# Colloids in Two-Dimensional Active Nematics: Conformal Cogs and Controllable
Spontaneous Rotation

Alexander J.H. Houston  School of Mathematics, University of Leeds, Leeds, LS2
9JT, United Kingdom  School of Physics, Engineering and Technology, University
of York, Heslington, York, YO10 5DD, United Kingdom  Gareth P. Alexander
<EMAIL_ADDRESS>Department of Physics, Gibbet Hill Road,
University of Warwick, Coventry, CV4 7AL, United Kingdom.

###### Abstract

A major challenge in the study of active systems is to harness their non-
equilibrium dynamics into useful work. We address this by showing how to
design colloids with controllable spontaneous propulsion or rotation when
immersed in active nematics. This is illustrated for discs with tilted
anchoring and chiral cogs, for which we determine the nematic director through
conformal mappings. Our analysis identifies two regimes of behaviour for
chiral cogs: orientation-dependent handedness and persistent active rotation.
Finally, we provide design principles for active nematic colloids to achieve
desired rotational dynamics.

##  I Introduction

A broad class of both biological and synthetic materials are maintained out of
equilibrium by motile constituents which exhibit orientational ordering [1, 2,
3], including cell monolayers [4], tissues [5], bacteria in liquid crystalline
environments [6] and bacterial suspensions [7], and synthetic suspensions of
microtubules [8]. Such materials may be modelled as active liquid crystals, of
which active nematics [3] have garnered the most attention, although
consideration has also been given to smectic [9, 10], cholesteric [11, 12] and
hexactic [13] phases. In active nematics the fundamental excitations are
topological defects [14, 15], which acquire self-propulsive [16, 17, 18, 19]
and self-orienting [20] dynamics. They have also been shown to be foci for
biological functionality, for example in epithelial cell apoptosis [5],
morphogenesis [21, 22] and the formation of bacterial colonies [23, 24, 25]
and biofilms [26]. Consequently, much effort has gone into controlling active
nematics through confinement [27, 28, 29], external fields [30], geometry and
topology [31, 32].

An ongoing challenge across all forms of active matter is to devise means of
rectifying the non-equilibrium dynamics of active systems, allowing for the
extraction of useful work and the development of active micromachines [33,
34]. Such a harnessing of active dynamics is strikingly typified by bacterial
ratchets [35, 36] – the spontaneous and persistent rotation of chiral cogs
immersed in a bath of bacteria, a phenomenon that is precluded in equilibrium
systems. These experiments were performed with an isotropic, unordered
collection of bacteria. In light of the many active systems which exhibit
nematic ordering, it is important to establish whether colloids are still able
to produce coherent dynamics from microscale activity and in particular if the
same active ratchet effects can occur.

There is a long and successful history of using colloids to modify the
behaviour of passive nematics [37, 38, 39], allowing the formation of
metamaterials [40, 41], the control of topological defects [42, 43, 39] and
the tuning of colloidal geometry to induce certain elastic distortions [44,
45]. However, the understanding of how to use colloidal inclusions to induce
prescribed dynamics is in comparative infancy. There has been work on the
dynamics of colloids in scalar active matter [46] as well as active droplets
in nematic environments [47, 48, 49] and various realisations of propulsive
[50, 51] and rotational [52, 53] colloids in active nematics, but a
theoretical understanding of how to design a colloid to produce a desired
dynamical response is lacking. Here we build upon our recent ‘active nematic
multipole’ framework for determining active responses to eluicidate the
connection between colloidal design and the induced dynamical response,
focusing on discs with tilted anchoring and chiral cogs. Of particular
relevance as a point of comparison for our work is the recent demonstration of
persistent rotation for chiral colloids in active nematics [53].

Our main findings are that spontaneous propulsion and rotation are generic
properties of colloids in active nematics and can be tuned by either anchoring
conditions or colloidal geometry. We illustrate the role of anchoring through
discs with a uniformly tilted boundary condition for the director, showing
that this tilt angle naturally determines the direction of propulsion or the
magnitude and sign of rotation. The effect of colloidal geometry is
demonstrated using chiral cogs, for which we find two regimes of active
response - orientation-dependent chirality for cogs with few teeth and
persistent rotation for those with more. We also provide optimal design
principles for generating active rotation via cogs.

The remainder of this paper is organised as follows. In Section II we present
a summary of our previous work on active nematic multipoles [54], showing how
dipole distortions produce net active forces and propulsion while quadrupoles
lead to torques and rotation. In Section III we determine analytically the
energy minimising director in the presence of colloidal discs and show how the
multipole structure and hence active response can be determined by the
director anchoring. Additionally, this section provides an intuition for our
subsequent results concerning chiral cogs, since, in the limit of many teeth,
their behaviour converges onto the ‘classical limit’ of discs with tilted
anchoring. Section IV, in which we study cog colloids, forms the main part of
the paper. In it we first construct the conformal maps and principles for the
lowest energy director boundary conditions that we need to determine the
induced director distortion. Then we once again perform a multipole expansion
of the director to infer the active response of the cogs, before discussing
how this is augmented by the number of teeth, their angle and boundary
conditions. Section V contains a summary and discussion.

##  II Active Nematic Multipoles, Forces and Torques

We summarise briefly the description of two-dimensional active nematic
multipole states and their connection to active forces and torques [54]. In a
minimal analytical approximation the director field is taken to have its
equilibrium form and the effect of activity is captured through the flows
induced by the usual active stress,
$\boldsymbol{\sigma}^{\textrm{a}}=-\zeta{\bf nn}$, for that equilibrium
director (1). We consider director fields that can be linearised around a
(locally) uniformly aligned state, ${\bf n}={\bf e}_{y}+\delta n\,{\bf
e}_{x}$. Taking a one-elastic constant Frank free energy, the equilibrium
condition on the director field is that $\delta n$ should be harmonic,
$\nabla^{2}\delta n=0$. We write the general solution as a multipole expansion

$\delta
n=A\frac{\ln(r/R)}{\ln(a/R)}+\frac{1}{2}\sum_{l=0}^{\infty}a^{l}\left(c_{l}\partial^{l}_{z}\ln
r+\bar{c}_{l}\partial^{l}_{\bar{z}}\ln r\right),$ (1)

where $a$ is a characteristic length scale associated with the ‘core region’,
and $\partial_{z}=1/2(\partial_{x}-\text{i}\partial_{y})$,
$\partial_{\bar{z}}=1/2(\partial_{x}+\text{i}\partial_{y})$ are the usual
derivatives in the complex variable $z=x+\mathrm{i}y$. The first term in (1)
is a monopole distortion whose strength is set by a large lengthscale $R$. At
each order there are two distinct multipoles. For example the two dipoles may
be written as $x/r^{2}$ and $y/r^{2}$, or equivalently as the real and
imaginary parts of $\partial_{z}\ln r$. The complex coefficient $c_{l}$
specifies the weightings of the two multipoles at order $l$, such that
$\mathfrak{R}\left\\{c_{l}\right\\}$ provides the weighting of
$\mathfrak{R}\left\\{\partial_{\bar{z}}^{l}\ln r\right\\}$ and similarly for
the imaginary part.

Linearising in the director deformation $\delta n$, the active flows are given
by the continuity and Stokes equations

$\displaystyle\nabla\cdot{\bf u}=0,$ (2) $\displaystyle-\nabla
p+\mu\nabla^{2}{\bf u}=\zeta\bigl{[}{\bf e}_{y}\partial_{x}\delta n+{\bf
e}_{x}\partial_{y}\delta n\bigr{]},$ (3)

where ${\bf u}$ is the fluid velocity, $p$ the pressure and $\mu$ an isotropic
viscosity. The activity is extensile when $\zeta$ is positive and contractile
when $\zeta$ is negative. In the linearised Stokes equation (3) the activity
has the structure of a multipole expansion of derivatives on the monopole
source $\ln r$, obtained from the expansion of $\delta n$ (1), and the general
solution can therefore be given by acting with the same derivatives of the
fundamental response to a monopole [54]. In two dimensions this fundamental
active response is [54]

$\displaystyle\begin{split}\tilde{{\bf
u}}&=\frac{\zeta\phi_{0}}{8\mu\ln(a/R)}\biggl{[}\frac{x^{2}-y^{2}}{r^{2}}\bigl{(}-y\,\mathbf{e}_{x}+x\,\mathbf{e}_{y}\bigr{)}\\\
&\qquad+2\ln\frac{r}{R}\,\bigl{(}y\,\mathbf{e}_{x}+x\,\mathbf{e}_{y}\bigr{)}\biggr{]},\end{split}$
(4) $\displaystyle\tilde{p}$
$\displaystyle=-\frac{\zeta\phi_{0}}{\ln(a/R)}\frac{xy}{r^{2}},$ (5)

where $\phi_{0}$ is the angle of rotation of the director on the surface
$r=a$. This minimal analytic approach has been found to be effective in
characterising the active flows around defects in two dimensions [18, 55],
defect loops in three dimensions [19, 20], in active turbulence [56], and on
curved surfaces [57]. It may be considered to correspond to a limit of weak
activity, however, even when there are defects, their structure may remain
close to that in equilibrium.

We can gain general insight into the active response of the nematic by
considering the net contributions of the active stresses to the force and
torque when integrated over a circle of radius $r$. This active force and
torque are

$\displaystyle\begin{split}\int\zeta\,\mathbf{n}\mathbf{n}\cdot\mathbf{e}_{r}\,r\text{d}\theta&\approx\int\zeta\left\\{\frac{y\delta
n}{r}\mathbf{e}_{x}+\frac{x\delta
n}{r}\mathbf{e}_{y}\right\\}r\text{d}\theta\\\ &=\frac{\zeta
a\pi}{2}\left[\mathfrak{I}\left\\{c_{1}\right\\}\mathbf{e}_{x}+\mathfrak{R}\left\\{c_{1}\right\\}\mathbf{e}_{y}\right],\end{split}$
(6)
$\displaystyle\begin{split}\int\mathbf{x}\times\zeta\mathbf{n}\mathbf{n}\cdot\mathbf{e}_{r}\,r\text{d}\theta&\approx\int\zeta\frac{(x^{2}-y^{2})\delta
n}{r}\,r\text{d}\theta\\\ &=-\frac{\zeta
a^{2}\pi}{2}\mathfrak{R}\left\\{c_{2}\right\\},\end{split}$ (7)

and are determined, respectively, by the dipole and quadrupole coefficients
$c_{1}$ and $c_{2}$. We see that in two dimensions both dipoles will self-
propel if free to move and that there is a single chiral quadrupole which
produces a rotational response. Comparing with (1) we see that motion along
$\mathbf{e}_{x}$ and $\mathbf{e}_{y}$ is dictated by the imaginary and real
parts of $c_{1}$ respectively, such that the magnitude and phase of $c_{1}$
determine the speed and direction of motion. Rotational effects are governed
by $\mathfrak{R}\left\\{c_{2}\right\\}$, which we denote by
$C_{\circlearrowleft}$, where the arrow indicates the sense of rotation that
results due to a positive chiral quadrupole coefficient in an extensile
system. Consequently, much of the remainder of this paper is dedicated to
means of controlling this chiral quadrupole coefficient, considering first
discs with tilted boundary conditions and then chiral cogs with normal
anchoring. Since the discs with tilted anchoring are a high-side-number limit
of the chiral cogs, understanding their behaviour will provide intuitive
principles for the more complicated cog dynamics.

##  III Disc Colloids With Varying Anchoring Conditions

In this section we determine the energy-minimising director in the presence of
a colloid. We can then read off the relevant multipole coefficients and infer
the active response from the results of Section II. With the aim of
understanding both the dipole propulsive and quadrupole rotational active
response we consider a colloid first with a single companion bulk defect and
then with a pair of antipodal defects.

The director is specified by an angle $\phi$, which in a one-elastic-constant
approximation must be a harmonic function. To maintain consistency with our
far-field description $\mathbf{n}=\mathbf{e}_{y}+\delta n\mathbf{e}_{x}$ we
define the director angle such that
$\mathbf{n}=-\sin\phi\,\mathbf{e}_{x}+\cos\phi\,\mathbf{e}_{y}$, with $\phi$
vanishing asymptotically. In the far-field region the correspondence between
the exact solution and the multipole expansion of (1) is therefore given by
$\delta n=-\phi$.

###  III.1 Spontaneous Motion

We begin by considering a colloidal disc of radius $a$ accompanied by a single
bulk defect whose charge compensates the topological index of its boundary
condition. We take the boundary condition to be one of uniform tilt from
normal anchoring by an angle $\alpha$. This boundary condition has index $+1$
and so the colloid is accompanied by a $-1$ defect which we take to have a
position described by the complex number $z_{d}=r_{d}e^{\text{i}\theta_{d}}$,
with $\theta_{d}$ measured anticlockwise from the far-field direction,
$\mathbf{e}_{y}$. The director angle may be written as
$\phi=\mathfrak{I}\left\\{\ln\Phi\right\\}$, with $\Phi$ a meromorphic
function [58]. In particular, by using the image system appropriate to a disc,
one finds that the solution commensurate with the colloidal boundary condition
we have described is

$\phi=\mathfrak{I}\left\\{\ln\left[\frac{z^{2}}{(z-z_{d})(z-\frac{a^{2}}{\bar{z}_{d}})}\right]\right\\}+(\theta_{d}+\alpha)\frac{\ln(\sqrt{z\bar{z}}/R)}{\ln(a/R)}.$
(8)

This solution has one complex free parameter $z_{d}$, to be determined by
minimisation of the Frank free energy as described in the Appendix and
considered previously in the context of smectic C films [59].

Figure 1: Analytical director configurations and active flows for a disc with
a single bulk defect and varying anchoring angle $\alpha$. In the top row the
integral director curves are shown in black upon a phase field of the director
angle. The angular position of the bulk defect varies commensurately with the
anchoring angle, but with opposite sign. Below, the far-field active flow is
shown superposed on the pressure. The green arrows indicate the direction of
self-propulsion for a system with extensile active stresses, which is always
towards the bulk defect.

Expanding the logarithm in (8) we find that the dipole coefficient is given by

$c_{1}=-2\frac{r_{d}^{2}+a^{2}}{ar_{d}}\text{e}^{\text{i}\theta_{d}},$ (9)

such that as expected, the dipole strength is set by the distance of the bulk
$-1$ defect from the virtual $+2$ defect at the centre of the colloid, while
the character of the dipole is determined by the orientation of this
separation vector with respect to the far field. While all functions of the
form of (8) provide a minimum energy configuration for a given bulk defect
location, there is still the question of which defect position provides the
global energy minimiser for a specific colloidal anchoring condition. As
determined in the Appendix, the Frank free energy is minimised by
$\theta_{d}=-\alpha$ and $r_{d}=\sqrt{2}a$ [59], such that

$c_{1}=-3\sqrt{2}\text{e}^{-\text{i}\alpha}.$ (10)

As illustrated in Figure 1, this has the consequence that the direction of
colloidal self-propulsion is directly determined by the anchoring condition.
Note that, although the boundary condition is invariant under changing
$\alpha$ by $\pi$, the dipole distortion is not. There are therefore two equal
energy defect locations for any boundary condition, whose dipole coefficients
differ by a sign. This is shown in Figure 1 for normal anchoring, for which
the bulk defect may sit below or above the colloid, with the motion of the
colloid being towards the companion defect in an extensional system. However,
between these two configurations the position vector of the bulk defect and
the self-propulsion direction rotate in opposite senses so that generically
the colloid does not propel towards the defect. This counterrotation of the
propulsion direction compared to the bulk defect location has been observed in
the context of Janus colloids [50] and can be understood by making an
association between dipole distortions and pairs of $\pm 1/2$ defects [54]. In
contractile systems the colloidal motion is reversed, such that it is towards
the companion defect for tangential anchoring and away from it for normal
anchoring.

###  III.2 Spontaneous Rotation

Figure 2: Analytical director configurations and active flows for a disc with
two bulk defects and varying anchoring angle $\alpha$. The top row shows the
integral director curves superposed on a phase field of the director angle.
The angular positions of the bulk defects vary commensurately and in
opposition with the anchoring angle. Below, the far-field active flow is shown
upon the pressure. The red arrows indicate the direction of self-rotation when
the active stresses are extensile.

In the above case the dipole term always dominates. For a situation where the
quadrupole is dominant we consider a disc with two $-1/2$ bulk defects. Taking
these to be at positions $z_{d1}=r_{d1}\text{e}^{\text{i}\theta_{d1}}$ and
$z_{d2}=r_{d2}\text{e}^{\text{i}\theta_{d2}}$ the appropriate harmonic
functions have the form

$\begin{split}\phi&=\mathfrak{I}\left\\{\ln\left[\frac{z^{2}}{\sqrt{(z-z_{d1})(z-a^{2}/\bar{z}_{d1})(z-z_{d2})(z-a^{2}/\bar{z}_{d2})}}\right]\right\\}\\\
&+\left(\frac{\theta_{d1}+\theta_{d2}}{2}+\alpha\right)\frac{\ln(\sqrt{z\bar{z}}/R)}{\ln(a/R)}.\end{split}$
(11)

The determination of the global energy minimiser can again be found in prior
work [60, 61], although we provide the calculation in the Appendix, with the
result being that the defects sit at antipodal points with angles
$-\alpha\pm\pi/2$ and with a common radial displacement of $(7/3)^{1/4}a$.
With this in place, expansion of (11) gives the quadrupole coefficient as

$c_{2}=-\frac{10}{\sqrt{21}}\text{i}\text{e}^{-2\text{i}\alpha}.$ (12)

As before, the character of the nematic multipole is determined by the
director anchoring condition. That the quadrupole varies with $\alpha$ at
twice the rate of the dipole is a direct consequence of its inversion
symmetry, indeed for an order $l$ multipole
$c_{l}\sim\text{e}^{-\text{i}l\alpha}$. Consequently the net active torque
varies with anchoring angle as $\sin 2\alpha$. The active response is shown in
Figure 2. The chiral quadrupole coefficient $C_{\circlearrowleft}$ may be
inferred from the defect configuration as follows. When there is a vertical
mirror line $C_{\circlearrowleft}$ is zero, when they are placed upper right
and lower left $C_{\circlearrowleft}$ is positive and when positioned
conversely it is negative. This can also be discerned from the director
orientation, represented by the colour map. For a purely achiral distortion
the colour map forms a checkerboard pattern, while when $C_{\circlearrowleft}$
is non-zero the colours form a cross, with the colours inverted when the sign
of $C_{\circlearrowleft}$ changes. The active flow is purely radial for normal
and tangential anchoring, while the angular flow component is maximal for
anchoring angles of $\pm\pi/4$. Variation of the anchoring condition therefore
allows both the sign and magnitude of the colloidal self-rotation to be
determined.

##  IV Cog Colloids

From chirality via anchoring conditions on a disc, we now turn our attention
to controlling chiral active responses geometrically using chiral cogs. In
particular we seek to understand how active ratchet behaviour such as is
observed in bacterial baths [35, 36] might be generated in nematically ordered
systems, as recently demonstrated experimentally [53]. In Section II we
identified the chiral quadrupole as being responsible for the generation of a
net active torque and so persistent actively-driven rotation will require a
consistent sign of this multipole as the orientation of the colloid is varied.

Figure 3: A schematic of a conformal mapping from the exterior of the disc to
that of a cog. The blue and red points on the disc are mapped to the
corresponding vertices of the cog, with the blue ones being fixed points of
the map. The offset $\chi$ is dependent both on the number of cog teeth and
their angle.

We find that the minimum energy director configuration changes discontinuously
with the orientation of the cog whenever an edge passes through the far-field
director orientation. In particular, this can lead to changes in the sign of
the chiral quadrupole, a form of orientation-dependent chirality [62, 63]. In
light of our preceeding results this presents an apparent impasse in our
attempt to generate active ratchet effects. However, we show that as the
number of cog teeth is increased this orientation-dependence fades away, the
cogs map onto discs with tilted anchoring and persistent active rotation is
recovered.

###  IV.1 Conformal mappings

We construct a two-parameter family of chiral cogs from regular $n$-gons by
adding a right-angled triangle with base angle $\gamma\pi$ to each edge, as
illustrated in Figure 3 for a square-based cog with $\gamma=1/8$. The energy-
minimising director angle is provided by a harmonic function which is uniquely
determined a boundary condition corresponding to normal anchoring on the cog
surface along with an asymptotic value provided by the far-field alignment.

The solution for the cog is conveniently obtained via conformal mapping
$f:\Omega\to\Omega^{\prime}$ from the exterior of a disc ($\Omega$) to the
exterior of the polygon ($\Omega^{\prime}$). A holomorphic function $\phi(z)$
on $\Omega$ satisfying the appropriate piecewise boundary conditions on the
disc will then provide the solution $\phi(f(z))$ for the polygon [64].
Provided we can construct the appropriate mapping, we can therefore separate
the determination of the director field into two parts: finding conformal maps
from the exterior of the disc onto the exterior of our colloid and solving the
analogous Dirichlet problem for the director angle on the exterior of the
disc.

To facilitate the construction of explicit mappings we use the framework of
Schwarz-Christoffel transformations [65, 66]. The relevant elements of the
theory along with the derivation of the requisite conformal map are provided
in the Appendix. Here we simply present the result that a conformal map
$f:\Omega\to\Omega^{\prime}$ from the exterior of the disc ($\Omega$) to the
exterior of a cog ($\Omega^{\prime}$) is given by

$f(z)=\frac{\Gamma\left(\frac{1}{2}+\frac{1}{n}-\gamma\right)zF_{1}\left(-\frac{1}{n},\gamma+\frac{n-4}{2n},-\frac{1}{2}-\gamma,1-\frac{1}{n};\text{e}^{\text{i}n\psi}z^{-n},\text{e}^{\text{i}n(\psi+\chi)}z^{-n}\right)}{\Gamma\left(1-\frac{1}{n}\right)\Gamma\left(\frac{1}{2}+\frac{2}{n}-\gamma\right){}_{2}F_{1}\left(-\frac{1}{n},-\frac{1}{2}-\gamma,1-\frac{1}{n}-\frac{n-4}{2n}-\gamma,\text{e}^{\text{i}n\chi}\right)},$
(13)

where the angle $\psi$ incorporates rotations of the cog. Here ${}_{2}F_{1}$
and $F_{1}$ denote respectively Gauss’s hypergeometric function and the first
Appell hypergeometric series of two variables [67]. This transformation is
illustrated in Figure 3, with the vertices of the cog and their corresponding
prevertices on the disc highlighted. For the vertices of the base $n$-gon the
prevertices sit at the $n^{\text{th}}$ roots of unity, those of the teeth tips
being rigidly rotated by an angle $\chi$, dependent on both $n$ and $\gamma$.

###  IV.2 Director field and orientation-dependent handedness

Figure 4: Local picture for director winding at a vertex. Schematics of the
director, decorated with an orientation to aid our argument, around a vertex
of a colloid with normal anchoring and vertical far-field alignment. a) A
convex vertex aligned along the far-field orientation induces positive
director winding equal to the exterior angle. Rotating the vertex
anticlockwise, the director orientation is reversed on the left-hand edge as
it passes through the vertical. This rotation leaves the vertex pointing ‘into
the flow’ of the director winding, shown in b), with the induced winding now
negative and equal in magnitude to the interior angle, having reduced by
$\pi$. The story is similar for concave vertices, as shown in c) and d),
except with a reversal of the signs of winding. When orientated along the far-
field direction as in c) the winding is again equal to the exterior angle,
this time negative. Upon rotation the director orientation along the right-
hand edge flips, increasing the winding induced at the vertex by $\pi$.

Before discussing the boundary conditions relevant to cogs it will be
beneficial to consider the winding induced by an isolated vertex, as this will
give insight into how the nematic distortion changes as a colloid rotates. It
should be emphasised that we are treating the cog’s rotation adiabatically,
such that at every orientation the director adopts the minimal energy
distortion. The first point to make is that the imposition of normal anchoring
does not correspond to a unique Dirichlet boundary condition as at any point
the director angle may be changed by a multiple of $\pi$ with the director
still being normal to the colloid. However, the boundary condition appropriate
for the minimal energy texture is that where the director angle
$\phi\in\left[-\pi/2,\pi/2\right]$. Any other choice would correspond to
additional topological defects on the boundary of the colloid and greater
director distortion in connecting the value of $\phi$ on an edge to its
asymptotic value.

Consequently, a vertex generically exists in one of two states, as illustrated
in Figure 4, namely with the director on the two edges oriented in the same
sense, either inwards or outwards, as in a) and c) or in opposing senses, as
in b) and d). According to these circumstances we will have to rotate the
director on one edge either clockwise or anticlockwise around the vertex to
have it match that on the other side and hence the induced winding will be
given by either the exterior angle $\nu\pi$ or interior angle
$\mu\pi=(1-\nu)\pi$. The dividing line between these two regimes of behaviour
is, in light of the bounds on $\phi$, when one of the edges of the vertex is
vertical. As shown in Figure 4, when one of the edges of a vertex passes
through the vertical an inversion event occurs in which the orientation on the
edge is reversed and the director winding at the vertex changes by $\pm\pi$.
Although we have chosen an orientation for the director to aid our discussion,
all statements we make are of course independent of this choice.

This local description of the director winding at a vertex underpins the
nature of the nematic distortions around all the colloids that we consider. As
the orientation of the colloid is varied the distortion is rigidly rotated,
with the addition of a monopole term to maintain the anchoring condition,
along with director inversions whenever an edge is moved through the vertical
so as to maintain the appropriate bounds on $\phi$. The redistribution of
topological charge caused by these inversions provides a mechanism for
orientation-dependent chirality [62, 63], as we discuss further shortly.

Figure 5: Analytical solutions for cog polygons in a two-dimensional nematic.
The integral curves of the director are shown superposed on the phase field of
its angle and the base regular polygon is shown inside in grey. Four equally-
spaced orientations are shown for each cog, which sit at the locations of the
polygonal icons in the phase space in Figure 7. Row I $(n=3,\gamma=0.1)$: the
chiral quadrupole changes sign with rotation but is net negative, despite the
teeth angle $\gamma$ being positive. Row II $(n=4,\gamma=0.125)$: the chiral
quadrupole is positive on average but is not consistent in its sign. Row III
$(n=7,\gamma=0.28$): the chiral quadrupole is consistently positive. This
value of $\gamma$ gives the greatest average chirality for a cog with seven
teeth, but there is still considerable variation as the cog rotates. Row IV
$(n=16,\gamma=0.25)$: as the number of teeth increases the variation of the
distortion due to rotation becomes smaller and smoother, the average chirality
is maximised for $\gamma=0.25$ and this maximum value tends to $1/2$.

With this understanding in place the boundary condition for a specific cog is
not particularly illuminating and so rather than presenting it for each cog
that we consider we give the algorithmic process for its construction. Taking
the boundary condition for the base regular polygon as a starting point, the
addition of the cog teeth is captured by a square wave alternating between the
values $\gamma\pi$ and $-\pi/2$. This will typically result in a boundary
condition where the director angle $\phi$ lies outside $[-\pi/2,\pi/2]$ in
places and so the director orientation will need to be reversed along these
segments to bring $\phi$ back within this range. The precise form of the
boundary condition depends on the value of $n$ modulo $4$, but using the case
$n\equiv 0$ mod $4$ as a partial example, the first step gives, for $\psi=0$,

$\begin{split}\phi(\theta)&=\sum_{m=1}^{\infty}\frac{(-1)^{m}}{m}\sin(2m\theta)+\sum_{m=1}^{\infty}\frac{2}{mn}\sin(nm\theta)+\gamma\pi-\frac{(1+2\gamma)n\chi}{4}+\\\
&\sum_{m=1}^{\infty}\frac{\left(\frac{1}{2}+\gamma\right)\left(1+(-1)^{m}\right)}{m}\left[\sin\left[\frac{nm}{2}\left(\theta-\frac{n\chi}{2}\right)\right]-\sin\left(\frac{nm}{2}\theta\right)\right].\end{split}$
(14)

Any requisite reversal of director orientation may be achieved by using that

$\frac{\theta_{2}-\theta_{1}}{2}+\sum_{m=1}^{\infty}\frac{\sin(m(\theta-\theta_{1}))-\sin(m(\theta-\theta_{2}))}{m}$
(15)

increases $\phi$ by $\pi$ between $\theta_{1}$ and $\theta_{2}$.

Pairing the boundary conditions with the conformal map in (13) produces the
director solutions shown in Figure 5. A striking feature of the director
solutions for triangle- and square-based cogs shown in Figure 5 is the
reversal in sign of the chiral quadrupole, $C_{\circlearrowleft}$, an instance
of orientation-dependent chirality [62, 63]. This is the multipole associated
with net active torques and so this change in its sign strikes at the heart of
our ability to use cogs to generate persistent rotations in active nematics.
We therefore shall devote a little attention to understanding this behaviour.

In this paper we are using the induced director distortion, in particular the
chiral quadrupole, as a measure of chirality for the colloid. Assigning a
chiral measure is in general fraught with difficulty. There is not a unique
choice, but rather an infinite hierarchy of possible measures [68], with the
potential for different measures to determine distinct shapes as the ‘most
chiral’ [69]. For us, although higher-order terms could be invoked, the chiral
quadrupole coefficient is a natural choice of chirality measure becuase it
determines the active rotational response. It can also be connected to
previously used measures. The chirality pseudotensor [62] reduces in the case
of two-dimensional nematics to $\partial_{i}\phi$ [70]. Of this, the lowest-
order term without mirror symmetry is the chiral quadrupole.

Our observation of orientation-dependent handedness of the nematic distortion
raises a second issue with associating a chiral measure to a system – that of
chiral connectedness [71, 62]. This occurs when a system can be continuously
deformed from a chiral state to its mirror image without ever passing through
an achiral configuration. This necessitates the chiral measure, which takes
opposing signs for the mirror images, to pass through zero as its value is
continuously changed and hence falsely label an intermediate chiral state as
achiral. It is not inevitable that we encounter the paradox of chiral
connectedness, since the chiral quadrupole can vary discontinuously with cog
orientation. Nonetheless, we do find cog orientations for which the chiral
quadrupole vanishes. However, in our context this is not so clearly a flaw, as
it has a direct physical meaning – in such orientations a cog, despite being
chiral, experiences no net active torque.

There are two factors behind the variation of the chiral mode: the rigid
rotation of the distortion as the cog is rotated and inversions of the
director orientation along certain edges. Both of these can be reduced by
increasing the number of teeth, as we now demonstrate. With $n$ teeth the cog
returns to its original state after a rotation of $2\pi/n$, so the smooth
variation of the chiral mode can be made arbitrarily small. Sticking with the
case $n\equiv 0$ mod $4$ for concreteness there will generically be two
inversion events, each affecting a pair of antipodal edges. The exact length
of the arc of the preimage circle corresponding to each edge depends on $\chi$
but is at most $2\pi/n$ and so from (15) we can see that the modification to
the boundary conditions necessitated by such an inversion is bounded by

$\begin{split}\frac{2\pi}{n}&+\sum_{m=1}^{\infty}\frac{1+(-1)^{m}}{m}\left[\left(1-\cos\left(\frac{2m\pi}{n}\right)\right)\sin(m(\theta-\theta_{1}))\right.\\\
&\left.+\sin\left(\frac{2m\pi}{n}\right)\cos(m(\theta-\theta_{1}))\right],\end{split}$
(16)

with $\theta_{1}$ varying according to the location of the edges on which the
inversion is occurring. The discontinuous change brought about by these
inversions can therefore also be made arbitrarily small. In short, the
distortions induced by a cog converge onto the rotationally invariant disc
distortion, with anchoring angle $\gamma\pi$. The orientational dependence of
the chiral quadrupole is a signature of discretisation which grows ever
fainter with increasing side number.

This smoothing out of distortion variations means that the persistent rotation
of cogs observed in bacterial ratchets [35, 36] is achievable in
orientationally-ordered active matter. An explicit demonstration of the
consistent chirality of the induced distortion is given in Figure 5 for a cog
with 16 teeth. The correspondence with the disc limit is confirmed by noting
that the negative winding is situated on the first teeth to be tilted through
the vertical, that is at $2\pi k/n$ where $k$ is the smallest integer such
that $\pi/n+\gamma\pi+2\pi k/n>\pi/2$. In the large $n$ limit we may take
equality and find that the displacement of the negative winding from the
equatorial points tends to $-\gamma\pi$, in accordance with the quadrupole
distortion for anchoring angle of $-\gamma\pi$ on the disc, as established in
Section III.

Figure 6: The far-field flows induced by cogs in extensile active nematics.
Above, the square cog induces a flow whose rotational component changes sign,
preventing consistent active self-rotation. Below, the active flow induced by
the cog with sixteen teeth has a consistent sense of rotation, meaning that
the cog exhibits persistent self-rotation. The background colour represents
the pressure field, with the white arrows indicating the fluid flow. The red
arrows show the direction and magnitude of the net active torque, while the
black cog tooth is intended as a visual aid to make the changing orientation
of the cog clearer.

The consequences for active dynamics of this orientation-dependence of the
chiral quadrupole are illustrated in Figure 6. These far-field active flows
are attained in the same fashion as those shown for discs in Section III –
from our exact solutions for the director around the cogs we read off the
multipole coefficients to quadrupole order and can then determine the active
response from the appropriate derivatives of (4) and (5). For the square-based
cog, also shown in row II of Figure 5, the discontinuous changes in the
director field are of sufficient magnitude to change the sign of the chiral
quadrupole and hence reverse the rotational component of the active flow,
meaning the cog will not persistently rotate. By contrast, the consistent
chiral quadrupole induced by the sixteen-toothed cog shown in row IV of Figure
5 results in a consistent sense of rotation in the active flow, allowing
persistent active rotation of the cog. To be explicit, and as evidenced by the
expression for the net active torque in (7), the direction of rotation is
determined by the signs of both the chiral quadrupole and the activity. The
flows and rotations shown in Figure 6 correspond to extensile activity and
would be reversed in contractile systems.

###  IV.3 Active response

Having identified these two distinct modes of behaviour, let us investigate
more closely the relationship between the chiral response of the active
nematic and the parameters of the cog. Figure 7 shows the phase space for the
coefficient of the chiral quadrupole as a function of both the number of teeth
$n$ and the tooth angle $\gamma\pi$. It was generated by constructing the
boundary condition for a cog and then averaging the chiral quadrupole
coefficient over a full rotation. This averaging was performed for 100 evenly-
spaced values of $\gamma$ for each $n$ from $3$ to $16$. The distortions
produced by the cogs which sit at the locations of the polygonal icons are
shown in Figure 5. Between the two black lines is the region for which the
chiral quadrupole has a consistent sign under rotation, allowing for active
nematic ratchets. There are several regions we can identify. When $n$ is small
($3$ or $4$) there is no value of $\gamma$ which will produce a consistent
chiral response. The square icon denotes a typical example, shown in row II of
Figure 5; there is a modest net chiral quadrupole but its sign changes from
the first two panels to the last two. The region in the bottom-left of Figure
7 is even more extreme, as despite the angle of the cog teeth being such as to
ostensibly induce a negative chiral quadrupole, the averaged coefficient is
actually positive. This counter-intuitive state of affairs is illustrated by
the triangle-based cog in row I of Figure 5. The cog shown there is also
convex, something that is possible only for triangle-based cogs with
$\gamma<1/6$, but this is not necessary for the net chiral quadrupole to be
positive. For $n\geq 5$ a band of $\gamma$ values develops for which the
chiral response is of a consistent sign, with this band opening up with
increasing $n$. The heptagon-based cog in row III of Figure 5 provides an
example of the behaviour typical for intermediate values of $n$; the chiral
quadrupole is of a consistent sign but there is still significant variation in
the nematic distortion as the orientation of the cog is changed. Lastly, in
the high $n$ limit we can understand the chiral response of the nematic to the
cog by modelling it as a disc with anchoring angle $\gamma\pi$. Our discussion
of discs in Section III.2 predicts the dependence of the chiral quadrupole
coefficient on $\gamma$ to be

$C_{\circlearrowleft}\sim-\sin(2\gamma\pi)$ (17)

and this is an increasingly good approximation for the behaviour in the
high-$n$ region of parameter space. As demonstrated by the hexadecagon-based
cog in row IV of Figure 5, the variation of the distortion with cog
orientation grows smaller and smoother with increasing $n$, the maximum value
of the average chiral quadrupole tends to $1/2$ and the value of $\gamma$ for
which this is achieved tends to $1/4$. As far as using cogs to generate active
nematic ratchets the optimal design is therefore to have as many teeth as
feasible, in order to maximise the smoothness of the net active torque and the
range of $\gamma$ which can be utilised. The strength of the active torque can
then be tuned using (17).

Figure 7: The phase space of the chirality of distortions induced by cogs in
two-dimensional nematics. The temperature map represents the strength of the
chiral quadrupole $C_{\circlearrowleft}$ when averaged over all orientations
of the cog as a function of the number of cog teeth $n$ and the tooth angle
$\gamma$. The two black lines form the boundaries of the region for which the
chiral quadrupole has a consistent sign under rotation, corresponding to a
consistent active torque in active nematics. The polygonal icons correspond to
the locations in phase space of the cogs shown in Figure 5.

It should come as no surprise that the cogs which are ‘most chiral’, at least
in the sense of our definition, are those with intermediate values of
$\gamma$. Both $\gamma=0$ and the limit $\gamma\to 1/2$ correspond to achiral
shapes, the former regular polygons the latter tending to star-shaped polygons
(although the $\gamma=1/2$ limit itself does not yield a bounded polygon, as
the two edges of the cog teeth become parallel). The black lines in Figure 7,
which bound the region of consistent chirality, are intriguingly non-monotonic
and display sizeable jumps. The upper boundary exhibits an odd-even effect and
is given by the largest fraction with denominator $n$ that is less than $1/2$.
The lower boundary is less regular and seems to display a mod $4$ effect.
Neither of these is particularly surprising, given that the compatibility of
discrete boundaries with the symmetry of a quadrupole is inherently dependent
on $n$ mod $4$ [72]. We do not discuss the specifics of these threshold values
any further since they are tied to the precise cog construction that we have
employed in this paper and are thus of secondary interest compared to the
qualitative phenomenology we have described, which applies to the chiral
response of a nematic regardless of the exact geometry of the colloid.

###  IV.4 Role of anchoring conditions

We conclude by discussing the effect of anchoring conditions on the active
phenomenology of cogs. This is partly motivated by recent experimental
realisations of cogs in active nematics [53], which utilised tangential rather
than normal anchoring. Once again the results we described for discs in
Section III will prove insightful.

Figure 8: The effect of anchoring conditions on chiral active response. The
panels on the left show the nematic director around a cog along with the
induced active flow and pressure for different anchoring angles $\alpha$ and
rotation angles $\psi$. The arrows indicate the direction and magnitude of
active self-propulsion, dictated by the chiral quadrupole coefficient
$C_{\circlearrowleft}$. The graph on the right plots this chiral quadrupole
coefficient as a function of cog orientation for normal and tangential
anchoring.

Having determined an energy-minimising director configuration for one
anchoring condition we can find the minimal energy distortion for any other
anchoring condition by applying the appropriate local rotation to the
director. However, this transformation rotates the far-field direction, so to
keep this fixed we must globally rotate the system in the opposite direction,
changing the orientation of the cog. Denoting the anchoring and orientation
angles by $\alpha$ and $\psi$ and recalling from Section III that for a
director rotation by $\Delta$ the multipole coefficients obey
$c_{l}\sim\text{e}^{-\text{i}l\Delta}$ we have the relation

$c_{l}(\alpha,\psi)=\text{e}^{-\text{i}l\Delta}c_{l}(\alpha-\Delta,\psi+\Delta).$
(18)

Of particular interest is the different effect normal or tangential anchoring
has on the quadrupole coefficients. In this case (18) reduces to

$c_{2}(\frac{\pi}{2},\psi)=-c_{2}(0,\psi+\frac{\pi}{2}).$ (19)

This relationship is illustrated in Figure 8. The net active torque,
proportional to $\mathfrak{R}\left\\{c_{2}\right\\}$, as a function of
orientation for normal anchoring is connected to that for tangential anchoring
by reversing the sign and shifting by an angle of $\pi/2$. The cogs in Figure
8 have six-fold symmetry and so the angular shift appears as a displacement by
$\pi/6$. Due to the reversal in sign of the chiral quadrupole, a cog with
tangential anchoring will, all other properties being equal, rotate in the
opposite direction to one with normal anchoring. The anticlockwise rotation
illustrated in Figure 8 is in agreement with that seen in experiments [53].

For any cog there is an optimal anchoring angle which maximises the strength
of the net active torque. Following the principles laid out in Section III we
can see that this optimised anchoring will be such that the director tilt
angle and the cog tooth angle sum to approximately $\pi/4$. However, in cases
where it is possible to control the anchoring to this extent it is simpler to
just use disc colloids as there is no need to use colloidal geometry to induce
chirality.

##  V Discussion

We have built upon a general framework for understanding the behaviour of
distortions in active nematics [54] to show that not only propulsion but also
rotation are generic properties of colloidal inclusions. Further, we have
demonstrated how the direction and magnitude of these non-equilibrium effects
can be controlled through either anchoring conditions or colloidal geometry.
The later is illustrated for a form of chiral cog and our predictions for the
direction of persistent rotation match those of recent experiments [53]. Our
results allow us to go beyond predicting the existence of persistent rotation
to also optimise the cog geometry for rotation by maximising the net active
torque.

It is worth emphasising that although the rotational effects we predict seem
similar to those reported in bacterial baths [35, 36], and even share the
property of becoming smoother with increasing teeth number [36], their origin
is not the same. The previously observed bacterial ratchets resulted from
rectification of the swimming direction of the bacteria to produce coherent
motion from an incoherent, isotropic environment and the mechanism for
generating rotation is unchanged by rotation of the cog. The orientational
order of a nematic gives meaning to the orientation of the cog relative to a
far-field alignment and it was by no means guaranteed that the ratchet
behaviour would be replicable in this context, particularly given our
observation of orientation-dependent handedness [62] of the nematic
distortions that drive the rotation of the cog. It is gratifying then that
these effects melt away with increasing tooth number, allowing the active
ratchet effect to be recovered.

Our work paves the way to using colloids to induce desired dynamical responses
in active nematics, including as work-extracting micromachines [33, 34]. A
natural continuation would be to complement the far-field results presented
here with a theoretical understanding of how the motion of active nematic
defects is rectified by colloidal surfaces [73, 53], as this is key to using
inclusions to temper active nematic turbulence [7, 74].

This work was supported by the UK EPSRC through Grant No. EP/N509796/1.

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##  VI Appendix

###  VI.1 Free energy for disc colloids

Consider a nematic texture defined by a harmonic director angle

$\phi=\mathfrak{I}\left\\{\ln\Phi\right\\}+A\frac{\ln(\sqrt{z\bar{z}}/R)}{\ln(a/R)},\qquad\Phi=\prod_{j}(z-z_{j})^{s_{j}},$
(20)

where the $a$ is the colloid radius, $R$ is a large length scale at which
uniform alignment is recovered and $A$ is the strength of the monopole term
appropriate to enforce boundary conditions. The $z_{j}$ denote the positions
of topological defects of charge $s_{j}$, both in the bulk and image defects
in the interior of the colloid. The Frank free energy density is given by

$\begin{split}&f=4\partial_{z}\phi\partial_{\bar{z}}\phi=\left(\frac{A}{|z|\ln(a/R)}\right)^{2}\\\
&+\frac{\text{i}A}{|z|^{2}\ln(a/R)}\left[\bar{z}\sum_{j}s_{j}\frac{z-z_{j}}{|z-z_{j}|^{2}}-z\sum_{j}s_{j}\frac{\bar{z}-\bar{z}_{j}}{|z-z_{j}|^{2}}\right]\\\
&+\sum_{i,j}\frac{s_{i}s_{j}(\bar{z}-\bar{z}_{i})(z-z_{j})}{|z-z_{i}|^{2}|z-z_{j}|^{2}}.\end{split}$
(21)

Rewriting this in polar coordinates we have

$\begin{split}f&=\left(\frac{A}{r\ln(a/R)}\right)^{2}+\frac{2A}{r^{2}\ln(a/R)}\sum_{j}\frac{s_{j}\mathbf{r}\times\mathbf{r}_{j}}{|\mathbf{r}-\mathbf{r}_{j}|^{2}}\\\
&+\sum_{i,j}\frac{s_{i}s_{j}(\mathbf{r}-\mathbf{r}_{i})\cdot(\mathbf{r}-\mathbf{r}_{j})}{|\mathbf{r}-\mathbf{r}_{i}|^{2}|\mathbf{r}-\mathbf{r}_{j}|^{2}}.\end{split}$
(22)

The second of these terms integrates to zero by symmetry and in what follows
it will be possible to enforce $A=0$, which will hold for the minimal energy
configuration. We therefore consider only the final term and write

$F=\frac{K}{2}\int\text{d}^{2}\mathbf{r}\left[\sum_{i,j\neq
0}\frac{s_{i}s_{j}(\mathbf{r}-\mathbf{r}_{i})\cdot(\mathbf{r}-\mathbf{r}_{j})}{|\mathbf{r}-\mathbf{r}_{i}|^{2}|\mathbf{r}-\mathbf{r}_{j}|^{2}}\right].$
(23)

To perform the angular integral we use [75, eq. 2.559(2)]

$\begin{split}&\int\frac{A+B\cos x+C\sin x}{(a_{1}+b_{1}\cos x+c_{1}\sin
x)(a_{2}+b_{2}\cos x+c_{2}\sin x)}\text{d}x=A_{0}\ln\frac{a_{1}+b_{1}\cos
x+c_{1}\sin x}{a_{2}+b_{2}\cos x+c_{2}\sin
x}+A_{1}\int\frac{\text{d}x}{a_{1}+b_{1}\cos x+c_{1}\sin x}\\\ &\hskip
256.0748pt+A_{2}\int\frac{\text{d}x}{a_{2}+b_{2}\cos x+c_{2}\sin
x},\end{split}$ (24)

where

$\begin{split}&A_{0}={\cal N}^{-1}\begin{vmatrix}A&B&C\\\ a_{1}&b_{1}&c_{1}\\\
a_{2}&b_{2}&c_{2}\end{vmatrix},\qquad A_{1}={\cal
N}^{-1}\begin{vmatrix}\begin{vmatrix}B&C\\\
b_{1}&c_{1}\end{vmatrix}&\begin{vmatrix}A&C\\\
a_{1}&c_{1}\end{vmatrix}&\begin{vmatrix}B&A\\\ b_{1}&a_{1}\end{vmatrix}\\\
a_{1}&b_{1}&c_{1}\\\ a_{2}&b_{2}&c_{2}\end{vmatrix},\qquad A_{2}={\cal
N}^{-1}\begin{vmatrix}\begin{vmatrix}C&B\\\
c_{1}&b_{1}\end{vmatrix}&\begin{vmatrix}C&A\\\
c_{1}&a_{1}\end{vmatrix}&\begin{vmatrix}A&B\\\ a_{1}&b_{1}\end{vmatrix}\\\
a_{1}&b_{1}&c_{1}\\\ a_{2}&b_{2}&c_{2}\end{vmatrix},\\\
&\mathcal{N}=\begin{vmatrix}a_{1}&b_{1}\\\
a_{2}&b_{2}\end{vmatrix}^{2}-\begin{vmatrix}b_{1}&c_{1}\\\
b_{2}&c_{2}\end{vmatrix}^{2}+\begin{vmatrix}c_{1}&a_{1}\\\
c_{2}&a_{2}\end{vmatrix}^{2}\end{split}$ (25)

to show that

$\begin{split}&\int_{-\pi}^{\pi}\frac{(\mathbf{r}-\mathbf{r}_{i})\cdot(\mathbf{r}-\mathbf{r}_{j})}{|\mathbf{r}-\mathbf{r}_{i}|^{2}|\mathbf{r}-\mathbf{r}_{j}|^{2}}\text{d}\theta\\\
&=\frac{r^{2}-r_{i}r_{j}\cos(\theta_{i}-\theta_{j})}{r^{4}+r_{i}^{2}r_{j}^{2}-2r^{2}r_{i}r_{j}\cos(\theta_{i}-\theta_{j})}\\\
&\qquad\pi\left[\text{sign}(r-r_{i})+\text{sign}(r-r_{j})\right].\end{split}$
(26)

####  VI.1.1 Dipolar colloids

We now consider the particular configurations of interest. For a disc of
radius $a$ inducing a dipolar distortion with an anchoring angle of $\alpha$
the director angle is given by

$\phi=\mathfrak{I}\left\\{\ln\left[\frac{z^{2}}{(z-z_{d})(z-\frac{a^{2}}{\bar{z}_{d}})}\right]\right\\}+(\theta_{d}+\alpha)\frac{\ln(\sqrt{z\bar{z}}/R)}{\ln(a/R)},$
(27)

that is a bulk $-1$ defect at $z_{d}=r_{d}\text{e}^{i\theta_{d}}$, an image
$-1$ defect within the colloid and a $+2$ defect at the origin. Clearly the
energy is minimised by the choice $\theta_{d}=-\alpha$ such that the angle of
the bulk defect is determined by the anchoring condition. This angular
position of the bulk defect is the relevant part for determining the direction
of self-propulsion in an active system, but for completeness we determine the
radial position.

Using (27) in (23) and performing the angular integration gives

$\begin{split}F&=\pi
K\int\text{d}rr\left[-\frac{2}{r^{2}}+\frac{1}{r^{2}-a^{2}}+\frac{1}{r^{2}-a^{4}/r_{d}^{2}}\right.\\\
&\left.+\left(-\frac{2}{r^{2}}+\frac{1}{r^{2}-a^{2}}+\frac{1}{r^{2}-r_{d}^{2}}\right)\text{sign}(r-r_{d})\right].\end{split}$
(28)

To regularise the integral we integrate from $a+\epsilon$ to $R$, excluding
where necessary an annulus of width $2\epsilon$ centred on $r_{d}$. Doing this
and expanding the resulting logarithms to first order in $\epsilon$ and $1/R$
yields

$\begin{split}F&=\pi
K\left[\frac{a^{2}+r_{d}^{2}}{a^{3}-ar_{d}^{2}}\epsilon-\frac{2a^{2}+a^{4}/r_{d}^{2}+r_{d}^{2}}{2R^{2}}-\ln(\epsilon)\right.\\\
&\left.+\ln\left(\frac{r_{d}^{4}}{2a(r_{d}^{2}-a^{2})}\right)\right],\end{split}$
(29)

with the first two terms arbitrarily small in the limits of small $\epsilon$
and large $R$, the third term the isolated divergent contribution and the
final term minimised by $r_{d}=\sqrt{2}a$.

####  VI.1.2 Quadrupolar colloids

Again considering a disc of radius $a$ with anchoring angle $\alpha$ but now
inducing a quadrupole distortion, the director angle is given by

$\begin{split}\phi&=\mathfrak{I}\left\\{\ln\left[\frac{z^{2}}{\sqrt{(z-z_{d1})(z-a^{2}/\bar{z}_{d1})(z-z_{d2})(z-a^{2}/\bar{z}_{d2})}}\right]\right\\}\\\
&+\left(\frac{\theta_{d1}+\theta_{d2}}{2}+\alpha\right)\frac{\ln(\sqrt{z\bar{z}}/R)}{\ln(a/R)},\end{split}$
(30)

where the $-1$ bulk defect of the dipolar distortion has been split into two
$-1/2$ defects, each with image defects within the colloid. As the free energy
is symmetric under exchange of $r_{d1}$ and $r_{d2}$ the global minimiser must
occur when $r_{d1}=r_{d2}$. Similarly, from (26) we can see that the angular
positions of the defects will only appear in the combination
$\cos(\theta_{d1}-\theta_{d2})$ and so minimising gives the condition
$\theta_{d1}-\theta_{d2}=\pi$. It is natural that these restrictions are
required to take (30) into the form of the global energy minimiser as they
also ensure that the leading order distortion is a quadrupole rather than a
dipole. Taking this angular condition in conjunction with the constraint that
the monopole term vanishes, that is $\theta_{d1}+\theta_{d2}=-2\alpha$, gives
that the two defects sit at $-\alpha\pm\pi/2$.

With these simplifications in place the free energy is given by

$\begin{split}F&=\pi
K\int\text{d}rr\left[\frac{a^{4}(r^{4}r_{d}^{4}+a^{4}r^{4}-2a^{8}}{r^{2}(a^{4}-r^{4})(a^{8}-r^{4}r_{d}^{4})}\right.\\\
&\left.+\frac{r^{4}r_{d}^{4}+a^{4}(r^{4}-2r_{d}^{4})}{r^{2}(r^{4}-a^{4})(r^{4}-r_{d}^{4})}\text{sign}(r-r_{d})\right].\end{split}$
(31)

Integrating over $r$ while employing the same regularisation proceedure as
before and then expanding to lowest order in $\epsilon$ and $1/R$ we find

$\begin{split}F&=\frac{\pi
K}{2}\left[\frac{2(a^{4}+r_{d}^{4})}{a(a^{4}-r_{d}^{4})}\epsilon-\frac{(a^{4}+r_{d}^{4})^{2}}{2r_{d}^{4}R^{4}}-\ln\epsilon\right.\\\
&\left.+\ln\left(\frac{r_{d}^{7}}{4a^{2}(r_{d}^{4}-a^{4})}\right)\right].\end{split}$
(32)

As before the only relevant term is the final one and it is minimised by
$r_{d}=(7/3)^{1/4}a$.

###  VI.2 Schwarz-Christoffel Maps

Schwarz-Christoffel transformations are predicated on the notion that the
desired mapping has a derivative expressible as the product of some set of
canonical functions, that is [66]

$f^{\prime}(z)=\prod f_{k}(z).$ (33)

It naturally follows that $\text{arg}f^{\prime}=\Sigma\text{arg}f_{k}$ and so
by choosing the $f_{k}$ appropriately such that each is a step function we can
contrive to make $\text{arg}f^{\prime}$ a piecewise constant function with
prescribed jumps and hence map the real axis to a polygon. Through additional
modifications we can generate our desired transformations from the exterior of
the disc to the exterior of polygons.

To make this precise, let us begin by setting out our notion for polygons. We
define a polygonal colloid through a piecewise linear curve $\Gamma$ comprised
of $n$ vertices $w_{1},\dots,w_{n}$ and bounding an interior region $P$. The
vertices have interior angles $\mu_{1}\pi,\dots,\mu_{n}\pi$. The corresponding
exterior angles are $\nu_{k}\pi$, with $\nu_{k}=1-\mu_{k}$, and for $\Gamma$
to form a closed curve we require $\sum_{k}\nu_{k}=2$.

Figure 9: The notation for labelling polygons. The vertices $w_{i}$ have
interior angles $\mu_{i}\pi$ and exterior angles $\nu_{i}\pi$, with
$\nu_{i}=1-\mu_{i}$.

To determine the form of $f_{k}(z)$ in (33) we note that
$(z-z_{k})^{-\nu_{k}}$ has a constant argument on $\mathbb{R}$ save for a jump
of $\nu_{k}\pi$ at $z_{k}$ and is analytic in the upper half-plane $H^{+}$. It
follows that a conformal transformation that maps $\mathbb{R}$ onto $\Gamma$
and $H^{+}$ onto $P$ is provided by

$f(z)=A+C\int^{z}\prod_{k=1}^{n}(z^{\prime}-z_{k})^{-\nu_{k}}\text{d}z^{\prime}.$
(34)

The $z_{k}$ are the prevertices, preimages of the vertices under the map such
that $w_{k}=f(z_{k})$. That these prevertices are well-defined is a
consequence of the Carathéodory–Osgood theorem [66, 76]. The complex constants
$A$ and $C$ allow for scaling, translation and rotation of the image. Together
with the prevertices they form a set of parameters which are overdetermined to
the tune of the three degrees of freedom inherent in Riemann’s mapping theorem
[65]. The lower limit of integration is left unspecified as it only provides a
constant that may be absorbed into $A$. The proof that (34) truly is an
analytic function as required is based upon showing that

$\mathcal{F}(z)=\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}-\sum_{k}\frac{\nu_{k}}{z-z_{k}}$
(35)

is an entire function and may be found in [66, 77, 72].

There are two modifications we must make to (34) that we now consider in turn.
The first is to change the domain of the map from the upper half-plane to the
unit disc, $D$. This is achieved by composition of maps and application of the
chain rule. For $h(z):H^{+}\to P$ and $g(z):\Sigma\to H^{+}$ we have
$f(z)=h(g(z)):\Sigma\to P$ with [66]

$\begin{split}f^{\prime}(z)&=h^{\prime}(g(z))g^{\prime}(z)\\\
&=Cg^{\prime}(z)\prod_{k}\left[g(z)-g(z_{k})\right]^{-\nu_{k}}.\end{split}$
(36)

A map from the disc to the upper half-plane is provided by the Möbius
transformation $g(z)=\text{i}\frac{1+z}{1-z}$, giving

$f^{\prime}(z)=C\frac{2\text{i}}{(1-z)^{2}}\prod_{k}\left[\frac{2\text{i}(z-z_{k})}{(1-z)(1-z_{k})}\right]^{-\nu_{k}}.$
(37)

Using that $\sum_{k}\nu_{k}=2$ and absorbing various constants into $C$ we
integrate to obtain

$f:D\to P,\qquad
f(z)=A+C\int^{z}\prod_{k}\left(1-\frac{z^{\prime}}{z_{k}}\right)^{-\nu_{k}}\text{d}z^{\prime}.$
(38)

Next we must consider mapping to the exterior region $P^{\prime}$. The first
change that this necessitates is reversing the sign of the exterior angles. In
the same manner as before this leads us to

$\mathcal{F}(z)=\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}-\sum_{k}\frac{\nu_{k}}{z-z_{k}}$
(39)

as an entire function. The difference now is that since the image of $f$
contains the point at infinity it no longer follows that $\mathcal{F}(z)$ is
constant. Taking $f(0)=\infty$ implies that $f$ has a simple pole there and so
at the origin $f^{\prime\prime}/f^{\prime}=-2/z$ plus some analytic part. We
therefore modify (39) to give

$\mathcal{F}_{2}(z)=\frac{f^{\prime\prime}(z)}{f^{\prime}(z)}-\sum_{k}\frac{\nu_{k}}{z-z_{k}}+\frac{2}{z}=0.$
(40)

Integrating we find

$f:D\to P^{\prime},\qquad
f(z)=A+C\int^{z}z^{\prime-2}\prod_{k}\left(1-\frac{z^{\prime}}{z_{k}}\right)^{\nu_{k}}\text{d}z^{\prime}.$
(41)

Finally, we may make the replacement $z\to 1/z$ in (41) to attain our desired
conformal map from the exterior of the disc to the exterior of the polygon.

Having attained this general expression we now derive the particular mappings
appropriate to the polygonal cogs that we consider in this paper. Our cogs are
formed by attaching right-angled triangles with base angle $\gamma\pi$ to each
side of a regular $n$-sided polygon and we can use this rotational symmetry to
simplify the integral. We take the prevertices of the base $n$-gon to be the
$n^{\text{th}}$ roots of unity and indeed to be fixed points of the map. Due
to the cyclic nature of the roots of unity
$\prod_{k}\left(1-\frac{z^{\prime}}{z_{k}}\right)=1-z^{\prime n}$. A similar
relation holds for the prevertices of the tips of the cog teeth, with these
being rotated from the roots of unity by an angle $\chi$ which depends on both
$n$ and $\gamma$ and must be determined numerically. In addition we can set
$A=0$ and determine $C$ by requiring $f(1)=1$ such that

$f(z)=\frac{\int^{z}z^{\prime-2}(1-z^{\prime
n})^{\frac{4-n}{2n}-\gamma}\left(1-\frac{z^{\prime
n}}{\text{e}^{\text{i}n\chi}}\right)^{\frac{1}{2}+\gamma}\text{d}z^{\prime}}{\int^{1}z^{\prime-2}(1-z^{\prime
n})^{\frac{4-n}{2n}-\gamma}\left(1-\frac{z^{\prime
n}}{\text{e}^{\text{i}n\chi}}\right)^{\frac{1}{2}+\gamma}\text{d}z^{\prime}}.$
(42)

Making the substitution $t=z^{\prime n}/z^{n}$ brings the numerator to the
form

$\frac{1}{nz}\int^{1}t^{-\frac{1}{n}-1}\left(1-z^{n}t\right)^{\frac{4-n}{2n}-\gamma}\left(1-\frac{z^{n}}{\text{e}^{\text{i}n\chi}}t\right)^{\frac{1}{2}+\gamma}\text{d}t.$
(43)

Comparing with Picard’s single integral representation of Appell’s $F_{1}$
function [67, Eq. 16.15.1]

$\begin{split}&F_{1}(a,b_{1},b_{2},c;x,y)=\\\
&\frac{\Gamma(c)}{\Gamma(a)\Gamma(c-a)}\int_{0}^{1}t^{a-1}(1-t)^{c-a-1}(1-xt)^{-b_{1}}(1-yt)^{-b_{2}}\text{d}t,\end{split}$
(44)

we find that the numerator of (42) is given by

$\frac{1}{nz}\frac{\Gamma(-\frac{1}{n})}{\Gamma(1-\frac{1}{n})}F_{1}\left(-\frac{1}{n},\frac{n-4}{2n}+\gamma,-\frac{1}{2}-\gamma;z^{n},\frac{z^{n}}{\text{e}^{\text{i}n\chi}}\right).$
(45)

We note that the representation in (44) is only valid for $\mathfrak{R}a>0$,
whereas here $a=-1/n$, leading to a singularity at $t=0$. This singularity can
be handled via adaptation of the integration contour, introducing a prefactor
into (45). We leave this prefactor undetermined as it is common to both the
numerator and denominator of (42) and hence does not feature in the desired
mapping $f(z)$.

For the denominator we use [78, Eq. 5.10.10]

$\begin{split}F_{1}(a,b_{1},b_{2},c;1,z)&={}_{2}F_{1}(a,b_{1};c;1){}_{2}F_{1}(a,b_{2};c-b_{1};z)\\\
&=\frac{\Gamma\left(c\right)\Gamma\left(c-a-
b_{1}\right)}{\Gamma\left(c-a\right)\Gamma\left(c-b_{1}\right)}{}_{2}F_{1}(a,b_{2};c-b_{1};z),\end{split}$
(46)

with the same caveat concering a prefactor as for (45). Replacing $z$ with
$1/z$ and including rotations of the image polygon by an angle $\psi$ we reach
the conformal transformation from the exterior of the disc to the exterior of
a cog

$f(z)=\frac{\Gamma\left(\frac{1}{2}+\frac{1}{n}-\gamma\right)zF_{1}\left(-\frac{1}{n},\gamma+\frac{n-4}{2n},-\frac{1}{2}-\gamma,1-\frac{1}{n};\text{e}^{\text{i}n\psi}z^{-n},\text{e}^{\text{i}n(\psi+\chi)}z^{-n}\right)}{\Gamma\left(1-\frac{1}{n}\right)\Gamma\left(\frac{1}{2}+\frac{2}{n}-\gamma\right){}_{2}F_{1}\left(-\frac{1}{n},-\frac{1}{2}-\gamma,\frac{1}{2}+\frac{1}{n}-\gamma,\text{e}^{\text{i}n\chi}\right)}.$
(47)


$\displaystyle=\left|\frac{1/\varepsilon+1}{\lceil\Phi_{G}/\varepsilon\rceil}-\frac{1}{\Phi_{G}}\right|=\left|\frac{\Phi_{G}/\varepsilon+\Phi_{G}-\lceil\Phi_{G}/\varepsilon\rceil}{\lceil\Phi_{G}/\varepsilon\rceil\Phi_{G}}\right|$
$\displaystyle\leq\left|\frac{\Phi_{G}/\varepsilon-\lceil\Phi_{G}/\varepsilon\rceil}{\lceil\Phi_{G}/\varepsilon\rceil\Phi_{G}}\right|+\left|\frac{\Phi_{G}}{\lceil\Phi_{G}/\varepsilon\rceil\Phi_{G}}\right|\leq
2\varepsilon/\Phi_{G},$

where for the first inequality we use the triangle inequality, and for the
second inequality we use the first case. Now by setting
$\varepsilon\leftarrow\varepsilon/2$, we obtain the result. ∎

At last, we are ready to show the general case.

###### Proof of Section 1.3.

We recall the definition of the ideal relative loads: $\ell^{*}(e)$ is defined
recursively as follows.

  1. 1.

Let $\mathcal{P}^{*}$ be a packing with
$\rm{pack\\_val}(\mathcal{P}^{*})=\Phi$.

  2. 2.

For all $e\in E(G/\mathcal{P})$, set $\ell^{*}(e):=1/\Phi$.

  3. 3.

For each $S\in\mathcal{P}^{*}$, recurse on the subgraph $G[S]$.

We prove the lemma by induction on the depth of this recursive definition. If
the depth is $0$ then the minimum partition value $\Phi_{G}$ is exactly
achieved by the trivial partition $\mathcal{P}=\\{v_{i}\\}_{i=1}^{n}$, so the
base case is immediate. Suppose it holds up to depth $i$. Then we have for
each component $X\in\mathcal{P}^{*}$ that there exists a tree-packing
$\mathcal{T}_{X}$ such that $|\mathcal{T}_{X}|=\Theta(\Phi/\varepsilon)$ and

$|\ell^{\mathcal{T}_{X}}(e)-\ell^{*}(e)|\leq\varepsilon/\Phi,$

for all $e\in E(G[X])$. Note that this follows from the induction hypothesis
since $\Phi_{G[X]}\geq\Phi$, so we can set
$\varepsilon\leftarrow\tfrac{\Phi_{G[X]}}{\Phi}\varepsilon$. We also have a
tree-packing $\mathcal{T}_{\mathcal{P}*}$ on $G/\mathcal{P}^{*}$ such that
$|\mathcal{T}_{\mathcal{P}^{*}}|=\Theta(\Phi/\varepsilon)$ and

$|\ell^{\mathcal{T}_{\mathcal{P}^{*}}}(e)-\ell^{*}(e)|\leq\varepsilon/\Phi,$

for all $e\in E(G/\mathcal{P}^{*})$. Clearly
$\mathcal{T}=\mathcal{T}_{\mathcal{P}^{*}}\cup\left(\bigcup_{X\in\mathcal{P}^{*}}\mathcal{T}_{X}\right)$,
where each tree is a union of the respective trees, is a tree-packing of $G$
that satisfies the bounds on $\ell^{\mathcal{T}}$. ∎

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