Phase-Field Modeling of Kinetics of Diffusive Phase Transformation in Compositionally-Graded Ni-Based Superalloys

Abstract

Kinetics of the $\gamma$ /$\gamma ^{\prime}$ phase transformation and the interdiffusion phenomena in single-crystal Ni-based superalloys, under isothermal annealing and composition gradient, is investigated through the phase-field and continuum diffusion models. The employed models in the present work exploit CALPHAD-based thermodynamics and kinetics databases, in order to perform realistic simulations. We specifically predict the interdiffusion of elements for a hypothetical alloy AlCoCrTaNi/Ni diffusion couple, equivalent to the CMSX-10/Ni diffusion couple, at 1323 K. Accordingly, the phase fraction and morphology of $\gamma ^{\prime}$ precipitations in the $\gamma$ matrix is simulated as well. The implemented multi-component diffusion model takes into account vacancies and pore formation, reflecting Kirkendall effect. Furthermore, the time evolution of morphology parameters of the precipitate-depleted zone in the diffusive region (i.e., the position and the size) are estimated.

1 Introduction

Metallic alloys as one of the most widely used structural/functional materials are still under broad development and discovery. In this regard, one of the exciting applications is using compositionally-graded alloys in order to meet specific structural or functional properties.^[1] The idea of compositionally-graded alloys, as a subgroup of Functionally Graded Materials (FGMs), consist, for example, additively fabricated parts,^[2,3] coatings,^[4] welded dissimilar alloys^[5] and diffusion couples.^[6,7] Understanding this category of alloys demands examining a variety of complexities such as the study of kinetics of microstructure evolution during processing and service life. The associated kinetic events might include grain growth, coarsening of precipitates, particle shape changes, interdiffusion, phase transformations, Kirkendall porosity etc.. Considering the challenges for the experimental observation of all these phenomena, computational techniques could help as an efficient alternative to overcome the barriers for alloy developments or even interpreting the available experiments. The practical modeling of the involved kinetics, specially for multi-component multi-phase alloys requires coupling different simulation techniques. In the present work, we combine phase-field modeling and CALPHAD-type databases^[8] in order to give a comprehensive image of kinetics in Ni-based diffusion couples analogous to the concept of compositionally-graded complex alloys.

In the previous works in this context,^[4,9] CALPHAD-based 2D phase-field simulations were performed using MICRESS software^[10] to quantify microstructure evolution and interdiffusion profiles in Ni-based superalloys diffusion couples under isothermal/non-isothermal circumstances for binary/ternary systems. Zhang et al.^[9] inspected the rafting behaviour in various NiAl/Ni diffusion couple configurations under an external compressive force. A work by Ta et al.^[4] studied modeling microstructure evolution in bond coat/substrate Ni-Al-Cr systems by annealing under temperature gradients. The main model applied by them is based on the well-established multi-phase-field model proposed by Steinbach et al. and Eiken et al.^[11,12] Recently, Li et al.^[13] simulated the effects of temperature gradients on the interdiffusion microstructure evolution in NiAlCr/NiAlCr diffusion couples using CALPHAD-based phase-field model for a ternary alloy.^[14] In summary, the previous works in the direction of the interdiffusion modeling are limited to binary and ternary alloys and do not explore the Kirkendall void formations linked to vacancy diffusion.

The focus of the present research is specifically on the high-temperature isothermal interdiffusion, phase transformation and Kirkendall pore density/locations in diffusion couples of Ni-based superalloys under gradient of composition. We use the multi-phase-field model^[11,12] implemented in OpenPhase software,^[15] which is successfully applied in many studies^[16,17] to investigate the $\gamma -\gamma ^{{\prime}}$ transformation. The results of the experiments carried out on CMSX10/Ni couples^[18,19] reveal that the pores are created on the interface between $\gamma ^{\prime}$-free and $\gamma -\gamma ^{\prime}$ regions, where the diffusivity of components drastically changes. Then an accurate prediction of position and width of this region is of critical importance for the calculation of Kirkendall pore volume-fraction.

In the current work, a hypothetical alloy AlCoCrTaNi/Ni equivalent to CMSX10/Ni couple is investigated during the annealing at T=1323 K for different times. The multi-phase-field simulations in 3D and 2D illustrate the $\gamma ^{\prime}$ precipitate morphology in $\gamma$ matrix along the diffusion couple sample. The kinetics of dissolution of $\gamma ^{\prime}$ phases in $\gamma$ matrix is implemented as the function of composition-dependent interdiffusion coefficients and the size of precipitate-free area is monitored accordingly. Furthermore, the interdiffusion profiles of all elements as well as Kirkendall voids location and density along the diffusion couple are calculated by standard multi-component diffusion models.

2 Modeling Framework

The interdiffusion of the elements in the diffusion couple is simulated by two approaches in this paper. The first one is based on the phase-field technique, employing constant diffusion coefficients of components and omitting the off-diagonal terms of the diffusion matrix. The second approach is the standard multi-component diffusion modeling where the composition-dependent diffusion coefficients are calculated in each space position according to the actual composition of the alloy. Additionally, the “labyrinth” model with a homogenization function is used here in order to evaluate effective diffusion coefficients within the two-phase region.

2.1 Phase-Field Modeling

For the computational scheme of the current study, the phase-field model plays a pivotal role. The intricate characteristics of the microstructure such as size, shape, and spatial arrangement of the $\gamma ^{\prime}$-particles within Ni-based superalloys typically span from nanometers to micrometers. Achieving precise modeling of these features is very important, and the phase-field approach has proven to be very promising in this regard. The model of the $\gamma$-$\gamma ^{\prime}$ transformation in the AlNi and AlCoCrTaNi alloys applied in OpenPhase software is based on the multi-phase-field model.^{[11,20,21,22,23]} The equations of the model are based on the general free energy functional combining interfacial, chemical, and elastic energies of the system using a double-obstacle potential.

In the multi-phase-field model, each grain, crystal or domain $\alpha$ is assigned an order parameter $\phi _\alpha$ and properties of a thermodynamic phase p it contains. In our system, we have one order parameter responsible for the $\gamma$-phase and many order parameters indicating different crystals of the $\gamma ^{\prime}$-phase. Defining $\phi _\alpha \in [0, 1]$ leads to $\sum _{\alpha = 1}^{N} \phi _\alpha = 1$, where N is the local number of order parameters. We can also calculate a local phase fraction of a thermodynamic phase with an index p as ${\mathcal {P}}_p = \sum _{\alpha = 1; \alpha \in p}^{N} \phi _\alpha$, where the notation $\alpha \in p$ means that a domain $\alpha$ possesses properties of the thermodynamic phase p.

By minimization of the free energy functional, one obtains the evolution equation of an order parameter as,

$$\begin{aligned} {\dot{\phi }}_\alpha= & \frac{M_{\phi}}{N} \sum _{\beta \ne \alpha }^N \left\{ \sigma \left[ \nabla ^2 \phi _\alpha - \nabla ^2\phi _\beta + \frac{\pi ^2}{\eta ^2}(\phi _\alpha -\phi _\beta )\right] \right. \nonumber \\ & \left. + \frac{2\pi }{\eta }\sqrt{\phi _\alpha \phi _\beta } \left[ \Delta G^{\text{CH}}_{ \alpha \beta } + \Delta G^{\text{EL}}_{\alpha \beta } \right] \right\} , \end{aligned}$$

(1)

where $M_{\phi}$ is the interface mobility, $\sigma$ is the interfacial energy, which is assumed to be equal for all interfaces, and $\eta$ is the interface width. $\Delta G^{\mathrm{CH/EL}}$ refer to the chemical and elastic parts of the transformation driving force. The chemical driving force follows the work by Eiken et al.^[12,20,24]

The chemical contribution to the driving force is defined as^[12]

$$\begin{aligned} \Delta G^{\text{CH}}_{\alpha \beta } = \Delta S_{pq}\sum _i m^i_{pq} (x_p^i - x_{p}^{i,\mathrm eq}) , \,\, {\text { for } } \alpha \in p;\, \beta \in q. \end{aligned}$$

(2)

Here, $x_p^i$ is the current mole fraction of the element i in a thermodynamically stable phase p, which can be $\gamma$ or $\gamma ^{\prime}$, $x_{p}^{i, \mathrm eq}$ is the equilibrium mole fraction the element i in phase p, $\Delta S_{pq}$ is the entropy difference of phases, $m_{pq}^i$ are slopes of equilibrium lines, i.e., phase boundaries, of a linear phase diagram. It is worthy emphasizing that indexes p and q denote thermodynamic phases.

The elastic contribution to the driving force is defined as a simpler version in Ref. [20]

$$\Delta G^{\text{EL}}_{\alpha \beta }= \left( \epsilon ^{ij} - \epsilon ^{* ij} \right) C^{ijkl} \left(\epsilon ^{* kl}_p-\epsilon ^{* kl}_q\right)+\Delta G^{\text{EL,Mod}}, {\text { for } } \alpha \in p; \beta \in q.$$

(3)

Here, $\epsilon ^{ij}$ is a total strain in whole system, $\epsilon ^{* ij }$ is an effective eigenstrain while $\epsilon ^{* kl}{_p,_q}$ are the eingenstrain values in each phase calculated through the lattice misfit between $\gamma$ and $\gamma ^{\prime}$ phases. $C^{ijkl}$ indicates effective elasticity moduli. The expression "effective" refers to homogenized values of the relevant parameters along the interface between phases, through the techniques explained in Ref. [20]. The extra term $G^{\text{EL,Mod}}$, is arising due to difference in elasticity moduli in phases ($\gamma ^{\prime}$ is slightly stiffer than $\gamma$). The corresponding parameters are summarized in Table 1. The mechanical equilibrium equation is solved to get the strain and stress via $\nabla {\sigma _{ij}} = 0$. For more details see.^[20]

On the other hand, the diffusion equation for the current total mole fraction of an element i is formulated as^[12]

$$\begin{aligned} {{\frac{\partial x_i}{\partial t}}} = \nabla \cdot \left[ \sum _{p=1}^2 {\mathcal {P}}_p D_p^i \nabla (x_p^i - x_p^{i, \mathrm eq})\right] \end{aligned}$$

(4)

while the current mole fraction in a phase p follows

$$\begin{aligned} x^i_p = x_p^{i, \mathrm eq} + \frac{(x^i- \sum _{q=1}^2 x_q^{i, \mathrm eq} {\mathcal {P}}_q)}{\sum _{q=1}^2 K_{pq}^i {\mathcal {P}}_q}. \end{aligned}$$

(5)

Here, ${\mathcal {P}}_p$ are the local phase fractions of $\gamma$ or $\gamma ^{\prime}$ phases, $K_{pq}^i=\dfrac{m_{pq}^i}{m_{qp}^i}$ are partition coefficients. The off-diagonal terms of the diffusion matrix $D_p^i$ are omitted. The diagonal terms are defined by Eq 7 in next section.

Due to the fact that $\gamma ^{\prime}$ phase is stoichiometric (very steep slopes of equilibrium boundaries), it is reasonable to assume a constant composition $x^i_{\gamma ^{\prime} } = x_{\gamma ^{\prime} }^{i, \mathrm eq}$. Moreover, assuming $K_{pq}^i=1$ for the case of $\gamma$ phase, Eq 5 becomes

$$\begin{aligned} x^i_\gamma&= x_\gamma ^{i, {\text{eq}}} + \left( x^i- \sum _{q=1}^2 x_q^{i, {\text{eq}}} {\mathcal {P}}_q\right) \quad \text { if } \quad {\mathcal {P}}_\gamma >0,\nonumber \\ x^i_\gamma&= x_\gamma ^{i, \mathrm eq} \quad \text { if } \quad {\mathcal {P}}_\gamma =0. \end{aligned}$$

(6)

In order to satisfy the thin interface limit, and avoiding the recalculation of the interface mobility in each grid point at each time step, we assume constant diffusion coefficients of components in the phase-field simulations. Another additional advantage of this approach is better numerical stability.

2.2 Diffusion Modeling

The diffusion profiles of components in the diffusion couple experiment are calculated by the diffusion models proposed by Andersson and Ågren.^[25] The interdiffusion coefficients taking Ni as the reference element are expressed as

$$\begin{aligned} {\tilde{D}}_{ij}^{\text{Ni}}= \sum _{k=1}^{n-1} (\delta _{ki}-x_i) x_k M_k \left( \frac{\partial \mu _k}{\partial x_j} -\frac{\partial \mu _k}{\partial x_{{\text{Ni}}}}\right) , \end{aligned}$$

(7)

where n is the number of components, $\delta _{ki}$ is the Kronecker symbol, $x_k$ is the mole fraction of a component k, $M_k$ is the atomic mobility, $\mu _k$ is the chemical potential of element k, and $\frac{\partial \mu _k}{\partial x_j}$ is the thermodynamic factor. It must be mentioned that an important assumption behind this formulation is that partial molar volumes of all elements are supposed to be constant and equal to each other.

The thermodynamic factors are calculated using the molar Gibbs energy, $G_m$, of the fcc $\gamma$- phase:

$$\begin{aligned} \frac{\partial \mu _k}{\partial x_j}=\frac{\partial ^2 G_m}{\partial x_k\partial x_j}-\sum _l x_l\frac{\partial ^2 G_m}{\partial x_k\partial x_l}. \end{aligned}$$

(8)

We assume that the Gibbs energy of the fcc-phase depends on the composition according to a regular solution type model by extended Redlich–Kister polynomials.^[8]

The atomic mobility of a component i in an alloy is defined in the fcc $\gamma$- phase as

$$\begin{aligned} M_k= \frac{1}{RT} \exp \bigg (\frac{Q_k}{RT}\bigg ), \end{aligned}$$

(9)

where R is the ideal gas constant, T is the absolute temperature, and $Q_k$ is a general activation energy (which is negative by the present definition), whose composition and temperature dependency are expressed by Redlich–Kister polynomials with an extension to quaternary interactions^[26] similar to the Gibbs energy.^[25,27,28]

The composition distributions of components in the diffusion couple are calculated by solving the diffusion equations:

$$\begin{aligned} \frac{\partial x_i}{\partial t} = \frac{\partial }{\partial z} \left[ \sum _j^{n-1}\tilde{D}_{ij}^{\text{Ni}}\frac{\partial x_j}{\partial z}\right] . \end{aligned}$$

(10)

The Euler descretization method is applied with the grid space $\Delta x = 0.2\,\mu \hbox {m}$ and time step $\Delta t = 1\,\hbox {s}$, similar to the phase-field model.

To quantify the pore volume fraction along the diffusion couple experiment, we use the method proposed by Höglund and Ågren (HÅ).^[29,30] We assume that all vacancies generated in the diffusion couple due to the difference in the fluxes of components, condense into the pores, so the pore volume fraction is calculated as

$$\begin{aligned} f_{{\text{p}}} = \dfrac{q_{{\text{V}}}}{1+q_{{\text{V}}}}, \end{aligned}$$

(11)

where $q_{{\text{V}}}$ is the time integral over the maximum source of vacancies

$$\begin{aligned} q_{{\text{V}}}(t) =\int _0^t \dot{q}_{{\text{V}}}(t) \, dt, \text { if } \dot{q}_{{\text{V}}} > 0. \end{aligned}$$

(12)

Here, the maximum source of vacancies is calculated as

$$\begin{aligned} \dot{q}_{{\text{V}}} = -\frac{\partial }{\partial z} \left[ \sum _{j=1}^{n} \left[ \sum _{k=1}^{n} D^\text{I}_{kj} \right] \frac{\partial x_j}{\partial z}\right] , \end{aligned}$$

(13)

where

$$\begin{aligned} D_{kj}^{\text{I}} = x_kM_k\dfrac{\partial \mu _k}{\partial x_j} \end{aligned}$$

(14)

are the intrinsic diffusion coefficients.

The source of vacancies is produced by the gradient of the vacancy flux and can be expressed as

$$\begin{aligned} \dot{q}_{{\text{V}}} = -V_{{\text{m}}}\frac{\partial J_{{\text{V}}}}{\partial z}, \end{aligned}$$

(15)

where $J_{{\text{V}}} = -\sum _{k=1}^{n} J_k$ is the vacancy flux and

$$\begin{aligned} J_k = -\frac{1}{V_{{\text{m}}}}\sum _{j=1}^{n} D^\text{I}_{kj} \frac{\partial x_j}{\partial z} \end{aligned}$$

(16)

are the fluxes of components, $V_{{\text{m}}}$ is the average molar volume, which is assumed to be constant. The diffusivity in ordered stoichiometric phase $\gamma ^{\prime}$ at the investigated temperature is assumed to be zero.

In the present work, we apply a “labyrinth” homogenization model similar to our recent article,^[7] accounting for the precipitation of the $\gamma ^{\prime}$-phase using a $\psi$-function. This function is set to 1 within the region of equilibrium $\gamma -\gamma ^{\prime}$ mixture and 0 within the $\gamma$-phase as depicted in Fig. 1. To accurately represent the depletion zone, the function is given in a hyperbolic tangent form

$$\begin{aligned} \psi (z,t) = -\tanh \left( \dfrac{z+L_\gamma (t)}{\omega }\right) , \end{aligned}$$

(17)

where z is the distance from the Matano plane, $L_\gamma (t)$ is the width of the $\gamma ^{\prime}$-free zone (see Fig. 2a in Ref. [18]), and $\omega$ is the width of the depletion zone. $L_\gamma (t)$ is determined at each time step of the simulation by the criterion by which Al mole fraction in the $\gamma ^{\prime}$-free zone should be smaller than a critical value for the corresponding alloy, below which $\gamma ^{\prime}$ is not stable see Ref. [7]. The value of $\omega =40 \Delta z$ is assumed to be constant during the simulation.

We replace the atomic mobility $M_k$ inside the mixed region by an effective atomic mobility $M_k^{\text{eff}}$, which is defined utilizing the $\psi$-function as

$$\begin{aligned} M_k^{\text{eff}} =M_k\left[ 1.0 - f_{\gamma ^{\prime} }\psi \right] , \end{aligned}$$

(18)

where $f_{\gamma ^{\prime} }$ is the equilibrium volume fraction of the $\gamma ^{\prime}$ phase in the initial alloy (so called “labyrinth” factor).

2.3 Thermodynamic and Kinetic Parameters

The formulation mentioned above necessitates integration with reliable thermodynamic and kinetic databases to ensure accuracy. In this context, the CALPHAD method (CALculation of PHAse Diagrams)^[8,31] plays a crucial role through providing the essential data for precise predictions of corresponding phase transformations.

Thermodynamic data for the system are taken from databases established in seminal works such as.^[32,33,34] These databases supply foundational information regarding the energetics of phase transitions. Additionally, atomic mobility parameters, indispensable for modeling kinetic processes, have been documented in publications like.^{[26,28,35,36]} Incorporating these parameters into the diffusion modeling ensures a comprehensive treatment of the system’s kinetic behavior.

3 Simulations/Discussion

The adopted model parameters required for the phase-field/diffusion simulations of phase transformation in the diffusion couples of Ni-based superalloys at 1323 K are listed in Table 1. The initial mole fractions of components for three different alloys for examination are listed in Table 2. In all alloys (i.e., binary, multi-component (I), and (II)), two phases $\gamma$ and $\gamma ^{\prime}$ coexist on the left side of the couple. Based on calculations with Thermo-Calc software^[37] using database from,^[33] the equilibrium mole fractions of $\gamma ^{\prime}$ are 56 and 70% in multi-component (I) and multi-component (II), respectively. The diffusion couple is assumed to be a single-crystal oriented in $\langle 001\rangle$ orientation, which corresponds to the direction of the interdiffusion. In addition to the simulation of the phase transformation in the diffusion couple, we determine the interdiffusion profiles and pore volume-fractions via the proper diffusion models (refer to Eq 10 and 11).

Table 3, summarizes the simulation set-ups for 5 tests concerning above three selected alloys. Tests 1 and 2 are used for the comparison of phase-transformation kinetics based on 3D and 2D phase-field simulations in a binary AlNi/Ni couple. Tests 3 and 4 are applied for the investigation of 3D and 2D phase-transformations during the diffusion in AlCoCrTaNi/Ni couple, i.e., multi-component (I). Finally, test 5 only considers 2D phase-transformation in third alloy, i.e., multi-component (II). In this respect, the width of the $\gamma ^{\prime}$-free region, $L_{\gamma }$, as a function of time, as well as the $\gamma ^{\prime}$-phase fraction versus the distance from the Matano plane are monitored during the simulations. In all figures of the diffusion results, the Matano plane lies in the center of the simulation box.

Table 1 Model parameters for the diffusion couple AlCoCrTaNi/Ni

Table 2 Initial mole fraction of components in the diffusion couples AlNi/Ni and AlCoCrTaNi/Ni

Table 3 Phase-field simulation set-up: test number, dimensions, number of grid points, domain size in units of μm, annealing time, types of alloy

Figure 1 illustrates the results of test 1 related to 3D simulation of binary alloy showing Al-composition field (indicating $\gamma ^{\prime}$) after 2 and 4.5 h. The $\gamma ^{\prime}$-precipitates (red color) in each section of the sample are recognizable. As seen in the section near the Matano plane, $\gamma ^{\prime}$ precipitates are dissolved while they are unaffected at the left end of sample and the phase fraction is conserved as its initial stage circa 70%. Test 2 models the same binary alloy, this time in 2D but for bigger size of sample and longer time. Figure 2 shows the Al-composition field and the interdiffusion profiles of Al at various time steps. As it is noticeable in the early stages (5 h), cuboidal $\gamma ^{\prime}$ particles line up in straight lines in the direction of diffusion (along the sample axis), between which smooth paths (channels) are formed across the main diffusion flow.

Proceeding with multi-component alloys, test 3 in multi-component alloy (I) exhibits 3D microstructure evolution after 6 and 36 h of annealing, as shown in Fig. 3. Moreover, the Al-composition field and interdiffusion profiles of all components regarding tests 4 and 5 in multi-component alloys (I) and (II) for 2D case, at several times of annealing, are plotted in Fig. 4 and 5. The comparison of the microstructure in different tests of these two multi-component alloys reveals the forthcoming conclusions. In multi-component alloy (I) (tests 3 and 4) with the equilibrium fraction of $\gamma ^{\prime}$-phase of 56% (initial stage of left side of couple), wider channels are formed during the diffusion in the $\gamma ^{\prime}$-denuded zone along the diffusion couple. The formation of channels can be observed in both 2D and 3D cases. However, in multi-component alloy (II) (test 5) and binary alloy (test 2) with higher $\gamma ^{\prime}$-phase fraction of 70%, the channels are much narrower.

Figure 6 shows the interdiffusion profiles of Al and Ta in multi-component alloy (I) obtained by phase-field model in test 4 versus interdiffusion profiles calculated by the multi-component diffusion equations with “labyrinth” model described in section 2.2. It can be seen that according to diffusion modeling, Al diffusion profile exhibits a minor kink around the sample’s midpoint due to the thermodynamic interaction with Ta. The Al diffusion profile calculated by phase-field model does not have this kink because of assuming constant diffusion coefficients along the sample. In all other aspects, the profiles achieved by these two approaches are closely similar till about 200 h. After passing this time the diffusivity increase in the diffusion model leads to the decreasing slope of Al-profile, in comparison to phase-field model result (not shown here). This is due to the fact that the phase-field model employs constant diffusion coefficients, whereas the diffusion model uses the composition-dependent ones, calculated for each space position at every time step.

The current diffusion model framework provides other interesting information, as shown in Fig. 7. The magnitude of vacancy flux gradient in Fig. 7 significantly decreases for longer times of annealing. Their peaks are located on the boundary between $\gamma -\gamma ^{\prime}$ zone and $\gamma ^{\prime}$-free zone due to the change in the diffusion coefficients in presence of particles of $\gamma ^{\prime}$ phase. Figure 7(b) indicates that longer annealing time not only increases pore fraction but also shifts pore location to the left side of couple. The fluxes of components are calculated as represented in Fig. 7(c) and (d). It is evident that Al as the fastest diffuser in these systems is responsible for vacancy flux from right to left side. Furthermore, as time of annealing progresses, the Al flux profile has smaller maximum while it becomes more distributed on both ends of couple.

More importantly, the dissolution of $\gamma ^{\prime}$-particles can be quantified by the widths of the $\gamma ^{\prime}$-free zone, $L_\gamma$, and the growth rate of the $\gamma ^{\prime}$-free zone, $\xi _\gamma$, which obey the parabolic rate expression suggested in Ref. [18]

$$\begin{aligned} \xi _\gamma = \frac{L^2_\gamma (t)}{t}. \end{aligned}$$

(19)

This equation implies that the dissolution rate, i.e., $\frac{d L_\gamma }{dt}$, slows down during annealing since $\frac{d L_\gamma }{dt}=\frac{\sqrt{\xi _\gamma }}{2\sqrt{t}}$. In Fig. 8(a), the time evolution of the squared width of the $\gamma ^{\prime}$-free zone, denoted as $L^2_\gamma (t)$, is shown for tests 3, 4, and 5. The plot showcases a linear trend throughout the annealing process, representing a consistent growth pattern. The growth rate, $\xi _\gamma$, can be calculated as the averaged slope. Table 4 presents the estimated growth rate constants for various tests. Remarkably, these values align closely with the average diffusion coefficient of aluminum ($\tilde{D}_{{\text{Al}}}^{\text{Ni}}$) on the left side of the diffusion couples, nearly around $4.5\times 10^{-15}$ $\hbox {m}^2$ $\hbox {s}^{-1}$. In particular, the growth rate constant tends to be higher for multi-component alloy (I) compared to multi-component alloy (II), correlating with lower mole fraction of the $\gamma ^{\prime}$-phase in multi-component (I). One possible explanation is that wider distances between particles, i.e., case of multi-component alloy (I), promote faster motion of species from the left side to the right side of the diffusion couple, consequently expediting dissolution. Moreover, the kinetics of the $\gamma ^{\prime}$-phase dissolution for multi-component alloy (I) exhibits comparability between 3D and 2D simulations, albeit with a slightly higher rate constant in the 3D. This discrepancy could be attributed to different diffusive transport behaviors in 2D and 3D cases.

Figure 8(b) shows the mole fraction of $\gamma ^{\prime}$-phase as a function of the distance from the Matano plane, i. e., the $\gamma ^{\prime}$-dissolution profiles, estimated in test 5 for different time steps. The width of the $\gamma ^{\prime}$-free zone is identifiable while blue dark color is fading from right of middle line (Matano plane) to left. The values of the phase fractions are averaged through each $40\Delta z$ grid spacing. The deviation of this value from the mean value is around 10%. According to this result, the form of the dissolution profiles can be approximated by the hyperbolic tangent function suggested by Eq 17 in the “labyrinth” model.

Table 4 Calculated widths, $L_\gamma$, and growth rates, $\xi _\gamma$, for $\gamma ^{\prime}$-free zone thickening for different tests

4 Conclusion

The kinetic behaviour of the phase transformation in Ni-based superalloy diffusion couples during the annealing at 1323 K was examined rigorously by means of the phase-field modeling and diffusion models. The CALPHAD-type thermodynamic and kinetic databases are directly linked to these models to conduct realistic simulations. The final results include phase fraction and morphology, interdiffusion profiles of all elements, dissolution length, elements/vacancy fluxes and pore fraction/distribution. The width of the $\gamma ^{\prime}$-free zone is monitored during the phase-field simulations and plotted as the function of time for different alloys in 2D and 3D. Our investigation yielded several key insights:

• (1) The phase-field simulations reveal that the microstructure and kinetics of the $\gamma$-$\gamma ^{\prime}$ phase transformation during diffusion in couples exhibit similarity between 2D and 3D scenarios, irrespective of whether the alloys are binary or multi-component. However, dissolution of the $\gamma ^{\prime}$ phase occurs slightly faster in 3D due to faster diffusion through three directions.

• (2) Notably, the test-4 results for multi-component alloy (I) containing a smaller phase fraction of $\gamma ^{\prime}$ (56%), displays wider horizontal channels between $\gamma ^{\prime}$-particles, in contrast to narrower vertical channels. This phenomenon can be attributed to the directional diffusion along the diffusion couple axis.

• (3) The dissolution of $\gamma ^{\prime}$-particles follows a constant rate governed by the parabolic rate Eq 19, indicating the slow down of dissolution as annealing progresses ($\frac{d L_\gamma }{dt}=\frac{\sqrt{\xi _\gamma }}{2\sqrt{t}}$). Interestingly, this parabolic trend persists throughout the entire annealing process up to 300 h (see Fig. 8a).

• (4) The comparison of interdiffusion profiles achieved by phase-field model on one hand and diffusion models on the other hand shows a good agreement. When comparing the phase-field simulation with results of the diffusion calculations incorporating “labyrinth” model, we find a good agreement of average compositions up to 200 h, despite the fact that phase-field model used diagonal diffusion coefficients versus full matrix ones in diffusion model (see Fig. 6). Additionally, the phase-field model approach exhibits fluctuating profiles in two-phase region because of the explicit consideration of $\gamma ^{\prime}$-particles in the model, which is missing in diffusion model with homogenization “labyrinth” function.

• (5) As observed in the simulation results, we can state that the utilization of the diffusion model as a faster approach enabled us to calculate accurately diffusion profiles and estimation of pore fraction, based on the CALPHAD technique.

Fig. 1

Phase-field modeling of the microstrucrure evolution during the annealing in test 1 in binary alloy AlNi/Ni diffusion couple at 5 and 11 h: (left) 3D view of the Al-composition; (right) 2D horizontal cross sections

Fig. 2

Phase-field modeling of microstrucrure evolution (Al-composition) and diffusion profiles during the annealing in AlNi/Ni diffusion couple in test 2 at different time steps

Fig. 3

Phase-field modeling of the microstrucrure evolution during the annealing in test 3 in multi-component alloy (I) at 6 and 36 h of annealing: (left) 3D view of the Al-composition; (right) 2D horizontal cross sections

Fig. 4

Phase-field modeling of microstrucrure evolution and diffusion profiles during the annealing in test 4 in multi-component alloy (I) at different time steps

Fig. 5

Phase-field modeling of microstrucrure evolution (Al-composition) and diffusion profiles during the annealing in test 5 in multi-component alloy (II) at different time steps

Fig. 6

Comparison of the interdiffusion profiles of Al and Ta calculated by the diffusion modeling (DM), represented by smooth curves, and phase-field modeling (PFM), represented by fluctuating curves, in test 5 for multi-component alloy (II) at various time steps

Fig. 7

Diffusion modeling results obtained with the “labyrinth” model related to test 4, for multi-component alloy (I): (a) negative gradient of vacancy flux through the sample at different times calculated by means of diffusion equations; (b) pore volume fraction at different times; (c, d) fluxes of components and vacancies at different time steps

Fig. 8

(a) Time evolution of the squared width of the $\gamma ^{\prime}$-free zone, $L^2_\gamma$, for tests 3, 4, and 5. (b) Mole fraction of $\gamma ^{\prime}$-phase as a function of distance at different annealing times for test 5