Ido Lavi,1, 2 Ricard Alert,3, 4, 5 Jean-Fran¸cois Joanny,6, 7 and Jaume Casademunt11-Departament de F´ısica de la Materia Condensada,Universitat de Barcelona, Mart´ı i Franques 1, 08028 Barcelona, Spain,and UBICS (University of Barcelona Institute of Complex Systems)2-Center for Computational Biology, Flatiron Institute, 162 5th Ave, New York, NY 10010, USA3-Max Planck Institute for the Physics of Complex Systems, N¨othnitzerst. 38, 01187 Dresden, Germany4-Center for Systems Biology Dresden, Pfotenhauerst. 108, 01307 Dresden, Germany5-Cluster of Excellence Physics of Life, TU Dresden, 01062 Dresden, Germany6-Coll`ege de France, Paris, France7-Laboratoire PhysicoChimie Curie, Institut Curie,Universit´e Paris Sciences & Lettres (PSL), Sorbonne Universit´es, Paris, FranceActive nematics exhibit spontaneous flows through a well-known linear instability of the uniformly alignedquiescent state. Here we show that even a linearly stable uniform state can experience a nonlinear instability, resulting in a discontinuous transition to spontaneous flows. In this case, quiescent and flowing states may coexist. Through a weakly nonlinear analysis and a numerical study, we trace the bifurcation diagram of striped patterns and show that the underlying pitchfork bifurcation switches from supercritical (continuous) to subcritical (discontinuous) by varying the flow-alignment parameter. We predict that the discontinuous spontaneous flow transition occurs for a wide range of parameters, including systems of contractile flow-aligning rods. Our predictions are relevant to active nematic turbulence and can potentially be tested in experiments on either cell layers or active cytoskeletal suspensions.
(6) cal. Beyond the tricritical point (filled circle), we find a saddle-node bifurcation (open circles) where the stable and unstable branches join. Between the subcritical pitchfork and the saddle-node, including the rod-aligning regime ( < 1), both the quiescent ( = 0) and the stripe (a1) states are stable, with an intermediate unstable (a2) state (Fig. 2(a,c)). The system therefore displays bistability; it reaches either of the bistable states depending on initial conditions. To illustrate this behavior, we computationally integrate the dynamical problem (via a pseudo-spectral method detailed in ) to obtain the evolution of the system from the unstable state (a2) towards either stable stripe pattern (a1) (Movie 1) or the quiescent state (Movie 2).
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We impose periodic boundary conditions, (1) = (0) and (1) = (0), and focus on bend and splay solutions with the longest wavelength (k = 2). We utilize a dedicated shooting method to numerically solve Eq. (6), yielding the director angle profiles (y), as depicted in Fig. 2(a,b), along with their saturation angle sat. In Fig. 2(c,d), we map out the branches of both stable and unstable steady states by plotting sat against the flow alignement parameter . Each thin line corresponds to a different activity number, increased from A = Ac + (smallest sat) up to A = 1000 (thick lines). For contractile stresses (Fig. 2(c)), either increasing activity A or going towards more negative , the pitchfork bifurcation switches from supercritical to subcriti-
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For extensile stresses (Fig. 2(d)), in the rod-aligning regime ( < 1), the quiescent state is unstable to either bend or splay perturbations. The corresponding striped patterns (b1, b2) are stable (Fig. 2(b,d)). Their saturation angle asymptotically approaches the Leslie angle L (black lines) as activity increases (Fig. 2(d)). This angle is achieved as a balance between the director rotations due to flow alignment and vorticity under uniform shear. At high activity, the flow between the domain walls has indeed a nearly uniform shear (vx y, snapshots in Fig. 2(b)), and hence the saturation angle approaches L. Using our time-integration method , we obtain the evolution of the system from (t = 0) = /2 to the (b1) bend pattern (Movie 3), and from (t = 0) = 0 to the (b2) splay pattern (Movie 4). 3
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FIG. 2. Examples and bifurcation diagrams of striped solutions. In all panels, solid (dashed) lines indicate stable (unstable) solutions. The states dominated by splay (bend) distortions are shown in magenta (cyan). Black lines mark instances of the Leslie angle. Arrows illustrate the dynamic evolution of the system, shown in Movies 14. (a,b) Steady stripe patterns of wavelength = 1 for contractile (a) and extensile (b) stresses in the rod-aligning regime. Flat lines correspond to uniform quiescent states. In the 2D snapshots of states (a1, a2, b1, b2), the director n is indicated by the gray line-integralconvolution plot, the flow v is represented by the black arrows, and the splay (bend) deformation energy is proportional to the magenta (cyan) intensity. (c,d) Bifurcation diagrams showing the saturation angle of the stripe patterns (max for splay, min for bend) as a function of for increasing contractile (c) and extensile (d) activity. Each line corresponds to a different value o
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