Diffusion coefficient longitudinal

Perrin model and the Johansson and Elvingston model fall above the experimental data. Also shown in this figure is the prediction from the Stokes-Einstein-Smoluchowski expression, whereby the Stokes-Einstein expression is modified with the inclusion of the Ein-stein-Smoluchowski expression for the effect of solute on viscosity. Penke et al. [290] found that the Mackie-Meares equation fit the water diffusion data however, upon consideration of water interactions with the polymer gel, through measurements of longitudinal relaxation, adsorption interactions incorporated within the volume averaging theory also well described the experimental results. The volume averaging theory had the advantage that it could describe the effect of Bis on the relaxation within the same framework as the description of the diffusion coefficient. 

Here m(r, t) is the relative difference of the longitudinal magnetization M and its equilibrium value Mo m = (M — Mo)/Mo, D is the bulk diffusion coefficient and p is the bulk relaxation rate. The general solution to the Torrey-Bloch equation can be written as... 

Outer sphere relaxation arises from the dipolar intermolecular interaction between the water proton nuclear spins and the gadolinium electron spin whose fluctuations are governed by random translational motion of the molecules (106). The outer sphere relaxation rate depends on several parameters, such as the closest approach of the solvent water protons and the Gdm complex, their relative diffusion coefficient, and the electron spin relaxation rate (107-109). Freed and others (110-112) developed an analytical expression for the outer sphere longitudinal relaxation rate, (l/Ti)os, for the simplest case of a force-free model. The force-free model is only a rough approximation for the interaction of outer sphere water molecules with Gdm complexes. 

From Eq. 17.26 it is clear that the variance due to longitudinal diffusion is negatively influenced by the length of the capillary and the diffusion coefficient of the solute. However, it is positively affected by the applied potential and the apparent mobility of a solute. According to this equation, fast migrating zones will show less variance due to axial diffusion. 

Eq. 17.42 is the expression of the resolution for CE in electrophoretic terms. However, the application of this expression for the calculation of Rs in practice is limited because of D,. The diffusion coefficient of different compounds in different media is not always available. Therefore, the resolution is frequently calculated with an expression that employs the width of the peaks obtained in an electropherogram. This way of working results in resolution values that are more realistic as all possible variances are considered (not only longitudinal diffusion in Eq. 17.42). Assuming that the migrating zones have a Gaussian distribution, the resolution can be expressed as follows ... 

The Hl value is reduced by an increase in the viscosity of the solvent or by a decrease in the temperature. Longitudinal diffusion can thus be reduced by decreasing the diffusion coefficient and increasing the flow rate however, these two actions are counter-effective in liquid chromatography because of the mass transport term. 

Here D, D , and Dr are, respectively, the longitudinal, transverse, and rotational diffusion coefficients of the chain averaged over the internal degree of freedom, h an external field, and v and angular velocity of the chain induced by a flow field in the solution. Furthermore, I is the unit tensor and 91 is the rotational operator defined by... 

When the solution is dilute, the three diffusion coefficients in Eq. (40a, b) may be calculated only by taking the intramolecular hydrodynamic interaction into account. In what follows, the diffusion coefficients at infinite dilution are signified by the subscript 0 (i.e, D, 0, D10> and Dr0). As the polymer concentration increases, the intermolecular interaction starts to become important to polymer dynamics. The chain incrossability or topological interaction hinders the translational and rotational motions of chains, and slows down the three diffusion processes. These are usually called the entanglement effect on the rotational and transverse diffusions and the jamming effect on the longitudinal diffusion. In solving Eq. (39), these effects are taken into account by use of effective diffusion coefficients as will be discussed in Sect. 6.3. 

Using Eq. (C16) in Appendix C and a similar procedure to those mentioned above for Dx and Dr, we can derive for the longitudinal diffusion coefficient D [19, 109]... 

The jamming effect, i.e., the slowing down of the longitudinal diffusion of a polymer chain by the head-on collision with other chains, can be treated by a model similar to that proposed by Cohen and Turnbull [112] for self diffusion of small molecules in a fluid. This model assumes that if at least one surrounding polymer chain exists within the critical hole ahead of a test chain, both collide, and this prevents the test chain from diffusing longitudinally. With this assumption, we express the longitudinal diffusion coefficient Dp of the test chain as... 

The longitudinal diffusion coefficient D has been formulated by the hole theory in Sect. 6.3.2. If the similarity ratio X in this theory is chosen to be 0.025 for the rod with the axial ratio 50, Eq. (58) with Eq. (56) gives the solid curve in Fig. 16a. Though it fits closely the simulation data, the chosen X is not definitive because the change in D(l is small and the definition of the effective axial ratio is ambiguous. Though not shown here, Eq. (53) for D, by the Green function method describes the simulation data equally well if P and C, are chosen to be 1000 and 1, respectively. 

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