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