Analytic binary alloy volume-concentration relations and the deviation from Zen`s law

Alloys expand or contract as concentrations change, and the resulting relationship between atomic volume and alloy content is an important property of the solid. While a well-known approximation posits that the atomic volume varies linearly with concentration (Zen`s law), the actual variation is more complicated. Here we use an apparent size of the solute (solvent) atom and the elasticity to derive explicit analytical expressions for the atomic volume of binary solid alloys. Two approximations, continuum and terminal, are proposed. Deviations from Zen`s law are studied for 22 binary alloy systems.

determines, in many respects, the enthalpy of phase formation: the higher space filling factor, the higher the enthalpy of phase formation. The highest the space filling factor, superstructure contraction, and the enthalpy of formation are observed for stoichiometric NiAl (B2) structure: 0.785, -0.140, and -59 kJ/mol, respectively.
Klopotov et al. [8] have addressed the main crystallogeometrical parameters of compounds in the Ni-Ti system. It has been found out that the space filling factor, superstructural contraction, and enthalpy of phase formation increase simultaneously. The highest space filling factor, superstructure contraction, melting temperature and the enthalpy of formation are observed for stoichiometric Ni3Ti (D024) structure: 0.80, -0.086, 1400 ºC, and -38 kJ/mole, respectively.
Potekaev et al. [9,10] have established the explicit correlation between the type of evolution of the binary phase diagrams based on elements of VIIIA and IB groups of the Mendeleyev's Periodic Table and the nature (positive/negative) of the superstructure contraction.
The superstructure contraction, which reflects the deviation from Zen`s law, is a very important parameter of the binary alloy crystal lattice. In the next Section we discuss a theoretical basis, based on Lubarda's [11] elastic inclusion model, for analytical determination of the atomic volume of solid solutions. 4

Theoretical Background
Lubarda [11] derived an expression for the effective lattice parameter of binary solid solutions by using an elasticity inclusion model, in conjunction with an apparent size of the solute atom when resolved in the solvent matrix. Assuming the R1 is the Wigner-Seitz radius of the solvent material, 1 = The terminal solid solution at the other end of the phase diagram can be treated by reversing the role of two materials (a2 is the lattice constant of the solvent and a1 is the lattice constant of the solute). Thus, Eq. (12) is replaced with where = 1+ . (17) In that case, Eq. (15) becomes To use Eqs. (12, 18) one must calculate the effective shear (µ) and bulk (K) moduli. Lubarda [11] used Hill's self-consistent method, as presented by Nemat-Nasser and Hori, [13] that gives the following system of equations for the effective shear and bulk moduli: and the atomic volume for the alloys can be calculated as where =1+ In the present calculations we use the Voigt-Reuss-Hill approximation [14] to calculate the effective shear modulus, µVRH(x): Equations (22) and (23) are solved self-consistently. For a special case of the small atomic volume misfit, Lubarda [11] assumed: which allows rewriting Eq. (23): Lubarda wrote this expression in the slightly different form emphasizing that in the case of = �1 + In this paper several additional assumptions have been made to achieve a continuum solution for the alloy atomic volume within the whole composition range. As pointed out above, see Eqs. (10,15), Lubarda assumed that calculations should be performed for two opposite terminal solid solutions located on the ends of the binary phase diagram. In that case the calculated lattice constants typically have a discontinuity in the middle, at the equiatomic composition. To avoid this problem, we redefine (symmetrize) the coefficient γ1, Eq. (3), as well as assume that the effective coefficient, γ, defined by Eq. (11), should be recalculated at each alloy composition, x, and be expressed through the effective bulk, K(x), and shear, µ(x), moduli calculated by Eqs. (22,24): 8 In addition, we have rewritten Eq. (26) for the two terminal solid solutions: And the atomic volume of the alloy is defined as a function of composition, x, Lubarda [11] used the apparent Wigner-Seitz radius, Eq. (9), in calculations of the lattice parameters of the alloy. In our calculations we introduce two approximations for the atomic volumes of the alloy components which, in turn, are used as input parameters in Eq. (32).
1. Continuum approximation. In the case, where the field of the disordered solid solution spans throughout the whole composition range, we assume that the atomic volume of the solvent (Ω1(x)) changes linearly with composition from the real value, Ω1, x = 0, to its apparent value, 1 * , in the pure solute, x = 1.
Similarly, the atomic volume of the solute, Ω2(x), changes linearly with composition, from its apparent value in the pure solvent, 2 * , x = 0, to the real value, Ω2, x = 1.
2. Terminal approximation. In the case of limited mutual solubility of the alloy components, it is reasonable to consider the atomic volume of the solvent to be constant and equal to its real value, Ω1(2), The atomic volume of the solute, Ω2(1), undergoes a linear change with composition, x, The experimental (real) atomic volumes of selected elements at room temperature, together with the bulk and shear moduli are listed in Table 1. The atomic volumes correspond to the Wigner-Seitz radii reported in Table 1 of Ref. [11], and the elastic constants are the same as in Table 3 of Ref. [11]. In addition to the binary alloys studied in Ref. [11], we present data for Mg-Cd and Fe-Cr solid solutions. The volume size factors, ω1 and ω2, for the alloy systems under consideration are reproduced in Table 2. The values of the real and apparent atomic volumes for 22 alloys are listed in Table 3.  Table 2. The volume size factor data: ω1 is the volume size factor when the first element of the alloy system is the solute, and ω2 when the second element is the solute. Ref. [11,12].

Results.
In this section we report calculations of the atomic volume as a function of concentration for 22 binary alloy systems chosen because of the availability of experimental data. All calculations and data are at room temperature. Both continuum and terminal approximations are applied.

Al-Mg.
According to Refs. [11,17], maximum solubility of magnesium in aluminum is about 18 at.% at 450 º C, and that of aluminum in magnesium is about 12 at.% at 437 º C, Fig 3a. According to Lubarda [11], the lattice spacing of Al based alloys increases by introduction of larger Mg atoms and the lattice spacing of Mg based alloys decreases by introduction of smaller Al atoms.
Superposition of these results produces a positive deviation from Vegard's law for Al based alloys and the negative deviation from Vegard's law for Mg based alloys, which results in a significant jump of the lattice parameter at the equiatomic composition [11]. However, it is inappropriate to

Al-Zn.
According to Refs. [20,37], there is a slight solid solubility of aluminum in zinc and extensive solubility of zinc in aluminum extending to ~ 65 at.% Zn at 381 ºC, Fig. 6a. For Al-based Al-Zn solid solutions the lattice spacing has been measured for alloys containing up to 35 at.% Zn [37].

Cu-Fe.
According to Ref. [23], the solubility limit of copper in α-iron is small, ~ 1.9 at.%. at eutectoid temperature 840 ºC, the solubility of α-iron in copper at the same temperature is also small, 1.3 at %, Fig. 9a.
The calculated, within the continuum approximation, atomic volume of Cu-Ni solid solution is in a good agreement with experimental data, Fig. 10b, reproducing a slight negative deviation from Zen`s law. The heat of formation of the Cu-Ni solid solution is moderate positive within the composition range, [38], signaling that the entropy factor plays a decisive role in formation of a continuous solid solution above 365 ºC [38].

Fe-Cr.
According to Refs. [27,37] at.% of Cr, then drops to its value at ~ 19 at.% of Cr, and then gradually increases within the remaining compositional range, [37]. Fig. 13b also shows calculated, within the terminal approximation, volume of Cr based solid solution in the compositional range, 30 at.% -100 at.% of Cr, which are in an excellent agreement with experimental data [37] (and Zen`s law). The heat of formation of Fe-Cr solid solution, measured at 1327 ºC [38], is positive indicating that the entropy factor is responsible for formation of a continuous solid solution at elevated temperatures.

Fe-V.
According to Refs. [28,37], iron and vanadium form a continuous solid solution at elevated temperatures, Fig 14a. The calculated, within the continuum approximation, atomic volume of Fe-V solid solution is in a good agreement with experimental data measured above 1252 ºC [37], Fig.   14b, reproducing a significant negative deviation from Zen`s law. According to [38], the heat of formation of Fe-V solid solution, measured at 1327 ºC, is positive up to ~ 52 at.% of V and slightly negative in the remaining part of the composition range.

Ag-Au.
According to Refs. [29,37], silver and gold form a continuous solid solution, Fig 15a. The calculated atomic volume in the continuum approximation shows a significant negative deviation from Zen`s like one observed experimentally [37], Fig 15b. The heat of formation of the solid solution, measured at 527 ºC [38] is significantly negative.

Cr-W.
According to Refs. [36,37], chromium and tungsten form a continuous solid solution at high temperatures (above 1677 ºC) , Fig 22a. The calculated atomic volume of Cr-W solution in the continuum approximation shows a positive deviation from Zen`s law which is in an excellent accord with experimental data, Fig. 22b.

Fe-Co, Fe-V, and Fe-Cr.
As gives excellent agreement with experimental data (and Zen`s law). This is only the case where the terminal approximation works better than the continuum approximation for alloys with mutual solubility of the components.

Ag-Au and Ag-Mg.
Both Ag and Au are isostructural (FCC) metals and belong to the same subgroup of the Periodic

Nb-Ta, Ti-Zr, and Cr-W.
Both Ti and Zr are isostructural (HCP) metals and belong the IV subgroup of the Periodic Table. Cr and W are also isostructural (BCC) and belong to the VI subgroup of the Periodic Table. The composition dependence of the atomic volume is described by the continuum approximations well, although the almost perfect Zen`s law behavior is observed (and described) in Ti-Zr alloys and a slight positive deviation is observed (and described) in Cr-W alloys. The situation with Nb-Ta alloys is more complex. Both Nb and Ta metals are isostructural (BCC) and belong to the same V subgroup of the Periodic Table. According to the phase diagram [33], Nb and Ta form continuous solid solutions. However according to Ref. [37], the experimental data for atomic volume is available for two compositions of Nb-Ta alloys only: the negative deviation from Zen`s law is observed at ~ 34 at.% of Nb and the positive deviations from Zen`s law is observed at ~ 62 at.% of Nb. The continuum approximation shows the negative deviations from Zen`s law.

Ge-Si and Pb-Sn.
Both Pb and Sn belong to the same 4A group of the Periodic Table and form the eutectic phase diagram. At room temperature the mutual solubilities of the components are negligible small, and the composition dependence of the atomic volume is described well within the terminal approximation. Ge and Si also belong the same 4A group of the Periodic Table however, contrary to Pb-Sn system, Ge-Si alloys form continuous solid solutions. As we already mentioned, both continuum and the terminal approximations could not reproduce the very small negative deviation from Zen`s law, so we speculate that this failure is due to the non-metallic nature of both Si and Ge.

Cd-Mg.
In 1940 Hume-Rothery and Raynor [40] found that the experimental atomic volumes of Mg-Cd solid solutions are smaller than one calculated using the additivity rule (the negative deviation from Zen`s law formulated in 1956, Ref. [3]). Since then, the behavior of Mg-Cd disordered solid solutions become the subject of numerous investigations [41][42][43][44]. These works used the pseudopotential method in conjunction with the thermodynamic perturbation theory (Gibbs-Bogoliubov inequality) to calculate the equation of state of the disordered solid and liquid MgxCd1-x alloys. The calculated composition dependence of the equilibrium volume of the solid MgxCd1-x alloys [44] shows a negative deviation from Zen`s law but not to such an extent as was reported in the experiment, Ref. [40]. The calculations [41][42][43][44] have been performed within the local pseudopotential approximations which excluded the charge transfer between alloy components due to the difference of their electronegativity, see Ref. [45] for details. Incorporation of the apparent size of solute atom, suggested in Ref. [11], together with modifications suggested in the present study, Eqs. (28-36), allows, for the first time, describe the negative deviation form Zen`s law in Mg-Cd solid alloys

Conclusion.
We have derived an analytical expression for the atomic volume of the binary alloys at the arbitrary composition for use in the equation of state modeling. We wanted this expression to be robust and predictive even in the absence of experimental data at particular concentration. This paper tests our proposed expression by comparison with experimental data for the binary alloys.
Lubarda [11] introduced an apparent size of the solute atom in order to account for the electronic interactions between the outermost quantum shells of the solute and solvent atoms. This idea reflects, to some extent, the electron density rearrangement due to the charge transfer in order to cancel the chemical potential difference due to alloying [46][47][48][49]. Jacob et al. [49] came to conclusion that both Vegard's and Zen`s laws should be downgraded to an approximation which is valid in specific conditions. We agree with this upshot by describing numerous cases of the deviation of Zen`s law with a satisfactory way to describe (or predict) these deviations which is the primary motivation for this study.