We analyze a model of mutually-propelled filaments suspended in a
two-dimensional solvent. The system undergoes a mean-field isotropic-nematic
transition for large enough filament concentrations and the nematic order
parameter is allowed to vary in space and time. We show that the interplay
between non-uniform nematic order, activity and flow results in spatially
modulated relaxation oscillations, similar to those seen in excitable media. In
this regime the dynamics consists of nearly stationary periods separated by
"bursts" of activity in which the system is elastically distorted and solvent
is pumped throughout. At even higher activity the dynamics becomes chaotic.Comment: 4 pages, 4 figure

B. Chakraborty

J. D. Murray

L. Giomi

L. Mahadevan

M. F. Hagan

T. B. Liverpool

Physical Review Letters

English

arXiv

Crossref

Excitable Patterns in Active Nematics

We analyze a model of mutually propelled ﬁlaments suspended in a two-dimensional solvent. The system undergoes a mean-ﬁeld isotropic-nematic transition for large enough ﬁlament concentrations, and the nematic order parameter is allowed to vary in space and time. We show that the interplay between nonuniform nematic order, activity, and ﬂow results in spatially modulated relaxation oscillations, similar to those seen in excitable media. In this regime the dynamics consists of nearly stationary periods separated by ‘‘bursts’’ of activity in which the system is elastically distorted and solvent is pumped throughout. At even higher activity, the dynamics becomes chaotic.PhysicsAuthor's Origina

Giomi, Luca

Mahadevan, Lakshminarayanan

Chakraborty, Bulbul

Hagan, Michael

Harvard University - DASH

 Excitable Patterns in Active Nematics  (Article begins on next page)The Harvard community has made this article openly available.Please share how this access benefits you. Your story matters.Citation Giomi, Luca, Lakshminarayanan Mahadevan, Bulbul Chakraborty,and Michael F. Hagan.  2011. Excitable Patterns in ActiveNematics. Physical Review Letters 106(21): 218101.Published Version doi:10.1103/PhysRevLett.106.218101Accessed February 19, 2015 8:53:26 AM ESTCitable Link http://nrs.harvard.edu/urn-3:HUL.InstRepos:5128482Terms of Use This article was downloaded from Harvard University's DASHrepository, and is made available under the terms and conditionsapplicable to Open Access Policy Articles, as set forth athttp://nrs.harvard.edu/urn-3:HUL.InstRepos:dash.current.terms-of-use#OAPExcitable Patterns in Active NematicsL. Giomi,1,2 L. Mahadevan,1 B. Chakraborty,2 and M. F. Hagan21School of Engineering and Applied Sciences, Harvard University, Cambridge, MA 02138, USA2Martin A. Fisher School of Physics, Brandeis University, Waltham, MA 02454, USA(Dated: November 18, 2010)We analyze a model of mutually-propelled laments suspended in a two-dimensional solvent. Thesystem undergoes a mean-eld isotropic-nematic transition for large enough lament concentrationsand the nematic order parameter is allowed to vary in space and time. We show that the interplaybetween non-uniform nematic order, activity and ow results in spatially modulated relaxationoscillations, similar to those seen in excitable media. In this regime the dynamics consists of nearlystationary periods separated by \bursts" of activity in which the system is elastically distorted andsolvent is pumped throughout. At even higher activity the dynamics becomes chaotic.Colonies of motile microorganisms, the cytoskeleton andits components, cells and tissues have much in commonwith soft condensed matter systems (i.e. liquid crystals,amphiphiles, colloids etc.), but also exhibit phenomenathat do not appear in inanimate matter. These uniqueproperties arise when the constituent particles are ac-tive: they consume and dissipate energy to fuel inter-nal changes that generally lead to motion. When activeparticles have elongated shapes, as seen in cytoskeletallaments and some cells, they undergo orientational or-dering at high concentration to form liquid crystallinephases. The theoretical and experimental study of ac-tive materials has disclosed a wealth of emergent behav-iors, such as the occurrence of giant density uctuations[1], the emergence of spontaneously owing states (some-times referred to as ocking) [2], unconventional rheolog-ical properties [3] and new spatiotemporal patterns notseen in passive complex uids [4]. Recently Schaller et al.[5] observed collective motion, large-scale uctuations indensity and the degree of polar order, and swirling mo-tions in a motility assay consisting of highly concentratedactin laments propelled by immobilized molecular mo-tors in a planar geometry. However, the range of be-haviors realizable in active materials and the connectionbetween material characteristics and system dynamics re-main incompletely understood. In this letter, we showthat active systems exhibit behaviors similar to thoseof excitable systems, showing relaxation oscillations thatcouple activity to spontaneous pulsatile ow with quies-cent periods in between, similar to biological pumps. Wefurthermore show that systems which undergo nematicordering can give rise to large-scale swirling motions re-sembling those observed in Ref. [5] even in the absence ofpolar order. These behaviors were not seen in previousinvestigations because they arise only when the degree ofnematic order is allowed to uctuate in space and time.Our model consists of mutually-propelled elongatedparticles suspended in a solvent conned to two dimen-sions, as seen for example in recent motor-lament assays[5]. The dynamical variables in such a system are par-ticle concentrations c, the solvent ow eld v and thenematic tensor Qij = S(ninj   12ij), with S the nematicorder parameter and n the director eld, all of which areallowed to vary in space and time. We assume that thesuspension of rod-like active particles have length ` andmass M in a solvent of density solvent. The total densityof the system  = Mc+solvent is conserved, so the uidis assumed to be incompressible. The total number ofactive particles is also constant, thus the concentration cobeys the continuity equation:@tc =  r  (jp + ja); (1)where jp and ja are respectively the passive and activecontributions to the current density. The passive cur-rent density has the standard form jpi = cvi   Dij@jcwhere Dij = D0ij + D1Qij is the anisotropic diusiontensor, while the active current can be constructed phe-nomenologically to be of the form jai =  1c2@jQij orderived from microscopic models [6]. Here the factor c2reects the fact that activity arises from interactions be-tween pairs of rods while the constant 1 describes thelevel of activity and is proportional to the concentrationof motors and the rate of adenosine-triphosphate (ATP)consumption.The ow velocity obeys an active form of the Navier-Stokes equation@tvi = vi   @ip + @jij ; (2)with  the viscosity, p the pressure, and the active stresstensor ij given by:ij =  SHij + QikHkj   HikQkj + 2c2Qij (3)Here rv = 0, and the rst three terms in Eq. (3) repre-sent the elastic stress due to the liquid crystalline natureof the system, with Hij =  F=Qij the molecular ten-sor dened from the two-dimensional Landau-De Gennesfree energy:F=K =ZdA[12(r  Q)2 + 14(c   c) trQ2 + 14c(trQ2)2]where K is the splay and bending stiness in the one-constant approximation. At equilibrium, above the crit-arXiv:1011.3841v1  [cond-mat.soft]  16 Nov 20102FIG. 1: (Color online) The velocity led (left) and the direc-tor eld (right) superimposed to a density plot of the concen-tration and the nematic order parameter for 2 = 0:4 (top)and 2 = 3 (bottom). The colors indicate regions of large(green) and small (red) density and large (blue) and small(brown) nematic order parameter. For moderate values of 2the ow consists of two bands traveling in opposite directionswith the director eld is nearly uniform inside each band. Forlarge 2 the ow is characterized by large vortices that spanlengths of the order of the system size and the director eldis organized in grains.ical concentration c, S =p2trQ2 =p1   c=c con-sistent with hard-rod uid models where the isotropic-nematic (IN) transition is driven only by the concentra-tion of the nematogens. The last term was rst intro-duced in Ref. [8] and represents the tensile/contractilestress exerted by the active particles in the direction ofthe director eld n with 2 a second activity constant.Finally, the nematic order parameter Qij satises ahydrodynamic equation that can be obtained by con-structing all possible traceless-symmetric combinationsof the relevant elds, namely the strain-rate tensor uij =12(@ivj + @jvi), the vorticity tensor !ij = 12(@ivj   @jvi)and the molecular tensor Hij [7], so that[@t+vr]Qij =  1Hij +Suij +Qik!kj  !ikQkj (4)where  is an orientational viscosity, and the additionalterms on the right-hand side describe the coupling be-tween nematic order and ow in two dimensions, with the ow-alignment parameter which dictates how thedirector eld rotates in a shear ow and aects the owand rheology of active systems [2, 3].The dynamics of such an active nematic suspensionis governed by the interplay between the active forcing,whose rate  1a is proportional to the activity parameters1 and 2, and the relaxation of the passive structures,the solvent and the nematic phase, in which energy isdissipated or stored. The response of the passive struc-tures, as described here, occurs at three dierent timescales: the relaxational time scale of the nematic degreesof freedom `2=( 1K), the diusive time scale `2=D0,the dissipation time scale of the solvent L2=, whereL is the system size. While the presence of three dimen-sionless parameters makes for a very rich phenomenology,we temporarily assume that the three passive time scalesare of the same magnitude p. When a  p, the activeforcing is irrelevant and the system is just a traditionalpassive suspension. On the other hand, when a  p,the passive structures can balance the active forcing lead-ing to a stationary regime in which active stresses are ac-commodated via both elastic distortion and ow. Finally,when a  p the passive structures will fail to keep up,leading to a dynamical and possibly chaotic interplay be-tween activity, nematic order and ow. In the rest of thepaper, we quantify these dierent regimes. We rst makethe system dimensionless by scaling all lengths using therod length `, scaling time with the relaxation time of thedirector eld p = `2=( 1K) and scaling stress using theelastic stress  = K` 2.We start with a linear stability analysis of the hy-drodynamic equations about the homogeneous solutionletting ' = fc; Qxx; Qxy; 2!xyg = '0 + '(t) with'0 = fc0; S0=2; 0; 0g, and consider solutions with peri-odic boundary conditions on a square domain of the form'(x;t) =P1n= 1P1m= 1 'nm(t)exp[2iL (nx + my)].With this choice the linearized hydrodynamic equationsreduce to a set of coupled of linear ordinary dieren-tial equations for the Fourier modes 'nm: @t'nm =Anm'nm. The rst mode to become unstable is thetransverse excitation (n; m) = (0; 1) associated with theblock-diagonal matrix:A01 =a01 00 b01;where:a01 =  22L2 (2D0   D1S0) 42L2 1c20c4c0 S0  c0S20   42L2!and:b01 =  42L212(1   )S042L2 2c20   164L4 S0(1   )  42L2!:The mechanism that triggers the instability of the ho-mogeneous state relies on the coupling between local ori-entations and ow embodied in the matrix b01, whosediagonalization yields a lower critical value of 2:2 =42[2 + S20(1   )2]c20L2S0(1   ): (5)We see that the critical value of the activity required forinstability falls with increasing system size and increases3FIG. 2: (Color online) The hydrodynamic elds c, S,  andvx along the y axis for 2 = 0:4 The yellow region indicatesthe band visible in the top panels of Fig. 1with the viscosity of the solvent. In general, shear owcauses the director eld to rotate for  6= 1, which gener-ates elastic stress. For small activity, the elastic stinessdominates and suppresses ow, while above 2 we ob-serve collective motion.To go beyond this simple analysis, we numerically inte-grated the hydrodynamic equations on a two-dimensionalperiodic domain with initial congurations of a homoge-neous system whose director eld was aligned along thex axis subject to a small random perturbation in densityand orientation, with 1 = 2=2,  = c = D0 = D1 = 1, = 0:1 and L = 10. As predicted by the linear sta-bility analysis, at low activity the system relaxes to astationary homogeneous state with vx = vy = 0 andS =p1   c=c. Above the critical value 2, the systemforms two bands owing in opposite directions. The solu-tion is constant along the ow direction (see Fig. 1 top)while the direction of the streamlines (in this case alongthe x direction) is dictated by the initial conditions. Asshown in Fig. 2, the optima in the ow velocity corre-spond to the maximal distortion of the director eld n.Variations in concentration c and the nematic order pa-rameter S are of order 2% with a minimum in S at thecenter of a owing band due to the balance between dif-fusive and active currents. We note that these densitymodulations parallel to the nematic director are charac-teristic of systems with nematic order, while the densitybands orthogonal to the direction of alignment observedin Ref. [5] are consistent with polar order [2].Upon increasing the value of the activity parameter2, the spontaneously owing state evolves into a pul-satile spatial relaxation oscillator. Fig. 3 (top left panel)shows a plot of the x and y component of the ow veloc-ity in the center of the box for 2 = 1:5. In this regimethe dynamics consists of a sequence of almost station-ary passive periods separated by active \bursts" in whichthe director switches abruptly between two orientations.During passive periods, c and S are nearly uniform, thereFIG. 3: (Color online) Dynamics of active \burst" for 2 =1:5. The ow velocity at the point x = y = L=3 is shown asa function of time over the course of a director eld rotation(top left) and the director eld is shown for the three labeledtime points. Between two consecutive bursts the system is ho-mogeneous and uniformly aligned. During a burst, nematicorder is drastically reduced in the whole system and the di-rector undergoes a distortion with a consequent formation oftwo bands owing in opposite directions. After a burst, a sta-tionary state is restored with the director eld rotated of 90with respect to its previous orientation.is virtually no ow and the director eld is either paral-lel or perpendicular to the x direction. Eventually thisconguration breaks down and the director eld rotatesby 90 (see Fig. 3). The rotation of the director eld isinitially localized along lines, generating a band of owsimilar to those in Fig. 1 (top). The ow terminatesafter the director eld rotates and a uniform orientationis restored. The process then repeats.This rotation of the director eld occurs through a tem-porary \melting" of the nematic phase. As shown in Fig.3, during each passive period the nematic order param-eter is equal to its equilibrium value S0 =p1   c=c,but drops to  25S0 during rotation. The reduction oforder is system-wide, but, as shown in the middle in thebottom-left panel of Fig. 3, is most pronounced along theboundaries between bands. Without transient melting,the distortions of the director eld required for a burstare unfavorable for any level of activity.To illustrate the origin of the relaxation oscillationsit is useful to construct a simplied version of the hy-drodynamic equations by retaining the minimal featuresresponsible for the oscillatory phenomenon: the couplingbetween active forcing and the uid microstructure and4FIG. 4: (Color online) (Left) The average nematic order pa-rameter hSi and the total shear stress xy are shown overseveral bursts for 2 = 1:5. (right) The frequency of bursts isshown as a function of 2.the variable nematic order embedded in the Landau-DeGennes free energy. This can be achieved by approxi-mating Q2xx as a constant and uxx  0. Then, lettingu =  uxy and Q = Qxy, and dropping the coupling be-tween Qij and !ij, Eqs. (2) and (4) can be expressed inFourier space as:_ Q = aQ   bQ3   u (6a)_ u = k2(Q   u) (6b)where k is a wave number of an arbitrary spatial modeand a and b are constants. Eqs. (6) has the form of theFitzHugh-Nagumo model for excitable dynamical sys-tems. For  < 13(2a + k2) the system rapidly relaxesto a state characterized by a nite strain-rate that bal-ances the active stress: u = Q = p(a   )=b. For > 13(2a + k2), this state becomes unstable and thetrajectory converges to a limit cycle with a frequency  k2. As anticipated, when the active and passivetime-scales are comparable the active forcing is accom-modated by the microstructure leading to a distortionof the director eld and a steady ow. However, whenthe active forcing rate is increased, the microstructuredynamics lag, resulting in oscillatory dynamics. Clearly,the full system exhibits a much richer behavior than thatcaptured by Eqs. (6), but the qualitative trends persist.For instance, the increase in the slope of  shown in Fig.4 is not simply associated with the excitation of a secondspatial mode of larger wave-number and is more likelydue to the dynamics of the concentration eld, whichhas been ignored in Eqs. (6).When the activity 2 is further increased, the sequenceof low activity periods and bursts becomes more complexand eventually chaotic. We emphasize that the oscilla-tory dynamics and the chaotic ows discussed below donot occur in our equations unless the magnitude of thenematic order parameter is allowed to uctuate. Fig. 1(bottom) shows a typical snapshot of the ow velocityand the director eld superimposed to a density plot ofthe concentration and the nematic order parameter re-spectively. The ow is characterized by large vorticeswith positions related to \grains" in which the directoreld is uniformly oriented. The grain boundaries are ofthe order of the system size and are the fastest ow-ing regions in the system. Thus the dynamics in thisregime is characterized by sets of grains of approxima-tively uniform orientation that swirl around each otherand continuously merge and reform.In summary, we analyzed the hydrodynamics of ac-tive nematic suspensions in two dimensions. By allowingspatial and temporal uctuations in the nematic orderparameter we have observed a rich interplay between or-der, activity and ow. Signicantly, we nd that allowinguctuations in the magnitude of the order parameter Squalitatively changes the ow behavior as compared tosystems in which S is constrained to be uniform. Finally,we note that both the ip-op dynamics (Fig. 3) and theswirling motion (Fig. 1 bottom) resemble behavior ob-served in the motility assay experiments of Schaller et al.[5]. Our analysis suggests that these classes of patternscan emerge even in the absence of polar order.We gratefully acknowledge support from the NSFBrandeis MRSEC (LG, BC, and MFH), the NSF Har-vard MRSEC (LG,LM), NIAID R01AI080791 (MFH),the Harvard Kavli Institute for Nanobio Science & Tech-nology (LG, LM), and the MacArthur Foundation (LM).[1] V. Narayan, S. Ramaswamy and N. Menon, Science 317,105 (2007). J. Deseigne, O. Dauchot and H. Chat e, Phys.Rev. Lett. 105, 098001 (2010).[2] R. Voituriez, J. F. Joanny and J. Prost, Europhys. Lett.70, 118102 (2005). D. Marenduzzo, E. Orlandini, M. E.Cates, and J. M. Yeomans, Phys. Rev. E 76, 031921(2007). L. Giomi, M. C. Marchetti and T. B. Liver-pool, Phys. Rev. Lett. 101, 198101 (2008). S. Mishra, A.Baskaran and M. C. Marchetti, Phys. Rev. E 81, 061916(2010).[3] M. E. Cates, S. M. Fielding, D. Marenduzzo, E. Orlan-dini and J. M. Yeomans, Phys. Rev. Lett. 101, 068102(2008). A. Sokolov and I. S. Aranson, Phys. Rev. Lett.103, 148101 (2009). L. Giomi, T. B. Liverpool and M. C.Marchetti, Phys. Rev. E 81, 051908 (2010). D. Saintillan,Phys. Rev. E 81, 056307 (2010).[4] D. Saintillan and M. J. Shelley, Phys. Rev. Lett. 100,178103 (2008). H. Chat e, F. Ginelli and R. Montagne,Phys. Rev. Lett. 96, 180602 (2006). F. Ginelli et al. Phys.Rev. Lett. 104, 184502 (2010)[5] V. Schaller, C. Weber, C. Semmrich, E. Frey and A. R.Bausch, Nature 467, 73 (2010).[6] A. Ahmadi, M. C. Marchetti and T. B. Liverpool, Phys.Rev. E 74, 061913 (2006). T. B. Liverpool and M. C.Marchetti, Hydrodynamics and rheology of active polar l-aments, in Cell Motility, P. Lenz ed. (Springer, New York,2007).[7] P. D. Olmsted and P. M. Goldbart, Phys. Rev. A 46, 4966(1992).[8] T. J. Pedley and J. O. Kessler, Annu. Rev. Fluid Mech.24, 313 (1992).

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Excitable Patterns in Active Nematics

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