Influence of Morphology of Cementite on Kinetics of Austenitization in the Binary Fe–C System

Abstract

When a binary Fe–C alloy with the ferrite (α) and cementite (θ) two-phase microstructure is isothermally annealed at a certain high temperature for the single-phase region of the austenite (γ) phase, the γ phase is produced at the α/θ interface by the reactive diffusion between the α and θ phases. Usually, this phenomenon is called austenitization. Owing to austenitization, the θ phase will completely dissolve into the γ phase at sufficiently long annealing times. For the flat plate of the γ phase produced between the α and θ lamellae, the one-dimensional diffusion of C occurs along the direction normal to the α/γ and γ/θ interfaces. In contrast, for the spherical particle of the θ phase distributed in the matrix of the α phase, the θ phase particle is covered with a spherical shell of the γ phase. In such a case, the three-dimensional diffusion of C in the spherical coordinate system occurs along the radial direction. The kinetics of the C diffusion is different from each other between the one-dimensional and three-dimensional coordinate systems. Consequently, the morphology of the θ phase will influence the growth behavior of the γ phase. To examine such influence, the dissolution of the θ phase was theoretically analyzed using kinetic models under various assumptions. On the basis of the analysis, the time-temperature-dissolution (TTD) diagram was constructed for each shape of the θ phase. This diagram provides quantitative information on the relationship between the dissolution time and the annealing temperature. According to the TTD diagram, the dissolution of the θ phase into the γ phase takes place much faster for the spherical morphology than for the flat one.

Fig. 14 The ratio t_d-S/t_d-P (Φ) versus the temperature T shown as dotted, dashed and solid curves for r₀ = 0.15, 0.25 and 0.50 µm, respectively.

1. Introduction

Binary Fe–C alloy with eutectoid composition shows single-phase microstructure of the austenite (γ) phase at temperatures of T > T_e and indicates two-phase microstructure consisting of the ferrite (α) and cementite (θ) phases at temperatures of T < T_e. Here, T is the temperature, T_e is the eutectoid temperature of 1000 K (727°C), and the concentration of C for the eutectoid composition is 0.76 mass% C or 3.46 at% C.¹⁾ If the eutectoid Fe–C alloy with the α + θ two-phase microstructure is isothermally annealed at a certain temperature of T > T_e, the γ phase will form at the α/θ interface owing to the reactive diffusion²⁾ between the α and θ phases. Such a phenomenon is usually called austenitization. During austenitization, the α/γ interface moves towards the α phase, and the γ/θ interface migrates towards the θ phase. At sufficiently long annealing times, the α and θ phases completely dissolve into the γ phase, and thus the γ single-phase microstructure is realized for the eutectoid alloy.

The kinetics of austenitization was investigated by many researchers.^3–16) For instance, the dissolution of the θ phase into the γ phase in the ternary Fe–Cr–C system was experimentally observed by Liu et al.¹⁴⁾ In their experiment, a ternary Fe–2.06 at% Cr–3.91 at% C alloy with the γ + θ two-phase microstructure was isothermally annealed at 1008 K (735°C) for various times up to 1.08 Ms (300 h). During annealing, the θ phase gradually dissolves into the γ phase. Using a kinetic model, they also quantitatively analyzed the observation.¹⁴⁾ The quantitative analysis could satisfactorily reproduce the kinetic behavior of the θ phase dissolution. For such austenitization, the single-phase microstructure of the γ phase is realized by the dissolution of the θ phase into the γ phase. Thus, in their kinetic model,¹⁴⁾ only the migration of the θ/γ interface was taken into consideration.

Various kinetic models for the single-interface migration^10,11) were conveniently used to analyze experimental observations of austenitization.^3–9) If the shrinkage of the α phase contributes to austenitization, however, such kinetic models cannot be reliably applicable to the quantitative analysis. The reliability of the quantitative analysis will be improved by kinetic models for the double-interface migration.^4,12,13,16) When a flat plate of the γ phase forms between lamellae of the α and θ phases, the one-dimensional diffusion of C occurs along the direction perpendicular to the α/γ and γ/θ interfaces. On the other hand, spherical particles of the θ phase may be distributed in the matrix of the α phase for the eutectoid alloy with spheroidizing annealing. In such a case, each particle of the θ phase is covered with a spherical shell of the γ phase, and thus the α/γ and γ/θ interfaces possess curvature. Consequently, the three-dimensional diffusion of C in the spherical coordinate system takes place. Since the kinetics of the C diffusion is different from each other between the one-dimensional and three-dimensional coordinate systems, the morphology of the θ phase will influence the growth behavior of the γ phase. Kinetic models for the one-dimensional and three-dimensional coordinate systems were proposed by Akbay et al.¹⁶⁾ In their study, however, attention was focused on the θ dissolution and austenitization in the one-dimensional coordinate system. Therefore, the influence of the morphology of the θ phase on the growth behavior of the γ phase is not understood well yet. In the present study, the dissolution behavior of the θ phase with planar and spherical shapes was quantitatively analyzed using different kinetic models. An analytical solution of the diffusion equation was used for the planar shape, and a numerical technique was adopted for the spherical shape. The influence of the morphology on the kinetics of austenitization was discussed on the basis of the analysis. Through the discussion, an analytical equation to describe accurately the dissolution behavior of the θ phase with the spherical shape was successfully obtained from the numerical result.

2. Methods for Analysis

2.1 Diffusion equations

Let us consider a binary Fe–C alloy with the α + θ two-phase microstructure. An example of microstructure for such an alloy is schematically depicted in Fig. 1. In this figure, a spherical particle of the θ phase with the initial radius of r = r₀ is located in the matrix of the α phase. If the binary Fe–C alloy is isothermally annealed at a temperature of T > T_e for a certain period of t > 0, a thin layer of the γ phase will form at the original α/θ interface. Here, r is the radius of the spherical θ phase, t is the annealing time, T is the annealing temperature, and T_e is the eutectoid temperature. Owing to isothermal annealing, the γ phase grows and hence the θ phase shrinks. A phase diagram in the binary Fe–C system is schematically drawn in Fig. 2(a), and the concentration profile of C across the γ phase is shown in Fig. 2(b). In the present study, the following assumptions were adopted to simplify the analysis: (1) the local equilibrium is realized at the θ/γ and γ/α interfaces; (2) the concentrations of C in the θ and α phases are uniform and coincide with that in the θ phase at the θ/γ interface and that in the α phase at the γ/α interface, respectively; (3) the diffusion coefficient of C in the γ phase is independent of the concentration of C in the γ phase; and (4) the molar volume of each phase is constant independently of the chemical composition and the temperature and equivalent to one another among the θ, γ and α phases; and (5) the capillarity effect due to the curvature of the interface is negligible. Owing to Assumption 3, Fick’s second law is expressed by the following equation.¹⁷⁾   

\begin{equation} \frac{\partial c^{\gamma}}{\partial t} = \frac{D^{\gamma}}{r^{n}}\frac{\partial}{\partial r}\left(r^{n}\frac{\partial c^{\gamma}}{\partial r} \right) \end{equation} (1)

Here, c^γ is the concentration of C in the γ phase, D^γ is the diffusion coefficient of C in the γ phase, r is the distance, t is the annealing time, n is the exponent of the power function rⁿ. The dimensions of r, t, c^γ and D^γ are m, s, mol/m³ and m²/s, respectively, and n is dimensionless. Furthermore, n = 0 for the diffusion of C in the one-dimensional coordinate system, n = 1 for that along the radial direction in the two-dimensional polar coordinate system, and n = 2 for that along the radial direction in the three-dimensional spherical coordinate system.¹⁷⁾ In the present study, however, attention is focused on the cases with n = 0 and 2. The origin of r is the center of a flat sheet of the θ phase for n = 0 and that of a spherical particle of the θ phase for n = 2. We consider the diffusion of C along the direction perpendicular to the flat sheet for n = 0 and that of C along the radial direction for n = 2. Thus, r is the distance for n = 0 but the radius for n = 2. The flux balance at the moving interface is described as follows.^2,18–20)   

\begin{equation} (c^{\theta \gamma} - c^{\gamma \theta})\frac{\mathrm{d}r^{\theta \gamma}}{\mathrm{d}t} = - J^{\gamma \theta} = D^{\gamma}\left(\frac{\partial c^{\gamma}}{\partial r} \right)_{r = r^{\theta \gamma}} \end{equation} (2a)

  

\begin{equation} (c^{\gamma \alpha} - c^{\alpha \gamma})\frac{\mathrm{d}r^{\gamma \alpha}}{\mathrm{d}t} = J^{\gamma \alpha} = - D^{\gamma}\left(\frac{\partial c^{\gamma}}{\partial r} \right)_{r = r^{\gamma \alpha}} \end{equation} (2b)

Here, r^θγ and r^γα are the locations of the θ/γ and γ/α interfaces, respectively, on the r axis; c^θγ and c^γθ are the concentrations of C in the θ and γ phases, respectively, at the θ/γ interface with r = r^θγ; c^γα and c^αγ are those in the γ and α phases, respectively, at the γ/α interface with r = r^γα; and J^γθ and J^γα are the diffusional fluxes of C in the γ phase at the θ/γ and γ/α interfaces, respectively. Due to Assumption 2, c^θ = c^θγ and c^α = c^αγ, and thus J^θγ = 0 and J^αγ = 0. Here, c^θ and c^α are the concentrations of C in the θ and α phases, respectively, J^θγ is the diffusional flux of C in the θ phase at the θ/γ interface, and J^αγ is that in the α phase at the γ/α interfaces. Therefore, as shown in eqs. (2a) and (2b), the migration rate v^θγ = dr^θγ/dt of the θ/γ interface is determined only by J^γθ, and that v^γα = dr^γα/dt of the γ/α interface is governed merely by J^γα. The initial and boundary conditions for Fig. 2 are expressed as follows.   

\begin{equation} c^{\theta}\ (0 \leq r < r_{0},\ t = 0) = c^{\theta\gamma} \end{equation} (3a)

  

\begin{equation} c^{\alpha}\ (r_{0} < r \leq \infty,\ t = 0) = c^{\alpha\gamma} \end{equation} (3b)

  

\begin{equation} r^{\theta\gamma}\ (t = 0) = r_{0} \end{equation} (3c)

  

\begin{equation} r^{\gamma\alpha}\ (t = 0) = r_{0} \end{equation} (3d)

  

\begin{equation} c^{\theta}\ (0 \leq r < r^{\theta\gamma},\ t > 0) = c^{\theta\gamma} \end{equation} (4a)

  

\begin{equation} c^{\gamma}\ (r = r^{\theta\gamma},\ t > 0) = c^{\gamma\theta} \end{equation} (4b)

  

\begin{equation} c^{\gamma}\ (r = r^{\gamma\alpha}, t > 0) = c^{\gamma\alpha} \end{equation} (4c)

  

\begin{equation} c^{\alpha}\ (r^{\gamma\alpha} < r \leq \infty,\ t > 0) = c^{\alpha\gamma} \end{equation} (4d)

As mentioned earlier, the origin of r is the center of a flat sheet of the θ phase for n = 0 and that of a spherical particle of the θ phase for n = 2. Hence, r₀ is half of the initial thickness of the flat sheet for n = 0 or the initial radius of the spherical particle for n = 2. Under the initial and boundary conditions of eqs. (3a)–(3d) and (4a)–(4d), respectively, eq. (1) will be solved for n = 0 and 2 by appropriate techniques in the present study.

Fig. 1

Schematic process of austenitization for spherical particle of cementite (θ).

Fig. 2

(a) Schematic phase diagram in the binary Fe–C system, and (b).

2.2 Analytical solution for one-dimensional diffusion

For one-dimensional diffusion with n = 0, an analytical solution of eq. (1) for n = 0 is obtained as follows under the initial and boundary conditions of eqs. (3a)–(3d) and (4a)–(4d), respectively.^2,18–20)   

\begin{equation} c^{\gamma} = c^{\gamma \theta} - \frac{c^{\gamma \alpha} - c^{\gamma \theta}}{\text{erf}(\lambda^{\gamma \alpha}) - \text{erf}(\lambda^{\theta \gamma})}\left\{\text{erf}(\lambda^{\theta \gamma}) - \text{erf}\left(\frac{r - r_{0}}{\sqrt{4D^{\gamma}t}} \right) \right\} \end{equation} (5)

Here, λ^θγ and λ^γα are the dimensionless proportionality coefficients for the θ/γ and γ/α interfaces, respectively. Using these parameters, r^θγ and r^γα are expressed as functions of t by the following equations.^2,18–20)   

\begin{equation} r^{\theta \gamma} = r_{0} + \lambda^{\theta \gamma}\sqrt{4D^{\gamma}t} \end{equation} (6a)

  

\begin{equation} r^{\gamma \alpha} = r_{0} + \lambda^{\gamma \alpha}\sqrt{4D^{\gamma}t} \end{equation} (6b)

As previously mentioned, r₀ is half of the initial thickness of the flat sheet of the θ phase for n = 0. Inserting eqs. (5), (6a) and (6b) into eqs. (2a) and (2b), we obtain the following equations.   

\begin{equation} \omega^{\theta \gamma} = \sqrt{\pi} \lambda^{\theta \gamma}\exp \{(\lambda^{\theta \gamma})^{2}\} \{\text{erf}(\lambda^{\theta \gamma}) - \text{erf}(\lambda^{\gamma \alpha})\} \end{equation} (7a)

  

\begin{equation} \omega^{\gamma \alpha} = \sqrt{\pi} \lambda^{\gamma \alpha}\exp \{(\lambda^{\gamma \alpha})^{2}\} \{\text{erf}(\lambda^{\gamma \alpha}) - \text{erf}(\lambda^{\theta \gamma})\} \end{equation} (7b)

Here, ω^θγ and ω^γα are the supersaturations for the θ/γ and γ/α interfaces, respectively, defined by the following equations.   

\begin{equation} \omega^{\theta \gamma} \equiv \frac{c^{\gamma \theta} - c^{\gamma \alpha}}{c^{\theta \gamma} - c^{\gamma \theta}} \end{equation} (8a)

  

\begin{equation} \omega^{\gamma \alpha} \equiv \frac{c^{\gamma \theta} - c^{\gamma \alpha}}{c^{\gamma \alpha} - c^{\alpha \gamma}} \end{equation} (8b)

According to Assumption 1, the C concentration c^φψ (φ, ψ = θ, γ, α) at the φ/ψ interface coincides with the corresponding C concentration of the two-phase φ-ψ tie-line at the annealing temperature T in the phase diagram of the binary Fe–C system. As a result, at a given value of T, ω^θγ and ω^γα in eqs. (7a) and (7b) are readily determined from eqs. (8a) and (8b), respectively. In such a case, λ^θγ and λ^γα are only unknown parameters in eqs. (7a) and (7b). Consequently, two unknown parameters λ^θγ and λ^γα can be evaluated from two simultaneous equations, eqs. (7a) and (7b). Since eqs. (7a) and (7b) are implicit functions of λ^θγ and λ^γα, however, the evaluation should be conducted by an appropriate numerical technique. For the numerical evaluation, the Newton-Raphson method²¹⁾ was used in the present study. Inserting the evaluated values of λ^θγ and λ^γα into eqs. (6a) and (6b), we can quantitatively estimate the values of r^θγ and r^γα as functions of t. The migration distance l^φψ of the φ/ψ interface is defined as follows.   

\begin{equation} l^{\theta \gamma} \equiv r_{0} - r^{\theta \gamma} = - \lambda^{\theta \gamma}\sqrt{4D^{\gamma}t} \end{equation} (9a)

  

\begin{equation} l^{\gamma \alpha} \equiv r^{\gamma \alpha} - r_{0} = \lambda^{\gamma \alpha}\sqrt{4D^{\gamma}t} \end{equation} (9b)

According to the concentration profile of C in Fig. 2(b) and the flux balance in eqs. (2a) and (2b), we may readily expect that the θ/γ interface migrates towards the θ phase and the γ/α interface moves into the α phase. Consequently, in eqs. (9a) and (9b), r^γα > r₀ > r^θγ, l^θγ > 0 and l^γα > 0, and hence λ^θγ < 0 and λ^γα > 0. Equations (9a) and (9b) can be rewritten as follows.   

\begin{equation} (l^{\theta \gamma})^{2} = 4(\lambda^{\theta \gamma})^{2}D^{\gamma}t = K^{\theta \gamma}t \end{equation} (10a)

  

\begin{equation} (l^{\gamma \alpha})^{2} = 4(\lambda^{\gamma \alpha})^{2}D^{\gamma}t = K^{\gamma \alpha}t \end{equation} (10b)

Equations (10a) and (10b) are usually called a parabolic relationship, where K^θγ and K^γα are the parabolic coefficients. Since λ^θγ and λ^γα are dimensionless, the dimension of K^θγ and K^γα coincides with that of D^γ. On the other hand, the migration rate v^φψ = dr^φψ/dt of the φ/ψ interface is obtained from eqs. (6a) and (6b) as follows.   

\begin{equation} v^{\theta \gamma} = \frac{\mathrm{d}r^{\theta \gamma}}{\mathrm{d}t} = \lambda^{\theta \gamma}\sqrt{\frac{D^{\gamma}}{t}} \end{equation} (11a)

  

\begin{equation} v^{\gamma \alpha} = \frac{\mathrm{d}r^{\gamma \alpha}}{\mathrm{d}t} = \lambda^{\gamma \alpha}\sqrt{\frac{D^{\gamma}}{t}} \end{equation} (11b)

Since λ^θγ < 0 and λ^γα > 0, v^θγ < 0 and v^γα > 0 in eqs. (11a) and (11b), respectively. Inserting r^θγ = 0 into eq. (6a), we obtain the annealing time t_d for the complete dissolution of the θ phase in the α phase as follows.   

\begin{equation} t_{\text{d}} = \frac{1}{4D^{\gamma}}\left(\frac{r_{0}}{\lambda^{\theta \gamma}} \right)^{2} \end{equation} (12)

Hereafter, t_d is called the dissolution time. From eq. (12), it is obvious that t_d is proportional to the square of r₀ for given values of λ^θγ and D^γ.

2.3 Approximate solution for three-dimensional diffusion

Analytical solutions of eq. (1) for three-dimensional diffusion with n = 2 under various initial and boundary conditions are extensively discussed by Carslaw and Jaeger.²²⁾ On the basis of the analytical solution for precipitation of a spherical particle in a matrix with infinite size, three different approximations were compared with the analytical solution by Aaron et al.²³⁾ They are stationary-interface, stationary-field and linear-gradient approximations. Various approximations were also proposed by many researchers.^{10,11,16,24,25)} On the other hand, unlike the precipitation, no analytical solution of eq. (1) is known for dissolution of a spherical particle with finite size in a matrix with infinite size. Thus, stationary-field and stationary-interface approximations were used to evaluate the kinetics of the dissolution by Aaron et al.²³⁾ The following relationship is assumed for the stationary-field approximation.^16,23,25)   

\begin{equation} \frac{\partial c^{\gamma}}{\partial t} = 0 \end{equation} (13)

From eqs. (1) and (13), we obtain the following equation.¹⁶⁾   

\begin{equation} c^{\gamma} = A\int_{r^{\gamma\theta}}^{r}\frac{\mathrm{d}r}{r^{n}} + B \end{equation} (14)

The integration constants A and B in eq. (14) for n = 2 are calculated as follows under the initial and boundary conditions of eqs. (3a)–(3d) and (4a)–(4d).¹⁶⁾   

\begin{equation} A = \cfrac{c^{\gamma \theta} - c^{\gamma \alpha}}{\cfrac{1}{r^{\theta \gamma}} - \cfrac{1}{r^{\gamma \alpha}}} \end{equation} (15a)

  

\begin{equation} B = c^{\gamma \theta} \end{equation} (15b)

From eqs. (2a)–(2b), (14), (15a) and (15b), we obtain the following equations for n = 2.¹⁶⁾   

\begin{equation} (r^{\theta \gamma})^{3} = (1 + \varOmega)(r_{0})^{3} - \varOmega (r^{\gamma \alpha})^{3} \end{equation} (16a)

  

\begin{align} t &= \frac{1}{2D^{\gamma}\omega^{\gamma \alpha}\varOmega}\{(1 + \varOmega)(r_{0})^{2} - \varOmega (r^{\gamma \alpha})^{2} \\ &\quad - [(1 + \varOmega)(r_{0})^{3} - \varOmega (r^{\gamma \alpha})^{3}]^{\frac{2}{3}}\} \end{align} (16b)

  

\begin{equation} \frac{\mathrm{d}r^{\theta \gamma}}{\mathrm{d}t} = \frac{D^{\gamma}\omega^{\theta \gamma}r^{\gamma \alpha}}{r^{\theta \gamma}(r^{\theta \gamma} - r^{\gamma \alpha})} \end{equation} (16c)

  

\begin{equation} \frac{\mathrm{d}r^{\gamma \alpha}}{\mathrm{d}t} = \frac{D^{\gamma}\omega^{\gamma \alpha}r^{\theta \gamma}}{r^{\gamma \alpha}(r^{\gamma \alpha} - r^{\theta \gamma})} \end{equation} (16d)

Here, Ω is defined as¹⁶⁾   

\begin{equation} \varOmega \equiv \frac{\omega^{\theta \gamma}}{\omega^{\gamma \alpha}} = \frac{c^{\gamma \theta} - c^{\gamma \alpha}}{c^{\theta \gamma} - c^{\gamma \theta}}\frac{c^{\gamma \alpha} - c^{\alpha \gamma}}{c^{\gamma \theta} - c^{\gamma \alpha}} = \frac{c^{\gamma \alpha} - c^{\alpha \gamma}}{c^{\theta \gamma} - c^{\gamma \theta}}. \end{equation} (17)

Furthermore, like eqs. (9a) and (9b), l^θγ and l^γα are defined as follows.   

\begin{equation} l^{\theta \gamma} \equiv r_{0} - r^{\theta \gamma} \end{equation} (18a)

  

\begin{equation} l^{\gamma \alpha} \equiv r^{\gamma \alpha} - r_{0} \end{equation} (18b)

As mentioned earlier, r₀ is the initial radius of a spherical particle of the θ phase for n = 2. In contrast, the dissolution time t_d for n = 2 corresponding to eq. (12) is estimated by the equation¹⁶⁾   

\begin{equation} t_{\text{d}} = \frac{(r_{0})^{2}}{2D^{\gamma}\omega^{\gamma \alpha}}\left[\frac{1 + \varOmega}{\varOmega} - \left(\frac{1 + \varOmega}{\varOmega} \right)^{\frac{2}{3}} \right]. \end{equation} (19)

From eq. (19), it is evident that t_d is proportional to the square of r₀ for given values of Ω, ω^γα and D^γ.

2.4 Numerical technique for three-dimensional diffusion

For n = 2, eq. (1) is rewritten as follows.   

\begin{equation} \frac{\partial c^{\gamma}}{\partial t} = \frac{D^{\gamma}}{r^{2}}\frac{\partial}{\partial r}\left(r^{2}\frac{\partial c^{\gamma}}{\partial r} \right) = D^{\gamma}\frac{\partial^{2}c^{\gamma}}{\partial r^{2}} + \frac{2D^{\gamma}}{r}\frac{\partial c^{\gamma}}{\partial r} \end{equation} (20)

Under the initial and boundary conditions of eqs. (3a)–(3d) and (4a)–(4d), respectively, eq. (20) can be numerically calculated using a technique reported in previous studies.^26–28) For the numerical calculation, a Crank-Nicolson implicit method²⁹⁾ was combined with a finite-difference technique.³⁰⁾ The calculation procedure was explained in detail elsewhere,³¹⁾ where the kinetics of the reactive diffusion in various coordinate systems was theoretically analyzed by a numerical technique.

3. Results and Discussion

3.1 Temperature dependence of supersaturation

As previously mentioned, the dimension of c^φ (φ = θ, γ, α) is mol/m³. Furthermore, c^φ is related with the mol fraction x^φ of C in the φ phase as follows.   

\begin{equation} c^{\varphi} = \frac{x^{\varphi}}{V_{\text{m}}^{\varphi}} \end{equation} (21)

Here, $V_{\text{m}}^{\varphi }$ is the molar volume of the φ phase with the dimension of m³/mol, and x^φ is dimensionless. According to Assumption 4, $V_{\text{m}}^{\theta }$, $V_{\text{m}}^{\gamma }$ and $V_{\text{m}}^{\alpha }$ are constant independently of c^φ and T, and $V_{\text{m}}^{\theta } = V_{\text{m}}^{\gamma } = V_{\text{m}}^{\alpha }$. Thus, c^φ is automatically replaced with x^φ in all the relevant equations mentioned above.

The mol fractions x^φψ and x^ψφ for the φ/ψ tie-line in the binary Fe–C system were calculated with Thermo-Calc Ver. S using SSOL4 database in the temperature range of T = 1000–1173 K. The calculations are shown as solid curves in Fig. 3. As reported in a previous study,²⁸⁾ the temperature dependence of x^φψ is reliably described by the following equation in the temperature range of T = 1000–1173 K.   

\begin{equation} x^{\varphi \psi} = a_{0} + a_{1}T + a_{2}T^{2} + a_{3}T^{3} \end{equation} (22)

The values of the coefficient a_i (i = 0, 1, 2, 3) for x^αγ, x^γα, x^γθ and x^θγ are listed in Table 1. Utilizing the values of x^αγ, x^γα, x^γθ and x^θγ shown as the solid curves in Fig. 3, ω^θγ and ω^γα were calculated from eqs. (8a) and (8b), respectively. The calculated values of ω^θγ and ω^γα are plotted as dashed and solid curves, respectively, against the temperature T in Fig. 4. As can be seen, both ω^θγ and ω^γα increase with increasing temperature T. As T increases from 1000 K to 1173 K, ω^θγ merely increases from 0 to 0.3. On the other hand, ω^γα increases from 0 to 10 with increasing value of T from 1000 K to 1140 K. Hence, the dependence of ω^γα on T is much more remarkable than that of ω^θγ on T.

Fig. 3

The mol fractions x^αγ, x^γα and x^γθ of C versus the temperature T calculated with Thermo-Calc Ver. S using SSOL4 database.

Table 1 The values of the coefficient a_i (i = 0, 1, 2, 3) in eq. (22) for x^αγ, x^γα, x^γθ and x^θγ in Fig. 3.

Fig. 4

The supersaturations ω^θγ and ω^γα versus the temperature T shown as dashed and solid curves, respectively.

3.2 Analysis for one-dimensional diffusion

According to Assumption 3, D^γ is independent of c^φ or x^φ. Thus, D^γ is only a function of T. In such a case, the dependence of D^γ on T is usually expressed as follows.   

\begin{equation} D^{\gamma} = D_{0}^{\gamma}\exp \left(-\frac{Q^{\gamma}}{RT} \right) \end{equation} (23)

Here, $D_{0}^{\gamma }$ is the pre-exponential factor, Q^γ is the activation enthalpy, and R is the gas constant. The values of $D_{0}^{\gamma } = 1.50 \times 10^{ - 5}$ m²/s and Q^γ = 135 kJ/mol are reported by Bhadeshia and Honeycombe.³²⁾ Their parameters were adopted to estimate the value of D^γ. Using the result in Fig. 4, the dependencies of λ^θγ and λ^γα on T were calculated from eqs. (7a) and (7b), respectively. The calculations of λ^θγ and λ^γα are shown as dashed and solid curves, respectively, in Fig. 5. As T increases from 1000 K to 1173 K, λ^θγ decreases from 0 to −0.13. In contrast, λ^γα increases from 0 to 1.5 with increasing value of T from 1000 K to 1165 K. Therefore, the dependence of λ^γα on T is much more remarkable than that of λ^θγ on T. Such a tendency is attributed to the dependencies of ω^θγ and ω^γα on T in Fig. 4. The positive value of λ^γα shows the migration of the γ/α interface towards the α phase, and the negative one of λ^θγ indicates that of the θ/γ interface towards the θ phase.

Fig. 5

The dimensionless proportionality coefficients λ^θγ and λ^γα versus the temperature T shown as dashed and solid curves, respectively.

According to eq. (12), t_d is proportional to the square of r₀ for given values of λ^θγ and D^γ. Thus, from eq. (12), t_d was calculated to be 1.9 s for r₀ = 0.5 µm at T = 1073 K. The value r₀ = 0.5 µm for n = 0 corresponds to the initial thickness of 1 µm for the flat sheet of the θ phase. Utilizing the result in Fig. 5, r^θγ and r^γα were calculated from eqs. (6a) and (6b), respectively, as functions of t within t = 0–1.9 s for r₀ = 0.5 µm at T = 1073 K. The calculations of r^θγ and r^γα are indicated as solid and dashed curves, respectively, in Fig. 6. From eqs. (18a) and (18b), the thickness l for the flat layer of the γ phase is calculated as follows.   

\begin{equation} l = l^{\gamma \alpha} + l^{\theta \gamma} = r^{\gamma \alpha} - r^{\theta \gamma} \end{equation} (24)

The dependence of l on t is also represented as a dotted line in Fig. 6. In this figure, the ordinate shows the logarithms of r^θγ, r^γα and l, and the abscissa indicates the logarithm of t. As shown in Fig. 5, λ^γα is positive, but λ^θγ is negative. Thus, as t increases from 0 to 1.9 s, r^γα increases from 0.5 to 4.5 µm, and r^θγ decreases from 0.5 to 0 µm. Consequently, after isothermal annealing for t = 1.9 s at T = 1073 K, the flat sheet of the θ phase with the initial thickness of 1 µm completely disappears, and the flat layer of the γ phase grows up to 4.5 µm in thickness on both sides of the θ phase.

Fig. 6

The distances r^θγ and r^γα and the thickness l of the γ phase for n = 0 versus the annealing time t shown as solid, dashed and dotted curves, respectively.

As expressed by eqs. (10a) and (10b), the growth of the γ phase and the disappearance of the θ phase obey the parabolic relationship. From these equations, we obtain the following equations.   

\begin{equation} K^{\theta \gamma} = 4(\lambda^{\theta \gamma})^{2}D^{\gamma} \end{equation} (25a)

  

\begin{equation} K^{\gamma \alpha} = 4(\lambda^{\gamma \alpha})^{2}D^{\gamma} \end{equation} (25b)

Using the result in Fig. 5, K^θγ and K^γα were calculated as functions of T from eqs. (25a) and (25b), respectively, at T = 1000–1173 K. The calculations of K^θγ and K^γα are shown as dashed and solid curves, respectively, in Fig. 7. As can bee seen, both K^θγ and K^γα monotonically increase with increasing value of T. Since the absolute value is greater for λ^γα than for λ^θγ in Fig. 5, K^γα is greater than K^θγ in Fig. 7. The dependencies of K^θγ and K^γα on T are indicated as dashed and solid curves, respectively, also in Fig. 8. In this figure, the ordinate shows the logarithms of K^θγ and K^γα, and the abscissa represents the reciprocal of T. It may be anticipated that the dependencies of K^θγ and K^γα on T are expressed by the following equation of the same formula as eq. (23).   

\begin{equation} K^{\varphi \phi} = K_{0}^{\varphi \phi}\exp \left(- \frac{Q_{K}^{\varphi \phi}}{RT} \right) \end{equation} (26)

Here, $K_{0}^{\varphi \phi }$ is the pre-exponential factor, and $Q_{K}^{\varphi \phi }$ is the activation enthalpy. According to the result in Fig. 8, however, eq. (26) does not necessarily hold in the whole temperature range of T = 1000–1173 K. Such irregular temperature dependence of the parabolic coefficient is reported also for numerical analysis of carburization in the binary Fe–C system.²⁸⁾

Fig. 7

The parabolic coefficients K^θγ and K^γα versus the temperature T shown as dashed and solid curves, respectively.

Fig. 8

The logarithms of K^θγ and K^γα versus the reciprocal of T shown as dashed and solid curves, respectively.

According to eqs. (25a) and (25b), the dependency of K^φψ on T is attributed to those of λ^φψ and D^γ on T. Thus, in the same manner as Fig. 8, the dependencies of |λ^θγ| and λ^γα on T in Fig. 5 are represented as dashed and solid curves, respectively, in Fig. 9. In this figure, the ordinate indicates the logarithms of |λ^θγ| and λ^γα, and the abscissa shows the reciprocal of T. Although both |λ^θγ| and λ^γα are monotone increasing functions of T in Fig. 5, they increase in irregular manners with increasing value of T in Fig. 9. However, the temperature dependencies of |λ^θγ| and λ^γα may be approximately expressed by the following equations in a certain temperature range.   

\begin{equation} |\lambda^{\theta \gamma}| = |\lambda_{0}^{\theta \gamma}|\exp \left(- \frac{Q_{\lambda}^{\theta \gamma}}{RT} \right) \end{equation} (27a)

  

\begin{equation} \lambda^{\gamma \alpha} = \lambda_{0}^{\gamma \alpha}\exp \left(- \frac{Q_{\lambda}^{\gamma \alpha}}{RT} \right) \end{equation} (27b)

Here, $\lambda _{0}^{\theta \gamma }$ and $\lambda _{0}^{\gamma \alpha }$ are the pre-exponential factors, and $Q_{\lambda }^{\theta \gamma }$ and $Q_{\lambda }^{\gamma \alpha }$ are the activation enthalpies. In the temperature range of T = 1063–1173 K, the pre-exponential factor and the activation enthalpy were estimated from the calculations in Fig. 9 by the least-squares method. The estimation provides $|\lambda _{0}^{\theta \gamma }| = 10.3$ and $Q_{\lambda }^{\theta \gamma } = 42$ kJ/mol for |λ^θγ| and $\lambda _{0}^{\gamma \alpha } = 6.40 \times 10^{3}$ and $Q_{\lambda }^{\gamma \alpha } = 81$ kJ/mol for λ^γα. On the other hand, the corresponding estimation at T = 1063–1173 K from the calculations in Fig. 8 gives $K_{0}^{\theta \gamma } = 6.76 \times 10^{ - 3}$ m²/s and $Q_{K}^{\theta \gamma } = 220$ kJ/mol for K^θγ and $K_{0}^{\gamma \alpha } = 2.95 \times 10^{3}$ m²/s and $Q_{K}^{\gamma \alpha } = 299$ kJ/mol for K^γα. Inserting eqs. (23), (26), (27a) and (27b) into eqs. (25a) and (25b), we obtain   

\begin{align} K^{\theta \gamma} &= 4|\lambda_{0}^{\theta \gamma}|^{2}D_{0}^{\gamma}\exp \left(- \frac{Q^{\gamma} + 2Q_{\lambda}^{\theta \gamma}}{RT} \right) \\ &= K_{0}^{\theta \gamma}\exp \left(- \frac{Q_{K}^{\theta \gamma}}{RT} \right) \end{align} (28a)

  

\begin{align} K^{\gamma \alpha} &= 4(\lambda_{0}^{\gamma \alpha})^{2}D_{0}^{\gamma}\exp \left(- \frac{Q^{\gamma} + 2Q_{\lambda}^{\gamma \alpha}}{RT} \right) \\ &= K_{0}^{\gamma \alpha}\exp \left(- \frac{Q_{K}^{\gamma \alpha}}{RT} \right), \end{align} (28b)

and thus   

\begin{equation} Q_{K}^{\theta \gamma} = Q^{\gamma} + 2Q_{\lambda}^{\theta \gamma} \end{equation} (29a)

  

\begin{equation} Q_{K}^{\gamma \alpha} = Q^{\gamma} + 2Q_{\lambda}^{\gamma \alpha}. \end{equation} (29b)

As mentioned earlier, eqs. (26), (27a) and (27b) hold merely approximately in a certain temperature range. Nevertheless, at T = 1063–1173 K, $Q^{\gamma } + 2Q_{\lambda }^{\theta \gamma } = 135 + 2 \times 42 = 219$ kJ/mol against $Q_{K}^{\theta \gamma } = 220$ kJ/mol in eq. (29a) and $Q^{\gamma } + 2Q_{\lambda }^{\theta \gamma } = 135 + 2 \times 81 = 297$ kJ/mol against $Q_{K}^{\gamma \alpha } = 299$ kJ/mol in eq. (29b). Thus, it is concluded that irregular temperature dependencies of K^θγ and K^γα in Fig. 8 are attributed to those of λ^θγ and λ^γα, respectively, in Fig. 9.

Fig. 9

The logarithms of |λ^θγ| and λ^γα versus the reciprocal of T shown as dashed and solid curves, respectively.

3.3 Analyses for one-dimensional and three-dimensional diffusion

Equation (20) for n = 2 was numerically calculated under the initial and boundary conditions of eqs. (3a)–(3d) and (4a)–(4d), respectively, by the technique reported in a previous study.³¹⁾ One of the results is shown in Fig. 10. In this figure, like Fig. 6, the dependencies of r^θγ, r^γα and l on t for r₀ = 0.5 µm at T = 1073 K are indicated as solid, dashed and dotted curves, respectively. As the annealing time increases from t = 0 s to t = 0.17 s, r^θγ decreases from 0.5 to 0 µm, but r^γα increases from 0.5 to 1.2 µm. Thus, t_d = 0.17 s in this case. This value is almost one order of magnitude smaller than t_d = 1.9 s in Fig. 6. According to eq. (24), l = r^γα for r^θγ = 0 µm. Hence, at t_d = 0.17 s, a spherical γ particle with radius of l = 1.2 µm is produced in the α matrix owing to the complete dissolution of the θ particle with r₀ = 0.5 µm. The values of t_d with n = 2 were obtained for r₀ = 0.15, 0.25 and 0.5 µm at T = 1000–1173 K in a similar manner to Fig. 10.

Fig. 10

The distances r^θγ and r^γα and the thickness l of the γ phase for n = 2 versus the annealing time t shown as solid, dashed and dotted curves, respectively.

On the other hand, the dissolution time t_d for n = 0 corresponding to eq. (19) for n = 2 is described as follows¹⁶⁾ using the parameter Ω defined by eq. (17).   

\begin{equation} t_{\text{d}} = \frac{(r_{0})^{2}(1 + \varOmega)}{2D^{\gamma}\omega^{\gamma \alpha}\varOmega^{2}} \end{equation} (30)

For n = 0, the dependence of t_d on T was calculated from eq. (30) for r₀ = 0.15, 0.25 and 0.5 µm at T = 1000–1173 K. The result is shown in Fig. 11(a). In contrast, Fig. 11(b) indicates the corresponding result calculated from eq. (12) using the values of λ^γα and λ^θγ in Fig. 5. Furthermore, the dependence of t_d on T with n = 2 was calculated from eq. (19). The result is represented in Fig. 12(a). Figure 12(b) shows the corresponding result for the numerical calculation mentioned above. In Figs. 11 and 12, the ordinate and the abscissa indicate T and t_d, respectively, and dotted, dashed and solid curves represent the results of r₀ = 0.15, 0.25 and 0.5 µm, respectively. Hereafter, Figs. 11 and 12 are called time-temperature-dissolution (TTD) diagram. As can be seen in Fig. 11(b), t_d monotonically decreases with increasing temperature T. At each temperature, the smaller r₀ is, the smaller t_d becomes. In Fig. 11(a), however, t_d is not a monotone decreasing function of T. Thus, eq. (30) is not reliable approximation of eq. (12). In contrast, Fig. 12(a) provides the result similar to Fig. 12(b). For each value of r₀ and T, t_d is almost one order of magnitude smaller in Fig. 12 than in Fig. 11(b). Hereafter, t_d is designated t_d-P, t_d-A and t_d-S for Figs. 11(b), 12(a) and 12(b), respectively. Here, t_d-P is the t_d value for n = 0 calculated from eq. (12) of the analytical model, t_d-A is that for n = 2 estimated from eq. (19) of the approximate model proposed by Akbay et al.,¹⁶⁾ and t_d-S is that for n = 2 evaluated from eq. (20) of the numerical model.

Fig. 11

The time-temperature-dissolution (TTD) diagrams for n = 0: (a) eq. (30), and (b) eq. (12).

Fig. 12

The time-temperature-dissolution (TTD) diagrams for n = 2: (a) eq. (19), and (b) eq. (20).

The ratio Ψ = t_d-A/t_d-S was calculated for each curve in Fig. 12(a) and 12(b). The results for r₀ = 0.15, 0.25 and 0.5 µm are shown as dotted, dashed and solid curves in Fig. 13. In this figure, the ordinate and the abscissa indicate Ψ and T, respectively. As can be seen, all the curves coincide with one another. This means that Ψ does not depend on r₀. At T = 1000 K, Ψ = 0.99. As the temperature increases, Ψ deceases and then takes the minimum value of 0.91 at T = 1100 K. On the other hand, at T > 1100 K, Ψ increases with increasing temperature and then reaches to 0.98 at T = 1173 K. Thus, unlike eq. (30) for n = 0, eq. (19) for n = 2 is rather acceptable approximation for evaluation of t_d. According to the result in Fig. 13, the evaluation error of eq. (19) is less than 10%.

Fig. 13

The ratio t_d-A/t_d-S (Ψ) versus the temperature T shown as dotted, dashed and solid curves for r₀ = 0.15, 0.25 and 0.50 µm, respectively.

In contrast, the ratio Φ = t_d-S/t_d-P was calculated for each curve in Fig. 11(b) and 12(b). The results for r₀ = 0.15, 0.25 and 0.5 µm are indicated as dotted, dashed and solid curves in Fig. 14. In this figure, the ordinate and the abscissa indicate Φ and T, respectively. Like Fig. 13, all the curves coincide with one another also in Fig. 14. Hence, Φ is insensitive to r₀. Although Φ gradually increases with increasing temperature, it takes values close to 0.1 at T = 1000–1173 K. The dependence of Φ on T is expressed by the following equation of the same formula as eq. (22).   

\begin{equation} \varPhi = b_{0} + b_{1}T + b_{2}T^{2} + b_{3}T^{3} \end{equation} (31)

The values of the coefficient b_i (i = 0, 1, 2, 3) estimated from the result in Fig. 14 are listed in Table 2. From eq. (12) and the definition of Φ, we obtain the following equation.   

\begin{equation} t_{\text{d-S}} = \varPhi t_{\text{d-P}} = \frac{\varPhi}{4D^{\gamma}}\left(\frac{r_{0}}{\lambda^{\theta \gamma}} \right)^{2} \end{equation} (32)

As previously mentioned, t_d-P and t_d-S are the values of t_d for n = 0 and 2, respectively. Combination of eq. (32) with eq. (31) provides an analytical technique for evaluation of t_d-S for n = 2 at T = 1000–1173 K in the binary Fe–C system.

Fig. 14

The ratio t_d-S/t_d-P (Φ) versus the temperature T shown as dotted, dashed and solid curves for r₀ = 0.15, 0.25 and 0.50 µm, respectively.

Table 2 The values of the coefficient b_i (i = 0, 1, 2, 3) in eq. (31) estimated from the result in Fig. 14.

3.4 Comparison with observation

The values of t_d in the binary Fe–C system were experimentally determined by various researchers.^3,9,33) Hereafter, the experimental value of t_d is denoted by t_d-E. For instance, Li et al.³³⁾ reported the t_d-E value for n = 0 at T = 1073 K with r₀ = 0.01 µm, Miyamoto et al.⁹⁾ published that for n = 2 at T = 1073 K with r₀ = 0.5 µm, and Bain³⁾ gave out that for n = 2 at T = 1118 K with r₀ = 4.5 µm. Table 3 summarizes their results. The corresponding values of t_d-P and t_d-S are also shown in Table 3. For n = 0 at T = 1073 K with r₀ = 0.01 µm, t_d-E = 3.5 s³³⁾ and t_d-P = 0.76 ms. Thus, t_d-E is more than three orders of magnitude greater than t_d-P. According to the observation by Li et al.,³³⁾ the γ phase grows owing to the migration of the interface between the γ phase and the lamellar-pearlite region. Hence, under their experimental conditions,³³⁾ the growth of the γ phase occurs due to the reverse pearlite reaction but not the reactive diffusion between the α and θ phases. In such a case, the interface has to migrate over long distances to complete the reverse pearlite reaction, and thus t_d-E becomes much greater than t_d-P. In contrast, for n = 2 at T = 1073 K with r₀ = 0.5 µm, t_d-E = 6.7 s⁹⁾ and t_d-S = 0.17 s. Hence, t_d-E is more than one order of magnitude greater than t_d-S. For n = 2 at T = 1118 K with r₀ = 4.5 µm, t_d-E = 8–15 s³⁾ and t_d-S = 5.3 s. In this case, t_d-E is several times greater than t_d-S. For spherical particles of the θ phase dispersed in the matrix of the α phase, there exists a certain distribution of the particle size. In such a case, t_d-E is predominantly determined by the maximum particle size but not by the mean particle size. In Table 3, r₀ = 0.5 and 4.5 µm are obtained from the mean particle size. If information on the maximum particle size is available, we may estimate a value of t_d-S closer to t_d-E.

Table 3 Experimental t_d-E value for n = 0 reported by Li et al.³³⁾ and those for n = 2 published by Bain³⁾ and Miyamoto et al.⁹⁾ Corresponding values of t_d-P and t_d-S are also shown.

4. Conclusions

The influence of the morphology of the cementite (θ) phase on the growth behavior of the austenite (γ) phase during austenitization of the binary Fe–C alloy with the ferrite (α) + θ two-phase microstructure was quantitatively analyzed using the different kinetic models. The various assumptions were adopted to simplify the quantitative analysis. For the flat plate of the γ phase produced between the α and θ lamellae, the one-dimensional diffusion of C takes place along the direction normal to the α/γ and γ/θ interfaces. In such a case, the analytical solution of the diffusion equation is available under the present assumptions. On the other hand, for the spherical particle of the θ phase distributed in the matrix of the α phase, the θ phase particle is covered with a spherical shell of the γ phase. As a result, the three-dimensional diffusion of C in the spherical coordinate system occurs along the radial direction. In this case, no analytical solution is known for the diffusion equation, and hence the numerical and approximate techniques were adopted for the quantitative analysis. The analysis provides the time-temperature-dissolution (TTD) diagrams for the flat sheet and the spherical particle of the θ phase. According to these TTD diagrams, the dissolution time t_d of the θ phase into the γ phase is much smaller for the spherical particle than for the flat sheet. Combining the analytical equation for the flat sheet with the result of the numerical calculation for the spherical particle, we can evaluate the TTD diagram for the spherical particle in an analytical manner.

REFERENCES

• 1)  T.B. Massalski, H. Okamoto, P.R. Subramanian and L. Kacprzak: Binary Alloy Phase Diagrams, Vol. 1, (ASM International, Materials Park, OH, 1990) p. 843.
• 2)   M.  Kajihara: Acta Mater. 52 (2004) 1193–1200.
• 3)  E.C. Bain: Alloying Elements in Steel, 2nd ed., (ASM International, Materials Park, OH, 1961) pp. 88–122.
• 4)   R.R.  Judd and  H.W.  Paxton: Trans. Metall. AIME 242 (1968) 206–215.
• 5)   G.R.  Speich and  A.  Szmirmae: Trans. Metall. AIME 245 (1969) 1063–1075.
• 6)   S.  Matsuda and  Y.  Okamura: Tetsu-to-Hagané 60 (1974) 226–238.
• 7)   K.  Aihara: CAMP-ISIJ 6 (1993) 1689.
• 8)   J.  Cai,  J.  Lin,  J.  Wilsius and  T.  Anthony Dean: Proc. Metal Forming 2008 2 (2008) 183–189.
• 9)   G.  Miyamoto,  H.  Usuki,  Z.-D.  Li and  T.  Furuhara: Acta Mater. 58 (2010) 4492–4502.
• 10)   M.J.  Whelan: Met. Sci. J. 3 (1969) 95–97.
• 11)  Y. Saito: Ph.D. Thesis, Kyoto University, Kyoto, (1986) pp. 42–48.
• 12)   M.  Hillert,  K.  Nilsson and  L.-E.  Torndahl: J. Iron Steel Inst. 209 (1971) 49–66.
• 13)  J. Crank: Free and Moving Boundary Problems, (Oxford Univ. Press, New York, 1984) pp. 101–110.
• 14)   Z.-K.  Liu,  L.  Höglund,  B.  Jönsson and  J.  Ågren: Metall. Trans. A 22 (1991) 1745–1752.
• 15)   Z.-K.  Liu and  J.  Ågren: Metall. Trans. A 22 (1991) 1753–1759.
• 16)   T.  Akbay,  R.C.  Reed and  C.  Atkinson: Acta Mater. 42 (1994) 1469–1480.
• 17)  H. Mehrer: Diffusion in Solids, (Springer-Verlag, Berlin, 2010) pp. 31–33.
• 18)  W. Jost: Diffusion of Solids, Liquids, Gases, (Academic Press, New York, 1960) p. 68.
• 19)   G.B.  Gibbs: J. Nucl. Mater. 20 (1966) 303–306.
• 20)  R.F. Sekerka and S.-L. Wang: Lectures on the Theory of Phase Transformations, ed. by H.I. Aaronson, (TMS, Warrendale, PA, 1999) pp. 231–284.
• 21)  G. Dahlquist and A. Björck: Numerical Methods, (Prentice-Hall, Englewood Cliffs, NJ, 1974) pp. 248–253.
• 22)  H.S. Carslaw and J.C. Jaeger: Conduction of Heat in Solids, 2nd ed., (Clarendon Press, Oxford, 1959) pp. 230–254.
• 23)   H.B.  Aaron,  D.  Fainstein and  G.R.  Kotler: J. Appl. Phys. 41 (1970) 4404–4410.
• 24)   H.B.  Aaron: Met. Sci. J. 2 (1968) 192–193.
• 25)  H.I. Aaronson, M. Enomoto and J.K. Lee: Mechanisms of Diffusional Phase Transformations in Metals and Alloys, (CRC Press, London, 2010) pp. 299–303.
• 26)   Y.  Tanaka and  M.  Kajihara: Mater. Sci. Eng. A 459 (2007) 101–110.
• 27)   A.  Furuto and  M.  Kajihara: Mater. Trans. 49 (2008) 294–303.
• 28)   M.  Kajihara: J. Mater. Sci. 44 (2009) 2109–2118.
• 29)  J. Crank: The Mathematics of Diffusion, (Oxford University Press, London, 1979) pp. 144–145.
• 30)   R.A.  Tanzilli and  R.W.  Heckel: Trans. Metall. AIME 242 (1968) 2313–2323.
• 31)   M.  Kajihara: unpublished work.
• 32)  H.K.D.H. Bhadeshia and S.R. Honeycombe: Steels Microstructure and Properties, 3rd ed., (Elsevier, Oxford, 2006) p. 12.
• 33)   Z.-D.  Li,  G.  Miyamoto,  Z.-G.  Yang and  T.  Furuhara: Metall. Mater. Trans. A 42 (2011) 1586–1596.