Fick-Jacobs equation for channels over three-dimensional curves

Area of Science:

• Physics
• Physical Chemistry
• Chemical Engineering

Background:

• Diffusion processes are fundamental in many scientific and engineering fields.
• Understanding diffusion in confined geometries, like narrow channels, is crucial for applications such as microfluidics and materials science.
• The Fick-Jacobs equation is a common model for diffusion in such systems, but generalizations are needed for complex geometries.

Purpose of the Study:

• To derive a novel formula for the effective diffusion coefficient in narrow three-dimensional channels.
• To extend the generalized Fick-Jacobs equation to account for complex channel geometries.
• To analyze the influence of channel cross-section geometry and curvature on diffusion.

Main Methods:

• Dimensional reduction of a three-dimensional diffusion equation via projection along a central curve.
• Integration of the diffusion equation across channel cross-sections.
• Derivation of the effective diffusion coefficient formula based on geometric moments and curvature.

Main Results:

• A new formula for the effective diffusion coefficient of the generalized Fick-Jacobs equation in narrow 3D channels was developed.
• The formula explicitly links the diffusion coefficient to the geometric moments of the channel's cross-sections and the curvature of the channel's central curve.
• The impact of rotating cross-sections with an offset on the effective diffusion coefficient was demonstrated.

Conclusions:

• The derived formula provides a more accurate description of diffusion in complex narrow channels.
• Channel geometry, including cross-sectional shape and curvature, significantly influences effective diffusion.
• This work offers a valuable tool for modeling and predicting transport phenomena in confined systems.