Description of Thermal Vacancies in the CALPHAD Method

Keywords:
point defect,
substitutional solid solution,
sublattice model,
thermodynamic database,
mono vacancy

2018 Volume 59 Issue 4 Pages 580-584

Details

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 investigated^{1}^{)} 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 vacancy^{7}^{)} 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}^{)} +30*T*,^{11}^{)} +0.2*RT*,^{12}^{)} +2.3*RT*^{7}^{)} 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 solution^{2}^{)} as

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

\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) |

\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) |

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

\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) |

\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) |

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

\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) |

\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) |

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) |

\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) |

\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) |

\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) |

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) |

\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) |

In the CALPHAD-type assessments^{11}^{)} 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 > Δ*G*_{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}^{)}).

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) |

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) |

\begin{equation} L_{\text{A,Va}}^{(0)} = (10 + a)RT_{\text{m}} - bRT - {}^{0}G_{\text{m}}^{\text{Va}}. \end{equation} | (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^{−1}K^{−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, +10.3*RT*_{m} ± 0.6*RT*_{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^{–1}K^{−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) |

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}^{)}

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) |

\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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