Theoretical Predictions of Thermodynamic Behavior for Cd-Tl Melts at Different Temperatures

Nabin Kumar Roy; Anadi Prasad Singh; Rajendra Prasad Chaudhary; Jagdhar Mandal; Indu Shekhar Jha

doi:10.4236/wjcmp.2026.163004

World Journal of Condensed Matter Physics > Vol.16 No.3, August 2026

Theoretical Predictions of Thermodynamic Behavior for Cd-Tl Melts at Different Temperatures

Nabin Kumar Roy¹, Anadi Prasad Singh², Rajendra Prasad Chaudhary³, Jagdhar Mandal^4*, Indu Shekhar Jha³
¹University Department of Chemistry, T.M. Bhagalpur University, Bhagalpur, India.
²Marwari College, T.M. Bhagalpur University, Bhagalpur, India.
³Department of Physics, M.M.A.M. Campus, Biratnagar, Nepal.
⁴University Department of Physics, T.M. Bhagalpur University, Bhagalpur, India.
DOI: 10.4236/wjcmp.2026.163004 PDF HTML XML 6 Downloads 50 Views

Abstract

The molecular interaction volume model is utilized to analyze the composition dependence of thermodynamic properties like Gibbs excess free energy of mixing, free energy of mixing, activity coefficients, activity, microscopic properties such as concentration fluctuations, short-range order parameter and diffusivity ratio of Cd-Tl melts at 750 K. The required interaction parameters are evaluated by Newton-Raphson method using the observed data of activity coefficients of constituent atoms in the infinite dilute solution region. The computed data for excess free energy of mixing, Gibbs free energy of mixing, activity, activity coefficients and concentration fluctuations exhibit excellent harmony with the analogous experimental data for Cd-Tl melts at 750 K. The observed asymmetries in excess free energy of mixing, free energy of mixing and concentration fluctuations for Cd-Tl melts at 750 K are successfully explained. The short-range order parameter and diffusivity relative to the composition for Cd-Tl melts at 750 K are theoretically predicted. The results illustrate the segregating character of Cd-Tl melts at 750 K. Further, the thermodynamic, microscopic and transport properties of Cd-Tl melts at 650 K, 750 K, 850 K, 950 K and 1050 K are predicted theoretically. The segregating behavior of Cd-Tl melts decreases with the elevation in temperature from 650 - 1050 K.

Keywords

Excess Free Energy of Mixing, Activity Coefficients, Concentration Fluctuations, Short-Range Order Parameter, MIVM Model

Share and Cite:

Roy, N.K., Singh, A.P., Chaudhary, R.P., Mandal, J. and Jha, I.S. (2026) Theoretical Predictions of Thermodynamic Behavior for Cd-Tl Melts at Different Temperatures. World Journal of Condensed Matter Physics, 16, 55-72. doi: 10.4236/wjcmp.2026.163004.

1. Introduction

The Cd-Tl and Cd-Tl-based multicomponent alloys like Sn-Cd-Tl, Bi-Cd-Tl, Pb-Cd-Bi-Tl etc., have potential applications in industries, as semiconductors, as ceramic compounds, as microsensor in technological process, etc. [1] [2]. The phase diagram [3] shows that the composition dependence of thermodynamic functions like excess free energy of mixing, $G_{M}^{E}$ , free energy of mixing, $G_{M}$ and enthalpy of mixing, $H_{M}$ are asymmetric while the entropy of mixing, $S_{M}$ is symmetric for Cd-Tl melts at 750 K at x = 0.5. In the entire composition region x = 0.1 to 0.9, $G_{M}^{E}$ and $H_{M}$ have positive values at each composition. The activity of both the components i.e. $a_{C d}$ and $a_{T l}$ illustrates the positive departures from the ideal mixing behaviour. Therefore, Cd-Tl system at 750 K in molten state has a segregating character [4].

The thermodynamic properties and wide range of applications indicate the importance of the Cd-Tl melts for the researchers. Nevertheless, there is scarcity of data related to the study of thermophysical properties of Cd-Tl in the literature. Earlier, some attempts have been made to explore the thermodynamic properties like $H_{M}$ and activity of the components Cd and Tl, and phase diagram by the optimization process on using CALPHAD approach [5] and $G_{M}$ , concentration fluctuations, $S_{c c} (0)$ , diffusivity, $D$ and viscosity of Cd-Tl melts at 750 K by quasi-lattice theory [6]. Despite these, Cd-Tl melts at 750 K still requires further investigation.

In present work, MIVM model is utilized for the analysis of $G_{M}^{E}$ , $G_{M}$ , activity coefficients, $γ_{υ}$ , activity, $S_{c c} (0)$ , short-range order parameter, $α_{1}$ (SRO), excess stability function, $E^{X S}$ and transport property like diffusivity ratio, $\frac{D_{M}}{D_{i d}}$ of Cd-Tl melts at 650 K, 750 K, 850 K, 950 K and 1050 K. The MIVM model [7] is a fluid-based model based on the statistical thermodynamics. It is necessary to point out that MIVM model is successfully employed for the analysis of the thermodynamic properties of several binary alloys (Zn-Bi, Al-Ga, Ni-Pd, Al-In, Zn-Cd, etc.) and multi-component alloys (Au-Sn-Bi, Au-Sn-Zn, Zn-In-Sn, Al-Sn-Zn, Zn-Cu-Sn-In, etc.) [8]-[15].

2. Formalism

2.1. Thermodynamic Properties Like Excess Free Energy of Mixing, $G_{M}^{E}$ , Free Energy of Mixing, $G_{M}$ , Activity Coefficient, $γ_{υ}$ and Activity, $a_{υ} (υ = i, j)$ of Binary Molten Alloys

MIVM model is established by Tao [7] which is a fluid-based model and deduced from statistical thermodynamics, the fundamental notion of non-random interchange of liquid molecules and fluid-phase equilibria. For a binary melt $i - j$ , the molecular excess free energy of mixing is expressed as [7]

$\begin{matrix} \frac{G_{M}^{E}}{R T} = x_{i} \ln (\frac{V_{m i}}{x_{i} V_{m i} + x_{j} V_{m j} A_{j i}}) + x_{j} \ln (\frac{V_{m j}}{x_{j} V_{m j} + x_{i} V_{m i} A_{i j}}) \\ - \frac{x_{i} x_{j}}{2} (\frac{Z_{_{i}} A_{j i} \ln A_{j i}}{x_{i} + x_{j} A_{j i}} + \frac{Z_{j} A_{i j} \ln A_{i j}}{x_{j} + x_{i} A_{i j}}) \end{matrix}$ (1)

where, the compositions of the element $i$ and $j$ represented by $x_{i}$ and $x_{j}$ , $A_{i j}$ and $A_{j i}$ refer to pair-potential energy interaction parameters, respectively and the first coordination numbers of the constituents $i$ and $j$ are respectively $Z_{i}$ and $Z_{j}$ . The pair potential energy interaction parameters may be expressed as [7]

$A_{j i} = exp [- \frac{ε_{j i} - ε_{i i}}{k T}]$ and $A_{i j} = exp [- \frac{ε_{i j} - ε_{j j}}{k T}]$ (2)

where, $ε_{i i}$ , $ε_{j j}$ and $ε_{j i}$ stand for potential energies for the pairs $i - i$ , $j - j$ and $i - j$ respectively, temperature is represented by T and k represents Boltzmann constant. It is noted that $ε_{i j} = ε_{j i}$ .

The partial molar energy and excess molar free energy are related as

$G_{M}^{\tilde{E}} = R T \ln γ_{υ} = G_{M}^{E} + δ {(\frac{\partial G_{M}^{E}}{\partial x_{i}})}_{T, P, x [i, N]} - \sum_{j = 1}^{N} {(\frac{\partial G_{M}^{E}}{\partial x_{i}})}_{T, P, x [j, N]}$ (3)

where, $γ_{υ} (υ = i, j)$ represents the activity coefficient of binary melts.

It is noted that Equation (3) follows the condition that for $i \neq N$ , $δ = 1$ ; and $i = N$ , $δ = 0$ . Again, the variables $x_{j}$ and $x_{N}$ determined from $x [i, N]$ with $x_{N} = 1 - \sum_{j = 1}^{N - 1} x_{j}$ . Hence, the activity coefficients of components $i$ and $j$ of a binary melt are respectively, represented by

$\begin{matrix} \ln γ_{i} = \ln (\frac{V_{m i}}{x_{i} V_{m i} + x_{j} V_{m j} A_{j i}}) + x_{j} (\frac{V_{m j} A_{j i}}{x_{i} V_{m i} + x_{j} V_{m j} A_{j i}} - \frac{V_{m i} A_{j i}}{x_{j} V_{m j} + x_{i} V_{m i} A_{i j}}) \\ - \frac{x_{j}^{2}}{2} (\frac{Z_{i} A_{j i}^{2} \ln A_{j i}}{{(x_{i} + x_{j} A_{j i})}^{2}} + \frac{Z_{j} A_{i j} \ln A_{j i}}{{(x_{j} + x_{i} A_{i j})}^{2}}) \end{matrix}$ (4)

and

$\begin{matrix} \ln γ_{j} = \ln (\frac{V_{m j}}{x_{j} V_{m j} + x_{i} V_{m i} A_{i j}}) - x_{i} (\frac{V_{m j} A_{j i}}{x_{i} V_{m i} + x_{j} V_{m j} A_{j i}} - \frac{V_{m i} A_{i j}}{x_{j} V_{m j} + x_{i} V_{m i} A_{i j}}) \\ - \frac{x_{i}^{2}}{2} (\frac{Z_{j} A_{i j}^{2} \ln A_{i j}}{{(x_{j} + x_{i} A_{i j})}^{2}} + \frac{Z_{i} A_{j i} \ln A_{j i}}{{(x_{i} + x_{j} A_{j i})}^{2}}) \end{matrix}$ (5)

For the component $i$ , the first co-ordination number is determined by the following relation [7]

$Z_{i} = \frac{4 \sqrt{2 π}}{3} (\frac{r_{m i}^{3} - r_{o i}^{3}}{r_{m i} - r_{o i}}) ρ_{i} r_{m i} \exp (\frac{Δ H_{m i} (T_{m i} - T)}{Z_{C} R T_{m i}})$ (6)

where, $T_{m i}$ represents the melting temperature in kelvin, $Δ H_{m i}$ , the enthalpy at melting temperature, $ρ_{i} = N_{i} V_{i} = \frac{0.6022}{V_{m i}}$ represents the molecular number density, $r_{m i}$ and $r_{o i}$ are initial value and first peak value for radial distribution function of metal $i$ in molten state near $T_{m i}$ , $R$ stands for molar gas constant and $Z_{C}$ represents the co-ordination number for closed packed structure which is usually taken to be 12. The radial distances may be represented by

$r_{o i} = 0.918 d_{cov, i}$ and $r_{m i} = σ_{i}$ (7)

where, $d_{cov}$ represents atomic covalent diameter and $σ_{i}$ the atomic diameter.

In infinite dilute solution region, such as $x_{i}$ or $x_{j} \to 0$ , the activity coefficients of the constituents $i$ and $j$ may be represented by

$\ln γ_{i}^{\infty} = 1 - \ln (\frac{V_{m j} A_{j i}}{V_{m i}}) - \frac{V_{m i} A_{j i}}{V_{m j}} - \frac{1}{2} (Z_{i} \ln A_{j i} + Z_{j} A_{i j} \ln A_{i j})$ (8)

and

$\ln γ_{j}^{\infty} = 1 - \ln (\frac{V_{m i} A_{i j}}{V_{m j}}) - \frac{V_{m j} A_{j i}}{V_{m i}} - \frac{1}{2} (Z_{j} \ln A_{i j} + Z_{i} A_{j i} \ln A_{j i})$ (9)

The activity and activity coefficients for the constituents of the binary melts may be expressed as [4]

$a_{i} = γ_{i} x_{i}$ (10a)

$a_{j} = γ_{j} x_{j}$ (10b)

2.2. Microscopic Functions Like Concentration Fluctuations, $S_{c c} (0)$ and SRO, $α_{1}$ of a Binary Molten Alloy

The $S_{c c} (0)$ and $α_{1}$ are found to be very important microscopic parameters to explore the interatomic interactions in a liquid system. If $S_{c c} (0) < S_{c c}^{i d} (0)$ then the system is ordered i.e. dissimilar ( $i - j$ ) atoms or molecules coupled together as the closest neighbors. Similarly, $S_{c c} (0) > S_{c c}^{i d} (0)$ describes the segregating character of the binary melt and like (i.e. $i - i$ or $j - j$ ) atoms or molecules link together as the closest neighbors [4].

The $G_{M}$ and $S_{c c} (0)$ are related as [4] [16]

$S_{c c} (0) = R T {(\frac{\partial^{2} G_{M}}{\partial x_{i}^{2}})}_{T, P, N}^{- 1} = R T {(\frac{\partial^{2} G_{M}}{\partial x_{j}^{2}})}_{T, P, N}^{- 1}$ (11)

where,

$G_{M} = G_{M}^{E} + G_{M}^{i d}$ (12)

$\Rightarrow G_{M} = G_{M}^{E} + R T [x_{i} \ln x_{i} + x_{j} \ln x_{j}]$ (13)

Using Equations (1) and (13) in (11), one can obtain

$S_{c c} (0) = \frac{x_{i} x_{j}}{1 + x_{i} x_{j} f (x_{i} x_{j})}$ (14)

where,

$\begin{matrix} f (x_{i}, x_{j}) = \frac{V_{m j} A_{j i} - V_{m i}}{x_{i} V_{m i} + x_{j} V_{m j} A_{j i}} + \frac{V_{m i} A_{i j} - V_{m j}}{x_{j} V_{m j} + x_{i} V_{m i} A_{i j}} + \frac{V_{m j} A_{j i} (V_{m j} A_{j i} - V_{m i})}{{(x_{i} V_{m i} + x_{j} V_{m j} A_{j i})}^{2}} \\ + \frac{V_{m i} A_{i j} (V_{m i} A_{i j} - V_{m j})}{{(x_{j} V_{m j} + x_{i} V_{m i} A_{i j})}^{2}} + [\frac{Z_{i} A_{j i}^{2} \ln A_{j i}}{{(x_{i} + x_{j} A_{j i})}^{3}} + \frac{Z_{j} A_{i j}^{2} \ln A_{i j}}{{(x_{j} + x_{i} A_{i j})}^{3}}] \end{matrix}$ (15)

For an ideal mixing, $G_{M} = G_{M}^{i d}$ hence Equations (14) and (15) yield

$S_{c c}^{i d} (0) = x_{i} x_{j}$ (16)

Again, the activity and $S_{c c} (0)$ are related as

$S_{c c} (0) = a_{i} x_{j} {(\frac{\partial a_{i}}{\partial x_{i}})}^{- 1} = a_{j} x_{i} {(\frac{\partial a_{j}}{\partial x_{j}})}^{- 1}$ (17)

The $S_{c c} (0)$ determined from Equation (17) on utilizing the experimental data of activity is considered as the experimental data [4] [16] [17].

The SRO, $α_{1}$ initiated by Warren-Cowley [18] [19] is related to $S_{c c} (0)$ as [4] [20]

$α_{1} = \frac{S - 1}{S (Z - 1) + 1}$ with $S = \frac{S_{c c} (0)}{S_{c c}^{i d} (0)}$ (18)

where, $Z$ refers to the coordination number in first neighbor shell.

Equation (18) exhibits that $α_{1}$ = positive for segregating system, $α_{1}$ = negative for ordered system and $α_{1} = 0$ for ideal mixture.

2.3. Excess Stability Function, $E^{X S}$ and Diffusivity, $D$ of a Binary Molten Alloy

The excess stability function, $E^{X S}$ initiated by Darken [21] [22] may be expressed as

$\frac{E^{X S}}{R T} = S_{c c} (0))^{- 1} - {(S_{c c}^{i d} (0))}^{- 1}$ (19)

Obviously, $E^{X S}$ = positive for ordering alloy, $E^{X S}$ = negative for segregating alloy and $E^{X S} = 0$ for ideal alloy depending upon the conditions that $S_{c c} (0) < S_{c c}^{i d} (0)$ , $S_{c c} (0) > S_{c c}^{i d} (0)$ , and $S_{c c} (0) = S_{c c}^{i d} (0)$ respectively.

The transport property, such as diffusivity is very useful to describe the mixing behavior of a binary solution at the microscopic level. For a binary liquid system, the diffusivity ratio is expressed as [4] [23]

$\frac{D_{M}}{D_{i d}} = \frac{S_{c c}^{i d} (0)}{S_{c c} (0)}$ (20)

where, $D_{M}$ and $D_{i d}$ represent the mutual diffusion coefficient of the system and intrinsic diffusion coefficient of ideal system respectively. Now, $D_{i d}$ is represented by

$D_{i d} = x_{i} D_{j} + x_{j} D_{i}$ (21)

where, $D_{i}$ and $D_{j}$ are self-diffusion coefficients of pure elements $i$ and $j$ respectively.

Equation (20) illustrates that $D_{M} / D_{i d} > 1$ for ordered system, $D_{M} / D_{i d} < 1$ for segregating system and $D_{M} / D_{i d} = 1$ for ideal system.

3. Results and Discussions

3.1. Excess Free Energy of Mixing, $G_{M}^{E}$ , Free Energy of Mixing, $G_{M}$ , Activity Coefficient, $γ_{υ}$ and Activity, $a_{υ} (υ = i, j)$ of Cd-Tl Melts at 750 K

The analytical equations for $G_{M}^{E}$ , $G_{M}$ , $γ_{μ}$ and $a_{μ} (μ = i, j)$ are employed to assess the thermodynamic properties of Cd-Tl melts at 750 K. The necessary parameters of pure Cd and Tl are presented in Table 1 [24]. The coordination numbers $Z_{i}$ and $Z_{j}$ of the constituents of the alloy are determined from Equation (6) which are presented in Table 2. The parameters $A_{j i}$ and $A_{i j}$ are evaluated on solving Equations (8) and (9) simultaneously by Newton-Rapson method using the values of infinite dilute activity coefficients i.e. $γ_{i}^{\infty}$ (1.932) and $γ_{j}^{\infty}$ (3.826) [3]. The evaluated values of $A_{j i}$ and $A_{i j}$ are 0.7107 and 0.8815 respectively. The approximated values of $A_{j i}$ and $A_{i j}$ are slightly altered in order to achieve a good harmony between the theory and experiment [3] for $G_{M}^{E}$ of Cd-Tl melts at 750 K. The reasonable values of $A_{j i}$ and $A_{i j}$ i.e. 0.7120 and 1.0990 respectively, are also incorporated in Table 2.

Table 1. Input parameters for the pure metals [24].

Metal, i	$Δ H_{m i}$ (KJ/mole)	$σ_{i}$ (×10⁻⁸ cm)	$r_{o i}$ (×10⁻⁸ cm)	$V_{m i}$ (cm³/mole)
Cd	6.40	3.00	2.54	14.0 [1 + 1.6 × 10⁻⁴ (T-594)]
Tl	4.31	3.22	2.74	18.0 [1 + 1.15 × 10⁻⁴ (T-575.65)]

Table 2. Values of $A_{i j}$ , $A_{j i}$ , $Z_{i}$ and $Z_{j}$ for the constituents of Cd-Tl melts at required temperature.

$i - j$	T (K)	$A_{i j}$	$A_{j i}$	$Z_{i}$	$Z_{j}$
	650	0.6757	1.1151	9.86	9.53
	750	0.7120	1.0990	9.71	9.42
Cd-Tl	850	0.7410	1.0869	9.56	9.32
	950	0.7648	1.0774	9.41	9.22
	1050	0.7846	1.0698	9.27	9.12

The values of $G_{M}^{E} / R T$ calculated from Equation (1) as a function of Cd for Cd-Tl alloys at 750 K show excellent agreement with the corresponding experimental values [3] as presented in Figure 1. In composition region, $x_{C d} = 0.1$ to 0.9, the value of $G_{M}^{E}$ is positive at each concentration. The computed and experimental [3] values of $G_{M}^{E} / R T$ are in excellent harmony with maximum values at $x_{C d} = 0.6$ i.e. 0.2478 (Theory) and 0.2504 (Experiment). The RMS error is found to be 0.00533% on employing Equation (22). Hence, the present theoretical investigation successfully reveals the symmetry in $G_{M}^{E} / R T$ and segregating nature of Cd-Tl melts at 750 K.

$RMS Error = \sqrt{\frac{\sum_{i = 1}^{n} {(x_{i} - y_{i})}^{2}}{n}}$ (22)

where,

$x_{i}$ = theoretical (calculated) values;

$y_{i}$ = experimental values;

$n$ = no. of data points.

Figure 1. $G_{M}^{E} / R T$ vs $x_{C d}$ for Cd-Tl melts at 750 K.

Equation (13) is used to compute the values of $G_{M} / R T$ for Cd-Tl melts at 750 K which are illustrated in Figure 2 along with experimental values [3]. A well agreement is seen. In the whole composition region, $x_{C d} = 0.1$ to 0.9, $G_{M} / R T$ is negative with minimum values at $x_{C d}$ = 0.4 i.e. $G_{M} / R T ({min}^{m}) = - 0.4481$ (Theory) and −0.4518 (Experiment). The RMS error computed from Equation (22) is found to be 0.00542%. Thus, the symmetry in $G_{M} / R T$ is successfully described by MIVM model.

Figure 2. $G_{M} / R T$ vs $x_{C d}$ for Cd-Tl melts at 750 K.

The activity coefficients $γ_{C d}$ and $γ_{T l}$ are evaluated from Equations (4) and (5) respectively in terms of the concentration of Cd. The theoretical and experimental [3] data of $γ_{C d}$ and $γ_{T l}$ are in excellent agreement as shown in Figure 3. The RMS errors for $γ_{C d}$ and $γ_{T l}$ computed from Equation (22) are found to be 0.029441% and 0.022264% respectively.

Figure 3. ( $γ_{C d}$ & $γ_{T l}$ ) vs $x_{C d}$ for Cd-Tl melts at 750 K.

The theoretical data of $γ_{C d}$ and $γ_{T l}$ are used to calculate the concentration dependence of $a_{C d}$ and $a_{T l}$ respectively from Equations (10a) and (10b) for Cd-Tl melts at 750 K, which are presented in Figure 4.

Figure 4. ( $a_{C d}$ & $a_{T l}$ ) vs $x_{C d}$ for Cd-Tl melts at 750 K.

A well concord is noticed between theory and experiment [3] with RMS errors for $a_{C d}$ and $a_{T l}$ computed from Equation (22) are found to be 0.008040% and 0.008330% respectively. The positive departures of $a_{C d}$ and $a_{T l}$ from ideal behavior refer to the segregating character of Cd-Tl melts at 750 K. It should be pointed out that the activity is a fortunate thermodynamic function which can be directly measured from the experiment. Therefore, the well harmony between theoretical and experimental data for $a_{C d}$ and $a_{T l}$ validates the reasonable values of $A_{j i}$ and $A_{i j}$ .

3.2. Microscopic Properties Like $S_{c c} (0)$ , $α_{1}$ , $E^{X S}$ and $D_{M} / D_{i d}$ of Cd-Tl Molten Melts at 750 K

Equations (14) and (15) are employed to compute the concentration dependence of $S_{c c} (0)$ for Cd-Tl melts at 750 K. The theoretical data of $S_{c c} (0)$ are incorporated in Figure 5 along with $S_{c c}^{i d} (0)$ and experimental values [3].

Figure 5. $S_{c c} (0)$ vs $x_{C d}$ for Cd-Tl melts at 750 K.

A well uniformity is noticed between the theory and experiment having the RMS error of 0.0118705% which is computed from Equation (22). It is observed that $S_{c c} (0) > S_{c c}^{i d} (0)$ in the composition region, $0.1 \leq x_{C d} < 0.95$ and $S_{c c} (0)$ exhibits ideal behavior of Cd-Tl melts at 750 K in the regions $0 \leq x_{C d} \leq 0.1$ and $0.95 \leq x_{C d} < 1.0$ . The maximum values of $S_{c c} (0)$ are at $x_{C d} = 0.60$ i.e. $S_{c c} (0) ({max}^{m}) = 0.520$ (Theory) and 0.518 (Experiment). Hence, the present investigation reveals that Cd-Tl melt at 750 K possesses segregating character in the region $0.1 \leq x_{C d} < 0.95$ .

Equation (18) is employed to achieve the concentration dependent values of $α_{1}$ for Cd-Tl melts at 750 K on using Z = 10. The theoretical data of $α_{1}$ in terms of the concentration of Cd are presented in Figure 6. In the full range of composition i.e. $x_{C d} = 0.1$ to 0.9, $α_{1}$ is positive at each composition and having maximum value of 0.0576 at $x_{C d} = 0.7$ , which reveals the segregating behavior of Cd-Tl melts at 750 K [25].

The concentration dependence of $E^{X S} / R T$ for Cd-Tl alloys at 750 K computed from Equation (19) is illustrated in Figure 7. Obviously, $E^{X S} / R T$ is negative at each composition of Cd in the concentration region $x_{C d} = 0.10$ to 0.90 for Cd-Tl melts at 750 K. This exhibits the segregating nature of Cd-Tl melts at 750 K [22] [26].

Figure 6. $α_{1}$ vs $x_{C d}$ for Cd-Tl melts at 750 K.

Figure 7. $E^{X S} / R T$ vs $x_{C d}$ for Cd-Tl melts at 750 K.

The diffusivity ratio, $D_{M} / D_{i d}$ as a function of the composition of Cd for Cd-Tl melts at 750 K is calculated from Equation (20) and the theoretical data of $D_{M} / D_{i d}$ are depicted in Figure 8. It is noticed that the diffusivity ratio, $D_{M} / D_{i d}$ has minimum value of 0.4554 at 750 K at $x_{C d} = 0.7$ . This confirms the segregation in Cd-Tl melts at 750 K [4].

Figure 8. $D_{M} / D_{i d}$ vs $x_{C d}$ for Cd-Tl melts at 750 K.

3.3. $G_{M}^{E}$ , $G_{M}$ , $a_{F e}$ , $a_{M n}$ , $S_{c c} (0)$ , $E^{X S}$ , $α_{1}$ and $D_{M} / D_{i d}$ of Cd-Tl Melts at Different Temperatures

For the theoretical analysis of the mixing behavior of Cd-Tl melts at different temperatures, the needed input parameters are presented in Table 1 and Table 2. The values of $A_{j i}$ and $A_{i j}$ at desired temperature, T K are determined via the following equations [7].

$- \frac{ε_{j i} - ε_{i i}}{k T} = T \ln A_{j i} (T) = T_{0} \ln A_{j i} (T_{0})$

$A_{j i} (T) = \exp (\frac{T_{0} \ln A_{j i} (T_{0})}{T})$ (22)

and

$- \frac{ε_{i j} - ε_{j j}}{k T} = T \ln A_{i j} (T) = T_{0} \ln A_{i j} (T_{0})$

$A_{i j} = \exp (\frac{T_{0} \ln A_{i j} (T_{0})}{T})$ (23)

where, T₀ = 750 K.

The computed data of $G_{M}^{E} / R T$ and $G_{M} / R T$ for Cd-Tl melts assessed via Equations (1) and (13) at 650 K, 750 K, 850 K, 950 K and 1050 K are presented in Figure 9 and Figure 10 respectively. Figure 9 illustrates that the segregating nature of Cd-Tl melts degrades when temperature is upgraded from 650 K to 1050 K. The maximum values of $G_{M}^{E} / R T$ are 0.2767 (at 650 K), 0.2478 (at 750 K), 0.2234 (at 850 K) all $x_{C d} = 0.6$ while $G_{M}^{E} / R T ({max}^{m}) = 0.2033$ (at 950 K) and 0.1863 (at 1050 K) both at $x_{C d} = 0.5$ . Thus, Cd-Tl melts exhibit asymmetric behavior at 650 K, 750 K and 850 K, and symmetric character at 950 K and 1050 K relative to the composition of Cd. Again, it is noticeable from Figure 10 that the negative values of $G_{M} / R T$ for Cd-Tl melts increase due to rise in temperature from 650 K to 1050 K. The values of $G_{M} / R T$ are minimum at $x_{C d} = 0.4$ for temperatures 650 K and 750 K i.e. −0.4256 and −0.4481, respectively while $G_{M} / R T$ shows minima at $x_{C d} = 0.5$ for temperatures 850 K, 950 K and 1050 K i.e. −0.4703, −0.4899 and −0.5068 respectively in terms of $x_{C d}$ . Thus, Cd-Tl melts indicate asymmetry at 650 K and 750 K and symmetry at 850 K, 950 K and 1050 K with respect to $x_{C d}$ .

Figure 9. $G_{M}^{E} / R T$ vs $x_{C d}$ for Cd-Tl melts at 650 K, 750 K, 850 K, 950 K & 1050 K.

Figure 10. $G_{M} / R T$ vs $x_{C d}$ for Cd-Tl melts at 650 K, 750 K, 850 K, 950 K & 1050 K.

The thermodynamic activities of Cd and Tl i.e. $a_{C d}$ and $a_{T l}$ are computed from Equations (10a) and (10b) respectively on applying the theoretical data of $γ_{C d}$ and $γ_{T l}$ determined from Equations (4) and (5) for Cd-Tl melts at 650 K, 750 K, 850 K, 950 K and 1050 K relative to $x_{C d}$ . The theoretical results for $a_{C d}$ and $a_{T l}$ at the temperatures ranging from 650 K to 1050 K are shown in Figure 11. The results reveal the decrease in segregation for rise in temperature from 650 K to 1050 K for Cd-Tl melts. However, the decrease in the segregation is weak in the temperature region 650 - 1050 K.

Figure 11. ( $a_{C d}$ _& $a_{T l}$ ) vs $x_{C d}$ for Cd-Tl melts at 650 K, 750 K, 850 K, 950 K & 1050 K.

The temperature dependence of $S_{c c} (0)$ for Cd-Tl melts at 650 K, 750 K, 850 K, 950 K and 1050 K relative to composition Cd is presented in Figure 12. For the calculation of $S_{c c} (0)$ , Equations (14) and (15) are applied. The $S_{c c} (0)$ - $x_{C d}$ isotherms indicate the degradation in segregating character of Cd-Tl melts for the rise in temperature from 650 K to 1050 K. The peak value of $S_{c c} (0)$ is noticed at $x_{C d} = 0.6$ i.e. 0.618, 0.520, 0.466 and 0.427 respectively at 650 K, 750 K, 850 K, 950 K while a minor shift in peak value of $S_{c c} (0)$ is observed at 1050 K i.e. $S_{c c} (0) = 0.405$ at $x_{C d} = 0.53$ .

Figure 12. $S_{c c} (0)$ vs $x_{C d}$ for Cd-Tl melts at 650 K, 750 K, 850 K, 950 K & 1050 K.

The $α_{1}$ - $x_{C d}$ isotherms for Cd-Tl melts are depicted in Figure 13. For this, the theoretical data of $α_{1}$ are computed with the help of Equation (18) at temperatures 650 - 1050 K. Obviously, the segregation in Cd-Tl melts goes on decreasing for the rise in temperature from 650 K to 1050 K. Again, the magnitude of $α_{1}$ is small which illustrates that Cd-Tl melts is a weakly segregating system [4] [26] in the temperature range 650 - 1050 K.

Figure 13. $α_{1}$ vs $x_{C d}$ for Cd-Tl melts at 650 K, 750 K, 850 K, 950 K & 1050 K.

The excess stability function of several binary alloys at a certain temperature has been discussed earlier [27] [28]. Nonetheless, the limited data is available regarding the theoretical exploration of $E^{X S}$ at different temperatures [22] [26], [29] on employing the optimization procedure. In present study, MIVM model is applied to explore the stability of Cd-Tl melts at various temperatures.

The variation of $E^{X S} / R T$ with respect to composition of Cd at 650 K, 750 K, 850 K, 950 K and 1050 K for Cd-Tl melts determined via Equation (19) is presented in Figure 14. Obviously, $E^{X S} / R T$ exhibits negative value at each composition in the region $x_{C d} = 0.10$ to 0.90 at each temperature under consideration. Again, the negative values of $E^{X S} / R T$ declines with the upgradation in temperature from 650 K to 1050 K. This is a sign of decrease in the segregation of Cd-Tl melts [22] [26] from 650 K to 1050 K.

The $D_{M} / D_{i d}$ relative to the concentration of Cd for Cd-Tl melts at 650 K, 750 K, 850 K, 950 K and 1050 K are computed using Equation (20) and the theoretical data are depicted in Figure 15. It is noticed that $D_{M} / D_{i d} < 1$ at the temperatures under consideration which confirms the segregating character of Cd-Tl melts. Again, $D_{M} / D_{i d}$ has minimum values at $x_{C d} = 0.7$ i.e. 0.3679 and 0.4554 respectively for 650 K and 750 K and again, has minimum values at $x_{C d} = 0.6$ i.e. 0.5145, 0.5621 and 0.6022 for 850 K, 950 K and 1050 K respectively. This indicates lowering of segregating character of Cd-Tl melts at the undertaken temperatures.

Figure 14. $E^{X S} / R T$ vs $x_{C d}$ for Cd-Tl melts at 650 K, 750 K, 850 K, 950 K & 1050 K.

Figure 15. $D_{M} / D_{i d}$ vs $x_{C d}$ for Cd-Tl melts at 650 K, 750 K, 850 K, 950 K & 1050 K.

4. Conclusions

In present work, the concentration dependent properties such as $G_{M}^{E}$ , $G_{M}$ activity coefficient, activity, $S_{c c} (0)$ , $α_{1}$ , $E^{X S}$ and $D_{M} / D_{i d}$ for Cd-Tl melts at 750 K have been successfully explored. The results exhibit the segregating nature of Cd-Tl melts at 750 K. The asymmetries in $G_{M}^{E}$ , $G_{M}$ and $S_{c c} (0)$ for Cd-Tl melts at 750 K are successfully explained. Again, MIVM model is employed for the theoretical predictions of $G_{M}^{E}$ , $G_{M}$ activity, $S_{c c} (0)$ , $α_{1}$ , $E^{X S}$ and $D_{M} / D_{i d}$ for Cd-Tl melts at 650 K, 750 K, 850 K, 950 K and 1050 K. The results indicate that the segregating character of Cd-Tl melts degrades when temperature rises from 650 K to 1050 K. Further, Cd-Tl melt is a feebly segregating alloy in the temperature range 650 - 1050 K.

Therefore, MIVM model is a relevant and reliable model to explore the thermodynamic behavior of segregating alloys at different temperatures.

Author Contributions

Conceptualization, I. S. Jha and J. Mandal; methodology, N. K. Roy; validation, I. S. Jha, J. Mandal and R. P. Chaudhary; formal analysis; R. P. Chaudhary; investigation, A. P. Singh; writing original draft preparation, A. P. Singh; writing-review and editing, N. K. Roy; visualization, I. S. Jha; supervision, J. Mandal. All authors have read and agreed to published version of the manuscript.

Conflicts of Interest

The authors declare no conflicts of interest regarding the publication of this paper.

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