J. CHEM. SOC. FARADAY TRANS., 1992, 88(11), 1561-1565 1561

Analysis of some Extrapolation Methods to derive KO for Polymers in Mixed Solvents

Vicente Soria, Clara M. Gomez, Rosa Garcia and Agusti'n Carnpos* Departament de Quimica Fisica, Universitat de Valencia, E-46100 Burjassot, Valencia, Spain

Unperturbed dimensions of a polymer have been computed from several graphical methods all based on approximate expressions for the expansion factor. These include Stockmayer-Fixman (SF) and Kurata- Stockmayer (KS) plots. However, deviations from linearity have been noticed when viscosity data for polymer- good solvent systems at large molar mass are available. These deviations are more emphasized in polymer-mixed solvent systems. Thus, a modification of t h e former equations has been proposed accounting for the dependence of t h e viscosimetric interaction parameter, €3 (computed from t h e slope), on the molar mass through t h e second virial coefficient. Second virial coefficients have been calculated from experimental data on intrinsic viscosities. Experimental data on linear polymers in mixed solvents, namely polystyrene, poly(methy1 methacrylate) and poly(dimethylsiloxane), have been used to test t h e validity of t h e modified equations. Good quantitative agreement is found between reported and experimental unperturbed dimension parameters, and a 'numerical factor', C, is shown to be important. Modified SF and KS plots show good linear correlation, even at large polymer molar masses in good solvents.

The size and shape of a polymer chain are of considerable interest to the polymer scientist, so it is very useful to know how these factors can be assessed, much of the information being derived from studies of dilute solutions. Several papers'-3 have been devoted to the study of the unperturbed dimensions of a polymer molecule chain in dilute solution, such as the root-mean-square distance between chain ends ( r i ) '/, or the root-mean-square radius of gyration ( s ~ ) ' ~ ~ . Experimental values of these magnitudes can be obtained in a 8 solvent from measurements in dilute solution, namely light scattering and osmometry where the second virial coefficient, A , , vanishes. However, in practice, a single 8 solvent (ideal solvent) is not available for every polymer, for example some water-soluble polymer^.^ To obviate this drawback, several extrapolation methods have been developed which allow esti- mation of the unperturbed dimensions in a non-ideal (good) s01vent;~ the method proposed by Stockmayer and Fixman (SF)6 directly related to the viscosity constant KO is the most useful of these. Nevertheless, the SF plot curves downward at large molar masses, M , for polymer-good solvent system^,^ namely binary polymer systems (BPSs). Moreover, curvature at low polymer molar mass is often observed owing to the non-Gaussian character of short flexible chains.'-''

A common procedure to obtain the unperturbed dimen- sions from experimental measurements in far from 8 condi- tions is to dissolve the polymer in a binary solvent, i.e. a ternary polymer system (TPS). In such cases, SF plots also deviate from linearity at large M , the deviation being more emphasized than in BPSs, and the extrapolated values of the 'unperturbed dimensions' for a given polymer are different from that computed for a single 0 solvent. KO values either higher or lower than in a single solvent have been obtained, and this is related to the excess (either positive or negative) free enthalpy of the solvent mixture.' ' , 1 2

For all these reasons, a broad range of KO data have been reported in the literature for polymers,' mostly depending upon the range of molar masses available for the extrapo- lation and on the excluded volume equation a ~ p l i e d . ~

The aim of this work is to modify two commonly used excluded volume theories : SF6 and Kurata-Stockmayer (KS),14 avoiding the curvature at large M . The key observa- tion is the decrease of the slope with increasing M when SF and KS equations are plotted, this being more emphasized in TPSs than in BPSs. Thus, the methodology followed is based

on the dependence of the parameter B, related to the slope of both SF and KS plots, on the polymer molar mass. This dependence is also more pronounced for TPSs than for BPSs. Note that, commonly, B is considered as a constant for a given polymer-solvent system.

SF and KS equations have been modified taking into account a previously reported15 expression for B as a func- tion of A , , and relating A , to the intrinsic viscosities of the binary and ternary polymer mixtures.? An in depth analysis of the intercepts (to compute KO) and slopes (to corroborate the consistency of the numerical factor, C), for both SF and KS plots, has been accomplished. In order to test the validity of the modified expressions, experimental data of linear homopolymers such as polystyrene (PS),' 6-1 poly(methy1 met hacry late) ( PMMA),'-, and poly(dimethylsi1oxane) (PDMS)23.24 in mixed solvents have been chosen.

Theory The SF plot is based on an excluded volume equation which correlates the intrinsic viscosity, ['I9 with the molar mass, M , of a polymer sample through the expression?

-- ['I - KO + 0.51@oBM'12 M I / ,

where KO is the viscosity constant in 0 conditions related to the unperturbed mean-square end-to-end distance of the polymer, ( r z ) , through K , = Qo ( ( r z ) / M ) 3 / 2 , B is a visco- metric interaction parameter accounting for long-range inter- actions, @' is the Flory-Fox factor assuming a constant value of 2.5 x (mol- ') for linear, flexible random coils,' though some objections about the universality of the factor Qo have recently been made for PS25 and PMMA.26 0.51 is a purely numerical factor and is denoted C throughout the paper. From a plot of [q ] /M"2 against M'/ , , K , can be obtained by extrapolation back to A4 = 0. However, as has been stated before for good solvent-polymer systems the plot curves downward at large M , and an 'uncertain' KO value is obtained, since the slope decreases as M increases.'g2 733 The same deviation^,^ have been noticed when the behaviour of the KS equation26 is examined. Thus, both SF and KS plots

t See eqn. (21) of J . Chem. SOC., Faraday Trans., 1992,88, 1555.

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1562 J. CHEM. SOC. FARADAY TRANS., 1992, VOL. 88

overestimate KO when only data in the range of large M are available. Several studies have focused on the modification of previous theories. By relating the viscosimetric expansion factor, a,, to the excluded volume parameter, z, Yamakawat obtained new empirical equations, namely

C‘l -- M1/2 - 1.05K0 + 0.287@0BM’/2; for O < a; < 2.5 (3)

where the numerical factor C # 0.51, and in eqn. (3) there is a coefficient for KO without physical meaning. Eqn. (2) and (3) give a reliable estimate of KO and the constant B, whereas the original SF equation underestimated B and overestimated KO. The same problem arises with different excluded volume theories. Thus, the Kurata,’ plot has been recommended for computation of B from second virial coefficients, even though different values have been obtained, and the problem of the variability of B remains un~olved.~’

On the other hand, consider a polymer solution formed from three components : solvent( l)-solvent(2)-polymer(3) (TPS). For a good solvent mixture, KO values obtained from the SF plot given by eqn. (l), conventionally named ‘apparent’ KO, will be higher than those obtained under 8 conditions, considered as ‘true’ values. This plot deviates more from linearity than in BPSs, thus a coefficient for KO higher than 1.05 and a numerical factor C lower than 0.287 [eqn. (3)] will yield reliable estimates of and B in TPSs. There is ample experimental evidence to support the devi- ation from the original plots for TPSs, for example, PDMS in a good solvent mixture, namely nonane(NONX1)-methyl ethyl ket0ne(MEK)(2)-PDMs(3)~’ at 293 K, and at volume fraction composition of mixed solvent = 0.50 has been chosen. B = 2.686 x cm3 mol g P 2 has been computed from the Kurata plot,31 through experimental [ q ] and A, data,33 and the insertion of this value into eqn. (1) yields KO = 8.4 x lo-, cm3 moll/, gP3/,. Nevertheless, a ‘true’ K , = 7.8 x l op2 cm3 g-3/2 for PDMS has been r ep~r t ed . ’~ Relating this ‘true’ value to the ‘apparent’ one and B from the Kurata plot with the one computed through eqn. (l), a modifed SF equation leads to

[‘I -- M1/2 - 1.08K0 + 0.169@0BM1/2; for 0 < a,” < 1.73 (4)

Eqn. (4) has been developed for a particular TPS, and con- tains stronger modifications than eqn. (2) and (3), and even more than the original SF equation [see eqn. (l)].

The previous results provide the framework for modifi- cation of the SF and KS equations. Note that: (a) the SF plot deviates from linearity at large M; (b) different B values have been computed through SF and Kurata plots; (c) different KO extrapolated values and a set of numerical factors C are obtained depending upon the range of M plotted, in contrast to the universality of both KO and C . For all these reasons, an approach to confirm the stated universality has been devel- oped.

Results and Discussion Accounting for the dependence of B on A, ,” namely B =

2A,/NA, where NA is Avogadro’s number, eqn. (1) reads :

[‘I 2A -- MI/, - KO + COO A M112 N A

Plotting [ v ] / M ’ / ~ against A, MI/,, KO and C can be obtained from the intercept and slope, respectively. The usefulness of eqn. (5 ) depends on the extent of the experimental knowledge of [ q ] and A, at the same solvent composition and M. It is well known that A , data for TPSs are scarce in the literature, even more so when [ q ] data for the same system are also

Thus, when only [ q ] data are available, A, can be evaluated through a recently proposed equation :

(098A[q] + 1.516 1 Y!ky[fJi3$i) (6) i = 1 . 2

where [‘Ii3 (i = 1,2) is the intrinsic viscosity of polymer (3) in the respective solvent i, A[‘] is the ‘excess’ of the intrinsic viscosity over the average of [‘Ii3 at a given volume fraction, $i (i = 1, 2) for the mixed solvents, that is A[‘] = [‘IT - [‘Il3 - $,, where [‘IT is the experimental

intrinsic viscosity of the polymeric sample in a solvent mixture. Y y denotes the interpenetration function of polymer (3) in the solvent i (i = 1, 2), assuming the approach of KY for the excluded volume. In this context, YLy can be expressed as?

( x2 --4--~41)’~0~46 - ‘1 6 . 0 4 ~ ~

(7) where

112.43

x = (*) Owing to the dependence of Yy upon through x, cal-

culation of A, from eqn. (6) cannot be directly made. To obviate this drawback, an iterative procedure has been applied to generate a self-consistent K , as follows: (1) A K, trial value is introduced into eqn. (6), from which A, is com- puted for every polymer and molar mass. (2) A new K, is computed from the extrapolation back to M = 0 in eqn. (5), with the former A, data. (3) This procedure is repeated using different K , trial values until the difference between both of the KO values (trial and calculated) is lower than 0.5%.

In Table 1, trial and calculated K , values, and the slope (denoted by P = 2C@,/NA) from eqn. (5), as well as the rela- tive KO error in YO for ethyl acetate(EA)<yclohexane(Ch)- PS,16 are gathered. In this example, the self-consistency is raised for KO = (8.13 f 0.06) x lo-, cm3 mol’l2 gP3I2, with a

Table 1 g - 3/2) for the ethyl acetate-cyclohexane-polystyrene ternary system

Trial and calculated [eqn. ( 5 ) ] KO values (in cm3 mo1’/2

lo2 K , lo2 K, (trial) (calc.) slope relative error (%)

7.07 7.23 7.38 7.54 7.69 7.84 8.00 8.15 8.30 8.45 8.61

7.29 7.41 7.53 7.66 7.79 7.92 8.08 8.13 8.15 8.25 8.35

0.439 0.439 0.439 0.439 0.438 0.438 0.437 0.437 0.436 0.436 0.435

3.1 1 2.48 2.48 2.03 1.59 1.30 1 .OO 0.25 1.81 2.37 3.02

t See eqn. (41.8) and (41.9) of ref. 7, p. 384. t See eqn. (16) of J. Chem. SOC., Faraday Trans., 1992,88, 1555.

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J. CHEM. SOC. FARADAY TRANS., 1992, VOL. 88 1563

slope P = 0.437, corresponding to a value of C = 0.52, which is very close to C = 0.51 reported by Stockmayer and Fixman in the original equation.

Fig. 1-4 depict plots of [q]M-’/’ against A 2 M 1 / ’ from eqn. (5) for EA-Ch-PS,16 Ch-dimethyl ketone(DMK)-PS,” d~decane(DoD)-MEK-PDMs’~ and acetonitrile(AcN)-( 1- chlorobutane)(ClBu)-PMMA,” respectively. A good linear correlation has been attained for every system, in contrast to the curvature observed at large M when the SF equation is plotted considering B as a constant. In Table 2, the KO and C values found from eqn. ( 5 ) for the above ternary systems, as well as for others for which plots are not shown here [decane(DEC)-M EK-PD M S, undecane(UND)-M EK- PD M S, ’ AcN-pen tyl acetate (P Ac)-PM MA,2 ’ AcN-car bon tetrachloride(CT)-PMMA” and dimethylformamide (DMF)-EA-PS”] are summarized. The average KO values are 8.07 x very close to the reported 8.05 x lop2, 5.40 x lo-’ and 7.9 x l op2 cm3 mol’/’ g-3/2 for PS, PMMA and PDMS, respectively. Note that the value KO = 10.50 x cm3 g-3/2 and C = 0.55 for DMF-EA-PS” is in considerable discrepancy with KO for PS and other C values. This discrepancy can be attributed to the high polarity of DMF, leading to specific interactions. For this reason, it has not been taken into account in order to calculate the average KO value. With

4.79 x lo-’ and 8.19 x

0.1 8

y 0.16 m I CI, c

E 0.14

0.0 0.1 0.2 0.3 A, M1l2/cm3 mol’/’ g-3/2

Fig. 1 Plot of eqn. (5) for the ethyl acetate+yclohexane-polystyrene ternary system, using [ q ] values from ref. 16 and A, calculated with eqn. (6); K , = 8.13 x lo-, cm3 mol’/2 g-312, C = 0.52

0.18

0.16

y 0.14

Y I 0)

.- - 0

,E 0.12 E *

0.10 2 s 0.08 n

0.0 0.1 0.2 0.3 A2M’ / ’ / cm3 g-3/2

Fig. 2 Plot of eqn. ( 5 ) for the cyclohexane-dimethyl ketone- polystyrene ternary system, using [q ] values from ref. 18 and A, cal- culated with eqn. (6); K , = 7.12 x lo-, cm3 mo1’I2 g-3/2, C = 0.52

i ! 0.12

0)

c - E m

E 0.10 c r

I

5 r U

0.08 [ I I I I 0.00 0.03 0.06 0.09 0.12

A,M1/ ’ /cm3 g-3/2

Fig. 3 Plot of eqn. ( 5 ) for the dodecane-methyl ethyl ketone- poly(dimethy1 siloxane) ternary system, using [ q ] values from ref. 23 and A, calculated with eqn. (6); K , = 8.18 x lod2 cm3 mol’/2 g-3/2, C = 0.51

regard to the numerical factor C, values gathered in Table 2 are in good quantitative agreement with C = 0.51.6

Other excluded volume equations have been used to evalu- ate KO values from the extrapolation procedure. In this paper, the KS e q ~ a t i o n ’ ~ has been chosen in order to apply the

0.14 t

c 0.12

c 0.10

E

E .q 0.08 c

$ 0.06

I CI,

m

r I

U

0.04 I 1

0.0 0.1 0.2 A,M1/ ’ /cm3 g-3/2

Fig. 4 Plot of eqn. ( 5 ) for the acetonitrile--(l-ch1orobutane)- poly(methy1 methacrylate) ternary system, using [ q ] values from ref. 20 and A, calculated with eqn. (6); K, = 4.73 x lo-, cm3 mol’l2 gP3I2, C = 0.51

Table 2 K, (in cm3 rno1’I2 gr3/,) and C values obtained from eqn. ( 5 ) for diverse ternary polymeric systems

ternary system lo2 K , C

EA-Ch-PS” EA-Ch-PSb

DM F-E A-PSd Ch-DMK-PS‘

AcN-ClBU-PMMA‘ AcN-CT-PM MAf AcN-PAC-PMM A‘ DEC-MEK-PDMS” UND-MEK-PDMS’ DOD-MEK-PDMS”

8.13 7.80 7.12

10.50 4.73 4.80 4.85 8.19 8.21 8.18

0.52 0.52 0.52 0.55 0.5 1 0.52 0.5 1 0.52 0.52 0.5 1

” See eqn. (21) of preceding paper. Ref. 16. ‘ Ref. 17. Ref. 18. Ref. 19. Ref, 20. Ref. 21. “ Ref. 22. ’ Ref. 23.

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1564 J. CHEM. SOC. FARADAY TRANS., 1992, VOL. 88

same methodology as for the SF equation. The KS equation can be written in the form,

where

So, following the same procedure as for the SF equation, the KS equation yields:

where C’ = 0.363 as in the original equation. The plot of [q]”’M- against A, g ( ~ r ) M ~ / ~ [ q ] - ‘ I3 , will yield Ki i3 and C’ values from the intercept and slope ( P = C’@,/N,), respec- tively.

Fig. 5-8 depict plots of eqn. (9) for the same systems as in Fig. 1-4. A good linear correlation has been attained even at large M . The computed KO and C‘ values are collected for the same TPS in Table 3. The average KO values are 7.62 x lop2, 4.74 x lop2 and 8.13 x lo-, cm3 gp3I2, slightly dif- ferent to those calculated from eqn. (5) , but within the range

0.31 - I c), ? - - z 0.27 ?

5 \ ? r 5 0.23 ? n E- Y

0.1 9 0.0 0.1 0.2 0.3 0.4 0.5

A 2g ( a,)M2/3 [ q ] - 113/cm5/3

Fig. 5 ternary system,I6 K, = 8.15 x lo-’ cm3 mol-’” g-’/’, C = 0.31

g - Plot of eqn. (9) for the ethyl acetate-cyclohexane-polystyrene

r

(5,

? 0.3

E ?

5 \ ? & 0.2 ? n E- u

0.1 0.0 0.2 0.4 0.6

A 2 g ( O ~ , , ) M ~ ‘ ~ [q] - ’ I3/cm5/j moll /3 g - ’ Fig. 6 Plot of eqn. (9) for the cyclohexane-dimethyl ketone- polystyrene ternary system,’* K , = 7.20 x lo-’ cm3 mol’/’ g-3’2, C’ = 0.31

0.25 I c),

? r - ,-’ 0.23 i?

5

2 0.21

\ ?

I

? n e u

0.19 0.0 0.2 0.4 0.6

A 2g ( c ~ , ) M ~ / ~ [ q ] - /3/cm 5/3 rnol l I 3 g -

Fig. 7 Plot of eqn. (9) for the dodecane-methyl ethyl ketone- poly(dimethylsiloxane) ternary system;23 K, = 8.20 x cm3 mol”’ g-’/’, C‘ = 0.33

0.25 r

I c),

5 0.22 z ?

--. 5 0.19

2 ? -

I

0.16 n e U

0.1 3 0.0 0.2 0.4

g - A g ( a,)M2/’ [ q ] - ’ /3/cm5/3 mol

Fig. 8 Plot of eqn. (9) for the acetonitrile-(1-ch1orobutane)- poly(methy1 methacrylate) ternary system;” K , = 4.70 x lo-’ cm3 rnoI’/’ g-3/2, C’ = 0.31

Table 3 K, (in cm3 mol’” gT3/’) and C values obtained from eqn. (9) for diverse ternary polymeric systems

ternary systems 10’ K, C‘

EA-C h-PS“ EA-Ch-PSb Ch-D M K-PS‘ DMF-EA-PSd AcN-CT-PM MA‘ AcN-ClBu-PMM A’ AcN-PAC-PMM A’ DEC-MEK-PDMS~

DOD-MEK-PDMS~ UND-MEK-PDMS’

8.15 7.50 7.20

10.50 4.68 4.70 4.85 8.00 8.20 8.20

~~

0.3 1 0.34 0.3 1 0.36 0.33 0.3 1 0.31 0.33 0.33 0.33

For footnotes see Table 2.

of reported KO. With regard to the numerical factor C , there is a mean deviation of 16% relative to the value C = 0.363 of Kurata and Stockmayer. The same discrepancy has been found when dealing with the system DMF-EA-PS.

Conclusions Two excluded volume plots, namely SF and KS, have been modified in order to obtain a unified set of values of the

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unperturbed dimensions K , and numerical factors C for TPSs. The modification is based on the dependence of the viscometric constant B on the molar mass M through the second virial coefficient. The modified equations need data on intrinsic viscosities only. Experimental data on linear (polar and non-polar) homopolymers have been used to test the validity of the modified equations. Both SF and KS modified equations give reliable estimates of the unperturbed dimen- sions and the numerical factor, avoiding the curvature of the plots at large M . Systems with a solvent mixture of high polarity yield K , values in considerable discrepancy with those obtained in 8 conditions, owing to specific interactions.

This work was partially funded by the Comision Inter- ministerial de Ciencia y Tecnologia (Spain) Grant No. MAT88-0192. We are also grateful to the Secretaria de Estado de Universidades (Spain) Grant No. OP90-0042, and to the Conselleria d’Educaci6 i Ciencia (Generalitat Valen- ciana, Spain) for financial support.

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Paper 2/00419D; Received 27th January, 1992

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