Relaxation and Diffusion Components

As in chronoamperograms, the fraction of the overall oxidation charge involved in relaxation processes is quite small in the absence of any external stress. The share of the overall current at every potential between electrochemical processes occurring under relaxation control and those driven by swelling-diffusion control can be observed in Fig. 66. I(r) has 

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Equations (37) and (38), along with Eqs. (29) and (30), define the electrochemical oxidation process of a conducting polymer film controlled by conformational relaxation and diffusion processes in the polymeric structure. It must be remarked that if the initial potential is more anodic than Es, then the term depending on the cathodic overpotential vanishes and the oxidation process becomes only diffusion controlled. So the most usual oxidation processes studied in conducting polymers, which are controlled by diffusion of counter-ions in the polymer, can be considered as a particular case of a more general model of oxidation under conformational relaxation control. The addition of relaxation and diffusion components provides a complete description of the shapes of chronocoulograms and chronoamperograms in any experimental condition ... 

Reference device, use of mercury for, 16 Relaxation and diffusion components in polymer formation, 397... 

Figure 66. Separation of the overall oxidation curve into its relaxation (/r) and diffusion (/ components. (ReprintedfromT. F. Otero, H.-J. Grande, andJ. Rodriguez,/. Phys. Chem. 101, 8525, 1997, Figs. 3-11, 13. Copyright 1997. Reproduced with permission from the American Chemical Society.)...

NMR spectroscopy is one of the most widely used analytical tools for the study of molecular structure and dynamics. Spin relaxation and diffusion have been used to characterize protein dynamics [1, 2], polymer systems[3, 4], porous media [5-8], and heterogeneous fluids such as crude oils [9-12]. There has been a growing body of work to extend NMR to other areas of applications, such as material science [13] and the petroleum industry [11, 14—16]. NMR and MRI have been used extensively for research in food science and in production quality control [17-20]. For example, NMR is used to determine moisture content and solid fat fraction [20]. Multi-component analysis techniques, such as chemometrics as used by Brown et al. [21], are often employed to distinguish the components, e.g., oil and water. 

Our approach is to use the two-dimensional relaxation and diffusion correlation experiments to further enhance the resolution of different components. It is important to note that the correlation experiment, e.g., the Ti-T2 experiment, is different from two experiments of and T2 separately. For instance, the separate Ti and T2 experiment, in general, cannot determine the T1(/T2 ratio for each component. On the contrary, a component with a particular Tj and T2 will appear as a peak in the 2D 7i-T2 and the Ti/T2 ratio can be obtained directly. For example, small molecules often exhibit rapid rotation and diffusion in a solution and Ti/T2 ratio tends to be close to 1. On the other hand, the rotational dynamics of larger molecules such as proteins can be significantly slow compared with the Larmor frequency and resulting in a Ti/T2 ratio significantly larger than 1. 

In summary, the new 2D experiments of relaxation and diffusion appear to offer a new method to identify and quantify the components in dairy products. The two components are well separated in the 2D maps while they can be heavily overlapped in the ID spectrum. We find that some microscopic properties of the products can be reflected in the relaxation and diffusion properties. These new techniques are likely to be useful to assist the characterization of the products for quality control and quality assurance. 

It should be noted that the decomposition shown in Eq. 3.7.2 is not necessarily a subdivision of separate sets of spins, as all spins in general are subject to both relaxation and diffusion. Rather, it is a classification of different components of the overall decay according to their time constant. In particular cases, the spectrum of amplitudes an represents the populations of a set of pore types, each encoded with a modulation determined by its internal gradient. However, in the case of stronger encoding, the initial magnetization distribution within a single pore type may contain multiple modes (j)n. In this case the interpretation could become more complex [49]. 

Many materials are complex mixtures of multiple molecular species and components and each component can be in multiple chemical or physical states. Realtime determination of the components and their properties is important for the understanding and control of the manufacturing processes. This paper reviews a recently developed technique of 2D NMR of diffusion and relaxation and its application to identify components of materials. This technique may have further applications for the study of biological systems and in industrial process control and quality assurance. 

Diffusivity correlates linearly with the ratio of temperature and viscosity. Therefore the diffusivity can also be expected to correlate with relaxation time because the latter correlates with temperature and viscosity according to Eq. (3.6.1). Figure 3.6.3 illustrates the correlation between relaxation time and diffusivity with the gas/oil ratio as a parameter [13]. The correlation between diffusivity and relaxation time extends to hydrocarbon components in a mixture and there is a mapping between the distributions of diffusivity and relaxation time for crude oils [17]. 

This bimodal dynamics of hydration water is intriguing. A model based on dynamic equilibrium between quasi-bound and free water molecules on the surface of biomolecules (or self-assembly) predicts that the orientational relaxation at a macromolecular surface should indeed be biexponential, with a fast time component (few ps) nearly equal to that of the free water while the long time component is equal to the inverse of the rate of bound to free transition [4], In order to gain an in depth understanding of hydration dynamics, we have carried out detailed atomistic molecular dynamics (MD) simulation studies of water dynamics at the surface of an anionic micelle of cesium perfluorooctanoate (CsPFO), a cationic micelle of cetyl trimethy-lainmonium bromide (CTAB), and also at the surface of a small protein (enterotoxin) using classical, non-polarizable force fields. In particular we have studied the hydrogen bond lifetime dynamics, rotational and dielectric relaxation, translational diffusion and vibrational dynamics of the surface water molecules. In this article we discuss the water dynamics at the surface of CsPFO and of enterotoxin. 

A similar technique is used to study the concentration - chemical potential relationships in nonstoichio-metric solids. In this case, -> solid materials are to be equilibrated with a gas phase, resulting in adsorption or desorption of a component the determination of compositional changes in the solid is based on the gas coulo-metric titration. The relaxation curves may be used to calculate the -> exchange currents and -> diffusion coefficients (see also -> Diffusion determination in solids). 

Alternatively, an equally powerful visualization of impedance data involves Bode analysis. In this case, the magnitude of the impedance and the phase shift are plotted separately as functions of the frequency of the perturbation. This approach was developed to analyze electric circuits in terms of critical resistive and capacitive elements. A similar approach is taken in impedance spectroscopy, and impedance responses of materials are interpreted in terms of equivalent electric circuits. The individual components of the equivalent circuit are further interpreted in terms of phemonenological responses such as ionic conductivity, dielectric behavior, relaxation times, mobility, and diffusion. 

In the present section, it is demonstrated how the linear response of an assembly of noninteracting polar Brownian particles to a small external field F applied parallel and perpendicular to the bias field Fo may be calculated in the context of the fractional noninertial rotational diffusion in the same manner as normal rotational diffusion [8]. In order to carry out the calculation, it is assumed that the rotational Brownian motion of a particle may be described by a fractional noninertial Fokker-Planck (Smoluchowski) equation, in which the inertial effects are neglected. Both exact and approximate solutions of this equation are presented. We shall demonstrate that the characteristic times of the normal diffusion process, namely, the integral and effective relaxation times obtained in Refs. 8, 65, and 67, allow one to evaluate the dielectric response for anomalous diffusion. Moreover, these characteristic times yield a simple analytical equation for the complex dielectric susceptibility tensor describing the anomalous relaxation of the system. The exact solution of the problem reduces to the solution of the infinite hierarchies of differential-recurrence equations for the corresponding relaxation functions. The longitudinal and transverse components of the susceptibility tensor may be calculated exactly from the Laplace transform of these relaxation functions using linear response theory [72]. 

The orientational van Hove self-correlation function corresponding to the relaxation of the backbone (top panel) and side (lower panel) vectors of glucose (see text for definition) at 220 K show that glucose rotation in the glass proceed by big jumps and does not have a continuous diffusion component. The side rotation, which involves the jump of the smaller glucose bead, the "exocyclic" B6, relaxes faster (ts 3 fjs) than the rotation involving the jumps of the big "backbone" beads B4 and B1 (ts 15 /is). 

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