Swarming behavior of gradient-responsive Brownian particles in a porous medium

Active targeting by Brownian particles in a fluid-filled porous environment is investigated by computer simulation. The random motion of the particles is enhanced by diffusiophoresis with respect to concentration gradients of chemical signals released by the particles in the proximity of a target. The mathematical model, based on a combination of the Brownian dynamics method and a diffusion problem is formulated in terms of key parameters that include the particle diffusiophoretic mobility and the signaling threshold (the distance from the target at which the particles release their chemical signals). The results demonstrate that even a relatively simple chemical signaling scheme can lead to a complex collective behavior of the particles and can be a very efficient way of guiding a swarm of Brownian particles towards a target, similarly to the way colonies of living cells communicate via secondary messengers.

Figure 1

The progress of a target localization process of Brownian particles (indicated by dots) in a porous medium. The target is indicated by a cross and the signal release area ${\Omega }_{\text{R}}$ is indicated by the light gray color. The concentration of the chemical signals in both the particles and the domain is indicated by the intensity of the gray (red) color. The parameter values are $\alpha =0.56\mu {\mathrm{m}}^2{\text{s}}^{-1}$, $c_{\text{R}}=0.5$, $c_{\text{T}}=0.7$, $\varepsilon =0.76$, and $\tau =2.2$ ($\varepsilon /\tau =0.35$). Scale bar, $50\mu \mathrm{m}$.

Figure 2

The target arrival time distribution evaluated from the simulation in Figure 1. The maximum corresponds to the mode arrival time ${\overline{t}}_{\text{A}}$ and the success rate $u\left(t\right)$ is the fraction of particles that have visited the target location within a given time $t$.

Figure 3

The dependence of the arrival time ${\overline{t}}_{\text{A}}$ (a), the success rate $u_{\infty }$ (b), and the target residence time ${\overline{t}}_{\text{R}}$ (c) on the diffusiophoretic mobility $\alpha$ for various values of the signaling threshold $c_{\text{R}}$ in a porous domain with $\varepsilon =0.76$ and $\tau =2.2$ ($\varepsilon /\tau =0.35$). Where present, the error bars signify the standard error based on five different random number sequences for otherwise identical parameter values.

Figure 4

Time evolution of the mean distance from the target $d_{\text{T}}\left(t\right)$ for various values of diffusiophoretic mobility $\alpha$ at high value of the signaling threshold $c_{\text{R}}=c_{\text{T}}=0.7$ in a porous domain with $\varepsilon /\tau =0.35$.

Figure 5

Example of anomalous behavior for large values of the diffusiophoretic mobility $\alpha$ and signaling threshold $c_{\text{R}}$ (with $\alpha =3\mu {\mathrm{m}}^2{\mathrm{s}}^{-1}$ and $c_{\text{R}}=c_{\text{T}}=0.7$). The target area is outlined by a black solid line and the target is indicated by a black cross. Scale bar, $25\mu \mathrm{m}$.

Figure 6

Dependence of the mode arrival time ${\overline{t}}_{\text{A}}$ (a), the success rate $u_{\infty }$ (b), and the target residence time ${\overline{t}}_{\text{R}}$ (c) on the diffusiophoretic mobility $\alpha$ for various values of the porous domain topology; $c_{\text{R}}=c_{\text{T}}=0.7$.

Figure 7

Various realizations of the random porous domain used to obtain the results depicted in Fig. 6. The black arrow indicates the steepest descent path connecting initial and target areas, based on the concentration gradient. Scale bar, $50\mu \mathrm{m}$.

Figure 8

Time evolution of the radius of gyration $r_{\text{G}}\left(t\right)$ for various values of the diffusiophoretic mobility $\alpha$ in an open domain, $\varepsilon /\tau =1$.