Introduction

Diffusion - which is the net movement of anything (including molecules and atoms, but also energy and charge) - is a central process in Materials Science and Engineering. It is one of the most common process that causes changes in the structures of materials. Of course, these structural changes then yields changes in various material properties - both desirable (e.g., changing strength) and undesirable (e.g., corrosion/degradation).

In previous chapters we've observed how simple models can yield emergent behaviors. For example, the Lennard-Jones Potential leads to close-packed crystal structures and the phenomenon of thermal expansion. Diffusion is another phenomenon that can be viewed from the perspective of emergence: the emergent macroscale behavior (observing changes in concentrations profiles of species over macroscopic scales) can be described independently from the microlevel behavior (how atoms and molecules move around). In this chapter we will use computational models to understand the micro-level behavior of diffusion and then use them to derive a powerful mathematical models of the macro-level behavior of diffusion.

Diffusion was first modeled mathematically by Adolph Fick in 1855. This was around the same time that atomic theory was being developed, but atomic theory was not used in Fick's work. His approach was "phenomenological", meaning that he created a mathematical model that related the measurable variables of the phenomenon to one another without explaining those connections based on a deeper theory. (Meher & Murch pg 3). This means he only modeled the macro-level observable phenomenon.

In our treatment, we will start with a micro-level model of atoms and then use it to derive the macro-level diffusion equation. In processing of materials, both of these models are important in different applications. We will then discuss materials-specific aspects of diffusion and outlined some of the limitations of the models we've developed and how they can be extended.