Effective diffusivity of a Brownian particle in a two-dimensional periodic channel of abruptly alternating width

We study diffusion of a Brownian particle in a two-dimensional periodic channel of abruptly alternating width. Our main result is a simple approximate analytical expression for the particle effective diffusivity, which shows how the diffusivity depends on the geometric parameters of the channel: lengths and widths of its wide and narrow segments. The result is obtained in two steps: first, we introduce an approximate one-dimensional description of particle diffusion in the channel, and second, we use this description to derive the expression for the effective diffusivity. While the reduction to the effective one-dimensional description is standard for systems of smoothly varying geometry, such a reduction in the case of abruptly changing geometry requires a new methodology used here, which is based on the boundary homogenization approach to the trapping problem. To test the accuracy of our analytical expression and thus establish the range of its applicability, we compare analytical predictions with the results obtained from Brownian dynamics simulations. The comparison shows excellent agreement between the two, on condition that the length of the wide segment of the channel is equal to or larger than its width.

Figure 1

Two-dimensional periodic channel formed by alternating wide and narrow segments of lengths $L$ and $l$ and widths $W$ and $w$, respectively.

Figure 2

Compartmentalized channel (upper panel) and the tortuosity in such a channel as a function of $\nu =w/W$ for $L/W=1$, 2, and 3 from top to bottom (lower panel). Solid curves are the dependences given by Eq. (21), and symbols are the tortuosity values obtained from our simulations.

Figure 3

Functions $f\left(\nu \right)$, Eq. (23), and $\nu f\left(\nu \right)$.

Figure 4

Tortuosity dependence on the length $l$ of the narrow segment at fixed values of the length of the wide segment $L$ and the segment width ratio $\nu =w/W$. Solid curves are theoretical dependences predicted by Eq. (14), symbols are simulation results. Two upper curves correspond to $\nu =0.05$. Two lower curves correspond to $\nu =0.1$.

Figure 5

Channel geometry used in our simulations. Particles start in the middle of the wide segment of the channel and are trapped by absorbing boundaries separated from the starting cross section by the period $\mathcal{L}=L+l$.

Figure 6

Effective one-dimensional description of particle diffusion in a two-dimensional channel shown in Fig. 1.

Figure 7

Illustration of the one-dimensional model used to find the mean first-passage time $\tau$.