Restricted internal diffusion model

Fig. 4. Restricted diffusion models for internal motions, (a) Bond rotation limited in angular amplitude to 7o. i is the angle between the rotational axis and the intemuclear vector, just as in the free internal diffusion model, (b) With the wobble-in-a-cone model, the diffiision is limited to the conic angle and the cone is tilted at angle from some reference axis.

The parameter t is given in Eq. A-9 in the Appendix, as a function of the correlation time, t associated with internal motion. One of the input parameters is the angle j3, formed between the relaxation vector (C—H bond) and the internal axis of rotation (or jump axis), namely the C-5—C-6 bond. The others are correlation times t0 and r, of the HWH model, obtained from the fit of the data for the backbone carbons. The fitting parameters for the two-state jump model are lifetimes ta and tb, and for the restricted-diffusion model, the correlation time t- for internal rotation. The allowed range of motion (or the jump range) is defined by 2x for both models (Eqs. A-4 and A-9). 

First, we consider internal rotational diffusion with restrictions on the amplitude of the diffusion. The limits on the diffusion can be established using a square well potential or a harmonic function that approaches the limit asymptotically. Although relatively little work has employed restricted diffusion models, those that do generally utilize a square well potential (Wittebort and Szabo, 1978 London and Avitabile, 1978). Two general models can be considered (see Fig. 4), one which is most useful for local motions (Fig. 4a) and the other most useful for segmental motions (Fig. 4b). 

It will be noted that the form of the spectral density in Eq. (32) is similar to that for free internal diffusion [Eq. (20)] and for restricted diffusion [Eq. (28)]. This model permits the relative populations of the jump states and B to be accounted for since the population in state A is proportional to the lifetime in state A. London (1978) has shown that Xg for relaxation to be substantially affected. In other words, the existence of a small amount of a minor configuration will have little inffuence on the measured relaxation parameters. 

A model of the ZFS coupling removing the restriction of its constant amplitude and allowing both processes, the stochastic variations of the internal coordinates and the rotational diffusion, to modulate the ZFS interaction was proposed by Westlund and co-workers (13,85,88,91). According to this model, the ZFS interaction provided the coupling between the electron spin variables, the stochastic time-dependent distortion coordinates and the reorientational degrees of freedom by the expression ... 

Apart from diffusion and migration, transport by convection can also take place due to different internal and external forces. Thus, natural convection due to gradients of density can occur when the electrode reaction provokes a significant local change in the solution composition or due to thermal variations. The modelling of this case is difficult and so electrochemical experiments are usually restricted to short time scales, low concentration of analyte, and thermostated cells such that the influence of natural convection is minimised. 

In the present work, a CFD model of PEM fuel cell with straight flow field channels is developed. The model developed in this chapter is different from those in the literature in that it is fully three-dimensional as opposed to two-dimensional (e g. [51]). It is accounts for liquid water transport through the gas diffusion layer as well as transport across the membrane, rather than restricted to transport of liquid water through GDL only (e.g. [60]). Furthermore, it is non-isothermal, rather than assumed constant cell temperature (e.g. [61], [64], and [65]). Also, the present model incorporates the effect of hygro and thermal stresses into actual three-dimension fuel cell model, rather than assumed simplified temperature and humidity profile, with no internal heat generation (e.g. [68] and [69]). 

The advances in time resolved techniques have fostered a reexamination of theories of the rotational motions of molecules in liquids. Models considered include the anisotropic motion of unsymmetrical fluorophores the internal motions of probes relative to the overall movement with respect to their surroundings, the restricted motion of molecules within membranes (e.g., wobbling within a cone), and the segmental motion of synthetic macromolecules [8]. Analyses of these models point to experimental situations in which the anisotropy can show both multi-exponential and none-exponential decay. Current experimental techniques are capable in principle of distinguishing between these different models. It should be emphasized, however, that to extract a single average rotational correlation time demands the same precision of data and analysis as fluorescence decay experiments which exhibit dual exponential decays. Multiple or non-exponential anisotropy experiments are thus near the limits of present capabilities, and generally demand favourable combinations of fluorescence and rotational diffusion times [48]. 

Specific models for internal motions can be used to interpret heteronuclear relaxation, such as restricted diffusion and site-jump models. However, model-free formal methods are preferable, at least for the initial analysis, since available experimental data generally are insufficient to completely characterize complex internal motions or to uniquely determine a specific motional model. The model-free approach of Lipari and Szabo for the analysis of relaxation data has been used for proteins and even for peptides. It attempts to reproduce relaxation rates by a weighted product of spectral density functions with different correlation times The weighting factors are identified as order parameters for the molecular rotational correlation time and optional further local correlation times r. The term (1-S ) would then be proportional to the amplitude of the corresponding internal motion. However, the Lipari-Szabo approach is based on the assumption that molecular and local correlation times are not coupled, i.e. they should be distinct enough (e.g. differing by at least a factor of 10 in time) to allow for this separation. However, in small molecules the rates of these different processes are of the same order of magnitude, and the requirements of the Lipari-Szabo approach may not be fulfilled. Molecular dynamics simulation provide a complementary approach for the interpretation of relaxation measurements. 

The second type of restricted difiiision is illustrated in Fig. 4b and has sometimes been dubbed the wobble-in-a-cone model because the orientation of the interaction vector (either DD vector or CSA tensor principal axis) is permitted free diffusion within the cone of half angle yo fhe cone is disposed toward a reference axis at a fixed angle fi. This model can be adapted for internal motions but is especially useful for characterizing segmental motions, in which case the reference axis can be the helix axis of the nucleic acid. Librational motions, such as predicted by molecular dynamics calculations on proteins (McCammon et al., 1977), are readily accommodated by the wobble-in-a-cone model (Howarth, 1979 Richarz et al, 1980), but the model has also accommodated other studies of dynamics, for example, the situation of halide ions bound to proteins (Bull etai, 1978). 

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