# Mermaid Cereal: Interactions and Pattern Formation in a Macroscopic
Magnetocapillary SALR System

Alireza Hooshanginejad  Center for Fluid Mechanics, School of Engineering,
Brown University, Providence, RI, US.  Jack-William Barotta  Center for Fluid
Mechanics, School of Engineering, Brown University, Providence, RI, US.
Victoria Spradlin  Center for Fluid Mechanics, School of Engineering, Brown
University, Providence, RI, US.  The Wheeler School, Providence, RI, US.
Giuseppe Pucci  Consiglio Nazionale delle Ricerche - Istituto di
Nanotecnologia (CNR-NANOTEC), Via P. Bucci 33C, 87036 Rende, Italy  Université
Rennes, CNRS, IPR (Institut de Physique de Rennes) UMR 6251, FR35000 Rennes,
France  Robert Hunt  Center for Fluid Mechanics, School of Engineering, Brown
University, Providence, RI, US.  Daniel M. Harris<EMAIL_ADDRESS>Center for Fluid Mechanics, School of Engineering, Brown University,
Providence, RI, US.

###### Abstract

When particles are deposited at a fluid interface they tend to aggregate by
capillary attraction to minimize the overall potential energy of the system.
In this work, we embed floating millimetric disks with permanent magnets to
introduce a competing repulsion effect and study their pattern formation in
equilibrium. The pairwise energy landscape of two disks is described by a
short-range attraction and long-range repulsion (SALR) interaction potential,
previously documented in a number of microscopic systems. Such competing
interactions enable a variety of pairwise equilibrium states, including the
possibility of a local minimum energy corresponding to a finite disk spacing.
Two-dimensional (2D) experiments and simulations in confined geometries
demonstrate that as the areal packing fraction is increased, the dilute
repulsion-dominated lattice state becomes unstable to the spontaneous
formation of localized clusters, which eventually merge into a system-spanning
striped pattern. Finally, we demonstrate that the equilibrium pattern can be
externally manipulated by the application of a supplemental vertical magnetic
force that remotely enhances the effective capillary attraction.

## Introduction

Self-assembly is the spontaneous formation of organized structures or patterns
that are governed by interactions between the individual constituents of a
system Whitesides and Grzybowski (2002). Methods for controlling self-assembly
at the microscale often focus on tuning the interactions between the
constituents of the system through internal Caplan  _et al._ (2000); Li  _et
al._ (2015a); Elsawy  _et al._ (2016) or external means Wei  _et al._ (2008);
Dave  _et al._ (2014); Li  _et al._ (2015b). Mathematically their interactions
can be described by a pair interaction potential, which represents the work
required to bring a pair of isolated particles from infinite separation to a
finite distance apart Donaldson and Alam (2008). In many cases, interaction
potentials are composed of competing attractive and repulsive forces, with
their relative strength and extent defining the possible structures achievable
in the self-assembly of soft materials Engel and Trebin (2007). One class of
competing interactions are defined by short-range attraction and long-range
repulsion (SALR) forces, and have been documented for proteins in low-salinity
solutions Tardieu  _et al._ (2001); Piazza (2004); Abramo  _et al._ (2012) and
charged or magnetic colloidal particles in suspension Wu  _et al._ (1998); Liu
and Xi (2019); Al Harraq  _et al._ (2022), among others. SALR interaction
potentials are sometimes referred to as mermaid potentials due to the
“attractive head” and “repulsive tail” constituting their energy landscape
Royall (2018). Overall, laboratory investigations of SALR systems are
typically reserved to the microscale. In the present work, we realize a
complementary macroscopic 2D SALR system that can be used to probe the
fundamentals of equilibrium self-assembly in an accessible tabletop platform.

In general, short-range attraction favors particle clustering while long-range
repulsion inhibits gelation Sciortino  _et al._ (2004); Campbell  _et al._
(2005); Zaccarelli (2007), and large systems of particles often arrange into
liquid-like states composed of finite equilibrium clusters Stradner  _et al._
(2004); Godfrin  _et al._ (2014); Campbell  _et al._ (2005). The patterns
observed for such microscopic systems of particles primarily depend on the
particle packing fraction and system temperature Pusey and van Megen (1986);
Godfrin  _et al._ (2014); Pȩkalski  _et al._ (2019). Regardless of the
physical origin of such competing forces in a system of SALR particles,
different structures can be predicted using models that apply techniques such
as molecular dynamics Reichhardt  _et al._ (2003); Olson Reichhardt  _et al._
(2004); Liu  _et al._ (2008) or Monte Carlo methods Dickinson and Euston
(1992); Tata and Ise (1998).

While commonly used to characterize interactions at the microscale, pairwise
interaction potentials are relevant at all scales – for instance, the
interaction of celestial bodies can be described by a pairwise gravitational
potential. Of relevance to the present study, two bodies resting at a mutual
fluid interface can be described by an interaction potential considering the
gravitational and surface energy of the system. Identical axisymmetric
particles tend to attract one another to minimize the total energy anomaly
introduced by their presence, an effect often referred to as capillary
attraction Gifford and Scriven (1971) or the “Cheerios effect” Vella and
Mahadevan (2005). This phenomenon has been theoretically rationalized in
various studies Gifford and Scriven (1971); Vella and Mahadevan (2005); Chan
_et al._ (1981); Kralchevsky  _et al._ (1992); Müller  _et al._ (2005);
Fournier (2007); Domínguez  _et al._ (2008); He  _et al._ (2013), and
experimentally measured at both the microscopic Di Leonardo  _et al._ (2008);
Park and Furst (2011); Carrasco-Fadanelli and Castillo (2019) and macroscopic
scales Rieser  _et al._ (2015); Ho  _et al._ (2019). Capillary attraction
between floating objects is a fundamental problem with relevance to
applications involving self-assembly at fluid interfaces in natural systems Hu
and Bush (2005); Bush  _et al._ (2007); Voise  _et al._ (2011); Peruzzo  _et
al._ (2013) and laboratory settings Bowden  _et al._ (1997); Liu  _et al._
(2018); Zeng  _et al._ (2022). To prevent complete particle aggregation under
capillary attraction, prior works have introduced repulsive magnetic dipole-
dipole interactions through the application of an external magnetic field
Golosovsky  _et al._ (2002); Vandewalle  _et al._ (2012); Grosjean  _et al._
(2018); Lagubeau  _et al._ (2016). However, previous work at the macroscale
has predominantly focused on short-range effects, with limited attention to
the consequences of the relative long-range nature of the magnetic repulsion
Vandewalle  _et al._ (2012). Furthermore, the region of approximately uniform
magnetic field in the Helmholtz coil geometry used in the prior experiments
sets practical limits on the number of particles and the size of the domain
that can be considered.

In this work, floating millimetric disks are embedded with permanent magnetic
dipoles giving rise to a tunable SALR interaction potential. We isolate the
possible pairwise equilibrium states in this macroscopic magnetocapillary SALR
system through a combination of experiment and mathematical modeling, with the
full landscape captured by two non-dimensional parameters. After establishing
the pairwise interaction landscape, we investigate the pattern formation of
our magnetocapillary disks experimentally and numerically under varying
packing fractions. We finally demonstrate external control of the self-
assembly process in our magnetocapillary system by adjusting the capillary
attraction strength remotely.

Figure 1:  Experimental setup and regimes of the magnetocapillary interaction
potential. (a) Schematics of two magnetocapillary disks of radius $a$ and mass
$m$ each embedded with a small permanent magnet of magnetic dipole $M$. (b) In
certain parameters regimes, two magnetocapillary disks find an equilibrium
state defined by a finite spacing. The scale bar indicates 2 mm. (c) Sample
magnetocapillary pairwise interaction potentials $U_{\rm{p}}$ versus interdisk
spacing $l$. The three characteristic regimes are shown here (corresponding to
the sequel snapshots in the inset): the locally attractive or Cheerios (red)
regime, the mermaid (violet) regime, and the fully repulsive (blue) regime.
For all three cases $a=3$ mm and $M=9$ A$\cdot$cm2 with increasing masses of
$m_{1}=0.080$ g, $m_{2}=0.092$ g, and $m_{3}=0.11$ g, respectively. Note that
if the disks are started sufficiently far apart, they will always repel, as
the magnetic repulsion decays at a slower rate than the capillary attraction.
(d) Interaction potential for the sample mermaid regime. Inset: experiment
shows that when the mean spacing in an annular 1D array drops below
$l_{\rm{cr}}$, disks begin to spontaneously pair.

## Results

We systematically investigate the pairwise energy potential defined by
capillary attraction and magnetic repulsion. We cast millimetric silicone-
based hydrophobic disks of radius $a$ and mass $m$, and embed each with a
vertically polarized cylindrical magnet of magnetic moment $M$ in its center.
We gently deposit a magnetocapillary disk on the surface of a water-glycerol
mixture followed by a second disk with distance $l$ from the first disk as
shown in Fig. 1(a). A complete description of the fabrication process for the
magnetocapillary disks and fluid details are provided in the Methods section.
Because the disks are hydrophobic, each disk rests in equilibrium on the fluid
surface under capillary flotation Vella  _et al._ (2006) with the contact line
pinned along the sharp bottom edge. In the absence of magnetic effects, two
disks are fully attracted to each other under the Cheerios effect Vella and
Mahadevan (2005). Although an exact analytical formula is not available for
the capillary attraction force between finite disks, it has been shown through
experiments, simulation, and scaling that the capillary attraction force
between them can be well approximated by a decaying exponential of the form Ho
_et al._ (2019)

$\mathbf{F_{\rm{c}}}=-F_{0}\>e^{-{l}/{\ell_{\rm{c}}}}\>\hat{e}_{l}.$ (1)

Here, $\hat{e}_{l}$ denotes the unit vector in the radial direction away from
the adjacent disk, and ${\ell_{\rm{c}}}=\sqrt{\gamma/\rho_{\rm{m}}g}$ denotes
the capillary length where $\gamma$ and $\rho_{\rm{m}}$ denote water-glycerol
surface tension and density, respectively. $\ell_{\rm{c}}$ takes a fixed value
of 2.4 mm for our experiments corresponding to the choice of working fluid.
$F_{0}$ is the characteristic attractive force found to be
$2(mg)^{2}a^{1/2}/\left(\pi^{2}\gamma\ell^{3/2}_{\rm{c}}[(a/\ell_{\rm{c}})^{2}+2a/\ell_{\rm{c}}]^{2}\right)$
for circular disks at the capillary scale (i.e. $a\gtrsim l_{c}$) Ho  _et al._
(2019). The exponential form is reminiscent of the shape of a 2D meniscus
determined by the linearized Young-Laplace equation Gennes  _et al._ (2004).
Since the magnet size is negligible compared to the disk center-to-center
spacing ($2a+l$), the magnets can be considered point dipoles with the
repulsive magnetic force approximated as

$\mathbf{F_{\rm{m}}}=\frac{3\mu_{0}M^{2}}{4\pi(l+2a)^{4}}\>\hat{e}_{l},$ (2)

where $\mu_{0}$ denotes the permeability of free space Abragam (1961). The
total pairwise interaction force is denoted by
$\mathbf{F_{\rm{p}}}=\mathbf{F_{\rm{m}}}+\mathbf{F_{\rm{c}}}$. Defining
$U_{\rm{m}}$ as the magnetic energy potential, and $U_{\rm{c}}$ as the
capillary energy potential where
$\mathbf{F}_{\rm{m}}=-\mathbf{\nabla}U_{\rm{m}}$ and
$\mathbf{F}_{\rm{c}}=-\mathbf{\nabla}U_{\rm{c}}$, the magnetocapillary
interaction can also be expressed as an interaction potential
$U_{\rm{p}}=U_{\rm{m}}+U_{\rm{c}}$, defined as

$U_{\rm{p}}=\frac{\mu_{0}M^{2}}{4\pi(l+2a)^{3}}-F_{0}\ell_{\rm{c}}\>e^{-l/\ell_{\rm{c}}}.$
(3)

As evident in Eq. (3), $\lvert\mathbf{F}_{\rm{m}}\rvert$ decays algebraically
with $l$ while $\lvert\mathbf{F}_{\rm{c}}\rvert$ decays exponentially.
Depending on the input parameters, $\mathbf{F}_{\rm{p}}$ can have 0, 1, or 2
roots (equilibria) for the physically relevant regime $l\geq 0$. As a result,
three characteristically different interaction regimes emerge. If
$\mathbf{F}_{\rm{p}}(l=0)<0$ (i.e. locally attractive), then there is only 1
root, and it represents an unstable equilibrium at $l_{\rm{cr}}$ where the
interaction switches from attractive to repulsive, and is a consequence of the
different decay properties of the two functions. Hence, the disks either repel
each other for $l>l_{\rm{cr}}$, or collapse into each other for
$l<l_{\rm{cr}}$. We refer to this regime as the Cheerios regime, since it
features the local (complete) collapse behavior characteristic of the Cheerios
effect Vella and Mahadevan (2005). Figure 1(c) shows $U_{\rm{p}}$ as a
function of $l$ for a sample Cheerios regime with the solid red line.

If $\mathbf{F}_{\rm{p}}(l=0)>0$ (i.e. locally repulsive),
$\mathbf{F}_{\rm{p}}$ can have either 0, or 2 roots for $l>0$. The former is
referred to as the fully repulsive regime, and an example is represented by
the solid blue line in Fig. 1(c). When $\mathbf{F}_{\rm{p}}$ has 2 roots, the
first root, $l_{\rm{eq}}$, is a stable equilibrium point, while the second
root, $l_{\rm{cr}}$, is an unstable equilibrium point with example as the
purple solid lines in Figs. 1(c) and 1(d). Under such a scenario,
$l_{\rm{eq}}$ represents a potential well, where the two disks will reach
equilibrium at a finite spacing as shown experimentally in Figs. 1(b) and 1(c)
inset. We refer to this regime as the mermaid regime since it presents a SALR
system most reminiscent of the SALR systems previously realized at the
colloidal scale Royall (2018). Note that the Cheerios regime also formally
represents a SALR regime, but the presence of static friction due to the
possibility of physical contact between the disks renders exploration of
pattern formation less predictable and reproducible. In the mermaid regime, a
pair of disks repel each other if they are positioned $l>l_{\rm{cr}}$ apart,
but fall into $l=l_{\rm{eq}}$ if $l<l_{\rm{cr}}$. The experimental
realizations shown in Figure 1(c) show that for a single disk size and
magnetic strength, the three regimes can be accessed by gradually increasing
the disk mass, therein increasing the strength of the capillary attraction
alone, effectively moving from the fully repulsive to mermaid to Cheerios
regimes.

Figure 1(d) inset images show a segment of a 1D array of magnetocapillary
mermaid disks confined in an annular channel of width 15 mm with the
corresponding pairwise energy potential presented in Fig. 1(d). Under low
packing fractions, the disks are all positioned above the $l_{\rm{cr}}$
threshold, securely in the repulsive tail of the energy potential. Given the
channel’s central radius, $R$, and the number of magnetocapillary mermaid
disks, $N$, the mean distance between the disks, $s$, can be estimated as
$(2\pi R-2Na)/N$. For the magnetocapillary mermaid regime demonstrated in Fig.
1(d), $l_{\rm{cr}}=7.5$ mm. As shown in Fig. 1(d) inset, when $R=6$ cm and
$a=6$ mm, for $N=19$ the disks are spaced uniformly with anticipated $s=7.8$
mm. However, by adding one more disk to the channel (i.e. $N=20$), the
anticipated spacing under uniform distribution is 6.8 mm, now lower than
$l_{\rm{cr}}$. Under this scenario, the new disk pairs up with the closest
disk at the equilibrium spacing position (Fig. 1(d) inset), breaking the
symmetry of the original 1D crystalline pattern.

Figure 2:  Magnetocapillary interactions. (a) The phase diagram for different
regimes as a function of capillary Bond number $\rm{Bo}$ and magnetocapillary
number $\mathcal{M}$. The left triangles ($\triangleleft$) show the Cheerios
regime, the right triangles ($\triangleright$) show the repulsive regime, and
the circles ($\circ$) show the mermaid regime, as observed in the experiments.
The symbol sizes correspond to three disk sizes, $a=2.5$ mm, $a=3$ mm, and
$a=3.5$ mm, hence three different Bo in ascending order. The phase diagram is
color coded based on $l_{\rm{eq}}^{*}$ computed from the model, while the
circles are color coded based on $l_{\rm{eq}}^{*}$ value measured in the
experiments. The symbols with solid outlines represent magnets with $M=9$
A$\cdot$cm2 while the dashed-line outlined symbols represent disks with
permanent magnets of $M=2.25$ A$\cdot$cm2. While Bo is only varied by $a$ in
our experiments, $\mathcal{M}$ is varied with $a$, $m$, and $M$. The dashed
lines indicate the model prediction for boundaries between different regimes.
The dimensional parameters and equilibrium spacing for each experiment is
provided in the supplemental material. (b) The equilibrium configuration in
experiment for fully repulsive disks with $\mathcal{M}=2.1$, Bo = 1.52, and
$\phi=0.1425$ ($a=3$ mm, $m=0.073$ g, $M=9$ A$\cdot$cm-2, $R=6$ cm, $N=57$).
(c) The equilibrium configuration in the experiments for mermaid disks with
$\mathcal{M}=1.22$, Bo = 1.52, and $\phi=0.1425$ ($a=3$ mm, $m=0.092$ g, $M=9$
A$\cdot$cm-2, $R=6$ cm, $N=57$).

We fabricated disks with different radius ($a$), mass ($m$), and magnetic
moment ($M$) to more exhaustively verify our magnetocapillary potential model,
and identify the limits of the various interaction regimes. To facilitate this
exploration, we first normalize $\mathbf{F}_{\rm{p}}$ by the characteristic
capillary force $F_{0}$, and the disk spacing $l$ by the disk diameter $2a$.
Hence, we arrive at a nondimensional interaction force

$F^{*}_{p}=\mathcal{M}(l^{*}+1)^{-4}-e^{-2\sqrt{\rm{Bo}}\>l^{*}},$ (4)

where the asterisks denote the nondimensionalized force and spacings. Two non-
dimensional parameters emerge that govern the interaction: the gravitational
Bond number ${\rm{Bo}}=(a/\ell_{\rm{c}})^{2}$ that represents the strength of
gravity to capillary forces, and a magnetocapillary number
$\mathcal{M}=3\mu_{0}M^{2}/64\pi a^{4}F_{0}$ that represents the ratio between
the magnetic and capillary forces, in a similar spirit to prior work
Vandewalle  _et al._ (2012) but adapted to our larger length scales.

We find the roots of $F^{*}_{p}$ for $l^{*}>0$ and varying Bo and
$\mathcal{M}$. For Bo $\leq 4$ three possible states exist, which we delineate
by the number of roots satisfying $l^{*}>0$. The transition between 0 roots to
2 roots marks the boundary between the fully repulsive regime and the mermaid
regime, and is demarcated by the curve
$\mathcal{M}=\frac{16}{\rm{Bo}^{2}}e^{-2\sqrt{\rm{Bo}}-4}$. The transition
from 2 roots to 1 root marks the boundary between the mermaid and Cheerios
regimes and is demarcated by $\mathcal{M}=1$. These predictions are summarized
in Figure 2(a) by the dashed lines, showing that the mermaid regime is only
possible in a finite range of parameter space. The color spectrum indicates
the predicted equilibrium spacing distance $l^{*}_{\rm{eq}}$ (when finite). At
$\rm{Bo}=4$, $\mathcal{M}=1$, a triple point is predicted to exist between the
three phases, and above Bo $\geq 4$ the mermaid regime is predicted to vanish
entirely. For Bo $\geq 4$, only the Cheerios and fully repulsive regimes
exists, switching over at $\mathcal{M}=1$.

The root corresponding to the inevitable transition to long-range repulsion
($l^{*}_{\rm{cr}}$) can be found analytically, and is given by

$l^{*}_{\rm{cr}}=-\frac{2}{\sqrt{\rm{Bo}}}\mathcal{W}_{-1}(\Theta(\mathcal{M},\rm{Bo}))-1.$
(5)

Here, $\mathcal{W}_{k}$ represents the $k$-th branch of the Lambert
$\mathcal{W}$ function, and its argument
$\Theta(\mathcal{M},\rm{Bo})=-\frac{2\mathcal{M}^{1/4}}{\sqrt{\rm{Bo}}}e^{-\sqrt{\rm{Bo}}/2}.$
As described above, $l^{*}_{\rm{cr}}$ is a positive finite value in both the
Cheerios and mermaid regimes. Furthermore, the second root ($l^{*}_{\rm{eq}}$)
can be expressed as

$l^{*}_{\rm{eq}}=-\frac{2}{\sqrt{Bo}}\mathcal{W}_{0}(\Theta(\mathcal{M},\rm{Bo}))-1,$
(6)

and takes a finite positive value only in the mermaid regime. For
$\mathcal{M}<1$ (i.e. in the Cheerios regime) the value of ($l^{*}_{\rm{cr}}$)
becomes negative, corresponding to a physically inaccessible equilibrium
location inside the disk. Steric contact forces between the disks thus balance
the residual local attraction in this regime, with the disks in solid contact
with one another. More details regarding exact solutions for the equilibrium
points and phase boundaries can be found in the supplementary material.

The overlaid points on Figure 2(a) show the experimentally observed
interaction regime under different combinations of parameters. The
experimental points in the mermaid regime are similarly color coded by their
measured equilibrium spacing. Complete details of the parameters associated
with each experimental symbol are presented in the supplemental material. The
model prediction for both the extent of the mermaid regime and the measured
$l_{\rm{eq}}$ value are in very good agreement with the experimental data,
validating the accuracy of our magnetocapillary SALR interaction model.
Further quantitative comparison of $l_{\rm{eq}}$ for the controlled parameters
is provided in the supplemental material.

We perform 2D experiments to investigate how the pairwise interactions in
magnetocapillary disks scale up in larger collections under the various
regimes presented in Fig. 2(a). A circular confinement corral is filled with a
water-glycerol mixture, and has a 1.5 cm opening connected to a smaller bath
where the disks are deposited and individually directed toward the main
confinement. The pattern is allowed to stabilize before the next disk is
introduced. The confinement corral is slightly underfilled, so the border
meniscus acts as a non-steric repulsive barrier. We control the
magnetocapillary disks areal packing fraction $\phi=Na^{2}/R^{2}$, where $N$
denotes the number of disks in a circular 2D confinement, and $R=6$ cm denotes
the confinement radius. In the absence of magnets, the disks collapse into a
single granular raft under unimpeded capillary attraction Protière  _et al._
(2017). As shown in Fig. 2(b), fully repulsive disks with $a=3$ mm, $m=0.073$
g, and $M=9$ A$\cdot$cm-2 ($\mathcal{M}$ = 2.1 and Bo = 1.52) exhibit a
hexagonal crystal lattice at $\phi=0.1425$. However, as shown in Fig. 2(c)
mermaid disks of the same size and same magnet dipole strength with slightly
higher mass at $m=0.092$ g ($\mathcal{M}$ = 1.22 and Bo = 1.52) form clusters
at $\phi=0.1425$. Due to confinement effects, disks first begin pairing along
the boundary. We use the pairwise interaction potential provided in Eq. 3 to
perform an $N$-body simulation on a confined 2D system of $N$ magnetocapillary
disks. In addition, a localized confining potential models the boundary of the
fluid container. The overdamped equation of motion for each disk $i$ is

$\frac{\mu(\pi
a^{2})}{H}\mathbf{\dot{x}}_{i}=\mathbf{F}_{{\rm{confining}},i}+\sum_{j\neq
i}\left(\mathbf{F}_{{\rm{c}},ij}+\mathbf{F}_{{\rm{m}},ij}\right),$ (7)

where $\mathbf{x}_{i}$ denotes the position vector for disk $i$ with respect
to the center of the circular confinement, the dot represents the derivative
with respect to time, and $H$ and $\mu$ are the liquid bath depth and
viscosity, respectively. The confining force,
$\mathbf{F}_{\rm{confining}}=-\nabla U_{\rm{confining}}$, is assumed to be
governed by a confining potential of form
$U_{\rm{confining}}(r)=\frac{\alpha}{2}\left[1+\tanh\left(\frac{r+a-R}{\ell_{\rm{c}}}\right)\right]$.
Although the exact functional form is not important here, the form used is
chosen to provide a repulsive barrier that decays away from the system
boundary over the capillary length scale, capturing the influence of the
repulsive meniscus barrier in a fluid container. Here, $r$ denotes the radial
distance from the disks center to the center of the circular confinement, $R$
denotes the confinement radius, and $\alpha$ is a constant corresponding to
the depth of the potential well tuned to be sufficiently large to prevent
particles from escaping the corral. The initial positions of the disks are
sampled from a uniform distribution with resampling occurring in the case of
disk-disk overlap. The positions of all disks are then evolved in time as 2$N$
first-order ODEs according to Eq. 7. The $\mathtt{ode45}$ solver in MATLAB is
used to iterate the system of ordinary differential equations. The underdamped
version of Eq. 7 (i.e. 4$N$ equations) gives indistinguishable results for the
equilibrium state, justifying the overdamped simplification for our particular
exploration.

Figure 3:  2D pattern formation in the magnetocapillary mermaid regime. (a)
Patterns formed by magnetocapillary mermaid disks with $\mathcal{M}$ = 1.22
and Bo = 1.52 ($a=3$ mm, $M=9$ A$\cdot$cm-2, $m=0.092$ g) for varying areal
packing fraction, $\phi$. The confinement radius $R$ is 6 cm. The top row
illustrates the experimental snapshots starting from the hexagonal lattice
state to clusters, stripes, and labyrinths. The bottom row shows sample model
results color coded based on corresponding $\phi$ values in the experiments.
The black scale bar shows 2 cm. (b) Simulation results of mermaid disks with
the same parameters as (a) in a large square domain of 17 cm $\times$ 17 cm
with a periodic boundary condition for varying packing fraction, $\phi$. (c)
The average radial distribution function RDF, $\overline{g}(r)$, from the
periodic boundary condition simulations in a square domain of 17 cm $\times$
17 cm as a function of $\overline{l}$ averaged over 25 independent simulations
for varying $\phi$. The error bars indicate one standard deviation of the
simulation results.

To further study the equilibrium pattern formation of the magnetocapillary
disks in the mermaid regime, we perform 2D experiments and simulations under
different areal packing fractions $\phi$. As shown in Fig. 3(a) top row for
$a\ll R$, the system exhibits a hexagonal lattice state at $\phi=0.05$. As we
increase $\phi$ in 0.05 increments, the magnetocapillary mermaid disks begin
pairing spontaneously, with clusters forming at $\phi=0.1$ and 0.15. At
$\phi=0.2$ the system shows stripe patterns until transitioning to labyrinths
at $\phi=0.25$. Figure 3(a) bottom row shows the simulation results under
confinement for the same physical parameters as in the experiments while
varying $\phi$, with good agreement. To explore any influence of geometrical
confinement on the pattern formation (a practical necessity in experiment),
simulations are also completed with periodic boundary conditions in a large
square domain (17 cm $\times$ 17 cm) that is 2.5 times larger than the
confinement area in the experiments. As shown in Fig. 3(b), similar patterns
and transitions arise, albeit with more spatial homogeneity due to the absence
of a true border.

To quantitatively characterize the simulated patterns in 2D, we construct a
radial distribution function (RDF) Zhao  _et al._ (2012). The RDF for disk
$i$, $g_{i}(r)$, is given by $g_{i}(r)=K_{i,[r,r+\Delta r]}/(2\pi r\Delta r)$
where $K_{i,[r,r+\Delta r]}$ denotes the number of disks located at a distance
of $r$ to $r+\Delta r$ from disk $i$. We can then obtain the average RDF for
the entire system as $g(r)=\frac{1}{N}\sum_{i}^{N}g_{i}(r)$. To characterize
the system’s phase under the mermaid potential, we average $g(r)$ over 25
independent simulations for each $\phi$ in a square domain of $17$ cm $\times$
17 cm with periodic boundary conditions. Figure 3(c) shows the average RDF
$\overline{g}(r)$ as a function of the distance from the disks normalized by
the equilibrium distance between two isolated disks, namely
$\overline{l}=(l+2a)/(l_{\rm{eq}}+2a)$. As indicated in Fig. 3(c),
$\overline{g}(r)$ is small at $\phi=0.05$ for $\overline{l}<2$, with a maximum
at $\overline{l}\approx 3.1$ which is the characteristic spacing in the
hexagonal lattice formation for this packing fraction. For $\phi=0.1$, the
strongest peak emerges at $\overline{l}\approx 1$ indicative of the
magnetocapillary mermaid stable equilibrium state and the appearance of
clustering. With increasing $\phi$, the clusters coalesce into longer chains
of disks, hence, more peaks appear at integer multiples of
$\overline{l}\approx 1$. The stripe and labyrinth phase exhibit similar RDFs
Zhao  _et al._ (2012). These phases undergo a topological transition and could
potentially be distinguished by alternative metrics, such as the corresponding
Betti numbers or average pore size.

Figure 4:  External control of equilibrium pattern. (a) When a magnetic strip
is placed below the bath, the disks are pulled downwards, increasing the
capillary attraction force. If tuned correctly, fully repulsive disks can
transition to mermaid disks under this additional force. (b) $l_{\rm{eq}}$ as
a function of $m$ from the magnetocapillary model for $a=2.5$ mm, and $M=9$
A$\cdot$cm2. The dashed-line shows the physical value of $m$ while the dotted
dashed-line shows the effective $m$ after applying the external magnetic force
based on the $l_{\rm{eq}}$ value extracted from the experiments shown in the
inset. (c) The hexagonal lattice state (on the left) transitions to the
striped pattern (right) by adding the magnetic strip below the bath for disks
with $a=2.5$ mm, $M=9$ A$\cdot$cm2, and $m=0.09$ g (Bo = 1.05,
$\mathcal{M}$=3.5) and $\phi=0.18$. The scale bar indicates 2 cm. (d) Sample
magnetocapillary simulations for $a=2.5$ mm, and $M=9$ A$\cdot$cm2 showing
phase transition from hexagonal lattice to stripes when $m$ changes from 0.09
g to 0.115 g (Bo = 1.05 with $\mathcal{M}$ decreasing from 3.5 to 1.9) for
$\phi=0.18$.

We further aim to showcase the tunability of our macroscopic system by
manipulating the system phase remotely. We place a large magnetic sheet below
the bath approximately 7 mm below the free surface as depicted in Fig. 4(a). A
pair of magnetocapillary disks with $a=2.5$ mm, $m=0.09$ g, and $M=9$
A$\cdot$cm2 (Bo=1.05, $\mathcal{M}$=3.5) show fully repulsive behavior as
shown in Fig. 4(b) inset. However, as we place the magnetic strip below the
bath, the disks are pulled down by the applied vertical magnetic force between
the embedded magnets and the magnetic sheet resulting in a larger depression
of the meniscus surrounding the disks, and hence a larger capillary force
between the disks as illustrated in Fig. 4(a). This force increase is such
that the fully repulsive disks in Fig. 4(b) transition to the mermaid regime
and fall into the potential well configuration.

In the absence of a detailed characterization of the magnetic sheet, the
increase in the capillary force is modeled by replacing the mass of the disk
with an effective (greater) mass that yields the equilibrium spacing
experimentally measured in a two-disk experiment under the additional uniform
vertical magnetic force. From the experimental $l_{\rm{eq}}$ under the
magnetic strip, we find the effective mass that yields the same $l_{\rm{eq}}$
using the model. Figure 4(b) shows the model solution with the solid line for
$l_{\rm{eq}}$ versus $m$ for $a=2.5$ mm, and $M=9$ A$\cdot$cm2. The dashed
line shows the disks original mass (0.09 g), and the dotted dashed line shows
the effective mass (0.115 g) corresponding to $l_{\rm{eq}}$ = 2.6 mm from the
experimental inset. Therefore, we can model the system before and after adding
the external magnetic strip by changing $m$ from 0.09 g to 0.115 g in the
magnetocapillary potential. Figures 4(c) and 4(d) show a 2D system of fully
repulsive disks with the same disk parameters from experiments and
simulations, respectively, for $\phi=0.18$. The disks form a hexagonal lattice
away from the boundary until the external magnetic strip is placed beneath,
upon which the system immediately transitions to a stripe state. A video of
this process is included with the supplementary material. As shown in Figs.
4(c) and 4(d), the simulation results with the effective mass is in good
qualitative agreement with the experiments.

## Conclusion

In summary, we have demonstrated that the competing interactions between the
attractive capillary force and the repulsive magnetic force can result in a
macroscopic SALR potential with rich self-assembly phenomenology reminiscent
of colloidal SALR suspensions McConnell (1989); Seul and Andelman (1995); de
Koker  _et al._ (1995); Malescio and Pellicane (2003). We have developed a
magnetocapillary potential model defined by two non-dimensional parameters
that rationalizes the pairwise interaction between the magnetocapillary disks,
and reproduces different phases of pattern formation for varying packing
fractions. In addition, we have showed that by manipulating the capillary
force, we can control the patterns externally. The current magnetocapillary
SALR system provides an accessible playground for investigating analogous
aspects of colloidal SALR systems where experiments are more challenging or
expensive, and could also be used in pedagogy when introducing modern topics
in self-assembly. To further establish connections with colloidal-scale SALR
systems, future studies will explore simulating the effect of thermal
fluctuations and Brownian motion via the addition of supercritical chaotic
Faraday waves Xia  _et al._ (2019); Thomson  _et al._ (2023).

## Methods

The disks were fabricated using OOMOO 30 which is a silicone rubber compound
comprised of part A and part B that should be mixed 1:1 in volume or 10:13 in
weight, respectively Barotta  _et al._ (2023). The density of the mixed
compound under this scenario was 1.34 $\pm$ 0.02 g$\cdot$cc-1. We mixed 30
milliliters of part A and 30 milliliters of part B for 30 seconds by hand with
a wooden craft stick. We then poured the compound into a 3D-printed mold that
was printed using a Formlabs Form 2 resin printer. After pouring the compound
into the mold, the mold was placed in a small vacuum chamber and degassed for
5 minutes to extract air bubbles trapped inside the mold. Next, we scraped the
surface of the mold using a blade to make a smooth and balanced surface on the
mold. The compound cured inside the mold for at least 6 hours in room
temperature. We then used a compressed air nozzle to remove the disks from the
mold.

In order to robustly achieve a smooth and axisymmetric three-phase contact
line when depositing disks, we designed the molds with a thin lip around the
disks edge that was 0.25 mm high and 0.25 mm thick. We also cast a cylindrical
recession in the center of the disk for placing magnets. We used Neodymium 50
magnets (Supermagnetman) of 1 mm height and two different diameters of 0.5 mm
and 1 mm (corresponding to $M=2.25$ A$\cdot$cm2 and $M=9$ A$\cdot$cm2,
respectively). To place a magnet inside a disk, we used the south pole of a
permanent magnet underneath the disks to get a magnet inside the hole. We then
removed the permanent magnet while manually holding the installed magnet in
the disk.

For the working fluid, we used a mixture of water-glycerol with density,
$\rho_{\rm{m}}=1139.0\pm 0.3$ kg$\cdot$m-3, measured with a density meter
(Anton Paar DMA 35A), surface tension, $\gamma$ = 0.068 $\pm$ 0.001
N$\cdot$m-1, and dynamic viscosity, $\mu=0.0083\pm 0.0004$ N$\cdot$s$\cdot$m-2
Takamura  _et al._ (2012). The fluid is gently poured into a circular
confinement corral made from laser-cut acrylic sheets of thickness 6 mm to a
depth of $H=5$ mm.

## Acknowledgements

The authors would like to acknowledge Ryan Poling-Skutvik for fruitful
discussions. A.H. acknowledges the Hibbitt Fellowship. J.-W.B. is supported by
the Department of Defense through the National Defense Science and Engineering
Graduate (NDSEG) Fellowship Program. G.P. acknowledges the CNR-STM and the
CNRS-Momentum Programs. This work is partially supported by the Office of
Naval Research (ONR N00014-21-1-2816).

## Author contributions

A.H., J.-W.B., G.P., and D.M.H. designed research, analyzed data, developed
models, and wrote the paper; A.H. and V.S. performed final experiments; A.H.
and R.H. analyzed the models; J.-W.B. developed and performed simulations; all
authors performed research, discussed the results, commented on the manuscript
and gave final approval for publication, agreeing to each be held accountable
for the work performed therein; D.M.H. supervised the project and secured
funding.

## Competing interests

The authors declare no competing interests.

## Additional information

Supplementary information 1 supplementary document and 5 supplementary videos
available. All code and data associated with the numerical simulations can be
found at https://github.com/harrislab-brown/MermaidCereal.

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