Hydrodynamic suppression of phase separation in active suspensions

We simulate with hydrodynamics a suspension of active disks squirming through a Newtonian fluid. We explore numerically the full range of squirmer area fractions from dilute to close packed and show that “motility induced phase separation,” which was recently proposed to arise generically in active matter, and which has been seen in simulations of active Brownian disks, is strongly suppressed by hydrodynamic interactions. We give an argument for why this should be the case and support it with counterpart simulations of active Brownian disks in a parameter regime that provides a closer counterpart to hydrodynamic suspensions than in previous studies.

Figure 1

Sketch of the disk geometry.

Figure 2

Schematic of effective description of the hydrodynamic interactions (HI in the figure), i.e., fluid velocities in the film generated by a point force on the disk boundaries. The fluid in the bulk is excited by motion in the film, and the Saffman length $\ell$ sets the extent of the bulk fluid set in motion. The region of bulk fluid excited by a disk of radius $R$ can be viewed as a “ghost” particle, and the size of the ghost particles controls the crossover between 2D and 3D nature of the effective interactions [30, 31, 32, 33]. When $\ell \le R$, the effective hydrodynamic interactions (HI) between the disks separated by distance $r$ are 3D-like and always scale to leading order as $1/r$. When $\ell >R$, the effective hydrodynamic interactions between disks depend on the ratio of the distance between the disks and the Saffman length $\ell$. For a pair of disks with centers at positions ${\mathit{r}}_i,{\mathit{r}}_j$, the hydrodynamic interaction (HI) between them is 3D-like if $r_{ij}=|{\mathit{r}}_i-{\mathit{r}}_j|\gg \ell$ and scales like $1/r_{ij}$ while it is 2D-like when $r_{ij}=|{\mathit{r}}_i-{\mathit{r}}_j|\ll \ell$ and scales like $ln{r}_{ij}$.

Figure 3

Active Brownian particles. Top: Average particle speed as a function of average particle area fraction. Expected spinodal shown by dotted line. Middle: Scaled number fluctuations. Bottom: Characteristic time ${\tau }_{\mathrm{o}}$ for reorientation of particle swim direction and interparticle collision time ${\tau }_{\mathrm{c}}$.

Figure 4

Snapshots for active Brownian particles, corresponding to circles (a)–(d) in Fig. 3.

Figure 5

Hydrodynamic squirmers (cylinders). Top: Average particle speed as a function of average particle area fraction. Expected spinodal shown by dotted line. Middle: Scaled number fluctuations. Bottom: Characteristic time ${\tau }_{\mathrm{o}}$ for reorientation of particle swim direction and interparticle collision time ${\tau }_{\mathrm{c}}$.

Figure 6

Snapshots for hydrodynamic squirmers, corresponding to circles (a)–(d) in Fig. 5.

Figure 7

Hydrodynamic squirmers (disks). Top: Average particle speed as a function of average particle area fraction. Middle: Scaled number fluctuations. Bottom: Characteristic time ${\tau }_{\mathrm{o}}$ for reorientation of particle swim direction and interparticle collision time ${\tau }_{\mathrm{c}}$.