Influence of interaction softness on phase separation of active particles

Using a minimal model of active Brownian particles, we study the effect of a crucial parameter, namely the softness of the interparticle repulsion, on motility-induced phase separation. We show that an increase in particle softness reduces the ability of the system to phase separate and the system exhibits a delayed transition. After phase separation, the system state properties can be explained by a single relevant length scale, the effective interparticle distance. We estimate this length scale analytically and use it to rescale the state properties at dense phase for systems with different interaction softness. Using this length scale, we provide a scaling relation for the time taken to phase separate which shows a high sensitivity to the interaction softness.

Figure 1

(a) Comparison of the local density distribution $P\left({\varphi }_{\text{loc}}\right)$ at $\varphi =0.7$ and $\mathrm{Pe}=150$ for different $\alpha$ values. (b) The peak value of the $P\left({\varphi }_{\text{loc}}\right)$ distributions observed in (a) are plotted as a function of $\alpha$. The bifurcation indicates the appearance of the dense phase as $\alpha$ is increased. (c) Scaling of the dense phase density ${\varphi }_l$ with $\alpha$ is plotted for $\mathrm{Pe}=100$ ($•$), 130 ($▸$), 150 ($⧫$), and 170 ($▾$), for $\varphi =0.7$. For convenience the ${\varphi }_l$ has been normalized by ${\varphi }_l(\alpha =4)$. For ${\varphi }_l\to r_c^2{\varphi }_l$, the $\alpha$ dependence disappears.

Figure 2

Study of the normalized pressure $\tilde{P}=P\varphi /P_{id}$. (a) and (b) show the heat maps of $\tilde{P}$ in the Pe-$\varphi$ plane for $\alpha =6$ and $\alpha =3$, respectively. (c) $\tilde{P}$ plotted as a function of $\varphi$ at $\mathrm{Pe}=150$ for $\alpha =2\to 6$. (d) Study of scaling behavior for the phase separating cases shown in (c). The $\varphi$ axis is scaled by $r_c^2{\alpha }^{\gamma }$, with $\gamma \approx 0.17$, while the pressure axis is scaled with a factor $r_c^2{\alpha }^{\delta }$ ($\delta \approx 0.45$). The inset shows the scaling of $\tilde{P}$ before phase separation, using the Boltzmann diameter $r_B$.

Figure 3

Quantification of global orientational order. (a)–(c) The global orientational order parameter ${\psi }_6$ is shown as a color map in the $\varphi$-Pe plane. Dynamics of phase separation as a function of $\alpha$. (d) ${\psi }_6\left(t\right)/{\psi }_6\left(\infty \right)$ $[{\psi }_6\left(\infty \right)$ is the steady state average] plotted as a function of time for $\alpha =4\to 6$ with $\mathrm{\Delta }\alpha =0.2$. Here $\mathrm{Pe}=150$ and $\varphi =0.7$. (e) Demonstration of temporal scaling. Introducing a scaling $t\to r_c^{\lambda }t$ leads to a collapse of all the different plots for $\lambda \approx 8.5$.

Figure 4

(a) Mean-square displacement $\langle \mathrm{\Delta }{\mathbf{r}}^2\left(t\right)\rangle$ of a particle inside the dense phase for different $\alpha$. (b) $\langle \mathrm{\Delta }{\mathbf{r}}^2\left(t\right)\rangle$ plotted for $\alpha =3$ in the case when the system does not phase separate to form a single large cluster. Here $\mathrm{Pe}=150$ and $\varphi =0.7$.

Figure 5

Effect of the overlap parameter $\alpha$ on the phase behavior. (a) Interaction potential $V_{ex}\left({\mathbf{r}}_{ij}\right)$ plotted against $r_{ij}$ for different values of $\alpha$ varied from $6\to 1$. (b)–(d) show the steady state snapshots indicating the phase behavior for a system of $N=40000$ particles, at $\varphi =0.7$, $\mathrm{Pe}=150$.

Figure 6

Local density distribution $P\left({\varphi }_{\text{loc}}\right)$ with varying Pe at $\varphi =0.7$ and $\alpha =3$. We notice that the system does not phase separate even at large value of activity ($\mathrm{Pe}=300$).

Figure 7

(a) Phase separation observed for $\alpha =3$ at very large densities ($\varphi =0.98$). He we set $\mathrm{Pe}=170$ and the simulation was run for a system of $N=40000$ particles. (b) shows the local orientational order ${\varphi }_{6i}$ as a color coded value superimposed on particle configurations.

Figure 8

Local orientational order ${\varphi }_{6i}$ as a color coded value superimposed on particle configurations in the x-y plane for different $\alpha =6$ and 4, respectively. Here we set $\varphi =0.7$ and $\mathrm{Pe}=150$. For $\alpha =6$, we notice that there are a few defect points of mainly 5–7 pairs present within the dense region [(see (a)]. Similar defect points are also observed in the dense phase for $\alpha =4$ but the number is higher. This is possibly caused by the large overlap between the particles in this case causing deviations from the sixfold symmetry locally.

Figure 9

(a) Values of ${\varphi }_l$ ($⧫$) and ${\varphi }_g$ ($•$) plotted as a function of $\alpha$ at $\varphi =0.7$ and $\mathrm{Pe}=150$. (b) To compare the trend we multiply ${\varphi }_l$ by a factor of $f=0.2$.

Figure 10

(a) ${\psi }_6$ plotted as a function of $\varphi$ at $\mathrm{Pe}=150$ for $\alpha =3\to 6$. (b) Study of scaling behavior. For the phase separating cases shown in (a), the $\varphi$ axis is scaled as $\varphi \to r_c^2{\alpha }^{0.17}\varphi$, while the ${\psi }_6$ is scaled as ${\psi }_6\to r_c^{-1.05}{\psi }_6$.

Figure 11

${\psi }_6\left(t\right)/{\psi }_6\left(\infty \right)$ plotted as a function of time for $\alpha =4\to 6$. (a) and (b) show the temporal scaling using functions $t\to t{\alpha }^{2.8}$ and $t\to t{e}^{14r_c}$, respectively, leading to a data collapse in both cases.

Figure 12

Mean-square displacement for the non-phase-separating case at $\varphi =0.30$ and $\mathrm{Pe}=150$ for $\alpha =3,5$, and 6, respectively. In all cases we notice a crossover from sub-$\mathrm{diffusion}\to \mathrm{ballistic}\to \mathrm{diffusion}$.