7 I
he thermodynamic properties of metallic systems T h a ve attracted the attention of metallurgists since Willard Gibbs first introduced the phase rule for equilibrium phase diagrams. Even now, the mmt extensive collection of thermodynamic information on alloys is embodied in the literature on phaseequilibria ( 7 1 , l O ) . Although a phase diagram reflects the dependence of the total free energy of the system on composition and temperature, it does not yield quantitative information to evaluate the thenncdynamic quantities. I n addition to providing a quantitative basis for equilibrium phase diagram, thermodynamic information is r e q u i d to specify the equilibrium conditions between solid or liquid alloys and a gaseous phase for reactions employed in the field of process metallurgy. More recently, the correlation of thermodynamic properties with other phydcal propedes, such as electrical, magnetic, and crystalchemical properties, has provided some insight into the relationship between electronic 28
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
1
THE THERMODYNAMIC PROPERTIES OF TRANSITION-METAL ALLOYS
APPLIED IWERMODYNAMICS SYMPOSIUM
JOSEPH B. DARBY, JR.
The impact of thermodynamic data on the theory of alloys has allowed the thermochemistry of metals and alloys to emerge as a field of its own.
Insight into the relationship between electronic structure and the thermodynamic properties is now possible.
structure and the thermodynamic properties of alloys (7, 34, 39). However, adequate theoretical models are not available to account for the energy changes that occur when an alloy is formed. The thermochemistry of metals and alloys has emerged as a field of its own even though the influence of the physical chemist is still apparent. This distinction arises from pronounced differences that set off the substances of metallurgical interest from those generally of interest to the physical chemist. The chemist can adequately describe the state of a sample, such as a gas or a liquid, in terms of its temperature, pressure, and composition. The three variables are insufficient to describe the complex state of a solid alloy, however, and, for the lack of other appropriate variables, a detailed history of the specimen is usually given for completeness. The need to characterize alloy samples in order to define the equilibrium state places additional demands on the experimentalist concerned with metallic materials. Alloy phases commonly are stable over a range of chemical composition, and when an alloy is prepared, segregation is likely to occur. Thus, homogenization anneals are required a t elevated temperatures to achieve sufficient atomic mobility and thus to eliminate the large chemical inhomogeneities. A metallographic examination by optical and x-ray techniques is required to assure that an equilibrium condition has been attained. An alloy phase may be in a disordered state-Le., the atoms of the different chemical species are randomly distributed on the crystalline lattice, or in an ordered state where each species occupies specific sites of the crystal lattice. Also, an alloy may become disordered at elevated temperatures as a result of the increase in vibrational energy. The order-disorder reaction, which may occur isothermally as the result of a diffusionless transformation or over a range of temperature, generally produces a change in the energy of the alloy.
A review will be made of the low- and high-temperature thermodynamic properties of binary transitionmetal alloys to indicate some of the experimental techniques employed currently and the progress in the interpretation of the results. A complete compilation of experimental results will not be attempted in view of the compilations (22, 29) that are becoming available on a routine basis. However, selected sets of data will be given to indicate the impact of thermodynamic results on the theory of alloys. low-Temperature Calorimetry
An important contribution of thermodynamic measurements to the theory of alloys has been made by lowtemperature specific-heat studies. The transition metals and alloys are of particular interest since the unfilled d bands, with a large density of states, make a substantial contribution to the specific heat at low temperatures. Furthermore, it will be shown that the low-temperature specific-heat coefficient, y, is directly proportional to the density of states at the maximum occupied energy, or Fermi, level. Thus, the structure of the d band of transition-metal alloys can be experimentally determined and compared with the theoretical models (56). The low-temperature specific heats of normal metals and alloys-i.e., neither superconductive nor magnetically ordered-generally can be considered to be composed of two terms, one linear and one cubic in temperature:
+ pT3
(11 where y is the electronic coefficient. The coefficient /3 represents the contributions from lattice vibrations and is related to the Debye characteristic temperature, 8, by C = yT
where R is the molar gas constant. VOL. 6 0
NO. 5
MAY 1968
29
A theoretical treatment by Sommerfeld (56), based on Fermi-Dirac statistics for a free-electron gas, showed that the specific-heat contribution from electrons a t low temperatures is given by
Sminles Steel Tub
-
3N
where C. is the specific heat, m is the free electron mas, and n is the number of electronsper unit volume. Equation 3 indicates that only the electrons near tIie Fermi level contribute to the total specific heat of the system. If the number of energy states with energy between e and e de is denoted by n(e)de, then
+
Liquid We hwir
(3)
\ If (o is the maximum energy of electrona at the top of the band and each energy state is occupied by two electrons, as required by the Pauli exclusion principle, then the elec!xonic specific heat per unit volume is given by
c.
-1
111 1111 1111 Y (W
Ti 4
30
Cr
5
6
7
fe
(0
8
9
INDUSTRIAL A N D ENGINEERING CHEMISTRY
=
2
- *'fi(eo)T = y T
3
(5)
Thus, the electronic specific.heat coe5cient 7 is a meamre of the density of states a t the Fermi level. The dic cussion thus far has been concerned with a parabolic band as indicated in Equation 4. A general treatment by Stoner (50) shows that the value of y for any band form is directly proportional r~ thc density of states at the F a m i level. H e m , the detCrmination of the y codticicnt is one of the m t dircct methods available for acquiring the denaity af state8 for metals and alloys. As noted later, aare must be exercised to amount fbr all p d b l e contributionato 7 to arrive a t the hue elecwonic specific heat. A schematic view of a low-temperature i a o t h d calorimeter employed successhrlly for alloys of transitiondements (54)is shown in Figure 1. The c y h d r h l aUoy sample has an axial hole to actommodate a ra sjstana thermometer and a resistance heatex. The thennometer is a '/matt carbon resistor. The sample containing tkc thenuometer-heater asmnbly is m ~ p d e d on a nylon thread in a vacuumtight brass container, and the entire assembly is submerged in a liquid helium bath. The bath temperature is controlled by a pumping system to attain tempcrlthrrcs down to approximately PK. The sample is cooled to the bath temperature with helium gas as the exchange medium; then the container is evacuated to isolate the sample fmm the surrounding bath. The measurements are carried out by introducing known quantities of cleatrieal in s u d v e steps and noting the temperature rise. In one applickon of this-* of &meter (6), tbc heat capacity of the Sample was obtained from measure-
3
E in eV
\
men@of the voltage and current supplied to the heater, the total time of the heating period,and the thermometer resistance as a function of time. The initial and final temperatures, T,and T,,were computed from the resistance measurements, and the heat capacity, C, was evaluated by the relationship
C=
En (TI- TO
(6)
where E is heater voltage, Z is the current, and t is the time of heating. The heat capacity is obtained for an average temperature between T,and T, It should be emphasized that while the calorimeter described, which employs a helium exchange gas as the “heat switch,’’ has been applied satisfactorily to metala and alloys wirh large variations in the specific heat d c i e n t , if is inadequate for measurements whve the changes in y values as a function of composition are small. Differential techniques of greater accuracy have been developed (33) that, combined with the use of a mechanical heat switch and improved calibration techniques (32,43)for resistance thermometers, permit reliable measurements to be made on alloys where the variations in the specific heats are small, particularly the noble metal alloys. The importance of low-temperature specific-heat measurements to the theory of transition-metal alloys can be demonstrated by the results (78) obtained for the bdy-centercd cubic solid solutions of the fust-longperiod Pansition elemen*i.e., the elements with unwed 3 d energy bands. Measurements of the electronic specific heat were made for a series of isostructural solid
solution alloys, and the values of the electronic specilkheat coefficient y are shown as a function of the elecwn concentration (the average number of electrons per atom outside of the closed argon shell) in Figure 2. The peak in the specific-heat curve to the right of chromium has aroused considerable interest. The high y values were confirmed with additional measurements on other binary and ternary alloys that have an average electron concentration in the same range. Since the peak is located in the concentration range where the onset of ferromagnetism is observed in chromium-iron alloys, a large magnetic contribution could be responsible foa the high 7 value observed. However, high-tempatwe specific-heat data of Schrirder (46) indicated the a k n c e of a magnetic contribution and supported the earlier conclusion that the peak in the y values is of electronic origin. Thus, the electronic specific heats for the body-centeKd cubic alloys among the 3 d transition elements can be described in t u m s of a more or less rigid band model. The extent of filling of this band by electrons appears to be determined essentially by the average electron concentration of a given alloy. This interpretation is valid even when the electron concentration of an d o y is averaged over three different species-i.e., ternary a h y k Since the electronic speciIic-heat coefficient is directly proportional to the density of states at the Fermi l e d , mperimental information on the density of states can be derived from they values. The density-of-states curves for each of the two electron spin dmtions vs. energy are plotted in Figure 3 for the body-centered cubic solid solutions of the fhtlong-period transition elements (78). The interpretation of the density-of-states cuwe, based on magnetic saturation moment data and the pcific-heat results, indicates that the first sub-band (-1.2 to 0 eV) contains a n q u a i number of electrons with positive and negative spin directions; this is consistent with the fact that alloys in this range have no permanent moment. The second sub-band (-0.4 to +1.35 eV) has electrons of one spin direction only. The atomic moments of an alloy in this region correspond to the number of electrons in this sub-band. The saturation moment value of 2.2 Bohr magnetons for iron can be understood in terms of the proposed density-of-states curves since the m n d subband begins at an electroil concentration of a p p p d mately 5.8 and the additional electrons added to this sub-band contribute to the magnetic moment of iron, which has a total of 8 electrons per atom. The third sub-band (+1.06 to +1.64 eV) extends from approximately 8.2 to 8.7 electrons per atom, the termination of the stability range for the body-centered cubic struoture. This band is filled by electrons with a spin direction opposite to that in the second sub-band, as suggesbxl by the Slater-Pauling curve-i.e., the saturation moment VOL 6 0 NO. 5
M A Y 1968
31
decreases with increasing number of electrons per atom. Thus the low-temperature specific-heat data provide a means of discerning the variation in the density of states in the energy bands of alloys as a function of electron concentration. While low-temperature specific-heat data are available for the entire series of alloys between first-longperiod transition elements, the equivalent information is needed for the other two long periods to assess the possible electronic contributions to the high-temperature thermodynamic quantities of these alloys. The apparent success in the interpretation of the results noted above has spurred a large effort to determine the specificheat coefficients for second- and third-long-period transition elements as well as for alloys among nontransition elements. Also, a substantial experimental and theoretical effort is in progress to define quantitatively the possible contributions to the specific-heat coefficient from electron-electron, electron-phonon, and electron-magnon interactions. While theoretical progress in the interpretation of specific heat results continues, nevertheless, an important contribution to the present understanding of the electronic structure of alloys has been derived from low-temperature specific-heat studies during the past decade. High-Temperature Thermodynamic Properties of Alloys
The majority of the thermodynamic data available on metallic systems pertains to the high-temperature region. This portion of the review will be primarily concerned with the thermodynamic properties related to the isothermal reaction involved in the formation of an alloy from its component elements, specifically, the relative free energy of formation ( A F ) , the relative enthalpy of formation ( A H ) , and the relative entropy of formation (AS). The three principal thermodynamic quantities are related by the Gibbs-Helmholtz equation : AF = AH - TAS
(7)
The free energy of formation of an alloy is a quantitative measure of its stability relative to the pure components and is a useful quantity to discern phase equilibria. However, the influence of various factors that contribute to the thermodynamic properties is usually more apparent in the enthalpy and entropy quantities. Before considering the thermodynamic properties of specific alloy systems, some of the high-temperature experimental techniques (37) currently employed in transition-metal alloy studies will be reviewed. The changes in the free energy, the enthalpy, and the entropy, upon the reaction of pure metallic elements to form an alloy, may be obtained by a suitable equilibrium technique. If the components of a binary alloy are sufficiently separated in the electrochemical series, a reversible galvanic cell can be assembled that consists 32
INDUSTRIAL A N D ENGINEERING CHEMISTRY
of an alloy electrode and an electrode of the less noble element. The electrodes are immersed in an electrolyte containing ions of the less noble component. If the cell reaction tends reversibly toward equilibrium, the partial free energy of formation of one component is given by A F = - z f E where f is the Faraday constant, zf is the number of coulombs that pass when the reaction proceeds to completion, and E is the electromotive force. The emf technique is a direct method of measuring the free energy quantity provided the reaction under consideration is the only one that generates the measured emf. Thus, localized concentration changes are not permitted during the measurement and, in the case of a solid alloy, the diffusion rate of the transported ion must be sufficient to avoid a change in composition at the electrode-electrolyte interface. The elevated temperatures necessary to attain adequate diffusion rates place severe limitations on the available number of suitable electrolytes. If the metal anion of the electrolyte can have more than one valence state, the experimentalist must ascertain that only a single valence state is present. Also, the affinity of the anion for the more electropositive component must be substantially less than its affinity for the other component; otherwise, the emf output of the cell will be a small and difficult quantity to measure. For this reason the emf method is not likely to be satisfactory for binary alloys with components close to one another in the electromotive series. Thus, satisfactory electrolytes for alloys of the transition metals are not readily available and other equilibrium techniques are required for these alloys. The vapor-pressure methods available for thermodynamic measurements fall into two categories, depending upon the magnitude of the vapor pressure involved. The high-pressure methods (lo3 to mm Hg) generally require a closed system in which a static equilibrium is established prior to the measurements, and pressure is determined by means of a manometer or a Bourdon gauge. Also, the dew-point method is suitable for vapor pressures in the range previously mentioned. Since the majority of the transition metals have vapor pressures that are in the low-pressure range
AUTHOR Joseph B. Darby, Jr., is a Metallurgist in the Metallurgy Division, Argonne National Laboratory, Argonne, 111. The author wishes to thank his associates, Drs. A . T . Aldred and K. M . Myles, for helpful discussions and comments and Profs. 0 . J . Kleppa and J . N . Pratt for fruitful discussions. A review of the manuscript by Miss Frances A d a m is gratefully acknowledged. This work was performed under the auspices of the U . S. Atomic Energy Commission.
. 1
compressed into compacts, A , and suspended within the furnace, B , surrounded by radiation shields, E. The furnace is enclosed in an aluminum block, F, and the entire assembly is suspended, J , in a vacuumtight container immersed in a thermostatically controlled water bath held at 25" C. The hot junctions, P, of a number of thermocouples in series are attached at representative positions on the surface of the aluminum block, and the cold junctions are attached to a brass block, Q . The compacts are heated by an electric furnace within the calorimeter until a spontaneous reaction is initiated. The electrical energy consumed is measured by a watthour meter. When the calorimeter has cooled to its initial temperature, a calibration experiment is repeated on the alloy formed in the previous thermal cycle. Sufficient electrical energy is supplied to the furnace to raise the calorimeter block to the maximum temperature achieved in the reaction experiment. The difference between the electrical energy consumed in the reaction and the calibration experiments is the heat of formation of the reaction. Reactions of a highly exothermic nature go to completion within a few minutes of the initiation. However, endothermic and small exothermic heats of reaction are difficult to measure since there is little or no self-heating of the compact; the calorimeter must be held at sufficiently high temperatures to ensure complete reaction. The final product of all reactions in this type of calorimeter must be examined thoroughly by metallographic techniques to verify that a homogeneous alloy was achieved. The most reliable heat-of-formation data are obtained by liquid-metal solution calorimetry (57). The advantages of this method include the following: The solution of a solid metal or alloy in a liquid metal is more rapid than the direct combination by a solid-state reaction; the final state of the process of solution is well defined; the reaction can be carried out at moderate temperatures with a suitable solvent; the heats of solution of metals in other liquid metals tend to be smaller than in aqueous solvents; and the reactions do not involve the formation of gaseous reaction products. High-temperature liquid-metal calorimeters have come into extensive use in recent years. Their principal use has been as a solution calorimeter for the determination of heats of formation of alloys. The heats of solution of the components as well as of the alloy in the solvent can be expressed as follows ( 3 7 ) : xA (solid, Ti)= x A (dissolved, conc. c, T,), H I
(10)
By (solid, Ti)= yB (dissolved, conc. c, Tf), Hz
(11)
A,B, (solid, T,) = xA
34
+ y B (dissolved, conc. c, T,),
INDUSTRIAL A N D E N G I N E E R I N G CHEMISTRY
H3
When the solutions on the right-hand side of the three equations are identical in temperature and concentration of all components, subtraction of Equation 12 frorn Equations 10 and 11 gives the reaction for the formation of the alloy from its components in their standard states :
+
xA (solid, T,) y B (solid, Ti)= A$, (solid, Ti) (13)
The heat of reaction for Equation 13 is the heat of formation of the alloy and this quantity can be calculated from the heats of reactions of Equations 10, 11, and 12 :
The heat of formation refers to the initial temperature of the alloy T iand is independent of the final temperature T,of the solvent. I n one design of a liquid-metal solution calorimeter (25, 38), the calorimeter proper consists of a single reaction cell surrounded by a massive constant-temperature jacket. A source of controlled heat is provided to maintain the assembly at the desired temperature. The total heat effect of a solution experiment is evaluated from : the observed change in the differential temperature between the reaction cell and the jacket, the measured energy equivalent of the calorimeter, and a correction for the heat transfer between the cell and jacket during the reaction period. This type of calorimeter is employed successfully in the determination of the heats of formation of alloys where the reaction period is short and where moderate heat effects are involved. However, the error increases substantially when the elements or alloys dissolve slowly in the solvent. Often, the error is due to the uncertainty in estimating the heat transfer correction. As noted previously, the transition metals such as cobalt or nickel dissolve very slowly in liquid tin at temperatures suitable for precision calorimetry (37). This has been confirmed in recent studies of the kinetics of solution of transition elements in liquid aluminum (8). These studies confirmed that, under otherwise identical conditions, the rate of solution is proportional to the solubility of the element in the solvent. A twin differential calorimeter, of the type developed by Calvet (4), is well suited for the precise determination of small heat effects generated over extended periods of time. This type of calorimeter consists of two calorimetric cells contained within two symmetrically located cavities in a metal-block thermostat. The liquidmetal solvent is contained in a suitable crucible within the calorimetric cells. A temperature difference between the outer surface of each calorimetric cell and the surface of the surrounding cavity is detected by a thermopile, which has an equal number of junctions in thermal contact with each of the two surfaces. The two
content of a material. The material must be a t a temperature above or below the temperature of the liquidmetal solvent so that a heat effect ia produced when the materid is inserted into this solvent. The material should be. inert-that. is, free of any interactions with a solvent. When it is convenient to introduce material .at the same temperature as the calorimeter bath, the established heat of solution of a pure inetal ma$ be utiliied for calibration purposes. The advantage of a calibration -.on the heat content or the heat of solution of a pure material is that,it more nearly duplicata, .the experimental conditions compared with the electrical ' & h o d . Alao, the temperature from which the calibration ma&al is dropped m a y be selected to produce an exothermic or an endothermic effect to match the e5ect produs+ by the aU~p under investigation. The use of the &e material ,br the liquid metal bath and for calibiation purp+ avoids the introduction of an additional species and the po%ible interactions with the solutes u n d b investigation. Liquid tin is a popular solvent for,.liquid-metal6 rimeters. Other pure metal solvents suitable pmvided they have the following pm.pgrti,& a n k n @ y low melting point, a low vapor pressure, good resistince to oxidation, and at least 3 to 5 atomic ?& solubility in the liquid phase for the wlutes to be investigated: While a number of twin-well dori'meters have been designed and p l a c d i n t o operatien in recent .years, a calorimeter designed specifically to in-ate trm& tion-metal &p has been described in the literature (8). A cross-sectional view of .this caloripetcr .is shown in F i b r e 5. The metal-block thermosat, A, is a cylinder of pure sil&, ideal because of. its.high thermal conduc~ v i vand high heat capacity. Two cplizldrical w& Are loeated symmetrically in:the blockj. an& a &rimetric ecll is centrally b t e d in each of th d . ,A calorimetric cell co&ts of a silver thimble, B, fitted snugly over the end of a s t a i n l m . a t d tube, C, which pac~e through ~ thetop lid of the b k k . a n r l e x b A s , t othe top of the watur-eooled vacuum chamw. tho^ tube mvw tbree important functions:. it providesa meansd .suspending the calorimetric cdl..withia thecavity Mads, a gastight environmentfor &e interior.of iJ-u & metric cell, and aCees.to the interior,o f , t h e d ' . Each thamopile, C, w&.of 176 j u n e t i o n S * ~ m bled froaa~chromcl-alumelwire;. half of:the.junctiom are attached toihe surface ofthc cdk a d the &a hal€ are attached to thei n n ~ r -afthe d surroundingdker block., The block is endosed by a. resistance. furnace, 11, surrounded by a . k e a ob radiition.shiclds,, and the entin assembly .is enclosed in a. vacuumtight eha~bar. A."push-pull" ty@eof stirring mechanism, D, is.providd to maintain a homogeneous liquid-metal ,bath. The upper temperature limit for the operaboM&calo* eter M determined by the crrcp properties of dvcr ad, ~
~
VOL 60
NO.
5
AIAY.1961
,35
thus, the maximum operating temperature is approximately 800' C. A higher temperature limit can be achieved by substituting a more refractory material, such as nickel, for the silver block and platinum-base thermoelectric wire for the chromel-alumel components. A liquid-metal solution calorimeter should give reliable results to a temperature of at least 1000" C. I n summary, a broad spectrum of techniques is available for the determination of the thermodynamic properties of transition-metal alloys. Most of the techniques were developed for alloys between the elements to the right of the transition series in the periodic table and, with the substitution of suitable materials of construction in the experimental apparatus, have been extended to the higher temperatures required for transition-metal alloys. I n principle, the three quantities in the GibbsHelmholtz equation (Equation 7) can be determined from the vapor-pressure techniques when applied over a range of temperature. I n practice, and when possible, a determination of the free-energy quantity by one of the vapor-pressure methods combined with a calorimetric investigation of the enthalpy of formation at the same temperature yields the most accurate data.
Theoretical Models of Metallic Solutions
The various models proposed to interpret the thermodynamic properties of solutions have been reviewed recently in detail (36) and only a few of the salient points will be noted in this discussion. The statistical models of solid solutions are of interest because the techniques of probability are employed to relate macroscopic thermodynamic properties to a microscopic interaction energy. The simplest statistical model of a solution (15) is the so-called quasichemical theory in which it is assumed that the total cohesive energy is the sum of all the energies of interaction between nearest neighbors in an alloy. The interaction energy of a pair of neighbors depends only on their species and is independent of the type and distribution of other neighboring atoms-Le., a random distribution of atoms on the lattice sites-and the interactions are independent of concentration. Higher order approximations of the quasichemical theory take into account the existence of nonrandom distributions of atoms. However, Oriani (36) has shown that the theoretical predictions of the higher order approximations are in contradiction with the experimental evidence for a number of alloy systems and the assumptions are drastically oversimplified. The influence of vibrational and volume changes that must occur on alloying are not considered in any of the higher order approximations. The attractive feature of statistical models has been that the local atomic arrangement 36
I N D U S T R I A L A N D ENGINEERING C H E M I S T R Y
assumed in a solid solution can be determined independently from diffuse x-ray scattering experiments. The average-potential model is another statistical treatment that considers the composition dependence of the pair-wise interaction energies as well as the vibrational and volume factors. However, the inadequacy of the model has been attributed to the inappropriateness of a central-field approximation for the pair potentials and the inclusion of only nearest-neighbor interactions (36). Friedel (73) has proposed models for calculating the limiting slope and the limiting curvature for the excess enthalpy of mixing in the dilute solution range of binary terminal solutions. The limiting slope for the enthalpy of mixing represents the enthalpy change in transforming a solute atom from its surroundings of like atoms into the pure solvent. A thermodynamic substitutional cycle was formulated in which the heat of solution appears as the difference between large numerical quantities ; hence, the correlation with experimental data has had limited success (26). The theory for the limiting curvature of the enthalpy of mixing indicates that if the atomic size difference between the solute and solvent is small and chemical interaction is modest, then the limiting curvature is determined by the difference in valence between the solute and solvent. If the solute has a higher valence than the solvent, an effective repulsion will exist between the solute atoms and a positive curvature is expected. When the reverse situation exists, attractive forces should dominate and a negative curvature is likely. Also, a proportionality between the valence difference and limiting curvature is predicted for different solutes in the same solvent, Experimental confirmation of the theoretical predictions was obtained by Kleppa (26) for the dilute solutions of cadmium, indium, and tin in silver. The electronic interactions that occur upon alloying have been considered in a theoretical treatment by Varley (52). A separate band of electron energy states is assumed for each of the components of a binary system and the energy contributions arising from the transfer of electrons from one energy band to another are calculated. The redistribution of electrons gives rise to negative heat effects. Varley's model also accounts for an atomic size disparity on the basis of classical elasticity theory, and a positive heat effect is associated with the resultant lattice strain energy. However, the use of classical elasticity theory as a basis for strain energy calculations was shown by Oriani (36) to be of questionable validity. The difficulties imposed by elasticity theory are avoided in the theoretical model proposed by Shimoji (48). A simplified Wigner-Seitz cellular approximation is employed to calculate the relative energy change on alloying in binary metallic solutions. Free electron behavior is assumed for the valence electrons and a
calculation of the energy of the lowest state in the valence band of an alloy is avoided by employing semiempirical equations of state b a d on the properties of the pure components. However,the qualitative treatment of the van der Waals force contributions and repulsive energies, which may represent major contributions to the thennodynamic quantities in some alloy rystems is a serious ddiciency. The lack of adequate theoretical models to account quantitatively for the various contributions to the thermodynamic propertien ofconantrated alloys docs deet the involved nature of the interactions associated with the in alloys. As a result of the vague bonding mechamma ' concepts, the variations in intaprrtationq and the poor agreement with experimental data that evolve from the applications of the available M e a , the expmknentalit seeks a qualitative confirmation of the thumodynamic behavior by cornpariaon with other physical propazies of the component ekmenta and alloys, e~p&iollyin the case of &tion-metal alloys The phenomendogid a p proaeh is b a d on the importame of relative atomic s u e , electron concentration, and electronegativity factors in interpreting binary phase diagram@, as indicated in the worL of Hume-RothQy (23) and many othcrs. While the three facton, are intemlated, binary systems do exist where one of the factors h dominant and can be definitely correlated with the thermodynamic p @ e s t h d . A mixing of atoms of unlike size will modify the vibrational spectrum by nducing the mean frequency, and positive deviations of the specific heat from NeumannKopp behavior will occur. As noted earficr, attempts to calculate excess thermodynamic quantities for binary solid dutions from clastic theory have not produced concordance with the expmknmtal o h a t i o n s . However, positive deviations of the thermodynamic quantities from ideality have been observed in systems where a size disparity in the atoms of the component elements was the
apparent dominating factor. For example, coppersilver is one system where the valence and electronegativity difference is zero, but the size difference is relatively larger. The excess thermodynamic quantities for this system have positive deviations from ideality (22). A difference in electronegativity-Le., the relative tendencies of atoms to acquire electrons-in the components of a binary system gives rise to negative departures from ideality. This is conlirmed by the properties of the gold4lver system (see Figure a), since the size difference between the atoms of the two species is negligible, the valence is the same, and a substantial dserence exists in the electronegativity for the two elements. The valence dilTerence between components of a binary metallic system has a significant idiuence on the sign, magnitude, and degree of asymmetry of the thermodynamic properties with composition. It is often difficult to eliminate the electronegativity factor since a large valence diffierence is accompanied by a sizable electronegativity factor that predominates and leads to a negative deviation. However, an increase in the valence difference results in an increase in the positive enthalpies and entropies of formation and the degree of asymmetry. The Group B metals have a restricted range of atomic size and electronegativity, and Kleppa (26)found that the thermodynamic properties of alloys formed between the divalent metals zinc, cadmium, and mercury with metals of higher valence do display the valence effect. The results for the zinc alloys are shown in Figure 7. While the integral excess thermodynamic properties reflect both the size and valence difference between the components, the correlation with the increase in valence difference in passing from cadmium to indium to tin is apparent. Exp.rimanlal Data on TmnsIHodetal Sys-s
The experimental data on metallic solutions extend over a broad area, and the data for binary systems alone are sufficient to fill a large volume (22). Several reviews have appeared recently that cover specific areas of metallurgical interest such as intermetallic compounds (44) and actinide materials (27,42). A majority of the detailed thermodynamic information on metallic materials pertains to alloys of nontransition metals. However, with the development of high-temperature experimental techniques of sufficient sensitivity, the thermodynamic properties of the transition-metal alloys are receiving renewed interest. While sufficient data are not available on which to base a general systematic discussion of the properties of these alloys, a number of binary satems have been investigated and the data exhibit features that are associated with the variations in the electronic structure of the transition-metal alloys ( 7 , 3 4 3 9 ) . The present review will examine some of the relevant thermodynamic measurements on transition-metal alloys. V O L 6 0 NO. 5 M A Y 1 9 6 8
37
The alloying behavior of the transition elements varies according to whether the incomplete electron levels are localized states around the individual ions or occur in the collective d-states of the crystal. The transition metals in the early groups of the periodic table generally have the more localized character and their alloys do not conform to a band model. I n contrast, alloys of elements of high group number do exhibit properties that may be described in terms of a band model and can be related to the variations of the Fermi level within a rigid band. However, the band structure of transition-metal alloys is indeed complex, as indicated earlier. The binary transition-metal alloys that have received the greatest attention in recent years are those in which an element to the right of the transition series in the periodic table is combined with a transition element near the end of the series. Large variations are observed in several properties of these alloys that can be related (assuming a rigid band model) to the variations of the Fermi level with the total number of s plus d electrons to be accommodated in the energy band (20). For example, the alloys of palladium with noble metals are typical systems in which a transition element with a high density of states is alloyed with an element having a low density of states. The Fermi energy level in the noble metal is at a higher energy, as shown schematically in Figure 8, and the electrons in s states can lower their energy by filling the vacant d states in the transition-metal component. Since palladium is paramagnetic, the paramagnetic sweptibility would be expeeted to decrease as the concentration of the noble metal is increased until the d band is filled and the diamagnetic state, characteristic of the noble metals, would be attained. This is observed experimentally, as shown in Figure 9, for the palladiumsilver (53),palladium-gold (53),and palladium-copper (47) systems. Similarly, the electronic specific heats for the Palladium-silver system (20),shown in Figure 10, are drastically reduced over the composition range in which the d band is filled. While the two properties are proportional to the density of states, the changes in electronic structure also would be expected to be d e c t e d in the relative thermodynamic properties of these alloys. The relative thermodynamic propaties AF,AH,and A S are related through the Gibbs-Helmholtz equation (seeEquation7). Since AFisafunctionofAHandM, it is more difficult to assess the influence of various factors on the free-energy quantity because of possible compensatory effects of deviations in AHand AS. Therefore, the contributions to the last two properties will be examined in detail.
Enlhalpy of Formation
The theoretical models for concentrated metallic solutions, mentioned earlier, have serious deficiencies that 30
INDUSTRIAL A N D ENGINEERING CHEMISTRY
1w Figure 8. A schematic rcprcrentation of the dmity-of-states mrue us. energy for a binary alloy with a transition metal and a noble metal as components. The d-bond of o transition mctol and the s-band of a noble mctol ore indicntcd togethrr with their respective Fermi Ieucls
0
-lwI
0
arise from an inadequate treatment of contributions to the strain energy, to the van der Waals force, and to the ion-ion repulsive energies. In the absence of a rigorous theoretical treatment, a qualitative approach is necessary. The cohesive energy calculation (74) for copper, which has a substantial ion-ion interaction, indicates three factors of prime importance. These include the energy of the lowest valence state, the Fermi energy, and the ion repulsive energy. Thus, the relative change in cohesive energy upon alloying would be as follows: AE
=
AEo
+ AEp + AE:..
I 02
I 0.6
I 0.8
IS
I”
Figure 9. Total magnetic wceptibiliry at room temperahm for the pallodim-doer (53),palladium-gold (53),and palladium-coppa (47) sysremr
.
(15)
where AE, is the relative energy change in the valence state, AEr is the relative total change in electron Fermi energy, and AEion is the relative change in the ion-core repulsion energy. In analogy with the cohesive energy argument, the heats of formation may include these contributions particularly when the effects are large. While an absolute calculation of the change in Fermi energy upon alloy formation is not possible at the present time, Pratt (39) has estimated the reduction in electronic energy on alloying for the palladium-silver system. He assumed a fixed rigid-hand model, a transfer of the excess s-electrons of the noble metal to the vacant holes in the d-band of palladium, and a density-of-states curve derived from low-temperature specific-heat results of Hoare el al. (20). The curve for the estimated total change in Fermi energy as a function of composition is shown in Figure 11. The absolute magnitude of the energies are uncertain, since electron-electron interactions and variations in the valence state are ignored. Nevertheless, the relative change in total Fermi energy as a function of composition for the palladium-silver system is similar to that observed for the enthalpy of formation.
I
04
0%
41
a
21
0
I Rh
so Composition
I Pd
.*I
1
s o #
- honk P~nial
I
Figure 10. The clectronic spe@c-heat coc&cnts silun andpalladium-rhodium olloyr (20)
for
palladium-
V O L 6 0 NO. 5 M A Y I960
39
L
-5.0 -6.0 0 0.1
3 0.4 0.5 0.6 0.7 0.8
1 I0
"A,
0
-m -400
-m -800
-5
I
L
E
-1000
-1m -1400 -1600
?I-1800 -2000 -2200 -2400
-2m
-2833 0
0.2
0.4
0.6
0.8
1.0
The experimental AH values for the palladiumailver (22), palladium-gold (7),and palladium-coppcr (22) systems are indicated in Figure 12. The asymmetry in the composition dependence of AH is essentially the eame for the three systems. While density-of-statesc u m are not available for the palladium-copper and palladiunr gold systems, a similarity in the composition dependence of the Fermi energy would be anticipated from the compositional dependence of the susceptibility plots. It appears that the reduction in electronic energy on alloy formation is the major factor that contributes to the characteristic compositional dependence of the AH values of the three systems. The order in which the maximum values of AH occur in Figure 12 has been attributed (7) to possible subtle differences in electronic structure and other factors such as the ion-core repulsion. The enthalpies may reflect differences in the Fermi level of the noble metals relative to that of palladium. While the ion-core repulsion would be expected to increase in the order Au > Ag > Cu on the basis of ionic sizes of the noble metals, the experimental order of the maximum AH values are in reverse order for gold and silver. If the minor contributions from these and other unidentified contributions were properly accounted for, perhaps better agreement with experiment might be achieved. However, the detailed nature and magnitude of the 'nteractions are difficult to assess. The enthalpies of formation for the gold-nickel, golciron, gold-platinum, and copprr-nickel (approximate estimate from the miscibility gap) systems are another interesting series for which data are available (22) (Figure 13). The AH values are all positive, and asymmetry in the compositional dependence is observed. O n the basis of elasticity theory, the positive heats of formation of gold-nickel and gold-iron systems are attributed to the strain energy arising from atomic misfit. Oriani (36)has indicated that the use of classical elastic models is questionable for the calculation of misfit energies. I n addition, the atomic sue factors for the coppernickel and gold-platinum systems are small and a negligible positive contribution would be expected from the model. I n order to properly account for the positive contributions, a model is required that employs a self-consistent field calculation of the charge distribution of the ions and a calculation of the overlap energy of the charge distribution as well. While the calculations have not been attempted for transition-metal alloys, they would be complex in view of the variations of the ion-electron configurations that probably exist. Another possibility (74) is to arbitrarily consider the energetic consequences of an ion-ion interaction on the basis of a pair-wise Born-Mayer interaction of
wtk) 40
INDUSTRIAL AND ENGINEERING C H E M I S T R Y
=
C exp1-d
- It - ~
J P
(16)
pisVn 13. Thc &pies of fanrorion for thegold4ckd. goldiron, gdbprorimmt, ond coppcmiCkd alloys (22)
where C and p are empirical constants, d is the distance between the ions, and r is the specific ionic radius. The quation indicates that if the ionic sizes retain their values, as in the pure components, and only the spacing varies in accordance with the experimental lattice parameters, a reduction in the total repulsive energy is expected in the palladium-copper, palladium-silver, palladium-gold, copper-nickel, and gold-platinum systems. However, if the transition-metal ion size approaches that of the noble metal as a result of electron transfer (Z),a large repulsive energy is liely to exist. T o generalize the qualitative argument, a large ionic size discrepancy and a small electronic contribution lead to endothermic or positive values for the enthalpy of formation while large electronic effects and small ionicsize Merences will yield exothermic behavior. A parrial confirmation of the generalization is found in the systems mentioned. The relative experimental AH valuer for the silver-palladium and copper-palladium systems would be predicted but the gold-palladium values are out of sequence. Also, large ionic or positive contributions are predicted for the gold-nickel and goldiron systems, as experimentally observed. Most of the experimental data for enthalpies of formation on alloys of transition elements are conftned to binary alloys between the elements of the first long period. The magnetic, electrical, and low-temperature specific-heat mlta indicate that these alloys have a far more complex electronic struchm than the alloys discussed earlier. F w 14 shows the AH curves for the iron-vanadium, irolreobalt, and iron-nickel systems. The iron-vana&urn results were obtained from torsion effusion experiments (35) and the iron-cobalt and iron-nickel values were determined from liquid-tin solution calorimetry (49). While the precise form of each curve may not be correct, the negative values for the iron-cobalt and ironnickel alloys may be compared with the apparent positive values €or the ironvanadium system. The similar atomic and ionic size of iron, cobalt, and nickel would not give rise to large size effect contributions in the ironcobalt or iron-nickel systems. The existence of a com-
plete series of solid solutions in iron-cobalt and ironnickel systems and the interpretation of saturation magnetization results in terms of a rigid-band model (18) indicate that negative contributions to the enthalpy would be expected, and this is observed experimentally. Myles (35)has suggested that the pooitive enthalpy of formatior in the iron-rich alloys may arisc from a misfit energ) contribution (consistent with the obmved minimum in the liquidubsolidus curves) to the vibrational enthalpy as well as a positive configurational enthalpy term. Enthalpies of formation for a series of alloys that in. . ed in Figure 15 volves chmmium or iron ~ + e The chromium-vanadium results were obtained from torsion-effusion studies (7). The binary alloys of c h m mium with iron, nickel, or molybdceum were investigated by means of direct-mactian calorimetry (72, 30), and the manganeeiron alloy work employed vapor pressure techniques and acid-solution calorimetry (24,45). The size difference between the components for all the systems is relatively small. Other factors offer little help in attempting to rationalize the relative values. In fact, the only d u e to the sign of the enthalpy values is the apparent trend that alloys of neighboring elements in the same period tend to form exothemkally while a group or period separation of the component elements, as in the chromium-molybdenum system, yields an endothermic behavior.
Figurs 74.
Thc anrhaljics of formatiion for
thc i r o r r d w n (35),
irowobalt (@), and imrrnickcl(49)sydems V O L 6 0 NO. 5 M A Y 1 9 6 0
41
entropy is negative. Unfortunately, measurements of C, values ovw the appropriate temperature ranges are usually not available. However, if low-temperature s p d c - h e a t data are available, estimates of ACp can be obtained by the Debye theory (70) of specific heats. While the estimates are generally reliible up to room temperature, they seldom agree with the aperimental determinations at elevated temperatures. The configurational contribution to the entropy is always positive and varies between zero and the ideal entropy of mixing ASa,where
asd = R[NAIn N A + N B In N B ]
EnkopV of Famallon
The entropy of fornution can be coarelated with the changa that occur in the lattice and in the electronic ahucturc of the component metab upm alloying as indicated by the vibrational, amfigweand magn a i c cburctaumu of the alloy. Thus the entropy change can be defined -follows:
..
AL$
Mvlb
+
+ A&
whue AC, represents the deviation of the lattice and electronic spcci6c heats from Neumann-Kopp behavior. The d a m m a r k a from the change in bond craength and in thc mystabpphic arran@ment of the a m . If the bond strength of the alloy is m n g e r than the amage bonds in the component dements, the vibrational 42
Departures from completely random mixing, whether the tendency is toward short-range ordering or clustering, result in entropy values that are less than ideal. A proper BsseSSment of this quantity usually requires a thorough examination of the alloy by x-ray or neutron diffraction techniques. The magnetic entropy contribution is particularly important when considering alloys containing a transition element. The magnetic entropy is dependent upon the degree of randomness in the orientation of the atomic magnetic moments that are associated with the atomic species in the alloy. The magnetic contribution to the entropy is equal to .fAC,,,,dT/T. A hypothetical heat capacity curve for a ferromagnetic material is shown in Figure 16 where the area under the peak represents the magnetic contribution to the specific heat. Applying classical Boltzmann statistics, Weisa and Tauer (55) derived an equation to determine the magnetic entropy for metals (76) that obey a localized electron model namely,
,S
= R l n (2 s
(17)
The change in vibrational entropy is the daerence betwecnthehcthamaltior~oftheawm~and electrons in the alloy and in the wmponent danenrs, and is exp d by the dationahip
I N D U S T R I A L AND ENGINEERING C H E M I S T R Y
(19)
-+ 1) E R l n 61+ 1)
(20)
where s is the vector sum of the individual moments and is equivalent to the average dective atomic moment p in Bohr magnetons. If there is a change from a random spin-spin orientation, then a change in S, results from a change in the atomic moment or the degree of magnetic order. The magnetic entropy of formation of an alloy is of the form
M-
RbbAdw N A ln b
+ 1) bndW+ 1) + 1)1 + 1) A
N E
&E
(21)
where p A and pE are the moments of the pure elements, and pEdw are the moments of the elements in the alloy, and N A and N B are the mole fraction of the component elements.
pA*
Elements such as nickel have nonintegral numbers of unpaired electrons and, therefore, a nonintegral one-half value for s. The S,, quantity then may be calculated by considering a nonintegral electron configuration by means of an appropriate mixture of integral configurations and by computing the change of entropy associated with the magnetic ion content. The magnetic entropy calculations discussed are applicable only to localized magnetic states, not to magnetic electrons in collective Stam.
While other contributions to the excess entropy may
arise from the changes in the electronic configurations of the components upon alloying, a theoretical model is not available to make quantitative calculations. Also, an atomic size disparity gives rise to positive contributions to ASFb while a difference in electronegativity gives rise to negative contributions (26). Kleppa has shown that a difference in the valence of the components often produccs an asymmetry in the plot of entropy m. composition (26). The data necessary to compare the calculated and experimental excess entropies of formation are l i i t e d . la fact; only in the silver-palladium (34,37),nickel-zinc (9, nickel-palladium (3), iron-aluminum (41), and iron-vanadium (35)systems have rigorous attempts been made to evaluate the various contributions and compare them with experimental values. The best example for which a complete analysis of the integral entropy has been carried out is the silverpalladium system (34). The electronic and lattice vibrational contributions were calculated from the lowtemperature specific-heat data of Hoare and Yates (21) by means of the equation S (electronic) = yT and the Debye theory of lanice specific heat (IO). No corrections were made for the (C, C,) terms or for the limitations in the Debye theory. AS (magnetic) was calculated by Myles (34) assuming spin pairing to occur in the 4 d band of palladium on the addition of silver such that the d
-
V O L 6 0 NO. 5 M A Y 1 9 6 8
43
band was filled at N A g = O.G. The localized spin approximation was employed together with an effective atomic moment of 1.44 Bohr magnetons (37) for palladium directly substituted for the nonintegral one-half spin. The estimated vibrational and magnetic contribution as well as the sum of the two are shown in Figure 17, together with the experimental excess entropy for the silver-palladium system (34). The estimated values are approximately an order of magnitude higher than the experimental values and the disagreement raises some doubt concerning the assumptions employed. This review has indicated the need for more systematic thermodynamic data from which theoretical models can evolve. The absence of appropriate models prevents a quantitative interpretation of the various nonconfigurational contributions to the thermodynamic quantities and the inability to calculate the enthalpies of formation is a particularly serious obstacle. This is especially true in the case of alloys of transition metals where the change in electronic structure provides an additional contribution. While an adequate theoretical model for calculating the absolute cohesive energy of an alloy has not been achieved, models for the relative change in thermodynamic properties should be an attractive field for renewed attention by the theoreticians. The determination of the low-temperature electronic specific heats is most important as a means of determining the electronic structure of alloys. T o establish the relationship between the experimental specific-heat coefficient y and the true density of states of alloys, various interactions, which contribute to the y value and are unrelated to the density of states, must be evaluated. While a quantitative understanding of the interactions has not been achieved, the pronounced theoretical and experimental interest in a quantitative assessment of these interactions suggests that the problem will be resolved. Although the integral free-energy quantity is most important to establish or confirm phase equilibria, it is not amenable to a fundamental interpretation because of the possible compensating effect of deviations in the enthalpy and entropy quantities. Accurate enthalpy and entropy data are essential for an examination of the role various factors have on the absolute values. I n principle, all of the integral properties can be acquired from equilibrium measurements determined over a range of temperature. However, the most reliable data are obtained from combining the free energy data with enthalpy values, determined from liquid-metal solution calorimetry, and calculating the entropy quantity by the Gibbs-Helmholtz equation. The interpretation of the entropy quantity for alloys is frequently limited by the absence of x-ray, magnetic, and heat capacity data to define the state of the alloy. The thermal contribution is certain only if heat capacity data are available over the complete composition range between the components and over the temperature range from 0’ K to the temperature to which the integral quantities are referred. This informaiton is seldom available. 44
INDUSTRIAL A N D ENGINEERING CHEMISTRY
The acquisition of high-temperature thermodynamic data is proceeding at a moderate pace. I t is clear that in spite of the number of transition-metal binary systems displaying a complete series of solid solutions, only a few systems are ideal or even regular in character. The influence of differences in atomic size, electronegativity, and valence of the component elements on the thermodynamic properties frequently can be discerned. The nonconfigurational factors that are related to the relative change in electronic structure upon alloy formation are often dominant. While changes in electronic structure appear to influence the enthalpy of formation directly, their contribution to the entropy is not so discernible. REFER ENCES (1) Aldred, A. T., Myles, K . M., Trans. A I M E 230, 736 (1964).
(2) Averbach, B. L., “Energetics in .Metallurgical Phenomena,” W.M. Mueller, Ed., Gordon and Breach, New York, 1965. (3) Bidwell, L. R., Speiser, R., Acta Met. 13, 61 (1965). (4) Calvet, E., Prat, H., “Microcalorimetrie,” Masson et cie, Paris, 1956. (5) Clougherty, E. V., Kaufman, L., Acta M e t . 11, 1043 (1963). (6) Corak, W. S., Garfunkel, hi. P., Satterthwaite, C. B., Wexler, A., Phys. Reo. 98, 1699 (1955). (7) Darby, J. B., Jr., Acto M e t . 14, 265 (1966). (8) Darby, J. B., Jr., Jugle, D. B., Kleppa, 0. J., Trans. A I M E 227, 179 (1963). (9) Darby, J. B., Jr., Kleb, R., Kleppa, 0. J., Rev.Sci. Instr. 37, 164 (1966). (IO) Debye, P., Z.Phys. 52, 555 (1912). (11) Elliott, R. P., “Constitution of Binary Alloys, First Supplement,” McGrawHill, New York, 1965. (12) Feschotte, P., Kubaschewski, O., Trans.Faraday SOC.60, 1941 (1964). (13) Friedel, J., Aduan. in Phys. 3, 446 (1954). (14) Fuchs, K., Proc. Roy. SOC.153A, 622 (1936). (15) Guggenheim, E. A., “Mixtures,” University Press, Oxford, 1962. (16) Guggenheim, E. A,, “Thermodynamics,” North-Holland, Amsterdam, 1949. (17) Gupta, K . P., Cheng, C. H., Beck, P. A,, J . Phys. Chem. Solids 25, 73 (1964). (18) Gupta, K . P., Cheng, C. H.,Beck, P. A., J . Phys. Rodzum 23,721 (1962). (19) Hansen, M., Anderko, K., “Constitution of Binary Alloys,” IvicGraw-Hill, New York, 1958. (20) Hoare,,,F. E “Electronic Structure and Alloy Chemistry of the Transition Elements, P. Beck, Ed., Interscience, New York, 1963. (21) Hoare, F. E., Yates, B., Proc. Roy. Soc. A240, 42 (1957). (22) Hultgren, R., Orr, R . L., Anderson, P. D., Kelley, K. K., “Selected Values o f Thermodynamic Propertlesof Metals and Alloys,” Wiley, New York, 1963. (23) Hume-Rothery, W “ T h e Structure of Metals and Alloys,” Institute of Metals (London), Monbgraph No. 1, 1936. (24) Kendall, W. B., Hultgren, R., Trans. Am. SOC.Metals 53, 199 (1960). (25) Kleppa, 0. J., J . Phys. Chem. 59, 175 (1955). (26) Kleppa, 0. J., “Metallic Solid Solutions,” Benjamin, New York, 1963. (27) Kubaschewski, O., Ed., “Atomic Energy Review,” International Atomic Energy Agency, Vienna, 1966. (28) Kubaschewski, O., Dench, W. A., Acta Met. 3, 339 (1955). (29) Kubaschewski 0 Evans, E. Ll., Alcock, C. B., “Metallurgical Thermochemistry,” PergAmo:, Oxford, England, 1967. (30) Kubaschewski, O., Heymer, G., Dench, W. A,, Z. Elektrochemie. 64, 801 (1960). (31) Kubaschewski, O., Hultgren, R., in “Experimental Thermochemistry,” Vol. 11, Interscience, N e w York, 1962. (32) Manchester, F. D., Can. J. Phys. 37, 989 (1959). (33) Montgomery, H.,Pells, G.P.,Wray, E. M., Proc. Roy.Soc. 301,261 (1767). (34) hlyles, K. M., Acta. Met 13, 109 (1965). (35) Myles, K. ?vi., Aldred, A . T., J. Phys. Chem. 68, 64 (1964). (36) Oriani, R. A., “Symposium Physical Chemistry of Metallic Solutions and Intermetallic Compounds,” Paper ZA, H.M.S.O., 1959. (37) Oriani, R. A,, Murphy, W. K., Acta M e t . 10, 879 (1962). (38) Orr, R. L., Goldberg, A., Hultgren, R., Reu.Sci. Instr. 28, 767 (1957). (39) Pratt, J. A’., Trans. Faraday $06. 56, 975 (1960). (40) Pratt, J. N., ALdred,A. T., J.Sci.Inrtr. 36, 365 (1959). (41) Radcliffe, S. V.,Averbach, B. L., Cohen, M., Acta Met. 9,169 (1961). (42) Rand, M. H., Kubaschewski, O., “ T h e Thermochemical Properties O f Uranium Compounds,” Oliver and Boyd, London, 1963. (43) Rayne, J. A,, Aust. J . Phys. 9, 189 (1956). (44) Robinson, P. M., Bever, M. B., in J. H. Westbrook, Ed., “Intermetallic COmpounds,” Wiley, New York, 1967. (45) Roy, P., Hultgren, R., Trans. A I M E 233, 1811 (1765). (46) Schroder, K., Phys. Rea. 125, 1207 (1762). (47) Sevensson, B., Ann. Physik 14, 699 (1932). (48) Shimoji, M., J. Phys. Soc. Japan 14, 1525 (1959). (49) (50) (51) (52) (53) (54) (55) (56)
Steiner, W., Krisement, O., Arch. Eisenhiittenw. 33, 877 (1962). Stoner, E. C . , Proc. Roy. Soc. (London) 47, 571 (1935). Ticknor, L. B., Bever, M. B., Trans. A I M E 194, 941 (1952). Varley, J. H. O., Phil. Mag. 45, 887 (1954). Vogt, E., Ann. Physik 14, 1 (1932). Wei, C. T., Cheng, C. H., Beck, P.A., Phys. Reu. 112,696 (1958). Weiss, R. J.,Tauer, K. J., Phys. Rev. 102, 1490 (1956). Wilson, A. H., “ T h e Theory ofhletals,” 2nd ed., Cambridge University, 1765.
