# From Disks to Channels: Dynamics of Active Nematics Confined to an Annulus

Chaitanya Joshi<EMAIL_ADDRESS>Department of Physics and
Astronomy, Tufts University,
574 Boston Avenue, Medford, Massachusetts 02155, USA  Martin Fisher School of
Physics, Brandeis University,
415 South Street, Waltham, Massachusetts 02453, USA  Zahra Zarei  Martin
Fisher School of Physics, Brandeis University,
415 South Street, Waltham, Massachusetts 02453, USA  Michael M. Norton  School
of Physics and Astronomy, Rochester Institute of Technology, Rochester, NY
14623  Seth Fraden  Martin Fisher School of Physics, Brandeis University,
415 South Street, Waltham, Massachusetts 02453, USA  Aparna Baskaran
<EMAIL_ADDRESS>Martin Fisher School of Physics, Brandeis University,
415 South Street, Waltham, Massachusetts 02453, USA  Michael F. Hagan
<EMAIL_ADDRESS>Martin Fisher School of Physics, Brandeis University,
415 South Street, Waltham, Massachusetts 02453, USA

###### Abstract

Confinement can be used to systematically tame turbulent dynamics occurring in
active fluids. Although periodic channels are the simplest geometries to study
confinement numerically, the corresponding experimental realizations require
closed racetracks. Here, we computationally study 2D active nematics confined
to such a geometry — an annulus. By systematically varying the annulus inner
radius and channel width, we bridge the behaviors observed in the previously
studied asymptotic limits of the annulus geometry: a disk and an infinite
channel. We identify new steady-state behaviors, which reveal the influence of
boundary curvature and its interplay with confinement. We also show that,
below a threshold inner radius, the dynamics are insensitive to topological
constraints imposed by boundary conditions. We explain this insensitivity
through a simple scaling analysis. Our work sheds further light on design
principles for using confinement to control the dynamics of active nematics.

###### pacs:

Valid PACS appear here

##  I Introduction

Active matter describes a collection of constituents that consume energy at
the level of the individual units to generate motion. Interactions between the
units combined with their motion result in emergent behaviors that span scales
much larger than those of the particles or their interactions, and would be
thermodynamically forbidden in equilibrium materials Marchetti  _et al._
(2013); Ramaswamy (2010); Toner  _et al._ (2005). Suspensions of such active
units, termed active fluids, are a paradigm to describe different biological
systems such as the cytoskeleton of cells Jülicher  _et al._ (2007), bacterial
colonies Wensink  _et al._ (2012) and tissues and cell sheets Duclos  _et al._
(2017); Saw  _et al._ (2017); Kawaguchi  _et al._ (2017). Enabled by the
continuous conversion of chemical energy into mechanical work due to
biomolecular processes, these systems exhibit myriad emergent functionalities
that are crucial for their organisms’ lifecycles, such as motility, division,
self-healing, and morphogenesis Giomi and DeSimone (2014); Maroudas-Sacks  _et
al._ (2020); Vafa and Mahadevan (2022); Khoromskaia and Salbreux (2023);
Guillamat  _et al._ (2022); Hoffmann  _et al._ (2022).

Active nematic liquid crystals, which are composed of motile energy-consuming
anisotropic units, provide a promising platform to achieve similar
capabilities in synthetic or biomimetic systems Doostmohammadi  _et al._
(2018). However, bulk active nematics exhibit chaotic turbulent-like flows
that lack long-range order Dombrowski  _et al._ (2004); Cisneros  _et al._
(2007); Wolgemuth (2008); Creppy  _et al._ (2015); Giomi (2015); Blanch-
Mercader  _et al._ (2018); Peng  _et al._ (2021). Thus, they are unable to
perform functions such as generating work or driving net material transport
without a means to suppress this turbulence.

Confinement can control active flows Voituriez  _et al._ (2005); Duclos  _et
al._ (2018); Theillard  _et al._ (2017); Woodhouse and Goldstein (2012); Wu
_et al._ (2017); Varghese  _et al._ (2020); Chandrakar  _et al._ (2020);
Edwards and Yeomans (2009); Giomi  _et al._ (2011, 2012); Wioland  _et al._
(2013); Ravnik and Yeomans (2013); Doostmohammadi  _et al._ (2016); Wioland
_et al._ (2016); Shendruk  _et al._ (2017); Gao  _et al._ (2017);
Doostmohammadi  _et al._ (2017); Van Teeffelen  _et al._ (2009), and enable
harnessing them for transport and other functions (e.g. Fürthauer  _et al._
(2012); Thampi  _et al._ (2016); Li  _et al._ (2021)). The simplest
confinement in 2D is that of an infinite channel of finite width,
computationally implemented using periodic boundary conditions. It has been
shown that, depending on the channel width and activity level, active nematics
in such channels exhibit a variety of emergent states including ‘dancing’
defect pairs and coherent flow along the channel Shendruk  _et al._ (2017);
Hardoüin  _et al._ (2019). However, in experiments, such a channel can only be
mimicked by physically joining its two ends, leading to a curved racetrack or
annulus geometry Wu  _et al._ (2017); Opathalage  _et al._ (2019); Hardoüin
_et al._ (2022, 2020), or by finite-length ‘lanes’ in which end-effects may
arise Hardoüin  _et al._ (2019); Opathalage  _et al._ (2019).

Figure 1:  2D active nematics confined to an annulus geometry (a) Snapshot of
the microtubule-kinesin active nematics system, obtained using fluorescence
microscopy. (b) Snapshot from a hydrodynamic simulation of the same system.
The lines indicate the director field lines while the color corresponds to the
scalar order parameter $S$. The magenta arrows indicate the positions and
orientations of $+\frac{1}{2}$ defects.

For instance, Fig. 1 shows a widely studied active matter system of
cytoskeleton-based 2D active nematics Sanchez  _et al._ (2012); Doostmohammadi
_et al._ (2018) confined to an annulus geometry. In bulk, this system exhibits
the widely studied phenomenon of low Reynolds number turbulence Giomi (2015);
Thampi  _et al._ (2016); Thampi and Yeomans (2016); Guillamat  _et al._
(2017); Lemma  _et al._ (2019); Doostmohammadi  _et al._ (2017); Opathalage
_et al._ (2019); Alert  _et al._ (2020, 2022); Martínez-Prat  _et al._ (2019,
2021). Experiments in the annulus geometry Hardoüin  _et al._ (2022, 2020) and
lane configurations Hardoüin  _et al._ (2019) demonstrate that confinement can
tame this turbulence to generate ordered flows. However, despite these
experiments and previous theoretical investigations of active nematics
confined in channels Shendruk  _et al._ (2017); Duclos  _et al._ (2018);
Hardoüin  _et al._ (2019); Varghese  _et al._ (2020); Chandragiri  _et al._
(2019, 2020); Li  _et al._ (2021); Wagner  _et al._ (2022), disks Norton  _et
al._ (2018); Opathalage  _et al._ (2019); Hardoüin  _et al._ (2022); Mirantsev
(2021), and annulus geometries Shendruk  _et al._ (2017); Chen  _et al._
(2018); Hardoüin  _et al._ (2022, 2020); de Oliveira  _et al._ (2023), the
influence of channel curvature and finite channel length on emergent behaviors
has yet to be studied in detail.

In this work we use hydrodynamic simulations to systematically investigate 2D
active nematics confined to the annulus geometry. We vary the confinement
shape by changing the inner radius of the annulus for a fixed width, going
from a ‘disk-like’ annulus, with a small pinhole at its center, to a ‘channel-
like’ annulus for which the radius is much larger than the channel width (Fig.
3). We vary the confinement size by changing the width of the annulus, and map
out a shape-size phase diagram of the dynamical steady states. This phase
diagram connects the previously observed steady states in disks and channels,
and reveals how the system behaves between these two limits. In particular,
the intermediate geometries reveal previously unobserved states exhibiting net
material transport, which are stabilized by the positive curvature of the
inner wall of the annulus.

Further, we find that below a threshold inner radius, annuli exhibit the same
dynamics as the corresponding true disks under a finite parallel anchoring on
the nematic director, despite the two systems being constrained to have a
different net topological charges by their boundary conditions. We provide a
theoretical explanation for this observation and an estimate of the threshold
pinhole size based on the competing anchoring and elastic energies at the
hole. Finally, we find that the net-transport state disappears at high
anchoring strengths. To understand these effects, we study the influence of
activity on the boundary layer at the walls Norton  _et al._ (2018); Hardoüin
_et al._ (2020), which uncovers a relationship between the anchoring strength
and the curvature. More broadly, these results advance our understanding of
how hydrodynamics and topological constraints of boundary conditions control
the behavior of a confined active nematic.

##  II Model Description

As in our previous work Norton  _et al._ (2018), we study a continuum model of
an active nematic described as a single incompressible fluid with internal
nematic symmetry Beris and Edwards (1994); Doostmohammadi  _et al._ (2018). It
has two fields: the nematic tensor order parameter $\mathbf{Q}=\rho
S[\mathbf{n}\otimes\mathbf{n}-(1/2)\mathbf{I}]$ and a flow field $\mathbf{u}$.
Here, $\mathbf{n}$ is the local orientation unit vector, $S$ is the scalar
order parameter, and $\rho$ is the density, which we assume to be uniform and
constant in the simulations throughout this paper. In the limits of low
Reynolds number and high Ericksen number Koch and Wilczek (2021), the dynamics
of this fluid is given by

$\eta\nabla^{2}\mathbf{u}-\nabla P-\alpha\nabla\cdot\mathbf{Q}=0$ (1)

along with the incompressibility constraint $\nabla\cdot\mathbf{u}=0$. Here,
$\eta$ is the dynamic viscosity and $P$ is the pressure. The third term is the
leading order contribution from the extensile active stress
$-\alpha\mathbf{Q}$, with $\alpha$ defined as the activity. We use no-slip
boundary conditions on the velocity throughout this article,
$\mathbf{u}|_{\partial\Omega}=0$.

The dynamical equation for $\mathbf{Q}$ is given by

$\partial_{t}\mathbf{Q}+\nabla\cdot(\mathbf{u}\mathbf{Q})=(\mathbf{Q}\Omega-\Omega\mathbf{Q})+\lambda\mathbf{E}+\gamma^{-1}\mathbf{H}$
(2)

Here, $\Omega_{ij}=(\partial_{i}u_{j}-\partial_{j}u_{i})/2$ is the anti-
symmetric vorticity tensor, $E_{ij}=(\partial_{i}u_{j}+\partial_{j}u_{i})/2$
is the symmetric strain rate tensor, and $\lambda$ is the flow alignment
parameter. The final term is the relaxation of the nematic field proportional
to the variation of the nematic free energy, $H_{ij}=-\delta\mathcal{F}/\delta
Q_{ij}$, with dissipation rate $\gamma^{-1}$. The nematic free energy for a
confined system consists of a bulk contribution and a boundary contribution.

The bulk contribution is the Landau de Gennes free energy Beris and Edwards
(1994) $\mathcal{F}_{\text{LDG}}$ given by

$\displaystyle\mathcal{F}_{\text{LDG}}=\int_{\Omega}\\!$
$\displaystyle\differential^{2}\mathbf{r}\left\\{C\quantity(-\frac{\beta_{1}}{2}Q_{ij}Q_{ji}+\frac{\beta_{2}}{4}(Q_{ij}Q_{ji})^{2})\right.$
$\displaystyle+\left.\frac{1}{2}K(\partial_{k}Q_{ij}\partial_{k}Q_{ij})\right\\}.$

In this work, we take the simple form, $\beta_{1}(\rho)=\rho-1$ and
$\beta_{2}(\rho)=(\rho+1)/\rho^{2}$ so as to set up the isotropic ($\rho<1$)
to nematic ($\rho>1$) transition. We fix $\rho=1.6$ throughout this article,
thus focusing on the far-from-critical nematic phase. At the boundary, we use
the Nobili-Durand anchoring energy Nobili and Durand (1992); Giomi and
DeSimone (2014)

$\mathcal{F}_{\text{ND}}=\oint_{\partial\Omega}\\!\differential\mathbf{r}\frac{1}{2}E_{\text{A}}(Q_{ij}-W_{ij})(Q_{ji}-W_{ji}),$

where $\mathbf{W}$ specifies the order and orientation at the boundary and
$E_{\text{A}}^{\prime}$ specifies the anchoring strength. Thus, the relaxation
term becomes

$\displaystyle\gamma^{-1}H_{ij}=$ $\displaystyle
D_{\text{r}}(\beta_{1}-\beta_{2}Q_{kl}Q_{lk})Q_{ij}+2D_{\text{E}}\partial_{k}\partial_{k}Q_{ij}$
$\displaystyle-E_{\text{A}}(Q_{ij}-W_{ij})|_{\partial\Omega},$

where $D_{\text{E}}=(L_{1}+L_{2})/2\gamma$,
$E_{\text{A}}=E_{\text{A}}^{\prime}/\gamma$ and $D_{\text{r}}=C/\gamma$.

We identify the time scale $T=1/D_{\text{r}}$ and the length scale
$L=\sqrt{D_{\text{E}}/D_{\text{r}}}$. We use these to non-dimensionalize the
equations, with the dimensionless operators
$\bar{\partial}_{t}=\partial_{t}/D_{\text{r}}$ and
$\bar{\partial}_{i}=\partial_{i}/\sqrt{D_{\text{E}}/D_{\text{r}}}$. This gives
the dimensionless equations

$\displaystyle\bar{\partial}_{t}\mathbf{Q}+$
$\displaystyle\bar{\nabla}\cdot(\bar{\mathbf{u}}\mathbf{Q})=(\mathbf{Q}\bar{\Omega}-\bar{\Omega}\mathbf{Q})+\bar{\lambda}\mathbf{\bar{E}}+\bar{\mathbf{H}}$
$\displaystyle\bar{H}_{ij}=$ $\displaystyle
D_{\text{r}}(\beta_{1}-\beta_{2}Q_{kl}Q_{lk})Q_{ij}+2\bar{\partial}_{k}\bar{\partial}_{k}Q_{ij}$
$\displaystyle-\bar{E}_{A}(Q_{ij}-W_{ij})|_{\partial\Omega},$
$\displaystyle\bar{\nabla}^{2}\bar{\mathbf{u}}-\bar{\nabla}P-\bar{\alpha}\bar{\nabla}\cdot\mathbf{Q}=0$

Here, the dimensionful quantities $\mathbf{u}$ and $P$ are re-scaled to
$\bar{\mathbf{u}}=\mathbf{u}\sqrt{D_{\text{E}}D_{\text{r}}}$ and
$\bar{P}=P/(D_{\text{r}}\eta)$. The dimensionful parameters get re-scaled as
$\bar{E}_{A}=E_{\text{A}}/D_{\text{r}}$, $\bar{\lambda}=\lambda/D_{\text{r}}$,
$\bar{\alpha}=\alpha/(\eta D_{\text{r}})$ and finally the system dimensions
$\bar{R}=R/\sqrt{D_{\text{E}}/D_{\text{r}}}$. In the following, we drop the
overbars on the parameters and implicitly assume this nondimensionalization.

We start the simulations with zero velocity and uniform director field with 1%
noise. We use the open source Python library FEniCS Alnæs  _et al._ (2015);
Logg and Wells (2010) to integrate the equations. The visualizations are made
using the open source visualization software Paraview James  _et al._ (2005)
or MATLAB MATLAB (2010). Unless specified otherwise, we use
$E_{\text{A}}=0.01$ (weak anchoring) and $\alpha=\alpha_{0}=5$ (strong
activity) in the simulations. For this activity, the system reaches steady
state within 500 simulation time units even for the largest geometries
investigated. Accordingly, we simulate all the systems for a total of
$t_{\text{f}}=1000$ time units and use the data from the last 10% of the time
for analysis. If we change the activity, we scale $t_{\text{f}}$ :
$t_{\text{f}}^{\prime}=(\alpha_{0}/\alpha)t_{\text{f}}$. For all simulations,
we set $\differential t=0.1$, and we use a mesh with average linear element
size $\Delta x\sim 0.2$.

Figure 2:  Dynamical steady states in active nematics confined to annulus
geometries. (top) The circulating state, as observed for $W=10$ and
$R_{\text{i}}/W=0.1$. A single defect going around the channel is shown with a
red circle to highlight the circulation. (middle) The dancing state, as
observed for $W=8$ and $R_{\text{i}}/W=1$. The highlighted region shows the
dancing behavior of a defect pair over time. (bottom) The corotating state, as
observed for $W=6$ and $R_{\text{i}}/W=1$. The highlighted defect shows the
corotating motion of the train of defects. The symbols correspond to those
used to identify these states in the phase diagram in Fig. 3. All times start
at an arbitrary point and are in the units of 0.1 simulation times.  Figure 3:
Shape-size phase diagram for active nematics in annuli. Steady states as a
function of width $W$ and the ratio of the inner radius to width $R_{i}/W$:
(i) Stationary state (grey diamonds), (ii) Corotating state (green circles),
(iii) Circulating state (indigo thin diamonds), (iv) Dancing state (blue
triangles), and (v) Chaotic state (magenta pluses). The indicated shapes for
various $R_{i}/W$ on the top are a guide to the eye and not to scale.
$R_{i}/W=0$ responds to a disk of radius $W$, whereas $R_{i}/W=\infty$
corresponds nominally to a channel simulation with width $W$, length $20W$,
and periodic boundary conditions at the channel ends. For these simulations,
we apply a no-slip boundary condition for the velocity, and weak parallel
anchoring for the director, with $E_{\text{A}}=0.01$.

##  III Results

###  III.1 Steady State Behaviors

We specify the size of the annulus by its width $W$, and its shape with the
ratio of the inner radius to the width, $R_{\text{i}}/W$. We use
$R_{\text{i}}/W$ values from $0.01$ (disk-like) to $3$ (channel-like), and
widths in the range $W\in[4,30]$ (see Fig. 2 for some examples). For
completeness, we also simulate active nematic disks and flat periodic
channels. Here, the disk of radius $R$ is thought of as an “annulus” with
$W=R$ and $R_{\text{i}}=0$, whereas a channel is thought of as an “annulus”
with width $W$ and $R_{\text{i}}\to\infty$. We thus get a shape-size phase
diagram for the 2D active nematic confined to an annulus (see Fig. 3). We
consider weak anchoring ($E_{\text{A}}=0.01$) and strong activity
($\alpha=5$). The system exhibits distinct dynamical steady-states that vary
as a function of the geometric parameters. Sequences of snapshots
characterizing the dynamics for some of these states are shown in Fig. 2. Fig.
3 gives a phase diagram of the observed steady states. The specific criteria
that we used to identify and classify these states are given in Appendix A.

#### Stationary State

At very high confinements, active instabilities are suppressed, resulting in a
class of stationary steady states, which we define as states for which the
director field and velocity profile are independent of time:
$\partial_{t}\mathbf{Q}=\partial_{t}\mathbf{u}=0$. The texture of the
stationary steady state in the disk-like limit depends on the confinement
geometry and boundary conditions, resulting in familiar dipolar states with
two $+\frac{1}{2}$ defects that closely resembles the behavior in a true disk
Norton  _et al._ (2018), as well as spiral states with a time-independent non-
zero velocity. We elaborate on the set of observed stationary states in
Appendix A. In the opposite high confinement extreme, i.e. the periodic
channel ($R_{\text{i}}/W\rightarrow\infty$), we observe a uniform nematic with
no flow.

#### Corotating State

Upon moving to weaker confinement by increasing $W$, the system transitions
into a corotating state (see Fig. 2 (bottom)), characterized by a train of
isolated $+\frac{1}{2}$ defects circulating along the annulus. These defects
always nucleate from the inner wall of the annuli. This is distinct from the
straight periodic channel of the same width, where the bend-instability occurs
from both boundaries resulting in a structure that is symmetric across the
channel, although also with a spontaneously chosen circulation direction (see
Movie S1). The weak anchoring strength at the boundary allows for the
corresponding $-\frac{1}{2}$ defects at the wall to exist. Thus, this state is
natural for Neumann boundary conditions on $\mathbf{Q}$ with no anchoring.
Fig. 3 shows that the corotating state occurs for a narrow range of $W$, with
phase boundaries that are essentially independent of $R_{\text{i}}$. This
reflects the fact that the corotating state occurs when only a single
$+\frac{1}{2}$ defect can form across the narrowest confinement dimension
(channel width in this case) Norton  _et al._ (2018).

The simplest corotating state is observed at very small inner radii and
consists of two $+\frac{1}{2}$ defects rotating around the center and no
$-\frac{1}{2}$ defects Norton  _et al._ (2018); Opathalage  _et al._ (2019).
Notably, this state is consistent with the net topological charge of $+1$
imposed by parallel anchoring in a true disk, but deviates from the net
topological charge of 0 for an annulus. This indicates that, despite changing
the topology, inserting a hole at the center of the disk does not affect the
steady-state behavior below a threshold radius. As will be discussed in
Section III.2, the threshold size of the hole is expected to depend on the
anchoring strength. For larger inner radii, the $-\frac{1}{2}$ defects at the
inner wall are stabilized by weak anchoring conditions and/or high inner
curvature ($R_{\text{i}}\rightarrow 0$). Thus, the corotating state is absent
for low inner curvatures when the anchoring strength is high (discussed in
Section III.3).

#### Circulating state

Figure 4: Distinction between the corotating and the circulating states. (a)
Average azimuthal speed ($\mathbf{u}\cdot\hat{\theta}$) as a function of time
for a representative corotating state (blue dashed line, $R_{\text{i}}=6,W=6$)
and a circulating state (orange continuous line, $R_{\text{i}}=1,W=10$). While
both have a steady circulation on average, there is significant variation in
the instantaneous flow of the circulating state, as opposed to the weakly
sinusoidal variation in the corotating state (the inset shows a zoomed-in plot
of the same). (b) Total number of defects as a function of time for the same
two datasets. The number of defects remains constant for the corotating case,
whereas it fluctuates for the circulating case, indicating annihilation and
proliferation.

Upon increasing the width of the annulus in the disk-like limit
($R_{\text{i}}/W\lesssim 0.1$), we observe a circulating state that is
different from the corotating state, in that it exhibits turbulent-like
nucleation and annihilation of defects in the steady state (see Fig. 2 (top)).
However, the circulating and corotating states have similar vortex structures,
with high circulation around a central vortex, albeit with significantly
larger perturbations of the velocity profile in the circulating state (see
Fig. 4).

#### Dancing state

For intermediate widths and relatively small curvatures, we observe the
‘dancing state’, which has been well studied in the channel geometry Shendruk
_et al._ (2017); Hardoüin  _et al._ (2019); Varghese  _et al._ (2020) (see
Fig. 2 (middle)). Here, pairs of $\pm 1/2$ defect pairs swim on a lattice of
vortices along the channel. We find that the dancing state persists for
annular channels with fairly high curvature (see Fig 3) as also noted in
Shendruk  _et al._ (2017).

In the limit of low curvature ($R_{\text{i}}/W\gtrsim 1$), the dancing state
in the annulus closely resembles that observed in straight channels. In
particular, there is usually an equal number of $+\frac{1}{2}$ defects
traveling in each direction along the channel axis, resulting in no net flow.
This symmetry can be broken by formation of a ‘drift-lattice defect’ Shendruk
_et al._ (2017), where two more $+\frac{1}{2}$ defects travel in one direction
than the other, resulting in a net drift. The probability of having such
drift-lattice defects increases with the length of the channel Shendruk  _et
al._ (2017) or correspondingly the annulus circumference.

Figure 5: Curvature-induced drift in the dancing state for $W=15$ and
$R_{\text{i}}/W=0.3$. In this case, due to the asymmetry of the inner and
outer curvature, the flow acquires a drift (see Movie S2) despite there being
an equal number of defects moving in each direction. The dotted green line
highlights the trajectory of a defect moving uninterrupted anti-clockwise,
while the red dotted line highlights the trajectory of a defect moving
clockwise, undergoing annihilation and nucleation periodically at the outer
edge. This asymmetry of defect motions creates a net flow in the counter-
clockwise direction.

However, at high curvatures, $R_{\text{i}}/W\lesssim 1$, the difference in the
outer and the inner curvature results in an asymmetry between the
$-\frac{1}{2}$ defects at those respective boundaries. This sometimes results
in an asymmetry in the $-\frac{1}{2}$ defect interactions of the clockwise and
anti-clockwise moving $+\frac{1}{2}$ defects. The defects moving in one
direction interact strongly with the outer $-\frac{1}{2}$ defects, which slows
their flow, whereas the defects traveling in the other direction move freely.
This results in a net ‘curvature-induced drift’, even with the same number of
defects moving in each direction (see Fig. 5 and Movie S2). This behavior is
unique to the annulus.

#### Chaotic state

Finally, above a threshold width, whose value depends on the curvature, the
system undergoes extensive defect proliferation resulting in turbulent -like
dynamics and no coherent motion or circulation.

###  III.2 Effect of topology

The similarity between the steady-states in the disk and pin-hole annuli
discussed for the corotating state extends for all widths (Fig. 3). That is,
the steady state dynamics are insensitive to the presence of a pin-hole below
a threshold value of the inner radius. The origin of this insensitivity can be
understood from the following. First, near the pin-hole, the energy cost for a
uniform director (resulting in the insensitivity) goes as
$E_{\text{uniform}}\sim E_{\text{A}}R_{\text{i}}$, whereas the energy for a
perfectly anchored ($+1$ defect) structure goes as $E_{\text{defect}}\sim
K\log(\rho_{\text{max}}/R_{\text{i}})$, where $\rho_{\text{max}}$ is a
characteristic defect spacing or confinement sizede and Prost (1993). The pin-
hole will only affect the topology of the director field if
$E_{\text{uniform}}>E_{\text{defect}}$. Thus, for sufficiently small holes,
the director structure will be indistinguishable from that of a disk. Second,
in the annulus, the no-slip boundary condition on the fluid at the boundary of
the hole imposes zero fluid velocity. While this requirement is absent in the
disk, the symmetry of the corotating state results in a vanishing velocity at
its center. Thus, the vortex structures of the disk and annulus with small
$R_{\text{i}}$ are very similar. These two effects combine to make the system
dynamics topologically insensitive to the pinhole.

###  III.3 Effect of anchoring strength

Figure 6: Phase diagram for strong anchoring conditions (with
$E_{\text{A}}=3.0$). The markers correspond to the same states as in Fig. 3:
(i) Stationary state (grey diamonds), (ii) Corotating state (green circles),
(iii) Circulating state (indigo thin diamonds), (iv) Dancing state (blue
triangles), and (v) Chaotic state (magenta pluses), except for the maroon
upward triangles that indicate the arrested bend instability state. This phase
diagram reveals the absence of the corotating state (green circles) for high
anchoring and low curvature. Figure 7: The corotating state disappears at high
anchoring strengths. Snapshots of the steady state for $W=6$,
$R_{\text{i}}/W=1$ for different anchoring strengths. (left) For low anchoring
strengths, the $-\frac{1}{2}$ defects are easier to form and we get the
corotating state. (middle) For intermediate anchoring, the bend instability is
arrested. (right) For high anchoring strengths, the bend instability is
completely suppressed. The width for the onset of the dancing state remains
independent of anchoring strength.

The results described thus far have focused on a relatively weak anchoring
strength $E_{\text{A}}=0.01$. Fig. 6 shows a corresponding phase diagram for
$E_{\text{A}}=3$. The key differentiating feature of this phase diagram is
that the corotating state disappears in the straight-channel limit. In this
regime, as $W$ increases, the system transforms from a uniform nematic with
azimuthal orientation, to an arrested bend instability, and then to the
dancing state. This can also be seen by increasing the anchoring strength for
a fixed $W$ and $R_{\text{i}}$ (Fig. 7). We note this observation is similar
to results from simulations of polar active fluids confined to channels, where
coherent transport through the channel is suppressed at high anchoring
strengths Gulati  _et al._ (2022). This suggests that it should be explainable
by a simple argument. To this end, we note that the altered phase diagram in
our system reflects that the formation of the corotating state in the channel
limit requires two conditions: a width that accommodates only one defect
($W\leq\ell_{\text{d}}$, section III.1) and a sufficiently small energy cost
to form the corresponding $-\frac{1}{2}$ defects at the inner wall. Above a
threshold value of $E_{\text{A}}$, this energy cost is too large for channel-
like annuli, and thus the bend instability is arrested before defect
nucleation can occur. For smaller inner radii, $-\frac{1}{2}$ defects are
stabilized by the curvature of the inner wall and thus incur a smaller energy.
Moreover, in the disk-like limit $R_{\text{i}}/W\lesssim 0.1$, the corotating
state forms without any $-\frac{1}{2}$ defects. Hence, the corotating state
emerges at smaller inner radii.

Figure 8: The effects of anchoring are renormalized by activity. (a) Snapshot
of a simulation with $\alpha=5$ and $E_{\text{A}}=3$. The $200\times 200$ box
has periodic boundary conditions in the $y$ direction and parallel anchoring
with strength $E_{\text{A}}$ in the x direction. The color shows $|n_{y}|$.
(b) $\expectationvalue{|n_{y}|}$ as a function of $x$ for the steady state,
averaged over time and $y$ for various activities, each for low (solid lines)
and high (dashed lines) anchoring cases. The dotted lines indicate the
thresholds for the definition of the alignment plateau length $x_{\text{p}}$
and decay length $x_{\text{d}}$. (c) $x_{\text{p}}$ vs. $E_{\text{A}}$ for
four different activities. The shaded region indicates the standard error. (d)
$x_{\text{p}}$ as a function of the active length scale $\sqrt{K/\alpha}$, for
various anchoring energies. The data in this plot is same as that in (c), with
the anchoring energies increasing with the darkness of the lines. Points
marked (i), (ii) and (iii) highlight specific parameter sets. (e) Steady
states obtained using the parameters in (d) (i), (ii), and (iii).

To understand the effect of the anchoring strength on the phase behavior and
in particular on the existence of the corotating state, we investigated how
the penetration length of the parallel anchoring boundary condition depends on
control parameters in an active nematic. To this end, we performed simulations
in a square channel geometry, with dimensions $200\times 200$, periodic
boundary conditions in the $y$ direction and parallel anchoring boundary
conditions in $x$ (See Fig. 8). Note that this domain is much larger than the
systems investigated in Fig. 3, and the active nematic exhibits turbulent-like
behavior everywhere except near the vertical boundaries.

To quantify the effect of anchoring in this system, Fig. 8 shows the extent to
which the alignment with the vertical wall persists into the channel interior.
In particular, we have plotted $\expectationvalue{|n_{y}|}_{y,t}$ as a
function of $x$, where $|n_{y}|$ is the magnitude of the vertical component of
the director field, and the average is performed over the vertical dimension
$y$ and time $t$. Strong parallel anchoring implies $|n_{y}(x=0)|\approx 1$,
whereas in the bulk we expect a uniform distribution of $n_{y}$, yielding
$\expectationvalue{|n_{y}|}_{y,t}(x\gg
1)=\expectationvalue{|\sin\theta|}=2/\pi\approx 0.64$. For convenience, we
define a parameter indicating the fractional drop in
$\expectationvalue{|n_{y}|}_{y,t}$ from 1 towards the bulk value $2/\pi$ as
$\Delta n_{y}(x)=\frac{1-\expectationvalue{|n_{y}|}_{y,t}(x)}{1-2/\pi}$.

As shown in Fig. 8, we observe that the variation of
$\expectationvalue{|n_{y}|}_{y,t}$ away from the vertical wall exhibits two
characteristic length scales. First, there is a plateau within which the
director is parallel to the wall $\expectationvalue{|n_{y}|}_{y,t}\approx 1$,
followed by an exponential decay to its bulk value. We define the width of the
plateau region, $x_{\text{p}}$, as the distance from the wall at which
alignment diminishes by 5%, $\Delta n_{y}(x_{\text{p}})=0.05$, and the decay
length $x_{\text{d}}$ as the distance corresponding to 50% of the alignment
decrease, $\Delta n_{y}(x_{\text{d}})=0.5$ (see Fig.8(b)).

Figure 9: Alignment decay length $x_{\text{d}}$ as a function of activity and
anchoring strength. Plot of $x_{\text{d}}$ (defined in Fig. 8) as a function
of anchoring strength and the active length scale, analogous to Fig. 8(c) and
(d), indicating the $x_{\text{d}}$ is independent of the anchoring strength.

We find that the overall decay length is primarily determined by activity. In
particular, both the plateau width $x_{\text{p}}$ and decay length
$x_{\text{d}}$ increase linearly with the active length scale
$\sqrt{K/\alpha}$ Giomi (2015) and thus diminish with increasing activity. In
contrast, $x_{\text{d}}$ is essentially independent of anchoring strength,
while $x_{\text{p}}$ increases moderately with $E_{\text{A}}$ until saturation
(Fig. 8(c,d), Fig. 9). We now compare the value of $x_{\text{p}}$ with the
channel width for the corotating state, $W=6$ (see Fig. 8(e) and points
(i-iii) on Fig. 8(d)). For the corotating state at $W=6$ and
$E_{\text{A}}=0.01$, $x_{\text{p}}\approx 0.5$, whereas for $E_{\text{A}}=1$,
$x_{\text{p}}\approx 1.86$. For the higher anchoring strengths, the plateau
region covers roughly $60\%$ of the entire annulus, thus hindering defect
formation, whereas for the lower anchoring case, it covers only $\sim 15\%$ of
the annulus, thus allowing for defect nucleation. Importantly, the anchoring
boundary layer only affects the director profile and not the velocity profile.
A similar (no-slip) layer in the velocity profile would have merely shifted
the width value at which the corotating state occurs, as opposed to
eliminating it. These observations thus directly explain the findings in Fig.
7. More broadly, the small value of $x_{\text{p}}$ for high activities
explains the observations from this work and Norton et al. Norton  _et al._
(2018) that strongly confined active nematics are insensitive to topological
constraints, and that the anchoring effects are restricted to a narrow region
near the boundary.

##  IV Discussion

Using an annulus confinement, we simulate the natural periodic boundary
conditions one can build in experiments, while studying the role of curvature
in the steady state dynamics. We map a shape-size phase diagram for annular
active nematics. At high confinement, we observe a set of stationary states
(meaning that the director field and velocity profiles are independent of
time, see Appendix A). As we progressively reduce confinement or increase
activity, we find four distinct dynamic steady states as follows. First, we
observe a corotating (coherently-flowing) state with a train of defects that
are stabilized by the $-\frac{1}{2}$ defects staying near the inner curved
wall. This state is unique to the annulus geometry. Second, we observe a
circulating state, which exhibits a vortex structure similar to a corotating
disk state but with significant perturbations. Third, we observe the dancing
state that is well known to occur in channel-like geometries Shendruk  _et
al._ (2017). However, within this regime we observe a behavior that is unique
to the annulus geometry, in which the curvature of the inner wall drives and
asymmetry of defect motions in opposite directions, leading to a net
‘curvature-induced drift’. Finally, at low confinement or high activity we
observe a chaotic state.

We observe that a sufficiently small hole in the disk, while changing the
topology, does not change the dynamical steady state of the active nematic. We
present scaling analysis that suggests that this threshold results from a
competition between the anchoring energy and elastic energy of the nematic.
Consistent with this analysis, the threshold pinhole size below which the
system dynamics is insensitive to topology decreases with increasing anchoring
strength. Thus, the behavior of a pinhole annulus can be globally switched by
tuning the boundary anchoring or its geometry (pinhole size). Lastly, we also
explore the effect of boundary anchoring on the steady states, finding that a
large anchoring strength destabilizes the $-\frac{1}{2}$ defects in the
corotating state. Probing the influence of the anchoring on the boundary layer
in bulk systems, we find that activity depletes the boundary layer. This
finding explains a previous observation that the director fields of active
nematics confined in disk geometries are insensitive to boundary conditions
Norton  _et al._ (2018), and furthermore extends that analysis to additional
geometries. Similar analyses can be applied to any geometry in which
topological constraints imposed by the boundaries complete with the preferred
global arrangement of the director field.

We note that our model does not capture all the states that are observed in
experiments of microtubule-kinesin active nematics in annulus geometries Zarei
_et al._ (2023). Additional features of the experimental system that could be
added to the model include the following. In the experiments the microtubule
nematic floats at an oil-water interface and is thus coupled to two isotropic
fluids, while the theory approximates the system as a single component fluid.
While the no-slip velocity boundary conditions of the theory are consistent
with the oil/water fluid, the microtubules can slip at the boundaries.
Further, the microtubule bundles have finite thickness and length, and thus
can act as material lines that exert non-local forces Opathalage  _et al._
(2019). In the continuum model, the director field only exerts stress locally.
Extending the model to incorporate these effects could address the
discrepancies between theory and experiment.

## Conflicts of interest

There are no conflicts to declare.

###### Acknowledgements.

This work was supported by the National Science Foundation (NSF) DMR-1855914
(CJ and MFH), DMR-2202353 (AB) and the Brandeis Center for Bioinspired Soft
Materials, an NSF MRSEC DMR-2011846 (CJ, AB, and MFH). Computing resources
were provided by the NSF XSEDE/ACCESS allocation MCB090163 (Stampede and
Expanse) and the Brandeis HPCC which is partially supported by NSF DMR-MRSEC
2011846 and OAC-1920147.

##  Appendix A Classification of steady-states

To aid the classification of the steady-states, we define a signed order
parameter indicating the degree of flow,
$\Phi(t)=\expectationvalue{\mathbf{u}\cdot\hat{\mathbf{e}}_{\theta}/|\mathbf{u}|}$
following Opathalage et al. Opathalage  _et al._ (2019). Using this order
parameter and other aspects of the dynamics, we categorize the steady states
as follows:

#### Stationary state

Figure 10: Various stationary states ($\partial_{t}\mathbf{Q}=0$) of active
nematics observed in disk-like annuli. The left panel shows the director field
with streamlines and the scalar order parameter $S$ with the color. The right
panel shows the corresponding flow field, also shown as directed streamlines
and the vorticity $|\nabla\times\mathbf{v}|$ with the color. (a) State with
$\mathbf{v}\sim 0$ ($|\nabla\times\mathbf{v}|\sim 10^{-3}$) and the director
in a $+1$ defect configuration. Here, $E_{\text{A}}=3,W=4,R_{\text{i}}/W=0.1$.
(b) Dipolar state with two defects and $v\sim 0$ ($|\nabla\times v|\sim
10^{-3}$), as also observed in disks in Ref. Norton  _et al._ (2018). Here,
$E_{\text{A}}=1,W=4,R_{\text{i}}/W=0.01$. (c) For low anchoring strength, we
can get a spiral configuration of the nematic, where we get a non-zero but
constant velocity ($\mathbf{v}\neq 0,\partial_{t}\mathbf{v}=0$). Here,
$E_{\text{A}}=0.01,W=4,R_{\text{i}}/W=0.1$.

A state with a time-independent director field and velocity profile
($\partial_{t}\mathbf{Q}=0=\partial_{t}\mathbf{u}$) is labeled as a stationary
state. Depending on the confinement and anchoring conditions, we observe
different textures of the nematic, which in turn result in different velocity
profiles (Fig. 10). High confinement and high anchoring strength result in a
circular nematic profile corresponding to the $+1$ defect anchoring condition,
with $\sim 0$ velocity, as seen in Fig. 10(a). For slightly lower confinement,
we observe the dipolar state with two $+\frac{1}{2}$ defects, as also observed
in disks in Ref. Norton  _et al._ (2018) (Fig. 10(b)). An interesting state
occurs at low anchoring, where the director aquires a spiral configuration,
resulting in a stationary but non-zero velocity, as shown in Fig. 10(c). This
state is also peculiar because it has not been observed in the experiments to
our knowledge.

#### Corotating state

A state with defects revolving around the center of the annulus in a steady
circulation is labeled a corotating state. Another defining property of the
corotating state is that its $\mathbf{Q}$ and $\mathbf{u}$ profiles are
constant in a rotating (or translating in the case of straight channels) frame
of reference with the appropriate angular (or horizontal) speed.

#### Circulating state

A state with net circulation ($\Phi(t)$ remaining far from 0 and not changing
sign, see Fig. 4) but with steady nucleation and annihilation of defects is
labeled a circulating state. This is unlike the corotating state where the
defects, once formed, remain in motion without annihilating.

#### Dancing state

A state with pairs of defects swimming along vortices, as extensively
characterized in Shendruk et al. Shendruk  _et al._ (2017).

#### Chaotic state

A state exhibiting proliferation (steady nucleation and annihilation) of
defects, with no net flow ($\expectationvalue{\Phi(t)}=0$) is labeled a
chaotic state.

#### Arrested bend instability

A state with the nematic undergoing bend instability, but unable to flow or
exhibit the dancing pattern due to high confinement.

##  Appendix B Defect detection and tracking

To locate the defects, we compute a map of the signed winding number
$w=1/(2\pi)\oint\nabla\theta\cdot\differential{\vec{s}}$ at every point in
space Kamien (2002); Norton  _et al._ (2018) with an integration ring of
radius of 5 pixels. The winding number is zero everywhere except at the defect
locations DeCamp  _et al._ (2015); Ellis  _et al._ (2018). To eliminate
spurious defects, we filter out regions with a non-zero winding number that
are smaller than 60 squared pixels in area.

To plot the defect trajectories in Fig. 5, the $+\frac{1}{2}$ defects are
tracked using the open source software Trackpy Allan  _et al._ (2021) using a
search_range value of 20 pixels. The trajectories thus obtained are further
filtered with a threshold of minimum three frames of survival.

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