Cylindrical Diffusion

This equation can be compared with Eq. (4.97) to see that for internal spherical diffusion the first term in brackets is multiplied by coth, and there is a change in sign in one term. Using Eq. (4.69) the mass transfer impedance is obtained  

The difference between Eqs. (4.106) and (4.98) is that one is replaced by coth in the denominator. It should be added that the difference in the denominator is always positive. 

For large values of or high fi-equencies coth(x) 1 and both equations become similar. On the other hand for low values of x, that is for small ro and low frequencies, l/[jc coth(jc) — 1] = 1/5 + 3/r and this equation simplifies to  

Cylindrical diffusion is observed, for example, during diffusion to a thin metallic wire or a carbon fiber electrode. Impedance of cylindrical electrodes was studied by Fleischmann et al. [165,166] and Jacobsen and West [167]. The partial differential equation describing diffusion to a cylinder, written for the oscillation concentration of an Ox form, is [144] 

4 Impedance of the Faradaic Reactions in the Presence of Mass Transfer 

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Analysis of neutron data in terms of models that include lipid center-of-mass diffusion in a cylinder has led to estimates of the amplitudes of the lateral and out-of-plane motion and their corresponding diffusion constants. It is important to keep in mind that these diffusion constants are not derived from a Brownian dynamics model and are therefore not comparable to diffusion constants computed from simulations via the Einstein relation. Our comparison in the previous section of the Lorentzian line widths from simulation and neutron data has provided a direct, model-independent assessment of the integrity of the time scales of the dynamic processes predicted by the simulation. We estimate the amplimdes within the cylindrical diffusion model, i.e., the length (twice the out-of-plane amplitude) L and the radius (in-plane amplitude) R of the cylinder, respectively, as follows ... 

Equation (7.15) shows that at small times the track develops as a collection of m isolated spheres, whereas at long times, V(f) increases linearly with t, which is characteristic of cylindrical diffusion. Using the definition of V(t) the solution of Eq. (7.13) may be given as... 

If the total current can be assumed to be limited by diffusion to the STM tip, Case III is similar to diffusion to a microdisk electrode (one electrode) thin-layer cell (63). Murray and coworkers (66) have shown that for long electrolysis times, diffusion to a planar microdisk electrode TLC can be treated as purely cylindrical diffusion, provided that the layer thickness is much smaller than the disk diameter (66). In contrast to the reversible case discussed above (Case I), the currents in this scenario should decrease gradually with time at a rate that is dependent on the tip radius and the thickness of the interelectrode gap. Thus, for sufficiently narrow tip/sample spacings, diffusion may be constrained sufficiently (ip decayed) at long electrolysis times to permit the imaging of surfaces with STM. 

Microcylindrical electrodes are easier to constract and maintain than microdisk electrodes [37]. Mass transport to a stationary cylinder in quiescent solution is governed by axisymmetrical cylindrical diffusion. For square-wave voltammetry the shape and position of the net current response are independent of the extent of cyhn-drical diffusion [38]. The experiments were performed with the ferri-ferrocyanide couple using a small platinum wire (25 pm in diameter and 0.5 -1.0 cm in length) as the working electrode [37]. 

Solution. Using Eq. 5.13, the quasi-steady-state current entering each dislocation by cylindrical diffusion is... 

A single DBP droplet is positioned in the vicinity of the microelectrode by the laser trapping technique, and the droplet-microelectrode (edge-to-edge) distance (L) is controlled arbitrarily in micrometer dimension. Knowing the oxidation potential of PPD in the water phase to be 30 mV, PPD is oxidized by a potential step method (100 mV) to induce the dye formation reaction. The anodic current relevant to oxidation of PPD reaches a steady-state value within a short electrolytic time (t) because of cylindrical diffusion of PPD to the microelectrode. The dye formation in the droplet can be easily confirmed by the color change from transparent to cyan or yellow. The dye formation reaction in a single microdroplet could be... 

An elegant solution to this problem has been proposed (Gough et al., 1985). By design, the mass transport of oxygen has been increased, relative to that of glucose, by cylindrical diffusion into the enzyme layer and the transport of glucose restricted to the linear diffusion through the distal end of the sensor (Fig. 7.14). In other words,... 

The porewall flux occurs as a boundary condition for the micro-scale as opposed to being a source term for the macroscale. For certain battery chemistries rectangular or cylindrical diffusion might be more relevant. 

Owen CS. Two dimensional diffusion theory Cylindrical diffusion model applied to fluorescence quenching. J. Chem. Phys. 1975 62 3204-3207. 

The electrode surface can be understood as an ensemble of independent cylindrical diffusion domains of radius Tq with the respective solid microparticle at the center (Figures 6.13 and 6.14). 

An example of cylindrical diffusion is diffusion toward a conducting wire. Solutions for cylindrical electrodes have been given by Fleischmaim et al. and Jacobsen and West. " The methods presented by both groups give the same results however, the latter is simpler. In this case the diffusion equation is similar to that for spherical diffusion [Eq. (70)]. The solution is shown here for the oxidized form only ... 

Figure 14. Faradaic (dashed line) and total (continuous line) impedance for a reversible reaction under conditions of semiinfinite cylindrical diffusion. Rs= 10 fl.

The function in Eq. (83) may be evaluated using Mathematica, Maple, or specific subroutines for complex modified Bessel functions. The corresponding complex plane plots are shown in Fig. 14. At low frequencies, cylindrical diffusion produces a constant imaginary impedance component. 

The considerable complexity of SECM theory is due to the combination of a cylindrical diffusion to the ultramicroelectrode (UME) tip and a thin-layer-type diffusion space. The time-dependent diffusion problem for a simple quasireversible reaction in cylindrical coordinates is as follows (2,3) ... 

Figure 8. Representative clinical applications using diffusing fiber light delivery, (a) multiple interstitial cylindrical fibers used to treat brain tumors (courtesy Dr T. Origitano, Chicago, USA), (b) cylindrical diffusing fibers being placed into the instrument channel of an endoscope for esophageal PDT (courtesy Dr. N. Marcon, Toronto, Canada), (c) three cylindrical diffusers in an umbrella configuration for endometrial irradiation (courtesy Drs B. Tromberg and Y. Tadir, Irvine, USA), (d) transcomeal irradiation of the retina from a laser diode coupled into a fundus camera (courtesy QLT Inc, Canada).

Figure 12. Treatment planning for PDT. This example is for interstitial treatment of prostate cancer, using multiple cylindrical diffusing fiber sources. The treatment volume is defined by trans-rectal ultrasound, (a) light fluence (rate) distribution for five fibers at specific locations and with equal power to each fiber, (b) corresponding threshold-dose boundary, (c) treatment boundary with the light fluences from each fiber adjusted to reduce the dose to the urethra. (Images courtesy CADMIT Inc, Canada).

There are mainly two types of Slurry seepage paths named cylindrical and surface-shaped, during the slurry diffusion process, cylindrical diffusion refers to slurry along the approximate space of the cylinder movement. The planar diffusion model of the slurry, is refers to the slurry along the approximate plane of the spatial movement... 

The first and simplest case corresponds to linear, spherical and cylindrical diffusion associated with the use of planar, (hemi)spherical and cylindrical electrodes. As can be inferred from Figure 1.2, in these diffusion fields all the points at a given distance from the electrode surface in the perpendicular coordinate x or r) are equivalent such that net flux of molecules... 

Figure 2.38 Experimental device chart. 1) thermometer 2) rotameter 3) tubular turbulent cylindrical (diffuser-confusor) device with jacket and 4) spectrophotometer Ind - indicator input point I - internal flow and II - circular flow...

A cylindrical immersion electrode (Pt wire of 0.5 mm diameter) was used as a working electrode, because assembling of flat microelectrode proved to be very hard in corrosive media such as fluoride melts. The upper limit of working transition times was then restricted by the effect of non-linear (cylindrical) diffusion. As described in Sect. 2.4.1, it was calculated to be 1.5 s. As for the lower limit, it was imposed by the effect of double layer charging. According to [23], it was evaluated... 

Mathematical Model Considering the Conditions of Spherical and Cylindrical Diffusion... 

Model of the Cylindrical Diffusion Around the Top Edge of a Surface Protrusion-Deposition to the Line... 

Unlike the conditions of spherical diffusion fulfilled around the tips of surface protrusions, the conditions of cylindrical diffusion are fulfilled around the top edges of the protrusions or crystals shown in Figs. 2.8b and 2.10a. 

Fig. 2.11 The cylindrical diffusion (a) geometry of the electrode and (b) schematic presentation of a protrusion growing inside the diffusion layer of the macroelectrode under the conditions for this type of diffusion (Reprinted from Ref. [34] with permission from Elsevier)...

Substitution of /l,c from Eq. (2.62) into Eq. (1.13), after rearranging, gives the Eq. (2.37) as in the case of spherical diffusion control, and the further derivation remains the same. Hence, the dendritic growth can be initiated at the same deposition overpotential under conditions of both spherical and cylindrical diffusion. 

The rate of growth of the protrusions under the conditions of spherical and cylindrical diffusion can be compared as follows. As shown earlier, the limiting diffusion current density at the tips of protmsions growing under the conditions of spherical diffusion (the needle-like dendrite or dendrite precursor), (L.tip inside the diffusion layer of the macroelectrode is given by ... 

Fig. 5 A cartoon illustrating development of a hemi-cylindrical diffusion profile near a band microelectrode in 3D, and a hemi-circular diffusion profile near a line electrode in 2D experiments. The two types of diffusion processes are analogous. Reduction of dimensionality converts the product of the electrode length (/) and a reagent bulk concentration (C ) appearing in the cylindrical diffusion equations into the surface concentration (T ) C / F (see text and Ref 46).

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