The diffusion coefficient

The diffusion coefficient describes the coarse-grained dynamics of particles in condensed systems (see Section 1.5.5). To get an explicit expression we start from (cf. Eq. (1.209)) 

This can be simplified by changing variables d = t — t, with 6 goes from z to 0 and dt = —dd. This leads to 

The integral is done over the shaded area in Fig. 6.1. Using this picture it is easily seen that the order of integration in (6.11) may be changed so as to give 

The correlation function Cy t) = (v(0) y t) vanishes at long time because velocities at different times become quickly (on a timescale of a few collisions) uncorrelated which implies (v(0). v(Z)) — (v(0))(v(Z)) = 0. For this reason the 

A time correlation function that involves the same observable at two different times is called an autocorrelation function. We have found that the self-diffusion coefficient is the time integral of the velocity auto-correlation function 

Equating the coefficients of dc/dx in the phenomenological equation (4.15) with that in Pick s law [Eq. (4.16)], is seen that 

is the diffusion coefficient a concentration-independent constant A naive answer would run thus 5 is a constant and therefore it appears that D also is a constant. 

However, expression (4.17) was obtained only because an ideal solution was considered, and activity coefficients were ignored in Eq. (3.61). Activity coefficients, however, are concentration dependent. So, if the solution does not behave ideally, one has, starting from Eq. (4.14), and using Eq. (3.63), 

Electrolyte Diffusion Coefficient D in Units of 10 cm at Concentration (in molarity) s 

Rigorously speaking, the diffusion coefficient is not a constant (Table 4.3). If, however, the variation of the activity coefficient is not significant over the concentration difference that produces diffusion, then (c,/ )(3 /9c,) 1 and for all practical purposes D is a constant. This effective constancy ofD with concentration will be assumed in most of the discussions presented here. 

Diffusion being a thermally activated process, the diffusion coefficient depends on the absolute temperature T according to an Arrhenius law 

In one space dimension, neglecting temperature gradients relative to concentration gradients, this can be rewritten 

Atom fractions Xt relate to volume concentrations through 

Neglecting variations in the molar volume makes the denominator constant and therefore the relative variations of concentrations and mole fractions are equal 

Even this more elaborated description of ion movements in response to gradients of chemical potential may turn out to be insufficient, in particular when uphill diffusion is active  

Since the majority of determinations of the recombination coefficient depend on the value of. lj2, it is important that we should be aware of the reliability to be placed on experimental values for this parameter. 

Lede and Villermaux [68] have reviewed the data available for hydrogen values of at different temperatures are given in Fig. 20. The 

Fick s law proportionality factor, Dn, is known as the diffusion coefficient or diffusivity. Its fundamental dimensions, which are obtained from equation (1-40), 

In order to use eqn 8.21 in practical cases the availability of data for two fundamental constants is needed (i) the partition coefficient, Kpp, of the migrating compound between the polymer P and the foodstuff or simulating liquid F and (ii) the diffusion coefficient. Dp, of the migrant in P. So far upper limits for migration amounts are needed from regulatory standpoints, predictions of worst case scenarios can start with the assumption of good solubility of the migrant in F and consequently A pp = 1 can be used. Much 

For polyolefins another semi-empirical diffiision model has been developed, by Limm and Hollifield (1996). From their studies it is possible to arrive at the following relationship  

a and K are empirical constants determined from experimental diffusion data. These values for polyolefins are given in Table 8.5. The constant In Dq 

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The value of coefficient depends on the composition. As the mole fraction of component A approaches 0, approaches ZJ g the diffusion coefficient of component A in the solvent B at infinite dilution. The coefficient Z g can be estimated by the Wilke and Chang (1955) method ... 

There is also a traffic between the surface region and the adjacent layers of liquid. For most liquids, diffusion coefficients at room temperature are on the order of 10 cm /sec, and the diffusion coefficient is related to the time r for a net displacement jc by an equation due to Einstein ... 

Here (D is the diffusion coefficient and C is the concentration in the general bulk solution. For initial rates C can be neglected in comparison to C/ so that from Eqs. IV-59 and IV-60 we have... 

Reference 115 gives the diffusion coefficient of DTAB (dodecyltrimethylammo-nium bromide) as 1.07 x 10" cm /sec. Estimate the micelle radius (use the Einstein equation relating diffusion coefficient and friction factor and the Stokes equation for the friction factor of a sphere) and compare with the value given in the reference. Estimate also the number of monomer units in the micelle. Assume 25°C. 

The state of an adsorbate is often described as mobile or localized, usually in connection with adsorption models and analyses of adsorption entropies (see Section XVII-3C). A more direct criterion is, in analogy to that of the fluidity of a bulk phase, the degree of mobility as reflected by the surface diffusion coefficient. This may be estimated from the dielectric relaxation time Resing [115] gives values of the diffusion coefficient for adsorbed water ranging from near bulk liquids values (lO cm /sec) to as low as 10 cm /sec. 

This complex Ginzburg-Landau equation describes the space and time variations of the amplitude A on long distance and time scales detennined by the parameter distance from the Hopf bifurcation point. The parameters a and (5 can be detennined from a knowledge of the parameter set p and the diffusion coefficients of the reaction-diffusion equation. For example, for the FitzHugh-Nagumo equation we have a = (D - P... 

The Turing mechanism requires that the diffusion coefficients of the activator and inlribitor be sufficiently different but the diffusion coefficients of small molecules in solution differ very little. The chemical Turing patterns seen in the CIMA reaction used starch as an indicator for iodine. The starch indicator complexes with iodide which is the activator species in the reaction. As a result, the complexing reaction with the immobilized starch molecules must be accounted for in the mechanism and leads to the possibility of Turing pattern fonnation even if the diffusion coefficients of the activator and inlribitor species are the same 62. 

Another class of instabilities that are driven by differences in the diffusion coefficients of the chemical species detennines the shapes of propagating chemical wave and flame fronts [65, 66]. 

If the diffusion coefficient of species A is less tlian tliat of B (D < D ) tlie propagating front will be planar. However, if is sufficiently greater than tire planar front will become unstable to transverse perturbations and chaotic front motion will ensue. To understand tire origin of tire mechanism of tire planar front destabilization consider tire following suppose tire interface is slightly non-planar. We would like to know if tire dynamics will tend to eliminate this non-planarity or accentuate it. LetZ)g The situation is depicted schematically in figure... 

We assume in the following that the ligand is bound in a binding pocket of depth 6 —a = 7 A involving a potential barrier AU = 25 kcal/mol, similar to that of streptavidin (Chilcotti et al., 1995). We also assume that the diffusion coefficient of the ligand is similar to the diffusion coefficient of the heme group in myoglobin (Z) = 1 A /ns) as determined from Mofibauer spectra (Nadler and Schulten, 1984). 

For the reasons explained in Chapter 2 we might expect the diffusion coefficients to be given by equations of the form (2.11) and (2.19)... 

One alternative approach to the calculation of the diffusion and other transport coefficier is via an appropriate autocorrelation function. For example, the diffusion coefficie... 

The friction coefficient determines the strength of the viscous drag felt by atoms as they move through the medium its magnitude is related to the diffusion coefficient, D, through the relation Y= kgT/mD. Because the value of y is related to the rate of decay of velocity correlations in the medium, its numerical value determines the relative importance of the systematic dynamic and stochastic elements of the Langevin equation. At low values of the friction coefficient, the dynamical aspects dominate and Newtonian mechanics is recovered as y —> 0. At high values of y, the random collisions dominate and the motion is diffusion-like. 

In these expressions, dp is the particle diameter of the stationary phase that constitutes one plate height. D is the diffusion coefficient of the solute in the mobile phase. 

McKillop and associates have examined the electrophoretic separation of alkylpyridines by CZE. Separations were carried out using either 50-pm or 75-pm inner diameter capillaries, with a total length of 57 cm and a length of 50 cm from the point of injection to the detector. The run buffer was a pH 2.5 lithium phosphate buffer. Separations were achieved using an applied voltage of 15 kV. The electroosmotic flow velocity, as measured using a neutral marker, was found to be 6.398 X 10 cm s k The diffusion coefficient,... 

Since the diffusion coefficient is constant for a given material, Eq. (2.63) shows that the time required for a displacement increases with the square of the distance traveled. This can be understood by thinking that the displacement criterion would be met by finding the diffused particle anywhere on the surface of a sphere of radius x after time t if it started at the origin. The surface area of a sphere is proportional to the square of its radius. 

Note that the diffusion coefficient for a polymer through an environment of low molecular weight molecules is typically on the order of magnitude of 10"" m" sec". If the first subscript indicates the diffusing species, and the second the surrounding molecules, and P stands for polymer and S for small molecules, we see that the order of diffusion coefficients is Ds g > Dp g > Dp P sequence which makes sense in terms of relative frictional resistance. 

At first glance, the contents of Chap. 9 read like a catchall for unrelated topics. In it we examine the intrinsic viscosity of polymer solutions, the diffusion coefficient, the sedimentation coefficient, sedimentation equilibrium, and gel permeation chromatography. While all of these techniques can be related in one way or another to the molecular weight of the polymer, the more fundamental unifying principle which connects these topics is their common dependence on the spatial extension of the molecules. The radius of gyration is the parameter of interest in this context, and the intrinsic viscosity in particular can be interpreted to give a value for this important quantity. The experimental techniques discussed in Chap. 9 have been used extensively in the study of biopolymers. 

In this expression, called Pick s first law, the proportionality constant D is the diffusion coefficient of the solute. Since J = (l/A)(dQ/dt) and c = Q/V, where Q signifies the quantity of solute in unspecified units, it follows that D has the units length time", or m sec in the SI system. The minus sign in Eq. (9.69)... 

Before pursuing the diffusion process any further, let us examine the diffusion coefficient itself in greater detail. Specifically, we seek a relationship between D and the friction factor of the solute. In general, an increment of energy is associated with a force and an increment of distance. In the present context the driving force behind diffusion (subscript diff) is associated with an increment in the chemical potential of the solute and an increment in distance dx ... 

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