Nigel Goldenfeld's Group:
Pattern formation on multiple scales 


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, suggesting that one could use many less grid points to do the calculation. In our detailed report, we checked that the quantitative properties of the patterns and their microstructures were accurately computed using the renormalization group approach. To make this a practical tool, we developed an adaptive mesh refinement algorithm, which adjusted the computational grid so that more grid points are placed in regions where the density amplitude and phase gradient are rapidly varying. Our method achieves this optimization without loss of accuracy, and permits a massive speedup in computational efficiency, described here. In the figure below, we show how this technique permits multiscale resolution of a grain boundary from the scale of thousands of nanometres down to atomic dimensions. The images show the computational grid and its multiscale resolution, in a computational domain of 722 nm square. The speed up over a conventional nonadaptive calculation was about a thousand. The effectiveness of this technique increases in three dimensions.
Our ongoing work includes using these techniques to speed up molecular dynamics calculations, to use the theory to derive the nonlinear elastic properties of materials, and thence to study plasticity and other nonlinear materials properties. Our preliminary results show that the statistical interactions of dislocations and other defects are wellcaptured by the phase field crystal method. 