The Mathematical Theory Of Diffusion In Solids: Time Dependent First Kind Boundary Conditions

TL;DR

This paper introduces a new analytical solution for one-dimensional diffusion with time-dependent boundary conditions, incorporating a surface saturation model to better describe diffusion processes reaching dynamic equilibrium.

Contribution

It provides a novel solution to the diffusion equation with time-variable boundary conditions, including a surface saturation model, applicable when diffusion times are comparable to surface saturation times.

Findings

Solution applicable to diffusion with surface saturation

Worked examples demonstrate model effectiveness for constant diffusion coefficient

Enhances understanding of diffusion dynamics near saturation

Abstract

A new solution to the mono-dimensional diffusion equation for time-variable first kind boundary condition is presented where the time-variable function at the surface is derived proposing a surface saturation model. This solution may be helpful in the treatment of diffusion processes where the overall time of diffusion is comparable with the time taken by the surface of the solid body to saturate achieving a dynamical equilibrium between the diffusing elements supplied by the external source and the ones transferred internally through the diffusion kinetic mechanisms. Worked examples for constant diffusion coefficient are presented and discussed.