( 381 ) THE STRUCTURE OF LIQUID METALS AND ALLOYS By J. R. WILSON,B.Sc.(Eng.),l\LMet.,Ph.D., A.I.l\1.* CONTENTS 1. INTRODUCTION 385 2. DIRECT MEASUREMENTS OF STRUCTURE 2.1. Principles: Coherent Scattering ofX-Rays and Neutrons 389 2.2. Incoherent Scattering ofNeutrons 393 2.3. Results 2.3.1. Pure liquid metals 394 2.3.2. Solid-solution systems 397 2.3.3. Eutectic systems: miscibility-gap systems 397 2.3.4. Compound-containing systems 398 2.3.5. Structural models 400 3. THERMODYNAMIC PROPERTIES 3.1. Principles 3.1.1. Specificheats 404 3.1.2. Mixtures. 405 3.2. Results 3.2.1. Pure metals 411 3.2.2. Liquid alloys: specificheats 414 3.2.3. Liquid alloys: The 'direct' method 415 A. Solid-solution systems 415 B. Eutectic systems 416 C. Miscibility-gap systems 420 D. Compound-containing systems 423 *Formerly in the Department ofPhysical Metallurgy,University ofBirming- ham; now in the Department ofMetallurgicalEngineering, Queen'sUniversity, Kingston,Ontario,Canada. METALLURGICAL REVIEWS, 1965, Vol. 10, No. 40. 29-M.R. XL 382 Wilson: The Structure of Liquid Metals and Alloys 3.2.4. Liquid alloys: The' indirect' method A. Free energies 426 B. Entropies . 429 C. Enthalpies. 429 3.2.5. Solubilities in liquid metals and alloys 434 3.2.6. Solution ofinterstitial atoms in liquid alloys 437 4. ATOMIC-TRANSPORT PROPERTIES 4.1. Principles 4.1.1. Viscosity 438 4.1.2. Self-diffusion 440 4.1.3. Thermal diffusion 442 4.2. Results 4.2.1. Viscosity ofpure metals 442 4.2.2. Self-diffusion . 445 4.2.3. Viscosity of alloys A. Solid-solution systems. 446 B. Eutectic systems 446 C. Miscibility-gap systems 447 D. Compound-containing systems 447 4.2.4. Diffusivities in alloys 450 4.2.5. Thermal diffusion 450 5. DENSITIES: VOLUMES OFMIXING 5.1. Principles . 450 5.2. Results 5.2.1. Pure metals 451 5.2.2. Alloys 452 5.2.3. Coefficients of expansion: compressibilities: vel- ocities of sound 454 6. ELECTRONIC PROPERTIES 6.1. Principles 6.1.1. Theories ofelectron transport in liquid metals 455 6.1.2. Optical properties , 463 Wilson: The Structure of Liquid Metals and AUoys 383 6.1.3. TheHall effectand thermoelectric power 464 6.1.4. Magnetic susceptibility: the Knight shift 465 6.1.5. Electrotransport in liquid metals 466 6.2. Results 6.2.1. Pure metals 467 t 6.2.2. Alloys A. General: dilute solutions 469 B. Solid-solutionsystems. 471 C. Eutectic systems 472 D. Compound-containing systems' 475 6.2.3. Thermal conductivity 482 6.2.4. Thermoelectric power 483 6.2.5. The Hall coefficient . 484 6.2.6. Optical properties 485 .6.2.7~Magnetic susceptibility: pure metals 486 6.2.8. Thefree-electron modelfor liquid metals 486 6.2.9. Magnetic susceptibility: alloys 488 6.2.10 Electrotransport 489 7. SURFACE PROPERTIES 7.1. Principles . 490 7.2. Results 7.2.1. Pure metals 493 7.2.2. Alloys 494 8. MELTING, SOLIDIFICATION, ANDSUPERCOOLING 8.1. Introduction 496 8.2. MeltingModels. 496 8.3. Pre- and Post-Melting and Pre-Freezing Effects 498 8.4. Supercooling 503 9. GENERAL CONCLUSIONS 9.1. Introduction 504 9.2 Pure Liquid Metals 504 384 Wilson: The Structure of Liquid Metals and Alloys 9.3. Solid-Solution Systems . 506 9.4. Eutectic Systems 507 9.5. Miscibility-Gap Systems: The SizeFactor inLiquid Alloys 509 9.6. Compound-Containing Systems 511 9.7. General Summary 513 10. GENERAL REFERENCES 514 11. TABLES AND REFERENCES TO TABLES 52"( LIST OF SYMBOLS Only the symbols frequently used are listed here. Others are defined where they occur. a(K) structure factor Op,Ov specific heat at constant pressure, volume d density D diffusivity En,ET} activation energies E~ enthalpy ofvaporization f coherent scattering factor e(r) g(r) defined as- eo GE excess free energy ofmixing GM free energy ofmixing HF enthalpy offusion HM enthalpy ofmixing I X-ray beam intensity K dimensio~ in K-spac~ k Boltzmann constant k diameter ofFermi sphere F m mass of one atom N number ofatoms N number ofelectrons/unit volume atomic radius Hall coefficient; gas constant excess entropy ofmixing entropy ofmixing entropy ofvaporization incoherent scattering factors Wilson: The Structure of Liquid Metals and Alloys 385 S(T), SL' Ss thermoelectric power SK EJrightshllt T temperature TB boiling temperature TF fusion temperature V molar volume VF volume change on fusion VJ molar free volume VlIf volume ofmixing Z valency; coordination no. GREEK a bulk coefficient ofthermal expansion, thermal diffusion coefficient aL,aS temperature coefficient of resistivity PA adiabatic compressibility PI isothermal compressibility y surface tension; ratio ofspecific heats interatomic bond energies eli, Etj C electronegativity viscosity 'YJ () angle ofincidence K thermal conductivity A wavelength; interaction parameter (equation 3.4) e(r) radial distribution function eo number density ofatoms eS,eL resistivity ofsolid, liquid 0'0 d.c. conductivity x magnetic susceptibility 1. INTRODUCTION THATmetals can exist in the molten state has been known to man for thousands of years, yet only in the last two decades has intensive interest been shown in their properties. The impetus for this has come from the organizations dealing with the design and construction of atomic reactors, both in Great Britain and in the United States'!. 2 More recently, interest has been growing in the possibilities of using liquid metals in magnetohydrodynamic applications,3(a) in fuel cells,4 (b) and various other applications.5-7 Liquid metals-normally mercury- have long been used onasmall scaleasboiler fluids, and it was clear that the high thermal conductivity of these materials made for good heat- 386 Wilson: The Structure of Liquid Metals and Alloys transfer properties when they were employed as coolants in nuclear- power generation. A number of reactors have now been built, using either liquid sodium or a liquid-potassium eutectic alloy as coolant. The possibility ofusing these and other liquid metals gave rise to avery considerable effortindetermining the physical properties ofliquid metals and alloys. The first results of these efforts were summarized in 1950 and 1952 by Lyon,l.2 The effort made was naturally rather specific, and did not cover the very wide range ofliquid metals and alloys ofno immediate use in the nuclear-reactor field. Awider field was surveyed in 1954 by Frost,S who attempted to relate the properties of liquid metals and alloys to their possible atomic structure. In most cases, however, far too little information wasavailable toallowthe formulation of a complete picture of structure. In the intervening decade much experimental work has been reported on liquid metals and alloys, and it isnow possible to present a more comprehensive picture ofthe struc- ture ofthese materials. This isparticularly soin the fields ofelectronic properties, transport properties (viscosity, diffusion), and direct struc- tural determinations, although in most areas interpretation lags far behind the experimental facts; little progress has been effected in the measurement orinterpretation ofsurface properties, which, in any case, are unable at present to provide any information about the structure of liquids. Considerable advances have been made in the theory ofthe electronic properties ofliquid metals and, very recently, ofalloys, but, in general, the fundamental understanding of the properties of liquid metals and alloys has remained at.a very qualitative stage of development. The interpretation of, for example, transport or thermochemical properties still requires, as in the solid state, a more detailed understanding of interatomic bonding in metals and a means ofapplying such an under- standing in afundamental theory ofthe property in question. Many of these problems have been partially solved in the field of non-metallic non-electrolytes, but, curiously, there appears to have been little extension of even well-established qualitative ideas from this area to that ofmetallic liquids. A small attempt is made in the present work to improve the situation. Anindication isgiven at the end ofthe review of some areas where more intensive effort appears to be necessary in both experimental and theoretical work. Frost's review suggested that the behaviour of the liquid was often closely related to the type ofsolid, eutectic or intermetallic compound, for example, which it formed on cooling. To examine this suggestion more closely, with a view to providing aframework on which to hang a discussion of physical properties, and thus reaching some conclusions about the actual structure of the liquid formed, all binary systems for Wilson: The Structure of Liquid Metals and Alloys 387 which any reliable thermodynamic or physical information is available have been classified, as indicated below, according to the type ofsolid- state phase diagram exhibited; reasons for this classification are dis- cussed later. (1) Simple solid-solution systems (SS). (2) Eutectic systems, subdivided into: (a) Systems with neither liquidus inflected (N1). (b) Systems with one liquidus inflected (S1). (c) Systems with both liquidi inflected (D1). (d) Systems with the eutectic composition at ..-....J0% of one com- . pohent, and with the liquidus inflected (0% S1). (3) Miscibility-gap (MG)systems, subdivided into: (a) Non-transition metal. (b) Transition metal. (4) Systems containing intermetallic compounds, subdivided into: (a) Systems containing electron compounds, but not exhibiting a liquidus maximum. (b) Systems containing electron compounds, and exhibiting a liquidus maximum. (c) Systems containing non-electron compounds, and exhibiting a liquidus maximum. (5) Miscellaneous systems. N- FIG. l.-Classification ofeutectic systems. 388 ' Wilson: The Structure of Liquid Metals and A.lloys (Electron compounds 45 are regarded as those whose phase boundaries in the binary system are determined primarily by the electron-to-atom (e/a) ratio at those compositions; examples are the fJ, y, and e phases in the copper-zinc system, which, with increasing zinc content, occur at roughly e/a =3/2, 21/13, and 7/4.) Examples of each type of eutectic system are shown in Fig. 1, and the classification allotted to all systems discussed below is indicated in Tables I-XIII pp. 527-537, together with the sizeand electronegativity * factors for each system. The review that follows is divided into sections. according to the property under discussion, and each section subdivided according to the type of system involved. The conclusions reached are summarized in Section 9 (p. 504), where some suggestions are made as to the type of structure that may be exhibited in various liquid-metal systems. Because avery large amount ofmaterial has necessarily been condensed into this review, readers lessconversant with the recent work inthe field may first like to read this summary. Alarge amount ofdata has been summarized in tabular form. It has not been practicable to indicate limits of experimental accuracy in the tables-such limits are often estimated, albeit optimistically, in the original papers, and there has been a very satisfactory tendency to provide a rigorous analysis of errors in all recently published experi- mental work. Only work that appears to be of reasonable quality, having regard to the difficulties of-measurement and to the accuracy usually achieved by other workers, has been quoted; an indication is given ofresults that are particularly doubtful. Where it has not been possible to make a clear choice, alternative ,sources are indicated in parentheses. Because only selected data are quoted, the bibliography isnot acomplete one, but it isintended to provide aselection ofthe best in the field. Much earlier work has been summarized by FrostS and is not necessarily repeated here. Earlier references will also usually be found in the papers quoted, ofcourse. . Most of the opinions expressed throughout the text are those of the writer, and it is hoped that they may prove controversial enough to promote discussion; the subject is still very much in its infancy, and * In this review, the sizefactor isdefined as SFOJc = 200 (fA""+ rB) o fA fB where fA, fB are the Goldschmidt radii of components A and B.45 The electro- negativity "factor" isdefined asthe difference in the electronegativities ofthe two components, expressed as a positive quantity. Most values of electronegativities have been taken from the paper by Teatum etal.9 The exact significance of the electronegativity factor isnot understood but ishere taken as aqualitative indica- 1 tion ofthe tendency oftwo elements to mix exothermically. Wilson: The Structure of Liquid Metals and Alloys 389 lively discussion isgreatly needed, even if it isbased on ideas that are subsequently proved to be ill·conceived! 2. DIRECT MEASUREMENTS OF STRUCTURE 2.1. PRINCIPLES: COHERENT SCATTERING OF X-RAYS AND NEUTRONS Direct structural investigation ofliquids can be carried out by means of X-ray, neutron, or electron diffraction. The first isby far the most common technique. A short-wave, monochromatic and collimated beam of X-rays is directed on to the carefully clean~d surface of the liquid alloy, contained in a wide, shallow crucible, usually heated by a smallfurnace in its base. The angle ofincidence, 8,isobserved, asisthe intensity (I) of the reflected beam. The crude results are usually presented as a plot of I vs. (sin ())/A, where A isthe wavelength of the incident beam. Atypical plot isshown in Fig. 2. This curve may be '2,500 10,000 7500 t J: 5000 2500 o 0·2 0·4 0·6 0·8 -S-inxO- _ Fig 2.-Typical plot ofI VB.Bin Of)•• obtained with great precision, using a spectrometer, but unfortunately provides no direct information about the distribution of atoms in the liquid. This can be obtained only by a method which introduces considerable possibility of calculational error. 390 Wilson: The Structure of Liquid Metals and Alloys The distribution ofatoms inaliquid may berepresented by the atomic radial distribution function (RDF), e(r), which is usually expressed as the density per unit volume of atoms at a specified distance rfrom an arbitrary reference atom. Thus, e(r) also represents the 'probability per unit volume' of finding an atom at a distance rfrom the reference atom; it may have values between 0and 2eo,where eoisthe constant average density ofatoms for the whole liquid. For values ofr less than. the atomic diameter, e(r)~O, and as r~ 00, so e(r)~eo. A typical variation ofg(r) =e(r) with r isshown in Fig. 3(a). e(r) may beobtained eo directly from a Fourier analysis of the I vs. (sin ())/),plot by use ofthe Zernicke-Prins formula: 10,596 4nr2e(r) =4nr2eo +-2r fClO8( -I -1) sin sr·ds-dr (2.1) no Nf2 4n sin() _. where 8 =--),-, the scattermg factor, f =coherent scattering factor, eo =mean number density (atoms/A3) ; and I =intensity. Q(r) (a) 3 'I• • Flo. 3(a).-Variation of the radial distribution function, e(r)/eo. with distance fromreferenceatom.