Phase-field modeling of γ′-precipitate shapes in nickel-base superalloys and their classification by moment invariants

Abstract

We develop a phase-field model for the simulation of precipitate microstructure pattern formation in nickel-base superalloys. The model accounts for the local effects from inhomogeneous and anisotropic elastic deformations, which mainly result from the lattice misfit between the precipitates and matrix phase. Further, in each time-step, we consider the chemical driving force for precipitate ripening to instantaneously equilibrate to a homogeneous value, leading to conserved phase volumes. The model is applied to study the equilibrium shape of a 2D single γ -particle embedded in the γ-matrix with varying lattice misfit and γ/γ interface energies. Further, we apply the method of moment invariants to quantify the resulting equilibrium shapes of precipitates, which turns out to be a size independent characterization of the particle shape. Resulting values for the 2D moment invariants of experimental as well as simulated particle shapes are discussed and compared. Considering ideally spherical particles, we find that large values for the γ/γ -interface width lead to systematic deviations in the resulting moment invariants.

Quantitative Shape-Classification of Misfitting Precipitates during Cubic to Tetragonal Transformations: Phase-Field Simulations and Experiments

“…A general way for the quantification of the shapes of precipitates is given by the method of invariant moments [ 19 ]. The advantage is that it provides a general framework for the quantitative shape classification and can be applied to arbitrarily shaped particles.…”

“…As most of the experimental micrographs are two-dimensional (2D) images, here, we restricted it to the 2D case. The calculation of a 2D moment of a precipitate, such as its barycenter, can be easily performed using an indicator or characteristic function, which takes the value one inside the precipitate and zero outside [ 19 ]. A 2D moment of a precipitate is defined as with the spatial coordinates , a reference point and the characteristic function , which equals 1 inside the precipitate and 0 in the matrix.…”

Phase-field modeling of γ/γ″ microstructure formation in Ni-based superalloys with high γ″ volume fraction

“…Precipitate aspect ratio as a function of L at ε3/ε1= 60 considering isotropic elasticity and no elastic interaction with neighboring precipitates. Exemplary elliptical precipitate shapes are shown together with the prediction of the analytical model given in Equation (25) and (26). Figure 7a) shows the influence of inhomogeneous elastic properties for precipitate and matrix phase on the precipitate shape as well as the influence of anisotropy of the elastic constants (see Table II).…”

“…The range in which realistic values for γ lie is indicated in gray (see Table III). Symbols are phase-field results and the lines are taken from the model given in Equation (25) and (26). b) Polar plots of the elastic energy function B ( n) for different orientations of the interface normal n in the (010) plane and different misfit ratios ε3/ε1.…”

“…The influence of elastic inhomogeneity [22] and periodic arrangements of precipitates with higher precipitate volume fractions [23] on precipitate shapes were studied using the boundary integral method. Comparison of simulated precipitate shapes to experimentally observed microstructures can be used to obtain realistic values of the interfacial energy density [24][25][26].…”