Crystallization in a Dense Suspension of Self-Propelled Particles

Using Brownian dynamics computer simulations, we show that a two-dimensional suspension of self-propelled (”active”) colloidal particles crystallizes at sufficiently high densities. Compared to the equilibrium freezing of passive particles, the freezing density is both significantly shifted and depends on the structural or dynamical criterion employed. In nonequilibrium the transition is accompanied by pronounced structural heterogeneities. This leads to a transition region between liquid and solid in which the suspension is globally ordered but unordered liquidlike “bubbles” still persist.

Figure 1

Cooling (solid lines) and melting curves (dashed lines) for (a) the orientational order parameter ${\psi }_6$ and (b) the long-time diffusion coefficient $D$ vs the potential strength $\Gamma$ for selected driving forces $f$. The crossings with the dashed horizontal lines define the position of the structural transition ${\Gamma }_S^{*}$ (${\psi }_6=0.45$) and the dynamical freezing ${\Gamma }_D^{*}$ ($D=0.086$), respectively. (c) Phase diagram in the $f\mathrm{\text{−}}\Gamma$ plane. The symbols mark the numerically estimated dynamical freezing line ${\Gamma }_D^{*}$ and melting line ${\Gamma }_L^{*}$ (see main text for definition). The thick dashed line indicates the structural transition ${\Gamma }_S^{*}$. Also plotted are the ${\psi }_6=0.67$ and ${\psi }_6=0.8$ “isostructure” lines along which ${\psi }_6$ is constant.

Figure 2

(a) Time dependence of the Lindemann parameter [Eq. (6)] for $f=8$ below and at the melting point ${\Gamma }_L^{*}\simeq 700$. (b) The plateau values of the Lindemann parameter ${\gamma }_L^{*}$ both measured (closed symbols) and from Eq. (7) (open symbols) as a function of applied force.

Figure 3

Top: Probability distributions for ${\overline{q}}_6$ at $f=0$ (solid lines) and $f=8$ (dashed lines) for three different global ${\psi }_6$ values. Bottom: Snapshots of particle configurations for both the equilibrium ($f=0$, left column) and driven ($f=8$, right column) suspension. The rows correspond to constant global ${\psi }_6$ values: from top to bottom ${\psi }_6\simeq 0(100,100)$, 0.45(240, 560), 0.67(360, 600), and 0.8(600, 760); cf. the phase diagram in Fig. 1c. Values for the coupling $\Gamma$ are given in parenthesis for $f=0$ and $f=8$, respectively. Particles are colored according to their ${\overline{q}}_6$ value. While liquid and crystal (top and bottom row) are indistinguishable, the transition region (middle rows) is marked by heterogeneous structure.