Entropic Transport: Kinetics, Scaling, and Control Mechanisms

We show that transport in the presence of entropic barriers exhibits peculiar characteristics which makes it distinctly different from that occurring through energy barriers. The constrained dynamics yields a scaling regime for the particle current and the diffusion coefficient in terms of the ratio between the work done to the particles and available thermal energy. This interesting property, genuine to the entropic nature of the barriers, can be utilized to effectively control transport through quasi-one-dimensional structures in which irregularities or tortuosity of the boundaries cause entropic effects. The accuracy of the kinetic description has been corroborated by simulations. Applications to different dynamic situations involving entropic barriers are outlined.

Figure 1

Schematic diagram of the tube confining the motion of the biased Brownian particles. The half-width $\omega$ is a periodic function of $x$ with periodicity $L$.

Figure 2

Numerically determined force dependence of particle current for a symmetric two-dimensional channel with the shape defined by the half-width $\omega (x)=[\sin (2\pi x/L)+1.02]/(2\pi )$, $L=1$, and for the values of $T/T_{\mathrm{room}}$: 0.01 (solid line), 0.1 (dashed line), 0.2 (dotted line), and 0.4 (dash-dotted line). The inset depicts the dependence of the particle current $⟨\dot{x}⟩/(L/\tau )$ on the dimensionless temperature $T/T_{\mathrm{room}}$ for the force value: $F/F_0=0.628$. Contrary to the case of energetic barriers, the particle current declines with increasing temperature.

Figure 3

The effective diffusion coefficient vs the external bias. The parameters for the various lines correspond to those detailed in Fig. 2. The inset depicts the effective diffusion coefficient $D_{\mathrm{eff}}/(L^2/\tau )$ vs dimensionless temperature $T/T_{\mathrm{room}}$.

Figure 4

Graph for the scaled nonlinear mobility. In the Langevin simulation the different symbols correspond to different values of $T/T_{\mathrm{room}}$: 0.01 (crosses), 0.1 (pluses), 0.2 (squares), 0.4 (triangles). The relative error of the simulation results is 0.01. The Fick-Jacobs results, Eq. (6), correspond to the solid lines. The inset depicts the long range behavior (the dotted line depicts the numerical results). The numerical values for the scaled nonlinear mobility approach, in the limit $f\to \infty$, to the value 1 (dashed horizontal line).

Figure 5

Same as in Fig. 4, but for the effective diffusion coefficient. The relative error of the simulation results is 0.1. The Fick-Jacobs results, Eq. (7), correspond to the solid lines. The inset depicts the long range behavior (the dotted line depicts the numerical results). The numerical values for the scaled nonlinear mobility approach, in the limit $f\to \infty$, to the value 1 (dashed horizontal line).