Summary and Terms

Summary

The importance of understanding the diffusion behavior cannot be understated within materials science and engineering - it central to how we engineer and process materials to have the structures we want and how we keep them in those useful arrangements.

The random walk, which emerged from observations of brownian motion in the 19th and 20th centuries, is one of the most powerful tools for modeling diffusion. The random walk treats diffusors as agents which are equally likely to move in any direction at a given time. Under the random walk, areas of high concentration relative to teir neighbors will naturally decrease in concentration due to the increased density of random walk events in those areas - it is more likely for the net flux out of those regions to be greater than the net flux in.

Mathematically, we can extract the flux at a given point (for a differentiable concentration profile) with Fick's first law. Furthermore, we can predict the change in the concentration profile over time with Fick's second law, a handy differential equation that includes $D$ the material-dependent diffusion coefficient which tells us about how fast diffusion occurs within the system.

We also took a look at two diffusion mechanisms in crystalline solids - vacancy-mediated diffusion and interstitial diffusion. Vacancy mediated diffusion, as the name suggests, occurs as lattice atoms switch positions with vacancies, effectively causing the vacancy to move in the opposite direction. We can think of this either in terms of the atoms themselves moving, or the vacancies moving opposite them, which is often more convenient considering there are often very few vacancies to keep track of. In the interstitial case, diffusing atoms move by hopping between interstitial sites. Because interstitial sites are typically much more numerous than vacancies, this process is typically significantly faster than vacancy-mediated diffusion.

We wrapped up by discussing the origin of the diffusion coefficient $D$ and its dependence on temperature and material. D is given by the following expression: (where we often take the natural log of both sides to get it in a linear form for ease of use)

$$ \begin{align} D &= D_{\text{0}}\,\text{exp}\left(- \frac{Q}{R T}\right)\\ \text{ln} D &= \text{ln} D_{\text{0}}- \frac{Q}{R}\left(\frac{1}{T}\right) \end{align} $$

In this expression, $D_0$ is a temperature independent, material dependent pre-exponential factor. $D$ is also related to the activation energy barrier $Q$ for a single diffusive event to occur, and this the energy available in the system as heat - at a higher temperature, atoms will be more likely to have enough energy to cross the diffusion barrier and complete the hop. We can see this in the Arrhenius plots in Figure 7.10.3 - as temperature increases (careful, the temperature axis is flipped on these!) we see that D also increases for a given material.

Terms

Brownian Motion: The tendency of small particles such as pollen suspended in water to move about randomly. Section 7.3.1

Random Walk Model: A model stating that particles in a system are equally likely to be moving in any direction at a given time. Over short timescales, the random walk model produces a gaussian for particles that all start at the same location, but on longer timescales the the model tends towards homogeneity. Section 7.4.1

Net Flux: The net flow of particles through some unit area per unit time. Net flux is generally defined as $J$. The amount of flux in a given region is related to the concentration difference between the region and its neighbors, or the first derivative of concentration. Importantly, net flux isn't a scalar - it has a direction associated with it. Question 7.5.1.2

Fick's First Law: An equation that returns the net flux for a point on a concentration profile based on the profile itself. This is given by the negative first derivative of concentration: $$J(x) \propto \frac{dC}{dx}$$ The sign is critical, as we expect particles to move opposite from areas of high concentration to low concentration in Fickian systems. Exercise 7.6.1

Fick's Second Law: A differential equation that describes how the concentration at a given point will change with time based on the concentration profile. This is given by the second derivative of concentration (or curvature) multiplied by a proportionality constant D (the diffusion coefficient) which is material dependent.

$$\frac{\partial C}{\partial t}\ = D\frac{\partial^2 C}{\partial x^2}$$

Fick's second law is an extremely powerful tool for modeling the behavior of applicable (fickian) systems. Section 7.8.4

Steady State: A state in which the system concentration profile does not change with time. In other words, (via Fick's second law) a concentration profile with no curvature. Many diffusional systems tend towards steady state on sufficiently long time scales. Section 7.8.2

Vacancy-Mediated Diffusion: A diffusion mechanism in crystalline solids by which atoms move around by swapping places with a vacancy. Section 7.9.2

Interstitial-Mediated Diffusion: A diffusion mechanism in crystalline solids by which diffusing species move through interstitial sites between lattice atoms, typically more rapidly than in the vacancy-mediated case due to the relative abundance of interstitial sites as compared to vacancies. Section 7.9.3

Diffusion Coefficient: A material-dependent proportionality constant that tells us how fast diffusion proceeds for a given system. Section 7.10.1

Activation Energy: An energetic cost associated with a diffusive event often represented by a maximum or "saddle point" in the system's potential energy. For a diffusion hop to occur, there must be enough energy in the system to cross this barrier $Q$. Section 7.10.5