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 processes that causes changes in the structures of materials, both during processing and during use. Of course, these structural changes then yield changes in various material properties. These may be 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 powerful mathematical laws that govern macroscopic-scale 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 macroscale-level observable phenomenon without any connection to the microscale.

In our treatment, we will start with a microscale model of atoms and then use it to derive the macroscale diffusion equations. In the processing of materials, both of these models are important in different contexts and applications. For example, if we know the materials we're using and we need to understand how atoms may be moving in that material, we can use macroscale approaches. For example, I need to understand the flux of $\ce{CO2}$ gas through PET to understand how quickly my Coca-Cola might go flat. On the other hand, if I want to engineer my material so that I can increase the time that my Coca-Cola will stay carbonated, I need a microscale understanding!

Outcomes

At the end of this module students should be able to:

• Recognize technological situations in which the understanding of diffusion of solids is important in determining materials performance.
• Broadly, describe the differences between vacancy and interstitial diffusion mechanisms with the understanding of how each mechanism applies in various scenarios.
• Derive the origins of Fick’s Laws using computational reasoning and random walk models. That is, connect atomic-level motion with macroscopic/continuous diffusional behavior.
• Utilize Fick’s First Law to predict diffusion flux qualitatively and quantitatively. Apply Fick’s First Law in the appropriate situations.
• Evaluate how concentration profiles will qualitatively evolve when following Fick’s Second Law (i.e., what does the differential equation say?).
• Evaluate how concentration profiles will quantitatively evolve when following Fick’s Second Law and when provided with a solution for the second law and other required information.
• Mathematically validate whether an equation satisfies Fick's Second Law.
• Utilize Fick’s First and Second Laws to compute diffusion coefficients and concentration profiles when supplied with sufficient data and boundary conditions.
• Evaluate how changes in structure, diffusing species, temperature, bonding, etc., may influence diffusion rates in solids.