3094
Macromolecules 1998, 31, 3094-3098
Entanglement Effects in Semiflexible Polymer Solutions. 2. Huggins Coefficient and Inclusion of Intermolecular Hydrodynamic Interactions Takahiro Sato* and Atsuyuki Ohshima† Department of Macromolecular Science, Osaka University, Machikaneyama-cho 1-1, Toyonaka, Osaka 560-0043, Japan
Akio Teramoto Department of Chemistry, Faculty of Science and Engineering, Ritsumeikan University, Noji-higashi, Kusatsu, Shiga 525-8577, Japan. Received July 8, 1997; Revised Manuscript Received January 2, 1998
ABSTRACT: A previously proposed viscosity equation for stiff-chain polymer solutions based on the fuzzy cylinder model theory was modified by incorporating empirically the effect of intermolecular hydrodynamic interaction (HI). The Huggins coefficient k′ derived from the modified viscosity equation consists of the contributions of the entanglement interaction (k′EI) and of the intermolecular HI (k′HI). While the former can be calculated using the fuzzy cylinder model theory, the latter remains as an adjustable parameter. For various stiff-chain polymer solutions, k′HI was estimated by subtracting the calculated k′EI from experimental k′. The results for dichloromethane (DCM) solutions of poly(n-hexyl isocyanate) (PHIC) showed that k′HI is much smaller than k′EI at 1 j N j 10, where N is the number of Kuhn’s statistical segments per molecule, but becomes important outside the above N range. Zero-shear viscosities calculated by the modified viscosity equation for concentrated DCM solutions of PHIC were favorably compared with experimental data for N up to 15.
1. Introduction Viscosities of polymer solutions exhibit remarkably strong concentration dependence. This strong dependence may be attributed to the effects of the entanglement and hydrodynamic interactions (HI) among polymer chains.1,2 Many years ago, Peterson and Fixman3 calculated the Huggins coefficient k′ for flexible polymer solutions by assuming that the intermolecular HI between two polymer chains would be identical with that between two spheres and also that two entangled polymer chains would behave like a dumbbell or an ellipsoid. Their theoretical result indicated that the contribution of the intermolecular HI to k′ is more important than that of the entanglement interaction for flexible polymer solutions. Afterward, Edwards, Freed, and Muthukumar2,4-14 calculated the contribution of the intermolecular HI to the zero-shear viscosity η0 of solutions of Gaussian chains and rigid rods, using the effective medium theory. Their results demonstrated that the intermolecular hydrodynamic effect (or the hydrodynamic screening effect) on solution viscosities is stronger for flexible polymer solutions than for rodlike polymer solutions,10 although the effective medium theory was not able to incorporate the entanglement effect on polymer solution viscosities. Recently, Sato et al.15,16 proposed a viscosity equation for stiff-chain polymer solutions from dilute through concentrated, which includes the effects of the entanglement interaction and the intramolecular HI but neglects the effect of the intermolecular HI. The equation was favorably compared with zero-shear viscosity η0 data of isotropic solutions of two rigid helical polysaccharides with persistence lengths q larger than 100 nm15 at † Present address: Central Laboratory, Rengo Co., Ltd., 1861-4, Ohhiraki, Fukushima, Osaka 553-0007, Japan.
polymer concentrations ranging from the dilute to (isotropic) concentrated regime. The success of this viscosity equation indicates the unimportance of the intermolecular HI effect in the viscosity of rigid polymer solutions, being consistent with the prediction of Muthukumar and Edwards.10 According to the results of the effective medium theory, the intermolecular HI effect may become more important in solution viscosities with increasing chain flexibility where the entanglement effect tends to be weaker. Thus it may be worth incorporating the intermolecular HI effect into the previous viscosity equation for stiff-chain polymers, when the equation is applied to solutions of semiflexible polymers with smaller q. In the present study, we have included empirically the contribution of the intermolecular HI into the previous viscosity equation (section 2). In the modified viscosity equation, the Huggins coefficient k′ consists of the contributions of the entanglement and of the intermolecular HI. The former is calculated theoretically (section 3), while the latter is estimated from experimental results of k′. The two contributions have been compared for various stiff-chain polymer solution systems (section 4). Finally, the modified viscosity equation has been compared with solution viscosity data of a semiflexible polymer, poly(n-hexyl isocyanate), which were reported in part 1 of this series17 (section 5). 2. Viscosity Equation and the Huggins Coefficient Sato et al.15,16 derived a viscosity equation for stiffchain polymer solutions by using the fuzzy cylinder model, a cylindrical smoothed density model for a wormlike cylinder of the contour length L, the diameter d, and the number N of Kuhn’s statistical segments.
S0024-9297(97)00999-6 CCC: $15.00 © 1998 American Chemical Society Published on Web 04/11/1998
Macromolecules, Vol. 31, No. 9, 1998
Entanglement and Hydrodynamic Interaction Effects 3095
kBT/D ˆ |0 ) (kBT/D|0)(1 + k′|[η]c)
Their viscosity equation is given by
c′kBT η0 ) η(S) + ηˆ (V) + χ2 10Dr
(1)
where
Dr/D ˆ r0 ) [1 + (6β)-1/2Le4fr(de/Le)c′(D ˆ r0/D|)1/2]-2
(2)
and
ˆ |0 exp(-Vex*c′) D| ) D
(3)
In eq 1, η0 is the (total) zero-shear viscosity of the polymer solution, which consists of the pure solvent viscosity η(S), the zero-shear viscosity ηˆ (V) induced by the friction between the polymer and solvent, and the zeroshear viscosity induced by the orientational entropy loss of polymer chains under the shear flow (the third term). The last term is expressed by the polymer number concentration c′, a hydrodynamic factor χ, and the diffusion coefficient Dr with respect to the end-over-end rotation of a polymer chain subject to the entanglement effect; kBT is the Boltzmann constant multiplied by the absolute temperature. In eq 2, D ˆ r0 is the rotational diffusivity when the entanglement effect is switched off, β is a numerical constant ()1350),18 Le and de are the length and diameter of the fuzzy cylinder, and D| is the longitudinal diffusion coefficient. The quantities Le and de are evaluated from the wormlike chain parameters L, d, and N.15,16 The function fr(de/Le) considers the entanglement release by fluctuation of the segment distribution in the fuzzy cylinder, which is given by16,17
1 fr(x) ) (1 + Crx) 1 - Crx 5 3
(
)
(4)
where the coefficient Cr is empirically expressed as
1 Cr ) {tanh[(N - N*)/∆] + 1} 2
(5)
with two adjustable parameters N* and ∆. The quanˆ |0 is tity D| appearing in eq 2 is given by eq 3, where D D| not perturbed by the jamming effect and Vex* is the excluded volume between the critical hole and a hindering chain; Vex* contains one adjustable parameter λ*, the similarity ratio between the critical hole and the fuzzy cylinder.15,16 Equations 2 and 3 were derived by the Green function formalism and by the hole theory, respectively. Previously, Sato et al. anticipated that the intermolecular HI (or the hydrodynamic screening effect) would be weak in stiff-chain polymer solutions and replaced ˆ |0, and ηˆ (V)/c′ by the hydrodynamic quantities D ˆ r0, D their infinite dilution values being affected only by the intramolecular HI. However, if the intermolecular HI is important, these quantities must be concentrationdependent.19 Considering the effect of the intermolecular HI up to the linear term of the polymer mass concentration c, we express those hydrodynamic quantities by
3 ηˆ (V) ) 1 - γχ2 [η]η(S)c(1 + k′η[η]c) 4
(6a)
kBT/D ˆ r0 ) (kBT/Dr0)(1 + k′r[η]c)
(6b)
(
)
(6c)
where [η], Dr0, and D|0 are the intrinsic viscosity and the rotational and longitudinal diffusion coefficients at infinite dilution, respectively, and γ is the hydrodynamic parameter relating Dr0 to [η] through c′kBT/Dr0 ) (15/ 2)c[η]η(S)γ.15 The coefficients k′η, k′r, and k′| represent the strengths of the intermolecular HI effects on the corresponding hydrodynamic quantities. Hydrodynamic calculations of these coefficients usually encounter mathematical difficulties due to the long-range nature of the HI3,5,20-22 and have not been properly performed for stiff-chain or semiflexible polymer solutions. Here, regarding these coefficients as adjustable parameters, we will evaluate them from experimental results for the Huggins coefficient k′ in section 4. The solution viscosity in a dilute region is expressed by the empirical Huggins equation:
η0 ) η(S)(1 + [η]c + k′[η]2c2)
(7)
A large number of experimental data of the Huggins coefficient k′ have been accumulated for various polymer solution systems so far.1,23 When eqs 2, 3, and 6a-c are inserted into eq 1 and the result is expressed as a power series of c, we can compare our viscosity equation with the Huggins equation, which gives
k′ ) k′HI + k′EI
(8)
3 k′HI ) k′η - γχ2(k′η - k′r) 4
(9)
where
and 4 3 2 -1/2 Le NA f (d /L )(F /F )1/2 k′EI ≡ γχ β 2 [η]ML r e e |0 r0
(10)
Here F|0 and Fr0 are factors relating to the effects of the intramolecular HI on D|0 and Dr0, respectively, which may be approximately calculated from the axial ratio p ()L/d) and N of the polymer chain.15 In ref 15, Sato et al. proposed to approximate the hydrodynamic parameters γ and χ to those for a straight spherocylinder24 with the same p as that of the polymer. 3. Entanglement Contribution k′EI to the Huggins Coefficient Figure 1 shows the chain-flexibility dependence of k′EI calculated by eq 10 at fixed axial ratios p. Here we have used the value of [η]M calculated by Yamakawa and Yoshizaki’s theory25,26 for the wormlike spherocylinder, and N* ) ∆ ) 4.17 At each p, k′EI is insensitive to the number N of Kuhn segments at N j 1 and decreases with increasing N at N J 1. This result demonstrates that the entanglement interaction becomes less effective with increasing chain flexibility (at a constant contour length). The solid curve in Figure 2 shows the N or molecular weight dependence of k′EI for poly(n-hexyl isocyanate) (PHIC) in dichloromethane (DCM), calculated by eq 10 with the parameters N, d, L, N*, and ∆, given in part 1.17 (Although the values of N* and ∆ were determined previously without considering the intermolecular hy-
3096 Sato et al.
Figure 1. Contribution k′EI of the entanglement interaction to the Huggins coefficient calculated by eq 10 as a function of Kuhn’s statistical segment number N at fixed axial ratios p. Both parameters N* and ∆ are chosen to be 4 according to part 1.17
Figure 2. Comparison between the contributions k′EI (the solid curve) of the entanglement interaction and k′HI (the circles) of the intermolecular HI to the Huggins coefficient k′ for dichloromethane solutions of poly(n-hexyl isocyanate).
drodynamic interaction, the dependence of k′EI on these parameters is so weak that alterations of the parameters do not essentially change the result of k′EI.) The calculated k′EI takes large values in an intermediate N region from 1 to 10. With increasing N above 10, k′EI becomes smaller, which reflects the chain flexibility effect; in the coil limit (N f ∞), k′EI decreases to 0.102. On the other hand, k′EI decreases also at N j 1, owing to the reduction of Le or the axial ratio p of the polymer. 4. k′EI vs k′HI According to eq 8, k′HI can be evaluated by subtracting the theoretical k′EI mentioned above from experimental k′. The circles in Figure 2 represent the estimated k′HI for PHIC samples in DCM that were previously studied.17 (The data point at N ) 0.37 was obtained from an unpublished result of k′.) The N dependence of k′HI obtained is opposite to that of k′EI shown by the solid curve, and the values of k′HI are much smaller than k′EI in an N range from 1 to 10. However, k′HI tends to increase outside this region and exceeds k′EI for the
Macromolecules, Vol. 31, No. 9, 1998
Figure 3. Axial ratio dependence of k′HI for various solutions of nearly rodlike polymers with N j 0.8; (0) PHIC-DCM;17 (O) PHIC-toluene;39 (b) PHIC-n-butyl chloride;40 (~) schizophyllan-water;41,42 (4) xanthan-0.1 M aqueous NaCl;43,44 (.) poly(γ-benzyl L-glutamate)-dimethylformamide;45 (]) poly[trans-bis(tributylphosphine)platinum 1,4-butadiynediyl]-heptane;46 (2) poly(terephthalamide-p-benzohydrazide)-dimethyl sulfoxide.47
PHIC samples with the smallest and largest N in Figure 2. As mentioned in the previous section, k′EI approaches 0.102 in the coil limit (without the excluded volume effect). This limiting value is much smaller than experimental results of k′ accumulated so far, ranging from 0.5 to 0.7 for flexible polymer solutions in the Θ condition.1 This implies that the contribution of the intermolecular HI is more important than the entanglement contribution for flexible polymer solutions, which is in qualitative agreement with the conclusion of Peterson and Fixman.3 Moreover, k′HI (0.4-0.6) estimated from experimental k′ and k′EI ) 0.102 for flexible polymers in the Θ condition is rather favorably compared with results of effective medium theories6,9,13 for the Gaussian chain model (0.38-0.695). Figure 3 shows the axial ratio p dependence of k′HI for various solutions of stiff or semiflexible polymers taking nearly rodlike conformations with N j 0.8. The value of k′HI was obtained by subtracting the theoretical k′EI from experimental k′ for each solution, and p was estimated from the sample molecular weight and literature values27-29 of the wormlike cylinder parameters for each system. The data points of k′HI almost follow a single composite curve. Several workers3,22,30-32 calculated k′HI for hard sphere solutions by different methods. Their k′HI results range from 0.69 to 1. The linear extrapolation of the data shown in Figure 3 to p ) 1 seems to give a value of k′HI that is consistent with those theoretical values. The present method of estimating k′HI may introduce to k′HI some errors coming from approximations to the hydrodynamic quantities appearing in eq 10 for k′EI, which were mentioned at the end of section 2. However, the above favorable comparisons of k′HI estimated by this method with the existing theories in both rod and coil limits indicate that such errors are quite small. The hydrodynamic calculation by Riseman and Ullman33 for rodlike polymer solutions indicated no dependence of k′HI on p and seem to be inconsistent with the results of Figure 3. However, their calculation contains an improper integration,20 and thus the results are not conclusive. Sakai34 extended the semiempirical theories of Brinkman35 and Roscoe36 to prolate ellipsoid solutions
Macromolecules, Vol. 31, No. 9, 1998
Entanglement and Hydrodynamic Interaction Effects 3097
Figure 4. Comparison of the modified Sato et al. viscosity equation with zero-shear viscosity data17 for dichloromethane solutions of poly(n-hexyl isocyanate): (solid curves) the results calculated by the modified viscosity equation including the effect of the intermolecular HI; (dotted curves) the results by the original Sato et al. viscosity equation.15
and obtained a negative dependence of k′on p; his k′ decreases from 0.7 to 0.538 with increasing p from 1 to 15. His p dependence of k′ seems to be weaker than our results shown in Figure 3.
However, if the intermolecular HI is weak enough, hˆ (c) may be expressed by
5. Comparison of the Modified Viscosity Equation with Experimental η0 up to High Concentrations
where k′HI can be calculated by eqs 8 and 10 along with experimental k′. As shown in section 4, the contribution of the intermolecular HI to the Huggins coefficient is weak for DCM solutions of PHIC with 1 j N j 10. Therefore, eq 13 may be applicable for these solutions. Although eq 13 considers only up to the term of order c, the intermolecular HI affects higher order terms in η0 given by eq 11 with eq 13. Equation 12 for D ˆ r0/Dr contains three adjustable parameters, N* and ∆ in fr(de/Le) (cf. eqs 4 and 5) and λ* in Vex*. In part 1, we chose N* ) ∆ ) 4 and λ* ) 0.06 to fit Sato et al.’s viscosity equation (without including the intermolecular HI) to η0 data for DCM solutions of PHIC.17 Since η0 is not so sensitive to N* and ∆, we have here used the values of N* and ∆ previously determined to search a value of λ*, which leads to a best fit of the modified viscosity equation to the same experimental data. Figure 4 compares the experimental η0 for the PHIC solutions17 with values calculated by the modified viscosity equation with λ* ) 0.03. For PHIC samples with N e 15 (in the left panel), the experimental data are successfully fitted by the calculated curve. This fit is better than the previous fit in part 1 for PHIC samples with N ) 8.0 and 15. The dotted curves in Figure 4 represent theoretical η0 with λ* ) 0.03 and k′HI ) 0, i.e., with no intermolecular HI effect. The difference between the solid and dotted curves is small at N e 4.3, while it is more appreciable at N g 8.0 in high concentration ranges. This shows that the intermolecular HI plays only a minor role in the viscosity of DCM solutions of PHIC with N e 4.3, but it is increasingly important at larger N. For the highest molecular weight PHIC sample (in the right panel of Figure 4), the fit of the same viscosity equation (the solid curve) is not as good as that for the
In section 2, we have modified Sato et al.’s equation for η0 by incorporating the intermolecular HI. Now, we compare this modified equation with η0 data of DCM solutions of PHIC up to high concentrations,17 which cannot be expressed by the Huggins equation (eq 7). The modified equation, given by eqs 1-3, includes the ˆ r0, and D|0, which three hydrodynamic quantities η(V)/c′, D are affected by the intermolecular HI. In a first approximation, these quantities are assumed to have an identical concentration dependence.37 Then, eq 1 is written as
{
} )
3 η0 ) η(S) 1 + c[η] 1 + γχ2[(D ˆ r0/Dr) - 1] hˆ (c) 4
(
(11)
where D ˆ r0/Dr is given from eqs 2 and 3 as
D ˆ r0 ) 1 + β-1/2(Le4/L)fr(de/Le)c′(F|0/Fr0)1/2 × Dr 1 exp Vex*c′ 2
[
(
2
)]
(12)
and hˆ (c) is a function of c, which expresses the concentration dependence of the three hydrodynamic quantities. It is noted that in eq 11 the contributions of the entanglement and intermolecular HI to η0 are intermingled and cannot be separated into two terms in a concentrated regime. We have no appropriate expression of hˆ (c) applicable over an arbitrary concentration range, because the intermolecular HI is hard to deal with theoretically.
hˆ (c) ≡ 1 + k′HI[η]c
(13)
3098 Sato et al.
lower molecular weight samples.38 Although experimental data for the sample with N ) 35 are lacking in an intermediate c range, there is a symptom of the disagreement between calculated and experimental values. There are two possible reasons for the disagreement at the large N. On one hand, the higher order terms of c in hˆ (c), neglected in eq 13, may become important at large N, where the intermolecular HI becomes strong; cf. Figure 2. On the other hand, the fuzzy cylinder model theory assumes that each polymer chain can freely change its conformation even in concentrated solutions.15 This assumption, however, may not be valid as the flexibility of the polymer chain increases. We might consider reptation-like motions of polymer chains with large N in concentrated solutions,2 although it is not easy to predict conditions of the onset of such motions. At present, we cannot specify which defect mainly causes the disagreement in the right panel of Figure 4. References and Notes (1) Bohdaneck, M.; Kova´, J. Viscosity of Polymer Solutions; Elsevier: Amsterdam, 1982. (2) Doi, M.; Edwards, S. F. The Theory of Polymer Dynamics; Clarendon Press: Oxford, U.K., 1986. (3) Peterson, J. M.; Fixman, M. J. Chem. Phys. 1963, 39, 2516. (4) Edwards, S. F.; Freed, K. F. J. Chem. Phys. 1974, 61, 1189. (5) Freed, K. F.; Edwards, S. F. J. Chem. Phys. 1974, 61, 3626. (6) Freed, K. F.; Edwards, S. F. J. Chem. Phys. 1975, 62, 4032. (7) Muthukumar, M.; Freed, K. F. Macromolecules 1977, 10, 899. (8) Muthukumar, M.; Edwards, S. F. Polymer 1982, 23, 345. (9) Muthukumar, M. J. Phys. A: Math Gen. 1981, 14, 2129. (10) Muthukumar, M.; Edwards, S. F. Macromolecules 1983, 16, 1475. (11) Muthukumar, M. J. Chem. Phys. 1983, 78, 2764. (12) Muthukumar, M.; DeMeuse, M. J. Chem. Phys. 1983, 78, 2773. (13) Muthukumar, M. J. Chem. Phys. 1983, 79, 4048. (14) Edwards, S. F.; Muthukumar, M. Macromolecules 1984, 17, 586. (15) Sato, T.; Takada, Y.; Teramoto, A. Macromolecules 1991, 24, 6220. (16) Sato, T.; Teramoto, A. Adv. Polym. Sci. 1996, 126, 85. (17) Ohshima, A.; Kudo, H.; Sato, T.; Teramoto, A. Macromolecules 1995, 28, 6095. (18) Teraoka, I.; Ookubo, N.; Hayakawa, R. Phys. Rev. Lett. 1985, 55, 2712.
Macromolecules, Vol. 31, No. 9, 1998 (19) Strictly speaking, the hydrodynamic parameter χ2 in eqs 1 and 6a may also be affected by the intermolecular HI. The possible concentration dependence of χ2 can be absorbed in the coefficients k′η and k′r in eqs 6a and 6b. (20) Saito, N. J. Phys. Soc. Jpn. 1952, 7, 447. (21) Yamakawa, H. Modern Theory of Polymer Solutions; Harper & Row: New York, 1971. (22) Batchelor, G. K.; Green, J. T. J. Fluid Mech. 1972, 56, 401. (23) Polymer Handbook, 3rd ed.; Brandrup, J., Immergut, E. H., Eds.; John Wiley & Sons: New York, 1989. (24) Yoshizaki, T.; Yamakawa, H. J. Chem. Phys. 1980, 72, 57. (25) Yamakawa, H.; Yoshizaki, T. Macromolecules 1980, 13, 633. (26) As mentioned by the original authors, Yamakawa and Yoshizaki’s theoretical results are applicable for polymers with d/2q j 0.2. Therefore, Figure 1 shows the results of k′EI under this condition. (27) Fujita, H. Polymer Solutions; Elsevier: Amsterdam, 1990. (28) Norisuye, T. Prog. Polym. Sci. 1993, 18, 543. (29) Itou, S.; Nishioka, N.; Norisuye, T.; Teramoto, A. Macromolecules 1981, 14, 904. (30) Bedeaux, D.; Kapral, R.; Mazur, P. Physica 1972, 76, 247. (31) Batchelor, G. K. J. Fluid Mech. 1977, 83, 97. (32) Freed, K. F.; Muthukumar, M. J. Chem. Phys. 1978, 69, 2657. (33) Riseman, J.; Ullman, R. J. Chem. Phys. 1951, 19, 578. (34) Sakai, T. J. Polym. Sci.: Polym. Phys. Ed. 1968, 6, 1535. (35) Brinkman, H. C. J. Chem. Phys. 1952, 20, 571. (36) Roscoe, R. Br. J. Appl. Phys. 1952, 3, 267. (37) Effective medium theories provide identical values for k′η, k′r, and k′| of spherical molecules32 but slightly different values for k′HI and k′r of Gaussian chains.11-13 There are no corresponding calculations for semiflexible polymers. (38) For the highest molecular weight PHIC sample (with N ) 97), the intramolecular excluded volume effect is still too weak to affect Le and de in eq 2, because its [η] agrees with the theoretical value calculated by the Yamakawa-FujiiYoshizaki theory25,48 for unperturbed wormlike cylinders. (39) Itou, T.; Chikiri, H.; Teramoto, A. Unpublished data, 1995. (40) Kuwata, M.; Murakami, H.; Norisuye, T.; Fujita, H. Macromolecules 1984, 17, 2731. (41) Yanaki, T.; Norisuye, T.; Fujita, H. Macromolecules 1980, 13, 1462. (42) Enomoto, H. Unpublished data, 1985. (43) Sato, T.; Norisuye, T.; Fujita, H. Polym. J. 1984, 16, 341. (44) Takada, Y. MS Thesis, Osaka, 1990. (45) Nakagawa, K. MS Thesis, Osaka, 1968. (46) Motowoka, M.; Norisuye, T.; Teramoto, A.; Fujita, H. Polym. J. 1979, 11, 665. (47) Sakurai, K.; Ochi, K.; Norisuye, T.; Fujita, H. Polym. J. 1984, 16, 559. (48) Yamakawa, H.; Fujii, M. Macromolecules 1974, 7, 128.
MA970999J
