Amplitude equations for polycrystalline materials with interaction between composition and stress

Abstract

We investigate the ability of frame-invariant amplitude equations ͓G. H. Gunaratne, Q. Ouyang, and H. Swinney, Phys. Rev. E 50, 2802 ͑1994͔͒ to describe quantitatively the evolution of polycrystalline microstructures and we extend this approach to include the interaction between composition and stress. Validations for elemental materials include studies of the Asaro-Tiller-Grinfeld morphological instability of a stressed crystal surface, polycrystalline growth from the melt, grain-boundary energies over a wide range of misorientation, and grain-boundary motion coupled to shear deformation. Amplitude equations with accelerated strain relaxation in the solid are shown to model accurately the Asaro-Tiller-Grinfeld instability. Polycrystalline growth is also well described. However, the survey of grain-boundary energies shows that the approach is only valid for a restricted range of misorientations as a direct consequence of an amplitude expansion. This range covers approximately half the complete range allowed by crystal symmetry for some fixed reference set of density waves used in the expansion. Over this range, coupled motion to shear is well described by known geometrical rules and a transition from coupling to sliding motion is also reproduced. Amplitude equations for alloys are derived phenomenologically in a Ginzburg-Landau spirit. Vegard's law is shown to be naturally described by seeking a gauge-invariant form of those equations under a transformation that corresponds to a lattice expansion and deviations from Vegard's law can be easily incorporated. Those equations realistically describe the dilute alloy limit and have the same flexibility as conventional phase-field models for incorporating arbitrary free-energy/composition curves. As a test of this approach, we recover known analytical expressions for open-system elastic constants ͓F.

Influence of short-range forces on melting along grain boundaries

“…To estimate the range of the appearing parameters, we start with typical interfacial energiesγ ∼ γ sl ∼ γ gb ∼ 10 [24]. These estimates demonstrate the applicability of the description to a wide class of metallic systems.…”

Structural short-range forces between solid-melt interfaces

“…As had been discussed in detail in Ref. 42 the presence of a lattice rotation makes the use of the full box operator mandatory, and still the description is only valid for small misorientations.…”

“…The definition ofǫ is used for convenience in the amplitude equations formulation, as this removes unpleasant numerical factors, as discussed in detail in Ref. 42. In equilibrium the chemical potential…”

“…This description predicts the correct anisotropy of surfaces energies 40 and elastic properties and contains naturally the linear theory of elasticity 42 . The form of these nonlinearities depends slightly on the underlying model: Above it is given for a PFC model, and there are some differences if these terms are derived from DFT using an equal weight ansatz.…”

“…From the classical density functional theory of freezing (DFT) a functional can be derived which expresses an emerging solid phase as density fluctuations appearing from the liquid state, whereas the (time-averaged) density is spatially constant in the melt phase [34][35][36][37][38][39][40][41][42] . To this end the spatial variations of the density field δψ( r), are expanded as a sum of density waves…”