Particle diffusion coefficients determination

Altenberger and Tirrell [11] utilized the Langevin equation for particle motion coupled with hydrodynamics described by the Navier-S takes equation to determine particle diffusion coefficients in porous media given by... 

An intriguing aspect of these measurements is that the values of D determined from NMR and from sorption kinetics differ by several orders of magnitude. For example, for methane on (Ca,Na)-A the value of the diffusion coefficient determined by NMR is 2 x 10 5 cm2 sec-, and the value determined for sorption rates only 5 x 10"10 cm2 sec-1. The values from NMR are always larger and are similar to those measured in bulk liquids. The discrepancy, which is, of course, far greater than the uncertainty of either method, remained unexplained for several years, until careful studies (267,295,296) showed that the actual sorption rates are not determined by intracrystalline diffusion, but by diffusion outside the zeolite particles, by surface barriers, and/or by the rate of dissipation of the heat of sorption. NMR-derived results are therefore vindicated. Large diffusion coefficients (of the order of 10-6 cm2 sec-1) can be reliably measured by sorption kinetics... 

Table 4 contains a collection of diffusion coefficients determined experimentally for a variety of adsorbate systems. It shows that the values may vary considerably, which is of course due to the specific bonding of the adsorbate to the surface under consideration. Surface diffusion plays a vital role in surface chemical reactions because it is one factor that determines the rates of the reactions. Those reactions with diffusion as the rate-determining step are called diffusion-limited reactions. The above-mentioned photoelectron emission microscope is an interesting tool to effectively study diffusion processes under reaction conditions [158], In the world of real catalysts, diffusion may be vital because the porous structure of the catalyst particle may impose stringent conditions on molecular diffusivities, which in turn leads to massive consequences for reaction yields. 

For adsorption onto colloidal particles two approaches are possible. The first is to determine vlscometrlcally the increase in the effective particle volume fraction upon adsorption, and the second is to measure the decrease in the particle diffusion coefficient. 

As was briefly outlined at the end of Sec. II, an average hydrodynamic thickness Lr of the polymer layer can be determined from the decrease in the particle diffusion coefficient due to polyelectrolyte adsorption. 

Jones [20] used a Smoluchowski approach to examine interacting spherical polymers. Jones predicted that, if one polymer species is dilute and labelled, the measured diffusion coefficient from QELSS is determined only by hydrodynamic interactions of the tagged polymers and their untagged matrix neighbors, and is the single-particle diffusion coefficient. The hydrodynamic approach culminated in analyses of Carter, et al. [21] and Phillies [22] of mutual and tracer diffusion coefficients, including hydrodynamic and direct interactions and reference frame issues. 

First, DLS measurements were conducted in the 1960s by analysing the intensity flucmations in terms of a frequency spectrum (frequency analysis— FA Cummins et al. 1964 Arrechi et al. 1967 Chu and Schones 1968 Dunning and Angus 1968). The width of the frequency spectrum is a measure of the relaxation time of the microstructural processes and can be employed for the determination of the particle diffusion coefficients (Pecora 1964). An alternative for evaluating the fluctuation of scattered light intensity is photon correlation spectroscopy (PCS), which has been used for the characterisation of colloidal suspensions since the end of the 1960s (lakeman and Pike 1969 lakeman 1970 Foord et al. 1970). PCS requires a different hardware than FA, but it can be shown that the results of both techniques are equivalent (lakeman 1970 Xu 2000, pp. 86-89). 

Hydrodynamic interaction mediated by the solvent originates from the movement of particles and occurs when particles are close to each other. It is determined by the viscous drag dependence on interparticle distance [16,58] and results in a decrease in the particle diffusion coefficient by the factor fi, i.e., D = DqP [59,60]. Here, Do is the diffusion coefficient of a particle in the infinitely diluted solution. For motion along the line of the particle centers, the exact expression for was obtained by Brenner [61] and was approximated by the following rational function [62] ... 

The method monitors transport over macroscopic distances (typically in the micrometer regime). Therefore, when the method is applied to the field of surface and colloid chemistry, the diffusion coefficients determined reflect the aggregate sizes and obstruction effects of the colloidal particles. This is the origin of the success the method has had in the study of microstructures of surfactant solutions and also forms the basis of its applications to emulsion systems. 

This equation can be exploited for determining the particle diffusion coefficient by measiuing experimentally the number of irreversibly adsorbed particle as a function of time. This is done under the diffusion transport conditions by eliminating all natural and forced convection ciurents [2,9,10,16]. However, due to the fact that particle flux decreases gradually with the time, diffusion-controlled adsorption becomes very inefficient for long times. Indeed, in these experiments, adsorption times reached tens of hours. 

These equations remain valid for a much broader coverage range (reaching 0.3) than for the convection-conirolled transport, which can be exploited for an accurate determination of the particle diffusion coefficient... 

In summary, the motion of the system of variables R on the effective potential V(R, ) are described by coupled generalized Langevln equations with mode frictions B(5,E(5)) associated with colllslonal excitation of the atoms Involved In that mode, and a friction which describes the relative motion of the two particles and Is closely associated with the particle diffusion coefficients. Processes such as IVR are determined by off-dlagonal potential and kinetic energy coupling terms Incorporated within the coupled Langevln equations. 

The first cumulant is related to the mean particle diffusion coefficient by = D)q. Once the mean diffusion coefficient is determined, the average hydrodynamic size may be calculated using the Stokes-Einstein relationship. The homogeneity exponent. A, is then determined from the kinetic... 

The shear viscosity is a tensor quantity, with components T] y, t],cz, T)yx> Vyz> Vzx> Vzy If property of the whole sample rather than of individual atoms and so cannot be calculat< with the same accuracy as the self-diffusion coefficient. For a homogeneous fluid the cor ponents of the shear viscosity should all be equal and so the statistical error can be reducf by averaging over the six components. An estimate of the precision of the calculation c then be determined by evaluating the standard deviation of these components from tl average. Unfortunately, Equation (7.89) cannot be directly used in periodic systems, evi if the positions have been unfolded, because the unfolded distance between two particl may not correspond to the distance of the minimum image that is used to calculate the fore For this reason alternative approaches are required. 

Because of the close similarity in shape of the profiles shown in Fig. 16-27 (as well as likely variations in parameters e.g., concentration-dependent surface diffusion coefficient), a contrdling mechanism cannot be rehably determined from transition shape. If rehable correlations are not available and rate parameters cannot be measured in independent experiments, then particle diameters, velocities, and other factors should be varied ana the obsei ved impacl considered in relation to the definitions of the numbers of transfer units. 

It is seen from equation (26) that the optimum velocity is determined by the magnitude of the diffusion coefficient and is inversely related to the particle diameter. Unfortunately, in LC (where the mobile phase is a liquid as opposed to a gas), the diffusivity is four to five orders of magnitude less than in GC. Thus, to achieve comparable performance, the particle diameter must also be reduced (c./., 3-5 p)... 

Parameters a and b are related to the diffusion coefficient of solutes in the mobile phase, bed porosity, and mass transfer coefficients. They can be determined from the knowledge of two chromatograms obtained at different velocities. If H is unknown, b can be estimated as 3 to 5 times of the mean particle size, where a is highly dependent on the packing and solutes. Then, the parameters can be derived from a single analytical chromatogram. 

While electrical conductivity, diffusion coefficients, and shear viscosity are determined by weak perturbations of the fundamental diffu-sional motions, thermal conductivity is dominated by the vibrational motions of ions. Heat can be transmitted through material substances without any bulk flow or long-range diffusion occurring, simply by the exchange of momentum via collisions of particles. It is for this reason that in liquids in which the rate constants for viscous flow and electrical conductivity are highly temperature dependent, the thermal conductivity remains essentially the same at lower as at much higher temperatures and more fluid conditions. 

The rate of MV formation was also dependent on pH. The bimolecular rate constant, as calculated from the first order rate constant of the MV build-up and the concentration of colloidal particles, was substantially smaller than expected for a diffusion controlled reaction Eq. (10). The electrochemical rate constant k Eq. (9) which largely determines the rate of reaction was calculated using a diffusion coefficient of of 10 cm s A plot of log k vs. pH is shown in Fig. 24. 

The temporal evolution of P(r,t 0,0) is determined by the diffusion coefficient D. Owing to the movement of the particles the phase of the scattered light shifts and this leads to intensity fluctuations by interference of the scattered light on the detector, as illustrated in Figure 9. Depending on the size of the polymers and the viscosity of the solvent the polymer molecules diffuse more or less rapidly. From the intensity fluctuations the intensity autocorrelation function... 

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