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EMU Notes in Mineralogy - volume 16
Mineral reaction kinetics: Microstructures, textures, chemical and isotopic signatures
(W. Heinrich and R. Abart, editors)
Chapter 9. Diffusion: Some mathematical foundations and applications in mineralogy
E. Petrishcheva and R. Abart
Diffusion equations studied in this chapter apply to many physical situations: mass transport, particle transport such as self-diffusion or binary diffusion, heat transfer, potential and current diffusion, and even expansion of biological populations. To be specific, in what follows we consider transport of matter that arises from the random movement of atoms, ions or molecules. Diffusion occurs in all states of aggregation, and it is the only mode of mass transfer in solids. Diffusion is one of the most fundamental kinetic processes underlying phase change. There are several excellent textbooks giving detailed accounts of diffusion including the mathematics of diffusion (Crank, 1975) and diffusion in solids (Glicksman, 2000; Mehrer, 2007). There are also a number of comprehensive reviews summarizing the current state of knowledge about diffusion in geological materials (Zhang and Cherniak, 2010).
In this chapter we present the mathematical framework that we consider indispensible for dealing with the complex diffusion problems in mineral and rock systems. This chapter is by no means a comprehensive presentation of the mathematics of diffusion, rather it is a selection of topics that we deem particularly relevant in the context of geomaterials research. We begin with a brief account of the historical evolution of diffusion theory and derive the basic relations from both the macroscopic and the microscopic point of view. We then make the important distinction between linear diffusion, where the diffusion coefficient is independent of composition, and non-linear diffusion, where the opposite is true. In recent work on mineral systems composition-dependent diffusivity has turned out to be the rule rather than the exception; ample room thus is given to this phenomenon. In a first step, non-linearity caused by relatively weak composition dependence of diffusivity that may arise from defects introduced with the diffusing species or from strongly contrasting self-diffusion coefficients in multicomponent diffusion are addressed. In a second step, the links between diffusion and thermodynamics are explored. Non-ideal thermodynamic behaviour of solution phases may cause effective diffusion coefficients to become negative leading to uphill diffusion and eventually producing phase separation. The basic mathematical concepts for describing diffusion phenomena in mineral and rock systems and for solving inverse problems such as extracting diffusivities from experiment or time scales from diffusion profiles in minerals are derived. Due to space restrictions we refrain from including phenomena such as the Kirkendall effect and reactive diffusion. These phenomena are addressed in the chapter on solid-state reactions in mineral systems by Gaidies et al. (2017, this volume).
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