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Mixedmode growth of a multicomponent precipitate in the quasisteady state regime
 Tohid Naseri^{1},
 Daniel Larouche^{1}Email authorView ORCID ID profile,
 Rémi Martinez^{2} and
 Francis Breton^{3}
 Received: 2 February 2018
 Accepted: 17 May 2018
 Published: 23 May 2018
Abstract
An exact analytical solution of the Fick’s second law was developed and applied to the mixedmode growth of a multicomponent ellipsoidal precipitate growing with constant eccentricities in the quasistationary regime. The solution is exact if the nominal composition, equilibrium concentrations and material properties are assumed constant, and can be applied to compounds having no limitations in the number of components. The solution was compared to the solution calculated by a diffusioncontrolled application software and it was found that the solute concentrations at the interface can be determined knowing only the nominal composition, the full equilibrium concentrations and the coefficients of diffusion. The thermodynamic calculations owing to find alternative tielines are proven to be useless in the mixedmode model. From this, it appears that the search of alternative tielines is computationally counterproductive, even when the interface has a very high mobility. A more efficient computational scheme is possible by considering that a moving interface is not at equilibrium.
Keywords
 Mixedmode
 Interfacial mobility
 Ellipsoidal precipitate
 Multicomponent diffusion
Introduction
The diffusional growth of precipitates in a supersaturated matrix is one of the most fundamental problems in metallurgy. Although this topic is fully addressed in textbooks (Christian, 1965; Aaronson et al., 2010; Kozeschnik, 2013a), accurate estimations of growth velocities are difficult because of the assumptions considered to obtain a solution. For the time being, as pointed out by E. Kozeschnik in his book, “there exist no general closed analytical solutions for the multicomponent growth of precipitates” (Kozeschnik, 2013b). The only solutions available today are those obtained for binary systems in the quasi steadystate regime. Zener (Zener, 1949) provided the solution for spherical precipitates in the diffusion control mode while Horvey and Cahn (Horvay & Cahn, 1961), as well as Ham (Ham, 1959), provided the solution in the same mode for the shape preserving growth of ellipsoidal precipitates. Larouche (Larouche, 2017) upgraded the solution of the latter precipitates for the mixedmode growth regime, where the growth is partly controlled by diffusion and the interface. The reason why it is difficult to obtain a closed analytical solution for multicomponent precipitates when one assumes local equilibrium at the interface is because there are too many constraints at the interface. To understand the problem, let us consider the formation of a precipitate A_{x}B_{y} in a binary system rich in species A. In the quasi steadystate regime, the boundary condition at infinity is c = \( \overline{c} \) where c is the molar concentration and \( \overline{c} \) is the nominal composition of solute. The problem is solved by imposing two boundary conditions at the interface. The first one is the Stefan condition, which ensures that the velocity of the boundary is equal to the velocity imposed by the flux of atoms coming from the matrix. The second boundary condition is the concentration of solute at the interface, which is imposed by the local equilibrium assumption in the diffusion controlled mode. These two boundary conditions are enough to find the two unknowns at a given time, which are the size of the precipitate and the matrix solute concentration gradient at the interface. Problems arise when one considers the growth of a precipitate B_{x}C_{y} in a ternary system rich in species A. In this case, one unknown is added, which is the matrix solute concentration gradient at the interface of solute C. However, this adds two new boundary conditions. These are the Stefan boundary condition and the local equilibrium assumption for solute C. So, in this case, one has to deal with a problem having 4 boundary conditions at the interface and 3 unknowns, which are the size of the precipitate and the matrix solute concentration gradients at the interface for solutes B and C. And for each species added in the composition of the precipitate, one has to add two new boundary conditions for only one extra unknown. To address this problem, Kirkaldy (Kirkaldy, 1958) and others suggested that there is only one independent solute concentration at the interface while the others are dependent to each other. Their determination depends on the operative tieline in the phase diagram, which evolves according to the solute diffusivity and the degree of advancement of the reaction. In general, the operative tieline differs from the conventional tieline (determined when the system is at equilibrium) and the computational scheme used to solve the set of equations is not straightforward. Moreover, a mapping of the multicomponent phase diagram is mandatory to calculate the operative solute partitioning, even under isothermal growth. The application software DICTRA (Andersson et al., 2002) is well known to provide a numerical procedure solving the moving boundary problem for onedimensional geometries. But one can testify about the difficulty of getting a solution with this software when the starting values cannot be found at the first time step. And this happens notably when the precipitate grows by the diffusion of two and more solute elements.
In this paper, an exact analytical solution of the shape preserving growth of a multicomponent ellipsoidal precipitate in the quasi stationary regime is presented. The solution is then applied for the growth of a β″ needle shaped particle growing in the AlMgSi system. The advantages of this approach, involving the interfacial mobility, are further highlighted and discussed. Finally, a comparison with the solution provided by the application software DICTRA is performed for a spherical Mg_{2}Si precipitate growing in an Aluminium matrix.
The role of the interfacial mobility in the early stage of the growth
Mathematical analysis
The detailed solution of the moving boundary problem of an ellipsoidal precipitate growing with constant eccentricities in the quasi stationary mode has been presented in (Larouche, 2017) for a binary alloy. Without repeating all the details contained in this paper, we will summarize the most important steps of the solution procedure.
Where D is the coefficient of diffusion and \( \overrightarrow{r} \) is the vector position. Assuming a shape conservative growth (eccentricities remain constant) and using the ellipsoidal coordinates, one obtains a general solution:
Where t is the time and ρ corresponds to an ellipsoidal surface in the Eulerian ellipsoidal system of coordinates. The constant k is called the interface migration coefficient because it was introduced with the travelling set of coordinates. At this stage, there are 3 variables that have to be determined for a given time. These are the size of the precipitate a_{1}, the concentration at the interface c^{*} and the constant k. This constant must be set in order to adapt the velocity of the reference frame V_{ r } to the velocity of the interface at the beginning of the mixedmode regime. Before this time (t_{ c }), the growth is assumed to be 100% controlled by the interface. The solution is valid only after this time.
 BC#1.
Mass conservation across the interface (Stefan condition)
\( \dot{N}=\underset{S}{\int }J\cdot dS=\left({c}_{\beta}^{\ast }{c}^{\ast}\right)\cdot \frac{dV}{dt} \), where S and V are respectively the surface and the volume of the precipitate
 BC#2.
Growth velocity governed by the mobility of the interface as expressed by Eq. (3)
With these 2 boundary conditions, one can determine a_{1} and c^{*} for any time > t_{ c }, once the constant k has been set. This procedure is sufficient for a binary alloy. But when two solutes and more diffuse during the growth of a precipitate, we have as many Stefan boundary conditions (BC #1) as there are species involved.
\( {\overline{c}}_i \) is the nominal molar fraction of solute i in the matrix,
D_{ i } (m^{2} s^{− 1}) is the coefficient of diffusion of solute i in the matrix phase,
The solution is then completed.
Application of the model
Chemical composition of the alloy used for the mixedmode growth modelling
Si  Mg  Al  

at.%  1.24  0.35  Bal. 

a_{1} = a_{ c } = 6 nm

a_{2} = a_{ 3 } = 0.67 nm
Equilibrium values calculated at 453 K and 101 kPa with the nominal composition given in Table 1. The Al matrix and the precipitate β″ were considered as the only active phases
Molar fraction of Mg in the precipitate: \( {\boldsymbol{c}}_{\boldsymbol{\beta}, \boldsymbol{Mg}}^{\ast} \)  0.4545 
Molar fraction of Si in the precipitate: \( {\boldsymbol{c}}_{\boldsymbol{\beta}, \boldsymbol{Si}}^{\ast} \)  0.5105 
Molar fraction of Al in the precipitate: \( {\boldsymbol{c}}_{\boldsymbol{\beta}, \boldsymbol{Al}}^{\ast} \)  0.0350 
Molar fraction of Mg in the matrix:\( {\boldsymbol{c}}_{\boldsymbol{eq},\boldsymbol{Mg}}^{\boldsymbol{\infty}} \)  7.06E5 
Molar fraction of Si in the matrix: \( {\boldsymbol{c}}_{\boldsymbol{eq},\boldsymbol{Si}}^{\boldsymbol{\infty}} \)  8.61E3 
Coefficient of diffusion of Mg in the matrix (m^{2}/s)  2.01E19 
Coefficient of diffusion of Si in the matrix (m^{2}/s)  1.35E19 
Molar volume of the precipitate (m^{3}/mol)  1.048E5 
Results
Discussion
For the first time, an analytical solution is provided for the mixedmode growth of a multicomponent precipitate, the latter growing with the simultaneous diffusion of several species in the matrix. In the example described above, 3 species were involved in the formation of the precipitate. The method can include as much species as wanted since it is always possible to reduce the number of equations to 1, with only 1 unknown to find. This unknown is the size of the precipitate at a given time. Once this variable is determined, the concentrations of all species at the interface can be calculated easily. The method can therefore be applied for the growth of any type of particle, under the condition that its composition is known and invariant. The applicability of the solution is limited to situations where each precipitate grows in a relatively large matrix since the solution is exact for one precipitate growing in an infinite matrix. This is the same limitation which applies with the popular Zener stationary approximate solution. However, the mixedmode growth solution takes into account the moving boundary problem under the assumption of a quasistationary growth, unlike the Zener stationary approximate solution, which ignores completely the term ∂c/∂t in the mass conservation equation.
The time evolution of the concentrations at the matrix interface (\( {c}_i^{\ast } \)) is certainly the most interesting result obtained as it can be considered as characteristic of the mixedmode growth. During the early stages of growth, the interface concentrations are equal to the nominal concentrations. After, they vary gradually with time toward their stabilized value at a rate controlled by the interfacial mobility. The offset between the final and equilibrium concentrations is produced by the difference existing between the coefficients of diffusion of the 2 solutes. Indeed, if one makes the calculations using the same coefficients of diffusion for the solutes, then one obtains a perfect match between the final and equilibrium values. Unlike the case where equilibrium is assumed at the interface, the time evolution of \( {c}_i^{\ast } \) is free of the drastic change occurring at the beginning of the growth. This gives a more realistic solution since the interfacial mobility is surely not infinite.
Equilibrium values calculated at 453 K and 101 kPa with the nominal composition given in Table 1. The Al matrix and the precipitateβMg_{2}Si were considered as the only active phases
Molar fraction of Mg in the precipitate: \( {\boldsymbol{c}}_{\boldsymbol{\beta}, \boldsymbol{Mg}}^{\ast} \)  0.6667 
Molar fraction of Si in the precipitate: \( {\boldsymbol{c}}_{\boldsymbol{\beta}, \boldsymbol{Si}}^{\ast} \)  0.3333 
Molar fraction of Mg in the matrix:\( {\boldsymbol{c}}_{\boldsymbol{eq},\boldsymbol{Mg}}^{\boldsymbol{\infty}} \)  2.14E6 
Molar fraction of Si in the matrix: \( {\boldsymbol{c}}_{\boldsymbol{eq},\boldsymbol{Si}}^{\boldsymbol{\infty}} \)  1.07E2 
Coefficient of diffusion of Mg in the matrix (m^{2}/s)  2.01E19 
Coefficient of diffusion of Si in the matrix (m^{2}/s)  1.35E19 
Molar volume of the precipitate (m^{3}/mol)  1.000E5 
Radius of precipitate at the start of the mixedmode regime: a_{ c } (nm)  0.4 
Radius of precipitate at the start of the DICTRA simulation (nm)  0.4 
Radius of the system in the DICTRA simulation (nm)  50 
Surface energy: γ:  0 
Conclusion
An exact analytical solution of the mixedmode growth of a multicomponent ellipsoidal precipitate having constant eccentricities was developed for the quasi stationary regime. The solution was applied to simulate the growth of a needle shape (prolate spheroid) AlMgSi compound and a spherical Mg_{2}Si compound growing in their aluminium based matrix. A comparison made with the diffusioncontrolled simulation software DICTRA showed that the solute concentrations at the interface in the early stages of the reaction can be found irrespective of local equilibrium considerations. The variables having a significant effect are the full equilibrium concentrations, the nominal composition and the coefficients of diffusion. With the mixedmode growth approach, multicomponent mapping of the phase diagram, allowing the determination of alternative tielines, is not necessary to solve the theoretical model.
Declarations
Acknowledgements
The authors thanks Martin Fortier of RioTinto and Denis Massinon of Linamar Montupet Light Metal Casting for their support.
Funding
This work was supported financially by the Natural Sciences and Engineering Research Council of Canada (Grant RDCPJ 468550–14) and the industrial partners RioTinto and Montupet.
Authors’ contributions
TN and DL made the theoretical developments and calculations presented in this paper. RM and FB analyzed the results and critically evaluated the relevance of the approach for the simulation of precipitation kinetics in multicomponent alloys. All authors read and approved the final manuscript.
Competing interests
The authors declare that they have no competing interests.
Open AccessThis article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
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