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

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 intermolecular interactions and enable simulation, prediction, and explanation of the behavior and properties of molecular systems. Various computational simulation studies, such as for protein folding, drug–receptor interactions, and material property prediction, are conducted using molecular pair potentials to ascertain the underlying mechanisms of molecular interactions, explore novel material properties, and optimize drug design. Materials science and surface science researchers have investigated molecular potentials to thoroughly understand phenomena such as material structure, stability, crystal defects, and surface adsorption. Understanding intermolecular interactions is essential in material design, catalyst development, and comprehending material interfaces and surface phenomena. Molecular 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 molecular 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 molecular 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 molecular pair potential. Herein, we establish a unified approach by exploiting the analogy between the computational procedure for the RDF in a pure gas and its application to the molecular pair potential in a binary liquid system. To substantiate our framework, we rigorously analyzed the RDF using 36 binary liquid alloys.

2. Thermodynamic Model

2.1. Regular Solution Model (RSM)

Hildebrand proposed the RSM in 1929 [8]. 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 [9]. The expression for the molar excess Gibbs free energy for a binary system is as follows; the activity coefficient of the component $i$ is $\ln {\gamma }_i$.

$$\mathsf{\Delta }{G}_m^E=\mathsf{\Delta }{H}_m={\mathsf{\Omega }}_{ij}{x}_i{x}_j$$

(1)

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

(2)

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.

2.2. Wilson model

In 1964, Wilson [10] introduced 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.

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

(3)

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

(4)

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 [11].

2.3. Nonrandom two-liquid (NRTL) model

The NRTL model, introduced by Renon and Prausnitz in 1968 [12], 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 $\ln {\gamma }_i$.

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

(5)

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

(6)

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

(7)

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

(8)

Here, $A_{ij}$ and $A_{ji}$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.

2.4. Molecular Interaction Volume Model (MIVM)

In 2000, Tao introduced the volume model of molecular interaction [13], which uses statistical thermodynamics and fluid phase equilibrium theory to describe the motion of liquid molecules. This model yields the following expression:

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

(9)

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

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

(10)

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.

3. Pair potential energy polynomials of the binary liquid

In an extremely diluted pure gas, the correlation between the intermolecular potential energy and RDF can be expressed as follows [14]:

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

(11)

However, in practical scenarios, the RDF and intermolecular 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 ${\rho }_0$:

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

(12)

In this case, we consider a pure gas system comprising three molecules labeled 1, 2, and 3 (Figure 1). These molecules afford six molecular distributions: 1–1, 1–2, 1–3, 2–2, 2–3, and 3–3. Among these distributions, the RDF of 1–2 represents the spatial distribution of molecule 1 around molecule 2. However, $g_{12}(r,{\rho }_0,T)$ is influenced by molecule 3 and is centered around this molecule, thereby affecting molecules 1 and 2 in $g_{12}(r,{\rho }_0,T)$. Consequently, based these observations, a connection can be established between the RDF and pair potential energy.

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

(13)

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 (13) is compared with Equation (12).

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

(14)

This knowledge is applied to binary liquid alloys based on the understanding of molecular interactions in pure gases. The molecular distribution in the binary liquid system is characterized by three scenarios: $i-i$, $i-j$, and$j-j$. The RDF $i-j$ describes the distribution of molecule $i$ around molecule $j$ in a binary system (Figure 1). However, the remaining molecule $i$ influences the $i$ molecule in the RDF $i-j$, while the remaining molecule $j$ affects the $j$ molecule in the RDF $i-j$. Unlike pure gas centered on the remaining molecules 3 affecting the RDF 1–2. In a binary system, the spatial positioning of the remaining molecules $i$ and $j$ influences the RDF $i-j$ for molecules $i$ and $j$. That is, the $i$ molecule in the RDF affecting $i-j$ is centered on the other $i$ molecules, while the $j$ molecule in the RDF affecting $i-j$ is centered on the other $j$ molecules.

If the RDF in a binary liquid system can be expanded as a polynomial based on number density, then Equation (14) takes the following form:

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

(15)

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:

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

(16)

Substituting Equation (16) into Equation (15), we obtain:

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

(17)

Then substituting Equation (17) into Equation (13), we obtain:

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

(18)

The relation between the RDF and potential energy can be obtained using Equation (18):

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

(19)

Pair potential energy between molecules can be accurately calculated using the RDF. This function represents the ratio of the probability of finding another molecule at a distance $r$ to the random distribution [15]. In the double distribution function, for a system with $N$ molecules and volume $V$, the probability of a molecule appearing in the element $d{r}_i$ is $(N/V)d{r}_i$, the probability of a molecule appearing at the distance $d{r}_j$ is $(N/V)d{r}_j$, and the probability of molecular pairs appearing at a distance $r$ is ${(N/V)}^{2}d{r}_{i}d{r}_j$. The double distribution function is given as follows:

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

(20)

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

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

(21)

However, in the RDF of binary systems, the probability of having molecule $i$ in $d{r}_i$ at $r_i$ and molecule $j$ 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:

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

(22)

Substituting Equation (19) into Equation (22) yields the potential energy between molecules:

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

(23)

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 using Equation (23) and the area of the first peak of the skewed RDF. The biased 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 the area between $r_0$ and $r_m$ (Figure 2). The trapezoidal method [16] is used to compute these areas (Equation (24)). Table 1 lists the references for the partial RDFs of 36 binary liquid alloys.

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

(24)

The RSM has a tunable parameter ${\mathsf{\Omega }}_{ij}$. The average coordination number $\overline{Z}$ [52] is obtained using the pure coordination number of the two components. Additionally, the temperature $T$ documented in literature and the calculated parameter ${\mathsf{\Omega }}_{ij}$ can be referred to calculate the parameter ${\mathsf{\Omega }}_{ij}\text{'}$ at the desired temperature $T\text{'}$.

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

(25)

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

(26)

$$T\ln {\mathsf{\Omega }}_{ij}=T\text{'}\ln {\mathsf{\Omega }}_{ij}\text{'}$$

(27)

The parameters $A_{ij}$ and$A_{ji}$ of Wilson equation are given. Additionally, the values of parameters $A_{ij}\text{'}$ and $A_{ji}\text{'}$ at the desired temperature $T\text{'}$ can be obtained using the temperature $T$ from the literature that was employed to calculate the corresponding parameters $A_{ij}$ and $A_{ji}$.

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

(28)

$$T\ln A_{ij}=T\text{'}\ln A_{ij}\text{'} \hspace{1em}T\ln A_{ji}=T\text{'}\ln A_{ji}\text{'}$$

(29)

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 ${\tau }_{ij}\text{'}$ and ${\tau }_{ji}\text{'}$, respectively, at the required temperature $T\text{'}$.

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

(30)

$$T\ln {\tau }_{ij}=T\text{'}\ln {\tau }_{ij}\text{'} \hspace{1em}T\ln {\tau }_{ji}=T\text{'}\ln {\tau }_{ji}\text{'}$$

(31)

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 $B_{ij}\text{'}$ and $B_{ji}\text{'}$ can be calculated at the desired temperature $T\text{'}$.

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

(32)

$$T\ln B_{ij}=T\text{'}\ln B_{ij}\text{'} \hspace{1em}T\ln B_{ji}=T\text{'}\ln B_{ji}\text{'}$$

(33)

4. Result analysis

4.1. Symmetry

Figure 3 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.

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

(34)

The symmetry increases as the S value decreases. At $S=0$, the system 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. Table 3 presents the symmetry degrees of the activity curves for the 36 systems based on the aforementioned definitions.

4.2. Asymmetric method for calculating the RDF

The asymmetric method uses the area between $r_0$ and $r_1$ presented in Figure 2 and Equation (23) to determine the parameters for each model. Table 4 and Table 5 and Figure 5 demonstrate that among the first 12 systems, the RSM performs better than other models for 7 systems. The average standard deviation (SD) is 0.033, and the average absolute relative deviation (ARD) is 8.9%. In case of the 12 systems with moderate symmetry, the RSM outperforms the other three models in 8 systems, resulting in a mean SD of 0.054 and a mean ARD of 13.7%. Notably, the MIVM also performs well, with a mean SD of 0.084 and an average ARD of 25.5%. For the 12 systems with low symmetry, the RSM outperforms the other three models in 6 systems, yielding an average SD of 0.115 and an average ARD of 36%. Additionally, the RSM outperforms the other three models in 5 systems when the systems have even lower symmetry, with an average SD of 0.103 and an average ARD of 41.5%. Considering the average performance across all 36 binary liquid alloy systems, 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 systems reveals a strong correlation between the RSM and symmetry.

4.3. Symmetric method for calculating the RDF

The methodology based on symmetry involves deriving parameters for each model using the region from $r_0$ to $r_m$ (Figure 6) and Equation (23). Analysis of data presented in Table 6 and Table 7 and Figure 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%. Conversely, the RSM exhibits an average SD of 0.111 and an average ARD of 44.6%. In the 12 systems with medium symmetry, the RSM outperforms the other three models, yielding a mean SD of 0.088 and an ARD of 32.9%. By contrast, the MIVM exhibits a mean SD of 0.118 and an ARD of 36.7%. For the 12 systems with low symmetry, the RSM and MIVM surpass the other three models. The RSM exhibits an average SD of 0.118 and an average ARD of 73.4%, and the MIVM shows a mean SD of 0.159 and a mean ARD of 58.4%. Each model exhibits distinct performance characteristics depending on the system representation. The data comparison indicates an increasing trend in ARD values with decreasing symmetry.

5. Conclusion

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 molecular distribution and unravel intermolecular 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 four 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.