Multi-phase-field analysis of short-range forces between diffuse interfaces

Abstract

We characterize both analytically and numerically short-range forces between spatially diffuse interfaces in multi-phase-field models of polycrystalline materials. During late-stage solidification, crystal-melt interfaces may attract or repel each other depending on the degree of misorientation between impinging grains, temperature, composition, and stress. To characterize this interaction, we map the multi-phase-field equations for stationary interfaces to a multi-dimensional classical mechanical scattering problem. From the solution of this problem, we derive asymptotic forms for short-range forces between interfaces for distances larger than the interface thickness. The results show that forces are always attractive for traditional models where each phase-field represents the phase fraction of a given grain. Those predictions are validated by numerical computations of forces for all distances. Based on insights from the scattering problem, we propose a new multi-phase-field formulation that can describe both attractive and repulsive forces in real systems. This model is then used to investigate the influence of solute addition and a uniaxial stress perpendicular to the interface. Solute addition leads to bistability of different interfacial equilibrium states, with the temperature range of bistability increasing with strength of partitioning. Stress in turn, is shown to be equivalent to a temperature change through a standard Clausius-Clapeyron relation. The implications of those results for understanding grain boundary premelting are discussed.

Influence of short-range forces on melting along grain boundaries

“…A repulsive interaction, V (W ) < 0, typical for large misorientations, gives raise to grain boundary premelting, whereas attractive interactions, V (W ) > 0, stabilise a dry grain boundary. While combinations of these two cases with extrema in the disjoining potential can occur and result from the structure of V (W ) as being a superposition of exponentially decaying contributions with different ranges [3][4][5], we focus here on the elementary case of monotonic disjoining potentials.…”

“…We therefore expect, that under these circumstances, both the mesoscopic description with the effective boundary condition x (0) = − tan φ ∞ (using α = 0) and the microscopic model with x (0) = − tan φ 0 (with α < 0), which covers all length scales, should lead to the same results on scales larger than δ. For that purpose, we have performed two sets of simulations using the full nonlinear model (5), and the results are shown in Fig. 3.…”

“…Grain boundary premelting was investigated recently e.g. in [3][4][5][6], showing that for low misorientation an attractive interaction between adjacent solid-melt interfaces can stabilise a grain boundary. Recent investigations of heterogeneous nucleation of liquid droplets in overheated grain boundaries link the short range interactions to nucleation processes [7].…”

Structural short-range forces between solid-melt interfaces

“…Hence the derivative −dV (W )/dW expresses the force between crystal-melt interfaces due to this overlap, which can be either repulsive or attractive depending on whether the sign of −dV (W )/dW is positive or negative, respectively. So far, there is little analytical knowledge on the short range contributions to these forces, with the exception of phase-field models 27 , which are based on phenomenological models that do not consider atomic structures, dislocation formation and elastic interactions. The pur- pose of the present article is therefore to gain analytical insights into the nature of these forces, based on a complex Ginzburg-Landau description.…”

“…On the theoretical side we mention in particular discrete lattice models 17 as well as molecular dynamics (MD) simulations [18][19][20][21][22][23] . 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.…”