Phase-ﬁeld modeling of γ/γ ′′ microstructure formation in Ni-based superalloys with high γ ′′ volume fraction Felix Schleifer, Markus Holzinger, Yueh-Yu Lin, Uwe Glatzel, and Michael Fleck ∗ Metals and Alloys, University of Bayreuth, Prof.-Rüdiger-Bormann-Str.1, 95447 Bayreuth, Bavaria, Germany The excellent mechanical properties of the Ni-based superalloy IN718 mainly result from coherent γ ′′ precipitates. Due to a strongly anisotropic lattice misﬁt between the matrix and the precipi- tate phase, the particles exhibit pronounced plate-shaped morphologies. Using a phase-ﬁeld model, we investigate various inﬂuencing factors that determine the equilibrium shapes of γ ′′ precipitates, minimizing the sum of the total elastic and interfacial energy. Upon increasing precipitate phase fractions, the model predicts increasingly stronger particle-particle interactions, leading to shapes with signiﬁcantly increased aspect ratios. Matching the a priori unknown interfacial energy density to ﬁt experimental γ ′′ shapes is sensitive to the phase content imposed in the underlying model. Considering vanishing phase content leads to 30 % lower estimates of the interfacial energy den- sity, as compared to estimates based on realistic phase fractions of 12 %. We consider the periodic arrangement of precipitates in diﬀerent hexagonal and rectangular superstructures, which result from distinct choices of point-symmetric and periodic boundary conditions. Further, non-volume conserving boundary conditions are implemented to compensate for strains due to an anisotropic lattice mismatch between the γ matrix and the γ ′′ precipitate. As compared to conventional bound- ary conditions, this speciﬁcally tailored simulation conﬁguration does not conﬂict with the systems periodicity and provides substantially more realistic total elastic energies at high precipitate volume fractions. The energetically most favorable superstructure is found to be a hexagonal precipitate arrangement. c  2020. The manuscript is made available under the license CC-BY-NC-ND 4.0 INTRODUCTION Ni-based superalloys have various applications at ele- vated temperatures, especially in stationary gas turbines and airplane engines. The main strengthening mecha- nism, which makes these alloys applicable for high tem- peratures is particle strengthening by coherent precipi- tations [1]. Apart from the most prominent case, the γ ′ strengthening of Ni-based superalloys for turbine blade materials, a series of alloys exist that are mainly strength- ened by the tetragonal metastable γ ′′ phase. These Nb- containing alloys, such as the well-known IN718, can be cast, forged, machined and welded which renders them ideal candidates for industrial applications. As IN718 contains up to 6% of the cubic L1 2 phase γ ′ , several authors modiﬁed its composition in order to increase the volume fraction of γ ′′ phase and to get rid of γ ′ precipitates while maintaining the composition of the matrix [2–5]. Table I shows the nominal composition of IN718 and a derivative IN718M without γ ′ forming ele- ments. Recent studies aim at deliberate co-precipitation of γ ′ and γ ′′ to exploit ripening inhibiting eﬀects [6, 7] and a dual lattice microstructure [8, 9]. IN718 powders also gain importance for additive manufacturing in the industrial environment [10–13]. It was recently found that alloys containing only one orientational variant of γ ′′ , a so-called single-variant microstructure, can be used to tailor creep resistant materials [14]. Figure 1a) shows a scanning electron microscope (SEM) image of a γ/γ ′′ microstructure in IN718M af- ter homogenization at 1423 K for 2h. The precipitates are found in the Nb-rich interdendritic region. Table I also shows the local composition measured by energy- dispersive x-ray spectroscopy. The precipitates are ar- ranged regularly in three spatial orientations perpendic- ular to one another and show a plate-shaped morphol- ogy. The volumetric γ ′′ phase fraction is ≥ 12 %. Fig- ure 1b) shows a dark-ﬁeld transmission electron micro- scope (TEM) image of γ ′′ preciptates in IN718M. Composition in wt. % Ni Cr Nb Mo Fe Al Ti IN718 nominal max. 55.0 21.0 5.5 3.3 bal. 0.8 1.2 IN718 nominal min. 50.0 17.0 4.8 2.8 bal. 0.2 0.7 IN718M nominal 58.0 18.0 5.0 3.0 bal. – – IN718M measured 56.8 17.6 6.6 3.2 15.8 – – Table I. Nominal composition in wt. % of IN718 and its deriva- tive IN718M together with the measured composition in the Nb-rich region of IN718M as shown in Figure 1a). The shapes and spatial arrangements of coherent mis- ﬁtting precipitates have been modeled assuming a sin- gle precipitate embedded in inﬁnite matrix [15–21]. The inﬂuence of elastic inhomogeneity [22] and periodic ar- rangements of precipitates with higher precipitate vol- ume fractions [23] on precipitate shapes were studied us- ing the boundary integral method. Comparison of simu- lated precipitate shapes to experimentally observed mi- crostructures can be used to obtain realistic values of the interfacial energy density [24–26]. The phase-ﬁeld method is widely considered to be a powerful tool for modeling solidiﬁcation as well as solid- state phase transformations based on a diﬀuse descrip- arXiv:2002.10815v1 [cond-mat.mtrl-sci] 25 Feb 2020