Effect of particle size oscillations on drift and diffusion along a periodically corrugated channel

We study diffusive transport of a particle in a channel with periodically varying cross-section, occurring when the size of the particle periodically switches between two values. In such a situation, the entropy potential, which accounts for the area accessible for diffusion particle, varies both spatially (along the channel axis) and temporally. This underlies the complex interplay between different timescales of the system and leads to novel dynamic regimes. The most notable observations are: emergence of directed motion (in case of asymmetric channel) and resonant diffusion, both controlled by the switching frequency. Resonantlike behaviors of the drift velocity and the effective diffusion coefficient are shown and discussed. Based on heuristic arguments, an approximate analytical treatment of the transport process is proposed. As a comparison with the results obtained from Brownian dynamics simulations indicates, this approach provides a satisfactory way to handle the problem analytically.

Figure 1

Schematic illustration of a 2D $L$-periodic channel, with bottleneck width $2a$. The channel walls confining the motion of a spherical Brownian particle are determined by Eq. (1). The particle radius alternates between two values, $b_{+}$ and $b_{-}$, every time interval $\tau$. The dashed and dotted lines indicate the limiting positions of the particle center within the channel in the + and $-$ states, respectively. The half-width of the effective region accessible for diffusion, $w_{\pm }\left(x\right)$, is defined by Eq. (3).

Figure 2

Drift velocity in units $D_{+}/L$ vs. scaled switching time for different values of $b_{+}/b_{-}$ indicated by figures. Points, marked by symbols, are obtained from 2D Brownian dynamics simulations (the error of the data near the maximums is smaller than the size of the symbols used). The lines are calculated from Eq. (19). The inset compares numerical and analytical results for $b_{+}/b_{-}=1.67$. The value of $b_{+}/L=0.25$ is fixed; the dimensionless diffusion coefficients are $D_{+}=1$ and $D_{-}=b_{+}/b_{-}$.

Figure 3

Effective diffusion coefficient in units of $D_{+}$ vs. scaled switching time for different values of $b_{+}/b_{-}$ indicated by figures. The results obtained from 2D Brownian dynamics simulations for asymmetrical, $k_1\ne k_2$, and symmetrical, $k_1=k_2$, channel are marked by filled and empty symbols, respectively (the statistical error of the data is of the order of the symbol size). The solid and dotted lines are calculated from Eq. (21) for asymmetrical and symmetrical channel, respectively. The dashed and dashed-dotted lines show the limiting large-$\tau$ and small-$\tau$ values of the dimensionless effective diffusion coefficient calculated from Eqs. (10) and (12), respectively. The difference between these values in the symmetric and asymmetric channels is negligible. The inset compares numerical and analytical results for $b_{+}/b_{-}=1.67$. The value of $b_{+}/L=0.25$ is fixed; the dimensionless diffusion coefficients are $D_{+}=1$ and $D_{-}=b_{+}/b_{-}$.