  A phase field theory of polycrystalline solidification is presented that is
able to describe the nucleation and growth of anisotropic particles with
different crystallographic orientation in three dimensions. As opposed with the
two-dimensional case, where a single orientation field suffices, in three
dimensions, minimum three fields are needed. The free energy of grain
boundaries is assumed to be proportional to the angular difference between the
adjacent crystals expressed here in terms of the differences of the four
symmetric Euler parameters. The equations of motion for these fields are
obtained from variational principles. Illustrative calculations are performed
for polycrystalline solidification with dendritic, needle and spherulitic
growth morphologies.
