Liquid-Liquid Equilibria in Multicomponent Polymer S ys ternsBY R. KONINGSVELDCentraal Laboratorium, DSM, Geleen, The Netherlands.Received 19th January, 1970The effect of polydispersity on liquid-liquid phase relationships in equilibrium is reviewed andfound to be similar for all the kinds of systems examined. As a rule, it is not allowed to identify acloud-point curve with a binodal (coexistence curve), nor the maximum in a cloud-point curve with acritical point. In any kind of system (solvent-polymer ; polymer 1-polymer 2 ; solvent-non-solvent-polymer ; solvent-polymer 1-polymer 2) measurement of the consolute state opens the way towardsaccurate determination of interaction parameters. Use of the critical opalescence for the latterpurpose is subject to complications.Liquid-liquid phase relationships in systems containing macromolecules are influ-enced by the polydispersity existing in virtually all polymers.It is the object of thepresent paper to review this effect in various types of systems. We shall first considerquasi-binary mixtures which consist of a single low-molecular-weight solvent and apolydisperse polymer. By checking the effect of an increase of the single-solventchain length up to values representative of (monodisperse) polymers we considerpolymer compatibility. Quasi-ternary mixtures represent a type of systems frequentlyencountered in polymer literature. They may either consist of a solvent, a non-solventand a polydisperse polymer, or contain a single solvent and two polymers differingin chemical structure.More complicated systems are conceivable, but we shallrestrict ourselves here to those mentioned above. It will appear that phase relation-ships in all types of systems are affected in a similar way by the polydispersity of themacromolecular constituents.When trying to derive values of molecular parameters from measured phaserelationships, the polydispersity of the samples must not be ignored since otherwiseerrors amounting to orders of magnitude may be introduced. Such quantitativetreatments are based on a free enthalpy (Gibbs free energy) of mixing function arisingfrom a statistical-mechanical treatment. In this paper we shall use the expressionderived by Flory 1* and Huggins 3-5 who employed the lattice model for the liquidstate.Flory,’ and Patterson.8 The original Flory-Huggins expression, however, has theadvantage of being much simpler than the functions arising from the later treatments.In addition, it provides a correct qualitative description of thermodynamic propertiesof macromolecular systems and can easily be made quantitative along empiricalTompa 9* l1 and Scott l2 formulated the Flory-Huggins expression for ternaryMore refined theories have been advanced; we mention those of Hugginslines.2, 5 .9, 10systems. Accounting for polydispersity,AGIRT = 4os; In 4o +x4irn; In 4i + c ~ + + ~ s i In $ j +where AG denotes the free enthalpy of mixing per mol of lattice sites ; R and T have144i j901404+90240$+g124J/, (1R. KONINGSVELD 1 45the usual meaning; #o, 4 and $ are the volume fractions of solvent and the wholepolymers 1 and 2, respectively ; 4, and $J those of polymer species i (in polymer 1) andj (in polymer 2); gol, gO2 and g12 represent free-enthalpy correction parametersaccounting for the solvent-polymer 1, solvent-polymer 2 and polymer 1 -polymer 2interactions ; so, m, and sj stand for the chain lengths of solvent-, species i- and speciesj-molecules, expressed as the number of occupied lattice sites.We shall deal with two-liquid-phase equilibrium states only.At equilibrium,phase-relationships can be calculated by equating the chemical potential in the twophases for every component. The numerical calculation procedures have been orwill be p~blished.l~-~' Application of eqn (1) requires that assumptions be made forthe molecular-weight distributions of the polymers.Use has been made of the fami-liar logarithmic normal and exponential functions, here defined as weight distributions(2)(3)where W = w(M)dM = total mass of polymer, normalized to 1 g. B2 = 2 In 6 ;b = MJM, ; il = (2- b)/(b- 1) ; z = (A+ l)/Mn. M,,M, = the number- andweight-average molecular weights. Distribution functions deviating in shape fromthose given by eqn (2) and (3) can be constructed by adding two or more exponentialor logarithmic normal functions. l4The difference in composition between two phases may decrease if the conditionsare changed. Eventually, when the two phases have become identical, the consoluteor critical state is reached.Such changes of the conditions can be brought about bychanging T at constant pressure or vice versa, or by varying the overall compositionof the system at constant p and T. Critical points are located on the spinodal, whichis the locus of points on the AG(41,42 . . ., 4h1,t,h2 . . .) surface (at constant p and T )where the curvature of the latter is zero in one direction. According to Gibbs l8 thespinodal is defined by (constant p and T )w(M) :log normal: W ( M ) = w~M,~B-'~-+M exp [ - B ~ In2 ( b 1 * 5 ~ ~ n - 1 ) J ,exponential : w(M) = w?+2[r(L+ 2)I-l MA+l exp [-(A+ 1) M/M,I, s,where xi and x, stand for the independent composition variables, which are so chosenas to be represented by the volume fractions qbi and $j of all polymer species.For the critical point,18J,, = 0in which Jcr is the determinant formed from Jsp by replacing the elements in anyhorizontal line bya J S P l a 4 1 , a J s P l a 4 2 - * a J s p l a $ l , aJSPla$z * * 'The conditions (4) and (5) have been worked out for eqn (1) in which the g may bearbitrary functions of the overall composition of the mixture.lg The resulting equa-tions are highly involved and instead of writing them out we shall refer to some specialcases in the next sections.Strictly speaking, using eqn (1) we exclude any type of macromolecules other thanlinear homopolymers.However, in a general sense, the following considerations maybe expected to have qualitative validity for other molecular structures as far as theeffect of polydispersity in chain length is concerned.2146 LIQUID-LIQUID EQUILIBRIAI .QUASI-BINARY SYSTEMSIf we introduce so = 1 and + j = 0 (allj), eqn (1) describes mixtures of a singlelow-molecular-weight solvent and a polydisperse polymer. We first consider a poly-mer containing two components differing in chain length. Tompa 9 p l1 has compre-hensively dealt with such ternary mixtures ; his phase diagrams are schematicallysummarized in fig. 1, which shows the extension of a miscibility gap in the temperature-composition space. Consideration of such ternary systems has the advantage thatthe phase diagram is still visualizable, and yet brings out the essential differencesbetween truly binary and quasi-binary mixtures of the present type whatever thenumber of macromolecular components.T4PIFIG.1.-Miscibility gap in a ternary system containing two macromolecular homologues P1 and P2and a single small-molecule solvent S. The chain lengths of P1 and P2 (mi and mz) are differentm2 > m AA2C5B, quasi-binary section (cloud-point curve) ; A2, precipitation threshold ; CC5C',locus of critical points ; DC5E, quasi-binary section of the spinodal surface (- - -).In the binary lateral face TSP2 we have the usual situation with the critical point Cat the maximum of the coexistence curve (binodal). Spinodal and binodal have acommon tangent at C. The binodal can be measured by changing the temperatureof a homogeneous system of known concentration and noting the temperature ofincipient phase separation (cloud point).If such a measurement is carried out on anon-binary polymer solution. e.g., on mixture X dissolved in S, one obtains the cloud-point czirve AA2C5B. The orientation of the tie lines (A2B2 ; AK ; Q'Q" ; LB)discloses that such a cloud-point curve should not be identified with a coexistencecurve. Quite generally, it can be shown 21 that the critical point (C,) in this kind ofquasi-binary system is shifted towards a concentration value higher than that of thR. KONINGSVELD 147maximum A2. Tompa 11* 22 proposed to denote this maximum by the termprecipita-tion threshold. The quasi-binary section DC5E of the spinodal surface has a commontangent with the cloud-point curve at the critical point.When plotting phase relationships in a quasi-binary system in a planar diagramthe graph paper in fact coincides with TSX.The coexisting-phase compositions (i.e.,Qr and Q”) lie outside TSX, but can be projected on this plane. On doing so fordifferent temperatures, one obtains coexistence curves whose location depends on theoverall polymer concentration.l** 23 Fig. 1 reveals a narrowing of the projectedcoexistence region upon an increase of the overall concentration. The phases B2 andK are on the verge of appearing and coexist with cloud-points at sub-critical concen-trations. Above the critical concentration we have L coexisting in the same way withB. Projecting these points onto TSX, we obtain the shadow curve, which is thecoexistence curve relating to the cloud-point curve. Shadow and cloud-point curvesintersect in the critical point, which also lies on the coexistence curve for an overallconcentration equal to the critical.For all other concentrations the coexistencecurves consist of two detached branches, representing the dilute and concentratedphases. If the whole-polymer volume fraction 4 is smaller than its critical value (bc,the dilute-phase branch ends in the cloud point at # and the concentrated-phasebranch in the shadow concentration. The reverse situation exists if +> 4=.These observations are quite general and independent of the number of compo-nents. Two-dimensional phase diagrams can be calculated for arbitrary molecular-weight distributions; fig. 2 gives some examples. Evidently, the shape of thediagram is governed by the polymer composition.An increase of a(= MJM,) isaccompanied by a distortion of the diagram (Mz = the z-average molecular weight ;The preceding theoretical considerations have found experimental verification,e.g., in the extensive work reported by Rehage et al. on the system polystyrene+cycl~hexane.~~~ 25 Fig. 3 is based upon their data.b = M,/M,,).05205148 LIQUID-LIQUID EQUILIBRIA0.53r( G055I005 0.1 0 0.1 54(4FIG. 2.Two-dimensional phase diagrams for quasi-binary systems. Cloud-point curves : - ;shadow curves, - - - ; coexistence curves, - - - ; (overall volume fractions 4 are indicated) ; spin-odals, -; critical points, 0 ; so = 1 ; gol independent of +; m/Mm (a) Whole polymerdistribution : sum of two exponentialfunctions, Mw = 1 3 1 .7 ~ lo3 ; b(= Mw/Mn) = 5 ; a(= Mz/Mw)= 2 ; 4c = 0.039 308 ; (b) whole polymer distribution : sum of two exponential functions, Mw =131.7 x lo3 ; b = 5 ; a = 7 ; cjC = 0.071 104 ; (c) whole polymer distributions : Mw = 131.7 x lo3 ;1, monodisperse (b = a = 1) ; 2, logarithmic normal (b = a = 2) ; 3, logarithmic normal (b = a =10)R . KONINGSVELD 149I.FIG. 3.-Experimental two-dimensional phase diagram for a polystyrene-cyclohexane system.Experimental data of Rehage et aLZ49 25 Cloud-point curve, -; shadow curve, - - - ; co-existence curves, - for 2 (0) ; 6( a), 10( A), 15 (V) and 20 (a) % whole polymer concentration(wt- %) ; critical point, determined with the phasevolume-ratio method, @. Characteristics ofpolystyrene sample, Mn = 2.1 x lo5 ; Mw = 3.46 x lo5 ; Mz = 5.5 x lo5 ; b = 1.6 ; a = 1.6.The spinodals and critical points in fig.2 were calculated with the equationsobtained by applying eqn (4) and (5) to eqn (l), subject to the conditions mentionedin the first paragraph of this section. The resulting equations are :I4CRITICAL POINT :6 ( a Y o l / a 4 ) c - 3(1 - 2 4 c ) ( a 2 g 0 1 1 a 4 2 ) c - 4 0 , c ~ c ( a 3 ~ o l l a 4 3 ~ c =so1(l-~,)-2--mz(4cm,)-2. (7)If gol is independent of 4, and so = 1, eqn (6) and (7) reduce to the well-knownexpressions previously derived by Stockmayer.26Eqn (6) reveals that distributions differing on any point except on weight-average,will share the same spinodal in a quasi-binary plot. This holds for the examples infigure 2 which show different cloud-point curves, touching the same spinodal atdifferent critical points (if M, differs, see eqn (7)).1411.COMPATIBILITY OF POLYMERSAn increase of so, the chain length of the solvent, broadens the miscibility gap andmoves it towards lower values of gol. At high values of so, a small positive value ofthe interaction parameter gol suffices to bring about incompatibility. Fig. 4 illus-trates this feature. In practice, the long-chain solvent, in fact, a polymer, also has amolecular-weight distribution. Cloud-point curves in mixtures of polymers, as faras they are measurable, may be expected to have widely differing shapes accordin150 LIQUID-LIQUID EQUILIBRIAas the distributions of the two constituents vary. That this must be so is indicated byfig. 5, which shows some spinodals calculated with the appropriate function.19(SPMOD AL)%I2 - 2(1--2+)%Z/a4 - o$a2g12/a42 = ($sw)-l + ( o m V Y 9 (8)0.60.060.1 R.KONINGSVELD 151J -- I I I T I0.2 0.4 0.6 0-84(4FIG. 4.-Calculated cloud-point (-) and shadow (- - -) curves for various values of the solventchain Iength SO. SpinodaIs, - ; criticaI points, 0 ; gol independent of 4. Molecular weightdistribution of constituent 2 (concentration 4) : sum of two exponential functions, Mw = 2 x lo5 ;b = 2 ; u = 2; m/M = lo-*. (a) SO = 1 ; (b) SO = 10 ; (c) SO = 100; (d) SO = 1000.The locations of the critical points were calculated with(CONSOLUTE STATE),where s, and s, are the weight- and z-average chain lengths of the ‘‘ solvent ” polymerand the interaction parameter is denoted by g12.The locations of spinodals and critical points indicate that the miscibility gap will,as a rule, be shifted towards the axis of the component with the shorter chains.If th152 LIQUID-LIQUID EQUILIBRIAopposite behaviour should be found, this must be attributed either to a large differ-ence between the a values of the two polymers or, perhaps more likely, to a concentra-tion dependence of g1 2 . Depending on the value of ag, Jaq5, the unstable region maybe found on either side of the diagram. From Fig. 6 substantializes this statement.00.10.2 40.30.400.10.20.30-4FIG. 5.-Spinodals calculated for various sets of sw/mw values, indicated in the figure. Drawncurves : g12 independent of 4; critical point for different uo/u ratios( 0); right-hand branch: 0.5;0.2 ; 0.1 ; 0.05 ; 0.02 ; 0.01 ; 0.001 (from left to right ; as far as they are indicated) ; left-hand branch,2 ; 5 ; 10 ; 20 ; 50 ; 100 ; 1000 (from right to left).The critical points 0 at the maxima refer toao/a = 1 ; uo and a stand for sz/sw and mz/mw. The spinodals - - - and - - - relate to a concentra-tion-dependent g12 (ag12/a+ = -0.1 and +0.1, respectively).this point of view one is inclined to attribute the unexpected locations of some cloud-point curves reported by Allen, Gee and Nicholson 27 to a strong concentration de-pendence of the interaction parameter in their system. They studied mixtures ofpolyisobutenes and silicones of relatively low molecular weight and found miscibilitygaps shifted towards the pure silicone axis.The silicones, however, had the longeraverage chain lengths of the two constituents.The maxima of the spinodals in fig. 5 are noteworthy. In truly binary mixturesthey coincide with the critical points. In the present quasi-binary ones the consolutestate may evidently also occur at the maximum of the spinodal, if the two polymershave special a values. If 9 1 2 does not depend on 4, these values must be equal. Ifg12 depends on 4 the two a values must either be equal to one (binary mixture) or berelated in a definite way given by the value of ag,,/a#. In this respect (critical pointat maximum of spinodal) a multicomponent system may behave as if it were binary.It remains to be ascertained whether this feature also exists in the other aspects inwhich polydispersity is revealedR.KONINGSVELD 1530005G0.1 00-150 0 40.0 6' 0 0 80.10g, = - 0.1I 1 1 1 1 10-2 0.4 0.64(40.4 06 0.8d(b)FIG. 6.-Calculated cloud-point (-) and shadow (----) curves and spinodals (-) for twoindicated values of agI2/8+( = g1 ; go is the concentration-independent term in gI2( d)). Long-chain solvent, so = 10 ; second constituent, polydisperse polymer (distribution : sum of two expo-nential functions, Mw = lo4, b = 5, a = 7). Critical points, 0.111. QUASI-TERNARY SYSTEMSQuasi-ternary systems come in two classes, viz., mixtures of a solvent, a non-solventand a polydisperse polymer, and mixtures of a solvent (good or poor) and two differentpolymers (mixtures of three polymers will be left out of consideration).Ternar154 LIQUID-LIQUID EQUILIBRIAdiagrams with monodisperse polymer components were calculated for either type ofsystem by Tompa 9 9 l1 and by Scott.12 We now discuss the effect of polydispersityon their diagrams. Calculations on this point have also been reported by Baysaland Stockmayer.28SOLVENT-NON-SOLVENT-POLYMERAddition of a second polymer component to a truly ternary mixture brings out theessential features in which polydispersity reveals itself. They are quite similar toquasi-binary systems. Since there are at least four components we must restrictourselves to an isothermal section; this is shown schematically in fig. 7.The twolateral faces NS-S-P1 and NS-S-P2 of the tetrahedron represent Tompa’s and Scott’sternary diagrams. The coexisting phases are located on a binodal surface within thetetrahedron and the tie lines will generally be so oriented as to fall outside a quasi-ternary section like ABC.SNSFIG. 7.-Miscibility gap in a quaternary system containing two macromolecular homologues P1 andPz, a solvent S and a non-solvent NS. The chain lengths of PI and P2 differ (mz >ml). ABC,cloud-point curve of polymer mixture X ; - 0 - - 0 -, critical line ; , coexisting-phase composi-tions relating to system A. Tie line, - - - -.Cloud-point curves in quasi-ternary systems can be measured by adding non-solvent to solutions of the polymer (say X) in solvent S and noting the amount yadded before phase separation sets in.Obviously, a plot of y against the wholepolymer concentration is identical to the quasi-ternary intersection NS-S-X of thebinodal surface. As a rule, the critical point will not coincide with the precipitationthreshold B at which y = yth. The coexisting-phase compositions can be projectedonto NS-S-X; this yields coexistence curves the location of which depends on thR . KONINGSVELD 155whole polymer concentration in a similar way to quasi-binary systems. Someexamples calculated by means of eqn (l), are collected in fig. 8 and 9. The extensionand location of the miscibility gap are determined primarily by the values of theinteraction parameters. The polydispersity of the polymer, however, reveals itself innon-negligible details, in the same way as it does in quasi-binary systems.0.1 AdFIG.8.-Coexistence curves in a quasi-ternary phase diagram calculated on the basis of eqn (1) for anexponential molecular weight distribution X (M, = 131.7 x lo3 ; b = 2 ; a = 1.5). Values of theinteraction parameters are indicated (gol~goz/glz). Coexistence 4 = 0.01, unless stated otherwise.curves, -, --- ; tie lines, -, - - -.SOLVENT-POLYMER ~ - P O L Y M E R 2Phase relations in solutions of two different polymers in a mutual solvent (good orpoor) may be very complicated. A quasi-ternary diagram (fig. 10) contains all thefeatures noted in $I and 11. In addition, however, the threefold interactions maygive rise to extra phenomena, not observed in the quasi-binary lateral faces.Werefrain from a complete description and only make some general qualitative remarks.A fuller treatment will be given in a forthcoming paper.19The two cloud-point curves SAIB1 and SA2B2 are intersections of the cloud-pointsurface within the prism with the lateral quasi-binary faces TSX1 and TSX2. Onthese faces, all features described in $1 are present. Hence, there must exist spinodaland shadow surfaces within the prism, while the coexisting phases can be projectedonto coexistence surfaces. Similar observations may be made on the third lateralface (4 11). Further, critical solution phenomena of the upper and lower consolutetypes may occur on either face. Consequently, phase behaviour of this type of quasi-ternary systems may vary considerably and be very complex156 LIQUID-LIQUID EQUILIBRIAFIG.9.-Calculated cloud-point (-),shadow (- - -) and coexistence (- - -)curves in a quasi-ternary system, go1 =0 ; gO2 = 0; g12 = 1. Polymer distri-butions: Mw = 1 3 1 . 7 ~ lo3; XI : expo-nential, b = 2, a = 1.5; X 2 : mono-disperse ; X3, logarithmic normal ;b = a = 10. Spinodals, - ; criticalpoints, 0. Some tie lines of the trulyternary systeni (A',) in fig. 9b areindicated (- - - - -).NS 0.05 0. I x4(bFIG. 1 1 .-Cloud-point surfaces in a quasi-ternary system composed of diphenylether, linear poly-ethylene (Mn = 7.9 x lo3 ; Mw = 7.6 x lo4) and crystallizable polypropylene ( M , = 5.2 x lo4 ;M~ = 6 . 4 ~ 105).To fucepage 157.R . KONINGSVELD 157T 1sFIG.10.-Quasi-ternary pliasc diagram for a solvent S and two polydisperse polymers XI and Xz.SAIBl and SA2B2, quasi-binary cloud-point curves ; 0 - - 0 -, critical line ; - , spinodals;- - -, shadow curves ; Al and Az, quasi-binary precipitation thresholds.Fig. 1 1 shows an experimental example referring to the system diphenylether-linearpolyethylene-crystallizable polypropylene. There are two miscibility gaps and these --_- _._-__ -I-.-- c,,,+t-- rpt, C-A - A ,. - - i - ~ : - . - i - . I - - - . --.--A- L-c:,, -r +t- --:---A C;UlIlt: VGly GlUbG 1 U ~ ~ l l I E ; l . lllt: IllSL, ill it 1t:lilLlVE;ly IUW C;UIlC;GlIl1illlUl~ Ul LUG IllliLGUpolymer, is the most obvious and arises from the cloud-point curves in the two quasi-binary systems solvent-polymer.Within the prism, however, this cloud-pointsurface, rises to a maximum (quasi-ternary precipitation threshold). Hence, attemperatures above the quasi-binary thresholds, quasi-ternary mixtures may still bepartially miscible. This effect may be due either to the molecular-weight distributionsof the polymers or, as pointed out by T ~ m p a , ~ to special values of the interactionparameters.The possible occurrence of such closed miscibility gaps jeopardizes the mutualsolvent method as a means for establishing polymer compatibility. Judged from thepresent example, phase separation of a solution of two polymers in a common solventdoes not necessarily involve that the polymers themselves are incompatible.The second miscibility gap in fig.11 has been found to be closed at higher temp-eratures (- 280°C). Whether it is also closed at higher concentrations of the mixedpolymer remains to be established. Its absence in the two quasi-binary solvent-polymer systems indicates that its origin must be sought in the polymer-polymerinteraction, whether or not in conjunction with the solvent.I V . DETERMINATION OF INTERACTION PARAMETERSThe examples presented so far indicate that measurable differences in the locationof miscibility gaps may accompany relatively small variations in the interactio158 LIQUID-LIQUID EQUILIBRIAparameters. Hence, measurement of phase relationships should conversely giveaccess to values of these parameters in a sensitive way. The experimental techniquesare simple ; with quasi-binary mixtures containing a low-molecular-weight solventthey have been applied freq~ently.~*~* O* 14* 9 * 2 0 p 2 3 * 29 Polymer mixtures presentproblems in that the viscosity of the systems soon becomes too high upon an increaseof the average molecular weights.Measurements on low-molecular-weight sampleshave been reported.27* 30 Quasi-ternary systems are generally well manageable, buttheir description involves considerable experimental effort. 9 2 9 9In all these studies the polydispersity of the samples must be taken into account ifthe conclusions are to have quantitative ~ignificance.~~ Accurate determination ofmolecular weight distributions, if possible at all, is a tedious Fortunately, thecritical point is a well-defined state in the phase diagram, being determined as it is bytwo average molecular weights only.Therefore, it provides the obvious key to theproblem.--I$ n-e.a Polystyrene -Cycl ohex drle(I:'- xic,'be0.2 00.180 16+ 0.10IN"10-8.I0.0 5 0.1 0 0.154FIG. 12.-Concentration dependence of the interaction parameter for two systems. The magnitudeof the estimated error in the ordinate is indicated. The abscissa in fig. 12a gives x*, the segment molfraction (Grundmolenbruch).As shown elsewhere, the consolute state in multicomponent systems can be ac-curately traced by determining the phase-volume ratio as a function of temperatureand overall polymer concentration. O* l4 These measurements, combined withcloud-point data, reveal the critical state in terms of 4c and T,, the critical volumefraction and temperature.Such data have, so far, been obtained only on quasi-binarysystems of which two examples are presented here.14* 34Having established $c and T, for a number of samples differing in M, and M,, wR . KONINGSVELD 159use eqn (7) to check the concentration dependence of the interaction parameter.Assuming the concentration dependence of Sol to have the form 9* 35SO1 = 90 +91+ f92#2 $- - - (10)the right-hand side of eqn (7), which contains measured quantities, will be : zero, ifgol is independent of + ; constant, if gol is linear in 46 ; linear in #, if gol is quadraticin 4, etc. In this approach it has been assumed that the temperature dependence ofgol is limited to go.Arguments supporting this assumption can be advanced lo* l4 ;however, a temperature dependence of the other gt coefficients can also be taken intoaccount, if necessary.Fig. 12 shows two examples of such an analysis, one (polystyrenefcyclo-hexane 34) pointing to a linear dependence of the right-hand side of eqn (7) on 4, theother (polyethylene + diphenylether) to a zero value within the experimental error.Eqn (6) can be used for calculating go and determining its temperature function from aplot of $qi0 against 1/Tc. This is illustrated in fig. 13 ; both plots show the usual lineardependence.I -. --LI.-- ---A- lo’ 3.30 3.35 3.40 3.45 350(2.50 d&-I .--- . L .A L---2.35 1.40 2.4 5 2.50 I ’ I 03-0103pFIG. 13.-Temperature dependence of the interaction parameter from critical miscibility data. Theblack dot denotes the 8 temperature taken from literature.The gO1(Q3,T) function thus obtained for polystyrene +cyclohexane is in goodagreement with those found by other methods (light-scattering, osmotic and vapourpressure measurements).In the description and quantitative prediction of phas160 LIQUID-LIQUID EQUILIBRIAbehaviour it is superior to the For polyethylene +diphenylether the g(T)function derived from critical miscibility data could so far only be checked againstChiang's value for the 9 temperature (161.4"C 37), which compares favourably withthe extrapolated value of 161.8"C. However, phase relationships could be predictedwith great accuracy.38Debye proposed the use of the critical opalescence, i.e., for determining the criticalcon~entration.~~ The angular dissymmetry of the light scattered at angles of 45 and135", for instance, should show a maximum at $c at the critical temperature. Thismethod was used with polymer solutions as well as with small-molecule mixtures byDebye et aL40 and by Borchard and Rehage 44 showed, however,that the maximum dissymmetry occurs at a concentration lower than the critical.The correctness of their finding can be verified by means of Debye et aZ. own data.Ifwe assume that the M, and M, values of their polystyrene samples were equal we cancalculate the critical whole-polymer volume fractions in cyclohexane their samplesshould have had, and compare these values with those at which they found maximumdissymmetry.We have used eqn (7) and the g(Q) function derived from criticalmiscibility data 34 for the purpose. The 4c values thus calculated are too smallbecause any deviation of M, from M, tends to shift 4c to higher values. Table 1 liststhe two sets of data and gives strong support to Borchard and Rehage's discovery.41-43TABLE 1 .-SYSTEM POLYSTYRENE+ CYCLOHEXANECritical whole-polymer volume fractions from critical opal-escence and from eqn (7)of ref. (40) dissymmetry * dc ( e w 7)A 0.068 > 0.0923 0.056 > 0.079C 0.050 > 0.073D 0.046 >0.061E 0.043 > 0.060F 0.028 > 0.043G 0.020 > 0.032data of Debye, Coll and W~errnann.~~sample notation maximumThese remarks do not suggest that the critical-opalescence method would beinvalid.It should, however, be adapted to the systems on hand ; these are not binaryand not strictly regular as those in the model employed by D e b ~ e . ~ ~ These deviationsmight be responsible for the discrepancy.Finally, a small value of M,/M, in a sample still does not necessarily justifyidentification of threshold and critical concentrations. 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