Defects at grain boundaries: A coarse-grained, three-dimensional description by the amplitude expansion of the phase-field crystal model

Abstract

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. arXiv:1803.03233v2 [cond-mat.mtrl-sci]

Closing the gap between atomic-scale lattice deformations and continuum elasticity

“…Periodic boundary conditions are used for all the boundaries of the simulation domains. To describe crystalline phases, the parameters entering the free energy are set to favor the crystal phase as follows: B x = 0.98, v = 1/3, t = 1/2 and ∆B = 0.02 [34,37,39].…”

“…Although some intrinsic limitations for large deformations and tilts exist [25], APFC has proved useful in the advanced modeling of materials as illustrated in studies of elasticity effects [20,25], compositional domains [33], binary alloys [34], dislocation dynamics [35,36], morphology and motion of dislocation networks at grain boundaries [37], and control of material properties [38][39][40]. However, the basic concept of APFC, namely the coarse-graining of an explicit lattice representation by fo-cusing on the complex coefficients of Fourier modes, can be readily applied to any atomistic description as obtained, e.g, from theoretical modeling, atomistic simulations, or experimental imaging.…”

A Three-Dimensional Continuum Simulation Method for Grain Boundary Motion Incorporating Dislocation Structure

“…Especially, the coupling motion of the grain boundary is associated with dislocation conservation during the motion of the grain boundary. The Cahn-Taylor theory and mechanisms of motion and reaction of the constituent dislocations have been examined by atomistic simulations and experiments [5,18,31,32,36,38,41,44,45]. It has been shown in Ref.…”

“…There are only limited studies in the literature for the three-dimensional coupling and sliding motions of grain boundaries. Grain boundary motion and grain rotation in bcc and fcc bicrystals composed of a spherical grain embedded in a single crystal matrix were studied by using three-dimensional phase field crystal model [45] and amplitude expansion phase field crystal model [36], and properties of grain boundaries and their dislocation structures as the grain boundary evolves have been examined. Although these atomistic-level phase field crystal simulations are able to provide detailed information associated with the coupling and sliding motions of grain boundaries in three dimensions, three-dimensional continuum models of the dynamics of grain boundaries incorporating their dislocation structures are still desired for larger scale simulations.…”

“…Using the obtained continuum model, we perform numerical simulations for the evolution of low angle grain boundaries by coupling and sliding motions, and compare the results with those of atomistic simulations using phase field crystal model [45] and amplitude expansion phase field crystal model [36] for validation of our continuum model. We also explain the anisotropic motion observed in these atomistic simulations based on our continuum model.…”