In this note, we model the formation of liquid cores during droplet crystallization as a Mullins Sekerka instability. We then compare the theoretical critical curve for the instability with simulation results.

Recall that in this post, we established a critical wavenumber for a perturbation to grow via Mullins-Sekerka instability. This expression was:

,

where:

- is the As flux during crystallization,
- is the temperature,
- is the rate prefactor,
- is the atomic spacing,
- eV is the surface-liquid interfacial energy,
- eV is the barrier for the diffusion of As in liquid Ga,
- are the detachment and attachment barriers of an As atom from and onto a flat GaAs substrate, respectively.

In order for this perturbation to arise within a liquid droplet of radius , we necessarily require that , for some constant of proportionality , otherwise modes of any perturbation would not have sufficiently small wavenumber (equiv. sufficiently large wavelength) to induce a instability. Recall we modeled the droplet radius as

where is constant of proportionality, monolayers/second is the flux at which Ga was deposited, eV is the energy barrier for Ga-on-Ga diffusion and is the power-law the radius obeys. Hence the Mullins-Sekerka condition may be written as:

Solving for , we obtain

Compare this to the expression for when surface nucleation occurs (without making any approximations):

.

Before, we had set eV (values obtained from simulations) and fit again to simulation data. We had argued that the barriers closely matched the attachment and detachment events illustrated below:

This is incorrect! Instead, we must consider the attachment of an As atom onto a flat substrate and its reverse event:

Here, eV and eV. We may then fit the nucleation barrier eV, a very small barrier to nucleation. Picking these values yield the following phase diagram:

This is a comparison between simulation data and our theoretical curves above. Red crosses correspond to simulations where liquid cores were observed. Blue crosses correspond to simulations where nucleation is present. The red line is the critical line for the Mullins-Sekerka instability, while the blue line is the theoretical critical curve for nucleation.

We observe an accurate fit. It is important to note: **only the energy was fit to simulation data**, and its value is consistent to the idea that the barrier to nucleation is very small. The other parameters were obtained from first principles.

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