Diffusion coefficient collective

Q2 behavior takes place at decreasing Q. The position of the crossover point Q (c) is a direct measure of the dynamic correlation length (c) = 1/Q (c). The plateau value at low Q determines the collective diffusion coefficient Dc. A simultaneous fit of all low Q spectra where a simple exponential decay was found led to the concentration dependence and the numerical values of Dc... 

Fig. 62. Collective diffusion coefficient Dc in semi-dilute PDMS/d-benzene systems at T = 343 K as a function of the monomer concentration c. (Reprinted with permission from [116]. Copyright 1982 J. Wiley and Sons, Inc., New York)...

As seen in the preceding section, the counterions play a crucial role in the mobility of the polyelectrolyte molecules. Even in the absence of an external electric field, the counterions exert an induced electric field in the immediate environment of a charged segment which in turn significantly modifies the collective diffusion coefficient of the polymer. This additional contribution is absent for uncharged polymers, where the cooperative diffusion coefficient Dc is given by the Stokes-Einstein law in dilute solutions. 

Therefore, the coupling of polymer segments to the counterion cloud, which is directly responsible for the term N in the above equation, dominates the collective diffusion coefficient. Since Rg N for salt-free solutions, Df is independent of N. 

For small chains in solution the translational diffusion significantly contributes to the overall decay of Schain(Q>0- Therefore precise knowledge of the centre of mass diffusion is essential. Combing dynamic light scattering (DLS) and NSE revealed effective collective diffusion coefficients. Measurements at different concentrations showed that up to a polymer volume fraction of 10% no concentration dependence could be detected. All data are well below the overlap volume fraction of (p =0.23. Since no -dependence was seen, the data may be directly compared with the Zimm prediction [6] for dilute solutions ... 

Fig. 6.16 NSE relaxation curves obtained from a 16% volume fraction poly(fluorosilicone) gel in acetone using tbe bigb resolution NSE spectrometer INI5 at tbe ILL, Grenoble. Tbe existence of plateaus that represent tbe level of excess scattering from static inhomogenieties at low wave vector Q is clearly visible. Tbe decay rates times of tbe dynamic parts yield tbe collective diffusion coefficient of tbe gel. (Reprinted with permission from [291]. Copyright 2002 American Chemical Society)...

At low Q the experiments measure the collective diffusion coefficient D. of concentration fluctuations. Due to the repulsive interaction the effective diffusion increases 1/S(Q). Well beyond the interaction peak at high Q, where S(Q)=1, the measured diffusion tends to become equal to the self-diffusion D. A hydrodynamics factor H(Q) describes the additional effects on D ff=DaH(Q)/S Q) due to hydrodynamics interactions (see e.g. [342]). Variations of D(Q)S(Q) with Q (Fig. 6.28) may be attributed to the modulation with H(Q) displaying a peak, where S(Q) also has its maximum. For the transport in a crowded solution inside a cell the self-diffusion coefficient is the relevant parameter. It is strongly... 

D, collective diffusion coefficient along the transverse direction... 

Fig. 25a, b. a. Collective diffusion coefficient D of a NIPA gel as determined by the kinetics of volume change, as a function of temperature. It diminishes at the critical point, b. collective diffusion coefficient as determined from the density fluctuations by use of photon correlation spectroscopy. The agreement between the results obtained from dynamics of microscopic fluctuations and from kinetics of macroscopic volume change is excellent considering the difficulty in the dynamic experiments... 

The results obtained by the present mechanical measurements are also consistent with the previous experimental results of the dynamic light scattering studies of the collective diffusion coefficient of gels and the rheological studies of the shear modulus of gels. The studies published by different researchers indicate that the concentration dependence of the collective diffusion constant of the polymer networks of gel and that of the elastic modulus are well represented by the following power law relationships [2, 3, 5]... 

The above experimental results as expressed in Eqs. (11) and (12), together with our experimental results Eq. (8), confirm the relationship between the collective diffusion coefficient, the elastic modulus, and the friction coefficient which is given in the Eq. (2). 

Nevertheless, the characteristic time constant is roughly proportional to the square of the typical size and to the inverse of collective diffusion coefficient D which is given by the modulus divided by the friction. The porous structures presented here are one of the solutions to achieve a high response material. 

The collective diffusion coefficient D, on the other hand, is very sensitive to the crosslink density, and TDFRS is well suited for its measurement since, because of the small scattering angles, there is no need to extrapolate for q—> 0. 

Fig. 21. Volume dependence of DsL Dg and D/D0. Collective diffusion coefficients D 1 20 and q 1 50 (TDFRS). Hard-sphere results for comparison (O [69,70], V [71], and theoretical predictions (dashed line) [72]). From Ref. [67]...

Gradient-driven diffusion phenomena at finite coverages are usually described in macroscopic terms via an adsorbate flux density in two dimensions. The chemical or collective diffusion coefficient D is accordingly defined through Fick s laws. The first of these empirical laws describes the diffusion flux density j across a borderline, which results from a coverage... 

The collective diffusion coefficient is thus relevant for the mass-transport at surfaces in systems, which are not in thermodynamic equilibrium. It generally depends on coverage. The above diffusion equation is widely employed to determine D, since the adsorbate concentration is a measurable quantity. In practice, frequently the decay of an adjusted coverage gradient is analyzed and diffusion equation is solved numerically or analytically for a given geometry. This task is considerably simplified when diffusion coefficients independent of coverage exist or may be assumed and ... 

Results from DLS measurements [Figure 5 (left)], show the collective diffusion coefficient Dc = T/q2 for the PNIPA gels swollen in aqueous solutions at different equilibrium concentrations ce of phenols, where T is the characteristic relaxation rate of the DLS correlation function. With this technique systems can be measured only below the phase transition. Samples equilibrated with phloroglucinol cannot therefore be studied above ca 30 mM at room temperature. All the systems display a similar response, with a noticeable, albeit limited, downturn as the limiting concentration is approached. 

Figure 5. Collective diffusion coefficient D c= Tlq1 (left) and Rayleigh ratio Rdyn (right) from dynamic light scattering intensity in PNIPA gels at 20°C at equilibrium with aqueous solutions containing different concentrations ce of phenols. Symbols are the same as in Figure 1.

Unlike the repulsive case, this important dependence of /on the energy correlation for attractive interactions is also expected to affect the collective diffusion coefficient. It is worth noting that diffusion will be significantly inhibited at low and high coverage except at intermediate 0. 

As with time-average light scattering, there is a need to introduce a stracture factor when calculating the average diffusion coeflflcient For comparative purposes, the collective diffusion coefficient D can be calculated, and this can be related to its value at inflnite dilution by [14],... 

At low ionic strength (kR 1), other effects connected with the finite diffusivity of the small ions in the EDL surrounding the particle are present. The noninstantaneous diffusion of the small ions (with respect to the Brownian motion of the colloid particle) could lead to detectable reduction of the single particle diffusion coefficient, Dq, from the value predicted by the Stokes-Ein-stein relation. Equation 5.447. For spherical particles, the relative decrease in the value of Dq is largest at k/ 1 and could be around 10 to 15%. As shown in the normal-mode theory, the finite diffusivity of the small ions also affects the concentration dependence of the collective diffusion coefficient of the particles. Belloni et al. obtained an explicit expression for the contribution of the small ions in Ac)... 

Without doing detailed quantitative analysis of the data, it can be stated that the polyion diffusion can be qualitatively described by two theoretical concepts. The first concept capable of qualitative explanation of the polyion diffusion is the concept based on considering polyions as interacting Brownian particles with direct interactions between polyions and hydrodynamic interactions. The short-time collective diffusion coefficient for a system of interacting Brownian particles treated by statistical mechanics is calculated from the first cumulant F of the dynamic structure factor S(q, t) as [15-17]... 

We note here also that, unlike collective diffusion coefficients, molecular self-diffusion coefficients are unaffected by critical effects, which is a significant advantage for systems with critical points, a typical situation in this context. We also note that self-diffusion studies are very general and that the NMR approach places little demand on the appearance of the sample (turbidity, color, rheology, etc.)... 

The apparent hydrodynamic diameters of the droplets (or the correlation length), as calculated using the Stokes-Einstein equation for a number of different systems, are given in Table 2. These early findings showed that the micelle sizes measured in near-critical and supercritical solutions were similar to those found for conventional water-in-oil microemulsions in liquid alkane. At lower fluid densities, DLS probes the combined effect of the collective diffusion coefficient of the micelle cluster and that of the individual micelles. 

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