Influence of Temperature Dependence of Solubility on Kinetics for Reactive Diffusion in a Hypothetical Binary System

Transcription

1 Materials Transactions, Vol. 49, No. 4 (2008) pp. 715 to 722 #2008 The Japan Institute of Metals Influence of Temperature Dependence of Solubility on Kinetics for Reactive Diffusion in a Hypothetical Binary System Masanori Kajihara* Department of Materials Science and Engineering, Tokyo Institute of Technology, Yokohama , Japan Influence of temperature dependence of solubility for each phase on kinetics of reactive diffusion has been theoretically analyzed for a hypothetical binary system consisting of two primary solid-solution phases ( and ) and one compound phase (). For the analysis, we consider that the phase is produced owing to the reactive diffusion between the and phases in a semi-infinite diffusion couple and the growth of the phase is controlled by volume diffusion. In such a case, the parabolic relationship holds good between the thickness l of the phase and the annealing time t as follows: l 2 ¼ Kt. Here, the parabolic coefficient K is mathematically expressed as a function of the interdiffusion coefficients and the solubility ranges of the, and phases. The temperature dependencies of the parabolic coefficient K, the solubility range y and the interdiffusion coefficient D of the ( ¼ ; ; ) phase are described by Arrhenius equations of K ¼ K 0 expð Q K =RTÞ, y ¼ y 0 expð Q =RTÞ and D ¼ D 0 expð Q D =RTÞ, respectively. The analysis indicates that Q K is close to Q D þ Q at Q D Q D and Q D Q D but greater than Q D þ Q at Q D > Q D or Q D > Q D. As a consequence, the temperature dependency of the parabolic coefficient is directly related with those of the interdiffusion coefficient and the solubility range of the compound phase in the former case but not in the latter case. [doi: /matertrans.mra ] (Received December 10, 2007; Accepted January 21, 2008; Published March 12, 2008) Keywords: computer simulations, reactive diffusion, activation enthalpy, intermetallics 1. Introduction Intermetallic compounds appear as stable phases in many binary alloy systems. 1) When a diffusion couple is prepared from two different pure metals in such a binary system and then isothermally annealed at an appropriate temperature T, some of the compounds may be observed as layers at the interface between the two metals after certain periods due to reactive diffusion. For various alloy systems, the kinetics of the reactive diffusion was experimentally studied by many investigators. 2 33) When the reactive diffusion is controlled by volume diffusion, the total thickness l of the compound layers is mathematically expressed as a function of the annealing time t by the parabolic relationship l 2 ¼ Kt. The parabolic coefficient K is usually described as a function of the temperature T by an Arrhenius equation of K ¼ K 0 expð Q K =RTÞ, where R is the gas constant. The pre-exponential factor K 0 and the activation enthalpy Q K can be evaluated by the least-squares method from experimental values of K obtained at different annealing temperatures. Most of the experimental studies indicate that the temperature dependence of K is reasonably expressed by the Arrhenius equation within experimental uncertainty. 2 33) Thus, we may expect that the Arrhenius equation of K provides representative properties of the interdiffusion in the diffusion couple. However, complex information of the temperature dependencies of the solubility ranges and the interdiffusion coefficients of the relevant phases is included in K 0 and Q K. Consequently, such derivation is not so straightforward. The reactive diffusion controlled by volume diffusion was theoretically analyzed using a mathematical model in a previous study. 34) In that analysis, a hypothetical binary alloy system composed of one intermetallic compound and two primary solid-solution phases was treated, and then the *Corresponding author, kajihara@materia.titech.ac.jp growth rate of the compound layer was evaluated for various semi-infinite diffusion couples initially consisting of the two primary solid-solution phases. That mathematical model was also used to analyze numerically the relationship between the temperature dependence of the interdiffusion in each phase and the kinetics of the reactive diffusion. 35) In the numerical analysis, the temperature dependence of the interdiffusion coefficient D was expressed by an Arrhenius equation of D ¼ D 0 expð Q D =RTÞ, and the following assumptions were adopted: (a) the molar volume, the solubility range and the pre-exponential factor D 0 are constant and equivalent for all the phases; and (b) the activation enthalpy Q D is equivalent for the solution phases but different between the compound and the solution phase. According to the numerical analysis, the equation K ¼ K 0 expð Q K =RTÞ is reliable enough to express the temperature dependence of experimental values of K but not completely exact. If Q D is smaller for the compound than for the solution phases, Q K is close to Q D of the compound. In this case, the temperature dependence of K corresponds well with that of D of the compound. Such a relationship no longer holds good unless Q D is smaller for the compound than for the solution phases. In order to examine whether these conclusions are universally valid, assumption (b) was eliminated in previous numerical analyses. 36,37) Nevertheless, attention was focused on the relationship between the temperature dependency of the kinetics and those of the interdiffusion coefficients of the constituent phases. As a consequence, assumption (a) still remained in these numerical analyses. Under such conditions, the validity of the relationship between Q K and Q D could be confirmed conclusively. 36,37) Many intermetallic compounds in binary alloy systems are line compounds. 1) For such a line compound, the solubility range is constant independently of the temperature. Even in such a binary system, the solubility range of the primary solid-solution phase may vary depending on the temperature. Furthermore, there are various binary alloy systems where

2 716 M. Kajihara the solubility range changes depending on the temperature not only for the primary solid-solution phase but also for the intermetallic compound. In this case, the temperature dependence of the solubility range as well as that of the interdiffusion coefficient will influence the kinetics of the reactive diffusion. Hence, in a previous study, 38) the influence of the temperature dependence of the solubility range on the kinetics was quantitatively analyzed for the reactive diffusion in the hypothetical binary system. For the analysis, Arrhenius equations were used to express the temperature dependencies of the solubility range and the interdiffusion coefficient. However, only limited combinations of the enthalpy of solution were adopted for the intermetallic compound and the primary solid-solution phases, and thus the influence could not be confirmed conclusively. In the present study, the analysis has been extensively carried out for various combinations of the enthalpy of solution. However, there exist too many parameters determining the kinetics. Thus, for simplicity, the same temperature dependence of the interdiffusion coefficient has been assumed for all the constituent phases. Even under such simplified conditions, valuable conclusions were drawn from the analysis. 2. Analysis A hypothetical binary A B system composed of one intermetallic compound and two primary solid-solution phases was considered in previous studies ) The phase diagram of this binary system is represented in Fig ) In this figure, the ordinate shows the temperature T, and the abscissa indicates the mol fraction y of element B. The and phases are the primary solid-solution phases of elements A and B, respectively, and the phase is the compound. The same binary system was adopted in the present study. Now, we consider a semi-infinite diffusion couple consisting of the and phases with initial compositions of y 0 and y 0, Fig. 1 Phase diagram in a hypothetical binary A B system. 34) respectively. The semi-infinite diffusion couple means that the thickness is semi-infinite for the and phases and the / interface is flat. In such a diffusion couple, the interdiffusion of elements A and B takes place unidirectionally along the direction perpendicular to the flat interface. This direction is hereafter called the diffusional direction. When the diffusion couple is isothermally annealed at T ¼ T 1 for an appropriate time, the phase will be produced as a layer at the interface due to the reactive diffusion between the and phases. If the local equilibrium is actualized at each migrating interface during annealing, the compositions of the neighboring phases at the interface coincide with those of the corresponding phase boundaries at T ¼ T 1 in the phase diagram of the binary A B system indicated in Fig. 1. In this case, the migration of the interface is governed by the volume diffusion in the neighboring phases. According to the phase diagram in Fig. 1, y and y are the compositions of the and phases, respectively, at the / interface, and y and y are those of the and phases, respectively, at the / interface. The compositions y, y, y and y provide the boundary conditions, and those y 0 and y 0 give the initial conditions. If the reactive diffusion is controlled by the volume diffusion, the positions z and z of the = and = interfaces are expressed as functions of the annealing time t by pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi z ¼ K 4D t ¼ K 4D t ð1aþ and pffiffiffiffiffiffiffiffiffiffi pffiffiffiffiffiffiffiffiffiffi z ¼ K 4D t ¼ K 4D t; ð1bþ respectively. 39,40) In eq. (1), D, D and D are the interdiffusion coefficients for volume diffusion in the, and phases, respectively, and K, K, K and K are dimensionless coefficients. The difference between z and z is equal to the thickness l of the layer, and hence eq. (1) deduces the following relationship describing l as a function of t. l 2 ¼ðz z Þ 2 ¼ 4D ðk K Þ 2 t ¼ Kt ð2þ Here, K is the parabolic coefficient defined as K 4D ðk K Þ 2 : The dimensionless coefficients are related to the initial and boundary conditions as follows: c c ¼ and c c ¼ K þ K þ ð3þ c 0 c pffiffiffi f1 erfðk Þg expf ð K Þ 2 g c c pffiffiffi ferfðk Þ erfðk Þg expf ðk Þ 2 g ð4aþ K c c pffiffiffi ferfðk Þ erfðk Þg expf ðk Þ 2 g c 0 c pffiffiffi f1 þ erfðk Þg expf ðk Þ 2 g: ð4bþ K Here, c is the concentration of element B measured in mol per unit volume. The initial and boundary conditions are indicated with the concentration c in eq. (4), but shown with

3 Influence of Temperature Dependence of Solubility on Kinetics for Reactive Diffusion in a Hypothetical Binary System 717 the mol fraction y in Fig. 1. However, y is readily converted into c by the equation c ¼ y=v m, where V m is the molar volume of the relevant phase. The following relationships are obtained from eq. (1): pffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ¼ K D =D ð5aþ and pffiffiffiffiffiffiffiffiffiffiffiffiffiffi K ¼ K D =D : ð5bþ According to eq. (5), only two of the four dimensionless coefficients are independent. In the present analysis, K and K are chosen as the independent variables. Insertion of eq. (5) into eq. (4) results in two independent equations. Consequently, the two independent variables are finally determined from the two independent equations. Inserting the values of K and K into eq. (3), we obtain the parabolic coefficient K. Since K and K are functions of c 0, c, c, c, c, c 0, D, D and D through eqs. (4) and (5), K is a function of these nine parameters. 3. Results and Discussion As already mentioned in Section 2, the composition is indicated with the mol fraction y in Fig. 1 but with the concentration c in eq. (4). However, y is readily converted into c by the equation c ¼ y=v m, where V m is the molar volume of the relevant phase. According to the analyses in previous studies, 34 38) the following assumptions have been adopted also in the present study: (A) V m is independent of the composition; and (B) V m is equivalent for all the phases. Due to assumptions (A) and (B), c 0, c, c, c, c and c 0 in eq. (4) are automatically replaced with y 0, y, y, y, y and y 0, respectively. Furthermore, we use the compositional parameters defined as and y y y 0 ; y y y ; y y 0 y y 0 ðy þ y Þ=2: ð6aþ ð6bþ ð6cþ ð6dþ Here, y and y 0 are the solubility range and the mean composition of the phase, respectively. The initial and boundary conditions are expressed with y 0, y 0, y 0, y, y and y. The following values were used in the present analysis: y 0 ¼ 0, y 0 ¼ 0:5 and y 0 ¼ 1. In the case of y 0 ¼ 0 and y 0 ¼ 1, y and y correspond to the solubility ranges of the and phases, respectively. The solubility range y of the ( ¼ ; ; ) phase is described as a function of the temperature T by the following equation. y ¼ y 0 expð Q =RTÞ ð7þ Here, y 0 and Q are the pre-exponential factor and the enthalpy of solution, respectively. In the present study, Q was varied from 0 to 50 kj/mol for the, and phases. On the other hand, y 0 was determined to yield y ¼ 0:1 at T ¼ 1000 K for each value of Q. Hence, y 0 ¼ 0:1, 0.333, 1.11, 3.69, 12.3 and 40.9 are obtained for Q ¼ 0, 10, 20, 30, 40 and 50 kj/mol, respectively. These values of y 0 and Q Fig. 2 Logarithm of y versus the reciprocal of T. result in temperature dependencies of y indicated as solid lines in Fig. 2. In this figure, the ordinate shows the logarithm of y, and the abscissa indicates the reciprocal of T. According to the solid lines, y ¼ 1:0 10 1, 5: , 3: , 2: , 1: and 7: for Q ¼ 0, 10, 20, 30, 40 and 50 kj/mol, respectively, at T ¼ 700 K. As already mentioned in Section 1, the temperature dependence of the interdiffusion coefficient D for the phase is usually expressed by D ¼ D 0 expð Q D =RTÞ; ð8þ where D 0 and Q D are the pre-exponential factor and the activation enthalpy, respectively. The parabolic coefficient K in eq. (3) is a function of y, y, y, D, D and D through eqs. (3) (5) for given values of y 0, y 0 and y 0. Therefore, there are too many parameters determining the value of K. In the present study, however, attention is focused on the relationship between the temperature dependency of the kinetics and those of the solubility ranges of the constituent phases. Hence, in order to simplify the analysis, values of D 0 ¼ D 0 ¼ D 0 ¼ 10 4 m 2 /s and Q D ¼ Q D ¼ Q D ¼ 50 kj/mol were used for the, and phases. With these parameters, K was numerically calculated as a function of T from eqs. (3) (5) at T ¼ 700{1000 K. The temperature dependence of K is expressed by the following equation. K ¼ K 0 expð Q K =RTÞ ð9þ The pre-exponential factor K 0 and the activation enthalpy Q K in eq. (9) were calculated in a manner similar to previous studies ) On the basis of the calculation, dependencies of K 0 on Q and Q were evaluated for various values of Q. The results are shown as solid curves with constant values of Q ¼ 0{50 kj/mol in Fig. 3. In this figure, the abscissa indicates Q, and the ordinate shows the logarithm of K 0. Figure 3(a), (b), (c), (d), (e) and (f) indicates the results for Q ¼ 0, 10, 20, 30, 40 and 50 kj/mol, respectively. At Q ¼ 0 kj/mol in

4 718 M. Kajihara Fig. 3 Dependencies of K 0 on Q and Q for (a) Q ¼ 0 kj/mol, (b) Q ¼ 10 kj/mol, (c) Q ¼ 20 kj/mol, (d) Q ¼ 30 kj/mol, (e) Q ¼ 40 kj/mol and (f) Q ¼ 50 kj/mol. Fig. 3(a), K 0 takes values around m 2 /s. For each solid curve, K 0 monotonically decreases with increasing value of Q. Since D 0 ¼ D 0 ¼ D 0 ¼ 10 4 m 2 /s and Q D ¼ Q D ¼ Q D ¼ 50 kj/mol as mentioned earlier, D ¼ D ¼ D at each temperature. Furthermore, y 0 ¼ 0, y 0 ¼ 0:5 and y 0 ¼ 1. Thus, the effects of Q and Q on the temperature dependence of K are equivalent each other. As a result, K 0 monotonically decreases also with increasing value of Q at a constant value of Q. On the other hand, at Q ¼ 10 kj/mol in Fig. 3(b), K 0 takes values around m 2 /s. Also in this case, however, K 0 monotonically decreases with increasing value of Q or Q. Such dependencies of K 0 on Q and Q are realized also at Q ¼ 20{50 kj/mol in Fig. 3(c) (f). However, K 0 takes values around , , and m 2 /s at Q ¼ 20, 30, 40 and 50 kj/mol, respectively. Therefore, it is concluded that K 0 is a monotonically decreasing function of Q and Q but a monotonically increasing function of Q.

5 Influence of Temperature Dependence of Solubility on Kinetics for Reactive Diffusion in a Hypothetical Binary System 719 Fig. 4 Dependencies of Q K on Q and Q for (a) Q ¼ 0 kj/mol, (b) Q ¼ 10 kj/mol, (c) Q ¼ 20 kj/mol, (d) Q ¼ 30 kj/mol, (e) Q ¼ 40 kj/mol and (f) Q ¼ 50 kj/mol. The evaluation of Fig. 3 simultaneously deduces dependencies of Q K on Q and Q for various values of Q. The results are shown as solid curves with constant values of Q ¼ 0{50 kj/mol in Fig. 4. In this figure, the ordinate and the abscissa indicate Q K and Q, respectively. The results for Q ¼ 0, 10, 20, 30, 40 and 50 kj/mol are shown in Fig. 4(a), (b), (c), (d), (e) and (f), respectively. As can be seen, Q K as well as K 0 is a monotonically increasing function of Q but a monotonically decreasing function of Q and Q.AtQ ¼ 0 kj/mol in Fig. 4(a), Q K is exactly equal to 50 kj/mol for Q ¼ Q ¼ 0 kj/mol. As already mentioned in Section 1, the relationship between the temperature dependency of the kinetics and those of the interdiffusion coefficients of the constituent phases was numerically analyzed for a constant solubility range of each phase in previous studies ) According to that analysis, Q K completely coincides with

6 720 M. Kajihara Q D in the case of D 0 ¼ D 0 ¼ D 0 and Q D ¼ Q D ¼ Q D.As long as Q D < Q D and Q D < Q D, Q K is close to Q D. If Q D > Q D or Q D > Q D, however, Q K becomes greater than Q D. The values y 0 ¼ y 0 ¼ y 0 ¼ 0:1 and Q ¼ Q ¼ Q ¼ 0 kj/mol result in y ¼ y ¼ y ¼ 0:1 independently of T according to eq. (7). Furthermore, D 0 ¼ D 0 ¼ D 0 ¼ 10 4 m 2 /s and Q D ¼ Q D ¼ Q D ¼ 50 kj/ mol are considered in the present study. Under such conditions, Q K completely coincides with Q D.35 37) This is just the case for the solid curve with Q ¼ 0 kj/mol at Q ¼ 0 kj/mol in Fig. 4(a). However, Q K gradually decreases with increasing value of Q or Q. This means that the temperature dependence of the solubility range violates the conclusions drawn in previous studies ) Although Q K is smaller than Q D at Q > 0 kj/mol or Q > 0 kj/mol, the difference between Q K and Q D is rather small. On the other hand, at Q ¼ 10 kj/mol in Fig. 4(b), Q K is greater than Q D even at Q > 0 kj/mol or Q > 0 kj/mol, and takes values around 60 kj/mol. According to Fig. 4(c), (d), (e) and (f), Q K is rather lose to 70, 80, 90 and 100 kj/mol at Q ¼ 20, 30, 40 and 50 kj/mol, respectively. Compared with the dependency of Q K on Q, those of Q K on Q and Q are less remarkable. Therefore, it is concluded that Q K is predominantly determined by the summation of Q D and Q. In order to examine the relationship between Q K and the summation Q D þ Q, the ratio r K is defined by the following equation. r K Q K =ðq þ Q D Þ ð10þ From eq. (10), dependencies of r K on Q and Q were estimated for various values of Q. The results are indicated as solid curves with constant values of Q ¼ 0{50 kj/mol in Fig. 5. In this figure, the ordinate and the abscissa show r K and Q, respectively. Figure 5(a), (b), (c), (d), (e) and (f) indicates the results for Q ¼ 0, 10, 20, 30, 40 and 50 kj/mol, respectively. As well as K 0 and Q K, r K is a monotonically decreasing function of Q and Q but a monotonically increasing function of Q. Nevertheless, r K is rather close to unity as shown in Fig. 5. At Q ¼ 0 kj/mol in Fig. 5(a), r K is equal to unity at Q ¼ 0 kj/mol for the solid curve with Q ¼ 0 kj/mol. The value r K ¼ 1 is indicated as a horizontal dashed line in Fig. 5. As Q increases from 0 to 50 kj/mol, r K increases from 1 to 1.23 at Q ¼ Q ¼ 0 kj/mol. On the other hand, r K decreases from 1 to with increasing value of Q from 0 to 50 kj/mol at Q ¼ Q ¼ 0 kj/mol. Since the effects of Q and Q on r K are equal to each other according to the reasons mentioned earlier, r K decreases from 1 to with increasing value of Q from 0 to 50 kj/mol at Q ¼ Q ¼ 0 kj/mol. This clearly indicates that the dependency of r K on Q is more remarkable than those of r K on Q and Q. With Q ¼ 0 kj/mol, r K takes values of , , , and at Q ¼ 0{50 kj/mol for Q ¼ 10, 20, 30, 40 and 50 kj/mol, respectively. As a result, there exist various combinations among Q, Q and Q to realize r K ¼ 1. Such combinations are obtained from intersections between the solid curves and the horizontal dashed line. The intersection provides Q ¼ 0, 6.68, 11.4, 14.7, 16.8 and 18.0 kj/mol for Q ¼ 0, 10, 20, 30, 40 and 50 kj/mol, respectively, at Q ¼ 0 kj/mol. Owing to the equality between the effects of Q and Q on r K, Q ¼ 0, 6.68, 11.4, 14.7, 16.8 and 18.0 kj/mol are obtained also for Q ¼ 0, 10, 20, 30, 40 and 50 kj/mol, respectively, at Q ¼ 0 kj/mol. According to eq. (10), the value r K ¼ 1 means Q K ¼ Q þ Q D. In such a case, the temperature dependency of the parabolic coefficient K is directly determined from those of the solubility range y and the interdiffusion coefficient D for the phase. In contrast, at Q ¼ 10 kj/mol in Fig. 5(b), the intersection gives Q ¼ 6:68, 15.7, 23.2, 28.7, 32.5 and 34.9 kj/mol for Q ¼ 0, 10, 20, 30, 40 and 50 kj/mol, respectively. At Q ¼ 20 kj/mol in Fig. 5(c), however, the intersection exists for the solid curves with Q ¼ 0, 10, 20, 30 and 40 kj/mol but not for that with Q ¼ 50 kj/mol within Q ¼ 0{50 kj/ mol. The intersection disappears for the solid curves with Q ¼ 30{50 kj/mol at Q ¼ 30 and 40 kj/mol in Fig. 5(d) and (e), respectively, and for those with Q ¼ 20{50 kj/mol at Q ¼ 50 kj/mol in Fig. 5(f). For the solid curve without the intersection, r K is always smaller than unity. In such a case, there is no combination among Q, Q and Q to realize the relationship Q K ¼ Q D þ Q. Even in this case, however, the difference between Q K and Q D þ Q is rather small, and thus the summation Q D þ Q approximately represents the temperature dependence of the parabolic coefficient K. A close relationship between Q K and Q D þ Q was experimentally observed by Taguchi et al. 15) As mentioned earlier, the relationship between the temperature dependency of the growth rate of the compound and those of the interdiffusion coefficients of the constituent phases was numerically analyzed for a constant solubility range of each phase in previous studies ) That analysis indicates that Q K is equal to Q D at Q D ¼ Q D ¼ Q D and close to Q D at Q D < Q D and Q D < Q D. However, Q K is greater than Q D at Q D > Q D or Q D > Q D. Combining these relationships with the results in Fig. 5, we may conclude that Q K is close to Q D þ Q at Q D Q D and Q D Q D but greater than Q D þ Q at Q D > Q D or Q D > Q D. The stable crystal structure of the compound is the ordered lattice in many binary systems. 1) In such a case, Q D is greater than Q D and Q D unless the melting temperature is much lower for the compound than for the primary solid-solution phases. For this combination of Q D, Q D and Q D, Q K is greater than Q D þ Q. In this case, the temperature dependency of the parabolic coefficient cannot be directly related with those of the interdiffusion coefficient and the solubility range of the compound. In order to analyze reliably the kinetics of the reactive diffusion, experimental information of the interdiffusion coefficients and the solubility ranges of the relevant phases is essentially important. 4. Conclusions The influence of the temperature dependence of the solubility range on the kinetics of reactive diffusion was theoretically analyzed for the hypothetical binary system composed of one compound phase () and two primary solidsolution phases ( and ). The growth rate of the phase during reactive diffusion in the semi-infinite diffusion couple initially composed of the and phases was expressed as a function of the solubility ranges and the interdiffusion coefficients of the, and phases using the mathematical model reported in a previous study. 34) The temperature

7 Influence of Temperature Dependence of Solubility on Kinetics for Reactive Diffusion in a Hypothetical Binary System 721 Fig. 5 Dependencies of r K on Q and Q for (a) Q ¼ 0 kj/mol, (b) Q ¼ 10 kj/mol, (c) Q ¼ 20 kj/mol, (d) Q ¼ 30 kj/mol, (e) Q ¼ 40 kj/mol and (f) Q ¼ 50 kj/mol. dependencies of the interdiffusion coefficient D and the solubility range y of the phase were described by Arrhenius equations of D ¼ D 0 expð Q D =RTÞ and y ¼ y 0 expð Q =RTÞ, respectively. Here, D 0 and y 0 are the pre-exponential factors, Q D is the activation enthalpy, Q is the enthalpy of solution, R is the gas constant, and stands for, and. For simplicity, however, the same vales of D 0 ¼ 10 4 m 2 /s and Q D ¼ 50 kj/mol were adopted for all the phases. On the other hand, Q was changed from 0 to 50 kj/mol, and y 0 was selected to deduce y ¼ 0:1 at T ¼ 1000 K for each value of Q. In the case of the reactive diffusion governed by volume diffusion, the square of the thickness l of the phase is proportional to the annealing time t as l 2 ¼ Kt. The temperature dependence of the parabolic coefficient K was described by an Arrhenius equation of K ¼ K 0 expð Q K =RTÞ, and then the pre-expo-

8 722 M. Kajihara nential factor K 0 and the activation enthalpy Q K were evaluated for various combinations of Q, Q and Q. Both K 0 and Q K are monotonically increasing functions of Q but monotonically decreasing functions of Q and Q. At Q ¼ Q ¼ Q ¼ 0 kj/mol, Q K is exactly equal to Q D.In such a case, representative properties of the interdiffusion in the phase may be derived from the temperature dependence of the parabolic coefficient K. As Q or Q increases, however, Q K becomes smaller than Q D. In contrast, at Q ¼ 10{50 kj/mol, Q K is greater than Q D even at Q > 0 or Q > 0. Under such conditions, there exist combinations among Q, Q and Q to realize the relationship Q K ¼ Q þ Q D. This relationship is reasonable approximation in the wide ranges of Q, Q and Q under the present conditions. Combining the results in previous studies 35 37) with those in the present study, it is concluded that Q K is close to Q D þ Q at Q D Q D and Q D Q D but greater than Q D þ Q at Q D > Q D or Q D > Q D. Acknowledgement The present study was supported by the Iketani Science and Technology Foundation in Japan. REFERENCES 1) T. B. Massalski, H. Okamoto, P. R. Subramanian and L. Kacprzak: Binary Alloy Phase Diagrams, (ASM International, Materials Park, OH, 1990), vol ) B. Lustman and R. F. Mehl: Trans. Met. Soc. AIME 147 (1942) ) D. Horstmann: Stahl Eisen 73 (1953) ) S. Storchheim, J. L. Zambrow and H. H. Hausner: Trans. Met. Soc. AIME 200 (1954) ) G. V. Kidson and G. D. Miller: J. Nucl. Mater. 12 (1964) ) K. Shibata, S. Morozumi and S. Koda: J. Jpn. Inst. Met. 30 (1966) ) K. Hirano and Y. Ipposhi: J. Jpn. Inst. Met. 32 (1968) ) M. M. P. Janssen: Metall. Trans. 4 (1973) ) G. F. Bastin and G. D. Rieck: Metall. Trans. 5 (1974) ) M. Onishi and H. Fujibuchi: Trans. JIM 16 (1975) ) EI-B. Hannech and C. R. Hall: Mater. Sci. Tech. 8 (1992) ) P. T. Vianco, P. F. Hlava and A. L. Kilgo: J. Electron. Mater. 23 (1994) ) M. Watanabe, Z. Horita and M. Nemoto: Interface Science 4 (1997) ) S. Choi, T. R. Bieler, J. P. Lucas and K. N. Subramanian: J. Electron. Mater. 28 (1999) ) O. Taguchi, G. P. Tiwari and Y. Iijima: Mater. Trans. 44 (2003) ) T. Yamada, K. Miura, M. Kajihara, N. Kurokawa and K. Sakamoto: Mater. Sci. Eng. A 390 (2005) ) T. Takenaka, S. Kano, M. Kajihara, N. Kurokawa and K. Sakamoto: Mater. Sci. Eng. A 396 (2005) ) K. Suzuki, S. Kano, M. Kajihara, N. Kurokawa and K. Sakamoto: Mater. Trans. 46 (2005) ) T. Takenaka, S. Kano, M. Kajihara, N. Kurokawa and K. Sakamoto: Mater. Trans. 46 (2005) ) M. Mita, M. Kajihara, N. Kurokawa and K. Sakamoto: Mater. Sci. Eng. A 403 (2005) ) Y. Muranishi and M. Kajihara: Mater. Sci. Eng. A 404 (2005) ) T. Takenaka, M. Kajihara, N. Kurokawa and K. Sakamoto: Mater. Sci. Eng. A 406 (2005) ) M. Mita, K. Miura, T. Takenaka, M. Kajihara, N. Kurokawa and K. Sakamoto: Mater. Sci. Eng. B 126 (2006) ) Y. Yato and M. Kajihara: Mater. Trans. 47 (2006) ) T. Takenaka, M. Kajihara, N. Kurokawa and K. Sakamoto: Mater. Sci. Eng. A 427 (2006) ) Y. Yato and M. Kajihara: Mater. Sci. Eng. A 428 (2006) ) T. Hayase and M. Kajihara: Mater. Sci. Eng. A 433 (2006) ) Y. Tanaka, M. Kajihara and Y. Watanabe: Mater. Sci. Eng. A (2006) ) A. Furuto and M. Kajihara: Mater. Sci. Eng. A (2006) ) D. Naoi and M. Kajihara: Mater. Sci. Eng. A 459 (2007) ) S. Sasaki and M. Kajihara: Mater. Trans. 48 (2007) ) K. Mikami and M. Kajihara: J. Mater. Sci. 42 (2007) ) M. Kajihara and T. Sakama: Proc. 13th Symp. Microjoining Assembly Tech. Electrn., Yokohama, Japan, Feb. 1-2, 2007, Microjoining Comm., Tokyo, 2007, ) M. Kajihara: Acta Mater. 52 (2004) ) M. Kajihara: Mater. Sci. Eng. A 403 (2005) ) M. Kajihara: Mater. Trans. 46 (2005) ) M. Kajihara: Defect Diffus. Forum 249 (2006) ) M. Kajihara: Proc. 3rd Asian-Pacific Cong. Computational Mechanics and 11th Int. Conf. Enhanc. Prom. Computational Methods Eng. Sci., Kyoto, Japan, Dec. 3-6, 2007, APACM & EPMESC, 2007, GS10, ) W. Jost: Diffusion of Solids, Liquids, Gases, (Academic Press, New York, 1960), p ) G. B. Gibbs: J. Nucl. Mater. 20 (1966)