I. INTRODUCTORYStructure and Properties of Simple Liquids and Solutions :a ReviewBY J. S. ROWLINSONDept. of Chemical Engineering and Chemical Technology, Imperial College ofScience and Technology, London, S.W.7.Received 15th December, 1969The structure of a simple liquid is determined primarily by the repulsive forces between the mole-cules. The effects of the attractive forces can be found by perturbation treatments, of which that ofvan der Waals is the prototype, and that of Barker and Henderson the most recent. The repulsiveforces are equally important in determining the structures of mixtures of spherical molecules ofdifferent sizes-structures which are quite different from those found in the unmixed liquids. RecentMonte Carlo calculations by Singer can be used as a standard by which theories of mixtures can bejudged, and it is shown that the only adequate theories are those which do justice to this differencein structures.It is suggested that the shape of molecules is often of less importance than their sizein determining the free energy, and hence that theories of solutions of n-alkanes or of linear polymersdo not, in their results, differ greatly from those of mixtures of spherical molecules of different sizes.If we are to discuss the properties of polymer solutions then we should, perhaps,first pay some attention to the properties of the solvents, and in this short review Iattempt to summarize some of the current ideas on the structure of liquids and ofmixtures of simple liquids. The mixtures that are most relevant to the subject ofthis meeting are those in which there is a substantial difference of size between themolecules, for such systems provide a natural bridge between polymer and non-polymer solutions.Their discussion may aid recent moves towards a reconciliationin the language and statistical methods used by the specialists in these two fields.LIQUIDSThe last five years have seen a simplification in our ideas on the structure ofliquids. A substance normally exists in at least three forms, one solid and two fluid.The subdivision of the fluid state into gas and liquid is, without doubt, a directconsequence of the existence of attractive forces between molecules. It has thereforebeen hard to admit that the form or even the existence of the attractive forces haslittle direct effect on the structure of a liquid, as described, for example, by the pairdistribution function g(r).The recent realization of this truth has followed theextensive studies by simulation on computers and, to a lesser extent, by direct experi-ment with niacroscopic models, of the properties of assemblies of hard sphereswithout attractive forces. These studies lead to a distribution function g ( r ) of whichan example is shown in fig. 1. The characteristic features of this function are astrong peak at r = 0, the collision diameter, and the oscillatory behaviour of thefunction about the value of unity for larger Y. This limiting value is that of a fluidin which there are no correlations.These features are entirely " geometric " in3J . S. ROWLINSON 31origin ; that is, they are the necessary consequence of the dense packing of hard spheresin a non-crystalline array. This can be put in another way by saying that in such asystem g(r) is a function of density but not of temperature. Nevertheless the formof g(r) has substantial thermodynamic implications. We have the well-knownequationspVINkT = 1 +(b/V)g(a+); b = +zNo3; (1)43nWb 2ILI 2 3 5rl0FIG. 1.-The radial distribution function g(r) for an assembly of hard spheres at a high density.The continuous curve shows the results of computer experiments (Wood l), and the histogram thesimulation of a fluid by an array of ball-bearings (Bernal and King ’).where g(a+) is the limiting value of g(r) as Y approaches CT from above.The largepositive value of g(a+) is thus both a geometrical consequence of the high density ofpacking, and also a measure of the amount by which the pressure exceeds the perfect-gas value of NkT/V.have shown that a system composed of hard spherescrystallizes if the density is sufficiently high (i.e., above about 0.6 of the close-packeddensity, or (b/V)> 1.9) but that, in the absence of attractive forces, there is no distinc-tion between liquid and gas-there is only the one fluid phase. There is, however,one molecular model with attractive forces, and hence with both liquid and gaseousThe computer studie32 PROPERTIES OF LIQUIDS AND SOLUTIONSphases, in which g(r) is still a function only of the density and not of the temperature.This is a system for which the pair potential u(r) is specified byu(r) = + 00where a is a positive constant.This is the potential of a pair of hard spheressurrounded by an attractive energy whose depth and slope tend to zero, and whoserange tends to infinity. Nevertheless, the configurational energy U is not zero.We haveU = 3- :I: u(r)g(r)4nr2dr (4)The product u(r) g(r) is zero for r< cr since the behaviour of g(r) is here dominated bya Boltzmann factor, exp [-u(r)/kT]. For r> 0 we can replace g(r) by unity sincethe integrand is sigiScantly different from zero for a<r<y, and over all but anegligible part of this rhllg: g(r) has its asymptotic value (see fig. 1). Therefore,by integration of (4),where n is the number density, N/Y.HenceU = -(+nNa)n (5)and by integration with respect to temperaturea na = +na. PnkT-- - f ( n ) - k T ' (7)The function of density in (7) is the constant of integration and is readily identifiedwith the equation of state of a system of hard spheres of diameter CT since this is thelimit to which (7) must reduce when a, and hence the attractive part of (3), is putequal to zero.In this model, molecules that are not in contact exert no forces on each other,since [du(r)/dr] is everywhere zero except for the &function at r = Q. Hence g(r)for r > a is the same for this model as for any assembly of hard spheres. It followsthat g(r), U and (PIT) are all independent of temperature and functions only ofdensity.These properties are just those of a van der Waals fluid and (7) is, in effect,his equation of state. He committed himself to a particular form off(n), namely,f ( n ) = (1 - bn)-lwhich we now know to be inaccurate. But we are at liberty to replace this approxi-mation by the function determined from the computer simulation of a system ofhard spheres. This function includes the first-order change to a crystalline phaseat high density. With this form, we have an equation of state (7) which generatesall three phases of matter, is of qualitively correct form and, which near the meltingline, is quantitatively a~curate.~ However, it is poor near the gas-liquid criticalpoint, since it falsely requires pressure to be an analytic function of density, and sorequires (Tc-T) to be a quadratic function of (nl-nc> = (nc-n*), where n1 andn g are the coexistent liquid and gas densitie~.~ This does not accord with the facts,and is a consequence of the infinite range of the chosen function (3).Potentials thatfall off more rapidly with r (e.g., r 6 ) are believed to lead to a non-classical criticaJ. S. ROWLINSON 33point in which p is singular at nc, Tc. However, no exact treatment is yet possiblein a 3-dimensional system.The qualitative success of this model has inspired more ambitious perturbationtheories. The attractive energy in (3) is everywhere small compared with kT, andthe equation of state (7) can be obtained from the configuration integral for thissystem by expressing the latter as a Taylor expansion about the (known) integralfor a system of hard spheres by using (a/y3kT) as the expansion parameter. Ontaking the limit in (3), it is found that in (7) all powers of (a/kT) beyond the first arezero.Real intermolecular potentials are not of zero depth, nor of idinite range,and at least qualitatively resemble the Lennard-Jones (12,6) potential,u(r) = 48[(;)12-(y]. (9)Nevertheless, the phase integral can again beexpanded about that for asystem withoutattractive forces in powers of (&/kT). Realistic potentials such as (9) do not haverepulsive potentials that are infiniteIy steep at r = 0. Again, a suitable perturbationtechnique can be devised to obtain the properties of the system with " soft " repulsiveforces from those of a hard sphere ; the parameter (v-l) now serves as a suitableexpansion parameter if the repulsive part of the potential varies as r-v.The most successful of such double perturbation expansions is that of Barkerand Henderson,' who have shown that if the reference potential is chosen with carethen one term in the expansion of powers of v-l and two in the expansion of powersof (&/kT) suffice for the quantitative prediction of the thermodynamic properties ofan assembly of Lennard-Jones molecules. This is shown in fig.2, in which theexperimental results are those for argon and the parameters in (9) are chosen to beThis choice has been shown by Monte Carlo calculations to represent a good efectivepair potential for argon at high densities.It is not important here that we knowthat (9) and (10) are not the true pair potential of the dilute gas.Elk = 119.8 K, CT = 3.405 A. (10)43-2-30 10 2 0(150.8 K/T)FIG. 2.-The phase diagram of argon. The full curves are the experimental results. The short-dashed curves are the results obtained by Barker and Henderson for the Lennard-Jones potential(9) and (10). The long-dashed curve is the melting line for an assembly of hard spheres with thesame collision diameter. The rise of the calculated melting pressure above this line at high tempera-tures is a consequence of the " softness " of the Lennard-Jones potential34 PROPERTIES OF LIQUIDS AND SOLUTIONSIt should be emphasized that these results, satisfactory though they are, are nota complete statistical theory of liquids.The basic result on which all perturbationmethods depend is a good knowledge off(n), the equation of state of an assembly ofhard spheres in both fluid and solid states. However, the good agreement withexperiment shown in fig. 2 does tell us much about the structure of a simple liquid.Since only two terms in the expansion of (&/kT) suffice (and even one gives quite goodresults) it follows that the resulting g(r) does not depart radically from that of a systemin which the forces are purely repulsive. In fact, the " softness " of the repulsiveforces at and near r = B has probably at least as great an effect on the form of g(r)as the presence of attractive forces. Why this perturbation expansion convergesso rapidly is still a mystery, since (&/kT) is about 2 at the triple point, and since therange of the potential (9) is not large compared with 0, nor with the range over which[g(r)- 11 is non-zero.However, the convergence is a fact and leads us to the con-clusion that the pair distribution function in a liquid is determined primarily by therepulsive forces between the molecules, and not by the attractive forces. The latterare less specific in their effects but are responsible for the large negative configurationalenergy and for generating the large internal pressure which maintains the high density.Internal pressure is a concept which cannot be used with precision except in avan der Waals fluid for which we can writekTnf(n) = p + a n2,total = external +internal pressure.The structure of a dense van der Waals fluid is determined only by the " totalpressure ", kTnf(n), however this is generated, whether by a high external pressureor by a high internal pressure.For real liquids we conclude that qualitatively thesituation is much the same although we do not then have the clear division shown inIn this review the computer experiments on hard spheres have been used as abasis for deciding what are the dominant factors in determining the structure ofsimple liquids. The methods of statistical mechanics provide, in principle, the meansfor determining a priori the properties of the hard sphere system, and so providef(n) without the need of computer experiments. This can be done in practice onlyafter the introduction of approximations into the formally exact equations ofstatistical mechanics.Such approximations were discussed fully at the GeneralDiscussion of the Society held at Exeter in 1967, and have been reviewed el~ewhere.~~There is little to add here since, after the burst of development between 1963 and 1967,there has since been little real progress. The position remains that the best of theapproximations, the second-order Percus-Yevick theory, can represent f(n) within1-2 % at all densities up to the transition point, that no theory can deal satisfactorilywith the transition, but that we have again an adequate treatment of the solid phaseat high densities.(1 1).MIXED LIQUIDSThe preliminary discussion of the properties of an assembly of hard spheres washelpful in formulating the factors that influence the structure of single liquids and soit is natural to see if this is equally true for mixtures.Moreover, since hard spherescan differ from each other only in their sizes, consideration of such mixturesemphasizes the important effects of differences of molecular size on the propertiesof real liquid mixtures, and hence paves the way for the discussion of polymersolutions.It is useful first to define the concept of randomness as it is applied to a mixture.g* lJ . S. ROWLINSON 35A mixture of N molecules, N a of species a etc., is said to be random with respect tothe chemical species if the frequency with which each distribution rl . . . ri . . .rNoccurs with species a at position i etc. is proportional to the product . . . (x& . . .,where xor is N,/N, or the mol fraction. In such a mixture the configurational integralis clearly equal to that obtained by taking an Q priori average of the configurationalenergy over the N assignments of the molecules to the N positions of each configura-tion. If we confine these configurations to those of a static lattice then the randommixture is one in which the neighbours of any one molecule are, on average, a randomselection from the remaining (N- 1) molecules. However, this definition of random-ness is not restricted to lattice models.The idea of a random mixture is an attractive one and if the potentials are all ofthe Lennard-Jones (v, p) type then it is possible to specify exactly the configurationalfree energy of such a mixture in terms of that of any one of the components and theappropriate set of parameters (gap, Gap).An assembly of mixed hard spheres is adegenerate case of a Lennard-Jones mixture and so the free energy can be so cal-culated. Unfortunately, the result is absurd.ll The mixture behaves as if all themolecules had diameters equal to that of the largest molecule present, however smallthe concentration of the species to which it belongs. Clearly if any position rl isto be able to accommodate any molecule then it must not be within a,,, of any otherposition, where omax is the diameter of the largest species.The random mixture is, however, a good approximation for mixtures of Lennard-Jones molecules of equal sizes and different energies since in this case departures fromrandomness are induced only by the differences in the Boltzmann factorsexp (- E~~ JkT) - a classical order-disorder problem much studied by physicists, andfor which there are adequate solutions, such as the quasi-chemical approximation.Since the approximation of random mixing is quite inappropriate for mixturesof hard spheres we cannot expect it to be satisfactory for mixtures of molecules ofdifferent sizes with steep Lennard-Jones potentials, e.g., v = 12.It is moreover sofar from the truth that it cannot serve satisfactorily as the starting point for perturba-tion schemes in which a measure of the departure from randomness is the expansionparameter.Clearly a statistical treatment of real mixtures must start from a position whichat least does justice to the simple case of mixtures of hard spheres.What are thefacts? If we mix two grades of sand or other powdered material, or if we mix ballbearings of different sizes we find a small contraction in the total volume. Computerstudies 13* l4 on mixtures of hard spheres at a constant pressure lead similarly tosmall negative excess volumes. Hence, since the configurational energy is zerowe have, in the last case, for the excess free energyGE= VEdp<O. 1:With real liquid mixtures it is difficult to determine experimentally that part ofthe observed GE which arises specifically from differences in molecular size, since wehave no means of choosing binary mixtures with gI1 = c12 = cZ2, whilst oll # c ~ ~ ~ .However, Monte Carlo simulation of a mixture of Lennard-Jones molecules withthese properties can provide us with the missing experimental results. Singer l5has studied a system at a temperature equivalent to 97 K, with ell, g12 and E~~equal to 133.5 K, with o12 = 3.596 A and a22 larger than all by up to 27 %, thatis, a molecular volume ratio of up to 2.1.012 = $a,, +;a, (Lorentz rule) (13)He choose36 PROPERTIES OF LIQUIDS AND SOLUTIONSand so an expansion of GE in powers of (ol - crZ2) starts with a term of second order.Let us writeGE = Ax1x2#f2,whereand croo is the diameter of a reference species, in this case argon (10).Singer'sresults show thatA = -200f100 J mol-l, (16)which is negative, smaller than RT (810 J mol-l) and much smaller than -uo, thenegative of the configurational energy of argon, which is 5500 J mol-'.This small negative value suggests that mixtures of Lennard-Jones molecules,like those of hard spheres, can pack together more economically than the puresubstances at the same external pressure.This is not surprising and, in extremecases, one can see how small molecules could fit into the holes between larger mole-cules. However, such structures are the antithesis of a random mixture, if theword random is defined as above.It has been shown that the generalized van der Waals equation (7) is of reasonablycorrect functional form to describe the behaviour of single liquids and, in particular,that it does justice to the dominant role of the repulsive forces in determining g(r).There is only one way to extend the equation to mixtures, and that is to allow theparameters a and b to be quadratic functions of composition :where aap and bprS are the parameters appropriate to the a-p interaction and axand bx are the parameters that describe a single fluid whose thermodynamic propertiesare those of the actual mixture, with the obvious omission of the ideal entropy ofmixing.A system of equations of this form is called a one-fluid approximation.16These equations have been applied, with success, to real mixtures both with theoriginal form of the van der Waals * equation,17~ l8 and with a more modern version* The adjective van der Waals has been used in this field with several different meanings.Wehave first the original equation of state which he proposed in 1873, viz., that obtained by substituting(8) into (7). There is then what may be called the generalized van der Waals equation? which isthe correct equation of state of an assembly of molecules whose intermolecular potential is given by(3). This equation of state is (7), wheref(n) is now the true equation of state of an assembly of hardspheres, or, sometimes, the Percus-Yevick approximation to it. Both the original and the generalizedvan der Waals equations lead to a classical critical point, that is, to one with the characteristicsdescribed briefly below eqn (8). This type of critical point is, therefore, itself often described as avan der Waals critical point even although its characteristics are a necessary consequence only of theexistence of an equation of state that is analytic in form, and not of the particular eqn (7); that is,(7) is a sufficient but not a necessary condition for a van der Waals critical point.Such a point isanalogous to the Weiss model of a Curie point and to the Bragg-Williams model of an order-disordertransit ion.The original 171 '* and generalized l9 van der Waals equations of state have both been extendedto mixtures by representing a and b as quadratic functions of mol fraction (17). However, theessence of (17) can be divorced from these two particular equations of state by writing the corre-sponding equations, (19), for the molecular parameters f and h of any conformal potential (e.g.,Lennard-Jones), and then using these with the actual properties of a reference substance, as in (20).It is the essential identity of (17) and (19) which justifies the name vun der Wuals approximation furmixtures for the set of equations (19)-(20) and their further extension 2o to (21)-(22)J .S . ROWLINSON 37in whichf(n) in (7) is taken from the Percus-Yevick re~u1ts.l~ However, it is perhapspreferable to extend (17) at once to any conformal mixture by writing it, not interms of a and 6, but of the equivalent molecular parameters (fh) and h, where h isthe ratio defined by (1 5) and f is a similar ratio for the energy,fap = Eabl&Oo* (18)We have then the following recipe l1 for the calculation of the Gibbs free energy of amixture from the parameters hs and ha, and a knowledge of Go, the free energyof a reference substance of parameters E~~ and goo.If all hap = 1 then this result is identical with that obtained for a random mixture(see below). It can be modified to include some of the energy-induced departuresfrom randomness by writing a two--uid approximation 2o in which G, is given byG ~ [ P , T ] = J,G,[PhaIfa, TIfal -RT In ha* (22)These equations can be compared with those for the random mixture 9* lo (aone-fluid approximation) for Lennard-Jones (v,~) molecules.Here G, is given by(20) andThe corresponding two-fluid approximation is obtained from (21)-(23), and is the" second refined version of the average potential model " of Prigogine and hiscolleagues 21* 22 (called here the average potential model, for short).We can now compare the performance of these four approximations, and of twoother widely used treatments 23-the regular solution of Hildebrand and the Flory-Huggins approximation-by calculating the value of GE for molecules that differonly in size. In each case it is convenient to adopt Lorentz's rule (13) and computethe coefficient A of (14).The method of calculating this for the first four approxi-mations is described elsewhere.1° For the regular solution of Hildebrand we have6, = -ua/va, (25)where zia and va are the molar configurational energy and volume of component a.If we write u1 = u2 = zi and v1 = v2(1 + 6) we obtain on expansionFor the Flory-Huggins approximation we havewhenceA = -$u. (26)(27) GE = RT[xl In vl +x2 In u2 -In ( q v , +x2v2)],A:= -+RT38 PROPERTIES OF LIQUIDS AND SOLUTIONSThese results together with their numerical values for argon at 97 K are collected intable 1.(For simplicity, certain terms of the order of pu have been omitted ; theseare negligibly small.)The conclusions to be drawn from this table are clear. The approximations ofboth the random mixture and of the average potential model are bad; the coefficientA has the wrong sign and is numerically too large. The regular solution of Hilde-brand is less inadequate, but also leads to a positive value of A. The remaining threeapproximations are in good agreement with the Monte Carlo results of Singer.TABLE 1 .-THE COEFFICIENT A OF (1 4) AT 97 K AND ZERO PRESSURE. IN THE RANDOM MIXTUREAND AVERAGE POTENTIAL MODEL THE INDICES OF THE POTENTIAL HAVE BEEN CHOSEN TO BEv = 12, p = 6.approximation A J mol-1Monte Carlo experiment - -2200fl00van der Waals (one-fluid) -3 RT - 270van der Waals (two-fluid) -(5/24)RT - 170random mixture (one-fluid) -2~-(19/12) RT +9900average potential (two-fluid) - U- (516) RT +4900regular solution (Hildebrand) -$ u + 1400Flory-Huggins -3 RT - 400Fig.3 shows that even for values of 42 as large as unity (that is, a volume ratio of 2)the van der Waals approximations accord well with the results. The Flory-Hugginsapproximation appears to lead to slightly too large a negative excess free energy,but the result is not unsatisfactory since this approximation was not intended foruse with such small volume ratios.In practice, it is found that the performance of the three theories that yield positivevalues of A is not as bad as might be expected from table 1.The reason is probablyto be found in an entirely different effect arising from the energies of interaction.Table 1 has been computed for the special case ell = c12 = E~~ in order to emphasizethe contribution of size effects to GE. This equation for the cross-energy e12 is aparticular case of the more general approximation,E l 2 = (EllE22)4 (Berthelot’s rule) (28)This rule is commonly used in making comparisons of theory and experiment formixtures of non-polar molecules ; indeed, it is a necessary rule in Hildebrand’s regularsolution.There is, however, good evidence from the properties of both gas andliquid mixtures that it yields values of c12 which are too large by 1-3 % for most non-polar 2o Such a departure from Berthelot’s rule produces a positivecontribution to GE which is first-order in the differences of E parameters, and whichis generally much larger than the small second-order term (14). Hence, most mixturesof non-polar molecules of different energies and sizes have positive GE. The failureof some of the theories to do justice to the size effects is, therefore, often compensatedby the choice of too large a value of e12.If the size ratio is appreciably greater than 2 then the negative contribution to thefree energy cannot be ignored. Thus, for the system carbon tetrachloride + octa-methylcyclotetrasiloxane (volume ratio, 3.2) Marsh 24 found GE to be about - 160J mol-l in the equimolar mixture at 45°C.Mixtures of n-alkanes have been studiedat sufficiently low pressures for GE to be measured in which the volume ratio is aslarge as 5. Such systems have negative excess free energies,25 and this result waJ . S . ROWLINSON 39expressed in quantitative form in the principle of congruence of Brarnsted andKoefoed.26 This stated that the configurational free energy of a mixture of n-alkanesis determined only by the ideal free energy of mixing and by the arithmetic mean ofthe number of carbon atoms per molecule in the mixture. This assumption requiresthat GE has the formGE = A'xlx2(nl -n2)2, (29)where nl and n2 are the numbers of carbon atoms in each n-alkane, and where theparameter A' is equal to -2.7 J moI-l at 20°C.This equation is strikingly similarr 0I -0.08FLODY - MUGGINS \\FIG. 3.-The excess free energy of mixtures of molecules that differ only in size, (13)-(16). The barsare the Monte Carlo results of Singer, and the curves the values calculated from the Flory-Hugginsequation (27), and the two van der Waals approximations, (19)-(20) and (19), (21)-(22).to (14) if A in the former is given the small negative value required by the MonteCarlo calculations and by the better theories, but the forms are not identical since(nl-n2)2 is proportional to t$2 only for a narrow range of n, and n2. Indeed, weshould not expect identity of form since n-alkanes which differ in the number ofcarbon atoms (n, # n,) have necessarily el, # E ~ ~ , and, moreover, since the propor-tions of CH3- to -CH2- groups is not the same in each component we haveno assurance that Berthelot's rule (28) is obeyed.The principle of congruence was put forward as an empirical generalization in1946, and although Longuet-Higgins 27 derived it in 1953 from the properties of theconfigurational integral for a mixture of chain molecules, he was forced to make someassumptions that would restrict the principle more narrowly than seems to be thecase in practice.However, the alkanes form a set of substances with obvious regu-larities in the thermodynamic properties and it has therefore been natural to seek 40 PROPERTIES OF LIQUIDS AND SOLUTIONSmore general principle under which both these regularities and the principle ofcongruence could be subsumed.This desire has led to an extension of the principleof corresponding states by Prigogine and his colleagues 21 which is justified onplausible but not rigorous arguments. This development is not described heresince Hijmans and Holleman 2 5 have recently reviewed it in detail, and since theargument does not readily lend itself to a short summary. Moreover this approachhas been fruitfully extended by Flory and his colleagues 28 by combining it with theuse of a reference equation of state of the generalized van der Waals form, and Ido not wish to trespass further on the field of our Spiers Memorial Lecturer.However, there is one aspect of the results above which should be consideredfurther, and that is the relative importance of the factors of size and shape of thelarger molecule in determining the free energy.The approximations compared intable 1 were, with the exception of the Flory-Huggins equation, devised primarilyfor mixtures of molecules which are spherical but of different size. Mixtures ofn-alkanes and solutions of linear polymers contain molecules that are of similardimensions in two directions but widely different in the third. Even if the chain isflexible and partially coiled it is not clear that it will resemble in its properties aspherical molecule of equal molar volume. I do not think that we know, for liquidsolutions, precisely how far this important difference may vitiate conclusions drawnfrom table 1.There are lattice studies which give us estimates of the number ofways of laying down molecules of fixed numbers of segments but of different shapes.The results are only an imperfect guide to the behaviour of liquid solutions, becausea lattice is not a good representation of a liquid, but they do suggest that shape isnot of prime importance. The free energy of solutions of triangles is close to thatof solutions of linear chains with 3 units, and that of solutions of tetrahedra close tothat of chains of 4 units, when both are calculated in the quasi-chemical approxi-mation. All are adequately described by the Flory-Huggins equation. However,more exact statistical treatments of the combinatorial problem in two dimensionsshow more substantial difference^.^^^ 30.For a cross-shaped pentamer in a solutionof monomers the system can separate into two phases 30* 31 ; a result which is in-compatible with the Flory-Huggins equation or with the small negative excess freeenergy of (14)-( 16). Differences of shape are exaggerated on a two-dimensionallattice and it is probable that the quasi-chemical results are a more adequate guide tothree-dimensional behaviour.There is other evidence, mathematically less precise but perhaps physically morerelevant, which suggests that shape is not of prime importance in this field. We knowthat some mixtures of n-alkanes and almost all polymer solutions separate into twoliquid phases at high reduced temperatures of the When this occurs thereis no longer a continuous locus of gas-liquid critical points connecting Tc,pc of onecomponent with Tc,pc of the other (or running towards TC of the other if it is chemicallyunstable at high temperatures).The locus is broken at temperatures near Tc of thesolvent by the appearance of a second liquid phase. Hydrocarbon mixtures canexhibit a variety of phase behaviour at high pressures and temperatures of which thisbroken critical locus is the simplest example. Patterson and Delmas 33 have analyzedthis behaviour in terms of the extension of the principle of corresponding states tochain molecules by Prigogine 21, and with the assumption of a generalized van derWaals equation of state, as used by Flory.28 Their model, which therefore takesexplicit account of the change of shape with increasing chain length, is able to accountfor the complicated phase behaviour of real systems.However, the same istrue,l0* 34* 35 at least qualitatively, for the original van der Waals equation if a1 # a22and a1 < (al This second result makes it unlikely that the complicated phasJ . S . ROWLINSON 41behaviour of hydrocarbon mixtures is a specific effect of the difference of chainlength. It is apparently primarily an energy effect.A more direct test of the effect of molecular shape on this aspect of the problemfollows from some recent (and incomplete) calculations made in this depa~tment.~~Solutions of n-alkanes in methane are completely miscible up to n-pentane.Forn-hexane and above there is a range of temperature in which there are two liquidphases. The free energy of such mixtures can be calculated from the one-fluidvan der Waals approximation (19)-(20) by taking the reference substance, 0, to bemethane and using the experimental results for G,[p,T] to represent the free energysurface for this substance. The parameters fll, hll, f22 and h22 can be obtainedfrom the Tc and pc of solute and solvent (i.e., substance 2 is here identical with thereference substance), whilst the cross-parameters fl and h12 can be eliminated by theLorentz-Berthelot rules, (13) and (28). This treatment leads to the result that n-octaneis the first alkane to be immiscible with methane, and so is in moderate agreementwith experiment.(The discrepancy between octane and hexane is probably a con-sequence of the use of the Berthelot, or geometric mean rule for f12.) The form of(20) implies the use of the simple principle of corresponding states, and so this cal-culation has been made on the false assumption that n-octane etc. obeys the samereduced equation of state as methane; that is, that its molecule is larger in the ratioof (Tc/pc) for solute to (TC/pc) for methane (5.5 for octane), but still approximatelyspherical.This assumption can be removed by representing the departure of the reducedequation of state of the solute from that of methane by means of a single additionalparameter which is a measure of the eccentricity of its molecules.37 The best knownof these, and the one used in these calculations, is the acentric factor o of Pit~er.~*If the calculation is now repeated then it is found that there is scarcely any change inthe p,T projection of the critical locus at temperatures near the critical point ofmethane, and hence in the region of liquid immiscibility.This is, in part, a con-sequence of the low concentration of solute along the critical locus at these tempera-tures, which means that it is the " shape " of the solute-solvent interactions that areimportant, not the solute-solute. But whatever the cause it seems to be the case thatshape is a less potent factor than energy and size in determining the range of immisci-bility in such solutions.Throughout the second part of this review I have considered only the free energy,and have discussed in detail only that part of it which arises from differences inmolecular size.This is a small and subtle effect which, outside the polymer field,had received little attention until recently. The reason for this neglect was thegreater attention paid to the effects of differences in the energies, gll, el, and eZZ,for in simple mixtures these usually account for the greater part of GE, HE, VE andeven SE. However, it is the study of the less easily comprehensible size effects which,I hope, will lead to the reconciliation of the methods used to discuss polymeric andnon-polymeric solutions. Our understanding of these size effects has improvedconsiderably in the last few years but it is still far from perfect.Nevertheless weneed not share the pessimism of Helmholtz, who wrote in a letter in 1891 that,39" thermodynamic laws in their abstract form can only be grasped by rigidly trainedmathematicians, and are accordingly scarcely accessible to the people who want todo experiments on solutions and their vapour tensions, freezing points, heats ofsolution, etc."I thank Dr. K. Singer for his permission to use his Monte Carlo results, shownin fig. 342 PROPERTIES OF LIQUIDS AND SOLUTIONSW. W. Wood (chap 5) and J. D. Bemal: and S. V. King (chap. 6) in Physics of Simple Liquids,ed. H. N. V. Temperley, J. S. Rowlinson and G. S. Rushbrooke, (North-Holland, Amsterdam,1968).M. Kac, Phys. Fluids, 1959,2,8 ; M. Kac, G. E. Uhlenbeck and P. C.Hemmer, J. Math. Phys.,1963, 4, 216, 229.H. C. Longuet-Higgins and B. Widom, Mol. Phys., 1964, 8,549.J. S. 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