Langmuir 1998, 14, 735-737
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Dynamics of a High Molecular Weight Polyelectrolyte Martial Pabon, Joseph Selb, and Franc¸ oise Candau* Institut Charles Sadron, 6, rue Boussingault, 67 083 Strasbourg Cedex, France Received September 8, 1997. In Final Form: November 13, 1997 We investigate the rheological behavior of semidilute salt-free solutions of a high molecular weight poly(acrylamide-co-sodium acrylate). The frequency dependence of the complex shear modulus G*(ω) was measured in a large concentration range. Results give support to a recent model based on the scaling theory. It is found in particular that the zero-shear rate viscosity varies as C3/2 in the entangled semidilute regime over 3 decades of concentration.
Introduction Considerable theoretical work has been devoted to polyelectrolyte solutions, but experimental studies on their dynamical properties are rather limited. Rheological experiments provide a powerful tool for the investigation of the dynamic behavior of polyelectrolytes. A rheological study of aqueous solutions of entangled poly(N-methyl2-vinylpyridinium chloride) was reported by Yamaguchi et al.1-3 It was found that the zero-shear viscosity of saltfree solutions follows a power law of the concentration with an exponent smaller than that for neutral systems. Moreover, a crossover was observed at rather high concentrations (C g 0.3 g‚cm-3) to a regime where the polymer solution behaves as a neutral one. These results are well accounted for by recent theoretical studies,4,5 which predict three regimes for salt-free semidilute solutions of polyelectrolytes in good solvent: (i) a semidilute unentangled regime in which the longest time of the stress relaxation decreases upon increasing the concentration, (ii) a semidilute entangled regime where the relaxation time is independent of concentration, and (iii) a high concentration regime where the charge effects are fully screened so that the system behaves as a neutral one. The terminal time of the stress relaxation is much shorter for entangled polyelectrolytes than for neutral chains of the same molecular weight and same concentration. Therefore the measurement of this relaxation time at relatively low concentrations requires use of very high molecular weight polymers. Such polymers can only be obtained through polymerization in dispersed media. For example, the free-radical polymerization of a water-soluble monomer dispersed in the aqueous phase of a microemulsion was shown to lead to a microlatex in which each particle contains on average a single macromolecule of ultrahigh molecular weight.6 With this process, the distribution of molecular weights is expected to be rather narrow due to the particular mechanism of polymerization. In this paper, we report on a rheological study of semidilute solutions of a poly(acrylamide-co-sodium acry* To whom correspondence may be addressed. (1) Yamaguchi, M.; Wakutsu, M.; Takahashi, Y.; Noda, I. Macromolecules 1992, 25, 470. (2) Yamaguchi, M.; Wakutsu, M.; Takahashi, Y.; Noda, I. Macromolecules 1992, 25, 475. (3) Takahashi, Y.; Hase, H.; Yamaguchi, M.; Noda, I. J. Non-Cryst. Solids 1994, 172, 911. (4) Rubinstein, M.; Colby, R. H.; Dobrynin, A. V. Phys. Rev. Lett. 1994, 73, 2776. (5) Dobrynin, A. V.; Colby, R. H.; Rubinstein, M. Macromolecules 1995, 28, 1859. (6) Candau, F.; Leong, Y. S.; Fitch, R. M. J. Polym. Sci., Polym. Chem. Ed. 1985, 23, 193.
late) of high molecular weight (Mw g 107)7 prepared by free radical microemulsion copolymerization. In the copolymer, the molar content of acrylate units is 16.9%. The experimental procedure has been described elsewhere.8,9 It was shown that this process leads to a copolymer homogeneous in composition with reactivity ratios close to unity, values significantly different from those obtained for the same copolymer in solution.10 Furthermore, the polymerization reaction was stopped at a low degree of conversion (10%) in order to ensure a distribution of molecular weights as narrow as possible. The frequency dependence of the complex shear modulus G*(ω) was measured for semidilute solutions in the concentration range 3 × 10-4 wt % e C e 4.4 wt %, and the results were compared to the theoretical predictions. Experimental Results The rheological device was a controlled stress rheometer Haake RS 100 using a cone-plate geometry with a diameter of 35 or 60 mm and an angle of 1°. Special care was taken to ensure that the rheological measurements were performed in the linear regime. The linearity tests consisted in measuring G′ and G′′ as a function of the stress amplitude σ0 for three given frequencies. The linear regime, characterized by a stress range for which G′ and G′′ are stress independent, was found to be σ0 e 10 Pa for a 1 wt % concentration sample and σ0 e 0.07 Pa for CP ) 10-3 wt %. Figure 1 shows typical frequency dependencies in log-log scale of the storage modulus G′ and of the loss modulus G′′. In the low-frequency range, G′(ω) and G′′(ω) follow, within the experimental accuracy, power laws of frequency with exponents equal to 2 and 1, respectively, as expected for Maxwellian liquids. The two curves G′(ω) and G′′(ω) cross each other at a circular frequency ωX, but in contrast to the behavior of Maxwellian fluids, the maximum of G′′(ω) does not appear at ωX and no well-defined plateau of the storage modulus is observed in the high-frequency range. Deviations from a Maxwellian behavior were also reported for semidilute solutions of neutral polymers, due to the occurrence of Rouse-Zimm-like modes.11 However, the characteristic frequency of these modes is high enough that they do not significantly modify the terminal relaxation associated with reptation motion. From the plots of G′, G′′(ω), one can determine the longest relaxation time τR ) ωX-1, a characteristic shear modulus G0 ) 2G′ (ωX) ) 2G′′(ωX) and the zero-shear viscosity η0. The latter is determined from the limit ω f 0 of the quantity G′′(ω)/ω. (7) The molecular weight was too high to be measured with accuracy by light scattering. (8) Candau, F.; Zekhnini, Z.; Durand, J.-P. J. Colloid Interface Sci. 1986, 114, 398. (9) Pabon, M. Thesis, Universite´ Louis Pasteur, Strasbourg, 1997. (10) Candau, F.; Zekhnini, Z.; Heatley, F. Macromolecules 1986, 19, 1895. (11) Doi, M.; Edwards, S. F. Theory of Polymer Dynamics; Clarendon Press: Oxford, 1986.
S0743-7463(97)01021-4 CCC: $15.00 © 1998 American Chemical Society Published on Web 01/16/1998
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Letters behavior to allow accurate measurements of G0 and τR. In the low concentration range (3 × 10-4 wt %e CP e 2 × 10-3 wt %) we have performed measurements of η0 by means of steady flow experiments using a low shear Contraves rheometer. The results thus obtained agree within the experimental accuracy with those determined from the limit ω f 0 of G′′/ω. The viscosity values obtained from η0 ) G0τR are found to be smaller than those obtained from the zero frequency extrapolation procedure. This is likely due to the uncertainty in determining the G0 and τR values from the crossing of G′(ω) and G′′(ω). From the data of Figure 2, one clearly sees a change in the polymer rheological behavior at a crossover concentration C2 ≈ 1 wt %. A second crossover is observed in Figure 2a at a much lower concentration C1 ≈ 10-3 wt %.
Discussion
Figure 1. Frequency dependences in log-log scale of the storage modulus G′(O) and of the loss modulus G′′(4) for a high molecular weight poly(acrylamide-co-sodium acrylate). Saltfree solution, 0.5 wt % in water.
The experimental results shown above differ significantly from those obtained for neutral systems where the exponents of G0(C), τR(C), and η0(C) are much larger in the semidilute regime. This is in agreement with the theoretical description of Rubinstein et al.4 The anomalous behavior of the dynamical properties of polyelectrolytes is partly related to the fact that the contraction undergone by the chains upon increasing the concentration in the semidilute regime is much larger for charged polymers: the radius of gyration varies such as C-1/4 for polyelectrolytes whereas it follows a C-1/8 law for neutral chains.12 As a result, there exists a larger range of concentrations C* < C < Ce where the polyelectrolyte chains overlap without making effective entanglements; C* is the crossover concentration between dilute and semidilutes regimes, and Ce is the concentration at which the chains become entangled. The latter occurs for a relative viscosity η/ηS ≈ 50 where ηS is the solvent viscosity.5 In this regime, the stress relaxation time decreases upon increasing the concentration and the viscosity follows a power law of the concentration with an exponent 1/2. In the entangled domain, Rubinstein et al.4 predict two regimes with the following power laws
Ce < C < CD; η0 ∼ C3/2, τR ∼ C0, G0 ∼ C3/2 where CD is the concentration beyond which the electrostatic blobs entangle and where the behavior is that of a neutral system
C > CD; η0 ∼ C15/4, τR ∼ C3/2, G0 ∼ C9/4
Figure 2. Concentration dependences in log-log scale of the viscosity (η0), the relaxation time (τR), and the plateau modulus (G0) for a high molecular weight poly(acrylamide-co-sodium acrylate) in salt-free water solution. Part a shows the viscosity obtained from the zero frequency extrapolation procedure (×) and from the relation η0 ) G0τR (+). The dashed lines refer to the behavior of neutral polymers. The solid lines are the predictions from Rubinstein et al.4 for polyelectrolytes. The results of the concentration dependences of τR, G0, and η0 are reported in Figure 2. In Figure 2a are also reported the data η0(C) deduced from the relation η0 ) G0τR for systems with a polymer concentration CP g 0.07 wt %. Below this concentration, and as discussed later, the viscoelastic spectra deviate too much from a Maxwellian
The experimental scaling behaviors shown in Figure 2 are quite in agreement whith the above predictions, assuming C2 ≡ CD. The solid lines in Figure 2 represent the theoretical predictions. The agreement with the scaling theory is quite good for the viscosity which is the parameter measured experimentally with the best accuracy. However, in the regime Ce < CP < CD, the best fit of the data is a power law with an exponent 7/4 instead of 3/2. The concentration dependencies of τR and G0 have been determined in a much narrower concentration range for reasons discussed below, but they still approch closely the theoretical predictions for polyelectrolyte solutions. It should also be noted that the crossover concentration CD ≈ 1 wt % is much lower than that obtained by Yamaguchi et al.3 (C2 ≈ 30 wt/v %). The concentration CD is related to the size of the electrostatic blob D through CD ∼ D-3. According to Rubinstein et al.,5 D ∼ R-18/7 for (12) Daoud, M.; Cotton, J.-P.; Farnoux, B.; Jannink, G.; Sarma, G.; Benoit, H.; Duplessix, R.; Picot, C.; de Gennes, P. G. Macromolecules 1975, 8, 804.
Letters
a polyelectrolyte with a hydrophilic backbone, where R is the ionization degree. The samples investigated by Yamaguchi et al.1-3 were fully charged whereas the sample studied here only contains 16.9 M % of acrylate units. This difference in the ionization degree can explain that found in the CD values. As for the lowest crossover concentration C1, one would expect that it corresponds to the Ce concentration where the system becomes entangled. In fact, the scaling law (η0 ∼ C3/2) predicted by Rubinstein et al.4 for the nonentangled semidilute regime describes the data within the experimental accuracy. The value of the viscosity at Ce should allow one to calculate n, the number of entanglements requested for affecting the chain dynamics. From the results of Figure 2, the viscosity at C1 is approximately equal to 2-3 ηS leading to n ≈ 1.5 (according to η ∼ n2ηS).5 This suggests that in contrast with the theoretical predictions and the results of Fernandez-Prini and Lagos13 the polyelectrolyte becomes entangled at concentrations slightly higher than C* for which η0 ≈ 2ηS. No transition regime can be observed around concentrations for which η0 ≈ 20-30 ηS. In fact, the scaling picture given by Rubinstein et al. to describe the behavior in the vicinity of C* has been recently questioned on the basis of simulation experiments14 and an analytical approach assuming a wavevector-dependent screening length.15 It is shown in these studies that there is a sharp crossover at C* but that the polyelectrolyte chains undergo a contraction at concentrations which are 2 orders of magnitude below C*. This might modify the regime between C* and Ce. Finally, one should comment on the observation that the rheological behavior of the samples deviates more and (13) Fernandez-Prini, R.; Lagos, A. E. J. Polym. Sci., Part A 1964, 2, 2917. (14) Stevens, M. J.; Kremer, K. J. Chem. Phys. 1995, 103, 1669. (15) Donley, J. P.; Rudnick, J.; Liu A. J. Macromolecules 1997, 30, 1188.
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more from a Maxwellian behavior as the concentration decreases. In fact, the frequency range over which the plateau modulus is observed depends critically on the ratio between the terminal time τR and the Rouse time τe of the entanglement length. For a neutral polymer in good solvent, this ratio increases strongly with concentration as τR ∼ C3/2 and τe ∼ C-5/2 and therefore the length of the plateau modulus becomes appreciable not too far beyond the entanglement concentration. For polyelectrolyte systems, τR is independent of concentration and τe ∼ C-3/2, so that the plateau modulus can be very short in a large range of concentrations beyond Ce. Furthermore some polydispersity effects might produce a smoothing on the ω frequency dependence of G′ and G′′. For these reasons, reliable measurements of τR and G0 could not be achieved in the low concentration range where the viscosity was still easily determined. In conclusion, the overall results reported here are in agreement with the model of Rubinstein et al.4 on the dynamical properties of semidilute solutions of polyelectrolytes. At high concentration (CD > 1 wt %), the rheological behavior of a partially charged poly(acrylamide-co-sodium acrylate) approches that of neutral polymers. Below CD, the concentration dependences of the zero shear viscosity, the plateau modulus, and the longest relaxation time are less than those for neutral systems in agreement with the theoretical model. In particular, the relaxation time is nearly independent of the concentration. The viscosity data indicate the existence of another crossover at a very small concentration (C1 ≈ 10-3 wt %) that should be quite close to C*. Acknowledgment. The authors wish to thank S. J. Candau (U.L.P., Strasbourg) for stimulating discussions. Financial support from Elf-ATOCHEM is also gratefully acknowledged. LA971021I
