Non-steady-state diffusion in two-dimensional periodic channels.

Area of Science:

• Physics
• Statistical Mechanics
• Physical Chemistry

Background:

• Particle diffusion in channels is often simplified using a 1D diffusion equation.
• Existing methods for relating channel geometry to diffusion coefficients have limitations.
• The potential of mean force U(x) is related to channel area A(x) by U(x) = -kBTln[A(x)].

Purpose of the Study:

• To derive a new expression for the spatially dependent diffusion coefficient D(x) in a 2D channel.
• To investigate the validity of the 1D diffusion equation for describing 2D particle dynamics.
• To explore the relationship between channel geometry and diffusion dynamics.

Main Methods:

• Analyzing non-steady-state single-particle diffusion in an open periodic channel.
• Developing a series expansion for D(x) in terms of the channel aspect ratio parameter ε.
• Comparing the derived D(x) expression with existing models (Zwanzig, Kalinay and Percus).

Main Results:

• The derived D(x) expression, when truncated at the leading term, recovers Zwanzig's formula.
• The first few terms of the expansion are consistent with the Kalinay and Percus model.
• The expansion converges rapidly for long-wavelength channels (ε≪1), providing an accurate 1D description.
• For short-wavelength channels, the expansion fails to converge, questioning the 1D approximation's validity.

Conclusions:

• The developed method provides an accurate description of 2D diffusion in long channels via a 1D diffusion equation.
• The validity of the effective 1D description is limited for short-wavelength channels.
• This work offers a new perspective on diffusion in geometrically constrained systems.