Structure of Sodium Carboxymethyl Cellulose Aqueous Solutions: A SANS and Rheology Study

We report a small angle neutron scattering (SANS) and rheology study of cellulose derivative polyelectrolyte sodium carboxymethyl cellulose with a degree of substitution of 1.2. Using SANS, we establish that this polymer is molecularly dissolved in water with a locally stiff conformation with a stretching parameter. We determine the cross sectional radius of the chain ( 3.4 Å) and the scaling of the correlation length with concentration (ξ = 296 c−1/2Å for c in g/L) is found to remain unchanged from the semidilute to concentrated crossover as identified by rheology. Viscosity measurements are found to be in qualitative agreement with scaling theory predictions for flexible polyelectrolytes exhibiting semidilute unentangled and entangled regimes, followed by what appears to be a crossover to neutral polymer concentration dependence of viscosity at high concentrations. Yet those higher concentrations, in the concentrated regime defined by rheology, still exhibit a peak in the scattering function that indicates a correlation length that continues to scale as. © 2014 The Authors. Journal of Polymer Science Part B: Polymer Physics Published by Wiley Periodicals, Inc. J. Polym. Sci., Part B: Polym. Phys. 2015, 53, 492–501


High q: chain diameter
At high q, we assume that no intermolecular effects are present and the single chain signal dominates [2,3]. The form factor of a wormlike chain for ql p > 1, where l p is the total persistence length of the chain, is given by the product of an infinitely long, thin wormlike chain P 0 = π bq and a term that accounts for its finite cross section (P cs ), here modelled a step function, corresponding to a cylinder of uniform density: where J 1 is a first order Bessel function of the first kind and r p defines the radius of the chain. The high q data are therefore fitted with: The constant I 0 is a function of the contrast factor, the polymer concentration and the monomer length (b = 5.15Å) for a glucose unit. S inc accounts for the q-independent scattering.

Intermediate q: correlation peak
At lower q values, inter molecular correlations become important and H(q) is no longer negligible. Hammouda et al. [4] proposed a Lorentzian function to fit polyelectrolyte peaks and related the peak intensity to the solvation quality, by analogy with neutral polymers, where the random phase approximation establishes a direct link between the high q intensity and the solvent quality. We find that, while a Lorentzian function can describe our data, it does so by forcing a constant slope at high q by background fitting.
If we use the value of S inc obtained by fitting a worm like chain to the high q region of our data, the Lorentzian fit becomes poor. We therefore opt to use an empirical function which correctly describes the peak profiles and is compatible with eq (1): which yields a descriptive fit to the data from which the q * , I(q * ), the peak position and intensity, and a sharpness parameter can be extracted. We assign no physical meaning to the fitting parameters m, d and k.

Low q upturn and fluctuations
We model the upturn by adding a power law term: where D and n are allowed to vary for each sample.

Fitting procedure
Data fitting was carried out by least squares minimisation using the error values estimated by Grasp or Mantid (the data reduction software for the D11/D22 and SANS2D beamlines respectively). Due to the high number of parameters in equations 1-3, the fitting is carried out sequentially. Equation 1 is fitted to the high q part of the data leaving I 0 , r p and S inc as free parameters. Fitting different q ranges, always for q ≥ q * we find greater consistency when fitting the data for q ≥ 1.5q * . The parameters I 0 and S inc vary linearly with φ and (1-φ) respectively, where φ is the volume fraction occupied by the monomers.
As explained in the main paper, a value of r p = 3.4Å was selected and the data was refitted with this new value and I 0 and S inc left as free parameters. The new values of I 0 and S inc did not change significantly. The water content and S pol inc were calculated using this last set of values. Next we fit the data around the peak position with equations 1 and 3 simultaneously allowing I 0 to vary (the value never deviates from the original by more 5%). The term for the upturn is finally added and the full set of data is fitted leaving D, n, d, m, k and I 0 free. The value of n is found to be close to 3.6 for all samples and the previous step is repeated with n fixed at 3.6.
Fitting of high q data using a helical form factor We employ the 'Series of Coaxial Shifted Infinitely Long Thick Helices with a Uniform Electron Density' form factor from ref [5], set to k = 1, corresponding to the case of a single helix. A cylindrical helix is defined by x = R H cos(t), y = R H sin(t), z = P/(2π)t, where R H is the radius and P is the pitch of the helix. Additionally, r H p , is the cross sectional radius of the helical chain. It is unclear to us the meaning of the pre-factor in of ref [5] expression as it is defined differently in two sections of the paper. In order to obtain the helical form factor in absolute units, we take the contour length of the helix to be N b, and calculate the z-projected contour distance using arclength = ((R H ) 2 + (P/2π) 2 ) 1/2 t, and require that the low q limit matches the form factor of a rod of equal mass per unit length, which is readily available in absolute units.
In the q-range studied, a number of combinations of r H p , R H and P yield a form factor identical to that of a cylinder. Of course, a helix will reduce to a cylinder when r H p or P tend to infinity. We first consider the case of a thin helix, setting r H p = 0 and allowing P to vary between 5 and 250Å, we obtain R H 2.5Å and P ≥ 40Å as valid fits. Setting r H p = 2, a more realistic value, and varying P to vary over the same interval, we get R 1 − 2Å. While our data is insufficient to discriminate between a linear conformation and a helical one, we can state that our data is compatible with a linear conformation or a conformation with small helicity. The lateral fluctuations (≈ R H + r H p 3.5 ± 1.0) are within the cylindrical fit value of r p 3.4 ± 1.0Å. The mass per unit length being 5% higher than the straight conformation case. The water content calculated from the helical form factor is 4% lower than for the straight chain.
Low q upturn Figure 1: Low-q upturn intensity D from power law fits I(q) = D/q 3.6 at low q as a function of concentration.
Low q upturn intensity. The upturn parameter D is plotted as a function of concentration in Fig. 1. We lack the q range or statistics in the low q region for a number of samples to estimate D, and therefore only plot D for concentrations at which it can be obtained with reasonable accuracy. Although it is not possible to deduce a clear functional dependence for the upturn intensity from our data, we find that D broadly increases with c. We note that all measurements are in the concentrated regime except for the 8 g/L sample which is in the semidilute region. is dissolved at all measurement timescales. In addition, we have found that c 1/2 dependence applies across the whole concentration range and that the solvent quality parameter B = 1, indicating that q * corresponds to a mesh of isotropic, locally stiff polymer segments.

Solution preparation and time dependent effects
These observations collectively establish that NaCMC D.S. = 1.2 is molecularly soluble in water within this concentration range.
The viscosity measurements were carried out between 1 and 4 days after sample preparation. We evaluated a possible time dependence in the viscosity by measuring one sample c 25g/L using a DVI-Prime Brookfield viscometer with a Couette geometry. We found that the viscosity decreases by about 2% from day one to day 2 and by a further 2% by     3 Crossover concentrations, comparison with literature data.

Overlap concentration
We noted in the paper that a significant disparity exists between the overlap concentration calculated from η sp (c * ) = 1 method, denoted c * v , and from solving eq (2) for c in main paper with R ee = ξ(c), denoted c * s . The ξ(c) relationships are obtained from SANS data.
The table below shows the values for c * obtained a number of systems using these two methods. The calculated volume fraction occupied by the chains at c * v (φ * pol ) assuming the end to end distance of the chains is given by  Table 1: Polyelectrolyte, M w , c * v , c * s and φ * . and All refer to aqueous systems except PMVP-Cl-55 for which the solvent is ethylene glycol. The reference column includes the source for η sp (c * ) = 1 and ξ(c). a taken from line of best fit to a wide range of M w in ref [8].
It thus appears that the discrepancy between the two methods is common to most systems and our result is within the typical range of reported values.

Crossover to the concentrated, correlation with intrinsic flexibility
We observe that the value of the correlation length at the crossover to the concentrated regime (ξ(c D )) is much larger than the thermal blob size (ξ T ) and of the order of the intrinsic Kuhn length (L K0 ) of NaCMC. Scaling theory assumes that polyelectrolytes are flexible above the monomer size and hence are collapsed or extended rather than rod-like at distances smaller than ξ T . With respect to the intrinsic persistence length l 0 = L K0 /2, it is often the case that ξ T ≤ l 0 , and in this situation we could expect polyelectrolytes to remain rigid at small length scales, if the covalent bonds or steric hinderance that give rise to intrinsic rigidity involve energies much greater than k B T . A crossover may then be expected at ξ = l 0 , rather than ξ = ξ T . In this section, we compile literature data on other polyelectrolyte systems to asses whether this conjecture is consistent for polyelectrolytes of different persistence lengths. Table 1 shows the intrinsic persistence length, c D in molar units and the correlation length at c D for a number of polyelectrolyte systems, along with the method used to estimate the crossover to the concentrated regime. For the majority of these systems, the crossover is identified from rheological methods, while for the first three it is identified from scattering measurements of the correlation length, where a change of the scaling of ξ with concentration from 0.5 to 0.25 is observed. For NaPSS, the crossover has been the persistence length of the neutral polymer has been used.   suggesting that, while there is a correlation between l 0 and ξ(c D ), it is not linear, and therefore the crossover to the concentrated is unlikely to correspond to a crossover between the two quantities.

Effect of intrinsic flexibility on unentangled viscosity
We next consider how the rheology of semiflexible polyelectrolytes may differ from that of flexible ones. The specific viscosity of polyelectrolytes in the semidilute unentangled regime can be described as: The best fit to our data is found for K = 6 and β = 0.68. Equation (4), commonly used to calculate B from viscosity data [9,50], which predicts β = 0.5 uses two simplifications: firstly, it does not include pre factors in the calculation of the Rouse time and Zimm times; secondly, it assumes the step length (or persistence length l p ) is purely electrostatic and equal to the correlation length, that is l p = l e = ξ.
Including the pre-factors in the Rouse and Zimm times affects K, but does not change the exponent β. We expect the agreement of K with the data to be qualitative, as it involves a number of parameters such as the friction factor, for which we do not have a precise form. We may however expect the scaling prediction for β to be more accurate, given the close match between theory and experiment for neutral polymer solutions and some flexible polyelectrolyte systems [51].
In contrast with the scaling assumption l p = ξ, for semiflexible polyelectrolytes, we should take into account the intrinsic rigidity of the chain. It is also important to note that neutron scattering experiments on NaPSS show that the electrostatic persistence length is approximately proportional but not equal to the correlation length. Spiteri [19] finds l e 0.6ξ, in agreement with theoretical predictions that l e ∝ κ −1 . It is of course possible for l e to have a more complex concentration dependence. For simplicity here we write: Davis [20] calculated the electrostatic persistence length of NaCMC using non linear electrostatic wormlike theory [52] for dilute solutions of different ionic strengths in the range 0.01-0.2M. Assuming the persistence length is just a function of the total ionic strength and extrapolating his values to our concentration range, we obtain γ = 0.54. We may expect this to be slightly lower given the assumption of Manning condensation and for our semidilute solutions, the charge density is lower than this estimate. We now revise the resulting exponent taking into account the intrinsic rigidity.

Free draining ideal chain
Assuming the chain to be free draining, the Rouse model applies, and the viscosity is then given by: where R is the end-to-end distance of the chain given by R 2 = N l l 2 p , l p is the step length, N l is the number of steps per chain (L/l p ) and ζ is the friction coefficient of a segment, which can be modelled by that of a sphere: ζ = 6πη s l p . Combining Eqs (1) and (2), we obtain β = 0.61, which compares more favourably with the experimentally measured β = 0.68 obtained for γ = 0.26.
The departure from the Fuoss exponent might be due to the intrinsic rigid of the chain, which results in a power law dependence of the step length smaller than 0.5, and in turn in a variation of the end to end distance of R ∼ c −1/5 , instead of the -1/4 dependence for flexible polyelectrolytes.

Rouse-Zimm chain
If we assume the chain is non draining up to the segment length and free draining above it, we can calculate the Zimm time of a segment (τ Z ∼ l 3 p ) and estimate the total relaxation time of the chain to be τ = τ Z ( N b lp ) 2 . The viscosity is then estimated by multiplying the relaxation time by the modulus (k B T per chain). This gives an exponent of β = 0.61, identical to the free draining case. The same chain conformation but assuming chains to be non draining up to the correlation length (instead of l p ), yields β = 0.72.
For all the previous calculations, if the intrinsic persistence length is set to zero, we trivially recover the Fuoss scaling of 0.5. We do not know the exact length scale of hydrodynamic screening, which is required for an accurate calculation of the viscosity. However, we find that for chains with some intrinsic rigidity, we can expect the power law in the unentangled regime to be higher those of flexible polyelectrolytes. Taking l 0 = 54Å and γ = 0.54, depending on the model used, the exponent is expected to be between 0.61 and 0.72, in reasonable agreement with the observed 0.68 ± 0.02.