Estimation of Two Component Activities of Binary Liquid Alloys by the Pair Potential Energy Containing a Polynomial of the Partial Radial Distribution Function

Abstract

An investigation of partial radial distribution functions and atomic pair potentials within a system has established that the existing potential functions are rooted in the assumption of a static arrangement of atoms, overlooking their distribution and vibration. In this study, Hill’s proposed radial distribution function polynomials are applied for the pure gaseous state to a binary liquid alloy to derive the pair potential energy. The partial radial distribution functions of 36 binary liquid alloy from literatures were used to obtain the binary model parameters of four thermodynamic models for validation. Results show that the regular solution model (RSM) and molecular interaction volume model (MIVM) outperform other models when the asymmetric method calculates the partial radial distribution function. RSM demonstrates an average SD of 0.078 and an ARD of 32.2%. Similarly, MIVM exhibits an average SD of 0.095 and an average ARD of 32.2%. Wilson model yields an average SD of 0.124 and an average ARD of 226%. Nonrandom two-liquid (NRTL) model exhibits an average SD of 0.225 and an average ARD of 911%. On applying the partial radial distribution function symmetry method, MIVM and RSM outperform the other models, with an average SD of 0.143 and an average ARD of 165.9% for MIVM. RSM yields an average SD of 0.117 and an average ARD of 208.3%. Wilson model exhibits average values of 0.133 and 305.6% for SD and ARD, respectively. NRTL model shows an average SD of 0.200 and an average ARD of 771.8%. Based on this result, the influence of the symmetry degree on the thermodynamic model is explored by examining the symmetry degree as defined by the experimental activity curves of the two components.

1. Introduction

Pair potentials play a fundamental role in comprehending the static and dynamic properties of gases, liquids, and solids [1]. The main objective of studying pair potentials is to develop accurate and reliable models that describe interatomic interactions and enable simulation, prediction, and explanation of the behavior and properties of atomic systems. Various computational simulation studies, such as for protein folding, drug–receptor interactions, and material property prediction, are conducted using atomic pair potentials to ascertain the underlying mechanisms of atomic interactions, explore novel material properties, and optimize drug design. Materials science and surface science researchers have investigated atomic potentials to thoroughly understand phenomena such as material structure, stability, crystal defects, and surface adsorption. Understanding interatomic interactions is essential in material design, catalyst development, and comprehending material interfaces and surface phenomena. Atomic pair potential models encompass widely used semiempirical potentials such as the Lennard–Jones potential [2], Morse potential [3], and Born–Mayer potential [4], as well as atomic force fields, quantum mechanical descriptions, and statistical mechanics methods. Different factors affect the selection and examination of these models, including application needs and accuracy considerations [5,6,7]. Accurate pair potentials provide a comprehensive description of energy, geometric structure, and spectral properties of molecules and serve as the basis for investigating collision and chemical-reaction dynamics, such as those for atomic collisions. Thus, an in-depth study of potential pair functions holds significant physical implications and offers broad applications. This study reveals a direct correlation between the radial distribution function (RDF) and atomic pair potential. This paper proposes a linkage between the radial distribution function and the atomic pair potential energy. This is applied to a binary liquid alloy system using a polynomial expression for the radial distribution function of a pure gas. The group element activity is estimated based on the symmetries outlined in this paper, utilizing the radial distribution functions of 36 binary liquid alloys from the literature into the model.

2. Thermodynamic Model and Symmetry

2.1. Miedema Model

In 1973, Miedema et al. suggested a semi-empirical theoretical model that effectively characterizes the heat of mixing in binary alloys. This model encompasses solid solubility, the mesostable positioning of metal ions following ion injection, surface aggregation phenomena, surface energy, the formation energy of single vacancies in metals and metal compounds, as well as the development of nonphenolic alloys. For any metal binary alloy, the expression is [8]

$$f_{ij}=\frac{2p{V}_{mi}^{{}^{2/3}}{V}_{mj}^{{}^{2/3}}\times [(q/p){(\Delta n_{ws}^{1/3})}^2-{(\Delta \phi )}^2-a(r/p)]}{{(\Delta n_{ws}^{1/3})}_i^{-1}+{(\Delta n_{ws}^{1/3})}_j^{-1}}$$

(1)

$$\Delta H_m^{}=f_{ij}\frac{x_i[1+{\mu }_i{x}_j({\phi }_i-{\phi }_j)]x_j[1+{\mu }_j{x}_i({\phi }_j-{\phi }_i)]}{x_i{V}_{mi}^{{}^{2/3}}[1+{\mu }_i{x}_j({\phi }_i-{\phi }_j)]+x_j{V}_{mj}^{{}^{2/3}}[1+{\mu }_j{x}_i({\phi }_j-{\phi }_i)]}$$

(2)

The activity coefficient is:

$$\ln {\gamma }_i=\frac{\Delta H_m^{}}{RT}\left\{1+(1-x_i)\left\{\begin{array}{l}\frac{1}{x_i}-\frac{1}{1 - x_i}-\frac{{\mu }_i({\phi }_i - {\phi }_j)}{1 + {\mu }_i(1 - x_i)({\phi }_i - {\phi }_j)} + \frac{{\mu }_j({\phi }_j - {\phi }_i)}{1 + {\mu }_j{x}_i({\phi }_j - {\phi }_i)}\\ -\frac{V_{mi}^{2/3}[1 + {\mu }_i(1 - 2x_i)({\phi }_i - {\phi }_j)] + V_{mj}^{2/3}[-1 + {\mu }_j(1 - 2x_i)({\phi }_j - {\phi }_i)]}{x_i{V}_{mi}^{2/3}[1 + {\mu }_i{x}_j({\phi }_i - {\phi }_j)] + x_j{V}_{mj}^{2/3}[1 + {\mu }_j{x}_i({\phi }_j - {\phi }_i)]}\end{array}\right\}\right\}$$

(3)

In the above equation, $x_i$ and $x_j$ represent the mole fractions of species i and j respectively. $V_{mi}$ and $V_{mj}$ denote the molar volumes of i and j, while ${(n_{ws})}_i^{}$ and ${(n_{ws})}_j^{}$ stand for the electron densities, ${\phi }_i$ and ${\phi }_j$ are electronegative. The variables $p$,$q$,$r$,${\mu }_i$,${\mu }_j$ and $a$ are empirical constants. For a binary alloy composed of a transition group metal and a multivalent non-transition group metal, the formula $f_{ij}$ is transformed to:

$$f_{ij}=\frac{2p{V}_{mi}^{2/3}{V}_{mj}^{{}^{2/3}}\times [(q/p){(\Delta n_{ws}^{1/3})}^2-{(\Delta \phi )}^2-\alpha {(r/p)}_i{(r/p)}_j]}{{(\Delta n_{ws}^{1/3})}_i^{-1}+{(\Delta n_{ws}^{1/3})}_j^{-1}}$$

(4)

2.2. Regular Solution Model (RSM)

Hildebrand suggested the RSM in 1929 [9]. According to this model, the molar excess volume and mixing entropy are zero, and the molar excess mixing Gibbs free energy is equal to the molar excess mixing enthalpy [10]. The expression for the molar excess Gibbs free energy for a binary system is as follows; the activity coefficient of the component $i$ is $\ln {\gamma }_i$.

$$\frac{G_m^E}{RT}=\Delta H_m={\mathsf{\Omega }}_{ij}{x}_i{x}_j$$

(5)

$$\ln {\gamma }_i={\mathsf{\Omega }}_{ij}{x}_j^2$$

(6)

Here, $x_i$ and $x_i$ are the mole fractions of components $i$ and $j$, respectively, and ${\mathsf{\Omega }}_{ij}$ is the interaction parameter between components $i$ and $j$. ${\mathsf{\Omega }}_{ij}$ is only related to temperature and does not change with the composition of components.

2.3. Wilson Model

In 1964, Wilson [11] suggested a semiempirical and semitheoretical model based on the local concept. This model assumes that “local concentrations” (represented as volume fractions) primarily determine molecular interactions. These concentrations are defined in relation to the Boltzmann distribution probability term for energy. The excess free energy associated with the concentrations can be expressed as follows.

$$\frac{G_m^E}{RT}=-x_i\ln (x_i+A_{ji}{x}_j)-x_j\ln (x_j+A_{ij}{x}_i)$$

(7)

$$\ln {\gamma }_i=-\ln (x_i+A_{ij}{x}_j)+x_j\left(\frac{A_{ij}}{x_i+A_{ij}{x}_j}-\frac{A_{ji}}{x_j+A_{ji}{x}_i}\right)$$

(8)

Here, $A_{ij}$ and $A_{ji}$ are the interaction parameters between components $i$ and $j$, which are only related to temperature and do not change with the composition of components [12].

2.4. Nonrandom Two-Liquid (NRTL) Model

The NRTL model, suggested by Renon and Prausnitz in 1968 [13], has been extensively used in correlating thermodynamic data, computing thermodynamic properties, and predicting phase equilibrium for diverse fluid systems in chemical processes. This model, based on the concept of local concentration, permits the determination of molar excess Gibbs free energy. The activity coefficient of the component $i$ is $\ln {\gamma }_i$.

$$\frac{G_m^E}{RT}=x_i{x}_j\left(\frac{{\tau }_{ji}{G}_{ji}}{x_i+x_j{G}_{ji}}+\frac{{\tau }_{ij}{G}_{ij}}{x_j+x_i{G}_{ij}}\right)$$

(9)

$$\ln {\gamma }_i=x_j^2\left[\frac{{\tau }_{ji}{G}_{ji}^2}{{(x_i+x_j{G}_{ji})}^2}+\frac{{\tau }_{ij}{G}_{ij}}{{(x_j+x_i{G}_{ij})}^2}\right]$$

(10)

$${\tau }_{ij}=\frac{g_{ij}-g_{jj}}{RT}\hspace{1em}\hspace{1em}\hspace{1em}{\tau }_{ji}=\frac{g_{ji}-g_{ii}}{RT}$$

(11)

$$G_{ij}=\exp (-\alpha {\tau }_{ij})\hspace{1em}\hspace{1em}\hspace{1em}{G}_{ji}=\exp (-\alpha {\tau }_{ji})$$

(12)

Here, $G_{ij}$ and $G_{ji}$ are energy parameters characterizing the interaction between components i and j; $\alpha$ is related to nonrandomness in the mixture, independent of temperature and composition of a solution. Moreover, the characteristic parameter of a solution depends on the solution type. When the mixture is entirely random, many binary system experimental data show that $\alpha$ varies from 0.2 to 0.47. In this study, $\alpha$ = 0.3.

2.5. Molecular Interaction Volume Model (MIVM)

In 2000, Tao suggested the molecular interaction volume model [14], which applied statistical thermodynamics and fluid phase equilibrium theory to describe the motion of liquid molecules. This model yields the following expression:

$$\frac{G_m^E}{RT}=x_i\ln \left(\frac{V_{mi}}{x_i{V}_{mi}+x_j{V}_{mj}{B}_{ji}}\right)+x_j\ln \left(\frac{V_{mj}}{x_j{V}_{mj}+x_i{V}_{mi}{B}_{ij}}\right)-\frac{x_i{x}_j}{2}\left(\frac{Z_i{B}_{ji}\ln B_{ji}}{x_i+x_j{B}_{ji}}+\frac{Z_j{B}_{ij}\ln B_{ij}}{x_j+x_i{B}_{ij}}\right)$$

(13)

The activity coefficients of the components $i$ is $\ln {\gamma }_i$.

$$\begin{array}{l}\ln {\gamma }_i=1+\ln \left(\frac{V_{mi}}{V_{mi}{x}_i + V_{mj}{B}_{ji}{x}_j}\right)-\frac{x_i{V}_{mi}}{V_{mi}{x}_i + x_j{V}_{mj}{B}_{ji}}-\frac{x_j{V}_{mi}{B}_{ij}}{V_{mj}{x}_j + x_i{V}_{mi}{B}_{ij}}\\ -\frac{x_j{}^2}{2}\left[\frac{Z_i{B}_{ji}{}^2\ln B_{ji}}{{(x_i + x_j{B}_{ji})}^2}+\frac{Z_j{B}_{ij}\ln B_{ij}}{{(x_j + B_{ij}{x}_i)}^2}\right]\end{array}$$

(14)

Here, $Z_i$ and $Z_j$ are the first coordination numbers of $i$ and $j$ pure substances, respectively, and $V_{mi}$ and $V_{mj}$ are the molar volumes of $i$ and $j$, respectively. $B_{ij}$ and $B_{ji}$ are the interaction parameters of $i-j$ and $j-i$, respectively. $k$ is the Boltzmann constant, and T is the temperature.

2.6. Symmetry

Figure 1 shows the molar fraction $x_i=x_j=0.5$ of the two components as the symmetry axis of their activity curve. The symmetry degree of the activity curve can be defined as the average absolute value of the activity difference between the two components at the same concentration. In this context, $S$ represents the measure of symmetry.

$$S_{ij}=\frac{{\displaystyle \sum _{l=1}^m{\left|{(a_i-a_j)}_{x_i=x_j}\right|}_l}}{m}$$

(15)

Figure 1. (a) Schematic diagram of symmetrical activity curves, (b) Schematic diagram of asymmetric activity curves.

The symmetry increases as the S value decreases. At $S=0$, the binary liquid alloy exhibits complete symmetry, where $a_i$ and $a_j$ denote the experimental activity of components i and j, respectively, and m represents the number of experimental activities. This definition also applies to similar geometric figures. The binary liquid alloy with low symmetry is shown in Figure 2. Table 1 presents the symmetry degrees of the activity curves for the 36 binary liquid alloys based on the definitions mentioned above.

Figure 2. (a) Schematic diagram of symmetrical Al-Zn activity curves, (b) Schematic diagram of asymmetrical Cu-Sb activity curves.

Table 1. Symmetry of the 36 binary liquid alloys from highest to lowest.

System	Co-Ni	Al-Zn	Cu-Ni	Al-Ni	Cu-Fe	Ge-Sn	Ag-Cu	Pb-Sb	Al-Si	Al-Co
$S_{ij}$	0	0	0.0028	0.0034	0.0048	0.007	0.0088	0.0096	0.0102	0.0114
System	Li-Mg	Sb-Sn	Cu-Zr	K-Na	Pb-Sn	Al-Mg	Cs-K	Li-Na	Au-Cu	Al-Li
$S_{ij}$	0.0114	0.0115	0.0119	0.0136	0.0158	0.0177	0.0228	0.0336	0.0364	0.0452
System	Nb-Zr	Ni-Pd	Cu-Mg	Al-Ca	Ni-Zr	Al-Sn	Al-Cu	Nb-Ni	Cu-Sn	Au-Si
$S_{ij}$	0.0473	0.0494	0.0597	0.0724	0.0807	0.0823	0.1116	0.1334	0.1342	0.139
System	Li-Sn	Fe-Si	Ag-In	Ge-Te	Al-Au	Cu-Sb
$S_{ij}$	0.14	0.1454	0.1483	0.1548	0.160	0.208

3. Pair Potential Energy Polynomials of the Binary Liquid

In an extremely diluted pure gas, the correlation between the interatomic potential energy and RDF can be expressed as follows [15]:

$$\underset{{\rho }_0\to 0}{\lim }{g}_{12}(r)=\exp \left(-\frac{{\epsilon }_{12}}{kT}\right)$$

(16)

However, in practical scenarios, the RDF and interatomic potential energy strongly depend on the number density of the system. The relation between the RDF and pair potential energy can be expanded using a polynomial expression in terms of the number density ${\rho }_0$:

$$g_{12}(r,{\rho }_0,T)=e^{-\epsilon /kT}\left[1+{\rho }_0{g}_1(R,T)+{\rho }_0^2{g}_2(R,T)+\cdots \right]$$

(17)

It is seen formula Equation (17), that there is connection between the RDF and pair potential energy. In the case of ${\rho }_0\ne 0$, 3 atoms in the pure gas system, labeled 1, 2 and 3, are shown in Figure 3a. These three atoms have six interactions: 1-1, 1-2, 1-3, 2-2, 2-3, 3-3, which influence each other. The radial distribution function for 1-2 is $g_{12}(r,{\rho }_0,T)$, i.e., the distribution of atom 1 centred on atom 2, but influenced by atom 3. This form is centred on atom 3 and influences atom 1 and atom 2 in $g_{12}(r,{\rho }_0,T)$.

$$\begin{array}{l}{g}_{12}(r,{\rho }_0,T)=e^{-{\epsilon }_{12}/kT}\exp \left\{{\rho }_0{\displaystyle {\int }_V[e^{-{\epsilon }_{13}/kT}-1]}[e^{-{\epsilon }_{23}/kT}-1]d{r}_3\right\}\\ =e^{-{\epsilon }_{12}/kT}\left\{1+{\rho }_0{\displaystyle {\int }_V[e^{-{\epsilon }_{13}/kT}-1]}[e^{-{\epsilon }_{23}/kT}-1]d{r}_3\right\}\end{array}$$

(18)

Figure 3. (a) Schematic diagram of pure gases or pure liquid atomics interactions, (b) Schematic diagram of binary liquid alloy atomics interactions.

Consequently, the expansion based on ${\rho }_0$ is $1+{\rho }_0{\displaystyle {\int }_V[e^{-{\epsilon }_{13}/kT}-1]}[e^{-{\epsilon }_{23}/kT}-1]d{r}_3$. Equation (18) is compared with Equation (17).

$$\begin{array}{l}{g}_1(r_{12},T)={\displaystyle {\int }_V[e^{-{\epsilon }_{13}/kT}-1]}[e^{-{\epsilon }_{23}/kT}-1]d{r}_3={\displaystyle {\int }_V[e^{-{\epsilon }_{13}/kT}{e}^{-{\epsilon }_{23}/kT}-e^{-{\epsilon }_{23}/kT}-e^{-{\epsilon }_{13}/kT}+1]}d{r}_3\\ ={\displaystyle {\int }_V{e}^{-{\epsilon }_{13}/kT}{e}^{-{\epsilon }_{23}/kT}d{r}_3-{\displaystyle {\int }_V{e}^{-{\epsilon }_{23}/kT}d{r}_3-{\displaystyle {\int }_V{e}^{-{\epsilon }_{13}/kT}d{r}_3+{\displaystyle {\int }_V}}d{r}_3}}\end{array}$$

(19)

The above is the influence of atomic interactions in pure gases; assume let the interaction of atoms is applied to the liquid metal system, and Equations (16) and (17) are applied to the liquid metal. The interactions between atoms in a pure liquid metal are similar to those in the pure gaseous state, both being interactions between the same atoms. However, there are differences in the binary liquid alloy and different interactions between different atoms, as shown in Figure 3b. There are three atomic interactions in binary liquid alloys: $i-i$, $i-j$, and $j-j$. The RDF $i-j$ describes atomic distribution $i$ around atom $j$ in a binary system. However, the remaining atom $i$ influences the atom $i$ in the RDF $i-j$, while the remaining atom $j$ influences the atom $j$ in the RDF $i-j$.

Unlike pure gas centered on the remaining atoms 3 affecting the RDF 1–2, as shown in Figure 3a. In binary liquid alloys, the remaining atoms i and j influence the atoms i and j in the radial distribution function $i-j$. That is, the atom $i$ in the RDF influencing $i-j$ is centered on the other atoms $i$, while the atom $j$ in the RDF influencing $i-j$ is centered on the other atoms $j$.

If the RDF in a binary liquid system can be expanded as a polynomial based on number density, then Equation (18) takes the following form:

$$g_1(r_{ij},T)={\displaystyle {\int }_V{e}^{-{\epsilon }_{ii}/kT}dr\times {\displaystyle {\int }_V{e}^{-{\epsilon }_{jj}/kT}}dr-{\displaystyle {\int }_V{e}^{-{\epsilon }_{ii}/kT}dr}}-{\displaystyle {\int }_V{e}^{-{\epsilon }_{jj}/kT}dr+(}{\displaystyle {\int }_V}d{r}_{ii}+{\displaystyle {\int }_V}d{r}_{jj})$$

(20)

Suppose that the subradial distribution function $g_1(r_{ij},T)$ of the principal RDF $g_{ij}(r,{\rho }_0,T)$ conforms to ${\rho }_0\to 0$, i.e., $i-j$ is the primary RDF, then $i-i$ and $j-j$ conform to ${\rho }_0\to 0$ and the subradial distribution function is:

$$g_{ii}(r)=\exp \left(-\frac{{\epsilon }_{ii}}{kT}\right)\hspace{1em}\hspace{1em}\hspace{1em}{g}_{jj}(r)=\exp \left(-\frac{{\epsilon }_{jj}}{kT}\right)$$

(21)

Substituting Equation (21) into Equation (20), we obtain:

$$g_1(r_{ij},T)={\displaystyle {\int }_V{g}_{ii}(r)dr}\times {\displaystyle {\int }_V{g}_{jj}(r)dr}-{\displaystyle {\int }_V{g}_{ii}(r)dr}-{\displaystyle {\int }_V{g}_{jj}(r)}dr+\left({\displaystyle {\int }_{V}d{r}_{ii}}+{\displaystyle {\int }_{V}d{r}_{jj}}\right)$$

(22)

Then substituting Equation (22) into Equation (18), we obtain:

$$\begin{array}{l}{g}_{ij}(r,{\rho }_0,T)=e^{-{\epsilon }_{ij}/kT}\exp \left\{{\rho }_0\left[\begin{array}{l}{\displaystyle {\int }_V{g}_{ii}(r)dr}\times {\displaystyle {\int }_V{g}_{jj}(r)dr}-{\displaystyle {\int }_V{g}_{ii}(r)dr}\\ -{\displaystyle {\int }_V{g}_{jj}(r)dr}+({\displaystyle {\int }_{V}d{r}_{ii}}+{\displaystyle {\int }_{V}d{r}_{jj}})\end{array}\right]\right\}\\ =e^{-{\epsilon }_{ij}/kT}\left\{1+{\rho }_0\left[\begin{array}{l}{\displaystyle {\int }_V{g}_{ii}(r)dr}\times {\displaystyle {\int }_V{g}_{jj}(r)dr}-{\displaystyle {\int }_V{g}_{ii}(r)dr}\\ -{\displaystyle {\int }_V{g}_{jj}(r)dr}+({\displaystyle {\int }_{V}d{r}_{ii}}+{\displaystyle {\int }_{V}d{r}_{jj}})\end{array}\right]\right\}\end{array}$$

(23)

The relation between the RDF and potential energy can be obtained using Equation (23):

$$-\frac{{\epsilon }_{ij}}{kT}=\ln \frac{g_{ij}(r,{\rho }_0,T)}{\left\{1+{\rho }_0\left[\begin{array}{l}{\displaystyle {\int }_V{g}_{ii}(r)dr}\times {\displaystyle {\int }_V{g}_{jj}(r)dr}-{\displaystyle {\int }_V{g}_{ii}(r)dr}\\ -{\displaystyle {\int }_V{g}_{jj}(r)dr}+({\displaystyle {\int }_{V}d{r}_{ii}}+{\displaystyle {\int }_{V}d{r}_{jj}})\end{array}\right]\right\}}$$

(24)

Pair potential energy between molecules can be accurately calculated using the RDF. This function represents the ratio of the probability of finding another atom at a distance $r$ to the random distribution [16]. In the double distribution function, for a system with $N$ atoms and volume $V$, the probability of a atom appearing in the element $d{r}_i$ is $(N/V)d{r}_i$, the probability of an atom appearing at the distance $d{r}_j$ is $(N/V)d{r}_j$, and the probability of atomic pairs appearing at a distance $r$ is ${(N/V)}^{2}d{r}_{i}d{r}_j$. The double distribution function is given as follows:

$$p^{(2)}(r)d{r}_{i}d{r}_j={(\frac{N}{V})}^{2}d{r}_{i}d{r}_j$$

(25)

In the system, the average potential energy $\epsilon$ between each atom is:

$$\epsilon ={\displaystyle \underset{V}{\iint }{\epsilon }_{ij}(r)}{p}^{(2)}(r)d{r}_{i}d{r}_j$$

(26)

However, in the RDF of binary systems, the probability of having atom $i$ in $d{r}_i$ at $r_i$ and atom $j$ in $d{r}_j$ at $r_j$ is $p^{(2)}(r)d{r}_{i}d{r}_j$. The potential energy is $\epsilon$, and the average value of $\epsilon$ is the sum of all possible times of the probabilities:

$$\begin{array}{l}\overline{{\epsilon }_{ij}}=\frac{1}{V^2}{\displaystyle \underset{V}{\iint }{\epsilon }_{ij}(r)}{g}_{ij}(r)d{r}_{i}d{r}_j=\frac{1}{V^2}{\displaystyle \underset{V}{\int }d{r}_i}{\displaystyle \int {\epsilon }_{ij}(r)g_{ij}(r)}d{r}_j=\frac{1}{V}{\displaystyle \int {\epsilon }_{ij}(r)g_{ij}(r)}4\pi r^{2}dr\\ =\frac{4\pi {\displaystyle \int {\epsilon }_{ij}(r)g_{ij}(r)}{r}^{2}dr}{4\pi {\displaystyle \int r^2}dr}=\frac{{\displaystyle \int {\epsilon }_{ij}(r)g_{ij}(r)}{r}^{2}dr}{{\displaystyle \int r^2}dr}\end{array}$$

(27)

Substituting Equation (24) into Equation (27) yields the potential energy between atoms:

$$-\frac{\overline{{\epsilon }_{ij}}}{kT}=\frac{{\displaystyle \int g_{ij}(r)}{r}^2\left\{\ln \frac{g_{ij}(r)}{1 + {\rho }_0\left[\begin{array}{l}{\displaystyle {\int }_V{g}_{ii}(r)dr} \times {\displaystyle {\int }_V{g}_{jj}(r)dr} - {\displaystyle {\int }_V{g}_{ii}(r)dr}\\ -{\displaystyle {\int }_V{g}_{jj}(r)dr} + ({\displaystyle {\int }_{V}d{r}_{ii}} + {\displaystyle {\int }_{V}d{r}_{jj}})\end{array}\right]}\right\}dr}{{\displaystyle \int r^2}dr}$$

(28)

The peak value of the RDF signifies the disparity between the local and bulk molar fractions. As the mole fraction increases, the contributions of the second and third RDF peaks diminish while the contribution of the first peak amplifies. The pair potential energy is then calculated by using Equation (28) and the area of the first peak of the skewed RDF. The area under the first peak was computed through graphical integration, as depicted in Figure 4. Notably, this approach differs from Wang’s utilization of the L-PPDF mathematical form with Gaussian fitting, which relies on fitting parameters u and v [17,18]. It is important to emphasize that this study does not incorporate any fitting parameters. The RDF used in this study is defined by three key coordinates: $r_0$ (which represents the starting point of g(r) before reaching zero), $r_m$ (the transverse coordinate of the first peak of g(r)), and $r_1$ (the transverse coordinate of the first valley of g(r)). The asymmetric method of calculating the RDF involves integrating the area between $r_0$ and $r_1$, while the symmetric method involves integrating twice the area between $r_0$ and $r_m$ (Figure 4). The trapezoidal method [19] is used to compute these areas (Equation (29)). Table 2 lists the references for the partial RDFs of 36 binary liquid alloys.

$${\displaystyle \underset{r1}{\overset{r0}{\int }}g(r)dr\approx \frac{b-a}{2N}{\displaystyle \sum _{n=1}^N\left[g(r_n)+g(r_{n+1})\right]}}=\frac{b-a}{2N}\left[g(r)+2g(r_2)+.......+2g(r_N)+g(r_{N+1})\right]$$

(29)

Figure 4. (a) Schematic diagram of the asymmetric method of graphical integration of a radial distribution function, (b) Schematic diagram of the symmetric method of graphical integration of a radial distribution function.

Table 2. Partial radial distribution functions for 36 binary liquid alloys in literatures.

System	Co-Ni [20]	Al-Zn [21]	Cu-Ni [22]	Al-Ni [23]	Cu-Fe [24]	Ge-Sn [25]	Ag-Cu [23]	Pb-Sb [26]	Al-Si [27]	Al-Co [28]
System	Li-Mg [29]	Sb-Sn [30]	Cu-Zr [31]	K-Na [32]	Pb-Sn [33]	Al-Mg [34]	Cs-K [35]	Li-Na [36]	Au-Cu [37]	Al-Li [38]
System	Nb-Zr [39]	Ni-Pd [40]	Cu-Mg [41]	Al-Ca [42]	Ni-Zr [43]	Al-Sn [44]	Al-Cu [45]	Nb-Ni [46]	Cu-Sn [47]	Au-Si [48]
System	Li-Sn [49]	Fe-Si [50]	Ag-In [51]	Ge-Te [52]	Al-Au [53]	Cu-Sb [54]

The RSM has a tunable parameter ${\mathsf{\Omega }}_{ij}$. The average coordination number $\overline{Z}$ is obtained using the pure coordination number of the two components. Additionally, the temperature $T$ documented in the literature and the calculated parameter ${\mathsf{\Omega }}_{ij}$ can be referred to calculate the parameter ${{\mathsf{\Omega }}_{ij}}^{\prime }$ at the desired temperature $T^{\prime }$.

$$\overline{Z}=\frac{1}{2}(Z_i+Z_j)$$

(30)

$${\mathsf{\Omega }}_{ij}=kT\left\{\overline{Z}\left[\overline{{\epsilon }_{ij}}-\frac{1}{2}(\overline{{\epsilon }_{ii}}+\overline{{\epsilon }_{jj}})\right]\right\}$$

(31)

$$T\ln {\mathsf{\Omega }}_{ij}=T^{\prime }\ln {{\mathsf{\Omega }}_{ij}}^{\prime }$$

(32)

The parameters $A_{ij}$ and $A_{ji}$ of Wilson equation are given. Additionally, the values of parameters ${A_{ij}}^{\prime }$ and ${A_{ij}}^{\prime }$ at the desired temperature $T^{\prime }$ can be obtained using the temperature $T$ from the literature that was employed to calculate the corresponding parameters $A_{ij}$ and $A_{ji}$.

$$A_{ij}=\frac{V_i}{V_j}\exp \left(-\frac{\overline{{\epsilon }_{ij}}-\overline{{\epsilon }_{jj}}}{RT}\right)\hspace{1em}\hspace{1em}\hspace{1em}{A}_{ji}=\frac{V_j}{V_i}\exp \left(-\frac{\overline{{\epsilon }_{ji}}-\overline{{\epsilon }_{ii}}}{RT}\right)$$

(33)

$$T\ln A_{ij}=T^{\prime }\ln {A_{ij}}^{\prime }\hspace{1em}\hspace{1em}\hspace{1em}T\ln A_{ji}=T^{\prime }\ln {A_{ij}}^{\prime }$$

(34)

The NRTL has two parameters ${\tau }_{ij}$ and ${\tau }_{ji}$. These parameters obtained using the temperature $T$ mentioned in the literature are employed to calculate the parameters ${{\tau }_{ij}}^{\prime }$ and ${{\tau }_{ij}}^{\prime }$, respectively, at the required temperature $T^{\prime }$.

$${\tau }_{ij}=-\frac{\overline{{\epsilon }_{ij}}-\overline{{\epsilon }_{jj}}}{kT}\hspace{1em}\hspace{1em}\hspace{1em}{\tau }_{ji}=-\frac{\overline{{\epsilon }_{ji}}-\overline{{\epsilon }_{ii}}}{kT}$$

(35)

$$T\ln {\tau }_{ij}=T^{\prime }\ln {{\tau }_{ij}}^{\prime }\hspace{1em}\hspace{1em}\hspace{1em}T\ln {\tau }_{ji}=T^{\prime }\ln {{\tau }_{ij}}^{\prime }$$

(36)

The MIVM involves parameters $B_{ij}$ and $B_{ji}$, representing the interaction parameters for i–j and j–i interactions, respectively. Using $B_{ij}$ and $B_{ji}$ values obtained at temperature $T$, the corresponding parameters ${B_{ij}}^{\prime }$ and ${B_{ij}}^{\prime }$ can be calculated at the desired temperature $T^{\prime }$.

$$B_{ij}=\exp \left(-\frac{\overline{{\epsilon }_{ij}}-\overline{{\epsilon }_{jj}}}{kT}\right)\hspace{1em}\hspace{1em}\hspace{1em}{B}_{ji}=\exp \left(-\frac{\overline{{\epsilon }_{ji}}-\overline{{\epsilon }_{ii}}}{kT}\right)$$

(37)

$$T\ln B_{ij}=T^{\prime }\ln {B_{ij}}^{\prime }\hspace{1em}\hspace{1em}\hspace{1em}T\ln B_{ji}=T^{\prime }\ln {B_{ij}}^{\prime }$$

(38)

4. Result Analysis

4.1. Asymmetric Method for Calculating the RDF

The asymmetric method uses the area between $r_0$ and $r_1$ presented in Figure 4 and Equation (28) to determine the parameters for each model, see Table 3. Table 4 demonstrates that among the first 12 systems, the RSM performs better than other models for six binary liquid alloys. The average standard deviation (SD) is 0.027, and the average relative deviation (ARD) is 7.7%. In the case of the 12 binary liquid alloys with moderate symmetry, the RSM outperforms the other three models in six systems, resulting in an average SD of 0.059 and an average ARD of 13.83%. MIVM also performs well, with an average SD of 0.091 and an average ARD of 25.7%. For the 12 binary liquid alloys with low symmetry, the RSM outperforms the other three models in three systems, yielding an average SD of 0.101 and an average ARD of 39.5%. Additionally, the MIVM outperforms the other three models in five binary liquid alloys when the binary liquid alloys have even lower symmetry, with an average SD of 0.131 and an average ARD of 45.3%. Considering the average performance across all 36 binary liquid alloys, the Wilson and NRTL models exhibit the poorest performance, displaying larger SD and ARD values than the other models. As shown in Figure 5, the SD and ARD values generally increase with decreasing symmetry, albeit not significantly. Further analysis of high, medium, and low-symmetry binary liquid alloys reveals a strong correlation between the RSM and symmetry.

Table 3. Parameters of four models of the asymmetric method.

System	MIVM		RSM	Wilson		NRTL
	${{\mathit{B}}_{\mathit{i}\mathit{j}}}^{\mathbf{\prime }}$	${{\mathit{B}}_{\mathit{i}\mathit{j}}}^{\prime }$	${{\mathbf{\Omega }}_{\mathit{i}\mathit{j}}}^{\prime }={{\mathbf{\Omega }}_{\mathit{i}\mathit{j}}}^{\prime }$	${{\mathit{A}}_{\mathit{i}\mathit{j}}}^{\prime }$	${{\mathit{A}}_{\mathit{i}\mathit{j}}}^{\prime }$	${{\mathit{\tau }}_{\mathit{i}\mathit{j}}}^{\prime }$	${{\mathit{\tau }}_{\mathit{i}\mathit{j}}}^{\prime }$
Co-Ni	0.964	0.977	0.343	0.964	0.977	−0.036	−0.023
Al-Zn	1.246	0.743	0.423	1.166	0.795	0.220	−0.297
Cu-Ni	1.086	0.863	0.374	0.999	0.938	0.082	−0.147
Al-Ni	1.493	1.649	−5.203	0.991	2.485	0.401	0.500
Cu-Fe	0.943	0.805	1.512	0.943	0.805	−0.059	−0.217
Ge-Sn	0.740	1.273	0.266	1.214	0.776	−0.302	0.241
Ag-Cu	0.675	1.194	1.225	0.471	1.709	−0.394	0.177
Pb-Sb	1.558	0.517	1.046	1.457	0.553	0.443	−0.660
Al-Si	1.405	1.077	−1.856	1.226	1.235	0.340	0.074
Al-Co	1.112	1.884	−4.213	0.802	2.612	0.106	0.633
Li-Mg	1.357	0.914	−1.095	1.469	0.844	0.305	−0.090
Sb-Sn	0.769	1.545	−0.847	0.735	1.618	−0.262	0.435
Cu-Zr	0.908	1.669	−2.277	1.782	0.851	−0.096	0.512
K-Na	0.700	1.176	1.018	0.366	2.252	−0.357	0.162
Pb-Sn	1.044	0.912	0.267	0.932	1.021	0.043	−0.092
Al-Mg	0.820	1.123	0.459	1.162	0.793	−0.198	0.116
Cs-K	1.204	0.635	1.310	0.801	0.955	0.185	−0.454
Li-Na	0.764	0.999	1.347	1.416	0.539	−0.270	−0.001
Au-Cu	1.163	1.444	−2.877	0.812	2.068	0.151	0.368
Al-Li	1.136	1.379	−2.356	1.485	1.055	0.128	0.321
Nb-Zr	0.814	1.255	−0.110	1.048	0.975	−0.206	0.227
Ni-Pd	1.168	1.087	−1.307	1.590	0.798	0.153	0.080
Cu-Mg	1.312	1.258	−2.781	2.575	0.641	0.272	0.230
Al-Ca	2.202	1.259	−5.761	5.826	0.476	0.789	0.230
Ni-Zr	1.522	1.730	−5.373	3.247	0.811	0.420	0.548
Al-Sn	0.624	1.440	0.598	1.024	0.877	−0.471	0.365
Al-Cu	1.568	1.530	−5.209	2.192	1.152	0.476	0.450
Nb-Ni	1.772	1.071	−3.558	1.070	1.774	0.572	0.069
Cu-Sn	1.987	0.849	−2.904	0.874	1.932	0.687	−0.163
Au-Si	1.224	1.684	−3.129	1.034	1.993	0.202	0.521
Li-Sn	3.396	2.186	−10.225	4.263	1.742	1.223	0.782
Fe-Si	5.678	1.776	−9.822	4.695	2.148	1.737	0.574
Ag-In	1.595	0.925	−2.227	2.476	0.596	0.467	−0.078
Ge-Te	0.409	1.891	1.285	0.912	0.848	−0.894	0.637
Al-Au	1.680	1.660	−5.740	2.287	1.219	0.519	0.507
Cu-Sb	1.804	0.881	−2.315	0.757	2.098	0.590	−0.127

Table 4. Deviations and relative errors of five models in the asymmetric method.

System	Miedema		MIVM		RSM		Wilson		NRTL
	SD	ARD/%	SD	ARD/%	SD	ARD/%	SD	ARD/%	SD	ARD/%
Co-Ni	0.013	3.852	0.036	10.8	0.029	8.8	0.003	0.8	0.015	4.4
Al-Zn	0.075	15.536	0.108	22	0.058	12.1	0.100	20.5	0.116	23.7
Cu-Ni	0.247	49.697	0.085	16.5	0.085	16.5	0.120	23.1	0.134	25.7
Al-Ni	0.210	1467.533	0.044	18.3	0.037	125	0.208	1472	0.398	4520
Cu-Fe	0.889	93.800	0.106	12.5	0.128	15.1	0.321	37.5	0.375	44
Ge-Sn	0.019	5.384	0.071	19.4	0.011	3.1	0.016	4.1	0.027	7.5
Ag-Cu	0.108	19.924	0.062	10.8	0.025	4.6	0.143	26.1	0.167	30.5
Pb-Sb	0.438	79.951	0.044	9.9	0.047	10.5	0.070	15.9	0.080	18.1
Al-Si	0.023	10.017	0.110	40.5	0.058	23.2	0.041	18.5	0.125	63.3
Al-Co	0.034	10.163	0.080	39.2	0.027	10.5	0.142	341	0.300	1085
Li-Mg	0.043	17.882	0.056	20.7	0.032	12.3	0.035	14.2	0.082	34.9
Sb-Sn	0.025	6.367	0.078	29.3	0.012	3.6	0.054	24.5	0.093	44.2
Cu-Zr	0.124	53.790	0.064	31.6	0.015	8.4	0.098	74.4	0.185	162
K-Na	0.019	3.619	0.081	15.3	0.012	2	0.144	27.8	0.145	28
Pb-Sn	0.015	1.847	0.016	3.2	0.009	1.7	0.063	14.3	0.087	19.6
Al-Mg	0.042	14.076	0.087	32.9	0.092	34.9	0.048	17	0.033	11.2
Cs-K	0.020	4.475	0.129	36	0.148	41.7	0.016	3.3	0.066	16.9
Li-Na	0.255	24.451	0.210	22.9	0.192	21.3	0.364	40.5	0.405	45.2
Au-Cu	0.028	7.697	0.076	34.5	0.038	12.1	0.105	101	0.219	255
Al-Li	0.034	17.993	0.048	17	0.037	12.2	0.118	115	0.209	236
Nb-Zr	0.036	8.347	0.089	21.1	0.066	15.1	0.058	13	0.056	12.4
Ni-Pd	0.115	72.225	0.044	15	0.039	14.1	0.090	53.7	0.140	91
Cu-Mg	0.022	8.778	0.054	30.2	0.038	12.4	0.104	103	0.226	286
Al-Ca	0.100	56.098	0.129	64.3	0.087	37.4	0.094	144	0.338	1081
Ni-Zr	0.080	20.755	0.076	80.2	0.085	266	0.225	1661	0.433	4415
Al-Sn	0.122	18.737	0.226	34.4	0.142	21.3	0.199	30.4	0.226	34.5
Al-Cu	0.099	228.621	0.076	43	0.075	30.1	0.158	516	0.343	2031
Nb-Ni	0.159	517.211	0.128	47.2	0.089	64	0.161	540	0.288	1478
Cu-Sn	0.073	38.233	0.131	51.2	0.103	33.3	0.135	48	0.524	544
Au-Si	0.106	43.309	0.116	44	0.090	45.6	0.147	279	0.290	847
Li-Sn	0.081	159.055	0.098	59.1	0.117	130	0.216	1347	0.596	7906
Fe-Si	0.095	67.563	0.259	82.9	0.137	58.3	0.088	186	0.516	5003
Ag-In	0.114	97.817	0.077	32.8	0.095	30.3	0.095	76	0.185	191
Ge-Te	0.240	319.161	0.099	27.3	0.333	496	0.195	239	0.150	148
Al-Au	0.159	53.722	0.149	55.9	0.139	42.7	0.168	405	0.333	1822
Cu-Sb	0.095	80.468	0.072	28	0.096	30.4	0.122	104	0.196	229
Ave	0.121	102.7	0.095	32.2	0.078	47.4	0.124	226	0.225	911

$\mathrm{SD}=\sqrt{\frac{{\displaystyle \sum {(a_{pre} - a_{\exp })}^2}}{N}}$; $\mathrm{ARD}=\frac{1}{N}{\displaystyle \sum \left|\frac{a_{pre}-a_{\exp }}{a_{\exp }}\right|}\times 100\%$. $a_{pre}$-calculated value of the activity; $a_{\exp }$ [55,56,57,58,59]-experimental activity value.

Figure 5. (a) is the SD of 36 binary liquid alloys, (b) is the ARD of MIVM and RSM models of 36 binary liquid alloys, (c) is the ARD of the Wilson equation, NRTL equation, and Miedema of 36 binary liquid alloys.

4.2. Symmetric Method for Calculating the RDF

The methodology based on symmetry involves deriving parameters for each model using the region from $r_0$ to $r_1$ (Figure 4) and Equation (28), see Table 5. Analysis of data presented in Table 6 reveals that among the 12 systems characterized by high symmetry, the MIVM outperforms the other three models with an average SD of 0.127 and an average ARD of 30.5%. The RSM exhibits an average SD of 0.111 and an average ARD of 44.6%. In the 12 systems with medium symmetry, the Miedema outperforms the other three models, yielding a average SD of 0.067 and an ARD of 22.8%. By contrast, the MIVM exhibits a average SD of 0.118 and an ARD of 36.7%, and the RSM exhibits a average SD of 0.088 and an ARD of 32.8%. For the 12 systems with low symmetry, the Miedema surpasses the other three models. The RSM exhibits an average SD of 0.116 and an average ARD of 70.2%. Each model exhibits distinct performance characteristics depending on the system representation. As shown in Figure 6, the data comparison indicates an increasing trend in ARD values with decreasing symmetry.

Table 5. Parameters of four models of the symmetric method.

System	MIVM		RSM	Wilson		NRTL
	${{\mathit{B}}_{\mathit{i}\mathit{j}}}^{\prime }$	${{\mathit{B}}_{\mathit{j}\mathit{i}}}^{\prime }$	${{\mathbf{\Omega }}_{\mathit{i}\mathit{j}}}^{\prime }={{\mathbf{\Omega }}_{\mathit{j}\mathit{i}}}^{\prime }$	${{\mathit{A}}_{\mathit{i}\mathit{j}}}^{\prime }$	${{\mathit{A}}_{\mathit{j}\mathit{i}}}^{\prime }$	${{\mathit{\tau }}_{\mathit{i}\mathit{j}}}^{\prime }$	${{\mathit{\tau }}_{\mathit{j}\mathit{i}}}^{\prime }$
Co-Ni	1.089	0.900	0.118	1.089	0.900	0.085	−0.106
Al-Zn	1.317	0.938	−1.162	1.232	1.003	0.275	−0.064
Cu-Ni	1.391	0.877	−1.138	1.280	0.953	0.330	−0.132
Al-Ni	1.155	2.099	−5.114	0.766	3.163	0.144	0.742
Cu-Fe	0.890	1.145	−0.103	0.890	1.145	−0.117	0.135
Ge-Sn	0.562	1.573	0.545	0.923	0.958	−0.576	0.453
Ag-Cu	1.309	0.582	1.538	0.914	0.833	0.269	−0.542
Pb-Sb	1.231	0.810	0.017	1.151	0.866	0.207	−0.211
Al-Si	1.534	1.140	−2.499	1.338	1.307	0.428	0.131
Al-Co	0.830	1.774	−2.218	0.551	2.673	−0.186	0.573
Li-Mg	1.087	0.988	−0.363	1.177	0.912	0.083	−0.012
Sb-Sn	0.825	1.576	−1.288	0.788	1.650	−0.192	0.455
Cu-Zr	1.206	1.206	1.206	1.206	1.206	1.206	1.206
K-Na	0.562	1.364	1.390	0.293	2.613	−0.577	0.311
Pb-Sn	1.601	0.830	−1.549	1.430	0.929	0.471	−0.186
Al-Mg	0.808	1.126	0.528	1.144	0.796	−0.213	0.119
Cs-K	1.642	0.660	−0.391	1.092	0.992	0.496	−0.416
Li-Na	0.728	1.069	1.245	1.350	0.577	−0.317	0.067
Au-Cu	2.423	0.515	−1.229	1.692	0.737	0.885	−0.664
Al-Li	0.813	1.374	−0.580	1.062	1.051	−0.208	0.318
Nb-Zr	0.671	1.414	0.279	0.864	1.098	−0.399	0.346
Ni-Pd	1.174	1.105	11.250	−1.445	1.599	0.811	0.159
Cu-Mg	0.882	1.538	−1.689	1.731	0.783	−0.126	0.430
Al-Ca	2.423	0.515	−1.229	1.692	0.737	0.885	−0.664
Ni-Zr	2.150	4.152	−12.151	4.587	1.947	0.766	1.424
Al-Sn	0.649	1.393	0.566	1.065	0.849	−0.433	0.332
Al-Cu	1.963	1.534	−6.647	1.472	2.180	0.714	0.453
Nb-Ni	3.602	1.064	−7.456	2.174	1.763	1.281	0.062
Cu-Sn	4.758	0.617	−5.972	2.091	1.403	1.560	−0.484
Au-Si	1.438	0.875	−0.992	1.214	1.036	0.363	−0.134
Li-Sn	2.077	2.672	−8.741	2.608	2.129	0.731	0.983
Fe-Si	1.534	1.843	−4.414	1.268	2.228	0.428	0.611
Ag-In	1.066	1.467	−2.560	1.654	0.946	0.064	0.384
Ge-Te	0.636	3.025	−3.272	1.419	1.356	−0.452	1.107
Al-Au	1.903	0.844	−2.370	0.799	2.011	0.643	−0.169
Cu-Sb	1.743	1.017	−3.207	1.743	1.017	0.556	0.017

Table 6. Deviations and relative errors of five models in the symmetric method.

System	Miedema		MIVM		RSM		Wilson		NRTL
	SD	ARD/%	SD	ARD/%	SD	ARD/%	SD	ARD/%	SD	ARD/%
Co-Ni	0.013	3.852	0.002	0.4	0.004	1.1	0.007	2.1	0.011	3.3
Al-Zn	0.075	15.536	0.233	45.8	0.201	40	0.128	26	0.085	17.5
Cu-Ni	0.247	49.697	0.252	46.9	0.220	41.3	0.147	28.3	0.107	20.8
Al-Ni	0.210	1467.533	0.081	31.7	0.039	133	0.257	2130	0.335	3370
Cu-Fe	0.889	93.800	0.365	42.8	0.359	42	0.351	41.2	0.347	40.7
Ge-Sn	0.019	5.384	0.135	35.4	0.045	13.2	0.007	1.6	0.039	10.8
Ag-Cu	0.108	19.924	0.063	10.2	0.087	16.3	0.115	21.2	0.175	31.8
Pb-Sb	0.438	79.951	0.054	17.9	0.164	62.6	0.038	13.3	0.004	0.9
Al-Si	0.023	10.017	0.147	50.8	0.089	34.2	0.027	12.1	0.142	72.6
Al-Co	0.034	10.163	0.074	39.6	0.070	130	0.161	411	0.258	852
Li-Mg	0.043	17.882	0.020	7.9	0.026	10.2	0.052	21.7	0.067	28.5
Sb-Sn	0.025	6.367	0.100	36.5	0.029	11.3	0.046	20.7	0.103	49.6
Cu-Zr	0.124	53.790	0.065	33	0.026	14.4	0.084	62.1	0.193	171
K-Na	0.019	3.619	0.130	24.3	0.076	15.2	0.154	29.6	0.156	29.9
Pb-Sn	0.015	1.847	0.244	51.4	0.194	42	0.105	23.7	0.049	11
Al-Mg	0.042	14.076	0.095	36.2	0.101	38.4	0.049	17.5	0.031	10.7
Cs-K	0.020	4.475	0.132	32.3	0.074	18.8	0.046	11.5	0.036	8.9
Li-Na	0.255	24.451	0.244	26.7	0.211	23.4	0.363	40.4	0.404	45.1
Au-Cu	0.028	7.697	0.117	40.7	0.067	58.8	0.135	137	0.168	181
Al-Li	0.034	17.993	0.093	86.2	0.110	106	0.151	156	0.171	182
Nb-Zr	0.036	8.347	0.092	21.6	0.032	6.3	0.052	11.5	0.065	14.7
Ni-Pd	0.115	72.225	0.046	15.2	0.036	11.3	0.088	52.4	0.143	93.2
Cu-Mg	0.022	8.778	0.065	22.3	0.051	40	0.136	144	0.202	245
Al-Ca	0.100	56.098	0.089	50.3	0.075	20	0.107	175	0.301	901
Ni-Zr	0.080	20.755	0.248	75.6	0.098	35.8	0.140	86.2	0.573	6633.0
Al-Sn	0.122	18.737	0.216	32.8	0.147	21.9	0.200	30.6	0.224	34.3
Al-Cu	0.099	228.621	0.113	57.6	0.078	39.9	0.114	289	0.417	2930
Nb-Ni	0.159	517.211	0.216	67.7	0.104	49.9	0.121	302	0.363	2270
Cu-Sn	0.073	38.233	0.255	75.4	0.163	56.7	0.076	28	0.200	167
Au-Si	0.106	43.309	0.070	79	0.136	240	0.192	425	0.234	591
Li-Sn	0.081	159.055	0.150	86.9	0.116	187	0.244	1730	0.564	7420
Fe-Si	0.095	67.563	0.138	45.8	0.105	79.9	0.165	693	0.358	2900
Ag-In	0.114	97.817	0.122	37.9	0.097	30.1	0.109	90.8	0.193	204
Ge-Te	0.240	319.161	0.245	73.9	0.114	29.1	0.135	118	0.211	268
Al-Au	0.159	53.722	0.149	55.9	0.139	42.7	0.168	405	0.333	1820
Cu-Sb	0.095	80.468	0.072	29.9	0.097	30	0.122	105	0.196	228
Ave	0.121	102.7	0.137	42.35	0.105	49.24	0.128	219.2	0.207	884.9

Figure 6. (a) is the SD of 36 binary liquid alloys, (b) is the ARD of MIVM and RSM models of 36 binary liquid alloys, (c) is the ARD of the Wilson equation, NRTL equation, and Miedema of 36 binary liquid alloys.

5. Conclusions

This study uses polynomials to describe the partial RDF in pure gases and extends this approach to binary liquid systems. The primary aim of this study is to characterize the atomic distribution and unravel interatomic interactions, which are essential for accurate thermodynamic calculations. The RDF exhibits irregularities when the symmetric method is used instead of the asymmetric one for RDF calculation. Notably, the estimation of binary liquid alloy activity favors using the asymmetric method, especially when considering the average results obtained from both methods for the 36 binary liquid alloy systems investigated in this study. We selected and compared five thermodynamic models based on their symmetry degree. Data analysis reveals that the RSM exhibits the highest dependency on the symmetry degree. Conversely, the MIVM demonstrates superior adaptability to symmetric and asymmetric systems.