Submersed Micropatterned Structures Control Active Nematic Flow, Topology and Concentration
Kristian Thijssen, Dimitrius Khaladj, S. Ali Aghvami, Mohamed Amine Gharbi, Seth Fraden, Julia M. Yeomans, Linda S. Hirst, Tyler N. Shendruk
aa r X i v : . [ c ond - m a t . s o f t ] F e b Submersed Micropatterned Structures Control Active NematicFlow, Topology and Concentration
Kristian Thijssen, ∗ Dimitrius Khaladj, ∗ S. Ali Aghvami, Mohamed Amine Gharbi, Seth Fraden, Julia M. Yeomans, Linda S. Hirst, † and Tyler N. Shendruk ‡ The Rudolf Peierls Centre for Theoretical Physics, Department of Physics,University of Oxford, 1 Keble Road, Oxford OX1 3NP, UK Department of Physics, University of California, Merced, CA, 95343, USA Qlibrium, Woburn, MA, 01801, USA Department of Physics, University of Massachusetts Boston, Boston, MA, 02125, USA Physics Department, Brandeis University, Waltham, MA, 02453, USA SUPA and School of Physics and Astronomy, The University of Edinburgh,Peter Guthrie Tait Road, Edinburgh EH9 3FD, United Kingdom
Coupling between flows and material properties imbues rheological matter with its wide-rangingapplicability, hence the excitement for harnessing the rheology of active fluids for which internalstructure and continuous energy injection lead to spontaneous flows and complex, out-of-equilibriumdynamics. We propose and demonstrate a convenient, highly tuneable method for controlling flow,topology and composition within active films. Our approach establishes rheological coupling via theindirect presence of fully submersed micropatterned structures within a thin, underlying oil layer.Simulations reveal that micropatterned structures produce effective virtual boundaries within thesuperjacent active nematic film due to differences in viscous dissipation as a function of depth. Thisaccessible method of applying position-dependent, effective dissipation to the active films presentsa non-intrusive pathway for engineering active microfluidic systems.
Active fluids are inherently out-of-equilibrium; theylocally transform internal energy into material stressesthat can result in spontaneous, hydrodynamic mo-tion. An increasing number of biophysical systems,including colonies of bacilliform microbes[1–4], cellularmonolayers[5–9], and subcellular filaments[10–12] displaysuch collective active motion, orientational order andtopological singularities. Controlling active dynamics isessential not only to fully understanding how such bio-logical systems employ self-generated stresses but also inorder to develop active-microfluidic devices.To this end, recent work considers how confiningwalls[13–15], arrangements of obstacles[16, 17] and thedynamics of topological defects[18] dictate active nematicflow. Control of active material concentration has beenstudied from the perspectives of co-existence of phasesin self-propelled rods[19–21] and motility-induced phaseseparation[22–24]. Controlled accumulation and deple-tion of active matter has been engineered in bacterialsystems to concentrate cells[25, 26] and to drive bacterial-ratchet motors[27–29]. Similarly, substrate gradientsmodify cellular motility driving density variation[30] anddirected migration[31, 32].In addition to varying concentration and flow, topol-ogy has been controlled by including externally drivenflows[33–35] and curvature[36, 37]. Recent workshows that locally altering activity modifies defectpopulation[38], and anisotropic smectic sublayers belowactive nematic sheets can constrain orientation[39]. Such ∗ These authors contributed equally † [email protected] ‡ [email protected] studies demonstrate how underlying sublayer propertieshave pronounced effects on active dynamics and suggestapproaches for engineering control of active matter.We propose a new micropattern-based method for con-trolling active nematic dynamics without contiguous con-tact with active films. By patterning oil-submersed solidsubstrates below 2D active nematic films with geomet-rical structures of differing height, we achieve effectivevirtual boundaries within active films that control topo-logical defect populations, collective flow and concen-tration of active nematic material without penetratingthe film. By implementing underlying submersed pat-terned microstructures, we tune the depth of the oillayer to adjust dissipation within the superjacent film andthereby generate a highly tuneable technique for control-ling the active dynamics. Presently, we introduce fourinitial submersed structures: micropatterned trenches( Figure 1 a-c), undulated substrates (
Figure S1 ), stair-ways (
Figure 1 d-f), and pillars (
Figure 1 g-i).To investigate how structures fully submersed in a layerof oil influence defect dynamics in the superjacent activefilm, we consider the trench geometry depicted in
Fig-ure 1 a. An active nematic microtubule network is gen-erated at an oil-water interface above a micropatternedtrench of depth ∆ t = 18 ± µ m and width w t = 327 ± µ mfabricated using photolithography ( Methods I B ). Weobserve that flows in the active nematic layer exhibitcoexistence of two distinct regions: one directly abovethe trench and another in the shallows surrounding thetrench (
Figure 1 b). These regions are separated bywell-defined virtual boundary lines located directly abovethe trench edges. The trench edges are visible as apair of parallel white lines due to stress-induced auto-fluorescence of the micropatterned photoresist in regions ch ie behf adg ee FIG. 1.
Submersed micropatterns control active nematic dynamics. (a-c)
Trench set-up.
An active film resides atthe oil-water interface above different substrate depths. The active flows drag the underlying oil layer but viscous dissipation isdepth dependent, affecting active nematic film dynamics. (b) Fluorescent image of the active nematic bundled-microtubule filmabove a submersed trench beneath. Scale bar = 250 µ m. (c) Simulation results for vorticity field within the superjacent activenematic layer. Distinct flow behaviours are found within the low friction region (between the dashed lines) and high friction(beyond the dashed lines). Plus-half (minus-half) defects denoted by dark green (magenta) symbols behave differently in thetwo regions. (d-f) Stairway set-up. (e) Fluorescent image of micromilled stairway and the superjacent bundled-microtubulefilm. Scale bar = 250 µ m. Step location indicated by dashed lines. The oil depth increases going from left to right. Thedifferences in oil depth alter the length scale of the active turbulence above each step. (f) Simulations results for discrete stepsin the effective friction (dashed lines). The effective friction coefficient decreases from left to right. (g-i) Pillar set-up. (h)Fluorescent image of the bundled-microtubule film above the SU-8 micropillar. Scale bar = 100 µ m. (i) Simulation results showthe active nematic concentration φ is depleted within the high friction region encircled by the pillar perimeter (dashed line).Active material is not observed above the pillar. of precipitous height change. Beyond the trench bound-aries, the active nematic retains the chaotic nature ofactive turbulence; however, within the boundaries, thetrench width establishes a local confining length scale within the superjacent active nematic film ( Movie M1 ).These virtual walls trap defects in the trench region andproduce active flow behaviours comparable to those ob-served in confining channels[14, 40–43]. This result illus-trates how such effective virtual boundaries can be usedto define areas of orderly flows and areas of active turbu-lence without penetrating the active film.The ± Methods I D ) demonstrate that -1/2 defectstend to be located in the vicinity of the virtual bound-ary (
Figure 2 a). Experimental observations of -1/2defect trajectories near the boundary reveal that theytend to linger over long intervals, contributing to peaks(
Movie M1 ). In contrast, +1/2 defects tend to be de-pleted from the vicinity of the trench boundary and areconfined within the trench region, moving along oscil-latory trajectories that do not typically approach theboundaries (
Movie M1 ). In the exterior region, far fromthe virtual trench, the defect density profile approachesa homogeneous distribution of positive and negative de-fects.The effective virtual boundaries arise from abruptsteps in fluid depth h ( r ) between the film and the un-derlying substrate at each point r . The fluid depth h increases from h s in the surrounding shallows to a trenchdepth h t = h s + ∆ t ( Figure 1 a). As activity drives flowswithin the nematic film, the underlying oil layer viscouslydissipates momentum due to the subjacent no-slip sub-strate, which can be described as a local effective friction γ ( r ) acting on each point within the superjacent activefilm[44]. Following from the lubrication limit, the effec-tive friction coefficient scales as γ ∼ η ′ /h ( r ), where η ′ isan effective viscosity of the film and surrounding fluids.The abrupt height change across the trench boundariesresults in sharp, virtual boundaries.We replicate the observed experimental phenomenawith 2D active nemato-hydrodynamic simulations, inwhich the submersed micropatterns are incorporatedvia an effective friction field ( Methods I E ). Numeri-cal results demonstrate that a step in effective frictioncan reproduce the experimentally observed active flows(
Figure 1 c) and introduce virtual boundaries in the ac-tive layer, which repel +1/2 defects (
Figure 2 a-b). Thisagreement between experimental and numerical defectdistributions demonstrates how effective friction is themechanism by which micropatterned structures createvirtual planar boundaries and introduce a confinementlength scale to the active nematic without penetratingthe film. The -1/2 defect density peak at the virtualboundary (
Figure 2 a) is consistent with work[14, 43]showing walls can act as catalysts for pair creation andunbinding: while newly created +1/2 defects move awaydue to self-propulsion, the -1/2 defects remain near theboundary.While the experimental -1/2 defect density peakssharply in the vicinity of the virtual boundaries(
Figure 2 a), it is broadened and peaked outside thetrench region in simulations (
Figure 2 b). To under-stand this difference, we consider the time-averaged di-rector orientation across the trench (
Methods I D ; Fig-ure 2 c), which reveals that the virtual boundaries intro-duce effective planar anchoring of the director, similar -300 -200 -100 0 100 200 30000.20.40.6 +1/2 defect-1/2 defects -2 -1 0 1 211.522.5-300 -200 -100 0 100 200 3000.10.20.3 -2 -1 0 1 20.10.20.3 bd ff -2 -1 0 1 20.20.40.60.81 f -300 -200 -100 0 100 200 300 x position ( m) N o r m a li z ed v e l o c i t y D e f e c t d i s t r i bu t i on P r ob o f a li gn m en t ea D e f e c t d i s t r i bu t i on P r ob o f a li gn m en t N o r m a li z ed v e l o c i t y x position/R FIG. 2.
Positive defects depleted at the trench in-terface
Panels with shaded grey backgrounds denote ex-perimental results, while non-shaded backgrounds denote nu-merical results throughout. (a-b) Distribution of +1/2 (darkgreen) and -1/2 (magenta) defects as a function of transverseposition x measured from experiments and simulations re-spectively. (c-d) The probability profile that the nematicdirector is oriented less than 10 ◦ from the direction paral-lel to the trench wall as a function of x . The director fieldhas a high probability of alignment with the virtual bound-ary. (e-f) Normalised root mean square fluid velocity profileacross the trench. The experimentally measured maxima are { . , . , . , . } µ m /s . to that seen for impermeable boundaries[14, 43]. Themodel captures this behaviour, showing that the prob-ability declines to a uniform distribution far from thetrench ( Figure 2 d). Experiments exhibit stronger pla-nar anchoring at the virtual boundaries than simulations,which is likely related to the model’s assumption of acontinuous fluid (
Methods I E ) compared to finite-sizedmicrotubule bundles, which act as material lines that pre-vent defects from crossing[14]. The stronger anchoring inthe experiments constrains the -1/2 defects to the regioninside the trench, while in simulations they are pushedto the outside of the boundaries (
Figure 2 a-b). In bothexperiments and simulations, +1/2 defects are trappedbetween the virtual boundaries.The submersed trench not only impacts the nematicfield but also generates a virtual boundary for the veloc-ity field (
Figure 2 e-f). Superjacent to the trench, veloc-ities are lower in the proximity of the trench boundaryand maximum at the trench centre. Since activity variesslightly between experimental realisations, we normalisethe flow profiles (
Figure 2 e) and compare to the decreasepredicted by the model (
Figure 2 f). The virtual bound-aries do not impose no-slip conditions but decrease thespeed to the slower value of the surrounding active turbu-lence. The decreasing flow profile explains the preferen-tial alignment of the microtubule bundles in the vicinityof the virtual boundaries (
Figure 2 c-d). Any orthog-onal bundle midway over the boundary is subject to alarge, axial, laminar flow inside and slow disorderly flowsoutside, which compete to produce an aligning torque.Comparing the faster orderly flow above the trench tothe slower active turbulence in the exterior region callsattention to the fact that active turbulence is a low-Reynolds number phenomenon[45]. Confinement andlow friction above the trench generate rapid-but-orderlyflows. On the other hand, the higher friction producesa smaller characteristic length scale[46], but also slowerspeeds in the shallows. Thus, the submersed micropat-terned trench segregates rapid laminar flow above thetrench and slow-but-disorderly active turbulence outside.Because the submersed micropatterned trench pro-duces virtual boundaries that introduce a confininglength scale, the competition between confinement andintrinsic active nematic length scale can be probed.
Movie M1 and
Figure 3 a experimentally demonstratethat a recurrent vorticity structure is established betweenthe virtual boundaries when active and confining scalescoincide[14, 17, 40] (
Figure 3 b and
Movie M2 ). Exam-ining the velocity autocorrelation functions quantifies thedifferent flow profiles above the trench (
Figure 3 e andf; blue curve). The correlation function exhibits repeti-tion between correlated and anti-correlated regions dueto repeating clock-wise and anti-clockwise vortices. Ac-tive turbulence exists outside of the virtual channel, ascharacterised by an immediate initial drop in the corre-lation (
Figure 3 f; dashed curve).In narrower or wider confinements, the flow transi-tions to other states (
Figure 3 c-f). In the narrow trench(
Figure 3 c), the flows are long-ranged, bidirectional andoscillatory with a preference for aligning with the bound-aries (
Figure 3 f). This oscillatory-streaming state oc-curs when the confining length scale w t is small com-pared to the low-friction intrinsic active nematic lengthscale[40]. The increased trench width allows active tur-bulence in both the area superjacent to the trench andthe shallow exterior regions ( Figure 3 d) but with differ-ing active nematic length scales (
Figure 3 f).The trench geometry demonstrates that submersedmicrostructure patterning can impose confining virtualboundaries and is a feasible technique for maintainingcoexistence of distinct flow behaviours simultaneously atdifferent locations in a single active nematic layer. Astrength of our submersed-micropatterned-structure ap-proach is that the boundaries act without physically pen-etrating the film and so active material does not first haveto saturate a cavity before confinement dynamics can beexplored[14]. Filling complex geometries with filament-based active material may be the prohibitive step inactive microfluidics[47]. The proposed micropatternedmethod circumvents these difficulties, opening possibili-ties for experiments involving more complex geometriesand fine-tuned positional control.While the trench geometry demonstrates that mi-cropatterns with precipitous edges can actualise abruptboundaries within superjacent active films, the abilityto gradually tune the effective friction through gradi-ents in oil-layer thickness allows our technique to gently guide defect dynamics. To demonstrate this, an undu-lating effective friction is produced by fully submersing amicropatterned 1D sinusoidal substrate (
Methods I C )characterised by amplitude ∆ u = 40 ± µ m and wave-length λ u = 150 ± µ m. Unlike the trench geome-try, the sinusoid system does not separate into distinctcoexisting flow states ( Movie M3 ). Rather, the re-sulting anisotropic friction gradients present a meansof orientation-control of motile defects. Self-propelled+1/2 defects orient and travel in trains above the troughs(
Figure S1 a). Motile defects move through the systemsubject to friction gradients when they have componentsperpendicular to the troughs, such that trajectories co-aligned with the troughs minimise dissipation, causingthe observed parallel/anti-parallel laning of +1/2 defects.However, such trains of +1/2 defects do not persist in-definitely since the trains produce nematically orderedregions, which are susceptible to the extensile-activenematic hydrodynamic-bend instability[48–50] causingpair creation events that inevitably destabilise the flow(
Movie M3 ). Since initially unbound +1/2 defects aretypically oriented perpendicular to the nematic orderedlanes (
Figure S1 b), this produces a crosshatched tra-jectory pattern. These dynamics are not nearly as pro-nounced in sinusoid systems with a larger wavelength( λ u = 500 µ m; Movie M4 ). Positive defects that arepartially oriented along the friction gradient exhibit thesame deflected motion as in the smaller wavelength sys-tem and so the motile defects show some alignment alongthe troughs but the crosshatched dynamics are indis-cernible. The sinusoid geometry demonstrates that sub-mersed micropatterned structures can fine-tune the ac-tive flow and nematic structure, thereby offering a meansto guide and control defect dynamics.We now present a substrate patterned as a submersedstairway (
Figure 1 d) designed to simultaneously observeactive turbulence and gradations in the characteristiclength scales (
Movie M5 ). Individual steps are mi-cromilled to possess horizontal width w s = 500 ± µ m andheight of ∆ s = 10 ± µ m ( Methods I C ). The fluid depthis h ( r ) = h +∆ s n ( x ), for step number n and initial fluiddepth h = 12 ± µ m, determined through confocal mi-crosopy ( Methods I C ). We focus on steps 5 ≤ n ≤ Figure 1 e; Movie M5 ). As thedepth increases with n , the effective dissipation withinthe oil-layer decreases, which we simulate via discretesteps in effective friction in the superjacent active film( Figure 1 f; Methods I E ).Above the step pattern, the active length scalesincrease with decreasing friction, as characterised bythe defect distribution (
Figure 1 e-f;
Figure 4 a; Movie M5 ). Within each step the distribution is flat.However, at each edge, the number density of -1/2 defectspeaks, while the density of +1/2 defects plunges. This isconsistent with defect densities at the edges of the trench(
Figure 2 a-b). Though the simulated edge peaks are lesspronounced than in experiments, numerical results show
Vortex latticeOscillating stream flowActive turbulenceShallow region b cd f Delay function V e l o c i t y c o rr Delay function V e l o c i t y c o rr Vortex latticeOscillating stream flowActive turbulenceShallow region fe FIG. 3.
Friction boundaries result in separate flow regions. (a) Instantaneous experimental vorticity superjacent thedeep trench region from PIV for a trench of width w t = 325 µ m. Panels with shaded grey backgrounds denote experimentalresults, 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 . R ( Methods I E ). Plus-half defects (dark green) are trapped between thevirtual boundaries, generating the repeating vortex structure along the centre line that is distinct from the active turbulencethat exists outside the virtual boundaries. (c) Decreasing the trench width to w t = 1 . R results in long-range, oscillatory,bidirectional streaming flow inside the trench. (d) Increasing the trench width to w t = 6 R results in active turbulence bothinside 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 ( δy ) = h v ( r ; t ) · v ( r + δy ˆ y ; t ) i / (cid:10) v (cid:11) for the experiment illustrated in (a),measured a distance δy along the x = 0 centre line of the trench. Due to the confinement effect, long-range flow structures areformed in the low-friction regime epitomised by the strong correlation-anti-correlation-correlation signal. (f) The autocorrelationfunction from the simulations shown in blue (b), red (c), yellow (d) and the shallow region outside the virtual channel (dashed).The blue curve displays pronounced correlation and anti-correlation indicating the counter-rotating vortex pattern (b), whichcorresponds to the behaviour observed in the presented experiments (a,e). The red curve is long-lived and fully correlated in thenarrow channels (c), while the yellow curve decorrelates to zero after an anti-correlation, signalling active turbulent behaviourboth within and outside the virtual boundaries. more clearly the decrease in defect density across multiplesteps. The change in defect density is modest, consistentwith studies demonstrating that increasing oil viscosityfive orders of magnitude only increases defect density bya factor of order unity[51, 52], which highlights the po-tential tunability of our method. Interestingly, we onlyobserve a continuously well-defined nematic field for oildepths that are much greater than h in both experi-ments and simulations ( Figure 1 e). For small n , theactive film exhibits disorderly nonhomogeneous texturesakin to those observed in experiments utilising high vis-cosity oils[51]. This suggests that submersed micropat-terned structures can do more than impact flow and ori-entational state: we now demonstrate how substrate mi-cropatterning can be used to control active matter con-centration via structures raised above the solid substrateyet still fully submersed in the underlying oil-layer.We consider fully submersed SU-8 pillar structures( Figure 1 g) of radius r p = 116 ± µ m and height h p = 6 . ± . µ m. As in the trench, sinusoid and stair- way geometries, the active nematic layer is subject to astep-change in the effective film friction. However, differ-entiating it from previous structures, the pillar’s virtualboundary forms a closed loop. The most prominent effectis a pronounced dilution of active material from the en-closed region above the pillar ( Figure 1 h; Movie M6 ),which is qualitatively recapitulated in the simulations(
Figure 1 i; Movie M7 ; Methods I E ). The phase-fieldactive material concentration φ ( r ; t ) ( Methods I E )demixes in the high effective friction region directly abovethe pillar. We confirm that the absence of active materialis not a result of the pillar intruding through the activenematic film by occasionally observing microtubule bun-dles moving above the apex of the pillar (
Movie M8 ).Further, we did not observed curvature of the superjacentactive layer in the vicinity of the micropillar, indicatingthat a finite oil depth lies between the subjacent pillartop and the nematic film.To understand the mechanism leading to the pillar-bound dilute phase of active matter, we consider a sim- x position ($ $m) D e f e c t den s i t y ( m) x position/ D e f e c t den s i t y ab FIG. 4.
Submersed micropatterned stairway allows si-multaneous coexistence of regions of separated activeturbulence with differing defect densities, while ATPconcentration remains constant.
Plus-half (dark green)and minus-half (magenta) defect distribution as a functionof position down the stairway x . (a) Measurements fromexperiments for steps of width w t = 500 µ m (grey shadedbackground). (b) Measurements from simulations for steps ofwidth of width w t = 5 R (non-shaded background). plified model of the active nematic. The effective frictionis locally large above the pillar, causing the flow speed todecrease in the enclosed area but remain non-zero beyondthe pillar border ( Figure 5 a). Since nematic orderingarises in active microtubule network films due to activity-induced motion, the sharp decrease in flow causes a cor-responding drop in scalar order S across the circular vir-tual border ( Figure 5 b). However, the abrupt gradientin S produces a radially outward average active force f ( r ; t ) ∼ h ∇ · Q ( r ; t ) i ≈ ∂ r ζ ( r ; t ) S ( r ; t ) ˆ r ( Equation 6 of Methods I E ), when the variation of the nematic or-der is dominated by the radial change in scalar order pa-rameter and bend-induced stresses are neglected. Thus,the active forcing is expected to be radially outward andsharply peaked about the interface as observed in simu-lations (
Figure 5 c).However, activity does not simply produce increasedpressure across the perimeter but rather is able to se-lectively deplete the concentration of active material φ .Since the active film is considered incompressible, fluidmass density is constant and hence divergence of the filmvelocity is zero ( Equation 3 of Methods I E ). Thus,depletion demands that outward advection is more fre-quent in regions where φ is larger on average. This isindeed the case because the activity depends on the lo- cal amount of active material present, ζ ( r ; t ) = ζ φ ( r ; t )( Equation 7 of Methods I E ), which causes the radi-ally outward forcing to be stronger in magnitude where φ is large. For this reason, if the surrounding active turbu-lence stochastically advects φ -rich active material acrossthe perimeter, the local active forces increase in kind.Hence, the active forces selectively repulse the materialfrom the dissipative region ( Figure 5 d; Movie M7 ).However, if φ -poor fluid enters the local active force de-crease allowing the fluid to more easily cross the perime-ter. Thus, the depletion of active matter above the pillaris a result of the high effective friction lowering the ve-locity. This causes nematic discomposure and selectivitydue to the direct dependence of activity on the local con-centration ( Figure 5 e).Noting that the highest steps in the stairway geometryalso fail to exhibit continuous nematic fields but are notdevoid of active matter (
Figure 1 d; Movie M5 ), we testif the curvature of the virtual barriers impacts depletionby simulating a rectangular pillar (
Figure S2 ). We ob-serve a comparable depletion of φ from the enclosed areaas in the circular pillar and so conclude that curvature isnot the critical difference. Secondly, we consider a circu-lar pit and find that active material accumulates in theenclosed area, consistent with the explanation of the de-pletion mechanism ( Figure S3 ). We conclude that accu-mulation or depletion of active material using submersedmicropatterned structures relies principally on two at-tributes: (i) The oil-layer must be thin (high effectivefriction) to suppress the active flows necessary to exhibitnematic order. (ii) An enclosed area must be circum-scribed by a virtual boundary to prohibit longitudinalactive streams through the incompressibility constraint.In addition to controlling concentration, submersedpillars interact with defects. We observe a greater fre-quency of -1/2 defects in the vicinity of the virtual bound-ary in our simulations (
Figure 5 f-g). The planar anchor-ing of the director field explains the distribution of defectsat the pillar boundary (
Figure 1 h-i). The resulting benddeformation around the perimeter drives hydrodynamicinstabilities to continually generate defect pairs, withnewly created self-motile +1/2 defects typically orientedradially away from the centre such that they swiftly moveaway from the interface (
Movie M9 ), leaving unboundimmotile -1/2 defects behind (
Figure 2 f-g). Submersedpillars can also serve as a virtual obstacle for defect tra-jectories (
Movie M9 ). Positive defects that approachthe pillars from the surrounding turbulence stall or aredeflected once in proximity to the pillar (
Figure 5 h-j). Deflected +1/2 defects first slow as they approachthe pillar, then scatter and regain speed as they moveaway from the submersed structure (
Figure 5 h-i). Pos-itive defects that stall as they approach the boundarytemporarily hold their position before annihilating withpillar-associated -1/2 defects (
Figure 5 j). While defectscan temporarily enter the depleted area, such infrequentevents are transient (
Movie M8 ) as the repulsive activeforce (
Figure 5 c) pushes such incursions radially out-
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Pillars cause local high friction regions which result in active matter depletion. (a) Due to the higherfriction, the flow in the active nematic film remains low. (b) The nematic is highly ordered far from the pillar but disorderedabove it because the speed is lower in the higher friction region. (c) The difference in nematic order at the friction interface resultsin a radial active force. (d) This radial active force pushes the active material concentration outwards resulting in depletioneffects. Panels with non-shaded backgrounds denote numerical results, while shaded grey backgrounds denote experimentalmeasurements. (e) Schematic of a-d. (f-g) +1/2 (dark green) and -1/2 (magenta) defect distribution as a function of radialdistance from a submersed pillar from experiments (f) and simulations (g). (h-j) xy trajectories of example ± ward.Using a combined experimental and simulation ap-proach, we have demonstrated that micropatterns fullysubmersed in an underlying oil layer can guide the flow,topology, and even concentration of active material in su-perjacent nematic films without direct contact. By im-posing changes in substrate depth, viscous dissipation inthe oil layer enacts a position-dependent effective fric-tion coefficient on the active material. Abrupt substrateheight steps can constitute sharp virtual boundaries inthe active matter layer, which can control flow and de-fect behaviour. As proof-of-concept systems, we pre-sented virtual channels exhibiting coexisting flow states,sinusoid substrates that gently guide defects, stairwaysseparating active turbulence with differing characteristic length, and a virtual enclosure depleted of active materialthat acts as an obstacle scattering nearby defects.The proposed technique of fully submersing micropat-terned structures facilitates new approaches for fabricat-ing complex active topological microfluidic devices. Forexample, complications associated with infiltration of ac-tive nematics into confined spaces could be avoided, andactive dynamics in various geometries at the same ac-tivity could be compared directly. Furthermore, locallyconcentrating or depleting active material could regulateactivity at constant levels of ATP or rheological proper-ties such as film viscosity or nematic elastic coefficientthrough this novel approach. I. METHODSA. Formation of the Active Nematic Network
An active nematic microtubule network is generatedfollowing the protocol previously reported by Sanchez etal. at an oil-water interface[53]. Prior to experiments,active premixtures are prepared in 3 . µ m aliquots con-taining biotin-labeled K401 kinesin motors, streptavadin,PKLDH, phosphoenol pyruvate (PEP) (used for ATP re-generation), 4mg/ml glucose, 0 . µ g/ml catalase and 2mM Trolox, and 6% (w/v) 20kDpolyethylene glycol (PEG) in M2b buffer (80mM PIPESpH6 .
8, 2mM MgCl , and 1mM EGTA). To prevent pho-tobleaching during imaging, premixtures contain 6 . − ◦ C for future use.To perform active nematic experiments, 1mM ATP (fi-nal concentration) is added to the 3 . µ m premixturealiquot followed by 2 µ l of 6mg/ml (3%) Alexa Fluor 647labeled GMPCPP microtubules (for a final concentrationof 1mg/ml). For fluorescence imaging, microtubules arefluorescently labelled with Alexa Fluor 647[53]. The re-sult is a suspended active nematic with a total volumeof 6 µ 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 procedure[18, 53]. Wefirst create a flow cell made from the glass substrate withpatterned structures, double-sided tape and a coversliptreated with a polyacrylamide brush ( Figure 1 ). Thepolyacrylamide brush prevents excess protein binding tothe coverslip.We flow in an oil/surfactant mixture (3M HFE7500with 1 .
8% PFPE-PEG-PFPE (perfluoropolyether) sur-factant) into the channel. Then, this mixture is ex-changed with the active microtubule network. This sys-tem forms a 3D, unconfined active microtubule network.Streptavidin can bind up to four biotin-labeled kinesinmolecules and when microtubules of opposing polaritiesalign parallel to each other, the kinesin molecules ori-ented at 180 ◦ to each other walk in opposite directionsalong those neighbouring microtubules. As the kinesinswalk, the filaments produce an extensile motion drivenby ATP hydrolysis. The ends of the flow cell are sealedusing a UV-curable glue (RapidFix). The active layer isthen centrifuged using a swinging bucket rotor for 42 minat 300rpm. B. Photolithography
The trench and pillar geometries are produced us-ing photolithography. SU-8 (MicroChem Corp.) is anegative tone epoxy-based photoresist that is used tocreate thin film plastics on substrates. SU-8 is com-posed of epoxy-based monomers and photo-acid gener- ators (PAGs) suspended in a solvent. Upon exposure toultra-violet (UV) light, the PAGs release acids servingas a catalyst for cross-linking available epoxy groups onthe monomer once heat is applied to the substrate. Priorto fabrication, glass substrates are thoroughly cleaned insoap and water followed by 30 minutes of sonication inacetone, methanol and ethanol in this order. The glasssubstrates are rinsed in nanopure water to ensure mini-mal presence of surface contaminants. The glass is thenplasma treated with oxygen for 2 minutes. To ensure theremoval of residual moisture left on the surface, the glasssubstrate is placed on a hotplate for 5 minutes at 200 ◦ C;the substrate is left to cool down to room temperaturefor 5 minutes in a humidity-controlled area.Upon thorough cleaning, a quarter-sized drop of SU-8 50 is deposited on the glass substrate. The SU-8 isspin-coated at 2000rpm for 45 seconds followed by a 10minute wait. The substrate is then soft baked at 65 ◦ C for12 minutes then at 95 ◦ C for 45 minutes on a hotplate toevaporate solvent. The substrate is again left to cool toroom temperature with another wait of 10 minutes. Thefilm is then exposed to 365nm UV light (500mJ/cm − )followed by a 10-minute wait step. To crosslink the ex-posed regions’ epoxy groups, the substrate undergoes apost exposure bake for 5 minutes at 65 ◦ C then for 15minutes at 95 ◦ C on a hot plate. After the 15 minutebake, the hot plate is turned off and allowed to cool toroom temperature without the removal of the coated sub-strate. This step is to avoid thermally shocking the filmwhich can result in cracks and poor adhesion. The sub-strate is developed for 30 minutes with gentle agitations.Once developed, the residual SU-8 developer is rinsedwith isopropanol and de-ionised water then dried withnitrogen gas. After development, the substrate is thenhard baked for 2 minutes at 150 ◦ C. The heights of themicrostructures are measured using profilometry.
C. Micro-milling
The stepped substrate and the sinusoid surface are pro-duced using the computer numerical controlled (CNC)micromilling. We use the milling machine TN5-V8-TC8 (MDA Precision) to fabricate the microstructuresfrom poly(methyl methacrylate), commonly known asacrylic. This milling machine is capable of handlingdrills and endmills as small as 50 µ m in diameter andhas a spindle rotational accuracy of around 2 µ m. Thefirst designed system has steps (height), each with arise of ∆ s = 10 µ m ± w s = 500 µ m ±
2. The system is composed of tensteps. The smaller of the two sinusoid systems has un-dulations with a peak-to-peak amplitude ∆ u = 40 ± µ mand a wavelength λ u = 150 ± µ m, while the larger hasa peak-to-peak amplitude ∆ u = 50 ± µ m and a wave-length λ u = 150 ± µ m. After milling, these featuresare used as a mold and transferred into a soft elastomer,polydimethylsiloxane (PDMS), which in turn is used asa mold again to transfer the features into a cyclic olefincopolymer (COC), using a thermopress. The COC sheetsof thickness ranging from 150 µ m to 350 µ m are then im-mersed for one hour in a solution of 10wt% of 8-Anilino-1-naphthalenesulfonic acid (ANS), a hydrophobic dye, dis-solved in a mixture of 85wt% ethanol and 15wt% de-calin. This treatment adds a fluorescent layer of ANSdye to the COC surface, which allows measuring the flu-orinated oil’s thickness under the active nematic, using aLeica SP8 UV/Visible Laser Confocal Microscope. Theconfined active nematic is imaged using a wide-field flu-orescent Nikon Eclipse Ti-E microscope with an AndorClara camera controlled by Micromanager open-sourcesoftware.
D. Data processing
To investigate how defect dynamics in the active layerare influenced by submersed structures, labeled micro-tubule bundles are imaged using fluorescence microscopy.Four-hundred-frame videos are collected at 1 frame persecond and processed using Fiji/ImageJ version 1.52asoftware. To acquire defect distributions, active nematicmicrotubule defects are identified and counted manuallyevery 10 frames for each video. Two-dimensional Carte-sian components for both x and y axes are acquired fromboth +1/2 and -1/2 defects using the
Click for positionplugin on Fiji/ImageJ. For the submersed trench geome-try, we use MATLAB to analyse the frequency and posi-tion for both +1/2 and -1/2 defects across the channel.Defects are organised and binned in 10 µ m horizontal in-crements across the field of view.For the stairway geometry and pillar geometries, weapply the same counting procedure to obtain +1/2 and-1/2 defect positional frequencies across all frames. Inthe stairway geometry, we apply the counting proceduresequentially to each step and, for the pillar, the videosare processed by centring the pillar in a 400 × µ m window. Similar to the analysis done with the submersedtrench, we use MATLAB to generate a 2D histogram torepresent the frequency of the +1/2 and -1/2 defects,positionally distributed in a 2D plane. Defects are or-ganised and binned every 13 . µ m in both horizontal andvertical increments.To measure the director orientation directly abovetrenches, the imaged active nematic is oriented with thelong axis parallel to the y-axis. Each pixel is convertedto micrometers (0 . µ m/pixel). Each pixel from eachframe containing a x-component, y-component and an-gle of the bundle-microtubule director is represented ona 2D grid and determined using MATLAB. Our processuses a nested loop and a conditional statement to de-termine if the angle is between 80 and 90 ◦ ; if a directorsatisfied this condition along the y-axis for the specifiedx-position, the total is summed then divided by the totalnumber of angles checked by the loop. This probability isappended to a new horizontal array for each probability in the x-position. The result is a time-averaged directororientation mapped across the trench, averaging over ally values. E. Simulations
To complement the experiments, we simulate the ac-tive nematic thin film using a 2D hybrid lattice Boltz-mann/finite difference approach. The incompressible ac-tive nematic film flows with velocity u ( r ; t ), has long-range orientational order described by the tensor orderparameter Q ( r ; t ) and varying concentrations of activematerial, which we take to be a phase field φ ( r ; t ) vary-ing from 0 to 1, coarsely describing the local amountof active materials (microtubules, kinesin complexes andATP). Four coupled equations describe the time evolu-tion of these continuous fields.The first is the Beris-Edwards equation for nematics,( ∂ t + u · ∇ ) Q − S = Γ Q H . (1)The co-rotation term S = ( ξ D + Ω ) (cid:0) Q + I (cid:1) + (cid:0) Q + I (cid:1) ( ξ D − Ω ) − ξ (cid:0) Q + I (cid:1) tr ( QW ) determinesthe alignment of the microtubules in response to gra-dients in the velocity field, with Ω the rotational partand D is the extensional part of the velocity gradienttensor W = ∇ v = Ω + D . The alignment parameter ξ is taken to be in the flow aligning regime and set to ξ = 0 .
5. The molecular field H = − ( δ F δ Q − I Tr δ F δ Q )is a functional derivative of free energy F , describ-ing the relaxation towards equilibrium at a rate Γ Q .The free energy depends directly on the nematic order Q and the active material concentration φ . The ne-matic part consists of a Landau-De Gennes contribution, F bulk,Q = A (cid:16) (cid:16) − ν (cid:17) tr( Q ) − ν tr( Q ) + ν tr( Q ) (cid:17) with A = 0 .
05, and deformation F grad,Q = K ( ∇ Q ) with K = 0 .
02. We set ν = 2 .
55, which favours theisotropic state in the absence of active flows[54] and isindependent of φ ( r ; t ). This choice allows nematic or-dering only due to activity-induced flows, in agreementwith experiments[53] and previous simulations[54, 55].The active material concentration evolves according toa Cahn-Hillard model. We assume that the film is in-compressible and active material concentration φ doesnot impact fluid mass density ρ . ∂ t φ + ∇ · ( u φ ) = Γ φ µ, (2)where µ = δ F δφ − ∇ · (cid:16) δ F δ ∇ φ (cid:17) is the chemical potential andΓ φ = 0 . F bulk ,φ = A φ φ ( φ − and a gra-dient term F grad ,φ = K φ ( ∇ φ ) with A φ = 0 .
03 and K φ = 0 .
1. The free energy minima are at φ = { , } .However, activity suppresses ordering and φ does notphase separate into high and low φ regions for sufficientlyactive flows. We initialise the concentration to φ = 0 . ρ (not active material con-centration φ ) leads to the incompressibility condition ∇ · u = 0 . (3)Because the experimental active nematic film lies ona 2D interface between two 3D fluids, the planar di-vergence of the velocity could conceivably be non-zero.However, non-zero divergent velocity fields would re-quire continuous circulation in the thin aqueous/oil layersabove/below the film. As we are unaware of evidence in-dicating such flows, we assume 2D incompressibility im-plying divergence-free flow. Since the fluid is taken tobe incompressible, any outward fluid mass flux into anenclosed area must be balanced by an inward flux andvice versa.The second equation is the Cauchy momentum equa-tion ρ ( ∂ t + u · ∇ ) u = − ∇ p + ∇ · Π − γ u , (4)where p is the pressure and Π is the stress tensor whichincludes the standard viscous stress Π visc = 2 η E for filmviscosity η = 2 /
3. Furthermore, it contains the elasticstress due to the nematic nature of the microtubules Π elastic = 2 ξ Q ( Q : H ) − ξ H · Q − ξ Q · H − ∇ Q : δ F δ ∇ Q + Q · H − H · Q , (5)where Q = Q + I /
3. The stress also contains the capilarystresses Π cap = ( F − µφ ) I − ∇ φ (cid:16) δ F δ ∇ φ (cid:17) due to differencesin concentration φ , and the active component Π act = − ζ ( r ; t ) Q = − ζ φ ( r ; t ) Q . (6) Here, we have taken the local activity ζ ( r ; t ) = ζ φ ( r ; t ) (7)to scale directly with active material concentration. Weset ζ = 0 . γ ( r ) u ( r ; t ). Since previousstudies have successfully modelled the dynamics ofmicrotubule/kinesin-based active nematic films withweak effective friction[56], we treat the friction as neg-ligible in the regions of the film superjacent to deepstructures. In the regions above shallows, we include lu-brication momentum dissipation via a non-zero effectivefriction coefficient. We set γ ( h s ) = 0 .
07 in the shal-lows for simulations of the trench system. We define acharacteristic confinement scale to be R ≡
20 LB nodes.The narrowest trench width is w t = 1 . R ( Figure 3 c)and the two wider trenches have widths of f w t = 1 . R ( Figure 3 b) and 6 R ( Figure 3 d), respectively. Alltrenches are simulated in 7 . R × R periodic systems.For the stairway, we simulate a 7 . R × R long sys-tem composed of the same number of steps as in theexperimental system (10 steps) of width w s = 5 R . Eachstep represents a different lubrication momentum dissipa-tion region with different non-zero effective friction coef-ficients set to γ ( h s , n ) = . n , where n ∈ { . . . } denotesthe different steps corresponding to different oil depths inthe experiments. In the main text, we present simulationresults from steps n = { , , , , } . For the pillars, weuse a radius r p = R . The friction coefficient is chosen tobe sufficiently large to clearly match the pronounced de-pletion observed experimentally in Figure 5 . We presentresults for the value γ ( h p ) = 0 . φ for γ ( h s ) = 0 .
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We thank Amin Doostmohammadi for helpful dis-cussions. We acknowledge generous funding from theNational Science Foundation, through several awards(DMR-1808926), NSF-CREST: Center for Cellular andBiomolecular Machines at UC Merced (HRD-1547848),and from the Brandeis Biomaterials facility MRSEC-2011486. K.T. acknowledges funding from the Euro-pean Union’s Horizon 2020 research and innovation pro-gramme under the Marie Sklodowska-Curie grant agree-ment no. 722497 (LubISS). T.N.S received funding fromthe European Union’s Horizon 2020 research and inno-vation programme (grant agreement no. 851196).
AUTHOR CONTRIBUTIONS
L.S.H conceptualised the project. L.S.H and T.N.Sdesigned the study. L.S.H, D.K, S.F and M.A.G all con-2tributed to the design of experiments. K.T designed themodel with contributions from T.N.S and J.M.Y. S.A.Aand D.K produced the surfaces. D.K and M.A.G per-formed experiments and K.T. implemented simulations.D.K and K.T performed data analysis. T.N.S and L.H.Swrote the paper with assistance from J.M.Y, D.K andK.T, and input from S.F and M.A.G. All authors dis-cussed and interpreted results and revised the paper.
COMPETING INTERESTS
The authors declare no competing interests.
III. SUPPLEMENTARY INFORMATIONA. Sinusoidal Microstructures
FIG. S 1.
Submersed micropatterned sinusoidal sub-strate.
Fluorescent image of the active nematic film superja-cent to an oil-submersed sinusoidal micropatterned structurewith a 150 µ m wavelength and 40 µ m amplitude. The undula-tions go from left to right such that the parallel troughs andcrests run in parallel up and down. Scale bar = 250 µ m. (a)A set of three motile +1/2 defects that move in a train upthe middle of a trough are highlighted in green. The trainaligns the director parallel to the trench. (b) 15 secondslater the initial train is destabilised by extensile-active ne-matic hydrodynamic-bend instability and the associated paircreation events in which +1/2 defects tend to unbind orientedperpendicular to the trough. To demonstrate that the method of submersing mi-cropatterned substrates in the oil layer below an ac-tive nematic film can guide defect dynamics, we pro-duced a series of micropatterned sinusoidal substrates(
Methods I C ). The smallest of these are characterisedby amplitude ∆ u = 40 ± µ m and wavelength λ u =150 ± µ m. As observed in Movie M3 and discussed inthe main text, the undulating effective friction producesorientation-control of motile defects for smaller wave-lengths. Trains of co-aligned +1/2 defects move along thetroughs (
Figure S1 a). However, these are intermittently disrupted by the extensile-active nematic hydrodynamic-bend instability. Regions of well-ordered nematic alignedalong the troughs result in pair creation events in which+1/2 defects unbind with an orientation perpendicularto the troughs (
Figure S1 b).
B. Rectangular Microstructure Pillar
FIG. S2.
Rectangular pillar causes depletion.
Same as
Figure 1 i but with the circular pillar replaced with a square,the perimeter of which is traced by dashed lines. We observea comparable degree of depletion of active material concen-tration φ , denoted by the colormap, for the same friction co-efficient γ = 0 . Simulations of rectangular pillar (
Figure S2 ) exhibita comparable depletion of concentration φ from the en-closed area above the pillar as the circular pillars consid-ered in the main text ( Figure 5 ). Hence, we concludethat a pillar curvature is not the critical property leadingto depletion in the pillar geometry. Rather, the abruptgradient in S that follows indirectly from the increasedeffective friction produces a outward average active forcethat can deplete active material from any enclosed areaof high effective friction. C. Microstructure Pit
In the main text, we argue that active material isdepleted from the enclosed region directly above a mi-crostructured pillar due to the increased effective viscousdissipation within the locally thinner oil-layer. Thus, onenaturally wonders if a microstructured pit in the sub-strate might act as an “anti-pillar” and lead to accumu-lation of active material within an enclosed region above3 vel mag radial distance/ gradient nematic mag concentration
FIG. S3.
Pits cause local lower friction regions whichresult in active matter accumulation.
We consider acircular pit of radius 2 R above which the frictional damping isnegligible but outside of which γ = 0 .
1. We measure the sameprofiles as
Figure 5 a-d for one instant after steady state hasbeen achieved. We observe accumulation of active material φ in the enclosed area. the structure. We consider an effective circular pit bysimulating a substrate with an effective friction exceptdirectly above the structure ( Figure S3 ) and observethat active material accumulates in the enclosed area.The velocity magnitude, nematic scalar order parame-ter, and concentration directly above the pit (
Figure 3 )are all comparable to their values far from the pillar(
Figure 5 ,a,b,d respectively). While the change in thescalar order parameter is positively peaked at the perime-ter of the pillar (
Figure 5 c), it is seen to dip at theperimeter of the pit (
Figure 3 ). This indicates a ra-dially inward average active force, consistent with themechanism which here leads to accumulation.
D. Movie Captions
M1 Micropatterned trench geometry.
Fluo-rescence microscopy video of a bundled micro-tubule/kinesin network at 1mM ATP at the oil-water interface above a submersed SU-8 substratewith a micropatterned trench. The trench possessesa depth 8 ± µ m and width w t = 327 ± µ m. Leftand right are shallow regions which exhibit activeturbulence, while the centre is the deep region, theedges of which form a well-defined virtual boundarythat traps defects. M2 Simulation results of a repeating lattice ofcounter-rotating vortices above a trench.
The trench width is w t = 1 . R . Plus-half de-fects (dark green) are trapped between the virtualboundaries, generating the repeating vortex struc-ture along the centre line that is distinct from theactive turbulence that exists outside the virtualboundaries. In the colormap, blue (red) denotesclock (anti-clock) wise rotating vortices. M3 Micropatterned sinusoid geometry.
Fluo-rescence microscopy video of a bundled micro- tubule/kinesin active nematic network above an oil-submersed sinusoidal microstructure with a 150 µ mwavelength and a 40 µ m amplitude. The undu-lations go from left to right such that the par-allel troughs and crests run in parallel up anddown. Train of +1/2 defects move up and downthe troughs but are unstable to intermediate paircreation and unbinding events oriented perpendic-ular to the troughs. M4 Micropatterned sinusoid geometry.
The sameas
Movie M3 but for a microstructure with a500 µ m wavelength and a 50 µ m amplitude. Scalebar = 250 µ m. Distinct trains of +1/2 are notas immediately apparent, though self-propelled de-fects tend to align with the troughs. M5 Micropatterned stairway geometry.
Fluo-rescence microscopy video of a bundled micro-tubule/kinesin film over an oil-submersed, mi-cropatterned stairway. Individual steps are 500 µ mwide and 10 µ m tall. The oil depth from left toright deepens from h − h = { , , , , } µ msequentially where the initial fluid depth is h =12 ± µ m. M6 Micropatterned pillar geometry.
Fluorescencemicroscopy video of a bundled microtubule/kinesinnetwork at the oil-water interface with 1mM ATPabove a substrate possessing a micropatterned pil-lar of radius r p = 116 ± µ m and height h p =6 . ± . µ m. The bundled microtubule/kinesin net-work in the film directly above the immersed pillaris fully depleted. Annihilation events can be ob-served as +1/2 approach the boundary and annihi-late with perimeter-associated -1/2 defects. M7 Simulation results of the active nematic filmabove a pillar illustrating depletion of activematerial.
Active matter concentration φ (denotedby colormap) is initialised uniformly but eventuallydepletes from the high friction region inside the cir-cle ( γ = 0 . M8 Micropatterned pillar with +1/2 defect en-tering depletion zone.
The same as
Movie M6 but exhibiting an instance of active material in thesuperjacent film crossing the boundary above thepillar.