Precipitation in powder-metallurgy, nickel-base superalloys: review of modeling approach and formulation of engineering methods to determine input data

The rate of nucleation of precipitates, J, is given by the following equation [4, 12, 26]:in which C denotes the solute content in the matrix expressed as an atomic fraction, D is the solute diffusivity in the matrix phase, a _o is the average lattice parameter of the matrix and precipitate phases (taken to be ~0.356 nm for most γ-γ′ superalloys), σ is the matrix-precipitate surface energy, k _B is Boltzmann’s constant (1.3806 JK⁻¹), T is the temperature in Kelvin, and t is the time. The critical radius of the precipitate, r*, is a function of σ, the chemical free energy of transformation, ∆G* (taken to be a positive quantity for a decrease in energy), and the elastic strain energy, ∆G _p, associated with the difference in lattice parameters of the matrix and precipitate phases, i.e.,

The nucleation rate J in Eq. (1) is related to the steady-state nucleation rate J _o through the term exp(−τ/t), which describes the initial nucleation transient during which a metastable distribution of embryos with sub-critical radii are formed. The so-called incubation time constant τ in this exponential term is given by the relation [26, 42]:in which R is the gas constant, and V _M is the molar volume of the precipitate.

Strictly speaking, Eqs. (1)–(3) apply to nucleation in two-component alloys. Methods and assumptions utilized to determine effective values for coefficients such as C, D, and ∆G* of multi-component alloys for use in the nucleation (as well as growth and coarsening) relations are discussed below.

In most theoretical treatments for γ′ precipitates, the particles are assumed to be spherical of radius r with their growth controlled by diffusion. The “exact” solution of the diffusion equation for the rate of growth in the presence of a finite matrix supersaturation is as follows [13, 14]:in which λ ² is related to the supersaturation Ω by the following relation:

The supersaturation Ω has its usual definition, i.e.,

Here, C _m, C _i, and C _p represent the compositions of the matrix far from the matrix-particle interface, the matrix at the matrix-particle interface, and the particle at the matrix-particle interface, respectively. The correction to the equilibrium (r = ∞) interface composition, C _i, due to the Gibbs-Thompson effect can be quantified using the following equation [43]:

The effect of soft impingement on the “far-field” matrix composition C _m is taken into account using the usual approximation derived from a mass balance [44]:in which C _o and f denote the overall alloy composition and the volume fraction of the precipitate.

As for the nucleation-rate relation (Eq. (1)), the diffusional-growth solution (Eq. (4)) is strictly applicable to a two-component system whose inter-diffusion coefficient is independent of composition. For multi-component alloys, D thus represents again an effective diffusivity (D _eff). Furthermore, it is often assumed in fast-acting simulations that the composition of the γ′ precipitate is constant and equal to the average value determined by phase extraction.

During continuous cooling at rates typical of production-scale components, the tendency for coarsening of γ′ precipitates (especially the larger secondary precipitates) is small due to retained supersaturation. On the other hand, coarsening during aging (and during service at high temperatures) can be quite substantial. In the present work, measurements of the isothermal, static-coarsening rate have been used to estimate the effective diffusivity needed in the expressions for both nucleation and growth. Specifically, the rate of increase of the average precipitate size was interpreted in terms of the classical Lifshitz-Slyozov-Wagner (LSW) theory modified to treat the coarsening of finite volume fractions of particles (15, 16, 21), viz.:

and

In Eq. (9), \( \overline{r} \) and \( {\overline{r}}_{\mathrm{o}} \) represent the average instantaneous and initial particle radii. In the expression for the modified LSW rate constant, K _MLSW, terms not previously defined include w(f), the factor to correct for the finite volume fraction of particles [17‐20], and C _γ and C _γ′, the equilibrium concentrations of the rate-limiting solute in the matrix and precipitate, respectively. The bracketed term in the denominator of Eq. (10) is the thermodynamic factor in which v denotes the activity coefficient for the rate-limiting solute in the γ matrix of specified composition.

Again, the formulation and application of Eqs. (9) and (10) for multi-component alloys implicitly assumes that coarsening is controlled by the diffusion of a single, rate-limiting solute. However, Kuehmann and Voorhees [24] demonstrated that rapidly diffusing solutes can also affect the coarsening rate and the related diffusional processes for ternary alloys. For multi-component alloys, their work suggests that an effective rate constant (K _eff) equal to the inverse of the sum of the inverse rate constants for the individual solutes can be defined as follows:

The key thermodynamic data consist of (1) the solvus approach curve (equilibrium fraction of γ′ as a function of temperature, f(T), for γ-γ′ superalloys), (2) the composition of the γ′ phase, and (3) the solute content in the matrix (C in Eq. (1)).

Two methods exist for determining the solvus approach curve. The first is the formal Calphad (thermodynamic) method based on the calculation of the fractions of γ and γ′ as a function of temperature using the Gibbs free energies of these phases [45]. For multi-component alloys such as commercial nickel-base superalloys, these calculations require an appropriate thermodynamic database typically obtained through the optimization of a number of model parameters from various measured binary, ternary, etc. phase diagrams. An alternate, semi-empirical method was developed by Dyson [46] and refined by Payton [47]. Specifically, it comprises fitting f(T) measurements to an analytical expression, whose application is especially attractive for fast-acting simulations of precipitation kinetics, viz.,

Here, T _γ′ denotes the γ′ solvus temperature, C* is the atomic fraction of γ′ formers in the alloy, and Q is a fitting parameter. Because the γ′ phase has a composition of Ni₃X, 4C* is approximately equal to the maximum volume fraction of γ′ in the alloy, i.e., the amount at 0 K. The value of the fitting parameter Q typically lies in the range of 60 to 75 kJ/mol.

The applicability of Eq. (12) is illustrated using IN-100, René 88, and LSHR as model alloys. From heat treatments at various temperatures followed by metallography, the solvus temperature for the three materials has been determined to be 1458, 1378, 1430 K, respectively. Typical compositions of the alloys are summarized in Table 1 (in weight percent) and Table 2 (in atomic percent). For the superalloys of interest here, the principal γ′-forming elements are Al, Ti, Nb, and Ta. From Table 2, the values of 4C* are thus 0.60, 0.38, and 0.53. These estimates are in reasonable agreement with published reports of the fraction of γ′ in fully heat-treated materials except for René 88, for which the measured volume fraction is 0.42, rather than 0.38 [30]. Differences between estimated and published magnitudes of the volume fraction of precipitate can be largely ascribed to the fact that the various γ′-forming elements do not partition totally to this phase, and the other alloying elements do not fully partition to the γ matrix. The specific alloying-element partitioning is illustrated by measurements of the average composition of γ′ determined by phase extraction (Table 3) [48]. This method weights the composition in favor of those precipitates with the larger volume fractions, i.e., the primary and secondary γ′, of course.

Using the above values of T _γ′ and 4C*, measurements of the equilibrium volume fractions of γ′ at various temperatures for each alloy have been fitted to Eq. (12) (Fig. 1). The measurements were performed via quantitative analysis of SEM micrographs taken on samples that had been heat treated for long times (4–24 h, depending on temperature) or isothermally forged to achieve equilibrium conditions. The best fits are obtained for Q = 75, 65, or 60 kJ/mol for IN-100, René 88, and LSHR, respectively. In Fig. 1b, the René 88 measurements and the fit using Eq. (12) also show good agreement with the solvus approach curve determined by a Calphad-type approach [49]. In addition to the precise shape of the free-energy curves as a function of temperature, the variation in the value of Q for the three alloys may be due to uncertainties in the measured volume fractions of γ′ used for fitting (approximately ±0.015) and the precise values of the maximum volume fraction of γ′ (assumed to be 4C*).

The solute content in the matrix (C in the nucleation Eq. (1)) depends on the volume fraction of precipitate and its composition. For supersolvus solution treatments, the majority of the secondary-γ′ nuclei form during a short time interval during which the matrix composition undergoes little change. Thus, C is relatively constant and equal to ~0.45–0.50 (Table 2). Furthermore, a sensitivity analysis in reference [50] has shown that C equal to 0.25, 0.5, or even the physically unrealistic value of 1.0 all yield similar precipitation predictions.

The principal driving force for nucleation is the reduction in chemical free energy, ∆G*, associated with the formation of γ′ from a supersaturated γ matrix. In addition to the formal Calphad approach, two somewhat simpler engineering methods can be used to determine ∆G* [50]. One is based on measurements of the specific heat and the γ′ solvus temperature. The other involves a thermodynamic calculation based on a pseudo-binary of nickel and chromium.

The first technique for determining ∆G* is strictly applicable to the matrix composition pertaining to supersolvus solution treatment, i.e., the overall alloy composition. Assuming that the enthalpy (∆H _avg) and entropy (∆S _avg) of formation (per mol) are constant, ∆G* varies linearly with temperature, i.e.,

The enthalpy of formation of the precipitate phase, ∆H _avg, is determined from the integral of the measured specific heat (C _p) over the on-cooling-transformation-temperature range (typically ~30 K below the solvus → ~1150 K for PM nickel-base superalloys). This integral is decremented by the heat content associated with temperature changes in the absence of transformation as quantified by extrapolations from low and high-temperature C _p behaviors, respectively. A schematic illustration of the construction, in which the enthalpy of transformation is shown as the cross-hatched region, is shown in Fig. 2. Knowledge of the fraction transformed (f _trans) yields the desired quantity:

At the γ′ solvus, ∆G* is equal to zero. Thus, the value of ∆S _avg is equal to ∆H _avg/T _γ′.

Specific-heat data measured during cooling from above the solvus at a rate of 20 K/min reveal a similar dependence on temperature for IN-100, René 88, and LSHR (Fig. 3) [51‐54]. Results for ME3, an alloy whose overall composition and γ′ composition are similar to those of LSHR (Tables 1, 2, and 3), are also included in Fig. 3c. All of the measurements indicate a finite undercooling below the solvus at which the transformation begins, a peak just below this onset temperature, and a rapid decrease in the magnitude of C _p thereafter. The rate of decrease after the peak can be ascribed to the decreasing rate of formation of γ′ inasmuch as the instantaneous fraction of precipitate follows the solvus approach curve, at least approximately, once nucleation starts to occur [50]. The estimates of ∆H _avg and ∆S _avg (Table 4) show very similar results for IN-100 and LSHR, but a measurable difference for René 88.

The second approach used to quantify ∆G* is based on classical expressions from solution thermodynamics. For the case in which the precipitate is enriched in solute (which is applicable for Al, Ti, Nb, Ta), ∆G* is given by [4]:

A similar relation applies when the precipitate is depleted in solute (e.g., Cr, Co), i.e.,in which all of the terms are the same as defined above.

Values of ∆G* for an undercooling of 30 K below the solvus temperature have been calculated for a supersaturated solid solution produced by supersolvus heat treatment of IN-100, René 88, and LSHR; i.e., C _m = solute content specified by the overall alloy composition. The calculations utilized Eq. (15) with the alloy/phase composition data in Tables 2 and 3 and estimates of the thermodynamic factor for each alloying element from the commercial code Pandat™ (CompuTherm LLC, Madison, WI) (Table 5). The equilibrium solute composition in the matrix (C _γ) was determined from the precipitate composition (Table 3) and the equilibrium volume fraction of γ′ per the solvus approach curves (Fig. 1). The results of the calculations (Table 5) reveal that the largest values of ∆G* correspond to those for chromium for all of the alloys. As suggested by the composition data in Tables 2 and 3, this behavior may be ascribed to a very high level of solute partitioning between the γ and γ′ phases for this element.

The importance of chromium in controlling the driving force for nucleation is confirmed by a comparison of the predicted variation of ∆G* with temperature from the methods based on either Eq. (13) or Eq. (15b) (for Cr solute) for a supersaturated solid solution produced by supertransus solution treatment. The comparison for IN-100 and LSHR shows excellent agreement (Fig. 4a). Although not formally appropriate, predictions from Eq. (15a), also plotted in Fig. 4a, show very good agreement with those from Eqs. (13) and (15b) as well. Furthermore, a comparison of the ∆G* dependence on temperature for supersolvus-solution-treated alloys LSHR and ME3, derived using Eq. (15b), shows essentially identical results. This behavior is as expected based on the very similar alloy and γ′ compositions, in particular with respect to the Cr content (Tables 2 and 3).

The principal retarding force for nucleation consists of the creation of γ-γ′ interfaces with their associated interfacial energy σ. The determination of σ is usually based on the temperature at which “noticeable” nucleation occurs. This onset temperature per se can be deduced using a variety of techniques including in situ (synchrotron) x-ray diffraction (e.g., reference [50]), differential thermal analysis (DTA), and on-cooling specific-heat measurements. The application of the latter technique is shown in Fig. 3, which reveals the temperature/undercooling at which the specific heat begins to rise noticeably. These temperatures are found to be 1431, 1345, and 1403.5 K (corresponding to an undercooling of 27, 33, or 26.5 K), respectively, for IN-100, René 88, and LSHR/ME3. The onset temperature and critical undercooling for LSHR from the specific heat are essentially identical to the values determined previously for LSHR via the synchrotron method [50]. Furthermore, the previous in-situ results indicated that the onset temperature is identical for cooling rates of 11 and 139 K/min.

Following an approach similar to that suggested by Doherty [4], the surface energy is chosen to yield a value of J _o (per Eq. (1) with a diffusivity based on static-coarsening measurements, as discussed below) that is discernable by post heat-treatment metallography. For cooling rates of the order of 0.2–2 K/s, a critical J _o of the order of 1/μm³s (10¹⁸/m³s) at the observed nucleation-onset temperature is reasonable. Examples of the determination of σ for IN-100, René 88, and LSHR by this means are summarized in Fig. 5. The surface energies so deduced are between 23 and 25 mJ/m². Because of the exponential dependence of nucleation rate on σ ³, the use of a different value for the “critical” value of J would have led to only a small difference in the appropriate choice of σ. For example, if the critical J were chosen to be 0.1/μm³s, the surface energy for the three alloys would have been between 24 and 26 mJ/m² per the plots in Fig. 5.

The values of σ in the present work are almost identical to that deduced by Sudbrack et al. [28] for a ternary Ni-Al-Cr alloy with levels of aluminum and chromium similar to those in IN-100 and LSHR. By contrast, the present value is considerably lower than the value quoted by Olson et al. [35] for LSHR (i.e., 31.5 mJ/m²). Perhaps, this difference can be rationalized in the context of values of ∆G* which may have been too high in the earlier prior work. For example, an examination of Eqs. (1) and (2) indicates that the nucleation rate depends on an exponential term whose argument includes a factor of σ ³/∆G*². Thus, to obtain an identical/observable nucleation rate at a given temperature, a surface energy that was high by a factor of approximately 35 pct would have required that ∆G* to be overestimated by 57 pct.

The misfit energy ∆G _p can also act as a retarding force through its effect in raising the overall system energy in much the same way that the interface energy does as precipitates are formed [55]. However, in most cases, ∆G _p is small relative to that of ∆G* and can be neglected to a first order or included implicitly with ∆G*. For example, Booth-Morrison et al. [56] determined that the misfit was ~0.2 pct for the ternary alloy Ni-5.2Al-14.2Cr (in atomic percent), i.e., a material whose levels of chromium and aluminum are similar to those in commercial PM superalloys. For this level of misfit, ∆G _p would be of the order of 1 J/mol. Such a value is two orders of magnitude smaller than the values of ∆G* for temperatures at which secondary and tertiary γ′ nucleation occurs.

The diffusivities of various solutes in nickel-base superalloys play a very important role in the precipitation process through their effect on particle growth per se as well as the rate of depletion of the matrix supersaturation controlling nucleation behavior. Because alloying elements such as Al and Ti partition in a sense opposite to that of Cr in the γ and γ′ phases and the off-diagonal terms in the diffusivity matrix are non-zero, the development of a concentration gradient for one alloying element may noticeably retard the overall diffusive flux of another. For example, Al and Cr in a nickel solid solution have a positive interaction such that Al can diffuse down a chromium concentration gradient [57‐59]. Thus, as a γ′ precipitate grows, the rate of diffusion of a given element to or away from the particle may be mitigated somewhat by its tendency to diffuse down the concentration gradient of a different alloying element.

Because of the complexity of the diffusion problem, it is therefore often simpler to determine an effective diffusivity for the alloying element whose behavior appears to be rate limiting and to which simple (pseudo-binary) diffusion analyses can be applied. For PM nickel-base superalloys such as those of interest here, the work of Campbell et al. [60] and Semiatin et al. [50] suggests that Cr diffusion is rate limiting in γ-γ′ superalloys.

In the present work, the specific method used to fit an effective diffusivity has been based on an analysis of the rate of coarsening of intragranular, secondary γ′ during isothermal heat treatments. In particular, experimental observations (e.g., Fig. 6 for IN-100 and Fig. 7 for René 88) were interpreted using Eq. (10), assuming that the process has been controlled solely by Cr diffusion, or Eqs. (10) and (11) to account for the influence of all of the solutes. In both analyses, the effective diffusivity of Cr as a function of temperature was taken equal to the product of the impurity diffusivity, D _B, of Cr in binary Ni-Cr alloys [61] (principally to obtain the activation energy/temperature dependence) and a fitting factor, A, to account for the finite concentration of Cr in the γ matrix and its interaction with other solutes, i.e.,

The diffusivities of the various other solutes in each alloy were taken to be in the ratios (relative to that of Cr) suggested by the work of Campbell et al. [60]. In addition, w(f) was taken from the research of Voorhees and Glicksman [20], and σ was assumed to be 23 mJ/m² per the results in Fig. 5. The molar volume V _M was calculated to be 7.20 × 10⁻⁶, 7.11 × 10⁻⁶, and 7.22 × 10⁻⁶ m³/mol for IN-100, René 88, and LSHR, respectively. The thermodynamic factors needed to apply Eq. (10) were derived from Pandat™ calculations (Table 6).

Predicted coarsening rate constants for IN-100, René 88, and LSHR at various temperatures and two different values of A (= D _eff/D _B), 0.33 and 0.8, are summarized in Table 7. For a given alloy, temperature, and value of A, the calculations reveal that the predicted coarsening rate based on Cr diffusion alone (Eq. (10)) is approximately 1.5 to 3 times that determined when accounting for the interaction among the solutes using Eqs. (10) and (11). Table 7 also summarizes measured rate constants based on SEM observations on 2D sections (e.g., those in Figs. 6 and 7). Two different methods were used to reduce such measurements. In one case (i.e., annotated as “SC”, or “stereological correction”), it was assumed that the true 3D diameters were 15 pct larger than those measured on the 2D sections. In the other case, no such SC was applied. This latter approach has been justified based on recent geometric analysis by Payton et al. [62] for distributions of spherical particles that have finite breadth, as is pertinent for precipitates undergoing coarsening. Specifically, it was deduced that the SC used in the first case overestimates the actual average 3D size and that no correction factor is actually needed for distribution shapes such as those developed during static coarsening.

Taken as a whole, the comparison of predicted and measured values of the coarsening rate constant (Table 7) shows two important trends. First, A = 0.8 is more appropriate for IN-100 and René 88, but a lower value, A = 0.33, gives a better fit for LSHR. Second, the predicted rate constants based on the interaction of all of the solutes are closer to the measurements than those based on diffusion of Cr alone. In fact, the agreement is remarkably good for the various alloys and different temperatures when the calculation incorporates the effect of all of the solutes and uses A = 0.8 for the first- and second-generation alloys and A = 0.33 for the third-generation alloy. The generality of this conclusion has been confirmed in a companion report [63] containing additional coarsening data and analysis for three third-generation PM alloys (LSHR, ME3, Alloy 10). In this other effort, the value of A was deduced to be 0.33 for all of three alloys, and, not surprisingly, the measured coarsening rates for LSHR and ME3 (having similar alloy and γ′ compositions) were identical. An inspection of the compositions of IN-100/René 88 and the various third-generation alloys (e.g., Tables 1 and 2) suggests that the tantalum addition is the differentiating element which plays a critical role in reducing the rate of diffusion which controls coarsening and likely the kinetics of precipitate nucleation and growth. In the latter regard, it was shown in reference [50] that A = 0.33 provides excellent predictions of the size and number density of secondary γ′ developed during continuous cooling of LSHR following supersolvus solution treatment.

Despite the generality of the present findings regarding the effective diffusivity, a word of caution is in order. That is to say, the companion effort [58] has shown that calculations assuming bulk-diffusion control, as embodied in Eq. (10), overestimate the measured coarsening rate by approximately a factor of five for temperatures below approximately 1050 K or those typical of service conditions for PM nickel-base superalloys. At such temperatures, the coarsening mechanism changes from bulk-diffusion control to more-sluggish trans-interface-diffusion control [64].