Deformable active nematic particles and emerging edge currents in circular confinements
DDeformable active nematic particles and emergingedge currents in circular confinements
Veit Krause
Institut f¨ur Wissenschaftliches Rechnen, TU Dresden, 01062 Dresden, Germany
Axel Voigt
Institut f¨ur Wissenschaftliches Rechnen, TU Dresden, 01062 Dresden, Germany andCenter for Systems Biology Dresden (CSBD), Pfotenhauerstr. 108, 01307 Dresden,Germany and Cluster of Excellence - Physics of Life, TU Dresden, 01062 Dresden,GermanyE-mail: [email protected]
January 2021
Abstract.
We consider a microscopic field theoretical approach for interacting activenematic particles. With only steric interactions the self-propulsion strength in suchsystems can lead to different collective behaviour, e.g., synchronized self-spinningand collective translation. The different behaviour results from the delicate interplaybetween internal nematic structure, particle shape deformation and particle-particleinteraction. For intermediate active strength an asymmetric shape emerges andleads to chirality and self-spinning crystals. For larger active strength the shape issymmetric and translational collective motion emerges. Within circular confinements,depending on the packing fraction, the self-spinning regime either stabilizes positionaland orientational order or can lead to edge currents and global rotation which destroysthe synchronized self-spinning crystalline structure.
1. Introduction
Active matter systems take energy from their environment and drive themselves outof equilibrium. This can lead to novel collective phenomena and provides hope touncover the physics of living systems and to find new strategies for designing smartdevices and materials. We refer to [1, 2, 3, 4, 5, 6] for various reviews. An importantexample of active matter is constituted by natural and artificial objects capable of self-propulsion. A fundamental challenge is to understand how such objects interact andlead to collective phenomena. Most of the microscopic modeling approaches in this fieldconsider active particles which have a fixed symmetric shape, and movement is definedalong a symmetry axis. This leads to motion along a straight line just perturbed byrandom, e.g., Brownian fluctuations. Both assumptions, on shape and symmetry, arerestrictive, as shape deformations as well as deviations from symmetry destabilize any a r X i v : . [ c ond - m a t . s o f t ] F e b straight motion and make it chiral, which would result in circular motion. As mostsystems are imperfect this should be the general case. While attempts exist to generalizeactive particle models in this direction, see, e.g., [7, 8, 9, 10] for imposed alignmentmechanisms, [11, 12] for anisotropic particle shapes and [13, 14] for shape deformations,multiphase-field models, e.g., [15, 16, 17], where each object is modeled by a phasefield variable, naturally allow for shape deformability and also provide the possibilityto incorporate asymmetry to enforce chirality. It has already been demonstrated thatcollisions of deformable objects can lead to alignment [18, 14, 19, 20]. As a result, thesemultiphase-field models do not require any explicit alignment interactions. A drawbackof such models is the huge computational effort for large numbers of interacting objects.We here consider an intermediate modeling approach. The approach considers a particledensity for all particles and combines it with internal nematic structure. We are onlyinterested in relatively dense systems and study the influence of activity in unconfinedand confined domains.The paper is organized as follows: In Section 2 we postulate a minimal model whichis capable of shape deformations and broken symmetry with respect to the direction ofmotion. The model is termed nematic active phase field crystal model. Besides themotivation, the evolution equations are explained and the numerical approaches forsolving and postprocessing are sketched. Section 3 analyses the model for a singleobject and identifies three different regimes: resting, circular or spinning motion andtranslation. Section 5 considers the emerging collective behaviour in unconfined andconfined geometries, and Section 5 discusses these results and relates the observedphenomena to that of other theoretical and experimental investigations.
2. Modeling
Microscopic field theoretical approaches for active system can be considered as acompromise between the full details of multiphase-field models and active particlemodels. They first have been introduced in [21] for active crystals and consider alocal particle density variation field ψ , a local polar particle orientation field P and aself-propulsion strength v . The model combines a phase field crystal model for freezing[22, 23] with a Toner-Tu model for self-propelled particles [24]. More recently thisapproach was also considered on surfaces [25] and has been extended by an active torqueand the interplay of self-propulsion and self-spinning of crystallites was investigated in[26]. A different path was followed in [27, 28] where the underlying phase field crystalenergy was modified to consider independent active particles [29, 30, 31, 32]. Theapproach allows to simulate a transition from a resting particle to a moving state byincreasing the self-propulsion strength. Within this transition, the particle deformsand elongates perpendicular to the direction of motion. Other phenomena, consideredfor more particles, are cell–cell collisions, oscillatory motion in confined geometries,collective migration and cluster formation in homogeneous systems [27], as well as therich dynamics of heterogeneous systems of active and passive particles, ranging fromhighly dilute suspensions of passive particles in an active bath to interacting activeparticles in a dense background of passive particles [28]. A common characteristicof these models is the presence of an underlying interaction potential for the densityvariation field ψ but no enforcement of aligning the polar particle orientation field P .Alignment results solely from inelastic particle deformations through their interaction[18]. Figure 1.
Particle density variation field ψ for a single density peak (color coding)with the 0 .
01 level-set indicating the shape of the particle. The internal polar (left) andnematic (right) structure is visualized by the director field. The figure further showsthe direction and strength of motion (arrows) and the location of topological defects(black points). For the polar model (left) one +1 defect is located on the symmetryaxis and for the nematic model (right) two +1 / For a single particle the interplay between a splay (or bent) instability of the polarparticle orientation field P , the particle shape and the strength of the self-propulsion v has been discussed [27] and corresponds to the same mechanism as in phase-fieldmodels for active droplets, see, e.g. [33, 34]. As a result of the vertical anchoring of thepolar particle orientation field P at the particle boundary, one +1 defect forms withinthe particle. With no further interaction, due to the isotropic properties of the particle,the resulting shape of the particle is symmetric with respect to the direction of motion,see Fig. 1(a). To incorporate chirality thus requires an additional active forces, as in[26], or a different particle orientation field. Adapting approaches of active nematicdroplets [35, 36], we propose a microscopic field-theoretical approach, which couples alocal particle density variation field ψ , a local nematic particle Q-tensor field Q and aself-propulsion strength v . Similar mechanisms, as described above, also follow for thismodel, but now the nematic properties lead to the presence of two +1 / v , to tune the richdynamics of the model. The proposed minimal model reads ∂ t ψ = M ∆ δ F vP F C δψ + v ∇ · ( ψ Q ∇ ψ ) (1) ∂ t Q = L ∆ Q − c (tr Q − Q − v (2 ∇ ψ ∇ ψ T − (cid:107)∇ ψ (cid:107) Id ) − β { ψ> } Q . (2)with particle density variation field ψ , nematic particle Q-tensor field Q and self-propulsion strength v . The first equation considers conserved dynamics for the freeenergy F vP F C = F P F C + (cid:82) H ( | ψ | − ψ ) d r , with F P F C = (cid:90) ψ r + (1 + ∇ ) ) ψ ) + ψ d r (3)the Swift-Hohenberg energy [37, 22, 23], with parameter r related to an undercooling,and an additional penalization term, with parameter H >
0. The penalization enforcesthe density variations to remain positive. This modifies the particle interaction andallows to phenomenologically describe independent particles [29, 30, 31, 32]. A detailedderivation of F P F C and its relation to classical density functional theory can be foundin [38, 39, 40]. The variational derivative reads δ F vPFC δψ = ( r + 1) ψ + 2 ∇ ψ + ( ∇ ) ψ + ψ + 3 H ( ψ | ψ | − ψ ). The parameter M sets a mobility and is responsible for thedeformability of the density peaks. The active contribution is considered in analogy tothe polar model [27], with Q ∇ ψ playing the role of the polar particle orientation field P . The second equation considers unconserved dynamics of a Landau-de Gennes energyin its one-constant approximation F LdG = (cid:90) L (cid:107)∇ Q (cid:107) + a Q + 23 b tr Q + c Q d r (4)with elastic constant L and entropic parameters b = 0 and a = − c . The activecomponent is constructed to ensure the Q-tensor properties and the last term restricts,in analogy to [27], the nematic particle Q-tensor field Q to be different from zero onlywithin the particles, with β > The coupled equations are reformulated as a set of second order equations andsolved using an operator splitting approach for ψ and Q in a semi-implicit manner.Discretisation in space is done by finite elements [41, 42] and adaptive refinementis considered to ensure a fine discretisation within the particles. The approach isimplemented in AMDiS [43, 44].We consider a square domain Ω = [ − d, d ] , with periodic boundary conditions,where d = π √ the lattice distance of the phase field crystal model. A circular confinementis enforced using an interaction potential to be added to F vP F C , which reads (cid:82) Bψ ϕ B d r with B > ϕ B a tanh-approximation of 1 Ω \ Ω c , with Ω c = {(cid:107) r (cid:107) < d } .The model parameters are fixed as r = − . M = 20, L = 0 . c = 0 . H = 10 , β = 10 and B = 10 . The self-propulsion strength v will be varied and specified below.Numerical parameters concerning grid resolution, time step and tanh-approximation arechosen to guarantee mesh-independency and stable behaviour.As initial condition we specify ψ = A (cid:80) Ni =1 (cos( √ (cid:107) r − r i (cid:107) ) + 1)1 (cid:107) r − r i (cid:107) < π/ √ withprefactor A such that (cid:82) ψ d r = Nd | Ω | (cid:112) ( − − r ) /
133 and particle initial positions r i for i = 1 , . . . , N with N the number of particles. As initial Q-tensor field we consider asymmetric field with one +1 defect in the center of each particle and vertical anchoringat the particle boundary. The symmetric Q-tensor field is perturbed by white noise.The +1 defects are unstable and immediately split into two +1 / i -th particle at time t n is computedas r ni = (cid:82) B i r ψ n d r / (cid:82) B i ψ n d r , with B i a small circle around the maximum of the i -thdensity peak. The radius of B i is related to d . The i -th particle velocity follows as v ni = ( r ni − r n − i ) / ( t n − t n − ) and the mean particle velocity magnitude is the averageover all v ni , computed as v n = N (cid:80) Ni =1 (cid:107) v ni (cid:107) .As in [19, 27] we define for every time t n the translational order parameter φ nT andthe rotational order parameter φ nR with φ T ( t n ) = 1 N (cid:107) N (cid:88) i =1 ˆ v ni (cid:107) , φ R ( t n ) = 1 N N (cid:88) i =1 (ˆ r ni ) T ˆ v ni and φ O ( t n ) = sin( 1 N N (cid:88) n =1 arctan( v ni ))where ˆ v ni = v ni / (cid:107) v ni (cid:107) is the unit i-th particle velocity vector and ˆ r ni = r ni / (cid:107) r ni (cid:107) the uniti-th particle position vector at time t n . In case of collective translation or collectiverotation, we get φ T,i ≈ | φ R,i | ≈
1, respectively. However, also collective orientationin synchronously spinning particles leads to φ T,i ≈
1. To distinguish translational andorientational order φ O measures synchronously changing orientation. The frequency ofthe oscillation in φ O ( t ) determines the collective angular spinning velocity.
3. Single particle
We first consider the situation of one particle. It is placed in the centre of the domainand we consider the effect of v . Fig. 2 shows the particle velocity, the eccentricity andthe asymmetry of the defect arrangement as a function of v . The eccentricity is definedas e ni = (cid:113) − ( r ni,min ) / ( r ni,max ) , where r ni,min and r ni,max are the minimal and maximaldistances between the center of mass and the 0 . ψ for particle i at time t n ,respectively. The 0 . a ni = |(cid:107) d ni, − r ni (cid:107)−(cid:107) d ni, − r ni (cid:107)| + (cid:12)(cid:12)(cid:12) ( d ni, − r ni ) T v ni (cid:107) d ni, − r ni (cid:107) − ( d ni, − r ni ) T v ni (cid:107) d ni, − r ni (cid:107) (cid:12)(cid:12)(cid:12) ,where d ni, and d ni, are the positions of the two +1 / i at time t n .Various approaches exist to determine defects in nematic liquid crystals, see [45] for acomparison of various methods. We here consider them as degenerate points of Q for Figure 2.
Particle velocity (top), eccentricity (middle) and defect asymmetry(bottom) of a single particle depending on self-propulsion strength v (left). Thevertical lines indicate the separation into three regimes. From left to right: (a) resting,(b) circular or spinning motion, and (c) translation. The error bars correspond tovalues at different times within the nonequilibrium steady state. For the circular orspinning regime the dynamically stable state is shown as a function of time (right) for v = 1 . v = 1 .
75 (blue) and v = 2 . which Q = Q = 0. This allows an easy detection of the position of a defect. Fora nematic liquid crystal in 2D two types of topological defects predominate +1 / − /
2. Considering the sign of δ = ∂Q ∂x ∂Q ∂y − ∂Q ∂y ∂Q ∂x allows to distinguish betweenthem. Due to the setting within a particle and the specified vertical anchoring only+1 / v and the defect positions are symmetric, (b) circularor spinning, the velocity fluctuates, which has an effect on the eccentricity and theasymmetry of the defect positions, and (c) translation, with increasing velocity, constantshape and symmetric arrangement of defects. The nonequilibrium steady state of thecircular or spinning regime is shown in Fig. 2(right) for different v . The oscillationsunderpin the correlation between velocity, eccentricity and defect asymmetry. Whilethey are strongest for v = 1 .
5, they decrease for v = 1 .
75 and are almost gone for v = 2 .
0, in accordance with the error bars in Fig. 2(left).To further highlight the connection between particle velocity, eccentricity andasymmetry of the defect positions Fig. 3(left) shows the particle shape together withthe principle eigenvector of the largest eigenvalue of Q (director field) and the defectpositions for various v . As the defects can also be located at the 0 . ψ ≤ Figure 3. (left) Particle shape, nematic liquid crystal field and position of +1 / v corresponding to the resting regime v = 1 .
25, the spinningregime v = 1 . , . , . v = 2 . , . , . , .
0. To highlightthe particle deformation a circular shape of the same area is plotted with the samecenter of mass. The arrows indicates the particle velocities. (right) Circular particlepath for v = 2 . regime the shape deformation is asymmetric with respect to the direction of movementand the defect asymmetry increases with v . The direction (up or down) depends on thesplitting of the +1 defect into two +1 / v the defects arelocated closer to the symmetry axis and the velocity of movement, which only slightlydeviates from the symmetry axis, increases. Fig. 3(right) shows a typical circularpath together with the corresponding director field and the velocity in the shown timeinstances. Due to the small radius of the circulation, which is almost independent onthe strength of activity v , we denote this motion as spinning in the following.
4. Collective behaviour
The behaviour in the resting and translation regimes essentially coincides with that ofthe polar active phase field crystal model [27]. This also remains true for the emergingcollective behaviour in unconfined and confined geometries, see Appendix A. We thusonly concentrate on the spinning regime in more detail. First, we characterize thebehaviour of interacting spinning particles in unconfined and confined geometries foran intermediate packing fraction of 0 .
57. To compute the packing fraction we considerthe 0.01-levelset of ψ to determine the area of the particles as A N = (cid:82) I { ψ> . } d r . Thearea of one particle A = A N /N ≈ . d with d = 4 π/ √ A N / | Ω | or A N / | Ω c | . We first consider 120 particles in the square domain Ω with periodic boundaryconditions. The self-spinning particles form crystalline structures with local triangularorder, with dislocations and regions with no particles, which dynamically rearrange.
Figure 4. (a) Three different time instances ( t = 100 , ,
900 from left to right)indicating the evolution to synchronized spinning for v = 2 . N = 120 in thesquare domain Ω with periodic boundary conditions. The particles are visualizedby the 0.15-levelset of ψ . The color corresponds with the direction of the arrowand indicates the direction of motion. The initial condition is a square lattice ofcircular particles with perturbed nematic fields. The perturbation leads to a randomdistribution of the resulting direction of motion. (b) The translational, rotationaland angular order parameters (from left to right) for corresponding simulations with v = 1 . v = 1 .
75 (blue) and v = 2 . t = 100 , ,
500 from left to right) indicating the evolution to synchronized spinningfor v = 2 . N = 100 in the circular domain Ω c . Visualisation and initial conditionsare as in (a). (d) The translational, rotational and angular order parameters (fromleft to right) for corresponding simulations with v = 1 . v = 1 .
75 (blue) and v = 2 . The particles are self-spinning and due to local interactions some particles also move topositions further away than the spinning radius. This is consistent for all considered self-propulsion strength v . However, only for v = 2 . φ T ≈
1, which indicates translational order or in the current context synchronizedspinning. This is confirmed by the angular order parameter φ O , which oscillates withfixed periodicity, see Fig. 4(a),(b). The behaviour in the circular confinement Ω c issimilar, see Fig. 4(c),(d). Also in this setting the translational order parameter φ T ≈ v = 2 . φ O oscillates with fixed periodicity. Thisnonequilibrium steady state is reached much faster than in the unconfined geometry.One could conclude that in this setting confinement helps to synchronise the particles.Deviations from synchronized spinning in the reached nonequilibrium steady state areonly found at the edge. This corresponds with regions with crystalline defects or noparticles. In the centre a crystal with perfect triangular lattice and synchronouslyspinning particles emerges.In contrast to the translational regime considered in Appendix A with translationaland rotational motion as nonequilibrium steady states, here both settings behave similar.In unconfined and confined geometries the initially independently spinning particlesundergo a transition to a nonequilibrium steady state of positional and orientationalorder, a synchronized spinning crystal. The simulations only confirm this for v = 2 . v remainsopen. All previous simulations consider a packing fraction of 0 .
57. We now vary this in thecircular confinement and observe different behaviour, see Fig. 5. For a smaller packingfraction of 0 .
48, at least within the considered time neither synchronized spinning norcrystal formation can be observed. Instead only small crystalline patches form anddynamically rearrange. Due to the available space particle interactions lead to localpositional rearrangements. Similar to the situation in unconfined geometries particlescan move to positions further away than the spinning radius. Fig. 5(b) shows thecoarse-grained trajectories of the particles (without the spinning component), and Fig.5(c) the bond number averaged over a larger time frame. The chaotic trajectories andthe low bond number for a packing fraction of 0 .
48 underpin the described behaviour.The bond number gives an indication of crystalline order and is computed for particle j as b n j = ( (cid:80) k ∈ N j e iθ njk ) /N j , with N j the nearest neighbors of particle j within a specifiedradius related to d and θ njk the angle between r nk − r nj and the x-axis. The consideredaveraged bond number ˆ b j accounts for the average over various times t n . ˆ b j = 0considers the situation of an isolated particle and ˆ b j = 1 that of a perfect triangularlattice, a particle with six neighbors. The nearest neighbors are constructed using aVoronoi-tesselation for the centers of mass. For the low packing fraction the system isin a fluid like regime with isolated particles which can easily change their positions.For packing fraction 0 .
57 the coarse-grained trajectories show more or lessstationary particles in the center and only small movements on the edge, see Fig. 5(b).This small edge currents differ from the behaviour in the unconfined geometry discussedabove. The emerging edge currents have an effect on the crystalline structure, which is0
Figure 5.
Varying packing fraction 0 . , . , . , . v = 2 . c . (a) Time instance t = 500.Visualisation and initial conditions as in Fig. 4. (b) Coarse-grained particlestrajectories in time interval (200 , , quantified by the averaged bond number, see Fig. 5(c). With ˆ b j ≈ .
66 and 0 . .
66 a triangular lattice still exist at least over some time spanbefore it gets rearranged, the averaged bond order for 0 . R from the center of the domain Ω c . Whilethere is almost no movement in orthoradial direction, the slight edge currents for packingfraction 0 .
57 in the radial component and their increased strength and extension towardsthe center for packing fractions 0 .
66 and 0 . Figure 6.
Kymographs corresponding to simulations in Fig. 5. (top) Orthoradialcomponent and (bottom) radial component of the particle velocity averaged over allparticles with distance R from center for varying packing fraction 0 . , . , . , . v = 2 . result of their spontaneous symmetry breaking. The majority decides on the emergingdirection at the edge, which persists towards the center.
5. Discussion
The proposed minimal model allows to explore different dynamical regimes by varyingone activity parameter only. The direct coupling of the self-propulsion strength v withthe internal nematic structure and the deformability of the particle leads to slightlydeformed resting states if v is below some threshold. It induces within a certainparameter range chirality, which leads to circular or spinning motion. If it is abovesome threshold a symmetric shape and translational motion emerges. All regimes havebeen investigated in unconfined and confined geometries. While the translational regimeis more or less identical with the behavior of the active polar phase field crystal model[27] and the observed rotational behavior in circular confinements reminiscent of variousexperiments, e.g., on highly concentrated bacterial suspension which self-organize into asingle stable vortex [46], collective behavior of self-circulating or self-spinning particlesare much less explored. Self-spinning particles are computationally considered in circularconfinements [47]. While fundamental issues differ, e.g., our particles are deformable,our spinning radius is significantly larger and our spinning velocity significantly lower,the emerging behaviour is similar. The competition between circular confinement, self-propulsion and steric interactions can lead to the emergence of edge flows and rotations.Within a wider perspective, similar edge flows and rotations have also been observed inchiral fluids [48]. It is shown that in systems of synchronously spinning particles bothparity (or mirror) symmetry and time-reversal symmetry are broken. Hydrodynamictheories with additional terms to account for these broken symmetries, e.g., rotationalviscosity tend to force the fluid as a whole to rotate with the angular velocity of thespinning particles. However, the motion of the fluid in the bulk is suppressed by friction.As a result, the fluid moves mostly at the boundary and the penetration depth ofthe vorticity of the fluid from the boundary into the bulk is controlled by the shearviscosity. These results are similar to the edge currents in our simulations and theirpropagation towards the center with increasing packing fraction. These similarities2with other systems which range from collective rotation of chiral molecules of a liquidcrystal [49], to sperm cells [50, 51], colloidal and millimeter scale magnetes [52, 53] androtating robots [54]. An interesting biological example is provided by Chlamydomonasreinhardtii (C. reinhardtii). This micron-sized unicellular algae is able to self-propel toperform translational motion, but also has the ability to self-rotate [55]. Rotation isused to sense the direction of light to optimize efficiency of phototaxis [56]. Figure 7. (a) Three different time instances ( t = 100 , ,
500 from left to right)indicating the evolution to collective migration for v = 4 . N = 120 in a squaredomain Ω with periodic boundary conditions. Visualisation and initial conditionsare as Fig. 4. (b) The translational, rotational and angular order parameters (fromleft to right) for corresponding simulations with v = 3 . v = 3 . v = 4 . t = 100 , ,
500 from left toright) indicating the evolution to collective migration for v = 4 . N = 100in the circular domain Ω c . Visualisation and initial conditions are as in (a). (d)The translational, rotational and angular order parameters (from left to right) forcorresponding simulations with v = 3 . v = 3 . v = 4 . Appendix A. Collective behaviour in translational/rotational regime
If the self-propulsion strength is above some threshold, a symmetric arrangement ofdefects and a symmetric shape of the particles is enforced. As a result, the particlesbehave as in the polar active phase field crystal model [27]. This behaviour leadsto collective translation in unconfined geometries and collective rotation in circularconfinements, see Fig. 7. It shows simulations for different values of v . Collectivebehaviour is only reached in the considered simulation time for the largest values, ϕ T ≈ v = 4 . | ϕ R | ≈ v = 3 , .
0. This behaviour is in qualitative agreementwith results in [27] and corresponding large scale simulations of multiphase-field models[61].
Acknowledgments
This work is funded by German Research Foundation within project FOR3013. Weused computing resources provided by JSC within project HDR06.
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