Liquid-liquid phase separation in multicomponent polymer systems. XXVII. Determination of the pair-interaction function for polymer blends

Polymer International 44 (1997) 356È364

Liquid–Liquid Phase Separation inMulticomponent Polymer Systems

XXVII. Determination of thePair-Interaction Function for Polymer

Blends¤

Ronald Koningsveld* & William J. MacKnight

Sylvio O. Conte National Center for Polymer Research, University of Massachusetts, Amherst, MA 01003, USA

(Received 11 April 1997 ; revised version received 24 April 1997 ; accepted 19 June 1997)

Abstract : It is demonstrated that estimation of the pair-interaction functionfrom polymer-blend cloud points cannot yield meaningful values if data arerestricted to a single blend. Additional information is required, such as cloudpoints in another blend made up of samples of the same two polymers di†eringin molar mass or, alternatively, scattering data on the original blend. Informa-tion obtained from melting points of a crystallizable constituent in the blend isinsigniÐcant in this respect. Molar-mass distributions in the constituents of theblend complicate the analysis, and must be known in considerable detail.

Polym. Int. 44, 356È364 (1997)No. of Figures : 8. No. of Tables : 3. No. of References : 37

Key words : phase separation, pair interaction, polymer blends, multicomponentsystems

INTRODUCTION

The molecular heterogeneity occurring in virtually allsynthetic polymers poses problems when mixtures ofsuch materials are subjected to a thermodynamicanalysis by means of the pair-interaction parameter.This parameter, usually denoted by s, depends at leaston temperature and concentration, and knowledge ofthis dependence permits predictions of equilibriumliquidÈliquid phase behaviour of the blend. It is practi-cally impossible to extract meaningful values for theinteraction function from a single measured cloud-pointcurve (temperature of incipient turbidity against blendcomposition) and additional data cannot be dispensedwith, even when the molecular heterogeneity is negligi-

¤ Dedicated to Professor Bob Stepto on the occasion of his60th birthday.* To whom all correspondence should be addressed.a Part XXVI : ref. 35.

bly small. The analysis is further complicated by thepresence of molar-mass distributions (MMD), whichcause the two cloud-point concentrations at a giventemperature not to represent coexisting phases, as theydo in the strictly-binary case.1h3

The present paper demonstrates that the extraction ofuseful values for the interaction function from a mea-sured cloud-point curve (CPC) not only requires quiteaccurate knowledge of the MMDs involved, but stillneeds additional information. Such extra data mayeither be cloud points in a blend of samples with di†er-ent MMDs, or spinodal scattering data on the originalblend. We illustrate this statement with examples ofpartial-miscibility behaviour and of measurements ofmelting-point depression, discussing the following :

(a) Experimental cloud points in the systempolyisopreneÈpolystyrene, in which both constit-uents have a very narrow MMD. The MMDscan be ignored for the present practical purpose,and such systems have, therefore, been termed

3561997 SCI. Polymer International 0959-8103/97/$17.50 Printed in Great Britain(

Pair-interaction function for polymer blends 357

practically-binary.4 The analysis reveals some ofthe subtleties inherent in polymerÈpolymer mis-cibility, and their incorporation into suitablemodels.5,6

(b) The e†ect of MMDs on the extraction of signiÐ-cant s values, which is investigated with the aidof calculated CPCs for well-deÐned MMDs. Weemploy FloryÈHugginsÈStaverman (FHS) ther-modynamics,7h13 in which s depends solely ontemperature, and show in passing that a possibleconcentration dependence of s does not alter theproblem essentially. Having Ðxed the twoMMDs and the temperature dependence of s,we have a priori knowledge on each of the twofeatures that determine the CPC, i.e. the MMDsand s(T ).

(c) The insensitivity of melting points in partiallycrystallizable blends to essential details of theinteraction function.

PRACTICALLY-BINARY BLENDS

Assuming the residual polydispersity in narrow-distribution polymer samples to be negligible, we mayapply the strictly binary version of the FHS expressionfor the free enthalpy (Gibbs free energy) of mixing, *G :

*G/NRT \ )1] )2] g(T , /2)/1/2 (1)

where is the volume fraction of component k, the last/k

term of the right hand side is the familiar Van-Laarinteraction function, and (k \ 1, 2).)

k\ (/

k/m

k)ln/

kThe system is thought to be built up of N identical basicvolume units (BVU), and is the number of BVUsm

koccupied by a molecule of component k. The symbols Rand T have their usual meaning.

The temperature and concentration dependence ofthe interaction function g may assume various forms,depending on the molecular model used for its deÐni-tion.6,14 For instance, the incorporation of HugginsÏorientational entropy contributions of *G (ref. 15) rep-resents one particular example, the application of whichto polymer blends has been discussed by Koningsveldand Stepto.16 We here represent all those di†erentexpressions in the form of a quadratic polynomial in

the coefficients of which will have di†erent mean-/2 ,ings for the di†erent models. Thus :

g(T , /2) \ g0] g1/2] g2/22 (2)

The spinodal condition and liquidÈliquid critical stateare deÐned by the following equations :17h19Spinodal

L 1] L 2[ 2g0] 2g1(1 [ 3/2)] 6g2/2(1 [ 2/2) \ 0 (3)

Consolute state

Q1 [ Q2[ 6g1] 6g2(1 [ 4/2) \ 0 (4)

where andLk\ 1/(m

k/k) Q

k\ 1/(m

k/

k2).

Experimental cloud points in practically-binary mix-tures of polyisoprene and polystyrene are collected inFig. 1. The samples are of relatively low molar mass, inthe range of which the chain lengths play an obviouslydominant role, determining not only the widely varyinglocation of critical temperatures, but also inÑuencingthe shape of the CPC. The bimodality of the miscibilitygap in the 2É7/2É1 system changes into a shoulder in the2É7/2É7 mixture, and has vanished altogether in the0É82/2É1 blend. The 2É0/2É7 data by other workers donot seem to be consistent at all with the other three sets.

It has been shown previously that the CPCs of Fig. 1cannot be described with the FHS expression in its orig-inal form with g independent of concentration (theng 4 s), and that the unusual bimodal shape can bereproduced qualitatively if eqn (2) is employed.21 Herewe extend the analysis somewhat further, assuming g1and to be independent of T , and the temperatureg2dependence of to be given by the usual form:g0

g0\ g0s] g0h/T (5a)

where and are constants. The enthalpic termg0s g0h g0hshould be of the order of 102K in order to be accept-able within the model (non-polar or weakly polar vander WaalsÏ interactions).

Fig. 1. Cloud points in practically-binary mixtures of anionicpolyisoprene (PI) and anionic polystyrene (PS) for indicatedvalues of the number-average molar mass in kg mol~1.M1Data taken from two sources : ref. 20, 2É0/2É7 ; ref. 21, 2É7/2É7,

2É7/2É1 and 0É82/2É1. Weight faction of PS : w2 .

POLYMER INTERNATIONAL VOL. 44, NO. 3, 1997

358 R. Koningsveld, W . J. MacKnight

Within the present scope we may identify the extremaof the CPCs with critical points, and use those of the2É7/2É1 system to determine and with eqn (4). Theg1 g2BVU may be treated as a mass unit, arbitrarily set at100 g/mol, so that the relative chain length arem

kobtained by dividing molar mass by 100, and the /values represent mass fractions. Critical points arelocated on the spinodal ; hence eqn (3) can be used tocalculate and its temperature dependence. Theg0 g0hvalue found on this basis with the measured coordinatesof the two maxima is 5131 K, too large a value to beacceptable. Keeping the two concentrations Ðxed at themeasured values, one would have to shift the criticaltemperatures to 120¡C and 150¡C, respectively, to bring

down to 292 K. Obviously, such a manoeuvre is notg0hjustiÐed by the data, and we might look for a better g(T ,

function. Alternatively, however, we may not strive/2)for a fully quantitative reproduction of the data, andcheck whether eqns (1), (2) and (5a), are capable ofcovering the sensitivity of location and shape of theCPC toward chain length in a general sense.

Some trial and error analysis of the data from ref. 20yields a set of parameters producing Fig. 2 which is inqualitative accord with the data. In view of the lattersÏlarge temperature range, eqn (5a) was found not to bequite adequate and had to be replaced by the betterapproximation.22,23

g \ g0s ] g0h/T ] g0k T (5b)

where is a constant. The analysis shows that theg0kextended FHS expression (eqns 1, 2 and 5b) correctlypredicts bimodality to appear at a given set of chainlengths, to vanish, or change into a shoulder, whenchain length is varied. Miscibility gaps in blendsshowing shoulders and, occasionally, a bimodal shape,have not infrequently been reported. The presentanalysis indicates that such phenomena should not tooreadily be discarded, or attributed to experimentalerrors, and may point to a marked dependence of s onconcentration.

The description obtained contains the above men-tioned subtle features at the cost of the concentrationsof the extrema that are too low, except for the right-hand maximum in system 2É7/2É1 and the shoulder in2É7/2É7. The intersection of the two CPC branches in2É7/2É1 indicates the occurrence of a (binary) non-variant equilibrium between the three phases a, b and c,in the experiment at about 126É5¡C and in the calcu-lation at 127É3¡C. The connection of the extensions ofthe binodals beyond the three-phase line abc are illus-trated in detail in Fig. 3, and have been discussed exten-sively in a previous paper.5

The CPC of the 2É0/2É7 system, reported by otherworkers,20 can now be predicted ; this results in a CPCthat is located about 100¡C higher than the experimen-tal data. This discrepancy illustrates an importanthazard of studies in polymer blend thermodynamics. In

Fig. 2. Phase diagram calculated with eqns (1), (2) and (5b) forindicated sets of values Heavy curves,m1/m2 (m

i\ 10~2M

i).

binodals ; light curves, spinodals ; critical points, three-L ;liquid-phase equilibrium, Interaction-parameter>È>È>.values : g0s \ [0É427 57, g0h\ 162É32 K, g1\ [0É1, g2\

0É091.

Fig. 3. Details of the equilibrium curves around three-phaseequilibrium abc in Fig. 2 Heavy solid(m1\ 27, m2\ 21).curves, stable equilibrium; dashed curves, metastable andunstable equilibrium. Light solid curve, spinodal ; critical

points, L.



the Ðrst place, the extreme sensitivity of the location torelatively small di†erences in chain length might suggesterrors in the determination of molar masses to bebehind the discrepancy. However, at the level of a fewthousand g/mol such errors cannot reasonably beexpected to have been all that large.

Other experimental problems suggest more obviousreasons. For instance, the polyisoprene samples mayhave di†ered in cis/trans conÐguration, and steric con-Ðgurations have been found to markedly inÑuence thetemperature location of blend miscibility gaps.24 Thismay well be the reason why, for instance, miscibilitydata of various authors on the system poly(vinyl methylether)/polystyrene cannot possibly be reconciled.25 It isfurther known that in research of this nature the purityof the constituents is critical,26,27 traces of admixturesalready being potentially detrimental for a meaningfulcomparison of data.

Quasi-binary blends

The above example illustrates some of the difficultiesencountered in the study of practically-binary systems.Non-negligible molar-mass distributions in the poly-mers cause additional problems. We may use a hypo-thetical system to analyse the situation, thus havingprecise a priori knowledge of the two MMDs togetherwith the interaction function, and so investigate the sig-niÐcance of conceivable evaluation methods for thelatter. We use FHS systems without concentrationdependence of the interaction parameter, giving thelatter some attention in passing. There is no loss of gen-erality involved, because if determination of the tem-perature dependence of s already calls for extensivee†orts, the usual concentration dependence can only beexpected to further complicate the issue.

The appropriate *G function is identical in form toeqn (1), except for the deÐnitions of and which)

k/k

now read.12,17h19

)k\ ; (/

ki/m

ki)ln /

ki/

k\ ; /

ki(6a)

where i is the running index for the components withinconstituent k. The deÐnitions of and also changeL

kQ

kand become:

Lk\ 1/(mwk

/k) Q

k\ mwk

/(mwk/

k2) (6b)

where and and are the weight-mwk\ mzk/mwk

, mwkmzk

and z-average BVU-numbers of polymer constituent k,respectively.

We deÐne each of the MMDs of the polymers as thesum of two SchulzÈZimm (SZ) distributions, con-structed to conform to chosen and valuesmn , mw mz

is the number-average number of BVUs occupied(mnkby polymer molecules k ; Settingmnk\ mwk

/mnk). mw1\and100, mn1 \ 2, mw1\ 1É75, mw2\ 500, mn2 \ 3, mw2\

the construction procedure30 leads to the distribution2,

TABLE 1. Molar-mass distribution characteristics

for set I

Polymer 1 Polymer 2

mw

¼100 mw

¼500

yn¼m

w/m

n¼2 y

n¼m

w/m

n¼3

yw

¼mz/m

w¼1·75 y

w¼m

z/m

w¼2

Constructed distributions :a

wa1

¼0·390 9; mwa1

¼55·22 wa2

¼0·5; mwa2

¼146·4

wb1

¼0·609 1; mwb1

¼169·8 wb2

¼0·5; mwb2

¼853·6

Binary polymers

w11

¼0·638 7; m11

¼34·86 w12

¼0·621 3; m12

¼109·6

w21

¼0·361 3; m21

¼215·1 w22

¼0·378 7; m22

¼1 140

Four-component polymers

Arbitrary

w11

¼0·304 6; m11

¼23·34 w12

¼0·25; m12

¼61·90

w21

¼0·304 6; m21

¼87·10 w22

¼0·25; m22

¼231·0

w31

¼0·195 4; m31

¼71·76 w32

¼0·25; m32

¼360·8

w41

¼0·195 4; m41

¼267·8 w42

¼0·25; m42

¼1 346

Bell-shaped

w11

¼0·076 6; m11

¼10 w22

¼0·304 3; m12

¼75

w21

¼0·440 5; m21

¼50 w22

¼0·313 5; m22

¼200

w31

¼0·421 6; m31

¼125 w32

¼0·281 6; m32

¼900

w41

¼0·061 3; m41

¼400 w42

¼0·100 6; m42

¼1 600

Six-component polymers

w11

¼0·01; m11

¼5 w12

¼0·01; m12

¼50

w21

¼0·055 3; m21

¼10 w22

¼0·286 4; m22

¼75

w31

¼0·440 8; m31

¼50 w32

¼0·320 8; m32

¼200

w41

¼0·440 7; m41

¼125 w42

¼0·286 8; m42

¼900

w51

¼0·048 2; m51

¼400 w52

¼0·091 1; m52

¼1 600

w61

¼0·005; m61

¼600 w62

¼0·005; m62

¼2 000

a Sum of two Schulz–Zimm distributions (yna

¼ynb

¼1·5;

ywa

¼ywb

¼4/3).

parameter sets given in Table 1. The weight fractions ofthe two SZ functions a and b for polymer k are andwak

their values and The and of thewbk , mw mwakmwbk

. mn mwpart distributions have each been taken as identical inthe two polymers. The continuous curve in Fig. 4 rep-resents the distribution curve thus constructed forpolymer 1. Other possible representations, all at thesame and are shown in Fig. 4 and Table 1.mw , mn mw ,They refer to binary, arbitrary quaternary, and “bell-shapedÏ quaternary distributions. The concentrations ofthe two added components to form the six-component“bell-shapedÏ set in Table 1 do not show up on the scaleof the drawing. The other distributions discussed belowexhibit quite analogous patterns.

Letting weg0s \ 0É05, g0h \[15 K, g1\ 0, g2 \ 0,deÐne the system to exhibit lower critical solutionbehaviour, and g to be independent of concentration.Procedures described elsewhere21,28h36 (and also thoseused for Fig. 2) allow calculation of cloud points interms of T against the volume fraction of the/2 ,second polydisperse polymer. The result is shown inFig. 5 (set I, Ðlled circles) for two SZ distributions ineach of the two polymers.



Fig. 4. Possible molar-mass distributions for polymer 1Curve, sum of two SZ distribu-(mw \ 100, mn\ 2, mw \ 1É75).

tions ; heavy solid lines, binary polymer distribution ; heavydashed lines, arbitrary quaternary distribution ; light solid

lines, “bell-shapedÏ quaternary distribution (Table 1).

We now consider these calculated cloud points asexperimental data and proceed to test the signiÐcance ofvarious procedures for extracting interaction-parametervalues from them. The simplest analysis involves com-plete neglect of the MMDs. The system is thus treatedas if it were strictly-binary and the extremum of theCPC (now regarded as a binodal) is identiÐed with thecritical point. Setting 100 and we Ðndm1 m2\ 500,

and with eqns (3) and (4) ;g0c\ 0É010 47 /2c \ 0É309

Fig. 5. Cloud points for two hypothetical polymer blends(…)(sets I and II), calculated with eqns (1) and (5) for g0s\ ]0É05,

Polymer MMDs represented byg0h\ [15 K, g1\ 0, g2 \ 0.a sum of two SZ functions. Critical points, Curves, strictly-L.binary reproductions of cloud points with estimated values of

and lower solid curve, and lowerg0s g0h , g1\ 0 g2\ 0 ;dashed curve, and upper solid and dashedg1D 0 g2 \ 0 ;curves, corresponding predictions for set II (see text).

the latter value is clearly not in agreement with thelocation of the minimum in set I, which occurs at /2\0É25. Ignoring this deviation we estimate the criticaltemperature at 105É8¡C and use it as one calibrationpoint for (eqn 5). From, for instance, the two con-g0(T )centrations at 112É5¡C (0É005 and 0É545), using calcu-lation procedures for co-existing-phasecompositions,21,28h36 we Ðnd them not to co-exist and,hence, to yield two di†erent values for Using theg0 .average of these two values as representative for112É5¡C, we obtain g0s \ 0É980 38, g0h \[33É183 Kwith eqns (3) and (4), and calculate the lower curve inFig. 5. The deviations between the curve and the simu-lated data points (set I) are relatively large (4¡C at /2\

with a total temperature range of the cloud points0É6,of about 7¡C for and so is the di†er-0É05 ¹ /2¹ 0É6),ence between the chosen and recovered values of g0sand g0h .

Whether or not one would accept such a datadescription, it has no value unless it allows correct pre-diction of data not used in obtaining the description, ordata yet unknown. Therefore, we simulate another set ofcloud points in the same manner as set I, but for mw2\

the other characteristics being kept the same (set100,II). A binary prediction on the basis of the andg0s g0hvalues obtained above, yields the upper solid curve inFig. 5 which is very far from the truth.

In a practical situation the thermodynamics of thesystem are not known a priori, and one might beinclined to blame the discrepancy on a possible concen-tration dependence of g. Setting withg \ g0] g1/2 ,both coefficients depending on T , and using an appro-priately adapted calculation procedure,21,28h36 we Ðndthe parameter values given in Table 2. As seen in Fig. 5,the description of set I has improved greatly, at least at

and would probably be considered satisfac-/2\ 0É6,tory if no other data were available (lower dashed curvein Fig. 5). However, if other data are available (set II),we note that the prediction has hardly improved (upperdashed curve). Evidently, neglect of the MMDs cannotlead to meaningful values of the interaction parameters,a conclusion that could not be drawn on the basis of setI alone. Neither can it be concluded too easily that aconcentration dependence of s prevails.

If the MMDs are included, there are two cloud-pointconcentrations at a given T that do not represent con-jugate phases,1h3 but the values so obtained can stillg0be calibrated against T , for instance with the cloudpoints of set I. If the MMDs of the simulation itselfwere used, one obviously would then recover g0s \ 0É05,

the values chosen originally. However, weg0h\ [15 K,usually do not know the MMDs all that accurately andmight tend to use binary approximations (w

k1, mk1 ; w

k2 ,being the weight fraction of component i inm

k2 ; wki

polymer k, i \ 1, 2)) that have at least the virtue ofbeing unambiguously determined by andmw , mn mw .30A quite acceptable description of set I is obtained (Fig.



TABLE 2. Values of interaction parameters and from cloud points forg0s

g0h

various distributions

g0s

g0h

(K)

Set I

Two SZ functions (model system) 0·05 É15

Binary polymers 0·079 091 É26·033

Quaternary polymers (arbitrary) 0·066 503 É21·260

Quaternary polymers (‘bell-shaped’) 0·056 757 É17·555

Six-component polymers 0·048 979 É14·595

Sets I and II



Set I and spinodal



Set I

Single-component polymers 0·098 038 É33·183

Single-component polymers (g1D 0) g

1¼É0·041 93 ½14·993/T 0·111 32 É38·372

6, lower dashed curve) but and still deviateg0s g0hgreatly from the chosen values and, accordingly, theprediction of set II is poor (Fig. 6, curve 1).

Knowing the “correctÏ and values, we realizeg0s g0hthat the reason for the discrepancy must be sought inthe approximation of the MMDs. Sets of four com-ponents in each of the polymers yield a description of

Fig. 6. Cloud-point data of Fig. 5. Lower dashed curve : rep-resentation of set I by two binary polymers and two quatern-ary polymers. Lower solid curve : two six-componentpolymers. Curves 1È4 : predictions of set II with increasinglyaccurate approximations of the polymersÏ MMDs. Critical

points, L.

set I that can hardly be distinguished from thatobtained with the binary MMDs (Fig. 6, lower dashedcurve). The prediction of set II, however, is about 45¡Ctoo low and demonstrates that the function is stillg0(T )too far o† (Fig. 6, curve 2 ; Table 2).

In the above example, the four weight fractions of theindividual components within a constituent polymerwere calculated quite arbitrarily, and do not reÑect thebell shape of the continuous distributions used for thesimulation of cloud points (see Table 1 and Fig. 4). Wemay adapt the sets of four components so that theycome a little closer to the bell shape (at the chosen mw ,

and values, Fig. 4). The e†ect is surprising, the gapmn mwbetween simulated and predicted set II being reducedby about 50% (Fig. 6, curve 3). Also the interactionfunction derived from the cloud points of set I comesconsiderably closer to the chosen one (Table 2). Addingminute amounts (not enough to show up in Fig. 4) oftwo more components at both ends of the MMDs,retaining the rough resemblance of the bell shape, weobtain considerable further improvement, the averagedi†erence between set II and its prediction being onlyabout 3¡C (Fig. 6, curve 4). The and values areg0s g0hnow very close to those set in the example (Table 2) andthe description of set I is excellent (lower drawn curve).

This numerical exercise indicates that the extractionof signiÐcant interaction parameter values from cloudpoints is no trivial matter and calls for quite accurateinformation about the MMDs involved, probably moreaccurate than is usually available. Moreover, the inclu-sion of additional data, such as set II in the example,appears to be essential for a reliable anchoring of the

function. To decide whether all this is sufficient,g0(T )we would still need one additional set of data which weobtain as before, simulating set III for mw1\ mw2\ 500



and retaining the other distribution characteristics. Theconstants and found by Ðtting of the combinedg0s g0hsets I and II are listed in Table 2 for binary and six-component (bell shaped) MMDs. Figure 7 shows that,again, the binary MMD representation does not suffice(dashed curves), and that the six-component bell-shapedMMDs yield the required result, both in description ofsets I and II, and prediction of set III (drawn curves).

The extra data do not need to be cloud points. Mea-surements could be limited to a single blend, if the cloudpoints are then supplemented by spinodal pointsobtained with scattering methods. Spinodal points canbe simulated easily with eqn (3) ; some are shown in Fig.7 for set I. Fitting to cloud points and spinodalg0(T )points for set I alone yields quite acceptable predictionsfor sets II and III, in spite of the considerable tem-

Fig. 7. Cloud points and spinodal points for three(…) (>)polymer blends (sets I, II, III). Interaction parametersobtained from cloud-point sets I and II. Dashed curves,description of sets I and II, and prediction of set III on thebasis of binary polymer distributions ; solid curves, same asdashed curves, but with six-component MMDs; dash-dotcurves, parameters obtained from cloud-point and spinodaldata on set I ; description of set I and prediction of sets II and

III. Critical points, L.

perature extrapolations involved (Fig. 7, dash-dotcurves). The inclusion of spinodal data being so obvi-ously beneÐcial may be because of the fact that scat-tering methods “measureÏ the second derivative of *Gwith respect to concentration, whereas cloud points aredetermined by chemical potentials, i.e. the Ðrst deriv-ative. Also, within the FHS framework, the spinodaldoes not depend on details of the MMDs, as do cloudpoints, but merely on the two (eqn 6b).mw

Melting-point depression

Melting-point depressions in a partially-crystallinepolymer blend might be expected to supply an alterna-tive extra source of information. If the second polymeris the crystallizable constituent we may write for abinary FHS blend)13

(*Hf, u/R)(1/T m20 [ 1/T )

\ (ln /2)/m2 ] (1/m2[ 1/m1)/1] (g0s] g0h/T )/12

(7)

where is the molar heat of fusion per mol of crys-*Hf, utallizable repeat units in polymer 2 ; and T are theT m20equilibrium melting points of polymer 2 in its pure stateand in the blend, respectively. Figure 8 shows somesolubility curves for polymer 2 (melting point againstvolume fraction), calculated with eqn (7) for m1\ 100,

Threem2\ 100, *Hf, u\ 1000 cal mol~1, T m20 \ 400 K.possible two-phase states of the liquid mixture are con-sidered and described as in the preceding section withthree sets representing lower-critical miscibility(g0s/g0h)above 425 K, and upper-critical miscibility below 375 Kand below 405 K (Table 3).

If the liquidÈliquid critical point is close to T m20(system 2) the melting-point depressions are very smallin the full composition range and, because miscibilitygap and solubility curve interfere, there is a non-variantthree-phase equilibrium.1 The solubility curve mustassume a sigmoidal shape in order to pass throughpoints that represent the two liquid phases on the three-phase line at 399É6 K. The spinodal passes throughmaximum and minimum of the very shallow metastablesolubility curve.37

The experimental determination of equilibriummelting points in polymers is extremely difficult and a

TABLE 3. Interaction-parameter values for Fig. 8

Tc

(K) g0s

g0h

(K)

System 1 425 0·155 06 É57·402

System 2 405 É0·228 39 100·60

System 3 375 É0·093 33 42·498

System 4 – 0 0

System 5 – 0 É50



Fig. 8. Solubility curves and miscibility gaps (heavy curves)calculated with eqns (1), (3È5), (7) for *Hf, u\ 1000 cal mol~1,

and various interaction-T m20 \ 400 K m1\ 100, m2\ 100,parameter values (Table 3). Three-phase equilibrium,

light curves, spinodals ; dashed curves, metastable>È>È>;equilibria ; error bar for ^1 K, I ; critical points, L.

reduction of the probable error to ^1 K would call forexceptionally good work. Even then, the error bar onthe solubility curves is too large to allow distinctionbetween the three types of demixing, and reliable deter-mination of parameters such as and is out of theg0s g0hquestion. Even complete miscibility in the liquid state

cannot be distinguished reliably. The(g0s \ 0, g0h\ 0)only types of blend in which the melting-point depres-sion might give an indication of the magnitude of theinteraction parameter are those in which very strongspeciÐc interactions are operative (see solubility curvefor The average s values isg0s \ 0, g0h \[50 K).[0É13 at 370 \ T \ 400, whereas it amounts toonly ] 0É02 for critical miscibility.

It is remarkable that experimental melting pointscannot distinguish at all between widely varying casesof demixing such as 1 and 3, the latter referring to meta-stable demixing of the liquid state. Within the presentexamples a rough distinction can only be made betweencomplete miscibility (4) and border-case 2 at /2 \ 0É5,but this is useless in the present context.

These results do not depend on the parameter valueschosen here and must be regarded as being of generalvalidity.

CONCLUSIONS

The present analysis demonstrates that :

(a) cloud-point data on a single blend cannotproduce signiÐcant information on the inter-action parameter and its dependence on tem-perature and concentration ;

(b) two, or better three, sets of cloud points for dif-ferent molar-mass distributions are needed toobtain s(T , with conÐdence ;/2)

(c) a single blend may still supply useful informa-tion provided the cloud points are supported byadditional data of a di†erent nature, forexample, spinodal points ;

(d) neglect of the two molar-mass distributions leadsto meaningless data on s(T , /2) ;

(e) melting-point depressions do not supply signiÐ-cant information on the interaction parameter.

These conclusions refer to model data, not a†ected byexperimental errors. In an actual experimental situationsuch unavoidable errors will further detract from thesigniÐcance of blend s parameter values. Moreover, acomparison with the aid of the thermodynamic modelobeyed by the system, as available here, is not possiblein experimental practice. Finally, the attainment ofequilibrium is usually much retarded in polymer blends,which introduces another source of inaccuracy. Inter-action parameter values on blends, as reported in theliterature, should therefore be carefully scrutinizedregarding the way in which they have been obtained.

ACKNOWLEDGEMENT

The authors are indebted to Professor Walter H. Stock-mayer (Dartmouth College, Hanover, N.H.) for stimu-lating discussions.

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Diagrams, Oxford University Press, in preparation.


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Liquid-liquid phase separation in multicomponent polymer systems. XXVII. Determination of the pair-interaction function for polymer blends