Kinetic Analysis for Contribution of Interface Reaction to Migration of γ/α Interface in Binary Fe–C System

Abstract

The kinetics for the isothermal carburization of pure iron (Fe) at a temperature of 1073 K (800°C) for times between 1.8 ks (0.5 h) and 32.4 ks (9 h) was experimentally observed by Togashi and Nishizawa. According to the observation, the austenite (γ) phase with the face-centered cubic (fcc) structure is produced on the Fe specimen of the ferrite (α) phase with the body-centered cubic (bcc) structure, and gradually grows into the α phase. The carbon (C) concentration in the γ phase at the moving γ/α interface is greater than that of the γ/(γ + α) phase boundary in a phase diagram of the binary Fe–C system. Although the former one gradually approaches to the latter one with increasing annealing time, their difference hardly vanishes even at the longest annealing time of 32.4 ks (9 h). The molar Gibbs energy of the γ phase was described by a two-sublattice model to evaluate the chemical driving force working at the moving γ/α interface. The evaluation provides that the chemical driving force monotonically decreases with increasing annealing time. This annealing time dependence of the chemical driving was used to calculate the migration distance of the γ/α interface as a function of the annealing time. The observation for the interface migration was satisfactorily reproduced by the calculation. According to the calculation, the migration of the γ/α interface is controlled by the interface reaction at the moving interface in the early stages, but it is governed mainly by the volume diffusion of C across the γ phase and partially by the interface reaction in the late stages. The experimental annealing times mostly belong to the transition stages between the rate-controlling processes of the interface reaction and the volume diffusion.

Fig. 5 The thickness l of the γ layer versus the annealing time t. Open circles show the observation reported by Togashi and Nishizawa,⁷⁾ a solid curve and a dashed line indicate the calculation from eq. (11), and a thin dashed line represents the extrapolation of the dashed line in the early stages.

1. Introduction

Owing to isothermal reactive diffusion at an appropriate constant annealing temperature, an intermediate phase may form as a layer in a diffusion couple consisting of different metals. If the diffusion couple is semi-infinite and the layer growth is controlled by volume diffusion across the constituent phases, the thickness of the layer increases in proportion to the square root of the annealing time. Such a relationship is called the parabolic relationship. Here, the semi-infinite diffusion couple means that the thickness of each metal is semi-infinite and the original interface between the different metals is flat. For such diffusion-controlled growth of the intermediate phase, the kinetics of the layer growth is mathematically described in an analytical manner.^1–3)

According to kinetic analysis using a mathematical model,³⁾ the growth rate of the intermediate phase is predominantly determined by the interdiffusion across the intermediate phase. In the early stages of the reactive diffusion, the layer thickness is very small, and thus the interdiffusion occurs considerably fast. In such a case, the interface reaction at the moving interface becomes the bottleneck of the reactive diffusion, and hence decelerates the layer growth. For the layer growth controlled by the interface reaction, the layer thickness is linearly proportional to the annealing time. Hereafter, this relationship is called the linear relationship. As the annealing time increases, the layer thickness gradually increases, and then the interdiffusion across the intermediate phase slows down. Consequently, the rate-controlling process alters from the interface reaction in the early stages to the interdiffusion in the late stages. This means that the transition of the rate-controlling process occurs between the early and late stages.

The transition of the rate-controlling process was experimentally observed for the reactive diffusion between Ta and a bronze in previous studies.^4,5) In those experiments, diffusion couples composed of Ta and a Cu–9.3 at% Sn–0.3 at% Ti alloy were isothermally annealed at temperatures of T = 973–1053 K (700–780°C) for various times up to t = 5.26 Ms (1462 h). Owing to isothermal annealing, a layer of Ta₉Sn forms at the original Ta/(Cu–Sn–Ti) interface. According to the observation, the thickness of the Ta₉Sn layer is proportional to a power function of the annealing time. The exponent of the power function is unity at annealing times of t < t_c, and becomes smaller than 0.5 at annealing times of t > t_c, where t_c is the critical annealing time for the transition. The values of t_c are 1.83, 0.463 and 0.598 Ms at T = 973, 1023 and 1053 K, respectively. Thus, the growth of the Ta₉Sn layer is controlled by the interface reaction at t < t_c but by the interdiffusion at t > t_c. Since the exponent is smaller than 0.5 at t > t_c, however, boundary diffusion as well as volume diffusion contributes to the interdiffusion.⁶⁾ For the growth of the Ta₉Sn layer, no reliable information on the diffusion coefficient and the phase stability is available for the constituent phases in the diffusion couple, and hence kinetic analysis cannot be conducted for the transition of the rate-controlling process.

In contrast, Togashi and Nishizawa experimentally observed carburization of Fe.⁷⁾ In their experiment, a pure Fe specimen of the ferrite (α) phase with the body-centered cubic (bcc) structure was isothermally annealed at 1073 K for various periods up to 9 h in a carburization atmosphere with the same activity of C as graphite. Owing to isothermal annealing, the austenite (γ) phase with the face-centered cubic (fcc) structure is formed on the α phase, and gradually grows into the α phase. For such growth of the γ phase, they experimentally determined the concentration $w_{\text{C}}^{\text{s}}$ of C in the γ phase at the moving γ/α interface. According to their experimental result, $w_{\text{C}}^{\text{s}}$ is greater than the concentration $w_{\text{C}}^{\text{e}}$ of C for the γ/(γ + α) phase boundary of a phase diagram in the binary Fe–C system, and asymptotically approaches to $w_{\text{C}}^{\text{e}}$ with increasing annealing time. Thus, typically, the local equilibrium is not realized at the moving γ/α interface during carburization. This means that the interface reaction at the γ/α interface surely contributes to the rate-controlling process for the growth of the γ phase. Fortunately, unlike the reactive diffusion in the Ta/(Cu–Sn–Ti) system,^4,5) information on the diffusion coefficient and the phase stability is available for the γ and α phases in the binary Fe–C system. On the basis of such information, the experimental result reported by Togashi and Nishizawa⁷⁾ was theoretically analyzed using a thermodynamic model in the present study. For this analysis, the molar Gibbs energy of the γ phase was mathematically described by a two-sublattice model. This mathematical description was used to estimate the chemical driving force working at the moving γ/α interface.

2. Analysis

As mentioned in Section 1, the carburization of Fe was experimentally observed by Togashi and Nishizawa.⁷⁾ To analyze theoretically their experimental result, we consider a semi-infinite diffusion couple initially composed of a binary Fe–C alloy with the concentration $y_{\text{C}}^{\alpha 0}$ of C and a carburization atmosphere with the activity a_C of C. Hence, the diffusion couple consists of the solid and gas phases. The diffusion couple is isothermally annealed at an appropriate constant temperature. At the annealing temperature, $y_{\text{C}}^{\alpha 0}$ is located in the α single-phase region of a phase diagram in the binary Fe–C system. The semi-infinite diffusion couple means that the lengths of the alloy and the atmosphere are semi-infinite and the interface between them is flat. If the value of a_C corresponds to that for the γ single-phase region in the phase diagram at the annealing temperature, a layer of the γ phase forms on the surface of the alloy. The concentration profile of C along the direction perpendicular to the γ/α interface in the alloy is schematically drawn in Fig. 1. In this figure, the vertical axis shows the concentration y_C of C, and the horizontal axis indicates the distance z measured from the surface of the alloy. Furthermore, l is the location of the γ/α interface and thus the thickness of the γ layer, $y_{\text{C}}^{\text{f}}$ is the value of y_C on the surface of the γ layer, and $y_{\text{C}}^{\gamma \alpha }$ and $y_{\text{C}}^{\alpha \gamma }$ are those of y_C in the γ and α phases, respectively, at the γ/α interface. If the local equilibrium is realized at the moving γ/α interface during carburization, the values of $y_{\text{C}}^{\gamma \alpha }$ and $y_{\text{C}}^{\alpha \gamma }$ coincide with those of y_C for the γ/(γ + α) and α/(γ + α) phase boundaries, respectively, at the annealing temperature in the phase diagram. Solid curves in Fig. 1 show the concentration profiles of C under such local equilibrium conditions. Within the experimental annealing times, however, the observation by Togashi and Nishizawa⁷⁾ indicates that $y_{\text{C}}^{\gamma \alpha }$ is typically greater than the concentration $y_{\text{C}}^{\text{e}}$ of C for the γ/(γ + α) phase boundary. This means that the local equilibrium is not actualized at the moving γ/α interface. Hence, the interface reaction at this interface contributes to the rate-controlling process for the growth of the γ layer. In Fig. 1, $y_{\text{C}}^{\text{s}}$ is the experimental value of $y_{\text{C}}^{\gamma \alpha }$. If $y_{\text{C}}^{\text{s}} > y_{\text{C}}^{\text{e}}$, the chemical driving force works at the γ/α interface. The following technique was used to evaluate the chemical driving force.

Fig. 1

Schematic concentration profile of C across the γ phase formed on the α phase of Fe during isothermal carburization.

The molar Gibbs energy $G_{\text{m}}^{\gamma }$ of the γ phase in the binary Fe–C system is expressed by the following equation according to the two-sublattice model proposed by Hillert and Staffansson.⁸⁾   

\begin{align} G_{\text{m}}^{\gamma} &= y_{\text{Va}}^{\gamma}{^{\text{o}}}G_{\text{Fe:Va}}^{\gamma} + y_{\text{C}}^{\gamma}{^{\text{o}}}G_{\text{Fe:C}}^{\gamma} \\ &\quad + RT(y_{\text{C}}^{\gamma}\ln y_{\text{C}}^{\gamma} + y_{\text{Va}}^{\gamma}\ln y_{\text{Va}}^{\gamma}) + {^{\text{E}}}G_{\text{m}}^{\gamma} \end{align} (1)

Here, $y_{\text{C}}^{\gamma }$ and $y_{\text{Va}}^{\gamma }$ are the site fractions of C (carbon) and Va (vacancy), respectively, in the interstitial site of the γ phase, T is the absolute temperature, and R is the gas constant. On the other hand, ${}^{\text{o}}G_{\text{Fe:Va}}^{\gamma }$ and ${}^{\text{o}}G_{\text{Fe:C}}^{\gamma }$ are the values of $G_{\text{m}}^{\gamma }$ for the γ phase, where the interstitial site is completely occupied with Va and C, respectively, and ${}^{\text{E}}G_{\text{m}}^{\gamma }$ is the excess molar Gibbs energy. In eq. (1), $G_{\text{m}}^{\gamma }$ corresponds to the value for 1 mol of Fe atoms, and ${}^{\text{E}}G_{\text{m}}^{\gamma }$ is described as follows.   

\begin{equation} {^{\text{E}}}G_{\text{m}}^{\gamma} = y_{\text{Va}}^{\gamma}y_{\text{C}}^{\gamma}L_{\text{Fe:Va,C}}^{\gamma} \end{equation} (2)

Here, $L_{\text{Fe:Va,C}}^{\gamma }$ is the interaction parameter between C and Va in the interstitial site. The chemical potential $G_{\text{C}}^{\gamma }$ of C in the γ phase is calculated by the following equation.⁹⁾   

\begin{equation} G_{\text{C}}^{\gamma} = \frac{\partial G_{\text{m}}^{\gamma}}{\partial y_{\text{C}}^{\gamma}} - \frac{\partial G_{\text{m}}^{\gamma}}{\partial y_{\text{Va}}^{\gamma}} \end{equation} (3)

Here, $y_{\text{C}}^{\gamma }$ and $y_{\text{Va}}^{\gamma }$ are considered mathematically independent each other, though the following relationship holds between $y_{\text{C}}^{\gamma }$ and $y_{\text{Va}}^{\gamma }$.   

\begin{equation} y_{\text{C}}^{\gamma} + y_{\text{Va}}^{\gamma} = 1 \end{equation} (4)

Inserting eqs. (1) and (2) into eq. (3), we obtain the equation   

\begin{equation} G_{\text{C}}^{\gamma} = \Delta{^{\text{o}}}G_{\text{C}}^{\gamma} + RT\ln \frac{y_{\text{C}}^{\gamma}}{1 - y_{\text{C}}^{\gamma}} + (1 - 2y_{\text{C}}^{\gamma})L_{\text{Fe:Va,C}}^{\gamma}, \end{equation} (5)

where $\varDelta {}^{\text{o}}G_{\text{C}}^{\gamma }$ is defined as   

\begin{equation} \varDelta{^{\text{o}}}G_{\text{C}}^{\gamma} \equiv {^{\text{o}}}G_{\text{Fe:C}}^{\gamma} - {^{\text{o}}}G_{\text{Fe:Va}}^{\gamma}. \end{equation} (6)

According to eq. (5), the chemical driving force $\Delta G_{\text{C}}^{\gamma }$ is evaluated as follows.   

\begin{align} \varDelta G_{\text{C}}^{\gamma} &= G_{\text{C}}^{\gamma}(y_{\text{C}}^{\text{s}}) - G_{\text{C}}^{\gamma}(y_{\text{C}}^{\text{e}})\\ & = RT\ln \frac{y_{\text{C}}^{\text{s}}(1 - y_{\text{C}}^{\text{e}})}{y_{\text{C}}^{\text{e}}(1 - y_{\text{C}}^{\text{s}})} + 2(y_{\text{C}}^{\text{e}} - y_{\text{C}}^{\text{s}})L_{\text{Fe:Va,C}}^{\gamma} \end{align} (7)

The migration rate of the γ/α interface, v = dl/dt, is related with the chemical driving force $\Delta G_{\text{C}}^{\gamma }$ by the following equation.   

\begin{equation} v = \frac{\mathrm{d}l}{\mathrm{d}t} = M\frac{\varDelta G_{\text{C}}^{\gamma}(t)}{V_{m}} \end{equation} (8)

Here, t is the annealing time, V_m is the molar volume of the γ phase for 1 mol of Fe atoms, and M is the mobility of the γ/α interface. At a constant temperature, $y_{\text{C}}^{\text{s}}$ is the only variable parameter for $\Delta G_{\text{C}}^{\gamma }$ in eq. (7). Thus, $\Delta G_{\text{C}}^{\gamma }$ is expressed as a function of t on condition that $y_{\text{C}}^{\text{s}}$ is described as a function of t.

3. Results and Discussion

3.1 Annealing time dependence of carbon concentration

Carburization of Fe was experimentally observed by Togashi and Nishizawa⁷⁾ as mentioned earlier. In that experiment, a pure Fe specimen of the α phase was isothermally annealed at T = 1073 K (800°C) for various times from t = 1.8 ks (0.5 h) to 32.4 ks (9 h) in a carburization atmosphere with the same activity of C as graphite. During isothermal annealing, a layer of the γ phase forms on the α phase, and gradually grows into the α phase. They experimentally determined the dependencies of the thickness l of the γ layer and the concentration $w_{\text{C}}^{\text{s}}$ of C in the γ phase at the moving γ/α interface on the annealing time t. According to the experimental result, the dependence of $w_{\text{C}}^{\text{s}}$ on t is expressed as follows.⁷⁾   

\begin{equation} w_{\text{C}}^{\text{s}} = w_{\text{C}}^{\text{e}} + \frac{m}{\sqrt{t}} \end{equation} (9)

Here, $w_{\text{C}}^{\text{e}}$ is the concentration of C for the γ/(γ + α) phase boundary, and m is the proportionality coefficient. Both $w_{\text{C}}^{\text{e}}$ and $w_{\text{C}}^{\text{s}}$ are measured in mass%. Since $w_{\text{C}}^{\text{s}}$ and $w_{\text{C}}^{\text{e}}$ take small values of about 0.3 mass%, we may use the following equation instead of eq. (9).   

\begin{equation} y_{\text{C}}^{\text{s}} = y_{\text{C}}^{\text{e}} + \frac{k}{\sqrt{t}} \end{equation} (10)

Here, k is the proportionality coefficient with the dimension of s^0.5. The value of m in eq. (9) reported by Togashi and Nishizawa⁷⁾ was converted into that of k = 1.019 × 10⁻² s^0.5 in eq. (10). According to the phase diagram in the binary Fe–C system thermodynamically estimated by Gustafson,¹⁰⁾ $y_{\text{C}}^{\text{e}} = 0.0159547$ at T = 1073 K (800°C). Using these values of k and $y_{\text{C}}^{\text{e}}$, $y_{\text{C}}^{\text{s}}$ was calculated as a function of t from eq. (10). The calculation is shown as a solid line in Fig. 2. In this figure, the vertical axis shows $y_{\text{C}}^{\text{s}}$, and the horizontal axis indicates the reciprocal of the square root of t. As previously mentioned, the experiment was conducted within the annealing times of t = 1.8–32.4 ks (0.5–9 h). Although $y_{\text{C}}^{\text{s}}$ may considerably vary depending on t for t < 1.8 ks according to eq. (10), no experimental information on $y_{\text{C}}^{\text{s}}$ is available for t < 1.8 ks. Furthermore, eq. (10) gives infinitely large values of $y_{\text{C}}^{\text{s}}$ at t = 0 s, even though $y_{\text{C}}^{\text{s}}$ cannot exceed $y_{\text{C}}^{\text{f}}$ as shown in Fig. 1. Thus, eq. (10) is not applicable to annealing times of t < 1.8 ks. Therefore, for convenience’ sake, we consider that $y_{\text{C}}^{\text{s}}$ takes a constant value of 0.01619 independently of t for t = 0–1.8 ks. This value of $y_{\text{C}}^{\text{s}}$ is indicated as a horizontal dashed line in Fig. 2.

Fig. 2

The concentration $y_{\text{C}}^{\text{s}}$ of C in the γ phase at the moving γ/α interface versus the reciprocal of the square root of the annealing time t at T = 1073 K (800°C).⁷⁾

The dependence of $y_{\text{C}}^{\text{s}}$ on t in Fig. 2 is represented in a different manner in Fig. 3. In this figure, the vertical axis shows $y_{\text{C}}^{\text{s}}$, and the horizontal axis indicates the logarithm of t. As the annealing time t increases, $y_{\text{C}}^{\text{s}}$ decreases considerably at shorter annealing times but slightly at longer annealing times. Even at t = 100 ks, however, $y_{\text{C}}^{\text{s}}$ does not coincide with $y_{\text{C}}^{\text{e}}$. This means that the local equilibrium is hardly realized at the moving γ/α interface during carburization within the experimental annealing times of t = 1.8–32.4 ks (0.5–9 h).

Fig. 3

The concentration $y_{\text{C}}^{\text{s}}$ of C in the γ phase at the moving γ/α interface versus the annealing time t at T = 1073 K (800°C).⁷⁾

3.2 Annealing time dependence of chemical driving force

At a constant annealing temperature, $y_{\text{C}}^{\text{s}}$ is the only variable parameter of $\Delta G_{\text{C}}^{\gamma }$ in eq. (7). Furthermore, the dependence of $y_{\text{C}}^{\text{s}}$ on t at T = 1073 K (800°C) is expressed by eq. (10). Therefore, at this annealing temperature, $\Delta G_{\text{C}}^{\gamma }$ is expressed as a function of t. The dependence of $\Delta G_{\text{C}}^{\gamma }$ on t was calculated from eqs. (7) and (10) using the value of $L_{\text{Fe:Va,C}}^{\gamma } = - 34671$ J/mol reported by Gustafson.¹⁰⁾ The calculation at T = 1073 K (800°C) is shown as a solid curve and a dashed line in Fig. 4. In this figure, the vertical and horizontal axes indicate the logarithms of $\Delta G_{\text{C}}^{\gamma }$ and t, respectively. Since $y_{\text{C}}^{\text{s}}$ is constant for t < 1.8 ks (0.5 h), $\Delta G_{\text{C}}^{\gamma }$ is also constant. On the other hand, for t > 1.8 ks (0.5 h), $\Delta G_{\text{C}}^{\gamma }$ monotonically decreases with increasing annealing time. Even at t = 100 ks, however, $y_{\text{C}}^{\text{s}}$ does not coincide with $y_{\text{C}}^{\text{e}}$ as previously mentioned, and thus $\Delta G_{\text{C}}^{\gamma }$ takes a certain positive value. Consequently, a certain amount of the chemical driving force is still necessary to make the γ/α interface migrate towards the α phase at t = 100 ks.

Fig. 4

The chemical driving force $\varDelta G_{\text{C}}^{\gamma }$ versus the annealing time t at T = 1073 K (800°C).

3.3 Growth behavior of γ layer

As previously mentioned, the dependence of the thickness l of the γ layer on the annealing time t was experimentally measured at T = 1073 K (800°C) for t = 1.8–32.4 ks (0.5–9 h) by Togashi and Nishizawa.⁷⁾ Their result is shown as open circles in Fig. 5. In this figure, the vertical and horizontal axes indicate the logarithms of l and t, respectively. As can be seen, the thickness l monotonically increases with increasing annealing time t. If l is expressed as a power function of t, the open circles might be located on a straight line.^3–6) However, the power function cannot be applicable to the experimental result in Fig. 5. This yields that the rate-controlling process for the growth of the γ layer varies depending on the annealing time.

Fig. 5

The thickness l of the γ layer versus the annealing time t. Open circles show the observation reported by Togashi and Nishizawa,⁷⁾ a solid curve and a dashed line indicate the calculation from eq. (11), and a thin dashed line represents the extrapolation of the dashed line in the early stages.

In contrast, the growth rate v of the γ layer is described by eq. (8). If M and V_m do not vary depending on $y_{\text{C}}^{\text{s}}$ thus on t, we obtain the following equation from eq. (8).   

\begin{equation} l = \frac{M}{V_{m}}\int_{t = 0}^{t = t}\varDelta G_{\text{C}}^{\gamma}(t)\mathrm{d}t \end{equation} (11)

Using the dependence of $\Delta G_{\text{C}}^{\gamma }$ on t in Fig. 4, l was calculated as a function of t from eq. (11). The value of V_m = 7.303 × 10⁻⁶ m³/mol¹¹⁾ was adopted for this calculation. According to eqs. (7) and (10), $\Delta G_{\text{C}}^{\gamma }$ is a rather complicated function of t. Thus, the integral of $\Delta G_{\text{C}}^{\gamma }$ on the right-hand side of eq. (11) was conducted by a numerical technique.¹²⁾ For the numerical calculation, M was chosen as the only fitting parameter to minimize the function f defined as   

\begin{equation} f = \sum_{i = 1}^{p}(l_{i} - l_{i}^{\text{e}})^{2}, \end{equation} (12)

where l_i^e is the experimental value of l at t = t_i and p = 7. From the seven open circles in Fig. 5, M = 1.72 × 10⁻¹⁵ m⁴/Js was obtained at T = 1073 K (800°C). The calculation with this value of M is shown a solid curve for t > 1.8 ks (0.5 h) and a dashed line for t < 1.8 ks (0.5 h) in Fig. 5. At first glance, the open circles for the two shortest annealing times seem to deviate rather remarkably from the solid curve. As previously mentioned, however, the minimization for the summation of the differences between l_i and l_i^e was conducted using the function f defined by eq. (12). In contrast, in Fig. 5, the vertical axis shows the logarithm of l. Thus, the deviation from the solid curve is visually exaggerated for small absolute values of l at the two shortest annealing times in Fig. 5. As can be seen in Fig. 5, the dependence of l on t is satisfactorily reproduced by the calculation. Consequently, eq. (11) is able to explain quantitatively the growth behavior of the γ layer during the carburization at T = 1073 K (800°C).

3.4 Rate-controlling process for growth of γ layer

The power relationship between l and t mentioned earlier is described as follows.^3–6)   

\begin{equation} l = p\left(\frac{t}{t_{0}}\right)^{n} \end{equation} (13)

Here, p is the proportionality coefficient with the same dimension as l, and n is the dimensionless exponent, and t₀ is unit time, 1 s, which is adopted to make the argument t/t₀ of the power function dimensionless. Assuming that eq. (13) holds at each annealing time t, we obtain the dependence of n on t for the solid curve and the dashed line in Fig. 5. The result is shown as a solid curve for t > 1.8 ks (0.5 h) and a dashed line for t < 1.8 ks (0.5 h) in Fig. 6. In this figure, the vertical and horizontal axes indicate n and logarithm of t, respectively. If the growth of the γ layer is controlled by volume diffusion, n = 0.5.³⁾ In contrast, n = 1 for the growth governed by interface reaction.^4,5) According to the dashed line and the solid curve in Fig. 6, n is equal to unity at t < 1.8 ks (0.5 h) but monotonically decreases with increasing annealing time at t > 1.8 ks (0.5 h), respectively. This means that the interface reaction at the moving γ/α interface is the rate-controlling process for the γ growth in the early stages with t < 1.8 ks (0.5 h). In the early stages, n = 1 as mentioned above. Inserting n = 1 into eq. (13), we obtain the following equation.   

\begin{equation} l = \frac{p}{t_{0}}t = v_{\text{c}}t \end{equation} (14)

Here, v_c is the kinetic coefficient with the dimension of m/s and thus the migration rate of the γ/α interface or the growth rate of the γ layer in the early stages. The value of v_c = 3.58 × 10⁻⁸ m/s was obtained as shown with a thin dashed line in Fig. 5. As the annealing time increases from t = 1.8 ks (0.5 h), the contribution of the interface reaction to the rate-controlling process decreases gradually. Since n becomes close to 0.5 at t = 100 ks, the volume diffusion of C across the γ layer predominantly contributes to the rate-controlling process in the late stages. Nevertheless, n is slightly greater than 0.5 within the experimental annealing times of t = 1.8–32.4 ks (0.5–9 h). Consequently, the contribution of the interface reaction does not necessarily disappear under the annealing conditions in the experiment by Togashi and Nishizawa.⁷⁾

Fig. 6

The exponent n versus the annealing time t for the solid curve and the dashed line in Fig. 5.

The mobility M is usually expressed as a function of the temperature T by the following equation.   

\begin{equation} M = M_{0}\exp\left(-\frac{Q_{M}}{RT}\right) \end{equation} (15)

Here, M₀ is the pre-exponential factor, and Q_M is the activation enthalpy. The values of M₀ and Q_M for the grain boundary in pure Fe were reported by Hillert.¹³⁾ These values were argued also by Hillert and Höglund.¹⁴⁾ The dependence of M on T was calculated from eq. (15) using their values of M₀ = 3.5 × 10⁻² m⁴/Js and Q_M = 147 kJ/mol.¹⁴⁾ The calculation is shown as a dashed line in Fig. 7. In this figure, the vertical axis indicates the logarithm of M, and the horizontal axis represents the reciprocal of T. On the other hand, for the γ/α interface in the binary Fe–C system, Krielaart et al.¹⁵⁾ reported M₀ = 4 × 10⁻⁷ m⁴/Js and Q_M = 140 kJ/mol. The calculation with these values is shown as a dotted line in Fig. 7. Furthermore, Wits et al.¹⁶⁾ reported M₀ = 1.7 × 10⁻⁵ m⁴/Js and Q_M = 140 kJ/mol for the γ/α interface in the binary Fe–X system, where X = Co, Cu, Mn, Cr and Al. A dashed-and-dotted line in Fig. 7 indicates the calculation with these values. In contrast, diffusion induced grain-boundary migration (DIGM) in the Fe(Zn) system was experimentally observed by Li and Hillert.¹⁷⁾ Here, DIGM is the phenomenon that a region with different composition is left behind a moving boundary due to grain boundary migration combined with the diffusion of solute atoms along the moving boundary.^17,18) Furthermore, the notation A(B) indicates that a solute B diffuses into either a pure metal A or a binary A–B alloy of the A-rich single-phase according to convention.¹⁸⁾ The values of M for the moving boundary of DIGM in the Fe(Zn) system reported by Li and Hillert¹⁷⁾ provide M₀ = 2.28 × 10⁻³ m⁴/Js and Q_M = 236 kJ/mol. The calculation with these values is shown as a solid line in Fig. 7. In contrast, the value of M = 1.72 × 10⁻¹⁵ m⁴/Js at T = 1073 K (800°C) for the γ/α interface in the binary Fe–C system obtained in the present study is represented as an open circle in Fig. 7. As can be seen, M is almost three order of magnitude smaller for the dashed-and-dotted line than for the dashed line and more than one order of magnitude smaller for the dotted line than for the dashed-and-dotted line. Such differences among the dashed, dotted and dashed-and-dotted lines were exhaustively discussed by Hillert and Höglund¹⁴⁾ and by Gamsjäger et al.¹⁹⁾ Furthermore, M is smaller for the solid line than for the dotted line. The value of M obtained in the present study is rather close to that of the grain boundary for DIGM in the Fe(Zn) system. This implies that the segregation of solute atoms in the moving interface or boundary considerably decelerates the migration of the interface or boundary, respectively. Such a deceleration effect is usually called the solute drag effect. The values of M for DIGM and DIR¹⁸⁾ (diffusion induced recrystallization) in various binary systems were extensively discussed in previous studies.^20–23)

Fig. 7

The mobility M versus the reciprocal of the annealing temperature T. An open circle shows the value of M for the solid curve and the dashed line in Fig. 5, a dashed line indicates that of M for the grain boundary in Fe,^13,14) a dotted line represents that of M for the γ/α interface in the Fe–C system,¹⁵⁾ a dashed-and-dotted line shows that of M for the γ/α interface in the Fe–X system,¹⁶⁾ and a solid line indicates that of M for the grain boundary of DIGM in the Fe(Zn) system.¹⁷⁾

4. Conclusions

The migration behavior of the γ/α interface during isothermal carburization of Fe at a temperature of T = 1073 K (800°C) for times of t = 1.8–32.4 ks (0.5–9 h) was experimentally observed by Togashi and Nishizawa.⁷⁾ Their observation indicates that the concentration $y_{\text{C}}^{\text{s}}$ of C in the γ phase at the moving γ/α interface is greater than the equilibrium concentration $y_{\text{C}}^{\text{e}}$. Although $y_{\text{C}}^{\text{s}}$ gradually approaches to $y_{\text{C}}^{\text{e}}$ with increasing annealing time t, $y_{\text{C}}^{\text{s}}$ is still slightly greater than $y_{\text{C}}^{\text{e}}$ even at the longest annealing time of t = 32.4 ks (9 h). The observation was theoretically analyzed using a thermodynamic model. According to the analysis, the chemical driving force working at the moving γ/α interface monotonically decreases with increasing annealing time t. On the basis of such dependence of $\Delta G_{\text{C}}^{\gamma }$ on t, the migration distance l of the γ/α interface was numerically calculated as a function of t. The calculation satisfactorily reproduces the observation. In the early stages, the interface reaction at the γ/α interface controls the migration of the interface. On the other hand, in the late stages, the interface migration is governed mainly by the volume diffusion of C across the γ phase and partially by the interface reaction. Consequently, the experimental annealing times almost belong to the transition stages between the rate-controlling processes of the interface reaction and the volume diffusion.

Acknowledgements

The present study was supported by the Iketani Science and Technology Foundation in Japan. The study was also partially supported by a Grant-in-Aid for Scientific Research from the Ministry of Education, Culture, Sports, Science and Technology of Japan.

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