Themo-Physical Properties of Al-Mg Alloy in Liquid State at Different Temperatures

Thermo-physical properties of Al-Mg alloys in molten state at 1073 K have been studied using thermodynamic modeling. Thermodynamic properties, such as free energy of mixing, heat of mixing, entropy of mixing, activities and structural properties, such as concentration fluctuation in long wavelength limit, Warren-Cowely short range order parameter have been studied at 1073 K, 1173 K, 1273 K and 1373 K on the basis of regular associated solution model. The surface properties such as surface concentrations and surface tension of the liquid alloys have been studied by using Butler’s model. A consistent set of model parameters have also been obtained by using optimization procedure based on statistical thermodynamics. Our analysis reveals that Al-Mg alloy is moderately interacting and it shows ordering nature at 1073 K. The nature of the alloys changes from ordering to segregating as the temperature increases.


Introduction
The properties such as light weight and excellent corrosion resistance make Al-based alloys suitable for developing engineering structures and their components. Because of these properties Al-based alloys have been extensively used in aircraft, automotive engines and architecture [1] [2] [3]. Apart from aforementioned properties, Al-Mg alloys have other very valuable properties, such as high Materials Sciences and Applications mechanical strength, good crack resistance, low cost maintenance, non-toxic, non-magnetic and much less inflammable than any other Al-based alloys. This is why a considerable attention has been given by metallurgist in developing Al-Mg alloys because of the possibilities of their uses in electrical industry, roofing sheets, vehicle paneling etc. [4] [5] [6] [7]. The Al-Mg alloys in liquid state had also been studied by theoreticians [8] [9] in the past. But these studies were only for a particular temperature; say 1073 K at which observed property is available.
In present work, we have studied the thermodynamic, structural and transport properties of Al-Mg liquid alloys at different temperatures. These properties are then assessed by optimization procedure which is the thermodynamic description based on statistical thermodynamics or polynomial expressions.
There have been several studies based on the multiple theoretical models for the study of the mixing properties of liquid alloys [8]- [17]. In present work, we study the thermodynamic and structural properties of liquid Al-Mg alloys at different temperatures using regular associated solution model (RASM) [16] [17] [18] [19]. The surface tension of the alloy has been studied by Butler's model [20]. In previous works, we used RASM to study the properties of binary liquid alloys having complexes A B µ υ with 1 µ ≥ and 1 υ = [16] [17] [18] [19] [21] [22]. The phase diagram of Al-Mg alloys [23] indicates that there exists an Al 3 Mg 2 complex in the mixture. So we also assume the existence of the same complex in Al-Mg liquid alloys near melting temperature. Here 3 µ = and 2 υ = . Thus the input parameters are obtained by using the expressions different from previous ones, which will be explained in section 2 of this article.
In RASM, the existence of complexes in binary solution is assumed and hence a binary alloy in molten state can be treated as a ternary mixture of monomers and complexes; all in chemical equilibrium [8] [16] [17] [19]. However, the interactions between both the unassociated atoms and the complex are not necessarily equal and therefore unassociated atoms may not interact equally with the complex [18] [19] [21] [22] [24]. The advantage of the RASM is that the most of the input parameters are computed by solving mathematical expressions [17] [18] [19] [24] [25] where as in most of the models the input parameters are estimated just by fitting methods.
The paper is organized as follows: In Section 2, the theoretical framework is presented. Section 3 gives the results and discussion of the work. Finally, the conclusions are outlined in Section 4.

Thermodynamic Properties
We consider a mixture of Al and Mg metals in molten state which on the basis of RASM [16] [17] [18] [19] leads to a pseudo-ternary mixture of Al-Al, Mg-Mg monomers and Al μ Mg υ complexes at equilibrium, where μ = 3 and υ = 2 [23]. On the basis of previous works [17] [18] [21] [25], it can be assumed that the thermodynamic properties of the liquid mixture is overruled by their true mole frac- Relations between true mole fractions and gross mole fractions for Al-Mg (= Al μ Mg υ ) alloys at molten state can be developed using regular associated solution model and is given as On the basis of RASM, the expression of free energy of mixing, ( M G ) can also be derived [16] where 12 ω , 13 ω , 23 ω are respective interaction energy parameters between the pairs A, B; A, A-B and B, A-B respectively. R is Universal gas constant, T the temperature and k the equilibrium constant to be estimated.
The equilibrium constant, k, is given as [16] [17] [25] A where 1 a and 2 a are activities of the pure components A and B respectively.
The expression for the mole fraction of the complex in terms of interaction energies and activity of pure components of the alloy can be obtained by using Equation (9) in Equations (4) and (5) along with algebraic simplification as fol- The heat of mixing, Using Equations (3) and (11) we get

Structural Properties
The structural properties of the molten alloys can be assessed by means of concentration fluctuation in long wavelength limit with M G given as [10] ( ) Using Equation (3) in Equation (14) implying the constrains   (15) where primes in true mole fractions denote the differentiations with concentrations. The observed value of the ( ) 0 CC S is obtained by the relations [10].
To measure the degree of ordering of binary alloys in liquid states, the know- ( ) where, Z is the coordination number of the alloy and it is taken as 10 for our calculation. The value of Z does not vary the nature of 1 α ; the effect is to vary the depth not the overall features.

Surface Properties
If the bulk and hypothetical surface of a solution is assumed to be in equilibrium (Butler's approach [20]) the surface tension of a liquid solution can be expressed as [20] [29] where , E s i G and The molar surface area of a pure component i in the liquid state can be expressed as [30] ( ) ( ) where N is Avogadro's constant, 0 i V is the molar volume of the pure component i at its melting temperature and f is the geometrical constant. The geometrical constant can be expressed as where b f is the volume packing fraction and s f is the surface packing fraction. Their values depend upon the type of crystal structure of the pure components of the alloys.

Optimization Procedure Thermodynamic, Structural and Surface Properties
A consistent set of model parameters in an analytical approach can be obtained by the optimization procedure which is the thermodynamic description based on statistical thermodynamics or polynomial expressions. The adjustable coefficients, used in the process, are estimated by least square method which gives an idea to extrapolate into temperature and concentration region in which the direct observed determination is unavailable.
The various thermodynamic properties, described by a power-series law whose coefficient are A, B, C, D, E… (say), are determined by least-square method [31]. The heat capacity can be expressed as From the thermodynamic relation, the heat of mixing and entropy of mixing are given by ( ) Using Equations (24) and (25) The coefficients K 1 is the function of the temperature same as that of G in Eq-

Thermodynamic Properties
The input parameter 12 RT ω was computed from the Equation (7)  were estimated by using Equations (7) and (8) in (10)  indicates that the system is moderately interacting and hence the affinity towards the compound formation is moderate.
On using Equation (12) and observed heat of mixing [23] the optimized value of the following parameters were found as   [23]. This low negative value suggests that the bonding between the species Al and Mg to form the complex is weak. The entropy of mixing for the alloys was also computed using Equation (13). Both the computed and observed values of entropy of mixing were found to be in good agreement (Figure 3).
The activities of the components of the alloys were computed from Equations (5) and (6) using the same input parameters which were used for the computation of free energy of mixing, heat of mixing and entropy of mixing. The computed and observed values [23] were found to be in good agreement (Figure 4).
The parameter, activity assesses the deviation of the system from the ideal behavior. Thus the knowledge of activities within a class of identical structure can be expected to offer, at least, a basis for correlation of the behaviour, which can then be used for extrapolation of the behaviour of a more complex system.

Surface Properties
The surface tension of the alloys in liquid state at 1073 K was calculated by using  The bulk partial excess free energy of mixing for the individual components of the alloys in liquid state at 1073 K is available in [23]. We considered the geometrical constant 1.061 f = [30] and melting temperature. Figure 7 shows that the surface tension of the solution is found to be less than ideal value in the entire range of concentrations at 1073 K.

Thermodynamic, Structural and Surface Properties at T ≥ 1073 K
In order to calculate the thermodynamic, structural and surface properties of alloys in liquid state at different temperatures we have made the following assumptions: 1) True mole fractions of the monomers and that of complex are temperature independent.
2) The pairwise interaction energies are temperature dependent.
3) The temperature dependence of pairwise interaction energies is linear above melting temperature.
Under these assumptions, the variation of interaction energy parameters with temperature can be expressed as where i, j = 1, 2, 3; i j ≠ . C is the mole fraction.
where, d   8. The plot clearly shows that the free energy of mixing becomes smaller in magnitudes as the temperature increases. Thus the ordering behavior of the alloys will be weaker and weaker as the temperature increases. The optimized value of partial excess free energy of mixing of the constituent X ( ≡ Al) and Y ( ≡ Mg) in Al-Mg liquid alloys are respectively given by where Al, Mg i = . The activities of the components of the alloys at different temperatures were then computed by using the calculated values of the corresponding activity coefficients and plotted in Figure 9.
We computed the ( ) 0 CC S at different temperatures and plotted in Figure 10. The  (20) following the similar procedure as that for temperature 1073 K. The compositional and temperature dependence of the surface tension of the liquid Al-Mg alloy at higher temperatures is shown ( Figure 11). It can be observed that the surface tension of the liquid alloy decreases with the increase in its temperature. The surface concentration of Al of the liquid alloy increases with increase in its temperature ( Figure 12). The surface concentrations of Mg atoms of the liquid alloy decreases with the increase in its temperature. This indicates that the atoms in the bulk tend to segregate and the atoms in the surface tend to move to the bulk as the temperature increases. Table 1. Calculated values of optimized coefficients A l , B l , C l and D l (l = 0 to 3) in liquid alloy Al-Mg.

Conclusions
From our study the following conclusions can be drawn: 1) The Al-Mg liquid alloy is moderately interaction in nature.
2) The Al-Mg liquid alloy is ordering alloy at 1073 K. Above this temperature the nature of the alloy shifts towards the ideal nature.
3) Al atoms remain in the bulk and Mg atoms segregate in the surface.
4) The surface tension of the alloy is lesser than ideal values at all temperatures.