Mechanics of Materials 35 (2003) 53–68 www.elsevier.com/locate/mechmat On the effective lattice parameter of binary alloys V.A. Lubarda * Department of Mechanical and Aerospace Engineering Sciences, University of California, 9500 Gilman Drive, San Diego, La Jolla, CA 92093-0411, USA Received 24 May 2002 Dedicated to the memory of Dr. Owen Richmond Abstract A study of the effective lattice parameter in binary alloys is given based on an analysis of the volume change produced by dilute resolution of the solute atoms in the solvent matrix. An apparent size of the solute atom is in- corporated in the analysis to approximately account for the electronic interactions between the outermost quantum shells of the solute and solvent atoms. The comparison with experimental data for various alloy systems and deviations from VegardÕs law are analyzed. The free energy of binary alloys and the ordering of solute atoms at higher solute concentrations are then discussed. Ó 2002 Elsevier Science Ltd. All rights reserved. Keywords: Binary alloys; Lattice parameter; VegardÕs law; Solute atoms; Apparent atomic radius; Nonlinear elasticity; Free energy 1. Introduction solution, such as elastic coefficients, thermal ex- pansion, chemical bonding, and electrical conduc- In the vicinity of each solute atom which is re- tivity, as well as to the onset of phase transition, solved in a solvent matrix there is a considerable occurrence of stacking faults, dislocation nucle- variation of interatomic spacing due to local dis- ation, etc. Analytical determination of the effective tortions of the crystalline structure. Away from the lattice parameter in terms of the properties of the solute, the atoms occupy positions of an ideal, constituting elements of the solution has been a regular lattice whose spacing represents an average long-standing research topic in the solid state spacing over several thousand atomic distances. physics and materials science. Vegard (1921) ob- The X-ray diffraction lines from a solid solution served that in the cases when two salts form a are thus almost as sharp as from pure metal, and continuous solid solution, the effective lattice pa- occur at angles corresponding to this average lat- rameter often varies almost linearly with the solute tice spacing. The lattice spacing is a fundamental concentration, as a weighted mean of the lattice quantity related to bulk properties of the solid spacings of the solute and solvent. This empirical rule has become known as VegardÕs law, although subsequent experimental and theoretical investi- * Tel.: +1-858-534-3169; fax: +1-858-534-5698. gations have shown that very few other, particu- E-mail address: [email protected] (V.A. Lubarda). larly metallic, solid solutions have lattice spacings 0167-6636/03/$ - see front matter Ó 2002 Elsevier Science Ltd. All rights reserved. PII: S0167-6636(02)00196-5 54 V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68 which agree with a linear relationship between the The reason for a limited success of continuum two end points of alloying elements. Nevertheless, elasticity models is not surprising, since in view of when studying solid solutions, it became a com- its atomistic scale the problem is fundamentally mon practice to speak about deviations or depar- one of the quantum-mechanics and physical tures from VegardÕs law (e.g., Hume-Rothery et al., chemistry. The electronic interactions between the 1969). outer-most quantum shells of the solute and sol- Among the factors that affect the structure and vent atoms depend on their electro-chemical formation of an alloy, the prominent include the characteristics, partially reflected by their positions difference in atomic size, the electro-chemical in the periodic table of elements, and are far too properties of the alloying elements, and their complex to be accounted for only by a simple size crystalline structures. The size effect has been often factor. For example, the alloy systems whose two studied from an idealized point of view by meth- end members have similar, if not identical, outer ods of continuum elasticity. A solid spherical in- electronic structures, show better agreement with clusion inserted into a spherical hole of a matrix predictions based on elasticity models and Ve- material was considered to be a model of substi- gardÕs law. King (1966) suggested that a tendency tutional or interstitial atom. The size disparity of metallic solid solution to deviate from VegardÕs between the solute and solvent atoms causes the law may actually be taken as a measure of the volume change of the solution, and this can be modification of the electronic environment of the related to the effective lattice parameter. Linear solute atom. At higher concentrations, sufficient elasticity was most frequently employed (Pines, amount of solute has been added to modify the 1940; Seitz, 1946; Lawson, 1947; Friedel, 1954, electronic environment of solvent atoms, as well. 1955; Eshelby, 1954, 1956), although nonlinear Sarkisov (1960) proposed an electron-gas model to effects were also considered (Gschneidner and explain these deviations, while Steinwehr (1967) Vineyard, 1962; Park et al., 1975; Lubarda and combined purely geometrical and chemical inter- Richmond, 1999). For certain alloy systems this action effects, closely related to the difference in has led to satisfactory agreement with experimen- electronegativities of the solute and solvent atoms. tal results, but in many cases the predicted lattice Yet, none of these models was able to predict the spacing was significantly above or below the effective lattice parameter with a satisfactory ac- measured values. King (1966) indicated that in curacy for a large number of randomly selected alloys where the solute atom is smaller than the binary alloy combinations. solvent atom, the sign of the predicted change in There has been a significant amount of research atomic volume was always found to be in error. devoted to theoretical and experimental determi- Where the atomic volume of solute is greater, the nation of the effective lattice parameter in more predicted and observed values agreed to within 5% complex alloy systems. An accurate X-ray dif- in only half of the considered random selection fraction measurement of the lattice parameter of a of alloy systems. The results for many solutions ternary Cu99ÀxAuxFe1 alloy along the gold con- based on iron, copper, aluminum, and magnesium centration range was reported by Finkler et al. were poor. For Au–Ag alloy, the analysis pre- (1987). A positive deviation from the linearity of dicted an increase of the effective lattice parameter VegardÕs law was observed. Similar observations due to substitution of larger silver atom in the were reported for ternary layered intercalation solvent matrix of smaller gold atoms. The mea- compounds by Solin et al. (1989). The applicability surements, however, indicate a decrease of the ef- of VegardÕs rule in complicated oxides with spinel fective lattice parameter (Pearson, 1972). Similar and perovskite structures was also studied (e.g., situation was found with many other systems; the Talanov, 1983; Yashima et al., 1992; Ganguly lattice distortion of the nickel lattice with germa- et al., 1993). Nongkynrih et al. (1988) observed nium as the solute is opposite in sense than ex- two different types of linear behavior in the regions pected from the relative sizes of the elemental below and above the mid-point of the concentra- atoms (Chessin et al., 1963). tion range of the substitute in calcium and lead V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68 55 " # based apatits, which is a rather general prop- 1 D D 2 erty, since in most systems the linear trends at the C ¼ þ c ð2Þ two limiting ends of the concentration range are c2 R1 R1 quite different. A geometric model suggested by and Urusov (1992) involves the calculation of the atomic displacement around substitutional atoms 2 3l1 þ n1 9l2 þ n2 1 2 c2c ¼ À 1 À þ ðc2 À 1Þ at the level of the first and second nearest neigh- 3j2 3j2 2 bors. c2 l1 þ 2m1 In this paper we derive an expression for the À : ð3Þ c1 j1 effective lattice parameter of binary solid solutions by using an elasticity inclusion model, in con- The parameters c1 and c2 are defined by junction with an apparent size of the solute atom 4l 4l when resolved in the solvent matrix. The latter is c ¼ 1 þ 1 ; c ¼ 1 þ 1 : ð4Þ 1 3j 2 3j intended to approximately account for the elec- 1 2 tronic interactions between the outermost quan- The third-order elastic moduli of the solute mate- tum shells of the solute and solvent atoms. The rial are l1, m1, and n1 (Murnaghan, 1951; Toupin comparison with experimental results for various and Bernstein, 1961). In the case of small misfit alloy systems and deviations from VegardÕs law are and linear elasticity, the constant C in Eq. (2) re- considered. The free energy of binary alloys and duces to the ordering of solute atoms at higher solute 1 D concentrations are then discussed. C ¼ : ð5Þ c2 R1 The volume increase within S is, therefore, 2. Volume change due to solute atom Z 1 1 3 DV ¼ u1 Á ndS ¼ 4pR1C; ð6Þ Consider a solvent material of volume V en- S closed by the surface S, free of any external load where n is the outward normal to S. This follows or surface constraint. Remove a solvent atom, by the Gauss divergence theorem, recalling that imagined to be a small sphere of radius R1 deep for any m and the three-dimensional vector r, inside the volume V, and replace it by a solute divðrmrÞ¼ðm þ 3Þrm. atom of radius R2 > R1 (an analogous derivation An auxiliary problem is next considered with im 1 proceeds in the case R2 < R1).