Modeling Elasticity in Crystal Growth

Abstract

A new model of crystal growth is presented that describes the phenomena on atomic length and diffusive time scales. The former incorporates elastic and plastic deformation in a natural manner, and the latter enables access to time scales much larger than conventional atomic methods. The model is shown to be consistent with the predictions of Read and Shockley for grain boundary energy, and Matthews and Blakeslee for misfit dislocations in epitaxial growth. DOI: 10.1103/PhysRevLett.88.245701 PACS numbers: 64.60.Cn, 05.70.Ln, 64.60.My, 81.30.Hd The appearance and growth of crystal phases occurs in many technologically important processes including epitaxial growth and zone refinement. While a plethora of models have been constructed to examine various aspects of these phenomena, it has proven difficult to develop a computationally efficient model that can be used for a wide range of applications. For example, standard molecular dynamics simulations include the necessary physics but are limited by atomic sizes (Å) and phonon time scales (ps). Conversely, continuum field theories can access longer length (i.e., correlation length) and time (i.e., diffusive) scales, but are difficult to incorporate with the appropriate physics. In this paper a new model is presented that includes the essential physics and is not limited by atomic time scales.To illustrate the features that must be incorporated, it is useful to consider two examples. First, consider the nucleation and growth of crystals from a pure supercooled liquid or vapor phase. In such a process, small crystallites nucleate (heterogeneously or homogeneously) and grow in arbitrary locations and orientations. Eventually, the crystallites impinge on one another and grain boundaries are formed. Further growth is then determined by the evolution of grain boundaries. Now consider the growth of a thin crystal film on a substrate of a different crystal structure. The substrate stresses the overlying film which can destabilize the growing film and cause an elastic defect-free morphological deformation [1,2], plastic deformation involving misfit dislocations [3], or a combination of both. Thus, the model must be able to nucleate crystallites at arbitrary locations and orientations and contain elastic and plastic deformations. While all these features are naturally incorporated in atomistic descriptions, they are much more difficult to include in continuum or phase field models.Historically, many continuum models have been developed to describe certain aspects of crystal growth and liquid/solid transitions in general. At the simplest level, "model A" in the Halperin and Hohenberg [4] classification scheme has been used to describe liquid/solid transitions. This model treats all solids equivalently and does not introduce any crystal anisotropy. Extensions to this basic model have been developed to incorporate a solid phase that has multiple states that represent multiple orientations [5,6] or, recently [7], an infinite number of orientations. Unfortunately, these models ...

Marginal stability analysis of the phase field crystal model in one spatial dimension

“…The relaxation time τ is assumed to be positive, because pure non-equilibrium contribution should lead to an increase of the free energy in comparison with (6). In general, non-equilibrium contribution (17) have a meaning of the kinetic energy as it has been shown in example of phase separation by mechanism of spinodal decomposition 33 .…”

“…As originally formulated in a parabolic form, the PFC model allows simulations on diffusive time scales which can be many orders of magnitude larger than molecular dynamics simulations 6,14 . More recently a hyperbolic 24 or modified 25 PFC model was introduced that includes faster degrees of freedom in a form of inertia and as such leads to the description of both fast and slow dynamics.…”

“…It can be related to other continuum fields theories such as classical density-functional theory 8,9 and the atomic density function theory 10 . The PFC-model may also be considered as a conserved version of the Swift-Hohenberg equation and provides an efficient method for simulating liquid-solid transitions 11,12 , colloidal solidification 13 , dislocation motion and plasticity 14,15 , glass formation 16 , epitaxial growth 6,17 , grain boundary premelting 18 , surface reconstructions 19 , and grain boundary energies 20 .…”

“…More specifically the patterns that emerge behind a front described by the phase field crystal (PFC) model 6 will be examined. The PFC model has been proposed to incorporate the physics naturally embedded on atomic length scales (elasticity, dislocation, etc.)…”

Structural short-range forces between solid-melt interfaces

“…Several continuum descriptions have been pushed forward, including those based on phase field models [24][25][26][27] with either an orientational order parameter 24,25 or multiorder parameter models 26,27 ; these order parameters are needed to distinguish between the different grain orientations. More recently, the phase field crystal (PFC) method has been introduced 28,29 , allowing to describe the atomic structure and thus the local lattice orientation via the crystal density field. GB premelting studies using these models have been performed in Refs.…”