Surface anchoring controls orientation of a 1 microswimmer in nematic liquid crystal 2

Microscopic swimmers, both living and synthetic, often dwell in 18 anisotropic viscoelastic environments. The most representative realization of such 19 an environment is water-soluble liquid crystals. Here, we study how the local 20 orientation order of liquid crystal affects the motion of a prototypical elliptical 21 microswimmer. In the framework of well-validated Beris-Edwards model, we show 22 that the microswimmer’s shape and its surface anchoring strength affect the swimming 23 direction and can lead to reorientation transition. Furthermore, there exists a critical 24 surface anchoring strength for non-spherical bacteria-like microswimmers, such that 25 swimming occurs perpendicular in a sub-critical case and parallel in super-critical 26 case. Finally, we demonstrate that for large propulsion speeds active microswimmers 27 generate topological defects in the bulk of the liquid crystal. We show that the location 28 of these defects elucidates how a microswimmer chooses its swimming direction. Our 29 results can guide experimental works on control of bacteria transport in complex 30 anisotropic environments. 31


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Here we examine how the stable swimming direction depends on physical and 84 geometrical parameters. Swimmers typically orient themselves in liquid crystal 85 (LC) either parallel or orthogonal to the far field director alignment n ∞ . In our 86 numerical studies, we consider n ∞ = i (horizontal, parallel to x-axis), and α(t) is the 87 counterclockwise angle between direction of the principal axis of the microswimmer and 88 n ∞ , as depicted in Figure 1. 89 The microswimmer is represented by a rigid body which moves with velocity 90 V (t) and rotates with angular velocity ω(t). Thus, a point of the microswimmer, 91 occupying location x at time t, has instant velocity V (t) + ω(t) × (x − x c (t)), where 92 x c (t) = x c (0) + t 0 V (τ ) dτ is the location of the center of mass of the microswimmer at 93 time t. Velocities V (t) and ω(t) are to be determined from the solution of the system 94 coupling the microswimmer and the liquid crystal dynamics, described below. 95 In order to capture effect of microswimmer's shape anisotropy, the microswimmer 96 is represented by an ellipse whose principal axis is given by unit vector 97 p = (cos(α), sin(α)) and orientation angle α (Figure 1). To model self-propulsion of the 98 microswimmer, we impose the velocity slip boundary condition on the microswimmer's Stokes flow (see, e.g., [26,5]). Swimming parameter β determines the microswimmer's 112 type, whether it is a pusher or a puller. If β < 0, then the microswimmer is a pusher 113 (e.g., Escherichia coli) such that the propulsion source is at the back. If β > 0, then the 114 microswimmer is a puller (e.g., Chlamydomonas reinhardtii) such that the propulsion 115 source is at the front. For β = 0 the microswimmer is a neutral one (e.g., Paramecium). 116 Conventionally, a liquid crystal (LC) is described by unit director field n(x) and 117 scalar order parameter q(x), as in the classical Ericksen- Leslie model [27,28,29]. 118 Director n(x) gives the average direction of the LC molecules around location x, whereas 119 order parameter q(x) can be thought of as the variation of LC molecular alignment from 120 the average direction. Since "head" and "tail" of an LC molecule are not distinguishable, 121 directions n and −n are equivalent. 122 We employ the Beris-Edwards model [30] to describe LC and consider a symmetric 123 traceless tensor Q(x) instead of fields n(x) and q(x):

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Here I is the d × d identity matrix, and d is the space dimension. The flow of LC is 126 described by velocity vector field v(x) and pressure p(x). The system for Q, v, and p 127 constitutes the Beris-Edwards model, described in Methods and Supplementary Note 1.

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To prescribe the director anchoring, we impose a Robin-type boundary condition 129 on the tensor Q at the microswimmer's surface: Here W is the strength of the surface anchoring, ν is the inward (relative to the 132 microswimmer) normal vector, and tensor Q anchor describes, in the sense of relation (2), 133 alignment in the direction n anchor ; q anchor is the corresponding scalar order parameter:

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Q anchor = q anchor (n anchor n anchor − 1 d I).
(4) , respectively, of converging to the limiting angle, either α → 0 (parallel to far field director n ∞ ) or α → π/2 (perpendicular to n ∞ ), for various swimming parameter β (c) and effective anchoring strength W/K (d). Markers, red crosses, and blue circles are from numerical integration, dashed lines sketch interpolation curves to visualize how the relaxation time depends on a parameter. As β → 0, the relaxation time diverges.

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We also performed computational studies for pullers with various values of 152 anchoring strength, see Figure 2(b). For various β and W/K, we computed the 153 relaxation time as orientation angle α(t) converges to its limiting value α ∞ = lim t→∞ α(t), 154 see Figure 2(c,d). In particular, our computations show that the convergence rate of 155 α(t) for pullers to π/2 as t → ∞ decreases as anchoring strength parameter W increases 156 with fixed K. This result indicates that the larger the anchoring between LC and the 157 spherical puller, the slower the puller turns to swim with orientation angle α = π/2.

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Our main finding is that for elongated (elliptical) microswimmers, the asymptotic 159 behavior of the orientation angle α(t) is dramatically different from the spherical case.

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Since an elongated passive rod with planar anchoring aligns along the nematic direction, 161 the tendency to align parallel to LC director is reinforced for an elongated pusher with 162 planar anchoring. Thus, no critical behavior is expected. In contrast, for a puller with planar anchoring, competition between propulsion (favoring perpendicular alignment) 164 and the elongated shape (favoring parallel alignment) leads to a critical transition. 165 First, we considered dynamics of α(t) for an elliptic microswimmer with no 166 anchoring, W = 0. We found that there is no symmetry with respect to the swimming 167 parameter β observed for a spherical microswimmer in Figure 2(a,c). Namely, the 168 relaxation time dependence is no longer close to an even function, as it can be seen 169 in Figure 3(a,c). Most importantly, we found that for a puller with positive anchoring 170 strength W > 0, the asymptotic orientation angle α ∞ depends on the value of anchoring 171 strength W , as shown in Figure 3(b,d). We also found that the relaxation time 172 apparently diverges as parameters β or W approach their critical values. For β = 0, 173 one sees from Figure 3(a) that α(t) increases and will eventually converge to π/2. On 174 the other hand, for W/K = 0.2 no angular dynamics is clearly observable, as depicted 175 in Figure 3(b). Since it is unlikely that there is a stable steady state for π/4, we expect 176 that in this case symmetry will eventually be broken as time evolves and α(t) will 177 asymptotically converge to either 0 or π/2.

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To further elucidate dependence of swimming orientation on anchoring strength, 179 we considered a puller (β > 0) for various values of elastic constant K, microswimmer's 180 aspect ratio , and swimming parameter β. The results of computational modeling 181 are summarized in Figure 4. We conclude that there exists a critical value of the 182 surface anchoring W crit > 0 which depends, in particular, on K, , and β. In the sub-183 critical case, W < W crit , the microswimmer orients perpendicular to LC orientation n ∞ , 184 α ∞ = π/2, whereas in super-critical case, W > W crit , the microswimmer eventually 185 orients parallel to n ∞ , α ∞ = 0. We note that for some values of parameters K, , and β 186 finding the critical surface anchoring W crit is not straightforward. Specifically, in these 187 cases, α(t) does not clearly converge neither to 0 nor to π/2 over the entire duration LC energy is minimized at parallel-to-LC orientation α = 0 and thus this orientation 206 is stable, see e.g. [24]. It also implies that an elongated pusher with planar surface 207 anchoring always reorients parallel to LC and does not possess a critical anchoring 208 strength.

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A somewhat opposite situation realizes for microswimmers with the homeotropic 210 anchoring instead of planar one, i.e., nearby LC molecules orient perpendicular to the 211 microswimmer's surface (n anchor = ν in (4)). In this case, the perpendicular orientation α = π/2 becomes more preferable than the parallel one with α = 0. Thus, an elongated puller reorients perpendicular to LC director whereas an elongated pusher is expected 214 to possess a critical anchoring: for small W it reorients parallel to LC director and 215 for large W it reorients perpendicular. Our numerical modeling for elongated pusher 216 and homeotropic boundary conditions confirms this expectation, see Figure 5  We show that there exists the critical value of anchoring strength for fixed initial 233 angle α| t=0 = π/4. We performed numerical simulations for various initial angles α| t=0 234 and we observed that effective dynamics is of the form  Equation (5) as well as the existence of critical anchoring strength can also be 249 explained as follows. In [5], it was justified that LC exerts the effective torque on 250 spherical planar puller T eff ∝ v 0 β sin(2α), wherein T eff is independent of W . Thus, a puller, for any value β > 0 and W , eventually orients itself perpendicular to LC 252 director, that is, α| t 1 ≈ π/2 for any initial angle 0 < α(0) ≤ π/2. In this work,  Furthermore, if we add written above quantities to which both torques, T eff and T stab , 269 are proportional, then we obtain the right-hand side of equation (5) for some γ and W crit .

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The above results were obtained for a relatively small self-propulsion parameter v 0 .

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As v 0 increases, the steady-state swimming may become unstable and the microswimmer 272 may generate topological defects in its wake. It is well-known that due to the anchoring   axis of the microswimmer, see Figure 8(g).

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The difference in geometry of topological defect locations can be explained by the 297 velocity field generated by the microswimmer, depicted in Figure 9(a,d) and sketched in 298 Figures 9(b,e). Let us assume that defects are passive objects dragged along streamlines.
299 Figures 9(b,e) show that defects will likely end up moving along the red dashed curves 300 that come out from side points for the flow generated by a puller, and from front 301 and rear for a pusher. This is consistent with our numerical modeling presented in trajectories [10,9] and thus affect interactions between microswimmers in LC.

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In conclusion, in this work we demonstrate that surface anchoring and shape

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Computational model

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The system for Q, v, and p constitutes the Beris-Edwards model: ∇ · v = 0.  is imposed on Q. We define the term F exter by [9]:

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where ζ an is the alignment strength, R π/2 is the matrix of rotation by angle π/2, The second term in the definition of E Q accounts for the surface anchoring energy and 363 its form is similar to the classical Rapini-Papoular condition [37].

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We write the force and torque balance for a microswimmer in an over-damped limit 365 (i.e., microswimmer's inertia is neglected): Here  p for x = ±R and y = ±R. We assume that the Reynolds number is small, and thus 389 we neglect the left-hand side of (7) by equating it to 0. and ω are computed using force and torque balances with the updated v and Q. 401 We use an adaptive triangular mesh for FEM so that the mesh is finer close to the The data that support the findings are available from the corresponding author upon 414 request.

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Code availability 416 The code to carry out the simulations is available from the corresponding author on 417 reasonable request. The authors declare no competing interests.