Brownian motion time-dependent diffusion coefficient

In fact, the validity of Eqs. (90) and (91) is not restricted to the simple (i.e., nonretarded) Langevin model as defined by Eq. (73). These formulas can be applied in other classical descriptions of Brownian motion in which a time-dependent diffusion coefficient can be defined. This is for instance, the case in the presence of non-Ohmic dissipation, in which case the motion of the Brownian particle is described by a retarded Langevin equation (see Section V). 

Recent advances in percolation theory and fractal geometry have demonstrated that Dc is not a constant when diffusion occurs as a result of fractional Brownian motion, i.e., anomalous diffusion (Sahimi, 1993). The time-dependent diffusion coefficient, D(t), for anomalous diffusion in two-dimensional free space is given by (Mandelbrot Van Ness, 1968),... 

Another use of Brownian dynamics is to compute the time-dependent rate coefficient k t) of diffusion-controlled reactions. In this case particles start on a boundary near the active site and undergo Brownian motion until they either react or their lifetimes exceed some preset cut-off. The starting positions on this boundary are assigned according to the distribution /c(ro)exp[— f/(ro)]. In this case /c(ro) is the space-dependent intrinsic bimolecular rate constant, is k T), and t/(ro) is the potential of mean force between the two particles. 

In order to examine the nature of the friction coefficient it is useful to consider the various time, space, and mass scales that are important for the dynamics of a B particle. Two important parameters that determine the nature of the Brownian motion are rm = (m/M) /2, that depends on the ratio of the bath and B particle masses, and rp = p/(3M/4ttct3), the ratio of the fluid mass density to the mass density of the B particle. The characteristic time scale for B particle momentum decay is xB = Af/ , from which the characteristic length lB = (kBT/M)i lxB can be defined. In derivations of Langevin descriptions, variations of length scales large compared to microscopic length but small compared to iB are considered. The simplest Markovian behavior is obtained when both rm << 1 and rp 1, while non-Markovian descriptions of the dynamics are needed when rm << 1 and rp > 1 [47]. The other important times in the problem are xv = ct2/v, the time it takes momentum to diffuse over the B particle radius ct, and Tp = cr/Df, the time it takes the B particle to diffuse over its radius. 

Photon correlation spectroscopy (PCS) has been used extensively for the sizing of submicrometer particles and is now the accepted technique in most sizing determinations. PCS is based on the Brownian motion that colloidal particles undergo, where they are in constant, random motion due to the bombardment of solvent (or gas) molecules surrounding them. The time dependence of the fluctuations in intensity of scattered light from particles undergoing Brownian motion is a function of the size of the particles. Smaller particles move more rapidly than larger ones and the amount of movement is defined by the diffusion coefficient or translational diffusion coefficient, which can be related to size by the Stokes-Einstein equation, as described by... 

The Peclet number compares the effect of imposed shear (known as the convective effect) with the effect of diffusion of the particles. The imposed shear has the effect of altering the local distribution of the particles, whereas the diffusion (or Brownian motion) of the particles tries to restore the equilibrium structure. In a quiescent colloidal dispersion the particles move continuously in a random manner due to Brownian motion. The thermal motion establishes an equilibrium statistical distribution that depends on the volume fraction and interparticle potentials. Using the Einstein-Smoluchowski relation for the time scale of the motion, with the Stokes-Einstein equation for the diffusion coefficient, one can write the time taken for a particle to diffuse a distance equal to its radius R, as... 

Many excellent introductions to quasi-elastic light scattering can be found in the literature describing the theory and experimental technique (e.g. 3-6). The use of QELS to determine particle size is based on the measurement, via the autocorrelation of the time dependence of the scattered light, of the diffusion coefficients of suspended particles undergoing Brownian motion. The measured autocorrelation function, G<2>(t), is given by... 

In PCS, Brownian motion is used to measure the particle size. As a result of Brownian motion of dispersed particles, the intensity of the scattered hght undergoes fluctuations that are related to the velocity of the particles. As larger particles move less rapidly than their smaller counterparts, the intensity fluctuation (intensity versus time) pattern will depend on particle size, as illustrated in Figure 19.8. The velocity of the scatterer is measured in order to obtain the diffusion coefficient. 

One of the important issues is the possibility to reveal the specific mechanisms of subdiflFusion. The nonlinear time dependence of mean square displacements appears in different mathematical models, for example, in continuous-time random walk models, fractional Brownian motion, and diffusion on fractals. Sometimes, subdiffusion is a combination of different mechanisms. The more thorough investigation of subdiffusion mechanisms, subdiffusion-diffiision crossover times, diffusion coefficients, and activation energies is the subject of future works. 

When multiple scattering is discarded from the measured signal, DLS can be used to study the dynamics of concentrated suspensions, in which the Brownian motion of individual particles (self-diffusion) differs from the diffusive mass transport (gradient or collective diffusion), which causes local density fluctuations, and where the diffusion on very short time-scales (r < c lD) deviates from those on large time scales (r c D lones and Pusey 1991 Banchio et al. 2000). These different diffusion coefficients depend on the microstructure of the suspension, i.e. on the particle concentration and on the interparticle forces. For an unknown suspension it is not possible to state a priori which of them is probed by a DLS experiment. For this reason, a further concentration limit must be obeyed when DLS is used for basic characterisation tasks such as particle sizing. As a rule of thumb, such concentration effects vanish below concentrations of 0.01-0.1 vol%, but certainty can only be gained by experiment. 

Dynamic light scattering involves the study of time-dependent fiuctuations in the intensity of scattered light which are the result of the Brownian motion of the particles. The random movement of the particles causes the distances between the particles to fiuctuate, causing constantly varying constructive and destructive interference patterns. This time-dependent fiuctuation in the intensity can be correlated with itself to obtain the diffusion coefficient of the particles. This method is also called photon correlation spectroscopy (PCS) and quasi-elastic light scattering (QELS) [21, 22]. 

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