1 Introduction
Assume a simple concentration-dependent diffusion in the absence of external driving forces; in this introduction we chose to model the diffusive flux through Fick’s constitutive (first) law stated below, where \(D\) is the diffusion coefficient that may depend on position \(x\) and/or concentration \(c\), i.e. \(D\equiv{}D(x, c)\). It states that diffusive flux of an species has the composition gradient as a driver force; the negative sign indicates that diffusion occurs from high to low concentration regions.
\[ J=-D\dfrac{\partial{}c}{\partial{}x} \]
We can anticipate here that this can be the case for simple diluted systems, but interactions between species must be considered in concentrated solutions, what is generalized by the Irreversible Thermodynamics by Onsager (1931a) and Onsager (1931b). Here we emphasize again the word model: empirical observation has led scientists to represent reality through such an approximation; different models can be used for the same physics with their applicability limited to certain scenarios. Properly selecting an applicable model is the role of the researcher/engineer, what is outside our scope here; in what follows we will focus on how to solve balance equations for a given model.
Simply put, diffusion equation solves for a local mass balance; and mathematically, a local balance leads to a divergence operation. Following this idea, it can be shown that the time derivative of concentration \(c\) at one location is given by the divergence - which collapses to the spatial derivative - of the negative of mass flux given by Fick’s (first) law (or in general any other constitutive law/model describing the mass flux); sometimes this form is called Fick’s second law in the literature. Given the possible non-linearity introduced by diffusion coefficient \(D\), the derivative in the right-hand side is not expanded. In Cartesian coordinate system, the governing diffusion equation in one dimension writes:
\[ \dfrac{\partial{}c}{\partial{}t}= \dfrac{\partial{}}{\partial{}x} \left(D\dfrac{\partial{}c}{\partial{}x}\right) \]
This is the form of the diffusion equation that is discussed in what follows.