Here are some images that illustrate the problem of nucleation and grain growth in thin films. We do the calculation in two ways:

(1) Solving the phase field crystal equations of motion.

Note that there is a lot of fine structure at the atomic level in these pictures, all of which had to be resolved when we solved the equations numerically.

(2) Solving the renormalization group equations for the phase field crystal, then reconstructing the density field.

The results are essentially the same as solving the phase field crystal equations.  But although we have reproduced the structure at the atomic level, and simulated the growth of grains, the variables we calculated with (amplitude and phase of the density) were virtually constant except near a defect or grain boundary, allowing us to use many less grid points to do the calculation.  This is the point of adaptive mesh refinement --- and this is what we are currently implementing (early 2005).

Here are plots of the amplitude and phase gradient at the final time, to show this explicitly:

Our ongoing work is extending this method to three dimensions, exploring a whole host of interest nanotechnology problems, putting in multi-component systems, incorporating empirical correlations into the phase field crystal.