Diffusion in quasi-one-dimensional channels: A small system n, p, T, transition state theory for hopping times

Abstract

Particles confined to a single file, in a narrow quasi-one dimensional channel, exhibit a dynamic crossover from single file diffusion to Fickian diffusion as the channel radius increases and the particles can begin to pass each other. The long time diffusion coefficient for a system in the crossover regime can be described in terms of a hopping time, which measures the time it takes for a particle to escape the cage formed by its neighbours. In this paper, we develop a transition state theory approach to the calculation of the hopping time, using the small system isobaric-isothermal ensemble to rigorously account for the volume fluctuations associated with the size of the cage. We also describe a Monte Carlo simulation scheme that can be used to calculate the free energy barrier for particle hopping. The theory and simulation method correctly predict the hopping times for a two dimensional confined ideal gas system and a system of confined hard discs over a range of channel radii, but the method breaks down for wide channels in the hard discs case, underestimating the height of the hopping barrier due to the neglect of interactions between the small system and its surroundings.

Effects of Cage-Breaking Events in Single-File Diffusion on Elongation Correlation

“…As an example to illustrate our calculation scheme of elongation correlation C ε , going beyond the elastic approximation, here we take SFD with overtaking, [19][20][21][22] allowing the particles to hop out of the cage. Although there has been a number of studies on the crossover behavior of MSD (from √ t to t) in SFD with overtaking as a cage-breaking event, to the best of our knowledge, none of them have presented analytical calculation of space-time correlations to clarify the effect of overtaking on the collective motion.…”

Collective Motion of Repulsive Brownian Particles in Single-File Diffusion with and without Overtaking

“…Although accurate computation of overtaking is difficult and may require improvement in the numerical scheme, which is outside the scope of the present work, the numerical prefactor is interesting enough to motivate theoretical attempts to explain it. While V max in Equation (2) corresponds to the Helmholtz free energy barrier in quasi-1D systems governed by Equations (1a) and (6) [10,27], the hopping rate is rather related to a barrier in the Gibbs free energy [52]. The ρ 0 -dependent prefactor is also reminiscent of the escape probability in predator-prey problems on a 2D lattice [53].…”

The effect of soft repulsive interactions on the diffusion of particles in quasi-one-dimensional channels: A hopping time approach

“…When the channels are very narrow, the maximum is located very close to the geometrically defined transition state, but this is not always the case. 38 The particles studied here are circular, so the greatest degree of configurational restriction occurs when x c = 0, but as the channel becomes wider, the influence of the pressure-volume work become significant, and this can lead to the appearance of the maximum before the geometric transition state is reached. Nevertheless, it is important to recognize that it is the probability of being at the geometric transition state that contributes to the hopping time because the particles are unable to diffuse in the long time limit unless they exchange positions along the channel.…”

“…Here, we provide a brief description of the small system isobaric-isothermal ensemble in the context of its application to hopping times. 38 Figure 1(a) shows the system, consisting of n = 2 particles, at a constant external longitudinal pressure, p l , and fixed T . The small system has a length, L, and radius, R, with the center located at r 0 .…”

“…However, the activated nature of the hopping process also suggests τ hop can be obtained theoretically using barrier crossing methods, and a simple transition state theory has been shown to provide the correct scaling exponent for the hopping time as a function of channel radius 37 for two-dimensional hard discs. It was recently shown that a transition state theory (TST) qualitatively predicts the hopping time for the same system, 38 where the height of the free energy barrier for two particles attempting to pass was obtained using the small system isobaric-isothermal ensemble. 39,40 The goals of the current paper are to show that TST provides quantitative measures of the hopping time for particles confined to quasi-one-dimensional channels when both the barrier and prefactor are calculated and to demonstrate how the hopping time approach can be used to study the role of particle-particle interaction on diffusion in confined fluids.…”