface is quantified by the integrated OP profile,

    Γ(T) = ∫_0^∞ [Φ(z;T) - ϕ_b(T)]   z  ,

for which, upon inserting <ref>, one infers a critical power-law divergence <cit.>:

    Γ(T → T_c) ∼ξ_b^1-β/ν∼ |ϵ|^β - ν ,

provided that β < ν.
(In mean-field theories, where β = ν = 1/2, a logarithmic divergence is predicted).
These scaling predictions for the OP profile have been tested successfully in simulations of Ising lattices <cit.>.
Scaling has also been corroborated in simulations of one-component LJ fluids,
but the verification of the critical divergences has remained challenging <cit.>.

For fluids confined to a slit pore, surface effects due to both walls are present and, as a result, the distance D between the two walls enters the problem as another relevant length.
In particular, critical fluctuations are suppressed in the direction of the surface normal for ξ_b ≫ D, whereas the slit confinement is expected to be ineffective for ξ_b ≪ D, approximating the situation of a semi-infinite fluid delimited by a single surface.
The OP profile Φ(z) is now a function of the position z, the temperature T, and the pore width D;
it also depends on the strength of the surface interaction, and we may again use the bulk correlation length ξ_b(T) instead of T.
The dependence on the pore width D is accounted for by decorating the scaling function in <ref> with a second scaling variable, D̂ = D/ξ_b, which is further used to substitute ξ_b by D, i.e., we compare other quantities such as the position z to the pore width D.
In the strong adsorption limit (large surface field), the finite-size scaling form of the OP profile reads <cit.>:

    Φ(z;T,D) = φ_0 (D/ξ_0^+)^-β/ν P_2(z/D, D/ξ_b)  ,
         z ≫σ .

Thus, plotting
φ_0^-1(D/ξ_0^+)^β/νΦ(z;T,D) vs. z̅ = z/D for a fixed value of D̂ = D/ξ_b should yield a data collapse.
The new, two-parameter scaling function P_2 reproduces the behaviour near a single surface via
P_1(ẑ) = lim_D̂→∞D̂^-β/ν P_2(ẑ/D̂, D̂)
and it encodes the critical profile Φ(z; T_c, D) as the limit P_2(z̅, D̂→ 0).
The function P_2 also depends on the boundary conditions at the two surfaces; here, we focus on a slit pore with equal surfaces on both sides, namely symmetric, attractive boundary conditions.

We note that, due to the confinement, the critical divergence at the bulk critical values of temperature and composition is shifted towards a lower temperature and a more A-rich composition (assuming A-preferring, symmetric surfaces) <cit.>.
The phenomenon is related to capillary condensation and depends on the pore width D; it is particularly relevant for thin films, D ≪ξ_b.
More precisely, it may be expressed as a competition of the pore width D and the bulk correlation length ξ_b and can be incorporated in the scaling form (<ref>) by adding an additional scaling variable for the composition.
However, we will not explore these aspects here and restrict to T > T_c in the following, where T_c always refers to the critical temperature of the bulk fluid, which also enters <ref> through the scaling of ξ_b.


For a conserved OP field, the critical scaling of the OP profiles has been studied in depth recently within a mesoscopic theory based on the Ginzburg–Landau (GL) free energy functional, prescribing either a surface field <cit.> or the contact values at the two surfaces <cit.>. 
In these works, analytic calculations within a systematic perturbative approach yield OP scaling functions, which are corroborated by numerical solutions of the field equations;
in addition, these findings are supported qualitatively with data from Monte Carlo simulations for Ising lattices.
The obtained OP profiles are mean-field like, as is typical for GL theories, and suggest
that, for symmetric boundary conditions [see eq. (33) of Ref. ],

    Φ(z;T, D) ≃μξ_b^2 - Φ_m(T, D)  cosh(z - D/2/ξ_b)

in the inner part of the pore, viz., for ξ_b ≪ z ≪ D - ξ_b.

Here, we consider binary fluids at critical composition so that the chemical potential μ associated with the OP field is zero and the first term vanishes.
[A similar expression was obtained for Dirichlet boundary conditions at both surfaces [see eq. (24) of Ref. ]. However, the situations studied analytically in Refs.  and  differ from the present setup, which corresponds to the boundary conditions Φ(0) = Φ(D) = Φ_0 ≈ 1 and zero total OP, ϕ_tot = 0.]









With this, the prefactor Φ_m = Φ(z = D/2) equals the minimum value of the OP profile, which is attained in the central plane of the slit pore by symmetry.
Furthermore, the local conservation of the OP field implies a constraint on the total OP ϕ_tot, which is the spatial integral of Φ(z) and which vanishes in the mixed phase at critical composition:

    ϕ_tot := ∫_0^D Φ(z)   z = 0  .

Therefore and exploiting the symmetry Φ(z) = Φ(D-z) of the set-up, Φ_m < 0 is related to the excess adsorption Γ on each surface via

    Γ(T,D) := ∫_0^D/2 [Φ(z) - Φ_m]   z = -Φ_m D/2 .


For a finite pore width, D < ∞, the profile Φ(z; T, D) is expected to depend on the temperature T analytically near T_c, which implies that the scaling function P_2(z̅, D̂) is analytic in D̂^1/ν with D̂ = D / ξ_b [<ref>].
Hence, Φ_m(T, D) may be expanded for T ↓ T_c or, equivalently, for ξ_b(T) →∞:

    Φ_m(T, D) ≃φ_0 (D/ξ_0^+)^-β/ν[ P_2(1/2, 0) + C (D/ξ_b)^1/ν]  ,

with some constant C > 0.
In particular, Φ_m(T,D), and thus the excess adsorption Γ(T,D), do not diverge near the bulk critical point, which is a consequence of the OP conservation and at variance to the behaviour in the grand canonical ensemble <cit.> [<ref>].
Rather, for fixed D, the excess adsorption decreases linearly from its critical value at T_c as the temperature is increased (ξ_b(T) ≫ D):

    Γ(T ↓ T_c, D) + Γ(T_c, D) ∼ξ_b^-1/ν∼ T - T_c  .





§ SIMULATION MODEL AND METHOD


For the MD simulations, we use a symmetric binary liquid with well-known bulk properties <cit.>.
Specifically, we consider an equimolar mixture of two components A and B which interact symmetrically with each other via the Lennard-Jones (LJ) pair potential,

    u_LJ(r; ε,σ)=4ε [(σ/r)^12-(σ/r)^6]  ,

with parameters ε and σ.
The potential is shifted and smoothly truncated at a cut-off distance r_c=2.5σ, which renders very good numerical stability with respect to energy conservation <cit.>:

    U_μν(r)
      = [u_LJ(r; ε_μν, σ)-u_LJ(r_c; ε_μν,σ)]  f((r-r_c)/h)

for μ, ν∈{A, B}, h=0.005σ, and f(ζ≤ 0)=ζ^4/(1+ζ^4) and f(ζ >0)=0.
The interaction range σ is the same for all pairs μν, whereas the interaction strengths are chosen as ε_AA=ε_BB=2ε_AB=:ε.
For the units of length and energy, we use σ and ε, respectively, which implies the reduced temperature T^*:=k_B T/ε.
Of each species, we simulate N_μ particles such that species μ has concentration x_μ=N_μ/N and such that the total number density is fixed at ρ_b = N/V = 0.8σ^3,
in terms of N=N_A + N_B and the pore volume V.
We have previously determined the demixing phase diagram of this fluid
[We note that a force-shifted potential was employed in the studies in Refs. , which yields different values for non-universal properties, i.e., the critical temperature and the critical amplitudes.]
and the relevant critical behaviour in bulk <cit.>; in particular, the critical point was found to be
in the liquid phase at temperature T_c^* = 1.629±0.001 and concentrations x_A,c=x_B,c=1/2;
the non-universal amplitude of the correlation length ξ_b(T) was obtained as ξ_0^+ = (0.47 ± 0.02)σ,
and we have estimated for bulk OP amplitude φ_0 = 0.77 ± 0.02.

The liquid is confined in between two planar walls separated by a distance D.
The area of each wall is L× L and is set by the lateral dimensions of the simulation box, with periodic boundary conditions applied along the two directions parallel to the walls;
the volume of the slit pore is thus V=L^2 D.
The walls are located at nominal positions z=0 and z=D, with the z-axis of the Cartesian coordinates chosen normal to the walls.
The surface–fluid interaction is the same for both walls and is given by the LJ wall potential (see, e.g., Refs. ):

    U_sf^(μ)(z̃)=2πε_w/3[
        2/15(σ/z̃)^9
        - w_μ(σ/z̃)^3 ],

in terms of the distances z̃=z+σ/2 and z̃=D+σ/2 - z to each of the walls.
The offset σ/2 is introduced in the surface potential to better match the volume that is effectively available to the fluid particles with the nominal pore volume L^2 D.
The form of the potential results from an integral over the LJ pair potential, modelling the wall as an infinite half-space which is filled homogeneously with LJ particles.
The surface potential U_sf^(μ)(z̃) is further shifted and smoothly truncated at the cutoff distance r_c,sf=2.5σ as described above for the pair interaction.
The energy ε_w and the dimensionless parameter w_μ control the
overall attraction strength and the relative surface preference for one of the two fluid components, respectively;
U_sf^(μ)(z̃) has its minimum at z̃_min = (2/5w_μ)^1/6σ, for w_μ > 0,
with depth (2πε_w/9)√(10w_μ^3).
We use the same value ε_w for both walls of the pore, and we put w_A=1 and w_B=0, which amounts to symmetric boundary conditions with a preference for A particles.

All results presented here are obtained at or above the bulk critical temperature of the binary liquid, T ≥ T_c (after a quench from the mixed phase at T > T_c), and at the critical concentrations x_A=x_B=1/2,
i.e., for zero total OP, ϕ_tot = x_A-x_B=0.
Extensive MD simulations were carried out in the canonical ensemble with the Nosé–Hoover thermostat
(NHT) chain <cit.>, using an integration timestep of δ t=0.001t_0;
the unit of time is t_0=√(mσ^2/ε) in terms of the particle mass m, which is the same for A and B particles.
To mitigate the computational issues associated with the simulation of critical fluids, such as critical slowing down <cit.> and finite-size corrections <cit.>, we have used large system sizes for up to 
300000 particles over a long time span of e4t_0;
in addition, ensemble averages were taken over up to 20 independent initial configurations for each parameter set.
Specifically, we have used large simulation boxes of lateral length up to L=100σ and we have varied the pore width from D=10σ to 30σ.
The simulations were performed with the software HAL's MD package <cit.>, which exploits the high degree of parallelism of recent accelerator hardware <cit.>, e.g., graphics processors (GPUs), and which is an efficient and precise tool for the study of inhomogeneous fluids <cit.>.
For example, a simulation run for D=20σ over a period of 10^4 t_0  (e7 steps),
takes about 1h on a single GPU of type Nvidia A40 
and about 5.8h on the much older hardware generation, Nvidia Tesla K20Xm. 
















The simulation data were stored in the structured, compressed, and portable H5MD file format <cit.>.

Laterally averaged number density profiles ρ_μ(z) = N_μ(z) / (L^2 Δ z) were calculated in a post-processing step by counting the number N_μ(z) of particles of each species μ that are located in the interval [z - Δ z/2, z + Δ z/2) along the z-axis for a bin width of Δ z = 0.5σ.
For the total density profile ρ(z), shown in the inset of <ref>(b), we used a different technique to be described elsewhere, which is based on Fourier transforms, avoids the spatial binning and reduces the amount of data to be stored. The method includes a slight smoothing of the obtained profile ρ(z), here with a Gaussian filter of width σ/8.



§ RESULTS AND DISCUSSION






Within the MD simulations, we have obtained the surface adsorption profiles Φ(z) of binary liquids
confined to a slit pore which preferentially attracts component A at both of its walls with equal strength [<ref>(a)];
unless stated differently, we consider a strong surface attraction, ε_w=8ε [see <ref>].
The present study is restricted further to binary liquids at critical composition (x_A = x_B = 1/2) and in the mixed phase (T ≥ T_c), so that the temperature T controls the distance to the bulk critical point of the demixing transition.
Φ(z) serves also as the corresponding local OP, it is defined such that positive values indicate an excess of A particles.
Since the set-up exhibits a mirror symmetry, Φ(z; T, D) = Φ(D-z; T, D), with respect to the central plane (z=D/2), we will  often discuss merely the behaviour near one of the surfaces.

Two exemplary OP profiles Φ(z) for two temperatures, in the mixed phase and at criticality, are shown in <ref>(a).
At the higher temperature, T^*=5 ≈ 3 T_c^*, the profile is comparably flat in a large part of the pore but increases sharply upon approaching one of the surfaces (e.g., z → 0), the latter reflecting the expected surface enrichment of A particles.
Upon decreasing the temperature to T_c, the contact value Φ(z → 0) increases to its largest possible value, Φ(z → 0) = 1, which we attribute to the choice of a strong surface interaction (ε_w=8ε);
this value of Φ(z) corresponds to a pure layer of A particles next to the surfaces.
Concomitantly, the OP profile broadens as T ↓ T_c, reflecting the emergence of critical fluctutations.
However, the width of the surface enrichment layer remains finite (of the order of 3-4σ) despite the bulk correlation length ξ_b(T) diverging at T_c, which is a consequence of the confinement to a slit and of the aforementioned symmetry of Φ(z) at z=D/2 = 10σ.



Furthermore, the conservation of the OP implies that the integrated OP ϕ_tot does not change upon varying the temperatures; here, we use ϕ_tot=0.
It follows from <ref> that the OP value in the middle of the slit pore must be negative, Φ_m = Φ(z=D/2) < 0, to compensate the surface enrichment.
This is a specific feature of the canonical ensemble and unlike in the grand canonical ensemble, in which the local OP remains positive, Φ(z) ≥ 0, for symmetric surfaces both preferentially attracting component A.
For conserved OP, the value Φ_m thus serves as a measure of the total excess adsorption Γ(T,D) = -Φ_m(T,D)   D /2 [<ref>].
The latter is predicted to vary linearly with T near the critical temperature [<ref>],
which is corroborated by our simulation data (<ref>).

In contrast to the monotone decay of Φ(z) for 0 ≤ z ≤ D/2 (<ref>), the number density of the critical fluid (T=T_c) exhibits a pronounced layering near the surfaces (inset of <ref>):
the density profile ρ(z) shows rapidly decaying oscillations around the mean density ρ_b=0.8σ^-3, which have a large amplitude next to the surface at z=0 and which are barely visible in the middle of the pore, for z ≳ 4σ; the period of the oscillations is of the order of the molecular length σ.
Such oscillations hinder the scaling analysis of critical surface profiles, and their absence in the present OP profiles (i.e., in the concentration and thus the excess adsorption) underscores that, in this respect, binary liquids are favourable systems for the study of critical behaviour.
We note that the critical behaviour of the structural and dynamic properties of binary liquids is governed by two fluctuating fields, viz., the total number density and the composition.
The investigated binary liquid (with ρ_b=0.8σ^3) was demonstrated <cit.> to be
sufficiently far away from its liquid–vapor critical point and thus the density field
ρ(z) acts as a secondary field <cit.>, which does not exhibit critical enhancement
(see, e.g., the bulk structure factors in Figs. 4 and 5 of Ref. ).

In order to further elucidate the lateral structure of the layers near the surface, we have calculated the lateral structure factor S_ρρ(|k⃗|) of the density for each layer separately, by restricting the calculation to quasi-twodimensional slabs encompassing one layer each and such that the wave numbers point parallel to the wall surfaces, for details see Ref. .
Upon approaching the surface, layer by layer, the apparent compressibility ∝ S_ρρ(0) is reduced and the degree of local ordering increases, as follows from the growth and the sharpening of the first peak of S_ρρ(|k⃗|) (<ref>).
However, also the first fluid layer, which is in direct contact with the wall, does not exhibit crystallisation but remains in a fluid state.




Next, we investigate the dependence of the OP profile Φ(z) of the critical mixture on the pore width D for fixed temperature T=T_c.
For strong surface attraction (ε_w=8ε), the contact value Φ(z → 0) at the surface does not
exhibit any prominent D dependence [<ref>(a)]. 
As D is increased, the profile becomes deeper (|Φ_m| increases) and broadens: the thickness of the
surface adsorption layer (e.g., the distance to the surface at which Φ(z) changes its sign) increases.

In order to test the scaling property of these critical OP profiles, we have plotted the combination
φ_0^-1 (D/ξ_0^+)^β/νΦ(z,T,D) as a function of the scaling variable z̅ = z/D
for a range of pore widths D that is spread out by a factor of 3. 
<Ref>(b) demonstrates convincingly that this rescaling of the OP profiles of <ref>(a)  yields data collapse onto the scaling function
P_2(ẑ, 0); due to T=T_c, we have D̂ = D/ξ_b(T_c) = 0.
Deviations from this universal master curve occur for distances z ≲ 0.8σ, which corresponds to the non-universal microscopic regime.
We note that, in order to avoid spurious finite-size corrections to the scaling, we found it necessary to also fix the aspect ratio L/D=3.3 of the pore geometry; the ratio L/D enters the scaling form of the OP profile [<ref>] as an additional finite-size scaling variable (see, e.g., Refs.  and ).




The scaling function P_2 of the OP profile Φ(z;T,D) depends on two scaling variables, z̅ = z/D and D̂ = D/ξ_b(T) [<ref>].
For elevated temperatures, T > T_c, a similar data collapse as for the critical mixture can be obtained by considering a range of combinations of pore width D
and temperature T such that D̂ = D/ξ_b(T) = (D/ξ_0^+)ϵ^ν is kept constant.
Indeed, the rescaled data for the OP profiles with D̂ = 5.9 and for strong surface attraction (ε_w=8ε) collapse onto the master curve P_2(z̅, D̂ = 5.9) similarly well as in <ref>(b) for T=T_c.
However, repeating this analysis for weak surface attraction (ε_w=0.8ε), the quality of the data collapse is considerably worse [<ref>(b)]. In this situation, the OP profiles depend also on the strength ε_w of the surface field, which enters the scaling function in <ref> as another scaling variable, in addition to z/D, D/ξ_b, and L/D (see, e.g., Ref.  for details).



For a more systematic investigation of the dependence of the OP profile on the strength ε_w of the surface attraction, we consider the OP value close to the surface.
Within the mesoscopic GL theory employing a surface field h_1 <cit.>, it was predicted that Φ(z → 0)=Φ_0 displays a crossover from the scaling Φ_0 ∼ h_1 for weak fields, h_1 → 0,
to Φ_0 ∼ h_1^1/2 for strong fields, h_1 →∞.
Since there is no direct mapping between the surface field h_1 of GL theory and the surface interaction parameters (ε_w, w_A) used in the MD simulations, we check if the scaling of Φ_0 with ε_w is analogous to what has been predicted within GL theory for the dependence on h_1;
to this end, the mean-field exponents should be replaced by their counterparts in the 3D Ising universality class.
However, the contact value Φ(z→ 0) appears to be not suited for this task since universality of the surface adsorption profiles is expected to hold only for z ≫σ.
Hence, we define the surface value Φ_0 as an integral over the particle layer second-closest to the surface:

    Φ_0 = 1/0.5σ∫_σ^1.5σΦ(z)   z  .

Using this definition of Φ_0, the data from MD simulations of the critical mixture (T=T_c, D=20σ, and L=100σ)
are described the behaviour Φ_0(ε_w → 0) ∼ε_w^0.55 for weak surface interaction strengths, varying from ε_w=e-2ε to ≈ 0.2 ε [<ref>(a)].
Despite these low values of ε_w, the observed scaling is at variance with the expectation from perturbation theory for Ising and GL models that Φ_0 scales linearly with the surface field <cit.>.
For the the strong-surface field behaviour, one needs to account for the bound |Φ_0| ≤ 1 due to the definition of Φ(z), similarly as above for the excess adsorption Γ. Thus, we consider 1-Φ_0 in order to test the approach of Φ_0 to its saturation value.
Fitting a power law to the data points for the two largest surface interactions simulated (ε_w = 4ε and 8ε) suggests that
1-Φ_0(ε_w →∞) ∼ε_w^-0.57 [<ref>(b)].
We note that this value of the exponent is larger than the standard surface exponent <cit.> Δ_1 ≈ 0.46 and that, to the best of our knowledge, a theoretical prediction for the strong-surface field behaviour of Φ_0 under the simulated conditions is not available.




§ SUMMARY AND CONCLUSIONS


Understanding the surface critical behaviour of a fluid confined to a slit pore is demanding due to the presence of multiple length scales: there are the molecular size σ and the pore width D, the temperature dependence implies the bulk correlation length ξ_b(T),
the strength of the surface interaction ε_w adds another implicit length, and, in simulations, the finite lateral extent L of the slit pore cannot be ignored.
Here, we have used large-scale, off-lattice MD simulations to explore the scaling of the excess adsorption profiles Φ(z) of a binary liquid with locally conserved mass and concentration;
the liquid is kept confined to a slit whose walls have selective surface adsorption preferences.
For the demixing transition, the excess adsorption plays the role of the OP field, which develops critical fluctuations near the liquid–liquid critical point.
Relying on the corresponding scaling form [<ref>],
we have shown that the data for laterally averaged OP profile Φ(z;T, D) for a range of temperatures T and pore widths D collapse onto a master curve upon suitable rescaling (<ref>).
This master curve is given by the scaling function P_2 in the limit of strong surface adsorption and for fixed ratios D/ξ_b and L/D.
The obtained OP profiles behave monotonically near the surfaces, also in the presence of a pronounced layering in the number density profiles [<ref>(b)].
Furthermore, our data suggest that OP profiles obtained within the MD simulations are better described by the regime of strong surface attraction, even for low values of ε_w, which raises a question on the precise mapping between the particle model used in the MD simulations and GL theory (<ref>).
Overall, our results are in qualitative agreement with mean-field predictions from field-theoretical calculations <cit.>.

Whereas our study is based on the generic and somewhat artificial Lennard-Jones pair interaction,
we anticipate that our findings for critical adsorption are universal and can be transferred mutatis mutandis to real-world binary liquids, including water–lutidine mixtures.
In particular,
understanding the influence of surfaces on a confined fluid is important for the study of critical Casimir forces <cit.>. We hope this work will motivate future work on the non-equilibrium dynamics of near-critical confined fluids and on the dynamic response to a temperature quench <cit.>.


 We thank Siegfried Dietrich and Markus Gross for helpful discussions.
 Financial support by Freie Universität Berlin via a research stipend under the Future Development Concept: “International Network University” (S.R.)
 and by Deutsche Forschungsgemeinschaft (DFG, German Research Foundation) under Project No. 523950429 (F.H.) is gratefully acknowledged.






 Paper Title: Spatio-temporal patterns in Growing Bacterial Suspensions: Impact of Growth dynamics.


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            pratikshyajena.rs.phy20@itbhu.ac.insmishra.phy@iitbhu.ac.in§ ABSTRACT 
            
            The field of active matter explores the behaviors of self-propelled agents out of equilibrium, with active suspensions, such as swimming bacteria in solutions, serving as impactful models. These systems exhibit spatio-temporal patterns akin to active turbulence, driven by internal energy injection. While bacterial turbulence in dense suspensions is well-studied, the dynamics in growing bacterial suspensions are less understood. This work presents a phenomenological coarse-grained model for growing bacterial suspensions, incorporating hydrodynamic equations for bacterial density, orientation, and fluid velocity, with birth and death terms for colony growth. Starting with low density and random orientations, the model shows the development of local ordering as bacterial density increases. As density continues to rise, the  model captures four distinct phases—dilute, clustered, turbulent, and trapped—based on structural patterns and dynamics, with the turbulent phase characterized by spatio-temporal vortex structures, aligning with observations in dense bacterial suspensions.
            
            Spatio-temporal patterns in Growing Bacterial Suspensions: Impact of Growth dynamics
                Shradha Mishra^1
                August 5, 2024
            ====================================================================================
            
            
            
            § INTRODUCTION
            
            
            
            
            
            
            The emerging field of active matter deals with the analysis of the individual and collective properties of self-propelled agents in out of equilibrium <cit.>. Among these, the investigation of active suspensions has been an intriguing research subject for the past few years due to their unique behaviour. And the collection of swimming bacteria in a solution is a simplest realization of active suspension and has a substantial impact on the context of active matter. 
            The bacteria bath <cit.> are interesting model to study due to the simplicity in manipulating the experimental control parameters like concentration of bacteria, activity, and swimming behaviour. 
            
            
             Several research endeavors have been conducted on this model and reported a variety of novel nonequilibrium phenomena including emergence of collective motion <cit.>, pattern formation <cit.>, reduction in viscosity <cit.>, enhancement of diffusivity <cit.> and energy transfer <cit.>. 
            Apart from these interesting properties, the occurrence of spatio-temporal patterns resembling the features of active turbulence in such systems of micro swimmers is highly unexpected  and has been observed  by researchers both experimentally and theoretically <cit.>. In conventional turbulence,  the vortices of different sizes are developed when inertial force dominates over the viscous force and need a  supply of external energy to maintain it <cit.>. 
            
            Alternatively   in dense bacterial suspensions, the energy injection is internal and turbulence can occur in over-damp systems. This motivated the researchers in recent years to   
            explore the bacterial turbulence in greater details <cit.>. 
            
            
             Some studies  <cit.> have focused on the role of activity and there are other studies performed and   characterised the turbulence in dense bacterial suspensions <cit.>. 
            However, bacterial turbulence in the growing bacteria suspension are still poorly explored although the growth phenomena is always present in natural bacterial suspensions.
            In recent experimental work <cit.>, the authors studied the dynamics of growing bacteria at the oil-water interface and discovered the existence of four dynamical phases. 
            But a theoretical model to explore  the appearance of distinct phases in growing bacterial suspension is still lacking. This motivated us to present our current work. 
            
            
            We develop a phenomenological  coarse-grained model for a growing bacterial suspension by writing the hydrodynamic equations of motion for the bacterial local density,  orientation and fluid velocity.  An additional birth and death term is introduced to take care of growth of colony. 
            
            We started with low density and random orientations of bacteria and with time local ordering among bacteria develops as bacteria grows with time with increasing density. As density increases further, the locally ordered state transit to the   dynamical vortex state.   Further at high densities system becomes heterogeneous with slower dynamics.  Thus, based on the structural pattern and dynamics of the system, four distinct phases
            are identified; namely dilute, clustered, turbulent and trapped phase. Notably, the turbulent phase exhibits spatio-temporal vortex structures, which are typically observed in dense bacterial suspensions <cit.>. 
            
            
            The structure of the paper is organized as follows. In section <ref>, we provide a detailed description of our model and the numerical scheme used for the simulation. Next, section <ref> presents a comprehensive  results of the model. Lastly, we discuss the relevance of our work on the basis of results obtained,  summarizing in section <ref>. 
            
            
             
            
            § MODEL
            
             We consider a suspension of  growing bacterial colony in two-dimensions. The periodic boundary condition is implemented in both the directions. The system is modeled using coarse-grained equations of motion for the local density ρ( r, t), orientation   P( r, t) and fluid velocity v( r, t) at position r and time t.
              The hydrodynamic equation for the density ρ( r,t) is: 
            
                ∂ρ/∂ t+∇.(ρv)+v_0∇.(ρP)=D_ρ∇^2ρ+g(ρ)
            
            the corresponding equation for the local orientation P( r, t) is:
            
                ∂P/∂ t+λ_1(P.∇)P+λ_2∇|P|^2 + (v.∇)P = D_r(ρ/ρ_c-1) P-β|P|^2P-ω_αβP_β-v_1 v_αβP_β-σ_1∇ρ+D_T∇^2P
            
            and the  equation for the fluid velocity v( r, t) is
            
                ∂v/∂ t+v.∇v= η∇^2 v-∇ p+∇.σ_αβ^total
            
            
            The density equation <ref> is a continuity equation with additional term  g(ρ) = ρ(1-ρ/ρ_0) which incorporates birth and death of the bacteria. This type of term was first used by Malthus, <cit.> and consequently, the flocks with birth and death are also known as Malthusian
            flocks and later the same term is used in other models of polar flock with birth and death <cit.>.  
            
            We fix the ρ_0 = 10, such that the birth rate is much higher than the death rate. 
            The second term ∇.(ρv) in equation <ref> refers the convection with response to  the fluid flow and the  third term v_0∇.(ρP) is the convection due to the bacteria's own orientation. The v_0 is the self-propulsion speed of bacteria.
            The first term on the right hand side is the diffusion term, which tries to  homogenise the density of bacteria in the suspension.
            
             The  equation <ref> presents the dynamics of orientation of bacteria and is quite similar to the  Toner-Tu equation <cit.> for flocking with few additional terms due to the coupling of orientation field with background fluid velocity.  The second and third terms on left hand side λ_1(P.∇) P and λ_2∇|P|^2 are the convective nonlinearities present due to absence of Galilean invariance in these systems <cit.>. The last term on the left hand side is the convective non-linearity due to fluid velocity. The first two terms and the last term D_T∇^2P on the right hand side are the deterministic part of Time dependent Ginzburg Landau (TDGL) model <cit.> for orientable rod type bacteria's. In addition two  terms on the right hand side 
             ω_αβ=1/2(∂_α v_β-∂_β v_α) and v_αβ= 1/2(∂_α v_β+∂_β v_α)
             represent, how the flow affects the local orientation of bacteria and are called the vorticity and strain-rate tensors respectively. The term, ∇ρ is the inertial coupling in local orientation of bacteria in the suspension. 
            
            The equation for the fluid velocity equation <ref> contains the terms present in Navier-Stokes equation with the additional force term due to the active and passive stresses in the suspension.  The origin of the term ∇.σ_αβ^total is due to the stresses experienced by the fluid including the internal stress due to viscosity and also the external stress by the suspended bacteria.  In this case the total stress σ_αβ^total= σ_αβ+σ_αβ^a, σ_αβ and σ_αβ^a are the passive and active contributions respectively.
            The constitutive equation for stress tensor is formulated as:
            
                σ_αβ^total=2η_1 v_αβ+1/2ν(P_α h_β+P_β h_α-d/2(P_γ h_γδ_αβ))+ζ q_αβ
            
             The first term corresponds to the internal stress caused by the viscosity, the second term is the passive stress with the flow coupling coefficient ν attributable to any suspended passive particles. Additionally, the term ζ q_αβ is the active stresses where q_αβ = P_αP_β-1/dδ_αβ.  The positive and negative values of ζ correspond to the ex -tensile and contractile stresses respectively in most bacterial suspension <cit.>. For ζ=0, the hydrodynamic coupling is purely passive. The equation <ref> is solved using  the stream function approach <cit.> and taking the `curl' `×' on both sides to get  the equation for the vorticity field ω = ∇×v, 
             
                ∂ω/∂ t+(v.∇)ω=η∇^2ω+∇×(∇.σ_αβ^t)
            
            Further we get the Poisson's equation for a scalar field ψ using 
            
                v=(∂_y ψ-∂_x ψ)
            
                ∇^2 ψ=-ω
            
            The intrinsic length and time scales in the system is chosen by persistent length l_0 = v_0/D_r = 0.4 and τ = 1/D_r = 2 respectively. Later all the lengths and times can be defined in terms of l_0 and τ, such that after writing the re-scaled equations, all terms are dimensionless. To solve the system we perform the numerical integration of equations <ref>,<ref>,  <ref>, <ref> and <ref> using Euler’s scheme with grid size Δ x = 0.5 = 1.25 l_0 and Δ t = 0.0025 τ. The above choice of grid size lead to roughly each vertices of the square grid consist of approximately 75-100  E. Coli bacteria (keeping typical size of bacteria roughly 10 μ m) and the reorientation rate τ is approximately after 100 steps of persistent motion (with typical speed of bacteria of the order of 5 μ m /s<cit.>).  
            
            The system is initialized with random, isotropic and homogeneous state with initial number density ρ = ⟨ρ⟩± 0.05, where  initial mean density ⟨ρ⟩ = 0.1. The initial state for orientation is random with two components of P = (P_x, P_y) and scalar field ψ randomly chosen between ± 0.1 and ±0.05 respectively. The coefficients (D_ρ, λ_1, λ_2, β and D_T) of different terms are set to 1. The  strength of active coupling ζ = -2 v_0 and v_0 is fixed to 0.2. The results are also tested for the extensile case ζ > 0 and found qualitatively same as the contractile case.   
            The critical density ρ_c beyond which mean field theory suggest ordering among the particles is fixed to 0.5. 
              The coefficient of inertial term  σ_1 is set to 1 for most of the detailed results. To see the effect of inertia later it is varied from 1 to 100. Larger value of σ favours the stronger alignment of orientation P in the direction of gradient of density. The stability of the system is checked for the set of chosen parameter before start of the simulation.  
            
            The simulation is carried out for difrerent system sizes i.e, K = 256, 512, 600 for total simulation time t= 3000 = 1500 τ. All the real snapshots in the Results section are generated for K = 256.
            The one simulation step is counted when equations <ref>, <ref>, <ref>, <ref> are updated for all lattice points. Statistical averaging  is performed over 100 independent realisations for better statistics.   
            
            
            §  RESULTS
            
            Starting with the random homogeneous initial state and low mean density we first analysed the snapshots of the system at different times. Since, the density of bacteria is growing with time, all the measurements are done with respect to mean density. 
            In 
            Fig <ref>(a-d), we showcase  the snapshots of the local density with the local orientation at mean densities ⟨ρ⟩ = 0.12 , 2.94 , 0.75 , 10.00 respectively. The presence of g(ρ) term makes the mean density of bacteria to grow with time. In Fig. <ref>(a), when the density of bacteria is low ⟨ρ⟩ = 0.1    the bacteria are sparsely dispersed with almost homogeneous density in the system with negligible local clustering and local orientation.  Then with time as density grows, the particles starts to form  small clusters  and small local orientation among the bacteria develops as shown in Fig. <ref>(b) for ⟨ρ⟩ = 2.94. As the density grows further  vortex type pattern starts to develop in the orientation field of particles, as shown in  Fig. <ref>(c) at ⟨ρ⟩ = 7.5 and as density increases further system becomes heterogeneous  with  random orientation as shown in Fig. <ref>(d) for ⟨ρ⟩
            = 10.00. The varying spatial distribution of local density at different  times are depicted by the heatmaps and local orientation of the bacteria is shown by small black arrows. Here, the length of the arrows are proportional to the magnitude of local P.
            For clarity arrows are drawn with averaging the local P( r) in a cell size of 20 × 20. In the bottom panel of the Fig. <ref>(e-h) we show the zoomed plots of local density shown in Fig. <ref>(a-d) respectively. 
            
            
            The above snapshots are observed at four distinct times. Next, we focus on the  properties of Bacterial suspension with continuous increase of density. The Fig. <ref>(a-b) illustrates the root mean square fluctuations of the order parameter P_rmsf (⟨ρ⟩) and velocity v_rmsf(⟨ρ⟩) vs.⟨ρ⟩ respectively. The root mean square fluctuations in orientation P_rmsf and velocity v_rmsf is obtained by calculating the fluctuations in magnitude of global P and v, where global P and v is obtained by averaging the local orientation and velocity over all the points in space.  The plots suggest that  both  the P_rmsf(⟨ρ⟩) and v_rmsf(⟨ρ⟩) varies non-monotonically with mean density ⟨ρ⟩. 
            
            
             Observing  such non-monotonic density dependence of root mean square fluctuations and spatio-temporal snapshots  of local density shown in Fig. <ref>(a-h), we identified the four distinct phases in the system refereed  as (i) dilute, (ii) clustered, (iii) turbulent and  (iv) trapped  in chronological order.  The vertical lines in Fig. <ref>(a-b) are drawn to mark the distinct phases in the system.  The four snapshots in Fig. <ref>(a-d) are at four different densities in four regions marked by the blue short vertical lines in Fig. <ref>(a-b). 
            In the dilute phase, the local density of bacteria is low (Fig. <ref>(a,e)), and there is almost no ordering (random orientation) and  zero P_rmsf(⟨ρ⟩) and v_rmsf(⟨ρ⟩).  In contrast, in the clustered phase, low and high-density contrasts begin to develop due to the formation of clusters (Fig. <ref>(b-f)),  the local fluctuation in P and v starts to grow. On further increment of density, in the turbulent phase both  P_rmsf(⟨ρ⟩) and v_rmsf(⟨ρ⟩) attain a maxima and local density becomes more inhomogeneous  as shown in Fig. <ref>(c) and (g). Later, we will also observe that in this phase swirling structures develop in the system, and finally, the P_rmsf(⟨ρ⟩) and v_rmsf(⟨ρ⟩) starts to decreases as system becomes  heterogeneous in the trapped phase. We named this phase as trapped due to the weaker dynamics  present in the system, which we will discuss later in the paper. 
            
            The four phases we mentioned above can be  compared with the recent experiments on growing bacteria on the oil-water interface <cit.>. The plots in Fig <ref>(a-b) are obtained for K= 256. 
             We also investigated the same variables for system size of K = 512 and the results are consistent.  
             Till now, we have identified the four phases based on the static properties of the suspension using the  local density, orientational ordering and velocity fluctuations. Next, we characterise these phases based on  other structural and dynamical properties of the suspension. 
            
             
             We calculated the vorticity field ω( r) at the specific densities discussed before in Fig. <ref>(a-d). The corresponding zoom plots are shown in bottom panel Fig. <ref>(e-h).
            The colors represent the normalised vorticity  Ω( r) = ω( r)/ω̅, where ω̅ is the average vorticity in the system.  The local orientation of the bacteria P( r) is shown by arrows. The arrows are drawn by local coarse-graining of the P( r) as in Fig. <ref>.  In Fig. <ref>(a-b) and (d) and (e-f, h), the vorticity field's magnitude is  small, and the bacterial orientations appear quite random, as shown by the black arrows. This low magnitude of vorticity indicates a low level of swirling motion.
            Fig <ref>(c, g) clearly depicts formation of vortices in the system (the magnitude of Ω( r) increased substantially). Local inhomogeneity in Ω is developed with random orientation of bacteria and  defect type structures at the core of high and low vorticity field as can be seen by dark  spots in  Fig. <ref>(g). The animation of the vorticity field of the system starting from low to high mean density (⟨ρ⟩) is given in https://drive.google.com/file/d/1_jQe_aPVcbiaTp6xj5fXZaUjOHkn-O46/view?usp=sharingSM. 
            
             
            The presence of large scale vortex in bacterial suspension is one of the key characteristics of bacterial turbulence. Therefore, we characterise the statistics of the four phases by measuring the probability distribution function of the magnitude of local vorticity P(Ω). In Fig. <ref>, we depict the P(Ω) vs.Ω  for dilute, clustered, turbulent and trapped  phase on semi-log-y scale and the inset shows the same plot in log-log scale. The nature of distribution is very different in turbulent phase in comparison to other three phases. The P(Ω) decay as a power law with 1/Ω for a range of Ω, with an exponential tail at larger Ω. The appearance of power law distribution suggests the finite probability of having the vortices of large magnitude in turbulent phase, but such vortices are less probable in  other three phases. 
            
            
            
            
            
            
            Thus far, our observation is either on the local spatio-temporal pattern or the global order parameters of the system. We further, quantify the characteristics of the four phases by calculating the spatial and temporal correlations of bacterial density, orientation and fluid velocity.  The two-point spatial correlation functions are defined by C_f(r)= ⟨δ f(r_0) ·δ f(r+r_0)⟩_r_0 where, f= ρ, P and v. The δ f is the fluctuations from their respective mean values.  The angular bracket represents the spherical averaging and  average  over 100 independent realizations. In Fig <ref>(a-c), we present the spatial correlation functions of these three fields in sequence. The four plots in each window are for four different phases as marked by the different symbols. We observe that the orientation and velocity correlation increases from dilute to turbulent phase and subsequently decreases in the trapped phase as shown in  Fig. <ref>(b-c). But density correlation is maximum in the clustered phase and lowest in trapped phase  Fig. <ref>(a).   
            This implies, the density patterns formed in respective fields are more correlated in the dilute to clustered and the onset of turbulent phase but, the correlation decreases in the trapped phase.  Although in the turbulent phase, the flow is chaotic and has highly unpredictable motion at various scales, at the onset of turbulence, correlated structures are formed. That leads to increase in spatial correlations initially but  decay at latter times due to the destruction in ordered patterns.  
            In the trapped phase, the bacteria motion is hindered as the system has become densely packed. This suppresses the  ordering and as a result, the correlation decreases. Unlike the other three phases, in the turbulent phase the spatial correlation for density C_ρ(r) shows a large dip in the negative side, which represents the presence of high density of bacteria at the core of the vortices and low density around it. 
            
             Further, we calculate the characteristic length for the fluid velocity L_v(⟨ρ⟩) of the system to examine the average size of the structures or patterns generated in the system as the growth of bacteria progresses. It defined by the distance by which the the two-point correlation function C_v(r) decay to 0.5 of its first  value. Fig <ref>
            (d) depict the correlation length L_v(⟨ρ⟩) vs.⟨ρ⟩.
            
            From the Figure, it is evident that as soon as the system enters the turbulent region, the length L_v(⟨ρ⟩) has an increment initially because of the formation of vortices like ordered patterns.
            At later times, the turbulent phase becomes more chaotic, disrupting the formation of ordered domains. This leads to decay of correlations as well as the characteristic length. 
             This kind of sharp decay suggests random and chaotic nature of turbulent phase in comparison to other phase. As mentioned earlier, in trapped phase the local trapping of the bacteria leads to decrement of the length L_v(⟨ρ⟩) due to very high density of particles.  
            To investigate the effect of system sizes, we plot the correlation lengths for all the variables for system K= 64, 128, 256 as depicted in the Figure. We observed the results are consistent with different system sizes.
            
            We further characterise the local correlations among the three fields in the system by calculating the cross-correlations. 
            In Fig. <ref>(a-c) we show the cross correlation of density-orientation C_ρ P, density-fluid velocity C_ρ v and bacterial orientation and fluid velocity C_Pv respectively. The details of the three correlations are given in Appendix <ref>. The three fields are uncorrelated in dilute and clustered phase, with zero cross-correlations, whereas as soon as the system transit to the turbulent phase local correlations start to pick non-zero values. The local ordering in the turbulent phase appears at the cost of density inhomogeneity, with high ordered region having lower density and  vice versa as can be seen by big negative dip in C_ρ P in Fig. <ref>(a), which is also see by bigger negative dip in the spatial correlation C_ρ(r) in Fig. <ref>(a). The C_ρ v shows a  positive peak in the turbulent phase suggesting the strong local correlation among the bacteria density and fluid velocity. Similar trend also found in the correlation of bacterial density with vorticity. Additionally the local orientation and fluid velocity also show similar trends as density and fluid velocity with smaller peak in the turbulent phase. As we move towards the trapped phase, C_ρ P and C_ρ v approaches to zero, because density starts to become homogeneous whereas the local orientation and fluid velocity shows an anti-correlation, due to the local trapping of bacteria due to high global density and fluid velocity. 
             Further we  investigate the dynamics of different phases. 
            Therefore, we calculated  the fluid velocity auto-correlation function.  
            Fig <ref> depict the auto-correlation function for  fluid velocity C_v(t) sequentially in dilute, turbulent and trapped phases. The details of the calculation of auto-correlation functions in given in Appendix <ref>.
            
            
             It can be evident from the plot that the fluid velocity is weakly correlated in dilute phase, higher correlations are developed in turbulent phase and further the correlations suppresses in the trapped phase due to less dynamics present in the system. 
             Unlike the dilute phase, the auto-correlation shows a strong negative values in the turbulent and trapped phase suggesting the rotation is present in the system, which is highest in the turbulent phase and suppresses in the trapped phase. 
             Alternatively, we present a comparison of our results with the simple Navier-Stokes fluid and find an algebraic decay  with  1/t for the C_v(t) as shown by the solid line in Fig. <ref>(c) <cit.>. That represents that the presence of active Bacteria makes the fluid turbulent, otherwise the fluid itself is in the inertial regime.
            
             
             
             
             
             
            
             
            
             
             We further study the dynamics  of different phases placing a probe particle in the system.
             The dynamics of different phases using a probe particle:-
            
              The dynamics of the probe particle is governed by an over-damped equation with three contributions to its velocity coming from: local density gradient of bacteria, fluid velocity and local orientation of bacteria. The dynamics of probe is affected by the bacterial suspension, but the probe itself is passive and cannot influence the flow of surrounding fluid. Hence, the probe acts like a tracer to study the dynamics of the suspension <cit.>. We started with low density of bacteria in the suspension and placed 1000 probe particles and characterised the speed of probe particle as the density of bacteria in the suspension increases. The details of the governing equations of motion for the probe particles and calculation of speed of particles is given in Appendix <ref>. In the following Fig. <ref> we show the mean speed U of probe particles  vs.⟨ρ⟩. As the system goes from a dilute to a clustered phase and then to a turbulent phase, the speed increases, peaking during the turbulent phase, before decreasing in the trapped phase. This behavior mirrors the system's dynamics and is reflected in the plot of fluid velocity root mean square fluctuations v_rmsf, as illustrated in Fig. <ref>. Similar kind of observation is also found in a recent experimental work <cit.>, where the authors have studied the dynamics of tracer in bacterial suspensions at different densities.
            
            
            To further quantify the system's energetic, we examine the energy spectrum. So far, it is evident that the turbulent phase is highly dynamic, exhibiting persistent spatio-temporal patterns.  
            The energy spectrum of the fluid velocity is a standard approach to characterise the turbulence in normal fluid. We calculate the energy spectrum of fluid  defined by E(q)= ∫<v(r)·v(r+R)> exp(-i  q·R) dR. This is employed in order to quantify the spread of energy cascade through the system at different length scales. Fig. <ref> showcases the same for the different phases in sequence as described above for σ_1=1. The comparison of energy spectrum for other values of σ_1 = 10, 50, 100 is shown in Appendix <ref>.
             For the dilute phase, the positive slope at lower q with E(q) ∼ q^2 indicates increasing energy  towards intermediate q and the steep decay shows the quick dissipation of energy at small scales (large q). The form of the spectrum remains almost similar for the clustered phase as well. 
            The E(q) in the turbulent phase,  increases at lower q showing a positive slope of value 5 (E(q) ∼ q^5), which is higher than that of dilute and clustered phase. That indicates the rate of increase of energy is more in turbulent phase. The peak of energy density shifts towards lower q value showing the transfer of energy towards large scale from small scales. This is an indicative of inverse energy cascade in 2 - dimensional active turbulence <cit.>. Here, the source of energy is intrinsic and comes from the active nature of the bacteria in the system. In this phase, we find E(q) shows an intermediate range of q values where E(q) ∼  q^-2. 
             This  intermediate q range shows the range of different sizes of vortices, through which energy is dissipated and further energy dissipation happens sharply as we go for larger q values.
            Despite the inverse cascade, the energy dissipates at small scales also due to viscous dissipation or  other mechanism of dissipation showing a negative slope of E(q)∼ q^-7.2. 
             The length is different energy scales, where the inverse cascade plays a role can be tuned by tuning the inertia in the system. The bigger range of length scale is found when inertial term is small (small σ_1) and it is suppressed as we increase σ_1. For higher value of σ_1 when system is in turbulent phase as shown in the Fig. <ref>, the peak of the energy spectrum  shifts towards even lower value of q, representing the accumulation of energy at large scales from small scales is favoured by the inertia. 
            Similarly from the plot of the energy spectrum in the trapped phase, we get two different scaling regimes having different slopes. The peaks are at higher q values (smaller scales) in comparison to turbulent phase. That indicates the trapping of bacteria in this phase, the probability of energy transfer from small scales to large scale is less. Different from the turbulent phase, here for a range of intermediate q values energy spectrum E(q) increases with q as q^1.5, representing the energy input on relatively larger q values due to high density of bacteria. Further, the peaks of energy spectrum E(q) shifts towards lower q values with increasing value of σ_1 favouring the accumulation of energy at  the larger scales.
            
            
            
            Density fluctuation is one of the interesting quantity in active system. To observe the effect in growing bacteria suspension, we measure the same in the system.  
            We  calculate the density fluctuations of bacteria in cells of different sizes. The details of the density fluctuation calculation in given in Appendix <ref>.  
            In Fig <ref> we show the plot of  density fluctuation σ̅/√(N̅) vs.N̅, where N̅ is the mean number of particles in different cells of the same size.   For dilute and clustered suspension the density fluctuation is larger for small N̅, σ̅∝N̅ and attains the equilibrium limit for larger N̅, σ̅∝√(N̅). Hence, at larger distances the density is homogeneous in dilute and clustered phase.   As system approaches to the turbulent phase, the density fluctuations are much larger at small scales suggesting large density inhomogeneity in the system and further the density fluctuation suppresses at larger N̅ (σ̅/√(N̅) decays with N̅) showing the local trapping of bacteria at the core of the vortices can be seen in the snapshot shown in Fig. <ref>(c) and (g). In Fig. <ref> (b), we show σ̅/√(N̅) vs.N̅ for large inertia terms (σ_1 = 10, 50, 100) when system is in turbulent phase. As σ_1 increases, the dip in σ̅/√(N̅) shifts towards larger N̅, showing the large scale structures as can be observed in the snapshots in Fig. <ref> (b-d). Additionally, similar information can be analysed from the energy spectrum E(q) in Fig. <ref> (a). 
            
            
            
            § DISCUSSION
            
            We developed a coarse-grained model to study the statistical and dynamical properties of the growing bacterial suspension by allowing the growth of bacteria in the suspension with time  introduced by a birth and death term in the hydrodynamic equations of motion. 
            This model provides a realistic description of growing bacterial dynamics in suspension. In this work, we present a comprehensive results of the growing bacterial suspensions.
            
            
            The properties of the suspension is characterised by local density of bacteria, local orientation and fluid velocity. Staring from low density of bacteria, with time the density of bacteria grows and system transits from dilute to clustered and further to turbulent phase, and at very high density we obtain trapped phase. It is observed that local orientation and fluid velocity develop correlation in the turbulent phase with vortex type patterns. The turbulent phase has specific spatio-temporal patterns resembling the bacterial turbulence in the previous studies <cit.>. Additionally, the turbulent phase lead to the enhanced dynamics with large speed of probe particle. The energy spectrum shows the presence of dissipation at different length scales due to presence of vortices in the turbulent phase and absent in other phases. Further, we found the turbulent nature of the suspension is suppressed, if we increase the effect of inertia σ_1 and instead wave like patterns are developed <cit.>. 
            
            Our study identifies four phases—dilute, clustered, turbulent, and trapped—in the growing bacterial suspension and these phases overlaps with the phase found in recent study of bacterial suspension on oil-water interface <cit.>.
            Unlike the previous study  (theoretical model on dense bacterial suspension), the density of bacteria in our study continuously grow in time, which makes it more closer to many natural bacterial suspension.
            
            The present model and our results can be useful to understand the dynamical and statistical properties of growing bacterial suspension in natural systems. In the current study, the suspension evolves in the clean environment, it is interesting to explore the dynamics of the suspension in the presence of random impurities.
            
            P.J. and S. M. thank Dr. Sivasurender Chandran for useful discussions. P.J. gratefully acknowledge the DST INSPIRE fellowship  for
            funding this project. The support and the resources provided by PARAM Shivay Facility under the National Supercomputing Mission, Government of India at the Indian Institute of Technology, Varanasi are gratefully acknowledged by all authors. S.M. thanks DST-SERB India, ECR/2017/000659, CRG/2021/006945 and MTR/2021/000438  for financial support. P.J. and S.M. also thank the Centre for Computing and Information Services at IIT (BHU), Varanasi.
            
            
             
            
            
            § CROSS-CORRELATION
            
            
            The cross correlation function is defined as C_AB (⟨ρ⟩) = ⟨δ A( r)δ B( r)⟩_ r. To obtain the cross-correlation function, we calculate the product δ A( r)δ B( r) on each lattice point, where δ A and δ B are the fluctuations in the desired quantities. Furthermore, we take the average of this quantity over all the lattice points. Additionally, for better statistical average we take  averaged data of 50 ensembles and plot C_AB (⟨ρ⟩) against the global mean density (⟨ρ⟩).
            
            
            
            § AUTO-CORRELATION
            
            We define the auto-correlation C_v(t) = <δv(t_0) δv(t+t_0)>. And δ v=v(t)- v̅ , where v̅ is the mean value of v(t). Here, t_0 is taken as different reference points within the time span of each phase.
             By observing the snapshots for vorticity field, we decided different time span to obtain the auto-correlation in different phases.
            Next, we calculate the auto-correlation function by averaging the quantity over all the lattice points and time in a particular phase. 
            
            
            § DYNAMICS OF PROBE PARTICLE
            
            
             To examine the dynamical properties of different phases, we use probe particles. We update the position r = (x,y) by using the over-damped Langevin equation.
             The position update of the tracer is
            
             
                d r/dt= -∇ρ +ρP + v
            
            Here, the forces on the right hand side are local density gradient of bacteria, fluid velocity and local orientation of bacteria. Further, to calculate the  speed of the tag particle, we observed the trajectory of the particle by tracking it's co-ordinates.  Using the co-ordinates in every 10 unit time differences, we calculate the mean speed of the particle in each phase during it's motion. Further, the speed is averaged over the time span of each phase.
            
            
            § DENSITY FLUCTUATION
            
            We calculate the normalized density fluctuation σ̅, by defining σ̅ = <ρ_n^2-ρ^2>/ρ_nn . To calculate the quantity numerically, we divide the system in to n multiple small cells and assume these cells as n ensembles.
            Here, ρ_n is the mean density in each ensembles in the corresponding  phase and ρ_nn is the average of the mean density in the corresponding phase.  N̅ is obtained by ρ_n/ρ_nn.
             Finally, we plot the density fluctuation σ̅/√(N̅) vs.N̅ in Fig <ref>. 
            
            
            § EFFECT OF INERTIA
            
            Until now, the results presented here are for fixed inertial coupling. Now,
            by increasing the strength of this term we increase the response of P with respect to δρ.
            To elucidate the significance of inertia in the turbulent phase, we examine the growth of bacteria by varying the coefficient σ_1 from 1 to 100. In Fig <ref>, we showcase the real snapshots of density field in the turbulent phase by fixing σ_1 = 1, 10, 50, 100 respectively. For all cases, we make sure that the system is in turbulent phase by characterizing P_rmsf and V_rmsf. For lower values of σ_1 = 1, 10, vortices are generated in the system distinctively, but with increasing value, the vortices are not sharp and wave like structures are developed instead and move outward from their central location. This type of concentrated wave formation with the higher value of σ_1 also seen in the work of <cit.>.
            
            
            The Fig. <ref> (a) represent the effect of σ_1 on the energy spectrum in the Fourier space in turbulent  phase. It shows the peaks in the energy spectrum shift towards lower q values in case of higher values of σ_1 in turbulent phase. 
            Similarly, Fig. <ref> (b) σ̅_1/√(N̅) vs.N̅ plot in turbulent phase shows the shifting of the bending towards large N.
            
            
            
            
            
            
            
            
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 Paper Title: Substrate stiffness modulates bacterial adhesion and diversity of adherent phenotypes across growth stages.
















 















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Substrate stiffness modulates bacterial adhesion and diversity of adherent phenotypes across growth stages
    René Riedel^1, Garima Rani^1, Anupam Sengupta ^1,2^∗



^1Physics of Living Matter Group, Department of Physics and Materials Science,

University of Luxembourg, 162 A, Avenue de la Faïencerie, L-1511, Luxembourg

^2Institute for Advanced Studies, University of Luxembourg,

2, Avenue de l’Université, L-4365, Esch-sur-Alzette, Luxembourg

^∗E-mail: anupam.sengupta@uni.lu

    
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Surface-adhesion and stiffness of underlying substrates mediate geometry, mechanics and self-organization of expanding bacterial colonies. Recent studies have qualitatively indicted that stiffness may impact bacterial attachment and accumulation, yet the variation of cell-to-surface adhesion with substrate stiffness remains to be quantified. Here, by developing a cell-level Force Distance Spectroscopy (FDS) technique based on Atomic Force Microscopy (AFM), we simultaneously quantify the cell-surface adhesion alongside stiffness of the underlying substrates to reveal stiffness-dependent adhesion in phototrophic bacterium Chromatium okenii. As stiffness of the soft substrate, modelled via low-melting-point (LMP) agarose pad, was varied between 20 kPa and 120 kPa by changing agarose concentrations, we observe a progressive increase of the mean adhesion force by over an order of magnitude, from 0.21±0.10 nN to 2.42±1.16 nN. In contrast, passive polystyrene (PS) microparticles of comparable dimensions showed no perceptible change in their surface adhesion, confirming that the stiffness-dependent adhesive interaction of C. okenii is of biological origin. Furthermore, for Escherichia coli, the cell-surface adhesion varied between 0.29±0.17 nN to 0.39±0.20 nN, showing a weak dependence on the substrate stiffness, thus suggesting that the stiffness-modulated adhesion is a species-specific trait. Finally, by quantifying the adhesion of C. okenii population across different timescales, we report an emergent co-existence of weak and strongly adherent sub-populations, demonstrating a diversification of adherent phenotypes over the growth stages. Taken together, these findings suggest that bacteria, depending on the species and their physiological stage, may actively modulate cell-to-surface adhesion in response to the stiffness of soft surfaces. Our results suggest how bacteria could leverage stiffness-dependent adhesion, and the diversity therein, as functional traits to modulate initial attachment, colonization and proliferation on soft substrates during the early stages of biofilm development.

















§ INTRODUCTION


From living tissues to biomedical scaffolds, bacteria attach and colonize a wide range of surfaces, spanning orders of magnitude of stiffness <cit.>. Once attached, subsequent growth and accumulation the surface-associated bacterial populations are underpinned by mechanotransduction, ultimately leading to well-developed biofilm structures<cit.>. Surface sensing and mechano-response during initial stages of biofilm development <cit.> are crucial for the long term fate of biofilms. At the scale of individual cells, the surface energy of underlying substrates, i.e., their hydrophobicity and hydrophilicity, influence bacterial adhesion <cit.>. In addition, van der Waals and electrostatic forces could regulate the cell-surface interactions <cit.>, thereby influencing biofilm growth. Recent results have also accounted for local osmotic pressure and poroelastic attributes of substrates in controlling biofilm growth and morphology <cit.>.

Over the recent years, multiple studies have indicated at the possible role of the substrate stiffness on the bacteria-surface adhesion. Majority of these studies have been conducted on agarose and poly-ethylene glycol (PEG) hydrogels <cit.>, and on polymeric substrates including poly-dimethylsiloxane (PDMS) <cit.> and composite thin films <cit.>. While different bacterial species have been covered in these studies, a generalized framework – capturing the impact of stiffness on bacterial adhesion – is yet to emerge, in part due to observable inconsistencies across the reported trends <cit.>. Traditionally, agarose, a linear polymer composed of alternating D-galactose and 3,6-anhydro-L-galactopyranose monomers<cit.> has been frequently used in studying bacterial growth and biofilm assays. Agarose possesses high gel strength, transparency,<cit.> and non-toxicity, making them suitable for investigating bacterial adhesion, growth and self-organization on such surfaces <cit.>.

More recently, low-melting-point (LMP) agarose has gained attention as a special form of agarose <cit.> wherein the gelling temperature is decreased due to chemical modifications, for example via hydroxyethylation<cit.>. The ability to form thermoreversible gels at low temperatures makes LMP agarose a preferred choice where gentle gelation conditions are desired, for instance, for bacterial assays <cit.>, encapsulation of heat-sensitive biomolecules<cit.>, tissue engineering scaffolds<cit.>, drug delivery systems<cit.>, and 3D cell culture models<cit.>. 

Despite the growing relevance of LMP agarose for bacterial studies, a thorough quantification of their mechanical properties are largely lacking. Bulk and local elasticity, stiffness and adhesion of substrates are of particular interest in microbial ecology, as they not only influence the physiology and behavior of cells, but also impact the biophysical underpinnings of bacterial growth, feedback and emergent traits <cit.>. Elasticity is a fundamental mechanical property that characterizes the ability of a material to deform under an applied force and return to its original shape upon force removal. Various techniques, including bulk rheology and local atomic force microscopy (AFM) have been employed to investigate the elastic properties of such soft gels to understand their behavior under different loading conditions<cit.>. Adhesion, the ability of a material to stick to another material, may originate either due to mechanical interlocking or various physico-chemical interactions between surface molecules <cit.>. In the context of bacterial adhesion, diverse species- and trait-specific attachment mechanisms have been proposed, which ultimately underpin the growth and dynamic self-organization into bacterial colonies<cit.>. 

While recent studies have indicated that bacterial attachment and accumulation could be impacted by the stiffness of the underlying substrates, currently we lack methodologies which could allow simultaneous measurement of both parameters. Furthermore, how stiffness-dependent adhesion evolves with the age of a bacterial population remains unexplored. As described in the aforementioned studies, the impact of stiffness on cell-surface adhesion has received qualitative treatment so far, either by estimation of the surface density of the adhering cells, or via retention-assays which estimate the relative fraction of leftover cells following a washing protocol. Quantification of the cell-surface adhesion forces, alongside concomitant measurement of the underlying substrate stiffness is currently lacking. 

Motivated by the gaps in our current understanding, here we investigate the mechanical properties of LMP agarose gels prepared with different LMP agarose concentrations in Lysogeny broth (LB) medium, a nutrient-rich liquid used for growing bacteria. We measure their elasticity by Young's modulus, and the adhesion property, both native and with respect to different bacterial species, by means of cellular Force Distance Spectroscopy (FDS) technique based on an Atomic Force Microscopy (AFM), under the LB medium as well as deionized (DI) water. Focusing on the Gram negative bacterial species, Chromatium okenii, we measure their cell-surface adhesion to the LMP agarose and analyze how their adhesion develops over the physiological growth stages. We compare our results with another Gram negative species, Escherichia coli, as well as assess the possibility of active modulation of cell adhesion by contrasting the bacterial experiments against adhesion of passive polystyrene beads of 5μm and 20μm diameters to LMP agarose substrates. Overall, our findings reveal that that bacteria, depending on the species and their physiological state, can actively modulate cell-to-surface adhesion, as a response to changes in the stiffness of underlying soft surfaces. In doing so, bacteria may harness adherent phenotypes within a population to initiate attachment and optimize proliferation strategies on soft substrates across a range of stiffness. 



§ EXPERIMENTAL




 §.§ Force Distance Spectroscopy of bacterial cells on soft surfaces

By means of a Nanoscope® V controller (Veeco) AFM with Multimode 8 scanner, force distance curves on pure agarose gels were measured using probes with a spherical SiO2 tip (3.5 µm diameter, 0.2 N/m force constant, CP-CONT-SiO, nanoandmore GmbH, Germany). For cell adhesion measurement, tipless cantilever (0.2 N/m force constant, TL-CONT, nanoandmore GmbH, Germany) were used. Sample measurement was fulfilled in fluid (deionised (DI) water or LB medium) using a fluid chamber with inlets and rubber sealing. Deflection sensitivity was determined beforehand from the slope of the retract curve of a force distance spectroscopy measurement performed in measurement fluid on a silicon surface. To translate deflection to forces, the force constant measured by nanoandmore GmbH was used.
After removing the cover slip, the prepared sample holder was put in the AFM and a droplet of the measurement fluid was transferred onto the agarose gel surface. The prepared fluid chamber was installed and filled with measurement fluid through the inlets using syringes. Subsequently, the cantilever was manually approached to the surface but not brought to contact. During this procedure, the horizontal position of the cantilever was moved to the center position of the region of interest. 
After visually close tip and surface approximation, the probe was false engaged (i.e. tip-surface contact was simulated) and the sample was subsequently manually approached by means of the step motor control and continuously measured force distance curves until an increase in slope was detectable.
For cell and microparticle adhesion measurement, the cantilever tip region was first approached towards cell accumulation regions. A cell/microparticle was picked up by the tipless cantilever and used as cantilever tip. The subsequent measurements were executed at a cell/microparticle-free region on the sample surface.
Force spectroscopy curves were recorded on 16 × 16 equidistant spots around the initial tip-surface contact point. The distance between adjacent spots was 500 nm. The measurements were repeated at least 4 times on the same sample. Maximum applied force did not exceed 10 nN. 



 §.§ Optical Microscopy

To ensure that cells and microparticles were present on the tipless cantilever, the sample holder as well as the tip holder were both investigated under an optical microscope (Eclipse LV100N POL, Nikon Corporation) in reflected light mode right after measurement and careful removal of excess liquid with a soft absorbant tissue.




 §.§ Evalution of the Force Distance Spectroscopy data

AtomicJ (v 2.3.1, Pawel Hermanowicz) was used to extract the measured values from the recorded data files and process them by applying different theoretical contact mechanic models. A typical force distance curve is shown schematically in Figure <ref>A. It consists of a region of no contact between tip and substrate and a region of indentation. These two regions are connected by the Contact Point z_c. For proper calculation, precise determination of the position of the contact point is indispensable.
Contact mechanic models are applied to the contact area of the force distance curves. They are based on the Hertz model<cit.>, which describes the interaction between two spheres or a sphere and an endless, plane surface (see Figure <ref>B). According to Hertz, the generalised force-indentation relation can be described by:


    F=λδ ^β


where F is the applied force, δ is the indentation depth, λ and β are values that depend on the underlying model. For the basic Hertz model indentation of a spherical body into a plane surface, β is 3/2 and λ is given by the following equation:


    F = 4ER^1/2/3(1-υ^2)δ^3/2


Here, E is the elasticity modulus, R is the tip radius and υ is the Poisson Ratio. The latter is assumed to be ≈ 0.5 for very soft samples.

The Hertz model is a very basic model that does not consider adhesion forces and their subsequent contact area enhancement. Derjaguin, Muller and Toporov (DMT)<cit.> as well as Johnson, Kendall and Roberts, JKR <cit.>, have parallely created extended models based on the Hertz model to incorporate adhesion forces. These two models represent the limit values for the true value of elasticity. While DMT is more accurate for harder samples with low adhesion, soft samples with higher adhesion are better described by JKR.

In this paper we therefore focus on the evaluation using the JKR model via the following three equations:


    δ = α_JKR^2/R -4/3√(α_JKR F_ad/R K)


    α_JKR = [ R/K( √(F_ad) + √(F + F_ad))^2]^1/3


    F_ad = 3/2πγ R


    K = 4E/3(1-υ^2)


where α_JKR is the JKR-extended contact area between tip and sample, F_ad is the tip-sample adhesion force, K is the elastic constant of the sample, and γ is the interfacial energy. The adhesion force, F_ad, was evaluated from the minimum point of the force distance curve relative to its baseline. 

E can hence be calculated by plotting δ versus F and adjust K to minimize the least square error between the resultant plot and the measured data, starting from the tip-sample contact point. This latter is  ascertained numerically for best fit. More comprehensive details on force spectroscopy data evolution is well described by the works of Lin, Dimitriadis, and Horkay<cit.>. The evaluated data are represented as boxplots without whiskers. The inner 50 % of the measured values sorted in ascending order are represented by a coloured area containing the median as a horizontal line. Outliers were identified as data points that lie below Q_1 - 1.5 * IQR or above Q_3 + 1.5 * IQR, where Q_1 and Q_3 are the first and third quartiles, respectively, and IQR is the interquartile range (Q_3 - Q_1). Outliers are not shown here. In addition, the average value is shown as a point and the standard deviation as an error bar.




 §.§ Low-melting-point agarose gel preparation

Agarose gels with different concentrations (see Table <ref>) were prepared by slowly dissolving the respective quantity of low-melting agarose gel (Agarose, LMP, Preparative Grade for Large Fragments (> 1,000 bp), Promega, USA) in 5  ml LB medium in an autoclaved glass tube under occasional stirring on a heat plate (50 °C). When the powder was completely dissolved, the solution was left to cool down to room temperature and then stored in a fridge (4 °C).



Sample preparation is shown in Figure <ref>C. A ring-shaped sample holder with an inner diameter of 10 mm and a height of 2 mm was glued onto a glass coverslip with a diameter of 12 mm so that no glue was inside the ring. A magnetic plate was glued to the other side of the coverslip. The whole setup was dried for 24 hours and used as AFM sample holder.
Prior to sample preparation, the agarose gel was taken out of the fridge and slowly heated on a heater plate. In order to increase hydrophilicity of the sample holder glass, the sample holder glass surface was plasma-treated for 45 seconds using a plasma torch (Mode BD-20, Electro-Technic Products, Inc., USA). 200 µl of the warm and viscous agarose gel was transferred to the sample holder. A coverslip was placed onto the sample holder so as to enclose the agarose gel while avoiding to trap air inside. The agarose gel was left to cool down for 30 minutes, then the coverslip was slowly and horizontally removed just prior to measurement. 
The topographies of two samples with 2.2 % agarose concentration were measured (in LB medium environment and in water environment) by means of a Nanoscope® V controller (Veeco) AFM with Multimode 8 scanner in Tapping mode using probes with a spherical SiO2 tip (3.5 µm diameter, 0.2 N/m force constant, CP-CONT-SiO, nanoandmore GmbH, Germany).







 §.§ Adhesion of polystyrene beads to LMP agarose gel

Polystyrene microparticles with 5 µm and 20 µm were purchased from microparticles GmbH, Germany (product numbers PS/Q-R-KM650 and PS/Q-R-KM636 respectively). The stock solutions were diluted by factor 1 × 10^-3 until single particles were visible on the substrate after sedimentation. The agarose gel sample was prepared as described above. After removement of the coverslip from the agarose gel sample, a drop of the microparticles solution was put onto the agarose gel sample. The microparticles were allowed to settle down for 30 minutes before the drop was carefully removed. Subsequently, force distance curves were recorded as described above. This procedure was repeated three times. In order to measure the adhesion of the passive particles, the same measurements were repeated with agarose gel based on DI water as well. For each agarose concentration and particle size, the procedure was repeated three times.



 §.§ Preparation of bacterial assays

Adhesion forces of two different cell species Escherichia coli (strain NCM 3722 ΔmotA, referred to as E. coli) and Chromatium okenii (referred to as C. okenii) to LMP agarose gel with different concentrations (1.5 %, 2.2 % and 3.1 %) were determined.

 E. coli were stored in a freezer at -80 °C. Agar plates were prepared by dissolving 16 g of LB-agar (LB-Agar (Luria/Miller), CARL ROTH, Germany) in 400 ml deionized water with subsequent autoclaving and stored at 4 °C. The cells were taken out of the freezer and streaked on an agar plate in sterile environment at room temperature. The agar plate was placed in an incubator for 48 h at 30 °C. Subsequently, a single cell island was transferred from the agar plate into 8 ml LB medium using a bacterial loop in sterile environment. The cell solution was placed into the incubator at 30 °C and shaken at 180 RPM during 24 hours. Thereafter, 200 µl of this cell solution was transferred to 2 ml LB medium and kept in the incubator at 30 °C under continuous shaking at 180 RPM. After 160 min, the cell suspension was centrifuged at 2000 rpm for 5 min. Except for 100 µl, the centrifugate was slowly removed. The cells were resuspended and transferred onto a freshly prepared agarose gel surface sample (described above). The cells were allowed to settle down on the agarose gel surface for 30 minutes. The drop was then carefully removed and FDS was executed as described below, under the LB medium environment.

Phototrophic sulfur bacteria C. okenii was obtained from Lake Cadagno in the Piora Valley (46°33’N, 8°43’E) in the southern Swiss Alps, following the protocol described in Ref. <cit.>. They play a key role in sulfide removal from shallow sediments and stratified waters <cit.>. Upon transferring the cells to laboratory, they were grown and propagated in Pfennig's Medium I protocol with periodic spiking of hydrogen sulphide<cit.>. Cultivation was run under light/dark photoperiod (16/8 h) with a light intensity of 38.9 µmol m^-2 s^-1. After soft shaking, 100 µl of the cell solution was transferred onto the freshly prepared LMP agarose surface (described above). The cells were allowed to settle down for 30 minutes. Subsequently, the drop was carefully removed, and measurement was executed as described below under acqueous environment. 



 §.§ Quantification of C. okenii adhesion properties over growth stages


In order to track the evolution of cell-surface adhesion over time, FDS was carried after 2 weeks, 10 weeks and 14 weeks of inoculation of C. okenii in Pfennig's Medium I. Although motile in their natural lake ecosystem, under laboratory conditions, C. okenii gradually shifts to a sessile lifeform <cit.>. The physiology-dependent measurements were conducted on agarose concentration of 2.2 %, following the gel preparation protocol described previously. The cell solution was kept in a glass container enclosed with a septum so as to protect it from air contact. For each measurement, 0.8 ml of cell solution was taken using a 1 ml syringe and needle. The cells were collected either from the surface of the cell solution (top cells) or from the bottom of the glass container (bottom cells) and transferred to a 2 ml Eppendorf tube. After removal, the cells were centrifuged at 6000 RPM for 180 s. The pellet was carefully aspirated using a 10 µl pipette and finally transferred to the prepared LMP agarose gel substrate. The sample was then left to stand for 30 minutes so that the cells could settle on the substrate. The cantilever was wetted with a drop of DI water before the cantilever holder was installed. This gentle method minimised fluid flow which could interfere measurements within the sample solution. Each measurement series consisted of six measurements. Firstly, three measurements were carried out with top cells. Then three measurements were taken with bottom cells. Each series of measurements was completed within 3 days.










§ RESULTS




 §.§ Topography of LMP agarose gel surfaces

Figure <ref> shows the topography of 2.2 % agarose gel in LB medium (A) and DI water (B) environments. The sample surfaces showed overall low surface roughness, furthermore, no significant difference in roughness was observed between the two cases. 






 §.§ Mechanical properties of the LMP agarose gels

A representative example for a measured force distance curve is given in Figure <ref>A. The computed outcomes of Young's modulus and adhesion values of agarose gel samples without cells are presented in Table <ref>. Figure <ref> shows the corresponding measurements conducted in DI water and LB medium. The provided data includes both median and mean values (n > 500).








The Young's modulus, also known as the modulus of elasticity, is a measure of the stiffness of a material, defined as the ratio of stress to strain within the elastic limit of the material. Upon increasing the LMP agarose concentration, the Young's modulus increases, for both DI water and LB medium environments. With Young's moduli smaller than 100 kPa, the agarose substrate falls within the range of soft biomaterials (e.g., cells, organs) and soft synthetic polymers<cit.>. The observed trend can be elucidated by examining the underlying structure of agarose gels<cit.>. LMP agarose monomers consist β-D-galactose and 3,6-anhydro-α-L-galactose units. Upon cooling the agarose gel solution below the gelation temperature (24 - 28 °C), the equatorial hydrogen atoms of the 3,6-anhydro-α-L-galactose residues form hydrogen bonds with each other, inducing the formation of α-helices, forming a single chain or double chains. The cooling rate determines the ratio of one-chain and two-chains α-helices. This process results in the formation of a 3D secondary agarose gel structure. The gelation temperature and pore size of the gel primarily depend on the availability of hydrogen and oxygen in the agarose solution. A lower agarose concentration leads to fewer α-helices, resulting in reduced structural stability and larger pore sizes. Alongside α-helices, other types of physical cross-linking between the chains increase the stability of the gel matrix. The structural stability directly correlates with the gel's Young's modulus, explaining the increase in Young's modulus with higher agarose concentration. Low-melting agarose is produced by hydroxyethylation such that a portion of the available alcoholic side groups is substituted by ethyl groups and hence unable to form hydrogen bonds. This substitution reduces the density and strength of α-helices, leading to a lower Young's modulus<cit.>. Overall, the Young's moduli we measure are lower than those of normal melting agarose gels, as reported in literature<cit.>. Roberts et al. investigated the elastic storage modulus of 1 % normal agarose gel to be around 80 kPa<cit.>, with other reports corroborating their results <cit.>.

The mechanical properties of LMP agarose gels in LB medium have received limited investigation so far. Existing studies report varying results on mechanical properties across different agarose concentrations, though the overall range matches well with our measurements. Kontomaris et al. used a similar LMP agarose gel and measured an elastic modulus of 154 kPa (2.5 %)<cit.>, while Topuz et al. observed values in the same range as this study for LMP agarose gel<cit.>. Finally, Zamora-Mora et al. measured an Young's modulus of 400 Pa for 0.5 % agarose gel and 1.45 kPa for 2 % agarose concentration.<cit.>, similar to Kumachev et al.. who reported the range to be between 50 Pa (0.75 %) to 2.5 kPa (3.00 %) <cit.>. It is to noted that last two studies measured the shear modulus, though the Young's modulus is expected to be in the same range as well.








 §.§ Cell-surface adhesion force on LMP agarose gels

The evaluated values for adhesion of E. coli and C. okenii cells to LMP agarose gel with different agarose concentrations are listed in Table <ref> and plotted in Figure <ref>. Optical microscopy images (Figure <ref>A and B) reveal that cells were attached to the cantilever surface, confirming that the adhesion force between cell and agarose gel surface was measured. The force distance curves show a smooth adhesion characteristic, as has been observed for cell initial adhesion on hydrophilic surfaces<cit.>. For E. coli, the adhesion values are in weak sub-nN range. As can be seen, adhesion forces slightly increase as agarose concentration increases, which is in agreement with the adhesion to normal agarose gels<cit.>.



For C. okenii, however, the adhesion force increased by up to an order of magnitude compared to E. coli adhesion forces. Furthermore, the C. okenii are found to be more sensitive to changes in the LMP agarose concentration. While doubling the agarose concentration from 1.5 % to 3.1 % increases the median of adhesion force surface by factor 1.3 for E. coli cells, it is enhances by a factor of  15 for C. okenii. Furthermore, a detailed examination of the adhesion force data distribution for C. okenii reveals emergence of a bimodal distribution of adhesion force at LMP agarose concentrations of 2.2 %, which then becomes more pronounced when the agarose concentration is raised to 3.1 % (see Figure <ref>). In order to understand the co-existence of the adherent phenotypes, we separated the two peaks in the bimodal adhesion force distribution for 2.2 % and 3.1 % agarose concentrations, with dynamic threshold adhesion values, as indicated by the yellow and green regions in Figure <ref>. The bimodal distributions indicate the co-existence of two different adherent sub-populations: weakly adherent cells (yellow regions of the plot), and the strongly adherent cells (green regions of the plot), which emerge as the stiffness of the underlying substrate is modulated. At an LMP concentration of 1.5 %, such a bimodality could not be detected.


As shown in the boxplots, Figure <ref>, the lower range of C. okenii adhesion force distribution peaks around the same range as for E. coli cells, i.e., low adhesion force that depends weakly on the agarose concentration. In contrast, the higher adhesion force peaks seen in C. okenii, reveal a stronger dependence on LMP agarose concentration. The small adhesion noted in both E. coli and C. okenii corresponds with the general observation that bacteria with hydrophobic surfaces tend to adhere more readily to hydrophobic than hydrophilic surfaces. Nevertheless, the weak adhesion forces observed here, vary slightly as the agarose concentration is increased. To understand this, a bacteria can be modeled as solid body, with biopolymers on their surface represented by flexible chains<cit.>. During the separation of a cell from the surface, various forces including electrostatic forces, hydrogen bonds, dipole interactions, and Lifshitz-Van der Waals forces act to oppose detachment. Taken together, the interactions can be captured using the extended DLVO theory (xDLVO)<cit.>, however, in a more complex biological context (as the one studied here), the xDLVO theory might require further modifications, which are beyond the scope of this work. Another possible effect during the detachment of a bacterial cell is steric hindrance, due to the partial entanglement of the polymeric surface chains <cit.>. The minimum force required for complete separation corresponds to the adhesion force, which depends on the properties of the flexible surface chains and the structure and properties of the underlying network, which in turn, alter with agarose concentration, offering a possible reason for the slight enhancement of the weak adhesion forces with increase in the agarose concentration.








While E. coli showed weak adhesion forces across different LMP agarose concentrations tested, C. okenii showed different trend. The bimodal distribution indicates that a few cells bind strongly to the surface macromolecules, while others bind weakly. Similar behaviour had been observed also for Staphylococcus aureus on hydrophobic surfaces<cit.>. Increasing the LMP agarose concentration could provide more anchor points for the C. okenii cells, thus increasing the overall adhesion force. However, the exact binding mechanism is currently under investigation, and will be discussed elsewhere.









 §.§ Adhesion of polystyrene microparticles on LMP agarose gels

To better understand the adhesion force results and to see if there is an active biological modulation of adhesion, experiments were repeated with passive PS microparticles. As with the bacterial cells, the PS microparticles possessed hydrophobic surfaces. To maintain a comparable size range, microparticles with diameters of 5 µm and 20 µm were used. The system also consisted of an agarose gel based on LB medium as the substrate and DI water as the surrounding medium. To minimize the influence of ions present in the LB medium, these experiments were repeated in a system with agarose gel based on DI water in a DI water environment. The results are shown in Figure <ref>, while the corresponding values are provided in Table <ref>. The adhesion forces fall in the nN range, with marginal increase increase with the agarose concentration. A comparison with the results for E. coli and C. okenii shows that the adhesion force for PS beads is somewhat higher. Though our results cannot conclusively confirm if E. coli actively tune their adhesion with LMP agarose, C. okenii exhibits a distinct adhesion trend, suggesting a specific interaction between the substrate and the bacteria that manifests at a specific agarose concentration. This phenomenon aligns with previous findings on the adhesion of Staphylococcus aureus <cit.>, suggesting the presence of diverse active adhesion mechanisms in bacteria.









 §.§ Evolution of C. okenii adhesion over growth stages


As a final step of our experiments, we measured the evolution of the C. okenii adhesion forces during the course of laboratory inoculation. Recent report by the authors have demonstrated that, under laboratory conditions, C. okenii undergoes a lifeform shift from a free-living motile to biofilm forming sessile phenotypes <cit.>. Since bacterial adhesion to surfaces is central to biofilm formation, the final measurements are motivated by the question: Does the lifeform shift of C. okenii correspond to changes in their cell-substrate adhesion properties? Regardless of the exact mechanism of adhesion between C. okenii and the LMP agarose substrate, it is intriguing that C. okenii populations elicit wek and strong adherent phenotypes at higher substrate stiffness. Relative to the planktonic states with suppressed cell-surface adhesion properties, transition to sessile lifeforms could be triggered by amplified adhesion mechanisms (for instance, via secretion of exopolymeric substances under environmental stressors), driving cellular flocculation and surface attachment, ultimately leading to sessile lifeforms<cit.>. Thus, starting with a fresh sample of planktonic C. okenii isolated from their natural lake habitat, we analysed their cell-substrate adhesion forces, and compared them with lab-grown cultures over a period of 14 weeks (late stationary phase, <cit.>). Macroscopically, we observed two distinct C. okenii layers in the inoculation bottle: a bottom layer attached to the floor of the glass bottom, and a top suspended layer at the air-water interface, as visualized and labeled in Figure <ref>.


Figure <ref>B-D illustrates the adhesion values of both top and bottom cells to a 2.2 % agarose gel substrate over time since their removal from their natural habitat and maintenance under laboratory conditions. The combination of top and bottom cells from each time period is shown in Figure <ref>, including data for populations grown into late stationary phase (Figure <ref>B). Table <ref> lists corresponding values.

After two weeks of laboratory inoculation, i.e., when the C. okenii population is in its mid-exponential phase, the cell adhesion forces fell within a narrow range, with top cells exhibiting higher adhesion (0.36 ± 0.18 nN), relative to their bottom counterparts (0.12 ± 0.07 nN). The scenario changes after ten weeks, when the C. okenii population enters stationary phase: the bottom sub-populations have an enhanced adhesion, with an overall increase in the diversity of the adherent phenotypes (as indicated by the high standard deviations in our measurements, see Table <ref>). The difference between median values of upper and lower cells decreased. Finally, once the population reached late stationary phase (at 14 weeks), the bottom population is now considerably more adherent, with a larger diversity of the adherent phenotypes. The top population however maintains its adhesion. A closer examination of the separate measurements for bottom and top cells revealed a progressive evolution of adherent characteristics of the bottom cells, whereas the adhesion properties of the top cells appear to stabilize over the same period of time. Thus, not only do the mean and median adhesion forces increase, the entire range of adhesion forces, including the maximum adhesion force, enhance over time, accompanied by an overall decrease in the number of top cells. 













§ DISCUSSIONS AND CONCLUSION


In this work, we present a Force Distance Spectroscopic approach to simultaneously measure the stiffness and adhesion of low-melting-point agarose gel, both in its native state as well as in the context of cell-surface adhesion of bacterial species. Our results show that with increasing agarose concentration, the substrate stiffness goes up, however, remains below the typical value of normal melting agarose gels. The Young's moduli of LMP agarose gels increase from approximately 20 kPa to around 100 kPa as the LMP agarose concentration is doubled from 1.5 % to 3 %. The reported range of Young's moduli falls between ultralow-melting-point agarose and the normal-melting-point agarose, and thus offers a well-suited alternative to existing hydrogels used in biomedical research and engineering. No significant difference was observed depending on the choice of the liquid environments (LB versus DI water) within which the measurements were conducted. The stiffness of LMP agarose gel lies in the range of soft tissues such as muscle, epithelial and neural tissues<cit.>, making it potential material for organ-on-a-chip tissue engineering applications.  

FDS experiments with E. coli and C. okenii bacteria on LMP agarose gel with three different concentrations revealed a stiffness-dependent variation of the cell-surface adhesion. While E. coli showed marginal increment of the adhesion with increasing agarose content, the attachment of C. okenii showed two different adherent phenotypes: one with adhesion force in the same range as observed for E. coli (weak adhesion); and a second sub-population of strongly adherent cells as well as increased adhesion with the LMP agarose concentration. Previous works with diverse species and substrates have indicated inconsistent adhesion forces, underscoring the possibility of cofounding factors, beyond stiffness, to contribute to the dynamics of bacteria-substrate adhesion. For example, Thio et al. determined the adhesion forces of E. coli bacteria towards polystyrene and different polyamides to be in the range of 2.9 nN to 9.7 nN<cit.>, while Wang et al. reported that Staphylococcus aureus adhesion on polyacrylamide hydrogels decrease by three orders with increasing substrate stiffness<cit.>. While E. coli and P. aeruginosa show a similar adhesion trend on polydimethylsiloxane substrate (0.1 MPa to 2.6 MPa)<cit.>, E. coli and S. aureus show an increased adhesion to stiffer poly(ethylene glycol) dimethacrylate hydrogels, relative to softer ones<cit.>. This is also supported by Francius et al.<cit.> who observed a linearly increasing adhesion dependence of different E. coli strains on poly(allylamine hydrochloride)/hyaluronic acid hydrogels stiffness between 20 kPa and 700 kPa. Guégan et al.<cit.> measured the effect of the stiffness of agarose gel of two different concentrations (0.75 % and 3.0 %) on the adhesion capability of gram-negative Pseudoalteromonas sp. D41 and gram-positive Bacillus sp. 4J6 using retention assays. While Pseudoalteromonas sp. D41 showed higher adhesion for softer agarose substrate, adhesion of Bacillus sp. 4J6 increased with increasing stiffness.

The mechanism of this adhesion process depends on the properties of the bacteria, substrate and the local environment<cit.>. For C. okenii, the larger contact area of surface polymeric molecules on the substrate, as well as its surface chemistry, may promote the observed adhesion and the adherent diversity on the agarose gel substrates. Our experiments over the physiological growth stages of C. okenii indicate a progressive increase in adhesion (on 2.2 % agarose gel) over time, supporting the lifeform shift from planktonic to biofilm state<cit.>. The C. okenii biofilm formation could be observed stably after 14 weeks of inoculation, as shown in Figure <ref>. As reported previously by Di Nezio et al.<cit.>, such shifts in lifeforms trigerred by domestication of species, occur synergistically across multiple phenotypic traits, including morphology, cell density, intracellular attributes, and alteration of motility. The cell-substrate adhesion data supports the previously reported shift in the C. okenii lifeform, occurring over a period of weeks after laboratory inoculation. The phenotypic diversity of the adherent cells increase over time, furthermore suggesting an adaptive transformation which could allow the domesticated cells to optimise surface attachment once they are inoculated over extended periods under laboratory conditions. Alteration of cell adhesion following domestication has been reported also for Saccharomyces cerevisiae<cit.>, wherein such changes were mediated by modified expression of genes related to cell-cell adhesion and the production of EPS <cit.>. Similar pathways could be targeted in follow up studies to delineate how planktonic C. okenii undergo lifeform shifts, specifically in the context of stiffness- and physiology-dependent cell adhesion.  







§ ACKNOWLEDGMENTS

 We thank F. Di Nezio and N. Storelli for sharing Chromatium okenii cells and valuable discussions on their growth and physiology. We gratefully acknowledge the support from the Institute for Advanced Studies, University of Luxembourg (AUDACITY Grant: IAS-20/CAMEOS to A.S.) and a Human Frontier Science Program Cross Disciplinary Fellowship (LT 00230/2021-C to G.R.). A.S. thanks Luxembourg National Research Fund for the ATTRACT Investigator Grant (A17/MS/ 11572821/MBRACE) and a CORE Grant (C19/MS/13719464/TOPOFLUME/Sengupta) for supporting this work. 




§ CONFLICT OF INTEREST

The authors declare no conflict of interest.




§ AUTHOR CONTRIBUTIONS

Conceptualization, planning, administration, and supervision: A.S. Methodology: R.R., G.R. and A.S. Investigation, data and statistical analysis: R.R. with inputs from A.S. Writing: R.R. and A.S.



§ DATA AVAILABILITY STATEMENT

All data and supporting materials are included in the manuscript. Any additional supporting material for this study can be obtained from the corresponding author upon reasonable request.





Science
































 Paper Title: Thermodynamics of random walking vortex loops in counterflowing superfluids.





Institute of Thermophysics, Lavrentyev ave, 1, 630090, Novosibirsk, Russia

and Novosibirsk State University, Novosibirsk


§ ABSTRACT 
Based on the theory of the thermodynamic equilibrium in a system of quantum
vortices in superfluids in the presence of a counterflow, the influence of a vortex tangle on various thermodynamic phenomena in quantum liquids is studied. Using the early calculated partition function we study some of the properties of He II related to counterflow, such as the distribution of vortex loops in their length, the suppression of the superfluid density ρ _s and the shift T_λ. Good agreement with the early obtained results is a fairly strong argument in favor of the point of view that the gas of string-like topological excitations can indeed be considered as a additional kind of quasi-particles having the inner structure at high temperatures, especially near the phase transition. The application of the developed formalism to the theory of quantum turbulence is briefly discussed.


Thermodynamics of random walking vortex loops in counterflowing superfluids
    Sergey K. Nemirovskii
email address: nemir@itp.nsc.ru
    August 5, 2024
===========================================================================







§ INTRODUCTION AND SCIENTIFIC BACKGROUND


Long ago, in pioneering papers by Onsager <cit.> and by Feynman
<cit.> the idea that the quantum vortices might be
responsible for the superfluid phase transition had been suggested. Since
then, many authors developed this idea by studying the gas of string like
topological excitations in various media. Of course, the main achievements
were in the two-dimensional case, where the
Berezinskii–Kosterlitz–Thouless transition was predicted and thoroughly
investigated <cit.> As for 3D case, the situation here is far
from a more less developed theory. Nevertheless, there are quite a few works
that use various approximations <cit.>,<cit.>,
<cit.>, <cit.>, <cit.>,<cit.>
,<cit.>). In these papers, the effect of an ensemble of vortex
filaments on the phase transition, on the behavior of the superfluid
component, on the behavior of the heat capacity, and other issues were
discussed. It can be stated <cit.>,<cit.>,<cit.> that the theory of string-like structures can provide a
physical understanding of phenomena in superfluid liquids, superconductors,
models of cosmic strings, crystalline bodies (melting), pulsars (glitches).In the letter we apply this ideas to the counterflowing superfluids. There
are many works in which various properties of superfluidity in the presence
of counterflow are investigated. It is of interest to consider these
phenomena from the standpoint of an ensemble of quantum vortices.In the paper we will focus on the effects related to counterflow. In
particular we will study the distribution of vortex loops in the space of
their lengths, the vortex line density (VLD) ℒ total length per
unit volume, the Lamb impulse and the superfluid flow induced by vortices.
We also investigate suppression of superfluid density ρ _s and shift T_λ (which we associate with Hagedorn temperature T_H ) due
to the counterflow velocity. Additionally, we discuss possible
applications of thermodynamically equilibrium vortices to the theory of
quantum turbulence (QT).



§ GIBBS DISTRIBUTION AND PARTITION FUNCTION


In paper by the author <cit.> it was shown that a vortex
filaments in superfluids under the action of a random Langevin force in the
presence of a counterflow with a relative velocity 𝐯_ns come
into a state of thermodynamic equilibrium with the Gibbs distribution

    𝒫({𝐬(ξ )},t)=𝒩exp (-H{𝐬}/
    k_BT).

Here 𝐬(ξ,t) are the radius vectors of the
vortex line elements, ξ is a label parameter, in this case it coincides
with the arc length, and, accordingly, runs from 0 to the length of the
loop l; Quantity 𝒩 is a normalization factor. The Hamiltonian
in the presence of relative velocity has the form

    H{𝐬}=E{𝐬}-𝐏·𝐯_ns.

The energy E{𝐬} and the Lamb impulse 𝐏 are defined
as (see, for example, <cit.>)

    E(𝐬)=ρ_sκ^2/8π∫_0^l∫_0^l
    𝐬^'(ξ )𝐬^'(ξ ^')/|
    𝐬(ξ )-𝐬(ξ ^')|dξ dξ ^',


    𝐏 =ρ _sκ/2∫_0^l𝐬(ξ
    )×𝐬^'(ξ ) dξ ,

where κ is the quantum of circulation, and ρ_s is
superfluid density. Relations (<ref>) - (<ref>) should be used
to calculate the partition function (PF), and, accordingly, to determine the
various properties of the vortex tangle (VT). In the ordinary statistical
mechanics when 𝐏 is the true momentum of a particle (or
quasiparticle), Eqs. (<ref>-<ref>) are obvious and follows from
the Galilean transformation. Unlike this, the situation for vortex loops is
more complicated, since the Lamb impulse is not the ”real” momentum, and Eqs. (<ref>) - (<ref>) need
the verification, which had been accomplished in <cit.>
.In the next work by the author devoted to this topic <cit.>
the one loop PF Z_1, corresponding the stated problem (<ref>) - (
<ref>) was calculated. In this calculation the vortex lines were supposed
to be closed loops having a random walking structure on the cubic lattice
with edge  equal to a and obeying the Wiener distribution. The starting
expression for Z_1 reads:


    Z_1=∫ d𝐫∑_l( a/l) ∫_𝐬(0)=
    𝐫^𝐬(l)=𝐫D𝐬(ξ}exp[ -3
    /2a∮ (𝐬^'(ξ ))^2dξ +ln 5l/a]
    
    ×exp[ -βρ_sκ^2/8π
    ∫_0^l∫_0^l𝐬^'(ξ _1)·𝐬
    ^'(ξ _2)/|𝐬(ξ _1)-𝐬(ξ
    _2)|dξ _1dξ _2+β𝐯_nsρ
    _sκ/2∫_0^l𝐬(ξ )×𝐬^'(ξ
    ) dξ] .


The first line is the contribution into PF due to
configurations of vortex loops. The path integral take into account all
curves starting and ending in the same point 𝐫. Term (l/a)ln 5
appears due to fact that l/a links, placed on the cubic lattice, have (2D-1) options for which the string can move at each step, so the number of
line states N(l,a) increases with its length as N(l,a)=(2D-1)^l/a ( D
is the dimension of space). The quantity ln 5 can be considered as the
entropy (normalized by the Boltzman constant k_B) per link. It is
interesting that for polymer chains the entropy per leg, obtained in
numerical simulations (see <cit.>) is about 1.6, which is very
close to ln 5. The quantity a is the length parameter of model. For the
cubic lattice model, it is the cube edge; for polymer chains it is an
elementary step. In the case of quantum vortices, quantity a is the
coherence length (see for details, e.g., <cit.>,<cit.>, <cit.>). Further the integration over 𝐫 will be omitted, since the spacial homogeneity is assumed and,
accordingly, Z_1 is considered as it is taken per unit volume.In work <cit.>  the calculation of the PF was implemented on the
base of the local induction approximation (LIA) for the energy E(𝐬
). In LIA, the energy of the vortex loop is E_loc=ε _Vl,
where ε _V is the energy of a vortex line per unit length
(tension, see, e.g. <cit.>). The main achievement of the LIA is
that the path integral in <ref> becomes Gaussian and can be calculated
exactly. Realizing that a complete description should be carried out on the
basis of a complete expression for the energy E (<ref>), we would
like note, however, that the LIA is capable of describing many important
features of the vortex ensemble. It can be motivated by that the main part
of the dynamics is described precisely by the LIA (See, e.g. review article
<cit.>). And, more importantly, since the main goal of our
investigations is to study the effects related to the counterflow, the LIA
is convenient option for this purpose.The further calculations are close to the problem of the motion of a charged
particle in a constant magnetic field <cit.>,<cit.>, although there are some distinctions and a bit different
methods have been used. Result of according calculations (when v_ns is
directed along z) is that the following expression for the one loop PF
(see <cit.>)



    Z_1=∫dl/l( 3/2π la) ^3/2[ 
    sin (lκβ aρ _sv_ns/6)/lκβ aρ _sv_ns/6
    ] ^-1exp[ -βε̃l]



Here we have introduced quantity ε̃=(ε _V-
ln 5/β a). The striking difference from the Kleinert's result
is that we have an ordinary sine instead of a hyperbolic one. Formally this
difference arises from the imaginary unit in the (Euclidean) path integral
in the expression for the motion of a particle in a magnetic field.
Physically the sine is responsible for instability of the flow with respect
to formation of large loops. Indeed, the combination E{s} - P·𝐯_ns is just an analog of the Landau criterion (but not for
quasiparticles, but for large vortex loops).Developing the factor in square brackets into a power series in the variable
v_ns (up to second term, the first one is obviously equal to unity), we
split the one loop PF Z_1 (<ref>) in two parts - the velocity-free
PF Z_1free and the velocity dependent part Z_1Vns. The quantity Z_1 ts the PF for a single closed string of arbitrary
length. Consistently assuming that in LIA vortex loops  do not interact, we calculate the grand canonical PF Z for an ensemble of many loops using
the following simple formula


    Z=∑_N=0^∞Z_1^N/N!=exp[ Z_1] .

The quantities Z, Z_1, Z_1free and Z_1Vns  are the key ones
for calculating various properties of superfluids due to the presence of a
VT. Let's discuss a series of issues.



§ VARIOUS APPLICATIONS




    Two populations


Loosely speaking, there are two ingredients that affect PF. The first one is
associated with thermal activation. The dynamics of such vortices, which we
will call thermodynamic vortices, is determined mainly by the exponential
(Gibbs) factor in (<ref>), and their sizes are concentrated in the
vicinity of the correlation length a. The role of the velocity enclosed in
square brackets in (<ref>), is reduced to small corrections to the
structure of the VT and to the thermodynamic properties of the system.
There exists one more population of vortex loops, which we call hydrodynamic
vortices. Their appearance is quite similar to nucleation of free vortices
from bound pairs in the 2D case <cit.>. From the form of the
Hamiltonian H{𝐬} and from the saddle-point approximation, it
follows that this population is concentrated near the scales where the
energy E{ 𝐬} competes with the Lamb impulse 𝐏
multiplied by 𝐯_ns (see Eq. (<ref>)-(<ref>)). It
is seen that the most preferred scale is of the order κ /v_ns. This
result is close to what takes place in the theory of quantum turbulence. A
full consideration of such a situation seems very complicated and is beyond
the scope of this article. In this article, we will focus only on
thermodynamic vortices and their influence on the properties of superfluids.
Questions related to hydrodynamic vortices, as well as other questions
relating to the relationship of equilibrium quantum vortices with quantum
turbulence (QT), are of great interest and will be investigated in the
future.



    The Hagedorn temperature


It can be seen that for ε̃<0. the PF Z_1 (<ref>
) diverges. The corresponding limiting temperature T_H, when ε̃(T_H)=0 is called the Hagedorn temperature <cit.>
,<cit.>, <cit.>, <cit.> and is
defined as

    T_H=ε _Va/k_Bln 5.

In paper <cit.> it was shown that the T_H is practically
equal to the phase transition temperature (T_λ in the case of He
II). A small difference arises due to the finiteness of the system and
disappears at large volumes. The physical meaning of T_H is that the
configurational entropy (multiplied by temperature) is comparable to the
energy of a vortex filament link. Therefore, the energy entering the system
is spent on the increasing the length of the lines, and the phase transition
is accompanied by the appearance of very long filaments.



    Vortex line density. A velocity-free state 𝐯_ns=0.



A velocity-free state  describes thermodynamically equilibrium vortices in
helium at rest. Previously, this problem was studied under various
assumptions. The most popular were studies in which vortex filaments were
considered as ideal rings (see, for example, <cit.>,<cit.>,<cit.>). Another approach used in our article, when
vortex filaments were either flexible polymers with an effective link, or
lattice models were used <cit.>,<cit.>, <cit.>, <cit.>. One more direction, was the study a
statistics of topological defects of the underlying fields (see, e.g., <cit.>). In these papers the authors predicted that VLD ℒ
at the Hagedorn temperature T_H has a value of the order of the inverse
squared coherence length, that is, ℒ 126 (1/a)^2.
Then ℒ is exponentially suppressed below T_H. All these
facts are immediately follows from Eq. (<ref>) for PF (if we put 𝐯_ns=0). Indeed, the integrand is the production of the power
and exponential ingredients. Therefore, at the Hagedorn temperature T_H,
when ε̃=0 the loops distribution becomes a pure power
function n(l)=( 1/l) ^5/2( 3/2π a
) ^3/2, which indicates the random walking nature of thevortex line
structure. For the temperatures T<T_H, when ε̃>0, the
Gibbs factor prevails and long loops are exponentially suppressed.
Thus, we can state, that our results in the velocity-free state are in good
agreement with those previously obtained.



    Vortex line density due to counterflow ℒ(𝐯
_ns)


Let's come to effects related to presence of the counterflow, 𝐯
_ns≠ 0. We begin with the correction for the VLD ℒ(𝐯
_ns). It can be evaluated from the velocity dependent part Z_1Vns of
PF Z_1 (<ref>). The most important result is that the VLD is
proportional v_ns squared, i.e. ℒ_therm(v_ns)
=γ_therm^2v_ns^2.  It is noteworthy that this result is
also valid while using the sine (in Eq. <ref>) without a power series
developing. Quantity γ_therm, calculated from Z_1Vns
, as a function of the temperature, is shown in the inset of Fig. 2.

It is remarkable fact that γ_therm is close to the γ, which takes place in the theory of quantum turbulence, ℒ_turb=γ_turb^2v_ns^2 (see, e.g., <cit.>). However, in other aspects, the set of thermodynamic
vortex loops related to Z_1Vns has properties different from the loops
observed in the theory of quantum turbulence. In particular, the average
loop length ⟨ l⟩ is of the order of the coherence
length a, in contrast to the QT state, where ⟨ l⟩∼ 1/√(ℒ).This is understandable, since the main source of
length is the thermal effect, and k_BT/ε _V∽ a. Furthermore, the interline space 1/L_therm(v_ns)  far exceeds the
radius of curvature and sizes of loops (which are of the order of a), in
contrast to the case of quantum turbulence. The same can be said about the
distribution of the loops in space of there lengths. In the theory of
quantum turbulence, the loop sizes are gathered on scales close to the
intervortex distance 1/ℒ.



    The Lamb impulse


Let's study now the Lamb impulse 𝐏 and the effects associated
with it. The quantity ⟨𝐏𝐯_ns⟩ can be
evaluated from (<ref>), applying it to the velocity-dependent part of
PF Z_1Vns. In Fig.2 there are shown the quantity ⟨𝐏𝐯
_ns⟩ /v_ns^2 and energy ⟨ E({𝐬})
⟩ /v_ns^2   due to extra vortices induced by counterflow.
It is seen that they are of the same order as it should be. Formally, the
Lamb impulse 𝐏 and the flow induced by it appear due to the
nonzero polarization of the extra vortex loops arising from counterflow.




    Suppression of the superfluid density Δρ
_s(𝐯_ns)


An analysis of the average ⟨𝐏𝐯_ns⟩
shows that the VT induces the superfluid flow directed against the external
superfluid 𝐯_s. From macroscopic point of view, in particular
in hydrodynamic experiments, this additional superfluid mass current which
cancels the part of applied external velocity should display itself as a
suppression of the superfluid density Δρ _s(𝐯_ns).
Depletion of the superfluid density Δρ _s(𝐯_ns) in
counterflow was predicted in Landau's pioneering work <cit.> It
was calculated by Khalatnikov <cit.> on the base of the
phonon and roton model. The results are that Δρ _s(𝐯
_ns)/ρ _s(0) are estimated as some numerical factor multiplied by v_ns^2/c_1^2 for phonons (c_1is the first sound
velocity), and v_ns^2/(k_BT/p_0)^2 for rotons ( p_0 is the
momentum of the roton minimum). In paper <cit.> the velocity
dependence of the superfluid density near the λ-point  was
investigated. The result is that Δρ _s(𝐯_ns)/Δρ _s(0) scales as v_ns^2/(κ /a_λ)^2 (a_λ is the correlation length, singular neat T_λ ).Let's study what the vortex model gives us. The quantity ⟨𝐏𝐯_ns⟩ can be interpreted as Δρ _s(
𝐯_ns)v_ns^2. Developing ⟨𝐏𝐯
_ns⟩, extracted from (<ref>) in powers of velocity v_ns we get the first nonvanishing term is ∝ v_ns^2,
therefore there is a velocity independent contribution into Δρ
_s. The numerical value of Δρ _scan be extracted directly
from Fig.2. This contribution appears since  excessive (due to counterflow)
vortices may be considered as quasiparticles (additional to rotons and
phonons), contributing into the superfluid density (and other thermodynamic
properties, such as heat capacity).The next term of the expansion (fourth order v_ns^2) corresponds to
the v_ns^2 dependence discussed above. One can be obtain that the
relative shift Δρ _s(𝐯_ns)/ρ _s scales as v_ns^2/(κ /l_av)^2, where l_av is average size of loop,
close to correlation length a_λ. Literally, this result is close
to the one proposed in <cit.>, although with a small numerical
factor.



    Shift T_λ


There is a number of works, studying an impact of the counterflow on the
shift T_λ (see, e.g. review <cit.>). The physics of
this process is quite obvious. The kinetic energy of the counterflow changes
the free energy of the entire system, which naturally leads to a shift in
the transition point. This scheme had been used in various theoretical
models, such as the so called model F <cit.>,<cit.>,
the phenomenological theory of superfluidity of helium near the λ
point <cit.>, a dynamic new look at the lambda transition <cit.>. On the basis of these considerations as well as on
experimental data, cumulative formula was proposed for the relative shift T_λ-T_cr ,

    (T_λ-T_cr)/T_λ=(q/q_0)^1/2ν.

Here, the index ν =2/3 is the crucial quantity in the theory of the
lambda transition , and q_0 spans from 1000 W/cm^2 up to 10000
W/cm^2.Let's consider the impact of the studying here vortex model on the shift of
the λ point. The free energy F of vortices is related to the one
loop PF as F=-k_BTln Z=-k_BTZ_1 (see (<ref>)). The quantity Z_1 consists of velocity-free PF Z_1free and the velocity dependent
part Z_1Vns. The former leads to free energy, which is positive and the
latter, associated with counterflow is negative. Accordingly, a phase
transition occurs when they balance each other. To move on we have to take
all the parameters entering Eq. (<ref>) in vicinity of critical
temperature T_cr . We adopt the following relations for the superfluid
density and the correlation length near T_λ(see, e.g., <cit.>, <cit.>)

    a_λ=4.75∗ 10^-8 (T_cr-T)^-2/3,   ρ _sλ=0.2 (T_cr-T)^2/3.

Using these data we get Z_1free∝ (T_cr-T)^2 which agrees with
the behaviour of free energy near T_λ in the resting He II <cit.>. Thus, at T_cr the free energy associated with the
vortex system vanishes as it should be. The velocity dependent part of PF Z_1Vns changes the whole free energy. To move further we rid of the
relative velocity v_ns, expressing it via the heat flux q with the use
v_ns=ρ q/ρ _sST. Substituting then the values a_λ, ρ _sλ(<ref>) into Z_1Vns, we get Z_1Vns∝ q^2(T_cr-T)^-2/3. Equating  the obtained values Z_1free and Z_1Vns, and taking into account that at zero counterflow q=0, T_cr coincides with T_λ we finally have (see Fig. (<ref>))

    q=2.83∗ 10^3(T_λ-T_cr)^4/3.

This relation agrees with the cumulative formula (<ref>). As
both theoretical estimates and experimental data, although they give close
results with respect to the exponent (4/3), lead to a fairly large scatter
in the heat flux q_0. One of the possible reasons for this discrepancy
is related to the difficulty of interpreting experimental data, in which
both heat transfer and gravity have a significant effect.




§ DISCUSSION AND CONCLUSION


The problem of thermodynamic quantized vortices in the counterflowing
superfluids, has been studied. Based on the calculated PF, some
properties of superfluids due to counterflow  were properly investigated. The
results are in good agreement with the data obtained on the basis
of other formalisms and from experiment.

The first important conclusion that follows from this is that this can be
considered a fairly strong argument in favor of the point of view that
the gas of topological excitations near the phase transition can indeed  play
the same role of quasiparticles as photons and rotons at low temperatures. We
refrain here from a tempting discussion of the question of the possible
connection between rotons and brownian vortex loops.One more conclusion, having important physical consequence,
is that there exists another population of vortex loops, which we call
hydrodynamic vortices. Their evolution is mainly due to the relative
velocity 𝐯_ns, entering into Gibbs distribution (<ref>) -
(<ref>) as well as into partition function (<ref>). From the form of
the Hamiltonian H{𝐬} it follows that this population is
concentrated near sizes where the energy E{𝐬} competes with the
Lamb impulse 𝐏 multiplied by 𝐯_ns. Using the saddle
point approximation one can conclude that most favoured scale both size
loop size and for the mean radius of curvature is of the order κ
/v_ns. This result is close to what takes place in the theory of
superfluid turbulence - a very challenging topic, actively discussed at
present. Our result gives hope to understand the principal mechanisms
responsible for the formation the structure of the vortex tangle in
the stochastic regime. This question, as well as other
questions concerning the relationship of thermodynamic equilibrium vortices
with the quantum turbulence, is of great interest and will be investigated
in the future.


Study of Gibbs distribution and partition function had been conducted under
state contract with IT SB RAS (No. 17–117022850027–5), study of physical
applications was financially supported by the Russian Science Foundation
(Grant No. 23-22-00128).













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