CALTPP: A general program to calculate thermophysical properties

Introduction

Thermophysical properties are of fundamental importance in the field of materials science and engineering. Those properties include diffusion coefficient, interfacial energy, thermal conductivity, viscosity, molar volume, etc. Thermophysical parameters are required as key inputs for the ICME (Integrated Computational Materials Engineering) tools, such as diffusion simulation (DICTRA software [1,2]), phase field simulation (OpenPhase [3] and MICRESS [4]), FEM (finite element method) [5,6] and other microstructural modeling [7,8]. In simulations of various processing, such as carburization, precipitates tempering, grain growth, liquid phase sintering, solidification, heat treatment of large forging parts, various thermophysical properties are frequently utilized. Thermophysical properties are also indispensable to model mechanical properties. In many simulations, unreasonable values for thermophysical properties are used. For example, it is found that many researchers world-wide are using constant values for thermophysical properties in FEM calculations, however, the adoption of such values cannot reflect reality conditions since thermophysical properties normally depend on both composition and temperature [9].

Since experimental determinations of thermophysical properties involve complex procedures, and the number of parameters to test is immense, models to obtain thermophysical properties through theoretical derivations are therefore of great value. Thermophysical properties can be expressed as function of temperature, pressure and composition of the phases [10], which are suitable to be modeled by means of the CALPHAD (CALculation of PHAse Diagrams) approach. To the best of our knowledge, there is no report about a general program by now, which can calculate various thermophysical properties as function of temperature and composition. The present authors have an extensive experience using several commercial software packages. It was found that these packages either calculate individual thermophysical property or cannot guarantee the calculation accuracy due to the use of semi-empirical equations. These software packages do not have one special subroutine, which can calculate the presently targeted several thermophysical properties as function of temperature and composition. In the present work, a general program CALTPP (CALculation of ThermoPhysical Properties) is developed with the aim to provide the above mentioned thermophysical properties, which can be used as inputs for microstructure simulations and mechanical property predictions. In section ‘General structure of CALTPP program’, we briefly describe the general structure of CALTPP, followed by CALPHAD-type models of thermophysical properties in Section ‘CALPHAD-type models of thermophysical properties’. In Section ‘Case studies of CALTPP’, a few case studies for CALTPP are demonstrated. An outlook for the future development of modeling for thermophysical properties is proposed in Section ‘Outlook for the future development of modeling for thermophysical properties’. The conclusions are drawn in Section ‘Conclusion’.

General structure of CALTPP program

Various thermophysical properties including diffusion coefficient, interfacial energy, thermal conductivity, viscosity and molar volume can be calculated and/or optimized by means of CALTPP. The general structure of CALTPP program is presented in Fig. 1. The program contains the input module, calculation and/or optimization modules and output module, which is performed in the Matlab environment. The data used in CALTPP for optimization of model parameters include experimental data and first-principles computed values. As shown in Fig. 1, the input module presents the information required to obtain thermophysical properties according to various CALPHAD-type models. And these models will be presented in Section ‘CALPHAD-type models of thermophysical properties’. The calculation and optimization related to thermophysical properties are performed in the calculation and/or optimization module of CALTPP program. Then we can capture the output information about thermophysical properties and their variants which usually are described as function of temperature and composition, as displayed in the output module. The CALTPP program can perform the calculation of diffusion coefficient adopting some well-known methods as well as the presently proposed numerical inverse method. In addition, the optimization of atomic mobility and the simulation of diffusion behavior can also be operated. The solid/liquid, coherent solid/solid and liquid/liquid interfacial energies can be given in CALTPP with the thermodynamic parameters and volume information input. The model parameters in the expressions for thermal conductivity, viscosity and molar volume can be optimized by means of least-squares method. Then, these thermophysical properties can be predicted at an extended condition according to the models and the optimized parameters. The output information in CALTPP can be directly used as key inputs in the applications of microstructure simulations and mechanical property predictions, such as diffusion simulation, phase field simulation and FEM.

CALPHAD-type models of thermophysical properties

Various CALPHAD-type models for thermophysical properties, such as diffusion coefficient, interfacial energy, thermal conductivity, viscosity and molar volume, are implemented in CALTPP program. These models can be used to obtain thermophysical properties effectively and accurately. In the following, the fundamental equations of the models for these thermophysical properties are given. For the detail about the models, one can refer to the corresponding references.

Diffusion coefficient

Diffusion coefficient is a rate of diffusion, which is a proportionality constant between the molar flux due to molecular diffusion and the driving force for diffusion. The widely used methods (Matano-Boltzmann method [11,12] and Whittle and Green method [13]) to calculate interdiffusion or diffusion matrix are included in CALTPP program.

In a binary system, the Matano-Boltzmann method [11,12] calculates the binary interdiffusion coefficient by analyzing the composition profiles of one element. In principle, the Matano-Boltzmann method [11,12] works only when the molar volume of the phase varies ideally following Vergard’s law. In view of the volume deviation from the Vergard’s law, the Sauer-Freise [14] method, the Wagner method [15] and the Den Broeder method [16] are recommended for calculating interdiffusion coefficient in binary alloys. The above methods to obtain the binary interdiffusion coefficient are implemented in CALTPP program.

For a ternary system, it is generally believed that 4 concentration-dependent diffusion coefficients can be accurately determined at the position where two diffusion couples intersected [17]. The well-known Matano-Kirkaldy method [18] to obtain the diffusivity at the intersection point is included in CALTPP program. Due to the low efficiency of the Matano-Kirkaldy method [18], several methods [[19], [20], [21], [22], [23], [24]] have been developed with the aim to evaluate the interdiffusivities in ternary systems by using a single diffusion couple. The CALTPP program adopted a novel numerical inverse method to calculate the diffusion matrix along the whole diffusion path of a ternary diffusion couple. In the following, a brief introduction about this numerical inverse method is presented.

The interdiffusion coefficient in a hypothetical 1-2-3 ternary system can be expressed by an extended Fick's law [25] on the basis of Matano coordinates:

$\frac{\partial c_{i}}{\partial _{t}}=\frac{\partial}{\partial _{Z}}(\tilde{D}^{3}_{i1}\frac{\partial c_{1}}{\partial _{Z}})+\frac{\partial}{\partial _{Z}}(\tilde{D}^{3}_{i2}\frac{\partial c_{2}}{\partial _{Z}})(i=1,2)$ (1)

where z is the diffusion distance from Matano interface, t represents the time and c_i(i=1,2) is the concentration of component i. $\tilde{D}^{3}_{11}$ and $\tilde{D}^{3}_{22}$ are the main interdiffusion coefficients, $\tilde{D}^{3}_{12}$ and $\tilde{D}^{3}_{21}$ are the cross interdiffusion coefficients.

Inspired by Bouchet and Mevrel [20], the concept of basis functions is also employed in this work. Some kinds of basis functions could be adopted to describe the continuous function of interdiffusivity. The expression is written as follows,

$\tilde{D}^{3}_{ij}=\tilde{D}^{3}_{ij}(c_{1},c_{2},c_{3})=\sum^{N}_{k=1}α^{k}_{ij}\phi^{k}_{ij}(c_{1},c_{2},c_{3})$ (2)

where $α^{k}_{ij}$ is the parameter which is independent of the concentrations c_i(i=1,2). $\phi^{k}_{ij}$ (c₁,c₂,c₃) is a kind of linearly independent basis function. In CALTPP program, the following two kinds of basis functions are proposed:

$\{1,c_{i},c_{i}c_{j},...\}^{3}_{i,j=1}$ (3)

and

$\{exp(-c_{i}),c_{i}exp(-c_{j}),c_{i}c_{j}exp(-c_{i}c_{j}),...\}^{3}_{i,j=1}$ (4)

Some other kinds of basis functions, such as the trigonometric function, can also be customized when needed. The parameter $α^{k}_{ij}$ is to be evaluated through the minimization of the difference between the calculated concentrations $c^{i}_{cal}$ and the experimental concentrations $c^{i}_{exp}$. The numerical procedure is implemented via the integration of FEM [26] and simple genetic algorithm (SGA) [27]. FEM is used to solve forward problem and SGA is used to optimize the parameter $α^{k}_{ij}$. In addition, the deterministic algorithms, such as gradient descent and quasi-Newton method [28], can also be adopted in the procedure of optimization. It is noted that calculating the diffusion coefficient $\tilde{D}^{3}_{ij}$ along the whole diffusion path is converted to calculating the parameter $α^{k}_{ij}$. Therefore, the inversion problem of parameter can be assumed as the optimization problem, and the cost function corresponds to the following form:

$g=\sum^{2}_{i=1}\sqrt{\frac{1}{M}\sum^{M}_{m=1}(c^{exp}_{i}(z_{m})-c^{cal}_{i}(z_{m},α))^{2}}$ (5)

where z_m is the position of experimental concentrations data, M is the number of experimental sets, and α is the undetermined coefficient vector. When using genetic algorithms to optimize parameters, we generally choose the fitness function as 1/g.

CALTPP program can also optimize atomic mobilities in diffusion. In view of the absolute-reaction rate theory, Andersson and Ågren [29] divided the atomic mobility of element k, M_k, into a frequency factor $M^{0}_{k}$ and an activation energy -Q_k, i.e.

$M_{k}=exp(\frac{RTlnM^{0}_{k}}{RT})exp(\frac{-Q_{k}}{RT})$ (6)

where R is the gas constant, and T is the absolute temperature. Furthermore, RT ln$M^{0}_{k}$ and -Q_k can be merged into one composition dependence parameter, i.e. Φ_k. According to the CALPHAD idea, the composition dependence can be expanded using the Redlich-Kister polynomial for binary terms and a power series expansion for ternary one [30,31],

$\Phi_{k}=\sum_{i}x_{i}\Phi^{i}_{k}+\sum_{i}\sum_{j>i}x_{i}x_{j}[\sum^{r}_{r}\Phi^{ij}_{k}(x_{i}-x_{j})^{r}]\\ +\sum_{i}\sum_{j>i}\sum_{k>j}x_{i}x_{j}x_{k}[\sum_{s}{v^{s}_{ijk}}^{s}\Phi^{ijk}_{k}]$ (7)

where x_i, x_j and x_k are mole fractions of elements i, j and k, respectively. $\Phi^{i}_{k}$ represents the contribution from one endpoint in the composition space. ${}^{s}\Phi^{ij}_{k}$ is the ternary interaction parameter if i, j and k are not equal to each other. Otherwise, ${}^{s}\Phi^{ij}_{k}$ is the binary interaction parameter. And ${}^{s}\Phi^{ijk}_{k}$ is the interaction parameters in ternary system. The parameter $v^{s}_{ijk}$ is given by

$v^{s}_{ijk}=x_{s}+\frac{1-x_{i}-x_{j}-x_{k}}{3}(s=i,j,k)$ (8)

The interdiffusion coefficient $\tilde{D}^{n}_{ij}$ with n as the dependent species is modeled with the atomic mobilities by [29]:

$\tilde{D}^{n}_{ij}=\sum^{n}_{i=1}(\delta_{ik}-x_{k})x_{i}M_{i}(\frac{\partial μ_{i}}{\partial x_{j}}-\frac{\partial μ_{i}}{\partial x_{n}})$ (9)

where x_i, μ_i and M_i are the mole fraction, chemical potential and atomic mobility of the component i, respectively. δ_ik is the Kronecker delta (δ_ik =1 if i = k, otherwise δ_ik =0).

According to Eqs. (6) and (9), the following equation is suggested to perform the optimization of atomic mobility parameters.

$\mathop{}_{a,b}^{minf}=\sum^{s}_{s=1}\omega_{s}\sum^{n-1}_{k,j=1}|F_{kj}(a,b,C_{s})-\tilde{D}^{n}_{kj,s}|^{2}$ (10)

where S presents the total number of experimental sets, ωs is the weight of the s^th experimental set and $\tilde{D}^{n}_{kj,s}$ is the interdiffusion coefficient $\tilde{D}^{n}_{kj}$ calculated based on s^th experimental set. F_kj(a,b,C_s) is a metric function similar to Eq. (9). a and b are the model parameters to be optimized, and C_s is the concentration of solution required to calculate the interdiffusion coefficient $\tilde{D}^{n}_{kj}$ based on s^th experimental set.

The optimization of atomic mobility can be attributed to the multivariate nonlinear optimization problem. The algorithms in CALTPP program include the nonlinear least squares methods such as gradient descent method, quasi-Newton method, and Levenberg-Marquardt method [28]. In order to obtain the reliable global optimum parameters, a multi-initial search method is used, which can reduce the risk of falling into local extremum.

In addition, the effect of molar volume on the optimized and/or calculated diffusion coefficient will be demonstrated in the section of molar volume.

Interfacial energy

Interfacial energy describes the thermodynamic state of interface between phases (or grains), and it quantifies the excess interfacial free energy per unit interfacial area. The models of the solid/liquid, coherent solid/solid and liquid/liquid interfacial energies proposed in our previous work [32,33] are included in the current CALTPP program. The calculation of semi-coherent and non-coherent interfacial energies needs the use of atomistic methods such as first-principles calculation method and molecular dynamics. These calculations will be considered in our future work.

The solid/liquid interfacial energy (γ_SL) is modeled as the separated parts of chemical (γ_SL(c)) and structural (γ_SL(B)) contributions [32]:

$γ_{SL}=γ_{SL(c)}+γ_{SL(B)}$ (11)

The chemical contribution is generated between the molar free energy ($G^{L}_{m}$) of the atoms in an equilibrium two-phase mixture and their energy (${}^{E}G^{IF}_{m}$) when forced to exist together as a liquid of the composition xi. The above assumption makes it possible to write the chemical contribution as follows,

$γ_{SL(c)}=\frac{G^{L}_{m}-{}^{E}G^{IF}_{m}}{A_{m}}$ (12)

where A_m is the interfacial area per mole of interface atoms.

The structural contribution is considered to be equal to the interfacial energy between the pure solid phase and its melt:

$γ_{SL(B)}= k⋅T_{m}/A_{S}$ (13)

in which Tm is the melting temperature of solid phase, k = 4.22 J∙K^-1 ∙mol^-1 is an empirical constant, and A_S is the interfacial area per mole of atoms in solid phase.

A general model for obtaining coherent solid/solid and liquid/liquid interfacial energies (γ_SS(LL)) in binary and multi-component system has been proposed in our recent work [33]. This model can predict the interfacial energies between the disordered γ (fcc_A1) and ordered γ' (fcc_L12) phases, where the Gibbs energies of both γ and γ' phases are given by the same expression. While the other model in the literature [34] cannot capture some disordered/ordered interfacial energies, like the Al-Ni system. In this model, the molar Gibbs energy of the interface phase (IF) is estimated as the average of the corresponding two bulk phases (BI and BII) at any composition and temperature. Using a hypothetical A-B binary system as an example, the expression of γ_SS(LL) is as follows,

$γ_{SS(LL)}=2·\frac{G^{IF}_{m}-{}^{E}G^{IF}_{m}}{A^{o}_{A}(1-x^{IF}_{B})+A^{o}_{B}x^{IF}_{B}}$ (14)

in which $x^{IF}_{B}$ is the effective interfacial composition. ${}^{E}G^{IF}_{m}$ and $G^{IF}_{m}$ are the equilibrium molar Gibbs energy in an equilibrium two-phase mixture and the molar Gibbs energy of the interface phase when the composition is $x^{IF}_{B}$, respectively. $A^{o}_{A}$ and $A^{o}_{B}$ are the partial molar area of component A and B, respectively.

The composition of interface phase xBIF is given by an equation of the form:

$x^{IF}_{B}=\frac{(1/A^{BI}_{m})·x^{BI}_{B}+(1/A^{BII}_{m})·x^{BII}_{B}}{(1/A^{BI}_{m})+(1/A^{BII}_{m})}$ (15)

in which $A^{j}_{m}$ (j= BI and BII) is the molar area in phase j. $x^{j}_{B}$ (j= BI and BII) is the mole fraction of component B in phase j.

Thermal conductivity

Thermal conductivity describes the quantity of heat transmitted due to a unit temperature gradient in a unit time under steady conditions in a direction normal to a surface of the unit area. CALPHAD is an effective approach to obtain information about the thermal conductivity [35]. In CALTPP program, different equations are adopted for describing thermal conductivities of pure elements, stoichiometric phases, solid solutions, and alloys (in multi-phase region).

Thermal conductivity of pure elements and stoichiometric phases is described as a function of temperature using the following equation:

$λ_{0}=a+bT+Ct^{-1}$ (16)

where λ₀ is the thermal conductivity of pure elements or stoichiometric phases, T is the temperature, and a, b and c are the evaluated parameters according to the experimental data.

Thermal conductivity for the (α) solid solutions in a binary i-j or ternary i-j-k system is described by the function similar to Eq. (7), where the endpoint is the thermal conductivity of the corresponding pure element. The binary and ternary interaction parameters are described as ${}^{r}L^{α}_{ij}$ and $L^{α}_{ijk}$, respectively.

The thermal conductivity of two-phase region is described as follows,

$λ_{α+β}=n_{α}λ_{α}+n_{β}λ_{β}-n_{α}λ_{β}·\sum^{j}_{j=0}M_{α+β}(n_{α}-n_{β})^{j}$ (17)

where λ_α+β is the thermal conductivity of the alloy in a (α)+(β) two-phase region, n_p and λ_p (p=α and β) are the mole fraction and thermal conductivity of the phase p, respectively. j_M is the j^th interface scattering parameter.

Viscosity

Viscosity represents the ratio of the shearing stress to the velocity gradient, which can be utilized to describe the fluid resistance to flow. The models of viscosity for pure liquid, binary liquid and multicomponent liquid are included in CALTPP program. The Arrhenius equation describing the temperature dependence of the viscosity holds for pure liquid, which is expressed as follows,

$η=η_{0}exp(\frac{E}{RT})$ (18)

where η₀ is the pre-exponential and E is the activation energy. These two parameters could be obtained through fitting to the experimental values. R is the gas constant and T the temperature.

Various models were developed to predict the viscosity of binary system in the literature. Among these models, the Seetharaman-Du Sichen (SDS) equation [36], Kaptay (K) equation [37] and Schick equations [38] are included in CALTPP, and the expressions for these models are presented in the corresponding literature.

The CALPHAD-type model for viscosity of a i-j binary system was developed in our previous work [39], which is expressed by the Redlich-Kister polynomial [31]. And its expression is similar to Eq. (7), where the endpoint is the viscosity of the corresponding pure element. The binary interaction parameter ${}^{r}L^{α}_{ij}$ is assessed from the experimental data and ternary interaction parameter is set to be zero. The viscosity of a multicomponent liquid alloy is extrapolated from the parameters of binary systems, which implies if the excess viscosities of all the binary systems were known, the viscosity of a multicomponent system could be predicted.

Molar volume

Molar volume is the volume occupied by one mole of a substance at a given temperature and pressure. The CALTPP program incorporates the models for molar volume, which are used in the modeling of various thermophysical properties like diffusion modeling. The molar volume of nonmagnetic materials at 1 bar is given as follows [40,[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42]],

$V(T,P_{0})=V_{0}exp(V_{α})=V_{0}exp(\int_{T_{0}}^{T}\ 3αdT)\\ =V_{0}exp(\int_{T_{0}}^{T}\ (a+bT+cT^{2}+dT^{(-2)})dT)$ (19)

in which V₀ and V_α denote the molar volume at the reference temperature T₀ and the integrated thermal expansion, respectively. a, b, c and d are the parameters to be evaluated from the extensive experimental data for the coefficients of volumetric thermal expansion at 1 bar, i.e., 3α.

The molar volume of a ternary system is extrapolated from the parameters of binary systems, and it is defined like Eq. (7), where the endpoint is the volume of the corresponding pure element. The interaction parameter ${}^{r}L^{α}_{ij}$ is related to the i-j binary system and the ternary interaction parameter is set to be zero.

One important application of molar volume is its effect on the calculation of diffusivities with variable volume for a phase. Wagner [15] considered the variation of molar volume V_m with composition and derived the diffusivity extraction equation with a similar equation as the Sauer-Freise [14] method. Interdiffusion coefficient $ \tilde{D}$ under the consideration of volume is described as follows,

$\tilde{D}(c^{*})=\frac{V^{*}_{m}}{2t(dY_{A}/dz)_{z^{*}}}[(1-Y^{*}_{A})\int_{-∞}^{z^{*}}\frac{Y_{A}}{V_{m}}dz\\ +Y^{*}_{A}\int_{z^{*}}^{+∞}\frac{(1-Y_{A})}{V_{m}}dz]$ (30)

where z and t are the diffusion distance and diffusion time, respectively. $ V^{*}_{m}$ and $ Y^{*}_{A}$ are the molar volume and concentration ratio at point c^*, respectively, and Y_A is defined as [14]:

$Y_{A}=\frac{c_{A}-c^{-}_{A}}{c^{+}_{A}-c^{-}_{A}}$ (21)

where $c^{-}_{A}$ and $ c^{+}_{A}$ represent the concentration at the far left end and far right end of the diffusion couple, respectively.

Case studies of CALTPP

According to the CALPHAD-type models described in Section ‘CALPHAD-type models of thermophysical properties’, the corresponding thermophysical properties can be calculated and/or optimized in CALTPP program. In the following, a few case studies are demonstrated to show the features of CALTPP. The comparisons between the calculated results and measured data from the literature illustrate the reliability of CALTPP program.

Diffusion coefficient

Diffusion is the basis for various technological applications such as solidification, heat treatment, mass transfer, absorption and catalysis [43,[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45]], being the most important thermophysical property. The calculation of diffusion coefficient and optimization of atomic mobility in binary and multicomponent systems can be performed in CALTPP program.

Taking fcc Co-Ti-V and Cu-Ni-Sn alloys as examples, we show the calculation of diffusion matrix along the whole diffusion path of a ternary diffusion couple. In CALTPP, the constant diffusivities as well as the composition-dependent diffusivities expressed by basis function are captured using the measured concentration profiles. Subsequently, we can simulate the diffusion behavior according to the captured diffusivities together with the diffusion equations. For the targeted fcc Co-Ti-V and Cu-Ni-Sn alloys, the polynomial and genetic algorithm are adopted to express the composition-dependent diffusivities and optimize the undetermined parameters, respectively. Fig. 2(a) and (b) displays the calculated concentration profiles in fcc Co-Ti-V alloy compared with the measured data [46] of Co-8 V/Co-7Ti (at.%) at 1373 K for 120 h and Cu-Ni-Sn alloy compared with the measured data [47] of Cu-7.38Ni/Cu-4.28Sn (at.%) at 1023 K for 40 h, respectively. The measured concentration profile in Fig. 2(b) displays a complex character that the concentration distribution of Sn is not monotonous along the distance. This measured complex concentration profile can be well produced in CALTPP program with the composition-dependent diffusivities.

It worth mentioning that in most cases, the main diffusivities computed by CALTPP program agree well with the reliable ones from the Matano-Kirkaldy method [18] within 50%. Besides, the sign of cross diffusion coefficient resulting from this program is generally the same as that from the Matano-Kirkaldy method. Such a high computational accuracy cannot be obtained in the previous assessments.

In addition to the calculation of diffusion matrix, CALTPP can also optimize the atomic mobilities of the element in phase. The optimization of atomic mobilities for fcc Ni-Sn and Cu-Ni-Sn alloys will be presented here to demonstrate this feature of CALTPP. The thermodynamic descriptions and atomic mobilities obtained by means of DICTRA have been reported in Refs. [47,48]. The presently obtained atomic mobility parameters for fcc Cu-Ni-Sn alloys via CALTPP program, as well as those in boundary binaries from the literature [47,[49], [50], [51], [52]], are listed in Table 1.

Fig. 3 shows the comparison between the calculated composition-dependent interdiffusivities in fcc Ni-Sn alloys at 1223-1473 K and the experimental data[53]. The dashed and solid lines denote the calculated results from the previous assessment [47] and the optimization in CALTPP. In Fig. 4, the calculated main interdiffusion coefficients in fcc Cu-Ni-Sn alloys at 1023 and 1073 K based on the atomic mobilities assessed by CALTPP program are compared with the measured data in Refs. [47,54]. The dashed lines, which refer to the diffusivities with a differential factor of 2 or 0.5 from the calculated ones, are superimposed in the plot. It can be seen from these results that almost all the experimental data were reasonably reproduced by means of CALTPP within general error ranges, demonstrating the reliability of CALTPP for the calculation of diffusivities and optimization of atomic mobilities.

Interfacial energy

Interfacial energy plays an important role in various material science phenomena, such as nucleation, coarsening, wetting and adsorption. The solid/liquid interfacial energies for Al-Zn, Cd-Zn, Cu-Co and Cu-Zn alloys at different temperatures are calculated by means of CALTPP program. The calculated results are presented in Fig. 5 together with the experimental data for Al-Zn [55], Cd-Zn [56,57], Cu-Co [58] and Cu-Zn [59] alloys. In the Cd-Zn system, Meydaneri et al. [56] studied the solid-liquid interfacial energy between solid Zn and Cd-Zn liquid solution (Zn in Cd-Zn) by Gibbs-Thomson method, and Saatçi and Pamuk [57] investigated the solid-liquid interfacial energy for solid Cd in equilibrium with the Cd-Zn eutectic liquid (Cd in Cd-Zn). In the calculation, the thermodynamic descriptions of these binary systems [[60], [61], [62], [63]], the molar volume of liquid for pure metals [64] and crystallographic information [65] were used, and the model-calculated values show a good overall agreement with the experimental results, which verifies the rationality of CALTPP program for the calculation of solid/liquid interfacial energies. Meanwhile, microstructure simulations can be performed with key input parameters obtained using this model, such as grain growth simulations during liquid-phase sintering [32].

The Ga-Ni system is selected as one representative example to predict the interfacial energies of coherent solid/solid phases using CALTPP program. The Gibbs energies of the disordered γ and ordered γ' phases in the Ga-Ni system are taken from Ref. [66], where the γ and γ' phases are described by one single Gibbs energy expression. The volume/area information is calculated from the lattice parameters measured by the High-temperature XRD [67]. Fig. 6 shows the CALTPP-predicted γ/γ' interfacial energies of the Ga-Ni system in comparison with the experimental value [68] using back-calculation with trans interface diffusion-controlled theory [69] and Lifshitz-Slyozov-Wagner theory [70,71]. The comparison demonstrates that the presently model-predicted γ_SS(LL) can provide reasonable value for the γ/γ' interfacial energies taking the uncertainties of the back-calculation into account.

The calculated liquid/liquid interfacial energies for the Ga-Pb system using the thermodynamic parameters from Ref. [72] are displayed in Fig. 7. The molar volumes of liquid Ga and Pb reported by Iida and Guthrie [64] are adopted in CALTPP ignoring the excess interaction parameters. As can be seen from Fig. 7, the model-predicted interfacial energies in CALTPP program is a little lower than the measured interfacial tension [73,74], but the deviation is still acceptable. And both the predicted and measured data display the same tendency against the temperature.

Thermal conductivity

Thermal conductivity controls the temperature gradients occurring in materials, which plays an important role in the microstructure and the performance of alloys. The reliable thermal conductivity for solution phase and multi-phase alloys can be reasonably accounted for by the model predictions. Table 2, Table 3 list the relevant thermal conductivity parameters optimized via CALTPP. Figs. 8(a) and (b) show the calculated thermal conductivities of (Mg) solid solution in the Mg-Gd and Mg-Y alloys in comparison with the experimental data [75], respectively. The thermal conductivity parameters of the binary alloys in Table 3 were used to predict the thermal conductivity of the Mg-Gd-Y ternary alloys in the same phase region. Fig. 8(c) shows the calculated thermal conductivity of the Mg-2Gd-3Y, Mg-4Gd-3Y, Mg-6Gd-3Y (wt.%) alloy in (Mg) solid solution phase and compared with the measured experimental data [75]. Fig. 9 represents the calculated thermal conductivities of (Mg)+Mg₅Gd two-phase region in the Mg-Gd alloy together with the experimental data [75]. These experimental data is obtained from our recent work and details are in Ref. [75]. It can be seen from these figures that the calculated results are in a good agreement with the experimental data.

Viscosity

Viscosity is a very important physical property of melts for the solidification simulation of the industrial cast metals and the modeling associated with fluid flow. Using CALTPP program, the viscosities of the pure Ag, Au, Cu and binary Ag-Au, Ag-Cu and Au-Cu systems were evaluated via CALPHAD-type method described in section 'CALPHAD-type models of thermophysical properties' in order to establish the viscosity database of the Ag-Au-Cu system.

The viscosities of multicomponent liquid phase can be extrapolated from the viscosities of pure elements and interaction parameters in binary melts. Thus the viscosity database of the Ag-Au-Cu system is constructed based on the pure and binary parameters obtained by CALTPP program, which are shown in Table 4, Table 5. In order to verify the reliability of the presently obtained parameters, the comparisons between the calculated viscosities and the experimental ones for pure Ag, Ag and Cu [76], as well as binary Ag-Au, Ag-Cu and Au-Cu alloys [77,78] are conducted as shown in Figs. 10(a) and (b), respectively. The calculated viscosities agree quite well with the experimental ones [7678]. Furthermore, this database is used to predict the viscosities of the ternary Ag-Au-Cu alloys. Fig. 11 shows the predicted viscosity of the Ag-Au-Cu system from 1170 to 1600 K using the present database in comparison with the data by Gebhardt and Wörwag [79]. It is noted that no ternary parameter is introduced in the calculation. As shown in Fig. 11, for samples 1 and 2, the predicted viscosity curve shows an excellent agreement with the experimental data. While for sample 3, the predicted viscosity is a little higher than experimental data. In general, the maximum deviation of all the three samples is no more than 5% and the tendency of the predicted viscosities is consistent with that of the experimental data [79], indicating the great reproducibility of the presently established viscosity database of the Ag-Au-Cu system and CALTPP program.

Molar volume

Molar volume is an important parameter when one describes the kinetic behavior together with other thermo-physical properties, such as diffusivity, surface tension, thermal conductivity and viscosity. Using the models of molar volume mentioned in Section 'CALPHAD-type models of thermophysical properties', the database of molar volume can be constructed in a way as done in our previous work [80]. Information about volume is needed in the modeling of diffusion for phase with the variable molar volume.

Fig. 12(a) displays the calculated molar volumes of the Al-Fe alloy compared with the measured data [81,[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83]]. For the liquid phase, the calculation results are consistent with the density of the intermediate alloy measured by Plevachuk [81], while large deviation can be observed at the Al-rich end, which is caused by large volatilization of Al in the alloy at high temperatures. Therefore, the Al-rich end data has not been used in the calculation. Fig. 12(b) shows the calculated molar volumes of the molten Al-Cu-Si alloys with compositions along Al-Cu₅₀Si₅₀ and Cu-Al₅₀Si₅₀ sections at 1300 K. The presently calculated values can reproduce the experimental data [84] well. The volume parameters for the Al-Fe and Al-Cu-Si systems are listed in Table 6.

Fig. 13 shows the comparison of the interdiffusion coefficients calculated in CALTPP program and the measured binary interdiffusivities of B2 NiAl alloy from different reports [[85], [86], [87], [88], [89]] under the cases of constant molar volume and variable molar volume. The work reported by Liu et al. [90] is also superimposed. The diffusion couple composition is Al_46.1Ti_4.7Ni_49.2/Al₅₀Ni₅₀, and the molar volume data and thermodynamic parameters described in Ref. [90] are utilized in CALTPP. According to the comments by Liu et al. [90], the concentration of Al is continuous at the two sides of the diffusion couple, while the concentration of Ni stays almost steady on the left side of the diffusion couple and the diffusion distance of Ti is short on the right-hand side of the diffusion couple. Therefore, the left end and right end of the diffusion couple can be regarded as the Al-Ti pseudo-binary system and Al-Ni pseudo-binary system, respectively. It can be seen from Fig. 13 that the interdiffusivities calculated for the case of constant molar volume are somewhat higher than those obtained for the case of variable molar volume. And in both cases the interdiffusion coefficients increase with the increased Al mole fraction. Furthermore, Fig. 13 illustrates that the interdiffusion coefficients obtained in the Al-Ni pseudo-binary system are generally smaller than the literature values for the binary Al-Ni system [85,[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89]]. When the mole fraction of Al reaches 0.5, the interdiffusion coefficients from the Al-Ni pseudo-binary system approach the literature values [85,[1], [2], [3], [4], [5], [6], [7], [8], [9], [10], [11], [12], [13], [14], [15], [16], [17], [18], [19], [20], [21], [22], [23], [24], [25], [26], [27], [28], [29], [30], [31], [32], [33], [34], [35], [36], [37], [38], [39], [40], [41], [42], [43], [44], [45], [46], [47], [48], [49], [50], [51], [52], [53], [54], [55], [56], [57], [58], [59], [60], [61], [62], [63], [64], [65], [66], [67], [68], [69], [70], [71], [72], [73], [74], [75], [76], [77], [78], [79], [80], [81], [82], [83], [84], [85], [86], [87], [88], [89]]. The application of CALTPP to the Al_46.1Ti_4.7Ni_49.2/Al₅₀Ni₅₀ diffusion couple demonstrates that the molar volume data from the current program is successfully used to describe the diffusion considering volume effect.

Outlook for the future development of modeling for thermophysical properties

The above case studies of CALTPP to the calculation and/or optimization for diffusion coefficient, interfacial energy, thermal conductivity, viscosity and molar volume exhibit the functional diversity and applicability of CALTPP program and its reliability. It should be mentioned that it is a severe challenge to develop the microstructure-dependent thermophysical models along with related programs. For example, the calculations of semi-coherent interfacial energy and in-coherent interfacial energy as well as the multi-component database of thermophysical properties are of importance in the future version of CALTPP program. The coupling of the accurate thermophysical properties in CALTPP-type programs with commercial and open source software is the trend of materials design in the future. It can be prospected that the reliable thermophysical properties in conjunction with the ICME tools will play more important role in materials design.

The CALTPP can also be used for the sake of education for both undergraduates and postgraduates. For instance, a few methods implemented in CALTPP including the Matano-Boltzmann method and Whittle and Green method can be utilized for teaching diffusion. The students can apply these methods in CALTPP to calculate diffusion coefficients and to understand the detail of these methods.

Conclusion

In the present work, a general program CALTPP for the calculation and/or optimization of several thermophysical properties is developed. These thermophysical properties including diffusion coefficient, interfacial energy, thermal conductivity, viscosity and molar volume are the important inputs for microstructure simulations and mechanical property predictions. The structure of CALTPP program is described, and it contains the input module, calculation and/or optimization module and output module. Various CALPHAD-type models for thermophysical properties, such as diffusion coefficient, interfacial energy, thermal conductivity, viscosity and molar volume, are implemented in CALTPP program. In addition, several representative case studies about the calculation and/or optimization of thermophysical properties are presented to show the important features of CALTPP program. It is highly expected that the functional diversity and applicability of CALTPP program will make it widely applied in the field of materials design and education.

Acknowledgements

The financial supports from the National Natural Science Foundation of China (Grant Nos. 51671219 and 51429101) and National Key Research and Development Plan (Grant No. 2016YFB0701202) are greatly acknowledged. The work of GK was supported by nano-Ginop Project GINOP-2.3.2-15-2016-00027 in the framework of the Szechenyi 2020 program, supported by the European Union.

---