Diversity of phase transitions and phase separations in active fluids

Active matter is not only indispensable to our understanding of diverse biological processes, but also provides a fertile ground for discovering novel physics. Many emergent properties impossible for equilibrium systems have been demonstrated in active systems. These emergent features include motility-induced phase separation, a long-ranged ordered (collective motion) phase in two dimensions, and order-disorder phase coexistences (banding and reverse-banding regimes). Here, we unify these diverse phase transitions and phase coexistences into a single formulation based on generic hydrodynamic equations for active fluids. We also reveal a novel comoving coexistence phase and a multicritical point.

Figure 1

Phase coexistence in a one-dimensional active fluid. The mass density $\rho$ (upper panels) and total flux $J$ (lower panels) profiles of the four distinct types of phase coexistences: (a) a dilute (d) and disordered (D) phase coexisting with a condensed (c) and disordered (D) phase, denoted as dD-cD; (b) two possible dO-cO coexistences where condensed and dilute phases are transported in the same direction (blue) or in opposite directions (red); in both cases shown here, the condensed ordered phase is moving to the right as the magnitude of the flux in the condensed phase $|J_{\mathrm{c}}|$ is higher than that in the dilute phase $|J_{\mathrm{d}}|$; (c) two examples of dD-cO coexistence, where in both cases, the condensed ordered phase is moving to the right; and (d) dO-cD coexistence, the condensed disordered phase is here moving to the left due to differential adsorption at the two interfaces. The profiles shown here correspond to the stationary states of the hydrodynamic EOM. In the lower panels, the symbols correspond to the symbols shown in Fig. 2. Movies of these cases can be found in the Supplemental Material [38].

Figure 2

Phase diagram of a one-dimensional active fluid. The shaded area corresponds to the instability regime obtained from the linear stability analysis of the hydrodynamic model (6) and (7). The edges of the instability region correspond to the spinodal lines. Black lines correspond to the binodal lines (or coexistence lines), which were obtained via direct numerical simulations of the one-dimensional hydrodynamic model in (6) and (7). Along with the homogeneous disordered (D) and homogeneous ordered (O) phases, we observe four coexistence regions which are delimited by the binodal lines (and horizontal dashed lines): A coexistence of a dilute disordered phase with a condensed disordered phase (dD-cD, blue), a coexistence of a dilute disordered phase with a condensed ordered phase (dD-cO, green), a coexistence of a dilute ordered phase with a condensed disordered phase (dO-cD, red) and a coexistence of a dilute ordered phase with a condensed ordered phase (dO-cO, orange). The symbols (and their colors) denote the locations in phase space from which we extracted the profiles shown in Fig. 1.

Figure 3

Multicritical point. In this schematic, the dD-cD coexistence (or MIPS) occurs above the red surface (at high $K_1$), and the corresponding critical phase separation (indicated by the red line) belongs to the Ising universality class [48] (but see also Refs. [49, 50]). In contrast, an ordered phase coexists with a disordered phase (dD-cO or dO-cD) between the blue and gray surfaces, and a spatially homogeneous ordered phase (O) exists below the gray surface. The corresponding order-disorder critical line is indicated by the blue line [42]. When the blue line and the red lines are tangent to each other (indicated by the yellow dot), a putatively novel multicritical point emerges.