Barrier Brownian motion

Plotting U as a function of L (or equivalently, to the end-to-end distance r of the modeled coil) permits us to predict the coil stretching behavior at different values of the parameter et, where t is the relaxation time of the dumbbell (Fig. 10). When et < 0.15, the only minimum in the potential curve is at r = 0 and all the dumbbell configurations are in the coil state. As et increases (to 0.20 in the Fig. 10), a second minimum appears which corresponds to a stretched state. Since the potential barrier (AU) between the two minima can be large compared to kBT, coiled molecules require a very long time, to the order of t exp (AU/kBT), to diffuse by Brownian motion over the barrier to the stretched state at any stage, there will be a distribution of long-lived metastable states with different chain conformations. With further increases in et, the second minimum deepens. The barrier decreases then disappears at et = 0.5. At this critical strain rate denoted by ecs, the transition from the coiled to the stretched state should occur instantaneously. 

The equation derived by Troelstra and Kruyt is only valid for coagulating dispersions of colloids smaller than a certain maximum diameter given by the Rayleigh condition, d 0.10 A0. Equation 4 applies in cases where particles are transported solely by Brownian motion. Furthermore, the kinetic model (Equations 2 and 3) has been derived under the assumption that the collision efficiency factor does not change with time. In the case of some partially destabilized dispersions one observes a decrease in the collision efficiency factor with time which presumably results from the increase of a certain energy barrier as the size of the agglomerates becomes larger. 

Here, coR is the frequency of motion in the reactant well, and Eb is the height of the transition-state barrier. Xr is the effective barrier frequency with which the reactant molecule passes, by diffusive Brownian motions through the barrier region and is given by the following self-consistent relation... 

Filtration is a physical separation whereby particles are removed from the fluid and retained by the filters. Three basic collection mechanisms involving fibers are inertial impaction, interception, and diffusion. In collection by inertial impaction, the particles with large inertia deviate from the gas streamlines around the fiber collector and collide with the fiber collector. In collection by interception, the particles with small inertia nearly follow the streamline around the fiber collector and are partially or completely immersed in the boundary layer region. Subsequently, the particle velocity decreases and the particles graze the barrier and stop on the surface of the collector. Collection by diffusion is very important for fine particles. In this collection mechanism, particles with a zig-zag Brownian motion in the immediate vicinity of the collector are collected on the surface of the collector. The efficiency of collection by diffusion increases with decreasing size of particles and suspension flow rate. There are also several other collection mechanisms such as gravitational sedimentation, induced electrostatic precipitation, and van der Waals deposition their contributions in filtration may also be important in some processes. 

To bias the direction the macro cycle takes at each of the transformations, temporary barriers would be required in order to restrict Brownian motion in one particular direction. Such temporary barriers are intrinsically present in [3]catenane 20 (Fig. 8 and Scheme 10). Irradiation at 350 nm of , -20 causes counter-clockwise rotation of the light-blue macrocycle to the succinic amide ester (orange) station to give Z,E-20. The light-blue macrocycle cannot rotate clockwise because the purple macrocycle effectively blocks that route. 

Recently, Hernandez et al. reported the first catenane system in which unidirectional rotation can be achieved in either direction [60]. Catenane 21 works by biasing Brownian motion with the aid of chemically labile kinetic barriers. The system is a simple [2] catenane in which the route that the smaller macrocycle can take between two stations on the larger can be se-... 

Suppose that the interaction forces establish an energy barrier that retards the motion of particles both toward and away from the collector. If this barrier reduces the adsorption and desorption rates significantly, particles near the primary minimum will have time to achieve a balance between the interaction forces and Brownian motion, before their population changes. Integration of Equation (6) with j 0 and D = mkT leads Lo a Boltzmann distribution... 

The rate of deposition of particles onto a surface, in the presence of London, double-layer, and gravitational forces, is calculated in terms of the energy of interaction between cell and surface by assuming that Brownian motion over a potential energy barrier is the rate-determining step of the... 

Rates of deposition of cells in a gravitational field are calculated in the current paper by considering Brownian motion over a potential barrier formed by London and double-layer forces acting between the cells and the surface on which they deposit. 

In short, the rate of deposition of cells onto a surface has been evaluated in terms of the potential energy of interaction for cases in which Brownian motion of the cells over a potential barrier is the rate-determining step of the process. The rate is exponentially sensitive to the height of the barrier. However, accurate calculation of deposition rates in biological systems awaits an appropriate expression for the interaction forces that considers the complexities observed for such systems. 

However, Ruckenstein Prieve (1975a) pointed out that cells are able to overcome the potential barrier between the secondary and the primary minima by a stochastic process similar to Brownian motion. The time delay in the experiments of Weiss Harlos (1972) was explained by suggesting that this escape over the potential barrier is the time consuming step in the overall process of deposition, because only a small fraction of the cells are able, at a given time, to be carried over the barrier. 

The Brownian motion of a particle under the influence of an external force field, and its consequent escape over a potential barrier has to be treated, in general, using the Fokker-Planck equation. This equation gives the distribution function W governing the probability that a particle will be after time t at a point x with velocity u (Chandrasekhar, 1943). In one dimension it has the form ... 

In the same year Hendrik Kramers published his landmark paper [117] on the theory of chemical reaction rates based on thermally activated barrier crossing by Brownian motion [77], These two papers clearly mark the domains of two related areas of chemical research. Kramers provided the framework for computing the rate constants of chemical reactions based on the molecular structures, energy, and solvent environment. (See Section 10.4.1.) Delbriick s work set the stage for predicting the dynamic behavior of a chemical reaction system, as a function of the presumably known rate constants for each and every reaction in the system. 

The first theory giving the tj -induced decrease of the rate constant is the Kramers theory presented as early as in 1940. He explicitly treated dynamical processes of fluctuations in the reactant state, not assuming a priori the themud equilibrium distribution therein. His reaction scheme can be understood in Fig. 1 which shows, along a reaction coordinate X, a double-well potential VTW composed of a reactant and a product well with a transition-state barrier between them. Reaction takes place as a result of diffusive Brownian motions of reactants surmounting... 

In order for the reaction to take place with the mechanism in the Grote-Hynes theory as well as in the Kramers theory, the reactant must surmount over the transition-state barrier only by diffusional Brownian motions regulated by solvent fluctuations. In the two-step mechanism of the Sumi-Marcus model, on the other hand, surmounting over the transition-state barrier is accomplished as a result of sequential two steps. That is, the barrier is climbed first by diffusional Brownian motions only up to intermediate heights, from which much faster intramolecular vibrational motions take the reactant to the transition state located at the top of the barrier. 

Next, let us discuss whether the Grote-Hynes theory can be fitted to the rate constants kgi,. In this thoery. kgf,Jkfsr is related to the frequency (that is, the speed) n with which the reactant passes, by diffusive Brownian motions, through Ae transition-state-barrier region in Fig. 1, as... 

If the energy barrier to aggregation is removed (e.g., by adding excess electrolyte) then aggregation is diffusion controlled only Brownian motion of independent droplets or particles is present. For a monodisperse suspension of spheres, Smoluchowski developed an equation for this rapid coagulation ... 

For monodisperse or unimodal dispersion systems (emulsions or suspensions), some literature (28-30) indicates that the relative viscosity is independent of the particle size. These results are applicable as long as the hydrodynamic forces are dominant. In other words, forces due to the presence of an electrical double layer or a steric barrier (due to the adsorption of macromolecules onto the surface of the particles) are negligible. In general the hydrodynamic forces are dominant (hard-sphere interaction) when the solid particles are relatively large (diameter >10 (xm). For particles with diameters less than 1 (xm, the colloidal surface forces and Brownian motion can be dominant, and the viscosity of a unimodal dispersion is no longer a unique function of the solids volume fraction (30). 

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