Brownian motion equation

The discussion of Kapral s kinetic theory analysis of chemical reaction has been considered in some detail because it provides an alternative and intrinsically more satisfactory route by which to describe molecular scale reactions in solution than using phenomenological Brownian motion equations. Detailed though this analysis is, there are still many other factors which should be incorporated. Some of the more notable are to consider the case of a reversible reaction, geminate pair recombination [286], inter-reactant pair potential [454], soft forces between solvent molecules and with the reactants, and the effect of hydrodynamic repulsion [456b, 544]. Kapral and co-workers have considered some of the points and these are discussed very briefly below [37, 285, 286, 454, 538]. 

What is the size of the overlapping volume The complete independence of the constant in Equation 1 from the degree of polymerization shows that the overlapping volume always consists of the same portion of the volume of the polymer coil (12). This can be easily understood by assuming that two polymer coils are able to migrate nearly unhindered through each other. Then the mean depth of permeation and, therefore, the time of overlapping is determined only by the statistics of the free Brownian motion. Equation 1 is based on this assumption. 

As described by Hohreiter et al. [2], the idea of PIV dijfusometry is based on the theory of Brownian motion. Equation 1... 

This relation takes on a very simple form when the particles are very small and the fall velocity is given by Stokes law (Table 15.1). Introducing also the expression for the diffusion coefficient of Brownian motion (equation [15.17]), equation [15.22] is readily expressed in terms of the mass m of the spherical particle ... 

The Pedet number for sedimentation can be used to estimate the influence of gravity on a colloidal system. It spedfies the ratio between gravitational settling and Brownian motion (Equation 9.2) [23] ... 

In the general case, (A3.2.23) caimot hold because it leads to (A3.2.24) which requires GE = (GE ) which is m general not true. Indeed, the simple example of the Brownian motion of a hannonic oscillator suffices to make the point [7,14,18]. In this case the equations of motion are [3, 7]... 

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... 

We call the correlation time it is equal to 1/6 Dj, where Dj is the rotational diffusion coefficient. The correlation time increases with increasing molecular size and with increasing solvent viscosity, equation Bl.13.11 and equation B 1.13.12 describe the rotational Brownian motion of a rigid sphere in a continuous and isotropic medium. With the Lorentzian spectral densities of equation B 1.13.12. it is simple to calculate the relevant transition probabilities. In this way, we can use e.g. equation B 1.13.5 to obtain for a carbon-13... 

Doi, M. and Edwards, S.F., 1978. Dynamics of concentrated polymer systems 1. Brownian motion in equilibrium state, 2. Molecular motion under flow, 3. Constitutive equation and 4. Rheological properties. J. Cheni. Soc., Faraday Trans. 2 74, 1789, 1802, 1818-18.32. 

Figure 5 relates N j to collection efficiency particle diffusivity from Stokes-Einstein equation assumes Brownian motion same order of magnitude or greater than mean free path of gas molecules (0.1 pm at... 

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... 

Equation (17) indicates that the entire distribution may be determined if one parameter, av, is known as a function of the physical properties of the system and the operating variables. It is constant for a particular system under constant operating conditions. This equation has been checked in a batch system of hydrosols coagulating in Brownian motion, where a changes with time due to coalescence and breakup of particles, and in a liquid-liquid dispersion, in which av is not a function of time (B4, G5). The agreement in both cases is good. The deviation in Fig. 2 probably results from the distortion of the bubbles from spherical shape and a departure from random collisions, coalescence, and breakup of bubbles. 

Keilson J., Storer J. E. On a Brownian motion, Boltzmann equation and the Fokker-Planck equation, Quart. Appl. Math., 10, 243-53 (1952). 

When the test temperature is raised, the rate of Brownian motion increases by a certain factor, denoted Ox. and it would therefore be necessary to raise the frequency of oscillation by the same factor flx to obtain the same physical response, as shown in Figure 1.6. The dependence of Uj upon the temperature difference T—Tg follows a characteristic equation, given by Williams, Landel, and Ferry (WLF) [11] ... 

Dynamic mechanical measurements for elastomers that cover wide ranges of frequency and temperature are rather scarce. Payne and Scott [12] carried out extensive measurements of /a and /x" for unvulcanized natural mbber as a function of test frequency (Figure 1.8). He showed that the experimental relations at different temperatures could be superposed to yield master curves, as shown in Figure 1.9, using the WLF frequency-temperature equivalence, Equation 1.11. The same shift factors, log Ox. were used for both experimental quantities, /x and /x". Successful superposition in both cases confirms that the dependence of the viscoelastic properties of rubber on frequency and temperature arises from changes in the rate of Brownian motion of molecular segments with temperature. 

The friction coefficient is one of the essential elements in the Langevin description of Brownian motion. The derivation of the Langevin equation from the microscopic equations of motion provides a Green-Kubo expression for this transport coefficient. Its computation entails a number of subtle features. Consider a Brownian (B) particle with mass M in a bath of N solvent molecules with mass m. The generalized Langevin equation for the momentum P of the B... 

Brownian motion theory may be generalized to treat systems with many interacting B particles. Such many-particle Langevin equations have been investigated at a molecular level by Deutch and Oppenheim [58], A simple system in which to study hydrodynamic interactions is two particles fixed in solution at a distance Rn- The Langevin equations for the momenta P, (i = 1,2)... 

Increasing the radius of the suspended particles, Brownian motion becomes less important and sedimentation becomes more dominant. These larger particles therefore settle gradually under gravitational forces. The basic equation describing the sedimentation of spherical, monodisperse particles in a suspension is Stokes law. It states that the velocity of sedimentation, v, can be calculated as follows ... 

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 ... 

The fundamental theory of electron escape, owing to Onsager (1938), follows Smoluchowski s (1906) equation of Brownian motion in the presence of a field F. Using the Nemst-Einstein relation p = eD/kRT between the mobility and the diffusion coefficient, Onsager writes the diffusion equation as... 

In the most general case the diffusive Markov process (which in physical interpretation corresponds to Brownian motion in a field of force) is described by simple dynamic equation with noise source ... 

Equation (2.6) is called the Fokker-Planck equation (FPE) or forward Kolmogorov equation, because it contains time derivative of final moment of time t > to. This equation is also known as Smoluchowski equation. The second equation (2.7) is called the backward Kolmogorov equation, because it contains the time derivative of the initial moment of time to < t. These names are associated with the fact that the first equation used Fokker (1914) [44] and Planck (1917) [45] for the description of Brownian motion, but Kolmogorov [46] was the first to give rigorous mathematical argumentation for Eq. (2.6) and he was first to derive Eq. (2.7). The derivation of the FPE may be found, for example, in textbooks [2,15,17,18],... 

For small colloidal particles, which are subject to random Brownian motion, a stochastic approach is more appropriate. These methods are based on the formulation and solution of the diffusion equation in a force field, in the presence of convection... 

The main disadvantage of the perfect sink model is that it can only be applied for irreversible deposition of particles the reversible adsorption of colloidal particles is outside the scope of this approach. Dahneke [95] has studied the resuspension of particles that are attached to surfaces. The escape of particles is a consequence of their random thermal (Brownian) motion. To this avail he used the one-dimensional Fokker-Planck equation... 

For example, in the case of PS and applying the Smoluchowski equation [333], it is possible to estimate the precipitation time, fpr, of globules of radius R and translation diffusion coefficient D in solutions of polymer concentration cp (the number of chains per unit volume) [334]. Assuming a standard diffusion-limited aggregation process, two globules merge every time they collide in the course of Brownian motion. Thus, one can write Eq. 2 ... 

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