Brownian motion force

For each bead of the dumbbell we may write an equation of motion, indicating that the mass-times-acceleration equals the sum erf the hydro-dynamic drag force, the Brownian motion force, and the force through the connector5 ... 

Mobile ions of the Gouy layer are distributed under the influence of Brownian motion forces and electrostatic field of the interface intrinsic charge. Brownian motion forces are distributed uniformly and the forces of electrostatic field increase toward the charged surface, according to Simeon Denis Poisson (1781-1840) equation. In the description of adsorption-desorption processes on a flat surface it is possible to consider a xmi-form field only along the x coordinate. In this case the Poisson equation has the format ... 

Postulates other than Eq. (12 9) for the distribution m momentum space have been used. For example, some kinetic theonsts have used a Maxwellian distribution about the velocity v at the center of mass of the molecule. Another possible assumption is that of a skewed distribution, m which additional empirical parameters are introduced so that the smoothed Brownian motion force may be stronger in the chain backbone direction than in the transverse directions [9,20a, 20b, 20c, 21], [DPL, Sect. 13.7]. This idea has been proposed for describing the restricted motion of polymer chains in concentrated polymer solutions and in undiluted polymers. An extreme case of this is the reptation assumption [9,14a], in which there is no Brownian force at all m the transverse directions, and the polymer cham is required on the average to slither back and forth along its backbone (DPL, Sect 19.2b). 

Very small particles in liquid or gas streams have a random Broumian motion due to the thermal energy of the continuous phase molecules. If there is a concentration gradient of particles due to a particle sink, then there is a Brownian motion force on these particles ... 

We now focus on macroscopic particles and provide an erqrression for the total external force acting on a particle of mass fWp, density pp, radius Cp, charge Qp, velocity Vp and volume (fWp/pp). We have not included here the Brownian motion force P , nor any force due to thermophoresis, radiation pressure, acoustic force and the electrical force in a nonuniform electrical field given by (3.1.13). Although not generated by an external force field, coulombic types of interactions, London dispersion and electrokinetic forces in the double layer are included in the expression given below ... 

In the Zimm theory, the force on the ith bead has, besides the terms due to the hydrodynamic and restoring forces considered in the Rouse model, additional terms due to Brownian motion and hydrodynamic shielding. The Brownian motion force exerted on the beads is expressed by ... 

In order to explain the observations made with natural rubber and other elastomers, it is necessary to understand the behavior of polymers at the microscopic level. This leads to a model that predicts the macroscopic behavior. It is surprising that in one of the earliest and most successful models, called the freely jointed chain [2,3], we can entirely disregard the chemical nature of the polymer and treat it as a long slender thread beset by Brownian motion forces. This simple picture of polymer molecules is developed and embellished in the sections that follow. Models can explain not only the basics of rabber elasticity but also the qualitative rheological behavior of polymers in dilute soluhon and as melts. The treatment herein is kept as simple as possible. More details are available in the literature [1-7]. 

Brownian motion force is equivalent to an osmotic pressure force whose magnitude in one-dimensional flow is —kT d ac/dx) when there is three-dimensional flow, this expression is generalized to —kTV In c. Negleeting inertia, the force balance on bead 2 yields... 

Smoluchowski theory [29, 30] and its modifications fonu the basis of most approaches used to interpret bimolecular rate constants obtained from chemical kinetics experiments in tenus of difhision effects [31]. The Smoluchowski model is based on Brownian motion theory underlying the phenomenological difhision equation in the absence of external forces. In the standard picture, one considers a dilute fluid solution of reactants A and B with [A] [B] and asks for the time evolution of [B] in the vicinity of A, i.e. of the density distribution p(r,t) = [B](rl)/[B] 2i ] r(t))l ] Q ([B] is assumed not to change appreciably during the reaction). The initial distribution and the outer and inner boundary conditions are chosen, respectively, as... 

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

Molecular dynamics is a simulation of the time-dependent behavior of a molecular system, such as vibrational motion or Brownian motion. It requires a way to compute the energy of the system, most often using a molecular mechanics calculation. This energy expression is used to compute the forces on the atoms for any given geometry. The steps in a molecular dynamics simulation of an equilibrium system are as follows ... 

Factors which adversely influence the separation of veiy fine particle systems are brownian motion and London forces. However, it is possible to counter these forces by the use of dispersants, temperature control, and so on. 

Further support for this approach is provided by modern computer studies of molecular dynamics, which show that much smaller translations than the average inter-nuclear distance play an important role in liquid state atom movement. These observations have conhrmed Swalin s approach to liquid state diffusion as being very similar to the calculation of the Brownian motion of suspended particles in a liquid. The classical analysis for this phenomenon was based on the assumption that the resistance to movement of suspended particles in a liquid could be calculated by using the viscosity as the frictional force in the Stokes equation... 

Diffusion filtration is another contributor to the process of sand filtration. Diffusion in this case is that of Brownian motion obtained by thermal agitation forces. This compliments the mechanism in sand filtration. Diffusion increases the contact probability between the particles themselves as well as between the latter and the filter mass. This effect occurs both in water in motion and in stagnant water, and is quite important in the mechanisms of agglomeration of particles (e.g., flocculation). 

Mean airflow velocities approach zero as the inspired airstream enters the lung parenchyma, so particle momentum also approaches zero. Most of the particles reaching the parenchyma, however, are extremely fine (< 0.5 pm MMAD), and particle buoyancy counteracts gravitational forces. Temperature gradients do not exist between the airstream and airway wall because the inspired airstream has been warmed to body temperature and fully saturated before reaching the parenchyma. Consequently, diffusion driven by Brownian motion is the only deposition mechanism remaining for airborne particles. Diffusivity, can be described under these conditions by... 

Short-time Brownian motion was simulated and compared with experiments [108]. The structural evolution and dynamics [109] and the translational and bond-orientational order [110] were simulated with Brownian dynamics (BD) for dense binary colloidal mixtures. The short-time dynamics was investigated through the velocity autocorrelation function [111] and an algebraic decay of velocity fluctuation in a confined liquid was found [112]. Dissipative particle dynamics [113] is an attempt to bridge the gap between atomistic and mesoscopic simulation. Colloidal adsorption was simulated with BD [114]. The hydrodynamic forces, usually friction forces, are found to be able to enhance the self-diffusion of colloidal particles [115]. A novel MC approach to the dynamics of fluids was proposed in Ref. 116. Spinodal decomposition [117] in binary fluids was simulated. BD simulations for hard spherocylinders in the isotropic [118] and in the nematic phase [119] were done. A two-site Yukawa system [120] was studied with... 

In general, increasing the temperature within the stability range of a single crystal structure modification leads to a smooth change in all three parameters of vibration spectra frequency, half-width and intensity. The dependency of the frequency (wave number) on the temperature is usually related to variations in bond lengths and force constants [370] the half-width of the band represents parameters of the particles Brownian motion [371] and the intensity of the bands is related to characteristics of the chemical bonds [372]. 

It is our experience that to the first question, the most common student response is something akin to Because my teacher told me so . One is tempted to say that it is a pity that the scientific belief of so mat r students is sourced from an authority, rather than from empirical evidence - except that when chemists are asked question (ii), they find it not at all easy to answer. There is, after all, no single defining experiment that conclusively proves the claim, even though it was the phenomenon of Brownian motion that finally seems to have clinched the day for the atomists 150 or so years ago. Of course, from atomic forced microscopy (AFM), we see pictures of gold atoms being manipulated one by one - but the output from AFM is itself the result of application of interpretive models. 

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