J. CHEM. SOC. FARADAY TRANS., 1994, 90(2), 339-344 339

Modified Interpenetration Function accounting for the Excluded-volume Effects in Ternary Polymer Systems

Rosa Garcia, Clara M. Gomez, lolanda Porcar, Vicente Soria and Agusth Campos' Departament de Qulinica Fisica, Facultat de Quimica, Universitat de Valencia, E-46100 Burjassot, Valencia , Spain

The excluded-volume effect in polymer-mixed-solvent systems has been investigated in the light of the two- parameter theory. These multicomponent polymer systems exhibit some specific features quite different from polymer-single-solvent systems, involving ternary interactions and the synergic action of the mixed solvents at a given composition on polymer dimensions. Both effects are taken into account in order to derive a new expression for the interpenetration ternary function, YT , and the expansion factor, a,, both being quantitatively evaluated for ternary systems based on polystyrene and poly(methy1 methacrylate). In addition, a complemen- tary treatment of the excluded-volume effects, proposed by Wolf, based on the evaluation of the intra- and inter-molecular interactions has been carried out.

The effects of excluded-volume interactions upon the trans- port and equilibrium properties of a random-coil polymer chain in solution have received considerable attention in the literat~re. '-~ One of the major reasons is that the practical use of polymer materials demands knowledge of their physi- cal properties, which are strongly dependent on intra- and inter-macromolecular repulsive interaction, closely related to the so-called excluded-volume effect^.^ Up to date, much effort has been devoted to the prediction of these effects, the two most commonly used treatments being the so-called two- parameter2 and the renormalization-group (RG) t heo r i e~ ,~ together with scaling concepts to simplify the calculation^.^

We briefly summarize how the excluded volume can affect the macromolecular dimensions. In a dilute solution of a flex- ible polymer with degree of polymerization, N , the chain expands by the excluded-volume effect and the mean-square end-to-end distance, ( r 2 ) , is proportional to N2" where v is the Flory excluded-volume exponent.' The scaling t h e ~ r y ~ . ~ describes the concentration dependence of ( r 2 ) in the semi- dilute regime, considering the unit of so-called 'blob' within which the excluded volume effect is exerted and outside which the effect is screened, thus ( r 2 ) oc g2"/(N/g) K N c ( ~ " - ' ) ' ( ' - ~ " ) , where c is the polymer concentration and g is the number of monomeric units in the 'blob'.

The main purpose of this work is the investigation of the excluded-volume effects for dilute polymer solutions in the frame of the two-parameter theory to give an explanation to many experimental findings in ternary polymer systems (TPS). The basic assumption in a two-parameter theory is the smallness of the excluded-volume interaction yielding univer- sal functions for flexible polymer chains. However, it deserves to be mentioned that there are some aspects absent from the current theory :7 (i) the assumption of two-body interactions instead of three-body ones which become important near to the @-state for polymer-polymer-solvent8~9 and for polymer- mixed-solvent ternary systems ; (ii) the polydispersity of the polymer sample is and (iii) the chain stiffness and chain-ends effects, as has been very recently described by Yamakawa and co-workers,' '-I3 must be considered in order to explain accurately the experimental findings that are inconsistent with the two-parameter theory predictions.

We present here an application of the two-parameter theory extended to TPS near to and far from @-conditions using a modified Kurata-Yamakawa interpenetration func- tion, YKY, including three-body interactions evaluated from ternary solution data, following the recently reported meth- ~ d o l o g y . ' ~ . ' ~ In addition, a complementary treatment of

YKY, considering the intra- and inter-molecular interactions, according to Wolf's suggestions, l6 is included.

Theoretical Background We proceed to show briefly the formalism of the two- parameter theory adapted to TPS. Thus, for convenience, we begin by reproducing some of the previously derived equa- tions, mainly concerning the intrinsic viscosity and second virial coefficient, that are required for the discussion. Polymer-mixed-solvent systems possess an 'excess viscosity ' defined as :

where [qli3 and [qIT are the intrinsic viscosities of polymer (3) in a pure solvent i (i = 1, 2) and in a binary solvent mixture, respectively and #i is the volume fraction of solvent i. Since [qIE comes from the transport property, it is possible to establish a relationship between this property and ther- modynamic behaviour of the solvent mixture, usually denoted by the binary interaction parameter g12 in TPS or by a more fundamental thermodynamic magnitude, such as the excess Gibbs energy GE [g124142 x GE/RT; see eqn. (4) in ref. 151. Then:

where the factor C was determined to be 0.51;17 m0 is the Flory viscosity constant (2.5 x mol-' when [q] is in ml g-'); U3 the partial specific volume of the polymer; M the weight-average molar mass of polymer; N A the Avogadro constant and V, the molar volume of solvent 1.

By combination of eqn. (1) and (2) and after some rearrangement, the following equation is obtained : l4

in which a plot of the left-hand side vs. g12 4' allows us to obtain [ q ] 2 3 from the intercept and the (1 - 2~4,) constant from the slope. Note that [qIz3 values used to be found from experimental measurements; however, the extrapolation pro- cedure proposed here is advantageous when the polymer, 3, is not soluble in the liquid, 2 (i.e. polymer-precipitant system).

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340 J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90

On the other hand, the second virial coefficient for TPS in the framework of the Flory-Huggins polymer lattice solution theory modified by Pouchly18 can be expressed as:

u; - 2v1

A --

x [41 -k s 4 2 - 2x13 41 - 2sx23 4 2

+ 2(1 - 2a,)g12 41421 (4)

where s = V1/V2 the ratio of molar volumes of solvents and x i3 (i = 1, 2) the polymer-solvent interaction parameter. For the sake of simplicity and denoting the left-hand side of eqn. ( 5 ) by X , eqn. (4) can also be written as:18

= -sx23 + - 2a,)g1241 (5)

Plotting X against g I2 41, x 2 3 and (1 - 2a,) can be obtained from the intercept and slope, respectively.

The A , for TPS expressed in eqn. (4) can be easily trans- formed into A , for binary polymer systems (BPS) when boundary conditions for solvent composition, 4i -, 1, are applied. Since in polymer-mixed solvents there are two second virial coefficients, it is convenient to denote them by A2, i 3 , and hence A2, i3 = V2,/V1(1/2 - xi3). In addition, this binary second virial coefficient can be defined, in the frame- work of the two-parameter theory, as A2, i3 = K[qIi3 Yzy, where K = 1.516NJQokf and Yky the classical Kurata- Yamakawa interpenetration function accounting for the excluded-volume effects. From these A2, i 3 expressions, there can easily be found a correlation between xi3 and [qli3, and therefore the insertion of these two expressions into eqn. (4) holds :

In general, Y can be evaluated experimentally from light scattering and viscosity measurements because Y FZ A2[q] - Assuming the nomenclature introduced in previous paper^'^,'^ for a TPS, the second virial coefficient has the following functionality: A, = KY[q]. The selection of appro- priate values for both Y and [ q ] is decisive in order to obtain reliable A , data. Often, in the past, a binary interpenetration function, Yky, has been used for both BPS and TPS in com- bination with the experimental intrinsic viscosity, thus :

A2 = K%YCZ?l* (8)

however, as has been recently proposed,14 a new functional- ity for the interpenetration function, namely YFy, works better. Thus, one can write:

(9)

Consequently, the YFY values used throughout the paper have been obtained by means of the equation:14

To sum up, eqn. (3), (5) and (7) allow us to calculate (1 - 2a,) from experimental [& data and g12 values at a given composition for each TPS.

Results and Discussion The above formalism has been applied to the following ternary systems : ethyl acetate (EA)-cyclohexane (Ch)- polystyrene (PS);" EA-Ch-PS;20 Ch-dimethyl ketone (DMK)-PS2 and 1-chlorobutane (C1Bu)-acetonitrile (AcN)- poly(methy1 methacrylate) (PMMA),22 selected from the liter- ature due to the availability of [q]T and g1223 experimental values. As has been stated in the above section, the values of these magnitudes are the input data necessary to test the validity for TPS of the proposed approach.

YT Evaluation including Ternary Interactions

Fig. 1 depicts, as an example, the plots of (a) eqn. (3), (b) eqn. ( 5 ) and (c) eqn. (7) for the EA-Ch-PS-106 system.20 As can be seen, good linear fits are obtained in all cases, allowing accu- rate determination of [ q ] 2 3 and (1 - 2a,) values from the intercepts and slopes. The same trend and behaviour is observed for the remaining systems studied (not shown here). These values are gathered in Table 1, where it seems to be reasonable to adopt the (1 - 2a,) values from eqn. (3) as ref- erence values because they have been directly determined experimentally. In general, there is good agreement between the three sets of c ~ 7 - J ~ ~ and (1 - 2a,) values for each system, except for Ch-DMK-PS where the corresponding values for the higher polymer molar masses cannot be evaluated because the plots of eqn. (3), (5) and (7) yielded poor linear-fit correlations.

60 ,

Y W + v 30t 0

I

X

+2

-0.41

-0.43

+ m

250

- 200

?L 150

Z E 6 50

'I + "00 0.1 0.2 0.3 0.4 0.5 0.6 0.7

g l z h

Fig. 1 EA-Ch mixed solvent (data taken from ref. 20)

Plot of (a) eqn. (3), (b) eqn. ( 5 ) and (c) eqn. (7) for PS-106 in

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J. CHEM. SOC. FARADAY TRANS., 1994, VOL 90 34 1

Table 1 Values of and (1 - 2u,) obtained from eqn. 3). (5) and (7) for different ternary polymer systems

[V123/dl g - ' 1 - 2a,

solvent mixture polymer" eqn. (3) eyn. ( 5 ) eqn. (7) eqn. (3) eqn. (5) eqn. (7)

EA-Ch' PS-35 15.8 15.8 15.8 0.111 0.1 11 0.1 11 PS-113 28.6 28.4 28.4 0.120 0.1 19 0.1 19 PS- 1 77 36.3 36.6 36.6 0.120 0.119 0.120 PS-379 53.9 53.8 53.9 0.109 0.108 0.108 PS-6 19 67.5 67.5 67.4 0.107 0.106 0.107

EA--Ch' PS- 106 28.7 28.5 28.8 0.120 0.120 0.1 19 PS-294 49.1 49.1 49.2 0.112 0.1 12 0.1 12 PS-420 58.8 59.0 58.8 0.108 0.107 0.108 PS-640 73.6 72.7 73.3 0.101 0.101 0.100 PS-960 90.9 90.3 90.8 0.094 0.094 0.094

Ch-DMKd PS-50.8 12.8 12.9 12.9 0.079 0.079 0.079 PS- 140 20.3 20.4 20.3 0.065 0.064 0.064 PS-270 27.4 27.5 27.3 0.079 0.079 0.079

- __ - - - PS-626 PS-870

-

- __ - - - -

CIBU-AcN' PMMA-73.4 14.1 12.9 12.9 0.105 0.104 0.104 PM MA- 124 16.3 16.0 16.0 0.111 0.1 10 0.1 10 PMMA- 189 19.2 18.9 18.8 0.102 0.101 0.102 PMMA-232 21.3 20.6 20.7 0.099 0.098 0.099

' The number indicates the weight-average molar mass in kg. * Ref. 19; ref. 10: ref. 21; ref. 22.

The availability of A , experimental data for TPS is rare; however, [qIT experimental values have often been reported for these polymeric systems. This apparent drawback to the application of our formalism can be easily obviated because the evaluation of A , starting from [qIT is workable, as has been previously demon~tra ted '~ for some TPS using eqn. (8) or (9). However, as can be seen from the preceding section,

in:pection of the (1 - 2a,) values we observed in all cases thiit they are lower than those compiled in Table 1, so that all tht: values related to them will be underestimated. The eluci- dation of the most appropriate way to calculate ( 1 - 2a,) or A ; values. to be introduced within the proposed formalism, wi 1 be further presented.

In order to visualize the differences between both 'Piy and these equations differ only in the expression of the ikerpene- tration function and therefore the calculated A , values will also be different. As an example, Fig. 2 depicts the A , values evaluated theoretically using eqn. (8) and (9), 1's. for the tetraline (TET)-Ch-PS-294 ternary showing a strong discrepancy between both sets of values. The same plots (not shown here) have been performed for the other TPS listed in Table 1. In order to corroborate that YT is a more appropriate function than Y i y , the A , values calcu- lated from eqn. (8) have been used to recalculate (1 - 2u,) and indirectly [ v ] , ~ from ~ 2 3 . These values are compiled in Table 2 for the same systems reported in Table 1. From the

Y,, quantitatively, we can plot both functions us. the-cubed radius expansion factor, x:, according to the classical plots de-ived from the two-parameter theory. Thus, in Fig. 3 Y i y (sc lid line) and YT (symbols) have been plotted for various 41 va ues, as a function of a: for the systems (a) TET-Ch-PS24 a n l (b) EA-Ch-PS2' It appears that, at a given r:, the YT va ues are always higher than the corresponding binary ones, an 1 this deviation between them increases with increasing r:.

Table 2 .4,

Values of [ q ] 2 3 and (1 - 2a,) obtained from eqn. (5) using from eqn. (8) for different ternary polymer systems

sc lvent mixture polymer" Crll23ldl g - 1 - 2a, - EA-Chb PS-35 16.5 0.08 1

PS-113 32.2 0.080 PS- 177 41.5 0.078 PS-379 62.8 0.069

" 3 PS-619 84.7 0.067 0,

EA-Ch' PS- 106 29.6 0.082 PS-294 52.8 0.072 PS-420 64.6 0.068 PS-640 80.0 0.063 PS-960 99.6 0.059

Jh-DMKd PS-50.8 4.8 0.058 PS- 140 6.2 0.045 PS-270 14.7 0.053 PS-626

0 PS-870 - -

- -

0.0 0.2 0.4 0.6 0.8 1 .o 41

'IBu-AcN' PMMA-73.4 13.5 0.08 8 PMMA- 124 15.8 0.106 PM M A- 1 89 16.8 0.078 PMMA-232 18.1 0.099

l L - A

Fig. 2 Comparison between the second virial coefficient calculated with (0) eqn. (8) and (0) eqn. (9) as a function of the solvent com- position for the PS-294 polymer sample in the TET-Ch mixed- solvent system " 4s Table 1.

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342

0.4

0.3

0.2

0.1

> Y m % 8 $-

0.3

0.2

0.1

J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90

-

-

-

v

(b 1

-

-

-

0.0 Y 0 1 2 3 4

a,”

Fig. 3 Y, (symbols) and (solid line) plotted against a: for PS in: (a) TET-Ch and (b) EA-Ch. 4l = (a) 0.80, (0) 0.60, (H) 0.40 and (0) 0.20.

This particular behaviour of the (Y,, a,”) pairs of values is generally attributed to the co-solvent feature of ternary polymer systems, as has been justified prev i~us ly’~ when g,, > 0. We believe that the thermodynamic behaviour of the solvent mixture, represented by g12 or its equivalent GE, is the main factor accounting for the discrepancies between Y, and YEy observed in Fig. 3. This speculation can be validated by Fig. 4 where it is assumed that the term on gI2 in eqn. (10) is cancelled out, and consequently a universal plot or agree- ment between Yr and YEy values is obtained, independent of solvent composition.

As can be seen in Fig. 3 the Y, values are always higher than the YE’ ones, even when using [SIT in the expression for YE’, because the variation of this magnitude with a: is unique whichever viscosity data are used. Consequently, the corresponding A , values calculated from eqn. (9) (with Yr)

0.3

0.2 Y m * z $- 0.1

0.0 0 1 2 3 4 5

a,”

Fig. 4 Y, (symbols) and YEy (solid line) plotted against a,” for PS in TET-Ch solvent mixture considered as a single liquid (gI2 = 0). Symbols as in Fig. 3.

are also higher than those obtained from eqn. (8) (with Y;’) as observed in Fig. 2.

On the other hand, when YEy is used instead of YT, the A , values obtained from eqn. (8) do not agree with the experi- mental ones. This fact can be corroborated quantitatively from the comparison of [v],~ and (1 - 2a,) values for PS and PMMA reported in Tables 1 and 2. From inspection of both sets of data, a clear disagreement is evidenced, supporting the argument that eqn. (9) and (10) are the most suitable, in the frame of the two-parameter theory, to predict the excluded- volume effects in TPS. Moreover, in contrast to the erron- eous data of Table 2, the corresponding ones compiled in Table 1 can be used to make right the interconversion of [q]T

in A , or oice uersa by means of eqn. (9) when Y, is known. In this connection, it is interesting to note that the inclusion into eqn. (3) of an additional term has been suggested, namely AKO, to account for the deviation from ideality of [q],-. However, we believe that this correction can be neglected, at least in the systems studied here, because its contribution does not substantially affect the (1 - 2a,) and [ v ] , ~ sets of values, according to the same behaviour exhibited by other TPSs and recently reported by our group.”

Y, Evaluation including Intra- and Inter-molecular Interactions

We shall now analyse the excluded-volume effect in the light of the intra- and inter-molecular interactions clearly defined by Wolf.16 In this context, both types of interactions can be related to macroscopic properties of polymer solutions, as will be pointed out below. According to the above-mentioned formalism, the required transformation for TPSs has been recently p~bl i shed , ’~ and it follows that:

(11) A - A: + b ~ ( 0 . 5 - 4

where A , and A? are the second virial coeficients of a polymer in a given solvent at finite and infinite molar mass, respectively; b and a are constants at a given 41. Both A; and b, are not accessible experimentally, but can be easily related to the intra- and inter-molecular interaction param- eters, namely xi3 and xS3, following the original nomencla- ture reported by Wolf16 (the subscript ‘M3’, exclusively affecting this section, refers to the polymer, 3, immersed in a binary solvent mixture), as:

2 -

A ? = (1 - - xh3 +xi3) 5 2 2 Vm

and

b = ( x e , x”) ;;; where V, = V, V,/(41 V, + 4, Vl), the molar volume of solvent mixture and KO and K are the viscosity constants near to and far from 8 conditions. The following values have been used for PS and PMMA polymer samples: for the first system, U 3 = 0.923 (ml g-1);23 KO = 76 x loW3 (ml g-3/2)25 and K values depending on solvent composition are reported in Table 1 from ref. 20. For the second one, U 3 = 0.805 (ml g-1),23 KO = 62 x (ml molli2 g-3/2)25 and K values are reproduced from Fig. 6 in ref. 22.

In Fig. 5, eqn. (11) has been plotted for EA-Ch-PS,’ covering the same range of molar masses as in Table 1. The A , data used have been calculated using eqn. (9) inserting the values of the viscometric exponent, a, given in Table 1 of ref. 20. A very good linear fit has been attained, which sup- ports the validity of the proposed extension of eqn. (11) to

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.I. CHEM.

N

0,

0 - E 0

5 z --.

SOC. FARADAY TRANS.. 1994. VOL. 90 343

d...

2.8 i A

A ,1

A - / - ' . '

2.4

2 .o

1.6 0.10 0.12 0.14 0.16 0.18 0.20

M(0.5 - a )

Fig. 5 Plot of eqn. ( 1 1 ) for PS at different compositions of EA--Ch solvent mixtures. 4, = (A) 0.60. (0) 0.65, (.I 0.70. (C) 0.75 and ( * I 0.80.

TPSs, allowing accurate evaluation of A; and tt values from the intercept and slope.

Because all the magnitudes involved in eqn. (12) and 13) are known. we can now evaluate the intra- and the inter- molecular interactions affecting a polymer coil in a binar) solvent mixture, namely xi3 and x k 3 . Both z i 3 and x',? values are plotted in Fig. 6 where the intra- cs. intcr- molecular interaction parameters are depicted for binar) systems" (solid line) and for polymer-mixed-solvent systems (symbols). These last results are clearly in conflict with the universal character exhibited in BPS because, as can be easil! seen, very good correlation is associated with BPS data whereas for TPS no correlation is observed, and most of the data are positioned to the right of the solid line. In order t o give a theoretical explanation of this behaviour. we have assumed that the intramolecular interactions of a macro- molecular chain under the conditions reported here, are the same for BPS and TPS, that is x0 = xi3, and no dependence on solvent composition is considered. This assumption. strictly speaking, is suitable for uncharged homopolymers in

solvents of very low polarity. In this way, Fig. 7(a) shows the solvent dependence of the difference - xi3 which is a measure of the deviation between binary and ternary data. for the same systems as in Fig. 6. Convex upward curves are clearly evidenced in Ch-DMK-PS and CIBu-AcN-PMM A systems; however, a smoothed curve is obtained for the system EA-Ch- PS. A similar but more continuous function- ality, in the purely thermodynamic sense, corresponds to the solvent mixture composition dependence of GE:'RT. In order

0.8

3n=

0.4 I 0.2s 0.35 0.45 0.55 0.65

XI' or j(h3 Fig. 6 Dependence of the intramolecular interaction parameters 2 ' ' and x G 3 on the intermolecular interaction parameters z'' or ~ 6 , ~ . Solid line BPS; (A) EA-Ch-PS: (0) Ch-DMK-PS: (CI CIBu-AcN- PMMA.

0.05 I

c \

0.0 0 0.2 0.4 0.6 0.8 1

d1 Fit:. 7 Variation of (a ) mi iture composition. Symbols as in Fig. 6.

- & and ( h ) C;" RT with the solkent

to compare both functions. this last dependence has been pltltted in Fig. 7 ( h ) for the same TPSs. where a certain coin- cicence on the placement of the maximum of the respective cu .\es can be observed, which could support the idea that the sh,ft from ideality of the solvent mixture is mostly responsible foi the discrepancies between z)' and zL3 values. The connec- tion between both magnitudes can be established in the framework of the Flory-Huggins theory of polymer solutions co ipled to TPS at infinite molar mass. I n this regard, the second virial coeflicient A; can be defined as A; = ( 1 2 - l&)i;S, Ck where zG3 is the polymer-solvent mixture inter-

aci ion parameter at infinite molar mass. The relationship be* ween the phenomenological interaction parameter &.3

anat the xC3 and x E 3 ones can be described by using zh3 = [ x : , . ~ + %E3]!2. On the other hand. z,& has also been defined a s It'

wh;.re the last term of the right-hand side of eqn. (14) involves thc binary parameter g I 2 proportional to G'IRT [see eqn. (4) fro n ref. 15). Consequently, the magnitudes on the ordinates of =ig. 7 are closely related and hence a maximum in GE-.'R7' deliotes the same extreme condition for y 1 2 , so that xh3 acc.uires a minimum value. Following our argument, when z- is a minimum xL3 will also be a minimum and the differ- ente ( X I ' - ~ 5 , ) will be a maximum. This explanation is sup- po'ted by the dependence of both molecular interaction pal ameters and excess Gibbs energy on solvent composition, as +hewn in Fig. 7.

I 'inally, we expand somewhat the previous analysis14*' on Y, in order to show explicitly the contributions of intra- and intl:r-molecular excluded-volume interactions to the interpen- etr.ttion function as well as its dependence on solvent com- po!ition, d l . For this purpose, it is necessary to derive an exrression relating YT and the lI3 and l k 3 parameters, by coribination of eqn. (15) and (16) of our preceding paper." yie ding:

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0, .-. - -0- -0- -0- - , - 9. 6 '0

$- 0.25 -

0.20 - * * * * *

o . 1 5 + ' 1 " ' 1 ' 1 ' -

J. CHEM. SOC. FARADAY TRANS., 1994, VOL. 90

0.35 1

Note that we have written for convenience a normalized interpenetration function for the ratio YT/"iy because YEy is a constant at a given a:, and hence the functionality of YT will remain unchanged. Fig. 8 depicts the plot of YT, from eqn. (lo), us. 41 for various a: for the EA-Ch-PS2' system. As can be seen, there is no linear dependence of YT on +1

and the curves connecting the data points are convex in all cases. However, it is increasingly difficult to select the 41 value where the maximum takes place as a: decreases, approaching the unperturbed state (a: = 1). This observed YT dependence on is not considered in the original two- parameter theory or in the RG theory, where a single com- posite curve is postulated, therefore we believe that it is an important goal to seek an explanation of this behaviour of the excluded volume in TPS in the framework of the two- parameter treatment. With this aim, focussing our attention on eqn. (l5), it seems that the difference (xi, - xb,) as a func- tion of solvent composition exhibits a maximum by analogy with that shown in Fig. 7(a). In contrast, VJV, obviously depends on c $ ~ but does not show any extrema condition (maximum or minimum) at least for the TPSs tested here; and the remaining parameters of eqn. (15) are not dependent on solvent composition. Thus, it may be concluded that the observed behaviour of YT in Fig. 8 cannot be well explained quantitatively using the extended Wolf formalism for TPS. In order to clarify this unsolved problem, a similar study is now in progress starting with the Y and a dependence on a modi- fied scaled excluded-volume parameter, z, especially for polymer-mixed-solvent systems. Much experimental and theoretical work still needs to be done.

Conclusions A recently proposed formalism dealing with the excluded- volume effects in polymer-mixed-solvent systems has been tested using solution data reported by other author^'^-^^ for PS and PMMA. The new functionality proposed for the Kurata-Yamakawa interpenetration function has been shown to be better than the classical one for reproducing the experimental intrinsic viscosities and second virial coefficients for polymers in a solvent mixture.

The importance of the ternary interactions contributions is noted and the a, parameter has been exhaustively and quan- titatively evaluated for PS and PMMA, covering a whole range of molar mass and solvent composition. Moreover, because one component of the solvent mixture behaves as a non-solvent (precipitant), a virtual intrinsic viscosity, [ q ] 2 3 , for the above samples has also been calculated by means of eqn. (3), (5) and (7) (see Table l), showing good agreement between the values which supports the proposed formalism.

The values of YT for the systems studied are always greater than the corresponding ones evaluated using the classical interpenetration function performed for binary polymer solu- tions, Yi'. In addition, the YT values are strongly dependent on solvent composition, in contrast with the universal char- acter claimed by the original two-parameter theory (see Fig. 3 and 8). In the light of eqn. (10) it is plausible to believe that the g12 term, closely related to the excess Gibbs energy, GE, is responsible for the observed differences between both interpenetration functions. In fact, when the single liquid approximation is done, g I 2 = 0, a universal function is found (see Fig. 4).

An alternative analysis of the excluded volume in TPS has been carried out by means of an extension of the formalism developed by Wolf16 for BPS. The intra- and inter-molecular parameters have been evaluated; however, we believe that this formalism does not give a good quantitative explanation of the YT behaviour.

This work has been partially funded by the Direccion General de Investigacion Cientifica y Tecnica (Ministerio de Educacion y Ciencia, Spain) Grant No. PB91-0808. One of us (I.P.) is indebted to Ministerio de Educacion y Ciencia (Spain) for a predoctoral long-term fellowship.

References 1

2

3

4

5 6 7 8 9

10 11

12

13

14

15

16 17

18 19

20

21

22

23

24 25

26

P. J. Flory, in Principles of Polymer Chemistry, Cornell Uni- versity Press, Ithaca, New York, 1953. H. Yamakawa, in Modern Theory of Polymer Solutions, Harper and Row, New York, 1971. P. G. De Gennes, in Scaling Concepts in Polymer Physics, Cornell University Press, Ithaca, New York, 1979. K. F. Freed, in Normalization Group of Macromolecules, Wiley, New York, 1987. K. F. Freed, Acc. Chem. Res., 1985, 18,38. M. Daoud and G. Jannink, J . Phys. (Paris), 1976,37,973. Z. Y. Chen and J. Noolandy, Macromolecules, 1992,2S, 4978. L. Schafer and C. Kappeler, Colloid. Polym. Sci., 1990,268,995. Y. Guan, T. H. Lilley and T. E. Treffry, Macromolecules, 1993, 26, 3971. C. H. Kang and S. I. Sandler, Macromolecules, 1988,21, 3088. F. Abe, Y. Einaga, T. Yoshizaki and H. Yamakawa, Macro- molecules, 1993, 26, 1884. F. Abe, Y. Einaga and H. Yamakawa, Macromolecules, 1993, 26, 1821. H. Yamakawa, F. Abe and Y. Einaga, Macromolecules, 1993,26, 1898. V. Soria, C. M. Gomez, R. Garcia and A. Campos, J . Chem. Soc., Faraday Trans., 1992,88, 1555. C. M. Gomez, V. Soria, J. E. Figueruelo and A. Campos, J. Chem. SOC., Faraday Trans., 1993,89, 1765. B. A. Wolf, Macromolecules, 1985, 18,2474. W. H. Stockmayer and M. J. Fixman, J . Polym. Sci., Part C , 1963, 1, 137. J. Pouchly and A. Zivny, Makromol. Chem., 1983,184,2081. P. Munk, M. T. Abijaoude and M. E. Halbrook, J . Polym. Sci., Polym. Phys., 1978, 16, 105. A-A. A. Abdel Azim, S. S. Moustafa, M. M. El Dessouky, F. A- Rehim and S. A. Hassan, Polymer, 1986,27, 1406. H. Maillols, L. Bardet and S. Grom, Eur. Polym. J., 1978, 14, 101 5. M. G. Prolongo, R. M. Masegosa, I. Hernandez-Fuentes and A. Horta, Macromolecules, 1981, 14, 1526. A. Campos, R. Gavara, R. Tejero, C. Gomez and B. Celda, J . Polym. Sci., Polym. Phys., 1989, 27, 1599. A-A. A. Abdel Azim and M. B. H u g h , Polymer, 1983, 24, 1429. Polymer Handbook, ed. J. Bandrup and E. H. Immergut, Wiley Interscience, New York, 1975. A. Campos, B. Celda, J. Mora and J. E. Figueruelo, Polymer, 1984,25, 1479.

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