Three-dimensional imaging and phase-field simulations of the microstructure evolution during creep tests of 〈011〉 -oriented Ni-based superalloys

Three-dimensional imaging and phase-field simulations of the microstructure evolution during creep tests of 〈011〉 -oriented Ni-based superalloys

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Available online at www.sciencedirect.com

ScienceDirect Acta Materialia 84 (2015) 237–255 www.elsevier.com/locate/actamat

Three-dimensional imaging and phase-field simulations of the microstructure evolution during creep tests of h0 1 1i-oriented Ni-based superalloys ⇑

A. Gaubert,a,1 M. Jouiad,b J. Cormier,c Y. Le Bouard, and J. Ghighic,2 a

Department of Metallic Materials and Structures, Onera, 29 Avenue de la Division Leclerc, 92322 Chaˆtillon Cedex, France b Department of Mechanical and Materials Engineering, Masdar Institute of Science and Technology, PO Box 54224, Abu Dhabi, United Arab Emirates c Institut Pprime, UPR CNRS 3346, Physics and Mechanics of Materials Department, ISAE-ENSMA, 1 avenue Cle´ment Ader, Te´le´port 2, BP 40109, 86961 Futuroscope Chasseneuil Cedex, France d Laboratoire d’Etude des Microstructures, CNRS/Onera, 29 Avenue de la Division Leclerc, 92322 Chaˆtillon Cedex, France Received 3 March 2014; revised 20 August 2014; accepted 14 October 2014

Abstract—Microstructure evolution during tensile creep of h0 1 1i-oriented samples of first-generation Ni-based single-crystal superalloys was investigated both experimentally and numerically. Based on scanning electron microscopy and 3-D volume reconstruction using a dual-beam system, it was shown that the initially cuboidal microstructure first elongates along the cubic axis perpendicular to the tensile axis, and then coalesces along the two other cubic directions. We found that the normal to the platelets, initially close to a cubic axis, slightly rotates towards the tensile axis direction and the destabilization of the platelet microstructure appears to control the onset of the tertiary creep stage. In a second part, microstructure evolutions are analyzed using 3-D phase-field simulations. The inhomogeneous and anisotropic elastic and plastic driving forces are discussed, as well as the importance of a small misorientation of the tensile axis from the [0 1 1] direction. Surprisingly, contrary to the case of h1 0 0i-oriented sample, complete rafting is not obtained in the simulations, suggesting that an additional mechanism at the level of individual dislocations could be missing. Finally, we have developed an energetic model for a perfectly rafted microstructure, which proves that plasticity in the c phase is at the origin of the c=c0 interface rotation during creep. Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved. Keywords: Superalloy single crystals; c0 -rafting; Microstructure; Phase-field models; Serial sectioning methods

1. Introduction Nickel-based single-crystal superalloys are unique materials initially developed for aeronautic and industrial gas turbine applications. In particular, they are used in the hottest sections of aero engines as turbine blades and vanes where they have to withstand complex thermomechanical forces under severe conditions. The extensive use of monocrystalline superalloys for the design of such components is motivated by their excellent fatigue and creep properties at high temperature (up to 1100 °C) under aggressive environments such as corrosion and oxidation [1–3]. These interesting properties are inherited from their particular microstructure which consists of a high volume fraction of c0 strengthening precipitates (L12 structure) embedded in a c solid-solution matrix (face-centered cubic (fcc) structure) [4,5]. The c and c0 phases are coherent but exhibit

⇑ Corresponding author; e-mail: [email protected] 1 2

Current address: Snecma Groupe Safran, Villaroche, France. Current address: University of Cape Town, Department of Mechanical Engineering, Cape Town, South Africa.

different mechanical yielding, resulting in two different deformation mechanisms with respect to the phase considered. For instance, as the c phase has lower yielding, plastic deformation is the primary mechanism in this phase where dislocation sources (subgrain boundaries and interdendritic casting pores) [6–8] are activated. The impinging mobile dislocations initiate first in the c matrix, then propagate toward c0 precipitates, which play a key role in hindering dislocation motion, leading to the main hardening effect. In addition, during service operation, these c0 particles are subject to directional coarsening, the kinetics of which depends on both the temperature and mechanical loading [9–12]. This coarsening, known also as c0 rafting, has been extensively studied (e.g. [9,10,13–15]) especially for h0 0 1ioriented samples during isothermal and/or non-isothermal creep tests. Indeed, previous experimental and simulation investigations have led to a good understanding regarding the driving forces governing the rafting process. These driving forces result from various factors such as the difference in elastic constants between c and c0 phases, the c=c0 lattice mismatch and the plastic deformation in the c matrix [16–18]. Moreover, it was observed that when the c/c0

http://dx.doi.org/10.1016/j.actamat.2014.10.034 1359-6462/Ó 2014 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.

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coherency stresses are relaxed by a small amount of creep deformation, c0 rafting can subsequently proceed even without an applied stress [19]. However, only a few studies have been reported on the creep behavior of h0 1 1i-oriented samples [8,20–26]. It is worth noting that the dependence of c0 rafting on crystallographic orientation is of great interest when dealing with complex blade geometries (e.g. internally cooled blades) that are subject to multiaxial forces. Early investigations showed that c0 rafting is likely to occur at ±45° with respect to the mechanical loading direction [25], in the form of rods instead of platelets as usually observed for creep testing along a h0 0 1i direction and for negative misfit alloys. More recently, Sass et al. [24], Tian et al. [20] and Agudo et al. [8] also analyzed such a 45° rafting for near h0 1 1i-oriented Xsamples with the occurrence of one or two variants of directional coarsening [20,23,25]. From these studies, it seems that the c0 directional coarsening for creep conditions along a close h0 1 1i crystallographic orientation is highly sensitive to the effective orientation of the specimens and results in different kinds of precipitate morphologies. The aim of this study is to investigate the microstructure evolution during tensile creep tests carried out on h0 1 1i-oriented samples of first-generation Ni-based superalloy single crystals. A combined approach, consisting of novel experimental observations and simulations, was conceived. On the one hand, a dual-beam microscope including a high-resolution scanning electron microscope (SEM) equipped with a focused ion beam (FIB) was used to make direct observations of the 3-D microstructure obtained during creep tests along the h0 1 1i axis of superalloy single crystals. Since the pioneering work of Lund et al. [27] using mechanical polishing, the interest of the scientific community working on superalloys for serial sectioning has been growing fast as the SEM/FIB technique enables a much more efficient characterization of 3-D microstructures [28]. In particular, this technique was recently extended to investigate the evolution of c0 precipitates during isothermal and non-isothermal creep tests [29]. In the present study, the spatial resolution of the 3-D reconstruction was improved using recent developments of the method [30]. On the other hand, creep tests have been simulated with the phase-field method. This method has proven to be very a powerful tool to understand microstructural evolutions, in particular in Ni-based superalloys. For instance, phase-field models have been intensively used for predicting c0 rafting in Ni-based single-crystal superalloys with a h0 0 1i orientation (e.g. [31]) or particle coarsening in bimodal microstructures during aging of polycrystalline superalloys (e.g. [32]). Indeed, many effects such as elasticity and plasticity can be accounted for in a simple and consistent framework. Recently, phase-field modeling has highlighted the importance of the c phase plasticity in controlling the c0 rafting process and microstructure evolution in the case of h0 0 1i-oriented samples [33–36]. In this case, both elastic and plastic driving forces were shown to contribute to rafting. In the present paper, a different loading axis is considered (close to h0 1 1i). Because of the strong anisotropy of both elasticity and plasticity in Ni-based superalloys, there is no reason why the consequences of these two driving forces should be similar. Moreover, even if plasticity increases the rafting kinetics

in h0 0 1i-oriented samples, it has been shown to slow down the transformation in other alloys [37]. In this context, a detailed study of the role of the c phase plasticity in promoting the c0 rafting process is clearly needed for h0 1 1i-oriented samples. In addition, in the present paper, special attention has been paid to the role of a small misorientation from the perfect h0 1 1i orientation on the resulting c0 directional coarsening. Finally, a simplified energetic model has been developed to discuss the precise orientation of the rafts and its evolution.

2. Experimental procedure 2.1. Materials and creep experiments Creep experiments have been carried out on AM1 and MC2 Ni-based single-crystal superalloys, which are both first-generation alloys. Their chemical compositions, listed in Table 1, are very similar, implying that similar microstructure evolutions are expected in these two alloys. Standard heat treatments have been applied to the specimens leading to the well-known cuboidal microstructures presented in Fig. 1. The precise orientation of the specimens was determined before the tests. For the AM1 sample, a 3° deviation from the [0 1 1] crystal orientation has been measured with the Laue diffraction technique. MC2 samples were machined from a test bar having a 6° secondary misorientation from a perfect h0 1 1i orientation (Fig. 1). Two different experiments were carried out depending on the sample considered. For the AM1 specimens, tensile creep loadings were conducted at 1050 °C under 150 MPa for 115 h. The tested specimens have a cylindrical shape with a gauge length of 14 mm and a cross-section diameter of 8 mm. The heating was performed by induction. The test was conducted using a 4000 daN electrohydraulic Schenk machine. A conventional extensometer device was used to record the specimen deformation during the test. An additional cyclic loading at 950 °C for about 25 h was also applied to the specimen in order to assess the influence of rafting on the mechanical behavior of the material. Since microstructure observations were made at the end of the entire loading sequence, we assume that the 950 °C loading has not affected significantly the microstructure obtained after the 1050 °C creep loading sequence. This assumption is based on previous results obtained during similar tests performed on h0 0 1i-oriented specimens under the same temperature and loading conditions [38]. For the MC2 specimens, multi-interrupted isothermal creep tests at the same temperature (1050 °C) were performed under 140 MPa, after an initial creep test at 1200 °C under 145 MPa for 30 s. This first very short creep test modifies neither the creep behavior nor the c0 microstructure evolution for h0 1 1i-oriented MC2 samples at 1050 °C [22]. After such an overheating, the c0 microstructure remains cuboidal as in Fig. 1, with a very low volume fraction (below 3%) of ultrafine tertiary c0 nucleating in the c channels that are below 10 nm in size. The sequential creep test was performed using a radiant furnace and a sample with a 14 mm gauge length and a load-bearing section of 1.5  4 mm2. Special attention was paid to cooling the sample down to room temperature as fast as possible

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Table 1. Chemical composition of the AM1 and MC2 single-crystal Ni-based superalloys (wt.%). Element

Ni

Co

Cr

Mo

W

Ta

Al

Ti

C

Fe

AM1 MC2

Bal. Bal.

6.5 5.0

8.0 8.0

2.0 2.0

6.0 8.0

8.0 6.0

5.2 5.0

1.2 1.0

<0.01 <0.01

<0.2

Fig. 1. (Left) Heat-treated microstructure in the (1 0 0) plane of the [0 1 1]-oriented MC2 specimen. (Right) Crystallographic orientation of the MC2 specimen tensile axis.

during each interruption of the creep test by switching off the radiative heating (cooling rates over 10 °C s1 in the 850–1050 °C temperature range). This procedure prevents any growth of the c0 particles during cooling due to the nucleation of tertiary c0 particles that are likely to coalesce with the coarsest particles in the case of slow cooling rates, and then change their morphology [39]. Additional details about the experimental procedure and sample surface analyses are indicated elsewhere [40]. Fig. 2 shows the creep curves of AM1 and MC2 samples recorded during the tests in terms of strain as a function of time. They exhibit quite similar mechanical behavior with an incubation period without deformation in the first stage of creep, which is in good agreement with Refs. [8,23], and subsequent comparable strain rates in the secondary stage. Both tests were interrupted at the beginning of the tertiary creep regime.

Fig. 2. Creep curves of the h0 1 1i-oriented specimens of AM1 and MC2 superalloys at 1050 °C under 150 and 140 MPa, respectively.

2.2. Microstructural observations 2.2.1. Interrupted creep test observations In this section, the interrupted creep tests on the MC2 alloy are analyzed in terms of microstructure evolution. SEM micrographs were obtained after 5, 15 and 25 h of creep. SEM observations were performed using a JEOL JSM 7000-F field emission gun microscope operating at 25 kV. Fig. 3 presents the microstructure evolution along a (1 0 0) crystal plane of the MC2 alloy in the dendrite cores. It can be seen that the coalescence, in the [0 0 1] and [0 1 0] directions, starts in the early stages of the creep test under the aforementioned conditions of stress and temperature. In addition, some L-shaped c0 particles with both variants of directional elongation are observed in Fig. 3a and b. It is also observed that after 15 h of creep deformation, c0 particles tend to align themselves along one main orientation. After 25 h of creep, significant microstructure degradation is observed (Fig. 3c). This degradation, characterized by local c/c0 interface rotation and branching of c0 particles, leads to less clear raft orientations. Note finally that the interface rotations are such that their normals are tilted towards the tensile axis direction. 2.2.2. 3D-imaging of microstructure using SEM/FIB SEM/FIB observations of the bulk AM1 samples were acquired on a Helios 650 dual-beam system using the slice-and-view technique. This technique consists of serial sectioning using a high-energy Ga ion source (30 keV) and imaging using a high-resolution in-lens detector. The piezo-controlled movement of the sample stage allows a fully automated slicing and imaging with very high precision. The image acquisition was performed during sequential sectioning in the Z direction. The whole process consists of three phases as follows: (1) protecting the cross-sections

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Fig. 3. c/c0 microstructure evolutions in dendrite cores of MC2 alloy at 1050 °C/140 MPa after (a) 5 h, (b) 15 h and (c) 25 h. Observations performed on the (1 0 0) plane.

of interest by platinum deposition; (2) setting up slicing and imaging conditions; and (3) collecting serial images and data processing. Additional adjustments such as image drift correction [30,28,29,41] and surface sample cleaning are required to avoid image misalignment. The 3-D image reconstruction is performed using AvizoÒ software by selecting the typical microstructure features. Fig. 4a illustrates the region of interest (ROI) of the AM1 sample before sectioning. The surface of the crosssection was first protected with platinum deposition at low beam current to avoid degradation of the ROI. The trenches surrounding the ROI were made by ion milling to allow collection of secondary electrons (SEs) impinging from the 52° inclined surface. Hundreds of SE images were then recorded and typical microstructure features (Fig. 4b) were selected to reconstruct the typical volume. The obtained result is given in Fig. 4c and d. The 3-D microstructure clearly reveals that coalescence of the c0 precipitates first occurs along the [1 0 0] direction in the case of the tensile [0 1 1]-oriented AM1 sample. A similar behavior was observed in the MC2 samples and has also been reported in other superalloys [8]. In addition, as in the case of the MC2 samples, coarsening in the (1 0 0) planes is also visible here, leading to a microstructure that comprises platelets rather than rods (see Fig. 4d). Moreover, even if the ROI is quite small, one can notice a trend of formation of predominantly one family of platelets. The c0 precipitates in the (1 0 0) plane exhibit a pronounced shape destabilization in the AM1 specimen. This observation is consistent with Fig. 2c given that both microstructures were observed at the beginning of the tertiary creep stage. 2.2.3. Quantitative analysis Stereological parameters were used to analyze c/c0 features observed during creep test interruptions performed on the MC2 alloy. These parameters are extracted from SEM images using specific image analysis procedures described elsewhere [42,43]. They are then used to study the evolution of c0 particles (length and thickness) and of the c channel width. Fig. 5a shows that the initial microstructures in the dendrite cores and the interdendritic spacings are rather similar. During creep, the c0 particles in the interdendritic spacings extend faster and become thicker (Fig. 5b). This strongly suggests a slightly higher c0 volume fraction in the interdendritic spacings in comparison to the primary dendrite arms, but only a 3-D microstructure characterization (such as the one presented in Fig. 4) of a sufficiently large volume would be fully conclusive on that point.

Finally, we note that no significant change is measured in the matrix channel widening (Fig. 5c), despite the progressive degradation of the c0 rafting previously observed (see Fig. 3c). 2.2.4. Summary of microstructure evolutions Our microstructural evolutions can be summarized as follows. Tensile creep along the [0 1 1] crystallographic orientation first coarsens the precipitates along the [1 0 0] direction to form rods. Coarsening then proceeds along two [0 1 0] and [0 0 1] directions in the early stages of creep deformation (i.e. up to 15–20 h of creep deformation at 1050 °C/140 MPa for the MC2 alloy). For both alloys, increasing the creep time tends to favor one type of directional coarsening and, at the onset of the tertiary creep regime, a progressive destabilization of the c0 rafting during which the c/c0 interfaces are tilted. The observed tilt is such that the normals to the interfaces come closer to the tensile axis. In good agreement with our results, Agudo et al. report the formation of platelets along two types of h1 0 0i directions during high-temperature h0 1 1i tensile creep of LEK94 alloy [8]. The authors also pointed out the influence of a very low misorientation from [0 1 1] in controlling the transition from directional coarsening along two h1 0 0i directions to elongation along one single h1 0 0i direction. In fact, if the loading is perfectly aligned with the [0 1 1] crystal orientation, both [0 1 0] and [0 0 1] orientations are equivalent in terms of elastic and plastic driving forces. In that case, the microstructure is expected to fulfill, on a macroscopic scale, the symmetry with respect to the (0 1 1) and (0 1 0) planes. A misalignment away from the perfect h0 1 1i crystallographic direction necessarily breaks this symmetry and is consistent with the observed c0 rafting along only one of the two h1 0 0i directions. Most interestingly, only a small misorientation of a few degrees seems to be enough to trigger a specific direction for c0 elongation. The relative importance of plastic and elastic driving forces in this microstructure evolution is still unknown and this issue will be addressed in the subsequent section using phase-field modeling. 3. Phase-field model The phase-field model used here has been previously proposed in Ref. [35]. In this model, the ordered nature of the c0 phase is accounted for, as well as the inhomogeneous and anisotropic elastoviscoplastic behavior of the

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Fig. 4. Setup of the slice-and-view technique in a dual-beam microscope. (a) SEM micrograph of AM1 cross-section at a milling position of 52°: the SE image is taken at tilt = 38°. Platinum protection of the region of interest is visible. (b) Typical selected microstructure, (c) reconstructed volume using AvizoÒ software. (d) Perspective view NB: The 3-D reconstruction in (c) and (d) of the c=c0 microstructure does not show small tertiary precipitates observed only for the first slices in (b). These precipitates most likely nucleate upon cooling from the high-temperature testing and are not representative of the high-temperature state.

coexisting phases. The main equations and assumptions are briefly recalled in the following section. 3.1. The elastoviscoplastic phase-field model Phase-field models describe microstructures at a mesoscale with continuous fields. In the case of the c/c0 microstructures, four fields at least are needed: an aluminum atomic concentration field cðr; tÞ and three long-range order (LRO) parameter fields gi ðr; tÞ, which describe the four variants of the c0 ordered phase. Indeed, in the model, multicomponent industrial alloys such as AM1 and MC2 are effectively treated as Ni–Al binary alloys. In the following simulations, our aim was to model a generic first-generation superalloy (e.g. AM1 or MC2) but not to focus on the slight differences between AM1 and MC2 superalloys. In other words, the numerical parameters used below will be representative of a first-generation Ni-based superalloy, and the simulation results will be relevant for a qualitative comparison with the microstructure evolution observed in both AM1 and MC2 alloys. The evolution of the fields are controlled by the Cahn– Hilliard equation for the conserved concentration field and the Allen–Cahn equation for the non-conserved LRO fields, respectively:

@c dF ¼ Mr2 @t dc @gi dF ¼ L dgi @t

ð1Þ ð2Þ

where the mobility coefficient M and the relaxation kinetic coefficient L are taken as constants. In these equations, the functional derivative of the free energy of the system with respect to the fields represents the driving force for microstructural evolution. The free energy is usually assumed to be an additive decomposition of different contributions coming from the bulk, the interfaces, elasticity or any other contributions: F ¼ F GL þ F el þ F vp

ð3Þ

These terms are described in the following subsections. 3.1.1. Ginzburg–Landau free energy The chemical part of the free energy accounts for the energy associated with phase transformation and interfaces between phases. The classical Ginzburg–Landau functional is used in the present model: Z 3 k bX F GL ¼ f homo ðc; fgi gÞ þ jrcj2 þ jrgi j2 dV ð4Þ 2 2 V i¼1

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Fig. 5. Evolution of the c/c0 microstructure during tensile creep of the MC2 alloy at 1050 °C/140 MPa along a crystallographic orientation close to [1 1 0]. Both dendrite cores and interdendritic spacings are investigated: (a) c0 particle length; (b) c0 particle thickness; (c) c channel width.

where k and b are gradient energy coefficients and f homo ðc; fgi gÞ is the free energy density of an homogeneous system. The homogeneous part is approximated by a Landau polynomial expansion with respect to the order parameters. More precisely, according to Boussinot et al. [44], the homogeneous free energy density is written as:

"

X 1 B ðc  cc Þ2 þ ðc2  cÞ g2i 2 6 i¼1;3 # X C D  g1 g2 g3 þ g4 3 12 i¼1;3 i

f homo ðc; fgi gÞ ¼Df

ð5Þ

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Df is the energy density scale of the model and c2 is an arbitrary concentration chosen between the equilibrium concentrations cc and cc0 . The coefficients B; C and D are constants related to those concentrations and the equilibrium amplitude of LRO g0 : 2 B ¼ 2 ðcc0  cc Þ g0 6 C ¼ 3 ðcc0  cc Þðc2  cc Þ g0 6 D ¼ 4 ðcc0  cc Þðcc0 þ 2c2  3cc Þ g0

ð6Þ

ð8Þ

The latter is associated with the lattice parameter mismatch between the two phases. Vegard’s law assumes a linear dependence of this stress-free strain on the concentration field. In the present model, the eigenstrain tensor also accounts for the plastic activity: e 0 ðrÞ ¼ e T DcðrÞ þ e p



ð9Þ

where DcðrÞ is the difference between the local concentration field and the average concentration, e p is the plastic   1 and 1 is the identity strain tensor, e T ¼ d= cc0  cc  matrix. The c/c0 lattice misfit d is the relative difference between the equilibrium lattice parameters of the c and c0 phases. The inhomogeneity between elastic moduli of the two phases is taken into account in the model by assuming a linear dependence of the elastic moduli tensor with respect to the concentration field:  þ C 0 DcðrÞ C ðrÞ ¼ C ð10Þ 



The average value of the deformation field  e is given by  the boundary conditions. In the case of an applied stress a r ; e is the solution of:   dEst ¼ raij deij

ð13Þ

   eij ¼ he0ij i þ Sijkl C 0klmn hDcðrÞe0mn ðrÞihDcðrÞdemn ðrÞiÞ þ rakl

where e is the average total strain and C the local elastic  moduli tensor. The first part in Eq. (7) accounts for the a work of the applied stress r . The elastic strain tensor e el  can be deduced from the total strain tensor e ðrÞ and the eigenstrain tensor e 0 ðrÞ : e el ðrÞ ¼ e ðrÞ  e 0 ðrÞ

  @2 @ h  uk ðrÞ ¼  C ijkl þ C 0ijkl DcðrÞ ekl  e0kl ðrÞ @rj @rj @rl i ð12Þ þC 0ijkl DcðrÞdekl ðrÞ

which reads:

3.1.2. Elastic energy In the framework of small deformations, the elastic energy reads: Z 1 a  F el ¼ V r : e þ C : e el : e el dV ð7Þ   2 V   



 ijkl C

243



This inhomogeneity has been shown to significantly impact the microstructure evolution in multiphase alloys [45–48]. During diffusive phase transformation, it can be assumed that mechanical equilibrium is always fulfilled since elastic degrees of freedom equilibrate much faster than the characteristic diffusion time. Decomposing the displacement field into its average value  e and deviation d e ,  the mechanical equilibrium is solved in Fourier space using a fixed-point iteration method [35,44,51]. More precisely, d e is related to the heterogeneous displacement field u by:  1 T ð11Þ d e ¼ ru þ ðruÞ 2 The mechanical equilibrium can be expressed in terms of the heterogenous displacement field [35]:

ð14Þ where h:i stands for the spatial average. At each time step, Eqs. (12) and (14) are solved numerically using the fixedpoint iteration method. 3.1.3. Viscoplasticity It is well known that plastic activity mostly takes place in the c channels and that the c-phase plasticity is the main driving force for c0 rafting [16,19]. Plastic deformation in the c phase is described in the framework of phenomenological viscoplasticity although the size of the matrix channels is small, especially with a c/c0 cuboidal microstructure (see Figs. 1 and 3). This point is extensively discussed in Ref. [35]. In addition, a crystal plasticity formulation is used to take into account the non-isotropic nature of plastic deformation in superalloys [52]. In that context, the plastic strain is the sum of the slip over all the slip systems: X s e p ðrÞ ¼ cs ðrÞ m ð15Þ   s s where m is the orientation tensor of the slip system s. As  explained in Ref. [35], both octahedral and cubic slip systems have been taken into account. Norton’s flow rule describes the evolution of the slip:  s n j s  xs j rs signðss  xs Þ ð16Þ c_s ¼ k s where hai is the positive part of a; ss ¼ r :m is the   resolved shear stress on the slip system s, xs is the kinematic hardening and rs is the isotropic hardening. The latter is assumed constant and equal to rs0 . A nonlinear kinematic hardening is employed:

xs ¼ cs a s

with

a_ s ¼ c_s  d s j c_s j as

ð17Þ

The viscoplastic material parameters k; n; c ; d s used in Eqs. (16) and (17) are phase dependent. Therefore, in the model, these parameters are made dependent on the local concentration:

 þ A0 tanh h DcðrÞ AðrÞ ¼ A ð18Þ cc0  cc rs0 ;

s

where A stands for any of the viscoplasticity material  ¼ ðAc þ Ac0 Þ=2 and A0 ¼ ðAc  Ac0 Þ=2. parameter, A In the microstructure evolution considered here, plastic activity mostly takes place in the c channels. To ensure this behavior in the model, a very high value of the isotropic hardening constant rs0 in the c0 phase is used. In this situation, the parameters k; n; cs and d s in the c0 are not relevant and, for convenience, their values are taken equal to

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the ones in the matrix. A consequence of this choice is that the viscoplastic free energy F vp does not explicitly depend on the concentration and LRO fields and therefore F vp does not contribute to the driving forces in Eqs. (1) and (2). 3.2. Implementation and numerical inputs of the model Equations are spatially discretized on a regular cubic grid of size d. Periodic boundary conditions are assumed. The Cahn–Hilliard and Allen–Cahn equations are solved in the Fourier space using a semi-implicit scheme [53]. Viscoplasticity evolution equations are solved with a firstorder explicit Euler scheme. Mechanical equilibrium is solved through a fixed-point algorithm described in Ref. [35]. Numerical calibration of phase-field models is still a challenging issue since data at the mesoscale are needed for the simulations. The parameters used here are representative of a first-generation commercial superalloy at 1050 °C. Table 2 sums up the numerical inputs used to run the simulations presented in this paper. The mobility M is chosen to recover the interdiffusion coefficient in the c phase [54]. The relaxation coefficient L is chosen much larger than D=d 2 to ensure that the interface motion is a diffusion-controlled process. The phase diagram data are taken from Ref. [32]. Calibration of the interface energy is explained in Ref. [35]. Concerning elasticity and viscoplasticity numerical inputs, the inverse procedure developed to obtain values for elastic moduli and viscoplasticity parameters for both phases is detailed in Ref. [35]. The numerical value for the lattice mismatch is consistent with synchrotron X-ray diffraction measurements on the AM1 superalloy [32,55].

Table 2. Phase-field numerical inputs. Diffusion Concentrations

Interface energy Misfit c phase elastic moduli

c0 phase elastic moduli

c phase viscoplastic param.

c0 phase viscoplastic param.

D0 [m2 s1] DU [eV] cc cc0 c2 rexp [mJ m2] d C 11 [MPa] C 12 [MPa] C 44 [MPa] C 11 [MPa] C 12 [MPa] C 44 [MPa] roct 0 [MPa] coct [MPa] d oct k oct [MPa s] [MPa] rcub 0 ccub [MPa] d cub noct =ncub roct 0 [MPa] coct [MPa] d oct k oct [MPa s] rcub [MPa] 0 ccub [MPa] d cub noct =ncub

1.45 104 2.8 0.16 0.231 0.18 5 0.002 185.1 134.0 84.2 185.6 129.4 90.6 35 20000 500 500 50 500000 2000 4.3 1000 20000 500 500 1000 500000 2000 4.3

4. Phase-field simulation results 4.1. Simulation results for h0 1 1i perfectly oriented specimens Creep along the [0 1 1] orientation is simulated in 3-D on a regular grid containing 643 points with a grid spacing equal to 4.12 nm and periodic boundary conditions. The initial configuration consists of one cubic precipitate of 222 nm in edge length leading to an initial c0 phase volume fraction around 0.6. Both precipitate size and volume fraction have been chosen to be representative of superalloy microstructure even if they are slightly smaller than the experimental values. Indeed, we have chosen to limit the grid spacing in order to prevent the decrease in c0 volume fraction which has been reported in Ref. [35]. Moreover, because we perform time-consuming 3-D simulations, a c0 volume fraction smaller than the experimental one has been chosen to avoid narrow c channels, which imply the use of a very small grid spacing. Both elastic and elastic–viscoplastic simulations have been run in order to assess the relative influence of elasticity and plasticity. The results are presented in Figs. 6 and 7. Fig. 6 shows that the elastic phase-field simulation predicts the coarsening of the precipitates along the [1 0 0] crystal orientation when applying a tensile creep load along a [0 1 1] direction. The final microstructure is a cylinder with a section close to a rounded square. It should be emphasized that the system has reached a stable configuration. Indeed, the width of the c channel, equal to 47.11 nm at a time t ¼ 4860 s, does not evolve during the following 4000 s. As shown in Fig. 7, a precipitate coarsening along the [1 0 0] direction is also observed when plasticity in the matrix phase is accounted for. However, one can notice that the shape of the precipitate is less regular in the elastic–viscoplastic simulation than in the purely elastic one. The differences appear clearly in a section of the precipitate along a (0 1 0) plane. In the elastic case, the section is straight, whereas in Fig. 7, a wavy shape is generated by the plastic deformation field. A similar behavior was previously reported in Ref. [35] for microstructure evolution during h1 0 0i creep. Moreover, in the (1 0 0) section of the elastic–viscoplastic simulation, the two types of precipitate corners, denoted a and b on Fig. 7, have a different curvature. The a corners, perpendicular to the loading axis, are rounder than the b corners. On the same picture, it can also be observed that the two c/c0 interfaces, namely the initially (0 1 0) and (0 0 1) interfaces, are no longer perpendicular. The aspect ratio R ¼ l2 =l1 has been defined to quantify the shape change of the precipitate, where l1 and l2 are 1, respectively the precipitate length along [0 1 1] and ½0 1  (see Fig. 7). At t = 4860 s, this aspect ratio is significantly above 1 (R = 1.1), which implies that, after subsequent creep with the elastic–viscoplastic model, the initially (0 1 0) and (0 0 1) interfaces are no longer perpendicular and that their normals tend to move closer to the [0 1 1] loading axis. This trend is similar to the one observed in the experimental results of Section 2 where it has been shown that c/c0 interfaces show similar local rotations at the beginning of the tertiary stage of creep (Fig. 3c). Elastic and elastic–viscoplastic simulations have also been compared in terms of coarsening kinetics along the [1 0 0] direction. No significant difference has been observed between the simulations. In both cases, the precipitate has

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245

Fig. 6. Elastic phase-field simulation result for a perfectly [0 1 1]-aligned loading in creep conditions (1050 °C/150 MPa). The figure at the top is a contour plot of the concentration field with c = 0.2 the partitioning concentration. The figures at bottom are 2-D maps of the concentration field.

coalesced with its periodic replica after 300 s of simulation. However, no clear conclusion can be drawn on the influence of plasticity on coarsening kinetics since a decrease in the c0 volume fraction is observed in the elastic–viscoplastic case, the final volume fraction being equal to 0.54. This value is in relative good agreement with the c0 volume fraction for AM1 and MC2 at 1050 °C [7,21] but significantly smaller than in the elastic simulation where its value is equal to 0.62. It is indeed classically known that elasticity changes the equilibrium concentrations [56], but it has also been shown that plasticity impacts those quantities [35], which explains the present decrease in the c0 volume fraction.

The shape of the c0 precipitate can be linked to the plastic activity. The von Mises invariant of the plastic strain tensor is plotted on Fig. 8. This quantity has no physical meaning in a crystal plasticity context but allows an easy visualization of plastic activity in the simulation. As can be observed, the plastic activity is confined in two types of c-channels (i.e. so-called roof channels under h0 1 1i loading direction), consistent with the experimental results [25]. A parallel can be drawn with tensile h0 0 1i creep loading simulations. It has been shown that plastic deformation concentrates in matrix channels perpendicular to the loading direction which are the remaining channels after the rafting process [35,57]. In that case, the dislocations at c/c0

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Fig. 7. Elastic–viscoplastic phase-field simulation result for a perfectly [0 1 1]-aligned loading. The figure at the top is a contour plot of the concentration field with c = 0.2 the partitioning concentration. The figures at bottom are 2-D maps of the concentration field.

interfaces have been shown to relax the misfit stresses [58]. In the present case, the remaining c channels are also the ones that exhibit the most pronounced plastic activity. The precipitate corners shape can also be linked to the plastic activity. The rounder corners (denoted a on Fig. 7) are located where the plastic activity reaches a maximum. A deeper investigation of the simulated plastic activity reveals that, in agreement with the Schmid factors associated with a h0 1 1i loading, four slip systems are mainly activated in the simulation as reported in Table 3, with a value of 0.408. It is interesting to note that, because of the stresses generated by the microstructure, two of these systems are activated in the [0 1 0] channels and the two others in the

[0 0 1] channels. Concerning the cubic slip systems, four of them mainly contribute to plastic activity, namely the systems defined by the (0 1 0) and (0 0 1) glide planes. Their spatial activation is different from the octahedral slip systems. The latter are activated in the matrix channels, whereas the cubic slip systems are activated at the corners of the precipitates. 4.2. Influence of a slight misorientation from the [0 1 1] loading direction In order to investigate the influence of a misorientation of the crystal lattice away from the perfect [0 1 1]

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Table 3. Activity of the octahedral slip systems reported in the elastic– viscoplastic phase-field simulation of a perfectly [0 1 1]-oriented specimen under creep.

Fig. 8. Visualization of plastic activity by plotting the equivalent von Mises invariant of the plastic deformation tensor at t = 100 s for a perfect [0 1 1] loading.

orientation on c/c0 microstructural evolution, we have simulated the MC2 specimen with 6° misorientation from

Slip system 1 (1 1 1)[1 0 1] 2 (1 1 1)[0 1 1] 3 (1 1 1)[1 1 0] 4 (1 1 1)[1 0 1] 5 (1 1 1)[0 1 1] 6 (1 1 1)[1 1 0] 7 (1 1 1)[0 1 1] 8 (1 1 1)[1 1 0] 9 (1 1 1)[1 0 1] 10 (1 1 1)[1 1 0]  0 1] 11 (1 1 1)[1 12 (1 1 1)[0 1 1]

[1 0 0] channel

[0 1 0] channel + +++

+ +

+ + + +++

[0 0 1] channel +++ + + + + +++

+ +

+ + +

+

[0 1 1] (see Fig. 1). The applied creep stress has been taken equal to 140 MPa as in the MC2 experiment. In the elastic simulation, a two-step coarsening is observed (Fig. 9). During the first step, the precipitate

Fig. 9. Elastic phase-field simulation results with 6° misalignment from the perfect [0 1 1] orientation, corresponding to the MC2 samples presented in Section 2. Figures are contour plots of the concentration field using c = 0.2 as the partitioning concentration.

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Fig. 10. Elastic–viscoplastic phase-field simulation results at t = 100 s and t = 5470 s with 6° misalignment from the perfect [0 1 1] orientation, corresponding to the MC2 samples presented in Section 2. The figure is a contour plot of the concentration field using c = 0.2 as the partitioning concentration.

coarsens along the [1 0 0] direction in accordance with the perfectly [0 1 1]-oriented simulation presented in Section 4.1. The difference with the previous simulation arises after 3500 s of simulation when coarsening along the [0 0 1] direction occurs. Therefore, the simulation results in a platelet normal to the [0 1 0] orientation. An elastic–viscoplastic simulation has also been performed for this misoriented specimen. Results are presented in Fig. 10. As in the perfectly oriented case, the microstructure leads to the formation of a needle along the [1 0 0] axis. However, the subsequent coarsening along the [0 0 1] direction obtained with the elastic model is not observed here. In terms of coalescence kinetics along the [1 0 0] direction, no significant difference is observed between the elastic and the elastic–viscoplastic simulations or between the perfectly [0 1 1] oriented and the misoriented simulations. In these four simulations, the coalescence occurs at t ¼ 300 s. Note also that in the elastic–viscoplastic simulations, the precipitate shape is very similar in the perfectly oriented [0 1 1] and misoriented simulations. For example, the aspect ratio at t = 5470 s is equal to R ¼ 1:1 in both simulations. To understand the precipitate shape difference between the elastic and the elastic–viscoplastic simulations we have analyzed the stress fields at an early time (t = 100 s). In Fig. 11, the equivalent von Mises stress in the (1 0 0) plane is presented for both simulations. First, it can be observed that the values of the equivalent stress in the c channels are lower when accounting for plasticity. Therefore, in the present situation, plasticity significantly relaxes the stress generated by the misfit. Moreover, as shown by the difference between the red and blue curves in Fig. 11, the stress

difference between the two types of channels is significant in the elastic simulation. In other words, the slight misorientation creates a small anisotropy between the channels, which ultimately generates the coarsening along the [0 0 1] direction. In the elastic–viscoplastic case, the anisotropy is also present but it is much lower than in the purely elastic case. Accordingly, the driving force to coarsen along [0 0 1] is much lower. Concerning the description of plasticity, the conclusions drawn in Section 4.1 on the perfect [0 1 1] creep simulation in terms of activated slip systems and plastic deformation field are still valid for the misoriented simulation. In addition, the time evolution of the macroscopic plastic strain is very similar in both cases. Therefore, even if the crystal plasticity framework used here is able to reproduce the anisotropic characteristics of plastic deformation in superalloys, it appears that the small misorientation considered here (6°) does not significantly impact the combined evolution of plasticity and microstructure evolution. This prediction differs from the conclusion drawn from the SEM micrographs presented in Fig. 3 where the formation of a single type of h1 0 0i platelets was related to the small misorientation of the sample. 5. Discussion The simulations of microstructure evolution under [0 1 1] tensile creep start by a coarsening along the [1 0 0] orientation leading to a needle-type microstructure. This result is in good agreement with the SEM/FIB observation made

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Fig. 11. Von Mises equivalent stress at t = 100 s obtained in the elastic (top) and elastic–viscoplastic (bottom) simulations for the 6° misoriented sample. (Left) Maps in the (1 0 0) center plane of the simulation box. (Right) Profiles in the horizontal (red squares) and vertical (blue circles) matrix channels. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

on the AM1 specimen and with recent experiments on a close-to-perfectly [0 1 1] oriented specimen presented in Ref. [8]. Such an evolution is expected because the (1 0 0) channels (sometimes called “gable” channels) are not equivalent to the (0 1 0) and (0 0 1) channels (the so-called “roof” channels). The second step of the microstructural evolution observed in commercial superalloys is a coarsening along the [1 0 0] or [0 1 0] directions leading to the formation of platelets. In perfectly oriented alloys, the energetic equivalence between the two directions results in the coexistence of two platelets families [8]. In slightly misoriented specimen, one of the two platelet families becomes dominant. This behavior appears clearly in the experiments conducted by Tian et al. [20] on a slightly misoriented specimen (4° from the [0 1 1] axis) as well as in our experiments on the MC2 specimen (6° from the [0 1 1] axis) presented in Fig. 3b. The present phase-field simulations succeeded in predicting this platelet formation only in the case of the elastic simulation of a misoriented specimen, indicating that at least part of the driving force leading to the platelet formation is of elastic origin. In all other simulations, the final microstructure is a needle-like microstructure. This result is a clear consequence of the difference in driving

forces between the coarsening along the [1 0 0] direction and the other cube directions. At first sight, the discrepancy between the final microstructures in the phase-field simulations and the observed microstructures is surprising because this model has been shown to correctly reproduce the c0 -rafting of [1 0 0]-oriented specimen [35]. Indeed, the plastic driving forces for rafting, as well as its anisotropy, are accounted for by the crystal plasticity of the c phase. In addition the elastic driving force for rafting is also included because the model takes into account the elastic inhomogeneity. Several reasons can be put forward to explain the discrepancy between the final microstructures in the phase-field simulations and the observed microstructures. First of all, the precipitate volume fraction considered here is slightly lower than the values in AM1 and MC2 superalloys. This necessary leads to larger c channels in the simulations, and therefore to a more difficult rafting process of the c0 precipitates. However, we have checked using 2-D simulations that an increase in the precipitate volume fraction could not explain the absence of coarsening along the [1 0 0] or [0 1 0] directions. It is possible that the initial configuration considered in the present paper is too simple (a cubic c0 precipitate in a

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cubic simulation box). Indeed, because of the high symmetry of the initial configuration, a large driving force may be necessary to break the symmetry and form a platelet. In real microstructures, the precipitate shapes are less symmetric and alignments are not perfect. Therefore, the microstructure can more easily find a way to break the symmetry and achieve an optimum configuration. Thanks to fast-evolving modern supercomputers, the necessary large 3-D simulations needed to confirm this point could be done in the future, but these simulation are not possible now because of the large memory and computer time required. However, preliminary 2-D simulations using a similar model and an initial configuration containing about 30 precipitates have been unable to reproduce the [0 0 1] or [0 1 0] rafting [59], suggesting that an important aspect of the [1 1 0] rafting is still missing in the simulations. As a conclusion of these simulations, we point out that an important result of this work is that, even with a phase-field model including bulk diffusion, inhomogeneous and anisotropic elasticity as well as crystal plasticity, the microstructural evolution under [0 1 1] tensile creep cannot be fully reproduced. This strongly differs from the evolution under [1 0 0] tensile creep where this model (and even cruder models) predicts microstructure evolution in rafts. The understanding of the microstructural evolution is therefore more subtle under [0 1 1] tensile creep than under [1 0 0] tensile creep. However, it is important to keep in mind that our simulations do not include plastic activity at the level of dislocations. If the evolution of dislocation networks at the interfaces or pipe diffusion along dislocation cores are relevant for the [1 1 0] rafting, these mechanisms are not included in our mesoscale model. A way to increase the accuracy of the model could be to use a dislocation density based crystal plasticity model as recently proposed by Cottura et al. [48]. A coupling between microstructure evolution and dislocations dynamics could also be relevant, provided that the thermally activated deformation mechanisms (cross-slip, climb) are accounted for. To our knowledge, such a coupled model remains to be created, even if advances towards this goal can be found in the recent literature [49,50]. In this work, we have chosen to complement the phasefield simulations by directly analyzing the elastic energy stored in platelet-like structures in the presence of homogeneous plastic strains in the c matrix. Within the above geometrical assumption, the calculations detailed below include the stress fields generated by the misfit of the c phase and the plastic strain in the c matrix. This analysis will not give information on the evolving microstructures, but will be able to address the question of the optimal microstructure without being stuck in local minima. More precisely, we have followed the mean-field approach for elasticity developed in Ref. [60] and applied to c0 rafting under h0 0 1i creep in Ni-based single-crystal superalloys in Ref. [44]. In this approach, the problem of inhomogeneous elasticity under external load is mapped into a problem of homogeneous elasticity without external load and characterized by an effective eigenstrain. In the present work, this approach is extended to include a plastic strain in the c phase. The development of the mean-field approach goes back to Eqs. (12) and (14) governing mechanical equilibrium. Assuming a small inhomogeneity between elastic constants of the matrix and precipitates phases, the term:

i @ h 0 C ijkl DcðrÞdekl ðrÞ @rj

ð19Þ

can be neglected in Eq. (12). Moreover, the plastic deformation tensor is assumed to be constant, equal to e p0 , in the c phase and equal to zero in the c0 phase. It results in the following expression for the plastic deformation as a function of the concentration field: e p ðrÞ ¼ 

ep0 cc0  cc

DcðrÞ þ

cc0  c p0 e cc0  cc 

ð20Þ

Eq. (12) is written replacing the concentration field by the variable: /ðrÞ ¼

cðrÞ  cc cc0  cc

ð21Þ

which exhibits the property /2 ðrÞ ¼ /ðrÞ as a sharp interface limit. Consequently, with the above assumptions, Eq. (12) can now be written: 2

! p0  ijkl eT dkl  ekl C cc0  cc

  cc0  c cc0  cc C 0ijkl ekl  cc0  cc ! #     @/ðrÞ ep0 kl 0 T 0 0 þC ijkl e dkl  cc  cc cc þ cc  2c cc0  cc @rj

 ijkl @ uk ðrÞ ¼ C @rj @rl

"

ð22Þ which is equivalent to: " ! @ 2 uk ðrÞ ep0 kl T   ¼ C ijkl C ijkl e dkl  @rj @rl cc0  cc

cc0  c p0 0 C ijkl ekl  e cc0  cc kl # ! p0   e @DcðrÞ 0 kl cc0 þ cc  2c þC ijkl eT dkl  @rj cc0  cc ð23Þ This equation has a form similar to the classical equation for mechanical equilibrium in the case of homogeneous elastic constants [44]. The configuration-dependent elastic energy can be written: Z d 3q 1 2 Econf ¼  q rH Gjk ðqÞrH ð24Þ kl ql j cðqÞj 2 q–0 ð2pÞ3 i ij where Gij represents the Green’s operator, and: ! ep0 kl H T  rij ¼ C ijkl e dkl  cc0  cc

  c  cc p0  C 0ijkl ekl  eT dkl cc0 þ cc  2c  ekl : cc0  cc ð25Þ An effective eigenstrain can be defined using the compliance  ijkl : Sijkl deriving from the elastic tensor C H  eH ij ¼ S ijkl rkl

The final expression for the effective eigenstrain is:

ð26Þ

eH ij ¼

ep0 ij

eT dij 

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!

cc0  cc

  c  cc p0 0 T   S ijkl C klmn emn  e dmn cc0 þ cc  2c  e cc0  cc mn ð27Þ

Moreover, as detailed in Ref. [44], if the condition 2 ra  C 0 eT cc0  cc is fulfilled, the average strain tensor can be approximated by eij  Sijkl rakl . The above equations are used to determine the habit planes of the microstructure. The Green’s function of homogeneous elasticity is used in this approximation to solve elastic equilibrium; the coupling between the microstructure and the external load appears through the effective eigenstrain. As detailed in Ref. [61], the minima of elastic energy give the elastic soft directions. Note that in case of cubic to tetragonal transformation, the analytical solution for those habit planes has been derived previously [61]. First, a purely elastic situation is considered. Assuming that e p0 ¼ 0, the effective eigenstrain reduces to:    0 T  ð28Þ eH emn  eT dmn cc0 þ cc  2c ij ¼ e dij  S ijkl C klmn  In the case of an uniaxial tensile loading along the [0 1 1] direction, and using numerical values from Table 2, one obtains: 0 1 0:02370612 0 0 B C eH ¼ @ 0 0:0252649 0:00612087 A  0

0:00612087

0:0252649 ð29Þ

Since the effective eigenstrain is not tetragonal, the minima of elastic energy are computed numerically. The density plot of the elastic energy in stereographic projection is presented in Fig. 12a. This plot displays only two deep minima, with the same value, one near the [0 0 1] direction (more precisely n1 =[0 0.151 0.988]) and the other one near the [010] direction (more precisely n2 =[0 0.988 0.151]) direction. The elastic energy is much larger in the [1 0 0] direction, indicating that interfaces parallel to the [1 0 0] axis are energetically very costly. The minimization of elastic energy consequently leads to a microstructure containing

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only interfaces normal to the [0 0 1] and [0 1 0] directions. More precisely, we expect the formation of either needlelike precipitates elongated in the [1 0 0] direction (consistently with the elastic phase-field simulation presented in Fig. 6) or a coexistence between two families of platelets perpendicular to n1 and n2 . The analysis of the competition between these two configurations is beyond this model because the shape changes related to the interfacial energy are not included. The effective eigenstrain tensor has also been computed for the misoriented MC2 specimen under the elasticity assumption: 0 1 0:02382219 0:00096173 0:0009346 B C e H ¼ @ 0:00096173 0:0253138 0:00563069 A  0:0009346 0:00563069 0:02522285 ð30Þ In that case, two deep minima are still observed, but these minima are no longer equal (see Fig. 12b). The absolute minimum is n3 = [0.025 0.99 0.139], which is very close to the [0 1 0] cubic axis. According to this model, the resulting c0 microstructure after creep deformation of a sample misoriented 6° away from the [0 1 1] tensile direction consists of plate-like precipitates perpendicular to a direction close to the [0 1 0] cubic axis. This is consistent with the results of the phase-field simulation presented in Fig. 9. Consequently, the mean-field approach gives the same trends as the phase-field simulations: a perfect [0 1 1] loading leads to two soft directions. If the loading is slightly misoriented, one can observe the selection of one specific soft direction, and the model proposed here is able to predict this direction. The case where plasticity is accounted for is now considered. In the present model, we consider that the c phase is the only plastic phase and that the plasticity is mainly due to the external stress. In other words, we neglect the contribution of the stress generated by the microstructure on the plastic activity. This assumption, questionable for the initial cuboidal microstructure, is relevant when considering plate-like microstructures where c=c0 interfaces are relaxed by dislocation networks. If the loading is perfectly aligned with the [0 1 1] axis, the plastic deformation tensor has the following shape:

Fig. 12. Maps of elastic energy computed with the mean-field model in the elastic case: (a) for a perfect [0 1 1] orientation; and (b) for the 6° misoriented sample. The energy is increasing along the color sequence blue–green–yellow–red (color online). The white dots highlight the localization of the minima of elastic energy. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

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0

1 0 B ep0 ¼ ep0 @ 0 1=2 0

1=2

1

0 C 1=2 A

ð31Þ

1=2

3

p0

Tacking e ¼ 10 as a reasonable value for plastic activity in the c matrix in the first stage of creep leads to the following numerical value for the effective eigenstrain: 0 1 0:0106633 0 0 B C eH ¼ @ 0 0:0318225 0:0142107 A ð32Þ  0

0:0142107 0:0318225

Plasticity does not change the general shape of the effective tensor compared to the purely elastic case. Consistently, the minimization of the elastic energy results in two equivalent directions n4 = [0 0.35 0.93] and n5 = [0 0.93 0.35] (Fig. 13a). Therefore, as in the elastic case, the predicted microstructure also consists of either needles along the [1 0 0] direction or the coexistence of two families of platelets perpendicular to n4 and n5 . One can also notice that the habit planes are further away from the cubic axis than in the elastic case and that the direction normal to the habit planes moves towards the tensile loading axis. This point is in a good agreement with the phase-field simulation as well as with the microstructures presented in Ref. [8]. The value of ep0 has been increased to 6  103 to assess the role of plasticity in the final stages of creep, in agreement with our experiments presented in Fig. 2. In that case, we find that the two directions that minimize the elastic energy in the case of a small plastic strain merge into a

single orientation close to [1 1 1] in Fig. 13b. Because of the symmetry of the configuration with respect to the (1 0 0) plane, this means that habit planes normal to both [1 1 1] and [ 1 1 1] are expected. The present calculation suggests that when plastic activity is important in the matrix channels, typically at the beginning of the tertiary creep stage, the habit planes may evolve towards complex orientations which are not perpendicular to the [1 0 0] direction. Note that the SEM/FIB technique presented in Section 2.2.2 could be successfully used to characterize, in three dimensions, the habit plane rotations during the final stage of creep. Finally, the case of the 6° misoriented loading with plasticity is addressed. As before, the plastic strain field corresponds to the one obtained in a uniaxially loaded cphase, however the shape of the plastic strain tensor is no longer known by symmetry arguments and has to be computed numerically. Considering ep0 = 103, the plastic deformation tensor can then be written: 0 1 0:0009664 0:00016435 0:0002039 B C e p0 ¼ @ 0:00016435 0:00054956 0:0004832 A ð33Þ  0:0002039 0:0004832 0:0004168 The corresponding elastic energy plot, presented in Fig. 14a, displays two minima, but contrary to the perfectly oriented situation, the minimum close to [0 1 0] is deeper than the other one. This behavior is similar to what was obtained in the purely elastic case. Therefore, one should expect also in the present situation a plate-like microstructure perpendicular to the [0 1 0] direction. In addition, the

Fig. 13. Maps of elastic energy computed with the mean-field model for a perfect [0 1 1] orientation with a plastic strain ep0 ¼ 103 (a) and ep0 ¼ 6  103 (b). The white dots highlight the localization of the minima of elastic energy. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

Fig. 14. Maps of elastic energy computed with the mean-field model for the 6° MC2 misorientation with a plastic strain ep0 ¼ 103 (a) and ep0 ¼ 6  103 (b). The white dots highlight the localization of the minima of elastic energy. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)

A. Gaubert et al. / Acta Materialia 84 (2015) 237–255

mean-field model predicts that the direction normal to the habit plane moves towards the tensile loading axis. Even if this plate-like configuration was not achieved in the phase-field simulations, a rotation of the habit plane was observed in the same direction. More importantly, however, the plate-like configuration predicted by the mean-field model is in qualitative agreement with the experimental results for the MC2 alloy (see Fig. 3c). This results also suggests that the selection of the platelets normal (either in a direction close to [0 1 0] or close to [0 0 1]) could be deduced from a purely elastic model. However, calculations with other misorientations are needed to be conclusive on that point. When increasing ep0 , the mean-field model predicts a tilt of the platelets similar to the one obtained for the perfect [0 1 1] loading case with a normal close to the [1 1 1] and [ 1 1 1] directions (Fig. 14b). Consequently, a similar creep degradation is expected in both cases. In the tertiary stage (Fig. 3c), the angle between the precipitates and the loading direction tends to reduce from 45° to about 30°. Some sections of the precipitates are even perpendicular to the [0 1 1] direction. All these orientations, observed in a slice of the microstructure perpendicular to the [1 0 0] direction, can be qualitatively explained by the elastic energy plots presented in Fig. 14. This means that the simple criterion of elastic energy minimization in the presence of plasticity is a useful tool to analyze the microstructure evolution during creep. Additionally, another possible analysis of the numerical results along with the experimental ones is that the onset of the tertiary creep stage for h0 1 1i tensile-loaded Ni-base single-crystal superalloys is the result of progressive destabilization of the c/c0 interfaces. In fact, it is predicted by the model that a progressive tilting of the c/c0 interfaces is unavoidable to reduce the elastic energy, and it is observed experimentally that this degradation leads to an accelerated creep strain rate (see Fig. 2). Hence, this degradation is verylikely to control the onset of the tertiary creep stage. Note finally that specimens are likely to undergo crystallographic rotation [22] during subsequent creep deformation above 1%. This process, not considered in the present paper, will directly affect the elastic energy analysis performed here and care needs to be taken at such high creep strains.

6. Conclusions In this paper, both experimental and phase-field techniques have been used and compared to investigate microstructural evolutions arising during [0 1 1] creep in Ni-based single-crystal superalloys. On the experimental side, SEM/FIB 3-D reconstructions, complemented by interrupted creep observations, have given a 3-D insight into microstructures and microstructural evolutions during [0 1 1] creep. On the phase-field side, 3-D simulations accounting for the plastic activity in c matrix channels have been performed to obtain a physical understanding of the driving forces responsible for the microstructural evolution. Finally, we have developed a simplified energetic model to analyze the c=c0 interfaces rotations during creep. The following main conclusions can be drawn: When applying a tensile [0 1 1] creep loading, experimental results have first shown coarsening of the c0 precipitates in the [1 0 0] crystal direction. Then, in both AM1

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and MC2 samples, it was observed that coarsening in the (1 0 0) plane along the cubic directions [0 1 0] and [0 0 1] leads to the formation of platelets. We have highlighted that one type of platelet is favored during creep. Finally, the observations have given an insight into the microstructural degradation linked with the onset of the tertiary stage of creep. Three-dimensional phase-field simulations have shown that, regardless of the precise crystal orientation, coarsening first occurs along the [1 0 0] direction as a result of a strong elastic driving force. Then, when neglecting plastic activity, phase-field simulations predict that, for a misoriented sample, a subsequent coarsening occurs in the (1 0 0) plane, leading to a microstructure made of one type of platelets oriented along the [0 1 0] or [0 0 1] cubic direction. When accounting for plasticity, a small rotation of the c=c0 interfaces is observed in the same direction as the one observed in the experiments. However, the misalignment of the sample does not qualitatively change the microstructure evolution: a needle-like microstructure is predicted, which is very similar to the one obtained in the case of a perfect [0 1 1] loading. Therefore, inhomogeneous and anisotropic elasticity and plasticity within a continuous framework are not able to fully explain the raft formation of h0 1 1i-oriented samples, in contrast to the wellknown case of rafting of h1 0 0i-oriented samples. This result suggests that another mechanism, probably acting at the level of individual dislocations, may be relevant to describe the rafting when the tensile loading axis is close to a h0 1 1i direction. A mean-field model for inhomogeneous elasticity has been applied to analyze the orientation of the final rafted microstructure observed in experiments. By calculating the elastic energy stored in platelet-like microstructures in the presence of plasticity in the c phase, it is shown that the orientation of the platelets strongly depends on the amount of plastic strain. The normal to the platelets, initially close to the [0 1 0] and the [0 0 1] directions, rotates towards the loading axis, in agreement with the observed rotation of the rafts, indicating that plasticity in the c phase is at the origin of the observed rotations. Finally, we have shown that our mean-field model is also able to predict which family of platelets (with a normal close to [0 1 0] or [0 0 1]) will prevail in the case of misoriented samples. In addition, the SEM/FIB technique appears to be a relevant technique to investigate microstructural evolutions in three dimensions, in particular during creep loading along complex directions for small volumes. Hence, it is of great interest to pursue investigations using this technique to obtain large representative volumes in order to acquire good statistics of 3-D microstructure features. The resulting observations combined with the state-of-the-art phase-field models as shown in this paper will allow significant progress in the understanding of the physical mechanisms controlling microstructure evolution in Ni-base superalloys under complex loading conditions. Acknowledgements SAFRAN companies (Snecma and Turbomeca) are gratefully acknowledged for providing AM1 and MC2 alloys. The authors are grateful to Dr. Rachid Sougrat for his valuable help on AvizoÒ Software.

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