Brownian diffusion forces

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. 

We group the forces that control the suspension rheology into two main categories colloidal forces and viscous forces. The colloidal forces include Brownian diffusion forces and the surface forces of electrostatic repulsion and van der Waals attraction. In order to define the dimensionless scaling parameters that characterize the relative magnitude of these forces we assume the particles are separated by a distance of the order of the particle radius a, which is in turn assumed to be close to the smallest particle separation... 

Air movement indoors is much slower than outdoors, but it is usually enough to ensure that concentrations are fairly uniform in a room. Convection from heating appliances gives air speeds typically in the range 0.05-0.5 m s-1 (Daws, 1967). However, to undergo deposition, vapour molecules or particles must be transported across the boundary layer, typically a few millimetres thick, of almost stagnant air over surfaces. This may be achieved by sedimentation, molecular or Brownian diffusion, or under the action of electrostatic or thermophoretic forces. 

Deposition other than in rain is termed dry deposition, and this includes sedimentation of particles, molecular and Brownian diffusion to surfaces, impaction on roughness elements and deposition under electrical or thermophoretic forces. The velocity of deposition is defined... 

In solution things are more complex. The reaction partners are no longer free in their translational motion as they are in the gas phase they have to move in a condensed medium, and their motion is governed by other physical phenomena which for economy of exposition we shall not consider in detail. It is sufficient to recall that the physical models for the most important terms, Brownian motions, diffusion forces, are expressed in their basic form using a continuum description of the medium. 

In the case of Brownian diffusion and interception, particle capture is enhanced by London attractive forces and reduced by electrostatic double layer repulsive forces. 

Particle capture occurs through an interception mechanism. Because of the strong electrostatic forces operating in the experimental system, the contribution of Brownian diffusion to particle capture is negligible. 

The external electric field is in the direction of the pore axis. The particle is driven to move by the imposed electric field, the electroosmotic flow, and the Brownian force due to thermal fluctuation of the solvent molecules. Unlike the usual electroosmotic flow in an open slit, the fluid velocity profile is no longer uniform because a pressure gradient is built up due to the presence of the closed end. The probability of the particle position is obtained by solving the Fokker-Planck equation. The penetration depth is found to be dependent upon the Peclet number, which is a measure of significance of the convective electroosmotic flow relative to the Brownian diffusion, and the Damkohler number, which is a ratio of the characteristic diffusion-to-deposition times. 

Each is discussed in Sec. 17 of this handbook under Gas-Solids Separations. The effectiveness of conventional air-pollution-control equipment for particulate removal is compared in Fig. 22-25. These fractional efficiency curves indicate that the equipment is least efficient in removing particulates in the 0.1- to 1.0-pm range. For wet scrubbers and fabric filters, the very small particulates (0.1 pm) can be efficiently removed by brownian diffusion. The smaller the particulates, the more intense their brownian motion and the easier their collection by diffusion forces. Larger particulates (>1 pm) are collected principally by impaction, and removal efficiency increases with particulate size. The minimum in the fractional efficiency curve for scrubbers and filters occurs in the transition range between removal by brownian diffusion and removal by impaction. 

The diffusion force, giving the local action of the Brownian motion on the deposition of the microparticle. A modified Peclet... 

To analyse bond breakage under steady loading, we take advantage of the enormous gap in time scale between the ultrafast Brownian diffusion (r 10 — 10 s) and the time frame of laboratory experiments ( 10 s to min). This means that the slowly increasing force in laboratory experiments is essentially stationary on the scale of the ultrafast kinetics. Thus, dissociation rate merely becomes a function of the instantaneous force and the distribution of rupture times can be described in the limit of large statistics by a first-order (Markov) process with time-dependent rate constants. As force rises above the thermal force scale, i.e. rj-t> k T/x, the forward transition... 

The physics behind this relation is the fluctuation-dissipation theorem the same random kicks of the surrounding molecules cause both Brownian diffusion and the viscous dissipation leading to the frictional force. It is -instructive to calculate the time scale t required for the particle to move a... 

In this chapter, we consider Brownian diffusion, sedimentation, migration in an electric Reid, and thermophoresis. The last term refers to particle movement produced by a temperature gradient in the gas. We consider also the London-van der Waals forces that are important when a particle approaches a surface. The analysis is limited to particle transport in stationary —that is. nonllowing— gases. I ransporl in flow systems is discussed in the chapters which follow. 

We consider particle transport from a gas to a body with a flat bounding surface by Brownian diffusion under the influence of van der Waals forces exerted by the body. The relative contributions of the two mechanisms can be estimated as follows The total flux normal to the surface is given by the x component of the flux... 

In the approach adopted in my first edition, the derivation and use of the general dynamic equation for the particle size distribution played a central role. This special form of a population balance equation incorporated the Smoluchowski theory of coagulation and gas-to-panicle conversion through a Liouville term with a set of special growth laws coagulation and gas-to-particle conversion are processes that take place within an elemental gas volume. Brownian diffusion and external force fields transport particles across the boundaries of the elemental volume. A major limitation on the formulation was the assumption that the panicles were liquid droplets that coalesced instantaneously after collision. 

A reduction of particle size As mentioned above, if R is significantly reduced (to values below 0.1 pm), the Brownian diffusion can overcome the gravitational force and no sedimentation will occur. This is the principle of the formation of nanosuspensions. 

Case (a) represents the situation for small droplets (<0.1 pm, i.e., nanoemulsions), whereby the Brownian diffusion kT (where k is the Boltzmann constant and T is the absolute temperature) exceeds the force of gravity (mass x acceleration due to gravity, g) ... 

Since the gravity force is proportional to R, then if R is reduced by a factor of 10, the gravity force is reduced by 1000. Below a certain droplet size (which also depends on the density difference between oil and water), the Brownian diffusion may exceed gravity and creaming or sedimentation is prevented. This is the principle of formulation of nanoemulsions (with size range 20-200 nm) that may show very little or no creaming or sedimentation. The same applies for microemulsions (size range 5-50 nm). 

Most fornmlations undergo creaming or sedimentation as a result of the density difference between disperse phase particles and medium [1]. This situation is particularly the case with most practical systems that contain particles with radii R that are large (>1 pm), whereby the Brownian diffusion is not sufficient to overcome the gravity force, that is... 

The coefficients a(p, c) and tj(p, c) describe chemical and physical effects on the kinetics of deposition. The transport of particles from the bulk of the flowing fluid to the surface of a collector or media grain by physical processes such as Brownian diffusion, fluid flow (direct interception), and gravity are incorporated into theoretical formulations for fj(p, c), together with corrections to account for hydrodynamic retardation or the lubrication effect as the two solids come into close proximity. Chemical effects are usually considered in evaluating a(p, c). These include interparticle forces arising from electrostatic interactions and steric effects originating from interactions between adsorbed layers of polymers and polyelectrolytes on the solid surfaces. 

Having a model that has a good theoretical basis, that has been validated in laboratory experiments, and that is consistent with field observations, it is advisable to make some predictions about particle deposition in systems of interest. An example is presented in Figure 3, adapted from the work of Tobiason (1987). The travel distance in an aquifer required to deposit 99% of the particles from a suspension is termed Lgg and is plotted as a function of the diameter of the suspended particles for two different values of a(p, c), specifically 1.0 (favorable deposition) and 0.001 (deposition with significant chemical retardation of the particle-collector interaction, termed unfavorable deposition ). Assumptions include U = 0.1 m day"1, T= 10°C, dc = 0.05cm, e = 0.40, pp= 1.05 gem"3, and H=10 2OJ. These results indicate the dependence of the kinetics of deposition on the size of the particles in suspension that has been predicted and observed in many systems. Small particles are transported primarily by convective Brownian diffusion, and large particles in this system are transported primarily by gravity forces. A suspended particle with a diameter of about 3 /im is most difficult to transport. Nevertheless, in the absence of chemical retardation, a travel distance of only about 5 cm is all that is needed to deposit 99% of such particles in a clean aquifer, that is, an aquifer that has not received and retained previous particles. 

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