Force Generation in Confined Active Fluids: The Role of Microstructure

We experimentally determine the force exerted by a bath of active particles onto a passive probe as a function of its distance to a wall and compare it to the measured averaged density distribution of active particles around the probe. Within the framework of an active stress, we demonstrate that both quantities are—up to a factor—directly related to each other. Our results are in excellent agreement with a minimal numerical model and confirm a general and system-independent relationship between the microstructure of active particles and transmitted forces.

Figure 1

(a) Electron scanning microscope image (top) and schematic picture (bottom) of the probe particle used. The scale bars are $3\mu \mathrm{m}$ each. (b) Schematic illustration of the experimental situation. The probe particle at $x_p$ and surface-to-surface distance to the circular wall $d$ is trapped by an optical tweezers ($x_{tr}$ denotes the center of the trap). The smaller spheres with the dark caps correspond to the APs.

Figure 2

(a) Experimental snapshot showing APs (dark points) and the optically trapped probe particle inside the circular cavity. Measured AP density distributions are displayed for different probe-wall distances: (b) $d^{\prime }=0.46$, (c) 1.02, (d) 1.59, (e) 2.78, (f) 6.30. Scale bars are $10\mu \mathrm{m}$ each.

Figure 3

(a) Distribution $P(\mathrm{\Delta }x)$ at the center of the circular confinement in the presence of active (squares) and passive (circle) particles. The solid lines are Gaussian fits. (b) Distribution $P(\mathrm{\Delta }x)$ with the probe at $d^{\prime }=1.02$ (stars), 1.59 (squares), and 6.30 (circles). The solid lines correspond to fits with a Gaussian distribution. The error bars show the standard deviation of experimental data calculated over five independent measurements.

Figure 4

(a) Experimentally measured values of $F_x$ (dark symbols) and $F_y$ (light symbols) as a function of normalized distance $d^{\prime }=d/(2r_a)$ of the probe from the wall of the circular confinement with radius $r_{\mathrm{con}}=50\mu \mathrm{m}$. The error bars correspond to the standard deviations calculated over five independent measurements. The inset shows $F_x$ (symbols) together with the prediction according to Eq. (4) from the measured number density $\rho$ (red solid line). (b) Density $\rho (\phi )$ of APs (normalized by the global number density ${\rho }_0$) around the probe for different values of $d^{\prime }$ [being labeled with identical symbols (colors) to those in (a)].

Figure 5

Simulation results for active Brownian particles. (a) Force on the probe in the $x$ (open circles) and $y$ (open triangles) directions as a function of normalized distance $d^{\prime }$ for a circular confinement of radius $r_{\mathrm{con}}=50\mu \mathrm{m}$. The error bars correspond to the standard deviation calculated over five independent simulation runs. The inset shows the measured force $F_x$ (symbols) together with the prediction of Eq. (4) from the measured number density $\rho$ (red line). (b) Density $\rho (\phi )$ of APs (normalized by the global number density ${\rho }_0$) around the probe on a circle of radius $r_b=r_p+2.2r_a$ for different $d^{\prime }$. $F_x$ values corresponding to the $\rho (\phi )$’s shown in (b) are marked with the same symbols in (a).