Diffusion relaxation frequency

The diffusion relaxation frequency, in turn, is determined by the diffusion coefficient, adsorption and bulk concentration of the surfactant... 

The high frequency relaxation is attributed in part to the modulation of intermolecular dipolar interactions by the translational diffusion. The cutoff frequency (60 MHz at 55°C) corresponds to the local diffusive jump frequency that is estimated from measurements of the diffusion coefficient (D 10"6 cm2/sec at 55°) (19, 21). This cutoff frequency also varies in temperature with the same activation energy (Eact 0.25 eV) as the diffusion frequency. 

Physical Mechanisms. The simplest interpretation of these results is that the transport coefficients, other than the thermal conductivity, of the water are decreased by the hydration interaction. The changes in these transport properties are correlated the microemulsion with compositional phase volume 0.4 (i.e. 60% water) exhibits a mean dielectric relaxation frequency one-half that of the pure liquid water, and ionic conductivity and water selfdiffusion coefficient one half that of the bulk liquid. In bulk solutions, the dielectric relaxation frequency, ionic conductivity, and self-diffusion coefficient are all inversely proportional to the viscosity there is no such relation for the thermal conductivity. The transport properties of the microemulsions thus vary as expected from simple changes in "viscosity" of the aqueous phase. (This is quite different from the bulk viscosity of the microemulsion.)... 

Here (0 is the oscillation frequency, and the parameter cOb is the characteristic frequency, which is inverse proportinal to the diffusion relaxation time Xd given in Eq. (35). This characteristic frequency exists also for any transient relaxation processes. The interfacial response functions for a number of transient relaxations were discussed recently by Loglio et al. (2001). Among these, the trapezoidal area change is the most general perturbation which contains area changes such as the step or ramp type and the square pulse as particular cases. 

The first problem is to exclude geometrical relaxation. When an electric field is applied, charges are localized on the sample under the influence of both the field and the diffusion gradient. On reversal of the field, charges find a new equilibrium so that the macroscopic dipole, the dimensions of which are those of the sample, and the new polarization are opposed to the previous one. The resulting Debye-like relaxation frequency depends on the sample thickness. 

Note that, according to Equation (3.66), any significant portion of the current out of phase with respect to the field can be interpreted macroscopically as a large real part of the dielectric constant of the suspension. From our earlier qualitative description, the slowest processes are the diffusion fluxes originated by concentration polarization. At low frequencies, a high dielectric constant is thus expected for the suspension. As the frequency increases, these slow processes cannot follow the field as a consequence, they are frozen and the dielectric constant decreases. This is the a- (or volume diffusion-) relaxation of the suspension, which will occur for o>... 

The dielectric (e" and M") spectra and the relaxation frequencies of the p relaxation are similar to the mechanical spectra (J" and E", respectively) and the corresponding relaxation rate over a wide temperatures range (Muzeau et al. 1991 Perez et al. 1999). These observations suggest that the underlying mechanisms for the local electrical and mechanical relaxation processes in PMMA are similar. Clearly, this is not always the case for polymers, since all modes of motion of a polymer chain are not dielectrically active. When rotational diffusion occurs about a variety of different axes among which only a few reorient a dipole, the shape of the relaxation and the average rates of relaxation in a dielectric measurement may and will differ from those in a mechanical test. Dielectric, dynamic mechanical, and DSC glass transition... 

These results indicate that the effect of the liquid crystal ordering potential is to decrease the relaxation frequency for end-over-end rotation (t ) and increase it for rotation about the molecular long axis. If anisotropy in the rotational diffusion constants is included, then the relaxation time Tqo is further retarded. 

The rotational diffusion constant and relaxation frequency depend on temperature mainly through the temperature dependence of the viscosity which... 

Fig. 1.1 Typical spectral density of states for liquids covering the intermolecular region of the spectrum. The low-frequency shaded area represents the region of the spectrum corresponding to diffusive relaxation processes, as the energy increases we move into nondiffusive inertial-type motions. The inset represents a microscopic examination of the same spectral lineshape in which the spectrum is constructed of a distribution of nuclear configurations giving rise to much narrower lineshapes within an inhomogeneous lineshape vs. the homogeneous limit in which these configurations rapidly interconvert...

Fig. 1.2 The experimentally determined spectral density of states of liquid water using the optical Kerr effect -linear response) is shown for comparison [3], Water is one of the most structured liquids in which there is large separation in timescales. The frequency range between 300 and 1,000 cm (1) is related to librational motions (note that the spectrum represented here is artificially truncated at 600 cm due to laser bandwidth hmitations), the peak at 170 cm (2) is hindered translational motion of the heavy O atoms, the 60 cm mode (3) is transverse or shear motion, and below 25 cm (4) corresponds to diffusive relaxation and hydrogen bond breaking [8]. The issue of inhomogeneous to homogeneous interpretations of the dynamic structure of liquid water still holds despite this additional structure. The agreement is quite good with the theoretical calculation of the corresponding spectral density of states for water but the calculations are rather insensitive to basis with respect to this observable. Reprinted with permission from [3]. Copyright 1994, American Chemical Society...

Figure 8 Effects of spin diffusion. The NOE between two protons (indicated by the solid line) may be altered by the presence of alternative pathways for the magnetization (dashed lines). The size of the NOE can be calculated for a structure from the experimental mixing time, and the complete relaxation matrix, (Ry), which is a function of all mterproton distances d j and functions describing the motion of the protons, y is the gyromagnetic ratio of the proton, ti is the Planck constant, t is the rotational correlation time, and O) is the Larmor frequency of the proton m the magnetic field. The expression for (Rjj) is an approximation assuming an internally rigid molecule.

A possible approach to interpretation of a low-frequency region of the G ( ) dependence of filled polymers is to compare it with a specific relaxation mechanism, which appears due to the presence of a filler in the melt. We have already spoken about two possible mechanisms — the first, associated with adsorption phenomena on a filler s surface and the second, determined by the possibility of rotational diffusion of anisodiametrical particles with characteristic time D 1. But even if these effects are not taken into account, the presence of a filler can be related with the appearance of a new characteristic time, Xf, common for any systems. It is expressed in the following way... 

Although long-time Debye relaxation proceeds exponentially, short-time deviations are detectable which represent inertial effects (free rotation between collisions) as well as interparticle interaction during collisions. In Debye s limit the spectra have already collapsed and their Lorentzian centre has a width proportional to the rotational diffusion coefficient. In fact this result is model-independent. Only shape analysis of the far wings can discriminate between different models of molecular reorientation and explain the high-frequency pecularities of IR and FIR spectra (like Poley absorption). In the conclusion of Chapter 2 we attract the readers attention to the solution of the inverse problem which is the extraction of the angular momentum correlation function from optical spectra of liquids. 

In this case one determines the spectral intensity solely in the centre, not over the whole frequency range. Therefore the analysis often refers not to the spectrum as a whole, but to relaxation times Tg,i or and their dependence on rotational relaxation time tj [85]. This dependence contains much information and can be easier to interpret. It enables one to determine when free rotation turns into rotational diffusion. 

---