Part IV – April 1968 - Papers - A Reformulation of Fick's First Law for Solid-State Diffusion

A theoretical development is presented which reformulates Fick's first law for diffusion in the solid state. The diffusion flux of component i in a multi-component system, Ji, is related to its local concentration, ci, and local jump frequency, ri : where a is a geometric factor and d is the jump distance. The product, ciri , is defined as the local concentration of atomic jumps for component i. The analog to Fick's second law is also derived. A solution to this equation is presented, which permits the experimental determination of the variation of atom jump frequency with composition in multicomponent systems. HISTORICALLY, the description of diffusion in the solid state has been presumed to be best approached as an application of more general diffusion laws, known as the phenomenological equations. These equations relate the flux, Ji, of the various components, expressed in atoms, grams, or volume, to the driving force involved, expressed as either a compositional or chemical potential gradient, sumption that the flux is simply proportional to the driving force is the foundation of this approach. The phenomenological equations take the form: where the concentration gradients are assumed to be the driving force, and: if the chemical potential gradient is taken as the driving force. The coefficients Dij in Eq. [1.] are the "diffusion coefficients"; the elements of the corresponding matrix in Eq. [2], the Mij, are called the "mobilities". Eqs. [I] and [2] may be considered to be definitions of these phenomenological coefficients. A formidable body of literature has been generated about these equations; this literature dates back to their original formulation for the simplest case of binary diffusion in a fluid, by Adolf Fick, in 1855.' This description has been shown to be self-consistent. However, the "constants of proportionality", the Dij or ,Mij, are not constant: they vary with composition in all but the simplest systems. Even in the case of tracer diffusion, wherein it can be shown that the diffusion coefficient is constant, the mobility, which is considered in irreversible thermodynamics to be the more physically meaningful quantity, varies with the concentration of the tracer involved.' In more complex systems, it is necessary to evaluate the entire matrix of phenomenological coefficients, as a function of composition, in order to describe diffusion locally. Further, in order to determine this matrix experimentally, it is necessary to produce diffusion couples that have different sequences of compositions; the matrix may be evaluated only at points where these "composition paths" cross.* Finally, and perhaps most