Interchain Hydrodynamic Interaction and Internal Friction of Polyelectrolytes

Polyelectrolytes (PE) are polymeric macromolecules in aqueous solutions characterized by their chain topology and intrinsic charge in a neutralizing fluid. Structure and dynamics are related to several characteristic screening length scales determined by electrostatic, excluded volume, and hydrodynamic interactions. We examine PE dynamics in dilute to semidilute conditions using dynamic light scattering, neutron spinecho spectroscopy, and pulse field gradient NMR spectroscopy. We connect macroscopic diffusion to segmental chain dynamics, revealing a decoupling of local chain dynamics from interchain interactions. Collective diffusion is described within a colloidal picture, including electrostatic and hydrodynamic interactions. Chain dynamics is characterized by the classical Zimm model of a neutral chain retarded by internal friction. We observe that hydrodynamic interactions are not fully screened between chains and that the internal friction within the chain increases with an increase in ion condensation on the chain.

P olyelectrolytes (PE) are widespread in nature like RNA or intrinsically unfolded proteins and widely used in technical applications. 1Beyond polymer architecture and solvent affinity, mainly the intrinsic charge with the surrounding screening aqueous solvent determines the polymer conformation between a collapsed coil conformation for high salt and a rodlike, at least strongly expanded, conformation if the polymer charges are not screened.Additional hydrogen bonding, dipolar interactions, ion condensation, or chain connectivity influence the collective behavior of the polymer chains.The polymer conformation and the resulting interchain interactions, that may also be entangled at higher PE concentrations, affect macroscopic properties as viscosity, turbidity, collective diffusion, or the interaction with surfaces.General relationships between selfand collective diffusion coefficients D s and D c , radius of gyration R g , viscosity η are described within the double screening model of Muthukumar 2 or the scaling model of Dobrynin 3 considering screening of electrostatics, excluded volume and hydrodynamic interactions (HI).The respective scaling relations present different behavior dependent on low or high salt conditions and concentration in dilute, semi dilute and entangled regimes also dependent on assumptions as hydrodynamic screening between chains. 4This quite general relation connects the observed collective diffusion with forces kT/S(Q) at thermal energy kT between particles or chains expressed in S(Q) and is related to "de Gennes narrowing", describing the reduced mobility at the correlation maximum.For colloidal spherical particles Ackerson et al. introduced a reduced mobility at increased concentrations. 9Later the δγ-expansion of Beenakker and Mazur quantified this and introduced the hydrodynamic function H(Q) describing the reduced mobility of hard spheres with increasing concentration splitting into a Qdependent distinct part H d (Q) and a self-part as the reduced self-diffusion D s /D 0 . 10,11−20 The dynamics of a neutral chain on a molecular level is theoretically depicted as Rouse like if hydrodynamic screening is assumed, or Zimm like if hydrodynamic interactions are relevant. 2,21,22A direct access to the PE dynamics on a molecular level is possible using neutron spinecho spectroscopy (NSE) which is a unique technique for assessing the polymer chain on the nanometer and nanosecond scales.It has been successfully used to examine polymer dynamics, 23 protein domain motions, 14,24 IDP 16,25 but is rarely applied for PE dynamics. 8,26n this Letter, we address the relationship between the macroscopic properties, in particular collective diffusion from dynamic light scattering (DLS) and self-diffusion from pulsed field gradient (PFG) NMR, and chain dynamics on a segmental level observed by NSE for a model PE in aqueous solution.The chain dynamics are described by the Zimm model including internal friction (ZIF).The transition to collective diffusion is governed using the δγ-expansion introducing a HI screening radius R HI.We demonstrate that the dynamics of charged PE can be fully described assuming a decoupling of local chain dynamics from polymer interchain interactions related to HI and charges for dilute to semidilute PE solutions.
We use PSS acid of molecular weight M w = 39 kg/mol (39k) and M w = 17.5 kg/mol (17.5k) purchased from Polymer Source Inc., Canada, with M w /M n = 1.03, respectively 1.09 and sulfonation degree > 90% after dialysis to remove impurities.PSSNa samples were titrated with NaOD to pD 7 in D 2 O while PSSH samples were dissolved in D 2 O resulting in pD 2.3 for 10 mg/mL PSSH and pD 1.8 for 30 mg/mL.NaCl was added to increase the ion concentration c I .Using DLS (Zetasizer Nano ZS, Malvern), we determined the collective diffusion coefficient D c of PSS. 27,28For low c I the signal splits into a fast component due to collective diffusion and a slow component attributed to aggregates.In the further discussion, we always refer to the fast component.Figure 1 shows respective D c for PSS 39k at different salt concentrations c s with c I between 10 mM and 2 M together with related selfdiffusion coefficients D s measured by PFG NMR.For 17.5k, see Supporting Information (SI).D s was measured by using a 600 MHz Varian spectrometer equipped with a diffusion probe head.
For a high c s of 500 mM and 1000 mM, we find that extrapolating D c to dilute conditions fits well to D s , while for low c s , we observe scaling relations that cannot be used to extrapolate.For low c s , D c are by a factor of up to 10 larger compared to D s .For the intermediate c s 100 mM, at larger c P , the slope equals the scaling found for the 21 mM presenting a crossover to the respective power law regime.D s increases with an increasing ion concentration ∼ c I 0.2 (left inset of Figure 1).For a constant number of segments N, segment length l, and solvent viscosity η s , the Zimm model with D z = (0.196kT)/(η s R e ) implies a reduced end-to-end distance R e = lN ν with decreasing ν for larger c I .Electrostatic repulsion between chain segments leads to a more extended R e for low c I .Monomer overlap concentrations Nc p * calculated from D s are shown in the right inset of Figure 1.For c I < 100 mM at least the larger c p are above c p * and semidilute while others are dilute.Similar D s increase was found for PSS 17.5k, but all concentrations were in the dilute region c p < c p * (see SI).
Figure 2 presents small-angle X-ray scattering (SAXS) data for PSS 39k (17.5k and fit parameters, see SI).The scattered intensity is and structure factor S(Q,c p ).Because the PE dissociated ions contribute to charge screening, the formfactor depends on c P .To extract F(Q,c p ) and S(Q,c p ) a self-consistent simultaneous fit is used.The chain formfactor is a Gaussian chain 30 with excluded volume parameter ν describing the chain extension complemented by a disc like cross section as we find a clear signature of a wormlike shape (see SI).At intermediate Q, we find a Q −2 power law, expected for Gaussian chains, followed by a clear decay described by a disc like cross section.The increase at even larger Q is due to diffuse scattering d(Q) resulting from deviations from a smooth wormlike shape.The respective disc radius of ≈0.8 nm equals the extension of PSS sidechains.Data from small angle neutron scattering (SANS) present a flatter power law (see SI).We conclude that the different contrast in SAXS and SANS is due to Na + ion condensation at the chain.Adsorbed Na + influences the neutron contrast less, which is dominated by hydrogens.Ion condensation also affects the effective charge of the PE chain leading to effective dissociation of ≈8%. 29(Q) is described by a two-Yukawa (2Y) SF 31 with screened electrostatic repulsion on a scale of a few nm and a second short-ranged attraction (<0.3 nm) (see Figure 2 and SI).We attribute the attraction to the hydrophobic backbone, to attraction mediated by Na + ions condensed at the chain or excluded volume screening.2 We observe for increasing ion concentration, respectively, electrostatic screening a weaker SF. . c  has contributions from added salt, ions from water dissociation according to the measured pD and counterions.Because of ion condensation on the chain, an effective dissociation of f* ≈ 8% is assumed.29 Right inset: Monomer overlap concentration Nc p * was calculated from D s (see SI).The line indicates the largest polymer concentration used for NSE.
The 2Y SF was the only SF that reproduced the experimental PE concentration also at high ion concentration.
Figure 3 presents the dynamic structure factor of a PE chain measured by NSE 32 in a time range from 0.1 ns to 100 ns together with a synthetic data set matching the DLS measured D c .To get an easier overview and later respective models, we determine an effective diffusion coefficient D eff by fitting ∼ exp(−Q 2 D eff t) which is shown for samples PSS 39k in Figure 4 and for PSS 17.5k in SI.At lower Q, we observe the crossover from collective diffusion to chain dynamics.The relaxation at short times ∼ 1−10 ns corresponding to high Q, allow in detail to examine the segmental relaxation.
To describe the transition from low Q collective dynamics to high Q in detail, we use a model combining the H(Q)/S(Q) correction due to interchain interactions with a classical polymer model complemented by internal friction.We assume the decoupling of chain dynamics from interchain interactions.The Zimm model describes neutral polymers by a bead-spring model of N beads connected by segments of length l including bead friction with a solvent of viscosity η s and HI between beads. 22The intermediate scattering function (ISF) for a single chain is with center of mass diffusion for a single chain D = D z = 0.196kT/(η s R e ), end-to end distance R e = lN ν and wavevector Q.The parameter ν describes polymer−solvent interactions and is typically between 0.5 and 0.6 for a Θ to good solvents.For charged chains, a stretching of the chain conformation is expected leading to even larger values υ > 0.8. 1,4The Zimm and depends on the chain expansion through R e .Considering friction between neighboring beads (ξ int ) leads to the ZIF model. 33Mode relaxation times = + are additive slowed down by the internal friction time τ int that stronger affects higher order modes observed at larger Q.This model was successfully used for single chain nanoparticles and IDP. 16,34Below the overlap concentration chain dynamics may be modified through interchain interactions. 35o describe the transition between D c and intermediate Q NSE data, we extend the ZIF model to account for interchain interactions such as the correction for colloidal spherical particles as D(Q) = D Z H(Q)/S(Q).This implies the assumption that S(Q) and H(Q) describe a kind of configurational ensemble average and the decoupling of center of mass diffusion and intrachain dynamics.The only analytical method to calculate the hydrodynamic function H(Q) = H d (Q) + D s /D 0 for spherical particles of radius a is the δγexpansion of Beenakker and Mazur 10,11 with the distinct contribution  (3) and the self-part describing the change in self-diffusion x is the angle between wave vectors Q and k, S γ is a known function given by Genz and Klein. 36Particle correlation and the associated interactions enter the distinct part of H d through the SF S(Q).The HI enters as the mobility of a sphere with the geometrical radius a.The corresponding HI volume fraction is Φ = n4πa 3 /3 with the number density n.For stronger interactions, e.g., charged spheres, the radius a and the interaction length scale are already separated.We describe the screened HI by an equivalent HI screening radius defining R HI = a like attributing a hydrodynamic radius R h to polymers or aspherical objects with an equivalent sphere diffusion coefficient.For strong HI screening, R HI tends to be very small.For ideal Gaussian chains, R h ≈ 0.66R g describing the single chain friction with the solvent, 22 which is different from R HI that describes the effective HI between chains, but R h might be an upper limit.
To fit the full NSE spectra, we use the SAXS measured S(Q) and calculate H(Q).The parameters for the ZIF model are fixed by the polymer dimension (l = 0.24 nm; 37 39k: N = 212; 17.5k N = 95 with p < 25).We fit as free parameter ν that determines D Z and reflects chain expansion, R HI quantifying the amount of HI screening, and τ int , which describes the slowing down due to internal friction.All functions are available in the used free software Jscatter. 38e find in general excellent fits of the full NSE spectra shown in Figure 3, and SI. Figure 4 compares D eff of measured data, fits, and model calculations for different c s .At Q ≲ 0.5nm −1 for lower c s the collective effect of the H(Q)/S(Q) correction is visible as an upturn, resulting in a much larger D c compared to D s .For larger c s the difference vanishes as S(Q) becomes weaker.Above Q ≈ 0.5nm −1 we find in general a H(Q)/S(Q) ≈ constant whereby we detect self-diffusion.We observe a Zimm like region D ∼ Q up to Q ≈ 1nm −1 with a slope close to the Zimm prediction (dotted line) for relaxation rate 22 The linear increase in D eff reflects D Z and additional low mode contributions that are less influenced by internal friction.For larger Q deviations from Zimm dynamics due to internal friction slow down more local modes.We see for all ion concentrations that τ int significantly reduces D eff compared to that of pure Zimm.Seemingly τ int increases with growing ion concentration but is still small compared to the Zimm time τ Z .For larger ion concentrations, the Zimm like region becomes narrower as τ int increases.Inspecting Figure 3 shows that NSE is sensitive to changes on some nanosecond scale for larger Q.Examining ν, we find that with increasing ion concentration ν is decreasing from > 0.8 to smaller values that indicates the conformational change due to stronger screening.
PSSNa 0 mM for 39k and 17.5k shows a smaller difference between D c and D s compared to PSSH 0 mM or PSSNa 21 mM.This is validated by the concentration series from DLS by a reduced power law and smaller values of D c and is not related to concentration or crossing c p *.For the 100 mM ion concentration for both, 39k and 17.5k (see SI), R HI shows negligible values (<0.3nm) compared to significant larger values at lower and higher salt concentrations.R HI values for 0 mM and > 100 mM are comparable.The stronger HI results in a noticeable effect from the self-part D s /D 0 correction visible as difference between solid and dashed green line.Additional, R HI /R g is close to the value R h /R g = 0.66 expected for a Gaussian polymer, which might be a limiting case for strong HI compared to the weaker HI intermediate ion concentrations.A weaker charge of PSSNa 0 mM compared to 21 mM can explain the reduced difference between D c and D s for 0 mM.This larger adsorption at lowest ion concentrations is surprising but also visible in the DLS data (see Figure 1) presenting a different power law, but it is not observed in D s .The differences can be explained by a larger counterion adsorption of Na + ions on PSS at very low and high ion concentration.Na + adsorption induces stronger coupling between neighboring segments, leading to larger internal friction and larger τ Z for 0 mM and larger ion concentrations.
To examine the influence of temperature, we measured PSSH 0 mM, PSSNa 0 mM and PSSH 500 mM additionally at 40 and 60 °C.Resulting viscosity scaled D eff are shown in Figure 5.The low Q values show a perfect scaling with solvent viscosity.For τ int , we see a decrease with an increasing temperature (see SI). D z and τ Z are both related to solvent viscosity, while τ int is related to bond rotation, steric hindrance, charge repulsion, or Na + binding between segments.The activation energy of τ int is close to the water activation energy; thus, viscosity scaling also compensates for changes in τ int (see SI).
In summary, we studied the dynamics of a classical PE extending observations on macroscopic length scales to segmental chain dynamics.Different from previous hypotheses, the hydrodynamic interactions between chains are not fully screened and are essential to understand the connection between collective diffusion, self-diffusion, and chain dynamics.We observe that the dynamics of charged PE on a segmental level follows a neutral chain Zimm model with internal friction (ZIF).The PE charges result in an increased R e .For intermediate Q, the chain dynamics follows the Zimm prediction, while at larger Q, the internal friction between neighboring monomers significantly slows segment motion.The source of internal friction seems to be related to counterion condensation.

■ ASSOCIATED CONTENT
Hayter et al. examined polystyrene sulfonate (PSS) as the classical PE and discussed the correlation D(Q)= kTμ(Q)/ S(Q) between PE structure factor (SF) S(Q), diffusion coefficient D(Q) and single chain mobility μ(Q) for low salt concentration.

Figure 1 .
Figure 1.Collective diffusion coefficient D c measured by DLS (fast mode, Q = 0.0264nm −1 , solid symbols) and self-diffusion D s (open symbols, conc.= 10 mg/ml) measured by PFG-NMR for PSS 39k.PSS type (H or Na) and added NaCl concentration c s as given in the legend.Lines describe power laws with the indicated powers.Left inset: PFG-NMR measured self-diffusion coefficient plotted against the ion concentration c I = 2c s + f *Nc p + 2•10 −pD.c I has contributions from added salt, ions from water dissociation according to the measured pD and counterions.Because of ion condensation on the chain, an effective dissociation of f* ≈ 8% is assumed.29 Right inset: Monomer overlap concentration Nc p * was calculated from D s (see SI).The line indicates the largest polymer concentration used for NSE.

Figure 2 .
Figure 2. SAXS data for PSS 39k after background correction and scaled by concentration (c s shifted consecutively by 10 for visibility).A stronger SF results mainly in the decreased intensity at a low Q for increased c p .Concentrations were 5, 10, 20, 30 mg/mL except for 21 mM with 8.3, 16.6, 33.2 and 50 mg/mL (colors red to green, along arrow).Black lines correspond to fits as described.For the two highest NaCl concentrations the repulsive part in 2Y SF was fixed to a negligible contribution.Broken lines indicate the extension of a Q −2 power law to a high Q to demonstrate the deviation from a Gaussian chain.Measured at in-house instrument SAXSpace (Anton-Paar).

Figure 3 .
Figure 3. NSE measured intermediate scattering function I(Q,t)/ I(Q,0) up to 100 ns for PSS 39k 500 mM at 30 mg/mL measured at IN15, ILL, Grenoble.Q values are shown in Figure 4. We add a synthetic Q = 0.0264 nm −1 data set that represents the measured D c from DLS at the same concentration (relaxation time of ≈36 μs).Solid lines represent the combined fit using the H(Q)/S(Q) corrected ZIF model with, in general, small X 2 .Other ion concentrations and data for PSS 17.5k are shown in SI.

Figure 4 .
Figure 4. Effective diffusion coefficients D eff for PSS 39k c p M w = 30 mg/ml with respective parameters from the full fit model: experimental (points); ZIF with full H(Q)/S(Q) correction (black line); ZIF only with self-part correction D s /D 0 (green dashed); same but τ int = 0 (Zimm) (dotted); ZIF without correction (green line); DLS extrapolated concentration 0 mg/mL (red square), 30 mg/mL (circle); PFG-NMR self-diffusion (blue diamond).It should be noted that the self-part corrected ZIF (green dashed line) fits well to D s .The self-part D s /D 0 correction is noticeable as a difference between solid and dashed green lines.

Figure 5 .
Figure 5. Viscosity scaled D eff for PSS 39k at different temperatures (color coded as indicated: experimental (points); ZIF with H(Q)/ S(Q) correction (solid line); same but τ int = 0 line).All values are scaled by solvent viscosity to 20 °C.Full NSE spectra with fits and an Arrhenius plot of τ int are shown in SI.