Uphill diffusion and a new nonlinear diffusion equation in ternary non-electrolyte system

Abstract

Diffusion in silicate melts often shows uphill diffusion, the diffusion of a component up its own concentration gradient. This paper presents a new nonlinear diffusion equation which can describe uphill diffusion in a ternary system and provide a criterion in what conditions uphill diffusion will take place. Cross coefficients D ij ( i≠ j) in a multi-component diffusion equation describe the strength in coupling between diffusional flows, which may cause uphill diffusion. Irreversible thermodynamics shows that cross coefficients are dependent on concentration, although their functional form is unknown. To estimate it, this paper combines an irreversible thermodynamic approach by Miller [Miller, D.G., 1959. Ternary isothermal diffusion and the validity of the Onsager reciprocity relations. J. Phys. Chem., 63, pp. 570–578] and an eigenvector analysis by Gupta [Gupta, P.K., Cooper, Jr., A.R., 1971. The [ D] matrix for multi-component diffusion. Physica, 54. pp. 39–59]. The functional forms satisfying the two conditions above are: D 1 2= φ 1( D 1 1− D 2 2) and D 2 1= φ 2( D 22− D 1 1), where φ 1 and φ 2 are concentrations in mole fraction of components 1 and 2, respectively, and D 1 1 and D 2 2 are proper diffusion coefficients. These cross coefficients give a nonlinear diffusion equation which satisfactorily describes uphill diffusion. Perturbed solutions together with numerical solutions of the nonlinear diffusion equations are presented and discussed in detail to clarify the nature of uphill diffusion.