Diffusion fields

Microelectrodes with several geometries are reported in the literature, from spherical to disc to line electrodes each geometry has its own critical characteristic dimension and diffusion field in the steady state. The difhisional flux to a spherical microelectrode surface may be regarded as planar at short times, therefore displaying a transient behaviour, but spherical at long times, displaying a steady-state behaviour [28, 34] - If a... 

All points on the boundary of a growing nucleus are coupled by the diffusion field, and the nucleus can differ from its inital compact form as is shown in Fig. 5. This will be explained further in Secs. IVE and V [7,15,17,101]. 

In this section we discuss the basic mechanisms of pattern formation in growth processes under the influence of a diffusion field. For simphcity we consider the sohdification of a pure material from the undercooled melt, where the latent heat L is emitted from the solidification front. Since heat diffusion is a slow and rate-limiting process, we may assume that the interface kinetics is fast enough to achieve local equihbrium at the phase boundary. Strictly speaking, we assume an infinitely fast kinetic coefficient. 

The opposite mechanism, for the increase of cell spacings (or annihilation of an existing cell), could occur through the competition of neighboring cells for the diffusion field, such that finally one cell moves at a slightly lower speed than the neighboring cells and consequently will be suppressed relative to the position of the moving front. 

In our treatment of directional solidification above, only one diffusion field was treated explicitly, namely the compositional diffusion. If a simple material grows dendritically (thermal diffusion) one may worry about small amounts of impurities. This was reconsidered [132], confirming a qualitative previous result [133] that impurities may increase the dendritic growth rate. Recently some direct simulation results have been obtained with two coupled diffusion fields, one for heat and one for matter, but due to long computing times they are not yet in the state of standard applications [120,134]. 

The diffusion field just ahead of the solid front can be thought of as containing two ingredients a diffusion layer of thickness associated with global solute rejection, and modulations due to the periodic structure of the solid of extent A (A averaging approximation by Jackson and Hunt [137] seems justified. 

Y. Saito, M. Uwaha. Fluctuation and instability of steps in a diffusion field. Phys Rev B 49 10677, 1994. 

Johans et al. derived a model for diffusion-controlled electrodeposition at liquid-liquid interface taking into account the development of diffusion fields in both phases [91]. The current transients exhibited rising portions followed by planar diffusion-controlled decay. These features are very similar to those commonly observed in three-dimensional nucleation of metals onto solid electrodes [173-175]. The authors reduced aqueous ammonium tetrachloropalladate by butylferrocene in DCE. The experimental transients were in good agreement with the theoretical ones. The nucleation rate was considered to depend exponentially on the applied potential and a one-electron step was found to be rate determining. The results were taken to confirm the absence of preferential nucleation sites at the liquid-liquid interface. Other nucleation work at the liquid-liquid interface has described the formation of two-dimensional metallic films with rather interesting fractal shapes [176]. 

FIG. 1 (a) Schematic of the hemispherical diffusion-field established for the steady-state diffusion-... 

The effect on the current-time behavior of varying Kg while keeping the kinetics of the interfacial process high and nonlimiting is shown in Fig. 11, for a typical tip-interface distance log(T) = —0.5. The general trends in Fig. 11 can be explained as follows. At short times, the diffusion field at the UME tip is not of sufficient size to intercept the interface, and there is thus no perturbation of the interfacial equilibrium. In this time regime,... 

Most electrochemical studies at the micro-ITIES were focused on ion transfer processes. Simple ion transfer reactions at the micropipette are characterized by an asymmetrical diffusion field. The transfer of ions out of the pipette (ejection) is controlled by essentially linear diffusion inside its narrow shaft, whereas the transfer into the pipette (injection) produces a spherical diffusion field in the external solution. In contrast, the diffusion field at a microhole-supported ITIES is approximately symmetrical. Thus, the theoretical descriptions for these two types of micro-ITIES are somewhat different. 

The theory has been verified by voltammetric measurements using different hole diameters and by electrochemical simulations [13,15]. The plot of the half-wave potential versus log[(4d/7rr)-I-1] yielded a straight line with a slope of 60 mV (Fig. 3), but the experimental points deviated from the theory for small radii. Equations (3) to (5) show that the half-wave potential depends on the hole radius, the film thickness, the interface position within the hole, and the diffusion coefficient values. When d is rather large or the diffusion coefficient in the organic phase is very low, steady-state diffusion in the organic phase cannot be achieved because of the linear diffusion field within the microcylinder [Fig. 2(c)]. Although no analytical solution has been reported for non-steady-state IT across the microhole, the simulations reported in Ref. 13 showed that the diffusion field is asymmetrical, and concentration profiles are similar to those in micropipettes (see... 

The first density correction to the rate constant depends on the square root of the volume fraction and arises from the fact that the diffusion Green s function acts like a screened Coulomb potential coupling the diffusion fields around the catalytic spheres. 

The incorporation of discreet nucleation events into models for the current density has been reviewed by Scharifker et al. [111]. The current density is found by integrating the current over a large number of nucleation sites whose distribution and growth rates depend on the electrochemical potential field and the substrate properties. The process is non-local because the presence of one nucleus affects the controlling field and influences production or growth of other nuclei. It is deterministic because microscopic variables such as the density of nuclei and their rate of formation are incorporated as parameters rather than stochastic variables. Various approaches have been taken to determine the macroscopic current density to overlapping diffusion fields of distributed nuclei under potentiostatic control. 

The second factor, electronic conduction through an assembly of redox centers, involves electron hopping between adjacent sites. It is analyzed so as to highlight the characteristics of this transport in terms of equivalent diffusion. Field effects will also be addressed. Although bearing some... 

Some of these stability issues can be addressed by the use of protective barrier membranes, at the risk of aggravating another fundamental challenge reactant mass transfer. Typical reactants present in vivo are available only at low concentrations (glucose, 5 mM oxygen, 0.1 mM lactate, 1 mM). Maximum current density is therefore limited by the ability of such reactants to diffuse to and within bioelectrodes. In the case of glucose, flux to cylindrical electrodes embedded in the walls of blood vessels, where mass transfer is enhanced by blood flow of 1—10 cm/s, is expected to be 1—2 mA/cm. ° Mass-transfer rates are even lower in tissues, where such convection is absent. However, microscale electrodes with fiber or microdot geometries benefit from cylindrical or spherical diffusion fields and can achieve current densities up to 1 mA/cm at the expense of decreased electrode area. ... 

Equations (7.8) to (7.11) show that the exponent j8 in the i oc function may vary from 1 to 3. Therefore, all models presented here predict that the current density i increases for all time (Eig. 7.4). This is not what is observed experimentally and thus is unacceptable. A new model that takes into account the effect of overlap between diffusion fields around nuclei (Eig. 7.3) is required. This is the subject of the next section. 

Ignoring direct interactions, neighbouring steps do not influence each other if the dynamics is dominated by evaporation-recondensation or by step-edge diffusion. In either of these cases, the single step results derived in Section 2 (i.e. Eq. (22) and (26)) then hold. However, if the dynamics is mediated by terrace diffusion, neighbouring steps influence each other through the diffusion field on the terraces, and a coupled set of Langevin equations must be solved, as shown below (see also [13-17]). 

From a physical perspective, the m term is due to the diffusion field between the two steps, and the q term is due to atomic diffusion on the infinite terraces on the outer sides of the two steps. When the steps are close, and if the sticking coefficients are not too small (m a 5r/i)we can ignore the linear term in 1 1, in which case, the eigenvector matrix U is given by. 

At a spherical electrode, one must consider a spherical diffusion field as discussed in Sect. 2.4. Fick s second law is then written... 

PB, the value of P will depend on the precise geometrical arrangement of the component phases. The problem is not analytically tractable, however, except in simple or idealised cases. Consequently, a large number of formulae of varying degrees of approximation and different physical connotations have been developed in various fields. The relations best known in the diffusion field appear in reviews by Barrer 88) and Crank 89) but appreciation of their relative merits and physical significance is as yet very limited. Ideally, one would like to know which formula is appropriate for what type of composite membrane structure or, inversely, to deduce structural information about the membrane from measurements of P as a function of vA. 

To solve the surface smoothing problem in Fig. 3.7, Eq. 3.72 can be simplified further by setting dcA/dt equal to zero because the diffusion field is, to a good approximation, in a quasi-steady state, which then reduces the problem to solving the Laplace equation... 

Point sources of components 1 and 2, containing JVi and N2 atoms, respectively, are located at the origin r = 0 in a large piece of pure component 3. Solve for the resulting three-dimensional diffusion field, assuming that the diffusivities Dn, D21, and D22, are independent of concentration. 

The quantity ( D) is the average effective diffusivity, which describes the overall diffusion in the system. The diffusion in the system therefore behaves macroscopically as if bulk diffusion were occurring in a homogeneous material possessing a uniform diffusivity given by Eq. 9.4. The situation is illustrated schematically in Fig. 9.4a, and experimental data for diffusion of this type are shown in Fig. 9.5. This diffusion regime is called the multiple-boundary diffusion regime since the diffusion field... 

Figure 11.11 Vacancy diffusion fields in cylindrical cells (of radius R) around...

The vacancy diffusion field around the toroidal loop will be quite complex, but at distances from it greater than about 2RL, it will appear approximately as shown in Fig. 11.12a. A reasonably accurate solution to this complex diffusion problem may be obtained by noting that the total flux to the two flat surfaces in Fig. 11.12a will not differ greatly from the total flux that would diffuse to a spherical surface of radius d centered on the loop as illustrated in Fig. 11.126. Furthermore, when d > Rl, the diffusion field around such a source will quickly reach a quasi-steady state [20, 26], and therefore... 

As in Fig. 11.13, the loop can be represented by an array of point sources each of length R0. Using again the spherical-sink approximation of Fig. 11.126 and recalling that d Rl Ro, the quasi-steady-state solution of the diffusion equation in spherical coordinates for a point source at the origin shows that the vacancy diffusion field around each point source must be of the form... 

The solution of the diffusion equation for the quasi-steady state in cylindrical coordinates shows that each dislocation line source will have a vacancy concentration diffusion field around it of the form... 

At a distance from the two closely spaced dislocations appreciably greater than l, the concentration due to their superimposed diffusion fields will then be... 

The situation becomes quite different when the a//3 interface is no longer capable of maintaining the concentration of B atoms in its vicinity at the equilibrium value c 0. If the concentration there rises to the value ca0, the instantaneous quasi-steady-state current of atoms delivered to the particle by the diffusion field (obtained from Eq. 13.22) will be given by... 

The second term inside the brackets is seen to be an initial transient that falls off as t 1/2. It is associated with the establishment at early times of a steep concentration gradient in the diffusion field over a distance from the particle equal to about R. 

At distances from the boundary along x greater than about half the dislocation spacing (i.e., d/2), the contours will be unaffected by the fine structure of the boundary and will essentially be planes running parallel to the boundary. Nearer to the dislocation cores, the contours will be concentric cylinders. A reasonable approximation is then to represent the diffusion field as shown in... 

---