Translational Diffusion Models

In recent times, the translational diffusion model with a constant capture radius has been given new elan by Russell [97, 119, 130, 131, 214, 215], With computer simulations Russell has shown that a translational diffusion model can account for almost all experimentally observed phenomena (see section 2.3.5). He applied the Smoluchowski model (equation 2.15) in which the diffusion coefficients A were defined according to  

A concise description of the above model can also be found in [227]. 

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Fig. 8. The water-proton spin-lattice relaxation rates vs. magnetic field strength plotted as the Larmor frequency at 282 K for hexacyanochromate(II) ion ( ), trioxalatochromate(III) ion ( ), and trimalonatochromate(III) ion (A). The lines were computed using translational diffusion models developed by Freed with and without the inclusion of electron spin relaxation effects 54,121).

Polnaszek and Bryant (1984a,b) measured the frequency dependence of water proton relaxation for solutions of bovine serum albumin reacted with a nitroxide spin label (4.6 mol of nitroxide per mol of protein). The relaxation is dominated by interaction between water and the paramagnetic spin label. The data were best fit with a translational diffusion model, with the diffusion constant for the surface water in the immediate vicinity of the nitroxide being five times smaller than that for... 

The Stokes-Einstein Dgff is proportional to temperature both directly and through its effect on rj. Yet on an absolute scale, temperature increases are relatively small for deteriorative reactions in foods. For example, when increasing temperature from 20°C (293 K) to 45°C (318 K), the theoretical diffusivity by Stokes-Einstein increases only 8%. Since deteriorative food reactions typically increase by 200 to 1000% per 10°C increase (Qio = 2 to 5), a simple local translational diffusion model would be insufficient. 

This again uses the random translational diffusion model for the dipolar interaction and the pulse model for the scalar interaction. 

These models are still limited by the restriction to rotational tumbling of both the electron and the nucleus. Miiller-Warmuth and coworkers developed combined rotational and translational diffusion models (reviewed in [39]) for dipolar and scalar interactions, assuming independent diffusion of the molecules. They also developed a pulse diffusion model assuming occasional collisions between molecules, described via a Poisson process [39]. Later Hwang et al. [53] used force-free pair correlation functions to account for translational diffusion and ionic interactions for dipolar interactions, leading to the now commonly used equation ... 

An appropriate value of 7 for a system modeled by the simple Langevin equation can also be determined so as to reproduce observed experimental translation diffusion constants, Dt in the diffusive limit, Dt is related to y hy Dt = kgTmy. See [22, 36], for example. 

Models for description of liquids should provide us with an understanding of the dynamic behavior of the molecules, and thus of the routes of chemical reactions in the liquids. While it is often relatively easy to describe the molecular structure and dynamics of the gaseous or the solid state, this is not true for the liquid state. Molecules in liquids can perform vibrations, rotations, and translations. A successful model often used for the description of molecular rotational processes in liquids is the rotational diffusion model, in which it is assumed that the molecules rotate by small angular steps about the molecular rotation axes. One quantity to describe the rotational speed of molecules is the reorientational correlation time T, which is a measure for the average time elapsed when a molecule has rotated through an angle of the order of 1 radian, or approximately 60°. It is indirectly proportional to the velocity of rotational motion. 

Under the condition that the Stokes-Einstein model holds, the translational diffusion coefficient, D, can be represented by Eq. (8.3). the diffusion time, Xd, obtained through the analysis is given by Eq. (8.4). 

Studies of the effect of permeant s size on the translational diffusion in membranes suggest that a free-volume model is appropriate for the description of diffusion processes in the bilayers [93]. The dynamic motion of the chains of the membrane lipids and proteins may result in the formation of transient pockets of free volume or cavities into which a permeant molecule can enter. Diffusion occurs when a permeant jumps from a donor to an acceptor cavity. Results from recent molecular dynamics simulations suggest that the free volume transport mechanism is more likely to be operative in the core of the bilayer [84]. In the more ordered region of the bilayer, a kink shift diffusion mechanism is more likely to occur [84,94]. Kinks may be pictured as dynamic structural defects representing small, mobile free volumes in the hydrocarbon phase of the membrane, i.e., conformational kink g tg ) isomers of the hydrocarbon chains resulting from thermal motion [52] (Fig. 8). Small molecules can enter the small free volumes of the kinks and migrate across the membrane together with the kinks. 

Fig. 70. NSE spectra for 2% linear h-PI in deuterated n-decane at Q/A 1 values of 0.064 ( ), 0.089 ( ) and 0.115 ( ). The solid lines represent a common fit with the dynamic structure factor of the Zimm model (see Table 1) neglecting possible effects of translational diffusion. (Reprinted with permission from [174]. Copyright 1993 The American Physical Society, Maryland)...

Nuclear magnetic resonance provides means to study molecular dynamics in every state of matter. When going from solid state over liquids to gases, besides mole- cular reorientations, translational diffusion occurs as well. CD4 molecule inserted into a zeolite supercage provides a new specific model system for studies of rotational and translational dynamics by deuteron NMR. 

The non-collective motions include the rotational and translational self-diffusion of molecules as in normal liquids. Molecular reorientations under the influence of a potential of mean torque set up by the neighbours have been described by the small step rotational diffusion model.118 124 The roto-translational diffusion of molecules in uniaxial smectic phases has also been theoretically treated.125,126 This theory has only been tested by a spin relaxation study of a solute in a smectic phase.127 Translational self-diffusion (TD)29 is an intermolecular relaxation mechanism, and is important when proton is used to probe spin relaxation in LC. TD also enters indirectly in the treatment of spin relaxation by DF. Theories for TD in isotropic liquids and cubic solids128 130 have been extended to LC in the nematic (N),131 smectic A (SmA),132 and smectic B (SmB)133 phases. In addition to the overall motion of the molecule, internal bond rotations within the flexible chain(s) of a meso-genic molecule can also cause spin relaxation. The conformational transitions in the side chain are usually much faster than the rotational diffusive motion of the molecular core. 

For the dependence of the translational diffusion parameter we assumed a model of an unfolded polymer in a good solvent (upper limit) where Rg MW3 5. It should be noted that the figure should only be read qualitatively, as the results for the NOE-based parameters will be influenced to a large degree by spin diffusion. 

But p decreases with salt concentration with an apparent exponent of k which changes from 0 at low salt concentration to — at high salt concentrations. The N-independence of p arises from a cancellation between hydrodynamic interaction and electrostatic coupling between the polyelectrolyte and other ions in the solution. It is to be noted that the self-translational diffusion coefficient D is proportional to as in the Zimm model with full... 

Pulsed field gradient (PFG)-NMR experiments have been employed in the groups of Zawodzinski and Kreuer to measure the self-diffusivity of water in the membrane as a function of the water content. From QENS, the typical time and length scales of the molecular motions can be evaluated. It was observed that water mobility increases with water content up to almost bulk-like values above T 10, where the water content A = nn o/ nsojH is defined as the ratio of the number of moles of water molecules per moles of acid head groups (-SO3H). In Perrin et al., QENS data for hydrated Nation were analyzed with a Gaussian model for localized translational diffusion. Typical sizes of confining domains and diffusion coefficients, as well as characteristic times for the elementary jump processes, were obtained as functions of A the results were discussed with respect to membrane structure and sorption characteristics. ... 

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