Diffusion of particles

The phenomena in which the effect of Brownian motion afq ears most clearly is diffusion small particles placed at a certain point will spread out 

If there is an external potential U x), Pick s law must be modified. The potential U x) exerts a force 

The constant is called the friction constant and its inverse is called the mobility. If the particle is sufficiently large, can be obtained from hydrodynamics. For example, if the particle is a sphere of radius a, and the viscosity of the solvent is then the hydrodynamic calculation indicates  

If the particle is not spherical, the formula for the velocity is not simple (see Section 3.8 and Chapter 8), but the linear relationship between the force and the velocity always holds provided that the force is weak. 

The average velocity of the particle gives an additional flux cv, so that the total flux will be 

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Brownian diffusion (Brownian motion) The diffusion of particles due to the erratic random movement of microscopic particles in a disperse phase, such as smoke particles in air. 

Diffusion of particles The transfer of small particles and gas molecules into the surrounding air due to concentration difference. 

Equation 2.101 enables calculation of local average quantities such as moments of the particle size distribution. Baldyaga and Orciuch (2001) review expressions for local instantaneous values of particle velocity and diffusivity of particles, Z)pT, required for its solution and recover the distribution using the method of Pope (1979). 

In order to discuss the various techniques we must distinguish between diffusive and non-diffusive systems (J8). Diffusive systems, such as liquids, are characterized by the eventual diffusion of particles over all of the available space non-diffusive systems such as solids, glasses and macromolecules with a definite average structure are characterized by time independent average positions around which the atoms fluctuate. 

The velocity, viscosity, density, and channel-height values are all similar to UF, but the diffusivity of large particles (MF) is orders-of-magnitude lower than the diffusivity of macromolecules (UF). It is thus quite surprising to find the fluxes of cross-flow MF processes to be similar to, and often higher than, UF fluxes. Two primary theories for the enhanced diffusion of particles in a shear field, the inertial-lift theory and the shear-induced theory, are explained by Davis [in Ho and Sirkar (eds.), op. cit., pp. 480-505], and Belfort, Davis, and Zydney [/. Membrane. Sci., 96, 1-58 (1994)]. While not clear-cut, shear-induced diffusion is quite large compared to Brownian diffusion except for those cases with very small particles or very low cross-flow velocity. The enhancement of mass transfer in turbulent-flow microfiltration, a major effect, remains completely empirical. 

K. W. Kehr, K. Mussawisade, T. Wich-mann 1998, (Diffusion of particles on lattices), in Diffusion in Condensed Matter, eds. J. Karger, P. Heitjans, R. Hab-erlandt, Vieweg/Springer, Braunschweig, Berlin. 

Particulate diffusion does not play a significant role in the deposition of pharmaceutical aerosols. However, it is worth noting the mechanism by which diffusion of particles occurs in the lungs. The principle of Brownian motion is responsible for particle deposition under the influence of impaction with gas molecules in the airways. The amplitude of particle displacement is given by the following equation ... 

Diffusion of particles in the polymer matrix occurs much more slowly than in liquids. Since the rate constant of a diffusionally controlled bimolecular reaction depends on the viscosity, the rate constants of such reactions depend on the molecular mobility of a polymer matrix (see monographs [1-4]). These rapid reactions occur in the polymer matrix much more slowly than in the liquid. For example, recombination and disproportionation reactions of free radicals occur rapidly, and their rate is limited by the rate of the reactant encounter. The reaction with sufficient activation energy is not limited by diffusion. Hence, one can expect that the rate constant of such a reaction will be the same in the liquid and solid polymer matrix. Indeed, the process of a bimolecular reaction in the liquid or solid phase occurs in accordance with the following general scheme [4,5] ... 

The above kinetic scheme of the bimolecular reaction simplifies physical processes that proceed via the elementary bimolecular act. To react, two reactants should (a) meet, (b) be oriented by the way convenient for the elementary act, and (c) be activated to form the TS and then react. Hence, not only translational but also rotational diffusion of particles in the solution and polymer are important for the reaction to be performed. So, the more detailed kinetic scheme of a bimolecular reaction includes the following stages diffusion and encounter the reactants in the cage, orientation of reactants in the cage due to rotational diffusion, and activation of reactants followed by reaction [5,13]. 

Rotational diffusion of particles occurs in polymer much slowly than in liquids. Therefore, the observed difference in liquid (k ) and solid polymer (ks) rate constants can be explained by the different rates of reactant orientation in the liquid and polymer. The EPR spectra were obtained for the stable nitroxyl radical (2,2,6,6-tetramethyl-4-benzoyloxypiperidine-l-oxyl). The molecular mobility was calculated from the shape of the EPR spectrum of this radical [14,15], These values were used for the estimation of the orientation rate of reactants in the liquid and polymer cage. The frequency of orientation of the reactant pairs was calculated as vor = Pvrot> where P is the steric factor of the reaction, and vIol is the frequency of particle rotation to the angle equal to 4tt. The results of this comparison are given in Table 19.2. 

Einstein s work on the diffusion of particles (1906) led to the well known Stokes-Einstein relation giving the diffusion coefficient D of a sphere ... 

The diffusion of particles can be described by the second-order differential equation (Pick s second law) ... 

In Eqns. (4.41) and (4.42), we should have marked z and c with an index k, designating the chemical nature of the diffusing particles (components). This is necessary since diffusion of particles of the sort k occurs in a solvent and the system consists of at least two components. In the previous section, we showed that under isothermal and isobaric conditions, the diffusive flux of particles of type k in the solvent is... 

Comparing its form with Eq. 3.83 shows that Eq. 19.10 is equivalent to the onedimensional diffusion of particles in cluster-size space under the influence of both a concentration gradient and a force field derived from a potential gradient. 

In studying processes of accumulation of the Frenkel defects, one uses three different types of simple models the box, continuum, and discrete (lattice) models. In the simplest, box model, which was proposed first in [22], one studies the accumulation of complementary particles in boxes having a certain capacity, with walls impenetrable for diffusion of particles among the boxes. The continuum model treats respectively a continuous medium the intrinsic volume of similar defects at any point of the space is not bounded. In the model of a discrete medium a single cell (e.g., crystalline lattice site) cannot contain more than one defect (v or i). 

Experiments with the metal-metal bonded polymers showed that the degradation rates depended on the curing time. For example, experiments showed that an increase in curing time of polymer 7 led to a decrease in the rate of photodegradation.62 This result was attributed to an increase in the fraction of the sample that is crystalline. It is well known that an increase in polymer crystallinity leads to a decrease in diffusion of particle or radicals in the polymer. It was hypothesized that the resulting decrease in diffusion (and the consequent increase in radical-radical recombination Scheme 10) leads to a decrease in the net rate of degradation. 

For dilute suspensions, particle-particle interactions can be neglected. The extent of transfer of particles by the gradient in the particle phase density or volume fraction of particles is proportional to the diffusivity of particles Dp. Here Dp accounts for the random motion of particles in the flow field induced by various factors, including the diffusivity of the fluid whether laminar or turbulent, the wake of the particles in their relative motion to the fluid, the Brownian motion of particles, the particle-wall interaction, and the perturbation of the flow field by the particles. 

In this section, we will describe the results of the previous section within a probabilistic framework based on stochastic dynamics [8,9]. Furthermore, the discussion will be extended to the diffusion of particles with an interaction potential U (r) depending on the distance r between the particles. An example will be the electrostatic potential associated with the interaction of ions. 

The development of the kinetic theory made it possible to obtain a solution of the problem on the self-consistent description in time and in an equilibrium state of the distributions of interacting species between the sites of homogeneous and inhomogeneous lattices. This enables one to solve a large number of matters in the practical description of processes at a gas-solid interface. The studied examples of simple processes, namely, adsorption, absorption, the diffusion of particles, and surface reactions, point to the fundamental role of the cooperative effects due to the interaction between the components of the reaction system in the kinetics of these processes. 

So far we have concentrated on the behavior of particles in translational motion. If the particles are sufficiently small, they will experience an agitation from random molecular bombardment in the gas, which will create a thermal motion analogous to the surrounding gas molecules. The agitation and migration of small colloidal particles has been known since the work of Robert Brown in the early nineteenth century. This thermal motion is likened to the diffusion of gas molecules in a nonuniform gas. The applicability of Fick s equations for the diffusion of particles in a fluid has been accepted widely after the work of Einstein and others in the early 1900s. The rate of diffusion depends on the gradient in particle concentration and the particle diffusivity. The latter is a basic parameter directly... 

The simple form in Eq. (7) can be maintained by replacing the Brownian diffusion coefficient in the expression kc = /-An /5 by the shear-induced hydrodynamic diffusion coefficient for the particles, Ds. Shear-induced hydrodynamic diffusion of particles is driven by random displacements from the streamlines in a shear flow as the particles interact with each other. For particle volume fractions between 20 and 45%, Ds has been related to... 

A complete theory of turbulence is still lacking, so we must restrict our discussions to two general cases of interest to us (a) the diffusion of particles from point or line sources where the turbulence may be said to be isotropic, and (b) the behavior of particles near large land surfaces— as for example, dust storms. We shall begin our discussion with an explanation of the meaning of eddy-diffusion, which is characteristic of the conditions to be more fully discussed later. 

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