Unrestricted diffusion

Surface diffusion — unrestricted diffusion along nanospaces over or under surfaces in bonded surface continua, over membranes (at both sides), and inside a membrane (Figure 1.14a). 

Electroless plating rates ate affected by the rate of reduction of the dissolved reducing agent and the dissolved metal ion which diffuse to the catalytic surface of the object being plated. When an initial continuous metal film is deposited, the whole surface is at one potential determined by the mixed potential of the system (17). The current density is the same everywhere on the surface as long as flow and diffusion are unrestricted so the metal... 

Fig. 11a and b. Decay of the alignment echo height as a function of the mixing time x2 for different motional mechanisms, a Tetrahedral jumps as a model for conformational changes b Diffusive motion, the solid lines correspond to unrestricted rotational diffusion, the dashed lines to diffusion restricted to an angular region of 8°. Note the strong dependence of the decay curves on the evolution time t, in case of diffusive motion... 

Fig. 12a and b. Calculated 2H spin alignment spectra for diffusive motion, a unrestricted rotational diffusion for different mixing times x2 b diffusion restricted to angular regions as indicated for long mixing times t2... 

Fig. 1.15 Left propagator for unrestricted self- tained in an experiment, 5(q), plotted semi-diffusion. The propagator P(R, A) is shown for logarithmically over q2. In this representation, increasing encoding times A and becomes the slope of the decaying function is equal to broader with increasing A, while its intensity at (4 jt)2AD, so that the diffusion coefficient D zero displacement is reduced due to the re- can be obtained directly by comparing at least quirement that the area remains normalized to two measurements taken at different values unity. Right signal function as would be ob- of q.

The size of a surface available for field ion microscope study of surface diffusion is very small, usually much less than 100 A in diameter. The random walk diffusion is therefore restricted by the plane boundary. For a general discussion, however, we will start from the unrestricted random walk. First, we must be aware of the difference between the chemical diffusion coefficient and the tracer diffusion coefficient. The chemical diffusion coefficient, or more precisely the diffusion tensor, is defined by a generalized Fick s law as... 

The Debye-Hiickel approximation to the diffuse double-layer problem produces a number of relatively simple equations that introduce a variety of double-layer topics as well as a number of qualitative generalizations. In order to extend the range of the quantitative relationships, however, it is necessary to return to the Poisson-Boltzmann equation and the unrestricted Gouy-Chapman theory, which we do in Section 11.6. 

Although the assignment of a value to DM is arbitrary, it is by no means unrestricted. From its definition (Equation 20.5) it is clear that DM must be positive. To find the upper limit for DM, one need only substitute the Einstein definition [6] of the diffusion coefficient,... 

The most difficult step in the performance of PFG-measurements is the analysis of the experimentally measured R-values. For unrestricted diffusion (i.e. the quantity V2DA is much smaller than the distance between the barriers) the PFG-NMR echo attenuation is given by ... 

Eq. 1 showed that in the case of unrestricted diffusion the echo attenuation value R depends upon the durations 8 and A. This is also true in the case of restricted diffusion, although in a different manner. The dependence of the R-value upon these two parameters is shown in Fig. 5. This figure clearly shows that the echo attenuation factor R steadily decreases with increasing A in the case of unrestricted diffusion, but becomes independent of this parameter in the case of restricted diffusion. It may be deduced from this figure that it is necessary to determine the parameters of the log-normal droplet size distribution R as a function of A or by measuring R as a function of 8 for a fixed large value of A. Measurement of only one R-value, at a chosen 8 or A, is not sufficient for a careful determination of the droplet size distribution in Fig. 5 a given In R-value can be found on more than one In R versus A-5/3 curve. This means that the In R-values have to be determined for different values of A and/or 8. 

Figure 5. Echo attenuation R versus the time interval A between field gradient pulses for different widths 5 of the field gradient pulses in the case of unrestricted (A) and restricted (B) diffusion.

As shown above, the pulsed field gradient NMR technique was first described by Tanner and Stejskal [1,2]. In addition to their work on unrestricted diffusion they also performed theoretical analyses of restricted diffusion and tested their results on octanol-in-water emulsions stabilized by surfactants. 

Studies of metal compound diffusion in porous media have consistently demonstrated that the rate of diffusion within the microporous material is less than would be observed in an unrestricted medium. This discrepancy, observed for all liquid diffusion processes in pores of small diameter is related to hydrodynamic phenomena. The proximity of the molecule to the pore wall increases the frictional drag on the diffusing species when the... 

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