Ni-based single-crystal superalloys are one type of the most important structural materials for advanced aircraft gas turbine blades. It is well known that these alloys can be optimized by a series of thermal heat treatments which produce a high volume fraction of hard cubical γ ′ precipitate phase embedded coherently in a softer γ matrix phase. Generally, superalloy blades are cast with the nominal [001] orientation parallel to their length, which can reduce the incidence of thermal fatigue from temperature variations encountered in operation [1]. The shape of the γ ′ precipitates is cubical with roughly {001} face, and they are approximately aligned along the common [001] direction. It was found that in modern single-crystal superalloys, the γ ′ precipitate fraction can reach 70 vol% or even higher, and the γ ′ precipitate particle size of about 0.45 μ m appears to be optimum for the yield strength and creep strength, and also for hot hardness [2, 3]. Moreover, due to the lattice mismatch of the γ and γ ′ phases with similar face-centered cubic (fcc) structure, an eigenstress field at the phase interface is presented. After a long time aging under the combined influence of loading and temperature, the dislocation network is formed at the γ /γ ′ phase interfaces [4, 5]. The evolution of the interfacial dislocation network has a close relationship with crack propagation of interface [6], γ ′ rafting and creep mechanical properties of alloys [7, 8, 9]. Thus, the yield strength and creep strength of alloys are not only dependent on the morphology of the γ ′ precipitates, but also dependent on the evolution of the dislocation network at the γ /γ ′ phase interfaces.

In strength theory, material failure is determined by stresses and strains. The failure analysis is directly related to the stress or strain concentration, because a material in a localized region slips when the stress reaches the yield strength, and fails when the stress concentration exceeds the failure strength of the material. Many research results have suggested that the microstructure features and its evolutions in a localized region are closely related to the stress field in this region [10, 11, 12, 13]. The morphological evolutions of the γ ′ precipitate and dislocation network induce the change of stress field of alloys and then result in the change of the yield strength and creep strength. Using molecular dynamics (MD) to simulate the evolution of dislocation network and atomic stress distribution at the γ /γ ′ phase interface, the construction of an atomic model has great influence on the microstructure evolution and calculation accuracy of atomic stress. In the previous works [7, 8, 14], the evolution of dislocation network at the γ /γ ′ phase interface was studied by MD simulation, and the relation between the mechanical properties of the phase interface and the evolution of the dislocation network was explored. However, it lacks the analysis of atomic stress field in the process of the evolution of dislocation network at the γ /γ ′ phase interfaces. Moreover, MD simulations in these work were carried out by using a simple two-layer lamellar structure model, which did not consider the cubical γ ′ precipitate with a fraction of 70 vol% embedded in the γ matrix, resulting in an inaccurate calculation of the atomic stress field, and it is also hard to give a real image of geometry and physical processes of dislocation evolution. Since the mechanical properties are strongly dependent on the morphological evolution of the γ ′ precipitates and dislocation network, it is important to investigate the microstructure evolution and relation with the local stress field, which can be a great help for further understanding of the physical mechanisms responsible for the better creep resistance of the Ni-based single-crystal superalloys.

In this work, MD simulations were performed to investigate the atomic local stress field and evolution of dislocation network at γ /γ ′ phase interface of Ni-based single-crystal superalloys under 0, 100 and 300 K. The MD model used a three-dimensional cubic unit and ensures that the fraction of γ ′ is about 70 vol%, which is similar to the structure of the actual materials. The objective of the present work was to explore the stress distribution characteristics of Ni-based single-crystal superalloys during the evolution of dislocation network under temperature and to offer an explanation for the influence of the temperature on the evolution of dislocation network and stress distribution.

In the present study, MD simulations were performed for the idealized Ni-based superalloy composed of Ni₃Al cuboidal precipitates (L1₂γ ′ phase) and the pure Ni matrix (fcc γ phase). It is natural for real materials that the γ matrix is not pure Ni, and the γ /γ ′ microstructure has small amounts of other elements such as W or Co. However, the model presented was simplified in order to extract the fundamentals of dislocation behavior. As we know, the lattice mismatch corresponds to the deformation of an unconstrained lattice, and the mismatch δ was defined as the normalized difference of the lattice parameter of the γ and γ ′ phases, given by

$$\delta = \frac{{2(\alpha_{{\gamma^{{\prime }} }} - \alpha_{\gamma } )}}{{\alpha_{{\gamma^{{\prime }} }} + \alpha_{\gamma } }}, \ \ (1)$$

where α _γ is 0.352 nm [15] and \(\alpha_{{\gamma^{{\prime }} }}\) is 0.3573 nm [16] or 0.362 nm [17], i.e., δ for the γ /γ ′ phase system is 1.5 or 2.8%.

When the mismatch exceeds the elastic limit, misfit dislocations will be formed on the phase interface to reduce the strain energy of the system. Considering the concept of coincidence site lattice on the misfit phase interface, the misfit is described by

$$n\alpha_{{\gamma^{{\prime }} }} = (n + 1)\alpha_{\gamma }, \ \ (2)$$

where n is the n-fold of the lattice parameter. For the current γ /γ ′ phase system, this yields

$$n = \frac{{\alpha_{\gamma } }}{{\alpha_{{\gamma^{{\prime }} }} - \alpha_{\gamma } }}, \ \ (3)$$

When α _γ = 0.352 nm and \(\alpha_{{\gamma^{{\prime }} }}\) = 0.3573 nm, n = 66; α _γ = 0.352 nm and \(\alpha_{{\gamma^{{\prime }} }}\) = 0.362 nm, n = 35. It indicates that within this range of the misfit a phase interface formed by 66 γ ′ and 67 γ or (35 γ ′ and 36 γ ) lattice distances should relax the stress induced by the different lattice parameters.

Based on the idea of constructing the γ /γ ′ interface [17, 18], it can be understood easily that a γ ′ structural unit of 66 × 66 × 66 (or 35 × 35 × 35) (defining the side length of the cuboidal γ ′ precipitate) has almost the same volume and shape as a γ structural unit of 67 × 67 × 67 (or 36 × 36 × 36). Then by substituting the γ ′ structural unit for the γ structural unit, which has more atoms, but approximately the same size, in the center of a 75 × 75 × 75 (or 40 × 40 × 40) γ block with the same lattice directions, the initial unit cell of the γ /γ ′ interface model can be obtained, as shown in Fig. 1. In this model, the cuboidal γ ′ precipitate will not be squeezed by the γ matrix, and the γ ′ fraction of the model is (67/75)³ ≈ 71 vol% or (36/40)³ ≈ 73 vol%, which is in agreement with the experimental value (about 70 vol%). Moreover, the orientations of crystal lattices in the models were all arranged along the [100], [010] and [001] directions as indicated inFig. 1.

Fig. 1

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	Fig. 1 Geometry for atomistic calculations

In order to eliminate the surface effect, the initial model was relaxed by MD calculation under the periodic boundary conditions. The Voter-Chen-type (VC) embedded atom method (EAM) potential is used [16], which is successfully applied to study the misfit dislocation motion and evolution in the Ni/Ni₃Al interface [18, 19, 20, 21, 22].

Since the lattice misfit causes the stress field at the interface, the coherent phase interface is unstable. Based on the principle of minimum energy, the atoms on the phase interface will rearrange in order to minimize the elastic stress field between γ and γ ′ phase. To study the effects of temperature on the stress field and microstructure deformation, three different temperatures 0, 100 and 300 K were applied. The MD simulations were carried out by integrating Newton’ s equations of motion for all atoms using a time step of 3.0 × 10^-15 s. The temperature was kept constant during the simulations, which was achieved by scaling all atoms’ instantaneous velocities with the appropriate Maxwell-Boltzmann distribution at a specified temperature. The open-source MD code LAMMPS [23] and the visualization tools AtomEye [24] were used in the atomistic simulations. The atomic configurations and their evolutions were analyzed by centro-symmetry parameter (CSP) proposed by Kelchner et al. [25], which can provide details of the dislocation motion and evolution in the Ni/Ni₃Al interface.

To study the nanoscale mechanical behavior and investigate the atomic stress field of alloys during the evolution of dislocation network at the γ /γ ′ phase interface, the atomistic stress definition was employed in this work. The atomic stress at an atom i is a stress quantity at the atomic scale. It is a strength measurement of the interatomic interactions of the atom with its neighboring atoms. This atomic stress tensor is defined as [26] $$\sigma_{\alpha \beta } \left( i \right) = - \frac{1}{{2\varOmega_{i} }}\sum\limits_{j \ne \left( i \right)}^{N} {f_{\alpha } } \left( {i, j} \right)r_{\beta } \left( {i, j} \right), \ \ (4)$$

where N is the number of atoms in a region around atom i within an EAM potential cutoff distance (which is 4.80 Å for Ni and 4.04 Å for Al), f_α (i, j) is the vector component form of the interaction force exerted by atom j on atom i, r_β (i, j) is the vector component form of the relative position from atom j to atom i, and Ω _i is the volume of atom i given by

$$\varOmega_{i} = \frac{4}{3}\pi R_{i}^{3}, \ \ (5)$$

where R_i is the radius of the atom i.

Taking an average over the volume around atom i within the potential cutoff distance, the average atomic stress tensor \(\bar{\sigma }_{\alpha \beta } \left( i \right)\) at atom i is given by [27] $$\bar{\sigma }_{\alpha \beta } \left( i \right) = \frac{1}{N}\sum\limits_{j = 1}^{N} {\sigma_{\alpha \beta } \left( j \right)}, \ \ (6)$$

A regular three-dimensional misfit dislocation network (mosaic structure) appears on the γ /γ ′ phase interfaces at T = 0 K after MD relaxation, the shape of dislocation network independent on the mismatch parameter δ , which only causes the change in the side length of the dislocation network.

(1) Under the influence of the temperature, the mosaic structure of dislocation network gradually becomes irregular till damage. Meanwhile, with increase in the temperature, dislocations emit in the γ matrix phase and cut into the γ ′ phase from the γ /γ ′ phase interfaces where the dislocation network is damaged.

(2) The dislocation motion is closely related to the stress level in the γ matrix phase, the dislocation firstly emits in the γ matrix phase, the number of dislocation increases when the stress caused by the temperature increases, and the peak stress occurs at the γ /γ ′ phase interfaces where the dislocation network is formed.