Active Particles with Soft and Curved Walls: Equation of State, Ratchets, and Instabilities

We study, from first principles, the pressure exerted by an active fluid of spherical particles on general boundaries in two dimensions. We show that, despite the nonuniform pressure along curved walls, an equation of state is recovered upon a proper spatial averaging. This holds even in the presence of pairwise interactions between particles or when asymmetric walls induce ratchet currents, which are accompanied by spontaneous shear stresses on the walls. For flexible obstacles, the pressure inhomogeneities lead to a modulational instability as well as to the spontaneous motion of short semiflexible filaments. Finally, we relate the force exerted on objects immersed in active baths to the particle flux they generate around them.

Figure 1

The density (a) and current (b) of noninteracting ABPs near the right edge of the system with the hard wall potential described in the text with $v=D_r=24$, $D_t=0$, $L_p=0.5$, $A=0.5$, and $\lambda =1000$. The red dashed curve corresponds to $x_w(y)$. (c) Three cross sections of the particle density taken at the three horizontal dashed lines in (a). The vertical lines correspond to $x_w(y)$. (d) Pressure normal to the wall, normalized by Eq. (6), as a function of $y$, in the hard wall regime.

Figure 2

(a) For a periodic wall, the probability flux is periodic along $y$ and vanishes in the wall so that ${\int }_0^{L_p}{J}_{x}dy=0$ in steady state. (b) The ratio between the maximal and minimal values of $P_x(y)$ on hard sinusoidal walls as a function of $v/D_{r}R$, where $R=L_p^2/(4{\pi }^{2}A)$ is the radius of curvature, measured for a variety of combinations with $v\in [0.05,40]$, $A\in [0.3,12]$, $D_r\in [0.1,20]$, $L_p\in [0.3,80]$, and $\lambda =1000–3000$. (c) Mean pressure versus its theoretical prediction Eq. (6). Data were collected from 70 simulations of ABPs, RTPs, passive Brownian particles, and combinations thereof, with symmetric and asymmetric, hard and soft, periodic walls.

Figure 3

(a) Equipotential line ($V=5$) of the asymmetric potential (7). (b) Ratchet current and shear stress as functions of $x$. Equation (9) is verified numerically within 1%. Simulation of ABPs with $x_0=10$, $\lambda =2$, $A=0.3$, $L_p=0.5$, and $v=D_r=24$.

Figure 4

Snapshots of a flexible interface in an active bath, not shown for clarity. The stretching constant controls the threshold (a),(b) while the bending constant controls the wavelength of the instability (b),(c). Snapshots were taken at $t=30$ after starting from a straight filament. The characteristic length $L_c$, defined as the first zero of the autocorrelation function of the transverse monomer displacement, slowly coarsens as time goes on (d). Parameters are ${\rho }_0=1$, $v=10$, $D_r=1$, $r_1=r_0=0.3$, ${\kappa }_s=1000$, and ${\kappa }_b=250$.

Figure 5

(Left) Typical configurations for filaments of varying length for ${\kappa }_b=250$, $v=10$, $D_r=1$, ${\rho }_0=1$, ${\kappa }_s=1000$, $r_1=r_0=0.3$, and box size $50\times 50$ with periodic boundary conditions. (Right) Diffusivity of the filament as a function of its length.