Interwoven Structure & Size Selection

    We have discussed before about the refining effect of surface stress.  The following figure compares two evolution sequences with/without surface stress.  The sequences are for evolution from t = 0 to t = 1.0E6.  The time unit in the figures is t.  The size of each figure is 256b´256b.  In the simulation,  and Q are taken to be 2.2 and 1, respectively.  The sequence on the left (a) has isotropic surface stress while the sequence on the right (b) has no effect of surface stress.  The average concentration is taken to be 0.5.  The initial condition is set to fluctuate randomly within 0.001 from the average.  The initial condition is the same for the two sequences.

It is observed that in sequence (a), the phases form interwoven structures with width of the stripes in nanoscale.  The width of the stripes is almost invariant when it reaches certain size after short time of evolution.  This size is determined by the competition of coarsening and refining, reflecting the intrinsic length scale of the system.  In sequence (b), we can also observe similar interwoven structures during the evolution.  However, these structures do not have any stable size – they always coarsen.  The only restriction is the calculation cell size, periodic boundary condition, and mass conservation.  We can imagine the system will coarsen into a state that minimizes the area of phase boundary within the restriction.  Such a state would be a configuration that one phase takes half of the cell and the other phase takes the rest.  This is confirmed by the figure of sequence (b) at t=1.0E6.  The comparison of sequence (a) with (b) clearly demonstrates the refining effect of surface stress.

Non-uniform concentration in the epilayer generates a strain in the substrate, which decays exponentially along the  direction.  The following figure shows the strain concentration in the substrate.