††thanks: These authors contributed equally††thanks: These authors contributed
equally

# Submersed Micropatterned Structures Control Active Nematic
Flow, Topology and Concentration

Kristian Thijssen  The Rudolf Peierls Centre for Theoretical Physics,
Department of Physics, University of Oxford, 1 Keble Road, Oxford OX1 3PU, UK
Dimitrius Khaladj  Department of Physics, University of California, Merced,
CA, 95343, USA  S. Ali Aghvami  Physics Department, Brandeis University,
Waltham, MA, 02453, USA  Department of Molecular Metabolism, Harvard
University, Boston, MA, 02115, USA  Mohamed Amine Gharbi  Department of
Physics, University of Massachusetts Boston, Boston, MA, 02125, USA  Seth
Fraden  Physics Department, Brandeis University, Waltham, MA, 02453, USA
Julia M. Yeomans  The Rudolf Peierls Centre for Theoretical Physics,
Department of Physics, University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK
Linda S. Hirst<EMAIL_ADDRESS>Department of Physics, University of
California, Merced, CA, 95343, USA  Tyler N. Shendruk<EMAIL_ADDRESS>SUPA, School of Physics and Astronomy, The University of Edinburgh, Peter
Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom

###### Abstract

Coupling between flows and material properties imbues rheological matter with
its wide-ranging applicability, hence the excitement for harnessing the
rheology of active fluids for which internal structure and continuous energy
injection lead to spontaneous flows and complex, out-of-equilibrium dynamics.
We propose and demonstrate a convenient, highly tuneable method for
controlling flow, topology and composition within active films. Our approach
establishes rheological coupling via the indirect presence of fully submersed
micropatterned structures within a thin, underlying oil layer. Simulations
reveal that micropatterned structures produce effective virtual boundaries
within the superjacent active nematic film due to differences in viscous
dissipation as a function of depth. This accessible method of applying
position-dependent, effective dissipation to the active films presents a non-
intrusive pathway for engineering active microfluidic systems.

Active fluids are inherently out-of-equilibrium; they locally transform
internal energy into material stresses that can result in spontaneous,
hydrodynamic motion. An increasing number of biophysical systems, including
colonies of bacilliform microbesDell’Arciprete  _et al._ (2018); You  _et al._
(2018); Li  _et al._ (2019); van Holthe tot Echten  _et al._ (2020), cellular
monolayersDuclos  _et al._ (2017); Saw  _et al._ (2017); Kawaguchi  _et al._
(2017); Duclos  _et al._ (2018); Pérez-González  _et al._ (2019), and
subcellular filamentsZhang  _et al._ (2017); Huber  _et al._ (2018); Kumar
_et al._ (2018) display such collective active motion, orientational order and
topological singularities. Controlling active dynamics is essential not only
to fully understanding how such biological systems employ self-generated
stresses but also in order to develop active-microfluidic devices.

To this end, recent work considers how confining wallsNorton  _et al._ (2018);
Opathalage  _et al._ (2019); Hardoüin  _et al._ (2020), arrangements of
obstaclesReinken  _et al._ (2020); Zhang  _et al._ (2020) and the dynamics of
topological defectsTan  _et al._ (2019) dictate active nematic flow. Control
of active material concentration has been studied from the perspectives of co-
existence of phases in self-propelled rodsBertin  _et al._ (2015); Nagel  _et
al._ (2020); Bär  _et al._ (2020) and motility-induced phase separationvan der
Linden  _et al._ (2019); Caprini  _et al._ (2020); Großmann  _et al._ (2020).
Controlled accumulation and depletion of active matter has been engineered in
bacterial systems to concentrate cellsMahmud  _et al._ (2009); Katuri  _et
al._ (2018) and to drive bacterial-ratchet motorsAngelani  _et al._ (2009);
Sokolov  _et al._ (2010); Pietzonka  _et al._ (2019). Similarly, substrate
gradients modify cellular motility driving density variationSunyer and Trepat
(2020) and directed migrationNovikova  _et al._ (2017); Schakenraad  _et al._
(2020).

In addition to varying concentration and flow, topology has been controlled by
including externally driven flowsSokolov  _et al._ (2019); Norton  _et al._
(2020); Rivas  _et al._ (2020) and curvatureEllis  _et al._ (2018); Pearce
_et al._ (2019). Recent work shows that locally altering activity modifies
defect populationsZhang  _et al._ (2021); Shankar and Marchetti (2019); Tang
and Selinger (2021); Mozaffari  _et al._ (2021), and anisotropic smectic
sublayers below active nematic sheets can constrain orientationGuillamat  _et
al._ (2017). Such studies demonstrate how underlying sublayer properties have
pronounced effects on active dynamics and suggest approaches for engineering
control of active matter.

We propose a new micropattern-based method for controlling active nematic
dynamics without contiguous contact with active films. By patterning oil-
submersed solid substrates below 2D active nematic films with geometrical
structures of differing height, we achieve effective virtual boundaries within
active films that control topological defect populations, collective flow and
concentration of active nematic material without penetrating the film. By
implementing underlying submersed patterned microstructures, we tune the depth
of the oil layer to adjust dissipation within the superjacent film and thereby
generate a highly tuneable technique for controlling the active dynamics.
Presently, we introduce four initial submersed structures: micropatterned
trenches (Fig. 1a-c), undulated substrates (Fig. S1), stairways (Fig. 1d-f),
and pillars (Fig. 1g-i).

Figure 1: Submersed micropatterns control active nematic dynamics. (a-c)
Trench set-up. An active film resides at the oil-water interface above
different substrate depths. The active flows drag the underlying oil layer but
viscous dissipation is depth dependent, affecting active nematic film
dynamics. (b) Fluorescence microscope image of the active nematic bundled-
microtubule film above a submersed trench. Scale bar $=250\upmu\textrm{m}$.
(c) Simulation results for vorticity field within the superjacent active
nematic layer. The flow behaviors within the low-friction region (between the
dashed lines) are distinct from the behaviour in the high friction region
(beyond the dashed lines). Plus-half (minus-half) defects denoted by dark
green (magenta) symbols behave differently in the two regions. (d-f) Stairway
set-up. (e) Fluorescence microscope image of micro-milled stairway and the
superjacent bundled-microtubule film. Scale bar $=250\upmu\textrm{m}$. Step
location indicated by dashed lines. The oil depth increases from left to
right. The differences in oil depth alter the length scale of the active
turbulence above each step. (f) Simulations results for discrete steps in the
effective friction (dashed lines). The effective friction coefficient
decreases from left to right. Colourbar is shared with (c). (g-i) Pillar set-
up. (h) Fluorescence microscope image of the bundled-microtubule film above
the SU-8 micropillar. Scale bar $=100\upmu\textrm{m}$. (i) Simulation results
show the active nematic concentration $\phi$ is depleted within the high
friction region encircled by the pillar perimeter (dashed line).

### Trench

To investigate how structures fully submersed in a layer of oil influence
defect dynamics in the superjacent active film, we consider the trench
geometry depicted in Fig. 1a. An active nematic microtubule network is
generated at an oil-water interface above a micropatterned trench of depth
$\Delta_{t}=18\pm 1\upmu\textrm{m}$ and width $w_{t}=327\pm 2\upmu\textrm{m}$
fabricated using photolithography (Materials and Methods). We observe that
flows in the active nematic layer exhibit coexistence of two distinct regions:
one directly above the trench and another in the shallows surrounding the
trench (Fig. 1b). These regions are separated by well-defined virtual boundary
lines located directly above the trench edges. The trench edges are visible as
a pair of parallel white lines due to stress-induced auto-fluorescence of the
micropatterned photoresist in regions of precipitous height change. Beyond the
trench boundaries, the active nematic retains the chaotic nature of active
turbulence; however, within the boundaries, the trench width establishes a
local confining length scale within the superjacent active nematic film (Movie
S1). These virtual walls trap defects in the trench region and produce active
flow behaviours comparable to those observed in confining channelsShendruk
_et al._ (2017); Doostmohammadi  _et al._ (2017); Chandragiri  _et al._
(2019); Opathalage  _et al._ (2019); Hardoüin  _et al._ (2019). This result
illustrates how such effective virtual boundaries can be used to define areas
of orderly flows and areas of active turbulence without penetrating the active
film.

The $\pm$1/2 topological defect distributions across the trench (Materials and
Methods) demonstrate that $-1/2$ defects tend to be located in the vicinity of
the virtual boundary (Fig. 2a). Experimental observations of $-1/2$ defect
trajectories near the boundary reveal that they tend to linger over long
intervals, contributing to peaks (Movie S1). In contrast, $+1/2$ defects tend
to be depleted from the vicinity of the trench boundary and are confined
within the trench region, moving along oscillatory trajectories that do not
typically approach the boundaries (Movie S1). In the exterior region, far from
the virtual trench, the defect density profile approaches a homogeneous
distribution of positive and negative defects.

The effective virtual boundaries arise from abrupt steps in fluid depth
$h\left(\boldsymbol{r}\right)$ between the film and the underlying substrate
at each point $\boldsymbol{r}$. The fluid depth $h$ increases from $h_{s}$ in
the surrounding shallows to a trench depth $h_{t}=h_{s}+\Delta_{t}$ (Fig. 1a).
As activity drives flows within the nematic film, the underlying oil layer
viscously dissipates momentum due to the subjacent no-slip substrate, which
can be described as a local effective friction $\gamma(\boldsymbol{r})$ acting
on each point within the superjacent active filmPismen and Sagués (2017).
Following from the lubrication limit, the effective friction coefficient
scales as $\gamma\sim\eta^{\prime}/h\left(\boldsymbol{r}\right)$, where
$\eta^{\prime}$ is an effective viscosity of the film and surrounding fluids.
The abrupt height change across the trench boundaries results in sharp,
virtual boundaries.

Figure 2: Positive defects depleted at the trench interface  Panels with
shaded grey backgrounds denote experimental results, while non-shaded
backgrounds denote numerical results. (a-b) Distribution of $+1/2$ (dark
green) and $-1/2$ (magenta) defects as a function of transverse position $x$
measured from experiments and simulations respectively. (c-d) The probability
profile that the nematic director is oriented less than $10^{\circ}$ from the
direction parallel to the trench wall as a function of $x$. The director field
has a high probability of alignment with the virtual boundary. The gap in data
at the trench edge is explained in Materials and Methods. (e-f) Normalised
root mean square fluid velocity profile across the trench. Experimental
results in (e) are calculated from two different identical trenches. The
experimentally measured maxima are
$\\{4.00,1.68,1.57,1.00,1.12\\}\upmu\textrm{m}/s$. Colours in (c) and (e)
denote different experimental iterations.

We replicate the observed experimental phenomena with 2D active nemato-
hydrodynamic simulations, in which the submersed micropatterns are
incorporated via an effective friction field (Materials and Methods).
Numerical results demonstrate that a step in effective friction can reproduce
the experimentally observed active flows (Fig. 1c) and introduce virtual
boundaries in the active layer, which repel $+1/2$ defects (Fig. 2a-b).
Integrating the defect density across the channel in Fig. 2b gives net zero
charge; however, it is more challenging to accurately identify minus defects
than positive in experiments. As a result, this labeling bias causes a slight
net positive charge in Fig. 2a. The qualitative agreement in behaviour between
experimental and numerical defect distributions demonstrates how effective
friction is the mechanism by which micropatterned structures create virtual
planar boundaries and introduce a confinement length scale to the active
nematic without penetrating the film. The $-1/2$ defect density peak at the
virtual boundary (Fig. 2a) is consistent with workOpathalage  _et al._ (2019);
Hardoüin  _et al._ (2019) showing walls can act as catalysts for pair creation
and unbinding: while newly created $+1/2$ defects move away due to self-
propulsion, the $-1/2$ defects remain near the boundary.

While the experimental $-1/2$ defect density peaks sharply in the vicinity of
the virtual boundaries (Fig. 2a), it is broadened and peaked outside the
trench region in simulations (Fig. 2b). To understand this difference, we
consider the time-averaged director orientation across the trench (Materials
and Methods; Fig. 2c), which reveals that the virtual boundaries introduce an
effective alignment of the director as the probability of aligning to the
interface becomes different from the bulk, similar to that seen for
impermeable boundariesOpathalage  _et al._ (2019); Hardoüin  _et al._ (2019).
This is because any orthogonal bundle midway over the boundary is subject to a
large axial laminar flow inside and slow disorderly flows outside, which
compete to produce an aligning torque. The model captures this behaviour,
showing that the probability declines to a uniform distribution far from the
trench (Fig. 2d). Experiments exhibit stronger planar alignment at the virtual
boundaries than simulations. This is likely related to the model’s assumption
of a continuous fluid (Materials and Methods), which is in contrast to the
network of finite-sized microtubule bundles that act as material lines
preventing defects from crossingOpathalage  _et al._ (2019), resulting in an
accumulation.

This is reminiscent of simulation studies of passive nematic tactoids where
defects travel outside the interface and become virtual unless an anchoring
term is includedMetselaar  _et al._ (2017). Future research is needed to more
fully understand if an effective term can be devised to account for this
effect in continuum models that do not contain sharp $\phi$ interfaces. The
stronger alignment in the experiments constrains the $-1/2$ defects to the
region inside the trench, while in simulations they are pushed to the outside
of the boundaries (Fig. 2a-b). In both experiments and simulations, $+1/2$
defects are trapped between the virtual boundaries.

The submersed trench not only impacts the nematic field but also generates a
virtual boundary for the velocity field (Fig. 2e-f). Superjacent to the
trench, velocities are lower in the proximity of the trench boundary and
maximum at the trench centre. Since activity varies slightly between
experimental realisations, we normalise the flow profiles (Fig. 2e) and
compare to the decrease predicted by the model (Fig. 2f). The virtual
boundaries do not impose no-slip conditions but decrease the speed to the
slower value of the surrounding active turbulence, which can be seen
explicitly for an experimental instance (blue dots; (Fig. 2e)) and in
simulations (Fig. 2f). The decreasing flow profile explains the preferential
alignment of the microtubule bundles in the vicinity of the virtual boundaries
(Fig. 2c-d). Any orthogonal bundle midway over the boundary is subject to a
large, axial, laminar flow inside and slow disorderly flows outside, which
compete to produce an aligning torque.

Comparing the faster flow profile above the trench to the slower, disorderly
active turbulence in the exterior region calls attention to the fact that
active turbulence is a low-Reynolds number phenomenonAlert  _et al._ (2020).
Confinement screens chaotic flows that would otherwise develop on scales
larger than the trench width, while the low friction allows the rapid-but-
steady flow profiles superjacent to the trench (Fig. 1b-c). On the other hand,
the higher friction produces a smaller characteristic length scaleThijssen
_et al._ (2020a), but also slower speeds in the shallows. Thus, the submersed
micropatterned trench segregates rapid laminar flow above the trench and slow-
but-disorderly active turbulence outside.

Figure 3: Friction boundaries result in separate flow regions. (a)
Instantaneous experimental vorticity superjacent the deep trench region from
Particle image velocimetry (PIV) for a trench of width
$w_{t}=325\upmu\textrm{m}$. Panels with shaded grey backgrounds denote
experimental results, while non-shaded backgrounds denote numerical results.
(b) Simulation snapshot of a repeating lattice of counter-rotating vortices
above a trench of width $w_{t}=1.7R$ (Materials and Methods). Plus-half
defects (dark green) are trapped between the virtual boundaries, generating
the repeating vortex structure along the centre line that is distinct from the
active turbulence that exists outside the virtual boundaries. (c) Decreasing
the trench width to $w_{t}=1.2R$ in simulations results in long-range,
oscillatory, streaming flow inside the trench. (d) Increasing the trench width
to $w_{t}=6R$ in simulations results in active turbulence both inside and
outside the trench region, but with differing intrinsic length scales due to
the different effective frictions. (e) The velocity-velocity autocorrelation
function $C_{v}(\delta
y)=\left\langle\boldsymbol{v}(\boldsymbol{r};t)\cdot\boldsymbol{v}(\boldsymbol{r}+\delta
y\ \boldsymbol{\hat{y}};t)\right\rangle/\left\langle v^{2}\right\rangle$ for
the experiment illustrated in (a), measured a distance $\delta y$ along the
$x=0$ centre line of the trench. Due to the confinement effect, long-range
flow structures are formed in the low-friction regime epitomised by the strong
correlation-anti-correlation-correlation signal. (f) Autocorrelation functions
calculated from the simulations inside the channel shown in (b) (blue), (c)
(red) and (d) (yellow), and the shallow region outside the virtual channel
(dashed). The blue curve displays pronounced correlation and anti-correlation
indicating the counter-rotating vortex pattern in (b), which corresponds to
the behaviour observed in the presented experiments of (a,e). The red curve is
long-lived and fully correlated in the narrow channels of (c), while the
yellow curve decorrelates to zero after an anti-correlation, signalling active
turbulent behaviour both within and outside the virtual boundaries. (g) The
director angle probability density at the friction boundaries of (b-d). The
solid (dashed) lines correspond to measurements at the right (left) friction
boundary. The black lines are the analytical Leslie angles
($\theta_{\text{L}}=22^{\circ}$).

Because the submersed micropatterned trench produces virtual boundaries that
introduce a confining length scale, the competition between confinement and
intrinsic active nematic length scale can be probed. Movie S1 and Fig. 3a
experimentally demonstrate that a recurrent vorticity structure is established
between the virtual boundaries when active and confining scales
coincideShendruk  _et al._ (2017); Opathalage  _et al._ (2019); Zhang  _et
al._ (2020) and simulations underscore the periodicity of counter-rotating
vorticies (Fig. 3b and Movie S2). Examining the velocity autocorrelation
functions quantifies the different flow profiles above the trench (Fig. 3e and
f; blue curve). The correlation function exhibits repetition between
correlated and anti-correlated regions due to repeating clock-wise and anti-
clockwise vortices. Active turbulence exists outside of the virtual channel,
as characterised by an immediate initial drop in the correlation (Fig. 3f;
dashed curve).

In narrower or wider confinements, the flow transitions to other states (Fig.
3c-f). In simulations of the narrow trench (Fig. 3c), the flows are long-
ranged and oscillatory with a preference for aligning with the boundaries
(Fig. 3f). This oscillatory-streaming state occurs when the confining length
scale $w_{t}$ is small compared to the low-friction intrinsic active nematic
length scaleShendruk  _et al._ (2017). Increasing the trench width predicts
active turbulence in both the area superjacent to the trench and the shallow
exterior regions (Fig. 3d) but with differing active nematic length scales
(Fig. 3f). Equivalently higher levels of activity decrease the active length
scale and we have experimentally observed instances of disorderly flow states
in both regions. The structure of the different flow states impacts the
effective director alignment observed in Fig. 2d. For streaming flow which is
dominated by a simple shearing flow with small fluctuations, we find the most
likely angle the director makes with the virtual boundary, as seen in Fig. 3g.
These angles are distributed about the Leslie angle
$\theta_{\text{L}}=22^{\circ}$ (Materials and Methods), furthering our
argument that the preferred orientation is due to flow alignment. In the wider
trenches, the director varies widely due to defect creation eventsHardoüin
_et al._ (2019), but we still see a symmetry breaking as perpendicular
alignment to the interface is disfavoured due to the aligning torque. The
relative position of the $-1/2$ defect peak to the interface does not change
as we alter the trench width.

The trench geometry demonstrates that submersed microstructure patterning can
impose confining virtual boundaries and is a feasible technique for
maintaining coexistence of distinct flow behaviours simultaneously at
different locations in a single active nematic layer. A strength of our
submersed-micropatterned-structure approach is that the boundaries act without
physically penetrating the film and so active material does not first have to
saturate a cavity before confinement dynamics can be exploredOpathalage  _et
al._ (2019). Filling complex geometries with filament-based active material
may be the prohibitive step in active microfluidicsKempf  _et al._ (2019). The
proposed micropatterned method circumvents these difficulties, opening
possibilities for experiments involving more complex geometries and fine-tuned
positional control.

### Sinusoid Substrates

While the trench geometry demonstrates that micropatterns with precipitous
edges can actualise abrupt boundaries within superjacent active films, the
ability to gradually tune the effective friction through gradients in oil-
layer thickness allows our technique to gently guide defect dynamics. To
demonstrate this, an undulating effective friction is produced by fully
submersing a micropatterned 1D sinusoidal substrate (Materials and Methods)
characterised by amplitude $\Delta_{u}=40\pm 2\upmu\textrm{m}$ and wavelength
$\lambda_{u}=150\pm 2\upmu\textrm{m}$. Unlike the trench geometry, the
sinusoid system does not separate into distinct coexisting flow states (Movie
S3). Rather, the resulting anisotropic friction gradients present a means of
orientation-control of motile defects. Self-propelled $+1/2$ defects orient
and travel in trains above the troughs (Fig. S1a). Motile defects move through
the system subject to friction gradients when they have components
perpendicular to the troughs, such that trajectories co-aligned with the
troughs minimise dissipation, causing the observed parallel/anti-parallel
laning of $+1/2$ defects. Similar trains have been observed in active nematic
layers above smecticsGuillamat  _et al._ (2016a) but, since the spacing
between smectic layers is significantly smaller, the defect trains in that
configuration are not caused by gradients in effective friction; rather, they
are due to uniform anisotropic friction.

However, such trains of +1/2 defects do not persist indefinitely since the
trains produce nematically ordered regions, which are susceptible to the
extensile-active nematic hydrodynamic-bend instabilityAditi Simha and
Ramaswamy (2002); Giomi  _et al._ (2011); Martínez-Prat  _et al._ (2019)
causing pair creation events that inevitably destabilise the flow (Movie S3).
Since initially unbound $+1/2$ defects are typically oriented perpendicular to
the nematic ordered lanes (Fig. S1b), this produces a crosshatched trajectory
pattern. These dynamics are not nearly as pronounced in sinusoid systems with
a larger wavelength ($\lambda_{u}=500\upmu\textrm{m}$; Movie S4). Positive
defects that are partially oriented along the friction gradient exhibit the
same deflected motion as in the smaller wavelength system and so the motile
defects show some alignment along the troughs but the crosshatched dynamics
are indiscernible. The sinusoid geometry demonstrates that submersed
micropatterned structures can fine-tune the active flow and nematic structure,
thereby offering a means to guide and control defect dynamics.

### Stairs

Figure 4: Submersed micropatterned stairway allows simultaneous coexistence of
regions of separated active turbulence with differing defect densities, while
adenosine triphosphate (ATP) concentration remains constant. Plus-half (dark
green) and minus-half (magenta) defect distribution as a function of position
down the stairway $x$. (a) Measurements from experiments for steps of width
$w_{t}=500\upmu\textrm{m}$ (grey shaded background). (b) Measurements from
simulations for steps of width of width $w_{t}=5R$.

We now present a substrate patterned as a submersed stairway (Fig. 1d)
designed to simultaneously observe active turbulence and gradations in the
characteristic length scales (Movie S5). Individual steps are micromilled to
possess horizontal width $w_{s}=500\pm 2\upmu\textrm{m}$ and height of
$\Delta_{s}=10\pm 1\upmu\textrm{m}$ (Materials and Methods). The fluid depth
is $h\left(\boldsymbol{r}\right)=h_{0}+\Delta_{s}n\left(x\right)$, for step
number $n$ and initial fluid depth $h_{0}=12\pm 3\upmu\textrm{m}$, determined
through confocal microsopy (Fig. S2; Materials and Methods). We focus on steps
$7\leq n\leq 9$ for which the microtubule network forms a well-defined,
continuous nematic field (Fig. 1e; Movie S5). As the depth increases with $n$,
the effective dissipation within the oil-layer decreases, which we simulate
via discrete steps in effective friction in the superjacent active film (Fig.
1f; Materials and Methods).

Above the step pattern, the active length scales increase with decreasing
friction, as characterised by the defect distribution (Fig. 1e-f; Fig. 4a;
Movie S5). Within each step the distribution is flat. However, at each edge,
the number density of $-1/2$ defects peaks, while the density of $+1/2$
defects plunges. This is consistent with defect densities at the edges of the
trench (Fig. 2a-b). Though the simulated edge peaks are less pronounced than
in experiments, numerical results show more clearly the decrease in defect
density across multiple steps. The change in defect density is modest,
consistent with studies demonstrating that increasing oil viscosity five
orders of magnitude only increases defect density by a factor of order
unityGuillamat  _et al._ (2016b); Martínez-Prat  _et al._ (2021), which
highlights the potential tunability of our method. Because the effective
friction is inversely proportional to oil depth, changes to the defect density
become less pronounced with step number and, in the large-oil-depth limit,
substrate features become indiscernible within the nematic flows.

Interestingly, we only observe a continuously well-defined nematic field for
oil depths that are much greater than $h_{0}$ in both experiments and
simulations (Fig. 1e). We could reliably resolve the nematic order and measure
defect density with low experimental uncertainty for steps $n\geq 7$. For
small $n$, the active film exhibits disorderly nonhomogeneous textures akin to
those observed in experiments utilising high viscosity oilsGuillamat  _et al._
(2016b). This suggests that submersed micropatterned structures can do more
than impact flow and orientational state: we now demonstrate how substrate
micropatterning can be used to control active matter concentration via
structures raised above the solid substrate yet still fully submersed in the
underlying oil-layer.

### Pillars

We consider fully submersed SU-8 pillar structures (Fig. 1g) of radius
$r_{p}=116\pm 2\upmu\textrm{m}$ and height $h_{p}=6.8\pm 0.3\upmu\textrm{m}$.
As in the trench, sinusoid and stairway geometries, the active nematic layer
is subject to a step-change in the effective film friction. However,
differentiating it from previous structures, the pillar’s virtual boundary
forms a closed loop. The most prominent effect is a pronounced dilution of
active material from the enclosed region above the pillar (Fig. 1h; Movie S6),
which is qualitatively recapitulated in the simulations (Fig. 1i; Movie S7;
Materials and Methods). The phase-field active material concentration
$\phi\left(\boldsymbol{r};t\right)$ (Materials and Methods) demixes in the
high effective friction region directly above the pillar. We confirm that the
absence of active material is not a result of the pillar intruding through the
active nematic film by occasionally observing microtubule bundles moving above
the apex of the pillar (Movie S8). Further, we did not observe curvature of
the superjacent active layer in the vicinity of the micropillar, indicating
that a finite oil depth lies between the subjacent pillar top and the nematic
film.

To understand the mechanism leading to the pillar-bound dilute phase of active
matter, we consider a simplified model of the active nematic. The effective
friction is locally large above the pillar, causing the flow speed to decrease
in the enclosed area but remain non-zero beyond the pillar border (Fig. 5a).
Since nematic ordering arises in active microtubule network films due to
activity-induced motion, the sharp decrease in flow causes a corresponding
drop in nematic scalar order $S$ across the circular virtual border (Fig. 5b).
However, the abrupt gradient in $S$ produces a radially outward average active
force
$\boldsymbol{f}(r;t)\sim\left\langle\boldsymbol{\nabla}\cdot\mathbf{Q}(\boldsymbol{r};t)\right\rangle\approx\partial_{r}\zeta(r;t)S(r;t)\boldsymbol{\hat{r}}$
(Eq. 6 of Materials and Methods), when the variation of the nematic order is
dominated by the radial change in scalar order parameter and bend-induced
stresses are neglected. Thus, the active forcing is expected to be radially
outward and sharply peaked about the interface as observed in simulations
(Fig. 5c). Interestingly, this argument suggests that contractile systems
still deplete from the pillar region: While the $\zeta$ term will change sign,
so will the $\partial_{r}S(r;t)\boldsymbol{\hat{r}}$ since contractile flows
lower the nematic orderThampi  _et al._ (2015). This is observed in
simulations (Fig. S3).

However, activity does not simply produce increased pressure across the
perimeter but rather is able to selectively deplete the concentration of
active material $\phi$, which enables depletion in incompressible films. Since
the active film is considered incompressible, fluid mass density is constant
and hence divergence of the film velocity is zero (Eq. 3 of Materials and
Methods). Thus, for depletion to occur, it is demanded that outward advection
is more frequent in regions where $\phi$ is larger on average. This is indeed
the case because the activity depends on the local amount of active material
present, $\zeta(r;t)=\zeta_{0}\phi(r;t)$ (Eq. 7 of Materials and Methods),
which causes the radially outward forcing to be stronger in magnitude where
$\phi$ is large. For this reason, if the surrounding active turbulence
stochastically advects $\phi$-rich active material across the perimeter, the
local active forces increase in kind. Hence, the active forces selectively
repulse the material from the dissipative region (Fig. 5d; Movie S7). However,
the local active forces decrease when $\phi$-poor fluid enters. This allows
the concentration to more easily cross the perimeter. In this way, the
depletion of active matter above the pillar is a result of the high effective
friction lowering the velocity. This causes nematic discomposure and
selectivity due to the direct dependence of activity on the local
concentration (Fig. 5e).

Noting that the highest steps in the stairway geometry also fail to exhibit
continuous nematic fields but are not devoid of active matter (Fig. 1d; Movie
S5), we test if the curvature of the virtual barriers impacts depletion by
simulating a rectangular pillar (Fig. S4). We observe a comparable depletion
of $\phi$ from the enclosed area as in the circular pillar and so conclude
that curvature is not the critical difference. Secondly, we consider a
circular pit where the friction is zero for radius $r<R$ and $\gamma=0.1$ for
$r>R$. We find the active forces point inwards and observe an accumulation of
$\phi$ (Fig. S5). These inward active forces can even be used to trap two
$+1/2$ defects for small radii, resulting in circulatory motion reminiscent of
an active nematic in circular confinement (Fig. S6)Opathalage  _et al._
(2019).

We conclude that accumulation or depletion of active material using submersed
micropatterned structures relies principally on two attributes: (i) The oil-
layer must be thin (high effective friction) to suppress the active flows
necessary to exhibit nematic order. (ii) An enclosed area must be
circumscribed by a virtual boundary to prohibit longitudinal active streams
through the incompressibility constraint. Hence, we only find minute changes
in $\phi$ to occur in the trench and stair geometries. Lastly, we point out
that material does not deplete from the pillar centre in simulations on
timescales investigated when a pillar radius is much larger, since the
material does not diffuse to the interface where it could be selectively
depleted.

In addition to controlling concentration, submersed pillars interact with
defects. We observe a greater frequency of $-1/2$ defects in the vicinity of
the virtual boundary in our simulations (Fig. 5f-g). The planar alignment of
the director field explains the distribution of defects at the pillar boundary
(Fig. 1h-i). The resulting bend deformation around the perimeter drives
hydrodynamic instabilities to continually generate defect pairs, with newly
created self-motile $+1/2$ defects typically oriented radially away from the
centre such that they swiftly move away from the interface (Movie S9), leaving
unbound immotile $-1/2$ defects behind Fig. 5f-g).

Submersed pillars can also serve as a virtual obstacle for defect trajectories
(Movie S9). Positive defects that approach the pillars from the surrounding
turbulence stall or are deflected once in proximity to the pillar (Fig. 5h-j).
In Fig. 5h and j, we exhibit nine representative trajectories out of the 38
analyzed, and we show four from a total of 360 in Fig. 5i to illustrate the
observed interactions. Deflected $+1/2$ defects first slow as they approach
the pillar, then scatter and regain speed as they move away from the submersed
structure (Fig. 5h-i). Positive defects that stall as they approach the
boundary temporarily hold their position before annihilating with pillar-
associated $-1/2$ defects (Fig. 5j). While positive defects that directly
approach the pillar can temporarily enter the depleted area by driving active
material ahead of them, such infrequent events are transient (Movie S8) as the
repulsive active force (Fig. 5c) pushes such incursions radially outward.

Figure 5: Pillars cause local high friction regions which result in active
matter depletion.  (a) Due to the higher friction, the flow in the active
nematic film remains low. (b) The nematic is highly ordered far from the
pillar but disordered above it because the speed is lower in the higher
friction region. (c) The difference in nematic order at the friction interface
results in a radial active force. (d) This radial active force pushes the
active material concentration outwards resulting in depletion effects. Panels
with non-shaded backgrounds denote numerical results, while shaded grey
backgrounds denote experimental measurements. (e) Schematic of a-d. (f-g)
$+1/2$ (dark green) and -1/2 (magenta) defect distribution as a function of
radial distance from a submersed pillar from experiments (f) and simulations
(g). Dark green and magenta lines denote $+1/2$ and $-1/2$ defects,
respectively. (h-j) xy trajectories of example $\pm$1/2 defect dynamics in the
vicinity of the pillar. Time along the trajectory is displayed as circular
markers coloured blue at the initial time and changing to red at the final
instant. (h-i) $+1/2$ defects deflecting from the pillar; experiments (h) and
simulations (i). (j) Experimental xy trajectories of $+1/2$ defects absorbing
to the pillar boundary through defect annihilation with $-1/2$ residing in the
vicinity of the perimeter.

## Conclusions

Using a combined experimental and simulation approach, we have demonstrated
that micropatterns fully submersed in an underlying oil layer can guide the
flow, topology, and even concentration of active material in superjacent
nematic films without direct contact. By imposing changes in substrate depth,
viscous dissipation in the oil layer enacts a position-dependent effective
friction coefficient on the active material. Abrupt substrate height steps can
constitute sharp virtual boundaries in the active matter layer, which can
control flow and defect behaviour. As proof-of-concept systems, we presented
virtual channels exhibiting coexisting flow states, sinusoid substrates that
gently guide defects, stairways separating active turbulence with differing
characteristic length, and a virtual enclosure depleted of active material
that acts as an obstacle scattering nearby defects.

The proposed technique of fully submersing micropatterned structures
facilitates new approaches for fabricating complex active topological
microfluidic devices. For example, complications associated with infiltration
of active nematics into confined spaces could be avoided, and active dynamics
in various geometries at the same activity could be compared directly.
Furthermore, locally concentrating or depleting active material could regulate
activity at constant levels of ATP or rheological properties such as film
viscosity or nematic elastic coefficient through this novel approach.

## Materials and Methods

### Formation of the Active Nematic Network

An active nematic microtubule network is generated following the protocol
previously reported by Sanchez et al. at an oil-water interfaceSanchez  _et
al._ (2012). Prior to experiments, active premixtures are prepared in
$3.73\upmu\textrm{L}$ aliquots containing biotin-labeled K401 kinesin motors,
streptavadin, Pyruvate Kinase/Lactic Dehydrogenase (PKLDH), phosphoenol
pyruvate (PEP) (used for adenosine triphosphate (ATP) regeneration),
$4\textrm{mg/ml}$ glucose, $0.27\textrm{mg/ml}$ glucose oxidase,
$47\upmu\textrm{g/ml}$ catalase and $2\textrm{mM}$ Trolox, and $6\%$ (w/v)
$20\textrm{kD}$ polyethylene glycol (PEG) in M2b buffer ($80\textrm{mM}$
piperazine-N,N’-bis(2-ethanesulfonic acid) (Pipes) $\textrm{pH}6.8$,
$2\textrm{mM}$ MgCl2, and $1\textrm{mM}$ ethylene glycol-
bis($\beta$-aminoethyl ether)-N,N,N’,N’-tetraacetic acid (EGTA)). To prevent
photobleaching during imaging, premixtures contain $6.65\textrm{nM}$
dithiothreitol (DTT), an antioxidant solution. Once the premix is prepared,
the aliquots are flash frozen in liquid nitrogen and promptly stored at
$-80^{\circ}\textrm{C}$ for future use.

To perform active nematic experiments, $1\textrm{mM}$ ATP (final
concentration) is added to the $3.73\upmu\textrm{L}$ premixture aliquot
followed by $2\upmu\textrm{l}$ of $6\textrm{mg/ml}$ ($3\%$) Alexa Fluor 647
labeled guanylyl-(alpha, beta)-methylene-diphosphonate (GMPCPP) microtubules
(for a final concentration of $1\textrm{mg/ml}$). For fluorescence imaging,
microtubules are fluorescently labelled with Alexa Fluor 647Sanchez  _et al._
(2012). The result is a suspended active nematic with a total volume of
$6\upmu\textrm{L}$. We use an ATP concentration at saturation, i.e. the local
microtubule extension speed was maximised. To confine the active nematic at an
oil/water interface, we follow the previously published procedureSanchez  _et
al._ (2012); Tan  _et al._ (2019). We first create a flow cell made from the
glass substrate with patterned structures, double-sided tape and a coverslip
treated with a polyacrylamide brush (Fig. 1). The polyacrylamide brush
prevents excess protein binding to the coverslip.

We flow in an oil/surfactant mixture (Novec HFE 7500 Engineered Fluid with
$1.8\%$ PFPE-PEG-PFPE (perfluoropolyether) surfactant) into the channel. Then,
this mixture is exchanged with the active microtubule network. We ensure that
the amount of oil in the channel is constant over time after filling by
sealing the flow cell after preparation. For filling consistency, each cell is
prepared using identical spacers and volumes of oil; however, some variation
in oil thickness is expected due to microfabrication tolerances. This system
forms a 3D, unconfined active microtubule network. Streptavidin can bind up to
four biotin-labeled kinesin molecules and when microtubules of opposing
polarities align parallel to each other, the kinesin molecules oriented at
$180^{\circ}$ to each other walk in opposite directions along those
neighbouring microtubules. As the kinesins walk, the filaments produce an
extensile motion driven by ATP hydrolysis. The ends of the flow cell are
sealed using a UV-curable glue (RapidFix). The active layer is then
centrifuged using a swinging bucket rotor for 42 min at $300\textrm{rpm}$. The
dynamic viscosity of the oil layer is $1.24\times
10^{-3}\textrm{Pa}\cdot\textrm{s}$ and the film viscosity of the active
nematic gel has previously been found to be $\sim
10^{-3}\textrm{Pa}\cdot\textrm{s}\cdot\textrm{m}$Guillamat  _et al._ (2016b).

### Photolithography

The trench and pillar geometries are produced using photolithography. SU-8
(MicroChem Corp.) is a negative tone epoxy-based photoresist that is used to
create thin film plastics on substrates. SU-8 is composed of epoxy-based
monomers and photo-acid generators (PAGs) suspended in a solvent. Upon
exposure to ultra-violet (UV) light, the PAGs release acids serving as a
catalyst for cross-linking available epoxy groups on the monomer once heat is
applied to the substrate. Prior to fabrication, glass substrates are
thoroughly cleaned in soap and water followed by 30 minutes of sonication in
acetone, methanol and ethanol in this order. The glass substrates are rinsed
in nanopure water to ensure minimal presence of surface contaminants. The
glass is then plasma treated with oxygen for 2 minutes. To ensure the removal
of residual moisture left on the surface, the glass substrate is placed on a
hotplate for 5 minutes at $200^{\circ}\textrm{C}$; the substrate is left to
cool down to room temperature for 5 minutes in a humidity-controlled
environment.

Upon thorough cleaning, a quarter-sized drop of SU-8 50 is deposited on the
glass substrate. The SU-8 is spin-coated at $2000\textrm{rpm}$ for 45 seconds
followed by a 10 minute wait. The substrate is then soft baked at
$65^{\circ}\textrm{C}$ for 12 minutes then at $95^{\circ}\textrm{C}$ for 45
minutes on a hotplate to evaporate solvent. The substrate is again left to
cool to room temperature with another wait of 10 minutes. The film is then
exposed to $365\textrm{nm}$ UV light ($500\textrm{mJ/cm}^{-2}$) followed by a
10-minute wait step. To crosslink the exposed regions’ epoxy groups, the
substrate undergoes a post exposure bake for 5 minutes at
$65^{\circ}\textrm{C}$ then for 15 minutes at $95^{\circ}\textrm{C}$ on a hot
plate. After the 15 minute bake, the hot plate is turned off and allowed to
cool to room temperature without the removal of the coated substrate. This
step is to avoid thermally shocking the film which can result in cracks and
poor adhesion. The substrate is developed for 30 minutes with gentle
agitations. Once developed, the residual SU-8 developer is rinsed with
isopropanol and de-ionised water then dried with nitrogen gas. After
development, the substrate is then hard baked for 2 minutes at
$150^{\circ}\textrm{C}$. Internal stresses within the substrate due to the
abrupt change in surface height are apparent near the edge of the trench as
parallel lines in Fig. 1b. This accounts for the gap in data at the trench
edge in Fig. 2c. We note that some local residual SU-8 is present at the base
of the pillar in Fig. 1h, but this material does not appear to affect the
active nematic flow dynamics. The heights of the microstructures are measured
using profilometry.

### Micro-milling

The stepped substrate and the sinusoid surface are produced using the computer
numerical controlled (CNC) micromilling. We use the milling machine TN5-V8-TC8
(MDA Precision) to fabricate the microstructures from poly(methyl
methacrylate), commonly known as acrylic. This milling machine is capable of
handling drills and endmills as small as $50\upmu\textrm{m}$ in diameter and
has a spindle rotational accuracy of around $2\upmu\textrm{m}$. The first
designed system has steps (height), each with a rise of $\Delta_{s}=10\pm
1\upmu\textrm{m}$ and a run (horizontal width) of $w_{s}=500\pm
2\upmu\textrm{m}$. The system is composed of ten steps. The smaller of the two
sinusoid systems has undulations with a peak-to-peak amplitude
$\Delta_{u}=40\pm 2\upmu\textrm{m}$ and a wavelength $\lambda_{u}=150\pm
2\upmu\textrm{m}$, while the larger has a peak-to-peak amplitude
$\Delta_{u}=50\pm 2\upmu\textrm{m}$ and a wavelength $\lambda_{u}=150\pm
2\upmu\textrm{m}$. After milling, these features are used as a mold and
transferred into a soft elastomer, polydimethylsiloxane (PDMS), which in turn
is used as a mold again to transfer the features into a cyclic olefin
copolymer (COC), using a thermopress. The COC sheets of thickness ranging from
$150\upmu\textrm{m}$ to $350\upmu\textrm{m}$ are then immersed for one hour in
a solution of 10wt% of 8-Anilino-1-naphthalenesulfonic acid (ANS), a
hydrophobic dye, dissolved in a mixture of 85wt% ethanol and 15wt% decalin.
This treatment adds a fluorescent layer of ANS dye to the COC surface, which
allows us to use a 944 Leica SP8 UV/Visible Laser Confocal Microscope to
measure the fluorinated oil’s thickness under the active nematic. The confined
active nematic is imaged using a wide-field fluorescence Nikon Eclipse Ti-E
microscope with an Andor Clara camera controlled by Micromanager open-source
software.

### Data processing

To investigate how defect dynamics in the active layer are influenced by
submersed structures, labeled microtubule bundles are imaged using
fluorescence microscopy. Four-hundred-frame videos are collected at 1 frame
per second and processed using Fiji/ImageJ version 1.52a software. To acquire
defect distributions, active nematic microtubule defects are identified and
counted manually every 10 frames for each video. Two-dimensional Cartesian
components for both x and y axes are acquired from both +1/2 and -1/2 defects
using the Click-Coordinates-Tool on Fiji/ImageJ. For the submersed trench
geometry, we use MATLAB to analyse the frequency and position for both $+1/2$
and $-1/2$ defects across the channel. Defects are organised and binned in
$10\upmu\textrm{m}$ horizontal increments across the field of view.

For the stairway geometry and pillar geometries, we apply the same counting
procedure to obtain $+1/2$ and $-1/2$ defect positional frequencies across all
frames. In the stairway geometry, we apply the counting procedure sequentially
to each step and, for the pillar, the videos are processed by centring the
pillar in a $400\times 400\upmu\textrm{m}^{2}$ window. Similar to the analysis
done with the submersed trench, we use MATLAB to generate a 2D histogram to
represent the frequency of the $+1/2$ and $-1/2$ defects, positionally
distributed in a 2D plane. Defects are organised and binned every
$13.4\upmu\textrm{m}$ in both horizontal and vertical increments.

The nematic director field inside the channel was calculated using a Fourier
transform-based method reported recently by our groupTan  _et al._ (2019). To
measure the director orientation directly above trenches, the imaged active
nematic is oriented with the long axis parallel to the y-axis. Each pixel is
converted to micrometers ($0.9434\upmu\textrm{m}$/pixel). Each pixel from each
frame containing a x-component, y-component and angle of the bundle-
microtubule director is represented on a 2D grid and determined using MATLAB.
Our process uses a nested loop and a conditional statement to determine if the
angle is between $80$ and $90^{\circ}$; if a director satisfied this condition
along the y-axis for the specified x-position, the total is summed then
divided by the total number of angles checked by the loop. This probability is
appended to a new horizontal array for each probability in the x-position. The
result is a time-averaged director orientation mapped across the trench,
averaging over all y values.

### Simulations

To complement the experiments, we simulate the active nematic thin film using
a 2D hybrid lattice Boltzmann/finite difference approach. The incompressible
active nematic film flows with velocity
$\boldsymbol{u}\left(\boldsymbol{r};t\right)$, has long-range orientational
order described by the tensor order parameter
$\boldsymbol{Q}\left(\boldsymbol{r};t\right)$ and varying concentrations of
active material, which we take to be a phase field
$\phi\left(\boldsymbol{r};t\right)$ varying from 0 to 1, coarsely describing
the local amount of active materials (microtubules, kinesin complexes and
ATP). The total free energy includes nematic bulk (LdG) and deformation (FO)
terms, as well as a binary mixture bulk (DW) and an interfacial (I) term;
$\mathcal{F}\left[\boldsymbol{Q},\phi\right]=\int
d^{2}\boldsymbol{r}\left(f_{\text{LdG}}+f_{\text{FO}}+f_{\text{DW}}+f_{\text{I}}\right)$.
Four coupled equations describe the time evolution of the continuous fields.

The first is the Beris-Edwards equation for nematics,

$\displaystyle\left(\partial_{t}+\boldsymbol{u}\cdot\boldsymbol{\nabla}\right)\boldsymbol{Q}-\boldsymbol{S}$
$\displaystyle=$ $\displaystyle\Gamma_{Q}\boldsymbol{H}.$ (1)

The co-rotation term
$\boldsymbol{S}=\left(\xi\boldsymbol{D}+\boldsymbol{\Omega}\right)\left(\boldsymbol{Q}+\frac{1}{3}\boldsymbol{I}\right)+\left(\boldsymbol{Q}+\frac{1}{3}\boldsymbol{I}\right)\left(\xi\boldsymbol{D}-\boldsymbol{\Omega}\right)-2\xi\left(\boldsymbol{Q}+\frac{1}{3}\boldsymbol{I}\right)\text{tr}\left(\boldsymbol{Q}\boldsymbol{W}\right)$
determines the alignment of the microtubules in response to gradients in the
velocity field, with $\boldsymbol{\Omega}$ the rotational part and
$\boldsymbol{D}$ is the extensional part of the velocity gradient tensor
$\boldsymbol{W}=\boldsymbol{\nabla}\boldsymbol{v}=\boldsymbol{\Omega}+\boldsymbol{D}$.
The alignment parameter $\xi$ is taken to be in the flow aligning regime and
set to $\xi=0.5$. The resulting Leslie angleThijssen  _et al._ (2020b);
Thijssen and Doostmohammadi (2020) is
$\theta_{\text{L}}=\tfrac{1}{2}\cos^{-1}\left(\tfrac{\xi(3S+4)}{9S}\right)$.

The molecular field
$\boldsymbol{H}=-(\frac{\delta\mathcal{F}}{\delta\boldsymbol{Q}}-\frac{1}{3}\boldsymbol{I}\;\text{Tr}\frac{\delta\mathcal{F}}{\delta\boldsymbol{Q}})$
is a functional derivative of free energy density $\mathcal{F}$, describing
the relaxation towards equilibrium at a rate $\Gamma_{Q}$. The free energy
depends directly on the nematic tensor $\boldsymbol{Q}$ and the active
material concentration $\phi$. The nematic part consists of a Landau-De Gennes
contribution,
$f_{\text{LdG}}=A_{0}\left\\{\tfrac{1}{2}\left(1-\tfrac{\nu}{3}\right)\text{tr}\left[\boldsymbol{Q}^{2}\right]-\tfrac{\nu}{3}\text{tr}\left[\boldsymbol{Q}^{3}\right]+\tfrac{\nu}{4}\text{tr}\left[\boldsymbol{Q}^{2}\right]^{2}\right\\}$
with $A_{0}=0.05$, and Frank-Oseen deformation
$f_{\text{FO}}=\tfrac{K}{2}\left(\boldsymbol{\nabla}\boldsymbol{Q}\right)^{2}$
with $K=0.02$. We set $\nu=2.55$, which favours the isotropic state in the
absence of active flowsThampi  _et al._ (2015) and is independent of
$\phi\left(\boldsymbol{r};t\right)$. This choice allows nematic ordering only
due to activity-induced flows, in agreement with experimentsSanchez  _et al._
(2012) and previous simulationsThampi  _et al._ (2015); Srivastava  _et al._
(2016). Throughout the simulations, the nematic order $S$ is found to be
between $0$ and $0.3$. For the trench data of Fig. 3c, we find $S=0.18$ and
calculate a Leslie angle of $\theta_{\text{L}}=22^{\circ}$.

The active material concentration evolves according to a Cahn-Hillard model.
We assume that the film is incompressible and active material concentration
$\phi$ does not impact fluid mass density $\rho$.

$\displaystyle\partial_{t}\phi+\boldsymbol{\nabla}\cdot\left(\boldsymbol{u}\phi\right)$
$\displaystyle=$ $\displaystyle\Gamma_{\phi}\nabla^{2}\mu,$ (2)

where
$\mu=\tfrac{\delta\mathcal{F}}{\delta\phi}-\boldsymbol{\nabla}\cdot\left(\tfrac{\delta\mathcal{F}}{\delta\boldsymbol{\nabla}\phi}\right)$
is the chemical potential and $\Gamma_{\phi}=0.1$ is a mobility coefficient.
The double-well free energy density depends on phase as
$f_{\text{DW}}=\tfrac{A_{\phi}}{2}\phi^{2}\left(\phi-1\right)^{2}$ and an
interfacial term
$f_{\text{I}}=\tfrac{K_{\phi}}{2}\left(\boldsymbol{\nabla}\phi\right)^{2}$
with $A_{\phi}=0.03$ and $K_{\phi}=0.1$. The free energy minima are at
$\phi=\\{0,1\\}$. While positive $A_{\phi}$ favours phase separation, activity
suppresses demixing and $\phi$ does not phase separate into high and low
$\phi$ regions for sufficiently active flows. In our model, active flows above
high friction regions (such as within the perimeter of the pillar geometry)
can become sufficiently small for spinodal decomposition to occur. This does
not happen in the low friction region. We have verified that depletion in the
pillar geometry also occurs for a binary mixture free energy modelled using a
single-well, which would not thermodynamically phase separate. We initialise
the concentration to $\phi=0.5$. All reported phenomena are due to dynamical
interactions, since the free energy does not favour global nematic ordering
nor phase separation.

Lastly, the system obeys the Navier-Stokes equations for the velocity field
within the active film. Assuming constant fluid mass density $\rho$ (not
active material concentration $\phi$) leads to the incompressibility condition

$\displaystyle\boldsymbol{\nabla}\cdot\boldsymbol{u}$ $\displaystyle=$
$\displaystyle 0.$ (3)

Because the experimental active nematic film lies on a 2D interface between
two 3D fluids, the planar divergence of the velocity could conceivably be non-
zero. However, non-zero divergent velocity fields would require continuous
circulation in the thin aqueous/oil layers above/below the film. As we are
unaware of evidence indicating such flows, we assume 2D incompressibility
implying divergence-free flow. Since the fluid is taken to be incompressible,
any outward fluid mass flux into an enclosed area must be balanced by an
inward flux and vice versa. If this constraint is relaxed, we allow flow
sources in our film and non-selective outward pointing forces
$\zeta\partial_{r}S(r;t)\boldsymbol{\hat{r}}$, where the $\zeta$ is
independent of $\phi$, will also deplete the material.

The second equation is the Cauchy momentum equation

$\displaystyle\rho\left(\partial_{t}+\boldsymbol{u}\cdot\boldsymbol{\nabla}\right)\boldsymbol{u}$
$\displaystyle=$
$\displaystyle-\boldsymbol{\nabla}p+\boldsymbol{\nabla}\cdot\boldsymbol{\Pi}-\gamma\boldsymbol{u},$
(4)

where $p$ is the pressure and $\boldsymbol{\Pi}$ is the stress tensor which
includes the standard viscous stress
$\boldsymbol{\Pi}^{\textmd{visc}}=2\eta\boldsymbol{E}$ for film viscosity
$\eta=2/3$. Furthermore, it contains the elastic stress due to the nematic
nature of the microtubules

$\displaystyle\boldsymbol{\Pi}^{\text{elastic}}$
$\displaystyle=2\xi\boldsymbol{\mathcal{Q}}(\boldsymbol{Q}:\boldsymbol{H})-\xi\boldsymbol{H}\cdot\boldsymbol{\mathcal{Q}}-\xi\boldsymbol{\mathcal{Q}}\cdot\boldsymbol{H}$
$\displaystyle\quad-\boldsymbol{\nabla}\boldsymbol{Q}:\frac{\delta\mathcal{F}}{\delta\boldsymbol{\nabla}\boldsymbol{Q}}+\boldsymbol{Q}\cdot\boldsymbol{H}-\boldsymbol{H}\cdot\boldsymbol{Q},$
(5)

where $\boldsymbol{\mathcal{Q}}=\boldsymbol{Q}+\boldsymbol{I}/3$. The stress
also contains the capilary stresses
$\boldsymbol{\Pi^{\text{cap}}}=(\mathcal{F}-\mu\phi)\boldsymbol{I}-\boldsymbol{\nabla}\phi\left(\frac{\delta\mathcal{F}}{\delta\boldsymbol{\nabla}\phi}\right)$
due to differences in concentration $\phi$, and the active component

$\displaystyle\boldsymbol{\Pi^{\text{act}}}$
$\displaystyle=-\zeta\left(\boldsymbol{r};t\right)\boldsymbol{Q}=-\zeta_{0}\phi\left(\boldsymbol{r};t\right)\boldsymbol{Q}.$
(6)

Here, we have taken the local activity

$\displaystyle\zeta\left(\boldsymbol{r};t\right)=\zeta_{0}\phi\left(\boldsymbol{r};t\right)$
(7)

to scale directly with active material concentration. We set $\zeta_{0}=0.09$.
While the velocity and director fields can in principle develop out-of-plane
components in the nematic film, in experiments and for the parameter values
used in this study, no out-of-plane components developDoostmohammadi  _et al._
(2017). Comparison to strictly two-dimensional simulations has confirmed the
qualitative agreementHemingway  _et al._ (2016).

In the Navier-Stokes equation, we also include an effective friction
$\gamma\left(\boldsymbol{r}\right)$. Employing an effective friction that
varies with oil-layer depth neglects any recirculation effects within the oil
layer, which is justified by the facts that the micropatterned structures are
characterized by small height/width ratios and the Reynolds number of flows
within the oil layers are infinitesimal. Hence, typical recirculation vortices
around pillars are neglectableBaykal  _et al._ (2015). Since previous studies
have successfully modelled the dynamics of microtubule/kinesin-based active
nematic films with weak effective frictionZhou  _et al._ (2021), we treat the
friction as negligible in the regions of the film superjacent to deep
structures. In the regions above shallows, we include lubrication momentum
dissipation via a non-zero effective friction coefficient. We set
$\gamma(h_{s})=0.07$ in the shallows for simulations of the trench system. We
define a characteristic confinement scale to be $R\equiv 20$ lattice Boltzmann
(LB) nodes. This is comparable to the intrinsic active nematic length scale
and, as a result, the reported defect densities are of order unity. The
narrowest trench width is $w_{t}=1.2R$ (Fig. 3c) and the two wider trenches
have widths of f $w_{t}=1.7R$ (Fig. 3b) and $6R$ (Fig. 3d), respectively. All
trenches are simulated in $7.5R\times 30R$ periodic systems. For the stairway,
we simulate a $7.5R\times 50R$ long system composed of the same number of
steps as in the experimental system (10 steps) of width $w_{s}=5R$. Each step
represents a different lubrication momentum dissipation region with different
non-zero effective friction coefficients set to
$\gamma(h_{s},n)=\frac{0.1}{1+n}$, where $n\in\\{0\ldots 9\\}$ denotes the
different steps corresponding to different oil depths in the experiments. In
the main text, we present simulation results from steps $n=\\{1,2,3,4,5\\}$.
For the pillars, we use a radius $r_{p}=R$. The friction coefficient is chosen
to be sufficiently large to clearly match the pronounced depletion observed
experimentally in Fig. 5. We present results for the value $\gamma(h_{p})=0.5$
but observe depletion of $\phi$ for $\gamma(h_{s})=0.07$, corresponding to the
choice used for the trench geometry. The pillar is simulated in a $10R\times
10R$ periodic system.

Defects are detected by calculating the winding numberHobdell and Windle
(1997). If the average nematic order $S$ in a range of 3 LB nodes around a
defect is lower then $0.05$, we assign this defect to be virtual and ignore
it.

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## Acknowledgements

We thank Amin Doostmohammadi for helpful discussions. We acknowledge generous
funding from the National Science Foundation, through several awards. LSH and
DAK thank DMR-1808926 and NSF-CREST: Center for Cellular and Biomolecular
Machines at UC Merced (HRD-1547848), LSH, DAK, SF and MG are grateful to the
Brandeis Biomaterials facility MRSEC-2011486. K.T. acknowledges funding from
the European Union’s Horizon 2020 research and innovation programme under the
Marie Sklodowska-Curie grant agreement no. 722497 (LubISS) and the EPSRC grant
EP/T031247/1. T.N.S received funding from the European Union’s Horizon 2020
research and innovation programme (grant agreement no. 851196).

## Supplementary Information

### Sinusoidal Microstructures

To demonstrate that the method of submersed micropatterned substrates in the
oil layer below an active nematic film can guide defect dynamics, we produced
a series of micropatterned sinusoidal substrates (Materials and Methods). The
smallest of these are characterised by amplitude $\Delta_{u}=40\pm
2\upmu\textrm{m}$ and wavelength $\lambda_{u}=150\pm 2\upmu\textrm{m}$. As
observed in Movie Smov:undulSmallnd discussed in the main text, the undulating
effective friction produces orientation-control of motile defects for smaller
wavelengths. Trains of co-aligned +1/2 defects move along the troughs (Fig.
S1a). However, these are intermittently disrupted by the extensile-active
nematic hydrodynamic-bend instability. Regions of well-ordered nematic aligned
along the troughs result in pair creation events in which +1/2 defects unbind
with an orientation perpendicular to the troughs (Fig. S1b).

FIG. S 1: Submersed micropatterned sinusoidal substrate. Fluorescent image of
the active nematic film superjacent to an oil-submersed sinusoidal
micropatterned structure with a $150\upmu\textrm{m}$ wavelength and
$40\upmu\textrm{m}$ amplitude. The undulations go from left to right such that
the parallel troughs and crests run in parallel up and down. Scale bar
$=250\upmu\textrm{m}$. (a) A set of three motile +1/2 defects that move in a
train up the middle of a trough are highlighted in green. The train aligns the
director parallel to the trench. (b) $15$ seconds later the initial train is
destabilised by extensile-active nematic hydrodynamic-bend instability and the
associated pair creation events in which +1/2 defects tend to unbind oriented
perpendicular to the trough.

### Confocal Microscopy Measurement of the Oil Thickness

We report on the positional defect density above submerged acrylic micro-
milled steps. Consistent oil layer thickness is controlled indirectly through
the height of the spacers and the oil’s filling volume. The initial thickness
of fluorinated oil layer in the stairway geometry is $h_{0}\approx 12\pm
3\upmu\textrm{m}$ followed by $10\upmu\textrm{m}$ deep increments along the
x-direction. Confocal microscopy provides a 3D reconstruction of the active
film to demonstrate that the active layer is without deformation. This
verifies that the active layer itself is consistently flat, even above
micropatterned structures. The solid substrate supporting the fluorinated oil
is also observed to be smooth and flat in the absence of micropatterns. The
difference between the positions of the oil interface and solid substrate,
together with the refractive index of the oil, allows us to estimate the
thickness of oil.

FIG. S 2:  Confocal microscopy measurement of the fluorinated oil’s thickness.
(a) Confocal fluorescent images of the active nematic film, (b) the solid
substrate supporting the fluorinated oil, and (c) the active nematic film near
a step. The thickness of oil $h_{0}\approx 12\pm 3\upmu\textrm{m}$ is obtained
by measuring the difference between the heights of the active nematic film and
the solid substrate $(z_{top}-z_{bottom})\approx 8.75\upmu\textrm{m}$,
multiplied by the refractive index of the oil $n_{oil}\approx 1.3$. (d) Near
the steps, the 3D reconstruction of the active nematic confirms the film is
flat without any deformation. Scale bars $=100\upmu\textrm{m}$.

### Depletion in Contractile Active Nematic Systems

In the main text, we explain the selective depletion of active material $\phi$
from the region superjacent to the pillar by considering the active force
$\boldsymbol{f}(r;t)\sim\left\langle\boldsymbol{\nabla}\cdot\mathbf{Q}(\boldsymbol{r};t)\right\rangle\approx\partial_{r}\zeta(r;t)S(r;t)\boldsymbol{\hat{r}}$
over the radial interface. For the extensile active nematic bundled-
microtubule films considered in the text, the positive activity $\zeta$ and
actively ordered surroundings, produce a radially outward force. In
contractile systems ($\zeta<0$), it has been established that the nematic
order $S$ will decrease due to active flows. Hence, both factors change sign
and our argument still predicts depletion for such contractile systems because
the product does not change sign. We verify this argument numerically (Fig.
S3) in simulations with activity $\zeta=-0.03$ and free energy parameter
$\nu=3$, which results in nematic order in the absence of activity. We find
that the nematic order $S$ decreases across the pillar perimeter (Fig. S3; red
line) and that depletion from the region above the pillar still occurs (Fig.
S3; purple line).

FIG. S 3: Contractile stresses still result in depletion. A contractile system
($\zeta=-0.03$) with a free energy parameter $\nu=3$. The friction is
$\gamma=0.5$ for $r<20$ and zero outside. The purple line is the active
nematic concentration $\phi$ and the red line is the nematic order $S$. With
this value of free energy parameter $\nu$, the equilibrium nematic ordering
$S_{eq}\neq 0$ and decreases due to contractile flows outside the pillar. The
blue dashed line is a visual guide to help the reader identify the change in
$S$.

### Rectangular Microstructure Pillar

Simulations of rectangular pillar (Fig. S4) exhibit a comparable depletion of
concentration $\phi$ from the enclosed area above the pillar as the circular
pillars considered in the main text (Fig. 5). Hence, we conclude that a pillar
curvature is not the critical property leading to depletion in the pillar
geometry. Rather, the abrupt gradient in $S$, which follows indirectly from
the increased effective friction, produces an outward average active force
that can deplete active material from any enclosed area of high effective
friction.

FIG. S 4: Rectangular pillar causes depletion. Same as Fig. 1i but with the
circular pillar replaced with a square, the perimeter of which is traced by
dashed lines. We observe a comparable degree of depletion of active material
concentration $\phi$, denoted by the colormap, for the same friction
coefficient $\gamma=0.5$.

### Microstructure Pit

In the main text, we argue that active material is depleted from the enclosed
region directly above a microstructured pillar due to the increased effective
viscous dissipation within the locally thinner oil-layer. Thus, one naturally
wonders if a microstructured pit in the substrate might act as an “anti-
pillar” and lead to accumulation of active material within an enclosed region
above the structure. We consider an effective circular pit by simulating a
substrate with an effective friction except directly above the structure (Fig.
S5) and observe that active material accumulates in the enclosed area. The
velocity magnitude, nematic scalar order parameter, and concentration directly
above the pit (Fig. S5) are all comparable to their values far from the pillar
(Fig. 5,a,b,d respectively). While the change in the scalar order parameter is
positively peaked at the perimeter of the pillar (Fig. 5c), it is seen to dip
at the perimeter of the pit (Fig. S5). This indicates a radially inward
average active force, consistent with the mechanism which here leads to
accumulation.

FIG. S 5: Pits cause local lower friction regions which result in active
matter accumulation. We consider a circular pit of radius $2R$ above which the
frictional damping is negligible but outside of which $\gamma=0.1$. This
friction term is lower than for the pillar geometry presented in the
manuscript. This is necessary to ensure spinodal decomposition does not occur
outside the geometry. We measure the same profiles as Fig. 5a-d for one
instant after steady state has been achieved. We observe accumulation of
active material $\phi$ in the enclosed area.

### Microstructure Pits Can Trap Defects

Depletion occurs due to radially outwards active forces, such that the region
superjacent to micropatterned pillars can act as obstacles for approaching
defects. On the other hand, the pit geometry produces a radially inward
average active force (Fig. S5). Together, these two facts suggest that pits
may trap defects within their perimeters. By varying the pit radius such that
it is comparable to the intrinsic active nematic length scale in simulations,
we indeed find a regime where two $+1/2$ defects can be trapped in the
interior of the pit for extended periods and rotate around each other (Fig.
S6). Similar behaviour has been observed for pairs of defects trapped by
impermeable circular confining walls.

FIG. S 6: Defects can be trapped in the pit area due to the inwards active
forces. If the pit radii is sufficiently small compared to the intrinsic
active nematic length scale, we find two trapped $+1/2$ circulating around
each similar to active nematics confined in circular geometries.

### Movie Captions

Movie S1. Micropatterned trench geometry. Fluorescence microscopy video of a
bundled microtubule/kinesin network at $1\textrm{mM}$ ATP at the oil-water
interface above a submersed SU-8 substrate with a micropatterned trench. The
trench possesses a depth $8\pm 1\upmu\textrm{m}$ and width $w_{t}=327\pm
2\upmu\textrm{m}$. Scale bar $=250\upmu\textrm{m}$. Left and right are shallow
regions which exhibit active turbulence, while the centre is the deep region,
the edges of which form a well-defined virtual boundary that traps defects.

Movie S2. Simulation results of a repeating lattice of counter-rotating
vortices above a trench. The trench width is $w_{t}=1.7R$. Plus-half defects
(dark green) are trapped between the virtual boundaries, generating the
repeating vortex structure along the centre line that is distinct from the
active turbulence that exists outside the virtual boundaries. Minus-half
defects are shown in magenta. In the colormap, blue (red) denotes clock (anti-
clock) wise rotating vortices.

Movie S3. Micropatterned sinusoid geometry. Fluorescence microscopy video of a
bundled microtubule/kinesin active nematic network above an oil-submersed
sinusoidal microstructure with a $150\upmu\textrm{m}$ wavelength and a
$40\upmu\textrm{m}$ amplitude. The undulations go from left to right such that
the parallel troughs and crests run in parallel up and down. Red lines
indicate the crest location. Scale bar $=150\upmu\textrm{m}$. Train of +1/2
defects move up and down the troughs but are unstable to intermediate pair
creation and unbinding events oriented perpendicular to the troughs.

Movie S4. Micropatterned sinusoid geometry. The same as Movie
Smov:undulSmallut for a microstructure with a $500\upmu\textrm{m}$ wavelength
and a $50\upmu\textrm{m}$ amplitude. Red lines indicate the crest location.
Scale bar $=250\upmu\textrm{m}$. Distinct trains of +1/2 are not as
immediately apparent, though self-propelled defects tend to align with the
troughs.

Movie S5. Micropatterned stairway geometry. Fluorescence microscopy video of a
bundled microtubule/kinesin film over an oil-submersed, micropatterned
stairway. Individual steps are $500\upmu\textrm{m}$ wide and
$10\upmu\textrm{m}$ tall. The oil depth from left to right deepens from
$h-h_{0}=\\{50,60,70,80,90\\}\upmu\textrm{m}$ sequentially where the initial
fluid depth is $h_{0}=12\pm 3\upmu\textrm{m}$.

Movie S6. Micropatterned pillar geometry. Fluorescence microscopy video of a
bundled microtubule/kinesin network at the oil-water interface with
$1\textrm{mM}$ ATP above a substrate possessing a micropatterned pillar of
radius $r_{p}=116\pm 2\upmu\textrm{m}$ and height $h_{p}=6.8\pm
0.3\upmu\textrm{m}$. The bundled microtubule/kinesin network in the film
directly above the immersed pillar is fully depleted. Annihilation events can
be observed as +1/2 approach the boundary and annihilate with perimeter-
associated -1/2 defects.

Movie S7. Simulation results of the active nematic film above a pillar
illustrating depletion of active material. Active matter concentration $\phi$
(denoted by colormap) is initialised uniformly but eventually depletes from
the high friction region inside the circle $(\gamma=0.5)$. The director field
is illustrated with black lines and the plus-half defects and minus-half
defects are denoted by dark green and magenta circles, respectively.

Movie S8. Micropatterned pillar with +1/2 defect entering depletion zone. The
same as Movie Smov:pillarExput exhibiting an instance of active material in
the superjacent film crossing the boundary above the pillar.

Movie S9. Simulation results of defect dynamics around a pillar that acts as a
virtual obstacle. Plus-half defects (dark green) deflect from the depleted
high-friction region or are absorbed by minus-half defects (magenta) when
approaching the pillar structure.