Tracer diffusion in active suspensions

We study the diffusion of a Brownian probe particle of size $R$ in a dilute dispersion of active Brownian particles of size $a$, characteristic swim speed $U_0$, reorientation time ${\tau }_R$, and mechanical energy $k_s{T}_s={\zeta }_a{U}_0^2{\tau }_R/6$, where ${\zeta }_a$ is the Stokes drag coefficient of a swimmer. The probe has a thermal diffusivity $D_P=k_{B}T/{\zeta }_P$, where $k_{B}T$ is the thermal energy of the solvent and ${\zeta }_P$ is the Stokes drag coefficient for the probe. When the swimmers are inactive, collisions between the probe and the swimmers sterically hinder the probe's diffusive motion. In competition with this steric hindrance is an enhancement driven by the activity of the swimmers. The strength of swimming relative to thermal diffusion is set by ${\text{Pe}}_s=U_{0}a/D_P$. The active contribution to the diffusivity scales as ${\text{Pe}}_s^2$ for weak swimming and ${\text{Pe}}_s$ for strong swimming, but the transition between these two regimes is nonmonotonic. When fluctuations in the probe motion decay on the time scale ${\tau }_R$, the active diffusivity scales as $k_s{T}_s/{\zeta }_P$: the probe moves as if it were immersed in a solvent with energy $k_s{T}_s$ rather than $k_{B}T$.

Figure 1

Active diffusivity of the probe as a function of the ratio of the pair-diffusion time to the advection time ${\text{Pe}}_s={\tau }_D/{\tau }_{\text{adv}}=U_0{R}_c/D^{\text{rel}}$, where $U_0$ is the swim speed, $R_c$ is the center-to-center separation distance of the probe and swimmer upon contact, and $D^{\text{rel}}$ is the relative thermal diffusivity of the probe-swimmer pair. The ratio ${\tau }_D/{\tau }_R$ indicates the strength of Brownian motion relative to the reorientations of the swimmers. The active diffusivity is nondimensionalized by the probe's SES diffusivity $D_P$ times the active volume fraction ${\varphi }_{\text{act}}=(4\pi /3)n^{\infty }{R}_c^{2}a/2$, where $a$ is the swimmer size and $n^{\infty }$ is the number density of swimmers.

Figure 2

Active diffusivity of the probe nondimensionalized by $(k_s{T}_s/{\zeta }_P)(R/R_c){\varphi }_{\text{act}}$ as a function of the ratio of the diffusion time to the swimmer reorientation time ${\tau }_D/{\tau }_R=R_c^2/{\tau }_R{D}^{\text{rel}}$ for various values of the mechanical to thermal energy, $k_s{T}_s/k_{B}T$, where $k_s{T}_s={\zeta }_a{U}_0^2{\tau }_R/6$.

Figure 3

Active diffusivity of the probe nondimensionalized by $U_{0}a{\varphi }_{\mathrm{act}}$ as a function of ${\text{Pe}}_s={\tau }_D/{\tau }_{\text{adv}}=U_0{R}_c/D^{\text{rel}}$. The ratio ${\tau }_R/{\tau }_{\text{adv}}=U_0{\tau }_R/R_c=\ell /R_c$ reflects the speed of reorientation relative to advection. The inset shows the total $O\left({\varphi }_{\text{act}}\right)$ change in the probe's diffusivity, nondimensionalized by $D_P{\varphi }_{\text{act}}$, where $D_P$ is the bare diffusivity of the probe.

Figure 4

Depiction of the model system. There is a Brownian probe of size $R$ immersed in a dispersion of ABPs with size $a$ at number density $n^{\infty }$. The ABPs swim in a direction $\mathit{q}$ with speed $U_0$, and they reorient with a characteristic time ${\tau }_R$.