Diffusion surface relaxation

Surface Relaxation and Pore Size Distribution 3.6.6.1 Fast Diffusion Limit... 

The key to obtaining pore size information from the NMR response is to have the response dominated by the surface relaxation rate [19-26]. Two steps are involved in surface relaxation. The first is the relaxation of the spin while in the proximity of the pore wall and the other is the diffusional exchange of molecules between the pore wall and the interior of the pore. These two processes are in series and when the latter dominates, the kinetics of the relaxation process is analogous to that of a stirred-tank reactor with first-order surface and bulk reactions. This condition is called the fast-diffusion limit [19] and the kinetics of relaxation are described by Eq. (3.6.3) ... 

Natural rocks seldom have a single pore size but rather a distribution of pore sizes. If all pores are in the fast-diffusion limit, have the same surface relaxivity and have no diffirsional coupling, then the pores will relax in parallel with a distribution of relaxation times that corresponds to the distribution of the pore sizes. The magnetization will decay as a sum of the exponentials as described by Eq. (3.6.4). 

M. D. Hurlimann, K. G. Helmer, L. L. Lator, C. H. Sotak 1994, (Restricted diffusion in sedimentary rocks. Determination of surface-area-to-volume ratio and surface relaxivity),/. Magn. Reson., Ser. A 111, 169-178. 

For a complete definition of Eq. (53) we need to determine the constants Cnk from the conditions (17)-(19) and then calculate the Fourier integral Eq. (1) for the echo signal. To avoid the tedious algebra we compare the three published solutions numerically, but first reproduce these solutions here using our notation. Two of these solutions resulted from a calculation that included the effect of surface relaxation. To make a correct comparison we eliminate from the equations the terms due to relaxation. Then we have the following formulae for the echo intensity for diffusion in a sphere with radius a and reflecting walls ... 

For more realistic modeling, it is necessary to consider the surface relaxation of the growing film due to the diffusion of the deposited particles. This process is not a specific feature of ALD and can be considered independently in the model in the same way as in PVD modeling similarly to the surface tension in liquids, selective diffusion [95] is introduced leading to the relaxation of the surface and the reduction of its area. 

Tlie classical interatomic potential can be used to carry out MD simulations of fast film growth on a substrate. Although the MD growth rates are several orders of magnitude faster than the experimental rates, the MD-deposited films and their surfaces can be characterized in detail and compared with experimental measurements. The main aim of such MD simulations is a fundamental mechanistic understanding and comprehensive identiheation of chemical reactions that occur on the deposition surfaces, as well as analysis of surface diffusion and relaxation mechanisms. Reaction identification is a very important part of the computational hierarchy it is the key to interpretation of various experimental observations and construction of the list of reactions needed for KMC simulation of film growth. Tlie identified set of reactions can be analyzed further to contribute... 

For example, the surface self-diffusion coefficient can be measured by etching a periodic surface profile (e.g., sinusoidal) into a single-crystal surface. The amplitude of the profile is measured as a function of time via the intensity distribution of a laser diffraction pattern generated by the profile itself [52]. The self-diffusion coefficient can be evaluated from the change of the profile amplitude A (/) with time as the surface relaxes into its equilibrium surface structure upon heating ... 

Later work on diffusion rates led Wu et al. [86] to reject diffusion as the rate-limiting step on Si(lOO). They fit a potential surface to results of their calculations and used Monte Carlo transition state theory to calculate rate constants. Surface relaxation effects that were neglected in their first prin-... 

However, there are a number of important limitations to this method, which will be discussed in the next subsections. First, the mathematical analysis to obtain the distribution of relaxation times is itself a problem. Secondly, the surface relaxivity parameter is required in order to obtain a pore size distribution. Thirdly, the model assumes that diffusion within the pore is rapid and that interpore coupling can be neglected. Finally, differences in magnetic susceptibility between the solid and fluid phase complicate the interpretation of... 

Calculations of the spin-echo intensity are complicated by the fact that surface relaxation may play a significant role. A general formalism for calculating PFG spin-echo attenuation for restricted diffusion in isolated pores has recently been proposed that allows for wall relaxation effects. Expressions have been obtained for the cases of diffusion within a sphere, and for planar and cylindrical geometries.These show that diffraction effects are still apparent even when surface relaxation is rapid. Also, the locations of the minima in the spin-echo intensities are not particularly affected by varying the surface relaxation parameter, Analysis of PFG spin-... 

The relaxation of hydrogen nuclei in a fluid confined in pores is determined by the self-diffusion of the molecules within the pores, the bulk relaxation of the liquid, and the surface relaxation at the pore walls (Valckenborg et al. 2000, 2001). For NMR spin-echo times of a few ms, the diffusion length of the water molecules in broad pores is about 1 J.m, and hence the surface relaxation is the dominating process at pore sizes below 0.1 pm (fast diffusion limit Brownstein and Tarr 1979). In this case, a one to one correspondence exists between the observed distribution of relaxation times and the PSD (Halperin et al. 1989, Valckenborg et al. 2000). Since the transverse relaxation time of... 

The obtained results have also been conhrmed by studies on water dynamics and water interaction in bovine serum albumin (BSA p) suspension performed on 300% swollen albumin NP by analysis of H NMR relaxation curves and self-diffusion measurements that showed the presence of two well-separated relaxation rates (Bellotti et al. 2010). These rates have been accounted for a surface-limited relaxation regime of water molecules inside albumin NP (60 wt% of the total water content) and a diffusion-limited relaxation of water molecules into the meso-cavities between the packed NP (40 wt% of the total water content). At higher water content, the appearance of slowly relaxing components was assigned to free water molecules external to the NP, supernatant on the top of the sample. Therefore, the non-freezing and freezing water fractions up to Wiot=300%, determined by DSC measurements, may be associated to the water molecules inside and between the packed NP, respectively. While, at higher water content free molecules external to the NP were also present (Bellotti et al. 2010). 

On the basis of this model and the equivalent circuit shown in Figure 4.5.67, the changes and differences, depending on the used anode in the fuel cell (Pt/C or PtRu/C) in the impedance spectra during the experiment, are dominated by the changes of the charge transfer resistance of the anode (Raj), the surface relaxation impedance (Rg, tg) and the finite diffusion impedance (Z ). 

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