Self-diffusion in confined systems

TL;DR

This work develops a microscopic kinetic theory for self-diffusion in a hard-sphere fluid confined between parallel plates with $h\,\sim\sigma$. Starting from a confinement-modified Boltzmann equation and employing Zwanzig–Mori projection, it derives a closed diffusion equation in the plane with a height-dependent self-diffusion coefficient $D(oldsymbol{\ell})$, where $oldsymbol{\ell}=(h-\sigma)/\sigma$. An explicit, parameter-free expression $D=D_0 D^*(\boldsymbol{\ell})$ is obtained (with $D_0$ set by density, temperature, and particle properties) and a Maxwellian-based eigenfunction approximation yields the analytical form of $D^*(\boldsymbol{\ell})$ for all heights, capturing the quasi-2D to 3D crossover. MD simulations across the height range show excellent agreement with the theory in the low-density regime, confirming the method's accuracy and the absence of adjustable parameters. The results provide a microscopic bridge between quasi-two-dimensional diffusion and bulk three-dimensional diffusion in confined systems and lay groundwork for extensions to higher densities and more complex wall interactions.

Abstract

The self-diffusion process of a hard sphere fluid confined by two parallel plates separated by a distance on the order of the particle diameter is studied. The starting point is a closed kinetic equation for the distribution function that takes into account the effects of the confinement and that is valid in the low-density limit. From it, the Boltzmann-Lorentz equation that describes the dynamics of some tagged particles when the whole system is in equilibrium is derived. An equation that describes the diffusion in the directions parallel to the walls is deduced by applying the Zwanzig-Mori projection technique to the Boltzmann-Lorentz equation, obtaining an explicit expression for the self-diffusion coefficient that depends on the height of the system. A very good agreement between its theoretical prediction and Molecular Dynamics simulation results is obtained for the whole range of heights.

Figures (9)

• Figure 1: Sketch of the system. The particles are hard spheres of diameter $\sigma$ confined between two parallel plates separated a distance $h\sim \sigma$.
• Figure 2: Sketch of the collision between two particles. The tagged particle is represented by a solid line circle, while the colliding particle is represented by a dashed line circle. The vectors $\bm{g}$ and $\bm{g}^\prime$ are the relative velocities before and after the collision respectively. The spherical coordinates in which $\widehat{\bm{\sigma}}$ is expressed are also shown.
• Figure 3: Diagram of the possible solid angles for two different heights of the tagged particle (represented by a solid line circle) for $\sigma < h \leq 2 \sigma$. The dashed line circles represent particles colliding with the tagged particle with the maximum and minimum value of $\theta$. It can be seen that, wherever the tagged particle is, the two walls restrict the orientation of the possible collisions.
• Figure 4: Diagram of the possible solid angles for three different heights of the tagged particle (represented by a solid line circle) for $2\sigma \leq h \leq 3 \sigma$. The dashed line circles represent particles colliding with the tagged particle with the maximum and minimum value of $\theta$. If the center of the tagged particle is in the intermediate region between the two dashed lines ($h-3\sigma/2 \leq h \leq 3\sigma/2$), the two walls restrict the possible angles (right side). If the center of the tagged particle is between the dashed line and the closest wall, only that wall restricts (center and left side).
• Figure 5: Diagram of the possible solid angles for three different heights of the tagged particle (represented by a solid line circles) for $h \geq 3 \sigma$. The dashed line circles represent particles colliding with the tagged particle with the maximum and minimum value of $\theta$. In the intermediate region between the two dashed lines ($3\sigma/2 \leq h \leq h- 3\sigma/2$), there is no restriction on the possible angles (right side). If the center of the tagged particle is between the dashed line and the closest wall, only that walls restricts (center and left side).