Diffusion in a sphere

Figure 6 Apparent elastic incoherent structure factor A q(Q) for ( ) denatured and ( ) native phosphoglycerate kinase. The solid line represents the fit of a theoretical model in which a fraction of the hydrogens of the protein execute only vihrational motion (this fraction is given by the dotted line) and the rest undergo diffusion in a sphere. For more details see Ref. 25.

Two physically reasonable but quite different models have been used to describe the internal motions of lipid molecules observed by neutron scattering. In the first the protons are assumed to undergo diffusion in a sphere [63]. The radius of the sphere is allowed to be different for different protons. Although the results do not seem to be sensitive to the details of the variation in the sphere radii, it is necessary to have a range of sphere volumes, with the largest volume for methylene groups near the ends of the hydrocarbon chains in the middle of the bilayer and the smallest for the methylenes at the tops of the chains, closest to the bilayer surface. This is consistent with the behavior of the carbon-deuterium order parameters,. S cd, measured by deuterium NMR ... 

To make contact with the diffusion-in-a-sphere model, we have defined the spherical radius as the root-mean-square fluctuation of the protons averaged over 100 ps. The varia-... 

Figure 13.9 The production and actions of nitric oxide (NO). The influx of calcium through either calcium channels or NMDA receptors triggers NOS to convert L-arginine to NO. L-NAME and 7-NI inhibit this process. NO, once produced, can diffuse in a sphere and then can activate guanylate cyclase...

Diffusion in a sphere may be more common than that in a cylinder in the pharmaceutical sciences. The example we may think of is the dissolution of a spherical particle. Since convection is normally involved in solute particle dissolution in reality, the dissolution rate estimated by considering only diffusion often underestimates experimental values. Nevertheless, we use it as an example to illustrate the solution of the differential equations describing diffusion in the spherical coordinate system [1],... 

Three analytical expressions for the spin-echo intensity as a function of the gradient in a pulsed field gradient NMR experiment for spins diffusing in a sphere with reflecting walls are reinvestigated. It is found that none of the published formulas are completely correct. By numerical comparisons the correct formula is found. 

Since the Murday-Cotts paper in 1968 much progress has been made toward the theoretical description of the spin-echo intensity E g, A) for spins diffusing in well-defined geometries. Tanner and Stejskal derived already in 1968 the exact expression of ii(g. A) for spins diffusing in a rectangular box. The derivation of an exact expression for E g, A) for diffusion in a sphere with reflecting walls is not a trivial mathematical problem and it took between 1992 and 1994 when three expressions were published. All three expressions are only valid in the short-gradient-pulse approximation (see below). 

When we wanted to numerically fit experimental PFGE data of water diffusion in a water-in-oil emulsion, we found that for a beginner in this field the literature is quite confusing. First, all three expressions for diffusion in a sphere with reflecting walls are somewhat different and lead to very different fitting results, especially when the formulas are combined with a radius distribution function. Since the derivation of the published expressions needs some tedious algebra (which has not been published), it is not trivial to check the derivation in order to establish which expression is the correct one. Here we use a numerical approach to decide which expression is correct. 

For diffusion in a sphere the propagator should be independent of the azimuth angle (p, then we can write for R ... 

For a complete definition of Eq. (53) we need to determine the constants Cnk from the conditions (17)-(19) and then calculate the Fourier integral Eq. (1) for the echo signal. To avoid the tedious algebra we compare the three published solutions numerically, but first reproduce these solutions here using our notation. Two of these solutions resulted from a calculation that included the effect of surface relaxation. To make a correct comparison we eliminate from the equations the terms due to relaxation. Then we have the following formulae for the echo intensity for diffusion in a sphere with radius a and reflecting walls ... 

Gav, the means value of Gx, depends only on the assumed geometry of diffusion 4.00660 for radial diffusion in a sphere, 3.29506 for radial diffusion in a cylinder, and 2.15821 for diffusion across a plane sheet (Dodson, 1986). Equation (19) may be used in conjunction with f b s for multiple mineral-isotopic systems, or for multiple grain sizes of the same mineral-isotopic system, in order to determine cooling histories for rocks. 

F. DIFFUSION IN A SPHERE WITH FAST REACTION - SINGULAR PERTURBATION THEORY ... 

F. Diffusion in a Sphere with Fast Reaction - Singular Perturbation Theory ... 

Figure 15. Half width at half maximum of the broadened component of the neutron quasielastic spectra obtained with an acid Nafion membrane containing 15% H20 by weight. The points are the widths of the best fit Lorentzian lines from spectra obtained with an incident wavelength of 10 A (Q), 11 A and 13 A (A). The full line is the theoretical width predicted by the model with diffusion in a sphere (with D = 1.8 X 10 5 cm2/s and a = 4.25 A). The two theoretical asymptotes for Q O and Q 00 are also shown (compare with Figure 2 of Reference 2 for more details). Half width at half maximum of the best fit Lorentzian lines to the spectra obtained with bulk water at 28° C (incident wavelength 10 A) is denoted by +. The straight line passing through the points (+) is the theoretical width predicted by the simple self diffusion model with Dt = 2.5 X 10-5 err /s. Note the different vertical scales for the Nafion sample and the bulk water sample.

This Equation 7.24 is the general equation of radial diffusion in a sphere when the volume of liquid is so large that a change in dimension occurs. 

Sorption uptake rate curves of ben ne in silicalite-1 (0 x) and HZSM-5 ( 0,+) at 395 K, 0.826 foir. (0>0) denote adsorption processes and (x,- ) denote desorption processes. Lines were calculated using the solution for diffusion in a sphere for silicalite and a cylinder for HZSM-5 from a well-stined solution of limited volume with fractional uptakes of 0.46 and 0.33, respectively (see eqn. (7)). 

With dp/2 as the characteristic length for diffusion in a sphere, we get for the ratio of the characteristic times of pore diffusion and reaction ... 

Ramachandran and Smith (7) assumed the surface area per unit volume of the pore, given by tCq/2, is equal to that of the solid particles from which the experimental data are obtained. Furthermore the effective length of the pore L is chosen so that the Thiele modulus at zero time given by eq. (19) is equal to that characterizing reaction and diffusion in a sphere of radius R with first order kinetics. It then follows that... 

In transient diffusion, the concentration of a species varies, as we have seen, with both time and distance. The underlying process of diffusion may take place in isolahon or it may be accompanied by a chemical reaction, by flow, or by both reaction and flow. These more complex cases are not taken up here, and we limit ourselves instead to the consideration of purely diffusive processes. Furthermore, with one or two exceptions, the treatment is confined to a single spatial coordinate represented by the Cartesian x- (or z-) axis or by the radial variable r. The last is used in formulating diffusion in a sphere, or in the radial direction of a cylinder. Pick s equation for these three cases can be deduced from the general conservation equation (Equation 2.24a) and Table 2.3. They are as follows ... 

FIGURE 4.3 Concentration profiles for unsteady mass diffusion in a sphere of radius R. Cx(R, t) = Caoc. CaCu 0) = Cao. (Reprinted by permission of the publisher from Crank, 1956.)... 

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