Insights from Single-File Diffusion into Cooperativity in Higher Dimensions

Abstract

Diffusion in colloidal suspensions can be very slow due to the cage effect, which confines each particle within a short radius on one hand, and involves large-scale cooperative motions on the other. In search of insight into this cooperativity, here the authors develop a formalism to calculate the displacement correlation in colloidal systems, mainly in the two-dimensional case. To clarify the idea for it, studies are reviewed on cooperativity among the particles in the one-dimensional case, i.e. the single-file diffusion (SFD). As an improvement over the celebrated formula by Alexander and Pincus on the mean-square displacement (MSD) in SFD, it is shown that the displacement correlation in SFD can be calculated from Lagrangian correlation of the particle interval in the one-dimensional case, and also that the formula can be extended to higher dimensions. The improved formula becomes exact for large systems. By combining the formula with a nonlinear theory for correlation, a correction to the asymptotic law for the MSD in SFD is obtained. In the two-dimensional case, the linear theory gives description of vortical cooperative motion. Special Issue Comments:This article presents methodological insights into mathematical treatment of particle cooperativity in colloidal liquids. This article is related to the Special Issue articles [in Biophys. Rev. Lett.] about mathematical approaches to single-file diffusion 1 and Fourier-based analysis of quasi-onedimensional systems.

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

“…The calculated displacement correlation revealed collective motions behind the slow diffusion in SFD, in contrast to free diffusion in which R i R j vanishes (unless i = j). The formalism for analytical calculation of the displacement correlation can be extended to the case of two-dimensional (2D) colloidal liquids [10,21], which reproduces some numerical findings in 2D systems, such as vortical cooperative motion, with negative velocity autocorrelation being a manifestation of the cage effect. One of the delicate points in this extension is that the cage effect in 2D liquids cannot be infinitely strong, in the sense that eventually the particles can escape from the 2D cage.…”

“…In search of insight into the theoretical treatment of such collective motions, here, we take note of the one-dimensional (1D) system illustrated in Figure 1b, following several authors who studied it as a simplified model of the cage effect [4][5][6][7][8][9][10]. The slow dynamics of such a 1D system are known by the name of single-file diffusion (SFD).…”

“…The MSD in the ideal SFD is known to grow subdiffusively as R 2 i ∝ √ t [11][12][13][14] (for 1D systems, we write R i instead of R i ). The subdiffusion in SFD emerges from collective motion of particles [5,10,15] and is also detected as a negative longtime tail in the velocity autocorrelation [16,17], indicating that the particle is pushed back by its neighbors. The importance of the collective motion is understood by considering the origin of the effective stochastic equation for a single particle in SFD: the one-body equation (yielding the negative velocity autocorrelation) is actually based on the collective dynamics described in terms of the fluctuating density field [18].…”

“…Focusing on the collective motions in SFD, the group of the present authors has noticed the usefulness of the displacement correlation R i R j [4,10,19]. It is a kind of four-point space-time correlation that probes both the time scale and length scale; the definition of the displacement includes t, while the spatial scale is included as the mean distance between the two particles (i, j).…”

“…It is a kind of four-point space-time correlation that probes both the time scale and length scale; the definition of the displacement includes t, while the spatial scale is included as the mean distance between the two particles (i, j). In the ideal SFD in which the particles are forbidden to overtake each other, the displacement correlation has been calculated both analytically and numerically [4,10,20]. The calculated displacement correlation revealed collective motions behind the slow diffusion in SFD, in contrast to free diffusion in which R i R j vanishes (unless i = j).…”