Geometric Ginzburg-Landau theory for faceted crystals in one dimension: From coarsening to chaos through a driving force.

We consider the dynamic behavior in driven phase transitions dominated either by attachment-detachment or by surface diffusion mass transport mechanisms. As the driving force increases, we numerically demonstrate for both cases that the spatiotemporal faceted structure of the surface undergoes a sequential transition from slow coarsening turning to accelerated coarsening followed by fixed length scale structures before finally becoming spatiotemporally chaotic. For the attachment-detachment dominated phase transition problem we compare in the accelerated coarsening regime the simulation results with an intrinsic dynamical system governing the leading-order piecewise-affine dynamic surface (PADS), which can be obtained through a matched asymptotic analysis. The PADS predicts the numerically observed coarsening law for the growth in time of the characteristic morphological length scale L_{M} . In particular we determine the prefactor of the scaling law which allows for quantitative predictions necessary for any use of the theory in preparing patterned surfaces through modifications of the driving force.