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Solution properties of poly(N-vinyl carbazole)—II. Thermodynamic properties in various solvents
European Pol~,mcr Jourmd. Vol. 15. pp. 29 t,.~ 34 .C~ Pergamon Pre~,s Lid 1979. Printed in Greal Bril~tin
(~)14-31)57 7'J 0101-0~29S()2 0~) I)
SOLUTION PROPERTIES OF POLY(N-VINYL CARBAZOLE)--II THERMODYNAMIC
PROPERTIES
IN VARIOUS
SOLVENTS
L. M. LEON, 1. KATIME and MATILDE RODRIGUEZ Departamento de Qu'lmica-F'~sica, Universidad de Bilbao, Apartado 644, Bilbao. Espafia
(Receired 30 May 19781 Abstract--Thermodynamic properties of dilute solutions of Poly-N-vinyl carbazole obtained by cationic polimerization have been studied by viscometry in various solvents. The validities of some excluded volume theories have been tested and the unperturbed dimensions have been calculated. The dependencc of molecular dimensions, (s'-). on molecular weight and viscometric equations have also been determined.
viscosities of the samples in tetrahydrofuran (THF) at 298 K, and the [r/]M values of narrow distribution polystyrenes in THF at 298 K. The validity of this method to obtain weight average molecular weights of (PNVCI has been proved by several authors[9,10]. Gel permeation chromatography measurements were made using THF as solvent at 298 K and an elution rate of 1 ml/min. The viscometric measurements on filtered solutions (6 concentrations covering the range 0.2q3.8 g.dl ~) were performed in a suspended level modified Ubbelhode viscometer [1 I] kept at _+0.01 C. Kinetic energy corrections were applied although normally they were very small. The intrinsic viscosity, [q], was obtained by double extrapolation of ~l~p/cand Inr/,/cto zero concentration. Molecular weights, M~, and intrinsic viscosities of the samples are listed in Table 1. The polydispersities of the samples lay between 1.2 and 1.3.
INTRODUCTION The u n p e r t u r b e d dimensions of macromolecular solutions can be determined by extrapolation to ideal conditions for various theories using some perturbed parameters. This value together with some other conformational a n d t h e r m o d y n a m i c parameters are of great interest in the study of other properties I-I,3]. In this work, several Poly-N-vinyl carbazole (PNVC) samples obtained by cationic polymerization have been used in order to analyse the validity of some excluded volume theories, and to calculate the unperturbed dimensions, as well as some t h e r m o d y n a m i c and conformational parameters. EXPERIMENTAL
Samples
RESULTS
Poly-N-vin3,1 carbazole (PNVC} samples were obtained by polymerization at 273 K, using dichloromethane as solvent and (C6Hs)3CSbF6 as initiator. Monomer and initiator were purified by standard procedures [4,5]. Polymerizations were carried out under nitrogen and gentle stirring. In all cases, full conversion was achieved in a few seconds, The polymers were repeatedly washed with methanol and then dried under vacuum at 298 K fer 12 hr. Finally the samples were freeze-dried from benzene. Reagent-grade 1,3-dichlorobenzene, 1,2-dichloroetbane and tetrahydrofuran were purified by standard procedures [6] and freshl~ distilled before use.
AND
DISCUSSION
Unperturbed coil dimensions For long enough chains, the intrinsic viscosity is customarily related to the spatial configuration and molecular weight of the chain by the equation [12]: [rl]o = 6 3 2q~o((.Soe~/M)3 2M1 2 = K,,M 1 2
(I)
[q] = zc3Er/],, = K,,M' 2~.3
(2)
and
Table 1. Molecular weights and intrinsic viscosities of (PNVC) at 298 K
Characterization Molecular weights of the samples and molecular weight distributions were determined b3 gel permeation chromatography. The use of this technique requires calibration of the column for the polymer. W e used the method of Benoit et al. [7] who stated that [q]M~, which is a measure of the hydrodynamic volume, governs the chromatographic column retention. This method, based on Florv's equation (Eqn 3). can be applied only to random coil polymers which have values of the Mark-Houwink [8] exponent between 0.5 and 0.8. In the present work it is shown that the exponents for (PNVCt fall in this range. Molecular weights and molecular weight distributions of our samples were obtained from the gel permeation chromatograms obtained using a Waters gel permeation chromatography instrument, as well as from the intrinsic
M~. 10- 5 (Daltons) 4.90 4.31 4.09 3.54 3.30 2.63 2.06 1.77 1.59 0.72 0.45 29
[r/] (dl/g) 1,3-dichlorobenzene 1,2-dichloroethane 1.030 ..-0.792 0.730 -0.523 0.440 0.241 --
0.760 0.732 0.660 0.530 0.445 0.410 -0.163
L. M. LEbN, I. KATIMEand MATILDERODRIGUEZ
30
Table 2. Ko values of (PNVC) for various solvents from several theories at 298 K
where I-r/]0 is the intrinsic viscosity when the expansion coefficient, ~,~ is unity, q~o is the Flory universal constant, ~bo = 2.87.102t, with (s~) in cm 2 and D1] in dl.g-~, ~, being different from • obtained from equilibrium properties, ~, ~< ~: similarly to Eqn (1): with
[r/] = KM 1'2
(3)
K = qS((s2)6/M) 3/2
(4)
Solvent 1,3-Dichlorobenzene 1,2-Dichloroethane Bromobenzene Chlorobenzene Nitrobenzene
A combination of these Equations gives: ~ = ~o(7,/~) 3. Flory's parameter. ~. therefore is not a constant due to the fact that 0~, ~ co. This behaviour originates in the fact that the influence of the excluded volume is greater on the statistical than on the hydrodynamic radius of the molecule. Analysis of the viscosity measurements implies knowledge of the dependence of ~, on the excluded volume parameter: we will perform such analysis for the following theories.
(C)
5.85 5.96 5.96 5.89 5.92
5.50 5.87 5.78 5.76 5.64
5.04 5.06 4.70 5.06 5.87
2.36 4.40 1.96 2.34 4.21
D112 3M-1 3 = K~3 + 0.857 K~ 3 ~boB M [ q ] - l (7) As can be seen in Fig. 3, application of this theory leads to very different Ko values. This result agrees with the fact that this theory is not applicable to good solvents or at temperatures far from the theta temperature. (d) Cowie (C)[16]
(5)
[q]M-1 2 =
where g(~,) = [-3~3/(2=~ + 13]3 2. To apply this equation, an iterative procedure is required starting with =,~ = 1, calculating 9(a,) and K0 and from this a better value for =,; convergence is fairly rapid. Plots of [q]2 3M-1 3 vs g(~,I)M2 3 [ q ] - t 3 are given in Fig. 1. Application of this theory in b o t h cases gives excellent results over the whole molecular weight interval studied, resulting in similar Ko values.
KdO/C~o)
+ O.0925(q~/Oo)Kd(s~)/M)-21 2OB~ lOM7 20 (8) where q~ is the Flory constant obtained using the Ptitsyn-Eizner approximation [17]. q~ = q~o(l - 2.63~ + 2.86E2)dl'cm-3'mo1-1 being E = ( 2 a - 1 ) / 3 and a, the exponent in the Mark-Houwink equation [8]. Plots of [ , / ] M - 1 2 vs M 7 20 are given in Fig. 4. Ko values depend on the solvent as in the Flory-Fox theory. Table 2 shows the Ko values obtained as a result of the application of these theories in both solvents studied and those obtained in bromobenzene, chlorobenzene and nitrobenzene, for which experimental data of I-q] and M were reported previously[18]. From Table 2 it can be deduced that (KS) theory leads to very similar Ko values.
(b) Stockmayer-Fixman (SF) [14] The equation is rather similar to the above, [ q ] M - t 2 = Ko + 0.15 ~boB M ~ 2
Ko" 104 (dl/g) (SF) (FF)
(c) Flory-Fox (FF) [15]
(a) Kurata-Stockmayer (KS) [13] DI]2 3 M - I 3 = K~ 3 + 0 . 3 6 3 q~oBg(~,~)M2 3[}/]-1 3
(KS)
(6)
Plots of [q]M -1 2 vs M 1 2 are given in Fig. 2. As can be seen, this theory leads to values of Ko slightly different and therefore to different unperturbed dimensions.
1.4
o
o
~o
1.0
0.6
i
0
200
I
i
400 600 g {i'~ ) M 2/:~','/]113104
I
800
Fig. 1. Kurata-Stockmayer theory for (PNVC) at 298 K (O: 1.3-dichlorobenzene, O: 1,2-dichloroethane).
Solution properties of poly(N-vinyl carbazolek--II
31
15 0
/e
1.0 ('4 "3-
0.5
I
I
200
I ,
400 1,4I/2
600
I
8OO
Fig. 2. Stockmayer-Fixman theory for (PNVCt at 298 K. Symbols as in Fig. I.
13
/
/ o° 0/0
.J / 0
~
0.9
0.5
I
I
I
I
200
400
600
800
M [7 ]"110"3 Fig. 3. Flory-Fox theory for (PNVC) at 298 K. Symbols as in Fig. I.
32
L . M . LEbN, I. KATIME and MATILDE RODRIGUEZ
O ¸
,o~ 1.2
...,~/o
/e.~e.e
~
0.9
~o N'"
0.6 0.3 I
210
/.0
I
610
80
M7120 Fig. 4. Cowie theory, for (PNVC) at 298 K. Symbols as in Fig. 1.
/..£
~o
j
e/
"T--.. i
i
÷
#
_I' 1..2 I
S I
I
I
I
I
0.18
0.20
0.22
0.2/.
0.26
(a-0.5) Fig. 5. Kamide-Moore method for (PNVC) in various solvents at 298 K. tO: 1,3-dichlorobenzene, bromobenzene, A: chlorobenzene, A: nitrobenzene, Iq: 1,2-dichloroethane).
I:
33
olu~on properties of poly(N-vinyl carbazole~ 11
-12.C
I
4.7
50
I
I
5.3
56
log Fig. 6. (s z) As function of molecular weight in 1,2-dichloroethane and 1,3-dichlorobenzene. Symbols as in Fig. I.
A study of molecular expansion of (PNVC) in these solvents through a calculation of the expansion coefficients, :t, following the expression proposed by Yamakawa and Kurata[19], confirms that (KS) theory is the most appropriate in order to calculate the unperturbed dimensions of this polymer, the value for which is ( < S 2 ) / M ) 1 2 = 2 . 4 1 ' 1 0 - 9 c m in good agreement with the reported data [9,18,20]. The application of the K a m i d e - M o o r e [ 2 1 ] method - l g K + lg[1 + 2[(a - 0.5)-' - 2] - l ] = (a - 0.5)lgMo - lgKo
(9)
using the values K and a from the Mark-Houwink equation [8], in Table 3, give a straight line, Fig. 5, which indicates that there is no dependence of the unperturbed dimensions on the solvent. However, the K,, values are somewhat overestimated.
Molecular dimensions The intrinsic viscosity for heterogeneous polymer solutions may be represented by
factor for molecular weight heterogeneity and excluded volume effect. If the Schulz-Zimm [22,23] molecular weight distribution function is assumed, q may be approximated by (h + 2)3/zf(h + 1) 0.634 q = (h + l)F(h + 1.5) ~'
(11)
where ~,, is the expansion factor defined as % = [[q]/[t/],~] 1/3, /" the gamma function and h = M , / t M w - M,). The dependence between the radius of gyration and molecular weight for our systems are given in Fig. 6. The obtained relations were: (s 2) = 6.99" 10 -19 M l2s (s 2) = 14.54-10 -19 M 1'2°
(I,3-dichlorobenzene) (l,2-dichloroethane)
The obtained values for the Mark Houwink constants (a, KI, as well for the exponent :~ of the equation which relates the molecular dimensions with molecular weight are given in Table 3. These values indicate that all these solvents are good solvents for (PNVC). bromobenzene being the best.
63]2{b(s2)3/2 In] -
(1o)
qM
REFERENCES
where 4) is the Flory constant obtained using the Ptitsyn-Eizner approximation[17] and q a correction Table 3. Values of Mark-Houwink constants (a, KI and :((exponent of M ( s : ) relationship) of (PNVC)in several solvents at 298 K Solvent Bromobenzene 1,3-Dichlorobenzene Chlorobenzene Nitrobenzene 1,2-Dichloroethane
t+.ej.
15
l +c
a 0.76 0.75 0.74 0.69 0.68
K. 105 5.14 5.71 5.93 9.25 1 1.04
1.29 1.28 1.27 1.20 1.20
I. C. David, M. Piens and G. Geuskens, Europ. Polym. J. 8, 1291 (1972), 2. W. Klopffer and D. Fischer, J. Polvm. Sci., Symp. no. 40, 43 (1973). 3. J. Y. C. Chu and M. Stolka, J. Polym. Sci., Pol),m. Chem. Ed. 13, 2867 (1975). 4. M. A. Harold, M. Nowakowska and P. H. Plesch, Makromolek. Chem. 132, 1 (1970). 5. C. E. H. Bawn, C. Fitzsommons, A. Ledwith, J. Penfold, D. C. Sherrington. and J. A. Weightman, Polymer, Lond. 12, 119 (19711. 6. J. A. Riddick and W. B. Bunger, Organic Solvents, Vol. 2. Wiley Interscience, New York (1970). 7. Z. Grubisic, P. Remp and H. Benoit, J. Polym. Sci. (B) 5, 753 (1967).
34
L. M. LE~)N. 1. KA~IME and MATILIIE RODRIGUEZ
8. H. Mark, derfeste Korper. p. 103. Hirzel. Leipzig (1938). 9. G. Sitaramaiah and D. Jacobs, Polrruer 11, 165 (1970). IO. K. Lejonagoitia, Tesis de Licenc&ura. University of Bilbao (1977). I I. 1.Katime and A. Roig, A.R. Sot. rup. Fis. Quiru. 69B. 1217 (1973). 12. P. J. Flory. Principles of’ Pol.wer Chernisrr!,. Cornell University Press. Ithaca. New York (1953). 13. M. Kurata and W. H. Stockmayer. Forrschr. HochpoIyrn Forth. 3. 196 (1968). 14. W. H. Stockmayer and M. Fixman. J. PO/VII. Sci. Cl. 137 (I 964). 15. P. J. Flory and T. G. Fox. J. Am. cheru. Sm. 73, 1904 (1951).
16. J. M. G. Cowie, Po/ymer, Land. 7, 487 (1966). 17. 0. B. Ptitsvn and Yu. E. Eizner. Sorier P/IIX reck Ph~s. 4. I OiO (I 960). 18. L. M. Leon, 1. Katime. M. Gonzalez and J. Figueruelo. Europ. PO/W. J. 14. 671 (1978). 19. M. Kurata and H. Yamakawa. J. chnrl. Ph~,s. 29, 31 I (1958). 20. K. Ueberreiter and J. 2. Springer. Phys. Chem. 36. 299 (1963). 21. K. Kamide and W. R. Moore. J. Pol~m Sci. B2, 809 ( 1964). 22. G. V. Schulz. Z. ph~x Cheru. B43. 25 (1939). 23. B. H. Zimm. J. them. Phjx 16, 1099 (1948).
(~)14-31)57 7'J 0101-0~29S()2 0~) I)
SOLUTION PROPERTIES OF POLY(N-VINYL CARBAZOLE)--II THERMODYNAMIC
PROPERTIES
IN VARIOUS
SOLVENTS
L. M. LEON, 1. KATIME and MATILDE RODRIGUEZ Departamento de Qu'lmica-F'~sica, Universidad de Bilbao, Apartado 644, Bilbao. Espafia
(Receired 30 May 19781 Abstract--Thermodynamic properties of dilute solutions of Poly-N-vinyl carbazole obtained by cationic polimerization have been studied by viscometry in various solvents. The validities of some excluded volume theories have been tested and the unperturbed dimensions have been calculated. The dependencc of molecular dimensions, (s'-). on molecular weight and viscometric equations have also been determined.
viscosities of the samples in tetrahydrofuran (THF) at 298 K, and the [r/]M values of narrow distribution polystyrenes in THF at 298 K. The validity of this method to obtain weight average molecular weights of (PNVCI has been proved by several authors[9,10]. Gel permeation chromatography measurements were made using THF as solvent at 298 K and an elution rate of 1 ml/min. The viscometric measurements on filtered solutions (6 concentrations covering the range 0.2q3.8 g.dl ~) were performed in a suspended level modified Ubbelhode viscometer [1 I] kept at _+0.01 C. Kinetic energy corrections were applied although normally they were very small. The intrinsic viscosity, [q], was obtained by double extrapolation of ~l~p/cand Inr/,/cto zero concentration. Molecular weights, M~, and intrinsic viscosities of the samples are listed in Table 1. The polydispersities of the samples lay between 1.2 and 1.3.
INTRODUCTION The u n p e r t u r b e d dimensions of macromolecular solutions can be determined by extrapolation to ideal conditions for various theories using some perturbed parameters. This value together with some other conformational a n d t h e r m o d y n a m i c parameters are of great interest in the study of other properties I-I,3]. In this work, several Poly-N-vinyl carbazole (PNVC) samples obtained by cationic polymerization have been used in order to analyse the validity of some excluded volume theories, and to calculate the unperturbed dimensions, as well as some t h e r m o d y n a m i c and conformational parameters. EXPERIMENTAL
Samples
RESULTS
Poly-N-vin3,1 carbazole (PNVC} samples were obtained by polymerization at 273 K, using dichloromethane as solvent and (C6Hs)3CSbF6 as initiator. Monomer and initiator were purified by standard procedures [4,5]. Polymerizations were carried out under nitrogen and gentle stirring. In all cases, full conversion was achieved in a few seconds, The polymers were repeatedly washed with methanol and then dried under vacuum at 298 K fer 12 hr. Finally the samples were freeze-dried from benzene. Reagent-grade 1,3-dichlorobenzene, 1,2-dichloroetbane and tetrahydrofuran were purified by standard procedures [6] and freshl~ distilled before use.
AND
DISCUSSION
Unperturbed coil dimensions For long enough chains, the intrinsic viscosity is customarily related to the spatial configuration and molecular weight of the chain by the equation [12]: [rl]o = 6 3 2q~o((.Soe~/M)3 2M1 2 = K,,M 1 2
(I)
[q] = zc3Er/],, = K,,M' 2~.3
(2)
and
Table 1. Molecular weights and intrinsic viscosities of (PNVC) at 298 K
Characterization Molecular weights of the samples and molecular weight distributions were determined b3 gel permeation chromatography. The use of this technique requires calibration of the column for the polymer. W e used the method of Benoit et al. [7] who stated that [q]M~, which is a measure of the hydrodynamic volume, governs the chromatographic column retention. This method, based on Florv's equation (Eqn 3). can be applied only to random coil polymers which have values of the Mark-Houwink [8] exponent between 0.5 and 0.8. In the present work it is shown that the exponents for (PNVCt fall in this range. Molecular weights and molecular weight distributions of our samples were obtained from the gel permeation chromatograms obtained using a Waters gel permeation chromatography instrument, as well as from the intrinsic
M~. 10- 5 (Daltons) 4.90 4.31 4.09 3.54 3.30 2.63 2.06 1.77 1.59 0.72 0.45 29
[r/] (dl/g) 1,3-dichlorobenzene 1,2-dichloroethane 1.030 ..-0.792 0.730 -0.523 0.440 0.241 --
0.760 0.732 0.660 0.530 0.445 0.410 -0.163
L. M. LEbN, I. KATIMEand MATILDERODRIGUEZ
30
Table 2. Ko values of (PNVC) for various solvents from several theories at 298 K
where I-r/]0 is the intrinsic viscosity when the expansion coefficient, ~,~ is unity, q~o is the Flory universal constant, ~bo = 2.87.102t, with (s~) in cm 2 and D1] in dl.g-~, ~, being different from • obtained from equilibrium properties, ~, ~< ~: similarly to Eqn (1): with
[r/] = KM 1'2
(3)
K = qS((s2)6/M) 3/2
(4)
Solvent 1,3-Dichlorobenzene 1,2-Dichloroethane Bromobenzene Chlorobenzene Nitrobenzene
A combination of these Equations gives: ~ = ~o(7,/~) 3. Flory's parameter. ~. therefore is not a constant due to the fact that 0~, ~ co. This behaviour originates in the fact that the influence of the excluded volume is greater on the statistical than on the hydrodynamic radius of the molecule. Analysis of the viscosity measurements implies knowledge of the dependence of ~, on the excluded volume parameter: we will perform such analysis for the following theories.
(C)
5.85 5.96 5.96 5.89 5.92
5.50 5.87 5.78 5.76 5.64
5.04 5.06 4.70 5.06 5.87
2.36 4.40 1.96 2.34 4.21
D112 3M-1 3 = K~3 + 0.857 K~ 3 ~boB M [ q ] - l (7) As can be seen in Fig. 3, application of this theory leads to very different Ko values. This result agrees with the fact that this theory is not applicable to good solvents or at temperatures far from the theta temperature. (d) Cowie (C)[16]
(5)
[q]M-1 2 =
where g(~,) = [-3~3/(2=~ + 13]3 2. To apply this equation, an iterative procedure is required starting with =,~ = 1, calculating 9(a,) and K0 and from this a better value for =,; convergence is fairly rapid. Plots of [q]2 3M-1 3 vs g(~,I)M2 3 [ q ] - t 3 are given in Fig. 1. Application of this theory in b o t h cases gives excellent results over the whole molecular weight interval studied, resulting in similar Ko values.
KdO/C~o)
+ O.0925(q~/Oo)Kd(s~)/M)-21 2OB~ lOM7 20 (8) where q~ is the Flory constant obtained using the Ptitsyn-Eizner approximation [17]. q~ = q~o(l - 2.63~ + 2.86E2)dl'cm-3'mo1-1 being E = ( 2 a - 1 ) / 3 and a, the exponent in the Mark-Houwink equation [8]. Plots of [ , / ] M - 1 2 vs M 7 20 are given in Fig. 4. Ko values depend on the solvent as in the Flory-Fox theory. Table 2 shows the Ko values obtained as a result of the application of these theories in both solvents studied and those obtained in bromobenzene, chlorobenzene and nitrobenzene, for which experimental data of I-q] and M were reported previously[18]. From Table 2 it can be deduced that (KS) theory leads to very similar Ko values.
(b) Stockmayer-Fixman (SF) [14] The equation is rather similar to the above, [ q ] M - t 2 = Ko + 0.15 ~boB M ~ 2
Ko" 104 (dl/g) (SF) (FF)
(c) Flory-Fox (FF) [15]
(a) Kurata-Stockmayer (KS) [13] DI]2 3 M - I 3 = K~ 3 + 0 . 3 6 3 q~oBg(~,~)M2 3[}/]-1 3
(KS)
(6)
Plots of [q]M -1 2 vs M 1 2 are given in Fig. 2. As can be seen, this theory leads to values of Ko slightly different and therefore to different unperturbed dimensions.
1.4
o
o
~o
1.0
0.6
i
0
200
I
i
400 600 g {i'~ ) M 2/:~','/]113104
I
800
Fig. 1. Kurata-Stockmayer theory for (PNVC) at 298 K (O: 1.3-dichlorobenzene, O: 1,2-dichloroethane).
Solution properties of poly(N-vinyl carbazolek--II
31
15 0
/e
1.0 ('4 "3-
0.5
I
I
200
I ,
400 1,4I/2
600
I
8OO
Fig. 2. Stockmayer-Fixman theory for (PNVCt at 298 K. Symbols as in Fig. I.
13
/
/ o° 0/0
.J / 0
~
0.9
0.5
I
I
I
I
200
400
600
800
M [7 ]"110"3 Fig. 3. Flory-Fox theory for (PNVC) at 298 K. Symbols as in Fig. I.
32
L . M . LEbN, I. KATIME and MATILDE RODRIGUEZ
O ¸
,o~ 1.2
...,~/o
/e.~e.e
~
0.9
~o N'"
0.6 0.3 I
210
/.0
I
610
80
M7120 Fig. 4. Cowie theory, for (PNVC) at 298 K. Symbols as in Fig. 1.
/..£
~o
j
e/
"T--.. i
i
÷
#
_I' 1..2 I
S I
I
I
I
I
0.18
0.20
0.22
0.2/.
0.26
(a-0.5) Fig. 5. Kamide-Moore method for (PNVC) in various solvents at 298 K. tO: 1,3-dichlorobenzene, bromobenzene, A: chlorobenzene, A: nitrobenzene, Iq: 1,2-dichloroethane).
I:
33
olu~on properties of poly(N-vinyl carbazole~ 11
-12.C
I
4.7
50
I
I
5.3
56
log Fig. 6. (s z) As function of molecular weight in 1,2-dichloroethane and 1,3-dichlorobenzene. Symbols as in Fig. I.
A study of molecular expansion of (PNVC) in these solvents through a calculation of the expansion coefficients, :t, following the expression proposed by Yamakawa and Kurata[19], confirms that (KS) theory is the most appropriate in order to calculate the unperturbed dimensions of this polymer, the value for which is ( < S 2 ) / M ) 1 2 = 2 . 4 1 ' 1 0 - 9 c m in good agreement with the reported data [9,18,20]. The application of the K a m i d e - M o o r e [ 2 1 ] method - l g K + lg[1 + 2[(a - 0.5)-' - 2] - l ] = (a - 0.5)lgMo - lgKo
(9)
using the values K and a from the Mark-Houwink equation [8], in Table 3, give a straight line, Fig. 5, which indicates that there is no dependence of the unperturbed dimensions on the solvent. However, the K,, values are somewhat overestimated.
Molecular dimensions The intrinsic viscosity for heterogeneous polymer solutions may be represented by
factor for molecular weight heterogeneity and excluded volume effect. If the Schulz-Zimm [22,23] molecular weight distribution function is assumed, q may be approximated by (h + 2)3/zf(h + 1) 0.634 q = (h + l)F(h + 1.5) ~'
(11)
where ~,, is the expansion factor defined as % = [[q]/[t/],~] 1/3, /" the gamma function and h = M , / t M w - M,). The dependence between the radius of gyration and molecular weight for our systems are given in Fig. 6. The obtained relations were: (s 2) = 6.99" 10 -19 M l2s (s 2) = 14.54-10 -19 M 1'2°
(I,3-dichlorobenzene) (l,2-dichloroethane)
The obtained values for the Mark Houwink constants (a, KI, as well for the exponent :~ of the equation which relates the molecular dimensions with molecular weight are given in Table 3. These values indicate that all these solvents are good solvents for (PNVC). bromobenzene being the best.
63]2{b(s2)3/2 In] -
(1o)
qM
REFERENCES
where 4) is the Flory constant obtained using the Ptitsyn-Eizner approximation[17] and q a correction Table 3. Values of Mark-Houwink constants (a, KI and :((exponent of M ( s : ) relationship) of (PNVC)in several solvents at 298 K Solvent Bromobenzene 1,3-Dichlorobenzene Chlorobenzene Nitrobenzene 1,2-Dichloroethane
t+.ej.
15
l +c
a 0.76 0.75 0.74 0.69 0.68
K. 105 5.14 5.71 5.93 9.25 1 1.04
1.29 1.28 1.27 1.20 1.20
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