Revista Latinoamericana de Metalurgia y Materiales, Vol 5, N° 1, 1985

ARTICULO INVITADO / INVITED PAPER

The Factors Influencing Solid Solubili ty in Metallic Alloys

Julio A. Alonso

Departamento de Física Teórica, Universidad de Valladolid, Valladolid, Spain

Solid solubility is a phenomenon of metallurgical interest. The extent of equilibrium solid solubility of a metal A into a host metal B..depends on several factors which control the free energy of formation of the solid solution and competing intermediate phases. One ofthose factors is the entropy of formation. Then, there are several elemental properties of the components which control the enthalpy offormation: atomic size, electronegativity, generalized 'valence, electro n density at the boundary of bulk atomic cells, and structure.Thosefactors are discussed separately and then integrated into a semiempirical theory ofsolid solubility. The theory is applied to seve-ral types of alloys and the comparison with experiment is satisfactory.

1. INTRODUCTION

The solid part of the phase diagram of a binary sys-tem formed by two metal s A and B normally contains(although not necessarily) one or several intermediatephases, in addition to terminal solid solutions of A in Band oí B in A. In a solid solution, the solute atoms entersubstitutionally into the host lattice preserving the crys-talline structure of the host, Interstitial solid solutionsalso exist but those are less common than substitutionalsolutíons. On the other hand, an intermediate phase nor-maJly has a narrow composition range and a crystalstructure different from those of the component metals.Solid solutions and intermediate phases are competitive.Progress in the understanding of the factors governingthe occurrence of the different alloy phases has been grow-ing in recent years, but a complete theoretical descrip-tion of phase diagrams, and in particular of solidsolutions is a very difficult task. In fact, the general opi-nion of workers in this field is that "the prediction ofter-minal solid solubilities is a formidable problem and itremains a rnajor unsolved question in metallurgy" [1].For this reason, semiempirical treatments, focussing ona few fundamental factors, have been traditionalIyemployed, and in order to do this one must know the fac-tors influencing solid solubility.

In this review I consider the different íactors whichinfluence solid solubility. It will become clear that a sin-gle factors is not enough and that progress can only beachieved when the different factors are considered toge-ther. The conclusion is, as I hope to make clear along thepaper, that substantial solid solubility only occurs whenthe components of the alloy are chemically similar. Thusthe objective oí the paper is to explain how this concept oíchemical similarity can be made quantitative,

2. THERMODYNAMIC DESCRIPTION

The extent oí equilibrium solid solubility of a solutemetal A in a solvent metal B at a temperature T can be

calculated fromthe free energy offormation LlGs (xA• xB; TIof the substitutional solid solution as a function ofcomposítíon

LlGs (XAf XB; T) = Gs (xA, XB; T)-- XA GA(T) - XB GB(T), (1)

where GA and GB are the free energies of the pure metalsand Gs (xA, xB; T) is the free energy of the solid solutionwith atornic concentrations XA and XB = 1 - XA. A neces-sary condition for the thermodynamic of the solid solu-tion is

(2)

Nevertheless. (2) is not a sufficient condition, because ofthe possible existence of competing phases, normally anordered intermetallic compound of fixed compositionA"BJJ with free energy of formation AGc (a, f3; T). Con-sequently, the extent of solid solubility at a temperatureT is calculated by computing LlGs (xA, XB; T) as a func-tionof concentration and comparing it with LlGc (a, (3; T)using the tangent method [2]. If intérrnediate phases donot exist, then the competing phase is a solid solution ofB in A (íf the crystal structures of A and B are dífferent).The free energy of formation can be separated as

AG= AH - TAS, (3)

where AH is the enthalpy of formation and AS is theentropy of formation.

The entropy of formation of a substitutional solidsolution is positive [3]. Its calculation is a difficult mat-ter. The simplest approximation is to consider idealmixing (when the enthalpy of formation is nearly zero).Therí the entropy of random mixing can be used towrite [4]

(4)

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LatinAmerican Journal of Metallurgy and Materials, Vol 5, N° 1, 1985

where K is the Boltzmann constant. More sophisticatedapproximations can also be used [5]for departures fromideal behaviour. In contrast, the entropy of formationaSe of ordered intermetallic compounds is normally zeroor negative [3, 6-9] (magnetic effects sometimes induceexceptions to this rule [7]. Therefore, the entropic termfavors the existence of solid solutions.

Most efforts to understand solid solubility havebeen devoted to investigate the factors which control theenthalpy of formation aR..

3. SIZE FACTOR (FIRST HUME-ROTHERYFACTOR)

In the early 1930's Hume-Rothery proposed severalfactors controllíng solid solubility, which form the basisof our present understanding [lO, 11].The first one is thefamous size rule, stating that solid solutions are not to beexpected ifthe atomic sizes of solvent and solute differ bymore than 15%,and that solid solutions may form if thesize difference is less than 15%, provided other factorsare favorable. To apply this rule, the atomic size of ametallic element can be measured by the radius r of asphere with a volume V equal to the volume per atom ofthe pure metal. Waber and coworkers [12] have appliedthe size rule to 1423 terminal solid solutions. Of the sys-tems predicted to be insoluble, 90%were found to exhíbitJimited solid solubility (the distinction between a limitedand an extensive terminal solid solution alloy was takenat5 atomic %).However, ofthe systems predicted to haveextended solid solutions only 50%were so found. In otherwords, a favorable size factor is a necessary but not a suf-ficient condition for the formation ofextensive solid solu-tions. Clearly other factors also playa roleo Since thesolid solution preserves the crystal structure of theparent metal it is understandable that atoms with a sizevery different from the size of the host atoms introducelarge lattice distortions, with a corresponding high elas-tic energy cost in aHs, which oposes the formation of asubstitutional solid solution.

In 1954 Friedel [13] used the eleastic continuumtheory to derive a quantitative expression for the sizefactor. He calculated the elastic energy needed to intro-duce a sphere of radius rA (representing the solute atom)into a spherical hole of radius rB(with rB= (3V¡J47T)1/3,the atomic radius of the host). Then assuming (a) com-plete disorder and (b) that the enthalpy offormation of asolid solution is dorninated by the elastic energy effect,Friedel found that there is a large solid solubility only ifthe following condition is fulfilled

I~rl < (KTBXB)1/2r 7TJ\

In this expression TBand XBare the melting tempe-rature and the compressibility of the host respectively,.ó.r= rA- rB'and f is theaverage atornic radius, Ther. h. sof (5) defines a specific size factor for each host metal, incontrast with the Hume-Rothery rule. A reasonable ave-rage value oí TBXBr~3= 1014C.G.S. leads to the condi-

tion Idr I / r < 0.15, in· agreement with the Hume-Rothery size rule. Independently, Eshelby [14] provedthat the contribution to the heat of solution from size mis-match can be written

where JlB is the shear modulus of the host and XAis thecompressibility of the solute metal. The heat of solutionAh, (A in B) is related to the heat of formation.ó.Hs (x¿ xB)of a concentrated solid solution by

From (6) Eshelby then derived another expressionfor the size factor (SF)

(8)

where aB is the Poisson ratio of the host and R is the gas'constant, This size factor was also used to justify Hume-Rothery's size rule. Values of the individual size factors(5) and (8) have benn tabulated by Gschneidner [15].Those values areclose to 15%for simple metals and be-tween 10% and 15%for transition metals.

The size factor has been further discussed by Egamiand Waseda [16] using elasticity theory [14]. These au-thors point out that a substitutional impurity introducesa volume strain on the host which increases Iinearly withsolute concentration. If the str.ain exceeds a certain pointthen the solid solution becomes topoligically unstableand the system transforms to a different crystallinestructure (or upon rapid quenching freezes into a glass).The reason for this topoligical transformation is thatlocal pressure and local coordination number are di-rectly related [17]. Then, Egami and Waseda have shownthat, at a critical composition, the local elastic energydue to the local pressure can be reduced by a change inthe local coordination number. This critical concentra-tion has been found to correlate with the minimum soluteconcentration necessary to obtain a stable amorphousphase by rapid quenching from the melt [16].

4. ELECTROCHEMICAL FACTOR (SECONDHUME-ROTHERY FACTOR)

(5)The second Hume-Rothery rule states that the elec-

trochemical nature of the two elements must be similar ifsolid solution formation is to be expected. If, however,their electrochemical natures are different, compoundformation is likely to occur. A measnre of the electroche-mical nature of the elements is provided by the "electro-negativity" 0, a concept introduced by Pauling [18].Thisauthor defined the electronegafu - as he power of anatom in a molecule or compo d aí:tl"act electrons toitself. In the modern view, electro ega - -ty is minus the

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Revista Latinoamericana de Metalurgia y Materiales. Vol 6. N° 1. 1986

denivative dE(N) / dN ofthe energy ofthe atom wíth res-pect to the total number of electrons N [191..Differentempirical electronegativity scales exist, derived fromthermochemical data, like those due to Pauling [18] orMiedema [20],orfrom free atom data. For instance Mulli-ken proposed [21]

IP+EAcP = 2 '

where IP is the first ionization potential and EA is theelectro n affinity of the atom. In fact, Mulliken's formulais a finite difference approximation to - dE(N) / dN.The physical significance of the electrochemical factorfor alloy formation is the following one: a differencedcp = cPA - CPBin the electronegativity of the two atomsinduces a transfer of electronic charge from the less elec-tronegative to the 'more electronegative atom until theelectronegativities are equalized in the alloy. Thischarge transfer (Ch T) contributes to the enthalpy ofalloy formation with a term

(10)

proportional to - (dCP? [18,20]. This term then favorsthe formation of the alloy. The proportionality factor is a.function ofthe area of contact between the atomic cells ofsolute and solvent [20].The total area of contact betweendisimilar atomic cells is larger in ordered compoundsthan in random substitutional solutions. Then the elec-trochemical factor favors the formation of ordered com-pounds. Only if dcp is small the entropy of randommixing is able to overcome the tendency to form orderedcompounds and to stabilize the solid solution. This is thereason for the need of similar electronegativities for solidsolution formation.

5. DARKEN-GURRY MAPS

In the early 1950's Darken and Gurry (DG) took animportant step in the predietion of solid solubilit~ whenthey were able to use simultaneously both the size andthe electronegativity factors [22]. These authors eons-tructed a map where the two coordinates are the electro-negativity and ·the atomic radius. In this map all theelements whieh líe close to the point representing thehost exhibit high solid solubility in that host. They madethis eorrelation more quantitative by constructing anellipse centered on the host and with an axis of± 15%ofthe solvent's size and another axis of ± 0.4 of a Paulingelectronegativity unit of the solvent's electronegativityvalue. The ellipse then encloses most ofthe solutes whichhave extensive solubility in the host and exeludes thosehaving limited solid solubility. A DG plot for Pb alloys isshown in Figure 1. Waber et al [121made an extensiveanalysis of the DG method, eoncluding that this methodeorrectly prediets limited solid solubility in 85% of thecases and extended solubility in 62%of the cases. Thusthe DG method yields an improvement in predictingsolidsolutions, with a slight deerease in the reliability of pre-

(9)

2.6

2.2

>-~>

1.6'....~us'Zo sew...JW

• SOLUBILI TY ~ 0.5 %

.0.01'1..$ SOLUBILlTY s 0.5%a SOLUBILITY i 0.01%

Ag O NOT MEASURED

PbD HOST

Hg BiTaNb

LatinAmerican Journal of Metallurgy and Materials, Vol 5, N° 1, 1985

>-f-J•....•>

RbO Oes

esubstitutionalo non su bs.

f--

Revista Latinoamericana de Metalurgia y Materiales. Vol 5. N° 1. 1985

7. ELECTRON DENSITY AT THE ATOMICCELL BOUNDARY AND MIEDEMA MODELOF ALLOY FORMATION

In Sections 3 to 6 we have presented the traditionalHume-Rothery factors, that have served as a basis to dis-cuss solid solubility. Other factors which have beenintroduced recently are now discussed.

In 1979 Chelikowsky [23] presented solubility plotsin which the two coordinates are the electronegativity (j)and the electron density n at the atomic cell boundary.Chelikowsky applied the new coordinates to the case ofdivalent hosts and he demonstrated that those coordina-tes give more accurate predictions than the Darken-Gurry coordinates. He was able to draw an ellipsecontaining most metal s which are soluble in a given host.The position of the host metallies inside the sllipse, al-though its precise location varies from host to host, Ásolubilityplot for Fe-alloys i~~iven in Fi81lre4.-(j) and n

el> (V)

6 ,-------,--------,--------,--,e SOLUBILlTY) 15% I Os I• 15%> SOLU,BILlTY > 5% ?t.;í~r u

o 5%) SOLUBILlTY) 1°;';° PdRh e° 1%>SOLUBILlTY eTe eb. HOST Ni¡e )

A¡jl Co·5 - b.As. B Fe .W

G~ 5;·° ee'M'o

/•• Cu Cr /

5bo °Ag M~ /Si HgAl Be eV

P b8s~4:.0.-0~TOoCd o UDNbIn Puo

.TiMg °Hf

° °ZrYooOTh

Loo Se

LiooNoCoooSr

oK °Bo

4'- -

3f-

o2 "-Rb

I I1

0.5 1.0 1.5 2.0n 1/3

l"ig'-.4. Chelikowsky's plot for so lid solubility in Iron alloys [35J. Thetwo coordinates are the electronegativity 1> and the electrondensity parametern ofMiedema's theory. An ellipse has beendrawn separating soluble and insoluble elements, The solubi-lit Y refers to the maximum solid solubility (at a temperaturenot higher than the melting temperature of the pure solute).See ref [35] for details. cp is given in Volts and nl/:l in (densityuníts)':".

are fundamental ingredients in a serniernpirical model ofheats of alloy formation developed by Miedema andcoworkers [20,36]. In this model, the heat of formation of

an intermetallic cornpound AoBfJ can be written(a = XM f3 = xB)

where

(12)

x~ and x~ are atomic-cell-surface area concentrations,

defined

(13)

-

Miedema's model assumes that the heat of formation isproportional to the area of contact between di similar ato-mic cells in the alloy. This is the reason for the prefactorXA x~ [1 + 8(xÁ X~)2]in (11). For a random distribution ofatoms, as in a liquid alloy, the factor is simply XA x~. Thisis corrected by the factor [1 + 8(xÁ X~)2] which has beenintroduced by Miedema to account for the fact that in anordered solid compound the total area of contact be-tween disimilar atomic cells is larger than for a randomalloy, P and Q in (12) are positive constants. The firsttermin (12), proportional to - P(~(j)2, is a chargetraris-fer term which accounts for the electrochernical factor ofSection 4. On the other hand, the second contribution,proportional to (~nli:l)2, takes into account the increasein electronic energy needed to smooth out the differencein electron density at the interfaces between A and B ato-mic cells. The recommended values of n and (j), as well asa fuI! discussion of the model, are given in Miedema'spapers [20,37]. A third electronic contribution to ~ho hasbeen neglected in writing (12). We have left it out for themoment since this contribution is not needed .to discussChelikowsky's plots, That contribution, only presentfor alloys of a transition metal and a polivalent non-transition metal, will be considered in Section 8.

The success of Chelikowsky's plots can now beunderstood from eqns (11) and (12). The two coordinatesin the plots control the heat of formation of ordered com-pounds, which are competitors of solid solutions. IfP(~(j)2 » Q(~nI!3)2, then ~Hc « O and urdered com-pounds are likely to form, On the other hand, ifP(~(j)l « Q(~nl/3)2, then ~Hc » ()/and alloys do notform at al!. Finally, if P(~(j)}2 :::,,:Q(~nl/3)2 then .ó.Hc:::":Oand a disordered solid solution can be the stable phasedue to entropy eontributíons.

Figure4 also shows that a rough correlatíon existsbetween (j) and n1/:1. The points are scattered about astraight line with slope (Q/P)I/2, which is the value of~(j) / ~n1ll dividing compounds with positive heats offormation from those with negative ones. If the values of

-

-

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LatinAmerican Journai of Metallurgy and Materials, Vol 5, N° 1, 1985

rP lie exactly on the line (Q / p)l/2 nl/3, all energies of corn-pound formation would be exactly zero. Therefore. thedeviation of the parameter cp from this line is the signifí-

. cant feature [38] and it means that cp and nlf3 are notequivalent coordinates.

Plots using cp and n as coordinates have also beenemployed in other problems. In fact, the first applicationwas to metastable alloys produced by low dose implanta-tion in Berillium [39, 40]. A perfect separation was foundbetween substitutional and interstitial implants in Be.Other recent applications to implantation and to amorp-hous alloys produced by rapid quenching have al so be enperformed [41-45].

8. TERMOCHEMICAL PLOTS

A point to notice is that the Darken-Gurry plots, alt-hough less successful than-Chelikosky's plots, provide agood first approximation for the prediction of solid solu-bilities. Both maps have a coordinate in common, theelectronegativity 1>. The fact that Pauling electronegati-vity [18] is normally used in the DG plots, whereas Mie-dema electronegativity was used by Chelikowsky, is ofminor importance. There is a good correlation betweenboth scales [36]. The second coordinate is different: ato-mic size V in one case and cell boundary electron densityn in the other. Noticing that V and n are not equivalentcoordinates [35, 36, 46] Alonso and Simozar 135] propo-sed in 1980 a new scheme that takes into account thethree coardinates 1>, n and V. To preserve the simplicityof two-dimensional map these authors combined 1> and nin a new coordinate: the chemical part of the heat of solu-tion as given in Miedema's theory. That chemical part,which excludes size mis match effects and peculiar struc-tural effects (tú be considered in Section 10) is given by~ho of eqn (12). In writting (12) a term was neglected.Then we now write the complete expression for the che-mical part of Ah,

(14)

where the new term - R is different form zero only foralloys formed by a transition metal (TM) and a polivalentnon-transition metal (PM). Miedema interprets this termas due to hybridization between the d-electrons of thetransition metal and the p-electrons of the polivalentneighbours. In first order, this (negative) contributiondepends not much on which d-metal is alloyed with whichp-metal partner. The values of Rhave been given by Mie-dema for the different TM-PM combination. This meansthat the thermochemical coordinates (V and Llh~)containmuch more than the combined effect ofV, n and 1>. Theyalso contain hybridization effects.

Aetually, the thermochemical coordinates weresimultaneously and independently praposed by Giessenand Whang [47] for the prediction of amorphous alloyformation. The thermochemical coordinates have been

5 /'Vl I¡e Boe \:J

:§-,

eCo1,'\

Ul-,

:J 15e 15 > sol. > 101 > sol.+ Host

o 80. 150.I1h; in Cu (kj/moI8solut8l

Fig. 6. Solubility map Ior Copper alloys using thermochemical COOl'-dinates [49[. Data correspond to room temperature solubili-tieso Contours of increasing solubility have been drawn.

The success ofthe thermochemical plots [49] and theform of the boundaries separating soluble and insolubleimpurities is easily understood, because the V axis givesa measure of the size mismatch contribution to the heatof solution and the Ll~ axis accounts for the chemicalpart of ~hs' These are the two main contributions to Llh,.Only if those two contributions are not too different fromzero a solid solution can exis since:

a) if ~~» O, thenb) if~« 0.0

o do not form at all,compounds are expected,

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Revista Latinoamericana de Metalurgia y Materiales, Vol 5, N° 1, 1985

e) if Ll~ > O (positive, but small), then the deci-ding factor is Llh:!ze. Only if Llh:!ze is small(Llh:ize~ O) can a substitutional solution existdue to the stabilizing effect of the entropy ofrandom mixing,

d) if Ll~ < O (not too negative), the deciding fac-tor is again Llh:!ze. If this term is small, theentropy of random mixing again favors a sub s-titutional solid solution. But if Llh!ize» O,orde-red compounds are expected (except if Llb; isvery nearly zero). This is because in an orderedcompound the crystal structure and the concen-tration of the components are optirnized in sucha way that elastic size mismatch effects arevitually eliminated.

u

- ---~

.• sol. > 15.-15>501.>1

I/! 01 > sol.'2 4 + Host~ ~----------~a

oCaNo°- o/

K

Zr Hf

OAg

Eo~o

-80 O 80 160

lthc in F," (kj / mal," 50Iut,")sFig. 7. Solubility map for Iron alloys using thermochemical coordi-

nates [491. Data eorrespond to room temperature solubilities.Contours of inereasing solubility have been drawn.

We close this Seetion with sorne cornrnents aboutche relative valence effect (see Section 6). First we recal!that Gscheneidner discovered 1281 that low solubilityoccurs when one eomponent of the alloy is a d-metal andthe other is an sp-metal. We observe that this effect isreflected in the ter-m - R of eqn (14). This contribution isnegntive and leads to substantially negative values ofAh". As already discussed, this favors the existenee ofordered eompounds.

9. SIMPLE SEMIEMPIRICAL THEORYOF SOLID SOLUBILITY

The ingredients presented so far allow us to esta-blish a simple theory of solid solubility [57]. The heat ofsolution Ah, (A in B) is obtained by adding the chemicalpart of eqn (14) to the size rnismateh part of eqn (6). Then,for a concentrated solid solution the enthalpy of forma-tion can be written, .

The chemical part is calculated from Miedema's theory(see Sections 7 and 8)

f'l.H~ (x\, XI;) = X,\ x~ Llh~ (A in. B). (16)

On the other hand, if the solute concentration is srnall,Eshelby [14] has shown that LlH:¡'''(xAo xu) is given by

(17)

(y - l)y ] ~h"'''(A in B)(y* - l)y*' ,

where

y=l+~.3XJ:

(18)

v" = 1+41-'-1:3 K\

(19)

Adding to (15) the entropy part of eqn (4) an expressionfor the free energy of formation of dilute solid solu-tions results,

This theory has been applied to a very simple classof alloys, those formed by metals with very low mutualsolid solubility and without intermediate compounds.AlIoys ofthis kind are: Se W,Se Ta, Se Nb, Sc Cr, Se V, VTh. Vi Th. Ti Y and Ti La 157J. In these cases, both f'l.h:""and Ah; are large and positive. Figure 8 is a plot of thecalculated free energy of forrnation of dilute solid solu-tions ofNb in Se and Se in Nb, at three different tempera-tures: 1;100K. 1400 K and 1500 K. At these temperatures,t-.G is negutive at very low solute concentration. This isdue to the entropy of mixing sinee AH, (XI' XI) is posi-tive, Then. after reaching a minimum. f'l.G, raises to posi-tive values at an still low solute concentration. Figure 8shows that the solubility of Sc in Nh 01' Nb in Sc is verylow. The extcnt of solubility evidently increases with T.The c.ilculated solubilities of these and other alloys aregiven in Table 1. where the predictions are eompared toexperiment. The expeJ'imental values are not very pre-cise in so me cases ancl calculated solubilities are verysmall, in :tgl'eement with measured limits.

I O f----f------f------------+---____\_______\_::\-113001400§t::)-

~ -001'h

G~~~

':;,g...•-002

O~~--~~~~~~·~--~~--~~~

O997 Oggg 1.0CNb(af%}~

Fig. 8. Calculated free energy of formation ~G, of dilute solid solu-tions of Nb in Se and of Se in Nb at several temperatures [57].~G, has been calculated as explained in Section 9.

1500

Due to the limited validity of eqn (17) the theory hasbeen applied to dilute alloys. Recently, the size mismatchcontribution has been extended to the eoneentratedregime 162J. Here we simply outline the method. The for-

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LatinAmerican Journai 01Metallurgy and Mtüerials, Vol 5, N' 1, 1985

mation of a eoneentrated alloy is divided into severa]steps. In the first step a dilute alloy is formed and eqn (17)is used to calculated the elastic size mismatch contribu-tion. At this point, the dilute alloy is considered as aneffective pure metal with atoms having the average ato-mic volume and average elastic constants. Then, a fewmore atoms are added to this effective host. The elasticenergy cost in this second step is again computed fromeqn (17), noticing that the host is now the effective host.The process continues in this way and any concentrationcan be reached after a sufficient number of steps. Thetheory was applied to the alloys Se Y, Se Zr, Se Rf, Ti Rf,VCr, VNb, VMo andVWatT = 1000 K, and itwasfound[62] that those eight alloys have a negative near-zerovalue of ~G, over the whole concentration range at thatternperature. This faet is eonsistent with the experimen-tal phase diagrarns ofthose alloys [6]. which show conti-nuous solid solubility at that temperature,

10. CRYSTAL STRUCTURE FACTOR

The erystal structure of the two pure metal s alsoinfluenees the extent of solid solubility. A substitutionalsolid solution preserves the crystal structure ofthe host.Then, it is clear that complete solid solubility (fromx" = Oto x,\ = 1) can only oeeur ifthe erystal structure ofthe two pure components is the same. Gsehneidner [28]has analyzed the influenee of crystal strueture in anempirical way. His diseussion is based on the distinctionbetween a) eommon metallie structures (fcc, bcc, hcp,double hcp, Sm-type and fct-In type structures) and b)other structures. Gschneidner observed that solid solu-bility is low when the crystal structures of the eompo-nents are different. In other case, structural effeets donot play an important role and the other factors are thedeciding ones. For the purposes of applying this rule al!the eommon metal!ic structures are considered to bethe same.

Even if structural effects are smal! when the twoeomponents have one of the common metallic structures,those effects are not negligible. This has been demons-trated reeently by Miedema and Niessen [63.64]. Theseauthors have studied the structural contribution to theenthalpy of solution of alloys of two transition metals.This contribution reflects the fact that there exists a pre-ferenee for the transition metals to crystallize in one ofthe main crystallographic struetures bcc, fcc or hcp,dependingon thenumberofvalenee (s + d) eleetrons peratorn. It is then reasonable to assume thatalso structuredependent energies in transition metal solid solutionswill vary systematically with the average number ofvalenee electrons per atom (z) if the two metals form aeommon band of d-type electronie states. Consequently,disolving metal A in the host B changes the energy thatstabilizes the crystal structure of the matrix metal, Thisenergy ehange will be proportional to the difference(z" - zB)' For a more quantitative description, the varia-tion of stability of the particular crystal struetures withrespect to z is needed. Combining experimental [65] andtheoretical data [66-69] for the energy differencesE¡,.e- Ereeand E¡,ep- Ercebetween bcc, fce and hcp structu-

res, Niessen and Miedema derived the variation of theabsolute stability Eu(z) (O' = bcc, fcc, hep) of eaeh of themain erystal structures. Their results are shown inFigure 9, where the referenee state is the average of theelose-paeked struetures and the looser paeked bee lat-tiee, that is

(~)20

t---bcc---- fcc

hcp'..~\ .•..

:J •:, \~,¡ \./¡ ,':

J ".I \! \O~----~~----~r-----------~

-10

-20

Fig. 9. Absolute stability ofthe bcc, fcc and hcp crystal structures of4d ánd 5d transition metal s as a function of the number of(s + d) electrons. The reference state in defined by eqn (20).From reference [641.

l!;hCJ\'(~) + Efcc(z) + E () = O2 I~c Z • (20)

Solid metals with z = 5,6,8 and 9 have a struetural stabi-lit y which is large compared to that for solid metal s withz = 3,4. 7 01' 10. This suggests that the entropy of fusionof the former will be higher than the average value formetals, Sueh an increase has indeed been observed forNb, Ta, Mo, W and Ru [64, 70].

Using these structural enthalpy differences, theehange in structural stability of the matrix upon solutioncan be predicted. The structural eontribution to the ent-halpy of solution of A in B can be written

i1h:"(A(O'A) in B(O'B» = ((z - z ) dEuA(z) +A Il dz

This equation contains two contributions, One of them(EaB(B) - EaA(A) comes from thedifference in structuralenthalpy of the two pure metals. The second term(z, - Zll) dEuB(z) / dz is the change of struetural stabilityofthe O'B structure due to thechange in averagevalence.For example, for hcp-Hf (z = ) and hcp-Re (z = 7). the

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Revista Latinoamericana de Metalurgia y Materiales, Vol 5, N° 1, 1985

matrix is destabilized upon dissolving either Hf in Re orRe in Hf (see Fig. 9). Niessen and Miedema have calcula-ted the structural contribution L1h:tr to the heat of solu-tion for all combinations of two transition metal s [64,71]showing that L1h:trrepresents a substantialcontributionto .ó.h,; in some alloys.

The theory presented in Section 9 can now be impro-ved by adding the structural contribution L1H~t. to theheat of formation .ó.Hs of a solid solution

(22)

For a very dilute alloy (x, « 1) , L1H:tr may be written

Evidently this formula is not valid in the concentratedregime. Then we apply a similar idea to that used in Sec-tion 9. The concentrated alloy is formed in several steps.In the first step a small amount of solute is added to thehost and eqn (23) can be used to compute the structuralcontribution. At this point the host is considered as aneffective pure host with atoms having the concentrationaveraged valence. Then a few more atoms are added tothis effective host and the corresponding structural con-tribution is cornputed again from (22). The process is con-tinued and any concentration can be reached after asufficient number of steps,

The applications ofthis theory to différentclasses ofalloys will be published elsewhere [72]. Here we onlyshow a sample of results. Figure 10 is a plot of the free

ZrSc

-O

~ 1. f---,~~

'\ 1V) I

LatinAmerican Joumal oi Metallurgy and Materials, Vol. 5, N' 1, 1985

oPd V

-10.oE

Revista Latinoamericana de Metalurgia y Materiales. Vol. 5. N° 1. 1985

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