A Grand-Potential Based Phase-Field Approach for Simulating Growth of Intermetallic Phases in Multicomponent Alloy Systems

An efficient and quantitative phase-field model for elastically heterogeneous two-phase solids based on a partial rank-one homogenization scheme

“…This is similar to the classical approach of Kim et al [42], and is useful because it is more intuitive to set an initial condition based on composition rather than diffusion potential. Particularly, in case of non-dilute and non-ideal alloys since composition cannot be analytically expressed as a function of diffusion potential [37,69]. In Appendix E , we provide additional implementation details and the source code to calculate the diffusion potential using Eq.…”

“…To calculate the phase mole fractions X θ r for non-dilute and non-ideal alloys, the interested reader can refer to Refs. [70], [69]. It is apparent from Eq.…”

“…Here, ω θ chem ( μ) is defined as the ratio of the molar grand-potential to molar volume, i.e., Ω θ m ( μ)/V m . It should be noted that the molar grand-potential can be calculated from molar Gibbs energy G θ m using Ω θ m = G θ m − n−1 k=1 μk X θ k , provided the relation between phase mole fraction and diffusion potential is invertible [37], [68], [69].…”

“…It should be emphasized that X β k (µ) and X α k (µ) are prerequisite input properties that are needed as functions of diffusion potentials. Moreover, these properties can be calculated either analytically [37] or numerically [69] depending on the solution model.…”

“…where μrj is the nodal value of the diffusion potential variable of the r th chemical component at node j and χ θ=β,α kr = ∂X α,β k /∂ μr are the components of the susceptibility matrix [37]. This matrix can be calculated by inverting the thermodynamic factor matrix [69], and is implemented as a MOOSE Material object. Second, the components of the Jacobian matrix associated with the variable φ h are…”