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Modeling of precipitation kinetics in multicomponent systems: Application to model superalloys
Acta Materialia 100 (2015) 169–177
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Acta Materialia journal homepage: www.elsevier.com/locate/actamat
Modeling of precipitation kinetics in multicomponent systems: Application to model superalloys M. Bonvalet ⇑, T. Philippe, X. Sauvage, D. Blavette Normandie Université, Groupe de Physique des Matériaux (GPM), UMR CNRS 6634 BP 12, Avenue de l’Université, 76801 Saint Etienne du Rouvray, France
a r t i c l e
i n f o
Article history: Received 4 March 2015 Revised 23 July 2015 Accepted 15 August 2015
Keywords: Precipitation kinetics Modeling Multicomponent systems Ni–Cr–Al Superalloys
a b s t r a c t A new general model dealing with nucleation, growth and coarsening simultaneously has been developed for the simulation of precipitation in non-dilute multicomponent alloys. Nucleation is implemented using the Zeldovich theory that includes regression effects. Growth and coarsening are modeled using the recently developed growth law in multicomponent alloys that accounts for capillarity, mass balance at the interface matrix-precipitate and diffusion-flux couplings. Numerical results are confronted to atom probe tomography (APT) experiments on model NiCrAl superalloys and to rigid lattice kinetics Monte Carlo (LKMC) simulations and are found in very good agreement with both APT experiments and LKMC simulations. We emphasize this work on the evolution of phase concentrations. The temporal evolution of the mean precipitate composition is found to be non-monotonic during the phase transformation, and phase composition does not follow the tie line due to a complex interplay between capillarity and diffusion process. The widespread availability of both thermodynamic and mobility databases makes this new model very suitable for material design. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Introduction Modeling precipitation kinetics in multicomponent alloys is both of utmost interest from an academic point of view but also for the optimization of thermal treatments and alloy compositions in terms of mechanical or structural properties. Numerical experiments based on Kinetic Monte Carlo simulations are known to be very effective. Such approaches were applied successfully to precipitation kinetics in model Ni-based superalloys [1–3]. Numerical explicit models of precipitation where nucleation, growth and coarsening theories are simultaneously implemented are attractive alternative approaches [4] due to the widespread availability of both thermodynamic and kinetic databases that make such approaches quantitative. Models based on the particle size distribution of precipitates (PSD) were shown to be quite accurate to study precipitation kinetics [4]. This paper describes a model based on this approach. Kampman and Wagner [5] were the first to use a PSD model for the precipitation in dilute binary alloys. The influence of the curvature on the composition of precipitates was implemented (i.e. the Gibbs Thomson effect). In their numerical resolution, they used a Lagrange-like approach for the size distribution. In such approach, ⇑ Corresponding author. E-mail address: [email protected] (M. Bonvalet). http://dx.doi.org/10.1016/j.actamat.2015.08.041 1359-6454/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
it is the precipitate size in each population N(R) that is computed, the initial number of precipitates being controlled by nucleation process. In each population of precipitates are included precipitates that nucleate during the same time interval (t, t + dt). It is assumed that these precipitates (defined by a radius and a composition) evolve in the same way over time. Myhr and Grong [6] also used a PSD for the precipitation in Al–Mg–Si but employing an Euler-like approach. Therein the size classes of precipitates were fixed and the number density in each class evolved over time. The size distribution is not managed in the same way with both approaches (Lagrange and Euler). One approach (Lagrange) deals with evolution of mean radius in each class while the other (Euler) deals with evolution of number density of particles in each class of defined interval radius. Perez et al. [7] compared these two approaches and concluded to their equivalence. PSD models were also used in various contexts [8–11], it was shown that nucleation, growth and coarsening generally superimposes and therefore cannot be treated, in general, separately. It was also found that interfacial energy is a key parameter for such approach, having a strong influence on simulation results [10,11]. In multicomponent alloys, the description of precipitation processes is much more complex. In the past few years, a few models dealing with coarsening in multicomponent alloys were developed. The first notable coarsening theory in binary alloy was given by Lifshitz and Slyozov and Wagner [12,13]. They used the
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Gibbs–Thompson equation so that to determine interfacial compositions. However, in multicomponent system, local equilibrium is no longer sufficient do determine interfacial compositions. Mass balance equation must be taken into account. Kuehmann and Voorhees [14] were the first to describe Ostwald ripening in a ternary alloy but they did not include the influence of off-diagonal diffusion coefficients. Latter, Bjorklund et al. [15], Slyosov and Sagalovich [16], Umantsev and Olson [17], Morral and Purdy [18] developed different approaches for the coarsening theory each time using different assumptions (dilute solutions, ideal solutions, equilibrium precipitate composition, diagonal diffusivities). Recently, Philippe and Voorhees [19] proposed a description of Ostwald ripening in multicomponent alloys removing restrictions of past works based on both thermodynamic and atomic mobility databases. Recent experiments on model Ni–Cr–Al superalloys have shown that during precipitation the compositional trajectory of the coarsening phase does not follow the equilibrium tie-line [2,3]. Rougier et al. have recently developed a PSD model for the simulations of c0 precipitation in NiCrAl [20]. They have used a multicomponent diffusion model in order to calculate the growth rate. They have also included off-diagonal diffusivities. However, their model has been fully coupled with CALPHAD for the calculation of nucleation driving force and interfacial compositions. This coupling is very expensive computationally. In contrast, we propose here a general model for precipitation in multicomponent alloys using the new generalized coarsening theory of Philippe et al. [19]. As a main advantage, this new approach does not need any coupling with external CALPHAD modules, and therefore guaranties remarkable computation efficiency. Our approach is validated through an application of this model to Ni-based superalloys for which diffusion and thermodynamic data bases are available. Numerical results are compared with both experiments and lattice kinetics Monte Carlo (LKMC) simulations.
2. Governing equations
where n0 is the density of potential nucleation sites (m3), f c0 is pffiffiffiffiffiffiffiffi pffiffiffiffi the volume fraction of precipitates, Z ¼ X r=ð2pR2 kB T Þ is the 3 Zeldovich factor, with X the atomic volume (m ) of both phases c and c0 (assumed to be equal), R⁄ is the critical radius of nucleation (m), r is the interfacial energy (J m2), T the temperature (K) and kB the Boltzmann constant (J K1). b ¼ ð4pR2 =a4 Þ minðDcii C cii Þ is the rate of attachment of solute atoms on a critical precipitate (s1), with a the lattice parameter (m), Dcii and C cii are respectively the diagonal diffusion coefficient (m2 s1) and the average concentration in c phase of element i:
DG ¼
16p r3 3 Dg 2n
is the nucleation barrier (J) with Dgn the nucleation driving force (J m3). s ¼ pb2 Z2 is the Zeldovich incubation time. It is related to the fact that a nucleus enters the growth stage only when thermal fluctuations will no more involve the partial dissolution of nuclei [21]. Indeed, regressions may occur for nuclei with size close but larger than the critical size R⁄. In this theory, the size for stable nuclei enter their growth stage is expressed as:
1 R ¼R þ 2 0
2.1. Nucleation
rffiffiffiffiffiffiffiffi kB T
ð3Þ
pr
where the critical radius for nucleation is given by R* = 2r/Dgn. It has recently been shown that [22], in the limit of small supersaturation, the driving force for nucleation (in J mol1) in multicomponent alloys can be written in dyadic notation as:
Gc DC1 DGn ¼ ðDCÞ T
ð4Þ
where Gc is the Hessian of the molar free energy Gcm of phase c evala. uated at equilibrium compositions of element i in a C i 0 c c c c c 1 and DC ¼ C C for i = 2. . .N with C the matrix ¼C C DC i
The model developed in this work aims at describing the precipitation kinetics including nucleation, growth and coarsening regimes. It is based on a mean field approach (i.e. the matrix composition is uniform). The model is based on the evolution of particle classes during time using a so-called Lagrange-like approach in order to track the particle size distribution. Size classes are considered. In this approach, at each time step, a new class is created with its own radius, number density and precipitate concentration. This corresponds to the nucleation regime. The various classes related to different dates of birth evolve at each time step (growth or coarsening). The number density, precipitate sizes, concentrations and volume fraction of precipitates are calculated at each time step. In the Lagrange-like approach, the number density of each class is kept constant except when the class disappears because of coarsening. The precipitate radius and volume fraction evolve over time in each class. The evolution of these parameters and of the concentration in both matrix and precipitates are governed by nucleation, growth and coarsening that are treated simultaneously in this approach.
ð2Þ
i
i
i
i
composition of element i (hereafter, bold symbols refer to vectors or tensors). The hessian of the molar free energy of a given phase 2
expression of the driving force for nucleation (Eq. (4)) has been obtained by expressing the equilibrium condition for a critical nucleus with a curved interface in chemical equilibrium with the matrix and by expanding chemical potentials in the close vicinity of the equilibrium compositions. 2.2. Growth law The growth rate is determined by the diffusion field surrounding the particle within the approximation of local equilibrium at the particle/matrix interface. Since this is a multicomponent alloy, shift from equilibrium values is not only due to capillarity, as a result the interfacial compositions are also determined by the diffusion process. It has been shown that the growth rate is not only governed by the supersaturation but also depends on thermodynamics quantities [19]. Assuming small supersaturations and same molar volume for both phases, the growth rate in a multicomponent alloy can be written as [19]: 0
The nucleation rate of precipitation J (m using the classical nucleation theory [21]:
J ¼ n0 ð1 f c0 ÞZb exp
3
DG t 1 exp kB T s
1
s
) is calculated
ð1Þ
q
@ Gm q is the curvature and is defined by Gqm;ij ¼ @C . This analytical j @C i
c dR 1 T Gc DC1 2rV m ¼ ðDCÞ dt RðDCÞ R T M1 DC
! ð5Þ
where R is the radius, M is the mobility matrix. It can be shown that if partial molar volumes are assumed to be equal, we have D = MGc (the demonstration of this expression in given in Appendix B) with
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D the diffusion matrix in the volume-fixed frame of reference. We thus have M1 = GcD1. The precipitate will grow or will shrink, when R > RC or R < RC respectively, RC being the critical radius for which dR/dt = 0 and defined by: 0
Rc ¼
2rV cm T Gc DC1 ðDCÞ
ð6Þ
One should notice that this radius is equal to the critical radius for nucleation. 2.3. Mass balance The temporal evolution of the matrix concentration is calculated using the mass balance equation: 0
C ci ¼
c0
C 0i C c f t
ð7Þ
c0
1 ft c0
where f t ¼
P
0
j j f c0
is the overall volume fraction of precipitates.
C c is calculated by averaging concentrations over all precipitate classes of the system. We assume that there is no concentration gradient inside precipitates so that the interfacial composition is the precipitate composition. Defining the interfacial compositions requires the development of the Gibbs–Thomson equation. The composition of c0 for a precipitate of radius R is given by [19]:
c0 þ Gc01 Gc DC1 D1 DC R dR Cc0 ¼ C dt
ð8Þ
The concentrations of c0 are a function of both the driving force for precipitation and the radius of the precipitate. Eq. (8) is the generalization of the Gibbs–Thomson equations in a multicomponent alloy to first order in supersaturation and 1/R [19]. It is worth mentioning that even if the matrix composition at the interface is not explicitly used in the present work, it could be defined by the same way (i.e. by the development of the Gibbs–Thomson equation [19]). At the interface of the critical cluster (R = RC), it is equal to the far-field composition. The particle size distribution model developed here is based on equations described above and does not need a coupling with Thermo-Calc software for the determination of interfacial compositions. This is a major advantage compared to Rougier’s work [20]. Indeed, only free energy curves are needed to compute growth, coarsening and compositions in both phase equations. At each time step, nucleation rate (Eq. (1)) is calculated as a function of matrix supersaturation. A new generation of nuclei is then created. This population is characterized by a volume fraction R and a radius with a given number density given by N Vj ¼ Jdt. During the same time step, precipitates generated earlier (i.e. appeared at shorter time) evolve. The growth rate is calculated for each population of precipitates that nucleates at different time. Their radius, volume fraction and their compositions evolve over time. The total volume density and the total volume fraction is calculated by summing those of each population. The mean radius of precipitates is given by:
¼ R
P
j j j R NV t NV
with N tV ¼
ð9Þ P
j j NV
was used to simulate isothermal heat treatment of Ni–6.5 at%Al– 9.5 at%Cr, Ni–5.2 at%Al–14.2 at%Cr and Ni–7.5 at%Al–8.5 at%Cr (labeled hereafter as respectively 6.5Al9.5Cr, 5.2Al14.2Cr and 7.5Al8.5Cr) at 600 °C during 8 106 s in order to compare with atom probe tomography experiments performed by Sudbrack et al. [23], Mao et al. [2,3] and Booth-Morrison et al. [24]. During these studies, they have characterized the formation of c0 precipitates during annealing. The Gibbs free energy curves for both phases were computed using the compound-energy formalism, widely used in the CALPHAD approaches. Details are given in Appendix A. The free energies were required to compute both the driving force for nucleation and the equilibrium compositions, as well as the diffusivities. The diffusion tensor given in Table 1 was calculated using the expression D = MGc, with M calculated from [2], see Appendix B. The influence of the diffusion matrix and more specifically the role of the non-diagonal diffusion coefficients on the precipitation have both been studied by Rougier et al. [20]. They have shown that the implementation of cross-diffusion has a significant influence on the overall precipitation kinetics and on all parameters linked to this precipitation phenomenon. They have concluded on the importance of the cross-diffusion coefficients on the evolution of c0 phase parameters. In the present work the diffusion matrix is non-symmetric and therefore takes into account the diffusion couplings. The best-fit value of interfacial energy used in our model has been slightly changed for the 3 nominal compositions. The interfacial energy is the only free parameter that is adjusted in our model. The interfacial energies were adjusted so that to get the best fit of the experimental Nv(t) curves. Besides, the interfacial energy is known to be affected by interfacial phase compositions (proportional to the square function of the composition gradient). It is therefore expected to be different for the three alloys considered in this work. c0 are given in Fig. 1(a–c) The temporal evolution of N t , f and R V
t
for the three nominal compositions. The model fits well the experiments (the order of magnitude of time and number density). The total number density of precipitates (Fig. 1a) increases until the end of nucleation (1 h, 4 h and 4 h respectively for (the nominal compositions) 7.5Al8.5Cr, 5.2Al14.2Cr and 6.5Al9.5Cr). At each time step, new populations of precipitates are created in this nucleation regime. When nucleation stops, the total number density of precipitates remains constant for a while. Actually, during this time, there is neither nucleation nor Ostwald ripening (or at least not sufficient) to allow the total shrinking of some precipitates populations. The total number density remains almost constant. The start of the number density decreasing (4 h, 20 h and 30 approximately respectively for 7.5Al8.5Cr, 5.2Al14.2Cr and 6.5Al9.5Cr) corresponds to the time when the first populations of precipitates disappear. It continues to decrease as the number of populations disappearing increases and because there is no more nucleation. One can notice that the difference between experiments and simulation results is more pronounced for shorter ageing times. This may be first explained by uncertainties in atom probe tomography experiments. Indeed, as we can see in Fig. 1b precipitate sizes for this time range are very small (of few nm). The mean radius (Fig. 1b) is observed to increase continuously with time with a good agreement between experiments and simulations. The radius values are of the same order of magnitude and the slope values of R(t) are very close. During the nucleation
the total number density.
3. Results and discussion We have applied the model described above to model NiCrAl superalloys for which experimental data are available. The model
Table 1 Parameters used for NiCrAl alloys. n0 (m3)
X (m3) 28
9.14 10
1.09 10
29
Vm (m3 mol1)
a (m)
6.59 106
3.4 1010
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Fig. 1. Evolution of the number density (a), mean radius (b) and precipitate volume fraction (c) as a function of time for alloys 1–3 in simulations (straight red line) and APT experiments (black dots) [2,23,24]. Results of LKMC on alloy 2 are represented by gray triangles. (1: Ni–6.5 at%Al–9.5 at%Cr, 2: Ni–5.2 at%Al–14.2 at%Cr, 3: Ni–7.5 at%Al–8.5 at %Cr). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
regime, the critical radius of new precipitate population increases with time because of the decreasing supersaturation and also because almost all populations previously created grow. During growth and coarsening regimes, even if there is a shrinking of some populations, precipitate populations that are growing affect more strongly the mean radius. The Fig. 2 shows the evolution of
nucleation critical radius and mean radius over time. This result indicates that the values of these radii are different until the pure growth regime is finished. Once the nucleation critical radius reaches the mean radius, the regime reached is that of pure coarsening. Nucleation critical radius is thus equal to coarsening critical radius. They are also equal to mean radius of precipitates in the system (see Table 2). The volume fraction of precipitates (Fig. 1c) is observed to increase until it reaches its equilibrium value. Note that experiments for 5.2Al14.2Cr result in an asymptotic volume fraction greater than equilibrium volume fraction as derived from thermodynamic database [23]. We have calculated the driving force for nucleation for each nominal composition. We found that the alloy 7.5Al8.5Cr has the c0
Fig. 2. Evolution of the mean radius of precipitate (straight line) and of the critical radius for nucleation (dotted line) as a function of time in the 6.5Al9.5Cr alloy.
highest driving force for nucleation at t = 0 (largest f ) and the alloy 6.5Al9.5Cr the lowest. This is not surprising in view of the kinetics obtained. The highest the nucleation driving force, the highest the nucleation rate, the faster the end of nucleation and the faster the decreasing of first precipitate classes. The criterion used in the model to stop nucleation stage was a nucleation rate below to 0.1% of the maximum nucleation rate (we have checked that lower values of this threshold did not affect results). Numerical results are compared to lattice kinetic Monte Carlo (LKMC) simulations. Both thermodynamic and kinetic parameters
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Table 2 Equilibrium concentrations in both matrix and precipitates, Diffusion matrix and interfacial energy for the three different nominal composition of NiCrAl studied in this work (1: 6.5Al9.5Cr, 2: 5.2Al14.2Cr, 3:7.5Al8.5Cr). Alloy
c C Al
c C Cr
c0 C Al
c0 C Cr
DAlAl (1021 m2 s1)
DAlCr (1021 m2 s1)
DCrAl (1021 m2 s1)
DCrCr (1021 m2 s1)
r (mJ m2)
1 2 3
5.12 3.67 5.48
9.81 14.93 8.87
17.45 16.1 17.74
7.04 8.93 6.64
11.8 18.1 11.6
5.62 8.09 5.65
2.73 3.12 2.81
1.88 1.64 1.92
31 32 32
for the LKMC simulations are taken from [1]. The comparison is shown in Fig. 1a–c for the 5.2Al14.2Cr alloy. Both numerical methods are in very good agreement with experiments. Whereas the LKMC simulations also give the evolution of the precipitates morphology during the phase transformation, only the early stages are accessible since LKMC simulations are time consuming. Our new approach has the great advantage to predict kinetics over much longer times and to explicit the physical origin of kinetic effects observed. Fig. 3 pictures the evolution of phase concentrations as a function of annealing time in the 6.5Al9.5Cr alloy. It is worth mentioning that the concentration in c0 phase is the concentration for It should be noted that concentrations in precipitates do R ¼ R. not remain constant. For long ageing time, atomic fractions tend to equilibrium values given by the phase diagram. Concentrations in matrix evolve only slightly in the early stages up to the end of nucleation. The supersaturation is not yet affected. Note the nonmonotonic precipitate concentration evolution during nucleation and just after. Concentrations cross equilibrium values. Recent experiments on model NiCrAl investigated with atom probe tomography have shown that the Cr and Al precipitate compositions decrease with time [24]. This decreasing is also observed in our simulations but only after nucleation. Let us discuss the origin of this non-monotonic temporal evolution of the Cr composi0
tion C cCr .
0
First C cCr increases during nucleation (Fig. 3), then decreases when growth starts and finally very slightly increases during the coarsening regime. It should be kept in mind that Fig. 3 gives the which is not the mean conconcentration in a precipitate of size R, centration. Note that we have compared the evolution of the mean concentration and the evolution of the concentration in a precipi We find the same trend. tate of size R. The specific behavior of the temporal precipitate concentration evolution can be understood and explained from (Eq. (8)) that gives the composition of a c0 precipitate of size R. In this equation,
Fig. 3. Evolution of concentrations in both phases c (green and blue curves) and c0 (red and black curves) as a function of the time in the 6.5Al9.5Cr alloy (Al: red and blue curves, Cr: green and black curves). The inset is a zoom of the evolution of Cr concentration in phase c0 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
three parameters are time dependent: DC1 , the supersaturation, R, the radius and dR/dt, the growth rate. Thus, the term in brackets in Rd R=dt (Eq. (8)): DC1 D1 DC drives the evolution of the precipi R=dt. tate composition, with a competition between DC1 and Rd Rd R=dt Fig. 4 represents the temporal evolution of DC1 D1 DC for Cr. R=dt drives During nucleation, DC1 is almost constant, so that Rd c0 1 1 the evolution of C Cr , i.e. the increasing of DC D DCRdR=dt (first increases because of part of the curve in Fig. 4). During this stage, R
growth of precipitates (Fig. 2). Moreover, because Rc, the critical radius for nucleation, evolves very slightly during nucleation, the and Rc increases (Fig. 2). As a consequence, difference between R is negative for increases (note that DC dR=dt (Eq. (5) with R ¼ R) Cr). Once nucleation is completed, pure growth is observed for a while, (justified later in the manuscript). The evolution of the supersaturation is more pronounced and the critical radius for nucleation increases. Thus, the mean radius of precipitates tends to the critical radius of nucleation with increasing time. It follows 0
a decrease in C cCr . Finally, once the pure coarsening stage is reached, the mean radius is equal to the critical radius, i.e. dR/dt = 0, and the supersat0
uration drives the temporal evolution of C cCr . The Al composition shows a similar trend, but in the opposite direction. 3.1. Asymptotic behaviors One can study the asymptotic behavior for long ageing times. The long-term behaviors of the average precipitate radius, number density and concentrations in both phases are represented on Fig. 5 and Fig. 6. It has been recently shown by Philippe and Voorhees [19] that the temporal exponents for these three parameters are identical to the binary limit (the amplitudes are different). The number density has been found to vary linearly with t1 as shown by Philippe and Voorhees:
R dR as a function of the time in 6.5Al9.5Cr alloy. Fig. 4. Evolution of DC1 D1 DC dt The red dashed line indicates the time of the end of nucleation (t = 12,860 s). (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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controlled by the nucleation rate. One can observe that normalized density of the different classes first increases (until the nucleation peak) and then decreases. The populations that nucleate last have a smaller reduced radius (i.e. the evolution of the nucleation critical radius is slower than the evolution of radius with growth law). This analysis will be further supported in the following paragraph. With the increasing annealing times, the distribution tends towards the well-known LSW distribution [12,13]. This again shows the validity of the model. In order to understand which precipitates shrink or grow, we analyze the raw size distributions. We find that the population of precipitates that nucleate first keeps the largest mean radius throughout the kinetics. In addition, it is worth mentioning that until the end of nucleation, there is no coarsening of precipitates
Fig. 5. Asymptotic behavior of the number density (a) and mean radius (b) as a function of t1 and t respectively as given by the simulation (red curves) and by the asymptotic laws of coarsening [19] (black curves) in 6.5Al9.5Cr. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
c0
f eq Nv ðtÞ ffi 0:21 t1 K where K ¼
c0
8V m r
M1 DC 9ðDCÞ T
ð10Þ .
The mean radius of precipitates varies linearly with t:
3 ðtÞ ¼ Kt R
ð11Þ
Concentrations in both matrix and precipitates evolve respectively as: 1
T 1 3 2 ½ðDCÞ M DC t 13 c þ ð3rV c0 Þ3 DC C ðtÞ ¼ C m T Gc DC ðDCÞ c
ð12Þ
and c01 c 1 2 0 c0 þ ð3rV c0 Þ3 ½ðDCÞ T M1 DC 3 G G DC t 13 Cc ðtÞ ¼ C m T Gc DC ðDCÞ
ð13Þ
(Fig. 5), Cc and Cc0 (Fig. 6) in 6.5Al9.5Cr, are found, when N tV , R time tends to infinity, to fall well on the theoretical predictions of Philippe and Voorhees [19]. These results validate the approach used in this model and confirm its relevance for the description of precipitation kinetics in multicomponent alloys. Even with a model dealing at the same time with nucleation, growth and coarsening, we find for long annealing time, the classical behavior of pure coarsening. Fig. 7 shows the normalized size distributions at different ageing times. These normalized distributions are also normalized as a function of the mean radius of precipitates. During pure coarsening regime, precipitate populations with a reduced radius R½ ¼ R=R lower than 1 are shrinking and those with a reduced radius greater than 1 are growing. During nucleation, the distribution is
Fig. 6. Asymptotic behavior of the composition in both matrix and precipitates (for elements Al and Cr) as a function of t1/3 as given by the simulation (red curves) and with the asymptotic laws of coarsening [19] (black curves) in 6.5Al9.5Cr. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
M. Bonvalet et al. / Acta Materialia 100 (2015) 169–177
175
Fig. 7. Normalized size distributions for different ageing times in 6.5Al9.5Cr. The radius R[] is normalized as a function of the mean radius of precipitates. D[] is the normalized number density so that the total area under the curve D[](R[]) is equal to unity. Circles correspond to distribution during nucleation regimes and stars correspond to times after the end of nucleation.
in this system. All the populations of precipitates created grow. Moreover, once nucleation is finished, pure growth of precipitates is observed for a while. This is explained by the fact that the supersaturation does not evolve a lot until this time (Eq. (5)) so the growth rate is positive for all precipitates classes. As shown in the ternary phase diagram given in Fig. 8, phase compositions do not follow the equilibrium tie-line during transformation, as it was reported experimentally [2,3,23,24]. However, as already mentioned the approach does not predict the global decrease of the average Cr and Al concentrations in c0 exhibited in atom probe tomography experiments. In the present approach, the precipitates are assumed to be in local equilibrium with the matrix and thus, their compositions are set by both capillarity and the diffusion process, as shown by the Gibbs–Thomson equation (Eq. (8)). Such chemical equilibrium leads to a complex temporal evolution of the average Cr concentration in c0 during the phase transformation, but does not lead to a monotonic decrease. As a consequence, the present study suggests that nonequilibrium effects, potentially due to the low Cr mobility, might be responsible for the decrease in Cr composition. Our model, however, does not permit to investigate such phenomena.
and f c0 as a function of the ageing Fig. 9. Influence of the interfacial energy on N TV , R time.
3.2. Influence of interfacial energy The model gives interesting features on the influence of the interfacial energy r on the precipitation in multicomponent alloys. We perform simulations for different values of r in order to determine the influence of interfacial energy on the overall kinetics. An important feature is that interfacial energy as a strong influence on kinetic pathway of precipitation. It is the only parameter that we can modify in our model but it is very sensitive. Fig. 9 shows its and f c0 as a function of ageing time. We use four influence on N t , R V
values of interfacial energy (30, 31, 32 and 33 mJ m2). One should notice that the smaller r, the bigger the maximum number density value and this one is reached for smaller time (smaller nucleation barrier, see Eqs. (2) and (4)). Actually, the nucleation rate is shifted to longer times when the interfacial energy is increased. Concomitantly, the volume fraction of precipitates increases faster during the nucleation and growth regimes, as the mean radius. However before entering the pure coarsening regime, mean radius is bigger for smaller r, due to a slow decrease in the supersaturation.
4. Conclusions
Fig. 8. Isothermal section of the ternary phase diagram of NiCrAl with compositional trajectories in the three alloys. The arrow indicates the direction of increasing time. Black circles, light gray circles, dark gray circles correspond respectively to 5.2al14.2Cr, 6.5Al9.5Cr and 7.5Al8.5Cr alloys.
A new model concomitantly dealing with nucleation, growth and coarsening has been developed for predicting precipitation kinetics in multicomponent alloys. Nucleation has been implemented using Zeldovich theory so that to include both regression effects and incubation time. Growth and coarsening has been modeled using the recently developed growth and coarsening laws in multicomponent alloys that account for capillarity, mass balance at the interface matrix-precipitate and diffusion-flux couplings. This new approach does not require any coupling with external
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CALPHAD modules, as a result, it guaranties remarkable computation efficiency as compared to past works. Numerical results have been confronted to atom probe tomography (APT) experiments on the NiCrAl system and to lattice kinetics Monte Carlo (LKMC) simulations. The results have been found in very good agreement with both APT experiments and LKMC simulations. As already shown, predictions have been shown to be very sensitive to the interfacial energy, particularly during the nucleation and growth regimes. The temporal evolution of the mean precipitate composition has been found to be non-monotonic during the phase transformation due to a complex interplay between capillarity and diffusion processes. The widespread availability of both thermodynamic and mobility databases makes this new model very suitable for heat treatment optimization and material design.
where higher order constituent array IZ have been introduced. IZ must not contain a component more than once for the same sublattice. I1 specifies that one sublattice contains two components but only one on the remaining sublattice. The interaction parameters LIZ are described using a Redlich–Kister polynomial for binary systems [27]:
LI1 ðYÞ ¼
X b ðysi0 ysi00 Þ :b GI1
i0 and i00 are two components occupying the sublattice s according to I1. b is the degree of the parameter bGI1 . As an illustration, the molar Gibbs energy of an A–B system having one sublattice is:
Gm ¼ yA :0 GA þ yB :0 GB þ RTðyA ln yA þ yB ln yB Þ þ yA yB LA:B Acknowledgment The authors are grateful for the financial support of CARNOT ESP Institute. Appendix A
with
Sideal m R
¼
s
ðA:1Þ
the ideal entropy of mixing:
X X as ysi ln ysi
ðA:2Þ
i
R is the gas constant, the superscript s denotes the sublattice under consideration. ysi is the site fraction of component i on sublattice s. as is the number of sites on the sublattice s per mole of formula units of the phase. The previous expression for the entropy assumes random mixing on each sublattice of the phase under consideration. The first sum is therefore over all sublattices and the second overall constituents on each sublattice. Gref m represents the state of reference for the Gibbs energy and can be expressed in a general form as:
Gref m ¼
X PI0 ðYÞ:0 GI0
ðA:3Þ
I0
Here was introduced the concept of constituent array. A constituent array I specifies one or more constituent i on each sublattice. The constituent arrays can be of different orders. The zeroth order refers to one constituent on each sublattice. Thus in the state of reference of the Gibbs energy, I0 is a constituent array of zeroth order specifying one constituent in each sublattice, PI0 ðYÞ represents the corresponding product of constituent fractions. 0 GI0 represents the Gibbs energy of the compound defined by I0. GEm is the excess Gibbs energy. While Gref m describes the interactions between neighboring atoms in different sublattices, GEm is mainly composed of interaction energies between different components in the same sublattice. The excess Gibbs energy can be written:
XX GEm ¼ PIZ ðYÞ:LIZ Z>0 IZ
In this case, the site fraction yi are also the molar fraction of component i in the phase under consideration. In the above expression, a component array is written with a colon ‘‘:” between components in the same sublattice. With two sublattices, the Gibbs energy becomes:
þRT½a1 ðy1A ln y1A þ y1B ln y1B Þ þ a2 ðy2A ln y2A þ y2B ln y2B Þ
The free energy densities of phases are described in the compound-energy formalism (CEF), which is widely used in the CALPHAD methods. The main concepts of the CEF are briefly reviewed in this section. Sundman and Agren [27] extended the regular solution model to phases with several components and sublattices. In the CEF, the molar Gibbs energy is given by:
Sideal m
ðA:6Þ
Gm ¼ y1A y2A :0 GA;A þ y1A y2B :0 GA;B þ y1B y2A :0 GB;A þ y1B y2B :0 GB;B
A.1. Coupling the model to physical databases
ideal Gm ¼ Gref þ GEm m TSm
ðA:5Þ
b¼0
ðA:4Þ
þy1A y1B y2A LA:B;A þ y1A y1B y2B LA:B;B þ y1A y2A y2B LA;A:B þ y1B y2A y2B LB;A:B
ðA:7Þ
þy1A y1B y2A y2B LA:B;A:B where a component array is written with a comma ‘‘,” between components in different sublattices. The last term in Eq. (A.7) is included in Eq. (A.4) for Z = 2. Despite the apparent complexity of the CEF, the extended regular solution models make the formalism suitable for computer applications and have been widely used in the CALPHAD method. In the CEF, the free energies are expressed in terms of site fractions. For a disordered phase, site fractions are equivalent to atomic fractions of the components in the phase under consideration. This is not the case for phases involving chemical ordering. In the present approach, it is assumed that chemical ordering is much faster than diffusion meaning that site fractions take their equilibrium values. The free energy of an ordered phase needs therefore to be minimized with respect to the site fractions with the constraint of mass conservation on the atomic fractions. Thus, as each set of atomic fractions corresponds a set of equilibrium site fractions, which is used to compute the free energy of the phase under consideration. Such minimization is performed using a downhill simplex method. As the procedure can be time consuming, the minimization of the free energy of an ordered phase is done prior the simulations and look-up tables for the free energy, as well as for its Hessian, are built and then read during the simulation. Such procedure does not require interfaces to Thermocalc or DICTRA software to calculate Gibbs energies or the other thermodynamic quantities. Note that in a multicomponent alloy, the use of parabolic approximations of the free energies is not advisable even for a ternary alloy due to complicated curvatures of the free energies. In this work, a published database [28] has been used for the computation of the thermodynamic quantities for the disordered phase (matrix c) and the ordered precipitates (c0 , L12) in the Ni–Cr–Al system. Appendix B B.1. Diffusion matrix in a multicomponent alloy Diffusion fluxes in the linear theory are linear functions of driving forces:
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M. Bonvalet et al. / Acta Materialia 100 (2015) 169–177 n X J k ¼ Lkj rlj
ðB:1Þ
j¼1
This expression is given in the lattice-framed of reference. Lkj are phenomenological parameters. rlj is the chemical potential gradient of component j. In the volume-fixed frame of reference, P there is no net flow of volume ( k J k ¼ 0 where the k are the components of the system). Andersson and Agren [25] have shown, that if the frame of reference is the volume-fixed frame of reference, then the diffusion matrix is given by:
D1kj
¼ Dkj Dk1 ðV j =V 1 Þ
where Dkj ¼
Pn
00 @½li ðV i =V 1 Þl1 i¼1 Lki @C j
ðB:2Þ is the diffusion matrix in the lattice
frame of reference and 1 is the majority element of the system. V i is the molar partial volume of element i. L00ki are the modified Onsager coefficients (defined in the lattice frame of reference and expressed hereafter). It should be noticed in this equation that li for i ¼ 1:::n are function of ðC 1 ; :::; C n Þ. If molar partial volumes of each element are equals then:
Dkj ¼
n X @½li l1 L00ki @C j i¼1
ðB:3Þ
and
D1kj
¼ Dkj Dk1
@½li l1 @C j
ðB:4Þ
in Eq. (B.3) could be rewritten as a function of the hessian of
the molar free energy of a. It is well-known that [26]:
li ¼ l1 þ Gam;i
ðB:5Þ
li l1 ¼ Gam;i ¼
@Gam @C i
ðB:6Þ
These equations are relations for the so-called diffusion poteni l1 , one has to differentiate Gam;i tial [17]. In order to determine @½l@C j with respect to C j using the result:
@f @f @f @f ¼ ¼ @C j @C j C j þC1 ¼cst @C j @C 1
ðB:7Þ
where f is a function of ðC 1 ; :::; C n Þ and f is a function of ðC 2 ; :::; C n Þ. These two functions are linked by the definition:
f ðC 1 ; :::; C n Þ ¼ f ðC 2 ; :::; C n Þ Pn
ðB:8Þ
with C 1 ¼ 1 i¼2 C i . Eq. (B.8) implies that if C j varies, f and f will vary in the same P way with the condition related to the mass balance i C i ¼ 1 (i.e. C j þ C 1 ¼ cst in the ðC 1 ; :::; C n Þ space).
We thus have: a
@Gm;i @ 2 Gam @ @ ¼ ¼ ½l l1 ½l l1 @C 1 i @C j @C j @C i @C j i If we now replace
D1kj ¼
@ @C j
ðB:9Þ
½li l1 in Eq. (B.3), we found that:
n X @ 2 Gam L00ki @C j @C i i¼2
ðB:10Þ
Eq. (B.10) can be written in standard dyadic notation:
D ¼ L0 G
ðB:11Þ
which is often written as D ¼ MG. In this expression, the matrix L00 (or M) is determined with the relation [25]:
L00ki ¼
n X n X Vr Vj djk xk Ljr dir xi V V m m j¼1 r¼1
ðB:12Þ
where Ljr are Onsager coefficients and dmn is the Kronecker delta (dmn ¼ 1 when m ¼ n and 0 otherwise). Once again, in our model, we assumed that molar partial volumes are equal. In this study, Onsager coefficients were taken from reference [2]. References [1] C. Pareige, F. Soisson, G. Martin, D. Blavette, Acta Mater. 47 (1999) 1889. [2] Z. Mao, C. Booth-Morrison, C.K. Sudbrack, G. Martin, D.N. Seidman, Acta Mater. 60 (2012) 1871. [3] Z. Mao, C.K. Sudbrack, K.E. Yoon, G. Martin, D.N. Seidman, Nat. Mater. 6 (2007) 210. [4] J.S. Langer, A.J. Schwartz, Phys. Rev. A 21 (1980) 948. [5] R. Kampmann, R. Wagner, Decomposition of Alloys: the Early Stages, 1984, p. 91. [6] O.R. Myhr, Grong, Acta Mater. 48 (2000) 1605. [7] M. Perez, M. Dumont, D. Acevedo-Reyes, Acta Mater. 56 (2008) 2119. [8] M. Serriere, C.A. Gandin, E. Gautier, P. Archambault, M. Dehmas, Mater. Sci. Forum (2002) 396. [9] P. Maugis, M. Gouné, Acta Mater. 53 (2005) 3359. [10] J.D. Robson, Acta Mater. 52 (2004) 1409. [11] J.D. Robson, P.B. Prangnell, Acta Mater. 49 (2001) 599. [12] I.M. Lifshitz, V.V. Slyosov, J. Phys. Chem. Solids 19 (1961) 35. [13] C. Wagner, Z. Elektrochem. 65 (1961) 581. [14] C.J. Kuehmann, P.W. Voorhees, Metall. Mater. Trans. A 27 (1996) 937. [15] S. Bjorklund, L.F. Donaghey, M. Hillert, Acta Metall. 20 (1972) 867. [16] V.V. Slyosov, V.V. Sagalovich, Sov. Phys. Solid State 17 (1975) 974. [17] A. Umantsev, G.B. Olson, Scripta Mettal. 29 (1993) 1135. [18] J.E. Morral, G.R. Purdy, Scripta Mettal. 30 (1994) 905. [19] T. Philippe, P.W. Voorhees, Acta Mater. 61 (2013) 4237. [20] L. Rougier, A. Jacot, C.A. Gandin, P. Di Napoli, P.T. Théry, D. Ponsen, V. Jacquet, Acta Mater. 61 (2013) 6396. [21] J. Zeldovich, J. Exp. Theor. Phys. (USSR) 2 (1942) 525. [22] T. Philippe, D. Blavette, P.W. Voorhees, J. Chem. Phys. 141 (2014) 124306. [23] C.K. Sudbrack, R.N. Noebe, D.N. Seidman, Acta Mater. 55 (2007) 119. [24] C. Booth-Morrison, Y. Zhou, R.N. Noebe, D.N. Seidman, Phil. Mag. 90 (2010) 219. [25] J. Andersson, J. Agren, J. Appl. Phys. 72 (1992) 1350. [26] M. Hillert, Phase Equilibria, Phase Diagrams and Phase Transformations, Their Thermodynamic Basis, Cambridge University Press, 2008. [27] B. Sundman, J. Agren, J. Phys. Chem. Solids 42 (1981) 297. [28] N. Saunders, Proc. Int. Symp. Super. (1996) 101.
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Acta Materialia journal homepage: www.elsevier.com/locate/actamat
Modeling of precipitation kinetics in multicomponent systems: Application to model superalloys M. Bonvalet ⇑, T. Philippe, X. Sauvage, D. Blavette Normandie Université, Groupe de Physique des Matériaux (GPM), UMR CNRS 6634 BP 12, Avenue de l’Université, 76801 Saint Etienne du Rouvray, France
a r t i c l e
i n f o
Article history: Received 4 March 2015 Revised 23 July 2015 Accepted 15 August 2015
Keywords: Precipitation kinetics Modeling Multicomponent systems Ni–Cr–Al Superalloys
a b s t r a c t A new general model dealing with nucleation, growth and coarsening simultaneously has been developed for the simulation of precipitation in non-dilute multicomponent alloys. Nucleation is implemented using the Zeldovich theory that includes regression effects. Growth and coarsening are modeled using the recently developed growth law in multicomponent alloys that accounts for capillarity, mass balance at the interface matrix-precipitate and diffusion-flux couplings. Numerical results are confronted to atom probe tomography (APT) experiments on model NiCrAl superalloys and to rigid lattice kinetics Monte Carlo (LKMC) simulations and are found in very good agreement with both APT experiments and LKMC simulations. We emphasize this work on the evolution of phase concentrations. The temporal evolution of the mean precipitate composition is found to be non-monotonic during the phase transformation, and phase composition does not follow the tie line due to a complex interplay between capillarity and diffusion process. The widespread availability of both thermodynamic and mobility databases makes this new model very suitable for material design. Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
1. Introduction Modeling precipitation kinetics in multicomponent alloys is both of utmost interest from an academic point of view but also for the optimization of thermal treatments and alloy compositions in terms of mechanical or structural properties. Numerical experiments based on Kinetic Monte Carlo simulations are known to be very effective. Such approaches were applied successfully to precipitation kinetics in model Ni-based superalloys [1–3]. Numerical explicit models of precipitation where nucleation, growth and coarsening theories are simultaneously implemented are attractive alternative approaches [4] due to the widespread availability of both thermodynamic and kinetic databases that make such approaches quantitative. Models based on the particle size distribution of precipitates (PSD) were shown to be quite accurate to study precipitation kinetics [4]. This paper describes a model based on this approach. Kampman and Wagner [5] were the first to use a PSD model for the precipitation in dilute binary alloys. The influence of the curvature on the composition of precipitates was implemented (i.e. the Gibbs Thomson effect). In their numerical resolution, they used a Lagrange-like approach for the size distribution. In such approach, ⇑ Corresponding author. E-mail address: [email protected] (M. Bonvalet). http://dx.doi.org/10.1016/j.actamat.2015.08.041 1359-6454/Ó 2015 Acta Materialia Inc. Published by Elsevier Ltd. All rights reserved.
it is the precipitate size in each population N(R) that is computed, the initial number of precipitates being controlled by nucleation process. In each population of precipitates are included precipitates that nucleate during the same time interval (t, t + dt). It is assumed that these precipitates (defined by a radius and a composition) evolve in the same way over time. Myhr and Grong [6] also used a PSD for the precipitation in Al–Mg–Si but employing an Euler-like approach. Therein the size classes of precipitates were fixed and the number density in each class evolved over time. The size distribution is not managed in the same way with both approaches (Lagrange and Euler). One approach (Lagrange) deals with evolution of mean radius in each class while the other (Euler) deals with evolution of number density of particles in each class of defined interval radius. Perez et al. [7] compared these two approaches and concluded to their equivalence. PSD models were also used in various contexts [8–11], it was shown that nucleation, growth and coarsening generally superimposes and therefore cannot be treated, in general, separately. It was also found that interfacial energy is a key parameter for such approach, having a strong influence on simulation results [10,11]. In multicomponent alloys, the description of precipitation processes is much more complex. In the past few years, a few models dealing with coarsening in multicomponent alloys were developed. The first notable coarsening theory in binary alloy was given by Lifshitz and Slyozov and Wagner [12,13]. They used the
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M. Bonvalet et al. / Acta Materialia 100 (2015) 169–177
Gibbs–Thompson equation so that to determine interfacial compositions. However, in multicomponent system, local equilibrium is no longer sufficient do determine interfacial compositions. Mass balance equation must be taken into account. Kuehmann and Voorhees [14] were the first to describe Ostwald ripening in a ternary alloy but they did not include the influence of off-diagonal diffusion coefficients. Latter, Bjorklund et al. [15], Slyosov and Sagalovich [16], Umantsev and Olson [17], Morral and Purdy [18] developed different approaches for the coarsening theory each time using different assumptions (dilute solutions, ideal solutions, equilibrium precipitate composition, diagonal diffusivities). Recently, Philippe and Voorhees [19] proposed a description of Ostwald ripening in multicomponent alloys removing restrictions of past works based on both thermodynamic and atomic mobility databases. Recent experiments on model Ni–Cr–Al superalloys have shown that during precipitation the compositional trajectory of the coarsening phase does not follow the equilibrium tie-line [2,3]. Rougier et al. have recently developed a PSD model for the simulations of c0 precipitation in NiCrAl [20]. They have used a multicomponent diffusion model in order to calculate the growth rate. They have also included off-diagonal diffusivities. However, their model has been fully coupled with CALPHAD for the calculation of nucleation driving force and interfacial compositions. This coupling is very expensive computationally. In contrast, we propose here a general model for precipitation in multicomponent alloys using the new generalized coarsening theory of Philippe et al. [19]. As a main advantage, this new approach does not need any coupling with external CALPHAD modules, and therefore guaranties remarkable computation efficiency. Our approach is validated through an application of this model to Ni-based superalloys for which diffusion and thermodynamic data bases are available. Numerical results are compared with both experiments and lattice kinetics Monte Carlo (LKMC) simulations.
2. Governing equations
where n0 is the density of potential nucleation sites (m3), f c0 is pffiffiffiffiffiffiffiffi pffiffiffiffi the volume fraction of precipitates, Z ¼ X r=ð2pR2 kB T Þ is the 3 Zeldovich factor, with X the atomic volume (m ) of both phases c and c0 (assumed to be equal), R⁄ is the critical radius of nucleation (m), r is the interfacial energy (J m2), T the temperature (K) and kB the Boltzmann constant (J K1). b ¼ ð4pR2 =a4 Þ minðDcii C cii Þ is the rate of attachment of solute atoms on a critical precipitate (s1), with a the lattice parameter (m), Dcii and C cii are respectively the diagonal diffusion coefficient (m2 s1) and the average concentration in c phase of element i:
DG ¼
16p r3 3 Dg 2n
is the nucleation barrier (J) with Dgn the nucleation driving force (J m3). s ¼ pb2 Z2 is the Zeldovich incubation time. It is related to the fact that a nucleus enters the growth stage only when thermal fluctuations will no more involve the partial dissolution of nuclei [21]. Indeed, regressions may occur for nuclei with size close but larger than the critical size R⁄. In this theory, the size for stable nuclei enter their growth stage is expressed as:
1 R ¼R þ 2 0
2.1. Nucleation
rffiffiffiffiffiffiffiffi kB T
ð3Þ
pr
where the critical radius for nucleation is given by R* = 2r/Dgn. It has recently been shown that [22], in the limit of small supersaturation, the driving force for nucleation (in J mol1) in multicomponent alloys can be written in dyadic notation as:
Gc DC1 DGn ¼ ðDCÞ T
ð4Þ
where Gc is the Hessian of the molar free energy Gcm of phase c evala. uated at equilibrium compositions of element i in a C i 0 c c c c c 1 and DC ¼ C C for i = 2. . .N with C the matrix ¼C C DC i
The model developed in this work aims at describing the precipitation kinetics including nucleation, growth and coarsening regimes. It is based on a mean field approach (i.e. the matrix composition is uniform). The model is based on the evolution of particle classes during time using a so-called Lagrange-like approach in order to track the particle size distribution. Size classes are considered. In this approach, at each time step, a new class is created with its own radius, number density and precipitate concentration. This corresponds to the nucleation regime. The various classes related to different dates of birth evolve at each time step (growth or coarsening). The number density, precipitate sizes, concentrations and volume fraction of precipitates are calculated at each time step. In the Lagrange-like approach, the number density of each class is kept constant except when the class disappears because of coarsening. The precipitate radius and volume fraction evolve over time in each class. The evolution of these parameters and of the concentration in both matrix and precipitates are governed by nucleation, growth and coarsening that are treated simultaneously in this approach.
ð2Þ
i
i
i
i
composition of element i (hereafter, bold symbols refer to vectors or tensors). The hessian of the molar free energy of a given phase 2
expression of the driving force for nucleation (Eq. (4)) has been obtained by expressing the equilibrium condition for a critical nucleus with a curved interface in chemical equilibrium with the matrix and by expanding chemical potentials in the close vicinity of the equilibrium compositions. 2.2. Growth law The growth rate is determined by the diffusion field surrounding the particle within the approximation of local equilibrium at the particle/matrix interface. Since this is a multicomponent alloy, shift from equilibrium values is not only due to capillarity, as a result the interfacial compositions are also determined by the diffusion process. It has been shown that the growth rate is not only governed by the supersaturation but also depends on thermodynamics quantities [19]. Assuming small supersaturations and same molar volume for both phases, the growth rate in a multicomponent alloy can be written as [19]: 0
The nucleation rate of precipitation J (m using the classical nucleation theory [21]:
J ¼ n0 ð1 f c0 ÞZb exp
3
DG t 1 exp kB T s
1
s
) is calculated
ð1Þ
q
@ Gm q is the curvature and is defined by Gqm;ij ¼ @C . This analytical j @C i
c dR 1 T Gc DC1 2rV m ¼ ðDCÞ dt RðDCÞ R T M1 DC
! ð5Þ
where R is the radius, M is the mobility matrix. It can be shown that if partial molar volumes are assumed to be equal, we have D = MGc (the demonstration of this expression in given in Appendix B) with
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M. Bonvalet et al. / Acta Materialia 100 (2015) 169–177
D the diffusion matrix in the volume-fixed frame of reference. We thus have M1 = GcD1. The precipitate will grow or will shrink, when R > RC or R < RC respectively, RC being the critical radius for which dR/dt = 0 and defined by: 0
Rc ¼
2rV cm T Gc DC1 ðDCÞ
ð6Þ
One should notice that this radius is equal to the critical radius for nucleation. 2.3. Mass balance The temporal evolution of the matrix concentration is calculated using the mass balance equation: 0
C ci ¼
c0
C 0i C c f t
ð7Þ
c0
1 ft c0
where f t ¼
P
0
j j f c0
is the overall volume fraction of precipitates.
C c is calculated by averaging concentrations over all precipitate classes of the system. We assume that there is no concentration gradient inside precipitates so that the interfacial composition is the precipitate composition. Defining the interfacial compositions requires the development of the Gibbs–Thomson equation. The composition of c0 for a precipitate of radius R is given by [19]:
c0 þ Gc01 Gc DC1 D1 DC R dR Cc0 ¼ C dt
ð8Þ
The concentrations of c0 are a function of both the driving force for precipitation and the radius of the precipitate. Eq. (8) is the generalization of the Gibbs–Thomson equations in a multicomponent alloy to first order in supersaturation and 1/R [19]. It is worth mentioning that even if the matrix composition at the interface is not explicitly used in the present work, it could be defined by the same way (i.e. by the development of the Gibbs–Thomson equation [19]). At the interface of the critical cluster (R = RC), it is equal to the far-field composition. The particle size distribution model developed here is based on equations described above and does not need a coupling with Thermo-Calc software for the determination of interfacial compositions. This is a major advantage compared to Rougier’s work [20]. Indeed, only free energy curves are needed to compute growth, coarsening and compositions in both phase equations. At each time step, nucleation rate (Eq. (1)) is calculated as a function of matrix supersaturation. A new generation of nuclei is then created. This population is characterized by a volume fraction R and a radius with a given number density given by N Vj ¼ Jdt. During the same time step, precipitates generated earlier (i.e. appeared at shorter time) evolve. The growth rate is calculated for each population of precipitates that nucleates at different time. Their radius, volume fraction and their compositions evolve over time. The total volume density and the total volume fraction is calculated by summing those of each population. The mean radius of precipitates is given by:
¼ R
P
j j j R NV t NV
with N tV ¼
ð9Þ P
j j NV
was used to simulate isothermal heat treatment of Ni–6.5 at%Al– 9.5 at%Cr, Ni–5.2 at%Al–14.2 at%Cr and Ni–7.5 at%Al–8.5 at%Cr (labeled hereafter as respectively 6.5Al9.5Cr, 5.2Al14.2Cr and 7.5Al8.5Cr) at 600 °C during 8 106 s in order to compare with atom probe tomography experiments performed by Sudbrack et al. [23], Mao et al. [2,3] and Booth-Morrison et al. [24]. During these studies, they have characterized the formation of c0 precipitates during annealing. The Gibbs free energy curves for both phases were computed using the compound-energy formalism, widely used in the CALPHAD approaches. Details are given in Appendix A. The free energies were required to compute both the driving force for nucleation and the equilibrium compositions, as well as the diffusivities. The diffusion tensor given in Table 1 was calculated using the expression D = MGc, with M calculated from [2], see Appendix B. The influence of the diffusion matrix and more specifically the role of the non-diagonal diffusion coefficients on the precipitation have both been studied by Rougier et al. [20]. They have shown that the implementation of cross-diffusion has a significant influence on the overall precipitation kinetics and on all parameters linked to this precipitation phenomenon. They have concluded on the importance of the cross-diffusion coefficients on the evolution of c0 phase parameters. In the present work the diffusion matrix is non-symmetric and therefore takes into account the diffusion couplings. The best-fit value of interfacial energy used in our model has been slightly changed for the 3 nominal compositions. The interfacial energy is the only free parameter that is adjusted in our model. The interfacial energies were adjusted so that to get the best fit of the experimental Nv(t) curves. Besides, the interfacial energy is known to be affected by interfacial phase compositions (proportional to the square function of the composition gradient). It is therefore expected to be different for the three alloys considered in this work. c0 are given in Fig. 1(a–c) The temporal evolution of N t , f and R V
t
for the three nominal compositions. The model fits well the experiments (the order of magnitude of time and number density). The total number density of precipitates (Fig. 1a) increases until the end of nucleation (1 h, 4 h and 4 h respectively for (the nominal compositions) 7.5Al8.5Cr, 5.2Al14.2Cr and 6.5Al9.5Cr). At each time step, new populations of precipitates are created in this nucleation regime. When nucleation stops, the total number density of precipitates remains constant for a while. Actually, during this time, there is neither nucleation nor Ostwald ripening (or at least not sufficient) to allow the total shrinking of some precipitates populations. The total number density remains almost constant. The start of the number density decreasing (4 h, 20 h and 30 approximately respectively for 7.5Al8.5Cr, 5.2Al14.2Cr and 6.5Al9.5Cr) corresponds to the time when the first populations of precipitates disappear. It continues to decrease as the number of populations disappearing increases and because there is no more nucleation. One can notice that the difference between experiments and simulation results is more pronounced for shorter ageing times. This may be first explained by uncertainties in atom probe tomography experiments. Indeed, as we can see in Fig. 1b precipitate sizes for this time range are very small (of few nm). The mean radius (Fig. 1b) is observed to increase continuously with time with a good agreement between experiments and simulations. The radius values are of the same order of magnitude and the slope values of R(t) are very close. During the nucleation
the total number density.
3. Results and discussion We have applied the model described above to model NiCrAl superalloys for which experimental data are available. The model
Table 1 Parameters used for NiCrAl alloys. n0 (m3)
X (m3) 28
9.14 10
1.09 10
29
Vm (m3 mol1)
a (m)
6.59 106
3.4 1010
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M. Bonvalet et al. / Acta Materialia 100 (2015) 169–177
Fig. 1. Evolution of the number density (a), mean radius (b) and precipitate volume fraction (c) as a function of time for alloys 1–3 in simulations (straight red line) and APT experiments (black dots) [2,23,24]. Results of LKMC on alloy 2 are represented by gray triangles. (1: Ni–6.5 at%Al–9.5 at%Cr, 2: Ni–5.2 at%Al–14.2 at%Cr, 3: Ni–7.5 at%Al–8.5 at %Cr). (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
regime, the critical radius of new precipitate population increases with time because of the decreasing supersaturation and also because almost all populations previously created grow. During growth and coarsening regimes, even if there is a shrinking of some populations, precipitate populations that are growing affect more strongly the mean radius. The Fig. 2 shows the evolution of
nucleation critical radius and mean radius over time. This result indicates that the values of these radii are different until the pure growth regime is finished. Once the nucleation critical radius reaches the mean radius, the regime reached is that of pure coarsening. Nucleation critical radius is thus equal to coarsening critical radius. They are also equal to mean radius of precipitates in the system (see Table 2). The volume fraction of precipitates (Fig. 1c) is observed to increase until it reaches its equilibrium value. Note that experiments for 5.2Al14.2Cr result in an asymptotic volume fraction greater than equilibrium volume fraction as derived from thermodynamic database [23]. We have calculated the driving force for nucleation for each nominal composition. We found that the alloy 7.5Al8.5Cr has the c0
Fig. 2. Evolution of the mean radius of precipitate (straight line) and of the critical radius for nucleation (dotted line) as a function of time in the 6.5Al9.5Cr alloy.
highest driving force for nucleation at t = 0 (largest f ) and the alloy 6.5Al9.5Cr the lowest. This is not surprising in view of the kinetics obtained. The highest the nucleation driving force, the highest the nucleation rate, the faster the end of nucleation and the faster the decreasing of first precipitate classes. The criterion used in the model to stop nucleation stage was a nucleation rate below to 0.1% of the maximum nucleation rate (we have checked that lower values of this threshold did not affect results). Numerical results are compared to lattice kinetic Monte Carlo (LKMC) simulations. Both thermodynamic and kinetic parameters
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Table 2 Equilibrium concentrations in both matrix and precipitates, Diffusion matrix and interfacial energy for the three different nominal composition of NiCrAl studied in this work (1: 6.5Al9.5Cr, 2: 5.2Al14.2Cr, 3:7.5Al8.5Cr). Alloy
c C Al
c C Cr
c0 C Al
c0 C Cr
DAlAl (1021 m2 s1)
DAlCr (1021 m2 s1)
DCrAl (1021 m2 s1)
DCrCr (1021 m2 s1)
r (mJ m2)
1 2 3
5.12 3.67 5.48
9.81 14.93 8.87
17.45 16.1 17.74
7.04 8.93 6.64
11.8 18.1 11.6
5.62 8.09 5.65
2.73 3.12 2.81
1.88 1.64 1.92
31 32 32
for the LKMC simulations are taken from [1]. The comparison is shown in Fig. 1a–c for the 5.2Al14.2Cr alloy. Both numerical methods are in very good agreement with experiments. Whereas the LKMC simulations also give the evolution of the precipitates morphology during the phase transformation, only the early stages are accessible since LKMC simulations are time consuming. Our new approach has the great advantage to predict kinetics over much longer times and to explicit the physical origin of kinetic effects observed. Fig. 3 pictures the evolution of phase concentrations as a function of annealing time in the 6.5Al9.5Cr alloy. It is worth mentioning that the concentration in c0 phase is the concentration for It should be noted that concentrations in precipitates do R ¼ R. not remain constant. For long ageing time, atomic fractions tend to equilibrium values given by the phase diagram. Concentrations in matrix evolve only slightly in the early stages up to the end of nucleation. The supersaturation is not yet affected. Note the nonmonotonic precipitate concentration evolution during nucleation and just after. Concentrations cross equilibrium values. Recent experiments on model NiCrAl investigated with atom probe tomography have shown that the Cr and Al precipitate compositions decrease with time [24]. This decreasing is also observed in our simulations but only after nucleation. Let us discuss the origin of this non-monotonic temporal evolution of the Cr composi0
tion C cCr .
0
First C cCr increases during nucleation (Fig. 3), then decreases when growth starts and finally very slightly increases during the coarsening regime. It should be kept in mind that Fig. 3 gives the which is not the mean conconcentration in a precipitate of size R, centration. Note that we have compared the evolution of the mean concentration and the evolution of the concentration in a precipi We find the same trend. tate of size R. The specific behavior of the temporal precipitate concentration evolution can be understood and explained from (Eq. (8)) that gives the composition of a c0 precipitate of size R. In this equation,
Fig. 3. Evolution of concentrations in both phases c (green and blue curves) and c0 (red and black curves) as a function of the time in the 6.5Al9.5Cr alloy (Al: red and blue curves, Cr: green and black curves). The inset is a zoom of the evolution of Cr concentration in phase c0 . (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
three parameters are time dependent: DC1 , the supersaturation, R, the radius and dR/dt, the growth rate. Thus, the term in brackets in Rd R=dt (Eq. (8)): DC1 D1 DC drives the evolution of the precipi R=dt. tate composition, with a competition between DC1 and Rd Rd R=dt Fig. 4 represents the temporal evolution of DC1 D1 DC for Cr. R=dt drives During nucleation, DC1 is almost constant, so that Rd c0 1 1 the evolution of C Cr , i.e. the increasing of DC D DCRdR=dt (first increases because of part of the curve in Fig. 4). During this stage, R
growth of precipitates (Fig. 2). Moreover, because Rc, the critical radius for nucleation, evolves very slightly during nucleation, the and Rc increases (Fig. 2). As a consequence, difference between R is negative for increases (note that DC dR=dt (Eq. (5) with R ¼ R) Cr). Once nucleation is completed, pure growth is observed for a while, (justified later in the manuscript). The evolution of the supersaturation is more pronounced and the critical radius for nucleation increases. Thus, the mean radius of precipitates tends to the critical radius of nucleation with increasing time. It follows 0
a decrease in C cCr . Finally, once the pure coarsening stage is reached, the mean radius is equal to the critical radius, i.e. dR/dt = 0, and the supersat0
uration drives the temporal evolution of C cCr . The Al composition shows a similar trend, but in the opposite direction. 3.1. Asymptotic behaviors One can study the asymptotic behavior for long ageing times. The long-term behaviors of the average precipitate radius, number density and concentrations in both phases are represented on Fig. 5 and Fig. 6. It has been recently shown by Philippe and Voorhees [19] that the temporal exponents for these three parameters are identical to the binary limit (the amplitudes are different). The number density has been found to vary linearly with t1 as shown by Philippe and Voorhees:
R dR as a function of the time in 6.5Al9.5Cr alloy. Fig. 4. Evolution of DC1 D1 DC dt The red dashed line indicates the time of the end of nucleation (t = 12,860 s). (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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M. Bonvalet et al. / Acta Materialia 100 (2015) 169–177
controlled by the nucleation rate. One can observe that normalized density of the different classes first increases (until the nucleation peak) and then decreases. The populations that nucleate last have a smaller reduced radius (i.e. the evolution of the nucleation critical radius is slower than the evolution of radius with growth law). This analysis will be further supported in the following paragraph. With the increasing annealing times, the distribution tends towards the well-known LSW distribution [12,13]. This again shows the validity of the model. In order to understand which precipitates shrink or grow, we analyze the raw size distributions. We find that the population of precipitates that nucleate first keeps the largest mean radius throughout the kinetics. In addition, it is worth mentioning that until the end of nucleation, there is no coarsening of precipitates
Fig. 5. Asymptotic behavior of the number density (a) and mean radius (b) as a function of t1 and t respectively as given by the simulation (red curves) and by the asymptotic laws of coarsening [19] (black curves) in 6.5Al9.5Cr. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
c0
f eq Nv ðtÞ ffi 0:21 t1 K where K ¼
c0
8V m r
M1 DC 9ðDCÞ T
ð10Þ .
The mean radius of precipitates varies linearly with t:
3 ðtÞ ¼ Kt R
ð11Þ
Concentrations in both matrix and precipitates evolve respectively as: 1
T 1 3 2 ½ðDCÞ M DC t 13 c þ ð3rV c0 Þ3 DC C ðtÞ ¼ C m T Gc DC ðDCÞ c
ð12Þ
and c01 c 1 2 0 c0 þ ð3rV c0 Þ3 ½ðDCÞ T M1 DC 3 G G DC t 13 Cc ðtÞ ¼ C m T Gc DC ðDCÞ
ð13Þ
(Fig. 5), Cc and Cc0 (Fig. 6) in 6.5Al9.5Cr, are found, when N tV , R time tends to infinity, to fall well on the theoretical predictions of Philippe and Voorhees [19]. These results validate the approach used in this model and confirm its relevance for the description of precipitation kinetics in multicomponent alloys. Even with a model dealing at the same time with nucleation, growth and coarsening, we find for long annealing time, the classical behavior of pure coarsening. Fig. 7 shows the normalized size distributions at different ageing times. These normalized distributions are also normalized as a function of the mean radius of precipitates. During pure coarsening regime, precipitate populations with a reduced radius R½ ¼ R=R lower than 1 are shrinking and those with a reduced radius greater than 1 are growing. During nucleation, the distribution is
Fig. 6. Asymptotic behavior of the composition in both matrix and precipitates (for elements Al and Cr) as a function of t1/3 as given by the simulation (red curves) and with the asymptotic laws of coarsening [19] (black curves) in 6.5Al9.5Cr. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
M. Bonvalet et al. / Acta Materialia 100 (2015) 169–177
175
Fig. 7. Normalized size distributions for different ageing times in 6.5Al9.5Cr. The radius R[] is normalized as a function of the mean radius of precipitates. D[] is the normalized number density so that the total area under the curve D[](R[]) is equal to unity. Circles correspond to distribution during nucleation regimes and stars correspond to times after the end of nucleation.
in this system. All the populations of precipitates created grow. Moreover, once nucleation is finished, pure growth of precipitates is observed for a while. This is explained by the fact that the supersaturation does not evolve a lot until this time (Eq. (5)) so the growth rate is positive for all precipitates classes. As shown in the ternary phase diagram given in Fig. 8, phase compositions do not follow the equilibrium tie-line during transformation, as it was reported experimentally [2,3,23,24]. However, as already mentioned the approach does not predict the global decrease of the average Cr and Al concentrations in c0 exhibited in atom probe tomography experiments. In the present approach, the precipitates are assumed to be in local equilibrium with the matrix and thus, their compositions are set by both capillarity and the diffusion process, as shown by the Gibbs–Thomson equation (Eq. (8)). Such chemical equilibrium leads to a complex temporal evolution of the average Cr concentration in c0 during the phase transformation, but does not lead to a monotonic decrease. As a consequence, the present study suggests that nonequilibrium effects, potentially due to the low Cr mobility, might be responsible for the decrease in Cr composition. Our model, however, does not permit to investigate such phenomena.
and f c0 as a function of the ageing Fig. 9. Influence of the interfacial energy on N TV , R time.
3.2. Influence of interfacial energy The model gives interesting features on the influence of the interfacial energy r on the precipitation in multicomponent alloys. We perform simulations for different values of r in order to determine the influence of interfacial energy on the overall kinetics. An important feature is that interfacial energy as a strong influence on kinetic pathway of precipitation. It is the only parameter that we can modify in our model but it is very sensitive. Fig. 9 shows its and f c0 as a function of ageing time. We use four influence on N t , R V
values of interfacial energy (30, 31, 32 and 33 mJ m2). One should notice that the smaller r, the bigger the maximum number density value and this one is reached for smaller time (smaller nucleation barrier, see Eqs. (2) and (4)). Actually, the nucleation rate is shifted to longer times when the interfacial energy is increased. Concomitantly, the volume fraction of precipitates increases faster during the nucleation and growth regimes, as the mean radius. However before entering the pure coarsening regime, mean radius is bigger for smaller r, due to a slow decrease in the supersaturation.
4. Conclusions
Fig. 8. Isothermal section of the ternary phase diagram of NiCrAl with compositional trajectories in the three alloys. The arrow indicates the direction of increasing time. Black circles, light gray circles, dark gray circles correspond respectively to 5.2al14.2Cr, 6.5Al9.5Cr and 7.5Al8.5Cr alloys.
A new model concomitantly dealing with nucleation, growth and coarsening has been developed for predicting precipitation kinetics in multicomponent alloys. Nucleation has been implemented using Zeldovich theory so that to include both regression effects and incubation time. Growth and coarsening has been modeled using the recently developed growth and coarsening laws in multicomponent alloys that account for capillarity, mass balance at the interface matrix-precipitate and diffusion-flux couplings. This new approach does not require any coupling with external
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M. Bonvalet et al. / Acta Materialia 100 (2015) 169–177
CALPHAD modules, as a result, it guaranties remarkable computation efficiency as compared to past works. Numerical results have been confronted to atom probe tomography (APT) experiments on the NiCrAl system and to lattice kinetics Monte Carlo (LKMC) simulations. The results have been found in very good agreement with both APT experiments and LKMC simulations. As already shown, predictions have been shown to be very sensitive to the interfacial energy, particularly during the nucleation and growth regimes. The temporal evolution of the mean precipitate composition has been found to be non-monotonic during the phase transformation due to a complex interplay between capillarity and diffusion processes. The widespread availability of both thermodynamic and mobility databases makes this new model very suitable for heat treatment optimization and material design.
where higher order constituent array IZ have been introduced. IZ must not contain a component more than once for the same sublattice. I1 specifies that one sublattice contains two components but only one on the remaining sublattice. The interaction parameters LIZ are described using a Redlich–Kister polynomial for binary systems [27]:
LI1 ðYÞ ¼
X b ðysi0 ysi00 Þ :b GI1
i0 and i00 are two components occupying the sublattice s according to I1. b is the degree of the parameter bGI1 . As an illustration, the molar Gibbs energy of an A–B system having one sublattice is:
Gm ¼ yA :0 GA þ yB :0 GB þ RTðyA ln yA þ yB ln yB Þ þ yA yB LA:B Acknowledgment The authors are grateful for the financial support of CARNOT ESP Institute. Appendix A
with
Sideal m R
¼
s
ðA:1Þ
the ideal entropy of mixing:
X X as ysi ln ysi
ðA:2Þ
i
R is the gas constant, the superscript s denotes the sublattice under consideration. ysi is the site fraction of component i on sublattice s. as is the number of sites on the sublattice s per mole of formula units of the phase. The previous expression for the entropy assumes random mixing on each sublattice of the phase under consideration. The first sum is therefore over all sublattices and the second overall constituents on each sublattice. Gref m represents the state of reference for the Gibbs energy and can be expressed in a general form as:
Gref m ¼
X PI0 ðYÞ:0 GI0
ðA:3Þ
I0
Here was introduced the concept of constituent array. A constituent array I specifies one or more constituent i on each sublattice. The constituent arrays can be of different orders. The zeroth order refers to one constituent on each sublattice. Thus in the state of reference of the Gibbs energy, I0 is a constituent array of zeroth order specifying one constituent in each sublattice, PI0 ðYÞ represents the corresponding product of constituent fractions. 0 GI0 represents the Gibbs energy of the compound defined by I0. GEm is the excess Gibbs energy. While Gref m describes the interactions between neighboring atoms in different sublattices, GEm is mainly composed of interaction energies between different components in the same sublattice. The excess Gibbs energy can be written:
XX GEm ¼ PIZ ðYÞ:LIZ Z>0 IZ
In this case, the site fraction yi are also the molar fraction of component i in the phase under consideration. In the above expression, a component array is written with a colon ‘‘:” between components in the same sublattice. With two sublattices, the Gibbs energy becomes:
þRT½a1 ðy1A ln y1A þ y1B ln y1B Þ þ a2 ðy2A ln y2A þ y2B ln y2B Þ
The free energy densities of phases are described in the compound-energy formalism (CEF), which is widely used in the CALPHAD methods. The main concepts of the CEF are briefly reviewed in this section. Sundman and Agren [27] extended the regular solution model to phases with several components and sublattices. In the CEF, the molar Gibbs energy is given by:
Sideal m
ðA:6Þ
Gm ¼ y1A y2A :0 GA;A þ y1A y2B :0 GA;B þ y1B y2A :0 GB;A þ y1B y2B :0 GB;B
A.1. Coupling the model to physical databases
ideal Gm ¼ Gref þ GEm m TSm
ðA:5Þ
b¼0
ðA:4Þ
þy1A y1B y2A LA:B;A þ y1A y1B y2B LA:B;B þ y1A y2A y2B LA;A:B þ y1B y2A y2B LB;A:B
ðA:7Þ
þy1A y1B y2A y2B LA:B;A:B where a component array is written with a comma ‘‘,” between components in different sublattices. The last term in Eq. (A.7) is included in Eq. (A.4) for Z = 2. Despite the apparent complexity of the CEF, the extended regular solution models make the formalism suitable for computer applications and have been widely used in the CALPHAD method. In the CEF, the free energies are expressed in terms of site fractions. For a disordered phase, site fractions are equivalent to atomic fractions of the components in the phase under consideration. This is not the case for phases involving chemical ordering. In the present approach, it is assumed that chemical ordering is much faster than diffusion meaning that site fractions take their equilibrium values. The free energy of an ordered phase needs therefore to be minimized with respect to the site fractions with the constraint of mass conservation on the atomic fractions. Thus, as each set of atomic fractions corresponds a set of equilibrium site fractions, which is used to compute the free energy of the phase under consideration. Such minimization is performed using a downhill simplex method. As the procedure can be time consuming, the minimization of the free energy of an ordered phase is done prior the simulations and look-up tables for the free energy, as well as for its Hessian, are built and then read during the simulation. Such procedure does not require interfaces to Thermocalc or DICTRA software to calculate Gibbs energies or the other thermodynamic quantities. Note that in a multicomponent alloy, the use of parabolic approximations of the free energies is not advisable even for a ternary alloy due to complicated curvatures of the free energies. In this work, a published database [28] has been used for the computation of the thermodynamic quantities for the disordered phase (matrix c) and the ordered precipitates (c0 , L12) in the Ni–Cr–Al system. Appendix B B.1. Diffusion matrix in a multicomponent alloy Diffusion fluxes in the linear theory are linear functions of driving forces:
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M. Bonvalet et al. / Acta Materialia 100 (2015) 169–177 n X J k ¼ Lkj rlj
ðB:1Þ
j¼1
This expression is given in the lattice-framed of reference. Lkj are phenomenological parameters. rlj is the chemical potential gradient of component j. In the volume-fixed frame of reference, P there is no net flow of volume ( k J k ¼ 0 where the k are the components of the system). Andersson and Agren [25] have shown, that if the frame of reference is the volume-fixed frame of reference, then the diffusion matrix is given by:
D1kj
¼ Dkj Dk1 ðV j =V 1 Þ
where Dkj ¼
Pn
00 @½li ðV i =V 1 Þl1 i¼1 Lki @C j
ðB:2Þ is the diffusion matrix in the lattice
frame of reference and 1 is the majority element of the system. V i is the molar partial volume of element i. L00ki are the modified Onsager coefficients (defined in the lattice frame of reference and expressed hereafter). It should be noticed in this equation that li for i ¼ 1:::n are function of ðC 1 ; :::; C n Þ. If molar partial volumes of each element are equals then:
Dkj ¼
n X @½li l1 L00ki @C j i¼1
ðB:3Þ
and
D1kj
¼ Dkj Dk1
@½li l1 @C j
ðB:4Þ
in Eq. (B.3) could be rewritten as a function of the hessian of
the molar free energy of a. It is well-known that [26]:
li ¼ l1 þ Gam;i
ðB:5Þ
li l1 ¼ Gam;i ¼
@Gam @C i
ðB:6Þ
These equations are relations for the so-called diffusion poteni l1 , one has to differentiate Gam;i tial [17]. In order to determine @½l@C j with respect to C j using the result:
@f @f @f @f ¼ ¼ @C j @C j C j þC1 ¼cst @C j @C 1
ðB:7Þ
where f is a function of ðC 1 ; :::; C n Þ and f is a function of ðC 2 ; :::; C n Þ. These two functions are linked by the definition:
f ðC 1 ; :::; C n Þ ¼ f ðC 2 ; :::; C n Þ Pn
ðB:8Þ
with C 1 ¼ 1 i¼2 C i . Eq. (B.8) implies that if C j varies, f and f will vary in the same P way with the condition related to the mass balance i C i ¼ 1 (i.e. C j þ C 1 ¼ cst in the ðC 1 ; :::; C n Þ space).
We thus have: a
@Gm;i @ 2 Gam @ @ ¼ ¼ ½l l1 ½l l1 @C 1 i @C j @C j @C i @C j i If we now replace
D1kj ¼
@ @C j
ðB:9Þ
½li l1 in Eq. (B.3), we found that:
n X @ 2 Gam L00ki @C j @C i i¼2
ðB:10Þ
Eq. (B.10) can be written in standard dyadic notation:
D ¼ L0 G
ðB:11Þ
which is often written as D ¼ MG. In this expression, the matrix L00 (or M) is determined with the relation [25]:
L00ki ¼
n X n X Vr Vj djk xk Ljr dir xi V V m m j¼1 r¼1
ðB:12Þ
where Ljr are Onsager coefficients and dmn is the Kronecker delta (dmn ¼ 1 when m ¼ n and 0 otherwise). Once again, in our model, we assumed that molar partial volumes are equal. In this study, Onsager coefficients were taken from reference [2]. References [1] C. Pareige, F. Soisson, G. Martin, D. Blavette, Acta Mater. 47 (1999) 1889. [2] Z. Mao, C. Booth-Morrison, C.K. Sudbrack, G. Martin, D.N. Seidman, Acta Mater. 60 (2012) 1871. [3] Z. Mao, C.K. Sudbrack, K.E. Yoon, G. Martin, D.N. Seidman, Nat. Mater. 6 (2007) 210. [4] J.S. Langer, A.J. Schwartz, Phys. Rev. A 21 (1980) 948. [5] R. Kampmann, R. Wagner, Decomposition of Alloys: the Early Stages, 1984, p. 91. [6] O.R. Myhr, Grong, Acta Mater. 48 (2000) 1605. [7] M. Perez, M. Dumont, D. Acevedo-Reyes, Acta Mater. 56 (2008) 2119. [8] M. Serriere, C.A. Gandin, E. Gautier, P. Archambault, M. Dehmas, Mater. Sci. Forum (2002) 396. [9] P. Maugis, M. Gouné, Acta Mater. 53 (2005) 3359. [10] J.D. Robson, Acta Mater. 52 (2004) 1409. [11] J.D. Robson, P.B. Prangnell, Acta Mater. 49 (2001) 599. [12] I.M. Lifshitz, V.V. Slyosov, J. Phys. Chem. Solids 19 (1961) 35. [13] C. Wagner, Z. Elektrochem. 65 (1961) 581. [14] C.J. Kuehmann, P.W. Voorhees, Metall. Mater. Trans. A 27 (1996) 937. [15] S. Bjorklund, L.F. Donaghey, M. Hillert, Acta Metall. 20 (1972) 867. [16] V.V. Slyosov, V.V. Sagalovich, Sov. Phys. Solid State 17 (1975) 974. [17] A. Umantsev, G.B. Olson, Scripta Mettal. 29 (1993) 1135. [18] J.E. Morral, G.R. Purdy, Scripta Mettal. 30 (1994) 905. [19] T. Philippe, P.W. Voorhees, Acta Mater. 61 (2013) 4237. [20] L. Rougier, A. Jacot, C.A. Gandin, P. Di Napoli, P.T. Théry, D. Ponsen, V. Jacquet, Acta Mater. 61 (2013) 6396. [21] J. Zeldovich, J. Exp. Theor. Phys. (USSR) 2 (1942) 525. [22] T. Philippe, D. Blavette, P.W. Voorhees, J. Chem. Phys. 141 (2014) 124306. [23] C.K. Sudbrack, R.N. Noebe, D.N. Seidman, Acta Mater. 55 (2007) 119. [24] C. Booth-Morrison, Y. Zhou, R.N. Noebe, D.N. Seidman, Phil. Mag. 90 (2010) 219. [25] J. Andersson, J. Agren, J. Appl. Phys. 72 (1992) 1350. [26] M. Hillert, Phase Equilibria, Phase Diagrams and Phase Transformations, Their Thermodynamic Basis, Cambridge University Press, 2008. [27] B. Sundman, J. Agren, J. Phys. Chem. Solids 42 (1981) 297. [28] N. Saunders, Proc. Int. Symp. Super. (1996) 101.