A1-L1_{2} interfacial free energies from data on coarsening in five binary Ni alloys, informed by thermodynamic phase diagram assessments
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DOI: 10.1007/s10853-011-5395-x
Abstract
The data on coarsening of γ′-type precipitates (Ni_{3}X, with the L1_{2} crystal structure) in Ni–Al, Ni–Ga, Ni–Ge, Ni–Si, and Ni–Ti alloys are re-evaluated in the context of recent (TIDC) and classical (LSW) theories of coarsening, with the objective of ascertaining the best values possible of interfacial free energies, σ, of the γ/γ′ interfaces in these five alloy systems. The re-evaluations include fitting of the particle size distributions, reanalyzing all the available data on the kinetics of particle growth and kinetics of solute depletion, and using thermodynamic assessments of the binary alloy phase diagrams to calculate curvatures of the Gibbs free energies of mixing. The product of the work is two sets of interfacial free energies, one set for the analysis using the recent TIDC theory and the other for the analysis using the classical LSW theory. The TIDC-based analysis yields lower values of σ by about a factor of 2/3. All the interfacial energies are considerably larger, by factors ranging from ~4 to 10, than those previously reported, which were for the most part calculated from data on coarsening assuming ideal-solution thermodynamics. In the TIDC theory the width of the interface, δ, is allowed to increase with particle size, r. A simple equation relating σ to the ratio of the gradient energy and δ is used to show that σ can remain constant even though δ increases with r. Published work supporting this contention is presented and discussed.
Introduction
Advances in one or more areas of research often impact the findings and conclusions drawn from analyses of data that predate these advances. The specific example in this work is the calculation of interfacial energies derived from the analyses of data on coarsening of precipitates. In the first theories of coarsening Lifshitz and Slyozov (LS) [1] and Wagner (W) [2] derived equations for the growth of a spherical particle of average radius 〈r〉. LS also derived an equation for the depletion of the small concentration of excess solute, \( X_{\alpha } - X_{{\alpha {\text{e}}}} \), that must accompany the growth of the average precipitate; X _{α} is the solute concentration in the matrix at time t and X _{αe} is its thermodynamic equilibrium value. The kinetics of the processes of growth and solute depletion were shown to obey the equations 〈r〉^{3} ≈ kt and \( X_{\alpha } - X_{{\alpha {\text{e}}}} \, \approx (\kappa t)^{ - 1/3} \), where k and κ are rate constants that depend on the thermo-physical parameters of the alloy system, including the chemical diffusion coefficient, \( \tilde{D} \), in the parent phase and the interfacial free energy, σ, between the precipitate and matrix phases. A remarkable ingredient of the LSW theory was an analytical equation describing the distribution of particle sizes (PSD). The PSD must exist as an essential component of the ensemble of particles, and its prediction was a significant advance over earlier theoretical efforts [3, 4] to describe the kinetics of growth of an ensemble of particles.
A brief history
Recent developments
Subsequent to the work published in 1995 [14] there have been several developments that directly impact the values of σ. Data on the kinetics of coarsening of γ′-type precipitates, including the kinetics of particle growth and solute depletion, have been published for Ni–Ga [15, 16] and Ni–Ge [17, 18] alloys. Thermodynamic models of the Ni-rich solid solutions have been published for all five binary alloy systems: Ni–Al [19–21], Ni–Ga [22], Ni–Ge [23], Ni–Si [24–26], and Ni–Ti [27]. There are also thermodynamic models of ternary Ni-rich solid solutions involving Al, Ga, Ge, Si, and Ti that are helpful in selecting data on G _{m} [28–33]. As discussed recently by Costa e Silva et al. [34], the deviation from ideality can have a significant effect on the values of σ derived from data on coarsening, invariably increasing them because \( G_{\text{m}}^{\prime \prime } \) increases as the departure from ideal solution behavior increases. Though this is quite evident from Eqs. 5 and 6, the discussions of Costa e Silva et al. on σ in Ni–Al and other alloys forcefully drive home the point.
Two other important factors have had a dramatic impact on the validity of the LSW theory itself under certain circumstances, and therefore whether it can be used without further modification to extract meaningful values of σ and \( \tilde{D} \) from data on coarsening. The first factor involves puzzling observations on the effect of equilibrium volume fraction, f _{e}, on the kinetics of coarsening in all the aforementioned binary Ni alloys—there is simply no effect of f _{e} when f _{e} exceeds ~0.08 or so.^{3} The conundrum arises from the theoretically sound expectation that when coarsening kinetics is diffusion-controlled, the rate constants k and κ should increase as f _{e} increases; discussions can be found in several review articles [35–38].
The second important factor is that the γ/γ′ interface in Ni–Al alloys is not sharp, but diffuse, the transition from the γ′ to the γ phase in planar interfaces occurring over a distance of ~2 nm. The first evidence for this was reported by Harada et al. [39]. This early observation has been confirmed by atomistic modeling [40], recent experimental observations using modern atom-probe tomography [41] and high-resolution transmission electron microscopy [42]. A reconciliation of these findings, i.e., the independence of the rate constants on f _{e} and the diffuse γ/γ′ interface, provided the stimulus for a new theory of coarsening by Ardell and Ozolins [43], who also showed that the interface is not only diffuse, but also quite ragged in structure. Ardell and Ozolins postulated that chemical diffusion through the interface controls the kinetics of coarsening when diffusion through the interface is slower than diffusion to the interface. This condition can prevail in Ni–Al alloys because diffusion in the ordered γ′ phase is generally much slower than diffusion in the disordered matrix [44–47]. The γ/γ′ interface thus becomes a bottleneck for diffusion, with significant consequences for the kinetics, leading to the so-called Trans-Interface-Diffusion-Controlled (TIDC) theory of coarsening. The consequences of the TIDC theory in obtaining values of σ from data on coarsening are described in the following sections.
The TIDC theory—quantitative predictions
A restriction of the TIDC theory is that it should no longer be valid for particles larger than a transitional particle size, r _{T}, defined by the condition \( r_{\text{T}} \ge \delta {{\tilde{D}} \mathord{\left/ {\vphantom {{\tilde{D}} {\tilde{D}_{\text{I}} }}} \right. \kern-\nulldelimiterspace} {\tilde{D}_{\text{I}} }} \) [43]. At such large sizes the flux of solute in the matrix to the interface is slower than the flux of solute through the interface, so the kinetics of coarsening become controlled by chemical diffusion in the matrix, i.e., LSW coarsening should prevail at larger particle sizes, or equivalently, longer aging times. In the Ni–Al, Ni–Ga, and Ni–Ti alloy systems the restriction does not apply because elastic interactions induce severe departures from equiaxed shapes at relatively small sizes (r < 20 nm). These interactions generally prevent meaningful average radii larger than this from being measured. Such is not the case for Ni–Si and Ni–Ge alloys, in which Ni_{3}Si and Ni_{3}Ge precipitates can grow large enough for the transition from TIDC to LSW kinetics to take effect. This will be evident in the following analyses of the data.
Examination of the data on binary Ni alloys
Values of the exponent n and average values of 〈u〉 = 〈r〉/r* resulting from fitting the PSDs measured experimentally in the Ni–Al, Ni–Ga, Ni–Ge, Ni–Si, and Ni–Ti alloys
Alloy |
n |
〈u〉 |
---|---|---|
Ni–Al |
2.424 ± 0.089 |
0.9537 |
Ni–Ga |
2.318 ± 0.164 |
0.9409 |
Ni–Ge |
2.385 ± 0.095 |
0.9492 |
Ni–Si |
2.444 ± 0.086 |
0.9559 |
Ni–Ti |
2.281 ± 0.069 |
0.9359 |
Ni–Al
The data on coarsening of γ′ precipitates in the context of the TIDC theory were analyzed recently by Ardell [48], so there is little to add here other than that the slightly different method of averaging produced a slightly higher value of n = 2.424 compared to the previous n = 2.4.
Ni–Ga
Ni–Ge
Ni–Si
There are several sets of data on the coarsening of Ni_{3}Si precipitates in binary Ni–Si alloys that can be used to calculate σ. Two of these, the aforementioned results of Rastogi and Ardell [52] and data of Polat et al. [53], were analyzed previously [54] and shown to yield results in reasonably good agreement, the data of Polat et al. producing somewhat larger values using an analysis based on LSW kinetics. The data on kinetics considered here are those of Rastogi and Ardell and Cho and Ardell [55].
The kinetics of solute depletion is shown in Fig. 11, plotted for consistency with the predictions of the TIDC theory. The fits with plots of X _{Si} vs. t ^{−1/3} are comparable. The data in Fig. 11 represent aging times for which 〈r〉 < 80 nm, so all the data are included in the linear fitting.
Ni–Ti
The kinetics of coarsening of Ni_{3}Ti precipitates has been investigated by Ardell [56] and Kim and Ardell [57]. These data were evaluated according to the predictions of the TIDC theory by Ardell et al. [49], so the figures will not be reproduced here. The main difference between the evaluation of the data by Ardell et al. and this work is that the value of n is somewhat lower; n = 2.281 cf. 2.375. This is due partly to the inclusion of the data for t = 4 h, partly to the way the average value of n was calculated and partly due to a minor change in the conversion of the original histograms to the representation shown in Fig. 2. The fits to the data for n = 2.281 are comparable to those seen in the paper by Ardell et al. [49].
Thermodynamics
Curvatures of the free energy functions
The calculations of σ for the five alloys using Eq. 12 require estimates of \( G_{\text{m}}^{\prime \prime } \). Values of σ can then be obtained from the experimental data on the rate constants using the appropriate rate constants from either the LSW or TIDC theories. Fortunately, the five alloy systems relevant here have been subjected to thermodynamic assessments with the objectives of describing the phase diagrams and their thermodynamic properties using a variety of experimental data; this is the essence of the CALPHAD method. These assessments require, as input, equations describing the Gibbs free energies of mixing, G _{m}, as functions of compositions for all the phases. For this work all that is needed are the equations describing G _{m} for the terminal γ solid solution phases as functions of composition and temperature. The necessary differentiations can then be performed to obtain \( G_{\text{m}}^{\prime \prime } \) evaluated at their appropriate equilibrium compositions.
The second derivative of G _{m,id}, Eq. 7, is valid for any value of X, not just at thermodynamic equilibrium. In all subsequent equations X represents the concentration of solute; subscripts identifying the solute will be added as needed.
The coefficients L _{ j } in the sums depend only on T and are obtained by fitting the free energy functions to the phase boundaries in the phase diagrams and other relevant available data.
The magnetic contribution to the free energy is expected to be small, since T _{C} for the Ni–Al, Ni–Ga, Ni–Ge, Ni–Si, and Ni–Ti alloys decreases sharply with composition (see [9, 16, 18, 52, 56] and references therein for representative dependencies of T _{C} on X), and the aging temperatures in the experiments on coarsening significantly exceeded T _{C}. Nevertheless, it is at least worthwhile to calculate \( G_{\text{m,mag}}^{\prime \prime } \) to see if neglecting its contribution is justifiable. It is shown in the Appendix 1 that this contribution is negligibly small.
Thermodynamic models and equilibrium solubility limits of the γ and γ′ phases
Equations for the L _{ j } (J/mol) of the f.c.c. solid solutions in the thermodynamic assessments of the binary Ni–Al, Ni–Ga, Ni–Ge, Ni–Si, and Ni–Ti phase diagrams
Alloy |
Source |
Case |
L _{ j } (J/mol) | |
---|---|---|---|---|
Ni–Al |
Ansara et al. [19] |
2 |
L _{0} |
–162,407.75 + 16.212965 T |
L _{1} |
73,417.798 – 34.914 T | |||
L _{2} |
33,471.014 – 0.837 T | |||
L _{3} |
–30,758.01 + 10.253 T | |||
Ni–Ga |
Yuan et al. [22] |
1 |
L _{0} |
–130,526 + 40 T |
Ni–Ge |
Liu et al. [23] |
2 |
L _{0} |
–91,312 + 11.452 T |
L _{1} |
120,929 – 45.241 T | |||
Ni–Si |
Tokunaga et al. [26] |
1 |
L _{0} |
–208,234.46 + 44.14177 T |
L _{1} |
–108,533.44 | |||
Ni–Ti |
De Keyzer et al. [24] |
1 |
L _{0} |
–98,143 + 6.706 T |
L _{1} |
–62,430 |
The remaining parameters needed to calculate σ using the equations of either the LSW or TIDC theories are the equilibrium solubility limits of both the γ and γ′ phases and the partial atomic volumes. As to the solubility limits, there are many options to choose from, since there are contributions to the phase diagrams from numerous investigators. The values of X _{γe} (= X _{αe}) chosen here were obtained from data on the kinetics of solute depletion, plotted according to the LSW version, Eq. 2, where extrapolation to t ^{−1/3} = 0 provides a value of X _{γe}. As pointed out by Rastogi and Ardell [14], the equilibrium solubility limits so obtained represent the solubilities of the coherent precipitates. These do not necessarily differ much from the incoherent solubility limits, but it seems a reasonable choice to use them. In the case of the L1_{2} form of the Ni_{3}Ti phase, the only other measurements of its solubility limits are those of Hashimoto and Tsujimoto [62], which agree well with those of Rastogi and Ardell. It is important to point out that when the same data on the kinetics of solute depletion are analyzed using the counterpart to Eq. 2 in the TIDC theory, the values of X _{γe} so obtained are very slightly smaller than those obtained from the LSW analysis; the typical difference is less than 0.01% and is ignored in this work.
Equations representing the solvus curves for the γ′-type phases in the Ni–Al, Ni–Ga, Ni–Ge, Ni–Si, and Ni–Ti alloy systems
Alloy |
X _{ γe} |
---|---|
Ni–Al |
5.5027 × 10^{−2} exp{8.124 × 10^{−4} T} |
Ni–Ga |
7.9824 × 10^{−2} exp{6.606 × 10^{−4} T} |
Ni–Ge |
5.8270 × 10^{−2} exp{7.186 × 10^{−4} T} |
Ni–Si |
4.6651 × 10^{−2} exp{8.911 × 10^{−4} T} |
Ni–Ti |
0.2566 exp{−1850/RT} |
Thermodynamic variables used in the calculation of σ at the various temperatures, T, used in the experiments on coarsening: V _{m} is the partial atomic volume; X _{ γe} and \( X_{{\gamma^{\prime } {\text{e}}}} \) are the equilibrium solute concentrations in the γ and γ′ phases, respectively; \( G_{\text{m}}^{\prime \prime } \) is the second derivative of the total Gibbs free energy of mixing in the γ phase, evaluated at the equilibrium compositions of the γ phase at each temperature
Alloy |
T (K) |
V _{m} × 10^{6} (m^{3}/mol) |
X _{ γe} |
\( X_{{\gamma^{\prime } {\text{e}}}} \) |
\( G_{\text{m}}^{\prime \prime } \) (J/mol) |
---|---|---|---|---|---|
Ni–Al |
898 |
7.0345 |
0.1141 |
0.2295 |
339,194 |
Ni–Al |
988 |
7.0633 |
0.1228 |
0.2269 |
346,735 |
Ni–Ga |
901 |
7.1968 |
0.1448 |
0.2327 |
249,480 |
Ni–Ga |
973 |
7.2145 |
0.1518 |
0.2327 |
246,038 |
Ni–Ge |
973 |
6.9477 |
0.1172 |
0.2396 |
591,570 |
Ni–Ge |
997 |
6.9590 |
0.1193 |
0.2388 |
584,914 |
Ni–Si |
923 |
6.7537 |
0.1062 |
0.2276 |
928,238 |
Ni–Si |
1048 |
6.7965 |
0.1187 |
0.2276 |
903,279 |
Ni–Ti |
965 |
6.8799 |
0.0978 |
0.2300 |
575,618 |
Ni–Ti |
993 |
6.8758 |
0.1005 |
0.2300 |
573,622 |
The partial molar (atomic) volumes of the solute atoms in the γ′-type phases were calculated from the equilibrium lattice constants at the relevant temperatures using the relationship \( V_{\text{m}} = a_{\text{o}}^{3} {{N_{\text{A}} } \mathord{\left/ {\vphantom {{N_{\text{A}} } 4}} \right. \kern-\nulldelimiterspace} 4} \), where N _{A} is Avogadro’s number. For Ni_{3}Al, Ni_{3}Ga, Ni_{3}Ge, and Ni_{3}Si the lattice constants were taken from the data of Kamara et al. [71]. These data take thermal expansion into account, but not the variations of equilibrium composition with T, so Vegard’s law was assumed to apply and corrections for variations with composition were assumed (Ardell, unpublished research). The values of V _{m} so obtained are summarized in Table 4. The numbers shown in Table 4 differ slightly from those published previously because of the variations of a _{o} with temperature and composition that were not considered in previous work.
The only measurements of the lattice constants of L1_{2} Ni_{3}Ti are those of Hashimoto and Tsujimoto [62]. Their reported lattice constants decrease slightly with increasing temperature over the range 600–900 °C. They suggest that this is due to a variation in composition of the Ni_{3}Ti phase from ~14.2% (900 °C) to 17.1% (600 °C) Ti, but their suggestion is based on very old data on the variation of lattice constant with composition. It is worth pointing out that at the aging temperature of 600 °C the data of Hashimoto and Tsujimoto indicate that the lattice mismatch between L1_{2} Ni_{3}Ti and the matrix phase is ~0.84%, which agrees well with the room-temperature value reported by Sass et al. [72]. This suggests that the compositions reported in their Fig. 4 are too low, and that the lattice constants they measured are close to their equilibrium values. Taking all these factors into account leads to the values of \( G_{\text{m}}^{\prime \prime } \) shown in Table 4.
Calculations of σ
Rate constants \( \kappa_{\text{T}}^{{ - {1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n}}} \) and k _{T} used in the calculations of σ assuming validity of the TIDC theory. The capillary length, \( \ell_{\rm T} \), is also shown
Alloy |
T (K) |
\( \kappa_{\text{T}}^{{ - {1 \mathord{\left/ {\vphantom {1 n}} \right. \kern-\nulldelimiterspace} n}}} \) (s^{1/n }) |
k _{T} (m^{ n }/s) |
\( \ell_{\text{T}} \left( m \right) \) |
σ (mJ/m^{2}) |
---|---|---|---|---|---|
Ni–Al |
898 |
0.17575 ± 0.00207 |
7.6826 ± 1.0685 × 10^{−26} |
7.6583 × 10^{−12} |
22.33 ± 1.31 |
Ni–Al |
988 |
0.04237 ± 0.00181 |
2.1414 ± 0.0874 × 10^{−24} |
7.2860 × 10^{−12} |
19.52 ± 0.90 |
Ni–Ga |
901 |
0.07591 ± 0.00238 |
5.3048 ± 0.0333 × 10^{−24} |
6.9054 × 10^{−12} |
11.19 ± 0.35 |
Ni–Ga |
973 |
0.01975 ± 0.00860 |
1.8441 ± 0.4633 × 10^{−23} |
3.0754 × 10^{−12} |
4.51 ± 2.02 |
Ni–Ga |
973 |
0.02036 ± 0.01678 |
2.6205 ± 0.1263 × 10^{−23} |
3.6899 × 10^{−12} |
5.41 ± 4.46 |
Ni–Ge |
973 |
0.02838 ± 0.01116 |
2.3645 ± 0.2120 × 10^{−23} |
9.2491 × 10^{−12} |
51.48 ± 20.34 |
Ni–Ge |
973 |
0.06693 ± 0.00818 |
2.9372 ± 0.1520 × 10^{−23} |
2.3890 × 10^{−11} |
132.97 ± 16.50 |
Ni–Ge |
973 |
0.03137 ± 0.01886 |
2.9543 ± 0.3972 × 10^{−23} |
1.1226 × 10^{−11} |
62.48 ± 37.72 |
Ni–Ge |
997 |
0.01887 ± 0.00462 |
6.2568 ± 0.3266 × 10^{−23} |
9.2473 × 10^{−12} |
49.71 ± 12.22 |
Ni–Ge |
997 |
0.03114 ± 0.00609 |
5.6927 ± 0.3231 × 10^{−23} |
1.4671 × 10^{−11} |
78.87 ± 15.55 |
Ni–Ge |
997 |
0.04148 ± 0.01320 |
5.8055 ± 0.4923 × 10^{−23} |
1.9701 × 10^{−11} |
105.91 ± 33.91 |
Ni–Si |
923 |
0.07113 ± 0.01370 |
1.3063 ± 0.0492 × 10^{−24} |
1.2011 × 10^{−11} |
104.84 ± 20.25 |
Ni–Si |
923 |
0.04321 ± 0.01694 |
1.4991 ± 0.0375 × 10^{−24} |
7.7182 × 10^{−12} |
66.73 ± 26.42 |
Ni–Si |
923 |
0.06530 ± 0.01651 |
1.1880 ± 0.0382 × 10^{−24} |
1.0606 × 10^{−11} |
92.57 ± 23.44 |
Ni–Si |
923 |
0.06597 ± 0.00663 |
1.4898 ± 0.0551 × 10^{−24} |
1.1754 × 10^{−11} |
102.60 ± 10.43 |
Ni–Si |
1048 |
0.00842 ± 0.00099 |
1.2740 ± 0.0179 × 10^{−22} |
9.2661 × 10^{−12} |
70.15 ± 8.24 |
Ni–Ti |
965 |
0.07558 ± 0.00376 |
2.5971 ± 0.0617 × 10^{−23} |
9.4801 × 10^{−12} |
57.27 ± 2.86 |
Ni–Ti |
993 |
0.04297 ± 0.01124 |
1.1684 ± 0.0720 × 10^{−22} |
1.0421 × 10^{−11} |
60.39 ± 15.88 |
Ni–Ti |
993 |
0.07203 ± 0.00795 |
2.3226 ± 0.2215 × 10^{−23} |
8.6027 × 10^{−12} |
49.85 ± 5.89 |
Ni–Ti |
993 |
0.09962 ± 0.01152 |
2.5075 ± 0.2665 × 10^{−23} |
1.2305 × 10^{−11} |
71.30 ± 8.89 |
Rate constants κ ^{−1/3} and k used in the calculations of σ assuming validity of the LSW theory. The capillary length, \( \ell \), is also shown
Alloy |
T (K) |
κ ^{−1/3} (s^{1/3}) |
k (m^{3}/s) |
\( \ell \left( m \right) \) |
σ (mJ/m^{2}) |
---|---|---|---|---|---|
Ni–Al |
898 |
0.08675 ± 0.00207 |
2.1379 ± 0.2565 × 10^{−30} |
1.1175 × 10^{−11} |
31.08 ± 1.45 |
Ni–Al |
988 |
0.02599 ± 0.00181 |
7.2526 ± 0.4011 × 10^{−29} |
1.0840 × 10^{−11} |
27.70 ± 2.00 |
Ni–Ga |
901 |
0.03932 ± 0.00111 |
2.3605 ± 0.1756 × 10^{−29} |
1.1278 × 10^{−11} |
17.19 ± 0.645 |
Ni–Ga |
973 |
0.01178 ± 0.00860 |
7.7129 ± 1.7371 × 10^{−29} |
5.0158 × 10^{−12} |
6.92 ± 3.10 |
Ni–Ga |
973 |
0.01194 ± 0.01678 |
1.4188 ± 0.0385 × 10^{−28} |
6.2276 × 10^{−12} |
8.59 ± 7.31 |
Ni–Ge |
973 |
0.01830 ± 0.00709 |
5.1503 ± 0.3141 × 10^{−28} |
1.4667 × 10^{−11} |
76.39 ± 29.66 |
Ni–Ge |
973 |
0.04050 ± 0.00479 |
7.9805 ± 0.5108 × 10^{−28} |
3.7565 × 10^{−11} |
195.67 ± 23.51 |
Ni–Ge |
973 |
0.01979 ± 0.01222 |
6.8513 ± 0.9493 × 10^{−28} |
1.7449 × 10^{−11} |
90.89 ± 56.32 |
Ni–Ge |
997 |
0.01051 ± 0.00250 |
2.4848 ± 0.1930 × 10^{−27} |
1.4240 × 10^{−11} |
71.51 ± 17.13 |
Ni–Ge |
997 |
0.01746 ± 0.00311 |
2.1185 ± 0.1830 × 10^{−27} |
2.2430 × 10^{−11} |
112.66 ± 20.90 |
Ni–Ge |
997 |
0.02304 ± 0.00728 |
2.1478 ± 0.2285 × 10^{−27} |
2.9721 × 10^{−11} |
149.28 ± 47.25 |
Ni–Si |
923 |
0.03214 ± 0.00619 |
1.0621 ± 0.0508 × 10^{−28} |
1.5219 × 10^{−11} |
126.98 ± 24.53 |
Ni–Si |
923 |
0.01928 ± 0.00774 |
1.1148 ± 0.0530 × 10^{−28} |
9.2797 × 10^{−12} |
77.43 ± 31.10 |
Ni–Si |
923 |
0.02927 ± 0.00759 |
1.0763 ± 0.0128 × 10^{−28} |
1.3923 × 10^{−11} |
116.17 ± 30.12 |
Ni–Si |
923 |
0.02976 ± 0.00306 |
1.1102 ± 0.0414 × 10^{−28} |
1.4303 × 10^{−11} |
119.34 ± 12.34 |
Ni–Si |
1048 |
0.00542 ± 0.00077 |
1.4558 ± 0.0593 × 10^{−26} |
1.3243 × 10^{−11} |
95.83 ± 13.70 |
Ni–Ti |
965 |
0.04273 ± 0.00231 |
6.7761 ± 1.0422 × 10^{−29} |
1.7421 × 10^{−11} |
96.36 ± 7.17 |
Ni–Ti |
993 |
0.02147 ± 0.00523 |
7.3544 ± 0.3924 × 10^{−28} |
1.9380 × 10^{−11} |
104.71 ± 25.57 |
Ni–Ti |
993 |
0.03560 ± 0.00360 |
5.8047 ± 0.4517 × 10^{−29} |
1.3784 × 10^{−11} |
74.48 ± 7.77 |
Ni–Ti |
993 |
0.03425 ± 0.00658 |
4.4362 ± 0.2509 × 10^{−29} |
1.2126 × 10^{−11} |
65.51 ± 12.85 |
Calculated values of the interfacial free energies, σ, of the γ/γ′-type interfaces in all five alloys
Alloy |
σ (mJ/m^{2})—TIDC |
σ (mJ/m^{2})—LSW |
---|---|---|
Ni–Al |
20.42 ± 1.05 |
29.92 ± 1.66 |
Ni–Ga |
10.96 ± 0.60 |
16.70 ± 1.09 |
Ni–Ge |
74.98 ± 17.74 |
110.40 ± 25.01 |
Ni–Si |
84.33 ± 13.00 |
109.18 ± 17.85 |
Ni–Ti |
56.16 ± 4.87 |
83.97 ± 9.56 |
The estimated errors in Table 7 are very conservative, because there are many other factors that can affect the calculations. These include the exponent n in the TIDC theory and the equilibrium concentrations of both phases. Additionally, there is some flexibility in choosing the thermodynamic model used to calculate \( G_{\text{m}}^{\prime \prime } \), and the reliability of the models themselves. With regard to the models selected, the curvatures of the Gibbs free energy of mixing of the Ni–Al solid solution calculated using the model of Du and Clavaguera [20] are about 25% larger than those from the model of Ansara et al. [19]. This would produce comparably larger values of σ. The model of Bellen et al. [27] of the Ni–Ti solid solution yields values of \( G_{\text{m}}^{\prime \prime } \) only slightly smaller than obtained from the model of De Keyzer et al. [30]. Matsumoto et al. [33] modeled the binary Ni–Ti phase diagram as part of their assessment of the ternary Nb–Ni–Ti system. The values of \( G_{\text{m}}^{\prime \prime } \) from their model are roughly 6–7% smaller than those from the model of De Keyzer et al. [30], so the models predict curvatures of the free energy of mixing that agree quite well. The values of σ reported in Table 7 for Ni–Al, Ni–Ga, Ni–Ge, and Ni–Ti alloys can therefore be considered to be as reliable as current analysis allows.
Discussion
It is not evident from the numbers in Table 7, but the influence of non-ideality of the solid solutions on the calculated values of σ is quite large, ranging from a factor of about 4 for Ni–Ga alloys to over a factor of 10 for Ni–Si alloys. For the reader who is not familiar with the data on coarsening of the γ′-type precipitates in these alloys, it should be obvious on viewing Figs. 5, 6, 7, 8, 9, 10, and 11 that there is no effect of initial alloy concentration on the kinetics of coarsening, which means that there is no effect of f _{e} either. For some of the data shown f _{e} varies by as much as a factor of 10, so if any of the theories noted in the review articles cited [35–38] were correct, the effect on kinetics would have to be considered in the calculations of σ. The TIDC theory completely obviates this issue since no effect of f _{e} is predicted.
In assessing the reliability of the calculated values of σ, independently measured or calculated values would be immensely helpful. Unfortunately, except for the important Ni–Al alloy system such calculations have not been made. Since the γ/γ′ interface is so technologically important, the Ni(Al)/Ni_{3}Al interfacial energy has been estimated by several groups using atomistic calculations. All these calculations assume that pure Ni is in equilibrium with stoichiometric Ni_{3}Al, so the calculated values represent upper limits of the energies at 0 K. The first such calculation was that of Farkas et al. [73] using embedded atom potentials. They calculated σ ≈ 22 mJ/m^{2}. Price and Cooper [74] used a different atomistic method and obtained values of 25 or 63 mJ/m^{2}, with the differences due to the treatment of magnetic effects. Mishin [40] used newer embedded atom potentials and found that σ varied depending on the orientation of the interface, being lowest for the (111) interface (12 mJ/m^{2}) and highest for the (100) interface (46 mJ/m^{2}). Somewhat similar results were obtained by Costa et Silva et al. [34], whose first-principles calculations also produced anisotropic values of σ, namely 39.6 and 63.8 mJ/m^{2} for the (100) and (110) interfaces, respectively. On comparing these theoretical estimates with the experimental results in this work, two factors need to be considered. The first is the theoretically predicted anisotropy of σ, and the second is effect of temperature, since the values of σ obtained from the data on coarsening are values representative of T ≫ 0 K.
There is no experimental evidence for an orientation dependence of the interfacial energy in Ni-base γ/γ′ alloys. Indeed, there is overwhelming evidence that the shapes of γ′-type precipitates are governed uniquely by the competition between elastic and interfacial energies. Review articles by Doi [75] and Fratzl et al. [76] show transmission electron micrographs of γ′ precipitates in ternary alloys with compositions deliberately selected to alter the elastic mismatch between the precipitate and matrix phases. The shapes of the γ′ precipitates are invariably spherical when the elastic mismatch is close to zero no matter how large the precipitates are, with no evidence whatsoever of faceting. There is currently no explanation for the discrepancy with Mishin’s atomistic calculations. Regarding the effect of temperature, the values of σ would be expected to decrease as T increases from 0 K. The interfacial energy would also decrease on taking the chemistry of the precipitates into account. The apparent agreement among the theoretically calculated values of σ for the Ni(Al)/Ni_{3}Al interface and those seen in Table 7 must clearly be regarded as fortuitous.
Inspection of Eq. 25, recalling Eq. 10, indicates that σ cannot remain approximately constant as r (hence δ) increases unless the increase in size is accompanied by an increase in χ. Over the range of average particle sizes of the precipitates analyzed in this work, Eq. 10 predicts that δ should increase by factors of about 1.4 (Ni–Ga), 1.7 (Ni–Al, Ni–Ge, and Ni–Ti), and 2 (Ni–Si). The question therefore is whether there is any evidence that χ can possibly increase by a comparable amount in order that the values of σ shown in Table 7 remain approximately constant, at least over the ranges of temperatures of the experiments on coarsening.
There are two studies that support the idea that χ can increase as r (hence δ) increases. One is found in a recent paper by Hoyt [79], who investigated the thermodynamic equilibrium conditions for Cu-rich droplets in the Cu–Pb system using molecular dynamics simulations. Hoyt compared the gradient energies of planar and curved solid Cu–liquid Pb interfaces and found that χ at 1000 K for a droplet 5.3 nm in diameter is 1.27 × 10^{−10} J/m compared to 2.18 × 10^{−10} J/m for a planar Cu–Pb interface. The factor of ~1.7 increase in χ is comparable to the increases in δ predicted by Eq. 10. This is a fortuitous, but nevertheless encouraging, result.
The other piece of evidence is found in Fig. 4 of a paper by Booth-Morrison et al. [80] on the coarsening of γ′ precipitates in a ternary Ni–Al–Cr alloy. They measured concentration profiles across interfaces using atom-probe tomography at aging times of 1, 4, and 4096 h. The maximum absolute values of the concentration gradients of Al and Ni, \( \left| {{{{\text{d}}X_{\text{Al}} } \mathord{\left/ {\vphantom {{{\text{d}}X_{\text{Al}} } {{\text{d}}y}}} \right. \kern-\nulldelimiterspace} {{\text{d}}y}}} \right|_{ \max } \,{\text{and}}\,\left| {{{{\text{d}}X_{\text{Ni}} } \mathord{\left/ {\vphantom {{{\text{d}}X_{\text{Ni}} } {{\text{d}}y}}} \right. \kern-\nulldelimiterspace} {{\text{d}}y}}} \right|_{ \max } \) in the notation used in Appendix 2, at 1 h of aging are a factor of 1.5 to 1.6 larger than those at 4096 h of aging. Assuming that these gradients are proportional to the average gradients, 〈dX _{Al}/dy〉 and 〈dX _{Ni}/dy〉, respectively (see Appendix 2), δ must increase as the interfacial concentration gradients increase, with increases comparable to those of the maximum gradients. The computer simulations of Hoyt [79] and the measurements of Booth-Morrison et al. [80] do not prove conclusively that χ varies with r, but their work provides clear evidence that it is reasonable to assume that σ can remain approximately constant while δ increases.
The interfacial free energies calculated using the TIDC theory are approximately 2/3 those calculated using the LSW theory, with the exception of the Ni–Si system where the ratio is ~3/4. The reason for the smaller difference in Ni–Si alloys is that the data on the 2 longest aging times in the results of Cho and Ardell [55] were omitted in the TIDC analysis but used in the LSW analysis. A similar treatment was accorded the data on the Ni–Ge alloys, but only one point was omitted and its omission has only a relatively small effect on σ calculated using the TIDC theory.
The rationale stated earlier for excluding the data at longer aging times is that 〈r〉 exceeds \( r_{\text{T}} > \delta {{\tilde{D}} \mathord{\left/ {\vphantom {{\tilde{D}} {\tilde{D}_{\text{I}} }}} \right. \kern-\nulldelimiterspace} {\tilde{D}_{\text{I}} }} \) ≈ 80 nm. There are no data on diffusion in Ni_{3}Si, so it is not possible to estimate \( \tilde{D}_{\text{I}} \), which is expected to be slightly larger than \( \tilde{D}_{{{\text{Ni}}_{3} {\text{Si}}}} \) though of course smaller than chemical diffusion in the matrix. It is, nevertheless, instructive to see if the data on chemical diffusion in Ni–Ge alloys can be used to rationalize the exclusion of the long-time data in this system. Chemical diffusion coefficients in both Ni_{3}Ge and the Ni–Ge solid solution have been measured by Komai et al. [81]. Using the empirical equations reported for the Ni–Ge solid solution (10% Ge) and Ni_{3}Ge (23.5% Ge), \( \tilde{D}_{{{\text{Ni}} - 10\% \,{\text{Ge}}}} = 6.29 \times 10^{ - 5} { \exp }\left( {{{ - 233,000} \mathord{\left/ {\vphantom {{ - 233,000} {RT}}} \right. \kern-\nulldelimiterspace} {RT}}} \right) \) and \( \tilde{D}_{{{\text{Ni}}_{3} {\text{Ge}}}} = 9.03 \times 10^{ - 5} { \exp }\left( {{{ - 248,000} \mathord{\left/ {\vphantom {{ - 248,000} {RT}}} \right. \kern-\nulldelimiterspace} {RT}}} \right) \), respectively, the ratio \( {{\tilde{D}} \mathord{\left/ {\vphantom {{\tilde{D}} {\tilde{D}_{\text{I}} }}} \right. \kern-\nulldelimiterspace} {\tilde{D}_{\text{I}} }} \) is about 4.5 at 700 °C. With δ = 2 nm the transition radius r _{T} is r _{T} ≈ 9 nm. This is about a factor of 10 smaller than the assumed transition radius, but it should be kept in mind that chemical diffusion in the Ni–Ge matrix is strongly dependent on concentration. Judging from the data of Komai et al. [81] reported in their Fig. 10 chemical diffusion in an alloy containing ~11.8% Ge, which is within the range of equilibrium concentrations in the matrix of the aged Ni–Ge alloys (see Table 4), \( \tilde{D} \) would increase by a factor of 3–4, leading to a concomitant increase in r _{T} by the same amount for this value of δ. Whereas these arguments do not fully justify the exclusion of data for which 〈r〉 > r _{T}, they provide sensible rationale for doing so.
The other factor involved in the transition from TIDC to LSW kinetics is the behavior of the PSDs. In principle, when the kinetics of coarsening is controlled by diffusion in the matrix phase the PSDs are broader than predicted by the original LSW theory owing to the effect of volume fraction [35–38]. In practice, experimental PSDs tend to be broader than the LSW PSD anyway, even if there is no effect of f _{e} on the kinetics. Given the relatively poor statistics involved in the measurements of the PSDs, it is impossible to characterize the PSDs discussed in this work accurately enough to distinguish any possible mechanism that might be responsible for broadening. In other words, any expected changes in the shape of the PSDs in the transition from TIDC to LSW kinetics are not detectable.
It is natural to conjecture about the relative magnitudes of σ for the different alloy systems. As noted some time ago [10] interfacial free energies in γ/γ′ alloys are dominated by second- and higher-order neighbor interactions. It is reasonable to conclude that these interactions for the covalent Group IV elements Si and Ge are stronger than those for the Group III elements Al and Ga. This is consistent with the larger Ni(Si)/Ni_{3}Si and Ni(Ge)/Ni_{3}Ge interfacial energies compared to the Ni(Al)/Ni_{3}Al and Ni(Ga)/Ni_{3}Ga interfacial energies. Moreover, since bond energies tend to decrease as the atomic number increases, i.e., bonding in Ge and Ga is weaker than bonding in Si and Al, respectively, the Ni(Al)/Ni_{3}Al interfacial energy should be larger than the Ni(Ga)/Ni_{3}Ga energy, and the Ni(Si)/Ni_{3}Si energy should exceed the Ni(Ge)/Ni_{3}Ge interfacial energy. These expectations are confirmed by the values of σ seen in Table 7, with the sole exception for the relative interfacial energies in the Ni–Si and Ni–Ge alloys calculated using the LSW analysis of the data, which produces roughly equal values of σ in the two systems.
The only independent measurements of thermodynamic quantities that support this argument are those of Martosudirjo and Pratt [82], who report that the heat of solution of Ge in Ni at 836 K is about 13 times larger than that of Ga at 778 K. These measurements indicate that first nearest-neighbor interactions in Ni–Ge are significantly larger than in Ni–Ga, and by inference the higher-order interactions are also larger. It is not possible to extend this argument to the transition metal Ti, but since the valence state of Ti is generally larger than that of Al and Ga it would be expected that larger values of σ for the Ni(Ti)/Ni_{3}Ti system would be observed, which is also consistent with the data in Table 7.
Given the uncertainties in the magnitudes of all the parameters associated with these estimates, it is concluded that the TIDC theory successfully describes the data on coarsening in all five alloys. The consideration of bonding also provides justification for the validity of the thermodynamic model of Tokunaga et al. [26] in describing the thermodynamics of the Ni–Si alloy system. The other two thermodynamic models used to calculate σ in Table 8 yield magnitudes that appear to be too small. Whether the values of σ obtained using the TIDC-based analyses are more “accurate” than those obtained from the LSW-based analyses awaits confirmation. This will come when atomistic calculations can properly account for the equilibrium compositions of the γ and γ′-type phases at temperatures in the range 800–1000 K, as well as the absence of an orientation dependence of σ in the absence of elastic mismatch.
In the original LSW theory no distinction was made between various kinds of diffusion coefficients. It was unnecessary because the solution was assumed to be very dilute, in which case the tracer and chemical diffusion coefficients are equal.
Ni_{3}Si and Ni_{3}Ti both exist with the L1_{2} crystal structure. The Ni_{3}Si phase is stable and is called β_{1}. Ni_{3}Ti is metastable. The stable phase is called η and has the hexagonal DO_{24} crystal structure.
The rate constants for coarsening in all five alloys actually decrease as f _{e} increases when f _{e} is very small (f _{e} < 0.05). This anomalous behavior awaits a satisfactory explanation.
The gradient energy χ differs by a factor of 2 in various treatments of decomposition from supersaturated solid solution. These distinctions are ignored here because the magnitude of χ is immaterial to the discussion in this paper.
Acknowledgements
I am very grateful to Dr. Nathalie Dupin, Calcul Thermodynamique, for her help in unraveling the mysteries (to me) of thermodynamic assessments of the Ni–Al phase diagram and, by inference, others. I also thank Professor Y.Q. Liu, China University of Geosciences, Beijing, for her help in dealing with the thermodynamic assessment of the Ni–Ge phase diagram. Professor Vidvuds Ozolins, UCLA, provided very helpful discussions of bonding in the five alloys, and his valuable insights are greatly appreciated.
Appendix 1
Appendix 2
In the derivation of Eq. 39 the gradient energy has been taken as constant. In fact χ generally varies with composition (see [86] for a representative evaluation), but for the relatively small difference between X _{ γ } and \( X_{{\gamma^{\prime } }} \) (~12%) the assumption of constancy of χ in the integral, Eq. 36, is reasonable. There is also a contribution to χ from the gradient in long-range order [87], but in phase-field simulations the magnitude of this contribution generally varies from 10 [87] to ~400 times [88] smaller than the chemical contribution.