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Monte-Carlo Simulation

Here you can see an applet which visualizes nucleation and precipitate coarsening in phase separating solids. Of course it is just two-dimensional, also I use a square lattice instead of a more realistic close packed structure. Nevertheless, it is ideally suited for demonstrating the basics.

The interface

The pink dots symbolize one sort of atoms, we call it A, the blue ones the other sort B. You can change the number of blue by typing in a number in the field paint atoms (positive numbers - more blue, negative numbers - more pink) and then clicking paint now!
The three following fields specify the energies in the system: T is the temperature, which is self-explaining, J is the energy which an A-B-bond costs (this can also be a negative value, then A-B-bonds are favoured), and U is the energy by which vacancy-A-bonds are favoured in contrast to vacancy-B-bonds. You can type in values in the text fields, by clicking update parameters, they will be applied.
Finally, we have the attempt frequency, which specifiees how often the vacancy tries to jump per second. Since each attempt needs some calculation, this is limited by the speed of your computer. With today's standard computers, this is rather accurate up until 1 MHz, requesting higher values leads just to a lower frame rate, but no faster progress of time.

Interesting things to try

The first thing you will want to do is increase the attempt frequency to something like 1000 or 10000, in order to see something happening. Since the interaction energies are initially low compared to the temperature, the configuration is rather random and gets further shuffled by the vacancy. Now you can decrease the temperature to 1. and see what happens. With an attempt frequency of 10000 after a few seconds blue precipitates will emerge.
You can now increase the attempt frequency to 1000000 and watch the precipitates growing. With T=J you will still have a rather high solubility of blue in pink, by decreasing the temperature to 0.7J, this drops significantly. You can try to find the phase transition where the solubility becomes so high, such that the particles vanish again. I think, this is somewhere in the vicinity of T=1.5J.
By increasing the rate of blue atoms, you follow this phase transition across the phase diagram, since the solubility should stay rather constant, the phase transition should move to higher temperatures. Another very interesting phenomenon occuring here is percolation, when you can't say any more if you see blue precipitates in pink or the other way around, because both the blue and the pink domain is connected. I think, this happens at a rate of about 0.4, but the scale of this simulation is too small to really be able to decide.
The last point is the influence of U on the coarsening mechanism: go again to a rate of 0.1 of blue atoms, set J to 1.0, T to 0.8, U to 1.0 and the attempt frequency to 1000000. Now watch the particles growing. If you single out a small precipitate, you will see that it becomes smaller and smaller with time, until it vanishes or becomes swallowed by a big particle. On the other hand, if you set U to -1.0, you will see that the particles themselves wobble around, and if you wait long enough, eventually two particles will merge, because the have met on their random walk. If you set the attempt frequency back to small values, you will see the reason: with positive U the vacancy is in the matrix, where it moves single solute atoms (blue). With negative U, however, the vacancy is inside the particles and makes the whole particle perform a random walk.
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