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Langmuir 1999, 15, 4120-4122
Concentration Dependence of Radius of Gyration of Sodium Poly(styrenesulfonate) over a Wide Range of Concentration Studied by Small-Angle Neutron Scattering† Yoshiaki Takahashi,*,‡ Naoki Matsumoto,§ Shinji Iio,§ Hidemi Kondo,§ and Ichiro Noda§ Center for Integrated Research in Science and Engineering and Department of Applied Chemistry, Graduate School of Engineering, Nagoya University, Nagoya 464-8603, Japan
Masayuki Imai and Yushu Matsushita Neutron Scattering Laboratory, The Institute for Solid State Physics, The University of Tokyo, Shirakata 106-1, Tokai, Naka 319-1106, Japan Received August 24, 1998. In Final Form: December 21, 1998 Concentration dependence of radius of gyration (Rg) of sodium poly(styrenesulfonate)s (NaPSS) in aqueous solutions without added salt was studied by small-angle neutron scattering measurements over a wide range of concentration. The Rg values are always higher than those of corresponding nonionic polymer, polystyrene, and the concentration (C) dependence of Rg of NaPSS in the semidilute region (C < 0.3 × 103 kg/m3) can be well expressed by the scaling relation (Rg ∝ C-1/4), while Rg values in the concentrated region are almost constant at the value almost equal to that at Θ condition. The crossover concentration from the semidilute to concentrated regions was also consistent with our previous studies of viscoelastic properties of polyelectrolyte solutions without added salts.
Introduction In the dilute region, flexible polymer chains are expanded in good solvents by excluded volume effects from their unperturbed dimension in θ solvents and the excluded volume exponent ν, relating radius of gyration Rg of polymer chains with molecular weight M, Rg ∝ Mν, increases from 0.5 in θ solvents to 0.6 in good solvents.1 With increase of concentration, however, Rg in good solvents decreases toward one in θ solvents since the excluded volume effects are screened at high concentrations as in bulk. According to the scaling concepts of de Gennes,2 the degree of screening of excluded volume effects is expressed by correlation (screening) length ξ. For nonionic linear polymers in good solvents in the semidilute region, its polymer concentration (C) dependence is given by
ξ ∝ C-ν/ (3ν-1)
(1)
and hence the concentration dependence of Rg is predicted as
Rg ∝ M1/2C(1-2ν)/2(3ν-1)
(2a)
Rg ∝ M1/2C-1/8
(2b)
which gives
in good solvents (ν ) 0.6). † Presented at Polyelectrolytes ’98, Inuyama, Japan, May 31June 3, 1998. ‡ Center for Integrated Research in Science. § Department of Applied Chemistry.
(1) Flory, P. J. Principles of Polymer Chemistry; Cornell University Press: Ithaca, NY, 1953. (2) de Gennes, P. G. Scaling Concepts in Polymer Physics; Cornell University Press: Ithaca, NY, 1979.
The concentration dependence of Rg of polystyrene (PS) in good solvents over a wide range of concentrations has been studied by small-angle neutron scattering (SANS).3,4 The experimental concentration dependence of Rg in the semidilute region (C < ca. 0.2 × 103 kg/m3) is consistent with the above scaling relation (eq 2b). For nonionic polymer in good solvents, it has been also reported that the concentration dependence of other properties, e.g., osmotic pressure,5,6 correlation length,7 and viscosity8,9 in the semidilute region, can be well explained by the scaling concepts. In polyelectrolyte solutions, on the other hand, the situation becomes complicated owing to strong electrostatic interactions, which depend on polymer concentration as well as charge density and added salt concentration. Although various theoretical studies on polyelectrolytes in the semidilute region have been published, the experimental studies so far published were limited and there are few systematic studies covering wide ranges of molecular weight and concentration as performed for nonionic polymer solutions.10,11 (3) Daoud, M.; Cotton, J. P.; Farnoux, B.; Jannink; Sarma, G.; Benoit, B.; Duplessix, R.; Picot, C.; de Gennes, P. G. Macromolecules 1975, 8, 804. (4) King, J. S.; Boyer, W.; Wignall, G. D.; Ullman, R. Macromolecules 1985, 18, 709. (5) Noda, I.; Kato, N.; Kitano, T.; Nagasawa, M. Macromolecules 1981, 14, 688. (6) Noda, I.; Higo, Y.; Ueno, N.; Fujimoto, T. Macromolecules 1984, 17, 1055. (7) Yukioka, S.; Higo, Y.; Noda, I.; Nagasawa, M. Polym. J. (Tokyo) 1986, 18, 941. (8) Takahashi, Y.; Isono, Y.; Noda, I.; Nagasawa, M. Macromolecules 1985, 18, 1002. (9) Takahashi, Y.; Noda, I.; Nagasawa, M. Macromolecules 1985, 18, 2220. (10) Dautzenberg, H.; Jaeger, W.; Ko¨tz, J.; Philipp, B.; Seidel, Ch.; Stscherbina, D. Polyelectrolytes Formation, Characterization and Application; Hanser Publishers: Munich, 1994. (11) Dobrynin, A. V.; Colby, R. H.; Rubinstein, M. Macromolecules 1995, 28, 1859.
10.1021/la9810861 CCC: $18.00 © 1999 American Chemical Society Published on Web 03/09/1999
Concentration Dependence of Radius of Gyration
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Recently, we studied12-15 viscoelastic properties of polyelectrolytes over wide ranges of polymer concentration, molecular weight, and added salt concentration Cs and reported that the viscoelastic properties of polyelectrolytes in the semidilute region (C < 0.3 × 103 kg/m3) can be well explained by our theory assuming that the chain length between entanglements is proportional to the correlation length which is determined by Donnan equilibrium, while they become similar to viscoelastic properties of nonionic polymers in the concentrated region (C > 0.3 × 103 kg/ m3), where the electrostatic interactions are screened. According to our theory the correlation length ξ in the semidilute region is given by
ξ ∝ [(i2c2 + 4Cs2)1/2 - 2Cs]-1/4c-1/4
Table 1. Molecular Characteristics of NaPSSa sample code sample 1 hSS32 dSS39 sample 2 hSS140 dSS144
MWPS (×104)
DS (%)
MW (×10-4)
N
1.81 2.20
90.3 93.4
3.24 3.91
174 196
7.46 8.04
94.7 90.3
14.0 14.4
717 718
a M PS, M of polystyrene obtained by gel permeation chromaW w tography; DS, degree of sulfonation.
(3)
where i and c are the effective charge of charged segments and the number of charged segments per unit volume, which is proportional to C, respectively. Equation 3 predicts simple scaling relations at two limiting conditions; ξ ∝ C-1/2 at salt-free condition (Cs , c), while ξ ∝ C-3/4 at high added salt condition (Cs . c) in the same manner as for nonionic polymers in good solvents. The scaling relation at the salt-free condition was also predicted by the theories of de Gennes et al.,16 Pfeuty,17 and Dobrynin et al.,11 assuming the crossover from rigid rods in the dilute region to Gaussian chains in the semidilute region. Using the scaling relations for ξ, we can predict the concentration dependence of Rg of polyelectrolytes in the semidilute region at the two limiting conditions as
Rg ∝ C-1/4 (at salt-free condition)
(4)
Rg ∝ C-1/8 (at high added salt condition)
(5)
It is interesting to examine the applicability of the theoretical relations to other properties such as Rg and ξ besides the viscoelastic properties over a wide range of polymer concentration, molecular weight, and added salt concentration. Because of the limitation in the ability of SANS instrument such as intensity and wave vector range and the experimental difficulties such as dialysis at high polymer concentrations, we studied the concentration dependence of Rg of sodium poly(styrenesulfonate)s (NaPSS) with relatively low molecular weights at the salt-free condition only in this work. We will report the polymer and added salt concentration dependencies of correlation length for high molecular weight samples at the high added salt condition elsewhere. Rg was measured over a wide range of polymer concentrations by using the contrast matching method,18 in which the average scattering length of a mixture of deuterium labeled and unlabeled polymers, ap, is equal to that of a mixture of D2O and H2O, a0, when the degrees of polymerization of both polymers are the same. Experimental Section Materials. Two pairs of ordinary (hydrogenated) and deuterated sodium poly(styrenesulfonate)s, designated as hSS and dSS, respectively, were used in this study. Ordinary and (12) Yamaguchi, M.; Wakutsu, M.; Takahashi, Y.; Noda, I. Macromolecules 1992, 25, 470. (13) Yamaguchi, M.; Wakutsu, M.; Takahashi, Y.; Noda, I. Macromolecules 1992, 25, 475. (14) Takahashi, Y.; Hase, H.; Yamaguchi, M.; Noda, I. J. Non-Cryst. Solids 1994, 172-174, 911. (15) Noda, I.; Takahashi, Y. Ber. Bunsen-Ges. Phys. Chem. 1996, 100, 696. (16) de Gennes, P. G.; Pincus, P.; Velasco, R. M.; Brochard, F. J. Phys. (Paris) 1976, 37, 1461. (17) Pfeuty, P. J. Phys., Collog. 1978, 39, C2-, 149. (18) Jahshan, S. N.; Summerfield, G. C. J. Polym. Sci., Polym. Phys. Ed. 1980, 18, 1859.
Figure 1. An example of plots of ln I(q) vs q for sample 2 at 0.561 × 103 kg/m3. deuterated PSs were prepared by anionic polymerization technique and then sulfonated by Vink’s method.19 Characterizations of these samples were performed by the same methods as in previous works15,20 and molecular characteristics of the samples are listed in Table 1. As shown in the Table 1, hSS and dSS in each pair have almost the same degree of polymerization (N), so that we can use the contrast matching method. The average scattering length density of a mixture of hSS and dSS at the 1:1 mixing ratio by weight, ap ) 0.423 × 10-15 m/A3, was calculated by using the scattering lengths of constituent atoms and the partial molar volumes of hSS and dSS which are assumed to have the same value (125.7 cm3/mol) determined for hSS.21,22 Then the mixing ratio of D2O to H2O was determined as D2O:H2O ) 0.655:0.345 by weight to make a0 equal to ap. Each of hSS and dSS samples was freeze-dried and further dried in vacuo to reduce the content of residual water to less than 1 wt %. The prescribed amounts of 1:1 mixtures (by weight) of hSS and dSS were directly dissolved in the mixture of D2O and H2O. The total polymer concentrations were determined by the UV adsorption method.12,20 Small-Angle Neutron Scattering. SANS measurements were performed at room temperature (about 25 °C) with the SANS-U spectrometer at the Neutron Scattering Laboratory of the Institute for Solid State Physics, The University of Tokyo in Tokai, Ibaraki, Japan. Sample cells made of quartz (sample thickness 2 mm) were used for all measurements. The sample to detector distances used were 12 or 8 m, and the wavelength, λ, was also changed (0.7, 0.8, or 1.0 nm) to cover the necessary range of low scattering vector appropriately and to measure the scattered intensity efficiently. The scattered intensity obtained on a two-dimensional detector was circularly averaged and corrected for background and incoherent scattering of the component polymers and solvent. Rg of NaPSS was evaluated by the Guinier plot.
Results and Discussion Figure 1 shows an example of plots of ln I(q) against q2 (Guinier plot). Here, I(q) is the scattering intensity and q is the scattering vector, where q ) (4π/λ) sin(θ/2) and θ is the scattering angle. In a low q range, the data can (19) Vink, H. Macromol. Chem. 1981, 182, 279. (20) Takahashi, Y.; Iio, S.; Matsumoto, N.; Noda, I. Polym. Int. 1996, 40, 269. (21) Eguchi, Y. Master Course Thesis, Nagoya University, 1964. (22) Kawaguchi, M.; Hayashi, K.; Takahashi, A. Macromolecules 1984, 17, 2066.
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Takahashi et al.
Table 2. Radius of Gyrations of NaPSS sample
C (103 kg/m3)
Rg (nm)
sample 1
0.113 0.168 0.222 0.338 0.442 0.909 0.129 0.210 0.278 0.397 0.561 0.816
7.1 6.1 5.7 5.5 5.3 4.7 12.7 11.7 11.4 10.8 9.1 9.7
sample 2
Figure 2. Double logarithmic plots of Rg vs C for NaPSS at the salt-free condition. Symbols are denoted in the figure.
be fitted to a straight line, the slope of which is equal to -Rg2/3 when the condition q‚Rg e 1.3 is fulfilled. In this case, the straight line persists up to q‚Rg ) 1.3. The Rg data are tabulated in Table 2. Figure 2 shows double logarithmic plots of Rg against C. It is clear that Rg for both samples slightly decreases with increase of concentration, but the concentration dependence becomes very small at higher concentrations. To discuss the overall behavior of the data in Figure 2 in more detail, they are replotted in double logarithmic form of Rg/N1/2 vs C in Figure 3, together with the data reported by Nierlich et al.,23 except their datum at the lowest concentration. The data of PS in good solvents3,4 are also shown in this figure for comparison. The N value for each sample used is the average value of h and d components. As mentioned in the Introduction, the Rg/N1/2 data of PS compose a single line irrespective of molecular weight, and the concentration dependence in the semidilute region can be expressed by eq 2b up to about 0.2 × 103 kg/m3, as shown by the broken line (slope ) -0.125), but it becomes almost constant to be equal to Rg in θ solvents24 in the concentrated region shown by the dotted line. As shown in Figure 3, Rg/N1/2 data of NaPSS compose a single line irrespective of molecular weight in the same manner as those of PS, though the former data are always larger than the latter data. Apparently, the concentration (23) Nierlich, M.; Boue´, F.; Lapp, A.; Oberthu¨r, R. J. Phys. (Paris) 1985, 46, 649. (24) Matsushita, Y.; Noda, I.; Nagasawa, M.; Lodge, T. P.; Amis, E. J.; Han, C. C. Macromolecules 1984, 17, 1785.
Figure 3. Double logarithmic plots of Rg/N1/2 vs C for the data in Figure 2, together with the data reported by Nierlich et al.23 and data for polystyrenes in good solvents reported by Daoud et al.3 and King et al.4 Circles and squares denote the data in this work (from Figure 2), while other symbols are denoted in the figure. Solid and broken lines denote slopes of -0.25 and -0.125 (eqs 4 and 2b), respectively, and dash-dot and dotted lines denote the constant values for NaPSS and PS in the concentrated region.
and molecular weight dependencies of Rg of NaPSS at the salt-free condition agrees with the scaling relation (eq 4) denoted by the solid line (slope ) -0.25) over a wide range of concentration up to say, 0.3 × 103 kg/m3. This result can be also explained by the polymer concentration dependence of persistence length calculated by the theory of Le Bret,25 as discussed by Nierlich et al.23,27 At concentrations higher than 0.3 × 103 kg/m3 the Rg data appear to deviate from eq 4 and become almost constant as shown by the dash-dot line. From this result alone we cannot have a definite conclusion about the existence of concentration region, but we may regard the region where C > 0.3 × 103 kg/m3 as the concentrated region, since the crossover from the semidilute to concentrated regions was also observed at ca. 0.3 × 103 kg/m3 in the viscoelastic data of poly(N-methyl-2-vinylpyridinium chloride)s14 at the salt-free condition as mentioned in the Introduction; the viscoelastic properties are explained by a uniform network model in the concentrated region (C > 0.3 × 103 kg/m3) implying that the electrostatic interactions are screened, while they can be explained by the theory based on eq 4 in the semidilute region (C < 0.3 × 103 kg/m3). Moreover, it is to be noted that the constant value (Rg/N1/2 ) 0.368) in the concentrated region is almost equal to that of NaPSS in θ solvents (Rg/N1/2 ) 0.343)26 but slightly larger than that of PS (Rg/N1/2 ) 0.292).24 In summary, for NaPSS solutions at the salt-free condition we conclude that (1) the concentration dependence of Rg in the semidilute region (C < 0.3 × 103 kg/m3) can be well expressed by the scaling relation (eq 4), (2) there is a crossover from the semidilute to concentrated regions at about 0.3 × 103 kg/m3, and (3) Rg is almost constant in the concentrated region and almost equal to that in θ solvents. LA9810861 (25) Le Bret, M. J. Chem. Phys. 1982, 76, 6243. (26) Raziel, A.; Eisenberg, H. Isr. J. Chem. 1973, 11, 183. (27) Nierlich, M.; Boue´, F.; Lapp, A.; Oberthu¨r, R. Colloid Polym. Sci. 1985, 263, 955.
