摘要:

GENERAL DISCUSSIONS OFTHE FARADAY SOCIETYDate1907190719101911191219131913191319141914191519161916191719171917191819181918i918191919191 92019201920192019211921192119211922192219231923192319231923192419241924192419241925192519261926192719271927SubjectOsmotic PressureHydrates in SolutionThe Constitution of WaterHigh Temperature WorkMagnetic Properties of AlloysColloids and their ViscoSityThe Corrosion of Iron and SteelThe Passivity of MetalsOptical Rotary PowerThe Hardening of MetalsThe Transformation of Pure IronMethods and Appliances for the Attainment of High Temperatures in aRefractory MaterialsTraining and Work of the Chemical Engineerosmotic PressurePyrometers and PyrometryThe Setting of Cements and Plast~rsElectrical FurnacesCo-ordination of Scientific PublicationThe Occlusion of Gases by MetalsThe Present Position of tho Theory of IonizationThe Examination of Materials by X-RaysThe Microscope : Its Design, Construction and ApplicationsBasic Slags : Their Production and Utilization in AgriculturePhysics and Chemistry of ColloidsElectrodeposition and ElectroplatingThe Failure of Metals under Internal and Prolonged StressPhysico-Chemical Problems Relating to the SoilCatalysis with special reference to Newer Theories of Chemical ActionSome Properties of Powders with special reference to Grading byThe Generation and Utilization of ColdAlloys Resistant to CorrosionThe Physical Chemistry of the Photographic ProcessThe Electronic Theory of ValencyElectrode Reactions and EquilibriaAtmospheric Corrosion.First ReportInvestigation on Oppau Ammonium Sulphate-NitrateFluxes and Slags in Metal Melting and WorkingPhysical and Physicu-Chemical Problems relating to TextiIe FibresThe Physical Chemistry of Igneous Rock FormationBase Exchange in SoilsThe Physical Chemistry of Stcel-Making ProcessesPhotochemical Reactions in Liquids and GasesExplosive Reactions in Gaseous MediaPhysical Phenomena at Interfaces, with special reference to MolecularAtmospheric Corrosion. Second ReportThe Theory of Strong ElectrolytesCohesion and Related ProblemsLaboratorycapillarityElutriationOrientationVolumeTrans. 33678999101011121213131314141414151516161616171717171818191919191920202020202121222223232GENERAL DISCUSSIONS OF THE FARADAY SOCIETYDate192819291929192919301930193119321932193319331934193419351935193619361937193719381938193919391940194119411942194319441945194519461946194719471947194719481948194919491 9491950195019501950195119511952195219521953195319541954SubjectHomogeneous CatalysisCrystal Structure and Chemical ConstitutionAtmospheric Corrosion of Metals.Third ReportMolecular Spectra and Molecular StructureOptical Rotatory PowerColloid Science Applied to BiologyPhotochemical ProcessesThe Adsorption of Gases by SolidsThe Colloid Aspects of Textile MaterialsLiquid Crystals and Anisotropic MeltsFree RadicalsDipole MomentsColloidal ElectrolytesThe Structure of Metallic Coatings, Films and SurfacesThe Phenomena of Polymerization and CondensationDisperse System in Gases : Dust, Smoke and FogStructure and Molecular Forces in (a) Pure Liquids, and (b) SolutionsThe Properties and Functions of Membranes, Natural and ArtificialReaction KineticsChemical Reactions Involving SolidsLuminescenceHydrocarbon ChemistryThe ElaCtrical Double Layer (owing to the outbreak of war the meetingThe Hydrogen BondThe Oil-Water InterfaceThe Mechanism and Chemical Kinetics of Organic Reactions in LiquidThe Structure and Reactions of RubberModes of Drug Action,Molecular Weight and Molecular Weight Distribution in High Polymers.(Joint Meeting with the Plastics Group, Society of Chemical Industry)The Application of Infra-red Spectra to Chemical ProblemsOxidationDielectricsSwelling and ShrinkingElectrode ProcessesThe Labile MoleculeSurface Chemistry.(Jointly with the Sociktc! de Chimie Physique atColloidal Electrolytes and SolutionsThe Interaction of Water and Porous MaterialsThe Physical Chemistry of Process MetallurgyLiDo-Proteinswas abandoned, but the papers were printed in the Transactions)SystemsBordeaux.) Published by Butterworths Scientific Publications, Ltd.crystal Growthchromatographic AnalysisHeterogeneous CatalysisPhysico-chemical Properties and Behaviour of Nuclear Acids Trans.46Spectroscopy and Molecular Structure and Optical Methods of In-vestigating Cell Structure Disc. 9Electrical Double Layer Trans. 47Hydrocarbons Disc. 10The Size and Shape Factor in Colloidal SystemsRadiation Chemistry 1211The Physical Chemistry of Proteins 13The Reactivity of Free Radicals 14The Equilibrium Properties of Solutions of Non-Electrolytes 15The Physical Chemistry of Dyeing and Tanning 16The Study of Fast Reactions 17Coagulation and Flocculation 18Volume2425252526262728292930303131323233333434353535363737383940414242 A42 BDisc. 12Trans. 43Disc. 34567OBNBRAL DISCUSSIONS OF THE FARADAY SOCIETYDate1955195519561956195719581957195819591959196019601961196119621962196319631964196419651965196619661967196719681968196919691970SubjcctMicrowave and Radio-Frequency SpectroscopyPhysical Chemistry of EnzymesMembrane PhenomenaPhysical Chemistry of Processes at High PressuresMolecular Mechanism of Rate Processes in SolidsInteractions in Ionic SolutionsConfigurations and Interactions of Macromolecules and Liquid CrystalsIons of the Transition ElementsEnergy Transfer with special reference to Biological SystemsCrystal Imperfections and the Chemical Reactivity of SolidsOxidation-Reduction Reactions in Ionizing SolvcntoThe Physical Chemistry of AerosolsRadiation Effects in Inorganic SolidsThe Structure and Properties of Ionic MeltsInelastic Collisions of Atoms and Simple MoleculmHigh Resolution Nuclear Magnetic ResonanceThe Structure of Electronically-Excited Species in the Gas-PhaseFundamental Processes in Radiation ChemistryChemical Reactions in the AtmosphereDislocations in SolidsThe Kinetics of Proton Transfer ProcessesIntermolecular ForcesThe Role of the Adsorbed State in Heterogeneous CatalysisColloid Stability in Aqueous and Non-Aqueous MediaThe Structure and Properties of LiquidsMolecular Dynamics of the Chemical Reactions of GasesElectrode Reactions of Organic CompoundsHomogeneous Catalysis with Special Reference to Hydrogenation andBonding in Metallo-Organic CompoundsMotions in Molecular CrystalsPolymer SolutionsOxidationFor current availability of Discussionvolumes, see back cover.Volume1920212223242526272829303132333435363738394041424344454647484

ISSN:0366-9033    

DOI:10.1039/DF970490X001

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

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. Meyer and R. Luhdeman, Helv. chim. Acta., 1935,18,307.G. V. Schulz, 2. phys. Chern. A, 1936,176,317.G. Gee and L. R. G. Treloar, Trans. Faraday Soc., 1942,38, 147.ti P. Stamberger, J. Chem. SOC., 1929, 2318.* K. H. Meyer, E. Wolff and Ch. G. Boissonnas, Helv. chim. A m . , 1940,23,43028 THERMODYNAMICS OF POLYMER SOLUTIONSK. H. Meyer, Z.phys. Chem. B, 1939,44,383 ; Helv. chinz. Acta, 1940,23,1063. K . H . Meyer,Natural and Synthetic Polymers, 2nd ed., (Interscience Publishers, Inc., New York, 1950),M. L. Huggins, J. Clzem. Phys., 1941, 9, 440 ; J.Chem. Phys., 1942, 46, 151 ; Ann. N. Y. Acad.Sci., 1942, 41, 1.P. J. Flory, J. Chem. Phys., 1941,9, 660; 1942, 10, 51.pp. 673-697.l 3 J. S. Rowlinson, Disc. Faraday Soc., 1970,49, 30.l4 P. J. Flory, Statistical Mechanics of Chain Molecules, (Wiley-Interscience, New York, 1969).l 5 L. Onsager, Ann. N. Y. Acad. Sci., 1949, 51, 627.I6 P. J. Flory, Proc. Roy. Soc. A , 1956, 234, 60; J. Polymer Sci., 1961,49, 105.A. R. Miller, Proc. Cambridge Phil. Soc., 1943, 39, 54 ; see also A. R. Miller The Theory ofSolutions of High Polymers, (Clarendon Press, Oxford, 1948).l 8 W. J. C. Orr, Trans. Faraday Soc., 1944, 40, 320.l9 E. A. Guggenheim, Proc. Roy. SOC. A, 1944, 183,203.'O P. J. Flory, J. Chem. Phys., 1949,17,223. '* H. C. Longuet-Higgins, Disc.Faraduy Soc., 1953, 15, 73.22 M. L. Huggins, J. Amer. Chem. Soc., 1948, 52, 248.23 C. Booth, G. Gee, M. N. Jones and W. D. Taylor, Polymer, 1964, 5, 353.24 D. Patterson, Macromolecules, 1969, 2, 672.2 5 P. J. Flory, Principles ofPolymer Chemistry, (Cornell University Press, Ithaca, New York, 1953).26 B. E. Eichinger and P. J. Flory, Trans. Faraday SOC., 1968, 64,2035,2053,2061, 2066." G. Gee and L. R. G. Treloar, Trans. Faraday Soc., 1942, 38, 147; G. Gee and W. J. C. Orr,Trans. Faraday Soc., 1946,42,507 ; G. Gee, J. B. M. Herbert and R. C. Roberts,PoZymer, 1965,6, 541.28 W. R. Krigbaum and P. J. Flory, J. Amer. Chem. Soc., 1953,75, 5254.29 W. R. Krigbaum and P. J. Flory, J. Amer. Chem. Soc., 1953,75, 1773.30 G. Delmas, D. Patterson and T. Somcynsky, J. Polymer Sci., 1962,57, 79.31 C. H. Baker, W. B. Brown, G. Gee, J. S. Rowlinson, D. Stubley and R. E. Yeadon, Polymer,32 P. J. Flory, J. L. Ellenson and B. E. Eichinger, Macromolecules, 1968,1,279.33 C. E. H. Bawn, R. F. J. Freeman and A. R. Kamaliddin, Trans. Faraday SOC., 1950,46,677,862.34 C. E. H. Bawn and M. A. Wajid, J. Polymer Sci., 1954, 12, 109.35 P. Doty, M. Brownstein and W. Schlener, J. Phys. Chem., 1949,53,213. M. J. Schick, P. Doty36 H. Hocker and P. J. Flory, to be published.37 K. J. Palmen, Thesis, (Technische Hochschule, Aachen, W. Germany, 1965).38 Th. G. Scholte, J. Polymer Sci., in press.39 W. R. Krigbaum and D. 0. Geymer, J. Amer. Chem. SOC., 1959, 81, 1859.40 N. Kuwahara, T. Okazawa and M. Kaneko, J. Polymer Sci. C., 1968,23, 543.41 H. Shih and P. J. Flory, unpublished.42 G. V. Schulz and H. Doll, 2. Elektrochem., 1953, 27, 301 ; G. V. Schulz, H. Baumann and43 R. Kirste and G. V. Schulz, 2. phys. Chem., (Frankfurt), 1961, 27, 301.44 H. Tompa, Polymer Solutions, (Academic Press, New York and London, 1956), pp. 174-182.45 M. L. McGlashan, K. W. Morcom and A. G. Williamson, Trans. Faraday Soc., 1961, 57,46 P. J. Flory, R. A. Orwoll and A. Vrij, J. Amer. Chem. Soc., 1964, 86, 3507, 3515.47 R. A. Orwoll and P. J. Flory, J. Amer. Chem. Soc., 1967,89,6814, 6822.48 I. Prigogine, The Molecular Theory of Solutions, (North Holland Publishing Co., Amsterdam,49 I. Prigogine and A. Bellemans, Disc. Faraday Soc., 1953, 15, 80.50 J. S. Rowlinson, Liquids and Liquid Mixtures, 2nd ed., (Butterworths Co. Ltd., London, 1969).51 J. H. Hildebrand and R. L. Scott, Regular Solutions, (Prentice-Hall, Englewood Cliffs, N.J.,5 2 R. L. Scott, Disc. Fardzy Soc., 1953,15,44, 113 ; J. Chem. Phys., 1956,25,193 ; H . Benninga5 3 I. Prigogine, N. Trappeniers and V. Mathot, Disc. Faraday Soc., 1953,15, 93 ; J. Chem. Phys.,54 I. Prigogine, A. Bellemans and C. Naar-Colin, J. Chem. Phys., 1957, 26, 751.5 5 J. Hijmans, Physica, 1961, 27, 433." P. J. Flory, J. Amer. Chem. Soc., 1965, 87, 1833.1962, 3, 215.and B. H . Zimm, J. Amer. Chem. Soc., 1950,72, 530.R. Darskus, J. Phys. Chem., 1966, 70, 3647.601.1957).1962).and R. L. Scott,J. Chem. Phys., 1955,23, 1911.1953, 21, 559, 560.L. Tonks, Phys. Rev., 1936, 50,'955PAUL J . FLORY 295 8 H. Eyring and J. 0. Hirschfelder, J. Phys. Chem., 1937, 41, 249 ; J. 0. Hirschfelder, D. P.59 R. L. Scott and D. V. Fenby, Ann. Reu. Phys. Chem., 1969,20, 111.6o J. H. Hildebrand and R. L. Scott, The Solubility of Non-Electrolytes, 3rd ed., (Reinhold61 A. Abe and P. J. Flory, J. Amer. Chem. Soc., 1965, 87, 1838.62 D. Patterson and G. Delmas, Disc. Faraday SOC., 1970, 49, 98 ; see also ref. (24).63 D. Patterson and G. Delmas, Trans. Faraday SOC., 1969, 65, 1708.64A. Bondi, J. Phys. Chem., 1964,68,441.H. Hocker and P. J. Flory, Trans. Farad~y SOC., 1968, 64, 1188.66 C. Price, J. Padget, M. C. Kirkham and G. Allen, Polymer, 1969, 10, 573. '' A. Abe and P. J. Flory, J. Amer. Chem. SOC., 1966,88, 2887.Stevenson and H. Eyring, J. Chem. Phys., 1937, 5, 896.Publishing Corp., New York, 1950)

ISSN:0366-9033    

DOI:10.1039/DF9704900007

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

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. Rowlinson, Liquids and Liquid Mixtures, 2nd ed. (Butterworth, London, 1969), chap. 3.R. W. Zwanzig, J. Chem. Phys., 1954,22,1420.J. S. Rowlinson, Mol. Phys., 1964,7,349; 8,107. ' J. A. Barker and D. Henderson, J. Chem. Phys., 1967,47,2856,4714; Mol. Phys., 1968,14,587.G. S. Rushbrooke, (chap. 2) and J. S. Rowlinson (chap 3.) of ref. (1).W. Byers Brown, Phil. Trans. A , 1957,250, 175,221.lo chap. 9, ref. (4).l1 T. W. Leland, J. S. Rowlinson and G. A. Sather, Trans. Fmahy Soc., 1968, 64,1447.l2 E. A. Guggenheim, Mixtures (Oxford Univ. Press, 1952).l3 E. B. Smith and K. R. Lea, Nature, 1960,186,714.l4 B. J. Alder, J. Chem. Phys., 1964,40,2724.l5 K.Singer, Chem. Phys. Letters, 1969, 3, 164, and private communication; I. R. McDonald,l6 R. L. Scott, J. Chem. Phys., 1956,25,193.l7 J. D. van der Waals, Die Kontinuitat des gasformigen und fliissigen Zustnndes (Barth, Leipzig,l 8 M. L. McGlashan, Trans. Faraday SOC., 1970, 66,18.l9 N. S. Snider and T. M. Herrington, J. Chem. Phys., 1967,47,2248.2o T. W. Leland, J. S. Rowlinson, G. A. Sather and I. D. Watson, Trans. Faraday SOC., 1969, 65,22 A. Bellemans, V. Mathot and M. Simon, Ado. Chem. Phys., 1967, 11,117.23 J. H. Hildebrand and R. L. Scott, Solubility of Nonelectrolytes (Reinhold, New York, 1950);24K. N. Marsh, Trans. Faraday Sac., 1968,64,883.25 J. Hijmans and Th. Holleman, Ado. Chem. Phys., 1969,16,223.26 J. N. Brmsted and J. Koefoed, Kgl. Danske Viaknskabs, Selskab, Mat. Fys. Medd., 1946, 22,27 H. C. Longuet-Higgins, Disc. Faraday Soc., 1953,15,73.28 P. J. Flory, R. A. Orwill and A. Vrij, J. Amer. Chem. SOC., 1964,86, 3507, 3515, 3563 ; P. J.Flory, J. Amer. Chem. SOC., 1965, 87, 1833; A. Abe and P. J. Flory, J. Amer. Chem. Soc.,1965,87,1838 ; H. Hocker and P. J. Flory, Trans. F'raday Soc., 1968,64,1188 ; B. E. Eichingerand P. J. Flory, Trans. Faraday SOC., 1968,64,2035,2053,2061,2066.(Warsaw, 1969), J. van Craen, J. Chem. Phys., in press.Chem. Phys. Letters, 1969, 3, 241.1900), vol. 2.2034.I. Prigogine, Molecular Theory of Solutions (North-Holland, Amsterdam, 1957).Regular SoZutions (Prentice-Hall, New Jersey, 1962).no. 17, 1.29 E. H. Lieb, J. Math. Phys., 1967, 8, 2339 ; R. J. Baxter, J. Math. Phys., 1968, 9, 650.30 A. Bellemans and J. van Craen, paper read at Int. Con$ Calorimetry and Thermodynamics,31 A. Bellemans and R. K. Nigam, J. Chem. Phys., 1967,46,2922.32 chap. 6 of ref. (4).33 D. Patterson and G. Delmas, Trans. Fizrahy Soc., 1969, 65,708.34 R. L. Scott and P. H. van Konynenburg, this Discussion, p. 87.35 M. L. McGlashan and I. R. Mackinnon, private communication, I. R. Mackinnon, Thesis,(University of Exeter, 1967).36 A. S. Teja, unpublished results.37 J. W. Leach, P. S. Chappelear and T. W. Leland, Amer. Inst. Chem. Eng. J., 1968, 14,568.38 K. S. Pitzer, J. Amer. Chem. SOC., 1955,77, 3427; K. S. Pitzer, D. Z. Lippmann, R. F. Curl,39 L. Koenigsberger, Hermann uon Helmholtz, trans. F. A. Welby (Oxford University Press, 1906),C. M. Huggins and D. E. Petersen, J. Amer. Chem. Soc., 1958,80,4793.p. 340

ISSN:0366-9033    

DOI:10.1039/DF9704900030

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Theory of the Single ChainBY S. F. EDWARDSThe Schuster Laboratory, Manchester University, lManchester, M 13 9PLReceived 19th February 1970The theory of a polymer chain is discussed with special emphasis on the role of entanglements inthe conformation and dynamics of a very long polymer.The single chain has always received much attention from polymer scientistssince it is not only the first thing to consider in any polymer problem, but also initself it can give rise to many of the problems which occur in polymerized materialsand solutions. The material has been reviewed many times, so I shall not attempta general survey here, but concentrate on aspects of the single chain problem whichhave not received so much attention as yet, mainly because the mathematicaltechniques required to solve them are unusual, and because experiments which canisolate the properties I wish to discuss are difficult, and also novel.The first pointto raise is one of mathematical technique. When discussing theoretically how apolymer can be constructed out of monomers one evidently will obtain differenceequations in many variables, i.e., be led to matrix equations. These equations tellone what configurations of bond angles and lengths are permitted, their probabilitiesand energies etc., and clearly all this detail is needed to understand local behaviour.However, over large distances, and possibly over large times, this detail may not benecessary, and in that case one may describe polymers by differential or integralequations and replace matrix algebra by the calculus.This leads to an enormoussimplification in that it becomes much easier to see the difficulties and tackle them,but it is not clear how valid this transition is. My belief is that until the continuummodels of polymers have been fully understood one will not obtain mastery over theproblems of real polymers and I shall concentrate on these models, and since acomprehensive treatment of what we know of real chain configuration has just beenpublished by Flory,l will refer to those problems only incidentally.The first question that arises is that of knowing the simplest possible descriptionof a chain. If the chain is made by adding monomers with a certain probabilityprescription p depending solely on the position of the last monomer (i.e., a Markovprocess), the probability of end points Ro, RN; P is given byso that, in Fourier transform,Defining an effective step length I byP(RN,RO) = Snd%p(R,,RN-l)p(R,-,,RN-,) p(R1Ro), (1)P(k) = pN&) = exp [ -N log p(k)J = exp [ - Nk2((a2p/(ak2)k2 = 01 - NO(k4).(a2p/ak2)kz = 0 = 12/6P(RN,Ro) = (3/21n)* exp [ - (3/22)(RN -Ro)2J.Here p need not be a smooth function.T h e result is true for S functions, i.e., R,-Ra-l having only a choice of fixed values. This model gives (RN-RJ2 = NZ2 = sl,444 TIJEORY OF rrrw S I N G L E CHAINs the arc length, and correctly characterizes the long-distance behaviour of a non-interacting chain, and thus a first sight would appear to give a complete theory nearthe Flory temperature.However, this is only true for conformation problems andno others. So it will be necessary to generalize the problem to that in which theprobability of RN depends not only on RN-1 but also on RN+1. In continuum terms,the curvature then enters. Now the random flight model is well known to satisfythe differential equationin the limit of large distances. This equation describes a system in which the tangentat each successive point is completely random, and is similar to the Schrodingerequation of quantum mechanics where the velocity at each successive interval of aparticle, whose position is known, is completely uncertain. When a probability isattached to the curvature of our polymer, it is similar to the transition from quantumto classical mechanics, and e.g., if a Boltzmann factor exp (-fK2/kT) is attached tothe polymer (IC the curvature, = (a2r/as2)' or in discrete terms I C ~ = @,+I-2R, + R,- ,)2/422, the result is a differential equation of the typea a 3- + v- + -v2 + as ar 21 6a2av2G(r,rr,v,v',s) = 6(r - r')6(v - v')6(s)where a is a convenient representation of the constants kT, f (and in general neednot depend on kT in any simple way) ; as before r is the position, v is the derivative.The end-to-end distance is now(RN - R,J2 = l{s- (a/2)[ 1 - exp (- 2s/a)]),which is as before for large s, but for small s is Zs2/a.Such seems to be the minimumcomplexity needed for chains to be treated in differential form.SELF INTERACTION OF A CHAIN ( I )We can learn more of principle by considering a simple chain cross-linked toitself, than from the effect of forces; so we consider first that one is at the Florytemperature, and that cross-links are introduced into a chain.We can arrangechemically that the cross-links are entirely randomly spaced along the chain (e.g.,direct substitution of reactive groups into an inert chain, at the low degree limit,then intramolecular linkage of the groups. Members of the research group arecurrently attempting to illustrate this point) and consider this case for simplicity.There will then be, say, M cross-links distributed at random along the chain.To discuss the size and shape of such a chain, we suppose the cross-links restrict it asif it were subjected to a harmonic well of radius w;', i.e., as if the diffusion equationwere 1 a i a 2 1 ( as 6ar2 2- + - - + - w $ r 2 P = O .The entropy of this system is readily determined to be bounded bywhere -woL/2 is the configuration entropy of the chain in the well and pwz loss ofentropy due to the cross-links, where p is the chemical potential of the N linksS/k = - W O L / ~ + ~ W ~M = -as/apS.F. EDWARDSso that since s is a bound,and wo = 3M/L.Now we consider the much more complicatedlinking agents which, it will be supposed, act fairlydS/dw, = 0,45problem of introducing cross-quickly and are then removed.Then the number of cross-links will be proportional to number of times that particularchain was within a certain distance of itself. When this probability is to be estimatedit will be seen that the basic trouble lies in entanglements of the chain.Manyconfigurations are rapidly interchangeable, but the presence of, say, a knot, willtake a long time to remove. Thus, it can be suggested that as a first approximationone should decompose the chain into those classes which for closed or infinite chainsare permanent, i.e., will not transform into one another as time goes by. We labelthese classes c, so thatP = CP,.C(3)In each c there will be probability of segments of the chain meeting one another, sothe number of links formed will be M,. The final ensemble probability densitywill then beP - , ~ ~ c . m , ,where P,,,, is the probability distribution of a chain in class c with M, cross-links.We now try a rough calculation of this effect.The label c will represent the values ofthe set of invariants characterizing the chain. The simplest of these is the self-angle,i.e., the angle swept out by the chain along itselfc = ~f(drlxdr2) . V(l/r12) (= 4n x +integer for closed curve)and we shall confine ourselves to this quantity for simplicity. (Higher invariantswhich treat the ordering of knots and so on do not alter things in principle.) Nowthe quantity c can be positive or negative corresponding to the existence of positiveand negative knots and to the fact that the equation r(s) is vectorial, i.e., it mattersto c which end of the chain is s = 0. If we calculate the mean value of c2 it divergesfor the random-flight diffusion equation, which is not surprising as the differentialequation permits total changes in direction over infinitesimal arc length and suchloci are infinitely entangled.But when the next order of accuracy is used, ie., eqn(2), a finite answer is obtained andwhere y is a numerical constant. Though not exact, it is reasonable to approximatethe distribution of c as(c2> = ul(L/al (4)since the c take on the values of 471 x +integer (for closed curves) and so can beapproximated uniformly by the continuum (- co < c < GO). The value of c buildsup rather like a one-dimensional random walk as one goes along the chain,positive and negative entanglements appearing at random. Entanglements can spanthe whole chain, but nevertheless the fact that the mean square exists and has thevalue (4) for large c suggests that the picture is a sensible one.One can thereforethink of decomposing the end-to-end probability,a,P(r,L) = ('>i.xp( 271Ll - 2Ll "> = zp,(r,L) , 2: -03 dcp(c,r,L)46 THEORY OF THE SINGLE CHAINwherep(c,r,L)dc = ( L y e x p ( - 2LZ "> s", 2nL2andThe simplest way to developp(c,r,L) is to use the Hermite form,p z (3/2Ll~)3(alqL~)* exp (- 3r2/2L1 - c2a/2qL)[1 - A(r2 - LZ)(c2 - qL/u)].(As with q, A is a number which can be obtained from fitting with the invariant, butspace precludes a derivation.) At this point, by analogy with the simplest case, theprobability of two points of the polymer being a distance I apart in spacewhere we have used the distribution derived for end-to-end probabilities also forintermediate probabilities of two points arc length s apart.Thus, we have M,and the calculation can proceed as before, with a chemical potential pc, and then afinal summation in eqn (3). Space does not permit the final calculation, but I wishto emphasize not so much the detail of the calculation, but the fact that this kind ofquestion can be raised, and answered at least approximately.This discussion leads to the question of whether a long polymer is localized inspace. If a polymer is long enough over a corresponding time scale, the variousinvariants are indeed invariant, and the entanglements can be considered permanent.This leads to the question : if at one time the polymer has a form rl(s), then will itsform r2(s) at a subsequent time have any relationship to rl(s) ? Or will rl(s) be essentiallyunrelated to r2(s) except in the sense that on working out the invariants they will bethe same? In other words, given a segment of the polymer cannot pass throughneighbouring segments, is the effect of neighbouring segments enough to box in thefirst segment, or is the system, in spite of the constraints virtually free to take up anyconfiguration.If the polymer were confined to a box of side much less than the" natural radius " (LZ)*, a single polymer will cross and recross the box many timesand the resulting system will not essentially different from a rubber or glass. Clearlythen a segment will be trapped by its neighbours. Will it also for a free polymer?One can try to answer in this way.If the probability distribution of the chain rl,isp([r,]), of r2 p([r2]), then the joint probability (before normalization) will bewhere 11, I, are the invariants of rl, r, and r the (Kronecker) delta function which isunity when 1, = I,, and zero otherwise. One can try to model SIIz2 which has a" mirco canonical " structure, by a canonical formand this in its turn by a constraint sufficiently simple to integrate, say,exp ( - w1-exp (- wgI(ry - ri)2ds).As before one can now evaluate the entropy of the system as a function of wo, andfind the minimum value. Again, I do not have the space to detail the calculationbut can only state the result : which is that for small a the system is indeed alwayS . F. EDWARDS 47localized but as a increases a critical value is reached after which there is no localiza-tion and the effects of the invariants can be obtained by treating them as a smallperturbation. We recall that the size of the polymer (R2) = LZ (L large) does notdepend on a, but the entanglement does.Thus, one can have two polymers of thesame mean square radius, one of which is fully entangled with itself like the manypolymers in a glass, whereas the other has very little constraint on its configurations.So far the discussion has been restricted to chains at the Flory temperature. Theeffect of constraints can be discussed above and below that temperature. In parti-cular, below the 6 point the self-attracting chain will tend to contract, being heldfrom collapse by the repulsive forces at small distances, and also, over a time scale,by the entanglements which will always work against the configurations of minimumvolume.These different mechanisms can be put together after the style of the van derWaals’ theory of a dense gas, and allowing for polymer solvent interaction, a theorydeveloped of the natural size and distribution of a polymer below the Flory tempera-ture. Once one goes any way below this temperature, polymers collapse and one isthen really dealing with a small piece of solid. We quote only the simplest form for thefree energy, We suppose the self-attraction confines the polymer to a box of side 9,and the van der Waals’ self attraction is v(T), excluded volume of step I is h2.ThenThe terms of F/kT are due respectively, to (i) the change in entropy due to confine-ment ; (ii) the effect of the excluded volume (which unlike the free chain can be assessedsince there is a reference volume g3); (iii) the net attractive potential effect for thepolymer below its &point ; (iv), (v) are polymer solvent effects.When u is large, Fhas a minimum for a definite 92 and the polymer is self confined. But as the tempera-ture increases one reaches a point where 9+co and there is no confinement. (Thederived expression is not accurate as B-+co, but serves as do van der Waal’s typetheories in general, to illustrate the point.)One cannot do justice to the case of the high temperature, T> 0 polymer in a briefdiscussion like this, so I shall leave that topic and discuss the motion of the chain.DYNAMICSTo discuss dynamics one needs to understand the behaviour at arbitrary timeintervals, so the entanglements of the polymers can never be neglected in principle ;they may not matter in special cases, but that has to be proved.To start this problemone needs to think of the probability distribution of the entire chain, not just the endpoints. Suppose that in (1) the probability from R, to R,,, can be approximatedby a gaussian. Strictly this cannot be true but it simplifies the mathematics withoutloss of generality. ThenP([R]) = N exp [-(3/2Z2)~(Rn-R,-J2].nN is the normalization factor. It is convenientcomponents,R,, = Z exp (2nimn/N)r(m),orR(s) = gdw exp (iws)r(w)in the continuum.to think in terms of Fourie48 THEORY OF THE SINGLE CHAINP([R]) is then N exp (( - 3/22)Z[ 1 - cos (2nmn/N)] I R, I 2 > ,a (diagonal) gaussian in the Fourier components.We now consider the motion of aparticle in a potential well V(r). In free space a particle will diffuse according toFick's equation- + - -2 p(r,t) = 0 (:z 2" as:>andp = constant is the equilibrium distribution. In the well one expects the equili-brium distribution in r to beE exp (- V(r)/kT)With the usual kind of assumptions of kinetic theory, the modification of Fick'sequation having the solution exp (- V/kT) can be shown to beand if V is an harmonic potential $q2r2(-+--(--+-+r))p=o. a I c a a 1at 2 ar ar kTThis equation has the Hermite functions for its solutions, and if at t = 0COP = C pnHen(r)n=Othenso that as t-m, all distributions tend to the equilibrium distribution n = 0.Thissuggests that the dynamics of the polymer may have a simple form in which eachr(w) acts like a particle in a potential well with equilibrium distributionexp (- (3/21)w2R(w)dw),i.e.,The symbol 6/6R means functional derivative to get the right number of dw, butthe reader unacquainted with this can treat it as a partial derivative and always thinkof (8) rather than (10). This equation can indeed be derived under certain circum-stances. For example,2 we suppose that the force acting on the polymer is that derivedfrom differentiating the free energy, and that this overcomes the friction of a solvent.ThuswhereF = TS = Tk log P so that (6F/JR) = (3/2Z)w2R(w).and 4 is the random fluctuating forces due to the solvent, and due to the fact that thethermodynamics only gives the mean force (about which there is a fluctuation whichin this case, unlike normal statistical physics, has a mean square larger than its ownuR(w) = [6F/6R(w)J + 4S.F. EDWARDS 49square mean). If it is considered to fluctuate instantaneously, this leads by standardarguments to a Rouse type equationIn this form the diffusion constant is independent of w. There are big assumptions inderiving this form and its range of validity is not clear since it would be surprising toget a complete solution without invoking the mechanism of molecular attachment.Perhaps a more realistic attempt is to consider the exchange of configurations acrosspotential barriers of the C-C type.Suppose we have a configurational idealized toovercoming a potential barrier Q. Then the rate at which this occurs in a thermalsystem according to Kramers isWAWC P(R,--+Rk+AR) = - exp (-Q I kT) = E ,2nPsay, where wA,wc are the frequencies associated with configurations (A) and (B)and p is the relaxation time of excitations of the bonds, i.e., characterizes the time inwhich energy is transported without the (A) +(B) transition.Butso that going into the continuumand we are led to the same form withAR = Rn+1-2Rn+R,-1k(w) = E W ~ R ( W ) + ~but E reflects the local structure and not the overall thermodynamics. We note thecurious law that whereas(R(s,t) -R(s',~))~ ~ ( S - S ' ) ~ ,(R(s,t)-R(s,t'))2-E(t - t')',or for FIory law excluded volume,(R(s,t)-R(s',t))2 N ( s - s ' ) ~ J ' ~(R(s,t)-R(s,t'))2 N ( t - t')6'11.So far no account has been taken of entanglements.I can, however, give a simplemodel to show how they can appear. We suppose our polymer is in two dimensionsheld at fixed points R(O), R(L) and suppose the " entanglement " simply means thatthe angle swept out around an obstacle (perpendicular to the plane) at the origin, isconstant, e.g., in fig. 1,FIG. 150 THEORY OF THE SINGLE CHAINWe now ask what is the dynamics of a chain which moves (as in the analysis above)but cannot get into the configuration, say, of fig. 2,FIG. 2.This means the dynamics has a Lagrange multiplier to ensure that Sd6(s) the integratedangle is that of fig. 1 and not that of fig. 2. One can show that the dynamical equationisE - +-w’R(w)+iAqR)])G, 1 = 0, (&+j%&+w) {&.) IwhereZ = - iAaA/as + i;ldy/ds x curl AA= (y(s)llR(s)l, -W/lR(dl)and if’ the angle swept out is 0, the final solution is 1 I m p (i;lO)G,dd3, playing the role of Lagrange multiplier. Evidently to solve such equations andextend them to three dimensions is a formidable task, but my point is that it ispossible to write these problems down, to define them, and from then on progress canbe made.CONCLUSIONI have deliberately presented a very unconventional emphasis on the problems ofa single chain, but this is a discussion paper so T think these really difficult problemsshould be aired. The work of this paper has mostly been generated by discussionwithin the polymer group at Manchester and I should like to thank the group fortheir stimulus, particularly to Prof. Allen and Prof. Gee.P. J. Flory, Statistical Mechanics of Chain Molecules, (Interscience, New York, 1969).P. G. de Gennes, Rep. Prog. Phys., 1969,32 ; Physics, 1967, 1,37

ISSN:0366-9033    

DOI:10.1039/DF9704900043

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Radius of Gyration of Stiff Chain Molecules as a Functionof the Chain Length and the Interactions with the SolventBY R. G. KIRSTEInstitute of Physical Chemistry, University of MainzReceived 8th December, 1969The radius of gyration and the second osmotic virial coefficient of a model chain molecule iscalculated numerically with a Monte Carlo procedure. The model is intended to resemble cellulosetrinitrate (CTN). The expansion coefficients of stiff chain molecules such as CTN or DNA are nearunity at high molecular weights and in good solvents.In fig. 1 the root-mean-square radius of gyration (r2)* of cellulose trinitrate (CTN)in acetone is shown as a function of the molecular weight M. The nitrogen content ofthe CTN was 13.9 %. The results are obtained by light scattering 1 s 2.As can beseen ( r 2 ) is nearly proportional to M. This occurs though the exponent in theviscosity relation [q] = KM" is exceptionally high and the solution is exothermic. Inorder to understand this result, ( r 2 ) and the second osmotic virial coefficient A2 of astiff chain model were calculated numerically as a function of the chain length and thedepth E of a square-well potential. E was varied widely so that the whole range ofpolymer-solvent interaction was covered.MFIG. 1.-Radii of gyration of cellulose trinitrate containing 13.9 % nitrogen in acetone at t = 25"C,r and M are weight-averaged, r is the root-mean-square ; 0, light-scattering measurements. Theslope of the line is 0.51 at the left and 0.53 at the right end.DESCRIPTION OF THE MODEL ( F I G .2)Nvectorsof length I are tiedtogether. Eachofthesevectorsformswithitsneighboursthe angle a, the cosine of which was put constant and equal to 19/21. The positionof a vector relative to the next but one depends in addition to a on the angle of rotation(fig. 3). The N-2 angles of rotation were determined by random numbers. It552 RADIUS OF GYRATION OF STIFF CHAIN MOLECULESmust be emphasized that a and q are not real valence angles and angles of rotation.They are only building materials of the model chain. With the use of a free-rotatingvalence-angle chain as a simple method of chain construction a stiff chain can beobtained only with a valence angle not far from zero. The statistical chain element ofour chain isl + cos a 1+19/211 - cos a ‘I- 19/21 = 201 -b = l The ends of the threads and the joining points of two successive vectors were chosenas centres of a square-well potential with spherical symmetry:00 if x < x10 if x > x2.E if x1 < x < x2It was assumed x1 = 0.6 I and x2 = 0.9 1.The potential u is related to the approach-ing of a second segment. In this respect it is irrelevant whether the two segmentsFIG. 2.-Projection of a random conformation of the used chain model, I = 17 8, is the distancebetween two potential centres, x1 = 0.6 I and xz = 0.9 I indicate the potential for the approachingof a chain fragment A to the chain fragment B (eqn (2)).(A and B in fig. 2) belong to the same chain or not. For the correspondence to theCTN-chain 1 = 17 A was chosen.This value was found in the following way.From the measurements which are shown in fig. 1 the unperturbed dimensions wereestimated by an extrapolation pr0cedure.l’ The statistical chain element of theseimagined unperturbed CTN-chains was found to be 340 long. Therefore we haveto take I = 340/20 = 17 A. The molecular weight of a 340 A fragment of the CTN-chain is = 19 600. The molecular weight belonging to a potential centre of themodel chain therefore emerges as MI = 980. Now the model is entirely defined.PERFORMANCE OF THE CALCULATIONLet us start with the equation 6-8A2 = - N/2 YMzJF1(1)F1(2) [expi - U(l,Z)/kT] - 11 d(1) d(2) (3R. G . KIRSTE 53Nis the Avogadros number and Vis the volume of the system.Fl(i) is the probabilityfor the conformation i, where i is representing all the coordinates which are necessaryto describe the position and the conformation of the molecule. U(1,2) is the intermol-ecular potential energy of a pair of molecules with the position- and conformation-coordinates 1 and 2. d(i) is the product of the differentials of the coordination i.FIG. 3.-The principle of thread construction, o( = valence angle, 'p = angle of rotation.In eqn (3) the integration over the coordinates s1 of the centre of mass of the firstmolecule yields V. Withone obtains thenFl(i) = exp { - U(i)/kT)/Jexp { - U(i)/RT)d(i- Si) (4)U(i) is the intramolecular potential energy and (i - s3 are the coordinates of conforma-tion and position without si.In order to calculate U(l), U(2) and U(1,Z) a model must be chosen.For a givenmodel the integrals in eqn ( 5 ) may be solved numerically. Tt would be hopeless totry this with one of the common methods of integration? for instance, with the rule ofSimpson. Such multiple integrals can be solved numerically by the following MonteCarlo method. Consider the integral J = Jf(xl,x2, . . .pN) dx, . . . dxN. 4, therange of integration, may be the interval 0 -= x1 <2n, 0 < x2 <2nY . . ., 0 e x , < 2n. Aset of random numbers r1,r2, . . ., r, each placed between 0 and 271 is now generatedand f(rl,r2 . . . rN) is calculated. Then (27~)~f(r~, . . . r,) is a first approximation ofthe integral. The procedure is repeated as often as possible and the average is taken.454 RADIUS OF GYRATION OF STIFF CHAIN MOLECULESIt converges towards the correct value of the integral. From the fluctuations of thesingle results one can estimate the uncertainty of the calculation.If eqn (5) is appliedto the model of thepolymer molecule, thevariables are the angles of rotation, azimuthand latitude of the k s t bond of each chain and the coordinates of the centre of mass ofthe second of the two chains.In the following way the calculation programme was worked. A random con-formation was constructed and was put with its centre of mass into the origin of thecoordinate system.U(i) = 21 . & (6)zf is the number of intramolecular contacts, i.e., the number of distances h,, betweentwo potential centres j and k with x1 <hjk<x2.Distances smaller than x1 are notallowed because this would mean U(i) = 00. As soon as such intramolecularoverlapping took place, the conformation was rejected and the construction of thechain was started with new random numbers. Because of the stiffness of the chainintramolecular overlappings did not happen often. At N = 300 with a probability ofabout 0.8, no ring closure occurs during the construction of a conformation.A second random coil with a random starting direction relative to the first wasconstructed and also put with its centre of mass into the origin of the coordinatesystem. Then it was shifted in small steps along the axes of the coordinate systemwithin the total range in which contacts between the two coils could occur.Thewidth Ad of the steps was about as big as the thickness of the chain. To each positionof the two coils relative to each other the intermolecular potential energy U(1,2) wascalculated. It is infinity if two centres of potential have a distance smaller than x1otherwiseHere z1,2 is the number of intermolecular contacts (x, < hlj2k < x2).With the knowledge of U(l),U(2) and U(1,2) the integrands in eqn (5) can be cal-culated for a given pair of random conformations in a specified position relative toeach other. For the integration with respect to the coordinates of the centre of masss2 in eqn (5) in each position of the second coil the integrand is multiplied with theproper volume element, which is 1/6 of a spherical shell with the thickness Ad and theradius d = distance between the centres of mass of the two coils.Together with A2 the mean-square-radius of gyration was calculated.In theaveraging r2 of any random conformation has to be multiplied with Fl(i) as a weightfactor. According to a2 = (r2)/<r:) the expansion coefficient CI is calculated. Thesubscript 0 corresponds to the unperturbed coil, i.e. to a solution in which A2 = 0.Radii of gyration and expansion coefficients of model-chains have been calculatedwith the Monte Carlo method already by Wall et aL9-15 In contrast to the presentpaper, Wall uses the sites of a lattice for the construction of his model chains.U(i) was calculated byU(1,2) = 21.2 . & (7)RESULTSVIRIAL COEFFICIENTSFrom eqn (5)-(7), A2 is obtained as a function of kT,k Parts of these A,-valuesare shown in fig.4. A2 becomes zero at about kT/e = -2.5. If &'is held constant,then from A2 = 0 a certain temperature T = 8 follows. If the temperature is fixedon the other hand, one obtains a certain depth of potential E~ at which A2 vanishes.For this discussion it is convenient to define the parameteR. G . KIRSTE 55- 6 -8 - 10kT/s00’ I,’ ,‘0 . ,, .’ I. . ,4 2FIG. 4.--A2 from Monte Carlo calculations against kT/e. N is the number of potential centres fromfig. 2. For comparison with CTN M = 980 N. 0 Q 0 = results from the calculation.Fro. 5.-A2 against T, results from the calculation56 RADIUS OF GYRATION OF STIFF CHAIN MOLECULESWith E = const, it follows z = 1 - O/T.At the &point, z = 0 ; in the athermicsolution z = 1, in the endothermic z < 1, and in the exothermic z> 1.In fig. 5 the calculated A,-values are plotted against z. Parameter is the chainlength. Fig. 4 and S give a rough impression of the accuracy of the calculation. Infig. 6 the numerically calculated A,-values are drawn against A4 together with experi-mental data?. The results are also compared with the formula of Casassa 7 *A2 = (N/2)(fi/Mi)h(z), with z = (4n)-3(P/Mi)(M*/Kt), (9)KO = (r$)/M, h(z) = [l - exp (- 5 . 6 8 ~ ~ ; 3j]/5.68za,3, and a; -a% = 2.043~.For KO the experimental value was used. With eqn (9) Az is obtained as a function ofM and fl/Mi.isthe excluded volume of a spherical segment. u(x) is the potential for the approachingof two segments.= 4nJ[1 -exp ( -u(x)/kT)]x2 dx (10)Mo is the molecular weight of a segment.FIG.6.-& against Mfor CTN and the model. Full lines, numerical calculation of the present paper(parameter 7) ; broken lines, theory of Casassa, eqn (9) (parameter PIM:). 0, measured values ofCI" (13.9 % nitrogen) in acetone ' S *One can see in fig. 6 that at high molecular weights the numerically calculatedcurves become more and more parallel to the curves of Casassa. But the distancebetween the curves is different indicating that eqn (9) is not applicable to our model or/3 is not proportional to 'c.In fig. 6 a discrepancy between experiment and the Monte Carlo model calculationoccurs if the heat of dilution is taken into account.The experimental values lyingnear the athermic curve ('c = 1) are obtained from exothermic solutions. Thecalculated curves can be shifted downwards if the thickness of the model chains islowered. With 10.2 A (= 0.61, I = 17 A) the thickness of the CTN-chains indeedmight have been overestimated. In an additional calculation, substituting, x1 = 8 Aand xz = 12 %L for N = 30, A,-values were obtained which are smaller by a factor ofabout 0.7 than those plotted in fig. 6R. G . KIRSTE 57RADII OF GYRATION AND EXPANSION COEFFICIENTSIn fig. 7 the radii of gyration obtained from the model calculation are plottedagainst N in a logarithmic scale with z as a parameter. The asymptote agrees fairlywell with the function (r:) = bL/6 (b = 20, L = N=(bL/6)O*’ = 1.83 No*’, theasymptote in fig.7 is 1.74 NO.9. For smaller N the calculated function shows theTABLE EXPANSION COEFFICIENTS a OF THE MODEL CHAIN AND OF CTN IN ACETONEmodel calculation CTN in acetone 1.N1030100300900aM T = 0.5 t = l r = 29 800 1 1 129 400 1.000 1 1.0002 1.o00398 000 1.003 1.005 1.007294 000 1.007 1.012 1.017882 000 1.03 1.04 1.055M 0:82 OOO 1.005150 000 1.016320 000 1.04740 000 1.051 660000 1.073 650 000 1.13transition behaviour between coil and rod. The main result of fig. 7 is, that the sec-ond virial coefficient has almost no influence on the radius of gyration. In table 1the numerically calculated expansion coefficients are listed together with experimentaldata. In agreement with fig.7 all the expansion coefficients are approximately equalto unity.ndE WLtocloo 10’ lo2 lo3 lo4NFIG. 7 . T h e radius of gyration as a function of chain length and the polymer-solvent interaction ;0 and 0, results from the calculation. For comparison with CTN : unit of length is I =17 A,L = 17Nand M = 980N.DISCUSSION OF ERRORSMonte Carlo calculations with respect to errors are to be treated like experiments.In the calculation of the standard deviation of A2 and (r2> one has to account for thedifferent statistical weight G of the trials. It isan58 RADIUS OF GYRATION OF STIFF CHAIN MOLECULES4The statistical weight of A2 and <r2> obtained from a group of q trials is thenandof trials. The standard deviations of these averages of groups were calculated byGi(A2)Gt(r2) respectively.The A2 and <r2)-values in fig. 4-7 are averages of groups4 i= 1i = 10 5 8a(fm) = (i i= 1 ~ i ~ f ; ) /C i= 1 ~ i * (13)In this way the standard deviations in table 2 were obtained. fmeans A2 or r2, fm isthe average from all groups,f' is the difference between this total average and theTABLE 2.-NUMBER OF MONTE CARL0 TRIALS AND STANDARD DEVIATIONS OF AND (r2>N number of trials*- 13 97010 986 * This is the number of trials for the30 1 058 determination of <r2>. Thenumberof pairs of threads for the determin-ation of Az is half as large. 100 422300 334900 914standard deviations for N = 9004 . 4 2 ) x 104 (cm3g-2)kT/& = -2.4 1.13 1.27 = 2.5 %kT/E = -5 0.24 0.38 = 0.75 %kT/& = 00 0.21 0.39 = 0.75 %kT/E = 2 1.16 0.39 = 0.75 %a(<rW.s) (unit : I )m 2 1 - ' lo5 2 4 6 106 2 4 6 10' 2 4 6MFIG.&-Radii of gyration of DNA against M. r is the root mean square and the z-average, M is theweight-average. Light-scattering measurements : 0, Doty (1 958)' ; , Cohen and Eisenberg(1966)" ; 0, Zimm et al. (1968)18 ; 0, new measurements on trout sperm DNA in 0.05 m trinatriumcitrate and 0.15 m NaC1.l9 Full line = eqn (14)R. G . KIRSTE 59average of the trials of a single group i, Gi is the statistical weight of a single group ands is the number of groups. Eqn (13) becomes invalid if the number of groups is toosmall or if the weight of few groups dominates too much.THE RADIUS OF GYRATION OF DESOXYRIBONUCLEIC ACIDIn fig.8 the root-mean-square radius of gyration of DNA in buffered aqueoussolution is shown as a function of M.16-19 The solutions have a positive secondosmotic virial coefficient. A straight line passing through the experimental datawould have a slope much greater than 0.5, but this is not a consequence of polymer-solvent interaction. The DNA-chain is so stiff that the samples used for the measure-ments lie in the transition range between rod and coil. A formula for the relationbetween t and M in this transition range can be derived using the persistent chainmodel of Kratky and Porod 2 o s 21 :a is the length of persistence and L the length of the extended chain. The full line infig. 8 was calculated with a = 1 100 and L = (M/200) A.G. V. Schulz and E. Penzel, Makromol. Chem., 1968, 112, 260.E. Penzel and G. V. Schulz, Makromoi'. Chem., 1968, 113,64.H. Eyring, Phys. Rev., 1932,39,746.H. Baumann, PoZymer Letfers, 1965, 3, 1069.R. G. Kirste, Int. Symp. MacromoZ. Chem. (Tokyo-Kyoto, 1966), vi, 163.B. H. Zimm, J. Chem. Phys., 1946,14, 164.E. F. Casassa, J. Chem. Phys., 1959,31,800.F. T. Wall, L. A. Hiller Jr. and D. J. Wheeler, J . Chem. Phys., 1954, 22, 1036.lo F. T. Wall, L. A. Hiller Jr. and W. F. Atchison, J. Chem. Phys., 1955,23,913 ; 1955,23,2314 ;and 1957, 26, 1742.l1 F. T. Wall and J. J. Erpenbeck, J . Chem. Phys., 1959,30,634; 1959,30, 637.lZ F. T. Wall, S. Windwer and P. J. Gans, J. Chem. Phys., 1962,37,1461; 1963,38,2220 and 2228.l3 P. J. Gans, J. Chem. Phys., 1965,42,4159.l4 S . Windwer, J. Chem. Phys., 1965, 43, 115.l6 P. Doty, Proc. Nat. Acad. Sci., 1958, 44,432.l7 G. Cohen and H. Eisenberg, Biopolymers, 1966, 4,429.l9 R. G. Kirste and B. E. Zierenberg, New Measurements (Mainz, 1969).'O 0. Kratky and G. Porod, Rec. Trav. Chim., 1949, 68, 1106.21 H. Benoit and P. Doty, J. Phys. Chem., 1953, 57,958.' E. F. Casassa and H. Markovitz, J. Chem. Phys., 1958,29,493.P. Mark and S. Windwer, J. Chem. Phys., 1967,47,708.J. A. Harpst, A. I. Krasna and B. H. Zimm, Biopoi'ymers, 1968, 6, 595

ISSN:0366-9033    

DOI:10.1039/DF9704900051

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Stable Conformations of Polymer Chains and ModelCompound MoleculesBY TAKEHIKO SHIMANOUCHIDepartment of Chemistry, Faculty of Science, University of Tokyo, Hongo, Tokyo,JapanReceived 5th January, 1970In order to have information about the conformational energy of polymer chains, the structuresof small molecules which have chemical configurations similar to those of polymer chains are studied.The results for polyethylene, poly(vinylchloride), polypropylene, 1,4-polyisoprene, 1 &polybutadiene,nucleic acid and polypeptides show that the conformations of small model molecules are closelycorreIated with those of polymer chains.For several years we have studied the structures of small molecules which havechemical configurations similar to those of various polymer chains.In the presentpaper I describe the basic idea underlying this group of studies, summarize the resultsso far obtained, and discuss the facts derived from these results.BASIC IDEAOne of the main purposes of the structural chemistry of high polymers is to knowthe conformation of polymer chains in various circumstances. The first step of thisproblem is to have knowledge about the energy of internal rotation for componentsingle bonds. This knowledge can be obtained by studying the energy of simplemolecules and by estimating the difference in the conformational energy between thesimple molecules and the polymer chains.In this process it is generally recognized that the energy of internal rotation about abond axis of a polymer chain is seriously affected by the conformations of neighbour-ing bonds.Accordingly, the energy is usually expressed by the two-dimensionalconformation map or diagram or by the statistical weight matrices for inter-dependentb0nds.l However, it is not easy to have accurate information about the energy ofsuch interactions because many kinds of factors are involved. This difficulty oftencauses the gaps between the structure study and the explanation of high polymerproperties.The purpose of our present research is to overcome the difficulty by studying theconformations of those small molecules which have two or more successive bond axessimilar to those found in polymer chains. We often encounter the cases in which theenergy is different from what we expect from the energies of independent bonds.This fact shows that the present kind of experimental work is important in the studyof high polymer conformations.NOTATIONSIn this paper the conformation is expressed with respect to the main chain andFlory’s notations,l t, gf and g-, are used.The notations g and g’ are also used, the6TAKEHIKO SHIMANOUCHI 619 --, , I , ,latter denoting the opposite g. Thus, gg represents g+g+ and g-g- and gg’ representsg+g- and g-g+.When we discuss the conformation of polyisoprene chain, we need c (cis) and s(skew).2 When the rotational angle is 0,60, 120, 180,240 and 300°, the conformationis expressed by t, s+, g f , c, g- and s-, respectively. We also use s’ which is oppositeto s. Thus, ss’ represents s+s- and s-S+ and gs‘ represents g+s- and g-s+.f - 0 0I I I t 1POLYETHYLENEThe conformational energy of a polymethylene chain has been studied by manyauthors The simplestmodel compound is n-pentane.The conformation of this molecule has been studiedby infra-red and Raman ~pectra.~‘~ They show the existence of t t , tg, and gg.However, the band due to gg’ has not yet been found.and the conformation diagram shown in fig. la is given.t1 I9- t 9 +09+t t 0The combination of t and g gives various shapes for the polymethylene chain.6is approximately given by For instance, the structure of the (CH& molecule 7*fig. 2 which is expressed byAs discussed by Newman and Kay, the structure ttggtggtt is a possible conformation ofa folded polymethylene chain and may furnish an explanation for the chain-folding inthe polyethylene crystal.’ The relative orientation of the two polymethylene planesin fig.2 is more or less different from that found for the neighbouring chains in poly-ethylene crystals. However, it may not be difficult to adjust the orientation by slightlychanging the angles of internal rotation.(t)l 29stgg(t)12s’s’tg’g’62 CONFORMATIONS OF MODEL COMPOUND MOLECULESFIG. 2.-Approximate conformation of (CH&+VINYL POLYMERSFor poly(viny1 chloride) chain there are two kinds of conformation diagrams.One expresses the interaction energy for the two bonds connected by a CH2 group andthe other for the two bonds connected by a CHCl group. A model compound,2,4-dichIoropentane CH3CHClCH2CHCICH3, has been studied both for meso andDL configuration^.^ Syndiotactic, isotactic and heterotactic 2,4,6-trichloroheptanesYwhich are related to both of the two diagrams, have also been studied spectroscopic-ally.lo* Schneider et al.also studied these compounds and 2-chloropentaneCH3CH2CHClCH2CH3.l2-l4 The results are shown in fig. 1.As for the conformation of a syndiotactic polymer chain, the problem is the stab-ility of tt and gg given in fig. lb. For PVC and its model compounds, tt is far inorestable than gg and the syndiotactic chain takes only (tt), conformation. With poly-propylene the situation is different. For the model compoundtt is identical with gg. This result explains the fact that the syndiotactic polypropylenechain takes the forms (tt), and (ttgg),.The (gg)n conformation is not stericallyallowed. The mixtures of tt and ttgg are allowed. An example is shown in fig. 3.CH3CH(CH3)CH2CH(CH3)CH3,For the isotactic chain PVC and polypropylene have the same situation. Themodel compounds take the forms tg or g t and the polymer chain the (tg)n or (gt)nhelix. The purely isotactic chain is allowed to have only one joint connecting theright-handed and the left-handed helices. The joint can migrate in the chain and thetransition from the right-handed helix to the left-handed one and the reverse transitioncan take place easily by this migration.There may be other conformations, which are similar to the gg' form of n-pentane,both for the syndiotactic and isotactic chains.However, they are far less stable andare not discussed hereTAKEHIKO SHIMANOUCHI 631,4-POLYISOPRENE A N D lY4-POLYBUTADIENEFor polyisoprene the model compounds shown in table 1 have been studied byinfra-red spectra.2 The vibrational frequencies were calculated for variousconformations ; they were compared with the observed infra-red bands in the regionTABLE STABLE CONFORMATIONS OF MODEL COMPOUND MOLECULES OF CIS- ANDTRANS-1,4-POLYISOPRENES amolecule solid liquidS+, s- s+, s-H CH2CHZCHgctct ( W + Y c9-1\ * // \c=c(s+ty s-t, s+g-, s-gf)CH3 Hin parentheses are less stable.a Conformations are expressed with respect to the carbon atoms with an asterisk. Conformations700-200 cin-l, and the stable conformations were determined.The results are shownin table 1 and a few examples of the spectra are shown in fig. 4.These results lead to the conformations for cis and trans 1,4-polyisoprenes shownin table 2. The fact that CH3-CH=C(CH3)CH,CH3 takes the forms cs+ and cs-(corresponding to the cis polymer) and tc, tsf and ts- (corresponding to the transTABLE 2.-ACCESSIBLE CONFORMATIONS OF CIS- AND TRANS- 1,4-POLYISOPRENESbond axis cis polymer trans polymerCH2-CH Sf, s- Sf, s-CH=C(CH,) C tCHZ-CHZ t, 9 + Y 9-2 t, s+, 9-C(CHS)-CH2 s+, s- c7 (S+Y s-laa skew is less stable.b trans and gauche are almost equal in energy for this structure. s+g+, s-g-, g+s+ and g-r arenot sterically allowed64 CONFORMATIONS OF MODEL COMPOUND MOLECULES1------II I I6 0 0 4 00 2 00U S6 00 4 00 2 00(4 cm-I (6)FIG 4.-Infra-red spectra of (a) 3-methyl-trans-2-pentene and (6) 3-methyl-cis-2-pentene.S and Ldenote the solid and liquid states, respectively. For (a) both the cis and skew forms coexist in theliquid state.TABLE 3.-sTABLE CONFORMA'MONS OF MODEL COMPOUND MOLECULES OF CIS- ANDTRANS-1 ,~-POLYBWTADIENES amolecules solid liquidH CHZCH3\ / c=c/ \H H\ / c=c/ \CH3 H\ // \C=CH HH CH2CH3\ / c=c/ \CHSCH2 Hc, s+, s-ts+, ts-cs+, cs-c, s+, s-tc, ts+, ts-cs+, cs-ctsf, cts-sfts-, s-tsc s+ts+, s-ts-s+ts-, s-ts+CH3CH2 CH2CH3C=CH Hs+csf, s-cs- s+cs+, s-cs-s+cs-, s-cs+ s+cs-, s-cs+\ // \a unpublished results by Yasuhide Alkai and the author.2TAKEHIKO SHIMANOUCHI 65polymer) and that tc is more stable than tsf and ts- may explain the difference betweennatural rubber and gutta-percha.At lower temperatures, the number of the acces-sible conformations for the cis polymer is larger than that for the trans polymer. Athigher temperatures, however, the number for the former is smaller than that for thelatter.TABLE 4.-ACCESSIBLE CONFORMATIONS OF CIS- AND TRANS-1,4-poIybutadienes abond axis cis polymer trans isomerCH=CH C tCH-CH2 Sf, s- c, s+-, s-CH2-CH2 t, Qf, 9- t, 9+, 9-a see notes of table 2.For polybutadiene the model compounds studied and the stable conformations areshown in table 3. The conformations for the cis and trans polymers are shown intable 4. From these results we can count the number of accessible conformations anddiscuss the relationship between the number and elasticity.Table 5 shows the averagenumber n, of accessible conformations per axis, which is defined byn, = ( n [ n i ) l ' P ,iwhere n1 is the number of stable conformations for each axis or for each set of axesfound in the unit polymer chain, andp is the number of axes in the unit. It providesthe intramolecular factor for rubber elasticity.TABLE 5.-AVERAGE NUMBER OF ACCESSIBLECONFORMATIONS PER AXIS apolymercis-l,4-polyisoprenetrans-l,4-polyisoprenepolyisobutylenepolyethylenepolytetrafluoroethylenepoly(viny1 chloride)isotactic polypropylenenc1.86H.T. 2.06L.T. 1.571.86H.T. 2.28L.T. 1.86-3H.T. 2.65L.T. 11110 H.T. and L.T.give the number at higher and lower temperatures, respectively. For L.T. wetake only the conformations which have the lowest energy.NUCLEIC ACIDThe only model compound for this substance so far studied is dimethylphosphateanion CH,O(PO ;)OCH3. l7 The infra-red and Raman spectra of barium dimethyl-phosphate were measured in the solid state and in aqueous solution. The normalvibration frequencies were calculated for the tt, tg and gg conformations. The resultshows that the anion takes only the gg form, which is also found for the phosphategroup in the DNA double helix. The conformation of the anion were confirmedafterwards by the X-ray diffraction analysis.l866 CONFORMATIONS OF MODEL COMPOUND MOLECULESPOLYPEPTIDESAs the model compounds of polypeptide chains, we have chosen acetylamino acidN-methylamides CH3CONHCHRCONHCH3, where R is the amino acid residueside chain.This compound was first synthesized by Mizushima et al. for some aminoacids, the near infra-red spectra in dilute carbon tetrachloride solution were measuredand the two conformations, one with an intramolecular hydrogen bond and onewithout it, were found.lg Recently, the model compounds were prepared forR=CH3 and DL), CH2CH3 a and DL), CH2CH2CH3 (L and DL), CH2CH2CH2CH3(DL), CH2CH2SCH3 (L and DL), CH2CH(CH3), (L and DL), CH2COOH (L) andCH2C6H5 (DL) and were studied.20TABLE 6.-CRYSTAL MODIFICATIONS OF ACETYLAMINO ACID N-METHYLAMIDESCH3CONHCHRCONHCH3 aR m.p. ("C) form A form B-CH3 L 182DL 162DL 161-CH;CHzCH3 L 197DL 160-CH2CH2SCHS L 181DL 133-CH2CH3 L 205-6-CH2CHzCH2CH3 DL 173-4-CH2CHCH3 L 165-6Ifrom melt aa -- aabaaaaa-------CH3 DL 152-3 from melt a-CHZCOOH L 190 from melt d-CH2CgH5 DL 183 C e, from melta In form A and form B, a, b, c, d and e mean that the crystal is obtained by the recrystallizationfrom ethylacetate, ethanol+ether, acetone, methanol + ether and water, respectively.The crystalobtained from melt is metastable.In the course of the study of infra-red spectra of these compounds we found thatthere are two kinds of crystalline modifications. For acetyl-L-alanine N-methylamide,for instance, we have " form B " when the crystal is grown from solutions and '' formA " when it is melted at high temperatures and the crystal grown from melt.Whenform A is kept at room temperature for a few days, it becomes form B, showing thatthe former is metastable. The situation is different when we chose a different aminoacid. It also depends on whether the compound is pure L or racemic. The result isshown in table 6. In many cases we have only form A.These two modifications are distinguished by the infrared spectra in the region700-500 cm-l, where the C-0 in-plane and out-of-plane bending vibrations (amide-IV and VI) appear (fig. 5). In this region the form B crystals have two definitely-separated bands near 630 and 600 cm-l, usually the former being weaker. On theother hand, the form A crystals have only one band or two almost overlapping bandsnear 600 cm-l.These two kinds of crystals are also distinguished by their physicalappearance, the latter being soft and fibrous and the former being hard and crystalline.The infra-red spectra of these modifications in the other wave-number regionsare also different from each other. We calculated the normal frequencies of theacetylalanine N-methylamide molecule for the various conformations, e.g., for thevarious values of II/ (the rotational angle for the Ca-CO axis) and + (that for thTAREHIKO SHIMANOUCHI 67NH-C'-axis) and compared the values with the observed frequencies of the infra-redbands. The result suggests that the conformation of the molecule in the form Acrystal is different from that in the form B crystal and that $ is near 300" and 4 is near60" for form B, and both II/ and 4 are near 120" for form A.Ichikawa and Iitaka 21studied the crystal structure of acetyl-DL-leucine N-methyl-amide which takes theform B modification. The result shows that the conformation of this molecule is$ = 319" and 4 = 86", thereby supporting our suggestion. The results are sum-marized in the conformation diagram shown in fig. 6. The model compound takesthe conformations corresponding to the stable forms of polypeptide chains.We have also studied the conformation of the acetylglycine N-methylamidemolecule CH3CONHCH2CONHCH3 .22 The compound has also two modifications,although the transition behaviour is not so definite as for the systems mentionedabove. The infra-red spectra and the calculated frequencies show that the conform-ation in one modification (form A) is t,b = 180" and $ = 120" and that in the order(form B) is $ = 0" and # + 120".In other words, $ is trans or cis, and $ is gauche.We have also studied the molecular conformations of N-methyl-propionamideCH3CH2CONHCH3, N-ethylacetamide CH3CONHCH2CH3, and N-methylchloro-acetamide ClCH2CONHCH3.23 For the last compound the X-ray crystal analysisForm A0004P e0, , I700 5 0 00I , ,700 5 0 0Form Bcm-IFIG. 5.-Infra-red spectra of form A and form B crystals. A, acetyl-L-alanine N-methylamide;B, acetyl-L-leucine N-methylamide ; C , acetyl-DL-leucine N-methylamide ; D, acetyl-I-aspertic acidN-methylamide ; E, acetyl-DL-phenylalanine N-methylamidc. The bands with a circle are the keybands68 CONFORMATIONS OF MODEL COMPOUND MOLECULESshows that the Cl atom is almost cis to the NH We are now doubtful of theaccepted concept that the inherent potential of internal rotation about the Ca-COaxis of polypeptide chains has three minima.The two minima of the trans and cismay explain the experimental results more ~traightforwardly.~~A P'iFIG. 6.-Conformation diagram of polypeptide chain. A and B are the suggested conformation forform A and form By respectively. 1 , a-helix; 2. antiparallel 6 : 3. acetyl-DL-leucine N-methylarnide.2On the other hand, the experimental results support the three potential minima(trans and gauche) for the NH-Ca axis.25 The fact that we often have the gaucheconformation may suggest that the gauche form is far more stable than the trans formfor this axis.23CONCLUSIONAll the above results show that the conformations of model compound moleculesare closely related with those of the polymer chains.This fact strongly suggests thatthe short-range forces are dominant when the polymer chain chooses its conformations.The intermolecular forces or the molecular packings may select one of the stableconformations when the short-range forces give more than one accessible conforma-tions. However, the intermolecular interactions cannot largely change the angles ofinternal rotation from those of the stable conformations, unless there are exceptionallystrong intermolecular forces or exceptionally small barriers to internal rotation.P. J .Flory, Statistical Mechanics of Chain Molecules (Interscience, New York, 1969).T. Shirnanouchi and Y. Abe, J. Polymer Sci. A , 2, 1968,6, 1419.S. Mizushima and H. Okazaki, J. Amer. Chem. Soc., 1949, 71,3411 ; N. Sheppard and G. J.Szasz, J. Chem. Phys., 1949, 17, 86.R. G. Snyder, J . Chem. Phys., 1967, 47, 1316.A. Tomonaga and T. Shimanouchi, Bull. Chem. SOC. Japan, 1968, 41, 1446.S. Mizushima and T. Shimanouchi, J. Amer. Chem. SOC., 1964, 86, 3521. ' B. A. Newman and H. F. Kay, J. Appl. Phys., 1967,38,4105.H. F. Kay and B. A. Newman, Acta Crysf. By 1968, 24, 615.T. Shimanouchi and M. Tasumi, Spectrochim. Acta, 1961, 17,755.lo T. Shimanouchi, M. Tasumi and Y. Abe, Makromol. Chem., 1965,86,43TAKEHIKO SHIMANOUCHI 69'l 2 D. DoskoEilova, J. Stokr, B. Schneidcr, H. Picovh, M. Kolinskjr, J. Petrhnek and D. Lim,l 3 B. Schneider, J. Stokr, D. DoskoEilovB, M. Kotinsk?, S. Slkora and D. Limy J. Polymer Sci. C,l4 A. Caraculacu, J. Stokr and B. Schneider, Coll. Czech. Chern. Comm., 1964, 29, 2783.l5 T. Shimanouchi, Y . Abe and M. Mikami, Spectrochim. Actu A , 1968,24, 1037.l6 T. Shimanouchi and Y. Abe, Kubunshi, 1968,17,727.l7 T. Shimanouchi, M. Tsuboi and Y. Kyogoku, Physical Properties of Biochemical Compoundsl 8 Y . Kyogoku and Y . Iitaka, Actu Cryst., 1966, 21, 49.l9 S. Mizushima, T. Shimanouchi, M. Tsuboi, T. Sugita, E. Kato and E. Kondo, J. Amer. Chern.Soc., 1951,73,1330 ; S . Mizushima, T. Shimanouchi, M. Tsuboi, T. Sugita and T. Yoshiinoto,J. Amer. Chem. Soc., 1954,76, 2479.T. Shimanouchi, Piire Appl. Chenz., 1966, 12, 287.J. PoZymer Sci. C, 1967, 16, 215.1968, 23, 3891.( A h . Cherrt. Phys., vol. VII), ed. J. Duchesne (Interscience, New York, 1964), p. 435.2o Y. Koyama, T. Shimanouchi, M. Sat0 and T. Tatsuno, Biupofymevs, to be published.21 T. Ichikawa and Y. Iitaka, Acta Cryst. B, 1969, 25, 1824.22 Y. Koyama and T. Shimanouchi, Biopolymers, 1968, 6, 1037.23 Y. Koyama, Thesis (University of Tokyo, 1970).24 Y . Koyama, T. Shimanouchi and Y. Iitaka, to be published.25 S. Mizushima and T. Shimanouchi, Adv. Enzymol., 1961, 23, 1.26 T. Shimanouchi, Y . Abe and Y . Alaki, Polymer J., to be published

ISSN:0366-9033    

DOI:10.1039/DF9704900060

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Method of Determining the Relative Stability of DifferentCon formational States of Biopolymer MoleculesBY 0. B. PTITSYNInstitute of Protein Research, the USSR Academy of Sciences, Poustchino, MoscowRegion, USSRReceived 27th February, 1970A method for estimating the relative stability of conformational states of biopolymer moleculesfrom the degree of sharpness of the transition between them has been suggested. The method isapplied to the estimation of stability of the DNA helical structure under physiological conditions aswell as to the estimation of stability of the a-helical structure of polypeptide chains in differentsolvents. The stability of an or-helical structure of polypeptide chains is relatively low not only inaqueous media but also in water-organic mixtures and even in inert organic solvents in which theenergy of intramolecular hydrogen bonds is essentially higher than in aqueous medium.This isan evidence that the formation of the helical structure in organic solvents must be accompanied bya noticeably greater decrease in the conformational free energy of monomer units than in water.This conclusion is confirmed by independent experimental data on free energies of initiation of helicalregions of polypeptide chains. For the explanation of this effect an assumption is made on the roleof dipole-dipole interactions of peptide groups of the backbone.PRINCIPLES OF THE METHODDepending on external conditions, biopolymer molecules exist in differentconformations, and cooperative transitions between them can occur.The study ofthe relative stability of such conformations is of interest, but under the conditionswhen one of the conformational states is considerably more favourable than the other,a direct measurement of the free energy difference of them is impossible. In thesecases it is necessary to change one of the parameters of the external medium so as tolevel the free energies of both conformational states. If this external parameter isdenoted as X , the free energy difference of the macroniolecule under the conditionsconsidered, X = X,, is equal towhereA@(Xo) CD2(X0)-@1(Xo) = -(&D2-6@1), (1)6% = @ I ~ ~ t ~ - - ~ ~ ~ ~ O ~ ~ (2)6% = @2(Xt) -@2(&),and X , is the value of X corresponding to the condition of A@(X,) = 0, i.e., to themidpoint of the conformational transition between the two states.Consequently,it is possible to determine A@(Xo), if the energies of transfer 6@ of the macromoleculebeing in two conformational states from X = Xo into X = X, conditions are known.The method suggested by Tanford of estimating the stability of the native structureof globular proteins by their denaturation with urea ( X is the urea concentration)and the method suggested both by Zimm and Rice and by Nagasawa and Holtzerof estimating the stability of the helical state of uncharged polypeptide moleules bythe helix-coil transition during their ionization ( X 3 -2.3 pH is the chemicalpotential of protons in solution) are based on this principle. In the first case, the70.B . PTITSYN 71energies of transfer are estimated on the basis of data on separate amino acid residuesand the spatial structure of a protein molecule ; in the second case they are deter-mined by the curves of potentiometric titration.2*suggested a methodof estimating A@(Xo) for those cases where a direct determintion of free energies oftransfer is not possible. The method is based on determining the X derivative ofA@ in the transition region and on estimating A@(Xo) by the equationRecently the author in collaboration with T. M. Birshtein 4*using physical considerations as to the type of dependence of the derivative aA@/sXon X. If, e.g., X is a chemical potential ,u of the " transforming agent ", i.e., asolvent component in different ways interacting with the macromolecule in twoconformational states, thenwhere An(p) is the difference of the number of the transforming agent molecules whichare bound with the macromolecule in the two conformational states; Consequently,amlax = am(p)/ap = -An@), (4)atA@(po) = [ PtAn(p)dp = [ An(a)da/a, ( 5 )J PO J a0where A@ and ,u are expressed in kT units, and a = e p is the transforming agentactivity.The An@*) value in the transition region can be either measured directly (whenthe transforming agent is hydrogen or metal ions), or obtained from calorimetricdata with the help of thermodynamic equality 4* :where AH(T,) is the heat of conformational transition and Tt is the transition tempera-ture, or, finally, determined from the degree of sharpness of the conformationaltransition according to eq~ation,~.where @,(X) and O,(X) are portions of molecules in the two conformational statesin the region of conformational transition. @,(X) and @,(X) values are easilydetermined experimentally when the transition occurs on the " all or none " principle,while in the general case their determination from experimental values requiresknowledge of the dimensions of the " cooperative region ", i.e., of the part of themacromolecule which accomplishes the transition as a whole.After An(a,) has been determined, the value of A@(Xo) can be estimated by theeqn (3) when the type of An(a) dependence is known.If, e.g., An is proportional toa (as in the binding of urea by proteins), thenIf, on the contrary, the molecule in both conformational states binds the maximumaccessible for it number of molecules of the transforming agent (as in the binding ofmetal ions by DNA molecules) then A n(a) = const.andIn @,(X)/@,(X) = -An@J(p--Pt) = -A.n(at) In (ala,), (7)AQ0 = An(a,)(a, - a,)/a,. (8)AB0 = An(a,) In (ar/ao). (9)THE STABILITY OF DNA DOUBLE HELIXWe made use of the above method for the determination of the DNA doublehelix stability in physiological conditiom6 Taking as a basis the microcalorimetri72 RELATIVE STABILITY OF CONFORMATIONAL STATESdata by Privalov on DNA denaturation heats in solution with different NaClconcentrations and using eqn (6) it was found that AnNaCI E -4NaCI mol/mol ofnucleotide pairs and that it practically does not depend on the NaCl concentrationin solution. Substituting this value of An into eqn (9), we find that under the physio-logical conditions (37"C, pH 7 and NaCl concentration -0.2 mol/l) the differenceof free energies of the denatured and native states of DNA Amo 12: 1.2 kcal/mol ofnucleotide pairs, i.e., almost an order of magnitude lower than that of the denatura-tion heat equal to -9.6 kcal/mol at pH 7 and ionic strength 0.2 moI/L7THE STABILITY OF THE CC-HELIX I N POLYPEPTIDE CHAINSLet us consider in more details the application of this method to the estimation ofa-helix stabilities of synthetic polypeptides in different solvents.For polypeptidescontaining ionizable groups and undergoing the helix-coil transition during theirionization, our method is reduced to the method by Nagasawa and H ~ l t z e r , ~ asin this case p = -2.3 pH, An(p) = NAcr(p) and consequently,In eqn (lo), N is the number of ionizable groups in the macromolecule, and Acr(pH)is the difference of degrees of the coil and helix ionization at given pH, which can bedetermined from the curve of potentiometric titration by the extrapolation of thoseparts of this curve which correspond to the titration of the helix and coil (see ref.(3)).We made use of this method in determining the stability of the helical state ofpoly-L-glutamic acid (PGA) molecules in 0.2 M NaCl aqueous solution and itsmixtures with dioxane at different temperatures. The results are given in table 1.TABLE 1THE STABILITY OF CI-HELICES A& (IN CAL~MOL) OF THE UNCHARGED POLY-L-GLUTAMIC ACIDIN WATER AND ITS MIXTURES WITH DIOXANEsolvent 0.2 M NaCl+dioxanetemp.*C 0.2 M NaCl 3 : 1 vol/vol 2 : 1 vol/vol9 240 460 (at 10°C) 49022 150 340 43040 100 280 36050 60 220 250- 245 60- 30 130 --In accordance with the data of other authors, (see, ref. (9)-(I 1)) table 1 showsthat the stability of a helical state of the uncharged poly-L-glutamic acid is lowand decreases rapidly with the increase of temperature. The temperature dependenceof values of A#,-, in aqueous medium given in table 1 leads to the values AH = 1.2kcal/mol and A S = 3.8 cal/mol deg. for the differences of the coil and helix enthalpiesand entropies. By addition of dioxane the stability of the helical state is increasedtwice or threefold remaining, however, comparatively low.The temperaturedependence of the stability also increases (AH = 1.9 kcal/mol, A S = 5.2 cal/mol deg.).These results seem unexpected as the energy of intramolecular hydrogen bondsin the mixtures of water with dioxane must be considerably higher than in water.Therefore, it is of interest to estimate the stability of the helical state of polypeptidechains in the inert organic solvents (e.g., dichloroethane), where the helix-coi0. B . PTITSYN 73The value An(a,) in this case transition can be caused by adding dichloroacetic acid.can be determined from the transition steepness by eqn (7).From the theory of helix-coil transitions it follows thatIn (O,/O,) = N/v (28- l)[O(l -O)]-*,where N is the number of monomers in the chain, v the number of monomers in thecooperative region, and 8 is the degree of helicity.From eqn (7) and (1 1) we obtain(20 - l)[8( 1 - @)I-+ = - vA,n(a,) In (ala,),where a is the activity of DCA, and An is the difference of numbers of DCA moleculesbound with one polypeptide nionomer unit in the random coil and helical states.We applied eqn. (12) to our experimental data 12* l3 on coil-helix transitions forfour polypeptides in the mixture of dichloroethane with DCA, substituting DCAactivity a by its volume fraction v. The values v, corresponding to the midpointof the transition, and the slopes of experimental curves vAn(vr) are given in the 2ndand 3rd columns of table 2. v values for these polypeptides are given in the 4thcolumn and the values of An(v,) obtained in the 5th column.TABLE 2.-AnDcA(ut) FOR SOME POLYPEPTIDES1 2 2 4 5polypeptide ut vAn(ut) V An(ut1poly-y -benzyl-L-glutamate 0.72 5 33 7014 0.47pol y-y -methyl-L-glutamate 0.70 32 -7012 -0.46poly- y -ethyl-L-glutamate 0.56 22 -701’ -0.31poly-E-carbobenzoxy-L-lysine 0.36 22 8315 0.26From table 2 it follows that An(vt) are approximately proportional to vr, which isalso confirmed by the data on transition of the p-structure-coil into poly-S-carbo-benzoxymethyl-L-cystein on addition of DCA to dichloroethane.The steepnessof transition in this polypeptide where 21, = 0.Og5 is considerably less than in theothers (vAn(vr) = 8.5). Since the cooperative region for the P-structure-coil transitioncannot be small, this means that at low ZJ, the value of An(vt) is also low.As DCAmust be bound mainly through NH and CO groups of the backbone, then it can beexpected that for every given polypeptide An will also be approximately proportionalto the DCA concentration. Taking this into account, from eqn (5) we obtain A&values in dichloroethane for all the polypeptides indicated which do not exceed 400callmol. Though this estimation is approximate it clearly shows the comparativelylow stability of the helix state of the polypeptide chain not only in waterforganicmixtures but also in a purely organic solvent.The free energy difference between random coil and helical states of the polypeptidechain can be represented asA40 = A#conf. +A4int., (1 3)where A#conf.( < 0) is the free energy decrease of monomer units due to their leavingthe helical conformation, and A&,t.(>O) is the free energy increase of monomerunits on account of the decrease of interaction between them (in the first place, dueto the rupture of hydrogen bonds). Since must be considerably greater ininert organic solvents than in water, then the comparative closeness of A& valuesin these two types of solvents can denote only a decrease of A$conf, (in absolute value)during the transition from water to organic solvents.This conclusion can be verified by independent experiments as the Acjconf, valu74 RELATIVE STABILITY OF CONFORMATIONAL STATESis closely connected with the cooperativity of transition (the free energy of initiationof the helical region is approximately equal to -2A4conf.17~ Is), so thatThus, it can be expected that the cooperative region of polypeptides will increase forthe transition from water to water + organic mixtures and organic solvents. Tocheck this assumption we determined the values v for PGA in 0.2 M NaCl aqueoussolutions and its mixture with dioxane (2 : 1, vol/vol) at different temperatures from8 to 50°C.The values of v were determined from the dependence of intrinsic viscosityof PGA on its helix degree by the method suggested by the author and A. M.Skvortsov 17* 1 9 9 2o taking into account the influence of Iong-range interactions onthe intrinsic viscosity. It was found that in an aqueous medium v 21 20 (in accordancewith the results of other authors 21) and practically does not depend on temperature.At the same time v increases in the mixture on water with dioxane from -20 to-40 with a decrease in temperature from 50 to 8°C.Thus, at low temperatures weobserve distinct increase of cooperative region during transition from water to water +organic mixtures. In purely organic solvents the cooperative region is still greater(see 4th column, table 2).Data on the temperature dependence of A&, and A+conf. permit us to understandthe nature of stability of the PGA helical state in aqueous medium. We found thatin this case A+o depends greatly on temperature (AH 21 1.2 kcal/mol, A S E 3.8cal/moldeg.). At the same time the v dependence on temperature denotes thatA&onf.is proportional to the absolute temperature T(AHconf. N 0, ASconf. 2: 6.2cal/mol deg.). This means that the helix-coil transition in aqueous medium is notaccompanied by a change of the conformational energy of monomer units but isconnected with a considerable increase in their conformational entropy. For changesof interaction of monomer units we obtain:AH,,,. = AH-AHConf. = 1.2 kcal/mol andASint, = AS-ASconf. = -2.4 cal/mol deg.The enthalpy increase of 1.2 Kcal/mol for the helix-coil transition is very close to thechange of enthalpy for the rupture of hydrogen bonds NH . . . CO in aqueous mediumaccording to Schellman’s estimation ( N 1.4 kcal/mo1)22. The decrease of entropyduring the rupture of intramolecular hydrogen bonds is evidently a result of the forma-tion of hydrogen bonds between peptide groups and water molecules.Thus, thedecrease of energy during the helix-coil transition for PGA in aqueous medium ispractically wholly a result of the rupture of intramolecular hydrogen bonds, and theincrease of entropy is the difference between the increase of conformational entropyof polymer units and the decrease of entropy of a solvent (water) during its binding withpeptide groups of the backbone.At present it is difficult to interpret unambiguously the reasons for the increaseI I in polypeptide chains during the transition from water to water+organicmixtures and organic solutions. It is possible, however, that an essential role isplayed here by the decrease of dielectric permeability of the solvent, which can leadto an increase in the role of dipole-dipole iiiteractions between the peptide groupsof the backbone.According to calculations by Flory et aZ.23 the increasing roleof these interactions leads to a shift of the energy minimum of monomer units fromthe region corresponding to the right a-helix into another region of angIes of internalrotation and thus can increase I I 0. R. PTITS’I” 75Ch. Tanford, J. Amer. Cheni. Sac,, 1964, 86, 2050.B. H. Zimni and S. A. Rice, &lo!. Phys., 1960, 3, 391.M. Nagasawa and A. M. Holtzer, J . Amer. Chem. Soc., 1964, 86, 538.0. B. Ptitsyn and T. M. Birshtein, Biopolynzers, 1969, 7, 435.T. M. Birshtein and 0. B. Ptitsyn, Mol. biologiya, 1969, 3, 121.P. L. Privalov, 0.B. Ptitsyn and T. M. Birshtein, Biopolymers, 1969, 8, 559.0. B. Ptitsyn, T. V. Barskaya and V. E. Bychkova, Bioptiysiku, 1971, in press.W. G. Miller and R. E. Nylund, J. Amer. Chem. Soc., 1965, 87,3542.l o J. J. Hermans, J. Phys. Chem., 1966, 70,510.l 1 D. S. Olander and A. M. Holtzer, J. Amer. Chern. Soc., 1968, 90,4549.l 2 T. V. Barskaya, I. A. Bolotina and 0. B. Ptitsyn, Mol. biologiya, 1968, 5, 700.l 3 N. G. Illarionova, I. A. Bolotina, B. Z. Volchek, A. T. Gudkov, Yu. V. Mitin and 0. B. Ptitsyn,Mol. biologiya, 1967, 4,544.l 4 G. D. Fasman, Poly-or-Amino Acids (Dekker, New York, 1967).M. Cortijo, A. Roig and F. G. Blanco, Biopolymers, 1969,7,315.l 6 E. V. Anufrieva, I. A. Bolotina, B. Z. Volchek, N. G. Illarionova, V. N. Kalikhevich, 0. Z.Korotkina, Yu. V. Mitin, 0. B. Ptitsyn, A. V. Purkina and V. E. Eskin, Biophysika, 1965,10,918. ’’ 0. B. Ptitsyn, in Conformations of Biopolymers, ed., G. N. Ramachandran (Academic Press,London, 1967), vol. 1, p. 381.l 8 A. M. Skvortsov, T. M. Birshtein and A. 0. Zalensky, Mol. biologiya, 1970, in press.l9 0. B. Ptitsyn and A. M. Skvortsov, Biophysika, 1965,10,909.2o A. M. Skvortsov and 0. B. Ptitsyn, Mol. biologiya, 1968, 2,71021 R. L. Snipp, W. G. Miller and R. E. Nylund, J. Amer. Chern. SOC., 1965, 87,3547.22 J. A. Schellman, Compt. rend. trav. Lab. Carlsberg. Sir. chim., 1955, 29,223.23 P. J. Flory, Stutisficnl Mechanics of Chain Molecirles (Interscience Publishers, 1969), chap. VTT.’ P. L. Privalov, Mol. biologiya, 1969, 3,690

ISSN:0366-9033    

DOI:10.1039/DF9704900070

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

GENERAL DISCUSSIONProf. R. L. Scott (University of California, Los Angeles) said : In the past some ofus have regarded concepts like the " regular solution ", the van der Waals mixture,and the Flory-Huggins equation for chain-molecules as arising from " random " modelsin the sense that these correspond closely to molecular distributions produced entirelyby the steep repulsive forces unperturbed by the weaker intermolecular attractions.However, I do not want to get involved in a semantic quibble; strictly speaking" random mixing " cannot be so interpreted and must be restricted to the definitiongiven in Rowlinson's paper.Nonetheless, it is worthwhile to consider the basic assumptions which lead to theScatchard-Hildebrand equation. Scatchard assumed " (1) that the mutual energyof two molecules is a function only of their relative position and orientations .. . anddepends not at all on the nature of the other molecules between or around them or onthe temperature; and (2) that the distribution function for any pair of molecules isalso independent of the temperature and the nature of the other molecules present."When this derivation is carried out explicitly in terms of pair distribution functions 2 9 3"Correct " 41 Irk 912rrrI .rrFIG. 1.-Radial distribution functions (schematic) for a dilute solution of large spheres in smallspheres (oZ2 = 30, The predictions of the random mixing and regular solution approximationsare compared with more nearly correct functions.G. Scatchard, Chem.Rev., 1931, 8, 321 ; Trans. Faraday SOC., 1937,33, 160.J. H. Hildebrand and S. E. Wood, J. Chem. Phys., 1933,1,817.J. H. Hildebrand and R. L. Scott, Regular Solutions (Prentice Hall, Englewood Cliffs, N.J.,1962), chap. 7.7GENERAL DISCUSSION 77it is clear that this is not a " random mixture ". Specifically, in the random mixturethe pair distribution function gas(r) is the same function for all a/? (e.g., 11, 12, and 22)in a particular mixture, but may vary with the composition (e.g., with mole fractionx,). Conversely the assumptions of Scatchard and Hildebrand require that gas beindependent of composition, although g, 1, g12, and g22 can be different ; one usuallyassumes that they scale with the collision diametersFig. 1 illustrates these differences schematically for a mixture of a dilute solute (2)of large spheres in a solvent (1) of small spheres.The various gas are sketched for the" random mixture " and the " regular solution " and compared with the " correct "function. At infinite dilution, the three gll are identical. The g12 and g22 for therandom mixture, the same as g,,, are patently absurd. The scaled g,, and g22 forthe regular solution are probably nearly right in the region of thc first maximum, butquite unrealistic at greater distances. The fact that the regular solution gas makessome allowance for differences in size may perhaps account in part for the somewhatbetter GE values in Rowfinson's table 1.The comparisons in Rowlinson's table 1 might be extended to include the excessentropy and enthalpy separately.For the van der Waals (one-fluid and two-fluid)and Flory-Huggins approximations the free energy coefficient A arises entirely fromentropy contributions, while that for the regular solution (more accurately thesolubility parameter) approximation is entirely an energy contribution. Singer'sMonte Carlo calculations suggest that the excess Gibbs free energy GE is the smalldifference between larger RE and TsE terms.such that ga&) = f(r/aag).Prof. G. Rehage (University of Ckausthal) said : Concerning the applicability of theHuggins-Flory-theory to non-polar, binary liquid mixtures with phase separation,the critical point of phase separation is theoretically given byV,, and VO2 are the molar volumes of the pure liquid components, #2,k is the volumefraction of the component 2 at the critical point.We assume that the component 2is an associated substance with unknown molecular weight. From the measuredvalues of $2,k and Vo, one can calculate Voz, when the theory is applicable. Onecan calculate from the known volume of the monomer the degree of association ofthe component 2. For the phase separation behaviour of sulphur in the solventschlorobenzene, mustard oil, benzylchloride, aniline ; p,p'-dichlorodiethylsulphide(mustard gas), benzene and toluene we found from the known atomic volume of thesulphur from the number of sulphur atoms in the molecule the value of 7.8+ 1.7.The critical points were in the temperature range 117-180°C. From cryoscopicand ebullioscopic determinations of the molecular weight in the range from -75to +277"C the molecular size of sulphur is S8 in all solvents.Thus, the Huggins-Flory-theory gives good accordance with the directly measured values. Incidentally,all investigated systems with sulphur had an upper critical p0int.lDr. K. N. Marsh (University of Exeter) said: Some recent calculations of theexcess functions of simple binary liquid mixtures lead to the conclusion that it is theprescription for extending the equations of state to the mixtures which primarilydetermines the excess functions and not the equation of state itself. The mostsuccessful prescriptions to date are those due to van der Waals as discussed byRowlinson. Young has discussed the adequacy of the v.d.W.prescription for theG. Rehage, unpublished results.K. N. Marsh, M. L. McGlashan and C. Warr, Trans. Furday SOC., submitted78 GENERAL DISCUSSIONa term in explaining the critical phenomena for mixtures of octamethylcyclotetra-siloxane (a large globular molecule) with small globular molecules. The van derWads prescription for the b term has been found to be less adequate for these mixtures.The equation of state for a hard sphere mixture given by Lebowitz and used in thetheory by Snider and Herrington can be considered as another prescription forthe b term. This prescription gives excess volumes for the above mixtures an order ofmagnitude better than those calculated using the v.d.W. b prescription. Calculationsalso show that for these systems, the hard sphere mixture equation gives a muchbetter representation of the composition dependence of the excess entropy comparedwith the v.d.W.prescription for the b term.Dr. M. Rigby (Queen Elizabeth College, London) said : Rowlinson has mentionedthe difficulty of assessing the effects of molecular shape. It is relevant to consider theequation of state of non-spherical hard molecules since it seems probable that thesewill provide a basis for describing the geometrical structure of real non-sphericalmolecules, as hard spheres do for spherical molecules. The scaled particle theoryhas been applied to systems of convex hard molecules of generalized shape andleads to an equation of state in which the molecular shape may be characterized by asingle parameter similar to that introduced by I ~ i h a r a .~It was not previously possible to investigate the accuracy of the scaled particletheory for such systems since no exact data were available either for the equation ofstate or for the higher virial coefficients. Recent calculations of the third andfourth virial coefficients for prolate spherocylinders have shown that the theoryappears to be applicable to non-spherical hard molecules with similar accuracy tothat found for hard spheres, and it may thus provide a useful basis for the study ofnon-spherical molecules. Preliminary calculations with an equation of state of thegeneralized van der Waals type using the scaled particle equation for the hard corecontributions show that the volumetric properties are not very sensitive to the valueof the shape parameter.Prof.C. Domb (King’s College, University of London) said : In his paper, Edwardsmakes the statement : “ My belief is that until the continuum models of polymershave been fully understood one will not obtain mastery over the problems of realpolymers. . . ”. I should like to comment on what information can be derived aboutthe continuum model from lattice models. One always has a slightly guilty feelingwhen working with a lattice model since the lattice is clearly an artificial construction.Nevertheless, if one allows walks to second- and higher-order neighbours of the lattice,it is possible to simulate any magnitude of excluded volume and to obtain a coherentseries of lattice approximations to any given continuum model.Some possibleexamples of this are illustrated in fig. 1.In practice, this has not been pursued very far but nevertheless calculations havebeen undertaken for a variety of different lattices. The particular lattice structurehad no physical significance, and hence any properties which depend on latticestructure cannot have any relevance for the continuum model. However, a numberof properties have emerged which are apparently independent of lattice structure,J. L. Lebowitz, Phys. Rev. A , 1964, 133: 895.N. S. Snider and T. M. Herrington, J. Chem. Phys., 1967, 47,2248.R. M. Gibbons, Mol. Phys., 1969, 17, 81.A. Tsihara and T. Hayashida, J. Phys. SOC. Japan, 1951,6,40.M. Rigby, J. Chenr. PJys., in pressGENERAL DISCUSSION 79and it is reasonable to expect that these same properties are valid for a continuummodel.for which parallelproperties have emerged as being independent of lattice structure, and we also havethe advantage of exact analytic solutions in two dimensions which confirm thisindependence.In table 1, I have drawn up a list of quantities which seem to be latticeAnalogy with the Ising model is helpful in this connection+a, allowed steps; 0, forbidden pointsFIG. 1.-Lattice models which approach more closely to a continuum.TABLE 1 .-SELF-AVOIDING WALK-LATTICE INDEPENDENT PARAMETERSking analogueno. of walks Cn ~ / i ’ ’ t l U (susceptibility)no. of polygons un -pn+ (specific heat)mean sq. length { R i ) WnY (range of coherence)probability distribution Pn(R) mexp - (R/C,)~ (spin pair correlation function)correlation between steps (ujui) -(I@, t)/n”-Y (critical equation of state)(s = iln, t = (n-j)/n)(and other moments of t,9(s, t ) )radius of gyration <Sn”)/<Rn2>independent and have listed the Ising analogues.The exponents a, p, and y havebeen known for some time (particularly y which was first introduced by Florymany years ago). However, the parameter 6 which represents the deviation from aGaussian distribution, is not so well known ; it has the value 4 in two dimensions and512 in three dimensions. More recently it has been suggested that the completeC. Domb, A&. Chem. Phys., 1969, 15,229.P. J. Flory, J. Chem. Phys., 1949, 17, 303.C . Domb and F.T. Hioe, J . Chem. Phys., 1969,51,1915, 192080 GENERAL DISCUSSIONcorrelation function between steps is also lattice independent and is parallel to thecritical equation of state for the Ising model. It is the moments of this function whichare related to ratios like ( S z } / ( R : } , and which are thus also lattice independent.Dr. J. L. Martin (King’s College, University of London) said: It has often beenconsidered that the behaviour of quite short self-avoiding walks on a regular latticewill provide a reasonable picture of a polymer chain in dilute solution, provided thereis no strong mutual attraction between remote parts of the chain. If this is true, thencorrelations between different parts of such walks are of interest.The simplest typeof correlation is that between the two extremities of a walk. To study this, the end-point distribution is required : cn(P) is the total number of self-avoiding walks of nsteps starting at the origin and finishing at the point P. Distributions for the commonlattices have been obtained by a combination of recurrence relations and directenumeration ; and the computation has been pushed almost to the limit of computercapacity. The work has been done in collaboration with a student, M. G. Watts.The accepted wisdom is that the mean-square end-to-end length of a walk of nsteps on a chosen latticeu, = (&@)r?)/(Cc,(~)) - nY . const.P Pas n--+ 00, where y depends only on the dimensionality of the lattice. We have made amuch more restrictive assumption (essentially that u, is a coefficient in a generatingfunction with a “ simple ” singularity) :un/un-l = 1 + y/(n - a) + O( 1 /n3).Such an assumption leads to(nn + 1 - ~ n > ( u n - un - 1)u,2-un+1un-1 *Yn == y+O(l/n2),whence we obtain a sequence of estimates for y.Some values of y- for the triangular and FCC lattices aretriangular FCCn Yn9 1.487910 1.4860I 1 1.487512 1.4874n Yn6 1.20577 1.20418 1.20269 1.2013These sequences suggest the estimates :with considerable confidence, since the yn are not varying rapidly.y(triangu1ar) - 1.487 ; y(FCC) somewhat less than 1.20,Prof. W.H. Stockmayer (Dartmouth College, U.S.A.) said: Monte Carlo methodswill be very useful in confronting the complicated dynamical problems posed bychain entanglements.Suitable formulations of chain dynamics for this purpose maybe of several kinds. For example, chains on simple lattices can diffuse by a successionof randomly selected local jumps, as in the studies of Verdier 1*2 and MonnerieP. H. Verdier and W. H. Stockmayer, J . Chem. Phys., 1962, 36,227.P. H. Verdier, J. Chem. Phys., 1966, 45, 21 18 and 2122GENERAL DISCUSSION 81and Geny. g 2 Alternatively, an appropriate Langevin equation for the familiarbead-and-spring model invites Monte Carlo treatment of the random Brownianforces. The latter method has already been applied to an entanglement problem(unwinding of a chain wrapped around a stick, as an idealization of DNA unwinding)by Simon and Zimm.4Prof.P. J. FIory (Stanford University) said: With reference to Domb's andMartin's calculations on self-avoiding random walks, I would point out that forpolymer chains convergence to an expression of the form (r2)ccnY, in which y = 6/5,is attained only for very large values of the number of bonds n. Treatment of theeffect of excluded volume in the smoothed density approximation clearly shows thisto be so. According to numerous experiments on polymer solutions, even at chainlengths n = 104-105 the empirical value of the exponent y falls appreciably below1.20. Certainly, a bond of a real chain should not be equated to one step of a randomwalk, but the facior relating them should be of the order of 10 and not 103-104.That the limiting form of the foregoing relationship can be ascertained with fewerthan 20 steps is diacult to reconcile with other evidence, both from experiment andfrom theory.Dr.R. F. T. Stepto (University of Manchester) said : In agreement with Flory's remarks,one thinks that since the advent and acceptance of the rotational-isomeric state modelfor real polymer chains, the problem of excluded volume would be approached moremeaningfully by using this model than by using lattice models. Using the rotational-isomeric state model and Lennard-Jones expressions for the energies of interactionbetween segments we have obtained preliminary results which indicate that the valueof the exponent y depends to a large extent on the actual Lennard-Jones parametersused.Dr. A. J. Hyde (University of StrathcZyde) said: In view of the different approachesused by Edwards and Domb, the one considering very long chains with equilibriumpopulations of knots and entanglements, and the other considering the enumerationof very short chains on lattices the shorter members of which are not long enough toshow excluded volume effects let alone knots, I would ask whether the agreementbetween the exponents in ((R2), n) relationships obtained by the different methods isa confirmation of the essential truth underlying both approaches, or is to someextent fortuitous.Prof. S.F. Edwards (University of Manchester) said: In reply to Hyde, in anequilibrium distribution the equilibrium number of entanglements will automaticallybe present, so that in a calculation of an (R2, n) relationship, topology can be com-pletely ignored.However, I would make this comment on Domb and Martin's work :I tried to estimate some while ago the molecular weight of a chain showing theasymptotic (R2, n) relationship. It came out very large, agreeing with Flory's paperon this subject. However, this is when one uses realistic forces and effective steplengths. Domb, in using a lattice, is using " forces " which are hard to translateinto chemical constants. The criterion for the use of R2Kn, or R 2 ~ n 6 / 5 (or what-ever it is), is the largeness or smallness of (Z/Z0)3'5 (l/n)'/'" where Z is the effective1 L. Monnerie and F. Geny, J. Chim. Phys., 1969, 66, 1691.F. Geny and L. Monnerie, J. Chim. Phys., 1969,66, 1708.R.Zwanzig, Ado. Chem. Phys., 1969,15,325.E. M. Simon and B. H. Zirnm, J. Statistical Phys., 1969, 1'4182 GENERAL DISCUSSIONstep length and Z, a length characteristic of the excluded volume. How precisely oneassigns these in a lattice is not clear, and I suspect that a lattice has an extremelystrong excluded volume effect relative to any known chemical constants.Prof. C. Domb (King’s College, Uiziversity of London) said: The essence of myargument is the analogy with the Ising model for which exact analytic solutions existin two dimensions. The self-avoiding walk configurations represent one contributionto the king model solution’but it is the dominant contribution, and if all other contri-butions are ignored, it gives a good approximation to the known results.It seemsreasonable to assume therefore that rates of convergence are the same in the twoproblems, and this is borne out by the smooth and steady behaviour of numericaldata (such as shown by Martin) and by the agreement between exact enumerationsand Monte Carlo results. I certainly agree that there are problems for which con-vergence is very slow (e.g., when strong attractive forces are present). But forpurely repulsive forces I think that convergence is quite rapid and this indicates thatknots do not contribute significantly to the (R2, n) relationship. However, chainsof length 15-20 on a tetrahedral lattice are quite long enough for the pure excluded-volume effect to be felt.Dr. R. UlJman (Ford Motor Co., Michigan) said: In eqn.2 of Edwards’ paper,the differential equation for the velocity and position of a polymer chain of restrictedcurvature is given. The second moment of this equation is the Kratky-Porod resultfor semi-flexible chains.l Does he have the solution for this differential equation?If a polymer chain is in a chain-folded arrangement in the solid, it will have lessthan the equilibrium number of knots which he would predict. Does he have anyidea of the time it would take to form these knots once the polymer is dissolved?Would this be long enough to permit an experiment in which the change in radius ofgyration with time could be determined?Prof. S. F. Edwards (University of Marzchester) said: In reply to Ullman, thedifferential equation quoted is the simplest which allows for curvature, i.e., the simplestmethod of introducing a persistence length a.It can be solved because it is a versionof Hermite’s equation, which has an inhomogeneous solution given by Mehler. If(~+$$+pw”n’ G(x, x’, t , t ’ ) = 6 ( ~ - ~ ‘ ) 6 ( t - t ’ ) , ) then0G == [ sinh ~ ( t - t’)]‘exp (-+(x’+x’~) coth ~ ( t - t ’ ) + x x ’ cosech o(t-l‘)f.This solution is readily extended to any differential equation of purely quadraticstructure.The calculation of the time to reach equilibrium from some totally remote state isextremely difficult even in conventional statistical mechanics (e.g., for a liquid dropsuddenly finding itself in a vacuum and becoming a low density vapour). However,a first step is to calculate the fluctuation time of particular configurations withinequilibrium and I think that problem is soluble and hope to get a solution to it.Prof.S. F. Edwards (University of Munchester) said: To include variationsA simplified in curvature of a polymer one must include torsion into the calculation.Rec. trav. chim., 1949, 68, 1106GENERAL DISCUSSION 83version of torsional energy is just I r"'(s) I 2. If one has a Boltzmann weight factorfor curvature and torsion one has a weight(where kT is included in the constants a, b) for each configuration. The probabilitythat a chain has r(s) = r, r'(s) = v, r"(s) = q, is then given by-+v-+q-+-v2+--q2+ a a 8 3 3a2 ~- 8:)p(r,v,q; s) = 0. { as ar av 21 21 a 2 b 2 . 6 a qThis is a soluble equation (see reply to Ullman).[r"' . (r' x r'')]2/[r"2]2.This still leads to a differential equation, but an awkward one :The exact form for torsion is more complicated, the energy being proportional to= 0.The kind of polymers discussed by North which are stiff but have occasional sharpbends can be described by differential equations likea a a 3 2 w-+v-+q-+----u + ar av 21where W(q) is a function with a principal minimum at the origin, but subsidiaryminima at the preferred kink directions. For the solution to such equations resultsin the Kramers-type expression, see my paper.Prof. C. Domb (King's College, University of London) said: I would suggest cautioni n regard to using direct enumeration and Monte Carlo methods in the presence ofattractive forces.The enumerations which Martin presented converge rapidly andsmoothly and the area of doubt is confined to whether the exponent should be 1.20 or1.195. In either case we can find a fairly accurate representation of the asymptoticbehaviour of the mean square length of a chain. However, when there are attractiveforces present, the rate of convergence is much slower, and it is more difficult to drawreliable conclusions about asymptotic behaviour.Both Fisher and Hileyl in early enumerations, and Mazur and McCracken2 intheir Monte Carlo work have suggested that when attractive forces are present, theexponents depend on the force of attraction; for example, y becomes a function ofJ/kT where J represents the force of attraction. Recent work by one of my formerst~dents,~ P.G. Watson, has cast doubt on this conclusion, and suggests that, althoughit may be valid as a representation of the mean-square end-to-end distance for acertain range of n, it is not valid asymptotically.Watson has drawn the following conclusions from some exact analytical work.(a) The exponents are unchanged in going from a normal self-avoiding walk to a self-avoiding walk with no near neighbour contacts. This suggests that exponents areunchanged by repulsive forces, a conclusion in agreement with Mazur and McCrackenbut at variance with Fisher and Hiley. (b) For a particular lattice in two dimensions,the extended Kagome lattice (fig. l), Watson has shown that an exact solution can' M. E. Fisher and B. J. Hiley, J. Chem. Phys., 1961,34,1253.J.Mazur and F. L. McCrackin, 1968, J. Chem. Phys., 1968,49,648.1'. G. Watson, J. Plrys. Chcm, 1969, L28, and further results privately communicated84 GENERAL DISCUSSIONbe obtained with attractive forces. If the generating function of closed polygons fora walk with no near neighbours isZU2&*" (1)then the generating function for walks with an interaction isL=u2,xn(1 + wx2)" [w = exp (J/kT)].From this and analogous formulae it is possible to show rigorously that this type ofattraction does not change the exponents.FIG. 1 .-Extended Kagome lattice.However, this corresponds only to a short-range attraction since the lattice hasthe peculiarity that a self-avoiding walk could not fold up along it. Watson hassubsequently extended his work to a Kagome lattice in which long-range attractionsare also taken into account.By using a star-triangle transformation, he is able toshow that the exponent of one lattice with a sufficiently weak attractive force isidentical to that of another lattice with a repulsive force.In fact, Watson's approach suggests an alternative method of dealing with attrac-tive forces on a general lattice, e.g., the triangle or face-centred-cubic. We start witha no-contact walk and insert contacts one at a time, taking account of their mutualattraction and repulsion. This is illustrated in fig. 2. A single contact can be intro-duced along every step of the walk. However, if we introduce two contacts, threedifferent cases of interaction arise, the first two corresponding to repulsion (orexclusion) and the third to attraction.Similarly, we can introduce three contacts, etc.If the generating function for no contacts is given byFdx) = Xcnd', (3)F(x) = ~ c n o ~ @ n ( x , w), (4)then the generating function with contacts is given bywhere the cc contact interaction 'I function is given bGENERAL DISCUSSION 85Any one familiar with Ising model expansions will recognize this type of pattern andmaking the usual assumption about asymptotic behaviour, we find thatwhere@n(& w> - [&, Wll",#(x, w) = 1 + 2 x ~ + x ~ [ ( - a ~ ~ - a ~ ~ ) w ~ + a ~ ~ w ~ ] + ... .(6)(7)When we pass to a folded chain corresponding to strong interaction, we know fromthe work of W. J. C . Orr that there is a finite entropy and no long-range order.Using our knowledge of the Ising model, this again tentatively suggests that there is nosharp phase transition.FIG.2 . 4 ~ 2 ) No-contact walk in which contacts can be inserted on each step (shown dotted).(b) Possible interactions between a pair of contacts.Any conclusions drawn from the analysis of the last paragraph must be tentative.But it suggests the possibility that the asymptotic exponent remains unchanged evenin the presence of attractive forces. However, since the asymptotic behaviour isdetermined by all roots of the equationit is likely that the expression becomes complicated for large w. However, thedominant asymptotic term for very large n is of the formThis opens up the possibility that for a single chain there is no single 8 temperatureat which the expansion factor is unity for all n.For a particular n there is such a8 temperature but this temperature is dependent on n.1lPo = XO(X, w) (8)(R: (w)} A 0 [a(w)]-W (9)Prof. T. Shimanouchi (University of Tokyo, Japan) said : The Raman spectra ofn-paraffin in the crystalline state show a progression of bands in the low-frequencyW. J. C. Orr, Trans. Faraday SOC., 1947, 43, 1286 GENERAL DISCUSSIONregion. They are assigned to the accordion-like vibration and its harmonics.'*The appearance of those longitudinal acoustic vibrations shows that the moleculestake the extended zigzag conformation in the crytalline state. This is the case evenfor C94H190, for which the progression is well displayed.The frequencies of the progression of bands are expressed byv = (m/21,n)J(E/P), (1)where m, lo, n, E and p are the order of the harmonics, the length of the CH2 unit,the number of carbon atoms, Young's modulus of the n-paraffin molecule and thedensity of the molecular rod, respectively. The frequency v is expressed as a functionof m/n.Schaufele and the author measured and assigned these acoustic vibrationbands for eight normal paraffins from C18H38 to Cg4HIg0. The obtained frequenciesare all on one line given by eqn (1) for small values of m/n and E = 3.58 x 10l2dyn/cm2. This line is in agreement with the v 5 curve (the CCC deformation vibration)of the frequency against phase difference diagram of the polyethyleneFrom the v 5 and vg curves of this diagram we can also obtain information about theelastic constants for the in-plane bending, the out-of-plane bending and the twistingmodes of the n-paraffin molecular rod (fig. 1).6 00500.-I 400\ 'Eh u 5 3 0 0$L?z2001000-- FIG. 1.-Dispersion curve for acousticvibrations of polymethylene chain. v5,CCC deformation mode; vg, CC tor-sional mode ; part A, longitudinal mode ;part B, in-plane bending mode ; part C,twisting mode ; part D, out-of-planebending mode.-0 TIphase differenceA similar treatment can be used for the molecular rods of polyvinyl chloride,polypropylene, polyoxymethylene, etc. Among them, the properties of the a-helixof polyamino acids are of interest. Itoh and the author obtained the frequencyagainst phase difference curve of the polyalanine a-helix and gave the value ofYoung's modulus and the relationship between the frequencies of the accordion-likevibrations and the lengths of the a-heli~.~ This relationship may be used for esti-mating the lengths of a-helixes in natural globular proteins from the Raman spectra.The elastic constants for the bending and twisting modes of the helix can also beobtained from the dispersion curve.S. Mizushima and T. Shimanouchi, J. Amer. Chem. SOC., 1949,71,1320.R. F. Schaufele and T. Shimanouchi, J. Chem. Phys., 1967,47, 3605.M. Tasumi, T. Shimanouchi and T. Miyazawa, J. MoZ. Spectr., 1962,9,261; 1963,11,422.K. Itoh and T. Shimanouchi, Biopolymers, 1970,9, 383

ISSN:0366-9033    

DOI:10.1039/DF9704900076

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

2. Static Properties of SolutionsVan der Waals and Related Models for HydrocarbonMixtures * -f-BY ROBERT L. SCOTT AND PETER H. VAN KONYNENBURG 1Received 6th January, 1970The van der Waals equation of state, in spite of its oversimplifications, gives useful qualitativeinformation about mixtures over a wide range of temperatures and pressures. When the Bronstedprinciple of congruence is used to evaluate the parameter a12 for the mixture, a wide range of prop-erties can be predicted : excess functions (including temperature and composition dependence) andphase eqiulibria (including lower critical solution phenomena at high temperatures), in good quali-tative agreement with experimental properties of mixtures of n-alkanes.Mixtures of n-alkanes serve, in a certain sense, as low molecular weight analoguesof polymer solutions and the extensive experimental data can be used to test varioustheoretical models. General qualitative features of alkane mixtures, which anysatisfactory model must produce, include : (a) the molar excess volume YE is negativeat all temperatures, increasingly so at higher temperatures, and with a minimum skewedin the direction of mixtures rich in the smaller component.(b) The molar excessenthalpy is small and positive at low temperatures and becomes negative at highertemperatures ; in a short intermediate range, the curve of RE against mol fraction isS-shaped. (c) For certain mixtures (e.g., CH4 + n-C6HI4), lower critical solutionphenomena occur near the gas-liquid critical point of the more volatile (smaller)component.Many of these properties can be interpreted in terms of the principle of congruencean extended theory of corresponding ~ t a t e s , ~ ' ~ or the new Flory equation of state.6*For several years we have been investigating the properties of van der Waals mixturesat elevated temperatures and pressures, particularly with respect to phase behaviour,8and have found that this model also reproduces-with a reasonable choice of para-meters-the qualitative behaviour of n-alkane mixtures.Because of the simplicityof the van der Waals equation and the physical reasonableness of its Q: and b parameters,it is useful to show how it leads to the observed behaviour of hydrocarbon mixtures.THE VAN DER WAALS EQUATIONThe equation of state of van der Waals was proposed in 1873 and was laterextended to binary mixtures lo in the formwhere p ,Allied Chemical Corporation.Angeles, California 90024, U.S.A.Pennsylvania 16802, U S A .p = RT/(P-b,,,)-a,/V2 (1)T, and R are the pressure, molar volume, thermodynamic temperature, and* This work has been supported by grants from the U.S.National Science Foundation and the=f Contribution No. 2522 from the Department of Chemistry, University of California, Los$ present address : Department of Chemistry, Pennsylvania State University, University Park,888 VAN DER WAALS MIXTURESmolar gas constant respectively. The parameters am and b, are usually written asquadratic averages for the binary mixture :a, = x~all+2xlx,a12+x,2a22, ( 2 )b, = xibl1+2~1x,b,z+xib,2, (3)where x1 and x2 are the mol fractions of the two components, and the parametersa l j and blj correspond to the appropriate pair interactions.In terms of modernpair potential energy functions u(r) for spherically symmetric molecules, a i j should beproportional to ~~~c~~ and b l j to clJ where E is the depth of the potential energy wellat its minimum and c the collision diameter. Recently, Leland, Rowlinson, andSather l 1 have used the Percus-Yevick approximation to support the argument thatthe prescriptions of eqn (2) and (3) are the most appropriate to use in a " one fluid "corresponding states treatment of mixtures.For mixtures of spherical molecules, for which o12 = (ell + 0 ~ ~ ) / 2 is a goodapproximation, the Berthelot combining rule b12 = (bt, + b$,)* would be appropriate,but for chain molecules the original van der Waals assumption that b, = (b, +bZ2)/2seems more reasonable.Then eqn (3) reduces tobm = Xlbll +xzbzz. (4)This leads to the conclusion that volumes are additive at the low temperaturelimit .As van der Waals showed, eqn (1) leads directly to an expression for the Helmholtzfree energy 2 for 1 mol of a binary mixture :~T~(T,Ex~)-.X,"(T,VO) = R T [ x ~ In x1 +xZ In xZ]-RT In [(T-bm)/Vo]-a,/E (5)where $(T,8") is thereference state of the unmixed components as ideal gases ofmolar volume vo. An equivalent development leads to an equation for the chemicalpotential p 1 of component 1 :p l ( ~ ~ x l ) - p ~ ( T , ~ o , l ) = RT In x,-RT In [ ( ~ - b m ) / ~ o ] +~ ~ [ b l 1 - ~ , 2 ( b 1 1 - 2 b l 2 + b2 2 ) I / ( r- bm) - 2(xla11+ x2a12)/ K (6)and a corresponding expression for component 2.The van der Waals equation is only an approximation at best and cannot beexpected to yield more than qualitative agreement with experiment, even for sphericalmolecules. It has the advantages, however, that (a) it is simple and its two parametershave a fairly clear physical meaning, (b) once the parameters are chosen, eqn (5) (or(6)) leads to explicit and unambiguous mathematical predictions, and (c) it yieldsqualitatively reasonable results over the entire range of temperature and pressure,including the coordinates of critical points (although not the correct form for functionsin the critical region).For mixtures of chain molecules, eqn (1-6) have the disadvantages that (a) withoutthe addition of a third parameter (e.g, the Prigogine-Hijmans-Flory c), they predictthat the critical compression factor ( p V/RT)" is invariably Q in disagreement with theexperimental evidence that for n-alkanes it decreases regularly with increasing chainlength, and (b) the special statistical effects of mixing chain molecules (e.g., the" Flory-Huggins " entropy) are not included, and there appears to be no entirely self-consistent way to add these.In the absence of correction terms, we may expectincreasing divergence between van der Waals predictions and experiment as the chain-length of the n-alkane increasesROBERT L . SCOTT A N D PETER H .VAN KONYNENBURG 89THE VAN DER WAALS PARAMETERSThe parameters a and b can be evaluated in various ways : from the properties ofthe gas at moderate pressures (e.g., the second virial coefficient B = b -a/RT) ; fromthe critical constants p", p, and T" : or from the properties of the liquid at low temp-erature and (essentially) zero pressure; e.g., the molar volume the coefficient ofthermal expansion a = (3 In r,aT),, the thermal pressure coefficient y = (aplaT),.Were the equation of state exact, all these methods would be equivalent and wouldlead to the same values for a and b. However, since the equations are only approx-imate the parameters selected will depend upon the method of evaluation. If one isinterested primarily in the dense fluid at low temperatures, it is preferable to use onlyproperties at high densities and low temperatures to determine a and b.Since we havebeen interested in the critical region as well we have elected to use a different method.Williamson and Scott l 2 have shown that the product of the liquid molar volume Pand the standard molar energy of vaporization AVOo, which should equal the van derWaals a, when evaluated at 25°C for n-alkanes from C5 to C16, closely fits the simpleequationwhere n is the number of carbon atoms in the n-alkane. An even better fit is obtainedby using the general quadratica = VAuDo = (81.065 kJ cm3 m0l-~)(n+0.879 5)2,a = (158.28+ 121.200 n+82.133 4 n2) kJ em3 mok2.(7)(8)Given these a and the experimental p a t 25"C, one may calculate the b from eqn(I), obtaining the simple linear fit,An even better fit is given by the quadratic equationb = (16.44 cm3 mol-l)(n+ 1.32) cm3 mol-l.b = (22.78+ 16.22 n+0.01 n2) cm3 mol-l.(9)(10)An alternative expression for b, V-R/y, evaluated at 25"C, yields an equationb = (16.5 cm3 mol-l)(n+ 1.33), with constants similar to those in eqn (9).Unfortun-ately, other procedures (e.g., using the critical constants) yield different results,showing more complex dependence of a and b upon n. For illustrative purpose weshall use the values of a and b given by eqn (7) and (9), but this uncertainty about the" best " values of a and b, reflecting the inexactness of the equation of state, meansthat detailed numerical agreement between theory and experiment is not to be expectedand, if found, may be at least partly fortuitous.If we adopt the principle of congruence, according to which the properties of ann-alkane mixture are set equal to the properties of the pure n-alkane of averagen = xlnl +x2n2, then the parameters a,n and b, are obtained by substituting ii into theappropriate equations for a and b.When this procedure is applied to eqn (7) and(9), and compared with the general eqn (2) and (3), one finds the geometric mean fora,, = (a11a2,)* and the arithmetic mean for b,, = (bll+b2,)/2 and thus eqn (4).The more nearly exact eqn (8), when combined with the principle of congruence, leadsto aI2 values slightly smaller than the geometric mean.-THE EXCESS FUNCTIONSFrom eqn (I) and (4) we can derive the various thermodynamic functions formixing at constant temperature and pressur90 VAN DER WAALS MIXTURESExcept for the entropy and the Gibbs free energy, where the ideal solution term mustbe subtracted, eqn (11) yields the excess functions themselves.While these are bestcalculated directly, it is useful for qualitative understanding to look at the variousexcess functions as power series expansions in temperature and pressure.cE = - A(a/b) - RTA In (pob2/a) - (RT)2A(b/a) -@T)'A(b2/a2) + . . .RE = - A(u/b) + (Rn2A(b/a) + 2(RT)3A(b2/a2) + . . .+p[(Ab+ RTA(b2/u) +2(RT)2A(b3/a2) + . . .] - . . . (12)(13)(14)+p[Ab - 2(RT)2A(b3/a2) - . . .] + . . .TSE = RTA In (pob2 la) +- 2(RT)2A(b/a) -k 3(RT)3A(b2/a2) -I- . ..-p[RTA(b2/a) + 4(RT)2A(b3/a2) + . . .] + . . . .PE = Ab+RTA(b2/a)+2(RT)2A(b3/a2)+ . . . -p[RTA(b4/a2)+ . . .I+. . . (15)where A( ), following eqn (1 l), means the difference between the quantity for themixture and the weighted average for the unmixed components, i.e. A(a/b) = (a,,,/b,)- xl(al 1) - ~ ~ ( a ~ ~ / b ~ ~ ) , etc. and where po is a reference pressure (here irrelevantsince it cancels out).The general behaviour of the excess functions will thus depend upon the propertiesof the A-functions, their signs, magnitudes, and concentration dependences. Formixtures of n-alkanes and probably for many other homologous series, all the A-functions are invariably negative and include a factor xIx2(nl -n2)2 which makes thefunction roughly parabolic.A further factor dependent upon x, n1 and n, skews thecurve (except for Ab, which if non-zero is simply parabolic) so that the minimumoccurs at a mixture richer in the component of smaller n. The " higher " A-functionsare somewhat more skewed than is A(a/b).If we consider the excess functions at low pressures where p-dependent terms arevirtually negligible, we conclude that : (a) the excess Gibbs free energy GE is invariablypositive for all x and increases with temperature. The function GE/RT decreaseswith temperature initially (from a hypothetical value of +co at T = 0) and thenincreases again. If we take GE/RT = 3 as a rough criterion for liquid-liquid phaseseparation,l this guarantees partial miscibility at sufficiently low temperatures (butprobably below the melting curve for many systems so not experimentally observed)and raises the possibility of two critical solution points in one system, an upper(" normal ") one at low temperatures and a lower one at higher temperatures, with aregion of complete miscibility between.(b) The excess entropy gE and the excessvolume VE are invariably negative for all x, and decrease (increase in magnitude) withincreasing temperature. The skewing of the curves becomes more marked as Tincreases. (c) The excess enthalpy is positive at low temperatures and decreases as Tincreases, be_coming negative at higher temperatures. In the low temperature limit(i,e., when V z b,) the leading term, - A(a/b) in eqn (13) is the only significant one.With the linear average of eqn (4) for b,, eqn (13) then reduces to the van Laar equa-tion :or, with eqn (7) and (9) for a and b,(17)xlb, 1 ~ 2 b 2 2 - (0.95 kJ mol-')(n, - n2)2x1~2(xlbll +x2bZ2) - (nl+1.32)(n2+1.32)(ii+1.32)'The right-hand side of eqn (17) must necessarily be positive, but the numerical valueis much too small to account for the magnitude of the positive RE observed for alROBERT L .SCOTT A N D PETER H . VAN KONYNENBURG 91n-alkane mixtures at low temperatures. Use of the more nearly accurate eqn (8)increases the small dimensional factor in eqn (17) to 8.6 kJ mol-l, but even this issomewhat too small to fit the heats of mixing at 25°C. This fact is in accord with theobservation that the principle of congruence leads to too small values of RE whenthe heats of vaporization of the pure n-alkanes are used.Of the A-functions, only Ab, which for eqn (9) is exactly zero but for eqn (10) isweaklynegative[Ab = - (0.1 cm3 mol-')(n, -n2)2x1x2], and A(a/b), whichisvery small,are extremely sensitive to the precise specification of a12 and b12.In view of this, itseems appropriate to regard the dimensional constant in eqn (17) for -A(a/b), andmaybe that in Ab also, as empirically adjustable to fit the low-temperature excessfunctions. The other A-functions, which determine the variation of the excessfunctions with temperature and pressure, are almost completely insensitive to smallvariations in a12 and b12 and can be evaluated from the simpler eqn (7) and (9).The second term in eqn (13) is proportional to ( R n 2 , there being no linear term, sothe excess enthalpy will decrease rapidly as it becomes negative.Moreover, sinceA(b/a) is more skewed than A(afb) there will be a narrow intermediate range oftemperature in which the curve of gE against x is S-shaped, with negative values formixtures rich in the component of smaller n and positive values for mixtures rich inthe other component.All of these predictions correspond qualitatively with the experimental data onhydrocarbon mixtures; the fact that the experimental excess properties change morerapidly with temperature than predicted is a measure of the inexactness of the vander Waals equation of state.PHASE EQUILIBRIAOur interest in the van der Waals equation applied to binary mixtures was initiallydirected almost entirely toward the problem of phase equilibrium at elevated temp-eratures and pressure.Since eqn (1)-(4) define analytic functions at all temperaturesand pressures, the usual thermodynamic conditions for equilibrium between phases ccand p [T" = Tb, pa = pB, pf = p?, and p; = pi] may be applied and the equationssolved explicitly. In particular, one may obtain complex equations defining thecritical lines in a three dimensional p,T,x diagram.Sixty years ago van Laar l4 investigated the phase behaviour of van der Waalsmixtures, but he was handicapped by the difficulties of hand-calculations and by hisself-imposed restriction to the geometric mean for a12. Now with the availability ofhigh speed computers it is possible to examine the full implications of the van derWaals equation, i.e., to determine phase equilibria for all possible values of the a and b.Van Konynenburg * has calculated a large number of p,T,x diagrams covering a widevariety of values of a, mostly for b, = b22, but some for b22 = 2bl ,.A brief r6sumCof his work must suffice here.We classify the various types of phase diagrams according to the nature of theirP,T projections,15 in particular by the presence or absence of three phase lines andazeotrope lines, and by the way critical lines connect with these. Critical lines mayend in various ways : at a one-component gas-liquid critical point C1(T;C,pE,x2 = 0)or C2(T&p&x2 -= l), at the limiting upper critical solution point C,(T,,p = CO,~,) in aclose-packed ( V = b,) system, at the end of a three-phase line (L1L2G) with an upperor lower critical end point (UCEP or LCEP).Azeotrope lines, if present, end tan-gentialiy on critical lines. In some cases a critical line is essentially gas-liquid(G-L, Vc M 3b,) or liquid-liquid (L-L, Vc = b,) over its entire course ; in other casesit changes continuously from one type to the other. Nine major types are distin-guished 92 VAN DER WAALS MIXTURES1, one critical line (G-L), C , to C2.I-A, same as I, with the addition of a negative azeotrope.11, two critical lines : CI to C2 (G-L) ; C, to UCEP (L-L).11-A, same as 11, with the addition of a positive azeotrope.111, two critical lines : C, to UCEP (G-L); C, to C2(L-L to G-L).111-HA, same as 111, except that the three phase line lies at lower pressures thaneither pure component, producing a “ heteroazeotrope ” type diagram.IV, three critical lines: C1 to UCEP (essentially G-L); LCEP to C,(L-L) tov, two critical lines : C, to UCEP (essentially G-L) ; LCEP to Cz (L-L toV-A, same as V with addition of a negative azeotrope.If we assume b,, = (bI1 + b2,)/2, but not the geometric mean for a, 2, any van derG-L; C , to UCEP (L-L).G-L).Waals mixture can be specified by three parameters, 5, c, and AWithin the limitations of the van der Waals equation, 5 is the difference of thecritical molar volumes divided by the sum, while 5 is the difference of the criticalpressures divided by the sum.The usefulness of the parameter A is that it is the onlyone which includes a12 and is proportional to the low-temperature limit for RE(i.e., containing the difference factor in - A(a/b)).Fig.1 summarizes van Konynenburg’s results for bl = b2, (5 = 0). Only positivevalues of c are shown, for the diagram is symmetrical with respect to reflection throughthe c = 0 axis. The dashed line represents those mixtures for which a,, = (al1aZ2)*,the geometric mean; only types I1 and I11 diagrams are found for such systems(when b , , = b22) ; one must relax the combining rule for a,, to find the other seven.The especially interesting type IV with both upper and lower critical solutionphenomena occurs only in a narrow region where (a, la22)* <al2 <(a, , + a2J/2.Systems for which b l l # b2, have not been investigated so thoroughly, but ourresearch programme is continuing.Results for 5 = 5. and for the special case ( = 0indicate that the general form of fig. 1 is topologically invariant, but that the positionsof the regions are shifted quantitatively. In particular, for > 1 and > 1, the wedge-shaped region for type IV moves across the locus of geometric means for a,,, so thatlower critical solutim phenomena are now possible for systems satisfying eqn (7).Since the n-alkane systems do not deviate markedly from the geometric mean fora12, we may use that relation to eliminate the parameter A from the set 5, c, A withrelationThen the behaviour of such systems can be represented by fig. 2. The coordinates ofseveral n-alkane systems are indicated on the figure.(All are calculated using eqn (7)and (9), although the experimental data on methane and ethane do not fit them verywell.)When the two alkanes do not differ too much in size (e.g., c6 + C , 6), the phasediagram will be type 11, but as increases, one passes through type IV, with its upperand lower critical solution temperatures, to type 111, i.e., from ‘‘ complete miscibility ”(1-A),+r2 == 1. (21ROBERT L . SCOTT AND PETER €3. VAN KONYNENBURG 93IIFIG. 1.-Types of phase equilibria for iiiixlures of molecules of equal size (E = 0). Nine majorregions of characteristic p,T,x phase diagrams (see text) are separated by the full lines. The dashedline is the locus of the geometric mean for a12 (eqn. (21)).Negative values of are not shown ; thefull diagram is symmetrical around the < = 0 axis.FIG. 2.-Types of phase equilibria for mixtures obeying the geometric mean for a12 (eqn. .(21)).Negative values of are not shown because the origin is a centre of symmetry. < and 4 coordinatesare shown for a few binary n-alkane mixtures94 VAN DER WAALS MIXTURESthrough " limited miscibility " to " absolute immiscibility " in the nomenclature ofDavenport and Rowlinson.16 The fact that the boundaries sketched in fig. 2 roughlyfit the methane systems (where C,+C, is type IV) is probably fortuitous, for theethane systems do not fit so well (experimentally 17* l8 CZ+Cl9 to C2+C22 are typeIV). For solutions of polymethylene in n-alkanes (e.g., C6 + Cm), the simple van derWaals theory predicts type I11 instead of the type IV 0b~erved.l~ This increasingdisagreement for very long chain lengths may be due to the disregard of correctionsfor chain entropy effects (e.g., both the parameter c and the configurational mixingterm).TI I1 I I I r I I02' 3:: 3.6 3 -7 3 8azzh 1FIG. 3.-The locus of critical end points for systems changing continuously from type 11 throughtype 111 to type IV.The points along the curve represent exact calculations of end points forsystems lying along the dotted line in fig. 1. The reduced temperature Tr = T/T,O.Fig. 3, calculated from the van der Waals model (for = 0), illustrates how theupper and lower critical end points coalesce as one crosses the type IV region fromtype I1 to type TIT.For most hydrocarbon mixtures the low temperature upper criticalend point is hidden below the melting curve, but it seems clear that the basic behaviouris type IV, not type V. Not only do the positive excess enthalpies at low temperaturerequire this, but upper critical end points have now been found in a number of methane+hydrocarbon systems 8 s 16* 2o ; fig. 4 shows how the two critical solution points movecloser together as the solubility parameter 6 (proportional to af/b) of the hydrocarbonincreases. There are different curves for C6, C,, and Cs because these systems havedifferent values of 5 ; since both alkanes and alkenes show the same general behaviourthe phenomenon is clearly not restricted to a special class of hydrocarbonsROBERT L.SCOTT AND PETER H. VAN KONYNENBURG 95Even more striking confirmation of this general form is provided by measurementson several " quasi-binary " mixtures of methane with a pair of CG-isomers. Fig. 5shows the variation of the critical end points with the proportions of 2-methylpentaneand 2-ethyl-1-butene in a ternary mixture with methane. (Similar results are obtainedfor the system methane + 3,3-dimethylpentane + 2-methylhexane.)200I90I eo170I60M L-150140I30I20I10~ - H E X A N E\ \ A I-HEXENE2,2-DIMETHYLPENTANE 4,2,4 - 0 I ME THY L PEN TAN E \\ \ \ o 2-METHYL-I-PENTENE12,2,4 - T RI M E THY L -\ \\ PENTANE 03,3-OIMETHYLPENTANE 0\FIG. 4.--Correlation of upper and lower critical end points in methanefhydrocarbon systems with thesolubility parameter 6 of the hydrocarbon.A Davenport et 2o ; 0, van Konynenburg.8 Thesolid lines are not the best fit of the data but represent experimental measurements okternary mix-tures such as that shown in fig. 5.OTHER MODELS AND POSSIBLE REFINEMENTSMost of the conclusions here developed from a van der Waals model have beenderived from other theoretical treatments. It is not our intention to argue that thevan der Waals equation of state is superior to others which have been used. Thereare better equations of state which, with reasonable choices of several adjustable par-ameters, should (and do) give better agreement between calculated and observedproperties. The importance of these van der Waals calculations is that with (a) avery simple equation of state, (b) a simple dependence of the two parameters a and bupon n, and (c) the principle of congruence, they lead to qualitatiw agreement withexperimental behaviour over virtually the whole range of temperature, pressure, andcomposition. From the convergence of the results of the various theoretical ap-proaches the phenomena observed are of quite general significance and the pre-dicted behaviour of alkane mixtures is closely related to the actual equation of stateof the pure hydrocarbons and should not be sensitive to the special parameters or thespecial forms of any particular theory96 VAN DER WAALS MIXTURESThis van der Waals model might be improved (in the sense of better fit with experi-ment and closer approach to other theories) by the following refinements.(a) Theintroduction of a special a,,'not in accord with the principle of congruence. Thisseems to be necessary in order to fit RE at low temperatures; Orwoll and Floryfound it necessary to introduce an empirically adjusted X , , for a similar purpose.FIG. 5.-Upper and lower critical end points in the ternary system methane+Zmethylpentane+2-ethyl-1-butene. f3 is the relative mol fraction x3/(x2 +x3). x indicates the gas-liquid criticaltemperature of methane.(b) The replacement of RT( r- b m ) in eqn (1) by c,RTf( v/bm), wheref( p/bm) is chosen 21to give a better fit to the known results for the equation of state of hard spheres, andc, is a function of fi which introduces a scaling parameter for the entropy of a chainmolecule.Unfortunately, Cm must in reality be a function of density, for it must havethe value unity in the perfect gas region, another value at the critical density, and stillanother value at normal liquid densities. (c) The replacement of the configurationalfree energy of ideal mixing RT[x, In xl +x2 In x,], in eqn ( 5 ) by the Flory expression,RT[xl In (xlbl /bm) +x2 In (x2b22/bm)]. Unfortunately this too, while an improvementat liquid densities, would be inappropriate for dilute gas mixtures.These refinements would effect marked improvements in the fit of theory andexperiment in certain regions of temperature, density, and composition. HoROBERT L. SCOTT AND PETER H . VAN KONYNENBURG 97seriously these would affect the predicted thermodynamic behaviour in other regions(e.g., the critical region) is unknown.At present we are surprised and gratified thatthe unrefined van der Waals model is so successful and informative.l J. N. Bronsted and J. Koefoed, Kgl. Danske Videnskab. Selskab, Fys. Med., 1946,22, no. 17.I. Prigogine, The MoZecuZar Theory of Solutions (North-Holland, Amsterdam, 1957), chap. 17.Th. Holleman and J. Hijmans, Physica, 1962, 28, 604 ; ibid., 1965, 31, 64.S. N. Bhattacharyya, D. Patterson and T. Somcynsky, Physica, 1964, 30, 1276.D. Patterson and G. Delmas, Trans. Furaday SOC., 1969,65,708. ' P. J. Flory, R A. Orwoll, and A. Vrij, J. Amer. Chem. SOC., 1964,86, 3507. ' R. A. Orwoll and P. J. Flory, J. Amer. Chem. Sac., 1967,89,6814.P. H. van Konynenburg, Critical Lines and Phase Equilibria in Binary Mixtures (Ph. D. Diss.,U.C.L.A.), 1968.J. D. van der Waals, On the Continuity of the Gaseous and Liquid States (Ph.D. Diss., Leiden,1873).lo J. D. van der Waals, Z. phys. Chem., 1890,5, 133.l1 T. W. Leland, J. S. Rowlinson and G. A. Sather, Trans. Faraday Soc., 1968, 64, 1447.l2 A. G. Williamson and R. L. Scott, Trans. Faraday SOC., 1970, 66,335.l3 J. L. Copp and D. H. Everett, Disc. Faraday SOC., 1953, 15, 174.l4 J. J. van Laar, Proc. Akad. Wetenschappen (Proc. Section Sci.), 1906,9,226 ; ibid., 1907,10,34.J. S. Rowlinson, Liquids and Liquid Mixtures, 2nd ed. (Butterworth, London, 1969), chap. 6. '' A. J. Davenport and J. S. Rowlinson, Trans. Faraday SOC., 1963,59,78.l7 J. P. Kohn, Y. J. Kim, and Y. C. Pau, J. Chem. Eng. Data, 1966,11, 333.l 8 A. B. Rodrigues and J. P. Kohn, J. Chem. Eng. Data, 1967, 12,191.l9 P. I. Freeman and J. S. Rowlinson, Polymer, 1960,1,20; C. H . Baker, W. Byers Brown, G.Gee, J. S. Rowlinson, D. Stubley, and R. E. Yeadon, Polymer, 1962, 3, 215; G. Allen andC. H. Baker, Polymer, 1965, 6, 181.E. A. Guggenheim, Mol. Phys., 1965,9,43, 199.2o A.. J. Davenport, J. S. Rowlinson and G. Saville, Trans. Faraday SOC., 1966, 62, 322

ISSN:0366-9033    

DOI:10.1039/DF9704900087

出版商:RSC    

年代:1970

数据来源: RSC

摘要:

Corresponding States Theories and Liquid ModelsBY D. PATTERSON AND G. DELMASChemistry Department, McGill University, Montreal, CanadaReceived 12th January, 1970The general corresponding states theory of Prigogine and collaborators, applicable to the thermo-dynamics of mixtures of quasi-spherical molecules and polymer solutions, is compared to the theoryof Flory, Abe, Orwoll and Vrij. The mixing functions are divided into two contributions : an inter-action term due essentially to the weakness of (1-2) interactions, and a term due to the dissimilarityof the free volumes of the two components. The latter term is small or negligible in mixtures of quasi-spherical molecules but dominant for polymer solutions. These terms are contrasted with the con-tact interaction term and the equation of state term of the Flory theory.The mixing functions arecalculated by a new approximate procedure, using several liquid models based on the cell partitionfunction (including the model used by Flory and collaborators). The results are similar and the modelsmake certain errors in common.Theories of polymer solution thermodynamics have usually had their origin inthe thermodynamics of mixtures of quasi-spherical molecules. Thus, the latticeapproximations for the combinatorial entropy of a polymer solution enabled thestrictly regular solution theory to be extended? giving the traditional treatments ofpolymer solution thermodynamics. Recently, theories of quasi-spherical moleculemixtures by Prigogine and collaborators,la Brown and Scott have included theeffect on the excess functions of volume changes occurring during mixing.AsMathot first pointed out, this effect becomes much more important in polymersolutions. The Prigogine-Trappeniers-Mathot theoryY5 valid for both types of solu-tion is in its simplest form a one-fluid corresponding states approach. The followingpaper restates this theory in order to facilitate its application to experimental data,and to compare it with the work of Flory and his collaborators which has beensuccessfully applied to many systems.The Prigogine corresponding states principle for polymeric liquids I b is madepossible by the concept of a division of the degrees of freedom of the chain-moleculeinto internal and external categories.The expansion or free volume of a liquid ischaracterized by the reduced temperature, essentially the ratio of the thermal energyof the 3c external degrees of freedom to the intermolecular contact energy q ~ * :U* T* = ~ T* , S*'The molar configurational quantities of a liquid at constant negligible pressure are thenrelated by reduction parameters to the dimensionless reduced quantities, functions ofT:V(n,T) = V*(n) v ( T ) ; V* = Nr(n) v*S(n,T) = S*(n) g(T) ; S* = Nc(n) k.Here N is Avogadro's number and k is the Boltzmann constant, The effectivenumbers of segments? Y, q and c are proportional to, respectively, the molecularvolumes, molecular surfaces and number of external degrees of freedom of the chainmolecule. However, it is only Y*, U* and S* which are obtained directly from thethermodynamic properties of the pure liquid.For high polymers it may be conven-ient to replace the molar quantities V, V*, etc. by specific quantities, so that Nr, the98U(n,T) = U*(n) o(T) ; U* = Nq(n) E* (2D. PATTERSON A N D G. DELMAS 99number of segments per mol of chains is replaced by Nr/M, the number per g, andsimilarly for the other quantities.When discussing mixtures, several composition variables are of importance.First, the segment fraction extensively used by Flory and collaborators,l r the Flory theory Gb the molecules of the two components are divided into equal-sized segments so that vT = v';5. The division of Y* into r and v* is for convenienceonly and the size of a segment has no absolute significance. On the other hand,Pnigogine and collaborators l b consider a molecular chain to be formed of sphericalsegments.Thus a segment is a section of chain of length equal to the cross-sectionaldiameter of the chain (measured either as the van der Waals diameter CT, or as Y*, thedistance between two non-bonded segments at the potential minimum, - E * ) . ThePrigogine theory of solutions involves the parameter p = rT2/rF1 - 1 characterizingthe difference in diameters of spherical molecules or of chains, whereas this effect doesnot explicitly enter the theory of Flory and co-workers. It is also convenient toconsider a new " contact energy fraction ", also directly accessible from the reductionparameters :and the fraction X , called by Prigogine,'" a molecular " surface fraction ".Tompa ' and Guggenheim * use, respectively, the symbols 5 and 4'.for this fraction.For high polymers, the mol fractions x in the concentration variables may be replacedby weight fractions and the U*', V':, S*, q and r placed on a per g basis. For chain-molecules, the quantity q is obtained from r in the Prigogine treatment by means of alattice model :(6)x 2 = x2q2/(xlql+ x2q2). ( 5 )q - z - 2 2 + -. r z rz---l n the Flory treatment,6b the surface fraction, called 8, is obtained from the dimensionsof the molecules or models. The quantity q is replaced by rs so that s corresponds tothe above ratio q/r.SOLUTION REDUCTION PARAMETERSs*, U", T*In the one-fluid, or " crude " model, the excess thermodynamic functions arefound assuming that the molar configurational properties of the solution are given byeqn (2) with reduction parameters obtained from those of the pure liquids. Theentropy reduction parameter S* of the solution is linear in the mol fractions of thecomponents,(7 1expressing the assumed lack of interaction between the external degrees of freedom ofmolecules of the two components.On the other hand, the quantity U': is equal to x, Ur + x2 U; oiily for a mixture ofoligomers of chemically identical segments.In other cases there is a net energetic" interaction " between components which alters U* due to : (i), the relative weaknessof the (1-2) interaction compared with the mean of the (1 -1) and (2-2) interactions and(ii), thc difference of diameters of the component molecules (cross-sectional diameters* = XIST + x,s;100 CORRESPONDING STATES THEORIESin the case of a chain-molecule).The total energetic interaction is expressed througha dimensionless parameter equal to ((&T1 + E ; ~ ) - ~ETJ/S:~ + 18p2 = v 2 in the averagepotential model of Prigogine when a (6,12) intermolecular potential is used.'" It isconvenient to call this whole term v2 for brevity. Then, according to theory (eqn(17.3.8) of ref. (1))It now seems evident SinceFlory and collaborators divide the molecules of the components into equal-sizedsegments, formally p = 0, and the diameter-difference effect is zero. The " van derWaals " combining rule used by Leland, Rowlinson and Sather is similar since 18p2is replaced by the much smalIer - ~ .S P ( E ; ~ / & ~ ~ - 1) for a Lorentz-Berthelot mixture.Combining eqn (7) and (S), Tfor the solution is given byIf the interaction parameter v2 = 0,Tis a contact energy fraction average of theof the components. If also ET' = &z2, as would be so for a mixture of chains of differ-ent numbers of identical segments, Tis given by the simpler average in surface fractionsX. The same expression for Tis used by Flory theory, i.e., eqn (27) of ref. (66), withX2 = 82 and y2 = X12/PT and the t j transformed to 4 through eqn (4). In spite ofthe formal identity of the Flory and Prigogine theories at this key point, there is still adifference in practice. For an equimolar mixture of spherical molecules of differentdiameter, p # 1, we have X2 = 3 in the Prigogine theory since rl = r2 = q1 = q2 =1.In the Flory theory, r2/r1 = ( l + ~ ) ~ and q2/ql = (rs),/(rs), = ( l + ~ ) ~ , 8, =u* = X' u; + x,u; - x1 UTX2V2. (8)that the magnitude of the p 2 term is far too great.= (+lTl++2T2)/(1 -$1X2V2). (9)(1 + P)2/(1 +(I +d"*THE VOLUME REDUCTION PARAMETERFlory and collaborators take the volume reduction parameter of the solution to belinear in mol fractions of the components, i.e., as for S* there is no interaction betweencomponents, orThe Prigogine average potential model approximation for Y* adds a large positivevolume " interaction " term in p2, which is probably incorrect. On the other hand,the van der Waals combining rule for spherical molecules givesv* = x l v z + x 2 v ; .(10)v* = xlv~+X2v;+x'X2[o.25(v;++ v;93- VT- v;] (1174(117b) = X' v : + x , v ; - x,x2(3p2/2 + 3p3/4)VT.Eqn (1 1 ,a) cannot be directly applied to systems where a chain-molecule is presentsince the third volume '' interaction " term becomes unrealistically large. ForV:/VjT = 3, the third term is II 5 % of the whole V* of an equimolar mixture. Forn-alkane mixtures, for instance, extremely large negative values of VE would bepredicted whereas the observed negative VE are small and in agreement with eqn (lo),(or the average potential model since here p = 0). In extending eqn (1 1) to polymersthe form (1 lb) might be used where, as in the Prigogine theory, p would refer to thecross-sectional diameter of the chain.We shall, however, use the simple eqn (10) inwhat follows.MIXING FUNCTIONSThe treatment is similar in obtaining the change on mixing of any quantity withenergy dimensions, e.g., AHM -N A U M , the non-combinatorial part of A(TS)M = T A S Mand of AGM. We denote the quantity by A . Then,AA, = U*A'(T)- (~1U:~(Ifi)+~2Uf~z(~z)), (12D . PATTERSON AND G . DELMAS 101or, using eqn (8) for U*,The first term in this equation for AH,, in v2, has been referred to by Flory 6b as the“ contact interaction term ”. He has called the second tErm the “ equation of stateterm ” since it can be associated with the change of Tor Vfrom their original valuesfor the pure components to their final values in the solution. On the other hand, fora quantity such as A V M or A S M , only the second, equation of state term appears sinceS* and V* are linear in the mol ffactions of the components.Thus, eqn (13) becomesand similarly for the non-combinatorial A&.However, the interaction parameter v2 also enters the equation of state term throughof the solution. In fact, the roles of the parameter v2 in the contact inter-action term and in the equation of state term are only distinguishable in a formal way.For instance, the contact interaction term is absent from ASM, but, according to eqn(13), it occurs in T A S M . It seems preferable to associate the mixing functions with adifferent pair of contributions. One contribution would exist if v2 = 0 for the system.It will be associated with the difference in free volumes of the components, and couldbe called a free volume dissimilarity term.The other term is the remainder of themixing function due to v2, and could be called an interaction term. Each mixingfunction may be separated into these two contributions in the following excellentapproximation. We define in eqn (13) a reduced temperature TA for the solutionsuch that :andThe quantity J(T)may be developed around TA ignoring terms of order ( ~ , b ~ X ~ v ~ ) ~ and($‘lTl +$2T2-F,).2 ThenAA,/‘(xl uT+x,U;) = (-J(TA)+ ~~(a~/a~A)]$1x1v2+(aA/a TA) (II/ 1 Tl + $2 T2 - TA)* (16)This equation, unlike eqn (13), involves v2 explicitly and only in the first term.results for AVM and ASM are almost identical :TheA Kl/(x 1 VT + x2 m = ( T/a %)I $2X2V2 + (a v a {$ 1 4; + $2 T 2 - Tv ) .(17)For ASM, S is substituted for V. Differentiations are at constant zero pressure. Theapproximation used in eqn (16) and (17) corresponds to an error in the calculatedmixing functions of 1 % at worst (for high polymer solutions) and usually much less.The first term in eqn (16) and (17) is the interaction term which is positive in thefour mixing functions. The ratios areThe interaction term is the first order term obtained lo by the conformal solutiontheory. It is similar to the contact interaction term of Flory but corresponds to thewhole effect of v2 (or X12/PT in the Flory theory).The second term in eqn (16) and (1 7) is the free volume term, the nature of whichin eqn (16) may be seen through an expansion of Fl and T2 around TA in powers ofAl -J2.Neglecting third and high powers, eqn (16) gives the following approxima-tion, good to - 4 % in the worst cases and usually much better :A G M : A H M : T A S M : A V M = - 0 : -0+TCp:TCp:TaV/a4f (18102 C OR R E S P 0 N D I N G STATES T I3 E 0 R I ESFurthermore, (Al - A2)/c32/dTA 2: ( Tl - T2) so that the free volume term of eqn (19)depends on the difference between the Tor the gof the two components, i.e., on theirdissimilarity of free volume or expansion volume. It has the sign of -a2x/i?Tz, theratios of the free volume terms in the mixing functions beingFor A G M the term is positive. At high temperature where a(?JaTis certainly positive,the term is negative in A H M and in T A S M .The free volume dissimilarity term may bedistinguished in the expansions of the Prigogine theory (eqn (9.5.4.) to (9.5.7)).Neither of the contributions in eqn (19) is symmetric in $ since the quantities multiply-ing $l$2y and particularly a2A/aFi, vary with the composition.Theaverage 2 of the pure components is @121 + t,h2A2 lying on a straight line between XIand x2 at the reduced temperature $lTl+$2T2. The 2 of the solution lies on thecurve Z(T) at T = (11/1 TI + t,b2T2)/(1 - i,b1 X2v2). Thus, both the curvature d2&3T2and v2 contribute to the change on mixing of a quantity of energy dimensions.The volume of mixing present. a new feature, however, The average p of thepure components, i.e., q51 Vl +q52 V2 now corresponds to a value of = &T1 + q52T2while the Tof the solution lies at ($1 TI + $2T2)/(1 - $1 X2v2).The value of AVM nowalso depends on the difference between the contact energy fractions @, and the segmentfractions 4, or on the difference between the contact energylsegmental volume ratios,i.e., P* of the two components. In the approximation for A V, corresponding to eqn(19) for AAM, the free volume term divides in two, givingAGM : A H M : T A S M = CplT : -aQaT : - C P p aCplaT (20)The two contributions may be visualized through a plot of 2 againstFor computational purposes this may be rewritten to eliminate 4, puttingh-41 = (p:-pN1~2I(p;~l +PT$2).$1$2+(t,bl 4 1 ) = (PZ291 +p:2$2)v91921(p:$I +p:VM2-(22)The last term is a negative free volume contribution corresponding to those for theother mixing functions in eqn (20).The new contribution in (Pf-Pz)( pl - r2)should also be negative for systems of quasi-spherical molecules since P* will tend tobe large when vand Tare \mall, and vice versa. However, it may be of either signfor polymer solutions since Vand Twill normally be small for the polymer component2. Then if P;<Pf, a positive contribution to A V M will be observed. Thus, thebenzene + polyisobutylene system (Pg <Pf) has a large positive value 6e of A VM whilethe cyclohexane + polystyrene system (Pz > PT) has an almost negligible value ofA VM.PARTIAL MOLAR QUANTITIESThe thermodynamics of polymer solutions is usually expressed in terms of partialmolar quantities. By differentiation of the mixing functions, eqn (13), the non-combinatorial parts of Apl, TAS, and ARl are given byWhen X2+0, as in a dilute polymer solution,(Az,lu;) = ( - 2 ( T ) + T(a~laT))x~v2+A(~)-A(~~)+(.fl: - T)(aA/dT). (23D.PATTERSON AND G. DELMAS 103wherez l-TTIT,*When X2--+l, a case of interest in GLC studies of the activity of a small moleculecomponent, 1, at high dilution in a polymer, 2,The partial molar quantities thus also show the two contributions mentioned whendiscussing the mixing functions. The quantity A Vl shows three contributionscorresponding to those in eqn (21) ; it will be omitted here.MODELS FOR THE CONFIGURATIONAL PROPERTIESThe corresponding states theory may be applied either using empirical data to givethe various configurational properties, e.g., u, cp, or a model of the pure liquid maybe used to predict these theoretically.Thus, Prigogine and collaborators used thecell partition function l a of Hirschfelder and Eyring plus a dependence of configura-tional energy on volume inspired by the Lennard-Jones (m,n) potential between pointcentresThis leads to the following equation of state,D = (-- n F m 1 3 + rn Fn~3)1(n - m). (27)where b is a packing factor, b = (m/n)ll(n--m), and gives r(T) at fi = 0. The reducedentropy is given byand cp may be found by differentiation, cv being zero. These configurationalquantities may be used with different (m,n) in eqn (16,17) to yield the mixing functionsor in eqn (24) for the partial molar quantities. Flory and collaborators have alsoused the cell partition function of Hirschfelder and Eyring plus the van der Waalsenergy-volume relation, u = - v-1.In a formal way at least this relation may be obtained from eqn (27) by puttingrn = 3, n+co so that the Flory equations for the mixing functions and partial molarquantities may be obtained from the general corresponding states equationsIgivenabove by making this particular choice of (m,n).The quantitative success of the theory with the configurational functions given bymodels depends critically on the validity of the model.The models with the differentvalues of (m,n) have been tested l2 against the precise thermal expansion coefficientand isothermal compressibility data for the normal alkanes as determined by Orwolland Flory.6d The principle of corresponding states is well obeyed.The modelsreproduce the data qualitatively, but with characteristic errors for three choices ofm,n : (3,co) (the Flory case), ( 6 , ~ ) and (6,12) (smoothed potential model of Prigogine,Trappeniers and Mathot). As or r a r e increased, the predicted thermal expansioncoefficient increases too rapidly and the isothermal compressibility too slowly. Theresult is that in order to compensate, the value of T* used must increase with temper-ature, or at constant Tit has an incorrect variation with chain length of the alkane.Similarly, P* must decrease with increase of 4: or at constant it incorrectly decreaseswith decreasing alkane chain length.s = 31n(r%-b) (29)-(30104 CORRESPONDING STATES THEORIESWe have used these three models in comparing predictions of the correspondingstates theory with experimental values of the mixing functions, most of which havebeen compiled by Abe and Flory.6c Some of the results for typical systems are pre-sented in table 1.Following Abe and Flory, the reduction parameters T*, Y*,U* = P* V* for the components are found from tabulated values of thermal expansioncoefficients, molar volumes, and isothermal compressibilities using 6, or givenby the three models. Then, a value of v2 is determined by fitting eqn (16) to theexperimental AH, for each model. The value of X, appearing in the interactionterm of eqn (16) was caiculated from & using the surface/volume data of Abe andFlory (s1/s2 ratios).The fitted value of v 2 is almost independent of the model.TABLE 1 .-INTERACTION AND FREE VOLUME CONTRIBUTIONS IN THE EQUIMOLAR MIXINGFUNCTIONSExperimentAHMcalmol-135271951411127530.53411,- 5.8TSE AvMcal cm3mol-1 mol-118 0.167 0.01117 0.655 0.141 -0.214 -0.18- 1 -0.546 -0.4915 -0.309* -0.50- -0.18AHt-.v.acalmol-1-0.01- 0.0 1-0.010.000.000.000.000.000.00-0.010.00 - 0.0 1- 1.2- 1.0 - 2.2 - 1.4 - 1.1 - 2.2-8.3- 10- 16-13- 12- 2616' 14- 29-8-6-11-6.1- 5.3-11.0T%V.calmol-1- 0.0 1-0.01-0.010.000.000.000.000.000.000.000.00 - 0.0 1- 3.0- 1.8-3.3-3.1 - 1.7-3.1- 23 - 13- 24- 36- 22- 40- 42 - 25- 45- 17-9 - 15- 14- 16- 8.6TheoryTSEca1mot-19.414101152827.27.8563.75.94.00.233.22.60.974.41.60.63.323204019-8 + 16-7- 10- 1-7222722a listed in the order of (3,00), ( 6 , ~ ) and (6,12) models.b estimated by Hacker and Flory 6f from variation of GE with temperature.c data 13 at equal volume fractions.AVf.".cm3mol-*0.000.000.000.000.000.000.000.000.000.000.000.00- 0.2 1-0.14-0.23-0.19-0.12 - 0.20- 1.1- 0.70-1.1-0.70-0.51- 0.99- 0.62-0.44-0.83-0.39 - 0.27- 0.52- 0.23-0.18 - 0.42*VMcm3mol-10.120.190.130.080.130.090.631 .o0.070.120.08-0.13- 0.02-0.13-0.08 + 0.05 - 0.07- 0.79- 0.78-0.71-0.55-0.28-0.80- 0.47- 0.23- 0.65- 0.22- 0.06-0.31- 0.23-0.18-0.390.68t"C252525- 189- 196- 18902070- 15730The free volume contributions to AHM, TASM, AVM, calculated using eqn (16) and (17)are listed in table 1 for each system in the order (3,c;o) or Flory theory ; ( 6 , ~ ) ; (6,12),or Prigogine smoothed potential model.The systems are in the order of increasingimportance of the free volume term in the total AH,. For the mixtures of quasi-spherical molecules, TSE is put equal to the total non-combinatorial TASM as foundfrom the theory. For hexane + hexadecane, benzene + diphenyl and the siloxanD . PATTERSON AND G . DELMAS 105mixtures, all chain molecules the Flory-Huggins combinatorial contribution to TSEis added, using volume fractions in the calculation.The contributions for thesystems are, respectively, 46, 21 and 36 cal/mol. The calculated total A V M and TSEare listed in table 1. Values with the ( 3 , ~ ) model are the same as given by Abe andFlory for the systems they treated.Table 1 shows that the results from the three models are similar, although the( 3 , ~ ) used by Flory and collaborators is probably best. It has been pointed that thepredictions of the Flory theory are superior to those originally made by Prigogine andcollaborators using their average potential model. This does not seem to be due toany marked superiority of the ( 3 , ~ ) model but rather to (l), the exaggeration by theaverage potential model of the effect of size differences (effect of p ) in the solutionreduction parameters, and (2), fitting the reduction parameters of the pure componentsto liquid state properties rather than using gas-phase data.For many systems of quasi-spherical molecules the free volume term is negligiblefor all mixing functions.(This contrasts with the Flory equation of state term whichis important due to the included effect of v2). It is only when the free volume dis-similarity between components becomes large that quasi-spherical molecule systemsshow an effect of the free volume term, and then mainly in ASM and AVM, where thethree models can give different results. The predominance of this term is almostlimited to chain-molecule mixtures, as found at the end of table 1, and to high polymersolutions.For most quasi-spherical molecule mixtures, therefore, the total mixingfunctions should follow the interaction term ratios given by eqn (1 8) which are roughlyindependent of the model. Thus AHM/TASM = 1 +(- UITC,), is, which for the(m,co) models equals 1 + (3/maT). The AHM/TSE ratio is therefore predicted to beabout four whereas the experimentalvalue is about two for the globular moleculesystems in table 1, and for many other such systems not presented here. This dis-crepancy is probably due to non-central forces.14A comparison of the eqn (23) and (24) with experimental values of A,ul and ARlfor high polymer solutions has been made and will be published later.We acknowledge the support of the National Research Council of Canada.(a) I. Prigogine (with the collaboration of A. Bellemans and V. Mathot), The MoZecuZur TheoryofSoZutions (North-Holland, Amsterdam, 1957), chap. 9 ; (b) ibid., chap. 16 ; (c) ibid., chap. 17.W. B. Brown, Phil. Trans. A, 1957,250, 175,221.R. L. Scott, J. Chem. Phys., 1956,25,193.V. Mathot, Compt. rend. reunion sur les changements dephuses (Paris, 1952), p. 95.I. Prigogine, N. Trappeniers and V. Mathot, Disc. Faraduy SOC., 1953, 15, 93.(a) P. J. Flory, R. A. Orwoll and A. Vrij, J. Amer. Chem. SOC., 1964, 86, 3507, 3515 ; (b) P. J .Flory, ibid., 1965,87,1833 ; (c) A. Abe and P. J. Flory, ibid., 1965,57,1838 ; (d) R. A. Orwolland P. J. Flory, ibid., 1967,89,6814; (e) B. E. Eichinger and P. J. Flory, Trans. Furaday Soc.,1968, 64,2053 ; (f) H. Hacker and P. J. Flory, ibid., 1968,64, 1188.E. A. Guggenheim, Mixtures (Oxford University Press, 1952), p. 218.T. W. Leland, J. S. Rowlinson and G. A. Sather, Trans. Faraday SOC., 1968, 64, 1447.’ H. Tompa, PoZymer Solutions (Butterworths, London, 1956), p. 84.lo H. C. Longuet-Higgins, Disc. Faraday SOC., 1953, 15, 73.l1 J. S. Ham, M. C. Bolen and J. K. Hughes, J . Polymer Sci., 1962, 57, 25.l2 D. Patterson and J. M. Bardin, Trans. Faraday SOC., to be published.I3 D. Patterson, S. N. Bhattacharyya and P. Picker, Trans. Furday Soc., 1968, 64, 648.l4 J. S. Rowlinson, Liquids and Liquid Mixtures (Butterworths, London, 1959), p. 329

ISSN:0366-9033    

DOI:10.1039/DF9704900098

出版商:RSC    

年代:1970

数据来源: RSC