The Difference between the Chemical Short Range Orders of Binary Liquid Alloys Using Different Models

Introduction

The electron microscopy and x-ray diffraction experiments are very useful tool to obtaining the structural information and thermodynamic properties of binary liquid alloys Singh [1]. In most cases, obtaining the experimental data needed for the calculation of specific thermodynamic properties are really available (except in some uncommon cases where the experimental data for some alloys may be difficult to obtain due to cumbersome task involved and experimental complexities). In principle, the chemical short range order ( )has connection with the Concentration- Concentration Fluctuation in the long wave length limit (Scc (0)). The Scc(0) can be experimentally determined from the knowledge of concentration-concentration partial structural factor, Scc (q), and the number-number partial structural factor SNN (q) Bhatia, et al. [2]. However, these structures are not easily measurable in most diffraction experiments. Hence is usually computed when experimental values are not available. Additionally, a direct experimental determination of Scc (0) is often avoided due to great deal of task often involved [3]. For this reason the options of thermodynamic models which have been used were employed and juxtaposed.

The focus of this study, therefore, is to compute the excess stability functions of two binary liquid alloys via Statistical Mechanical Model (SMM) [4], Quasi-Lattice Model (QLM), Two and Four Atoms Cluster models (TACM and FACM) [5]. The results comparisons would be made thereafter to ascertain to what extent it is safe to consider these approaches.as the two options serves as alternatives when information on concentration-concentration fluctuation is attached.

In doing the aforementioned, we applied these models to Cu-Pb, Li-Mg, Cd-Ga and Bi-Cd liquid alloys [6, 7] for the qualitative investigation of their thermodynamic properties. Ordering energy values determined from Scc (0) are recorded in Table 1. Programs were inscribed to generate data for thermodynamic expressions as functions of concentration, c, using ordering energy values, w, coordination number, Z, Boltzmann constant, K and temperature, T presented in Table 1.

Theory

The determination of ordering energy (w) or interchange energy can be linked to the calculation of the Chemical Short Range Order ( 1 α ) of binary liquid alloys. The variation of this quantity with composition is informative [8]. Thermodynamically, the relationship between short range order parameter 1 α and other thermodynamic properties had been sighted in the literature Khanna, et al. [9, 10]. Moreover, between w and 1 α and other properties they are given below. The following thermodynamic expressions are from different Models.

Expressions for Chemical Short Range Order (SMM)

$$ \left(1 + \frac {\alpha_ {1}}{c \left(1 - c\right) \left(1 - \alpha_ {1}\right) ^ {2}}\right) = \exp \left(\frac {2 w}{z K _ {B} T}\right) $$

2 1 exp 1 1 B

α w zK T c c

(1)

1 2 1 ( )( )

α Z is the coordination number for the first shell, w is the ordering energy, K is the Boltzmann constant, T is the temperature, c is concentration of atom A and 1-c is the concentration of atom B.

Expressions for Chemical Short Range Order (QLM)

( ) ( ) ( ) 1 2 1 1 exp 2 1 1 B C C w Z K T α (2)

$$ \frac {\alpha_ {1}}{\left(1 - \alpha_ {1}\right) ^ {2}} = C \left(1 - C\right) \left(\exp \left(2 w / Z K _ {B} T\right) - 1\right) $$ ( )

Results

Concentration-Concentration Fluctuation in the Long Wavelength Limit (TACM &FACM) ( ) (1 ) 0 (1 ( )(1 ) 2 c c Scc Z β β $$ ) = \frac {c (1 - c)}{(1 + \left(\frac {Z}{2 \beta}\right)) (1 - \beta)} $$ (3) $$ \text {W h e r e} \eta = \exp (w / Z K T) \tag {4} $$ and Z is the coordination number for the first shell, w is the ordering energy, K is the Boltzmann constant, T is the temperature, c is concentration of atom A and 1-c is the concentration of atom B.

$$ \mathrm {w h e r e} \beta = (1 + 4 c (1 - c) [ (\eta) (\eta) - 1 ]) ^ {0. 5} \tag {5} $$ η and β are thermodynamic parameters which are interwoven Excess Stability Function (ES) (TACM &FACM) $$ E S = \frac {R T}{S c c (0)} - \frac {R T}{c (1 - c)} [ 6 ] $$ Ideal Concentration-Concentration Fluctuation in the Long Wavelength Limit $$ S c c _ {c c} ^ {i d} (0) = c (1 - c) \tag {7} $$

R is molar gas constant.

(0) (1 ) id

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Discussion

Figures 1 & 2 displayed the relationship between the SROs and the concentrations of copper and lithium in their liquid phase. The figures portray homo-coordination in the nearest neigbour shell. The atomic distributions signifies like-atoms pairing in the neigbourhood shells. Figures 3 & 5 shows the plots of concentration-concentration fluctuation Scc (0) against concentration of element for Cd-Ga, and Bi- Cd liquid alloys at their melting temperatures. The Scc(0) of these alloys increases initially to a maximum (owing to the charge transfer between neighboring atoms) within the entire concentration range with distinct peaks at Ccd = 0.5 and CBi = 0.4, and, the remaining liquid alloy has some depression at the right side of the curve (owing to chemical alternation of positive and negative charges with length scale approximately twice the nearest neighbor distance). In Figure 3, the calculated Scc (0) is in not perfect agreement with ideal solution values This is because the is a near cancellation of the ionic potentials wile at large distances ionic potentials was screened. Figure 3 & 5 also, have their calculated values for Scc(0) of Cd-Ga and Bi-Cd above the ideal solution values which are in support of homocoordination or self coordination.

Figures 4 & 6 shows the plots of excess stability function versus concentration of element. The display in Figure 4 shows an initial decease in curve to a minimum (possibly when the disordered potential is too large) with a corresponding gradual increase and some fluctuations with concentration. The excess stability function has negative values, its falls downward to concentration Ccd=0.1 before ascending in a straight line between Ccd= 0.2 and 0.8 eventually a repetition of what was displayed at Ccd =0.1 was also displayed at ccd=0.9 with minimum excess stability function between Ccd = 0.8 and 1.0 which was lower than what was observed at the initial stage of the curve. The excess stability function displayed in Figure 6 shows initial decrease as the concentration increases and subsequent increase in concentration makes excess stability function reach the highest value. The short range repulsive potential which prevented the two atoms from reaching each other than the effective diameter has allowed the Es values completely negative throughout the concentration. In Figure 4, as the concentration increases there was corresponding decrease in excess stability function (due to interatomic potentials repelled by the central potential) between CBi = 0.2 and 0.6 with negative significance. Sharp increase in excess stability up to maximum value (due to pair correlation function thus formed at a distance a little greater than the effective diameter) in the concentration range CBi = 0.8-0.9 was observed before falling sharply to zero excess stability function at CBi = 1.0

Conclusion

Bi-Cd liquid alloys is a chemically strong interacting compound with chemical short-range order. The dip in Bi-Cd liquid alloy is an indication of slight formation. The narrow width and considerable height of excess stability function for Bi-Cd is described with strong stability and Cd-Ga liquid alloy is a weak interacting system with intermediate-range.