Competing effect of disorder on phase separation in active systems

Abstract

We investigate the impact of random pinned disorder on a collection of self propelled particles. To achieve this, we construct a continuum model by formulating the coupled hydrodynamic equations for slow variables, local density and momentum density of particles. The disorder in the system acts as pinning sites, effectively immobilizing the particles that come into contact with them. Our numerical results reveal that weak disorder leads to phase separation in the system at density and activity lower than the typical values for motility induced phase separation. We construct a phase diagram using numerical simulations as well as linearized approximation in the plane of activity and packing fraction of particles at weak disorder densities. On increasing disorder density the system shows the Micro phase separation, while at large disorder densities, the system becomes heterogeneous and eventually undergoes kinetic arrest. The structure factor tail deviates from the Porods law, indicating increased roughness at domain interfaces under strong disorder. Furthermore, we analyze the fractal dimension of the interface as a function of disorder density, highlighting the increasing irregularity of phase separated domains. We also found that disorder significantly suppresses number fluctuations in the system.

Figures (9)

• Figure 1: The panels (a-d) showcase the local fluctuation $\delta\rho$ at across four panels , each representing system with different disorder densities $\rho_d = 0.0, 0.05, 0.1, 0.5$ respectively at $t=200$. Within each panel, multiple figures are plotted from left to right for self-propulsion speed $v_0 = 0.2, 1, 3, 5, 7$ keeping the mean density $\rho_0 = 0.7$ fixed. The color on the heatmap represents the value of $\delta\rho$. The results are obtained for systems having $K = 256$.
• Figure 2: The panels (a-d) depict the local fluctuations in density, $\delta \rho$, at time $t = 200$ for systems with varying disorder densities, $\rho_d = 0, 0.05, 0.1, 0.5$ in sequence. Each panel consists of multiple snapshots, from left to right, representing systems with mean densities $\rho_0 = 0.2, 0.4, 0.6, 0.8, 0.9$ respectively by fixing the self-propulsion speed $v_0 = 4.0$. The color in the heatmap indicates the magnitude of the $\delta \rho$. The results are generated for systems having $K = 256$.
• Figure 3: The plot(a) presents semi-log $x-$ plot of $\Delta\rho$vs.$\rho_d$ with error bars. The inset shows a linear plot of $\Delta\rho$vs.$\rho_d$ depicting the non-monotonicity with respect to disorder. The plot(b) illustrates the phase diagram in the $\rho_0 v_0^2-v_{0}^2$ plane for different disorder densities: $\rho_d = 0.0, 0.001$ respectively. The blue (circles) and red (squares) solid lines are the boundary drawn from the numerical simulation for $\rho_d = 0.0, 0.001$ which is also mentioned in the legend. Additionally, the green solid line and magenta dotted line represent the analytical boundary obtained from linearized calculation. The shaded light-orange region shows the extra regime of phase separation in the presence of disorder. The plot (c) showcases $\Delta\rho$vs.$v_0$ depicting the transition from non-phase separation to phase separation for $\rho_d = 0.0-0.2$.
• Figure 4: The plot (a) showcases the log-log plot of characteristic length $L(t)$vs.$t$ for different disorder density $\rho_d$ ranging from $0-0.5$ in the system showing in the legends. The dashed orange line depicts the line of slope $0.33$. The plot (b) shows the saturation length $L_s$$vs.$$\rho_d$ for different system sizes $K=64,128,512$. The inset depicts the semi-log $x-$ plot for saturation time $T_s$$vs.$$\rho_d$. The lines are guide to eyes, both in the main and inset panel. The plot (c) displays semi-log $x-$ plot of the corresponding $1/z_{eff}$ with $\rho_d$. The error bars represent the standard deviations of the $1/z_{eff}$. The vertical line drawn here at the critical value of $\rho_d = 0.23$ (from Eq. \ref{['critical_rho']}) after which the phase separation is suppressed and is obtained from the linearized calculation. The plot (d) presents the static scaled correlation $g(x)$ for $\rho_d = 0 -0.5$.
• Figure 5: The plot (a) exhibits the plot of scaled structure factor $S(k)L^{-2}$vs.$kL$ for $\rho_d = 0- 0.5$ as shown in the legend. The maroon and orange dotted lines present the slope of line $-3$ and $-2.5$. The plot (b) displays $1-g(x)$vs.$x$ for different disorder density and the symbols have the same meaning as shown in (a). The legends present the slope of lines. (c) showcases the log-log plot of $d_f$vs.$\rho_d$. The error bars are of the size of the symbols used. The vertical dashed line is drawn at $\rho_d = 0.23$ and represents the same meaning as the solid line in Fig. \ref{['fig6']}(c). The plot (d) depicts the density fluctuation $\Delta \rho_n$vs.$\rho_n$ for different $\rho_d$. The symbols have the same meaning as in plot (a). The legends show the slope of lines as mentioned.