Diffusion Mechanisms in Crystalline Solids

So far, in our random walk model, we assumed that each atom can randomly step in any direction, unconstrained by other atoms. This is an okay assumption if we are modeling liquids and (especially) gases. Even then it is an obvious simplification, but it is an okay simplification to make, because atoms in liquids and gases are basically free to move around in this way. But what about in solids? Particularly in crystals in which atoms are relatively fixed in an orderly structure, how can atoms diffuse?

The two most common forms of diffusion in crystalline solids are:

1. Vacancy-mediated diffusion in which atoms diffuse by exchanging positions with vacancies in the lattice.
2. Interstitial-mediated diffusion in which small atoms (compared to the main lattice atoms) diffuse through interstitial sites.

Below are two models that extend the random walk model from Section 1.7 to showcase diffusion in solids.

Vacancy-mediated Diffusion

In vacancy-mediated diffusion, atoms diffuse by moving into vacancies in the crystal lattice and leaving behind new vacancies (or "exchanging" sites with the vacancy. In bulk materials at normal temperatures, vacancies spontaneously form. To form a vacancy, some bonds have to be broken. The energy to break these bonds comes from thermal vibrations resulting in some equilibrium concentration of vacancies in any given material at a given temperature. The higher the temperature, the more vacancies there will be.

NetLogo model 1.8.1 simulates vacancy diffusion on a square lattice of two different materials brought into contact. Vacancy diffusion typically takes place between atoms of similar or equal the sizes. In this model, the vacancies are positioned first between regions of blue and green, and then asked to perform random walk.

Given the number of vacancies is small compared to the number of atoms (generally 4 orders of magnitude smaller), it sometimes makes sense to think about the vacancies moving around instead of the atoms. These are just two ways of looking at the same phenomenon. If you look at the code of the model, it focuses on vacancies moving by "asking" empty patches to move a neighboring atom onto themselves.

In NetLogo model 1.8.1 is is clear that, with sufficient time, the random walk behavior of the vacancies has structural effects over time. Namely, our interface becomes more diffuse (less sharp) and the two regions mix. Indeed, with enough time (I get this after about 100K ticks), our model would yield a completely mixed region of green and blue atoms! This factor is often a factor in designing nanoscale materials. Their small dimensions can yield interdiffusion in such a way that two materials in contact may completely intermix due to this process.

Interstitial-mediated Diffusion

Interstitial-mediated diffusion occurs when atoms diffuse in the interstitial sites of a lattice. In previous chapters we've spent time exhaustively calculating the size, shape, and position of these interstitial sites. We can now consider how these interstitial sites serve as a network of jump sites for small diffusing atoms. While vacancy-mediated diffusion occurs when atoms can sit on substitutional sites, interstitial-mediated diffusion occurs when atoms are smaller and can fit in the interstitial sublattice without causing too much distruption.

NetLogo model 1.8.2 below simulates interstitial diffusion on a square lattice. The blue circles represent lattice atoms. In this model they don't move and are only for visualization. You can hide them with the "show/hide lattice atoms" button if you want. The small yellow circles represent diffusing atoms.