Particle diffusion interaction

The study of the behavior of reactions involving a single species has attracted theoretical interest. In fact, the models are quite simple and often exhibit IPT. In contrast to standard reversible transitions, IPTs are also observed in one-dimensional systems. The study of models in ID is very attractive because, in some cases, one can obtain exact analytical results [100-104]. There are many single-component nonequilibrium stochastic lattice reaction processes of interacting particle systems [100,101]. The common feature of these stochastic models is that particles are created autocatalytically and annihilated spontaneously (eventually particle diffusion is also considered). Furthermore, since there is no spontaneous creation of particles, the zero-particle... 

Solute molecules which diffuse into the stationary phase particles and interact with them are left behind by those molecules that bypass the stationary phase. 

The restricted access principle is based on the concept of diffusion-based exclusion of matrix components and allows peptides, which are able to access the internal surface of the particle, to interact with a functionalized surface (Figure 9.2). The diffusion barrier can be accomplished in two ways (i) the porous adsorbent particles have a topochemically different surface functionalization between the outer particle surface and the internal surface. The diffusion barrier is then determined by an entropy controlled size exclusion mechanism of the particle depending on the pore size of adsorbent (Pinkerton, 1991) and (ii) the diffusion barrier is accomplished by a dense hydrophilic polymer layer with a given network size over the essentially functionalized surface. In other words, the diffusion barrier is moved as a layer to the interfacial... 

In filtration, the particle-collector interaction is taken as the sum of the London-van der Waals and double layer interactions, i.e. the Deijagin-Landau-Verwey-Overbeek (DLVO) theory. In most cases, the London-van der Waals force is attractive. The double layer interaction, on the other hand, may be repulsive or attractive depending on whether the surface of the particle and the collector bear like or opposite charges. The range and distance dependence is also different. The DLVO theory was later extended with contributions from the Born repulsion, hydration (structural) forces, hydrophobic interactions and steric hindrance originating from adsorbed macromolecules or polymers. Because no analytical solutions exist for the full convective diffusion equation, a number of approximations were devised (e.g., Smoluchowski-Levich approximation, and the surface force boundary layer approximation) to solve the equations in an approximate way, using analytical methods. 

Unlike charges attract and like charges repel each other, so there is a high concentration of counterions attracted to the particle surface whilst co-ions (those with the same sign charge as that of the surface) are repelled. Thermal motion, i.e. diffusion, opposes this local concentration gradient so that the counterions are in a diffuse cloud around the particle. Of course particles which have a like charge will also repel each other but the interaction of the particle surfaces will be screened by the counterion clouds between the particles. The interaction potential is a function of the surface potential, i]/o, and the permittivity of the fluid phase, e = r80, where r is the relative permittivity.12,27... 

This result implicitly requires (a) that no strongly attractive/repulsive electrostatic interaction occurs as the particles approach each other, and (b) that a stationary diffusional concentration field surrounds the particle. [Note Einstein s result for spherical particle diffusivity i.e.,D = k Tlfm-qr, where is the Boltzmann constant, T is temperature in kelvin, 17 is the viscosity of the medium, and r is the radius) indicates that RuD will be approximately Ak TKyni].]... 

Attention has so far been focused on reaction between spherical particles diffusing in a hydrodynamic continuum with no forces acting between reactants. In this chapter, the most important force, the coulomb interaction, between ions in solution is included. The potential energy, [/(rj — r2),of ions at r (= y l5 zx) and r2 having charges z,e and z2e is... 

In earlier chapters we examined systems with one or two types of diffusing chemical species. For binary solutions, a single interdiffusivity, D, suffices to describe composition evolution. In this chapter we treat diffusion in ternary and larger multicomponent systems that have two or more independent composition variables. Analysis of such diffusion is complex because multiple cross terms and particle-particle chemical interaction terms appear. The cross terms result in TV2 independent interdiffusivities for a solution with TV independent components. The increased complexity of multicomponent diffusion produces a wide variety of diffusional phenomena. 

The solution of (2.3.69) is a purely mathematical problem well known in the theory of diffusion-controlled processes of classical particles. However, a particular form of writing down (2.3.69) allows us to use a certain mathematical analogy of this equation with quantum mechanics. Say, many-dimensional diffusion equation (2.3.69) is an analog to the Schrodinger equation for a system of N spinless particles B, interacting with the central particle A placed... 

In order to get a more realistic description of surface reactions energetic interactions must be taken into account. We introduced in Section 9.2.1 a general model which is able to handle systems which include mono- and bimolecular steps like adsorption, desorption, diffusion and reaction [38]. Here we apply this model to an extended version of the ZGB-model which incorporates particle diffusion and desorption [41]. 

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. 

The rate of deposition of Brownian particles is predicted by taking into account the effects of diffusion and convection of single particles and interaction forces between particles and collector [2.1] -[2.6]. It is demonstrated that the interaction forces can be incorporated into a boundary condition that has the form of a first order chemical reaction which takes place on the collector [2.1], and an expression is derived for the rate constant The rate of deposition is obtained by solving the convective diffusion equation subject to that boundary condition. The procedure developed for deposition is extended to the case when both deposition and desorption occur. In the latter case, the interaction potential contains the Bom repulsion, in addition to the London and double-layer interactions [2.2]-[2.7]. Paper [2.7] differs from [2.2] because it considers the deposition at both primary and secondary minima. Papers [2.8], [2.9] and [2.10] treat the deposition of cancer cells or platelets on surfaces. 

When van der Waals and double-layer forces are effective over a distance which is short compared to the diffusion boundary-layer thickness, the rate of deposition may be calculated by lumping the effect of the particle-collector interactions into a boundary condition on the usual convective-diffusion equation. This condition takes the form of a first-order irreversible reaction (10, 11). Using this boundary condition to eliminate the solute concentration next to the disk from Levich s (12) boundaiy-kyersolution of the convective-diffusion equation for a rotating disk, one obtains... 

This corresponds to mass-transfer limitation of the apparent surface reaction. Thus the combination of Eqs. p] and 2] is expected to give reasonable estimates of the rate of deposition for all particle-collector interaction profiles, provided the interactions are confined to a region which is thin compared to the diffusion boundary layer. 

The second chapter examines the deposition of Brownian particles on surfaces when the interaction forces between particles and collector play a role. When the range of interactions between the two (which can be called the interaction force boundary layer) is small compared to the thickness of the diffusion boundary layer of the particles, the interactions can be replaced by a boundary condition. This has the form of a first order chemical reaction, and an expression is derived for the reaction rate constant. Although cells are larger than the usual Brownian particles, the deposition of cancer cells or platelets on surfaces is treated similarly but on the basis of a Fokker-Plank equation. 

Coagulation Catalyst preparation cluster and polymer formation in chemical vapor deposition intermolecular interactions between particles diffusion-limited aggregation soot formation. 

In particular, for sources and sinks due to inelastic processes in the Bra-ginskii equations, and with fa = we see that integrals /o,o,-fo,i and /o,2 are involved, whereas the integrals /io,/i,i and /g2 (i.e., with the diffusion-or transport cross-sections) appear in the expressions from elastic neutral particle plasma interactions. 

The presence of the interface restricts the diffusive motion to a half space. More important is the presence of particle-surface interactions, which can modify the transport properties. 

Due to their close proximity, the electron-hole pair is bound by their mutual Coulomb interaction as shown in Fig. 8.7. When the potential is strong enough, the particles diffuse together, giving geminate recombination. Otherwise the electron and hole diffuse apart and any subsequent recombination is non-geminate. The Onsager (1938) model... 

For liquid-phase diffusion of large adsorbate molecules, when the ratio = r /r of the molecule radius r to the pore radius is significantly greater than zero, the pore diffusivity is reduced by steric interactions with the pore wall and hydrodynamic resistance. When < 0.2, the following expressions derived by Brenner and Gaydos [/. Coll. Int Sci, 58,312 ( 1977)] for a hard sphere molecule (a particle) diffusing in a long cylindrical pore, can be used... 

In the simplest possible model for an Ideal chain, the bonds between atoms In the backbone are treated as vectors connecting volumeless points which do not Interact. Such a model chain is depicted in fig. 5.2 the (fluctuating) distance between the end points is denoted as r. If, moreover, any orientation between two consecutive bonds is assumed to have the same probability, the conformational properties of long chains can be described by the universal random-Jlight model, first introduced by Kuhn l. Let the chain have N randomly oriented bonds, each of length t. Such a model chain contains IV + 1 backbone atoms. When these bonds are assumed to be fully Independent of each other, the conformation resembles the trajectory of a particle diffusing under the action of a random force, for which the solution is well known -S- ). The mean square displacement [Pg.614]

This model has been successful in describing flux decline during dead-end filtration of particulate suspensions, but is not appropriate for application to crossflow filtration where the feed solution continuously recirculates [158]. Also, neither the occurrence of macromolecules and colloidal particles diffusion nor the influence of solute-solute and solute-membrane interactions on flux decline is considered in this model [42,59,159]. 

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