Isotropic-nematic transition of self-propelled rods in three dimensions

Using overdamped Brownian dynamics simulations we investigate the isotropic-nematic (IN) transition of self-propelled rods in three spatial dimensions. For two well-known model systems (Gay-Berne potential and hard spherocylinders) we find that turning on activity moves to higher densities the phase boundary separating an isotropic phase from a (nonpolar) nematic phase. This active IN phase boundary is distinct from the boundary between isotropic and polar-cluster states previously reported in two-dimensional simulation studies and, unlike the latter, is not sensitive to the system size. We thus identify a generic feature of anisotropic active particles in three dimensions.

Figure 1

Isotropic-nematic (IN) transition of self-propelled rods (SPRs). The solid line and the filled region corresponds to soft-repulsive Gay-Berne (rGB) ellipsoids and the dashed line and symbols to hard spherocylinders (HSC). The particles, sketched on the right indicating their (approximate) length-to-width ratio, are propelled with constant velocity $v_0$ in the direction of their orientation ${\widehat{\mathbf{u}}}_i$ (red arrows). In both systems, the state points are given by the modified Péclet number ${\text{Pe}}^{\prime }=v_0/\sqrt{6D_{\text{T}}{D}_{\text{R}}}$ (swim speed rescaled by a particle-geometry dependent factor) and the packing fraction $\eta$, relative to the IN transition packing fraction ${\eta }_{\mathrm{IN}}^0$ in equilibrium. Beyond a certain threshold $\eta$ (as indicated by the label “smectic”), we find smectic clusters for HSC (more work is required to determine whether these are a separate phase).

Figure 2

Phase diagram of the 3D rGB model in the packing fraction–activity plane for $N=500$ particles. The stability of the isotropic (I), nematic (N), and local polar (P) states is determined by analyzing global order parameters. Typical snapshots are shown for three distinct state points, as indicated by triangles. The IN boundary (see Fig. 1 for a closeup) is insensitive to changes in the system size and separates true nonequilibrium phases. In contrast, the boundary separating I and P states is system-size dependent; the region P merely indicates where system-spanning polar clusters occur. The differently shaded squares near the IN phase boundary at $v_0^{*}=5,10,15$ denote the stable phase found in the larger system with $N=1000$ (see also the SM [47]).

Figure 3

Simulation snapshots of the isotropic (I) and nematic (N) phases for the active rGB system with $v_0^{*}=10$ and (a) $\eta =0.5$ and (b) $\eta =0.55$ and for the active HSC system (side view) at packing fractions just (c) below and (d) above the transition at ${\text{Pe}}^{\prime }=1.5$ (see Fig. 1). The color scheme serves to distinguish particles with different orientations.

Figure 4

Nematic $S$ and polar order parameter $P$ used to determine the phase behavior in the active rGB simulation. The state points indicated by a square and a circle correspond to the snapshots in Figs. 3 and 3, respectively. (a) Time average of $S$ (the inset shows $P$) for different velocities $v_0^{*}$ as a function of packing fraction $\eta$. The broken line indicates our threshold for determining the IN phase boundary. Around this threshold, the error bars are comparable to the symbol size, whereas they become significantly smaller in the nematic phase and are barely visible in the isotropic phase (see also the SM [47]). (b) Time evolution $S\left(t\right)$ and $P\left(t\right)$ for $\eta =0.55$ and $v_0^{*}=10$. The inset shows the same plots for a larger system with $N=1000$ particles (see also the SM [47]).

Figure 5

The nematic order parameter $S$ for HSC as a function of the packing fraction $\eta$ for various swimming speeds ${\mathrm{Pe}}^{\prime }$. The error bars indicate the standard deviation.