A New Analysis of the Microstructure of Ni‐Based Single‐Crystal Superalloys: Relevant Topological Parameters for Efficient Microstructural Modeling

Abstract

Two topological parameters derived from pattern formation theory are introduced, and their potential to accurately describe the elasticity-induced coarsening in the microstructure modeling of Ni-based singlecrystal superalloys at macroscale in 2D is investigated. For that purpose, and given the primordial role of the spatial arrangement of the precipitates on microstructure evolution, two-dimensional phasefield calculations of ideal microstructures undergoing slight topological defects have been carried out. The unstable and stable responses of the system respectively against Eckhaus and zigzag instabilites is discussed and quantitatively described by means of the developed topological parameters ψ x and ψ y .

Instabilities in the periodic arrangement of elastically interacting precipitates in nickel-base superalloys

“…These directions are elastically unfavorable, as can be deduced from the configurational force maps [7,25]. Note that the instability of the γ/γ microstructure with respect to this mode has already been reported in two-dimensional simulations [26]. In pattern formation, the long-wavelength longitudinal instability of lamellar patterns is called Eckhaus instability [27].…”

“…We have used the classical phase field model detailed in Refs. [26,37]. In this model, the two phases alloy is described by a single concentration field and a homogeneous free energy density approximated by a standard double-well potential.…”

“…Also we mention that the model is constructed in such a way that, under stress, interfaces perpendicular to the elastically soft directions adopt a predefined profile (here (1+tanh(2x/w))/2, where w is the interface width) with a predefined surface tension [37]. The parameters of the phase field model are taken from [26] and correspond to a simplified Ni-base superalloy at 950 • C. In this section, the elastic constants are C 11 = 197 GPa, C 12 = 144 GPa and C 44 = 90 GPa, and the precipitate volume fraction is f v = 37% as in §3.1. The eigenstrain is ε 0 = 0.059 and the interface energy, assumed isotropic, is σ = 10 mJ/m 2 .…”

“…The movement of the precipitates is analyzed using a new method proposed in Ref. [26]. It relies on the phase ψ x (r) (not to be confused with phase field φ(r)) which is a continuous equivalent of the discrete phase shift ψ x (R 0 ) = k x u x (R 0 ) induced by the movement of one precipitate along [100].…”