  We address a three-dimensional, coarse-grained description of dislocation
networks at grain boundaries between rotated crystals. The so-called amplitude
expansion of the phase-field crystal model is exploited with the aid of finite
element method calculations. This approach allows for the description of
microscopic features, such as dislocations, while simultaneously being able to
describe length scales that are orders of magnitude larger than the lattice
spacing. Moreover, it allows for the direct description of extended defects by
means of a scalar order parameter. The versatility of this framework is shown
by considering both fcc and bcc lattice symmetries and different rotation axes.
First, the specific case of planar, twist grain boundaries is illustrated. The
details of the method are reported and the consistency of the results with
literature is discussed. Then, the dislocation networks forming at the
interface between a spherical, rotated crystal embedded in an unrotated
crystalline structure, are shown. Although explicitly accounting for
dislocations which lead to an anisotropic shrinkage of the rotated grain, the
extension of the spherical grain boundary is found to decrease linearly over
time in agreement with the classical theory of grain growth and recent
atomistic investigations. It is shown that the results obtained for a system
with bcc symmetry agree very well with existing results, validating the
methodology. Furthermore, fully original results are shown for fcc lattice
symmetry, revealing the generality of the reported observations.
