The Diffusion Domain Approach

1 The Diffusion Domain Approach The so-called diffusion domain approach was first proposed by Amatore et al. [36], and has proved highly useful in several theoretically based reports on this subject to model the diffusion current at those randomly distributed spherical micro- (or nano-) particle arrays [35, 37-39]. 

These zones are approximated as being cylindrical, with the particle situated at the symmetry axis. If a random spatial distribution of microparticles is assumed, the respective diffusion domains (cylinders) are of different sizes, with a probability distribution function as follows [41]  

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Davies and Compton simulated a regular array of microelectrodes, varying the diffusion domain approach, as illustrated in Figure 6.20. 

The electrode surface was assumed to contain N electroactive metal or metal oxide centers, respectively, which can be not only uniformly but also (mimicking more realistic experimental conditions) randomly distributed an example is the results of atomic force microscopy (AFM) studies on microparticle electrodes [53]. Here, the diffusion domain approach (as described in Section 6.3.2.2.1) has been employed that is, the electrode surface is assumed to be an arrangement of independent diffusion domains of radius Fq. If all particles are of the same radius, rj, but are distributed in a random manner, then a distribution of diffusion domains with different domain radii, ro, follows. The local position-dependent coverage is given by T. The electroactive microparticle flat disks of the radius rj are located in the center of the respective diffusion domain cylinder. The simulated (linear sweep voltammetric) reaction follows a one-electron transfer, and species B is stripped from the electrode surface into the solution, forming A, or ... 

Godino, N., Borrise, X., Munoz, F.X. et al (2009) Mass transport to nanoelectrode arrays and limitations of the diffusion domain approach theory and experiment. Journal of Physical Chemistry C,113, 11119-11125. 

Fig. 12.14 The diffusion domain approach to (a) a square lattice and (b) a hexagonal lattice of UMDEs of radius a and a centre-to-centre separation of /...

An evaluation of the retardation effects of surfactants on the steady velocity of a single drop (or bubble) under the influence of gravity has been made by Levich (L3) and extended recently by Newman (Nl). A further generalization to the domain of flow around an ensemble of many drops or bubbles in the presence of surfactants has been completed most recently by Waslo and Gal-Or (Wl). The terminal velocity of the ensemble is expressed in terms of the dispersed-phase holdup fraction and reduces to Levich s solution for a single particle when approaches zero. The basic theoretical principles governing these retardation effects will be demonstrated here for the case of a single drop or bubble. Thermodynamically, this is a case where coupling effects between the diffusion of surfactants (first-order tensorial transfer) and viscous flow (second-order tensorial transfer) takes place. Subject to the Curie principle, it demonstrates that this retardation effect occurs on a nonisotropic interface. Therefore, it is necessary to express the concentration of surfactants T, as it varies from point to point on the interface, in terms of the coordinates of the interface, i.e.,... 

The last issue that remains to be addressed is whether the MBL results are sensitive to the characteristic diffusion distance L one assumes to fix the outer boundary of the domain of analysis. In the calculations so far, we took the size L of the MBL domain to be equal to the size h - a of the uncracked ligament in the pipeline. To investigate the effect of the size L on the steady state concentration profiles, in particular within the fracture process zone, we performed additional transient hydrogen transport calculations using the MBL approach with L = 8(/i — a) = 60.96 mm under the same stress intensity factor Kf =34.12 MPa /m and normalized T-stress T /steady state distributions of the NILS concentration ahead of the crack tip are plotted in Fig. 8 for the two boundary conditions, i.e. / = 0 and C, =0 on the outer boundary. The concentration profiles for the zero flux boundary condition are identical for both domain sizes. For the zero concentration boundary condition CL = 0 on the outer boundary, although the concentration profiles for the two domain sizes L = h - a and L = 8(/i - a) differ substantially away from the crack tip. they are very close in the region near the crack tip, and notably their maxima differ by less than... 

Equation (9.15) was written for macro-pore diffusion. Recognize that the diffusion of sorbates in the zeoHte crystals has a similar or even identical form. The substitution of an appropriate diffusion model can be made at either the macropore, the micro-pore or at both scales. The analytical solutions that can be derived can become so complex that they yield Httle understanding of the underlying phenomena. In a seminal work that sought to bridge the gap between tractabUity and clarity, the work of Haynes and Sarma [10] stands out They took the approach of formulating the equations of continuity for the column, the macro-pores of the sorbent and the specific sorption sites in the sorbent. Each formulation was a pde with its appropriate initial and boundary conditions. They used the method of moments to derive the contributions of the three distinct mass transfer mechanisms to the overall mass transfer coefficient. The method of moments employs the solutions to all relevant pde s by use of a Laplace transform. While the solutions in Laplace domain are actually easy to obtain, those same solutions cannot be readily inverted to obtain a complete description of the system. The moments of the solutions in the Laplace domain can however be derived with relative ease. 

This is inherently impossible in the time-independent approach because the wavefunction contains the entire history of the wavepacket. The real understanding, however, is provided by classical mechanics. Plotting individual trajectories easily shows the type of internal motion leading to the recurrences which subsequently cause the diffuse structures in the energy domain. The next obvious step, finding the underlying periodic orbits, is rather straightforward. 

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