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Article

Effect of salt on the ordinary-extraordinary transition in solutions of charged macromolecules Di Jia, and Murugappan Muthukumar J. Am. Chem. Soc., Just Accepted Manuscript • DOI: 10.1021/jacs.9b00562 • Publication Date (Web): 21 Mar 2019 Downloaded from http://pubs.acs.org on March 21, 2019

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Journal of the American Chemical Society

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Journal of the American Chemical Society 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Effect of salt on the ordinary-extraordinary transition in solutions of charged macromolecules Di Jia and Murugappan Muthukumar∗ Department of Polymer Science and Engineering, University of Massachusetts, Amherst, Massachusetts 01003, USA.

∗

Electronic address: [email protected]

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Abstract Using dynamic light scattering technique, we address the role of added salt at higher concentrations on the ‘ordinary-extraordinary’ transition in solutions of charged macromolecules. The ‘ordinary’ behavior has previously been associated with a ‘fast’ diffusion coefficient which is independent of salt concentration Cs and polymer concentration Cp if the ratio Cp /Cs is above a threshold value. The ‘extraordinary’ transition is associated with formation of aggregates, with a ‘slow’ diffusion coefficient, formed from similarly charged macromolecules. By investigating aqueous solutions of sodium poly(styrene sulfonate) and sodium chloride with variations in Cp , Cs , and polymer molecular weight Mw , we report the emergence of a new diffusive ‘fast’ relaxation mode at higher values of Cp , Cs , and Mw , in addition to the previously known ‘fast’ and ‘slow’ relaxation modes. Furthermore, we find that Mw plays a crucial role on the collective dynamics of polyelectrolyte solutions with salt, instead of just the Cp /Cs ratio as previously postulated. As Mw is progressively decreased, the salty solution exhibits dynamical transitions from three modes to two modes and then to one mode of relaxation. The emergence of the new fast mode and the dynamical transitions are in marked departure from the general premise of the ordinary-extraordinary transition developed over several decades.

In an effort to rationalize our experimental findings

we present a theory for the collective dynamics of polyelectrolyte solutions with salt by addressing the coupling between the relaxations of polyelectrolyte chains, counterions from the polymer and added salt, and red co-ions from the salt. The predictions are in qualitative agreement with experimental findings. The present combined work of experiments and theory forms the basis for accurately characterizing dynamics of charged macromolecules in salty solutions, which are ubiquitous in biological systems and polyelectrolyte-based technologies.

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I.

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INTRODUCTION

Diffusion of charged macromolecules in concentrated aqueous solutions and crowded environments is a ubiquitous phenomenon in broad contexts ranging from the biological world to various therapeutic formulations in medicine. Typically, dynamic light scattering technique is employed in measuring diffusion coefficients of macromolecules in solution, which in turn are used to characterize the size of the macromolecules. However, an adequate understanding of the collective movements of charged macromolecules continues to be elusive due to nonlinear couplings among the long-range forces arising from electrostatics and hydrodynamics, and long-range structural correlations arising from chain connectivity. Experimental observations on concentrated solutions of charged macromolecules, such as DNA and synthetic polyelectrolytes, exhibit many tantalizing puzzles [1-3] which are counterintuitive to the established laws of diffusion seen in uncharged systems [4-7]. A fundamental understanding of these puzzles is necessary in order to successfully implement the dynamic light scattering technique in characterizing charged macromolecules. None of the well-established laws for diffusion in solutions of uncharged polymers is observed for polyelectrolyte solutions [1, 8-35]. A dramatic example is the phenomenon of the ‘ordinary-extraordinary‘ transition observed in the diffusional behavior of polyelectrolyte solutions [1, 8-12, 15-38]. In the ‘ordinary’ behavior, represented by a ‘fast’ mode of relaxation, the measured diffusion coefficient D in dynamic light scattering (DLS) is independent of the molecular weight Mw and the concentration Cp of the polyelectrolyte over several decades of Mw and Cp , D ∼ (Cp Mw )0 .

(independence on Cp and Mw )

(1)

This behavior is in stark contrast with neutral solutions where D depends on Mw and Cp [5,6]. In the ‘extraordinary’ behavior, representing a ‘slow’ mode of relaxation, D is typically several orders of magnitude smaller than the ‘fast’ diffusion coefficient, and depends on Mw and Cp [1]. The fast mode is attributed to a collective relaxation arising from an electrostatic coupling between the fluctuations in the polyelectrolyte concentration and the counterion concentration [1, 36-46]. The slow mode is attributed to the presence of large aggregates arising from dipole-dipole interactions [1, 21-28, 38]. Although salt-free polyelectrolyte solutions have been of considerable interest in fundamental studies of electrostatic correlations in polymer systems, a study of concentrated soluACS Paragon 3Plus Environment

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tions of charged macromolecules containing additional low molar mass salt is of much more relevance to the real world such as the various biological systems and polyelectrolyte-based technologies. The primary objective of the present paper is to address the effect of added salt on the ordinary-extraordinary transition in concentrated polyelectrolyte solutions. The amount of added low molar mass salt is already well-known to influence the simultaneous occurrence of both the ‘ordinary’ diffusion and ‘extraordinary’ diffusion. Drifford and Dalbiez [1, 16, 17] made an empirical suggestion that the onset of ordinary-extraordinary transition occurs at a particular threshold value of the ratio of polymer concentration Cp to salt concentration Cs , Cp ' 5.7, (2) Cs in the sodium polystyrene sulfonate-sodium chloride system and that for Cp /Cs > 5.7, both the slow and fast modes are to be observed. More significantly the Drifford-Dalbiez ratio was also argued to be independent of molecular weight [16, 17]. For the system of quarternized poly(2-vinylpyridine) in water with potassium bromide, investigated by F¨orster et al. [22], the splitting transition from one mode into the ordinary and extraordinary modes is shifted to higher Cp approximately by one order of magnitude as Cs is increased by one order of magnitude, consistent with the results of Drifford and Dalbiez [16, 17], but now the threshold value being about unity. In these studies, the deduction of the fast and slow diffusion coefficients from DLS data has generally been implemented with the implicit assumption that polyelectrolyte solutions containing added salt is essentially a two-component system, although the salt is explicitly accounted for in colloidal solutions [3]. As in the salt-free condition, the slow mode has been attributed to aggregates and the fast mode is associated with the coupled dynamics of polyelectrolyte chains and the salty background. The observation of only one relaxation mode corresponding to the coupled dynamics in salty polyelectrolyte solution is surprising, particularly given that multiple relaxation modes are observed in colloidal systems with salt [3]. In the presence of added salt, there are three charged species: polyelectrolyte, counterion (carrying a charge opposite to that of the polyelectrolyte), and coion (carrying a charge of the same sign as the polyelectrolyte). Hence there are three coupled equations for describing the time evolution of local polyelectrolyte concentration. As a result, we should expect three decay rates. One of these is the plasmon mode representing the relaxation of the ion atmospheres surrounding the various charged species in the system, as was already argued in the previous studies [1, 2]. This mode is ACS Paragon 4Plus Environment


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non-diffusive and too fast to be measured in DLS [2]. The other two decay rates correspond to two fast diffusive modes in salty polyelectrolyte solutions. Of course these should be present in addition to the slow mode due to aggregates. Overall, there are a total of four modes. One is non-diffusive and is too fast to be observed in DLS. The other three are diffusive. Therefore, there should be an additional fast diffusion mode when added salt is present, in comparison with salt-free polyelectrolyte solutions. Experimental and theoretical considerations of the dynamical coupling among the polyelectrolyte, counterions, and co-ions form the main focus of the present paper. We report for the first time the observation of the additional fast diffusive mode in salty polyelectrolyte solutions in the range of high values of Cp , Cs , and Mw , and provide a theory for this new diffusive mode. The new mode is an extra intermediate fast mode emerging between the well-known ordinary-extraordinary modes. This intermediate mode is detectable using DLS at higher values of polymer concentration (Cp >> Cp∗ ), Mw , and salt concentration. In addition, we also find that the molecular weight of the polyelectrolyte plays a significant role in the relaxation behavior of charged macromolecules. Even when the ratio Cp /Cs is kept constant above the threshold value of about unity [1], the number of relaxation modes changes progressively from three to two and then to one as the molecular weight is decreased. Therefore, both Cp /Cs and Cp /Cp∗ (Mw ) control the nature of dynamical couplings among the polyelectrolyte chains, counterions, and co-ions. Proper recognition of the four modes, namely the plasmon mode, two diffusive fast modes (namely the conventional fast mode and the new intermediate mode), and a slow mode is necessary in interpreting the DLS data in order to adequately describe the collective dynamics of semidilute salty solutions of charged macromolecules, and hence facilitate the use of DLS in accurately characterizing charged macromolecules.

II.

EXPERIMENTAL SECTION

Materials. Sodium polystyrene sulfonate (NaPSS) of four molecular weights (126, 234, 587, 2270 kDa) with narrow distribution (Mw /Mn ) = 1.04, 1.11, 1.05, 1.07 for 126, 234, 587, 2270 kDa, respectively) was purchased from Scientific Polymers.

These samples

were characterized by the supplier using light scattering, GPC/MALLS, and titration. ACS Paragon 5Plus Environment

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The reported level of sulfonation is 98%. Sodium Chloride (NaCl) was purchased from Sigma-Aldrich. Deionized water was obtained from a Milli-Q water purification system (Millipore, Bedford, MA, USA). The resistivity of deionized water used was 18.2 MΩ cm. Hydrophilic Polyvinylidene Fluoride (PVDF) filters with two pore sizes 220 and 450 nm were purchased from Millex Company.

Sample preparation and dynamic light scattering (DLS) measurement. Since DLS measurement is extremely sensitive to dust, the DLS tubes were first washed several times with pure water and acetone separately. After they were dried in the oven overnight, aluminum foil was used to wrap up the tubes and then these tubes were further cleaned by distilled acetone through an acetone fountain setup [47, 48]. Before conducting the DLS measurement, all the sample solutions were slowly filtered through 200 nm or 450 nm PVDF hydrophilic filter (depending on Mw of polymers) into the tubes to remove the dust. All the sample preparation work was conducted in a super-clean bench to avoid dust. DLS measurement was performed on a commercial spectrometer equipped with a multi-τ digital time correlator (ALV-5000/E) using Argon laser light source (output power=400 mW) of wavelength 514.5 nm, and the scattered intensity was measured with a photo-multiplier tube attached to an ALV goniometer arm. The spectrometer has a high coherence factor because of a single-mode optical fiber. The LS cell is held in an index matching thermostat vat filled with purified and dust-free xylene. For each sample, the intensity at each of the scattering angles (30°, 40°, 50°, 60°, and 90°) was correlated and each data point was obtained by averaging over three samples. All the experiments were conducted at 25°C. Data acquisition and analysis. DLS measures the intensity-intensity time correlation function g2 (q, t) by means of a multi-channel digital correlator and related to the normalized electric field-field autocorrelation function g1 (q, t) through the Siegert relation [4], g2 (q, t) =

< I(0)I(t) > − 1 = β|g1 (q, t)|2 , < I >2

(3)

where β is the instrument coherence factor, t is the decay time and I(t) is the scattered intensity. q = (4πn/λ) sin(θ/2) is the scattering wave vector at the scattering angle θ, where λ = 514.5 nm, and n is the refractive index of the solution (1.332). β is around 0.7-0.9 depending on samples, indicating that the coherence is very high and signal-to-noise ratio is very good. The electric field-field correlation function g1 (t) is fitted with a sum of several ACS Paragon 6Plus Environment


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relaxation modes, g1 (q, t) = A0 +

X

Ai exp(−Γi t),

(4)

where A0 is the baseline, and Ai is the intensity weighting fraction of i-th mode, each having a characteristic decay rate Γi . Both multi-exponential fitting method and the CONTIN method [49] were used to analyze the data. CONTIN analysis fits a weighted distribution of multiple relaxation decay times by the inverse Laplace transform. The decay rate Γi is obtained from Eq.(4) for the multi-exponential fits. If the mode is diffusive, the decay rates are linear with q 2 for all scattering angles. From the slope of Γ = Dq 2 across all q, diffusion coefficient D is obtained [2-4]. The fitting residuals, which are obtained from the difference between the raw data and the fitting curve, are randomly distributed near the mean of zero and do not have systematic fluctuations about their mean, indicating that the fitting quality is high. The fitting results obtained from both multi-exponential fits and the CONTIN analysis are similar.

RESULTS AND DISCUSSION In a systematic DLS study of polyelectrolyte solutions with high values of polyelectrolyte concentration Cp , salt concentration Cs , and polyelectrolyte molecular weight Mw , we have observed the emergence of a new diffusive mode. This mode is in addition to the ‘ordinary’ and ‘extraordinary’ modes reported in earlier studies [1, 10, 21, 22, 24, 25, 38]. In order to set the stage and as a reference frame, we first present data on the ordinary-extraordinary behavior in salt-free solutions of a polyelectrolyte with high molecular weight. In salt-free solutions or with very small amount of added salt, we observe only two modes (fast and slow) even at very high Mw , consistent with all previous studies on lower molecular weights [1, 10, 21, 22, 24, 25, 38]. Next, we present data for the emergence of a new mode in addition to the ordinary and extraordinary modes in the presence of sufficient amount of added salt. The polymer concentration dependence of this effect is presented next. We then present data demonstrating the crucial role of Mw on the number of relaxation modes in the system. In an effort to rationalize the occurrence of the new mode, we next present a theory by addressing the mutual coupling between relaxations of fluctuations in the concentrations of the polyelectrolyte chains, counterions, and co-ions.

1. Ordinary-extraordinary behavior for high Mw polyelectrolyte in salt-free ACS Paragon 7Plus Environment

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solutions The normalized electric field correlation function g1 (t) with decay time t, obtained from DLS, for sodium poly(styrene sulfonate) (NaPSS) with Mw = 2270 kDa (labeled as PSS2770) at Cp = 40 mg/ml ( Cp,monomer =0.22 M) in salt-free solutions is given in Fig.1(a) at the scattering angles 30°, 50°, 60°, and 90°. The molecular weight of this sample is higher than the molecular weights of NaPSS used in the previous studies reported in the literature [1, 10, 21, 22, 24, 25, 38]. The field-correlation functions are analyzed with the CONTIN method as well as the multi-exponential fitting procedure. Both procedures show that there are two relaxation modes. The residuals between the raw data and the fitted exponentials are given as green open triangles for the scattering angle 30°, as a typical example. The residuals are naturally very close to zero and based on such good fits to the raw data the two decay rates Γ1 and Γ3 are obtained. Γ1 and Γ3 are plotted against q 2 in Figs.1(b) and 1(c), showing their linear relation. The slopes give the diffusion coefficients as D1 = (3.26 ± 0.29) × 10−6 cm2 /s and D3 = (1.34 ± 0.06) × 10−10 cm2 /s. The CONTIN analysis of the same data for the correlation function also shows two decay rates Γ1 and Γ3 as included in Figs.1(b) and (c). Both Γ1 and Γ3 are linear in q 2 with the corresponding diffusion coefficients given as D1 = (3.15 ± 0.24) × 10−6 cm2 /s and D3 = (1.29 ± 0.05) × 10−10 cm2 /s. The error bars are obtained only from three samples and hence only two significant figures are reliable in the values of the various diffusion coefficients. Both CONTIN and multi-exponential fitting procedure give essentially the same values of diffusion coefficients. Nevertheless we prefer the multi-exponential fitting procedure when several relaxation modes are present due to the transparency of the analysis and the readiness with which the quality of the residuals exhibit the accuracy of the fitting procedure. In view of this, the data analyses for all systems discussed below are performed using the multi-exponential fitting procedure, except for portraying the distribution function of decay rates, which is obtained from the CONTIN analysis. The values of D1 and D3 for salt-free PSS2270 at Cp = 40 mg/ml are given in Table I, along with data for Cp = 10 mg/ml ( Cp,monomer =0.055 M) and 5 mg/ml ( Cp,monomer =0.027 M) (Figure S1). As Cp is decreased, D1 decreases and D3 increases for fixed Mw = 2270 kDa in salt-free conditions. Although the molecular weight of this sample is higher than in the previous studies, the numerical values of our diffusion coefficients are consistent with prior reports in the literature [1, 10, 21, 22, 24, 25, 38]. In view of this, we identify the two ACS Paragon 8Plus Environment


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FIG. 1: Dynamic light scattering results for PSS2270 at 40 mg/ml (Cp,monomer = 0.22 M) in saltfree solution. (a) Normalized field-correlation function at different scattering angles. Green open triangles are the residuals between the fitting curve and raw data at the scattering angle 30°. (b), (c) q 2 dependence of the relaxation rate Γ of the fast mode (b) and slow mode (c). Both CONTIN and multi-exponential fitting methods give similar results for the diffusion coefficient.

diffusion coefficients measured in salt-free PSS2270 solutions as the ‘fast’ (D1 ) and ‘slow’ (D3 ) diffusion coefficients corresponding to the ‘ordinary’ and ‘extraordinary’ diffusion. Furthermore, even when 0.01 M NaCl is present in 40 mg/ml PSS2770 solution, we have observed two diffusive modes with D1 = 2.49 × 10−6 cm2 /s and D3 = 1.61 × 10−10 cm2 /s (Figure S2). As seen from Table I, these values of the diffusion coefficients at such small salt concentrations are close to the values in salt-free solutions. Therefore, we identify D1 and D3 as the usual fast and slow diffusion coefficients, respectively. There are only two modes observed in salt-free or at very low salt concentrations for PSS2770 solutions. In conformity with the prior advances [1-3, 38-46], the fast mode arises from the dynamical coupling between fluctuations in the polymer concentration and the neutralizing counterion concentration. The slow mode arises from the transient inter-chain aggregation due to longlived quadrupoles induced by attractive dipole-dipole interactions [1, 38]. ACS Paragon 9Plus Environment

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1

(M )

FIG. 2: (a), (c), (e) Field-correlation functions at different scattering angles for PSS2270 at 40 mg/ml with 0.1 M, 1 M and 2 M NaCl, respectively. Green open triangles are the residuals between the fitting curve and raw data at 30°. (b), (d), (f) Corresponding fitting results of Γ vs q 2 at all angles. (g) Relaxation time distribution function of PSS2270 at 40 mg/ml (Cp,monomer = 0.22 M) with different salt concentrations at scattering angle 30°. (h) Diffusion coefficient D as a function of salt concentration Cs for PSS2270 at 40 mg/ml. ACS Paragon10 Plus Environment


Page 12 of 29

TABLE I: Ordinary-Extraordinary diffusion coefficients for salt-free PSS2270 Cp (mg/ml) Cp,monomer (M) Cs (M)

D1 (cm2 /s)

D2 (cm2 /s)

D3 (cm2 /s)

40

0.22

0

(3.15 ± 0.24) × 10−6

-

(1.29 ± 0.05) × 10−10

10

0.055

0

(9.85 ± 0.08) × 10−7

-

(3.50 ± 0.12) × 10−10

5

0.027

0

(7.37 ± 0.11) × 10−7

-

(8.20 ± 0.13) × 10−10

40

0.22

0.01

(2.49 ± 0.09) × 10−6

-

(1.61 ± 0.06) × 10−10

2. Emergence of a new mode for polyelectrolyte solutions with high salt At higher salt concentrations in PSS2770 solutions at 40 mg/ml ( Cp,monomer =0.22 M), careful data analysis of the field-correlation function in our DLS measurements (Figs.2(a)(h)) reveals three relaxation modes, instead of the usual two modes of ordinary and extraordinary behavior. We find the decay rate of the extra new mode to be in between the fast and slow modes. The normalized field-correlation function g1 (t) for PSS2770 (Cp = 40 mg/ml, ( Cp,monomer =0.22 M)) at different scattering angles is given in Figs.2(a), 2(c), and 2(e), respectively, for 0.1 M, 1.0 M, and 2.0 M NaCl solutions. The fitted curves for different scattering angles quantitatively superpose on the raw data. As an illustration of the good quality of the fitting procedure, the residuals between the fitted curve and raw data are included as green open triangles for the scattering angle 30°. The only way such “zero” residuals can be obtained in our data analysis is by invoking three relaxation modes. The corresponding three decay rates Γ are plotted versus q 2 in Figs.2(b), 2(d), and 2(f) for Cs = 0.1, 1.0, and 2.0 M NaCl. The Γ versus q 2 plots are linear for each of the three modes demonstrating that they are all diffusive, with the corresponding diffusion coefficients summarized in Table II and Fig.2(h). As an additional support for the observation of the new mode, the distribution function f (t) of the decay time t at the scattering angle 30°, as obtained from the CONTIN analysis, is given in Fig.2(g) for Cs = 0.1, 1.0, and 2.0 M NaCl. This distribution function unambiguously demonstrates the presence of three modes. Also, it is evident from Fig.2(g) that the two fast modes are comparable in their weights. The three diffusion coefficients for different salt concentrations for PSS2770 at Cp = 40 mg/ml are summarized in Table II. As an example, when Cs =0.1 M, D1 = 8.63 × 10−7 cm2 /s, D2 = 5.25 × 10−8 cm2 /s, and D3 = 7.90 × 10−10 cm2 /s. The numerical values of D1 and D3 are very close to the values of the fast (D1 ) and slow (D3 ) modes, respectively, ACS Paragon11 Plus Environment

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TABLE II: Emergence of the co-ion mode for PSS2270

Cp (mg/ml) Cp,monomer (M) Cs (M)

D1 (cm2 /s), fast

D2 (cm2 /s), intermediate

D3 (cm2 /s), slow

40

0.22

0

(3.15 ± 0.24) × 10−6

-

(1.29 ± 0.05) × 10−10

40

0.22

0.01

(2.49 ± 0.09) × 10−6

-

(1.61 ± 0.06) × 10−10

40

0.22

0.1

(8.63 ± 0.11) × 10−7

(5.25 ± 0.79) × 10−8

(7.90 ± 0.18) × 10−10

40

0.22

1

(3.17 ± 0.08) × 10−7

(5.44 ± 0.31) × 10−8

(2.85 ± 0.21) × 10−9

40

0.22

2

(2.23 ± 0.08) × 10−7

(4.23 ± 0.28) × 10−8

(5.67 ± 0.25) × 10−9

for salt-free solutions or at very low salt concentration (which are included in Table II for easy comparison). Also, as evident from Figs.2(g) and 2(h) and Table II, the third diffusion coefficient D3 increases with an increase in salt concentration, consistent with the expected smaller aggregate size at higher salt concentration, which is a well established result for the slow mode [1, 38]. Based on these qualitative and quantitative comparisons, we identify the first and third relaxation modes as the well-known fast and slow modes, respectively. The second mode is the newly discovered extra mode with a decay rate intermediate between the usual fast and slow modes. As a result, we refer D1 , D2 , and D3 as Dfast , Dintermedite , and Dslow , as denoted in Table II. As evident from Table II, as Cs increases, Dfast decreases and Dslow increases, while Dintermediate remains essentially a constant at higher Cs values, but absent at lower Cs values. The decrease of Dfast with an increase in salt concentration, observed for PSS2770 at 40 mg/ml, is analogous to the behavior of the fast mode reviewed in Ref.(1). Also, the observed increase in Dslow as Cs increases is entirely consistent with the extraordinary mode reported in the literature[1, 10, 21, 22, 24, 25, 38]. While the new Dintermediate is insensitive to Cs as seen from Fig.2(h) and Table II, this second fast mode is not observed in our system if the salt concentration is very low as in the situations of Cs = 0.01 M and Cs = 0. The second fast mode was not reported altogether in the earlier studies as well. Based on the theory presented below, we assert that, for salty polyelectrolyte solutions, there ought to be three relaxation modes, corresponding to three electrostatically coupled charged species: polyelectrolyte, counterions from the polyelectrolyte and added salt, and co-ions from the added salt. These are the plasmon mode [1,2] which is unobservable in DLS, and two ACS Paragon12 Plus Environment


fast modes. These are on top of the slow mode. When the dynamics of three species are coupled, the three relaxation rates are collective phenomena and cannot be labeled as modes corresponding to specific individual species. Nevertheless, since Dfast is observed in the absence of added salt, we refer to it as the ‘polymer mode’. Since Dintermediate is absent in the absence of added salt, we refer to it as the ‘salt mode’ or interchangeably ‘co-ion mode’. As theorized below, the extent of coupling between the three species is directly related to the product of concentrations of possible combinations of pairs among the three species. Therefore, only at higher concentrations, the coupling among the three species is strong. The coupling is weak for low values of molecular weights, polymer concentrations, and salt concentrations used in the earlier investigations and in our study at very low Cs . This might be the reason for the ‘salt’ mode not being observed. Our experimental conditions of high Cp , Cs , and Mw enable strong coupling and hence the observation of the new additional ‘salt’ mode. Our results on the other two modes (D1 and D3 ) are indeed entirely consistent with earlier results in the literature [1, 21, 22, 25, 26, 38].

3. Role of polyelectrolyte concentration on the new mode. There are three relaxation modes for PSS2770 at Cs = 0.1M and Cp = 40 mg/ml as described above. By following the exactly same careful procedure, data analysis of measured g1 (t) for PSS2770 at Cp = 20 mg/ml ( Cp,monomer =0.11 M) and Cs = 0.1M also shows three modes. The deduced diffusion coefficients are given in Fig.3 and Table III. When Cp is lowered, but still high enough for aggregate formation, we observe only two modes and the intermediate fast mode is absent. At lower polyelectrolyte concentrations, the coupling between the polyelectrolyte and co-ions is so weak that the correlation among the co-ion dynamics and polyelectrolyte dynamics is essentially immeasurable in DLS. However, the other two diffusive modes (fast and slow) are fully consistent with the values reported previously in the literature.

4. Dependence of relaxation modes on molecular weight.

Previous studies on the diffusion of polyelectrolyte solutions have advanced the hypothesis that only the ratio Cp /Cs is relevant to the simultaneous occurrence of the ordinary and extraordinary modes and that the molecular weight of the polymer does not play a significant ACS Paragon13 Plus Environment

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D 1

P S S 2 2 7 0 w ith 0 .1 M

N a C l

1 0

-6

1 0

-7

1 0

-8

1 0

-9

D 3

(c m

2

/s )

D 2

D

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0

1 0

C

2 0 p

3 0

4 0

( m g /m l)

FIG. 3: Diffusion coefficients as a function of polymer concentration Cp for PSS2270 with 0.1 M NaCl. The error bars are within the size of the symbols. (40 mg/ml = 0.22 M monomer concentration)

TABLE III: Dependence on Cp for PSS2270

Cp (mg/ml) Cp,monomer (M) Cs (M)

D1 (cm2 /s)

D2 (cm2 /s)

D3 (cm2 /s)

40

0.22

0.1

(8.63 ± 0.11) × 10−7 (5.25 ± 0.79) × 10−8 (7.90 ± 0.18) × 10−10

20

0.11

0.1

(4.40 ± 0.05) × 10−7 (5.14 ± 0.22) × 10−8 (2.71 ± 0.04) × 10−9

10

0.055

0.1

(2.42 ± 0.01) × 10−7

-

(5.66 ± 0.28) × 10−9

1

0.005

0.1

(1.11 ± 0.05) × 10−7

-

(2.10 ± 0.08) × 10−8

role. We observe strong violation of this hypothesis and that the molecular weight indeed plays a very significant role. By maintaining the ratio Cp /Cs to be 2.2 (Cp,monomer =0.22 M, and Cs = 0.1M), which is above the threshold value of about unity [1], we have investigated the role of molecular weight with four NaPSS samples of Mw = 2770 kDa, 587 kDa, 234 kDa, and 126 kDa. As already discussed, there are three modes for Mw = 2270 kDa. As Mw is lowered to 587 kDa and 234 kDa, there are only two modes. The normalized field-correlation function g1 (t) for PSS587 and the corresponding distribution function f (t) of decay times, ACS Paragon14 Plus Environment


and the q 2 dependence of the decay rates are given in Figs.4(a), 4(b), and 4(c), respectively. The residuals for the fitting at the scattering angle 30° are included in Fig.4(a). It is evident from these figures that there are only two decay modes. The diffusion coefficients for these modes are D1 = (1.36 ± 0.07) × 10−6 cm2 /s and D3 = (1.58 ± 0.13) × 10−8 cm2 /s, as given in Table IV. These values are similar to the Dfast and Dslow values in the literature for similar molecular weights [1, 21, 22, 25, 26, 38]. Hence the intermediate second mode (Dintermediate ) is absent for PSS587 (Mw = 587 kDa). The same result of only two relaxation rates is observed for PSS234 (Mw = 234 kDa) at Cp = 40 mg/ml and Cs = 0.1M as well. The values of the diffusion coefficients for this system are included in Table IV. Again, the values of D1 and D3 for PSS234 are commensurate with Dfast and Dslow values in the literature for comparable molecular weights [1, 21, 22, 25, 26, 38]. When the molecular weight is lowered even further to 126 kDa (PSS126), but still keeping the ratio Cp /Cs as 2.2 above the previously claimed threshold value of about unity [1], we observe only one relaxation mode. The normalized field-correlation function g1 (t) along with residuals at the scattering angle 30°, and the distribution function f (t) of decay rates are presented in Figs.4(d) and 4(e). It is clearly obvious that there is only one mode for this molecular weight, although Cp /Cs is above the Drifford-Dalbiez ratio. The linear dependence of the decay rate on q 2 is demonstrated in Fig.4(f), where the diffusion coefficient is D1 = (1.48 ± 0.02) × 10−6 cm2 /s. For such lower molecular weights as for PSS126, with Cp /Cs = 2.2 (molar ratio), even the slow mode is absent. Therefore, we report dynamical transitions from three modes of relaxation to two modes of relaxation and then to only one mode of relaxation as the molecular weight is progressively reduced at fixed Cp and Cs . The absence of the salt mode for molecular weights below 2270 kDa in our system suggests that high molecular weights of the polyelectrolyte is required in addition to higher values of Cp and Cs in order to realize strong dynamical coupling among the three species. Moreover, addition of more salt to the lower molecular weight system does not promote the observation of the second fast mode. For example, the field-correlation function g1 (t) for PSS126 at Cp = 40 mg/ml ( Cp,monomer =0.22 M) and Cs = 1.6 M NaCl is given in Fig.5(a), along with the residuals. There is only one relaxation as shown in Fig.5(b), which is diffusive with the diffusion coefficient D1 = (3.94 ± 0.02) × 10−7 cm2 /s. Although the salt concentration and the polyelectrolyte concentration are held at higher values, the ‘salt’ (‘co-ion’) mode is not observed for PSS126. Therefore, the ability of polyelectrolyte chains ACS Paragon15 Plus Environment

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P S S 5 8 7 4 0 m g /m l 0 .1 M

1 .0 0 .8

o

4 0

o

5 0

o

N a C l

P S S 5 8 7 4 0 m g /m l 0 .1 M

1 .0

(b ) o

0 .6

0 .4

0 .4

0 .2

0 .2

0 .0

3 0

o

4 0

o

5 0

o

6 0

o

9 0

o

0 .0

1 E 7

1 E 8

1 E 9

1 E 1 0

2

-2

t q / (m s .c m

1 E 1 1

1 0

-3

1 0

-2

-1

P

S

S

6 0

5

8

7

4

F

i r s t

T

h

0

m

i r d

m

m

o

d

o

g

/ m

l

0

. 1

M

N

a

C

1

1 0

P S S 1 2 6 4 0 m g /m l 0 .1 M

1 .0

( c )

l

0

1 0 1 0 1 0 d e c a y tim e t / m s

)

8 0 e

d

0 .8

e

- 6

D

=

( 1

. 3

6

± 0

. 0

7

) × 1

0

2

c m

/ s

1

- 8

D

2 0

=

( 1

. 5

8

± 0

. 1

3

) × 1

0

3 0

o

4 0

o

5 0

o

2

1 0

N a C l

3

(d )

o


0 .6 4 0

g 1(t)

Γ( 1 /m s )

N a C l

0 .8

f(t)

g 1(t)

3 0


0 .6

0 .4

2

c m

/ s

3

0 .2 0 .0

0 0

1 x 1 0

1 0

2 x 1 0

1 0

3 x 1 0 2

q

1 0

4 x 1 0

1 0

5 x 1 0

1 0

6 x 1 0

1 E 7

1 0

1 E 8

1 E 9

1 E 1 0

2

-2

t q / (m s .c m

- 2

( c m

)

P S S 1 2 6 4 0 m g /m l 0 .1 M

1 .0

N a C l

8 0

(e )

7 0

0 .8

P

S

S

1

2

6

4

0

m

g

/ m

l

0

. 1

M

=

( 1

. 4

N

1 E 1 1

) a

C

l

. 0

2

( f )

6 0

Γ( 1 /m s )

f(t)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60


0 .6 0 .4

3 0

o

4 0

o

5 0

o

6 0

o

9 0

o

5 0 4 0 3 0

- 6

D

8

± 0

) × 1

0

2

c m

/ s

1

0 .2

2 0 1 0

0 .0

1 0

-3

1 0

-2

-1

0

1 0 1 0 1 0 d e c a y tim e t / m s

1

1 0

2

1 0

3

0 0

1 x 1 0

1 0

2 x 1 0

1 0

3 x 1 0

1 0

2

q

4 x 1 0

1 0

5 x 1 0

1 0

6 x 1 0

1 0

- 2

( c m

)

FIG. 4: DLS results for PSS587 and PSS126 at 40 mg/ml (Cp,monomer = 0.22 M) in 0.1 M NaCl solutions. (a), (d) Field-correlation functions g1 (t) at different scattering angles for PSS587 (a) and PSS126 (d). Green open triangles are the residues between the fitting curve and raw data at 30°. (b), (e) Relaxation time distribution function for PSS587 (b) and PSS126 (e). (c), (f) Fitting results of Γ vs q 2 at all angles for PSS587 (c) and PSS126 (f). The error bars are within the size of the symbols.

to interpenetrate heavily also appears to be a factor in the emergence of the new relaxation mode. 5. Theory We consider a semidilute solution of n polyelectrolyte chains in an aqueous solution of volume V containing zc : zc type salt. By modeling each chain as a Kuhn chain of N segments with ACS Paragon16 Plus Environment


TABLE IV: Dependence on Mw for Cp /Cs = 2.2.

D1 (cm2 /s)

Mw (KDa) Cp,monomer (M) Cs (M)

D2 (cm2 /s)

D3 (cm2 /s)

2270K

0.22

0.1

(8.63 ± 0.11) × 10−7 (5.25 ± 0.79) × 10−8 (7.90 ± 0.18) × 10−10

587K

0.22

0.1

(1.36 ± 0.07) × 10−6

-

(1.58 ± 0.13) × 10−8

234K

0.22

0.1

(1.36 ± 0.02) × 10−6

-

(1.11 ± 0.07) × 10−8

126K

0.22

0.1

(1.48 ± 0.02) × 10−6

-

-

P S S 1 2 6 4 0 m g /m l 1 .6 M

1 .0

N a C l

P

2 0

(a )

S

( b

S

1

2

6

4

0

m

g

/ m

l

1

. 6

M

N

a

C

l

)

0 .8

0 .4

3 0

o

4 0

o

5 0

o

Γ( 1 /m s )

1 5

0 .6

g 1(t)

1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60

Page 18 of 29

1 0

o


0 .2

5

- 7

D

=

( 3

. 9

4

± 0

. 0

2

) × 1

0

2

c m

/ s

1

0 .0 1 E 7

1 E 8

1 E 9 2

t q / (m s .c m

1 E 1 0 -2

1 E 1 1

0 0

1 x 1 0

1 0

2 x 1 0

)

1 0

3 x 1 0 2

q

1 0

4 x 1 0

1 0

5 x 1 0

1 0

6 x 1 0

1 0

- 2

( c m

)

FIG. 5: (a) Field-correlation function g1 (t) at different scattering angles for PSS126 at 40 mg/ml (Cp,monomer = 0.22 M) in 1.6 M NaCl solution. (b) q 2 dependence of the relaxation rate Γ for the same sample.

segment length `, let each segment bear an effective charge αzp e, where α is the degree of ionization of the chain. We take zc e as the charge of the counterion. The average number concentration of the polymer segments is nN/V and let the average number concentration of the salt be Cs0 . Assuming the counterions from the polyelectrolyte and the added salt to be identical species, the average number concentration of the i-th species, with i = 1, 2, and 3 denoting respectively the polyelectrolyte segments, counterions, and coins is given by the condition of electroneutrality as C10 =

nN , V

C20 = α|

zp 0 |C + Cs0 , zc 1

C30 = Cs0 .

(5)

The time evolution of the local concentration of the i-th species Ci (r, t) at the spatial location r and at time t is given by the continuity equation [2], ∂Ci (r, t) = Di ∇2 Ci (r, t) − ∇ · [Ci (r, t)µi E(r, t)], ∂t ACS Paragon17 Plus Environment

(6)

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where Di is the cooperative diffusion coefficient and µi is the electrophoretic mobility of the i-th species, and E(r, t) is the local electric field due to the ionic environment acting on the polyelectrolyte segment at r. In view of the Poisson equation and electroneutrality condition, the time evolution of the fluctuation in Ci (r, t) = Ci0 + δCi (r, t) about the average value Ci0 follows as [2] 3

∂δCi (r, t) µi X = Di ∇2 δCi (r, t) − Ci0 zj eδCj (r, t), ∂t 0 j=1

(7)

where 0 is the permittivity of vacuum and is the dielectric constant of the medium. For the counterions and co-ions, the electrophoretic mobility is given directly in terms of the diffusion coefficient [2], µi =

zi eDi . kB T

(i = 2, 3)

(8)

On the other hand, for polyelectrolyte chains in semidilute solutions, the electrophoretic mobility is given in terms of the self diffusion coefficient Dself whereas the diffusive part of Eqs.(7-8) is given in terms of the cooperative diffusion coefficient Dcoop . Writing Dcoop as D1 and Dself as γ1 D1 , the electrophoretic mobility of polyelectrolyte segments is given by µ1 =

αzp eγ1 D1 . kB T

(9)

Explicit expression for γ1 has already been derived in previous publications based on a selfconsistent theory [37] for the triple screening of topological correlations of chain connectivity, electrostatic interactions, and hydrodynamic interactions. γ1 is a rich function of C10 , Cs0 , and α, and the original references [36-38] may be referred to for more details. Using Eqs.(7-9) and taking the Fourier transform of δCi (r, t) to δCi (q, t), where q is the scattering wave vector, the three coupled equations for the polymer segments, counterions, and co-ions become ∂δC1 (q, t) αzp eγ1 D1 = −D1 q 2 δC1 (q, t)−C10 [αzp eδC1 (q, t)+zc eδC2 (q, t)−zc eδC3 (q, t)] (10) ∂t 0 kB T

αzc eD2 ∂δC2 (q, t) = −D2 q 2 δC2 (q, t) − C20 [αzp eδC1 (q, t) + zc eδC2 (q, t) − zc eδC3 (q, t)] (11) ∂t 0 kB T

∂δC3 (q, t) αzc eD3 = −D3 q 2 δC3 (q, t) + C30 [αzp eδC1 (q, t) + zc eδC2 (q, t) − zc eδC3 (q, t)] (12) ∂t 0 kB T ACS Paragon18 Plus Environment


Page 20 of 29

The three coupled equations have three decay rates (Γ1 , Γ2 , and Γ3 ) which are the roots of the cubic equation Γ3 + a2 Γ2 + a1 Γ + a0 = 0,

(13)

a0 = −D1 D2 D3 [q 6 + q 4 (γ1 q12 + q22 + q32 )]

(14)

where

a1 = {q 4 (D1 D2 +D1 D3 +D2 D3 )+q 2 [D1 D2 (γ1 q12 +q22 )+D1 D3 (γ1 q12 +q32 )+D2 D3 (q22 +q32 ]} (15)

a2 = −(D1 + D2 + D3 )q 2 − (D1 γ1 q12 + D2 q22 + D3 q32 )

(16)

with q12 =

(αzp e)2 C10 0 (zc e)2 C20 0 (−zc e)2 C30 , q2 = ,q = . 0 kB T 0 kB T 3 0 kB T

(17)

The three decay rates are calculated by finding the roots of the above equation for a given set of polyelectrolyte concentration, salt concentration and the degree of ionization. The time correlation function of δC1 (q, t) is a superposition of exponentials of these three decay rates. The actual weights of these modes are not provided in this calculation [2]. Here we illustrate the emergence of three modes (one plasmon and two fast modes) by considering a representative situation with the assumptions: D2 = D3 = D0 , D1 < D0 , zp = 1, zc = 1, e2 /(4π0 kB T )=0.7 nm, and `=0.5 nm. For small values of the scattering wave vector pertinent to DLS such that q 2

Page 1 ! + ! !! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! ! !! ! ! ! ! ! ! + + + + + + Polymer*segment

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