Bubble formation in active binary mixture model

TL;DR

This work introduces an active binary lattice model where solutes and solvents exchange positions and possess self-propulsion. It demonstrates that bubble formation during active phase separation is controlled by the asymmetry of activity between solute and solvent, with moderate asymmetry promoting a broad, power-law distribution of solvent bubbles and strong symmetry suppressing bubbles, allowing conventional critical analysis. Mean-field theory (both single-site and four-site) provides a mechanistic picture: interface polarity alignment and Lifshitz-point–driven instabilities explain bubble growth, while local correlations are essential to recover correct spinodal structure and critical points. In the symmetric limit, finite-size scaling indicates Ising-like critical behavior, suggesting a route to study universality in nonequilibrium phase transitions using bubble-suppressed active matter.

Abstract

Phase separation, the spontaneous segregation of density, is a ubiquitous phenomenon observed across diverse physical and biological systems. Within a crowd of motile elements, active phase separation emerges from the interplay of activity (i.e., self-propulsion) and density interactions. A striking feature of active phase separation is the persistent formation of dilute-phase bubbles within the dense phase, which has been explored in theoretical models. However, the fundamental parameters that systematically control bubble formation remain unclear in conventional self-propelled particle models. Here, we introduce an active binary mixture model in which solutes and solvents dynamically exchange positions on a lattice; both solutes and solvents are self-propelled particles, but solvents play a role analogous to empty space in typical dry active matter. Within this model, we find that spontaneous bubble formation of solvents can be tuned by activity asymmetry, which is the difference between the solute and solvent activities. Numerical simulations reveal that moderate solvent activity enhances bubble formation, while larger solvent activity, comparable to solute activity, suppresses it. By employing mean-field theory, which captures essential phase behaviors, we consider the mechanism for the enhancement of bubble formation induced by solvent activity. Beyond these findings, when solute and solvent activities are equal, we apply the finite-size scaling analysis to estimate the critical exponents for active phase separation under the suppression of bubbles. Our findings establish activity asymmetry as a key control parameter for active matter phase transitions, offering new insights into universality in nonequilibrium systems.

Figures (17)

• Figure 1: Active binary mixture model. Each site of a rectangular lattice with size $(L_x, L_y)$ is occupied by a solute with polarity (black particle with arrow) or a solvent with polarity (white particle with arrow). Each solute (solvent) stochastically (i) rotates its polarity by $\pm 90^\circ$ or (ii) exchanges its position with a neighboring solvent (solute). No exchange between two solutes or between two solvents is allowed. The polarity rotation rate is $\gamma$ [see (i)], the passive exchange rate is $1$ [see (ii-1)], and the active exchange rate for solute (solvent) is $\varepsilon_\mathrm{solute}$ ($\varepsilon_\mathrm{solvent}$) [see (ii-2)].
• Figure 2: Qualitative phase diagram for active phase separation. (a, b) Heatmap of $\Delta \rho$, the shifted difference in solute density between the dense and dilute regions, for $(L_x, L_y) = (100, 50)$ with (a) $\rho = 0.5$ or (b) $\rho = 0.8$. The brighter color indicates a higher degree of phase separation, and the white lines represent constant-$\Delta \rho$ lines. We show typical snapshots at several sets of activities $(\varepsilon_\mathrm{solute}, \varepsilon_\mathrm{solvent})$, where black and white colors indicate solutes and solvents, respectively. We also present several snapshots for a larger system size [$(L_x, L_y) = (400, 200)$]. (c) Typical time evolution of bubbles in the steady state for $(\varepsilon_\mathrm{solute}, \varepsilon_\mathrm{solvent}) = (10, 4)$. Bubbles form and grow within the bulk solute-rich phase and merge into the bulk solvent-rich phase. We conducted Monte Carlo (MC) simulations of the active binary mixture model (Fig. \ref{['fig_model']}) by sequentially updating the configuration according to the probabilities of rotation and exchange based on time discretization with interval $\Delta t$ for a single MC step. We used $\Delta t \simeq 0.0413$ for the case of (c), where time is measured in units of the inverse of the passive exchange rate [see Eq. \ref{['appeq_timestep']} in Appendix \ref{['app_implementation_of_stochastic_dynamics']} for the general parameter dependence of $\Delta t$].
• Figure 3: Bubble formation controlled by solvent activity. (a, b) Bubble size distribution divided by the total number of solvents, $N_\mathrm{bubble} (A) / N_\mathrm{solvent}$, at $\rho = 0.5$ and $\varepsilon_\mathrm{solute} = 10$. The brightness suggests solvent activity $\varepsilon_\mathrm{solvent}$, and the line style indicates the system size [$(L_x, L_y) = (100, 50)$ (dotted lines), $(L_x, L_y) = (200, 100)$ (dashed lines), and $(L_x, L_y) = (400, 200)$ (solid lines)]. For moderate activity ($\varepsilon_\mathrm{solvent} = 2, 4, 6$) and large enough system sizes, the curves approximately follow a power-law scaling $\sim A^{-1.75}$ (black dotted line). (c, d) Bubble fraction $f_\mathrm{bubble}$ at $\rho = 0.5$ as a function of (c) $\varepsilon_\mathrm{solvent}$ or (d) $L_x$, where brightness suggests $\varepsilon_\mathrm{solvent}$ as in (a). For moderate $\varepsilon_\mathrm{solvent}$, the bubble fraction follows an approximate scaling: $f_\mathrm{bubble} \sim {L_x}^\mathrm{0.5}$ [black dotted line in (d)]. (e-h) The corresponding plots at $\rho = 0.8$ and $\varepsilon_\mathrm{solute} = 10$.
• Figure 4: Schematic explanation of (a) single-site mean-field theory and (b) four-site mean-field theory. Black dots indicate lattice sites, and dotted lines represent the neglected correlations between (a) adjacent sites or (b) adjacent clusters.
• Figure 5: Phase behavior in mean-field dynamics. (a, b) Typical snapshots for different sets of $(\varepsilon_\mathrm{solute}, \varepsilon_\mathrm{solvent})$ with $(L_x, L_y) = (100, 50)$ at (a) $\rho = 0.5$ or (b) $\rho = 0.8$. Grayscale represents solute density $\rho_{i, \mathrm{solute}}$ (white for $\rho_{i, \mathrm{solute}} = 0$ and black for $\rho_{i, \mathrm{solute}} = 1$).