Diffusion in a two Phases System: Application to Ion Exchange in Silicate Glasses

TL;DR

This paper revisits the classical diffusion problem in two-phase systems, providing detailed solutions and discussing their application to ion exchange in silicate glasses, including assumptions and interface issues.

Contribution

It offers a detailed derivation of the general diffusion solution in two-phase systems and compares it with the erfc approximation, clarifying assumptions in ion exchange modeling.

Findings

The general solution reduces to the erfc under specific assumptions.

Interface and surface effects depend on diffusion coefficient ratios.

Instantaneous equilibrium assumption is justified for long immersion times.

Abstract

Diffusion in a two-phase system is a classical problem discussed in the literature. The general solution of the one-dimensional case to this problem is revisited and a detailed derivation is proposed. The solution is discussed in relationship with ion exchange in silicate glasses. The general solution is compared with the complementary error function (erfc) solution and the assumptions under which the general solution reduces to the erfc are identified. Interface or glass surface issues related to ion exchange are discussed in terms of the ratio between the diffusion coefficients of the diffusing alkali ions in the glass and in the ion reservoir phases respectively. Another relevant issue can be identified in the assumption of instantaneous equilibrium at the two-phase interface once the contact is established. Even though this last assumption is somehow unphysical it is the common approach used in the technological application of ion exchange in silicate glass which is justified when the contact time (immersion time) of glass with an ion reservoir is much larger than the time to achieve thermodynamic equilibrium at the surface.