Refinement and Experimental Validation of a Vacancy Model of Pore Annihilation in Single-Crystal Nickel-Base Superalloys During Hot Isostatic Pressing

Initially as-cast and homogenized single-crystals of nickel-base superalloy CMSX-4 were subjected to hot isostatic pressing at temperature of 1288 °C. Two series of experiments were performed: under the same pressure of 103 MPa but with different durations, between 0.5 and 6 h, and under different pressures, between 15 and 150 MPa, but for the same time of 0.5 h. The porosity annihilation was investigated metallographically and by high resolution synchrotron X-ray tomography. The obtained experimental results were compared with the predictions of the vacancy model proposed recently in our group. In this work the model was further refined by coupling with X-ray tomography. The model describes the evolution of the pore arrays enclosed in the 3D synchrotron tomograms during hot isostatic pressing and properly predicts the time and stress dependences of the pore annihilation kinetics. The validated model and the obtained experimental results can be used for selecting the optimal technological parameters such as applied pressure and processing time.


Introduction
Turbine blades are critical components of aircraft jet engines and gas power plants because they operate at high temperatures under mechanical loads of different types in an aggressive environment. Therefore, the turbine blades are cast from nickel-base superalloys possessing excellent service properties at high temperatures [1]. In order to improve their creep strength and corrosion resistance, hot section turbine blades are commonly produced as single-crystals by directional solidification [2], and are therefore free of damaging high angle grain boundaries. However, application of this advanced technology cannot avoid such a detrimental microstructural defect as microporosity. Microporosity forms in single-crystal blades during the different steps of their manufacturing as well as in service. The initial porosity forms during dendritic solidification of blades due to insufficient afflux of melt into the interdendritic regions [3,4]. These pores have an irregular shape, can branch and extend up to a few hundred microns in length. Fig. 1a shows a crack initiated at such a pore during low cycle fatigue test of a single-crystal of superalloy CMSX-4 at 700 °C [5], which eventually led to specimen failure. Such large pores are quite rare, but they are very dangerous. The next type of porosity forms during homogenization heat treatment. The formation of these pores was considered in [6][7][8], and it was shown [8] that the main mechanism of porosity growth during homogenization is vacancy flux toward the dissolving nonequilibrium eutectics compensating the counter-diffusion of fast aluminum atoms (Kirkendall effect). These pores have round shape with diameter of about 10 m, see Fig. 1b. The third type of porosity develops during long-term high temperature deformation (creep) which occurs under service conditions. This porosity is represented by small faceted pores of a few microns in size, see Fig. 1c. Because of their crystallographic shape, such pores as called as "negative crystals" [9]. The growth of these pores in single-crystal superalloys was reported in many papers [10][11][12][13] and it was shown [13] that they form by condensation of vacancies emitted by the edge parts of dislocations climbing along the /'-interfaces. It is proposed in [14] to call the types of pores mentioned above respectively as solidification (S), homogenization (H) and deformation (D) pores. S-and H-porosity in single-crystal nickel-base superalloys can be reduced by selecting the optimal parameters of directional solidification and homogenization heat treatment [4,15] but a complete removal of the pores is only possible by applying a hot isostatic pressing (HIP) treatment to the components. Effects of HIP on the mechanical properties and microstructure of single-crystal superalloys were investigated in [16][17][18][19][20][21][22][23][24]. It was found that removing the pores by HIP dramatically increases the fatigue life [16,17,22], while the other mechanical properties are hardly altered. The reason for the first effect is that under cycling loading the stress concentrations near the pores result in enhanced the local strain amplitudes leading to the nucleation of fatigue cracks. HIP of the single-crystal blades is usually performed in a narrow temperature window, between the '-solvus and the solidus, in which the strengthening '-phase is totally dissolved, and the blade material is very soft. However, such a high temperature process can damage a costly blade by recrystallization and incipient melting. Therefore, the parameters of commercial HIP (temperature T, pressure p, duration t) have to be carefully optimized under the conditions of full pore healing, avoidance of material damage, and minimal processing costs. The HIP parameters can be optimized using a reliable physical model validated by HIP-experiments performed under different HIP conditions. Two models, based on different pore closure mechanisms, were proposed: a model of the pore closure by dislocation creep [25] and a model of the pore dissolution by emission of vacancies [26]. More promising preliminary results were obtained by the vacancy model while the creep model significantly overestimated the porosity annihilation rate. Therefore, in this work the vacancy model was considered and applied. This vacancy model assumes that under applied pressure the low angle boundaries operate as sinks of vacancies emitted by the surface of shrinking pores. As a result of this process, the pores gradually shrink and heal. The aim of the present work is the experimental investigation of the porosity annihilation in singlecrystal superalloy CMSX-4 during HIP as well as refinement and validation of the proposed vacancy model. The residual porosity in HIPed samples has been investigated metallographically and by high resolution synchrotron X-ray computer tomography (XCT). In a previous work, we used XCT to investigate microporosity in single-crystal nickel-base superalloys in as-cast, heat treated and deformed (after creep) conditions, and these investigations demonstrated the high efficiency of this technique for porosity characterization [27]. Now we have applied this method to investigate porosity annihilation during HIP.

Experimental
The material under investigation is the single-crystal nickel-base superalloy CMSX-4 [28] developed by the Cannon & Muskegon Corporation, USA, and widely used for casting gas turbine blades. The alloy contains 3 wt. % Re (see Table 1) and according to the international classification belongs to the 2 nd generation of single-crystal superalloys. Single-crystal bars of CMSX-4 with axial orientation [001] were solidified by the Howmet Alcoa, USA. Some bars were left in the as-cast condition, others have been subjected to the standard solution heat treatment (homogenized). All HIP-experiments were performed at 1288 °C, which is the temperature at which HIP is commercially applied to CMSX-4 by the Howmet Alcoa. This temperature is slightly above the '-solvus temperature of CMSX-4 but significantly lower than its solidus temperature, respectively equal to 1282 and 1339 °C [2]. Two series of HIP-experiments were performed:  Residual porosity in the samples from the first HIP-series was previously characterized by optical microscopy (OM), scanning electron microscopy (SEM) and density measurement (DM) [21,22]. In the present work, it has been investigated by high resolution XCT. For tomographic imaging of microporosity, small needle-shaped samples have been cut by spark erosion. The samples have a diameter of 1 mm and a length of 10 mm. The tomographic measurements have been carried out at the beamline ID19 at the European Synchrotron Radiation Facility (ESRF) in Grenoble, France. A monochromatic X-ray beam with an energy of 70 keV at a current of 200 mA (7/8 multibunch mode) has been used. The radiographs have been recorded in absorption and phase contrasts with a sCMOS camera PCO EDGE 4.2 exhibiting a resolution of 2048 x 2048 pixel. The continuous 360° rotation of the sample enables an effective measuring field of 3700 x 3700 x 2048 voxel with spatial image sizes of 1265 x 1265 x 700 µm³. This corresponds to a sub-micron resolution of 0.347 3 µm³/voxel in the reconstructed 3D tomograms. During one complete rotation of the sample, about 5000 radiographs have been recorded with an exposure time of 50 ms/radiograph. Thus, the total measuring time is about 5 minutes per tomogram. The reconstructed tomograms have been evaluated with the defect analysis tool of the commercial visualization software VGStudioMax (Version 2.1) from the company Volume Graphics [29].
Processing of the porosity tomograms with this software provides arrays of data containing geometrical parameters for every pore, including in particular: the pore identification number , pore i position , pore volume , and the projection areas on the coordinate planes , , . The porosity volume fraction is determined as: is the analyzed specimen volume.

T V
In the refined vacancy model used in this work, a correction of the pore surface due to deviation i p S , of the pore shape from the spherical shape has to be considered. This effect can be accounted for by using the inverse of Wadell's sphericity factor [30] as a corrective term: , with . (2) However, the available software VGStudioMax (Version 2.1) represents the objects by voxels (as stepped surface) which leads to significant overestimation of and consequently of . For example, for a spherical object the software gives =1.5 instead of 1. In order to reduce the error 1   of -measurement it was proposed in [31] to approximate as: is the mean projection area defined by: This approximation has been used in our work. Residual porosity in the samples HIPed in the second series of HIP experiments was investigated in SEM. From every sample, 81 backscatter electron (BSE) and 81 secondary electron (SE) images have been taken covering the area of 2.9  3.9 = 11.3 mm 2 , that corresponds to about 90 dendritic cells. Simultaneous BSE and SE imaging of every area is necessary to distinguish the pores from other objects. The SEM images have been analyzed with the image processing program ImageJ [32]. Automatic image binarization reveals also objects different from pores, therefore every binarized image has been subsequently manually corrected to exclude these artifacts. It should be mentioned that the porosity has been quantified in sections, so that the area fraction of pores is the quantity that has been actually characterized. However, since the pores are randomly distributed, the surface area fraction of pores is equal to the volume fraction [33,34]. Therefore, from here and below the surface area fraction of pores measured by SEM and OM will be referred to as the volume fraction .

Vacancy model
The proposed vacancy model [26] assumes that under an external pressure applied during HIP, the p vacancy concentration at the cores of edge dislocations forming the low angle boundaries (LAB) LAB c decreases below the equilibrium value according to: is the atomic volume, the Boltzmann constant and the absolute temperature. On the a V k T other hand, the vacancy concentration is equal to its equilibrium value at the surface of the gas-0 c free pore. This results in the vacancy concentration gradient between the LAB and the pore, which activates a vacancy flux from the pore to the LAB, as shown in Fig. 2. The pore emits vacancies and gradually shrinks while the LAB operates as a vacancy sink. One can get an analytical solution for the kinetics of pore shrinkage assuming:  Spherical symmetry.  Stationary vacancy diffusion, .
, where and are respectively the radii of the pore and the subgrain.
It should be mentioned that the last assumption is reasonable for materials with low porosity, which is the case of single-crystal superalloys, where the porosity volume fraction is typically between p f 0.1 and 0.2 vol.%.
The following analytical solution was derived in [26]: is the self-diffusion coefficient of base element Ni, the correlation factor for diffusion in the fcc lattice and is the time elapsed since beginning of the HIP process. It is assumed in (6) that t the LABs are perfect vacancy sinks. In other words, the LAB is assumed to be equivalent to a free surface. One of the most important geometrical parameters of a pore is its volume . The pore volume is p V directly measured by XCT as opposed to the widely used equivalent radius (or diameter), which is a parameter derived from . Therefore, for coupling XCT with diffusion modeling it is reasonable to p V change the variable in Eq. (6) from the pore radius to the originally measured pore volume .
Performing this change, one gets: (7) is derived under the assumption that all pores in the material are spherical and have the same volume initially equal to the average pore volume in unHIPed material. However, as it   0 p V will be shown below, a prediction under this assumption overestimates the porosity annihilation rate. Therefore, for an adequate description of the porosity annihilation during HIP, the pore volume distribution has to be considered. It is seen that Eq. (7) is independent from the boundary conditions, and thus can be applied independently to every individual ith pore of volume in the porosity i p V , tomogram: However, Eq. (8) does not take into account that the pore shape can deviate from the spherical shape, which increases the pore surface emitting vacancies. This shape effect can be considered by i p S , introducing the inverse sphericity factor as defined in (2). The factor in (8)   In the calculations performed in this work, the following values for the parameters of Eq. (7) have been used: , m 2 /s with = 28700069.8T, J/mol [35], which gives .6210 -14 m 2 /s, = 1.2510 -29 m 3 [26] and = 0.781 [36,37].

Results
The 3D tomographic images in Fig. 3 show the evolution of microporosity in the initially homogenized CMSX-4 during HIP at 1288 °C and 103 MPa. It is seen that porosity continuously decreases with HIPing time t. First, numerous small pores annihilate followed by less numerous large pores. This can be observed in the sequence of Figs 3a, b, c, d and e corresponding to HIPing durations 0, 0.5, 1, 2 and 4 h. After 4 h HIP (duration of commercial HIP applied to CMSX-4 by Howmet Alcoa) only rare pores of relatively small size remain. Such an unclosed pore marked by an arrow in Fig. 3e is shown in Fig. 3f at higher magnification. It is seen that this pore has an irregular shape indicating that it originally is an S-pore. The pore length is of about 36 m ( =23.5 m), which is eq D much shorter than the length of a critical pore (a few hundred microns) shown in Fig. 1a.

  dt t df p
The prediction for can be improved by applying Eq. (8), thereby assuming that all pores are   t f p spherical but with different initial volumes. Such predictions are shown by the red dashed lines. It is seen that the predicted lines are now closer to the DM and XCT measurements but underestimate the porosity annihilation rate. The best agreement with the experimental results is obtained under the assumption that the pores originally have different volumes and that their shape can deviate from the spherical shape, as described by Eq. (9). It is seen that the red solid lines predicted with Eq. (9) are closer to the DM and XCT results than the predictions with Eq. (7) and (8). Fig. 5 shows the evolution of the histograms of the equivalent diameters during HIP of the eq p D initially homogenized CMSX-4. Fig. 5a presents the experimental histograms obtained by processing the porosity tomograms after 0, 0.5, 1, 2 and 4 h HIP at 1288 °C and 103 MPa, see Figs 3a-e. Fig. 5b presents the corresponding histograms predicted by applying Eq. (9) to each observed pore in tomogram in Fig. 3a and cumulating the results. One can recognize a clear similarity of the experimental and predicted histograms. At t= 0 h, the -distribution has a maximum at  10 eq p D eq p D m and it is positively skewed. But with increasing HIP the shape of the -distribution changes. eq p D The distribution peak progressively decreases and after 2 h HIP the -distribution flattens and eq p D becomes negatively skewed. Such a histogram evolution results from the size dependence of the pore shrinking rate. From equation (9) follows: It is seen from Eq. (10) that the relative reduction of the pore volume increases with decreasing initial pore volume . Therefore, the numerous small pores rapidly annihilate while ) 0 (   (2-4). The corresponding correction yields a correct value of the inverse sphericity =1 for a sphere, but for other 3D-shapes is generally underestimated. For example, for a 1 , cuboid with edges parallel to the coordinated axes x,y,z or a thin plate normal to one of these axes, is underestimated by a factor of 2/3. The degree of underestimation of is especially high for 1 , the non-convex shapes with developed surface, e.g. like a great icosahedron, which is also the case of large S-pores, see Fig. 1. Therefore, for reliable prediction of the evolution of large pores during HIP a precise determination of is needed. The similitude of the results from independent experiments testifies the reliability of the performed HIP tests and the following porosity characterization. The fitting of the SEM results obtained for the samples HIPed at RUB Bochum (blue circles) reveals a nearly linear dependence (blue line) on the applied pressure p. The red lines in Fig. 6 are predictions with the vacancy model performed under different assumptions. Again, it is observed that assuming equal volume spherical pores (red dashdotted line) leads to faster pore annihilation than in the experiment. The predictions under the assumptions of spherical pores with different initial volume (red dashed line) and pores of different shape and initial volume (red solid line) are generally in agreement with the experimental results. work is not possible for two reasons: First, it is very expensive. Second, it needs small samples, as described in chapter Experimental, and therefore cannot be applied to full-scale components. However, for practical purposes such a high resolution is unnecessary because the critical pores are large, see Fig.1a. Fractographic inspection of single-crystals of CMSX-4 ruptured at 700 °C during low fatigue tests performed by BAM Berlin [5] showed that in all samples the fatigue cracks nucleated from such large S-pores. Therefore, for practice it is suggested to combine lower resolution X-ray tomography of large samples with laboratory equipment with validated vacancy modeling. Based on the tomography characterization of the large pores, one could then simulate their evolution during HIP under different conditions and analyze the results to select the optimal HIP parameters for porosity healing. An additional benefit of this procedure is that the analysis of larger samples would improve the statistical characterization of large pores. In spite of the observed consistency between the theoretical predictions and the experimental results, the vacancy model obviously should be further refined, namely:  The effect of pore shape on the shrinking rate should be analyzed in detail. It is relevant for large pores of irregular shape, which are critical defects of single-crystal superalloys.  The effect of multicomponent diffusion should be understood, too. It is often considered by introducing the so-called "effective diffusion coefficient" , as discussed in [38]. However, following from the results of [35] and used in this work. In [38] it is proposed to calculate in two ways, including arithmetic averaging of the diffusion activation energies and Ni eff D alloy because it not only depends on the alloy composition but also on the specific diffusion problem under consideration. For example, it was shown in [40,41] that the alloying elements in nickel-base superalloys significantly interact during interdiffusion and therefore the overall diffusion process can accelerate or slow down depending on the mutual directions of the diffusion fluxes. Thus, it is suggested to assess the impact of multi-component diffusion on pore shrinkage by directly solving the multi-component diffusion problem with a diffusion simulation software like, e.g., Dictra [42]. In turn, from the results of such computationally more demanding simulations, specific effective diffusion coefficients can be identified for use in fast calculations.  Finally, additional pore closure mechanisms should be also considered. The most probable second mechanism of pore closure is dislocation creep, as modeled in [22,25]. Therefore, the dislocation creep model should be refined and coupled with the vacancy model, which would extend the predictive capacity of modeling to a wider field of the HIP parameters.

Conclusions
1. Annihilation of microporosity in initially as-cast and in homogenized single-crystals of nickel-base superalloy CMSX-4 during HIP at 1288°C has been characterized by metallographic methods and by high resolution sub-micron synchrotron tomography. The time and pressure dependencies of the annihilation kinetics have been obtained.
2. On the base of the obtained experimental results, the previously proposed vacancy model of porosity annihilation has been improved and validated. This model refinement was made possible by identifying the proper initial conditions via X-ray tomography measurements of the initial pore volumes and shapes. The model describes the evolution of the porosity array present in a tomogram during HIP and properly predicts the kinetics of pore annihilation and its pressure dependence.
3. The obtained experimental and numerical results have demonstrated that porosity healing in CMSX-4 during HIP at the temperature of 1288°C and pressures up to about 100 MPa occurs via pore dissolution by vacancy emission and diffusion. The experimental results and the vacancy model can be used for the optimization of the parameters (pressure, time) of commercial HIP of superalloy CMSX-4 or comparable alloys.