Viscosity of Semidilute and Concentrated Nonentangled Flexible

May 24, 2019 - We report viscosity data of nonentangled sodium polystyrene sulfonate (NaPSS) in salt-free aqueous solution as a function of polymer ...
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Viscosity of Semidilute and Concentrated Nonentangled Flexible Polyelectrolytes in Salt-Free Solution Carlos G. Lopez* and Walter Richtering Institute of Physical Chemistry, RWTH Aachen University, Landoltweg 2, 52056 Aachen, Germany

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S Supporting Information *

ABSTRACT: We report viscosity data of nonentangled sodium polystyrene sulfonate (NaPSS) in salt-free aqueous solution as a function of polymer concentration (c) and degree of polymerization (N). Different empirical equations are examined and found not to describe the semidilute solution viscosity over a wide concentration range and/or to yield values of [η] that do not match dilute solution measurements. Deviations from the scaling prediction of ηsp ∝ c1/2 (Fuoss’ law) are observed at high concentrations. Specifically, we find ηsp ≈ N1.26c1/2 e1.4c in the semidilute regime, which agrees with the scaling prediction only for c ≲ 0.02 M. The viscosity data presented in this study and in earlier reports show a high degree of consistency. A comparison with diffusion measurements for NaPSS in salt-free solution by Oostwal and co-workers suggests that the disagreement between the scaling theory and experiments does not arise solely from the concentration dependence of the monomeric friction coefficient. applications such as turbulent drag reduction,22−25 but many applications, in particular those which require fluids to have high viscosities, employ polyelectrolytes in the semidilute regime, where their effect on the shear and extensional rheology of fluids is much larger than in the dilute region.26−28 Although flexible neutral polymers entangle at approximately ce ≃ 10c*, high molar mass polyelectrolytes in salt-free solution can display values of ce/c* ≃ 102 to 104. The large semidilute, nonentangled region results in a unique viscosity−concentration profile and makes them interesting systems to study nonentangled polymeric dynamics.14,16,17,29 We next review the main results of the scaling and empirical approaches to polyelectrolyte dynamics and then present new results for sodium polystyrene sulfonate (NaPSS) in salt-free solution over a broad range of polymer concentrations. We show that common empirical equations do not provide a good description of polyelectrolyte viscosity over the entire concentration range studied and that the scaling model of Dobrynin et al.18 only agrees with experimental data over a limited concentration range. Discrepancies with theoretical expectations are discussed in Section 5. Previous studies have

1. INTRODUCTION Polyelectrolytes are polymers with ionic groups along their backbone, which can dissociate in solution.1−3 Counterion dissociation leads to a large entropic gain which confers polyelectrolytes good solubility in media with high dielectric constants, most importantly water. As a result of their high phase stability, good biocompatibility, and ability to form complexes with charged species, polyelectrolytes have found broad applications as thickening and structuring agents in paints, oil-drilling fluids, and food and cosmetic products.4−8 Beyond their importance as rheology modifiers and complexing agents, polyelectrolytes play an important role in biology. For example, they are relevant to the origin of life or joint lubrication,2,9 as well as important components of cartilage tissue.10−12 Numerous reports have analyzed the viscosity of dilute and nondilute polyelectrolytes in solution, aiming at understanding and predicting their rheological properties from molecular parameters.13−17 Most studies use either scaling models1,14,18 or empirical equations to describe the viscosity profiles of polyelectrolytes as a function of polymer and added salt concentration.19−21 Owing to their highly extended conformations in salt-free solution, polyelectrolytes display low overlap concentrations (c*). Dilute (c < c*) solution properties are important for polymer characterization, and some industrial © 2019 American Chemical Society

Received: April 1, 2019 Revised: May 23, 2019 Published: May 24, 2019 5626

DOI: 10.1021/acs.jpcb.9b03044 J. Phys. Chem. B 2019, 123, 5626−5634

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The Journal of Physical Chemistry B

2.2. Scaling Theory of Polyelectrolyte Solutions. De Gennes’ 1978 isotropic model54 gave the first theoretical explanation to Fuoss and co-workers’ experimental observations. According to this model (later extended by Pfeuty55 and Dobrynin et al.18), polyelectrolytes adopt a rodlike conformation in dilute solution, so that their end-to-end distance (R) scales with the degree of polymerization (N) as R ≃ b′N, where b′ is the effective monomer size and the overlap concentration scales as c* ≃ N/R3 ≈ N−2. Above the overlap concentration, polyelectrolytes display dilute-like statistics up to the correlation length ξ and Gaussian conformation for larger distances. The correlation length in salt-free solution is predicted to scale as ξ = (b′c)−1/2. This prediction agrees well with the experimental results up to the concentrated crossover (cD), beyond which a scaling of ξ ∝ c−1/4 is observed.56 Semidilute chains are random walks of correlation blobs and their end-to-end distance is R ≃ b′1/4c−1/4N1/2.16,18 The diffusion coefficient of a semidilute nonentangled polyelectrolyte chain is, according to the Rouse model,

suggested that entanglement or concentration dependence of the local friction coefficient may explain some of the inconsistencies between theory and experiments. A careful analysis of rheology and diffusion data suggests that these two phenomena cannot fully account for the observed behavior.

2. LITERATURE REVIEW 2.1. Concentration Dependence of the Viscosity of Polyelectrolyte Solutions. The viscosity of nonentangled polyelectrolytes was first studied by Fuoss and co-workers,19,30,31 who observed an increase in the reduced viscosity (ηred) with decreasing polymer concentration in salt-free and low added salt solutions. By analogy with simple electrolytes, they proposed expressing the viscosity of polyelectrolyte solutions according to what is now known as the Fuoss law ηred = A /(1 + Bc1/2) + D

(1)

where A + D is identified as the intrinsic viscosity and B is related to the charge density of the polymer.32,33 D is sometimes identified with the intrinsic viscosity under high salt conditions ([η∞]).33 Equation 1 predicts that the reduced viscosity is an increasing function of c in the c → 0 limit. Experimental results however show that for sufficiently low concentrations, the reduced viscosity of salt-free polyelectrolytes decreases with polymer concentration.34−36 Several expressions for the viscosity of dilute and semidilute polyelectrolyte solutions, including those of Liberti and Stivala,37 Schaefgen and Trivisonno,38 Yuan, Dougherty and Stivala,39 and Fedors,40 have been applied widely to ionic polymers, in particular polysaccharides.41−44 We discuss these in more detail in the Supporting Information. More recently, an empirical approach by Wolf and coworkers to model the viscosity of polyelectrolytes in salt-free solution or excess added salt, 45 has been extensively employed46−52 c[η] + λ(c[η])2 ln η/ηs = 1 + βc[η] + γ(c[η])2

DR =

kBT Nξζξ

(3)

where kB is Boltzmann’s constant, T is the absolute temperature, Nξ = b′N/ξ is the number of correlation blobs in a chain, and ζξ is the friction coefficient of a single correlation blob, given by Stokes’s law ζξ ≃ 6πηsξ

(4)

Chains relax following Zimm dynamics up to the correlation length, which marks the onset of hydrodynamic screening. The relaxation time of a correlation blob is therefore τξ ≃

ξ2 ≃ ηsξ 3/(kBT ) Dξ

(5)

where Dξ = kBT/ζξ is the diffusion coefficient of a correlation blob. The longest relaxation time of a chain can be approximated as that of a Rouse chain of correlation blobs

(2)

τR ≃ τξ(Nξ)2 ∝ N 2c −1/2

where ηs is the viscosity of the solvent; λ, β, and γ are free parameters; and [η] is the intrinsic viscosity of the polymer. Equation 2 with γ = 0 was first developed as an empirical formula that could closely follow the viscosity−concentration curves of dilute polyelectrolytes in both salt-free and excess salt solution.47 The quadratic term in the denominator was then added to broaden the applicability of eq 2 to higher polymer concentrations. The first studies aimed at establishing the suitability of eq 2 to model experimental data on salt-free polyelectrolytes, and the variation of [η], β, γ, and λ with molar mass and added salt, employed gravity-driven capillary viscometers to measure the viscosity of polyelectrolyte solutions. Comparison with literature data on NaPSS solutions of similar molar masses and concentrations16,53 makes it clear that at the applied shear rates (70−270 s−1), some of the data in ref 49 correspond to the shear thinning region. Note also that much of the literature that followed these original studies suffer from similar shortcomings,46−51 see, for example, refs17,53 for a discussion of the effects of shear thinning in the reported viscosities of carboxymethyl cellulose and polystyrene sulfonate, respectively. The conclusions drawn from such reports are therefore not directly applicable to the present study, which deals with the Newtonian limit of polyelectrolyte viscosity.

(6)

The terminal modulus of nonentangled polymer solutions (both neutral polymers and polyelectrolytes) is G = kBTc/N or kBT per chain. The specific viscosity of a polyelectrolyte solution can then be calculated as ηsp ≃ GτR/ηs ηsp ≃ b′3/2 Nc1/2

(7)

Equation 1 provides an interpolation function between the scaling predictions for dilute solution (ηsp ∝ c) and semidilute (ηsp ∝ c1/2). The crossover between these two regimes is expected to occur at c ≃ c* or when ηsp ≃ 1.14,53 Experimental values for the specific viscosity−concentration exponent in semidilute nonentangled solutions often deviate from the scaling prediction of 1/2. Semiflexible polyelectrolytes display larger exponents, of α ≃ 0.6−1.17,57−61 For flexible polyelectrolytes, the situation is less clear: agreement with the Fuoss exponent of 1/2 has been reported,16,29,62−64 but results often use data that do not correspond to the Newtonian region, see refs53,64 for a discussion. Boris and Colby53 measured a lower exponent of α ≃ 0.33 for a highmolecular-weight NaPSS sample in the low c and low added salt region which increased to the Fuoss exponent of α ≃ 0.5 at higher polymer concentrations. Ü zü m et al.65 observed 5627

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The Journal of Physical Chemistry B exponents (α ≃ 0.85) for several low molar mass NaPSS samples. Simulations by Liao et al.66 showed an increase in α for flexible polyelectrolytes at high concentrations, which was assigned to an increase in the monomeric friction coefficient. This observation is qualitatively consistent with the experimental observations of Takahashi et al. for concentrated NaPSS solutions.67 In this article, we seek to establish the value of the semidilute nonentangled viscosity−concentration exponent and to resolve the apparent deviations from the scaling theory reported in the literature. For this purpose, we measure the viscosity of NaPSS over a wide range of polymer concentration for samples of varying molar mass. Our results show that the viscosity− concentration exponent α increases with polymer concentration. Part of this deviation may be caused by an increase in the monomeric friction coefficient with polymer concentration, as was suggested in earlier works. The exponent of α = 1/2 is observed only for c ≲ 0.01 M.

1.2 M56,68) and the bulk (no solvent) concentration cbulk = 8.3 M.56

4. DATA ANALYSIS 4.1. Empirical Equations. Figure 2 plots the viscosity of the four samples studied in the Fuoss representation. Fits to eq

3. EXPERIMENTAL METHODS AND RESULTS NaPSS of varying molar mass was purchased from Polymer Standard Services (Mainz, Germany). The polymers were dialyzed by the manufacturer to remove ionic impurities. The polydispersity of the samples (estimated from the parent polystyrene) was around 1.1−1.2. Deionized (DI) water was from a Milli-Q source with a conductivity of 0.05 μS cm−1. Solutions were prepared by mixing appropriate masses of NaPSS and DI water, assuming a polymer density of 1.65 g/ mL and no change in volume upon mixing. The concentrations are given in moles of repeating units per liter. Viscosity measurements were performed on a stresscontrolled Kinexus Pro rheometer, using cone and plate geometries with an angle of 1° and diameters of 40 or 60 mm. The temperature was fixed at T = 298 K with a Peltier plate. A solvent trap was employed to minimize evaporation. All viscosity data correspond to the zero-shear rate limit. Figure 1 plots the specific viscosity of NaPSS as a function of polymer concentration for four different molecular weights. All data are above the overlap concentration (i.e., ηsp ≳ 114,17,53). Lines indicate the crossover to the concentrated regime (cD ≃

Figure 2. Fuoss plot for NaPSS in salt-free water. Lines at low concentrations are fits to eq 1 with D = 0. Values for A and B are collected in Table 1. Symbols have the same meaning as shown in Figure 1. Fits to Fedor’s equation are shown in the Supporting Information and the best fit parameters and range of applicability are compiled in Table 1.

1 with D = 0 describe the data well only at low concentrations, see Table 1. Leaving D as a free parameter increases the range of applicability, but the downturn observed beyond c ≃ 0.5 M cannot be accounted for by eq 1. The best fit parameters for the lines included in Figure 2 are compiled in Table 1. Figure 3 plots the reduced viscosity of the four NaPSS samples studied as a function of polymer concentration. All samples display an upturn in the reduced viscosity at low polymer concentrations. No peak is observed in the concentration range studied, in agreement with earlier studies, which find that the reduced viscosity of NaPSS reaches a maximum in the c ≃ 2.5 × 10−5 to 7 × 10−4 M range.36,53,69 Although Wolf’s model (eq 2) does not correctly describe the data over the entire concentration range studied (see Figure 3 and Table 1), it tracks experimental results over a much wider polymer concentration range than the other equations considered. As expected, the intrinsic viscosity increases with increasing molar mass. The parameter λ decreases at a similar rate so that an approximately constant value of λ[η] ≃ 2.1 ± 0.7 M−1 is observed. The parameters β and γ decrease and increase with increasing molar mass, respectively. We discuss the physical significance of the parameters collected in Table 1 in Section 5.5. 4.2. Scaling Analysis. Following Colby’s criterion14,17,53 of ηsp(c*) = 1, the overlap concentration c* is obtained for each molecular weight and listed in Table 1. An approximately constant value of c*N2 ≃ 2400 M is observed, in agreement with earlier experimental results for higher molar mass samples.53,70 Deviations from the scaling prediction of ηsp ∝ c1/2 are apparent in Figure 1 at high concentrations. Values of the power law exponent for ηsp with concentration (α) for our samples and other data collected from the litera-

Figure 1. Specific viscosity of NaPSS as a function of polymer concentration in salt-free aqueous solution for different molecular weights: 29.1 (○), 63.9 (△), 148 (□), and 261 kg/mol (◇). Triangles indicate a slope of 0.5 predicted by scaling theory. The dashed dotted line is the crossover to the concentrated regime (cD = 1.2 M)56,68 and the dashed line is the bulk concentration of NaPSS (≃8.3 M). 5628

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The Journal of Physical Chemistry B Table 1. Fit Parameters and Range of Applicability for Eqs 1, 2, and S3a Fuoss (eq 1)

Fedors (eq S3)

Wolf (eq 2)

Colby

sample

[η]F (M−1)b

B

rangec

[η]Fed (M−1)

cm

range

[η]W (M−1)

λ

β

γ

1/c* (M−1)

2.91 ×104 6.39 ×104 1.48×105 2.61×105

9.3 67 670 4700

0.066 0.11 0.055 0.022

c ≲ 5/[η]F c ≲ 12/[η]F c ≲ 54/[η]F c ≲ 91/[η]F

7.5 38 590 1000

0.64 0.072 0.057 0.0036

c ≲ 4/[η]Fed c ≲ 6/[η]Fed c ≲ 20/[η]Fed c ≲ 20/[η]Fed

13 45 200 970

0.14 0.035 0.013 0.0025

1.1 0.76 0.62 0.41

0 1.0 ×10−5 2.2 ×10−5 1.7 ×10−5

7.7 37 220 1000

a The last column estimates the reciprocal of the overlap concentration according to Colby’s criterion. bD is set to zero for all fits; therefore, we take [η] = A. cThe lowest concentration studied for all samples corresponds to that at which ηsp ≃ 1.

Colby’s sample for c ≳ 0.02−0.05 M. The value of 1/2 predicted by scaling theory is only observed for c ≲ 0.02 M. For higher polymer concentrations, α deviates to higher values. The data of Boris and Colby deviate to an exponent of α ≃ 1/3 for c ≲ 7 × 10−3 M, which is not observed for any other sample. Note that any residual salt present in the polymer samples modifies the viscosity of a polyelectrolyte solution but does not affect α.74 Figure 5a plots the experimental results for the specific viscosity divided by the scaling prediction ηsp = Nc1/2, where we have neglected N- and c-independent prefactors. The plot

Figure 3. Reduced viscosity of NaPSS as a function of polymer concentration in salt-free solution for different molecular weights. Lines are fits to eq 2, and best fit parameters are compiled in Table 1. Symbols have the same meaning as shown in Figure 1.

ture16,53,65,67,69,71,72 are plotted in Figure 4 as a function of polymer concentration. For each adjacent pair of data points

Figure 4. Power law exponent of specific viscosity with concentration as a function of polymer concentration for samples of different molar masses from this work and refs.16,53,65,67,69,71 Black line shows the scaling prediction of α = 1/2. Blue line is a guide to the eye. Figure 5. (a) Experimental results for specific viscosity of NaPSS in salt-free solution divided by scaling prediction. Data from this work and refs.16,53,65,69,71 Symbols have the same meaning as shown in Figure 4. Circles with patterned fill are for sample with Mw = 177 kg/ mol from ref 71. (b) Same as part abut each data set for a given Mw is shifted vertically by a factor of Aη instead of N−1, see Figure 7. The red line is a fit to an exponential function (ηspAηc−1/2 = 0.016 e1.4c), and the blue line is a fit to a sum exponential and a stretched exponential:

(ci,ηsp,i), (cj,ηsp,j), we compute the power law exponent as αi,j = log10(ηsp,j/ηsp,i)/log10(cj/ci). Values of αi,j are assigned the xcoordinate ci,j = 10[log10(cicj)]/2. Data points that are very close ( 10−4 M, which correspond to the low-salt regime.73 The exponent α is independent of molar mass for the data considered in Figure 4 expect for Boris and

2

(ηspAηc−1/2 = 0.016 e1.4c + 0.0007 e1.3c ). 5629

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The Journal of Physical Chemistry B of ηspc−1/2N−1 versus c fails to collapse data onto a single curve, probably because of the nonlinear dependence of ηsp on N, as reported in an earlier paper.16 We therefore plot ηspAηc−1/2 as shown in Figure 5b as a function of polymer concentration, where Aη is adjusted to make the lowest concentrations for each molar mass collapse into a single curve. We find a variation of Aη = 3.9N−1.26±0.06, consistent with ref 16, see also Figure 7. The data are well fitted by ηspAηc−1/2 ≃ 0.016 e1.4c + 2 0.0007 e1.3c over the entire concentration range. The use of stretched exponentials to describe solution viscosity data75−78 is discussed in the Supporting Information. We find that a power law and/or the product of a power law and an exponential, depending on the concentration range, fit the N and c dependence of the data better than stretched exponentials.

concentration. We multiply the data by AD to adjust the diffusion coefficients to the same value at c = 0.03 M. The parameter AD is proportional to the molecular weight (see Figure 7), in agreement with scaling theory. We further divide

5. DISCUSSION The viscosity data both from this study and earlier reports show deviations from the predictions of Dobrynin’s scaling model for the concentration and molar mass dependence of salt-free polyelectrolyte solutions. Next, we consider some possible causes for the observed disagreements between theory and data. 5.1. Monomeric Friction Coefficient. A simple way to explain the apparent disagreement with scaling theory is to assume an increase in the monomeric friction coefficient with increasing polymer concentration. Such behavior has been observed for a number of neutral polymers and polyelectrolytes in solution.66,79−82 This would result in an increase in τξ, the relaxation time of a correlation blob (eqs 4 and 5). The temperature dependence of the viscosity of NaPSS solutions and their thermal expansion coefficient could in principle be used to estimate the effects of an increase in local friction,83,84 but we find that evaporation prevents us from measuring the viscosity of solution over a sufficiently wide temperature range to use this approach. As we expect the effect of local friction to affect the viscosity and diffusion coefficient in a similar way, we next consider data for the translational diffusion coefficient of NaPSS in salt-free solution reported by Oostwal et al.71,85 Figure 6 plots the diffusion coefficient of NaPSS in salt-free D2O for different molecular weights as a function of

Figure 7. N dependence of AηN and AD/N. Lines are best fit power laws with the exponents indicated on the graph. Values of Aη are for data from this work, ref 88, and references listed in Figure 5. Errors for the power law exponents are 95% confidence interval calculated using linear regression, see ref 16. 2

the data by ψ(c) = 0.016 e1.4c + 0.0007 e1.3c (the blue line shown in Figure 5b). If the deviations from scaling theory observed in Figure 5 are due to a variation of the monomeric friction coefficient with concentration, the plot shown in Figure 6 should reduce all data into a single curve varying as c0N0. This behavior is not observed as the lowest and highest molar mass samples deviate upward and downward at high concentrations, respectively. The intermediate molar masses are approximately constant within the scatter of the data. As we have no reason to discard the 16 and the 370 kg/mol data sets, we conclude that interpreting the deviations of viscosity data from scaling theory as arising from a variation of the monomeric friction coefficient with concentration is inconsistent with current experimental data on NaPSS. 5.2. Entanglement Effects. An alternative way to resolve the observed disagreements could be to assume an increase in the terminal modulus of NaPSS solutions with increasing concentration, as would be expected for entangled solutions. The independence of the exponent α on the molar mass of the polymer for all samples (except for that of Boris and Colby) suggests that the data plotted in Figure 4 correspond to the nonentangled regime. Data by Boris and Colby53 show that the modulus of NaPSS with Mw ≃ 1.2 × 106 g/mol does not exceed kBT per chain for c < 0.5 M. All other data shown in Figure 4 are either below this concentration or correspond to a much lower molar mass and we can therefore rule out an increase in G. This is further confirmed by Takahashi’s et al.67 data discussed earlier, which suggest that all data shown in Figure 6 are nonentangled. 5.3. Deviations of Static Properties from Dorbynin’s Scaling. Dobrynin et al.’s model expects R ∝ N1/2c−1/4 in the semidilute regime. Using experimental values for the Kuhn length (lK)86 along with the scaling assumption that semidilute polyelectrolyte chains are Gaussian on length scales larger than lK, we obtain R = (LlK)1/2 ∝ (2 + 4.1c−1/2)1/2 and αR ≡ d(log R)/d(log c) ≃ −0.5/(2 + c), which agrees with Dobrynin et al.’s scaling prediction of αR = 0.25 only in the c → 0 limit.87

Figure 6. Diffusion coefficient for NaPSS in salt-free D2O. Data are multiplied by AD to remove the influence of molar mass and divided by ψ(c) (blue line shown in Figure 5). Data are from ref 85. 5630

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The Journal of Physical Chemistry B Alternatively, fitting a power law to SANS data in the 0.07 < c/ M < 4.3 concentration range gives Rg/N1/2 ∝ c−0.22±0.03, which is slightly lower than Dobrynin et al.’s prediction. Deviations to lower values of αR lead to higher values α. However, even in the limit of αR = 0 (analogous to the θ solvent condition), a value of α = 2 is expected. The small differences between the theoretical estimates and experimental measurements of NaPSS conformation do not therefore suffice to explain the results observed in Figures 4 and 5. 5.4. Do Nonentangled Polyelectrolytes Follow Rouse Dynamics? The data accumulated in the literature until now do not give a clear picture about the validity of the Rouse model for polyelectrolytes in salt-free solution. On the one hand, Takahashi et al.’s67 oscillatory shear data show that the mechanical relaxation spectra of nonentangled polyelectrolyte solutions at high concentrations are well described by the Rouse model. Further, Oostwal et al.’s diffusion data are consistent with the D ≈ N−1c0 scaling expected for ideal Rouse chains. On the other hand, the viscosity data presented in this study and in earlier literature show clear inconsistencies with the Rouse behavior: a plot of AηN versus N, shown in Figure 7, shows a power law of AηN ∝ N−0.26±0.06, in contrast to the scaling prediction of AηN ∝ N0 and in agreement with an earlier study which found ηsp ∝ N1.24±0.08 for nonentangled solutions of NaPSS in salt-free water.16 The failure of Figure 6 to reduce all data onto a single curve also suggests the nonRouse behavior. A further check may be carried out from Oostwal’s data. Scaling theory expects Dηsp /c ≈ R2/N ∝ c −1/2

In summary, diffusion and rheology data appear to rule out that either an increase in the modulus of NaPSS solutions or an increase in the monomeric friction coefficient are sufficient to explain the observed deviations from scaling theory. Deviations of experimental results from Dobrynin et al.’s theory for the static properties of NaPSS do not suffice to explain the observed discrepancies in the viscosity−concentration exponent either. From an experimental point of view, the viscosity data for NaPSS in salt-free solution from this study and other literature reports obtained for different samples and using different measuring techniques are overall highly consistent. By contrast, all available diffusion data are from a single study (using a single method, namely, pulsed field gradient nuclear magnetic resonance).89−91 Further studies on the diffusion coefficient of NaPSS in semidilute solution would be useful to understand the apparent disagreement between the viscosity and diffusion data. 5.5. Empirical Equations: Range of Applicability and Intrinsic Viscosity. The parameters collected in Table 1 show that the various empirical approaches discussed yield qualitatively similar results for the intrinsic viscosity of our samples. None of these equations describe the concentration dependence of the viscosity over the entire range studied and only Wolf’s model fits the data over a fairly wide range of concentrations. We note however that [η]W ≲ 1/c* is observed for the two highest molar mass samples, in contrast with the expectation that for salt-free polyelectrolytes, [η] > 1/c* because chains are contracted at the overlap concentration with respect to their conformation at infinite dilution. Of the models considered, only eq 1 appears to capture this feature correctly. Application Wolf’s equation to NaPSS in the dilute region yields [η]W > 1/c*,49 which suggests that the fits shown in Figure 3 use an artificially low value of [η]. Imposing [η]W > 1/c* significantly reduces the quality of the fits for the two high molar mass samples. In summary, we find that the various empirical equations analyzed describe semidilute viscosity over a limited concentration range and/or yield values for the intrinsic viscosity that do not agree with the infinite dilution limit.

(8)

where N- and c-independent prefactors have been dropped. For three of Oostwal’s samples, all quantities in eq 8, except R, are known, which allows us to compare the values of R inferred from dynamic data with direct measurements of the chain conformation by SANS. This comparison is shown in Figure 8.

6. CONCLUSIONS The scaling prediction that the specific viscosity of nonentangled polyelectrolytes in salt-free solution scales as ηsp ∝ c1/2 is observed for NaPSS in DI water only for c ≲ 0.02 M. At higher concentrations, the power law exponent increases with polymer concentration. Diffusion data by Oostwal et al. suggest that these results cannot be interpreted as arising solely from an increase in the monomeric friction coefficient with increasing polymer concentration. These results show that the Rouse model may not be appropriate to describe the nonentangled dynamics of polyelectrolyte solutions, but further experimental work on the diffusion of polyelectrolytes is necessary to confirm this. The range of applicability of several empirical equations to describe the concentration dependence of NaPSS in salt-free solution is evaluated. Wolf’s model provides a moderately good description in the semidilute range but requires values of the intrinsic viscosity that do not match with those obtained from measurements made at high dilution. We hope this work will stimulate further research into this topic.

Figure 8. Dηsp/c as a function of c−1/2 for Oostwal’s data.71 Line is Dηsp/c = 1.13 × 10−9c−1/2 where the prefactor is left as a free parameter and the relation is forced through the origin as expected by scaling theory.

The three molecular weights collapse onto a single curve and display an approximately linear relation, as expected by scaling theory. As chains tend to their θ dimensions at high concentrations, we expect Dηsp/c to tend toward a fixed value of R2/N ≃ 37 Å for low values of c−1/2. This is not observed in Figure 8, perhaps because of the limited concentration range considered. 5631

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The Journal of Physical Chemistry B

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trolyte Aqueous Solution (PSSNa): Correlation Between Scattering and Viscosity. J. Phys. Chem. B 2011, 115, 8951−8960. (16) Lopez, C. G.; Richtering, W. Conformation and Dynamics of Flexible Polyelectrolytes in Semidilute Salt-free Solutions. J. Chem. Phys. 2018, 148, 244902. (17) Lopez, C. G.; Colby, R. H.; Graham, P.; Cabral, J. T. Viscosity and Scaling of Semiflexible Polyelectrolyte NaCMC in Aqueous Salt Solutions. Macromolecules 2016, 50, 332−338. (18) Dobrynin, A. V.; Colby, R. H.; Rubinstein, M. Scaling Theory of Polyelectrolyte Solutions. Macromolecules 1995, 28, 1859−1871. (19) Fuoss, R. M.; Strauss, U. P. The Viscosity of Mixtures of Polyelectrolytes and Simple Electrolytes. Ann. N.Y. Acad. Sci. 1949, 51, 836−851. (20) Xiong, X.; Wolf, B. A. Intrinsic Viscosities of Polyelectrolytes: Specific Salt Effects and Viscometric Master Curves. Soft Matter 2014, 10, 2124−2131. (21) Tam, K. C.; Tiu, C. Improved Correlation for Shear-Dependent Viscosity of Polyelectrolyte Solutions. J. Non-Newtonian Fluid Mech. 1993, 46, 275−288. (22) Bonn, D.; Amarouchène, Y.; Wagner, C.; Douady, S.; Cadot, O. Turbulent Drag Reduction by Polymers. J. Phys.: Condens. Matter 2005, 17, S1195. (23) Wyatt, N. B.; Gunther, C. M.; Liberatore, M. W. Drag Reduction Effectiveness of Dilute and Entangled Xanthan in Turbulent Pipe Flow. J. Non-Newtonian Fluid Mech. 2011, 166, 25− 31. (24) Wagner, C.; Amarouchene, Y.; Doyle, P.; Bonn, D. TurbulentDrag Reduction of Polyelectrolyte Solutions: Relation with the Elongational Viscosity. Europhys. Lett. 2003, 64, 823. (25) Abdulbari, H. A.; Shabirin, A.; Abdurrahman, H. N. BioPolymers for Improving Liquid Flow in Pipelines: A Review and Future Work Opportunities. J. Ind. Eng. Chem. 2014, 20, 1157−1170. (26) Walter, A. V.; Jimenez, L. N.; Dinic, J.; Sharma, V.; Erk, K. A. Effect of Salt Valency and Concentration on Shear and Extensional Rheology of Aqueous Polyelectrolyte Solutions for Enhanced Oil recovery. Rheol. Acta 2019, 58, 145−157. (27) Jimenez, L. N.; Dinic, J.; Parsi, N.; Sharma, V. Extensional Relaxation Time, Pinch-Off Dynamics, and Printability of Semidilute Polyelectrolyte Solutions. Macromolecules 2018, 51, 5191−5208. (28) Rodríguez-Rivero, C.; Hilliou, L.; Martín del Valle, E. M.; Galán, M. A. Rheological characterization of commercial highly viscous alginate solutions in shear and extensional flows. Rheol. Acta 2014, 53, 559−570. (29) Dou, S.; Colby, R. H. Charge Density Effects in Salt-Free Polyelectrolyte Solution Rheology. J. Polym. Sci., Part B: Polym. Phys. 2006, 44, 2001−2013. (30) Fuoss, R. M.; Strauss, U. P. Polyelectrolytes. II. Poly-4vinylpyridonium Chloride and Poly-4-vinyl-N-n-butylpyridonium Bromide. J. Polym. Sci., Part A-1: Polym. Chem. 1948, 3, 246−263. (31) Fuoss, R. M. Viscosity Function for Polyelectrolytes. J. Polym. Sci. 1948, 3, 603−604. (32) Fuoss, R. M.; Maclay, W. N. Polyelectrolytes. VI. Viscosities of 4-polyvinylpyridine hydrochloride in methanol at 25°. J. Polym. Sci. 1951, 6, 305−317. (33) Strauss, U. P.; Smith, E. H. Polyphosphates as Polyelectrolytes. II. Viscosity of Aqueous Solutions of Graham’s Salts1. J. Am. Chem. Soc. 1953, 75, 6186−6188. (34) Fuoss, R. M.; Cathers, G. I. Polyelectrolytes. III. Viscosities of n-butyl bromide addition compounds of 4-vinylpyridine-styrene copolymers in nitromethane-dioxane mixtures. J. Polym. Sci. 1949, 4, 97−120. (35) Pavlov, G. M.; Okatova, O. V.; Gubarev, A. S.; Gavrilova, I. I.; Panarin, E. F. Strong Linear Polyelectrolytes in Solutions of Extreme Concentrations of One-One Valent Salt. Hydrodynamic Study. Macromolecules 2014, 47, 2748−2758. (36) Cohen, J.; Priel, Z.; Rabin, Y. Viscosity of Dilute Polyelectrolyte Solutions. J. Chem. Phys. 1988, 88, 7111−7116. (37) Liberti, P. A.; Stivala, S. S. On the Viscosity of Aqueous Polyelectrolytes. J. Polym. Sci., Part B: Polym. Lett. 1966, 4, 137−145.

ASSOCIATED CONTENT

S Supporting Information *

The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acs.jpcb.9b03044.

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Analysis of viscosity data according to Fedor’s equation and other empirical models from the literature and analysis of viscosity data using stretched exponentials (PDF)

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. ORCID

Carlos G. Lopez: 0000-0001-6160-632X Walter Richtering: 0000-0003-4592-8171 Notes

The authors declare no competing financial interest.

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REFERENCES

(1) Dobrynin, A.; Rubinstein, M. Theory of Polyelectrolytes in Solutions and at Surfaces. Prog. Polym. Sci. 2005, 30, 1049−1118. (2) Muthukumar, M. 50th Anniversary Perspective: A Perspective on Polyelectrolyte Solutions. Macromolecules 2017, 50, 9528−9560. (3) Dobrynin, A. Theory and Simulations of Charged Polymers: From Solution Properties to Polymeric Nanomaterials. Curr. Opin. Colloid Interface Sci. 2008, 13, 376−388. (4) Thiele, M. J.; Davari, M. D.; Hofmann, I.; König, M.; Lopez, C. G.; Vojcic, L.; Richtering, W.; Schwaneberg, U.; Tsarkova, L. A. Enzyme-Compatible Dynamic Nanoreactors from Electrostatically Bridged Like-Charged Surfactants and Polyelectrolytes. Angew. Chem., Int. Ed. 2018, 57, 9402−9407. (5) Ammala, A. Biodegradable Polymers as Encapsulation Materials for Cosmetics and Personal Care Markets. Int. J. Cosmet. Sci. 2013, 35, 113−124. (6) Sakamoto, K.; Lochhead, R.; Maibach, H.; Yamashita, Y. Cosmetic science and Technology: Theoretical Principles and Applications; Elsevier, 2017. (7) Meka, V. S.; Sing, M. K. G.; Pichika, M. R.; Nali, S. R.; Kolapalli, V. R. M.; Kesharwani, P. A Comprehensive Review on Polyelectrolyte Complexes. Drug discovery today 2017, 22, 1697−1706. (8) Siyawamwaya, M.; Choonara, Y. E.; Bijukumar, D.; Kumar, P.; Du Toit, L. C.; Pillay, V. A Review: Overview of Novel Polyelectrolyte Complexes as Prospective Drug Bioavailability Enhancers. Int. J. Polym. Mater. Polym. Biomater. 2015, 64, 955−968. (9) Yu, J.; Jackson, N. E.; Xu, X.; Morgenstern, Y.; Kaufman, Y.; Ruths, M.; de Pablo, J. J.; Tirrell, M. Multivalent Counterions Diminish the Lubricity of Polyelectrolyte Brushes. Science 2018, 360, 1434−1438. (10) Horkay, F.; Basser, P. J.; Hecht, A.-M.; Geissler, E. Chondroitin Sulfate in Solution: Effects of Mono- and Divalent Salts. Macromolecules 2012, 45, 2882−2890. (11) Geissler, E.; Hecht, A.-M.; Horkay, F. Scaling Behavior of Hyaluronic Acid in Solution with Mono- and Divalent Ions. Macromol. Symp. 2010, 291−292, 362−370. (12) Janmey, P. A.; Slochower, D. R.; Wang, Y.-H.; Wen, Q.; Cebers, ̅ A. Polyelectrolyte Properties of Filamentous Biopolymers and Their Consequences in Biological Fluids. Soft Matter 2014, 10, 1439−1449. (13) Hodgson, D. F.; Amis, E. J. Viscometric Titration of a Linear Polyelectrolyte. J. Chem. Phys. 1989, 91, 2635−2642. (14) Colby, R. H. Structure and Linear Viscoelasticity of Flexible Polymer Solutions: Comparison of Polyelectrolyte and Neutral Polymer Solutions. Rheol. Acta 2010, 49, 425−442. (15) Essafi, W.; Haboubi, N.; Williams, C.; Boué, F. Weak Temperature Dependence of Structure in Hydrophobic Polyelec5632

DOI: 10.1021/acs.jpcb.9b03044 J. Phys. Chem. B 2019, 123, 5626−5634

Article

The Journal of Physical Chemistry B (38) Schaefgen, J. R.; Trivisonno, C. F. Polyelectrolyte Behavior of Polyamides. II. Viscosities of Solutions of Linear and Multichain Polyamides in Formic Acid1. J. Am. Chem. Soc. 1952, 74, 2715−2717. (39) Yuan, L.; Dougherty, T. J.; Stivala, S. S. A Semi-Empirical Approach to the Viscosities of Polyelectrolyte Solutions. Journal of Polymer Science, Part A-2: Polymer Physics 1972, 10, 171−189. (40) Fedors, R. F. An Equation Suitable for Describing the Viscosity of Dilute to Moderately Concentrated Polymer Solutions. Polymer 1979, 20, 225−228. (41) Arinaitwe, E.; Pawlik, M. Dilute Solution Properties of Carboxymethyl Celluloses of Various Molecular Weights and Degrees of Substitution. Carbohydr. Polym. 2014, 99, 423−431. (42) Dragan, S.; Ghimici, L. Viscometric Behaviour of Some Hydrophobically Modified Cationic Polyelectrolytes. Polymer 2001, 42, 2887−2891. (43) Ghimici, L.; Nichifor, M. Transport properties of some cationic polysaccharides. J. Colloid Interface Sci. 2006, 302, 589−596. (44) Ghimici, L.; Avram, E. Viscometric Behavior of Quaternized Polysulfones. J. Appl. Polym. Sci. 2003, 90, 465−469. (45) Wolf, B. A. Coil Overlap in Moderately Concentrated Polyelectrolyte Solutions: Effects of Self-Shielding as Compared with Salt-Shielding as a Function of Chain Length. RSC Adv. 2016, 6, 38004−38011. (46) Chatterjee, A.; Das, B. Radii of Gyration of Sodium Carboxymethylcellulose in Aqueous and Mixed Solvent Media from Viscosity Measurement. Carbohydr. Polym. 2013, 98, 1297−1303. (47) Wolf, B. A. Polyelectrolytes Revisited: Reliable Determination of Intrinsic Viscosities. Macromol. Rapid Commun. 2007, 28, 164− 170. (48) Eich, A.; Wolf, B. A. Intrinsic Viscosities of Polyelectrolytes: Determination and Modeling of the Effects of Extra Salt. ChemPhysChem 2011, 12, 2786−2790. (49) Eckelt, J.; Knopf, A.; Wolf, B. A. Polyelectrolytes: Intrinsic Viscosities in the Absence and in the Presence of Salt. Macromolecules 2008, 41, 912−918. (50) Brunchi, C.-E.; Morariu, S.; Bercea, M. Intrinsic Viscosity and Conformational Parameters of Xanthan in Aqueous Solutions: Salt Addition Effect. Colloids Surf., B 2014, 122, 512−519. (51) Morariu, S.; Brunchi, C.-E.; Bercea, M. The Behavior of Chitosan in Solvents with Different Ionic Strengths. Ind. Eng. Chem. Res. 2012, 51, 12959−12966. (52) Bhattarai, A.; Das, B. Solvodynamic Properties of Sodium Polystyrenesulfonate in Methanol−Water Mixed Solvent Media in Absence and in the Presence of a Salt Using Viscometric Method; Macromolecular Symposia, 2012; pp 52−59. (53) Boris, D. C.; Colby, R. H. Rheology of Sulfonated Polystyrene Solutions. Macromolecules 1998, 31, 5746−5755. (54) De Gennes, P. G.; Pincus, P.; Velasco, R. M.; Brochard, F. Remarks on Polyelectrolyte Conformation. J. Phys. 1976, 37, 1461− 1473. (55) Pfeuty, P. Conformation des Polyelectrolytes Ordre dans les Solutions de Polyelectrolytes. J. Phys. Colloq. 1978, 39, C2−149. (56) Nishida, K.; Kaji, K.; Kanaya, T. High Concentration Crossovers of Polyelectrolyte Solutions. J. Chem. Phys. 2001, 114, 8671−8677. (57) Lopez, C. G.; Rogers, S. E.; Colby, R. H.; Graham, P.; Cabral, J. T. Structure of Sodium Carboxymethyl Cellulose Aqueous Solutions: A SANS and Rheology Study. J. Polym. Sci., Part B: Polym. Phys. 2015, 53, 492−501. (58) Lopez, C. G.; Colby, R. H.; Cabral, J. T. Electrostatic and Hydrophobic Interactions in NaCMC Aqueous Solutions: Effect of Degree of Substitution. Macromolecules 2018, 51, 3165−3175. (59) Lopez, C. G.; Richtering, W. Influence of Divalent Counterions on the Solution Rheology and Supramolecular Aggregation of Carboxymethyl Cellulose. Cellulose 2019, 26, 1517. (60) Yu, F.; Zhang, F.; Luan, T.; Zhang, Z.; Zhang, H. Rheological Studies of Hyaluronan Solutions Based on the Scaling Law and Constitutive Models. Polymer 2014, 55, 295−301.

(61) Dodero, A.; Vicini, S.; Alloisio, M.; Castellano, M. Sodium Alginate Solutions: Correlation Between Rheological Properties and Spinnability. J. Mater. Sci. 2019, 54, 8034−8046. (62) Di Cola, E.; Plucktaveesak, N.; Waigh, T. A.; Colby, R. H.; Tan, J. S.; Pyckhout-Hintzen, W.; Heenan, R. K. Structure and Dynamics in Aqueous Solutions of Amphiphilic Sodium Maleate-Containing Alternating Copolymers. Macromolecules 2004, 37, 8457−8465. (63) Di Cola, E.; Waigh, T. A.; Colby, R. H. Dynamic Light Scattering and Rheology Studies of Aqueous Solutions of Amphiphilic Sodium Maleate Containing Copolymers. J. Polym. Sci., Part B: Polym. Phys. 2007, 45, 774−785. (64) Krause, W. E.; Tan, J. S.; Colby, R. H. Semidilute solution rheology of polyelectrolytes with no added salt. J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 3429−3437. (65) Ü züm, C.; Christau, S.; von Klitzing, R. Structuring of Polyelectrolyte (NaPSS) Solutions in Bulk and Under Confinement as a Function of Concentration and Molecular Weight. Macromolecules 2011, 44, 7782−7791. (66) Liao, Q.; Carrillo, J.-M. Y.; Dobrynin, A. V.; Rubinstein, M. Rouse Dynamics of Polyelectrolyte Solutions: Molecular Dynamics Study. Macromolecules 2007, 40, 7671−7679. (67) Takahashi, Y.; Iio, S.; Matsumoto, N.; Noda, I. Viscoelastic Properties of Polyelectrolyte Solutions in Non-Entangled Concentrated Regions. Polym. Int. 1996, 40, 269−273. (68) Wang, L.; Bloomfield, V. A. Osmotic Pressure of Polyelectrolytes Without Added Salt. Macromolecules 1990, 23, 804−809. (69) Ganter, J. L. M. S.; Milas, M.; Rinaudo, M. On the viscosity of sodium poly(styrene sulphonate), a flexible polyelectrolyte. Polymer 1992, 33, 113−116. (70) Note that a value of c*N2 ≃ 2000 is reported in ref 53 because both SAXS and viscosity data are considered. There is a small discrepancy between these two methods, with the latter one yielding slightly higher values for c*. We will address this in more detail in a forthcoming paper. (71) Oostwal, M.; Odijk, T. Novel Dynamic Scaling Hypothesis for Semidilute and Concentrated Solutions of Polymers and Polyelectrolytes. Macromolecules 1993, 26, 6489−6497. (72) Results for sample with Mw = 177 kg/mol from ref 71 are not included due to the large scatter in the data. (73) A NaPSS concentration of c = 1 × 10−4 M corresponds to approximately 8 × 10−5 M free ions. Deionised water typically contains a salt concentration of cS ≃ 5 × 10−6 M from carbonic acid. (74) As long as the polymer to salt ratio is kept constant, eq 46 of ref 18 expects no change in the power-law exponent of ηsp versus c. (75) Phillies, G. D. J. Hydrodynamic Scaling of Viscosity and Viscoelasticity of Polymer Solutions, Including Chain Architecture and Solvent Quality Effects. Macromolecules 1995, 28, 8198−8208. (76) Phillies, G. D. J.; Quinlan, C. A. Analytic Structure of the Solutionlike-Meltlike Transition in Polymer Dynamics. Macromolecules 1995, 28, 160−164. (77) Phillies, G. D. J. Range of Validity of the Hydrodynamic Scaling Model. J. Phys. Chem. 1992, 96, 10061−10066. (78) Phillies, G. D. J.; Peczak, P. The Ubiquity of StretchedExponential Forms in Polymer Dynamics. Macromolecules 1988, 21, 214−220. (79) Von Meerwall, E. D.; Amis, E. J.; Ferry, J. D. Self-Diffusion in Solutions of Polystyrene in Tetrahydrofuran: Comparison of Concentration Dependences of the Diffusion Coefficients of Polymer, Solvent, and a Ternary Probe Component. Macromolecules 1985, 18, 260−266. (80) Bai, R.; Basser, P. J.; Briber, R. M.; Horkay, F. NMR Water SelfDiffusion and Relaxation Studies on Sodium Polyacrylate Solutions and Gels in Physiologic Ionic Solutions. J. Appl. Polym. Sci. 2014, 131, . DOI: 10.1002/app.40001 (81) Petit, J.-M.; Zhu, X. X.; Macdonald, P. M. Solute Probe Diffusion in Aqueous Solutions of Poly(vinyl alcohol) As Studied by Pulsed-Gradient Spin-Echo NMR Spectroscopy. Macromolecules 1996, 29, 70−76. 5633

DOI: 10.1021/acs.jpcb.9b03044 J. Phys. Chem. B 2019, 123, 5626−5634

Article

The Journal of Physical Chemistry B (82) In ref 66, a variation of ηsp ≈ N eKc1/3 was observed, which differs significantly from our experimental results. (83) Ferry, J. D. Viscoelastic Properties of Polymers; John Wiley & Sons, 1980. (84) Colby, R. H.; Fetters, L. J.; Funk, W. G.; Graessley, W. W. Effects of Concentration and Thermodynamic Interaction on the Viscoelastic Properties of Polymer Solutions. Macromolecules 1991, 24, 3873−3882. (85) Oostwal, M. G.; Blees, M. H.; De Bleijser, J.; Leyte, J. C. Chain self-diffusion in aqueous salt-free solutions of sodium poly(styrenesulfonate). Macromolecules 1993, 26, 7300−7308. (86) See ref 16 for a compilation of lK as a function of NaPSS concentration. (87) Since lK is larger than ξ, for example lK ≃ 1.2ξ for c ≲ 0.01 M and lK ≃ 2ξ at c = 1 M, it is unlikely that chains will be able follow Rouse-like dynamics on all length-scales above ξ. In any case, we expect that a change in the hydrodynamics screening length will affect eqs 3 and 6 by a similar factor. (88) Chen, S.-P.; Archer, L. A. Relaxation Dynamics of Salt-Free Polyelectrolyte Solutions Using Flow Birefringence and Rheometry. J. Polym. Sci., Part B: Polym. Phys. 1999, 37, 825−835. (89) The only other systematic datasets on the diffusion properties of NaPSS in salt-free solution are refs 90 and 91. These papers discuss diffusion coefficients of NaPSS at low concentrations (D0). Their values of D0N are larger than those reported by Odijk et al. for semidilute solutions with kMw > 31 g/mol and approximately the same for lower molar masses. (90) Xu, G.; Yang, J.; Zhao, J. Molecular weight dependence of chain conformation of strong polyelectrolytes. J. Chem. Phys. 2018, 149, 163329. (91) Xu, G.; Luo, S.; Yang, Q.; Yang, J.; Zhao, J. Single chains of strong polyelectrolytes in aqueous solutions at extreme dilution: Conformation and counterion distribution. J. Chem. Phys. 2016, 145, 144903.

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DOI: 10.1021/acs.jpcb.9b03044 J. Phys. Chem. B 2019, 123, 5626−5634