Brownian diffusion model

ESR spectra of NO2 adsorbed on X- and Y-type zeolites were observed in the temperature range 77-346 K. Based upon spectral simulation using a Brownian diffusion model, motional dynamics of NO2 adsorbed on zeolite surface were analyzed quantitatively. In the case of X-type zeolite, it was found that the ESR spectra below 100 K is near the rigid limit. Above 230 K, the average rotational correlation time decreased from 1.7 x 10 (230 K) to 7.5 x lO sec (325 K) with increasing temperature and its degree of anisotropy was very close to one (N = 1.25). On the other hand, the temperature-dependent ESR spectra of NO2 adsorbed on Y-type zeolite were observed to be somewhat different from that for X-type zeolite. 

The standard model for diffusive motion in polymers is Brownian diffusion, which occurs as a series of infinitesimal reorientational steps. This model is most appropriate for intermediate-to-large sized spin probes and spin-labeled macromolecules, where the macromolecule is much larger than any solvent molecules. Because of this broad applicability, the Brownian diffusion model is the most widely used. This type of rotational diffusion is completely analogous to the one-dimensional random walk used to describe translational diffusion in standard physical chanistry texts, with the difference that the steps are described in terms of a small rotational step 59 that can occur in either the positive or negative direction. In three dimensions, rotations about each of three principal axes of the nitroxide must be taken into account. A diffusion constant may be defined for each of these rotations motions, in a way that is completely analogous to the definition of translational diffusion constant for the one-dimensional random walk. 

The use of the Brownian diffusion model will give similar results for the Walden product. All the present theories are inadequate for the computation of the viscosity and yield values about half of the observed value when the viscosity is computed from the observed friction constants. 

The first question to be asked is why the Brownian diffusion model of Kirkwood should give reasonable results for the unlike-ion friction constants, as mentioned in Section 3.4, when the Coulomb potential is ignored and the experimental radial distribution function used. The assumptions in the Brownian diffusion model are difficult to evaluate but Douglass et have shown it to be a factor of njl greater than their result using a Gaussian autocorrelation function. Now from molecular dynamics Alder et have shown for hard spheres at high densities that the autocorrelation... 

Kramers H A 1940 Brownian motion in a field of force and the diffusion model of chemical reactions Physica 7 284-304... 

Analysis of neutron data in terms of models that include lipid center-of-mass diffusion in a cylinder has led to estimates of the amplitudes of the lateral and out-of-plane motion and their corresponding diffusion constants. It is important to keep in mind that these diffusion constants are not derived from a Brownian dynamics model and are therefore not comparable to diffusion constants computed from simulations via the Einstein relation. Our comparison in the previous section of the Lorentzian line widths from simulation and neutron data has provided a direct, model-independent assessment of the integrity of the time scales of the dynamic processes predicted by the simulation. We estimate the amplimdes within the cylindrical diffusion model, i.e., the length (twice the out-of-plane amplitude) L and the radius (in-plane amplitude) R of the cylinder, respectively, as follows ... 

Figure 4. The Brownian ratchet model of lamellar protrusion (Peskin et al., 1993). According to this hypothesis, the distance between the plasma membrane (PM) and the filament end fluctuates randomly. At a point in time when the PM is most distant from the filament end, a new monomer is able to add on. Consequently, the PM is no longer able to return to its former position since the filament is now longer. The filament cannot be pushed backwards by the returning PM as it is locked into the mass of the cell cortex by actin binding proteins. In this way, the PM is permitted to diffuse only in an outward direction. The maximum force which a single filament can exert (the stalling force) is related to the thermal energy of the actin monomer by kinetic theory according to the following equation ...

Another approach we can use to describe the stress relaxation behaviour and all the linear viscoelastic responses is to calculate the relaxation spectrum H. Ideally we would like to model or measure the microstructure in the dispersion and include the role of Brownian diffusion in the loss of structural order. The intermediate scattering... 

Schmitz et al (31) have proposed that the discrepancy between QLS and tracer diffusion measurements can be reconciled by considering the effects of small ions on the dynamics and scattering power of the polyelectrolyte. In this model, the slow mode arises from the formation of "temporal aggregates . These arise as the result of a balance between attractive fluctuating dipole forces coming from the sharing of small ions by several polyions, and repulsive electrostatic and Brownian diffusion forces. This concept is attractive, but needs to be formulated quantitatively before it can be adequately tested. 

A critical comparison by van Konyenberg and Steele [230] and Jones et al. [231] of extended diffusion models with Brownian motion and other continuum models strongly favours the former treatment. More detailed analysis is given by Berne and Pecora [232]. 

The validity of Eqs.(4.10)(4.12) probably extends well beyond the Rouse model itself [characterized by the specific set of rt values in Eq. (4.5)1 and it seems likely that they will apply, at least for small disturbances, whenever the elements supporting the stress are joined by sufficiently flexible connectors and configurational relaxation is driven by simple Brownian diffusion. One might speculate further that these same forms would apply even in concentrated systems, with Eq.(4.10) expressed in a somewhat more general form because of intermolecular interactions ... 

Later Tien and Payatakes (15), in their study of the deposition of colloidal particles, developed a mathematical model which included in one expression the contributions by Brownian diffusion, interception with London attraction, and gravitational field. The most general form of their expression is ... 

Chapter 8 by W. T. Coffey, Y. P. Kalmykov, and S. V. Titov, entitled Fractional Rotational Diffusion and Anomalous Dielectric Relaxation in Dipole Systems, provides an introduction to the theory of fractional rotational Brownian motion and microscopic models for dielectric relaxation in disordered systems. The authors indicate how anomalous relaxation has its origins in anomalous diffusion and that a physical explanation of anomalous diffusion may be given via the continuous time random walk model. It is demonstrated how this model may be used to justify the fractional diffusion equation. In particular, the Debye theory of dielectric relaxation of an assembly of polar molecules is reformulated using a fractional noninertial Fokker-Planck equation for the purpose of extending that theory to explain anomalous dielectric relaxation. Thus, the authors show how the Debye rotational diffusion model of dielectric relaxation of polar molecules (which may be described in microscopic fashion as the diffusion limit of a discrete time random walk on the surface of the unit sphere) may be extended via the continuous-time random walk to yield the empirical Cole-Cole, Cole-Davidson, and Havriliak-Negami equations of anomalous dielectric relaxation from a microscopic model based on a... 

Diffusion models of geminate pair combination connect the time-dependent pair survival probability, P t), with the macroscopic properties of the host solvent. Radicals are treated as spherical particles immersed in a uniformly viscous medium. The pair is assumed to undergo random Brownian movements that ultimately lead to either recombination or escape. The expression of P i) depends on the degree of sophistication of the theory chosen for analyzing the process. In the simplest theory,... 

Because the assumption of simple Brownian diffusion breaks down, the diffusion in biomembranes cannot be described by a single diffusion coefficient. For instance, FRAP experiments in the plasma membrane showed that the observed translational diffusion rates depend on the size of the initial photobleached spot, which is also inconsistent with a simple Singer-Nicolson model. 

To reconcile this apparent contradiction the membrane skeleton fence and anchored transmembrane picket model was proposed (54). According to this model, transmembrane proteins anchored to and lined up along the membrane skeleton (fence) effectively act as a row of posts for the fence against the free diffusion of lipids (Fig. 11). This model is consistent with the observation that the hop rate of transmembrane proteins increases after the partial removal of the cytoplasmic domain of transmembrane proteins, but it is not affected by the removal of the major fraction of the extracellular domains of transmembrane proteins or extracellular matrix. Within the compartment borders, membrane molecules undergo simple Brownian diffusion. In a sense, the Singer-Nicolson model is adequate for dimensions of about 10 x lOnm, the special scale of the original cartoon depicted by the authors in 1972. However, beyond such distances simple extensions of the fluid mosaic model fail and a substantial paradigm shift is required from a two-dimensional continuum fluid to the compartmentalized fluid. 

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