Crowding-Enhanced Diffusion: An Exact Theory for Highly Entangled Self-Propelled Stiff Filaments

We study a strongly interacting crowded system of self-propelled stiff filaments by event-driven Brownian dynamics simulations and an analytical theory to elucidate the intricate interplay of crowding and self-propulsion. We find a remarkable increase of the effective diffusivity upon increasing the filament number density by more than one order of magnitude. This counterintuitive “crowded is faster” behavior can be rationalized by extending the concept of a confining tube pioneered by Doi and Edwards for highly entangled, crowded, passive to active systems. We predict a scaling theory for the effective diffusivity as a function of the Péclet number and the filament number density. Subsequently, we show that an exact expression derived for a single self-propelled filament with motility parameters as input can predict the nontrivial spatiotemporal dynamics over the entire range of length and timescales. In particular, our theory captures short-time diffusion, directed swimming motion at intermediate times, and the transition to complete orientational relaxation at long times.

Figure 1

Mean-square displacement $⟨[\mathrm{\Delta }\mathbf{r}(t){]}^2⟩$ of a self-propelled stiff filament suspended in needle solutions with different number densities $n^{\star }=n{L}^3$. Here, the Péclet number is $\mathrm{Pe}=125$. Inset: Effective diffusivity $D_{\mathrm{eff}}$ as function of the number density $n^{\star }$ rescaled by the effective diffusivity in a free environment $D_{\mathrm{eff}}^0$. The solid black line indicates a quadratic increase.

Figure 2

(a) In the highly entangled regime (mesoscale), a self-propelled filament is confined to an effective tube of diameter $d$ created by neighboring self-propelled filaments with motility parameters Pe, $D_{\perp }$, $D_{\parallel }$, and ${\tau }_{\mathrm{rot}}$. At a later time, the filament is confined to a new tube tilted by an angle $\epsilon$ with respect to the initial tube. (b) Tube diameter $d$ as a function of density $n^{\star }$. (c) Orientational relaxation time ${\tau }_{\mathrm{rot}}$ extracted from simulations at different number densities $n^{\star }$ for several Péclet numbers. Inset: Rescaled effective diffusivities $D_{\mathrm{eff}}$ for finite Péclet numbers.

Figure 3

Intermediate scattering function $F(k,t)$, for (a),(b) a broad range of Péclet numbers Pe at fixed density $n^{\star }=1024$ and (c) different densities $n^{\star }$ at fixed Péclet number $\mathrm{Pe}=125$. The solid lines are the theoretical predictions of the tube model [Eq. (6)] and symbols are from simulations. Panels (a),(b), and (c) represent data for different wave numbers $kL=50$, 4, and ${10}^{-3}$, respectively, where $L$ denotes the filament length. The inset in (a) shows the data in log-log scale, where the dashed solid line indicates an algebraic decay $\sim t^{-1/2}$. The black dotted-dashed lines in (b) indicate the sinc function $\sin (vkt)/(vkt)$ and the dotted lines in (c) represent effective diffusion $\exp (-k^2{D}_{\mathrm{eff}}t)$.