Diffusion coefficients, effects Brownian motion

When the regular motion is simply uniform rotation of the absorption and emission dipoles with angular velocity to around the helix axis, one has p(t) - p(0) = cot. For the corresponding random motion, one might have m)2> = 2Dt, where D is the effective diffusion coefficient for Brownian rotation of the transition dipole around the helix axis. When these expressions are incorporated in Eqs. (4.31) and (4.24), the latter becomes a generalization of a relation recently derived using a more cumbersome approach/104-1... 

In this section we examine the flow of a suspension of particles, particularly the apparent viscosity coefficient of the suspension. Our interest is in calculating the convective mass flux of a suspension as distinct from the diffusive flux of Brownian motion. As previously, we shall assume a very dilute suspension in which each particle behaves as if it were in a liquid of infinite extent. To simplify the calculation, we neglect Brownian motion, although, as we discuss later, in the very dilute limit considered and for spherical particles it has no effect on the suspension viscosity. 

The hydrodynamic drag experienced by the diffusing molecule is caused by interactions with the surrounding fluid and the surfaces of the gel fibers. This effect is expected to be significant for large and medium-size molecules. Einstein [108] used arguments from the random Brownian motion of particles to find that the diffusion coefficient for a single molecule in a fluid is proportional to the temperature and inversely proportional to the frictional coefficient by... 

Diffusion of small molecular penetrants in polymers often assumes Fickian characteristics at temperatures above Tg of the system. As such, classical diffusion theory is sufficient for describing the mass transport, and a mutual diffusion coefficient can be determined unambiguously by sorption and permeation methods. For a penetrant molecule of a size comparable to that of the monomeric unit of a polymer, diffusion requires cooperative movement of several monomeric units. The mobility of the polymer chains thus controls the rate of diffusion, and factors affecting the chain mobility will also influence the diffusion coefficient. The key factors here are temperature and concentration. Increasing temperature enhances the Brownian motion of the polymer segments the effect is to weaken the interaction between chains and thus increase the interchain distance. A similar effect can be expected upon the addition of a small molecular penetrant. 

A probabilistic kinetic model describing the rapid coagulation or aggregation of small spheres that make contact with each other as a consequence of Brownian motion. Smoluchowski recognized that the likelihood of a particle (radius = ri) hitting another particle (radius = T2 concentration = C2) within a time interval (dt) equals the diffusional flux (dC2ldp)p=R into a sphere of radius i i2, equal to (ri + r2). The effective diffusion coefficient Di2 was taken to be the sum of the diffusion coefficients... 

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... 

It has been established that geometrical disorder has only a small effect on Brownian motion [S. Havlin, D. Ben Avraham (1987)]. Also, for thermally activated jumps, if the distribution of es and evv in a geometrically regular lattice is chosen to be Gaussian, as characterized by the variances as and crw, it has been ascertained [Y. Limoge, J. L. Bocquet (1990)] that there are two limiting diffusion coefficients ... 

At low ionic strength (kR 1), other effects connected with the finite diffusivity of the small ions in the EDL surrounding the particle are present. The noninstantaneous diffusion of the small ions (with respect to the Brownian motion of the colloid particle) could lead to detectable reduction of the single particle diffusion coefficient, Dq, from the value predicted by the Stokes-Ein-stein relation. Equation 5.447. For spherical particles, the relative decrease in the value of Dq is largest at k/ 1 and could be around 10 to 15%. As shown in the normal-mode theory, the finite diffusivity of the small ions also affects the concentration dependence of the collective diffusion coefficient of the particles. Belloni et al. obtained an explicit expression for the contribution of the small ions in Ac)... 

This interpretation of the effective diffusion in terms of individual trajectories of an ensemble of particles advected by the flow and a superimposed random Brownian motion, as described by the stochastic advection equation (2.34), can be extended further. The characteristic time for molecular diffusion across the channel td L2/D gives the correlation time of the longitudinal velocity experienced by a particle. Thus the longitudinal motion can be described as a collection of independent longitudinal displacements of typical length Utd over time intervals td- Thus, for long times, t td, the effective diffusion coefficient of such random walk can be estimated as Deff (Utd)2/td U2L2/D that is consistent with (2.51) when Pe > 1. 

Surface self-diffusion is the two-dimensional analogue of the Brownian motion of molecules in a liquid bulk. Measurements of self-diffusion have to be performed in complete absence of any Marangoni flow caused by surface tension differences. Such experimental conditions are best established in an insoluble monolayer where one part consists of unlabelled and the other of radio-tracer labelled molecules. The movement of molecules within the surface monolayer can be now observed by using a Geiger-Miiller counter. There are possible effects of liquid convective flow in the sublayer which was discussed for example by Vollhardt et al. (1980a). With e special designed apparatus Vollhardt et al. (1980b) studied the self-difihision of different palmitic and stearic acid and stearyl alcohol and obtained self-diffusion coefficients between l-i-4 lO cm /s. 

In a QELS experiment, a monochromatic beam of light from a laser is focused on to a dilute suspension of particles and the scattering intensity is measured at some angle 0 by a detector. The phase and the polarization of the scattered light depend on the position and orientation of each scatterer. Because molecules or particles in solution are in constant Brownian motion, scattered light will result that is spectrally broadened by the Doppler effect. The key parameter determined by QELS is the diffusion coefficient, D, or particle di sivity which can be related to particle diameter, d, via the Stokes-Einstein equation ... 

The effect of free volume on penetrant diffusion coefficients in polymers is often described using concepts from the Cohen and Turnbull model (7. This statistical mechanics model provides a simplistic description of diffusion in a liquid of hard spheres. A hard sphere penetrant is considered to be trapped in a virtual cage created by its neighbors. Free volume is defined as the volume of the cage less the volume of the penetrant. Free volume fluctuations, which occur randomly due to thermally-stimulated Brownian motion of neighboring hard spheres, provide opportunities for the penetrant to execute a diffusion step if the gap (i,e, free volume fluctuation) occurs sufficiently close to the i netrant to be accessible and is of sufficient size to acconunodate it. The diffusion coefficient of a penetrant is given by ... 

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. 

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