Slow mode concentration dependence

The slow diffusion coefficient is measurable only at high enough polyelectrolyte concentrations. The value of c at which the slow mode appears is higher if Cj is higher. When the ratio X = c/cg is about 1, the onset of the slow mode and the crossover between the smaller Df for A, < 1 and higher Df for A > 1 occur. Dj depends [33] on c strongly. 

A simple theory of the concentration dependence of viscosity has recently been developed by using the mode coupling theory expression of viscosity [197]. The slow variables chosen are the center of mass density and the charge density. The final expressions have essentially the same form as discussed in Section X the structure factors now involve the intermolecular correlations among the polyelectrolyte rods. Numerical calculation shows that the theory can explain the plateau in the concentration dependence of the viscosity, if one takes into account the anisotropy in the motion of the rod-like polymers. The problem, however, is far from complete. We are also not aware of any study of the frequency-dependent properties. Work on this problem is under progress [198]. 

In the case of PMMA dissolved in acetone, the change of LSI could be correlated both with the separation of fragments and, to some extent, with the lifetime of the intermediates which contribute to the main-chain scission [71]. The LSI decreased in two modes, probably due to the two pathways for the main-chain scission. The fast mode with a lifetime of about 20 ps was influenced neither in its extent nor in its rate by the addition of 02 or mercaptane. Therefore the first mode was ascribed to the diffusional separation of fragments which are generated by the main-chain scission through the direct decomposition of electronically excited or ionic intermediates. The slow mode with a lifetime of 6 ms was suppressed, to an extent, depending on the ( -concentration it was attributed to long-lived polymer radicals. The added 02 reacts with lateral polymer radicals to prevent their decomposition. 

Thus, in the fluid state, there are two relaxation processes, the a and the with relaxation times that scale with proximity to the critical point with differing exponents, -y and — l/2fl, respectively. For spherical particles, y — 2.58 and l/2a = 1.66 thus the a process is predicted to slow more dramatically as the transition is approached than the process. Figure 4-22 shows the relaxation times t and extracted from the relaxation data of Fig. 4-20 for the colloidal fluids. The power laws given by Eqns. (4-33) and (4-34) fit these experimental concentration dependencies well, supporting the mode-coupling theory of this transition. 

FIG. 22 Dependence of scattering amplitudes of the fast (o) and slow ( ) mode on solution ionic strength. Both added salt (NaCl) and free counterions are included in the calculation of ionic strength. Amplitudes represent excess scattering and are expressed in units of the scattering intensity of a benzene standard. Sodium poly(styrene sulfonate) (NaPSS), Mw = 5,000, polyion concentration c = 5 g/L, scattering angle 6 = 90°. (Adapted from Ref. 12.)... 

FIG. 16 Dependence of the fast and slow effective diffusion coefficients following the a-line (see Figure 15) as a function of NaPSS concentration (C) fast mode (full circle) slow mode (open circle). 

A bimodal decay is observed with the fresh solution and only a pseudo-monomodal correlation function is obtained with the old solution, which corresponds to the fast mode. Thus the time dependence of the contribution of the slow mode is an important phenomenon. Our results indicate that whatever these structures are, they must originate in the polyelectrolyte solution. The evolution of the slow mode with time seems to indicate that the fresh solution is not macroscopically in thermodynamic equilibrium with the presence of large clusters which disaggregate with time, indicating that a polyelectrolyte solution with a low external salt concentration (Cs/C 10 2) slowly tends to equilibrium in agreement with other observations [36,62,63], To quantify the evolution of the slow mode with the elapsed time from a fresh solution, we have measured the ratio of the amplitude of the autocorrelation function of the slow mode over the amplitude of the fast one AJAf. We observe a very slow decrease of this ratio as a function of time. Since the amplitude of the fast mode is time independent, the amplitude of the slow mode decreases to a value that is too small to be observed any longer. 

To examine the concentration dependence, a limiting value of T) q2 at q = 0, (T)lq2)q=0, which was obtained as the intercept of the lines in Figure 11a, is plotted as a function of polymer concentration in Figure lib. There is an increase in the ((T)/q2)g=0(= Df) value with concentration for the fast mode, but no discernible concentration dependence is noted for the slow mode (Ds). The concentration dependence of the fast mode may be compared with the concentration dependence of Kc/R(j in Figure 6. The (mutual) diffusion coefficient of nonaggregating polymer solutions is given by the generalized Stokes-Einstein relation [66,89,90],... 

The most surprising feature of the behavior of PS-PVP-PEO micelles with water-soluble PVP (protonized) and PEO blocks in acidic media is their aggregation in the region of low pH. Because it is a rather unexpected phenomenon, we studied it in more detail. The distributions of relaxation times obtained by DLS are bimodal (Eig. 8). Angular dependences (not shown) prove that both fast and slow relaxation modes correspond to diffusive processes. The intensity of the slow mode decreases with increasing pH and decreasing copolymer concentration. At very low copolymer and HCl concentrations, the slow mode disappears completely. The DLS measurements thus show that PS-PVP-PEO solutions contain two types... 

Figure 3.41. Concentration dependence of the diffusion coefficient Z), normalized by Dj = Dj at c -> 0 for polyvinylpyrrolidone in aqueous solutions from Figure 3.40 s data. Fast (/) and slow (2) modes. Curve 3 is the interdiffusion coefficient from the first cumulant ACi at q -> 0 (Burchard and Elsele, 1984) [Reprinted with permission from W.Burchard, M.Eisele. Pure and Appl. Chem. 56 (1984) 1379-1390. Copyright 1984 by the American Chemical Society]...

The associative behavior of the AMPS-Dodl m copolymer depends strongly on ybod (24,48). Figure 6 compares relaxation time distributions in QELS at varying concentrations for the copolymers with/ood = 2.5 and 10 mol %. The relaxation time distributions for the copolymer with /ood = 2.5 mol % are apparentiy unimodal. However, as the polymer concentration is increased, the distribution becomes bimodal, and the slow mode component increases with increasing the polymer concentration. In contrast, the copolymer with bcxl = % shows unimodal distributions... 

We now turn to the stretching exponents. Figure 2 shows typical c-dependence of p and Pf for several probe sizes. P(c) and pf(c) for other probe sizes can be found in ref. (26). Three distinct characteristic behaviors for p and p are apparent First, for large probes p is a concentration-independent constant, namely P = 1- Second, p from the small-probe slow mode and Pf from the large-i obe fast mode both decrease from 0.9-1 at low c to a 0.6-0.7 in 7 g/L HPC. Third, for small probes pf decreases from s 0.6 at low concentration to 0.2 in 6g/L HPC. At each concentration, the small-probe p and the large-probe Pf are substantially larger than the small-probe pf. 

Figure 3a shows 9 as a function of c for all probes. The large-probe and small-probe slow modes clearly have very different concentration dependences. For large probes 9 has a strong c-dependence, decreasing as 9oexp(-ac ). For small probes, 9 is c-independent. Fitting parameters are in ref (/). 

Figure 3 b shows the concentration dependence of 9f. In the solutionlike regime (c 7 g/L), 9f of each probe species is nearly independent of c. The large-probe 9f and the small-probe 9 thus have similar concentration dependences, consistent with our proposed incorporation of the large-probe fast mode and the small-probe slow mode into a single physical regime. 

FIGURE 5.3 Modeling of transient water flux data for Nation 117. (a) The relaxation of the experimental outlet vapor pressure (open circle) for Nafion 117 in LE mode at 50°C, flow chamber volume V = 0.125 L, flow rate V = 0.1 L min membrane area A = 2 cm, and saturation vapor pressure = 12336.7 Pa. Plotted for comparison are model simulations for a slow transport coefficient (dash dot), fast transport coefficient (dash), and a concentration-dependent transport coefficient (gray), (b) Water concentration profiles calculated in the model at different time. (Reprinted from Electrochem. Commun. 13, Rinaldo, S. G. et al. Vaporization exchange model for dynamic water sorption in Nafion Transient solution, 5-7, Figures 1 and 2, Copyright (2011) Elsevier. With permission.)... 

For the samples with C > Cf, q dependence of was entirely different from that of the slow mode described above. F was independent of q within an error of 20%, which means that the mode is the relaxation mode. For C > Cn, all samples were highly viscoelastic, so that the slow mode may be related to dynamical coupling between concentration fluctuation and elasticity of the viscoelastic network. 

Fig. 10 The diffusion constant of the fast mode (O) and slow mode ( ) of bimodal polydimethylsiloxane, crosslinked at both ends of the chain. The gel concentration dependence of intensity A of the fast mode is also shown.

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