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Effect of polydispersity on huggins coefficient of polystyrene solutions
Eur. Polym. J. Vol. 25, No. 6, pp. 567-569, 1989
0014-3057/89 $3.00 +0.00 Copyright ~ 1989 Pergamon Press pie
Printed in Great Britain. All rights reserved
EFFECT OF POLYDISPERSITY ON HUGGINS COEFFICIENT OF POLYSTYRENE SOLUTIONS K. K. Cr~E Department of Chemistry, University of Malaya, 59100 Kuala Lumpur, Malaysia (Received 26 September 1988)
Abstract--A recent model for the variation of the Huggins coefficient, k, with the molecular weight of a polymer is modified by considering the effect of polydispersity. It is shown that the value of k for polymer solutions is lowered significantly by the existence of a molecular weight distribution. The proposed equation is applicable to whole polymers of styrene in a wide variety of solvents.
[,I] = y w,[.]~
INTRODUCTION
(6)
i
The Huggins coefficient, k, for random coil polymers [1] is defined by
rl~p/C = [r/]
+
k[r/]2C,
and
(1) with C~ being the concentration of the ith polymeric species with intrinsic viscosity [r/]~ and Huggins coefficient k~, and w~being its weight fraction equal to the ratio of C, and C. Hereafter, the Huggins coefficients for monodisperse and polydisperse polymers are designated by k and ~, respectively. The denominator on the r.h.s, of equation (7) serves to normalize the variable w~. Introduction of equation (1) into equation (4) for the ith polymeric species with molecular weight M r leads to
where [7] is the intrinsic viscosity, and r/sp = ~//r/o- 1,
(2)
with q and r/0 being, respectively, the zero-shear viscosities of the polymer solution with concentration C, and the pure solvent. According to equation (1), the parameters [r/] and k are accessible by a linear plot of rlso/C against C. These quantities may be used to determine the molecular weight averages, unperturbed dimensions as well as the thermodynamic properties of the polymer solutions [2-5]. Several theories have been proposed for the k coefficient in both poor and good solvents [6-12]. Among them, a recent model of Gundert and Wolf [12] is of particular interest because of its theoretical origin and simplicity. By considering the uneven probability of inter- and intramolecular contacts between polymer segments, which are susceptible to solvent dilution, they obtain k = a o + bo[q]o/[q],
(8)
= a + t'[~]0/[~], where ?t = ao
• i Mi I
and 2 12+a T,=bo~w,M, i
Mv
(9)
wi i
l\
I
/( g , Mo Ew, ) . --a
- - ,'2
/\
2
i
(10)
/
Here, the viscometric parameters appearing in equations (9) and (10) are related to the well-known Mark-Houwink-Sakurada (MHS) relation given as
(3)
where a0 and b0 are the characteristic parameters of the model, and [r/]0 is the intrinsic viscosity under theta conditions. This particular approach and other contemporary theoretical interpretations of k deal exclusively with monodisperse polymers. However, in reality, there always exists a distribution of molecular weights in a polymer sample. In the present study, the influence of the polydispersity factor on k is pursued by using equation (3).
[r/] = K~S¢~,
(11 )
where K and a are empirical constants, and ~v is the viscosity-average molecular weight. Under theta conditions, we have a = 1/2, K = Ko and h,tv = )~0. Combining equations (1) and (8) results in a general expression correlating k and k, viz. = 7 k - (7 - 3)a0,
HUGGINS COEFFICIENT
(12)
where
OF POLYDISPERSE POLYMERS
Equation (1) can be readily adapted to polydisperse systems, yielding
~l~p/C = [rl]
+ ~:[r/]2C,
(4)
where
c = E c,,
(5)
i
567 EPA 25~
3 = fi/ao
(13)
7 = b/bo.
(14)
and In order to compare fc with k numerically, we first proceed to evaluate the factors/~ and 7. Applying the general form of Schulz's molecular weight distribution function [13] to equations (9) and
568
K.K. Crme 0.6
(a) 1.0
-g 0.5
~" 0.8
0.5
0.600.7 0.6
0.4
e ~
e
0.6 I
I
I
I
:(b)
I
oo~,ee
o
/
0.8
0.3
0.7 N
0
I t0
I
I I l I I I I I I 12 14 16 16 20
[r/]-~ X 103 ( g / m U
2
0.6
x
Fig. 2. Linear plot of k against [q ]-J for a PS fraction in a range of solvents [16].
m. i>,.
I °I 8
1
0.5 I
I
2
4
I
I
I
6
8
I0
infinity. Certainly, the present analysis is best tested by the relevant experimental data as elaborated below.
U
Fig. 1. Plots of (a) 7 and (b) (7 - fl) against U for a = 0.5, 0.6, 0.7 and 0.8 based on the Schulz molecular weight distribution function.
(10) yields
fl=2_~F(2;(2b2b+3)~ +3)
F(b+2)72
Lr~7_-g~z)j
(15)
and
), = 2-ol2 +al F(a + 2b + 3.5)[F(b + 2)]2 F(2b + 3 ) r ( b + 2.5)F(a + b + 2)'
(16)
where F(x) is the gamma function of x and b=
(2 - U)/(U - 1)
(17)
with U being the polydispersity index defined as the ratio of weight-average and number-average molecular weights. Figure 1 displays the polydispersity dependence of ~ and (7 - f l ) resulting from equations (15) and (16) at four levels of solvent quality. Clearly, equation (12) indicates that k is always smaller than the corresponding k if a0 t> 0, except for the ideal monodisperse case where U = 1 and k = k. The difference between them is greatly enhanced by increasing the degree of polydispersity and the goodness of solvent. However, the rate of this depression seems to diminish appreciably for very broad molecular weight distributions. Under theta conditions, the ratio of k and k is exactly equal to y, which in turn drops from unity to 0.85 at U = 2, and eventually tapers off at ~0.68 when the breadth of distribution approaches
HUGGINS COEFFICIENT OF POLYSTYRENE SOLUTIONS
The viscometric behaviour of dilute polystyrene (PS) solutions has been extensively investigated in the literature. According to equation (3), the model parameters a0 and b0 may be obtained from a linear plot of k against [q ]0/[t/] using well-fractionated polymer samples. In the present study, we take Ko = 0.083 ml/g [14] for PS to facilitate the estimation of [r/]0. Table 1 lists the estimates of a0 and b0 for PS fractions with molecular weights less than 3.1 x 10 6 in various good solvents obtained from the reliable data provided by Gundert and Wolf [12] and Einaga et al. [15]. In any event, the values of the correlation coefficient, R, are close to unity as shown in Table 1, confirming the linearity of equation (3). Unfortunately, these a0 and b0 data are rather scattered over a range of the MHS exponent a varying from 0.60 to 0.75. Attempts at deriving more information on this pair of coefficients corresponding to poor and intermediate solvents are unfruitful, as the literature data available hitherto are limited and do not conform to equation (3). However, this adverse situation may be overcome by the work of Streeter and Boyer [16] who have employed equation (1) to analyse the viscosity data of a PS fraction in a series of solvents. The value of [q ]0 of the sample is 50.5 ml/g, and the MHS exponent a of the PS solutions ranges from 0.50 to 0.76 [4]. In this connection, a plot of k against [t/]- J is linear with R = 0.9779 as shown in Fig. 2. This means that equation (3) is indeed valid for these solutions which differ considerably in terms of the solvent goodness. Hence, in the present context, a 0 and b0 are assumed to be indepen-
Table I. Viscometric parameters of polystrene solutions No.
Solvent/temperature
a
ao
bo
R*
t
Ref.
l 2 3 4 5 6
Cyclohexane/50° Carbon tetrachloride/20 ° Carbon tetrachloride/50 ° Ethylbcnzene/25° Ethylbenzene/65° Benzene/25 °
0.60 0.72 0.73 0.74 0.74 0.75
0.100 0.133 0.178 0.179 0.230 0.154
0.570 0.509 0.390 0.358 0.253 0.459
0.9789 0.9530 0.9413 0.9534 0.9668 0.9223
6 6 6 6 6 9
[12] [12l [12l [12] [12] [14]
*Correlation coefficient; ttotal number of data points analysed.
Effect of polydispersity on PS solutions 0.9
I1 0.7' "5
24a 08
0 05 0.4
O.2
-
I
I
I
I
I
I
0.3
0,4
0.5
0.6
0.7
0.8
O
Fig. 3. Plots of Huggins coefficient against the M H S
exponent a. Curves I, 11 and 11I refer to k and k-computed by equations (3) and (8) based on U = 1.0, 1.5 and 2.0, respectively. The data points are obtained from a whole polystyrene in various solvents designated by the numerical codes as follows [17]: (1) tetrahydrofuran (THF); (2) benzene; (3) chloroform; (4) carbon tetrachloride; (5) toluene; (6) cyclohexanone; (7) dichloromethane; (8) trichloroethylene; (9) dioxane; (10) xylene; (11) 1,2-dichloroethane; (12) THF-water (91.4:8.6); (13) butyl acetate; (14) butanone; (15) ethyl acetate; (16) methyl isobutyl ketone; (17), (18) toluene-methanol (76.9:23.1); (19), (22), (24) cyclohexane; (20) cyclohexanone-acetone (20:80); (21) diisobutyl ketone; and (23) methyl acetate. The temperatures are: (Q) 25°; (A) 27c'; (V) 28°; (/k) 30°; (V) 34°; and (©) 35 o.
dent of the thermodynamic quality of the solvent. Table 1 provides the average values of a0 and b0 respectively equal to 0.16 and 0.42, with their respective standard deviations being 0.04 and 0.11. These results are parallel to those generated by Fig. 2, which gives a0 = 0.18 and b0 = 0.40. As a result, the foregoing arithmetic means of the two model parameters are considered to be appropriate for PS in all solvents, and as such they are utilized for the ensuing analysis. A list of the Huggins coefficients and intrinsic viscosity data associated with a whole polymer of styrene has been collected by Dort [17]. This particular polymer was prepared by radical polymerization terminated by coupling, and its h4"0 was found to be 2 x 105. Hence, it may be adequately described by a Schulz's molecular weight distribution function [18, 19]. This would allow us to apply equation (8) to determine fc via equations (15) and (16) by setting b or U equal to a particular value. In this connection, we have U = 1.5 for the chain coupling vinyl polymerizaton. However, chain transfer, high conversion and autoacceleration effects would broaden the molecular weight distribution appreciably, leading to higher U. The most probable distribution corresponds to U = 2.0. Here, the MHS exponent a varying from 0.30 to 0.75 is computed by an iterative procedure established recently [4]. The value of k is readily obtained by equation (3).
569
Curves I, II and III in Fig. 3 represent the estimated k and J~ based on U = 1.0, 1.5 and 2.0 respectively, as functions of a. Also, the data points reported by Dort [17] are shown in Fig. 3. As expected, J~is consistently less than k over a wide range of the MHS exponent a. Specifically, the k value based on a monodisperse PS is reduced markedly by 20% and 12% in an extremely good solvent with a = 0.80 and a theta solvent respectively, due to a twofold increase in the polydispersity index. More important, curves II and III are closer to all the experimental observations, except for a few cases where curve I fits better to data points (8), (17) and (20) derived from trichloroethylene and some mixed solvents. Statistically, the standard errors of estimate o f k on a are estimated to be 0.053 (curve II), and 0.048 (curve III), whereas that of k on a is 0.091. Although exact information on the molecular weight distribution of the PS sample is unavailable, it is not unreasonable to assume 1.5 < U ~< 2.0 considering the possible termination modes and polymerization conditions. Hence, the estimated k, which allows for the polydispersity effect, is strikingly more satisfactory for the present systems of interest. This implies that, in practice, equation (8) is superior to equation (3) in predicting the Huggins coefficient of polymers, which basically are mixtures of molecules with different sizes. In conclusion, the effect of polydispersity on the Huggins coefficient is indeed noticeable, even in poor solvents. For example, referring to Fig. 3, the predicted values of k and k (curve III) are respectively 0.58 and 0.50 under theta conditions. Incidentally, the former datum approximates to the result of the theoretical calculation of Riseman and Ullman [20]. REFERENCES
1. 2. 3. 4. 5.
M. L. Huggins. J. Am. chem. Soc. 64, 2716 (1942). K. K. Chee. J. appl. Polym. Sci. 27, 1675 (1982). K. K. Chee. J. appl. Polym. Sci. 30, 1359 (1985). K. K. Chee. Polymer 2.8, 977 (1987), N. Sutterlin. In Polymer Handbook (Edited by J. Brandrup and E. H. Immergut), Chap. IV. Interscience, New York (1975). 6. M. Bohdanecky and J. Kovar. Viscosity of Polymer Solutions. Elsevier, Amsterdam (1982). 7. H, L. Bhatnagar, A. B. Biswas and M. K. Gharpurey. J. chem. Phys. 2,8, 88 (1958). 8. S. Imai. Proc. R. Soc. (Lond.) A308, 497 (1969). 9. M. Bohdanecky. Coll. Czech. Chem. Commun. 35, 1972 (1970). 10. M. Muthukumar and K. F. Freed. Macromolecules 10, 899 (1977). 11. F. A. H. Peeters and A. J. Staverman. Macromolecules 10, 1164 (1977). 12. F, Gundert and B. A. Wolf. Makromolek. Chem. 187, 2969 (1986). 13. L.H. Tung. In Polymer Fractionation (Edited by M. J. R. Cantow), Chap. E. Academic Press, New York (1967). 14. K. K. Chee. Polymer (Commun.) 27, 135 (1986). 15. Y. Einaga, Y. Miyaki and H. Fujita. J. Polym. Sci.; Polym. Phys. Edn 17, 2103 (1979). 16. D. J. Streeter and R. F. Boyer. Ind. Engng Chem. 43, 1790 (1951). 17. I. Dort. Polymer 29, 490 (1988). 18. G. Odian. Principles of Polymerization. McGraw-Hill, New York (1970). 19. K. K. Chee. Polymer 26, 581 (1986). 20. L Riseman and R. UUman. J. chem. Phys. 19, 578 (1951).
0014-3057/89 $3.00 +0.00 Copyright ~ 1989 Pergamon Press pie
Printed in Great Britain. All rights reserved
EFFECT OF POLYDISPERSITY ON HUGGINS COEFFICIENT OF POLYSTYRENE SOLUTIONS K. K. Cr~E Department of Chemistry, University of Malaya, 59100 Kuala Lumpur, Malaysia (Received 26 September 1988)
Abstract--A recent model for the variation of the Huggins coefficient, k, with the molecular weight of a polymer is modified by considering the effect of polydispersity. It is shown that the value of k for polymer solutions is lowered significantly by the existence of a molecular weight distribution. The proposed equation is applicable to whole polymers of styrene in a wide variety of solvents.
[,I] = y w,[.]~
INTRODUCTION
(6)
i
The Huggins coefficient, k, for random coil polymers [1] is defined by
rl~p/C = [r/]
+
k[r/]2C,
and
(1) with C~ being the concentration of the ith polymeric species with intrinsic viscosity [r/]~ and Huggins coefficient k~, and w~being its weight fraction equal to the ratio of C, and C. Hereafter, the Huggins coefficients for monodisperse and polydisperse polymers are designated by k and ~, respectively. The denominator on the r.h.s, of equation (7) serves to normalize the variable w~. Introduction of equation (1) into equation (4) for the ith polymeric species with molecular weight M r leads to
where [7] is the intrinsic viscosity, and r/sp = ~//r/o- 1,
(2)
with q and r/0 being, respectively, the zero-shear viscosities of the polymer solution with concentration C, and the pure solvent. According to equation (1), the parameters [r/] and k are accessible by a linear plot of rlso/C against C. These quantities may be used to determine the molecular weight averages, unperturbed dimensions as well as the thermodynamic properties of the polymer solutions [2-5]. Several theories have been proposed for the k coefficient in both poor and good solvents [6-12]. Among them, a recent model of Gundert and Wolf [12] is of particular interest because of its theoretical origin and simplicity. By considering the uneven probability of inter- and intramolecular contacts between polymer segments, which are susceptible to solvent dilution, they obtain k = a o + bo[q]o/[q],
(8)
= a + t'[~]0/[~], where ?t = ao
• i Mi I
and 2 12+a T,=bo~w,M, i
Mv
(9)
wi i
l\
I
/( g , Mo Ew, ) . --a
- - ,'2
/\
2
i
(10)
/
Here, the viscometric parameters appearing in equations (9) and (10) are related to the well-known Mark-Houwink-Sakurada (MHS) relation given as
(3)
where a0 and b0 are the characteristic parameters of the model, and [r/]0 is the intrinsic viscosity under theta conditions. This particular approach and other contemporary theoretical interpretations of k deal exclusively with monodisperse polymers. However, in reality, there always exists a distribution of molecular weights in a polymer sample. In the present study, the influence of the polydispersity factor on k is pursued by using equation (3).
[r/] = K~S¢~,
(11 )
where K and a are empirical constants, and ~v is the viscosity-average molecular weight. Under theta conditions, we have a = 1/2, K = Ko and h,tv = )~0. Combining equations (1) and (8) results in a general expression correlating k and k, viz. = 7 k - (7 - 3)a0,
HUGGINS COEFFICIENT
(12)
where
OF POLYDISPERSE POLYMERS
Equation (1) can be readily adapted to polydisperse systems, yielding
~l~p/C = [rl]
+ ~:[r/]2C,
(4)
where
c = E c,,
(5)
i
567 EPA 25~
3 = fi/ao
(13)
7 = b/bo.
(14)
and In order to compare fc with k numerically, we first proceed to evaluate the factors/~ and 7. Applying the general form of Schulz's molecular weight distribution function [13] to equations (9) and
568
K.K. Crme 0.6
(a) 1.0
-g 0.5
~" 0.8
0.5
0.600.7 0.6
0.4
e ~
e
0.6 I
I
I
I
:(b)
I
oo~,ee
o
/
0.8
0.3
0.7 N
0
I t0
I
I I l I I I I I I 12 14 16 16 20
[r/]-~ X 103 ( g / m U
2
0.6
x
Fig. 2. Linear plot of k against [q ]-J for a PS fraction in a range of solvents [16].
m. i>,.
I °I 8
1
0.5 I
I
2
4
I
I
I
6
8
I0
infinity. Certainly, the present analysis is best tested by the relevant experimental data as elaborated below.
U
Fig. 1. Plots of (a) 7 and (b) (7 - fl) against U for a = 0.5, 0.6, 0.7 and 0.8 based on the Schulz molecular weight distribution function.
(10) yields
fl=2_~F(2;(2b2b+3)~ +3)
F(b+2)72
Lr~7_-g~z)j
(15)
and
), = 2-ol2 +al F(a + 2b + 3.5)[F(b + 2)]2 F(2b + 3 ) r ( b + 2.5)F(a + b + 2)'
(16)
where F(x) is the gamma function of x and b=
(2 - U)/(U - 1)
(17)
with U being the polydispersity index defined as the ratio of weight-average and number-average molecular weights. Figure 1 displays the polydispersity dependence of ~ and (7 - f l ) resulting from equations (15) and (16) at four levels of solvent quality. Clearly, equation (12) indicates that k is always smaller than the corresponding k if a0 t> 0, except for the ideal monodisperse case where U = 1 and k = k. The difference between them is greatly enhanced by increasing the degree of polydispersity and the goodness of solvent. However, the rate of this depression seems to diminish appreciably for very broad molecular weight distributions. Under theta conditions, the ratio of k and k is exactly equal to y, which in turn drops from unity to 0.85 at U = 2, and eventually tapers off at ~0.68 when the breadth of distribution approaches
HUGGINS COEFFICIENT OF POLYSTYRENE SOLUTIONS
The viscometric behaviour of dilute polystyrene (PS) solutions has been extensively investigated in the literature. According to equation (3), the model parameters a0 and b0 may be obtained from a linear plot of k against [q ]0/[t/] using well-fractionated polymer samples. In the present study, we take Ko = 0.083 ml/g [14] for PS to facilitate the estimation of [r/]0. Table 1 lists the estimates of a0 and b0 for PS fractions with molecular weights less than 3.1 x 10 6 in various good solvents obtained from the reliable data provided by Gundert and Wolf [12] and Einaga et al. [15]. In any event, the values of the correlation coefficient, R, are close to unity as shown in Table 1, confirming the linearity of equation (3). Unfortunately, these a0 and b0 data are rather scattered over a range of the MHS exponent a varying from 0.60 to 0.75. Attempts at deriving more information on this pair of coefficients corresponding to poor and intermediate solvents are unfruitful, as the literature data available hitherto are limited and do not conform to equation (3). However, this adverse situation may be overcome by the work of Streeter and Boyer [16] who have employed equation (1) to analyse the viscosity data of a PS fraction in a series of solvents. The value of [q ]0 of the sample is 50.5 ml/g, and the MHS exponent a of the PS solutions ranges from 0.50 to 0.76 [4]. In this connection, a plot of k against [t/]- J is linear with R = 0.9779 as shown in Fig. 2. This means that equation (3) is indeed valid for these solutions which differ considerably in terms of the solvent goodness. Hence, in the present context, a 0 and b0 are assumed to be indepen-
Table I. Viscometric parameters of polystrene solutions No.
Solvent/temperature
a
ao
bo
R*
t
Ref.
l 2 3 4 5 6
Cyclohexane/50° Carbon tetrachloride/20 ° Carbon tetrachloride/50 ° Ethylbcnzene/25° Ethylbenzene/65° Benzene/25 °
0.60 0.72 0.73 0.74 0.74 0.75
0.100 0.133 0.178 0.179 0.230 0.154
0.570 0.509 0.390 0.358 0.253 0.459
0.9789 0.9530 0.9413 0.9534 0.9668 0.9223
6 6 6 6 6 9
[12] [12l [12l [12] [12] [14]
*Correlation coefficient; ttotal number of data points analysed.
Effect of polydispersity on PS solutions 0.9
I1 0.7' "5
24a 08
0 05 0.4
O.2
-
I
I
I
I
I
I
0.3
0,4
0.5
0.6
0.7
0.8
O
Fig. 3. Plots of Huggins coefficient against the M H S
exponent a. Curves I, 11 and 11I refer to k and k-computed by equations (3) and (8) based on U = 1.0, 1.5 and 2.0, respectively. The data points are obtained from a whole polystyrene in various solvents designated by the numerical codes as follows [17]: (1) tetrahydrofuran (THF); (2) benzene; (3) chloroform; (4) carbon tetrachloride; (5) toluene; (6) cyclohexanone; (7) dichloromethane; (8) trichloroethylene; (9) dioxane; (10) xylene; (11) 1,2-dichloroethane; (12) THF-water (91.4:8.6); (13) butyl acetate; (14) butanone; (15) ethyl acetate; (16) methyl isobutyl ketone; (17), (18) toluene-methanol (76.9:23.1); (19), (22), (24) cyclohexane; (20) cyclohexanone-acetone (20:80); (21) diisobutyl ketone; and (23) methyl acetate. The temperatures are: (Q) 25°; (A) 27c'; (V) 28°; (/k) 30°; (V) 34°; and (©) 35 o.
dent of the thermodynamic quality of the solvent. Table 1 provides the average values of a0 and b0 respectively equal to 0.16 and 0.42, with their respective standard deviations being 0.04 and 0.11. These results are parallel to those generated by Fig. 2, which gives a0 = 0.18 and b0 = 0.40. As a result, the foregoing arithmetic means of the two model parameters are considered to be appropriate for PS in all solvents, and as such they are utilized for the ensuing analysis. A list of the Huggins coefficients and intrinsic viscosity data associated with a whole polymer of styrene has been collected by Dort [17]. This particular polymer was prepared by radical polymerization terminated by coupling, and its h4"0 was found to be 2 x 105. Hence, it may be adequately described by a Schulz's molecular weight distribution function [18, 19]. This would allow us to apply equation (8) to determine fc via equations (15) and (16) by setting b or U equal to a particular value. In this connection, we have U = 1.5 for the chain coupling vinyl polymerizaton. However, chain transfer, high conversion and autoacceleration effects would broaden the molecular weight distribution appreciably, leading to higher U. The most probable distribution corresponds to U = 2.0. Here, the MHS exponent a varying from 0.30 to 0.75 is computed by an iterative procedure established recently [4]. The value of k is readily obtained by equation (3).
569
Curves I, II and III in Fig. 3 represent the estimated k and J~ based on U = 1.0, 1.5 and 2.0 respectively, as functions of a. Also, the data points reported by Dort [17] are shown in Fig. 3. As expected, J~is consistently less than k over a wide range of the MHS exponent a. Specifically, the k value based on a monodisperse PS is reduced markedly by 20% and 12% in an extremely good solvent with a = 0.80 and a theta solvent respectively, due to a twofold increase in the polydispersity index. More important, curves II and III are closer to all the experimental observations, except for a few cases where curve I fits better to data points (8), (17) and (20) derived from trichloroethylene and some mixed solvents. Statistically, the standard errors of estimate o f k on a are estimated to be 0.053 (curve II), and 0.048 (curve III), whereas that of k on a is 0.091. Although exact information on the molecular weight distribution of the PS sample is unavailable, it is not unreasonable to assume 1.5 < U ~< 2.0 considering the possible termination modes and polymerization conditions. Hence, the estimated k, which allows for the polydispersity effect, is strikingly more satisfactory for the present systems of interest. This implies that, in practice, equation (8) is superior to equation (3) in predicting the Huggins coefficient of polymers, which basically are mixtures of molecules with different sizes. In conclusion, the effect of polydispersity on the Huggins coefficient is indeed noticeable, even in poor solvents. For example, referring to Fig. 3, the predicted values of k and k (curve III) are respectively 0.58 and 0.50 under theta conditions. Incidentally, the former datum approximates to the result of the theoretical calculation of Riseman and Ullman [20]. REFERENCES
1. 2. 3. 4. 5.
M. L. Huggins. J. Am. chem. Soc. 64, 2716 (1942). K. K. Chee. J. appl. Polym. Sci. 27, 1675 (1982). K. K. Chee. J. appl. Polym. Sci. 30, 1359 (1985). K. K. Chee. Polymer 2.8, 977 (1987), N. Sutterlin. In Polymer Handbook (Edited by J. Brandrup and E. H. Immergut), Chap. IV. Interscience, New York (1975). 6. M. Bohdanecky and J. Kovar. Viscosity of Polymer Solutions. Elsevier, Amsterdam (1982). 7. H, L. Bhatnagar, A. B. Biswas and M. K. Gharpurey. J. chem. Phys. 2,8, 88 (1958). 8. S. Imai. Proc. R. Soc. (Lond.) A308, 497 (1969). 9. M. Bohdanecky. Coll. Czech. Chem. Commun. 35, 1972 (1970). 10. M. Muthukumar and K. F. Freed. Macromolecules 10, 899 (1977). 11. F. A. H. Peeters and A. J. Staverman. Macromolecules 10, 1164 (1977). 12. F, Gundert and B. A. Wolf. Makromolek. Chem. 187, 2969 (1986). 13. L.H. Tung. In Polymer Fractionation (Edited by M. J. R. Cantow), Chap. E. Academic Press, New York (1967). 14. K. K. Chee. Polymer (Commun.) 27, 135 (1986). 15. Y. Einaga, Y. Miyaki and H. Fujita. J. Polym. Sci.; Polym. Phys. Edn 17, 2103 (1979). 16. D. J. Streeter and R. F. Boyer. Ind. Engng Chem. 43, 1790 (1951). 17. I. Dort. Polymer 29, 490 (1988). 18. G. Odian. Principles of Polymerization. McGraw-Hill, New York (1970). 19. K. K. Chee. Polymer 26, 581 (1986). 20. L Riseman and R. UUman. J. chem. Phys. 19, 578 (1951).