On the Effective Lattice Parameter of Binary Alloys

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 incorporated 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 expansion, 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 Cu99xAuxFe1 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 1 þ ðc2 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 : 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). It will be assumed the image traction t ¼n Á r1 applied over the that the solute atom is a spherical inclusion whose external surface S of the finite body V with the elastic properties are equal to those of the solute inserted inclusion. The superposition of two material in bulk. The external surface S expands, problems makes the total traction over the surface causing an increase DV of the volume within S.As S equal to zero, and hence represents the solution in Eshelby (1954, 1956), this volume increase can of the original problem. Since the inclusion is small be calculated in two steps. First, the solute atom is comparing to volume V, in the image problem we considered to be inserted in an infinite medium of can neglect the fact that the material of the in- the solvent matrix. The points on an imagined clusion is different from the surrounding matrix. internal surface, having the size and shape of S, are Consequently, the volume change in the image far from the solute atom and, thus, neglecting local problem is Z crystal anisotropy, have displacements governed im 1 im 4l1 by DV ¼ r Á t dS ¼ DV 1: ð7Þ 3j1 S 3j1 R3 1 1 The last step follows from the relationship r1 ¼ u1 ðrÞ¼C ; ð1Þ 1 r2 1 1 2l11 , since 1 is a traceless tensor, and because 1 where (Lubarda and Richmond, 1999) the displacement u1 ðrÞ is a homogeneous function 56 V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68

) of degree 2. Therefore, upon summation of Eqs. 1 1 (6) and (7), the volume increase produced by in- R ¼ 1 R1 þ R2 c c sertion of the solute atom is 2 2 1 1 0 3 ¼ 1 R1 þ R ; ð10Þ DV ¼ 4pR c1C ð8Þ 2 1 c1 c1 0 with C given by Eq. (2). If C is given by Eq. (5), where from R2 ¼ R1 þ c1D=A. The fictitious misfit 0 0 implying a small misfit and linear elasticity, Eq. (8) is thus D ¼ R2 R1 ¼ c1D=c2, and from the cited 0 2 0 reduces to EshelbyÕs expression DV ¼ Vmisc1=c2, theorem DV ¼ Vmis ¼ 4pR1D ¼ c1Vmis=c2. 2 where Vmis ¼ 4pR1D is the misfit volume. In this If j1 ¼ j2, the volume change DV is equal to the 1 im 2 case DV ¼ Vmis=c2 and DV ¼ðc1 1ÞVmis=c2. misfit volume 4pR1D. Hence, in this case the in- The volume change can also be derived in a crease of the volume of the matrix material is ex- manner analogous to that used by Pines (1940). actly balanced by the decrease in the volume of the Consider a volume V0 bounded by a traction free inclusion (within linear elasticity model). The ac- external surface S and the surface of a small hole tual volume of the material does alter if the sec- of radius R1 under the pressure p, deep inside the ond-order elasticity effects are included in the surface S. As an auxiliary problem consider the analysis, as originally shown by Zener (1942) and same body under hydrostatic pressure p over verified by Seeger and Mann (1959). Gschneidner the external surface and the surface of the hole. By and Vineyard (1962) used ZenerÕs formula in their the Betti reciprocal theorem, the volume change second-order elasticity analysis of the VegardÕs 3 DV0 is 4pR1p=3j1. The change of the volume V rule in binary alloys. bounded by S (including the volume of the hole), produced by pressure p on the surface of the hole, 3. Volume change of a solid solution 2 is DV ¼ DV0 þ 4pR1u0, where u0 is the displacement of the points on the surface of the hole. If the To calculate the volume change associated with solid sphere of radius R1 þ D is to be inserted in the a random resolution of xN1 solute atoms in the hole of radius R1, the misfitting condition is solvent matrix, imagine that solute atoms are first u þ pR = 3j ¼ D, and therefore inserted into an infinite matrix. Since nonlinear 0 1 2 4 1 1 elasticity effects are localized to regions around DV ¼ 4pR2D þ R3p p: ð9Þ each solute atom, and assuming that these do not 1 3 1 j j 1 2 overlap (dilute distribution), we can calculate the Since the pressure required for insertion of a solid volume change DV 1 of the material within the sphere into a small hole is given by p ¼ð4l1= surface S by superposition as c2ÞD=R1, the substitution into Eq. (9) again leads 1 3 DV ¼ xN14pR1C: ð11Þ to Eq. (8), with C given by Eq. (5). The image traction tim ¼2l n Á 1 is then applied Yet another derivation of the expression for the 1 1 over the external surface S, associated with the volume change DV in the case of linear elasticity displacement field u1 ¼ R3Cr=r3 due to each in- was constructed by Friedel (1955), who employed 1 1 serted solute atom. Since the medium within the the elasticity theorem that the volume of a ho- surface S is now a heterogeneous medium, com- mogeneous medium does not alter upon intro- posed of the solvent matrix and inserted solute duction of an internal self-equilibrating state of atoms of concentration x, the effective shear and stress. Thus, imagine that in place of the solute bulk moduli (l and j) can be calculated by using atom with the bulk modulus j and the radius R , 2 2 the HillÕs self-consistent method (Nemat-Nasser we insert a fictitious solute atom with the bulk and Hori, 1999). This gives modulus j of the solvent matrix and with the 1 1 x x radius R0 adjusted so that the fictitious solute atom þ 2 1 4l=3j 1 4l=3j has the same radius R when inserted into the sol- þ 1 þ 2 1 x x vent matrix as does the original solute atom. Then, 5 þ þ 2 ¼ 0; ð12Þ we have 1 l=l2 1 l=l1 V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68 57

1 DV 1 x x 4 X ¼ X þ : ð17Þ j ¼ þ l: ð13Þ 1 N j1 þ 4l=3 j2 þ 4l=3 3 1 The volume change in the image problem is ac- Substituting Eq. (15) into Eq. (17), there follows cordingly 3 X ¼ X1 þ 4pR1xcC; ð18Þ 4l where x is the atomic concentration of solute. The DV im ¼ 1 DV 1: ð14Þ 3j radius of an atom is deﬁned as the Seitz radius, which is related to the atomic volume by X ¼ Upon summation of Eqs. (11) and (14), the total 4pR3=3. For example, assuming the small misﬁt, volume change produced by insertion of the solute 1 3 atoms is 4pR1c2C ¼ X2 X1 ð19Þ

3 4l1 and Eq. (18) becomes DV ¼ xN14pR1cC; c ¼ 1 þ : ð15Þ 3j c c If the solute atom is only slightly larger than the X ¼ 1 x X1 þ xX2: ð20Þ c2 c2 matrix atom, linear elasticity applies and the volume change becomes For c ¼ c2 this is the ZenÕs (1956) mixture rule of additive atomic (or molar) volumes of the solute c and solvent. DV ¼ xN1 Vmis: ð16Þ c2 If there are k atoms per unit lattice cell, the 3 When the effective bulk modulus j of the solution atomic volume is X ¼ #a =k, where a denotes the lattice parameter, and k equals to 1 for the simple is replaced by the bulk modulus j1 of the solvent itself, Eq. (16) reduces to the EshelbyÕs (1956) re- cubic, 2 for the body-centered cubic, 4 for the face- sult for dilute solutions and low concentration of centered cubic, 6 for the hexagonal close-packed, solute atoms. The use of the effective bulk modulus and 8 for the diamond-cubic lattice structure. For accounts, to a certain extent, for elastic interac- complex cubics, k takes larger values (e.g., k ¼ 58 tions among solute atoms in the case of more for the a-manganese). Furthermore, thep parame-ffiffiffi concentrated solutions. ter # ¼ 1 for cubic lattices, and # ¼ 3 3c=2a 4:2427 for an ideal hexagonal close-packed lattice. Therefore, from Eq. (18) the effective lattice pa- 4. Effective lattice parameter of binary alloys rameter of a binary alloy is k 1=3 a ¼ a3 þ 4pR3x 1 cC : ð21Þ In the vicinity of each solute atom, the atoms 1 1 # are not located at regular points of the ideal lattice. 1 On proceeding outwards, local distortions disap- In Eq. (21) it is assumed that the solution and pear and atoms take positions of an ideal, uni- solvent share the same lattice structures. If the formly expanded or contracted lattice. The Brag solution has a different lattice structure, the right- reflections from a solid solution are thus almost as hand side of Eq. (21) should be multiplied by 1=3 sharp as those from a pure metal, and occur at ðk#1=k1#Þ . Since the Seitz radius of an atom is angles corresponding to the lattice spacing aver- related to the lattice parameter by R ¼ð3#= 1=3 aged over approximately several thousand atomic 4pkÞ a, from Eq. (2) it follows that C ¼ Cða1; a2Þ distances. If the solvent material contains N1 at- in Eq. (21). oms of atomic volume X1, its volume is V ¼ N1X1. When xN1 randomly selected solvent atoms are 5. Apparent radius of the solute atom replaced by solute atoms of atomic volume X2, the volume V changes by DV . The effective atomic The following stipulation regarding the size of volume X of the resulting solid solution can be the solute atom will be implemented in the subse- defined by V þ DV ¼ N1X. This gives quent analysis. If the radius of the solute atom in 58 V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68 its parent lattice is R2, the radius of the solute tration the atomic volumes of solid solutions vary atom, when inserted into the solvent matrix, will linearly with composition. In many systems this partially change without any mechanical action by linear trend continues up to the phase boundary; the transfer of electrons between high-energy in other systems there is a limiting concentration quantum shells of the solute and solvent atoms. above which the volume change is no longer linear. This is due to the fact that the outer electron The electronic interactions between the solute clouds of atoms in a crystalline structure are more and solvent atoms also affect the compressibility of or less diffused than those in the free atomic state, the solute atom. We could then assume that the or those in a different atomic environment. The elastic bulk modulus of the solute in solvent en- often large differences between atomic and ionic vironment is different from that of the solute in its radii of elements would be an extreme illustration pure state. The consequences of this modification of this effect. This will be incorporated in the would not be particularly significant for the cal- elasticity analysis by assuming that a solute atom culation of the strain energy of an alloy (at low Ã of an apparent radius R2 ¼ g2R2 is substituted in concentration of solute), since this is dominantly place of the solvent atom of radius R1. The pres- stored in the solvent matrix, but it could be im- sure is only required to bridge the apparent mis- portant for the calculation of effective properties fit between the solute and solvent atom, DÃ ¼ of an alloy. For example, the building up of 3d Ã R2 R1, during which the inner electron shells electron shells, which may occur in the solute at- (inert-gas core) of the solute atom deforms, as oms of transition elements during alloying process, well. The coefficient g2 can be slightly greater or can suddenly increase the compressibility of the smaller than one. Its value depends on the atomic atom by an increased atomic cohesion. structures, relative valence and electronegativity difference between the solute and solvent (bond energy), ratio of the number of valence electrons to 6. Deviations from Vegard’s law the number of atoms in the unit cell (electron concentration), the coordination number, and the The effect of the parameter g2 will be included in crystalline structure of the solvent. In principle, the sequel by replacing the radius of the solute this could be determined or estimated by the quan- atom R2 in all previous expressions by g2R2. The tum mechanics or physical chemistry consider- apparent misfit to be bridged during the insertion ations (Pauling, 1956, 1960; Mott, 1962; Blandin, process is

1965; Weiss, 1990), which is beyond the scope of Ã D ¼ g2R2 R1: ð22Þ this paper. For example, the parameter g2 should be more nearly equal to one, more similar are the If only linear term in the misfit is retained in the outer electron structures of the solute and solvent. expression for C, the effective lattice parameter The apparent atomic radius of the solute atom becomes considered here is analogous to the apparent atomic radius used by Axon and Hume-Rothery c 1=3 a ¼ a3 þ 3x a2ðfa a Þ ; (1948). They calculated the apparent radius (based 1 c 1 2 1 on the closest distance definition) from the ap- 2 c 1 þ 4l =3j parent lattice spacing of the solute, which was ¼ 1 ; ð23Þ obtained by extrapolating the initial linear lattice c2 1 þ 4l1=3j2 spacing vs. composition variation at small con- where centration of solute to 100 at.% of solute. The sffiffiffiffiffiffiffiffiffi same idea was used by Massalski and King (1961), 3 #2k1 and King (1965, 1966), who defined the effective f ¼ g2 : ð24Þ #1k2 atomic volume of the solute by linear extrapola- tion of the initial volume plot to 100% solute. They By further straightforward approximation, the observed that at low values of the solute concen- lattice parameter becomes V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68 59 c c noticed that it approximately holds for many a ¼ 1 x a1 þ f x a2: ð25Þ mutually soluble pairs of ionic salts at constant c c 2 2 temperature. The observation has become known If the solute atoms are much more compressible as VegardÕs law or rule, although, when subse- quently applied to other solid solutions, particu- than the solvent matrix, c=c2 is small and the lattice parameter of the solution is nearly equal to the larly metallic alloys, the law was found to be seldomly obeyed (Hume-Rothery et al., 1969; lattice parameter of the solvent (a a1). Terminal solid solution at the other end of the Pearson, 1972). By comparing Eq. (25) with Eq. (31), a deviation from the VegardÕs law is phase diagram can be considered by reversing the roles of two materials. Thus, Eq. (21) is replaced V c c with a a ¼ 1 a1 1 f a2 x: ð32Þ c2 c2 1=3 This is equal to zero only if f 1 and c c , k 4l ¼ ¼ 2 a ¼ a3 þ 4pR3ð1 xÞ 2 cC ; c ¼ 1 þ 2 : i.e., if the solute and solvent share the same crys- 2 2 # 3j 2 talline structure (isostructural elements), if the ð26Þ apparent radius of the solute is taken to be the radius of the solute in its home environment, and if The parameter C is again defined by Eq. (2), in the elastic properties of the solute and solvent are which the indices 1 and 2 are interchanged, and the identical. As an illustration, for two isostructural apparent size difference is elements with different elastic properties, positive Ã V D ¼ g1R1 R2: ð27Þ deviation from VegardÕs law ða > a Þ occurs when ðj j Þða a Þ > 0. Thus, if the solute is more Consequently, Eq. (23) becomes 2 1 2 compressible than the solvent, positive deviation 1=3 occurs if the solute atom is smaller than the solvent 3 c 2 atom. Pines (1940) suggested that elements with a ¼ a2 þ 3ð1 xÞ a2ðfa1 a2Þ ; c2 larger atoms tend to be more compressible, and, c 1 þ 4l =3j consequently, the negative sign of deviation from ¼ 2 ; ð28Þ c2 1 þ 4l2=3j1 VegardÕs law is met more often than the positive sign. For many systems the sign of deviation is not where constant along the composition. Friedel (1955) sffiffiffiffiffiffiffiffiffi calculated deviations from VegardÕs law resulting 3 #1k2 from the difference in compressibilities of the sol- f ¼ g1 : ð29Þ #2k1 ute and solvent for a number of solid solutions. For nearly equal atomic volumes of the solute and Although in some alloy systems the agreement was solvent, the lattice parameter becomes good, for alloys of copper, silver and gold with each other, deviations were very different from the c c observed values (see, e.g., Table 1 of Mott, 1962). a f 1 x a 1 1 x a : 30 ¼ ð Þ 1 þ ð Þ 2 ð Þ King (1966) indicated that in alloys where the c2 c2 solute atom is smaller than the solvent, the sign of Neither of the derived equations (25) or (30) is in the predicted change in atomic volume was always agreement with a simple linear dependence be- wrong. Where the atomic volume of the solute is tween the effective lattice parameter and the solute greater, the predicted and observed values agreed concentration, to within 5% in only half of the considered random V selection of alloy systems. In the case of Cu–Ag a ¼ð1 xÞa1 þ xa2: ð31Þ and Cu–Au alloys, deviations were opposite in This empirical relationship was observed in an sign to observed deviations. The results for solu- early application of the X-rays diffraction to study tions based on iron, aluminum and magne- the crystalline structure by Vegard (1921), who sium were also poor. For Au–Ag alloy, both the 60 V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68

VegardÕs law and Eq. (25), with f ¼ 1, predict an Table 1 increase of the effective lattice parameter due to Lattice parameters and atomic radii (in units of AA) for selected elements at room temperaturea substitution of larger silver atom in the solvent matrix of smaller gold atoms. Precise measure- Element Lattice a Rc:d: RG RS c ments, however, indicate a decrease of the effective lattice parameter (see Table 2 of King, 1966, or Mg HCP 3.2094 1.6047 1.6016 1.7703 5.2107 (1.5985) Tables 4–10 of Pearson, 1972). The size effect by Al FCC 4.0496 1.4317 1.4317 1.5826 itself cannot explain such behavior. Similar situa- Si DC 5.4306 1.1758 1.3361 1.6844 tion is found in other systems. The lattice distor- Ti HCP 2.9506 1.4753 1.4616 1.6153 tion of the nickel lattice with germanium as the 4.6835 (1.4479) solute is opposite in sense to that expected from V BCC 3.0240 1.3094 1.3499 1.4889 Cr BCC 2.8850 1.2492 1.2878 1.4205 the relative sizes of the elemental atoms. Consid- Mn cubic 8.9140 1.1200 – 1.4286 ering the latter case, Chessin et al. (1963) pro- Fe BCC 2.8665 1.2412 1.2796 1.4114 posed a model in which the atom is regarded to Co HCP 2.5071 1.2535 1.2510 1.3827 consist of a hard incompressible core, surrounded 4.0686 (1.2484) by a compressible shell, smeared out through the Ni FCC 3.5240 1.2459 1.2459 1.3772 Cu FCC 3.6146 1.2780 1.2780 1.4126 space not filled by the hard cores (e.g., 0.26 of the Zn HCP 2.6648 1.3324 1.3944 1.5371 unit cell volume in FCC, and 0.32 in BCC struc- 4.9470 (1.4565) ture). Ge DC 5.6574 1.2250 1.3920 1.7548 Zr HCP 3.2316 1.6158 1.6026 1.7713 5.1475 (1.5894) Nb BCC 3.2986 1.4283 1.4725 1.6219 7. Results based on the apparent radius of solute Mo BCC 3.1470 1.3627 1.4048 1.5495 atom Ag FCC 4.0857 1.4445 1.4445 1.5969 Sn cubic 6.4892 1.4050 – 1.8685 Lattice parameters and atomic radii of selected Ta BCC 3.3030 1.4302 1.4744 1.6263 elements at room temperature are listed in Table 1. W BCC 3.1652 1.3706 1.4130 1.5585 Au FCC 4.0784 1.4420 1.4420 1.5939 In the absence of theory which can predict the Pb FCC 4.9502 1.7502 1.7502 1.9345 apparent radius of the solute atom in solvent en- a For FCC crystals the closest distance atomic radius is vironment solely in terms of the atomic properties pffiffiffi Rc:d: ¼ a 2=4, which is also equal to the Goldschmidtffiffiffi (C.N.12) of pure solute and solvent, the apparent radius of p radius RG. For BCC crystals Rc:dp: ¼ffiffiffi a 3=4, which is 3% less the solute atom will be estimated by using one than RG. For DC lattice Rc:d: ¼ a 3=8, 12% less than the cor- experimental information about the solid solution, respondingpffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiRG. For HCP (C.N.6,6) crystal Rc:d: is equal to a=2 2 2 i.e., the initial slope of the lattice spacing vs. or a =3 þ c =4=2, while RG is taken to be the arithmetic mean of these two values. The Seitz radius RS is calculated from the composition curve, ðda=dxÞx¼0. This is related to atomic volume, as explained in the text. the volume size factor x of King (1965) by 2 1 dX 3 da x2 ¼ ¼ : ð33Þ c X dx a dx Ã 2 1 x¼0 1 x¼0 R2 ¼ R1 1 þ x2 : ð34Þ 3c1 Numerical values of x2 for 469 metallic solid so- Ã lutions, based on the precision lattice parameter An analogous expression is derived for R1.Ifa data available in the literature, were listed by King nonlinear elasticity is used, the constant C is given (1966), and are reproduced for alloy systems con- by Eq. (2), in which D is replaced by DÃ. The ap- sidered in this paper in Table 2. By differentiation parent radius is then obtained by solving the re- Ã with respect to concentration x, from Eq. (18) we sulting quadratic equation for R2. In view of the 3 find that X1x2 ¼ 4pR1c1C. In the case of linear uncertainties about the magnitudes of the third- Ã elasticity, the constant C is equal to ðR2 R1Þ= order elastic constants for most metals (Landolt- R1c2, and the apparent radius of the solute atom Boornstein,€ 1979; Lubarda, 1997, 1999), the results can be calculated from based on the linear elasticity will only be pre- V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68 61

Table 2 Table 3 a The volume size factor data: x1 is the volume size factor Elastic constants for polycrystalline metals when the first element of the alloy system is the solute, and x2 Element j (GPa) l (GPa) when the second element is the solute Mg 35.6 17.3 Alloy x1 x2 Al 72.6 26.0 Al–Ag 0.0918 þ0.0012 Si 97.6 66.2 Al–Cu þ0.2000 0.3780 Ti 108.2 45.6 Al–Mg 0.3580 þ0.4082 V 157.9 46.7 Al–Mn þ0.1620 0.4681 Cr 160.0 115.1 Al–Ti 0.2009 0.1506 Mn 98.0 39.0 Al–Zn 0.0625 0.0574 Fe 169.6 81.4 Cu–Ag 0.2775 þ0.4352 Co 82.3 88.8 Cu–Au 0.2781 þ0.4759 Ni 183 80.0 Cu–Fe þ0.1753 þ0.0457 Cu 136.4 46.8 Cu–Ni þ0.0718 0.0845 Zn 69.6 41.9 Cu–Zn 0.5457 þ0.1710 Ge 75.0 54.9 Fe–Co þ0.0524 þ0.0154 Zr 94.0 30.0 Fe–V 0.1886 þ0.1051 Nb 170.3 37.5 Ag–Au 0.0064 0.0178 Mo 261.3 125.5 Ag–Mg 0.6342 þ0.0713 Ag 103.4 30.3 Si–Ge 0.2065 þ0.0468 Sn 58.2 18.4 Nb–Ta 0.0023 0.0026 Ta 196.5 69.0 Pb–Sn þ0.2905 0.0825 W 311.0 160.6 Ti–Zr 0.2233 þ0.3008 Au 170.7 27.5 Cr–W 0.2173 þ0.3735 Pb 45.9 5.6 a For Si, Ge, Au, W, Zr, Co, and Mn the constants are calculated from the single crystal constants by using the self-consistent method. sented. The isotropic second-order elastic moduli used in calculations are recorded in Table 3. The values of the apparent radius obtained from Eq. shear and bulk modulus of aluminum are assumed (34), in the case of 20 considered alloy systems, are to be those of an isotropic polycrystalline aggre- listed in Table 4. A detailed analysis of selected gate, i.e., 26 and 72.6 GPa (Table 3). The corre- alloy combinations is as follows. sponding values for copper are 46.8 and 136.4 GPa. The lattice parameter of aluminum at room 7.1. Al–Cu temperature is 4.0496 AA, and of copper 3.6146 A (Table 1; data from Barrett and Massalski, 1966, The equilibrium diagram for this alloy is and Massalski et al., 1990). The Seitz radius of the extremely complex between 30 and 50 at.% Al aluminum atom in its parent crystalline structure is (Hansen and Anderko, 1958). Maximum solid R1 ¼ 1:5826 AA, and of the copper is R2 ¼ 1:4126 AA. solubility of copper in aluminum is about 2.5 at.% The apparent atomic radius of aluminum in cop- Ã at 550 °C. This structure can be retained at room per lattice is R1 ¼ 1:5327 AA. This is obtained by temperature by rapid quenching (supersaturated using the volume size factor x1 ¼ 0:2 from Table aAl solution). Beyond this concentration, up to 2. The apparent radius of the copper in aluminum Ã about 33 at.% (54 wt.%) of copper, the alloy is lattice is R2 ¼ 1:4133 A (Table 4). Solid curve in (aAl þ h) two phase mixture. The h phase is an Fig. 1 shows the effective lattice parameter calcu- intermetallic compound Al2Cu (b.c. tetragonal, C- lated from Eqs. (23) and (28). Eq. (23) was applied 16 type structure), which provides dispersion for the concentration range below x ¼ 0:5, and Eq. strengthening. At the opposite end of the phase (28) above, which gives a jump in the value of the diagram, aluminum is soluble in copper to about effective lattice parameter at the midpoint of the 20 at.% (aCu phase). Additional phases occur in- concentration range. The experimental data for between, such as g2, f2 and d (Al2Cu3) phase. The the two terminal solid solutions is indicated by 62 V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68

Table 4 Ã Ã Atomic radii of the elements in their parent lattices ðR1; R2Þ, and apparent atomic radii (R1 ¼ g1R1, R2 ¼ g2R2) of the solutes for selected alloy systemsa Ã Ã Alloy R1 R2 g1 g2 R1 R2 Al–Ag 1.5826 1.5969 0.9745 0.9914 1.5422 1.5831 Al–Cu 1.5826 1.4126 0.9685 1.0005 1.5327 1.4133 Al–Mg 1.5826 1.7703 1.0565 1.0119 1.6015 1.8703 Al–Mn 1.5826 1.4286 0.9573 0.9494 1.5151 1.3563 Al–Ti 1.5826 1.6153 0.9403 0.9358 1.4881 1.5116 Al–Zn 1.5826 1.5371 0.9514 1.0096 1.5056 1.5519 Cu–Ag 1.4126 1.5969 1.0330 1.0258 1.4592 1.6380 Cu–Au 1.4126 1.5939 1.0191 1.0180 1.4396 1.6225 Cu–Fe 1.4126 1.4114 1.0627 1.0152 1.5011 1.4329 Cu–Ni 1.4126 1.3772 1.0010 0.9990 1.4140 1.3758 Cu–Zn 1.4126 1.5371 0.9333 0.9871 1.3184 1.5173 Fe–Co 1.4114 1.3827 0.9916 1.0281 1.3995 1.4216 Fe–V 1.4114 1.4889 0.9899 0.9821 1.3972 1.4623 Ag–Au 1.5969 1.5939 0.9958 0.9966 1.5901 1.5884 Ag–Mg 1.5969 1.7703 0.9349 0.9347 1.4927 1.6551 Si–Ge 1.6844 1.7548 0.9783 0.9770 1.6478 1.7145 Nb–Ta 1.6219 1.6263 1.0019 0.9964 1.6250 1.6205 Pb–Sn 1.9345 1.8685 1.0669 1.0077 2.0639 1.8829 Ti–Zr 1.6153 1.7713 1.0181 1.0084 1.6446 1.7861 Cr–W 1.4205 1.5585 0.9871 0.9980 1.4021 1.5553 a The radii are in the units of AA.

both ends of the concentration range. A dashed curve corresponds to the elasticity model which does not take into account the apparent size of the solute atom. A dotted line is a variation of the eﬀective lattice parameter according to VegardÕs law. Both latter predictions largely overestimate the eﬀective lattice parameter of Cu-rich alloy.

7.2. Al–Ag

Maximum solid solubility of silver in aluminum is about 20 at.% Ag, at the eutectic temperature (566 °C). Maximum solid solubility of aluminum in silver is about 20 at.% Al, and occurs over wider range of temperature (Hansen and Anderko, Fig. 1. Lattice parameter a (in AA) vs. concentration x for Al– 1958). The lattice spacing of Al based solution Cu alloy system.The solid curve corresponds to continuum elasticity model with incorporated apparent size of the solute remains practically unchanged up to 6 at.% Ag; atom. Dashed curve is the same without incorporation of the between 6 and 14 at.% there is an almost linear apparent size, while dotted line is the effective lattice parameter increase in spacing, while further additions up to according to VegardÕs law. Experimental values for two termi- about 27 at.% Ag cause little change in lattice nal solid solutions are indicated by circles. parameter (see Fig. 62, p. 351 of Pearson, 1958). The lattice parameter of Ag based solution is ba- superimposed circles. These are taken from Pear- sically linear with concentration up to about 10 son (1958, pp. 328 and 331). Good agreement be- at.% Al (Fig. 47, p. 263 of Pearson, 1958). The tween the calculations and experiment is evident at elasticity theory, without modification due to ap- V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68 63 parent radius of the solute atom, predicts the ef- Table 5 fective lattice parameter in close agreement with Original and apparent misfits of atoms for selected alloy systems (in units of AA) VegardÕs law, far overestimating the true lattice Alloy R R RÃ R R RÃ spacing. The incorporation of the apparent size 1 2 1 2 1 2 (solid curve in Fig. 2) results in good agreement Al–Ag 0.0143 0.0547 0.0006 with experimental data at both ends of the con- Al–Cu þ0.1700 þ0.1201 þ0.1693 Al–Mg 0.1878 0.1689 0.2877 centration range. For example, from Table 5 it is Al–Mn þ0.1540 þ0.0865 þ0.2262 seen that in the case of Al based solution the ap- Al–Ti 0.0328 0.1273 þ0.0710 parent misfit between the solute and solvent atom Al–Zn þ0.0455 0.0314 þ0.0307 is practically equal to zero, while the original Cu–Ag 0.1843 0.1377 0.2254 misfit, based on the sizes of two atoms in their Cu–Au 0.1813 0.1543 0.2100 Cu–Fe þ0.0012 þ0.0897 0.0203 parent lattices, is 0.014 AA. Cu–Ni þ0.0354 þ0.0369 þ0.0368 Cu–Zn 0.1245 0.2187 0.1047 7.3. Al–Mg Fe–Co þ0.0287 þ0.0168 0.0102 Fe–V 0.0775 0.0918 0.0509 For this alloy system, Fig. 3 indicates that ac- Ag–Au þ0.0030 0.0038 þ0.0084 Ag–Mg 0.1735 0.2777 0.0583 cording to VegardÕs law the lattice spacing of Al Si–Ge 0.0703 0.1070 0.0300 based alloy decreases with introduction of Mg Nb–Ta 0.0044 0.0013 þ0.0014 solute atoms, while the observed behavior is op- Pb–Sn þ0.0660 þ0.1953 þ0.0516 posite, i.e., the effective lattice parameter increases Ti–Zr 0.1559 0.1267 0.1708 by introduction of (larger) Mg atoms. Superim- Cr–W 0.1380 0.1563 0.1348 posed circles are the measured values, taken from pp. 351 and 728 of Pearson, 1958. Similar remarks apply for Mg-rich solution, where the effective lattice spacing decreases with introduction of Al solute, in contrast to VegardÕs law which predicts an increase of the effective lattice parameter. Therefore, for this alloy system there is a positive deviation from VegardÕs law in Al-rich alloy, and negative deviation in Mg-rich alloy. Maximum

Fig. 3. Lattice parameter a (in AA) vs. concentration x for Al– Mg alloy system.

solid solubility of magnesium in aluminum is about 18 at.% of Mg at 450 °C, and of aluminum in magnesium about 12 at.% of Al at 437 °C.

7.4. Cu–Au

Fig. 2. Lattice parameter a (in AA) vs. concentration x for Al– Copper and gold form a continuous solid Ag alloy system. solution at high temperatures, while at lower 64 V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68

Fig. 5. Lattice parameter a (in AA) vs. concentration x for Au– Fig. 4. Lattice parameter a (in AA) vs. concentration x for Cu– Ag alloy system. Au alloy system. temperatures superlattices are formed about Ã radius of silver atom in the gold matrix is R1 ¼ CuAu, Cu3Au, and possibly CuAu3. The circles in 1:5901 AA, which is smaller than R2. It should be, Fig. 4 correspond to measured values of the ef- however, pointed out that the silver an gold atoms fective lattice parameter in quenched alloys, re- in their parent lattices are nearly equal in size, and ported by Pearson (1958, pp. 411 and 601). that very precise measurements of the eﬀective Positive deviation from VegardÕs law persists lattice parameter are needed to accurately record throughout the concentration range. its variation with the solute concentration.

7.5. Ag–Au 8. Free energy of binary alloys Silver and gold form continuous series of solid solutions without formation of (superlattice) or- The strain energy of a binary solid solution due dered structures at temperatures between the soli- to solute atoms resolved in the solvent matrix gives dus and room temperature, although there is an a significant contribution to the energy or heat of indication for a partial, short-range ordering in the the formation of an alloy, which determines the 50 at.% alloy. Fig. 5 shows the variation of the degree to which two metals tend to alloy and is lattice parameter. Experimental data is from directly related to the structure of the corre- Pearson (1958, pp. 267 and 289). Negative devia- sponding phase diagrams. For a completely ran- tion from VegardÕs law persists throughout the dom (totally disordered) distribution of solutes, concentration range. A notable feature of this al- the configurational entropy is KN1½x ln x þð1 loy system is that the introduction of larger solute xÞ lnð1 xÞ, where K is the Boltzmann constant. It atoms of silver (R1 ¼ 1:5969 AA) in the solvent is assumed that the number xN is large enough for 1 matrix of smaller gold atoms (R2 ¼ 1:5939 AA) ac- SterlingÕs approximation to be valid in the statis- tually decreases the volume and the effective lattice tical derivation of the entropy expression. The free parameter of the resulting alloy. This is a conse- energy of formation of a binary alloy is then ap- quence of the electronic interactions between the proximately outer electron shells of the solute atom of silver F ¼ E þ KN T ½x ln x þð1 xÞ lnð1 xÞ; ð35Þ and surrounding solvent atoms of gold, which 1 shrinks the silver atom and makes it smaller than where T is the absolute temperature, and E is the the gold atoms. Indeed, from Table 4 the apparent strain energy. A general expression for the latter V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68 65 can be deduced from the results obtained by Lu- energy is E=N1 ¼ E0xð1 xÞ. Assuming this to barda and Richmond (1999). For example, in the apply throughout the concentration range, the case of small misfit the strain energy is corresponding free energy curve is concave down- ward, having the minimum at x ¼ 0:5 for all tem- c1 E ¼ N1E0x 1 x : ð36Þ peratures greater than T ¼ E =2K, which is in c g 0 2 agreement with the prediction based on a quasi- The self-energy of a single inserted solute atom is chemical model of regular solutions (Swalin, 2 1972). The solute and solvent components will l1 D 4 3 E0 ¼ 6V1 ; V1 ¼ R1p: ð37Þ therefore have a temperature range of complete c2 R1 3 miscibility if Tg is lower than the lowest tempera- The electro-chemical contribution to internal en- ture on the solidus curve in the alloyÕs phase dia- ergy due to charge transfer, which lowers the en- gram. Friedel (1955) used this analysis to ergy of an alloy (e.g., Varley, 1954; Girifalco and determine the limiting value of the size factor in Alonso, 1980), is not included in Eq. (35). Also binary alloys, and to deduce the Hume-Rothery neglected is a mixing entropy contribution due 15% rule (Hume-Rothery et al., 1969). to the change in atomic vibrations around in- The presented analysis applies to terminal solid serted solute atoms. The gradient of the specific solutions, or solid solutions with an unlimited free energy F =N1 with respect to the concentration solubility of two elements and one alloy phase x is the corresponding thermodynamic force (af- throughout the concentration range. For most finity) solutions, however, there is a range of concentra- 2 tion in which the free energy is lowered by order- l1 c1 D 1 x ing of solute atoms into appropriate aggregates. In X ¼ 6V1 1 2 x KT ln : c2 c2 R1 x this case there is a competition between a decrease ð38Þ of the strain energy due to clustering, and an increase of the free energy due to decrease of the The first term represents an increase of the strain configurational entropy, associated with the lost of energy produced by insertion of an additional complete randomness. The clustering zones appear atom in the existing solution of concentration x. to be spherical for very small misfits, and thin The last term is dominant at small x indicating a plate-like for large misfits (e.g., disk-shaped Gui- strong tendency for mixing of pure solvent in nier–Preston zones within (1 0 0) crystallographic contact with another substance. The location of a planes in cubic crystals). There are alloys that possible spinode (inflection point) of the free en- show complete solid solubility at high tempera- ergy is determined from the condition of the van- ture, whose solute atoms upon cooling organize in ishing rate of X with respect to x, which gives definite long-range order, superlattice structures. 2 For instance, a disordered equiatomic alloy of KT l1c1 D ¼12V1 : ð39Þ copper and gold upon cooling takes a superlattice xð1 xÞ c2 R 2 1 structure with alternate (1 0 0) planes occupied by The critical temperature Tg below which there is a copper and gold atoms. During formation of a miscibility gap in the phase diagram is obtained by superlattice in the solution of aluminum in iron, requiring that a spinode point coincides with a the aluminum atoms segregate to cube centers, minimum of the free energy curve. Thus, in addi- avoiding the cubes that are side by side (avoiding tion to Eq. (39), the affinity vanishes in Eq. (38). to be nearest neighbors). In some alloy phases, an The complete analysis requires the consideration ordered structure is made by a particular (trian- of the free energy for other terminal solution, as gular or rectangular) ordering in close-packed well, particularly when this has a different crys- crystallographic planes. Large difference in atomic talline structure. size increases a tendency for ordering, since this In the case of small misfit with the elastic prop- relaxes the lattice strain of random solution. In erties of the solute and solvent alike, the strain addition, large and small atoms tend to group in 66 V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68 order to achieve more efficient packing (size factor The nonlinear and nonlocal elasticity may prove to phases). Interstitial type phases may form and be a bridge between electronic/atomic models and these often have very complex structures (Laves continuum models. By adding nonlinearity and phases). The alloy may also change its structure to nonlocality in the vicinity of dilatation centers, avoid an undue increase in its electronic energy some of the essential features from the nanoscale (electron compounds). may be captured. For example, there has been a computationally intensive two-scale modeling in which the severely deformed regions are described 9. Discussion by an atomistic scale model, while outer regions are modeled by a local linear elasticity (Tadmor The most important extension of the present et al., 1996; Phillips, 2001). With rapidly improv- work is to provide a theoretical basis for the an- ing situation in computational quantum mechan- alytical determination of the apparent radius of ics, the coefficients of nonlinearity and nonlocality the solute atom in terms of the atomic properties could be computed from first principles, rather of alloying elements in their pure state. This will than being inferred from sensitive measurements. require considerations from metal physics and This will enable us to say more accurately, for physical chemistry. Separate consideration of vari- example, what additions of copper or magnesium ous combinations of alloying elements may be to an aluminum alloy should do to such quantities needed, depending on their position in the periodic as lattice volume, dislocation mobility, grain table and their electro-chemical characteristics. boundary energy, etc. Wide solid solubility of one metal in another is in general favored by similar atomic sizes and equal valences of two elements, albeit these are not suf- Acknowledgements ficient conditions for solid solubility. If atomic sizes and electronegativities are about the same, a This work has been motivated by a series of metal of lower valence usually dissolves better one communications with the late Dr. Owen Rich- of higher valence than vice versa. The importance mond. His insight to mechanics and physics of of various effects should be viewed having in mind solids has influenced my work over a number of that the metallic radius, valence and electroneg- years and is greatly appreciated. A research sup- ativity are not truly independent quantities, since port from the Montenegrin Academy of Sciences they all depend on the electronic configuration of and Arts is also acknowledged. an element and can be related to each other, for example, by an equation of the Gordon–Thomas type (Gschneidner, 1980). References Another extension of the present work is to study the intermediate phases in the equilibrium Axon, H.J., Hume-Rothery, W., 1948. The lattice spacings of phase diagram, since an unlimited solvability of solid solutions of different elements in aluminium. Proc. two elements, and the existence of only one alloy Roy. Soc. A 193, 1–24. phase throughout the concentration range has Barrett, C.S., Massalski, T.B., 1966. Structure of Metals, third been assumed in the present work. There has been ed McGraw-Hill, New York. Blandin, A., 1965. Some applications of quantum mechanical a significant amount of research devoted to cluster calculations to the electronic structure of alloys. In: Mas- formation and interactions through atomistic cal- salski, T.B. (Ed.), Alloying Behavior and Effects in culations and first-principles approach (e.g., Takai Concentrated Solid Solutions. Metall. Soc. Conf. 29, pp. et al., 1982; Gonis et al., 1989). A semi-empirical 50–78. cellular model was also developed in a series of Chessin, H., Arajs, S., Colvin, R.V., Miller, D.S., 1963. Paramagnetism and lattice parameters of iron-reach iron– papers by Miedema and coworkers, which allows a germanium alloys. J. Phys. Chem. Solids 24, 261–273. determination of the heat of formation in inter- Eshelby, J.D., 1954. Distortion of a crystal by point imperfec- metallic compounds (Miedema and Chaatel,^ 1980). tions. J. Appl. Phys. 25, 255–261. V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68 67

Eshelby, J.D., 1956. The continuum theory of lattice deﬀects. Massalski, T.B., Subramanian, P.R., Okamoto, H., Kacprzak, Solid State Phys. 3, 79–144. L., 1990. Binary Alloy Phase Diagrams, second ed., vol. 1– Finkler, D.K.-H., Maurer, A.E., Campbell, S.J., Heck, 3. Am. Soc. Metals Mater., Park, Ohio. T., Gonser, U., 1987. Precision determination of the Massalski, T.B., King, H.W., 1961. Alloy phases of the noble lattice parameters of CuAuFe alloys. Physica 145B, 335– metals. Prog. Mater. Sci. 10, 1–78. 341. Miedema, A.R., Chaatel,^ P.F., 1980. A semi-empirical approach Friedel, J., 1954. Electronic structure of primary solid solutions to the heat of formation problem. In: Bennett, L.H. (Ed.), in metals. Adv. Phys. 3, 446–507. Theory of Alloy Phase Formation. Metall. Soc. AIME, New Friedel, J., 1955. Deviations from VegardÕs law. Phil. Mag. 46, York, pp. 344–389. 514–516. Mott, N.F., 1962. The cohesive forces in metals and alloys. Ganguly, P., Shah, N., Phadke, M., Ramaswamy, V., Mulla, Rep. Progr. Phys. 25, 218–243. I.S., 1993. Deviation from VegardÕs law: Changes in the c- Murnaghan, F.D., 1951. Finite Deformation of an Elastic

axis parameter in La2xSrxCuO4d in relation to the Solid. John-Wiley and Sons, New York. insulator–superconductor-metal transition. Phys. Rev. 47, Nemat-Nasser, S., Hori, M., 1999. Micromechanics: Overall 991–995. Properties of Heterogeneous Materials (second revised ed.). Girifalco, L.A., Alonso, J.A., 1980. Energy-density approach to Elsevier, Amsterdam. the theory of alloy formation. In: Bennett, L.H. (Ed.), Nongkynrih, P., Rao, Y.S.T., Gupta, S.K., Rao, P.V.R., 1988. Theory of Alloy Phase Formation. Metall. Soc. AIME, New Crystal structure of the substituted apatites – deviation from York, pp. 218–261. VegardÕs law. J. Mater. Sci. 23, 3243–3247. Gonis, A., Turchi, P.E.A., Zhang, X.-G., Stocks, G.M., Park, M., Mitchell, T.E., Heuer, A.H., 1975. Subsolidus

Nicholson, D.M., Butler, W.H., 1989. Concentration equilibria in the TiO2–SnO2 system. J. Am. Ceramic Soc. dependent eﬀective cluster interactions in substitutional 58, 43–47. alloys. In: Vitek, V., Srolovitz, D.J. (Eds.), Atomistic Pauling, L., 1956. An electronic structure of metals and alloys. Simulations of Materials. Plenum Press, New York, pp. In: Theory of Alloy Phases. American Society for Metals, 15–28. Cleveland, Ohio, pp. 220–242. Gschneidner, K.A., Vineyard, G.H., 1962. Departures from Pauling, W., 1960. The Nature of the Chemical Bond, third ed VegardÕs law. J. Appl. Phys. 33, 3444–3450. Cornell University Press, Ithaca, New York. Gschneidner, K.A., 1980. L.S. (Larry) DarkenÕs contributions Pearson, W.B., 1958. In: Handbook of Lattice Spacings and to the theory of alloy formation and where we are today. In: Structures of Metals and Alloys, vol. 1. Pergamon Press, Bennett, L.H. (Ed.), Theory of Alloy Phase Formation. London. Metall. Soc. AIME, New York, pp. 1–39. Pearson, W.B., 1972. The Crystal Chemistry and Physics of Hansen, M., Anderko, K., 1958. Constitution of Binary Alloys, Metals and Alloys. John Wiley, New York. second ed McGraw-Hill, New York. Phillips, R.B., 2001. Crystals, Defects and Microstructures: Hume-Rothery, W., Smallman, R.E., Haworth, C.W., 1969. Modeling Across Scales. Cambridge University Press, The Structure of Metals and Alloys. Institute of Metals, Cambridge. London. Pines, B.J., 1940. On solid solutions. J. Phys. 3, 309–319. King, H.W., 1965. Atomic volume and size correlations on solid Sarkisov, E.S., 1960. Variation in the lattice constants of solid solutions. In: Alloying Behavior and Eﬀects in Concentrated solutions with composition and VegardÕs rule. Russian J. Solid Solutions. In: Massalski, T.B. (Ed.), Metall. Soc. Phys. Chem. 34, 202–205. Conf., 29, pp. 85–104. Seeger, A., Mann, E., 1959. Anwendung der nichtlinearen King, H.W., 1966. Quantitative size-factors for metallic solid Elastizitaatstheorie€ auf Fehlstellen in Kristallen. Z. Natur- solutions. J. Mater. Sci. 1, 79–90. forschg. 14a, 154–164. Landolt-Boornstein,€ 1979. Numerical Data and Functional Seitz, F., 1946. Color centers in alkali halide crystals. Rev. Relationships in Science and Technology. In: Hellwege, Mod. Phys. 18, 384–408. K.-H., Hellwege, A.M. (Eds.), Group III, vol. 11. Springer- Solin, S.A., Lee, S., Miyazaki, H., Mahanti, S.D., 1989. C-axis Verlag, Berlin (Also, vol. 2, 1969). lattice expansion in ternary intercalated layered solids. Synt. Lawson, A.W., 1947. On simple binary solid solutions. J. Chem Metals 34, 243–248. Phys. 15, 831–842. Steinwehr, H.E.V., 1967. Ursachen der Abweichungen von der Lubarda, V.A., 1997. New estimates of the third-order elastic Vegardschen Regel. Z. Kristallogr. 125, 360–376. constants for isotropic aggregates of cubic crystals. J. Mech. Swalin, R.A., 1972. Thermodynamics of Solids. John Wiley, Phys. Solids 45, 471–490. New York. Lubarda, V.A., 1999. Apparent elastic constants of cubic Tadmor, E.B., Ortiz, M., Phillips, R.B., 1996. Quasicontin- crystals and their pressure derivatives. Int. J. Nonlinear uum analysis of defects in solids. Phil. Mag. A 73, 1529– Mech. 34, 5–11. 1563. Lubarda, V.A., Richmond, O., 1999. Second-order elastic Takai, O., Doyama, M., Hisamatsu, Y., 1982. Pseudopotential analysis of dilute distribution of spherical inclusions. Mech. formulation of point interactions in metals. In: Takamura, Mater. 31, 1–8. J.-I., Doyama, M., Kiritani, M. (Eds.), Point Defects and 68 V.A. Lubarda / Mechanics of Materials 35 (2003) 53–68

Defect Interactions in Metals. University of Tokyo Press Vegard, L., 1921. Die Konstitution der Mischkristallen und die and North-Holland, Amsterdam, pp. 109–112. Raumfuullung€ der Atome. Z. fur Physik 5, 17–26. Talanov, V.M., 1983. On the conditions pertaining to the Weiss, R.J., 1990. Physics of Materials. Hemisphere Publ, New applicability of VegardÕs rule in spinel solid solution. J. York. Solid State Chem. 48, 86–92. Yashima, M., Ishizawa, N., Yoshimura, M., 1992. Application Toupin, R.A., Bernstein, B., 1961. Sound waves in deformed of an ion-packing model based on defect clusters to zirconia perfectly elastic materials––acoustoelastic eﬀect. J. Acoust. solid solutions: II. Applicability of VegardÕs law. J. Am. Soc. Am. 33, 216–225. Ceram. Soc. 75, 1550–1557. Urusov, V.S., 1992. A geometric model of deviations from Zen, E-an, 1956. Validity of ‘‘VegardÕs law’’. Am. Mineralogist Vegard rule. J. Solid State Chem. 98, 223–236. 41, 523–524. Varley, J.H.O., 1954. The calculation of heats of formation of Zener, C., 1942. Theory of lattice expansion introduced by cold- binary alloys. Phil. Mag. 45, 887–916. work. Trans. Am. Inst. Min. (Metall.) Engrs. 147, 361–368.