FOURTEENTH SPIERS MEMORIAL LECTUREStructural Theories of FluidsBY G. S. RUSHBROOKEDept . of Theoretical Physics, The University, Newcastle-upon-TyneReceioed 13th April, 1967To have the honour of giving the Spiers Memorial Lecture at this General Dis-cussion of the Faraday Society on the Structure and Properties of Liquids is a privilegewhich I prize : but to respond responsibly I must keep to the one aspect of the subjectwith which I have some familiarity, namely, the equilibrium structural theories asthey relate to monatomic fluids. In speaking on these I shall hope to give a convenientbackground to some of the papers with which we shall later be concerned in theearlier half of this meeting.The last Faraday Society Discussion concerned with the structure and propertiesof pure liquids was that held at Edinburgh in 1936.The date coincides, ratherprecisely, with the start of the transition from theories of interpretation to theoriesof prediction as regards the structure of simple fluids. Of course, there is still, andalways will be, room for theories of interpretation : it would be foolish, for example,to attempt to predict the structure of water from first principles. I am glad thatlater we have papers on so important a fluid. But for the really simple fluids,in particular (perhaps exclusively) for the inert gases, the last thirty years have seenincreasingly successful attempts to predict scattering intensities, X-ray or neutron,directly from assumptions about the interatomic forces. And for liquid metalsalso, there is hope that the attempt will be profitable.If we ignore multiple scattering, the scattering function i(s) is essentially theFourier transform of the pair correlation function h(r), or g(r) - I , where g(r) is theradial distribution function.For the customary assumption that the interatomicforces may be represented by additive pair potentials, all thermodynamic propertiescan be derived from this correlation function h(r), if it is known over an appropriaterange of density and temperature. The theory of the structure of simple liquidsis thus intimately enmeshed in the theory of the equation of state and, more par-ticularly, in the theory of phase-changes. This is very satisfactory for those theo-reticians, like myself, to whom the most remarkable thing about the liquid state is thatit exists.The three theories, customarily called Born-Green, hyperchain, and Percus-Yevick,of which any discussion of the theory of fluids must take account, had very differentorigins : though the last two are sufficiently alike in structure to be classed together.It is the Born-Green theory which dates back to the mid-thirties, its two equations,and9(1,2,3) = 9(1,21g(2,31g(1,3), (2)having been given, by Yvon and Kirkwood respectively, in 1935.The first,in which g(1,2) and g(1,2,3) denote the pair and triplet distribution functions and8 STRUCTURAL THEORIES OF FLUIDS4( 1,2) the interatomic potential, is an exact equation : one of a heirarchy of equationslinking successive distribution functions, it is a consequence of Boltzmann’s distribu-tion law.The second is an approximation, the superposition approximation,expressing the additivity of potentials of mean force, which can be argued on physicalgrounds. They were brought together by Yvon in 1937. Yvon also gave (1935)the pressure, or virial, equation,p = pk-T- +p2fr#’(r)g(r)d3r, (3)enabling us to pass from the distribution function g(r) to the equation of state.But the time was not yet ripe for the exploitation of these equations. Some tenyears later (1946) the theory was redeveloped, independently of Yvon’s work, bothby Bogolyubov and by Born and Green.5 Within four years (1950), Kirkwood,Maun and Alder had solved the equations, and those of the very similar theorydeveloped by Kirkwood himself, for hard spheres, on an I.B.M.computer : and withthis we enter the modern era. For without machine calculations, and machineexperiments, no progress in the structural theory of fluids is possible.Although quantitatively, the Born-Green theory is very much inferior to itslater rivals, perhaps no theory has yet had greater consequences. For it was thediscovery by Kirkwood, Maun and Alder that the equations did not have integrablesolutions for densities greater than pa3 = 0.95, where p is the number density andc the sphere diameter,* which led Alder and Wainwright to embark on machinecalculations, by the method now known as molecular dynamics, to produce an experi-mental equation of state for a hard sphere fluid.The results, first published in1957, and supported by independent Monte Carlo calculations by Wood andJacobson,* gave rather clear evidence of a transition from a fluid to a solid phase ata density approximately that predicted by the Born-Green theory. The Born-Greentheory gives appreciably toG low a pressure, perhaps 30 % too low at the limit pa3 =0.95 ; but the later theories, while they do justice to the experimental fluid isotherm,would not have predicted the fluid-solid transition.One may legitimately question the reliability of these machine calculations,inevitably based on the dynamics of a comparatively small number of particles. Butour confidence is restored by the close agreement between the fluid branch of thisexperimental equation of state and that calculated by the quite independent methodof evaluating successive virial coefficients.It is a sobering thought that whereasB~ltzmann,~ in 1899, using some calculations of von Laar, could writeP+ ap2 = k-Tp[l+ bp + 0-625b2p2 + 0.2869b3p3 + . . .(where b = 2na3/3), and regard it as a more precise form of van der Waals’s equation,even now, with the indispensable advantages of the Ursell-Mayer theory and high-speed computers, we can add only three more terms lo+0.1103b4p4+ 0.0386b5p5 +0*0138b6(with some uncertainty in the last digits). Nevertheless, extrapolation of the extendedseries by the method of Pad6 approximants, gives nice agreement with the fluidbranch of the experimental isotherm : which thus constitutes a firm, if artificial,base line for testing theories at high temperatures.And probably few of us todaywould disbelieve that freezing, or melting, is essentially an excluded volume problem :as was indeed foreshadowed by the considerations of Bernal at the Edinburgh Con-ference in 1936, in one of the earliest papers of the predictive type.* In these units, close packing corresponds to p03 = 1.414G . S . RUSHBROOKE 9But we must turn to the other theories. Perhaps their most attractive featureis that both of them incorporate the Omstein-Zernike l 1 equationh(1,2) = ~(1,2)+p[c(l,3)h(2,3)d3, (4)relating the total correlation function h(r), or g(r)- 1, to the direct correlation functionc(r). Essentially this equation provides the definition of one function in terms ofanother, and nothing more.It has the consequence, of course, that the compressi-bility equation of Zernike and Prins (1927) l2.Taplap = i+pJh(1,2)d2 = qo), (5)where I(0) is the structure factor, 1 + i(s), for limitingly small scattering angles, andwhich is a second route to thermodynamic behaviour, can be written in the alternativeform,-!- 2 = 1-p c(1,2)d2KT ap s This shows that for a highly compressible fluid, near its critical point, c(r) is short-ranged compared with h(r). But whether we have gained anything, whether c(r) isa useful physical concept, can be judged only by whether c(r) is more simply, ortransparently, related to the interatomic potential than is h(r). A definition is nota theory.The simplest possible theory, of hyperchain or Percus-Yevick type, is to assumethat c(r) is simply the Mayerf-function, i.e., to writeThis gives the direct correlation function the range of the interatomic potential, andincorporates a temperature dependence of the correct Boltzmann form.The assump-tion is intuitively appealing : and both Ornstein and Debye, in their work on criticalopalescence, argue very much along these lines. In the theory of liquids, however,it is not good enough. It leads,13 essentially, to the crudest form of the linearizedBorn-Green equations. Nevertheless, it corresponds precisely to what Montroll andMayer l4 did in the theory of ionic solutions, when they derived the Debye-Huckeltheory by summing over a certain class of interaction-graphs (in the Ursell-Mayerformalism) : namely the chain, or ring, diagrams. And it was the achievement ofMontroll and Mayer in showing that physically interesting theories could result fromsumming over subsets of interaction-graphs in a low density expansion which ultimatelyled to the hyperchain equation : though we must not lay the defects of this theoryat their door.The set of graphs over which we sum to obtain the hyperchain equation is dictatedfaute de mieux, rather than on physical grounds.It is just the largest subset, completewithin its topological class, which we can handle conveniently. But the result,obtained independently by many people some eight years ago,15 is a theory whichretains the Ornstein-Zernike equation together with a second equation for c(r),namely,c(r) = exp [ - #(r)/lcT] - 1.(7)c(r) = h(r) -In [ 1 + h(r)] - 4 ( r ) / ~ T . (8)The Percus-Yevick theory,16 although of the same structure, had an entirelydifferent origin. It could have arisen, as Stell later showed, out of the graphicalanalysis that led to the hyperchain theory : retaining a different, but equally plausible,set of interaction graphs. In fact, it arose out of an attempt to define collectiv10 STRUCTURAL THEORIES OF FLUIDScoordinates for a fluid, in a paper (1959) which I am probably not alone in findingobscure. Its best derivation is undoubtedly that which Percus later gave (1962)using the mathematical technique of functional differentiation recently introduced,in the theory of liquids, by Yvon.” This, which examines the change in densityat one point produced by a change in force field at another, brings us closest tothinking physically, and furthest both from the dilute gas and from arbitrary assump-tions of mathematical convenience. Fortunately, as Percus showed, all our theoriescan in fact be derived this way: though the Born-Green theory then appears assomething of a tour deforce.But however it is derived, the Percus-Yevick theoryjoins to the Ornstein-Zernike equation the second prescription for c(r),c(r) = [ 1 - exp ( 4 ( r ) / ~ T ) ] [ 1 + h(r)]. (9)We must now turn to the testing of these theories. They can be used in threerather different ways. Assuming a knowledge of 4(r), we can calculate the thermo-dynamic properties : or we can calculate the correlation fiinctions and structurefactors.Alternatively, we can use the theories to determine 4(r) from scatteringexperiments.Not unnaturally, the earliest calculations, with any theory, are the simplest :in this case calculating the predicted values of successive virial coefficients for hardspheres. These are not exact theories, nor are they entirely internally consistent.Different routes to any thermodynamic property will lead to different answers.The two routes most commonly used are based, respectively, on the virial equationand the compressibility equation. Ultimately we may have to be content with themost satisfactory prescription (preferably understanding why it succeeds) ; buta serious lack of internal consistency is certainly a defect in any theory.At first, for hard spheres, the test was mainly one of internal consistency, sincethe true virial coefficients were not known beyond the fourth.But now that we knowthe rigid sphere isotherm with some confidence, it is more satisfactory to comparethe theories directly with it rather than with low density expansions.18 Judged inthis way, the Percus-Yevick theory is incomparably the best: in respect of bothinternal consistency and agreement with the true isotherm. There is the addedadvantage, as Thiele l9 and Wertheim 2o have shown, that, for hard spheres, thePercus-Yevick equations can be solved exactly, in closed form. The compressibilityprescription accords with the known isotherm to within perhaps 2 % over the wholefluid range.But the thermodynamic properties show no evidence of a phase transi-tion : indeed the Thiele-Wertheim solutions show no singularity until we reach thequite unphysical density pa3 = 1.9, far greater than that of close packing (pa3 = 1.4).The Thiele-Wertheim solution, however, enables us also to find the correlation func-tions ; and we now know 21 that for pc3 >1-18, g(r) takes on negative values. Sincethis is physically impossible, it gives us an excuse for disregarding the more implausiblepredictions of the Percus-Yevick theory.A hard sphere gas, however, though it provides a useful high-temperature baseline, is far removed from a real liquid. To separate the fluid phase, at low enoughtemperatures, into gas and liquid regions, we must add an attractive potential:and we should soften the repulsive core.Quite appropriately, most further workhas been based on the Lennard-Jones potential4(r) = 4&[(a/r)12 - (a/r)7.For this potential, all three theories have been compared with a high-temperatureMonte Carlo isotherm, for T*(= KT/E) = 2-74, obtained ten years ago by Wood andParker 2 2 : there is quantitative agreement, or disagreement, of the same order as iG . S . RUSHBROOKE 11the hard sphere case. But this is still at twice the critical temperature ; and we mustnext turn to the critical constants themselves.These critical constants have now been found for all three theories, notably byL e v e s q ~ e , ~ ~ whose recent paper is the source of the numbers I shall quote.For thehyperchain theory, hitherto the most studied, Levesque’s estimates agree, withintheir limits of error, with those of Klein and Green,24 and de Boer et ~ 1 . ~ ~ Table 1shows T:, pccr3, and the ratio (p/prcT),, for the Lennard-Jones potential,TABLE 1B.G.h.c.P.Y.“ argon ”T:1 -451 -251.251-26PcQ30.400.260.290.32( P I P K T ) ~0-440.350.300.30together with values customarily quoted for argon, assuming the Lennard-Jonesparameters Elk = 120°, cr = 3.4A. Bearing in mind that Tz, for the theoreticalpredictions, is uncertain to perhaps 2 x, and that the uncertainty in p c is greater,we might well conclude that the modern theories were by no means inadequate.But this is not the whole story : first, because I have deliberately chosen the valuesobtained from the virial equation, and not from the compressibility equation ;secondly, because we do not really know what the true experimental values are fora Lennard-Jones gas.To take the second point first, Levesque has also determined the critical constantspredicted by the hyperchain and Percus-Yevick theories for the hard core with squarewell potential, +(r) = - E , o<r<1*5a, and the corresponding results are shown intable 2.This table shows also the predictions of Verlet’s P.Y.11 equation 26 (aTABLE 2h.c.P.Y.P.Y. I1M.D.Tr1.161.161 -281.28Pc030.250-2 10.320.33(PIPKTIc0.320.300-320.3 1natural extension of the Percus-Yevick theory, when approached through functionaldifferentiation), together with recent “ experimental ” results, for this potential,found by Alder from molecular dynamic studies (and quoted by Levesque).It nowlooks as if both hyperchain and Percus-Yevick theories do only scant justice to thecritical region, yielding T‘ to within perhaps 10 %. The P.Y.11 equation, however,seems to be appreciably more accurate.I shall not discuss the corresponding figures obtained by use of the compressibilityequation, save to say that the internal inconsistency in the theories is of the sameorder as their inaccuracy. It is already clear both that first-order theories give onlya moderately satisfactory account of thermodynamic properties in the critical region,and that there is need for caution before drawing conclusions for real fluids.At greater densities and, more particularly, at lower temperatures, in the trueliquid region not far from the triple point, the thermodynamic situation is muchworse: certainly if the virial equation is used to find the pressure.To take anextreme example, Gaskell 27 has shown that were we to use the hyperchain equation12 STRUCTURAL THEORIES OF FLUIDSto infer the pressure from accurate scattering experiments on liquid argon, essentiallyby solving (8) for 4(r) and substituting in (3), we should overestimate the pressure atthe triple point by a factor of about lo3. But it does not follow from this that thetheories are useless. We are far from being able to calculate a vapour pressure curvefrom first principles : but we may yet have a good theory of liquid structure.Thevirial equation is exceptionally sensitive to the precise form of g(r) in the liquid region.If we look at the internal energy the results are much more encouraging ; indeed, asLevesque shows, hyperchain and Percus-Yevick predictions are in very close agree-ment with Monte-Carlo results for the Lennard-Jones potential. And there is theobservation by Ashcroft and Lekner,28 that if the neutron diffraction data on alkalimetals are fitted, as regards the first peak, by the first peak of the Percus-Yevickhard sphere theory, which for Na at 100°C means choosing the density value po3 =0.85, then the compressibility from zero-angle scattering on Percus-Yevick theory isextraordinarily close to the experimental, thermodynamic, value (within 5 %).Whilethis may tell us more about metals than about our theories, it at least suggests thatthe Percus-Yevick theory does good justice to the excluded volume problem in theliquid state, even at effectively high densities.Perhaps our greatest present need is to be able to answer for the structural pro-perties the questions we have already largely answered for the thermodynamic ones,namely, over what regions of the fluid phase, gas, liquid, or supercritical, equationssuch as (8) and (9) adequately interrelate the structural properties, h(r) and c(r),or their Fourier transforms, and the interatomic potential, #(r). We may ask, ofcourse, adequately for what ? Here, surely, the most physically important questionis whether current theories enable us to infer properties of the potential 4(r) fromthe best experimental measurements of the structure factors of real fluids.I am notthinking of the oscillatory potentials found by Johnson, Hutchinson and March 29in this way from the structure factors of liquid metals, oscillations which have beenonly partially confirmed by the later work of Ascarelli 30 : though it is to the creditof the theories if they can point to significant differences between metals and inertgases. The issue, rather, is whether we can attach quantitative significance to suchpredictions of $ ( r ) for the inert gases themselves.This question can be answered only by computer studies of the kind on whichVerlet 31 has recently embarked. For a chosen 4(r), h(r) and c(r) must be determinedby machine experiments over a wide range of p and T : then equations such as (8)and (9) can be tested by seeing to what extent, and over what range of p and T,they successfully reproduce 4(r).A preliminary report from Verlet is encouraging,for although at large densities or low temperatures the shape of $(r) is not wellreproduced, nevertheless for temperatures of the order of T,, and for a range ofdensity extending well beyond pc, the depth of the potential minimum in +(r) is re-produced by (9) to within 2 %. The analysis, by Mikolaj and Pings,32 of theirexperimental measurements on argon, suggests that the hyperchain equation, (8),may have an approximately equal validity in this region.More importantly, if theseresults can be trusted, the finding by Mikolaj and Pings that the depth of the potentialminimum so deduced for argon varies linearly with p must afford evidence of theinadequacy of the assumption of additive pair potentials, and demonstrate that forreal liquids we are concerned also with multibody forces. If, and when, our theoriesof dense fluids enable us to say this with complete confidence, and with some quantita-tive precision, they will have been justified.Attempts to improve on the present theories usually bring us back to the problemof the triplet distribution function, g(1,2,3). This is inevitably true of the Born-Green theory, where it must be the superposition approximation that is inadequatG. S. RUSHBROOKE 13since the other, Yvon, equation is exact : and it is the very essence of Born-Greentheory to use this exact equation.Prof. Rice will later be telling of his work withLekner to replace eqn. (2) by a more adequate approximation. But it is true also ofthe hyperchain and Perms-Yevick theories, which successfully by-pass the tripletdistribution problem in their original forms. If we proceed systematically along thepath indicated by the functional differentiation approach of Yvon, Percus and Verlet,the Ornstein-Zernike equation is retained, but the correction to the second expressionfor c(r) involves g(1,2,3) : and the theory is not self-contained without an additionalassumption. Though a crude approximation may suffice for calculating a smallcorrection, and good though Verlet’s P.Y.11 theory appears to be, we do not knowhow sensitive the predictions are to the element of arbitrariness here, and may doubtif these second-order theories are yet in their final form.We are also faced with the triplet distribution function if we attempt to bringthree-body forces, or triplet potentials, into the theory of fluids.For although thecompressibility equation ( 5 ) or (6), is unmodified, the pressure equation (3), mustnow include the virial of the three-body forces: which implies a term involvingg( 1,2,3). Nevertheless, if we are content with predicting the scattering functions?triplet potentials can be introduced at the level of the hyperchain and Percus-Yevicktheories : though it is less certain that the theories are quite strong enough to standthe strain.On the hyperchain theory, the inclusion of triplet potentials is particularly simple.33For eqn. (8) is just a truncated form of the exact equationh(1,2) - c( 1,2) = In [ 1 + h(1,2)] + 4( 1,2)/rcT- x( 1,2) (10)where, in the graphical analysis? x(1,2) corresponds to the so-called “ elementary ”graphs : which are neglected in the hyperchain theory. With only two-body forces,the first elementary graph has two field-points, and is proportional to a term in p2.With triplet potentials? however, there is a new class of elementary graph havingonly one field-point? and summing to giveM , 2 ) = P [ f ( 1,2,3)e(ly3)e(2,3)d3where e(i, j) is the Boltzmann factor for the pair potential, exp [---4(i, j)/rcT] andf(i,j,k)is the Mayer function for the triplet potential, exp [ - 4(i,j,k)/rcT] - 1.Thus, at thelevel of hyperchain theory, the use of (8) to infer +(r) from h(r) and c(r) will lead notto the true pair potential 4(1,2) but to an effective pair potential4*(1,2) = 4(1?2) - KT.l(1?2),I would expect this result to have the same range of validity as the hyperchain theory :but shall not pursue the matter further since Prof. Pings will be describing an alterna-tive, and possibly sounder, approach to what I think are essentially the same con-clusions.But even if we refine our theories, by going to a higher order of approximationor including triplet potentials, there remain two outstanding problems : the problemof asymptotic behaviour and the problem of freezing.It is ironic that whereasOrnstein and Zernike introduced the direct correlation function c(r) to argue intui-tively that this is short-ranged compared with h(r), and consequently that for large rh(r) - exp ( - ccr)/v,where a+O at the critical point (or wherever the compressibility becomes infinite1 4 STRUCTURAL THEORIES OF FLUIDSit is just this classic conclusion that is today most open to attack.34 All our approxi-mate theories suggest that c(r) decays like 4(r), (more accurately, - $(r)/rcT), whichfor real fluids, in the absence of triplet potentials, means like l / r 6 : i.e., more slowlythan does h(r) on the Ornstein-Zernike theory. Even if we try to salvage, and refine,the Ornstein-Zernike result by supposing that perhapsh(r)-A&)+B exp (-ar) cos @+y)/r,(where the coefficients are functions of p and T, and a and p vanish at the criticalpoint), we know that the two-dimensional analogue of this is disproved by the workof Onsager and Kaufman on the Ising One technical comment may bepermitted.SinceI($) = 1 +i(s) = 1 + P i ( $ ) = 1/[1 -pE(s)] (1 1)[where &) is the Fourier transform of h(r), and in scattering experimentss = (4n/R) sin (0/2)] it is clear that 1 -pe'(s) cannot become negative for real s :we cannot have a negative intensity of scattered radiation. On the Ornstein-Zerniketheory, for p = pc and T = Tc the equation, 1 -pi.(s) = 0, has a double root at theorigin. It would seem that any heuristic theory must consider the roots of 1 -pE(s) = 0in the complex plane, without assuming that ?(s) is an analytic function of s2 nearthe origin : since this last assumption leads to the Ornstein-Zernike result.Fisher'shypothesis, that (in three dimensions)h(r)-exp (-ar)/rl+q, where O<q<lavoids this assumption : and its two-dimensional analogue can be made to cover theknown behaviour of the Ising model. Nevertheless, for van der Waals forces, itdoes not avoid the conceptual difficulty of a correlation function which decays morerapidly than do the interatomic forces whxh produce it.36 While this may not beinherently impossible, certainly the theory is not yet so firmly rooted as to force usto so surprising a conclusion.Writing eqn. (5) and (6) aswe see that near freezing, indeed anywhere in the true liquid region, when the fluid iscomparatively inconipressible, 1 - pZ(0) is large (though we are not now concernedwith infinities) and it is 1 +pz(O) which is small.The question whether our theoriesof liquids give us any real indication of the onset of freezing, or are capable in principleof so doing, is, I think, the most important with which we are at present faced. Is itaccidental, or significant, that the Born-Green theory suggested a phase-change fora hard-sphere gas? It was, of course, l(0) which ceased to exist : though withoutthe divergence which would indicate infinite compressibility. If freezing is essentiallyan excluded volume problem, can we approach a theory of freezing against a con-tinuum background, or must we discuss the excluded volume problem against thebackground of a lattice-gas ? Are there other, physically acceptable, solutions ofthe Percus-Yevick equations for hard besides that found by Wertheimand Thiele? I think we do not know the answers to these questions : and althoughthe non-equilibrium, or dynamic properties of fluids offer a richer experimental field,there is certainly still work to be done in the development of a satisfactory equilibriumtheoryG. S .RUSHBROOKE 151 J. Yvon, Actualites Sci. Ind., (Hermann et Cie, Paris, 1935), p. 203.2 J. G. Kirkwood, J. Chem. Physics, 1935, 3, 300.3 J. Yvon, Actualites Sci. Ind. (Hermann et Cie, Paris, 1937), 542, 543.4N. N. Bogolyubov, J. Phys. U.R.S.S., 1946,10, 257, 265.5 M. Born and M.S. Green, A General Kinetic Theory of Liquids (Cambridge University Press,6 J. G. Kirkwood, E. K. Maun and B. J. Alder, J. Chem. Physics, 1950,18,1040.7 T. Wainwright and B. J. Alder, Suppl. Nuovo Cimento, 1958,9, 116.B. J. Alder and T. Wainwright, J . Chem. Physics, 1957, 27, 1209 ; 1959, 31, 459.8 W. W. Wood and J. D. Jacobson, J. 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