FIFTEENTH SPIERS MEMORIAL LECTUREThermodynamics of Polymer SolutionsBY PAUL J. FLORYStan ford University, Stan ford, California, U. S . A.Received 1 1 th May, 1970The Faraday Society has materially aided and encouraged the study of liquidsand solutions by generously allocating pages of its Transactions to these subjects andby devoting a number of its inimitable Discussions over the years to topics in thisarea. These meetings have brought together the most important contributions fromthroughout the world. The present Discussion, another in this distinguished series,is especially timely and, by design or coincidence, is on a subject to which Prof.Geoffrey Gee, President of the Faraday Society, and his collaborators have mademonumental contributions. In view of these precedents, it is a very special privilegeto have been invited to deliver one of the main lectures, and, even more, for theFaraday Society to have attached the singular honour of designating it the FifteenthSpiers Memorial Lecture.The unsurpassed traditions of the Faraday SocietyDiscussions owe much to the initiative and vision of Frederick Solomon Spiers, thefirst secretary of the Society, to whose memory these Lectures are dedicated. I amdeeply indebted to the Faraday Society for according me the rare privilege of deliveringa lecture in his honour.The treatment of liquid solutions has long been dominated by theories designedprimarily for mixtures of molecules of equivalent size and shape, the fact notwith-standing that examples meeting these terms are comparatively rare.The entireframework of classical solution theory was developed with reference to mixtures ofinterchangeable species. Textbooks introduce such mixtures as the prototypes forsolutions in general; the ideal, or perfect, solution is represented as the norm, anddepartures therefrom are regarded in a manner befitting the connotations of “ non-ideal ” and “ imperfect ”. The importance of stipulations of equivalence of molecularsize, shape and fields of force in the derivation of the ideal solution laws was earlyrealized, but the matter is complicated by the axiom that all solutions of monodispersesolutes must, on the most general grounds, be asymptotically ideal with reference tothe solvent in the limit of infinite dilution of the solute.The range over which thisasymptotic conformance to Raoult’s law holds eludes specification in general terms,and, in the course of evolution of the subject, this absence of a clearly defined limitengendered the hope that the ideal law mi& in fact enjoy a much greater range ofvalidity than warranted either by its elementary derivation from statistical-moleculartheory or by the Gibbs-Duhem relationship in conjunction with Henry’s inexorablelaw for sufficiently dilute solutions.The laws of solutions framed by Raoult and van’t Hoff were hailed in the late1880’s as the basis for determining molecular weights of substances not easilyvapourized. The importance of extrapolating colligative measurements to infinitedilution was emphasized. Yet, application to polymers, the class of materials forwhich these methods alone were qualified, was held in abeyance by the doctrine of7”8 THERMODYNAMICS OF POLYMER SOLUTIONSthat time according to which colloidal materials constitute a special state of matterbeyond the scope of ordinary physio-chemical laws.Not only were molecularweights of macromolecular substances unknown or grossly underestimated for halfa century ; the equilibrium properties of their solutions, as manifested in solubility,swelling, miscibility, and vapour activities, were discussed and interpreted in termsremoved from developments in solution theory and especially in isolation from thedictates of thermodynamics. Thus, while investigators of solutions of low molecularsubstances often erred in disregarding the limitations of ideal solution theory in thecase of mixtures of components differing appreciably in molecular size, mixturescontaining polymers were deprived of even the limited elucidation that the thenexisting knowledge of solution theory could have provided.Hucke1,l in a comprehensive paper published in 1936, took up the question of theeffect of differences in molecular sizes of components on the equilibrium propertiesof solutions.He forcefully reasserted the precepts of solution theory, stressing thatthe ideal solution laws are inherently restricted to mixtures of interchangeable coni-ponents, or, if this condition is not met, to sufficiently dilute solutions. It isincorrect, he pointed out, to ascribe departures from ideality to special causes suchas solvation or association when the systems do not meet the underlying conditionsfor adherence to ideality. For a chain molecule, the negative deviation from Raoult’slaw should not be ascribed to semi-independent kinetic motion of chain segments forthe same reason, namely, that this law is inherently inapplicable to such a system.Mixtures of molecules differing in size by a factor of the order of two were discussedfrom a thermodynamic standpoint by Guggenheim in 1937.In the same year,Fowler and Rushbrooke investigated the configurations of mixtures of sphericalmonomers and dumbbell-shaped dimers with emphasis on the extremities of thecomposition range. Deviations from ideality were indicated to be small in both ofthese investigations.Thus, the ideal solution laws seemingly might yet survive asthe norm for liquid solutions in general. The studies cited did not, however, extendinto the range of large disparities in the molecular sizes of the components.On the experimental side, in 1934 Brarnsted and Colmont published a study ofthe activities of mixtures of benzene and of n-propyl bromide with long chain esters(e.g., di-n-butyl sebacate). Similar, more comprehensive studies on mixtures ofvarious mono- and di-esters of high molecular weight with volatile solvents werereported by Meyer and Luhdeman in 1935. Both investigations revealed substantialnegative deviations from Raoult’s law, i.e., at finite concentrations the long-chainester molecules depress the activity of the solvent to a greater degree than in proportionto their number. Early iiivestigations 6-9 yielded abundant evidence that polymersolutions exhibit much larger deviations from ideality, especially at higher concentra-tions. Meyer lo directed attention to the very large number of internal configurationsaccessible to a long-chain polymer.He concluded that the large negative deviationscharacteristics of solutions of polymers in low molecular solvents are to be explainedon this basis. The concept was represented by a polymer chain consisting of aconcatenated sequence of spherical segments assigned to a contiguous set of sites ofa regular lattice. The explanation misses the mark, as I shall have occasion to pointout, but the graphic model presented had its impact.Meyer lo did not undertaketo pursue its implications by attempting quantitative treatment of his lattice scheme,or model, preferring to regard it as a qualitative device. It was adopted by others *12shortly thereafter, however, for the derivation of the essential relation of polymersolution theory, namely, the expression for the combinatory entropy of a mixtureof long chains of segments with small, approximately spherical molecules of solvent.While the investigations briefly outlined above, and those which foffowed aPAUL J . FLORY 9discussed below, firmly established the importance of disparity of molecular sizesof the components of a solution, models advanced for interchangeable sphericalmolecules have retained pre-eminence as subjects for a theoretical science addressingitself to furtherance of the understanding of liquids and solutions in general.Majoradvances in theory in this area have been achieved within the past decade. Theywere the subject of a most timely Discussion of the Faraday Society in 1967. Thesignificance of these achievements have been vividly portrayed by Prof. Rowlinson l 3in his paper opening the present Discussion. Yet, it should not be supposed that thesuccessful treatment of the simplest of all classes of liquids leads straightforwardlyto the elucidation of much more complex liquids consisting of polyatomic moleculesof various sizes and spatial forms. The far more numerous liquids in the lattercategory present circumstances not shared by the comparatively rare examplesconsisting of spherical molecules.A model suitable for the preponderance of liquidsmust take account of molecular features absent in the most elementary exampleswhich have attracted the mainstream of attention. In particular, the mathematicaladvantages of spherical symmetry must perforce be abandoned. This is not todisparage the spectacular recent achievements in the theory of liquids and solutionsconsisting of monatomic or approximately spherical molecules. Rather, it is toassert that the conceptual scheme required for interpretation of common liquids andsolutions must depart from the theories advanced for the most erudite treatment ofspherical molecules.From a slightly different viewpoint, we may note that liquids of any kind constitutea state of matter fraught with complications shared neither by gases nor by crystallinesolids.To be effective, any conceptual scheme, or theory, of liquids and solutionsmust entail approximations either in model or in mathematical technique, even forthe simplest of real liquids. It is our contention that the best model or conceptualscheme and the associated approximations for treating simple liquids should not beexpected likewise to be most efficacious for liquids and solutions in general. Adifferent scheme, or model, will be required for the latter. One cannot proceeddeductively from the simple to the complex without drastic innovations. Compati-bility of viewpoints adopted for the comprehension of different types of liquids isessential, of course, but insistence on adherence to one scheme for all would eitherlimit the depth of treatment of the simplest examples, or it would seriously obstructcomprehension of the more complex cases.COMBINATORY ENTROPY FOR MIXTURES CONTAINING CHAIN MOLECULESTypically, a polymer solution comprises a low-molecular weight solvent and asolute consisting of long chain molecules having lateral dimensions commensuratewith those of the solvent but with lengths many times greater.The solvent moleculesmay often be regarded as approximately spherical, by comparison with those of thehighly anisometric solute at any rate. In order to retain some of the obvious advant-ages of interchangeability of constituents for purposes of enumeration of configura-tions, it is expedient to conceive of the polymeric solute molecules as catenationsof segments, a segment of the chain being so defined as to be interchangeable with amolecule of solvent insofar as their spatial requirements are concerned.If the solventmolecules are likewise chain-like, though of shorter length, they too may be subdividedinto segments in order to facilitate enumeration of configurations. In any event,the segment is so chosen as to be an acceptable subdivision of both components.The required enumeration is conveniently carried out by resort to a hypotheticallattice consisting of a three-dimensional array of identical sites each of which i10 THERMODYNAMICS OF POLYMER SOLUTIONSoccupied by one, and only one, segment of one or another of the component species.The succession of segments comprising a given solute molecule (or solvent molecule, ifit also is chain-like) must, of course, occupy a series of consecutively contiguouslattice sites.Huggins l 1 and the author l2 independently derived expressions on this basisfor the total number of configurations of a liquid mixture formed from n1 solventand n2 polymeric solute molecules. The essential step involves estimation of theexpected number vc of situations accessible to the [-th chain molecule added to thelattice, initially empty. Each molecule is assigned a location at random and itsinternal configuration (i.e., the configuration of the sites to which it is assigned) islikewise unconstrained.If, for simplicity, we take the solvent molecule to consist ofa single segment, the total number of sites is no = nl +m,, where r is the number ofsegments in a polymer chain or the ratio of molar volumes of the two components.The quantity vs is identified with the number of different sets of r contiguous sitesthat are vacant in the lattice having no - ([ - 1)r vacant sites. If, as a first approxima-tion, we treat the (c- 1)r segments previously added as uniformly distributed through-out the lattice, then the required expectation is just the product of the total numberof vacant sites (accessible to the first segment) multiplied consecutively by the expectednumber of vacant sites surrounding each of the remaining sites required for the [th-chain, i.e.,where z is the coordination number of the lattice ; hence, z - 1 is the number ofeligible sites for the ith segment exclusive of the neighbouring site occupied by the(i- 1)th segment.Taking the serial product of the vc for the total number of configura-tions,the latter expression having been obtained through introduction of Stirling’s approxi-mation and by subsequent rearrangement of factors. The configurational entropyfollows asS = k l n Q = kn2((r-1)ln[(z-l)/e]+1nr)-k(n,ln+,+n,In+,), (3)where k is Boltzmann’s constant and and +, are volume fractions of the respectivecomponents.The separation of eqn (2) for Q into the two factors in brackets, and the corre-sponding separation of the entropy into two terms as in eqn (3), is a feature of foremostimportance.The first factor in eqn (2) represents the number of configurations forthe undiluted polymer. Correspondingly, the first term of eqn (3) expresses theentropy of disorientation of the pure liquid polymer according to the lattice treatment.The second factor in eqn (2), and the second term in eqn (3), express the effect ofdilution. Thus, we designate the latter as the combinatory entropy of mixingA ~ c o m b = -k(n, In 41 +n2 In + 2 ) , (4)the adjective and subscript being included in the expectation of the need to acknow-ledge other contributions to the entropy, as discussed in a later section of this paper.The obvious generalization of eqn (4) for any number of polymeric components isstraightforward.Before elaborating the implications of the separation of factors, or terms, citedabove, it is appropriate to examine the origins of this feature and the extent to whicPAUL J .FLORY I 1it may be contingent upon artificialities of model and procedure. To this end wenote that the first factor comprises the two sub-factors (2- and (r/er-')?The former represents the total of the number of internal configurations for the setof n2 chains when separated from one another and hence out of range of mutualinterference. The latter sub-factor expresses the acute reduction of this very largenumber of configurations which occurs when the chains are brought together tofill the same space subject to the condition of volume exclusion, i.e., when theyintermingle within the same lattice each of whose sites may be occupied by one andonly one segment.Carrying the description a step further, we observe that the secondmajor factor in eqn (2), and correspondingly the second term in eqn (3), takes accountof the moderation of this severe attrition factor for volume exclusion when a fractionof the lattice sites is reserved for solvent and hence left unoccupied by segmentsconnected in polymer chains.Both sub-factors of the first quantity in brackets in eqn (2) are subject toinaccuracies inherent in the model. The latter one appears to be a reasonable estimateprovided that the segment is so defined as to have a length approximating the breadthof the chain. But the former sub-factor is altogether a figment of the lattice, for ittakes no account whatever of the configurational characteristics of the chain itself.This error is easily remediable.Although correction is not required for our purposes,a simple alternative procedure that avoids the commission of the error is outlinedbelow inasmuch as it serves also to clarify the basis for separation of the several factorsoccurring in R.Instead of assigning flexible chains to a lattice according to the procedure setforth above whereby internal configurations and the competition for space are takeninto account simultaneously, we may first fix the chain in one of its internal configura-tions and then assess the expected number of situations offered by the partially occupiedlattice to the chain in that configuration.The chain may then be rearranged toanother internal configuration and the process repeated etc., for all internal configura-tions. For the overwhelming preponderance of configurations, the expected numberof sites available to each successive segment of a chain will be given in satisfactoryapproximation by the same factors ( ( 2 - l)[n, - r(5 - l)]/no) appearing in eqn (1).The only exceptions are the comparatively " dense " configurations in which thesegments occur within a small volume with the consequence that the conditionalityof the probability of vacancy of a site (or volume element) is significantly affected bythe established vacancy of a succession of lattice sites available for preceding segments.Such configurations ordinarily are sufficiently rare to warrant their being ignored.Hence, the same intermolecular factors derived above must hold irrespective of thenature of the chain configurations (dense configurations excepted), and we haveonly to replace the spurious intramolecular factor (z - l)@--l) for each chain moleculeby a proper estimate of the effective number of its internal configurations.Theconfiguration partition function for the chain molecule furnishes this number. Itsformulation from bond parameters and rotational potentials according to theoreticalmethods l4 recently developed would carry us too far afield from the theme of thisDiscussion.The more immediate purpose of the exercise outlined in the paragraph above isto clarify the mutual independence of the internal configurations and their externalinteractions (i.e., volume exclusion, or competition for lattice sites in the idiom of thelattice). This independence is embodied in the assertion that the expected numberof configurations for a chain is, in good approximation, independent of itsinternal configuration (dense configurations excepted).The assertion is implicitin the original derivation; it is made explicit by the procedure described i12 THERMODYNAMICS OF POLYMER SOLUTIONSthe paragraph above. Arguments for its plausibility on physical grounds are bothself-evident and compelling ; they transcend artificialities of the lattice model.The separation of intra- and intermolecular factors in Q, and the correspondingseparation of terms in the expression for the configurational entropy has not beenseriously questioned, but its implications have by no means gained general acceptance.It follows from this separation that the thermodynamic properties of polymer solutionsshould be independent of the average configuration of the polymeric solute.Thus,the same kind of dependence of the activity on composition should be observed forstiff rods * as for random coils. Conversely, it follows with equal force that theaverage configuration of the polymer chain should be independent of dilution. Thisassertion has been challenged vigorously and is generally viewed with scepticism.It is at variance with superficial physical intuition according to which the difficultyof packing polymer chains to high density should force them to compromise theirconfigurational propensities. To be sure, at high concentrations the difficulties ofaccommodating a long chain molecule without violating the volume exclusion con-straint are acute, but it does not follow that some kinds of chain configurations willbe repressed to a greater degree than others.The volume exclusion constraint shouldin fact operate essentially without bias with respect to the preponderance of configura-tions. Correspondence of this physical argument to the analysis presented abovewill be apparent$If indeed the molecular configurations were appreciably perturbed by the stricturesof packing chain molecules to high density, then the thermodynamics of mixing athigh concentrations should reflect such effects.Thus, the order allegedly imposedin the liquid polymer must rapidly be dissipated by addition of solvent. The decreasein order should manifest itself in an enhanced entropy of dilution. Experimentalmeasurements of activities in concentrated polymer solutions give no clear evidenceof such an effect.Another feature of major significance is the absence of the parameter z in thecombinatory entropy as given by eqn (4). In the higher approximations of Huggins l1and of others 17-19 which take account of the conditionality of the probability ofvacancy of a site adjacent to one known to be vacant, AS,,,,, depends to a minordegree on z. The refinement probably is beyond the limits of reliability of the latticemodel, however.Experimental results seem to support this opinion ; significantimprovement in the agreement between experiment and theory is not generallyachieved for any realistic value of z.The absence of any parameter of the lattice in the expression for the combinatorialentropy suggests that the result is independent of the model used for its originalderivation. Alternative methods of derivation reinforce this inference. A methodintroduced in 1949 for the treatment of semi-crystalline polymers 2o is especiallyilluminating in this regard. It involves random combination of all molecules,solvent and solute alike, to form a single linear chain. The segments of this giantsuper-molecule are assigned consecutively to the lattice, or any other space, as setforth earlier.The artificial connections between molecules are then severed,whereupon the ends of molecules are relieved of the requirement to occur as adjacent* For stiff rods, high concentrations cannot be realized without imposing order on the rods. Theseparation of an anisotropic tactoidal phase with increase in concentration has been predicted ' ' 9 l6and is confirmed by experiments on solutions of solute molecules meeting this description.1- Specific local interactions with neighbour molecules may, of course, affect the configurationsthrough alteration of bond rotational potentials. Effects of this nature will be manifested moreprominently in the thermodynamics of mixing through the direct effects of interactions betweenspecies. They are peculiar to molecular components having polar or other groups prone to inter-actions of one kind or another, and should be of little importance in mixtures of non-polar moleculesPAUL J . FLORY 13pairs.The full equation (2) may be derived in this way.2o The entropy gained inthe final step compared with the corresponding gain for the pure liquid componentstreated in like manner is responsible for AScomb of eqn (4). This derivation obviouslyis independent of a lattice.21* 22The foregoing derivation reveals a very simple physical explanation for eqn (4),the basic mixing “ law ” for solutions of chain molecules. The combinatory entropyof mixing arises in its entirety from the greater space (volume) over which themolecules of both components are distributed in the mixture as contrasted to thepure components.The respective ratios of volumes are just the reciprocals of thevolume fractions (the small volume change on mixing being ignored). The additionof a molecule of polymeric solute contributes a far greater volume increase than theaddition of a molecule of solvent and this fact accounts for its effect, which, at finiteconcentrations, is much in excess of predictions from the ideal “law ”. Solventmolecules are far more numerous than those of polymers at intermediate compositions,and their dispersion over the greater volume of the solution makes the major contri-bution to the entropy of mixing (i.e., nl In +1 $n2 In #2 in eqn (4) for intermediatecompositions). Thus, the polymer chains contribute a large volume, the solvent alarge number of kinetically independent units.This elementary explanation of the basic polymer mixing relationship replacesMeyer’s intuitive one lo which attributed the marked deviations of polymer solutionsfrom ideality to the large number of configurations accessible to the polymer chain,an explanation which in fact continues to be the one usually offered at the presenttime.We have stressed that the configuration of the chain, be it a random coil ora stiff rod, is of little importance insofar as the mixing entropy for formation of therandom solution is concerned. It is the increase in volume over which moleculesmay distribute themselves that is all important.It should be borne in mind that the mixing expressions, eqn (2)-(4), have beenderived for mixtures of molecules differing in length but having approximately thesame cross section. This will be seen to be a necessary condition.In terms ofthe latter, and most direct, derivation of the polymer mixing law, the volume contri-buted by each species must be equally accessible to both species (or to end segmentsof both, if the derivation is followed literally). If, for example, the solute moleculesare large, dense, globular particles (e.g., dense spheres) instead of long chains, thenthe entropy of mixing will be substantially less than that for polymer chains of thesame mass. The interior of the globular particle is not freely accessible to othermolecules, as for a random coiled solute. While the mixing entropy for the solution ofglobular solute molecules may exceed that given by the ideal law, it will certainly beless than for a solute consisting of long chains.The contrast between a dense globular solute and a polymer chain becomes moremarked in the limit of infinite molecular weight for the solute.Macroscopic densespheres have a vanishing effect on the activity of solvent occupying interstices betweensuch particles, but a random polymer chain of macroscopic weight is readily permeableby the solvent. In the latter case, the second term of eqn (4) vanishes but the firstterm, which is the dominant one in any case except at very high solute concentrations,remains unimpaired. Thus, the activity al of the solvent in a polymer solution ofgiven volume fraction 0 @ c 1 rapidly approaches an asymptotic value a, < 1with increase in molecular weight of the solute.For a solution of dense globularparticles, a+l in the same limit.Polymer solution theory has frequently been applied to mixtures in which themolar volume ratio r of the components is comparatively small, e.g., in the range2-5. Much depends on the Here the suitability of the theory is less well defined14 THERMODYNAMICS OF POLYMER SOLUTIONSgeometrical forms of the molecular species. If they are homologous chain molecules,then polymer solution theory should hold. If their diameters, or cross-sections,differ notably, then a mixing entropy between the ideal and the polymer formulationsmay be expected. The entropies in excess of ideality are in any case small for volumeratios r in this range.13 Hence, the choice between the two entropy expressionsmay be marginal.COMPARISON OF CONVENTIONAL THEORY WITH EXPERIMENTThe partial molar entropy obtained by differentiation of eqii (4) isASl.comb = -“n (1 - 4 2 ) +(I - W M 2 1 (5)where R is the gas constant.If the free energy of mixing is taken to be the combina-tion of -TASComb with a van Laar enthalpy of mixing, one obtains the familarexpression( P I - P W T = In (1 - 4 2 ) + (1 - w 4 2 +x4f (6)for the chemical potential of the solvent, where x is the dimensio%less quantityformally defined byAwl, being the exchange interaction energy (per mol).Eqn (4)-(7) are the primary relationships of conventional polymer solution theory.The deductions following from them are well known.They succeed in accounting,approximately, for the principal thermodynamic properties of polymer solutions :the dependence of the activity on composition, the depression of the melting point ofcrystalline polymers by diluent, the solubility of crystalline polymers at high dilution,the swelling of cross-linked gels, liquid-liquid miscibility, and the dependence of thedimensions of random-coiled polymers on the solvent medium in a dilute solution.The agreement between theory and experiment is not quantitative, however, andcareful scrutiny of experimental results reveals a pattern of departure from theorywhich is manifested consistently by various polymer + solvent systems. We shallnot attempt to review comparisons between experiments and the theory in all itsramifications.Instead, we shall focus on the chemical potential of the solvent andits dependence on composition and temperature.For the stated purpose it is useful to treat x in eqn (6) as an empirical quantity,after the manner of Booth, Gee, Jones and Taylor,23 the definition of this quantityas the exchange enthalpy parameter according to eqn (7) being abandoned. Solvingeqn (6) for x we haveThus, chemical potentials obtained from activities, or other experimentally measuredquantities, yield empirical values of 2. If the conventional theory were strictlycorrect, these values would be independent of composition and inversely proportionalto temperature (on the assumption that the exchange interaction is exclusivelyenthalpic).Let the residual chemical potential ,uy be defined byin conjunction with eqn (8). It will be seen to be the analogue of the excess chemicalpotential, the ideal combinatorial entropy of dilution being replaced by the expressiongiven by polymer solution theory. Accordingly, x may be called the reduced residualciiemical pot en tial.x = zAw,~/RT, (7)x = (PI -P;)/Rw:-Cln ( 1 - 4 ~ ) + ( 1 - l / r ) ~ ~ l / ~ ~ . (8)x = P W w : (9PAUL J . FLORY 15The reduced residual enthalpy of dilution, or reduced enthalpy of dilution, is formallydefined byand the reduced residual partial molar entropy byXH = AWi/RT&$ = - T(axlaT), (10)-Obviously,Xtr+Xs = x. (12)The quantities xH and xs correspond * to K and $--$ in terminology used hereto-fore.25* 26 Dependence of x, and likewise of xtl and xs, on composition may beexpressed by forming a series in powers of the volume fraction (or segment fraction,cf.following). Thus,x = x l + X 2 4 2 + X 3 4 ; + - . * : (13)Osmotic measurements at low concentrations and vapour pressure determinationsat higher concentrations invariably yield positive values of x for non-polar polymer +solvent mixtures. Negative values of x are rare even for polar mixtures in whichthe components display strong mutual attractions. A number of examples arelisted in table 1, where the limiting values, xl, of x at q52 = 0 are given in the secondcolumn. Resolution of x into its enthalpic and entropic contributions [see eqn(10)-(12)], either from precise determinations of x over a range of temperature orfrom calorimetric measurements of the enthalpy of mixing, shows xs generally to bethe major contributor. It is especially large in those cases where xH is small ornegative.Recalling that the contribution, - Asl ,comb/Rq5$ of the combinatoryentropy of dilution to the reduced total chemical potential, (,ul -py)/RT&,2, is -3in the limit &2-+0 [see eqn ( 5 ) and (6)], we note that in two of the examples in table 1the residual contribution xs actually dominates the combinatory one of negative sign,i.e., x ~ ; ~ > J . Thus, the total entropy of dilution at small 42 is negative for theseexamples at small g52. Obviously, an adequate theory of polymer solutions mustaccount for the large negative contribution -R&; to the entropy of dilution.The prevalence of values of which are zero or negative in cases of closechemical similarity of solvent and solute is a further observation of note, althoughless marked than the one above.The interchange of neighbour species should makecontributions to xH which, though small, are positive in such instances. Negativecontributions from another source are indicated. The foregoing features are notpeculiar to dilute solutions. They persist at higher concentrations, and in factusually increase with 26* 36It should be noted that the results quoted in table 1 were obtained by identifying42 in equations given above with the " segment fraction ", a quantity whose precisespecification depends on the analysis described in the following section.Jf differsonly slightly from the volume fraction. Numerical values of x, xH and xs may exceedthose derived using volume fractions by ca. 0.02-0.05.Experimentally determined values of x (and also of xH and xs) usually increasewith 42, and in some cases the increase is substantial. Fig. 1-3 are illustrative.(Calculated curves in these figures are discussed in the following section.) The* The same symbols have been introduced by Patterson 24 for the enthalpic and entropic contri-butions to x when Awlz is treated as an exchange free energy. In keeping with the treatment of xas an empirical parameter, we use XH and xs in the more general sense of contributions to x fromwhatever source.They do not represent exchange interactions alone ; these may in fact be minorcontributorsTABLE 1 .-RESUMB OF THERMODYNAMIC PARAMETERS FOR REPRESENTATIVE POLYMERexperimental results at 4 2 = 0 x12 xs: 1 xs; 1 dxld 42 J cm-3 calc. polymerb+solvent X l XH; 1NR+C6H6PIBf C6H6PIB+C-C~HI~PIB+n-CsH12PIB+n-CsHzoPS+ CH3COC2H5PS+C2&C&PS+C6Hi2PDMS+CsHlzPDMS+ CsHSC1PMMA+ CHC13PM MA + heptanone-4PMMA+ tetrahydrofurane0.420.500.470.490.460.470.400.5050.420.470.3770.5090.4470.10.260.00- 0.42'-0.17' - 0.03- 0.020.30. 19e0.13-0.100.170.050.30.240.470.910.630.500.420.20.230.340.480.340.400.10.40.10.350.100.40.30.40.190.30642611 .7e4.4e298.8427.55.00.22c0.1 10.381.080.640.44c0.33c0.7c - 0.02- 0.03U A l l results refer to a temperature of 25°C.Experimental data have been treated according to eqn (8)-(bAbbreviations used for polymers are as follows : NR for natural rubber, PIB for polyisobutylene, PSc These values of xs; 1 have been calculatedwithout the aid of an additional parameter (Q12 ; see ref.dX12 has been calculated from VE/V in this instance. The value of XH; 1 calculated from XI, agreese In these cases the integral enthalpy of dilution was used to obtain X12. The values of XH; 1 in columnrather than the volume fraction. Where necessary, corrections have been introduced for departure fromsiloxane), and PMMA for poly(methy1 methacylate).differ therefore from calculated values quoted in the references in these instances.given in column three.enthalpy with the aid of theoryPAUL J .FLORY 171.5r- 1'4 -1.2-1.1 -PO -x 0 9 - x-0'X0.0-0.7 -00 04 0.2 0.3 0 4 05 06 07 08 09 ko9 2 x-@, 10°C;' 0, 25°C; 0,40"C; 0,24.5"C. XH-O.FIG. 1.-Reduced residual chemical potential X and partial molar enthalpy XH for the system poly-isobutylene+ benzene at 25OC plotted against the segment fraction & of polymer.26X0.290.100.0c-0 100.2 0 4 06 1.0XH9 2ma. 2.Parameters for the system polystyrene+methyl ethyl ketone at ca 25"C.318 THERMODYNAMICS OF POLYMER SOLUTIONSapproximate rates of increase of x with q52 are recorded in the fifth column of table 1.The increase in x with polymer concentration is reflected in liquid-liquid miscibilitybehaviour. The traditional treatment l2, 2 5 carried out on the basis of eqn (6)with x assumed constant yields relationships representing the critical compositionand temperature as functions of chain length which appear to be of the correct form,but with parameters whose magnitudes are somewhat in error. As Tompa 44 pointedout a number of years ago, inaccuracies in this regard are traceable to an increasein x with concentration.x0.71 I I I I I I I I I 10% -04 -1 I I I 1 1 I I I I I00 0.1 02 0.3 04 Q-5 0-6 0.7 08 0-9 10'p2FIG.3.-Reduced residual chemical potentials for the systems : 0, polyisobutylenc-t-n-octane 32 ;and 0, polyisobutylene+ cyclohexane,26 both at ca.25°C.Finally, the volume change occurring on mixing is beyond reach of the foregoingpolymer solution theory. Yet, in principle, it represents a property of mixturesequally as deserving of attention as the change in enthalpy. The change of volumeupon mixing a low molecular liquid with a polymer, also in the liquid state (asopposed to the glassy or crystalline states, which introduce other features extraneousto the present discussion), often is appreciable ; it may be of either sign (see columneight of table 1).INFLUENCE OF LIQUID STATE PROPERTIESThe departures of polymer solutions from theory find parallels in mixtures ofsmaller molecules. The latter exhibit deviations from regular solution theory, towhich the polymer solution theory is analogous. In both theories, a combinatoryentropy is joined with a van Laar enthalpy on the assumption that the only contri-bution to the mixing free energy in addition to -TAScomb is the exchange enthalpyassociated with the replacement of molecular neighbours of the same kind by mole-cules of the other species. Neither theory takes account of the liquid structure andproperties peculiar to the respective components.Only the exchange enthalpy ofthe given system is subject to characterization. The sole parameter of the theoryPAUL J . FLORY 19namely, AwI2, or the parameter x related to it according to eqn (7), serves this pur-pose.The importance of comprehending liquid state properties within the frameworkof an adequate theory of solutions becomes apparent upon considering the volume,the intermolecular enthalpy and the intermolecular entropy (i.e., the entropy exclusiveof AScomb), and in particular the changes these extensive quantities undergo uponmixing.The latter two depend strongly on the volume, and the volume in turn isdetermined by the condition that the Gibbs free energy be minimized at constantpressure and temperature. The three quantities are thus inter-related. Differencesin liquid structure lead to non-additivity of Y and of the intermolecular contributionsto H and S. Departures from additivity with respect to the composition of themixture are manifested in non-zero excess, or residual, quantities. The readilymeasured change in volume upon mixing is complemented by corresponding changesin the intermolecular enthalpy and entropy.These changes may, of course, becalculated from VE and the coefficients (aHlaV), and (aS/i?v>, for the mixture,but the changes thus calculated should not be identified with the “anomalous”contributions to H E and SE, i.e., with those contributions not attributable to exchangeinteractions. There is no reason to suppose that the various excess quantities aredirectly related to one another and hence that suppression of one of them (e.g.,suppression of YE through alteration of pressure after mixing) would eliminate theothers. Departures from additivity for the several quantities should not be expectedto be correlated in such a way. Experimental evidence on mixtures of n-alkanes 45-47shows unambiguously that liquid structure or ‘‘ equation-of-state ” contributionsare by no means eliminated through correction for the direct effects of VE in theforegoing manner.Several conclusions may be drawn from considerations along the lines indicated.First, the specific effects of liquid structure and properties cannot be taken into accountby any simple modification of regular solution theory or of its counterpart for polymersolutions.A more comprehensive approach is required. Secondly, the anomalouscontributions associated with liquid properties should not be looked upon solely asconsequents of the change in volume upon mixing, or of that elusive quantity calledthe “free volume ”. Finally, since the excess, or residual, quantities arise from non-additivity, they should depend upon differences between the liquid properties ofthe two components of the mixture.The fact that these differences tend to beextreme for solutions comprising a polymeric solute and a low molecular diluentappears to be responsible for the large residual quantities observed for such solutions.Early attempts to take account of liquid state characteristics in the treatment ofsolutions proceeded along two main lines : exploitation of the cell model as a basisfor formulating the properties of liquid 49 and application of correspond-ing states methods to selected classes of liquid mixtures.48* 5 0 The cell theoryutilizes the intermolecular potential as its primary ingredient, and hence it offersthe appeal of being derived from a fundamental molecular entity. However, iterrs in ascribing an excessive degree of order to the arrangement of the neighboursof a given molecule or segment of the liquid, as Hildebrand and Scott 51 haveemphasized.In consequence of this misrepresentation, the energy is predicted todepend on a higher power of the density than is 52 Further, thestereotyped description of a liquid in terms of discrete cells renders the model unsuitedfor mixtures of species disparate in size and non-spherical in shape. Prigogine andco-workers 4 8 9 53 have overcome this difficulty in a formal way for solute moleculeswhich can be represented as r-meric polymers of segments spatially equivalent tothe soivent. But, for polymer solutions in general, the cell model suffers the same20 THERMODYNAMICS OF POLYMER SOLUTIONSlimitation as for mixtures of small molecules differing in size and shape, even to aminor degree.The principle of corresponding states rests on the assumption that the intermole-cular potentials for solvent and solute are of equivalent form when expressed asfunctions of the distance between molecular centres, or of segment centres in theadaptation for r-meric liquids by Prigogine and coworkers.Specifically, theintermolecular potential is required to be parametric in the distance and in the energy.If the thermodynamic properties are known as a function of temperature and densityfor one reference liquid, those for any other of the family of corresponding liquidsare determined by two scale factors, one for the distance of separation of molecularcentres and the other for the magnitude of the intermolecular potential.Thus,the properties of a given liquid are determined by these two parameters, which maybe conveniently embodied in a characteristic temperature T* and a characteristicpressure p*. Prigogine and coworkers 48* 53 made the important step of introducingthe number of intermolecular degrees of freedom as a third parameter for r-mericliquids.Since the mathematical form of the intermolecular potential is unspecified, it iscustomary in applying the principle of corresponding states to resort to graphicalmethods in order to establish the required reference relationships as functions ofreduced variables T = T/T* and j = p/p*.The graphical procedures are subjectto errors of judgment, and they are more difficult to apply than analytical methods.Application of the principle of corresponding states to mixtures depends also onevaluation of T* and p* for the mixture from values for the components. Theprinciple does not afford a basis for such evaluation. One is obliged to turn elsewherefor means to this end. Moreover, its underlying premise concerning equivalenceof intermolecular potentials cannot be expected to hold for a wide variety of liquids.The form of the functional dependence of the intermolecular potential on the centre-to-centre distance must vary considerably, even amongst molecules for which theassumption of spherical symmetry of the field of force is legitimate.50 The principleof corresponding states has been applied successfully to inixtures of n - a l k a n e ~ .~ ~ ~ 5 4 9 5 5Its applicability to solutions in general appears to be limited.In quest of a means for taking into account the contributions of liquid stateproperties to the thermodynamics of mixtures in general, the author and his col-laborators 46* 56 explored the feasibility of commencing with a partition function ofsufficient versatility to encompass various molecular types, and of such simplicityas to permit its adaptation to mixtures. The partition function would at once definethe equation of state and the partial molar thermodynamic functions as well. Itmust, of course, offer a reasonably accurate representation of the equation of state,it should not be predisposed to spherical molecules, and in particular it should beapplicable, in principle at least, to polymer solutions where the disparity of propertiesof the two components is maximal.The number of arbitrary parameters shouldbe held to a minimum. Since the partition function would embrace both purecomponents, some of the parameters could be evaluated from their equations ofstate. In this way, the number of parameters subject to arbitrary adjustment couldbe reduced. Appeal to the vapour state for the evaluation of parameters (e.g., anintermolecular energy parameter) would thus be avoided, and it would suffice forthe partition function and equation of state to hold only within the range of highdensities characteristic of the liquid state.Inasmuch as the partition function wouldserve only for interpolation between the two pure liquids, high accuracy of representa-tion would not be necessary.The partition function formulated with these objectives in mind combines PAUL J . FLORY 21rudimentary factor for hard spheres (or other particles whose dimensions in differentdirectioiis are similar, if not equal ; cf. following) with an intermolecular energy ofthe van der Waals form. It is expressed by 46956Z = z,.omb[g(d - ~*+>~]'"'exp (rnsy/2vkT), (14)where Zcomb is the combinatory factor discussed earlier, v* is the hard-core volume ofa segment, v is the volume per segment, g is an inconsequential geometric factor,n is the number of molecules, r is the mean number of segments per molecule, c is theparameter introduced by Prigogine 4 8 * 5 3 for the mean number of external degreesof freedom per segment, and -sy/2v is the intermolecular energy per segment, sbeing the number of contact sites per segment and q the energy parameter charac-terizing a pair of sites in contact.The segment may be defined as the largest,approximately equi-dimensional, sub-division of hard core volume of the moleculeswhich is common to both species. The specification of its size is essentially arbitrary.However,it antecedes the cell model, having been derived earlier by Tonks 57 (1936) by firsttreating a one-dimensional liquid of incompressible molecules and then cubing theresults to obtain the expression for three dimensions.The treatment of the one-dimensional liquid is exact, but the latter step lacks rigour. On physical grounds,however, it appears to offer a plausible approximation for three dimensions. Deriva-tion by this path relaxes the restriction to spherical molecules, or segments, impliedby derivation through resort to the cell model, and it suggests in fact that appreciabledepartures from equality of dimensions in different directions could be toleratedwithout severe departure from the result given. The same factor was employed byEyring and Hirschfelder * (1 937).We have avoided the cell theory as a basis for eqn (14) not only in the interestsof relaxing the condition on sphericity of the particles, but also in view of the illusiveenergy it prescribes.Recent advances in the theory of hard sphere liquids andrelated computations have strengthened the justification for use of an energy of thevan der Waals form.13* 5 9 As Rowlinson l3 points out, these studies lead to theconclusion that the structures of liquids, expressed by the radial distribution functionfor spherical molecules, '' . . . is determined primarily by the repulsive forces betweenmolecules, and not by the attractive forces ". Hildebrand and Scott 51* 6o pointedout some years ago that a radial distribution function determined primarily by repul-sions of very short-range connotes an intermolecular energy for a dense fluid thatvaries approximately as the inverse of the volume, i.e., an energy of van der Waalsform.This assertion, which has gained support in the further work of 59is quite independent of the form of the attractive energy. The latter is largelyresponsible for the intermolecular energy. Thus, eqn (14) does not depend on acommitment as to the nature, the mathematical form, or spherical symmetry of theforces of attraction acting between pairs of molecules of the liquid. Neither theassumption of pairwise additivity nor the principle of corresponding states is invoked.The only novelty that can be claimed for eqn (14) is the particular combinationof its parts from the previous sources acknowledged above. In particular, whenc = 1 it reduces to the partition function of Eyring and Hir~chfelder.~~The factor (vlf - v**)~ is suggestive of the largely discredited cell theory.Rewriting eqn (14) in reduced form, we obtain 56which yields the reduced equation of statejc/T = v"+/(Cf - 1) - (CT)-'22whereTIIERMODYNAMICS OF POLYMER SOLUTIONSEqn (16) is identical in form to the reduced equation of state of Eyring and Hirsch-felder,58 but the definition of T* according to eqn (18) differs from theirs owing toincorporation of the primary parameter c.Whereas in the absence of this parameterT* is fixed by p* and v* (see eqn (18) and (19)), its presence permits D*, p* and T*to be specified independently. In practice, therefore, the formulation given heredeparts from that of Eyring and Hirschfelder.The expressions obtained from eqn (16) for the thermal expansivity a and thethermal pressure coefficient y at p = 0 yieldu"* = 1 + (aT/3)/( 1 + aT), (20)andp* = yTC2.Thus, the experimentally-determined value of a yields u" which, in conjunction withthe volume v per segment, gives v*.= 0 yields4: and T*. Finally, the observed thermal pressure coefficient furnishes the additionalnumerical information needed for evaluation of p* according to eqn (21). The threecharacteristic parameters u*, T* and p* are obtained in this way from the equation-of-state coefficients. The primary parameters sq and c are not required, but may beobtained from T* and p* by use of the eqn (18) and (19).The reduced equation of state, eqn (16), and quantities derived therefrom, givea good account of the isotherm.46* 47 They are much less satisfactory in theirrepresentation of the temperature dependence, as expressed in the isobar forexample.26* 47* 61 The inaccuracy of the theory in this regard manifests itself inchanges of the values of the characteristic parameters u*, T* and p* with the tempera-ture at which v, a and y are taken.It is the root of a major deficiency of the theory.The drift of the reducing parameters with temperature encumbers the treatment ofmixtures, but the difficulties are by no means insurmountable. They may be circum-vented by using values of the characteristic parameters applicable at the temperatureof the experiments carried out on mixtures.Adaptation to mixtures proceeds unambiguously from two premises, as follows :(i) core volumes of the components are additive, and (ii) the intermolecular energydepends on the surface areas of contact between molecules and/or segments.Theformer premises is implicit in the prescription of the partition function. Provisionfor the second was anticipated by expressing the energy as proportional to the surfacearea measured by the number of contact sites s. Its validity rests on the assumptionthat the intermolecular attractions are of short range compared with the moleculardiameters, a condition that should be fulfilled for most liquids, monatomic onesexcepted. Both of the premises (i) and (ii) depart from the various presciptionsfor adaptation of the cell model, and its corresponding states counterpart, to mixturesof molecules that differ in size.62Since the specification of a segment for each component remains arbitrary, it isconvenient to choose segments of equal size so that 07 = O; = D*.With thisstipulation, premises (i) and (ii) lead at once to the following relations fixing theparameters r, s and c applicable to the mixture 56 :Substitution of 5 in eqn (16) witPAUL 3 . FLORY 231/r = 4 1 / h + 4 2 h 9s = 4 r s l f 4 2 S 2 , (22)c = 41c1 f 4 2 C 2 ,where 41 and 42 are redefined as segment fractions, or core volume fractions, insteadof as volume fractions. Site fractions 8, and O2 required for formulation of theintermolecular energy on the basis of premise (ii) and the van der Waals form of theintermolecular energy are given byExpressions for the characteristic parameters for the mixture follow directly fromthe foregoing premises and eqn (22) and (23).They are6, = 1-81 = 42s2/s. (23)P* = 41PT+452P;-4l~lX129 (24)T* = (41 PT+ 4 2 P F - 4,62xl2>/(4lPTITT + 4 2 P W 9 , (25)X12 = (s1/2u*2)(rll + Y 2 2 4 Y 1 2 ) , (26)where X12 is the exchange interaction parameter formally defined byq l l , q22 and q12 being the energy parameters for the segment pairs indicated. Theparameter X , , is analogous to Aw12 of eqn (7), but with the dimensions of an energydensity instead of energy. We adhere to the identification of X,, as an energydensity as opposed to a free energy density. In this respect, practices in the usageof X,, and Aw12 differ. 47 we have introduced a separate parameter,when required, to represent the entropy contribution similarly due to contact inter-actions.Incorporating these relationships in the partition function, we obtain for theenthalpy and the residual entropy of mixing 56(27)andSR = -3nrv*{(4,p:/TT)ln [(a$- 1)/(5*- 1)]+(42p~/T~)ln[(5& l)/(i?- l)]), (28)where n = n, +n2 is the total number of molecules and 6 is the reduced volume ofthe mixture.The latter quantity is obtained from T, given by eqn (18) and (25),through the use of eqn (16) with the term in p set equal to zero for low pressures.The excess (or residual) volume VE is obtained at once fromwhere ijE is the reduced excess volume. The residual chemical potential of thesolvent is 56(30)The foregoing relationships are so framed as to be applicable both to solutionsof small molecules and to polymer solutions.The influence of liquid-state propertieson each of the thermodynamic functions HR (or HE), SR and is represented by" equation-of-state " terms that depend on the differences of reduced volumes, ortheir reciprocals, and on the characteristic parameters p* and T*. These termsdepend both on the difference between ijl and fi2 and on the excess volume throughfi.46* 47 They do not vanish in general for ijE = 0. Thus, the equation-of-statecontributions cannot be interpreted simply in terms of a change of " free volume ".The equation-of-state terms depend implicitly on X, through u", the latter quantitybeing determined by eqn (25) and (16) as noted above. The functions HR (or HE)AH, = HR = nrv*[4,pT(G,l-G-')+#,p~(ii, 1-5-1)+(4162/ij)X12],VE/V = uE/C = 1 -(4,u"1 +42G2)/u", (29),u; = p ~ r , ~ * ( 3 ~ ~ l n [ ( C ~ - l)/(G3- l)] +(iiT1 -G-l))+(r,v*Xl2/ii))e,224 THERMODYNAMICS OF POLYMER SOLUTIONSand py include also a term depending explicitly on X1, that represents the exchangeenthalpy contribution from neighbour interactions when the process is conductedat constant reduced volume 5.Superficially, these relationships and their application to solutions have featuresin common with cell theory and with corresponding states treatments.Reconciliationof the intermolecular energy in Prigogine’s cell theory 48 with the van der Waals formwe have used must yield eqn (16) for the equation of state. As Patterson and Delmaspoint out,62 this requires adoption of a 3, co potential (i.e., an attraction energyvarying as the inverse third power of distance, and a hard “ sphere ” repulsion).This is, of course, an absurd intermolecular pair potential, and the fact that its adoptionis required to achieve reasonable agreement with experiment is a serious indictmentof cell theory.The point of view we have adopted 46* 56 obviates specification of apair potential, as emphasized above. Incorporation of premises (i) and (ii) into thecell model scheme completes the establishment of equivalence to out theory. Therelationships given by cell theory after introduction of these drastic modifications 6 2must obviously conform to those given above. But premises (i) and (ii), like thevan der Waals energy, are alien to the precepts of cell theories of solutions.The use of characteristic parameters and reduced quantities finds abundantprecedent in corresponding states theory. However, our treatment and the para-meters it employs are not contingent upon adherence to the theorem of correspondingstates or upon an equivalence of intermoleclar pair potentials.The “ correspon-dence” lies rather in the form adopted for the partition function and the rulesemployed for adapting it to mixtures. Conventional corresponding states theorycan, of course, be rendered equivalent to the treatment reviewed above by adoptinga generalized van der Waals equation of state and premises equivalent to (i) and (ii)as a basis for assignment of characteristic parameters to the mixture.62* 6 5COMPARISON OF THEORY WITH EXPERIMENTTreatment of mixtures according to the theory summarized above requiresequation-of-state parameters-density, thermal expansivity and thermal pressurecoefficient-of sufficient accuracy for both components. The ratio s1 Is2 of molecularsurface areas (sites) for segments of the two species also is required.Inasmuch asthe segments are chosen to have the same core volume, this is just the ratio of thesurfaces per unit of core volume, which can be estimated from structural informationor from the tabulations of B ~ n d i . ~ ~ There remains the single parameter X , , tobe assigned arbitrarily for the given pair of liquids. It is usually so chosen to optimizeagreement with experimental enthalpies of mixing or of dilution. The excess volume,the chemical potential and the entropy of dilution, calculated from theory as functionsof composition, may then be compared with experiment. Other quantities such asLCST’s and excess compressibilities and excess thermal pressure coefficients may alsobe calculated and, if suitable experimental data are available, they offer further testsof the theory.56Abe and the author 56 treated the excess enthalpies, volumes and entropies ofmixtures formed from 23 pairs of low molecular non-polar liquids.Calculatedexcess volumes VE are in good agreement with experiment except for perfluorocarbon+hydrocarbon mixtures for which calculated values are only a little more than halfof those observed. For all other systems examined, the standard deviation betweentheory and experiment is only 0.11 cm3 mol-l.The agreement of the entropy isless satisfactory, but in most instances is within limits of errors in the several experi-mental quantities required for the calculation. The theory has even been appliePAUL J . FLORY 25with gratifying success to the following systems in which one of the components(Ar) is monatomic : Ar + CH4,56 Ar + 02,56 Ar + N2,56 and Ar + Kr? Althoughnot formulated for systems of such simplicity, as we have taken pains to point out,it appears to be competitive with more refined theories for simple spherical mole-c u l e ~ . ~ ~A mixture formed from two members of the same homologous series of chainmolecules, e.g., a mixture of two n-alkanes, may be regarded as a mixture of twokinds of segments : internal chain segments and terminal segments.If the primaryparameters u*, c and sq are defined in terms of these segments, all mixtures formedfrom homologues of the given series may be interpreted with a single set of parameters.The excess volumes, excess enthalpies and residual chemical potentials of mixturesof n-alkanes have been treated successfully in this manner.46* 47 The LCST exhibitedby several pairs comprising a low and a high member of the series have been approxi-mated within the same scheme, no additional parameters being req~ired.~' Thesesystems may be treated alternatively by corresponding states 5 4 9 5 5 * 62as we have pointed out above. The corresponding states methodology is not asstraightforward in application, and is more susceptible to ambiguities of interpretation.Solutions of high polymers in low-molecular-weight diluents place more stringentdemands on the theory owing to the greater differences usually occurring betweenthe properties of the two liquid components, as reflected in their characteristicparameters. Calculated and observed results for such solutions are compared intable 1.With one exception (rubber + benzene ; see footnote d to table l), valuesof X12 have been chosen to optimize agreement with experimental enthalpies.Entropic contributions, xSz1, to x1 calculated in the limit +2 = 0 are given in theseventh column. Observed and calculated excess volumes are tabulated in thefollowing columns.Calculated values of xSil are positive in nearly all cases ; the only exceptions arethe PDMS solutions, for which xSil ~0 according to calculation.For most of theother systems the calculated values compare favourably with those observed (fourthcolumn). Calculated excess volumes given in the penultimate column are of thecorrect sign, with a single exception, and the magnitudes generally approximatethe values observed. Again, departures from theory are most marked for the twoPDMS solutions. The differences between observed and calculated excess volumesfor these solutions (Le., VE observed< VE calculated) probably are related to thelower values of the residual entropies observed compared to those calculated (i.e.,xS,obs.>xs,calc.), but the reasons for these discrepancies are not clear. A clue tothe explanation may occur in the unusually low thermal pressure coefficient of thisand therefore the low values of its internal pressure and of its intermole-cular energy as reflected in sq/u*.From a structural viewpoint, the polymer isexceptional in consisting of a polar chain skeleton surrounded by methyl groupscharacterized by low dispersion forces.Curves representing and xH calculated for several systems as functions of compo-sition are compared with experimental results in fig. 1-3. Differences between thetwo curves give xs. Increases in these quantities with composition are correctlypredicted, but the theory appears to be inaccurate in its estimation of the magnitudesof the increases. It does not consistently over-estimate nor under-estimate thechanges, however.26* 36Improved agreement between calculated and observed values of xs may beachieved by making the reasonable postulate that exchange interactions contributealso to the entropy.The additional parameter thus made available permits agree-ment to be established with the observed value of xs at one composition. The exces26 THERMODYNAMICS OF POLYMER SOLUTIONSenthalpy and volume are unchanged. Although the values of xs and of x are shifted,their changes with composition are little affected. This means of improving agreementwith experiment has been avoided in all comparisons presented in this resumC. Itshould be borne in mind, however, that exchange interactions may contribute to theentropy, as well as to the enthalpy, and full agreement with experiment may requireinclusion of the contribution from this source.CONCLUDING REMARKSOn the basis of systems investigated to date, the theory under review appears tobe somewhat less successful in the treatment of polymer solutions than in its applica-tions to low molecular mixtures.61 Nevertheless, it overcomes the principal in-adequacies of the conventional theory of polymer solutions, which only takes accountof the combinatory entropy and of an exchange enthalpy due to neighbour inter-actions.The negative contributions (i.e., -xs) to the entropy, which are mainlyresponsible for the large positive values of x, are consequences of the differences inliquid-state properties of the polymeric solute compared with the low molecularsolvent.These differences stem primarily from the smaller number of externaldegrees of freedom for a segment of the polymer. Negative enthalpies (xH<O)for mixtures of a polymer with a chemically similar solvent (e.g., n-pentane+PIB)are of like origin. The contribution from exchange interactions in non-polar mixturesis positive, though small ; it may be dominated (see table 1) by the large equation-of-state contributions to the enthalpy for a polymer solution. The negative enthalpiesTABLE 2.-cOMPARISON OF EXCHANGE INTERACTION PARAMETERS FOR CHEMICALLY SIMILARSYSTEMSX I 2 J cm-387.34.41 .ob6424042428.85%58c56.5c5 9 3ref.613232266161263636666666660 PIB = polyisobutylene, PM = polymethylene, PS = polystyrene.b estimated by extrapolation from n-alkane+PIB systems.c All values quoted for hydrocarbon+perfluorocarbon systems were calculated from criticalmiscibility (UCST).often found for polymer solutions are thus explained 32 without invoking a negativeexchange interaction, which, with respect to our knowledge of intermolecular forces,would be implausible.Increases in x with composition, and in its components xH and xs as well, are crudely approximated by the theory. Volume changes oPAUL J . FLORY 27mixing are brought within the scope of therniodynaniic theory, albeit imperfectly.All of this is accomplished without increasing the number of arbitrary parametersbeyond the one traditionally allowed for the exchange interactions.The present theory is obviously rudimentary.It is also of the utmost simplicityin application. Better agreement with experiment doubtless can be achieved witha more refined theory, but the improvement thus gained may be at a considerablesacrifice of simplicity. If additional arbitrary parameters are required, the refinementwill be of questionable advantage.There remains the deeper question whether any manageable theory, or treatment,of solutions can be both comprehensive in scope and accurate in its representation ofequilibrium properties. Details of interactions at the molecular level may introduceeffects which can scarcely be covered by any general scheme applicable to a widevariety of liquid mixtures. Local correlations of molecular orientations may producespecific effects in this category, for example.Instead of seeking a theory which isaccurate in detail as well as comprehensive in scope, it may be more fruitful to adopta simpler treatment of reasonable generality at sacrifice of accuracy of representationof individual cases. Departures from such a scheme may then be interpreted,qualitatively at least, with reference to molecular characteristics of the particularcomponents.Finally, it is noteworthy that values found for the enthalpy exchange parameterXI, appear to be determined by the kinds of atoms and groups comprising themolecules of the components and not by molecular size or shape. The principalevidence in support of this somewhat tentative conclusion is presented in table 2.The values of XI, there included embrace both low-molecular weight and polymersolutions.Within the first group consisting of aliphatic-aliphatic pairs, XI isconsistently small irrespective of whether one of the components is polymeric. Onemember of each pair in the second group is aliphatic and the other aromatic. Valuesof X12 for these systems are large ; they do not differ from one another beyondexperimental error, and again values for polymer solutions conform with those forlow molecular mixtures. For the single example of an aromatic-aromatic mixturethe value of X , is again small. For the hydrocarbon + perff uorocarbon mixturesof the last group, the XI, are large,67 as expected. They are remarkably constantfor all of the systems in this group, major differences in molecular size from methaneto methylcyclohexane notwithstanding.The binary liquid mixtures for which XI, has been evaluated are too restrictedin number and variety for final conclusions, but they are sufficient to encourage thehope that the enthalpy exchange parameters thus derived from experiment may admitof rational interpretation in terms of molecular forces.The usual van Laar para-meters obtained without benefit of separation of equation-of-state contributionsdefy such interpretation. The consistency of values of X12, together with the factthat they are invariably positive for non-polar mixtures, lends credence to the theoryas a basis for treating liquid mixtures.E. Huckel, 2. Elektrochem., 1936,42,753.E.A. Guggenheim, Trans. Faraday Soc., 1937,33, 151.R. H. Fowler and G. S. Rushbrooke, Trans. Faraday SOC., 1937,33, 1272.J. N. Brransted and P. Colmont, 2. phys. Chern. A , 1934,168,381.K. H. 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