# Active Nematic Ratchet in Asymmetric Obstacle Arrays

Cody D. Schimming<EMAIL_ADDRESS>Theoretical Division and Center for
Nonlinear Studies, Los Alamos National Laboratory, Los Alamos, New Mexico,
87545, USA  C. J. O. Reichhardt  Theoretical Division and Center for Nonlinear
Studies, Los Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA
C. Reichhardt  Theoretical Division and Center for Nonlinear Studies, Los
Alamos National Laboratory, Los Alamos, New Mexico, 87545, USA

###### Abstract

We numerically investigate the effect of a periodic array of asymmetric
obstacles in a two-dimensional active nematic. We find that activity in
conjunction with the asymmetry leads to a ratchet effect or unidirectional
flow of the fluid along the asymmetry direction. The directional flow is still
present even in the active turbulent phase when the gap between obstacles is
sufficiently small. We demonstrate that the dynamics of the topological
defects transition from flow-mirroring to smectic-like as the gap between
obstacles is made smaller, and explain this transition in terms of the pinning
of negative winding number defects between obstacles. This also leads to a
non-monotonic ratchet effect magnitude as a function of obstacle size, so that
there is an optimal obstacle size for ratcheting at fixed activity.

Active nematics are anisotropic fluids that exhibit local orientational order
and generate macroscopic flows from microscopic forces [1, 2]. In large,
unconfined systems these flows are typically chaotic, leading to a phase
dubbed “active turbulence” [3, 4, 5, 6]. Additionally, orientational order in
the nematic allows the existence of topological defects, which may
spontaneously nucleate in the active turbulence phase and act as sources for
the flow [7, 8, 9, 10, 11]. In addition to the inherent interest in chaotic
flows and defect dynamics in active nematic turbulence, much recent activity
has focused on controlling the flows for potential technological and
biological applications such as microfluidic devices, wound healing, and
morphogenesis [12, 13, 14]. Proposed flow control methods include modifying
the boundary geometry, employing spatially varying activity, applying external
fields, and altering substrate properties [15, 16, 17, 18, 19, 20, 21, 22].
There has also been experimental work on the interaction of active nematics
with fabricated obstacle arrays [23, 24], where defect pinning was observed.

A ratchet effect can be used to control flows in systems coupled to an
asymmetric substrate under external ac driving or flashing of the substrate
[25, 26, 27, 28]. Ratchet effects have been demonstrated for colloidal
particles [29, 30] and superconducting vortices [31, 32], where ac driving
results in a net unidirectional flow of particles. In active matter systems
coupled to asymmetric substrates, ratchet effects can arise without external
driving due to the activity [33, 34]. Particle-based active matter ratchets
have been studied for biological systems such as swimming bacteria [35] as
well as active colloids [36, 34]. An open question is whether ratchet effects
also occur for active nematics coupled to an asymmetric substrate, and if so,
how the fluid flow and topological defects would be modified.

Here, we numerically study a two-dimensional active nematic interacting with a
periodic array of asymmetric obstacles of triangular shape. Topological
defects, which are known to generate flows [2, 11], spontaneously appear in
the system out of geometrical necessity due to the shape of the obstacles. We
show that when the gap between the asymmetric obstacles is sufficiently small,
an active nematic ratchet effect occurs in the form of unidirectional flow
along the asymmetry axis, something that does not occur for an array of
symmetric obstacles [37]. Ratcheting effects have been observed for rotational
flows in active nematics interacting with asymmetric inclusions and boundaries
[38, 39], but, to our knowledge, this is the first realization of a
translational active nematic ratchet. We demonstrate that the ratchet effect
is robust across a wide range of obstacle gap sizes and activity levels. By
tuning the gap size, a transition in the defect dynamics occurs, and the flow
speed is optimized at the transition point.

We model a two-dimensional active nematic using a well-established nemato-
hydrodynamics model in terms of the tensor order parameter
$\mathbf{Q}=S\left[\mathbf{n}\otimes\mathbf{n}-(1/2)\mathbf{I}\right]$, where
$S$ is the local degree of orientational order and the director, $\mathbf{n}$,
gives the local orientation of the nematic [40, 2, 41]. We measure lengths and
times in units of the nematic correlation length $\xi$ and the nematic
relaxation time $\tau$, respectively, so the dimensionless evolution equation
for $\mathbf{Q}$ is given by

$\frac{\partial\mathbf{Q}}{\partial
t}+\left(\mathbf{v}\cdot\nabla\right)\mathbf{Q}-\mathbf{S}=-\frac{\delta
F}{\delta\mathbf{Q}}$ (1)

where $\mathbf{v}$ is the fluid velocity, $\mathbf{S}$ is a generalized tensor
advection, and $F$ is a Landau-de Gennes free energy with a single elastic
constant [42, 43, 41]. We work in a free energy regime where the passive
nematic is in the nematic phase and the equilibrium defect diameter is unity.
The fluid velocity is generated from active stresses given by inhomogeneities
in the nematic, and is computed from the Stokes equation:

$\nabla^{2}\mathbf{v}=\nabla p+\alpha\nabla\cdot\mathbf{Q}$ (2)

where $p$ is the fluid pressure and $\alpha$ is the strength of active forces,
called the activity. We also assume the fluid is incompressible and enforce
the constraint $\nabla\cdot\mathbf{v}=0$.

Figure 1: (a) Schematic of the computational domain with a periodic array of
triangular obstacles. (b) Time snapshot of the nematic configuration. The
color represents the scalar order parameter $S$ while the white lines
represent the director $\mathbf{n}$. (c) Time snapshot of the vorticity and
velocity. The color represents the vorticity while the black lines represent
the flow field.

We discretize Eqs. (1) and (2) in space and time and solve them using the
MATLAB/C++ package FELICITY [44, 41]. We simulate on domains with a square
lattice of concave triangular obstacles that break the symmetry along the
$x$-axis, shown in Fig. 1(a). We use strong planar anchoring of the director
on the obstacles, as well as a no slip condition for the fluid velocity. On
the outer boundaries, we employ periodic boundary conditions. The distance
between obstacle centers is fixed at $a=14$ while we vary the size of the
obstacles so that the shortest gap between them, $d$, changes. Due to the
strong planar anchoring, the concave triangles each carry a topological charge
(winding number) of $-1/2$. The total topological charge of the system must be
$0$ due to the periodic boundary conditions, so a defect of charge $+1/2$ must
nucleate in the bulk nematic for each obstacle. We show in Fig. 1(b) a time
snapshot of the scalar order parameter $S$ and director field $\mathbf{n}$ for
a system with $\alpha=1$ and $d=4$, while Fig. 1(c) shows the corresponding
time snapshot of the velocity and vorticity field.

Figure 2: (a,b,c) Distribution of velocity directions $p(\theta_{v})$ for
varied gap width $d$ at activity $\alpha=1$. (a) $d=8$, (b) $d=6$, (c) $d=4$.
(c,d) Trajectories of 27 virtual particles over the course of a simulation
with varied $d$ at $\alpha=1$. The dots on the right side indicate the
starting positions of the particles. (c) $d=8$. (d) $d=4$.

At $\alpha=0$, there are no flows in the system and topological defects are
pinned to the obstacles. For $\alpha>0$, topological defects unpin and move
while additional defects continuously nucleate and annihilate. For all
obstacle gap sizes, we find that the average number of defects in the system
and the average magnitude of the flow velocity increases linearly with the
activity (Fig. S1). These are traditional measures of active turbulence in
active nematic systems [8, 45, 18], indicating that the system is in active
turbulence for $\alpha>0$. This differs significantly from our recent study on
active nematics in periodic arrays of symmetric obstacles, where multiple
phase transitions occurred when varying $\alpha$ [37].

Although the flow measurements in our system are consistent with active
turbulence, the detailed nature of the flow differs from traditional active
turbulent states, where the flow directions are distributed randomly. We find
that the flows through our asymmetric obstacles are distributed
anisotropically, as illustrated in Fig. 2(a-c) where we plot the distribution
of fluid flow directions $p(\theta_{v})$ in systems with $\alpha=1$. When the
obstacle gap is large, as shown in Fig. 2(a) for $d=8$, the flow directions
match the three-fold symmetry directions of the obstacle surfaces, suggesting
that the obstacles are merely locally modulating the flow. As $d$ decreases,
however, $p(\theta_{v})$ becomes strongly peaked along $\theta_{v}=\pi$, as
shown in Figs. 2(b,c) for $d=6$ and $d=4$, respectively. This indicates the
emergence of a directional or rectified flow in the absence of an external
drive. To visualize the rectification of the flow, in Figs. 2(d,e) we plot the
trajectories of $27$ virtual tracer particles that are initially placed near
the right outer boundary of the domain and are advected by the flow over the
course of a simulation. In Fig. 2(d), for the wide gap case of $d=8$ where
strong rectification is not present, the tracer particles generally remain
close to their starting points and have no coordinated motion. In contrast,
for $d=4$ in Fig. 2(e), the tracer particles tend to travel towards the left
side of the domain, as indicated by the appearance of a gradient in the
density of the trajectories and regions of aligned flow.

To further quantify the unidirectional flow we measure the space and time
averaged $x$-component of the flow velocity $\langle v_{x}\rangle$. Figure
3(a) shows $\langle v_{x}\rangle$ versus activity $\alpha$ for a range of $d$
values. When $d\geq 8$, $\langle v_{x}\rangle\sim 0$ for all $\alpha$,
indicating that there is no net flow in the $x$-direction. For $d\leq 7$, we
find that the magnitude of $\langle v_{x}\rangle$ increases linearly with
increasing $\alpha$. Since the sign of $\langle v_{x}\rangle$ is negative,
this indicates that there is a net flow to the left that becomes greater as
the activity increases. Interestingly, we find the magnitude of $\langle
v_{x}\rangle$ varies non-monotonically with $d$ at fixed $\alpha$, as
illustrated in Fig. 3(b). To identify the overall greatest magnitude of the
average flow, we plot a heatmap of $\langle v_{x}\rangle$ as a function of
$\alpha$ versus $d$ in Fig. 3(c), and find that the maximum ratchet effect
occurs for the largest simulated value of $\alpha$, $\alpha=2$, at $d=5$.

Figure 3: (a) Average $x$ velocity of the flow field $\langle v_{x}\rangle$ vs
activity $\alpha$ for $d=1$, 2, 4, 5, 6, 7, 8, and 10. (b) $\langle
v_{x}\rangle$ vs $d$ for representative $\alpha$ values of $\alpha=0.5$, 1,
1.5, and 2.0. (c) Heatmap of $\langle v_{x}\rangle$ plotted as a function of
$\alpha$ vs $d$.

There are previous studies that have shown that breaking rotational symmetry
in a circular or annular domain or along a circular inclusion may induce
unidirectional azimuthal flows [38, 39]. Further, it has been shown in channel
geometries for small values of the activity that unidirectional flows may
occur [46]; however, the flow direction is a spontaneously broken symmetry,
and may be in either direction along the channel. Additionally, at higher
activities, the emergence of either vortex lattices or active turbulence
destroy the unidirectional flow [46]. For the triangular obstacles studied
here, the direction of the active nematic ratchet flow is set by the asymmetry
of the obstacle. To our knowledge, this is the first observation of
translational active nematic ratcheting behavior. Further, below a critical
gap size, the ratchet effect is robust to activity level and obstacle gap
size, indicating that it would not be necessary to extensively tune the system
parameters to obtain ratcheting motion for microfluidic applications.

We also find that the defect dynamics change depending on the obstacle gap
size. For large $d$, the plot of the distribution $p(\theta_{+})$ of the
velocities of $+1/2$ winding defects in Fig. 4(a) for $d=8$ and $\alpha=1.5$
indicates that the $+1/2$ winding defects move in the same direction as the
flow. At the same time, Fig. 4(d) indicates that the $-1/2$ winding defect
velocity distribution, $p(\theta_{-})$, is much more isotropic. As the gap
size decreases, the ratchet effect emerges and the net flow velocity is
primarily along the $-x$ direction, but for $d=5$ the $+1/2$ defect velocities
break the up-down symmetry of the domain, as shown by the plot of
$p(\theta_{+})$ in Fig. 4(b). In this intermediate regime, the $-1/2$ defects
also break up-down symmetry and tend to move in the direction opposite to the
primary flow direction of the $+1/2$ defects, as illustrated by the plot of
$p(\theta_{-})$ in Fig. 4(e). In the limit of small $d$, shown in the plots of
$p(\theta_{+})$ and $p(\theta_{-})$ in Figs. 4(c) and (f) at $d=2$, the $+1/2$
defects tend to move either up or down with equal frequency, restoring the up-
down symmetry of the domain, while the $-1/2$ defects primarily move to the
left, in the direction of the ratcheting flow. We note that in this regime,
the $+1/2$ defects tend to move transverse to the fluid flow direction.

Figure 4: Defect velocity direction distributions (a-c) $p(\theta_{+})$ for
$+1/2$ defects and (d-f) $p(\theta_{-})$ for $-1/2$ defects at $\alpha=1.5$
for (a,d) $d=8$, (b,e) $d=5$, and (c,f) $d=2$.

To better understand the defect dynamics, in Fig. 5(a–c) we construct defect
density plots $N/N_{\rm max}$ as a function of position relative to the
obstacle for the $+1/2$ defects at different values of $d$, and show the
corresponding $N/N_{\rm max}$ plots for the $-1/2$ defects in Fig. 5(d–f).
Here, we first accumulate $N$, the local defect density around each obstacle,
on a grid surrounding the obstacle during the entire simulation. We then sum
this quantity over all obstacles and normalize it by the maximum value $N_{\rm
max}$ on the grid [41]. For $d=8$ in Figs. 5(a,d), both the positive and
negative defects are distributed throughout the interdefect region, with peak
values of $N/N_{\rm max}$ appearing close to the obstacle for the $+1/2$
defects. Thus, for large gap sizes, defects of both signs are freely moving in
the domain, but the $+1/2$ defects can become briefly pinned by the obstacles.
For $d=5$ in Figs. 5(b,e), $N/N_{\rm max}$ for the $+1/2$ defects breaks the
up-down symmetry of the obstacle and is largest along a line in interstitial
space connecting the left and right sides of the obstacle, indicating that the
defects are flowing horizontally. Further, there are no longer strong peaks in
$N/N_{\rm max}$ near the obstacle, indicating that the $+1/2$ defects no
longer become pinned (see Fig. S2 for the distribution of radial distances of
defects). At the same time, the distribution of negative defects becomes
highly concentrated in the region between the upper and lower sides of the
obstacles, indicating defect localization in this area. At $d=2$ in Figs.
4(c,f), $N/N_{\rm max}$ for positive defects mirrors the symmetry of the
obstacles but drops nearly to zero partway across the region connecting the
left and right sides of the obstacles, indicating that $+1/2$ defects are no
longer flowing horizontally. Meanwhile, the $-1/2$ defects become even more
strongly localized in the region between obstacles.

Figure 5: Distribution of defects around obstacles. The distribution around
each obstacle in a simulation is computed and then all are added together to
form the plots. (a–c) Distribution of $+1/2$ defects and (d–f) distribution of
$-1/2$ defects for simulations with (a,d) $d=8$, (b,e) $d=5$, (c,f) $d=2$ and
$\alpha=1.5$.

We find that at large gap sizes, the motion of the defects tends to mirror the
flow of the system, but that the defects may become pinned for a period of
time, reducing their flow speed. As the gap size decreases, the likelihood of
pinning diminishes and the defects can move more freely. At a critical gap
size of $d=5$, the negative defects become strongly localized in the vertical
gap between obstacles. At first this allows the positive defects to travel
more efficiently in the $-x$ direction by skirting the negative defects, but
as the gap size diminishes further, the positive defects begin to annihilate
with the negative defects and the $x$-direction flow is lost. Instead, the
$+1/2$ defects begin to travel transverse to the flow in lanes along the
$y$-direction, forming a smectic-like defect state. In Supplemental Movies 1–3
we show the nematic configuration and flow velocities for simulations in these
three regimes.

Since we find maximal $-x$ direction flow at $d=5$ for all values of $\alpha$,
the maximum does not result from a commensuration effect between $d$ and the
active length scale $\xi_{a}\propto 1/\sqrt{\alpha}$. Instead, the
commensuration occurs between $d$ and the characteristic size of topological
defects $\xi_{d}$, which we hold fixed in this study. It appears when $d$
reaches a length for which $-1/2$ defects become localized between the
obstacles, enhancing the overall flow.

Conclusion— We showed numerically that a periodic array of asymmetric
obstacles can produce translational ratchet flows in an active nematic. As the
gap distance between obstacles decreases, the flow velocity directions become
peaked along the asymmetry direction, but the average flow velocity varies
non-monotonically. The obstacle asymmetry induces a translational active
nematic ratchet that has not been described previously. The ratcheting effect
is robust over a large range of obstacle gap sizes and activity levels. We
also observed a transition in defect dynamics that is correlated with the flow
speed non-monotonicity. Positive winding defects follow the fluid flow for
large gap sizes, while for small gap sizes, pinned negative defects inhibit
the movement of positive defects along the flow and cause the positive defects
to travel transverse to the flow.

This work opens a variety of future directions for steering or patterning
active nematic flows and defect structures using ratchet geometries. Such
effects have potential microfluidic applications, including logic gate design
[12] or the creation of complex patterns [47]. It would be interesting to
explore other asymmetric obstacle geometries or lattice arrangements.
Different obstacle geometries may produce distinct topological defect
arrangements, while different lattices may generate novel flow patterns.

###### Acknowledgements.

This work was supported by the U.S. Department of Energy through the Los
Alamos National Laboratory. Los Alamos National Laboratory is operated by
Triad National Security, LLC, for the National Nuclear Security Administration
of the U.S. Department of Energy (Contract No. 89233218CNA000001).

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