Complex Spontaneous Flows and Concentration Banding in Active Polar Films
aa r X i v : . [ c ond - m a t . s o f t ] M a y Complex Spontaneous Flows and Concentration Banding in Active Polar Films
Luca Giomi, M. Cristina Marchetti, and Tanniemola B. Liverpool Physics Department and Syracuse Biomaterials Institute, Syracuse University, Syracuse, NY 13244, USA Department of Mathematics, University of Bristol, Clifton, Bristol BS8 1TW, U.K. (Dated: October 26, 2018)We study the dynamical properties of active polar liquid crystalline films. Like active nematicfilms, active polar films undergo a dynamical transitions to spontaneously flowing steady-states.Spontaneous flow in polar fluids is, however, always accompanied by strong concentration inhomo-geneities or “banding” not seen in nematics. In addition, a spectacular property unique to polaractive films is their ability to generate spontaneously oscillating and banded flows even at lowactivity. The oscillatory flows become increasingly complicated for strong polarity.
Active materials are a new class of soft materials main-tained out of equilibrium by internal energy sources.There are many examples in biological contexts, includ-ing bacterial colonies [1], purified extracts of cytoskeletalfilaments and motor proteins [2], and the cell cytoskele-ton [3]. A non-biological example is a layer of vibratedgranular rods [4]. The key property that distinguishes ac-tive matter from more familiar non-equilibrium systems,such as a fluid under shear, is that the energy input thatmaintains the system out of equilibrium comes from eachconstituent, rather than the boundaries. Each active par-ticle consumes and dissipates energy going through a cy-cle that fuels internal changes, generally leading to mo-tion. The experimental systems studied to date typicallyconsists of elongated active particles of two types: polarparticles, with a head and a tail, and apolar ones thatare head-tail symmetric. Active suspensions can thenexist in various liquid crystalline states, with novel struc-tural and rheological properties [5, 6]. Apolar particlescan form phases with nematic order, characterized by amacroscopic axis of mean orientation identify by a unitvector n and global symmetry for n → − n , as in equilib-rium nematic liquid crystals. Polar particles can order inboth nematic and polar phases. The polar phase is againcharacterized by a mean orientation axis p , but p = − p .The protein filaments which are the major component ofcell extracts are generally polar and these extracts cantherefore have both nematic and polar phases.Conventional liquid crystals exhibit a rich non-equilibrium behavior when subject to external forcing,such as shear or applied magnetic and electric fields. Thisincludes transitions to stable statically distorted defor-mations of the director field (Freedericksz transition [7]),shear banding [8], and even the onset of turbulent andchaotic behavior in the presence of shear [9]. Active liq-uid crystals exhibit a similar wealth of phenomena dueto internal forcing, i.e., spontaneously. A striking prop-erty of active nematic liquid crystal films is the onsetof spontaneous flow above a critical film thickness firstidentified by Voituriez et al [10]. This phenomenon isanalogous to the Freedericksz transition of a passive ne-matic in an applied magnetic field, but the flowing state is driven by the internal activity of the system - hence thename. This prediction was obtained by analytical stud-ies of the phenomenological hydrodynamic equations ofan an active nematic film in a one-dimensional geome-try. More recently, Marenduzzo et al [11] have employedhybrid lattice-Boltzmann simulations to study the activenematic hydrodynamics in both 1D and 2D geometry andhave mapped out the phenomenon in parameter space.In this letter we show that active polar fluids exhibitan even richer behavior. First, like active nematics, po-larized active liquid crystals exhibit steady spontaneousflow. Unlike active nematics, however, where the filamentconcentration remains practically uniform in the sponta-neously flowing state, spontaneous flow in polar fluidsis accompanied by “concentration banding”, i.e., a sharpgradient in the concentration of filaments across the film.The concentration banding is due to active couplings ofconcentration gradients to the polar director in the hy-drodynamic equations that are allowed only in fluids withpolar symmetry. Upon increasing the magnitude of these polar couplings, the steady state becomes unstable andthe system undergoes a further transition to a dynamicstate with bands of oscillating concentration and orienta-tion. In the oscillatory regime, travelling bands nucleateand oscillate from one end of the film to the other. Foreven larger couplings the oscillatory behavior becomesincreasingly complex with the appearance of multiple fre-quencies with incommensurate ratios between the periodsof the orientational and concentration oscillations.Hydrodynamic equations for a two component ac-tive suspension have been written down phenomenologi-cally [12] and derived from a microscopic model [14]. Therelevant hydrodynamic variables are the concentration c of filaments and the total density ρ and momentum den-sity g = ρ v of the suspension, with v the flow velocity.We consider an incompressible film, with ρ = constantand ∇ · v = 0, and macroscopic dimensionless polarity P = | P | p , with direction characterized by a unit vector p = (cos θ, sin θ ), the polar director.The hydrodynamic equations for two-component po-lar suspensions have been derived elsewhere by coarse-graining a microscopic model of the dynamics of inter-acting motors and filaments [14]. The active suspensionis an intrinsically non-equilibrium system and cannot bedescribed by a free energy. However, for clarity of pre-sentation we introduce here the equations phenomeno-logically and write all equilibrium-like terms (i.e., thoseterms that are also present in an equilibrium polar sus-pension) in terms of derivatives of a non-equilibrium ana-logue of a free energy, given by [15] F = Z r ( C (cid:18) δcc (cid:19) + a | P | + a | P | + K ∇ · P ) + K ∇ × P ) + B δcc ∇ · P + B | P | ∇ · P + B c | P | P · ∇ c (cid:27) , (1)with C the compressional modulus and K and K thesplay and bend elastic constants, both taken equal to K below. The last three terms on the right hand side ofEq. (1) couple concentration and splay and are presentin equilibrium polar suspensions ( B = B = B = B below). The dynamics of the concentration and of thepolar director is described by ∂ t c = − ∇ · (cid:20) c ( v − β ′ cℓ P ) + Γ ′ h − Γ ′′ ∇ (cid:16) δFδc (cid:17) (cid:21) , (2a) h ∂ t + ( v + βcℓ P ) · ∇ i P i + ω ij P j = λu ij P j + Γ h i − Γ ′ ∂ i (cid:16) δFδc (cid:17) , (2b)where ℓ is the length of the filaments, u ij = ( ∂ i v j + ∂ j v i ) and ω ij = ( ∂ i v j − ∂ j v i ) are the rate-of-strain andvorticity tensors, and h = − δF/δ P the molecular field.Here Γ, Γ ′ and Γ ′′ are kinetic coefficients and β and β ′ areactive parameters. The equation describing momentumconservation is written in the Stokes approximation as ∂ j σ ij = 0. The stress tensor σ ij is the sum of reversible,dissipative and active contributions, σ ij = σ rij + σ dij + σ aij .The reversible part is written in an equilibrium-like form, σ rij = − δ ij Π − λ (cid:2) P i h j + P j h i (cid:3) − λδ ij P · h + 12 (cid:2) P i h j − P j h i (cid:3) , with Π the pressure and λ and λ alignment parameters.The dissipative part of the stress tensor is written as σ dij = ηu ij assuming a single viscosity η . Finally, thereare additional stresses induced by activity given by σ aij = c ℓ Γ h − δ ij Π a + αP i P j + ℓ β ′′ ( ∂ i P j + ∂ j P i ) i . with Π a the active part of the pressure. There are twocontributions to the active stress tensor. The first ( ∼ α )describes active stresses that arise from contractile (if α >
0) forces induced by activity. This term is present
FIG. 1: (Color online) (a) Sketch of the film geometry. (b-d)Solutions of Eqs. (3a), (3b) for λ = 0 . ξ = 0 . D = 1, e C = 0 . η Γ = 0 . w = 0 . e α = 0 .
08 and variable e β . in both nematic and polar liquid crystals and its effectshave been studied before. The second term ( ∼ β ′′ ) arisesfrom “self-propulsion” of the active units and is exclusiveto polar systems. The same mechanism is also responsi-ble for the “convective” terms proportional to β and β ′ inEqs. (2a) and (2b). For a motor/filament mixture, all ac-tive contributions are proportional to the mean rate ∆ µ of ATP consumption, which is the internal driving forcefor the system. β -type terms have dimensions of veloc-ity and have been estimated in the microscopic model as β ∼ e mu , with e m a dimensionless concentration of motorclusters and u ∼ ∆ µ the mean velocity at which motorclusters step along the filaments [13]. Previous work onconfined active films [10, 11] has been limited to activenematics with all the β terms equal to zero. Here forthe first time we incorporate the polar active terms andanalyze their role in controlling non-equilibrium effectsin active films. Hereafter we will assume β ′ = β ′′ = β .We consider a two-dimensional active polar suspensionwith polarization of uniform magnitude and discuss thedynamics of the hydrodynamic fields c , v and p . For sim-plicity we set | P | = 1. The film sits on a solid plane at y = 0 and is bound by a free surface at y = L (Fig. 1a).The discussion below is easily extended to other bound-ary conditions. The film extends to infinity in the x di-rection and we assume translational invariance along x .The Stokes equation requires ∂ y σ yy = 0 and ∂ y σ xy = 0.The first of these two conditions fixes the pressure in thefilm. The second, together with the boundary condition σ xy ( L ) = 0, requires σ xy = constant = 0 throughout thefilm. We also assume no-slip boundary conditions at thesubstrate, so that v x (0) = 0.It is convenient to work with dimensionless quantitiesby introducing the time scale τ = ℓ / (Γ K ). Letting z = y/ℓ , τ = t/τ , φ = c/c and specializing Eqs. (2a) and FIG. 2: (Color online) On the left discrete Fourier transformsof θ ( zℓ = L/ , τ ) (center of the film) for λ = 0 . ξ = 0 . D = 1, e C = 0 . η Γ = 0 . w = 0 . e α = 0 . e β = 15 (a)18 (b) and 20 (c). On the right space-time plots of φ ( z, τ ). (2b) to our quasi-one-dimensional geometry, we obtain ∂ τ φ = ∂ z n e βφ sin θ + λ e u sin θ sin 2 θ + (cid:2) D (1 − ξ sin θ ) − w cos θ (cid:3) ∂ z φ o , (3a) ∂ τ θ = (1 − w cos θ ) ∂ z θ + w sin 2 θ ( ∂ z θ ) − φ e β sin θ∂ z θ + w cos θ∂ z φ − e u (1 − λ cos 2 θ ) , (3b)where e β = βc ℓτ , w = 2 ℓB/K , ξ = Γ ′ / (ΓΓ ′′ ), and D = Γ ′′ C/ ( c Γ K ). The dimensionless rate-of-strain e u =2 u xy τ can be obtained from the condition σ xy = 0, e u = − η Γ + λ sin θ n λw sin 2 θ sin θ ( ∂ z θ ) + h w (1 − λ cos 2 θ ) − λ e C sin θ i cos θ∂ z φ − e βc ℓ φ sin θ∂ z θ + e αφ sin 2 θ o , (4)where e α = αc ℓ / (Γ K ) and e C = ℓ Γ ′ C/ (Γ K ) . The hy-drodynamic equations for a two component active ne-matic suspension are obtained from the above by letting˜ β = 0 and w = 0. The terms proportional to w arealso present in passive polar fluids as they arise from thefact that the polar symmetry allows the coupling propor-tional to B between splay and density fluctuations. Theterms proportional to e β are intrinsically non-equilibriumpolar terms. Finally, the case of an incompressible one-component nematic fluid, investigated by Voituriez etal [10] and by Marenduzzo et al [11], can be recoveredfrom our equations by setting e β = w = 0 and assuminga constant concentration φ . Eqs. (3b) and (3a) are integrated numerically withboundary conditions θ (0 , τ ) = θ ( L/ℓ, τ ) = 0, ∂ z φ (0 , τ ) = ∂ z φ ( L/ℓ, τ ) = 0 (i.e. j y (0 , t ) = j y ( L, t ) = 0). The ini-tial conditions on θ and φ are chosen as random, withthe constraint h θ ( z, i = 0 and h φ ( z, i = 1 where h · i stands for a spatial average. Steady spontaneous flow:
Both the polar and the apo-lar systems exhibit a Freedericksz-like transition betweena state where the director field is constant and parallelto the walls throughout the channel to a non-uniformlyoriented state in which the system spontaneously flows inthe x -direction. The transition can be tuned by chang-ing either the film thickness or the activity parameter e α .Fig 1 shows a numerical solution of Eqs. (3a) and (3b)for fixed e α and variable e β , with L/ℓ = 10. As the activevelocity e β is increased, the maximum tilt θ m decreasesand the alignment is progressively restored. Remarkablythe variation in the concentration φ across the film issignificantly stronger than in the apolar case (solid greencurve in Fig. 1) with a relative difference between thehighest and the lowest values up to 50%. This “concen-tration banding” is a characteristic of polar active sys-tems. It is a consequence of the active β ′ coupling in Eq.(2a) resulting from self-propelled ‘convection’ of the ac-tive elements along the local polarization direction. Thevarying local polarization angle required for spontaneousflow therefore leads to an even stronger variation in thelocal concentration. Close to the transition, e α c ( e β ), thecoupling between the polar director and concentrationalso leads to an asymmetric director profile across thefilm. We also point out that in the absence of this polaractive term, there are equilibrium-like gradient couplingsbetween local director and density. These are, however,much weaker since they occur at higher order in gradi-ents. In contrast, the active nematic shows a negligibleconcentration gradient even for anomalously large valuesof the contractile activity parameter, e α ≫ e α c ( e β ). Spontaneous oscillations:
Upon further increasing e β ,spontaneous oscillations of φ ( z, τ ) and θ ( z, τ ) are ob-tained. The coupled dynamics of the two fields givesrise to travelling waves of concentration and orientationbands. Initially only one frequency is observed, but theoscillations become more complicated as e β increases. Us-ing Fourier decomposition we find that this is due to theappearance of additional frequencies at different valuesof e β for concentration and orientation bands (see Fig 2).A phase diagram in the ( e α, e β ) − plane is displayed inFig. 3. It shows transitions between stationary (S) flow,spontaneous steady flow (SF) and spontaneous periodic(oscillatory) flow (PF).We can understand the phase behavior close to astationary homogeneous state ( φ = φ , θ = 0, ˜ u =0) by expanding θ ( z ) = δθ + ( z ) e iωt + δθ − ( z ) e − iωt and φ ( z ) = φ + δφ + ( z ) e iωt + δφ − ( z ) e − iωt . The boundaryconditions require φ ± ( z ) = P ∞ n =1 a ± n cos ( nπℓz/L ) and FIG. 3: (Color online) Phase-diagram in the ( e α, e β ) − plane.The points have been obtained numerically using λ = 0 . ξ = 0 . D = 1, e C = 0 . η Γ = 0 . w = 0 .
13 and
L/ℓ =10. For e β < e β TP upon increasing e α the system undergoesa transition between stationary homogeneous state (S) andinhomogeneous steady flow (SF). Above the “tricritical point”(TP) the spontaneous flow becomes oscillatory (PF). θ ± ( z ) = P ∞ n =1 b ± n sin ( nπℓz/L ). The first unstable modesare a ± , b ± and a linear stability analysis shows that thereis a steady-state ( ω = 0) instability at e α c ( e β ) = η Γ(1 − w )1 − λ (cid:18) πℓφ L (cid:19) + w e β [ η Γ + (1 − λ ) ](1 − λ )( D − w ) , (5)to a steady spontaneously flowing state with concentra-tion banding. Oscillatory modes with frequency ω c ∼ φ ( ℓ/L )( ω e β ) / appear beyond a “tricritical point” (TP) e α TP = η Γ( π/φ ) ( ℓ/L ) ( D + 1 − w )1 − λ , e β TP = ( π/φ ) ( ℓ/L ) ( D − w ) w [1 + (2 η Γ) − (1 − λ ) ] . This linearized analysis predicts the positions of the“phase boundaries” S-SF and S-PF in quantitative agree-ment with the numerical solution. The boundary SF-PFhas been obtained only numerically. The appearance ofspontaneous oscillations results from the coupled motionof concentration and director orientation bands due toboth the convective active polar coupling ( β ) and thepassive polar coupling ( w ) of director and concentration.Upon increasing e β the oscillatory behavior becomes in-creasingly complex, but we have not been able to observefully fledged chaos for reasonable values of e β .We have studied the dynamical properties of thin filmsof active polar fluids and found a rich variety of com-plex behaviors which should be observable experimen-tally in polar active systems. Using microscopic modelsof motor-filament coupling, it was estimated in [13] that β ≫ α in microtubules/kynesin mixtures, while β ≤ α inactomyosin systems. This suggests that in-vitro micro-tubules/kynesin mixtures may be the best candidate for the observation of the oscillating bands predicted here.It should, however, be noted that filament treadmillingalso leads to terms with polar symmetry at the contin-uum level, where in this case β would be proportionalto the polymerization rate [16]. The intriguing possi-bility that our findings may be relevant to treadmillingacto-myosin systems and therefore have implications forlamellipodium dynamics will be explored elsewhere.MCM and LG were supported on NSF grants DMR-0305407 and DMR-0705105. MCM acknowledges thehospitality of the Institut Curie and ESPCI in Paris andthe support of a Rotschild-Yvette-Mayent sabbatical fel-lowship at Curie. TBL acknowledges the hospitality ofthe Institut Curie in Paris and the support of the RoyalSociety and the EPSRC under grant EP/E065678/1. LGwas supported on a Graduate Fellowship by the SyracuseBiomaterials Institute. Finally, we thank Jean-FrancoisJoanny and Jacques Prost for many useful discussions. [1] C. Dombrowski, L. Cisneros, S. Chatkaew, R.E. Gold-stein, and J.O. Kessler, Phys. Rev. Lett. , 098103(2004).[2] F. N´ed´elec, T. Surrey and E. Karsenti, Curr. Opin. CellBiol. , 118 (2003).[3] J. Howard, Mechanics of Motor Proteins and the Cy-toskeleton (Sinauer, New York, 2002).[4] D. L. Blair, T. Neicu and A. Kudrolli, Phys. Rev. E ,058101 (2002).[6] Y. Hatwalne, S. Ramaswamy, M. Rao and R. A. Simha,Phys. Rev. Lett. , 118191 (2004); T. B. Liverpool, M.C. Marchetti, Phys. Rev. Lett. , 268101 (2006).[7] P. G. de Gennes and J. Prost, The Physics of LiquidCrystals , 2nd ed. (Oxford Science, New York, 1993).[8] P. D. Olmsted, Rheol. Acta. , 283 (2008).[9] B. Chakrabarty, M. Das, C. Dasgupta, S. Ramaswamyand A. K. Sood, Phys. Rev. Lett. , 055501 (2004).[10] R. Voituriez, J. F. Joanny and J. Prost, Europhys. Lett. , 118102 (2005).[11] D. Marenduzzo, E. Orlandini and J. M. Yeomans, Phys.Rev. Lett. , 118102 (2007); D. Marenduzzo, E. Orlan-dini, M. E. Cates and J. M. Yeomans, Phys. Rev. E ,031921 (2007).[12] J. F. Joanny, F. J¨ulicher, K. Kruse and J. Prost, New J.Phys. , 422 (2007).[13] T. B. Liverpool and M. C. Marchetti, Europhys. Lett. , 846 (2005); A. Ahmadi, M. C. Marchetti and T. B.Liverpool, Phys. Rev. E , 061913 (2006).[14] T. B. Liverpool and M. C. Marchetti, in Cell Motility , P.Lenz, ed. (Springer, New York, 2007); and in preparation.[15] D. Blankschtein and R. M. Hornreich, Phys. Rev. B ,3214 (1985); W. Kung, M. C. Marchetti and K. Saunders,Phys. Rev. E , 031708 (2006).[16] K. Kruse, J.-F. Joanny, F. J¨ulicher and J. Prost Phys.Biol.16