Synchronization and Liquid Crystalline Order in Soft Active Fluids

We introduce a phenomenological theory for a new class of soft active fluids with the ability to synchronize. Our theoretical framework describes the macroscopic behavior of a collection of interacting anisotropic elements with cyclic internal dynamics and a periodic phase variable. This system can (i) spontaneously undergo a transition to a state with macroscopic orientational order, with the elements aligned, a liquid crystal, (ii) attain another broken symmetry state characterized by synchronization of their phase variables, or (iii) a combination of both types of order. We derive the equations describing a spatially homogeneous system and also study the hydrodynamic fluctuations of the soft modes in some of the ordered states. We find that synchronization can promote or inhibit the transition to a state with orientational order, and vice versa. We provide an explicit microscopic realization: a suspension of microswimmers driven by cyclic strokes.

Figure 1

(a) The phase variable $\varphi$ associated with the internal cycle. (b) Two elements with orientations ${\widehat{\mathit{u}}}_1$ and ${\widehat{\mathit{u}}}_2$ and phases ${\varphi }_1$, ${\varphi }_2$. To visualize the phase, we schematically represent it as the length of the arrow. Panels (c) and (d) represent, respectively, the orientationally disordered and polar (ordered) state of elements that have identical phases.

Figure 2

Elements that are aligned but have different phases. (a) and (b) contribute differently to ${\mathbf{\Pi }}_0·{\mathbf{P}}_0$.

Figure 3

Phase diagram, in the plane ($\zeta$, $b$): interplay between phase and orientation dynamics. We discuss three important cases. (I) Polar order promotes the transition to synchronization in a region that is forbidden if they are decoupled provided $\xi >0$. (II) Likewise, synchronization promotes the transition to polar order in a region that is forbidden to the decoupled system, provided $v>0$. In both cases, the region of both synchronization and polar order is widened. (III) The opposite situation arises when $\xi$, $v<0$: there, synchronization inhibits the transition to polar order; similarly, the polar order works against synchronization. As a result, the region with both synchronization and polar order is reduced compared to the uncoupled system. Other intermediate cases are also possible.