The fluid based model MIVM has been introduced by Tao [16] and deduced from statistical thermodynamics, fluid-phase equilibria and the fundamental concept of the non-random exchange of liquid molecules. The molecular excess free energy of mixing of a binary molten alloy i-j is expressed as [16]

where, x i and x j are the compositions (or mole fractions) of i and j components respectively; Z i and Z j represent the first coordination numbers for i andj molecules respectively. A i j and A j i refer to pair-potential energy interaction parameters, expressed as [16]

where, ε i i , ε j j and ε j i = pair potential energies for i-i, j-j and i-j respectively, in which ε j i = ε i j , k = Boltzmann constant and T = temperature in K of the liquid metal.

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

It is necessary to mention that the above equation is based on the condition such that when i ≠ N , δ = 1 ; and i = N , δ = 0 . Again, x [ i , N ] defines the variables x j and x N where, x N = 1 − ∑ j = 1 N − 1 x j . Therefore, the activity coefficients of i and j elements of a binary alloy are, respectively, expressed as

and

Now, the first co-ordination number of i^th component can be achieved on using the relation [16]

with Δ H m i is the enthalpy at melting temperature, T m i and Z C is taken to be 12 for closed packed structure; ρ i = N i V i = 0.6022 V m i molecular number density; r m i and r o i = initial and first peak values of radial distribution function for the molten metal i near melting temperature, respectively, and R is the molar gas constant. The radial distances are expressed as

where, σ i and d c o v i = atomic diameter and atomic covalent diameter respectively.

The equations for activity coefficients of the components of the alloy in infinite dilute solution range i.e. x i or x j → 0 , are respectively given by

and

Again, the expressions for the activity of the constituents i and j in terms of activity coefficients are, respectively given by [25]

The concentration fluctuations, S c c ( 0 ) and SRO, α 1 play a vital role to explore the interatomic interactions in a liquid melt. If S c c ( 0 ) < S c c i d ( 0 ) , then the alloy is ordering and dissimilar atoms or molecules (i.e.i-j) associate together as the closest neighbours while S c c ( 0 ) > S c c i d ( 0 ) represents the segregating nature of the system i.e. similar atoms or molecules (i.e.i-i or j-j) associate together as the closest neighbours [25].

S c c ( 0 ) may be represented in terms of G M by [25][26]

where,

Equations (1), (11) and (13) yield

where,

For the ideal mixing condition, G M E = 0 , hence Equation (12) reduces to G M = G M i d . Therefore, Equations (14) and (15) provide

which represents the ideal S c c ( 0 ) .

Again, concentration fluctuations in terms of activity can be expressed as [25]-[27]

The value of S c c ( 0 ) computed from above equation on employing the experimental values of activity is considered as the experimental value of S c c ( 0 ) [25]-[27].

The SRO, α 1 introduced by Warren-Cowley [14][15] in terms of S c c ( 0 ) is represented as [25][28]

where, Z is the coordination number in first neighbor shell.

It is obvious from Equation (18) that α 1 = positive for segregating alloys, α 1 = negative for ordered alloys and α 1 = 0 for ideal mixture.

The excess stability function, E X S described by Darken [29][30] can be expressed as

It is necessary to mention that 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 facts 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 analyse the mixing behaviour of a binary solution at the microscopic level. For a binary liquid system, the diffusivity ratio is expressed as [25][31].

where, D M and D i d represent mutual diffusion coefficient of the system and intrinsic diffusion coefficient for ideal system respectively. Now, D i d is given by

where, D i and D j = 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 alloy.

The analytical equations for G M E , G M , γ μ and a μ ( μ = i , j ) are utilized to determine the thermodynamic properties of Fe-Mn melts at 1863 K. The necessary parameters for pure Fe and Mn are presented in Table 1[32]. The coordination number Z i and Z j of the constituents of alloy are determined from Equation (6) which are presented in Table 2. The parameter A j i and A i j are evaluated on solving Equations (8) and (9) simultaneously by Newton-Raphson method. For this, γ i ∞ and γ j ∞ i.e. infinite dilute activity coefficients at 1863 K are required which have been presented in Table 2 [3]. The evaluated values of A j i and A i j are 0.9488 and 0.9410, respectively. In order to acquire good agreement between the theory and experiment [3] of Fe-Mn melts at 1863 K, the values of A j i and A i j are slightly altered. The best fit values of A j i and A i j are 0.9616 and 0.9863 respectively (included in Table 2). Obviously, values of A j i and A i j are increased through 1.34% and 4.8% respectively.

Table 1. Input parameters for the pure metals [32].

Table 2. Estimated values of A i j , A j i , Z i and Z j for the constituents of Fe-Mn molten alloys at various temperatures.

Work flow of Newton-Raphson Method for calculation of A_ij and A_ji

START → Input experimental data: T, x_i, x_j, γ_i (exp.), γ_j (exp.), molar volumes (V_mi & V_mj), coordination number (Z_i & Z_j) → Give initial guesses for A_ij and A_ji → Calculate γ_i (calc.) and γ_j (calc.) using the MIVM → Calculate residual functions: F' = γ_i (calc.) − γ_i (exp) & F" = γ_j (calc.) − γ_j (exp) → Compute the Jacobian matrix (partial derivatives of F' and F" with respect to A_ij and A_ji) → Solve the Newton-Raphson equation: J ΔA = −F → Update parameters: A_ij (new) = A_ij (old) + ΔA_ij & A_ji (new) = A_ji (old) + ΔA_ji → Convergence test: |F'| < ε and |F"| < ε → Result is No or Yes → If No then Repeat Iteration, If Yes i.e. Output final values of A_ij and A_ji

The values of G M E / R T calculated from Equation (1) as function of Fe for Fe-Mn alloysat 1863 K exhibit excellent uniformity with the corresponding experimental values [3] as shown in Figure 1. In composition range, x F e = 0.1 to 0.9, the value of G M E is positive at each concentration. The theoretical and experimental [3] data of G M E / R T are in excellent harmony with maximum values at x F e = 0.50 i.e. 0.0710 (Theory) and 0.0711 (Experiment). The RMS error for G M E is found to be 0.0000894% from Equation (22). Hence, the present theoretical investigation successfully represents the symmetry in G M E / R T and segregating character of Fe-Mn melts at 1863 K.

where,

x i = theoretical (calculated) values;

y i = experimental values;

n = no. of data points.

Equation (13) is utilized to compute the values of G M / R T for Fe-Mn melts at 1863 K, which are illustrated in Figure 2 along with correlated experimental data [3]. A well agreement is observed. In the entire composition region, x F e = 0.1 to 0.9, G M / R T is negative with minimum values at x F e = 0.5 i.e. G M / R T ( min m ) = − 0.6221 (Theory) and −0.6221 (Experiment). The RMS error for G M / R T is found to be 0.00064238%. Therefore, the symmetry in G M / R T is successfully described by MIVM Model. It is notable that the minimum value of G M / R T for Fe-Mn melts at 1863 K computed by self-association model [13] is −0.615 at x F e = 0.5 . Thus, the present theoretical model provides more accurate result in comparison to that obtained by self-association model [13].

Figure 1. G M E / R T vs x F e for Fe-Mn melts at 1863 K.

Figure 2. G M / R T vs x F e for Fe-Mn melts at 1863 K.

Figure 3. ( γ F e & γ M n ) vs x F e for Fe-Mn melts at 1863 K.

The activity coefficients γ F e and γ M n are determined from Equations (4) and (5) respectively in terms of concentration for Fe. The theoretical and experimental [3] data of γ F e and γ M n are in excellent agreement as shown in Figure 3.

The theoretical data of γ F e and γ M n are employed to calculate the concentration dependence of a F e and a M n respectively from equations (10a) and (10b) for Fe-Mn melts at 1863 K, which are presented in Figure 4.

Figure 4. ( a F e & a M n ) vs x F e for Fe-Mn melts at 1863 K.

A well harmony is noticed between theory and experiment [3]. The RMS error for a F e and a M n are 0.000252% and 0.000238% respectively. The positive deviations of a F e and a M n from ideal behaviour indicate the segregating nature of Fe-Mn melts at 1863 K. The agreement between theory and experimental data validates the estimated values of A j i and A i j .

Expressions (14) and (15) are employed to compute the concentration dependence of S c c ( 0 ) for Fe-Mn melts at 1863 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].

A well uniformity is observed between the theory and experiment. The RMS error for S c c ( 0 ) is found to be 0.000124%. It is noticed that S c c ( 0 ) > S c c i d ( 0 ) in the composition region, x F e = 0.1 to 0.9 and S c c ( 0 ) exhibits ideal behaviour of Fe-Mn melts at 1863 K in the region 0.0 ≤ x F e ≤ 0.1 and 0.9 ≤ x F e ≤ 1.0 . The similar behaviour of Fe-Mn melts is observed in the region 0.0 ≤ x F e ≤ 0.2 and 0.9 ≤ x F e ≤ 1.0 [13]. Thus, the present theoretical investigation confirms the segregating character of Fe-Mn melt at 1863 K. The maximum values of S c c ( 0 ) are at x F e = 0.50 i.e. S c c ( 0 ) ( max m ) = 0.293 (Theory) and 0.291 (Experiment). The theoretical values of S c c ( 0 ) for Fe-Mn melts at 1863 K available in the literature [13] exhibit some discrepancies with the experimental data in the region 0.2 ≤ x F e ≤ 0.5 . This indicates that the present theoretical model provides comparatively better results in comparison to the data available in the literature [13].

Figure 5. S c c ( 0 ) vs x F e for Fe-Mn melts at 1863 K.

Equation (18) is used to achieve the concentration dependent values of α 1 for Fe-Mn melts at 1863 K on using Z = 10 . The theoretical data of α 1 in terms of the concentration of Fe are presented in Figure 6. In the whole of composition i.e. x F e = 0.1 to 0.9, α 1 is positive at each composition and having maximum value of 0.0148 at x F e = 0.5 , which confirms the segregation in Fe-Mn melts at 1863 K. It is necessary to state that the present theoretical investigation suggests the symmetric behaviour of Fe-Mn melts at 1863 K which differs with the theoretical data of α 1 suggested by Ogundenji et al. [13] on using self-association model.

Figure 6. α 1 vs x F e for Fe-Mn melts at 1863 K.

The concentration and temperature dependence of G M , S c c ( 0 ) , α 1 and D M / D i d for Fe-Mn melts have been analysed by MIVM model. The necessary input parameters determined at 1763 K, 1863 K 1963 K and 2063 K are presented in Table 1 and Table 2. The values of pair potential parameters A j i and A i j are determined at 1763 K, 1863 K, 1963 K and 2063 K on using the following equations [16]

and

where, T₀ = 1863 K and T K = required temperatures.

The values of A j i and A i j determined at different temperature, are included in Table 2.

Now, Equation (1) is utilised to determine the concentration dependence of G M E / R T at 1763 K, 1863 K, 1963 K and 2063 K for Fe-Mn melts. The theoretical data of G M E / R T as a function of composition of Fe are illustrated in Figure 7. The theoretical and experimental data of G M E / R T for Fe-Mn melts at 1863 K are also presented in Figure 7. It is found that the values of G M E / R T decline on raising the temperature from 1763 K to 2063 K. However, the position of maxima remains same i.e. at x F e = 0.50 i.e. G M E / R T ( max m ) = 0.0763 , 0.0710, 0.0657 and 0.0614 at 1763 K, 1863 K, 1963 K and 2063 K respectively. This signifies the degradation of the segregating behaviour of Fe-Mn melts when the temperature rises from 1763 K to 2063 K.

Equations (1) and (13) are utilized to compute G M / R T at 1763 K, 1963 K and 2063 K for Fe-Mn melts. The theoretical data of G M / R T for Fe-Mn melts at 1863 K are shown in Figure 8 along with the theoretical and experimental data [3]. Obviously, the negative value of G M / R T increases with the increase in temperature from 1763 K to 2063 K, Nevertheless, the change in G M / R T is small in comparison to the rise in temperature and the position of minima remains same i.e. x F e = 0.50 . The values of G M / R T are minimum i.e. −0.6169, −0.6221, −0.6274 and −0.6317 respectively at 1763 K, 1863 K 1963 K and 2063 K. This shows the declination in the segregating behaviour of Fe-Mn melts with the rise in temperature.

Figure 7. G M E / R T vs x F e for Fe-Mn melts at 1763 K, 1863 K, 1963 K and 2063 K.

Figure 8. G M / R T vs x F e for Fe-Mn melts at 1763 K, 1863 K, 1963 K and 2063 K.

The concentration and temperature dependency of Fe-Mn melts have been analysed in terms of a F e and a M n . The Equations (10a) and (10b) are employed to compute a F e and a M n as a function of the composition of Fe at 1763 K, 1963 K and 2063 K. The theoretical data of a F e and a M n at 1763 K, 1863 K, 1963 K and 2063 K are depicted in Figure 9. Obviously, a F e and a M n exhibit positive deviations from linear law at each temperature under consideration. However, the deviations of a F e and a M n from linearity are found to be slightly decreased with the rise in temperature from 1763 - 2063 K. This indicates the lowering of the segregation in Fe-Mn melts due to rise in temperature from 1763 K to 2063 K.

Figure 9. ( a F e & a M n ) vs x F e for Fe-Mn melts at 1763 K, 1863 K, 1963 K and 2063 K.

The theoretical values of S c c ( 0 ) for Fe-Mn melts at 1763 K, 1963 K and 2063 K are ascertained from Equations (14) and (15) which are depicted in Figure 10 along with S c c i d ( 0 ) . For comparison, the experimental values of S c c ( 0 ) at 1863 K are also included in Figure 10. It is clear that S c c ( 0 ) > S c c i d ( 0 ) at each composition in full concentration range i.e. x F e = 0.10 to 0.90 at each temperature under consideration. However, the values of S c c ( 0 ) decrease due to elevation of temperature from 1763 K to 2063 K. This illustrates the decreasing behaviour of segregation in Fe-Mn alloys in the temperature range, 1763 - 2063 K. Further the symmetry in S c c ( 0 ) is maintained in the temperature region 1763 - 2063 K because the S c c ( 0 ) is maximum at x F e = 0.50 i.e. 0.295, 0.293, 0.288 and 0.286 at 1763 K, 1863 K 1963 and 2063 K.

Figure 10. S c c ( 0 ) vs x F e for Fe-Mn melts at 1763 K, 1863 K, 1963 K and 2063 K.

The concentration dependence of the excess stability function, E X S / R T for Fe-Mn alloys at 1763 K, 1863 K 1963 and 2063 K determined from Equation (19) are illustrated in Figure 11. Obviously, E X S / R T is negative at each composition of Fe in the concentration region x F e = 0.10 to 0.90 at each temperature under consideration. Nevertheless, the negative magnitude of E X S / R T degrades due to rise in temperature from 1763 K to 2063 K. This is a sign of decrease in the segregation of Fe-Mn melts from 1763 K to 2063 K.

Figure 11. E X S / R T vs x F e for Fe-Mn melts at 1763 K, 1863 K, 1963 K and 2063 K.

Figure 12 exhibits the concertation and temperature dependence of α 1 for Fe-Mn melts determined from Equation (18). At each concentration, α 1 is positive at each temperature under consideration and its magnitude decreases with the rise in temperature in the region 1763 - 2063 K which shows the lowering in segregation of Fe-Mn alloys in the temperature range 1763 K to 2063 K. Again, the magnitude of α 1 is small which illustrates that Fe-Mn melts are a weakly segregating system in the temperature range 1763 - 2063 K.

The diffusivity ratio, D M / D i d as a function of the composition of Fe for Fe-Mn melts at 1763 K, 1863 K, 1963 K and 2063 K are calculated from Equation (20) and the theoretical values of D M / D i d are depicted in Figure 13. It is noticed that diffusivity ratio, D M / D i d < 1 in the region 0.1 ≤ x F e ≤ 0.9 and has minimum values of 0.8434, 0.8543, 0.8654 and 0.8744 respectively at 1763 K, 1863 K, 1963 K and 2063 K around the equiatomic composition, x F e = 0.5 for Fe-Mn melts. Again, due to rise in temperature from 1763 K to 2063 K, the values of D M / D i d increase at each concentration which is a sign of degradation in the segregating behaviour of Fe-Mn melts due to elevation of temperature from 1763 K to 2063 K. Therefore, the present theoretical investigation suggests that the probability of phase separation is maximum at x F e = 0.5 which is also suggested by Ogundenji et al. [13].

Figure 12. α 1 vs x F e for Fe-Mn melts at 1763 K, 1863 K, 1963 K and 2063 K.

Figure 13. D M / D i d vs x F e for Fe-Mn melts at 1763 K, 1863 K, 1963 K and 2063 K.