Description of Thermal Vacancies in the CALPHAD Method

Abstract

Thermal vacancies in solids have not been treated explicitly in the CALPHAD-type thermodynamic assessments because it was considered that their contributions to the Gibbs energy were limited, even at the melting point. However, the vacancy fraction is necessary for dynamic simulations, such as precipitations and diffusion processes. In this paper, a procedure is proposed to set parameters in the CALPHAD-type assessments, to reproduce the temperature dependency of thermal vacancies in pure metals and solid solutions.

 

This Paper was Originally Published in Japanese in J. Japan Inst. Met. Mater. 81 (2017) 127–132. In order to more precisely explain how to set the Gibbs energy of the empty end-member, the 0GmVa term was explicitly described in eqs. (19) and (20). The reference 8) was changed. Eq. (21) was omitted. Eqs. (4), (5), (21), (22) and (23) were renumbered.

1. Introduction

Thermal vacancies in pure metals, and alloys, have been intensively investigated¹⁾ since they play an important role in precipitation during aging and in diffusion processes. Accordingly, the behavior of thermal vacancies has been studied from the perspective of thermodynamics.^2–4) In the CALPHAD method, structural vacancies in the B2 compounds^5,6) and vacancies in the description of the interstitial solid solutions,^7,8) have been considered. However, both the fraction of the thermal vacancy and the formation Gibbs energy of the thermal vacancy have not been considered in the CALPHAD-type thermodynamic assessments. In recent years, a third generation CALPHAD database^7–9) has been developed to expand the valid temperature range of the Gibbs energy functions of the phases. Using thermodynamic quantities estimated from theoretical calculations, the Gibbs energies at high temperatures were estimated, however, the effect of the thermal vacancy⁷⁾ was not included.

In most of the CALPHAD-type assessments, the Gibbs energy of the formation of thermal vacancies has been given as a constant value, which has no dependency on crystal structure and element types. For example, in the literature the values 0,¹⁰⁾ +30T,¹¹⁾ +0.2RT,¹²⁾ +2.3RT⁷⁾ are used, where R is the gas constant and T is temperature in Kelvin. Thus, previous thermodynamic analysis^7,10–12) are insufficient to describe accurately the effect of thermal vacancies in solids. The purpose of this study is to provide a formulation of the thermal vacancies in solids for the CALPHAD method.

2. Gibbs Energy with Thermal Vacancy

2.1 Gibbs energy of pure elements with thermal vacancy

In this study we considered the thermal vacancy in the substitutional solid solution²⁾ as   

\begin{equation} (\text{A},\text{Va})_{p}, \end{equation} (1)

where Va denotes a vacancy, and p is moles of the substitutional site. The Gibbs energy is given for one mole of element A as   

\begin{equation} G_{\text{m}} = \frac{1}{1 - y_{\text{Va}}}\left[ \begin{array}{l} y_{\text{Va}}{}^{0}G_{\text{m}}^{\text{Va}} + (1 - y_{\text{Va}}){}^{0}G_{\text{m}}^{\text{A}} \\ {}+{} RT\left\{ (1 - y_{\text{Va}})\ln (1 - y_{\text{Va}})+ y_{\text{Va}}\ln y_{\text{Va}} \right\}\\ {}+{} (1 - y_{\text{Va}})y_{\text{Va}}\displaystyle\sum_{k = 0}(1 - 2y_{\text{Va}})^{k} L_{\text{A,Va}}^{(k)} \end{array} \right], \end{equation} (2)

where y_Va is the mole fraction of the vacancy, ${}^{0}G_{\text{m}}^{\text{Va}}$ is Gibbs energy of an empty end-member which consists of only vacancies, ${}^{0}G_{\text{m}}^{\text{A}}$ is the Gibbs energy of pure element A, and $L_{\text{A,Va}}^{(k)}$ is the interaction parameter between A and Va. The prefix k denotes the k-th term of the Redlich-Kister (R-K) polynomial. At constant temperature and pressure, the mole fraction of vacancy at equilibrium is given at the minimum of the derivative with respect to the mole fraction of vacancy. The first derivative of eq. (2) is   

\begin{align} \frac{dG_{\text{m}}}{dy_{\text{Va}}} &= \frac{1}{(1 - y_{\text{Va}})^{2}}{}^{0}G_{\text{m}}^{\text{Va}} + RT\frac{1}{(1 - y_{\text{Va}})^{2}}\ln y_{\text{Va}} \\ &\quad+ \sum_{k = 0}\left(1 - \frac{2y_{\text{Va}}k}{1 - 2y_{\text{Va}}}\right)(1 - 2y_{\text{Va}})^{k} L_{\text{A,Va}}^{(k)}. \end{align} (3)

At equilibrium dG_m/dy_Va = 0, the mole fraction of vacancy for the ideal solution, where $L_{\text{A,Va}}^{(k)} = 0$ is   

\begin{equation} y_{\text{Va}} = \exp \left(-\frac{{}^{0}G_{\text{m}}^{\text{Va}}}{RT}\right). \end{equation} (4)

Combining eqs. (2) and (4), the change in the Gibbs energy due to vacancy formation is given as   

\begin{equation} \Delta G_{\text{m}} = RT\ln \left[1 - \exp\left(-\frac{{}^{0}G_{\text{m}}^{\text{Va}}}{RT}\right)\right]. \end{equation} (5)

Equations (4) and (5) suggest that the Gibbs energy decreases when a vacancy is included and the vacancy fraction increases with increasing temperature. As will be presented in section 3, in most solids the vacancy fraction is less than 10⁻³. Thus, the solution phase in eq. (1) can be treated as a dilute solution. From eq. (3), when $L_{\text{A,Va}}^{(k)} \ne 0$ and y_Va ≪ 1, the vacancy fraction in the dilute solution is given as   

\begin{equation} y_{\text{Va}} = \exp \left(-\frac{\displaystyle{}^{0}G_{\text{m}}^{\text{Va}} + \sum L_{\text{A,Va}}^{(k)}}{RT}\right). \end{equation} (6)

In eq. (6), the contribution of the excess Gibbs energy, the fourth term in the main bracket of eq. (2), is given by a linear summation as $\sum L_{\text{A,Va}}^{(k)} $. Higher terms of the R-K polynomial (k > 0) are neglected because of the difficulty in determining solutions. Thus, eq. (6) is rewritten as,   

\begin{equation} y_{\text{Va}} = \exp\left(-\frac{{}^{0}G_{\text{m}}^{\text{Va}} + L_{\text{A,Va}}^{(0)}}{RT}\right). \end{equation} (7)

From eqs. (2) and (7), ΔG_m is written as   

\begin{equation} \Delta G_{\text{m}} = RT\ln \left[1 - \exp \left(-\frac{{}^{0}G_{\text{m}}^{\text{Va}} + L_{\text{A,Va}}^{(0)}}{RT}\right)\right]. \end{equation} (8)

Using the Gibbs energy of vacancy formation, $G_{\text{Va}}^{f}$, the enthalpy of vacancy formation, $H_{\text{Va}}^{f}$, and the entropy of vacancy formation, $S_{\text{Va}}^{f}$, eq. (7) can be rewritten as   

\begin{align} y_{\text{Va}} &= \exp \left(-\frac{{}^{0}G_{\text{m}}^{\text{Va}} + L_{\text{A,Va}}^{(0)}}{RT}\right) = \exp \left(-\frac{G_{\text{Va}}^{f}}{RT}\right) \\ &= \exp \left(\frac{S_{\text{Va}}^{f}}{R}\right)\exp \left(-\frac{H_{\text{Va}}^{f}}{RT}\right). \end{align} (9)

2.2 Thermal vacancy in the binary substitutional solid solutions

In this section, we will discuss the thermal vacancies in an A-B binary solid solution, which can be described as   

\begin{equation} (\text{A},\text{B},\text{Va})_{p}, \end{equation} (10)

where the total amount of atoms is 1 mole. For this binary substitutional solution phase, the Gibbs energy is given by   

\begin{equation} G_{\text{m}}^{\text{A-B}} = \frac{1}{1 - y_{\text{Va}}}\left\{ \begin{array}{l} \displaystyle\sum_{i}y_{i}{}^{0}G_{\text{m}}^{i} + RT\sum_{i = \text{A,B,Va}}y_{i}\ln y_{i}\\ {}+{} \displaystyle\sum_{k = 0}\sum_{i = \text{A,B}}\sum_{j < \ne i}y_{i}y_{j}(y_{i} - y_{j})^{k}L_{i,j}^{(k)} \\ {}+{} y_{\text{A}}y_{\text{B}}y_{\text{Va}}L_{\text{A,B,Va}} \end{array} \right\}, \end{equation} (11)

where, L_A,B,Va is the ternary interaction parameter. The mole fraction of thermal vacancy can be derived by using the same procedure as discussed in section 2.1. For the case where y_Va ≪ 1, the first derivative is   

\begin{align} \frac{dG_{\text{m}}^{\text{A-B}}}{dy_{\text{Va}}} & = {}^{0}G_{\text{m}}^{\text{Va}} + RT\ln y_{\text{Va}} \\ & \quad- \sum_{k = 0}(1 + k)y_{\text{A}}y_{\text{B}}(y_{\text{A}} - y_{\text{B}})^{k}L_{\text{A,B}}^{(k)}\\ & \quad+ \sum_{k = 0}(y_{\text{A}}^{k + 1}L_{\text{A,Va}}^{(k)} + y_{\text{B}}^{k + 1}L_{\text{B,Va}}^{(k)}) + y_{\text{A}}y_{\text{B}}L_{\text{A,B,Va}}. \end{align} (12)

Using only the first term of the R-K polynomial ($L_{\text{A,Va}}^{(0)}$ and $L_{\text{B,Va}}^{(0)}$), the vacancy fraction at equilibrium $dG_{\text{m}}^{\text{A-B}}/dy_{\text{Va}} = 0$ is given by   

\begin{equation} y_{\text{Va}} = \exp \left[-\frac{G_{\text{m}}^{\text{Va}} + y_{\text{A}}L_{\text{A,Va}}^{(0)} + y_{\text{B}}L_{\text{B,Va}}^{(0)} - \displaystyle\sum_{k = 1}(1 + k)y_{\text{A}}y_{\text{B}}(y_{\text{A}} - y_{\text{B}})^{k}L_{\text{A,B}}^{(k)} + y_{\text{A}}y_{\text{B}}L_{\text{A,B,Va}}}{RT}\right]. \end{equation} (13)

3. Thermal Vacancy at the Melting Point

There are several studies published which have considered the enthalpy of vacancy formation in pure elements. They have shown that there is a linear relationship between the enthalpy of the vacancy formation, $H_{\text{Va}}^{f}$, and melting temperature, T_m, which is   

\begin{equation} H_{\text{Va}}^{f} = cRT_{\text{m}},\quad \text{Jmol$^{-1}$}, \end{equation} (14)

where c is a constant. This relationship is presented in Fig. 1¹⁾ where c = 9.7 is obtained from a regression analysis. Since the constant c can vary¹³⁾ depending on crystal structure, eq. (14) can be rewritten as $H_{\text{Va}}^{f} = + (10 + a)RT_{\text{m}}$, where a is a constant.

Fig. 1

The relationship between formation enthalpy of the thermal vacancy and the melting temperature. The experimental data were taken from the Ref. 1.

The vacancy fraction at the melting point, as a function of the melting point temperature, is presented in Fig. 2(a).¹⁴⁾ Although no clear relationship can be seen in the figure, the vacancy fractions at the melting point of pure metals are less than 10⁻³ in experiments. Figure 2(b)¹⁴⁾ is a plot of the entropy of vacancy formation as a function of the melting temperature. Again there is no particular relations that will fit this data; in this case $S_{\text{Va}}^{f}/R$ is approximately less than 5. The effect of the entropy term on the vacancy fraction will be examined in the next section.

Fig. 2

Relationship between the melting temperature and (a) vacancy fraction at the melting temperature, and (b) the formation entropy of thermal vacancy. The data were taken from the Ref. 14.

4. Gibbs Energy of Vacancy Formation in the Substitutional Solution Model

4.1 Gibbs energy of an empty end-member, ${}^{0}G_{\text{m}}^{\text{Va}}$

In this section the parameter ${}^{0}G_{\text{m}}^{\text{Va}}$ in eq. (2) is discussed. ${}^{0}G_{\text{m}}^{\text{Va}}$ is the Gibbs energy of an empty structure, which consists of only vacancies. Although the definition of this term has been discussed in many publications,^{7,12,15–18)} the term has not been validated. In Refs. 15, 17, 18 it is given as zero because it is an empty end-member. Oates et al.¹⁶⁾ discussed ${}^{0}G_{\text{m}}^{\text{Va}}$ as an compound energy. Rogal et al.⁷⁾ recently proposed an average value for ${}^{0}G_{\text{m}}^{\text{Va}}$, which can be applicable to any crystal structure and element. However, when ${}^{0}G_{\text{m}}^{\text{Va}}$ is taken as a constant, it is insufficient to describe properties of vacancies since there is a dependency on crystal structure and element. This fact suggests that ${}^{0}G_{\text{m}}^{\text{Va}}$ is not an appropriate parameter to describe properties of a thermal vacancy. ${}^{0}G_{\text{m}}^{\text{Va}} = 0$ would be the most physically sound definition, however, when ${}^{0}G_{\text{m}}^{\text{Va}} = 0$ (as obtained from eq. (9)), the Gibbs energy of vacancy formation is described by $L_{\text{A,Va}}^{(0)}$. Consequently, this parameter should be a large positive value, and might result in a miscibility gap.¹²⁾ This means that a hypothetical solid phase, which has very high vacancy concentration, becomes stable. The spinodal region can be defined by the negative value of the second derivative of the Gibbs energy with respect to the vacancy fraction. From eq. (2), it is   

\begin{align} \frac{d^{2}G_{\text{m}}}{dy_{\text{Va}}^{2}} &= \frac{2}{(1 - y_{\text{Va}})^{3}}{}^{0}G_{\text{m}}^{\text{Va}} \\ &\quad+ RT\left[\frac{2}{(1 - y_{\text{Va}})^{3}}\ln y_{\text{Va}} + \frac{1}{y_{\text{Va}}(1 - y_{\text{Va}})^{2}}\right] < 0. \end{align} (15)

where the excess Gibbs energy is given by the first term, $L_{\text{A,Va}}^{(0)}$ as in eq. (2). It is worth noting that eq. (15) does not depend on $L_{\text{A,Va}}^{(0)}$, but only ${}^{0}G_{\text{m}}^{\text{Va}}$. In other words, when ${}^{0}G_{\text{m}}^{\text{Va}} = 0$, the spinodal region exists for any value of $L_{\text{A,Va}}^{(0)}$. The condition to avoid the spinodal decomposition can be obtained from eq. (15) as   

\begin{equation} \ln y_{\text{Va}} + \frac{(1 - y_{\text{Va}})}{2y_{\text{Va}}} > -\frac{{}^{0}G_{\text{m}}^{\text{Va}}}{RT}. \end{equation} (16)

Since the left hand side of eq. (16) has a minimum at y_Va = 0.5, the condition is ${}^{0}G_{\text{m}}^{\text{Va}}/RT > \ln 2 - 1/2 \simeq 0.19$.¹²⁾ However, the vacancy fraction in the solid at equilibrium is still very high ($y_{\text{Va}} \simeq 0.83$), even though the spinodal decomposition is avoided. As presented in Fig. 2(a), the vacancy fraction in metallic solids at the melting point $y_{\text{Va}}^{T_{\text{m}}}$ is less than 10⁻³. Using eq. (4), the condition ${}^{0}G_{\text{m}}^{\text{Va}} \geq 7RT_{\text{m}}$ limits $y_{\text{Va}}^{T_{\text{m}}}$ to a value that is less than 10⁻³ at the melting point. However, it is not enough to avoid a high vacancy fraction in metastable solids at very high temperatures. It may result a high vacancy fraction in alloys if there is a large difference in the melting points of constituents. By using y_Va ≤ 10⁻³, then ${}^{0}G_{\text{m}}^{\text{Va}} \geq 7RT$ is obtained, which can avoid this problem at any temperatures. In addition, $L_{\text{A,Va}}^{(0)} = 0$ can be used to avoid stabilization of the vacancy-rich phase at high temperatures.

In the CALPHAD-type assessments¹¹⁾ optimized parameters are rounded off, where the Gibbs energy change must be less than ±1 Jmol⁻¹ at 1000 K. Using eq. (5), it is found that $^{0}G_{\text{m}}^{\text{Va}} > + 9.03RT$ for 0 > ΔG_m > −1 Jmol⁻¹. Therefore, ${}^{0}G_{\text{m}}^{\text{Va}} = + 10RT$ is the optimized option to avoid the spinodal region and the vacancy-rich phase, at high temperatures. This results in a vacancy fraction of y_Va = 4.5 × 10⁻⁵ at any temperature, and the Gibbs energy change due to the introduction of a vacancy is −0.4 Jmol⁻¹ at 1000 K. Previous works report higher vacancy fractions, such as y_Va = 1 (${}^{0}G_{\text{m}}^{\text{Va}} = 0$ Jmol^{−1 10)}), y_Va = 0.819 (${}^{0}G_{\text{m}}^{\text{Va}} = 0.2RT$ Jmol^{−1 12)}), and y_Va = 0.027 (${}^{0}G_{\text{m}}^{\text{Va}} = 30T$ Jmol^{−1 11)}).

4.2 $L_{\text{A,Va}}^{(0)}$ in Redlich-Kister polynomial

The temperature at which the vacancy fraction forms in pure metals depends on crystal structure and element type. As discussed in the previous section, ${}^{0}G_{\text{m}}^{\text{Va}}$ is the Gibbs energy of an empty end-member, and thus, cannot be used to describe properties of a thermal vacancy.

In this section, we will discuss $L_{\text{A,Va}}^{(0)}$ when ${}^{0}G_{\text{m}}^{\text{Va}} = 0$. Using the enthalpy and the entropy of vacancy formation in eq. (9), $L_{\text{A,Va}}^{(0)}$ is given as   

\begin{equation} L_{\text{A,Va}}^{(0)} = H_{\text{Va}}^{f} - S_{\text{Va}}^{f}T. \end{equation} (17)

$H_{\text{Va}}^{f}$ and $S_{\text{Va}}^{f}$ can be determined from measurements: for pure Al, the vacancy fraction was obtained in experiments as y_Va = 8.2 exp(−70434/RT).¹⁹⁾ From this equation, they can be given as $H_{\text{Va}}^{f} = + 8470R$ (Jmol⁻¹) and $\ S_{\text{Va}}^{f} = + 2.1R$ (Jmol⁻¹K⁻¹).

In the case where ${}^{0}G_{\text{m}}^{\text{Va}}$ has a non-zero value and one would like to keep this non-zero value, from eq. (7), the following relation can be used instead of eq. (17)   

\begin{equation} L_{\text{A,Va}}^{(0)} = H_{\text{Va}}^{f} - S_{\text{Va}}^{f}T - {}^{0}G_{\text{m}}^{\text{Va}}. \end{equation} (18)

There are many cases where vacancy fractions were measured only at the melting point. Inserting $H_{\text{Va}}^{f} = + (10 + a)RT_{\text{m}}$ into eq. (18), the following is obtained   

\begin{equation} L_{\text{A,Va}}^{(0)} = (10 + a)RT_{\text{m}} - bRT - {}^{0}G_{\text{m}}^{\text{Va}}. \end{equation} (19)

where, $S_{\text{Va}}^{f}/R$ is replaced by b. From eqs. (7) and (19) the constant b can be estimated as $b = \ln y_{\text{Va}}^{\text{T}_{\text{m}}} + 10 + a$, where the range of the term a would be 1 > a > −1.¹³⁾ Figure 3(a) is a plot of the vacancy fraction as a function of normalized temperature for various a (b = 0); as a increases the vacancy fraction decreases. The b term affects significantly the vacancy fraction, as presented in Fig. 3(b). Since a large b value (∼5) results in a steep increase in the vacancy fraction with increasing temperature, if a large b value is necessary to reproduce the experimental data, assessors should consider carefully the thermodynamic models and the parameters in the models.

Fig. 3

Temperature dependency of the vacancy fraction: (a) variation with a in eq. (19), and (b) with b in eq. (19).

5. Analysis of Vacancy Fraction

5.1 Vacancy fraction in pure Ag and Al

In Fig. 4 the measured vacancy fraction in pure FCC-Cu is presented.^20,21) From the function ln y_Va = 1.87 − 14240/T, the enthalpy and the entropy of vacancy formation are given as $H_{\text{Va}}^{f} = + 14240R$ (Jmol⁻¹) and $S_{\text{Va}}^{f} = + 1.87R$ (Jmol⁻¹K⁻¹), respectively. For pure FCC-Ag,²²⁾ the vacancy fraction $y_{\text{Va}}^{\text{Tm}} = 1.7 \times 10^{ - 4}$ at the melting temperature of 1235 K, and the enthalpy of vacancy formation, +10.3RT_m ± 0.6RT_m Jmol^–1,¹³⁾ were obtained. Using eqs. (7) and (19), the entropy of vacancy formation was estimated from these values as $S_{\text{Va}}^{f} = + 1.32R$ Jmol^–1K⁻¹. Consequently, parameters in eq. (2) for FCC-Cu and FCC-Ag are:   

\begin{equation} \begin{split} & {}^{0}G_{\text{m}}^{\text{Va}} = + 10RT \\ & L_{\text{Ag,Va}}^{(0)} = +12350R - 11.32RT\\ & L_{\text{Cu,Va}}^{(0)} = +14240R - 11.87RT \end{split} \end{equation} (20)

Using the Gibbs energy functions from the SGTE-Unary database,²³⁾ the melting temperature of FCC-Ag and FCC-Cu are 1234.93 K, and 1357.78 K, respectively. By introducing a vacancy, the solid phase is slightly stabilized, the melting temperature increases to 1235.12 K and 1357.99 K for the FCC-Ag and FCC-Cu, respectively. Thus, the change in the melting point due to the vacancy for both metals is negligible.

Fig. 4

Comparison between calculated vacancy fraction in the FCC Cu using eq. (9) and experimental data.^20,21)

Since the Gibbs energy functions in the Unary database include contributions of thermal vacancies, the effect of vacancies is counted twice in the Gibbs energy functions. As demonstrated in FCC-Ag and FCC-Cu, the effect of the vacancy on the phase equilibria is negligible. Therefore, it may not be necessary to modify the Unary database.^23,24)

5.2 Vacancy fraction in the Ag–Cu binary solution

The Gibbs energy functions of the FCC solid solution phase in the Ag–Cu binary system were taken from the assessment by Hayes et al.²⁵⁾ The eutectic temperature in their assessment is 1056.01 K. It increases slightly to 1056.03 K by introducing a vacancy. In Fig. 5, the dependency of the vacancy fraction on composition at 1000 K is presented, where the ternary parameter is assumed to be zero, L_Ag,Cu,Va = 0. The vacancy fraction becomes larger than the broken line (the broken line is a plot for $L_{\text{Ag,Cu}}^{(0)} = 0$) due to the repulsive interaction between Ag and Cu in the FCC lattice. This is the case because an extra vacancy is formed to avoid the formation of the Ag–Cu pair. In contrast, in the case of the attractive interaction between elements A and B, the vacancy fraction may decrease.

Fig. 5

Concentration dependency of the thermal vacancy in the FCC Ag–Cu solid solution at 1000 K. The broken line is for $L_{\text{Ag,Cu}}^{(0)} = 0$.

6. Conclusions

In this work, we examined the thermal vacancies in the CALPHAD method. The following results were obtained; for a vacancy in a pure A, the parameters in the substitutional solution model are,   

\begin{equation} L_{\text{A,Va}}^{(0)} = +(10 + a)RT_{\text{m}} - bRT - {}^{0}G_{\text{m}}^{\text{Va}}. \end{equation} (21)

If the vacancy-rich phase is stabilized at high temperatures, although ${}^{0}G_{\text{m}}^{\text{Va}} = 0$ is the most physically sound definition, it can be avoided by using   

\begin{equation} {}^{0}G_{\text{m}}^{\text{Va}} = + 10RT. \end{equation} (22)

Equations (21) and (22) were applied to the thermal vacancy in the FCC-Ag and FCC-Cu. Using the obtained parameters, the vacancy fractions were reproduced.

The obtained parameters for pure elements were used to estimate the vacancies in the Ag–Cu binary solution. It was found that the vacancy fraction depends on the interaction between elements A and B; the vacancy fraction increases because of the repulsive interaction between Ag and Cu. This suggests that it decreases in the case of an attractive interaction between the constituents.

Acknowledgements

TA thanks Prof. Hiroshi Numakura of Osaka Prefecture University for fruitful discussions and Dr. Cenk Kocer, University of Sydney, for the review of this paper.

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