The diffuse field

Upon solving the transfer equation in the context of various physical problems it is evident that the intensity of the diffuse radiation field is the quantity most easily handled in the equation, because the boundary conditions are much simpler to impose than for the case in which the intensity of the total radiation field is the dependent variable. In particular it is convenient to distinguish between the reduced incident radiation field from the Sun, which penetrates to the level tv without suffering any scattering or absorption processes, and the diffuse field, which has arisen as a result of one or more scattering and emission processes. 

In order to separate the diffuse field from the directly transmitted radiation we consider a collimated beam of radiation (sunlight is a fair approximation) of flux jtFq crossing a unit surface area normal to the beam. The magnitude of the flux in the downward direction crossing a unit area contained in a plane at the top of the atmosphere is fio Fq, where /aq is the cosine of the zenith angle of the point source, and this is [see Eq. (1.8.4)], 

The total intensity I(Vv, ii, / ) associated with Eq. (2.1.40) is the sum of the intensity Iv)(Tv, IX, (p) arising from the diffuse radiation field and the intensity directly transmitted from the point source to the level Tv. By analogy with Eq. (2.1.42) and the related discussion this latter intensity may be expressed by 

Upon dropping the subscripts D and v, Eq. (2.1.44) for the diffuse radiation field becomes 

Equation (2.1.47) is the basic equation of transfer considered in this book. Solutions to Eq. (2.1.47) are sought in the context of specific problems as they appear in the course of investigation. In the remainder of this chapter we first derive formal solutions, and then examine explicit solutions that are possible, either because certain approximations are invoked or because at some point numerical procedures are introduced. 

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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]. 

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. 

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. 

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. 

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. 

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... 

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... 

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... 

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... 

Assume that the diffusion field is in a quasi-steady state and that local equilibrium is maintained at the surface and in the volume at a long distance from the surface, where yv = 0 and yA has the value characteristic of a flat surface. 

For electrodes with total dimensions smaller than VDt, often called micro-voltammetric electrodes (see Chapter 12), roughness is less of an issue compared to mass transport. Such electrodes are typically less than 10 pm in diameter and exhibit radial rather than planar diffusion. In most cases (the exception being fast voltammetry) the diffusion field is thicker than both the electrode diameter and its surface roughness, and the diffusion-limited response is unperturbed by roughness. 

Overlapping of the diffusion fields of two particles slows the growth. From W. F. Hosford, Physical Metallurgy (Boca Raton CRC Press, 2004), p. 198, figure 10.18. 

It is worth highlighting that, when different diffusion coefficients are considered, the half-wave potential depends on the characteristics of the diffusive field (geometry and size of the electrode), as indicated in Sect. 2.5.1. The variation of the... 

In the preceding chapter, single pulse voltammetry and chronoamperometry were applied to the study of reversible electrode reactions of species in solution. Under these conditions, the surface concentrations fulfill Nemst equation and are independent of the duration of the experiment, regardless of the diffusion field... 

Equations (3.105)-(3.107) point out the existence of three different polarization causes. So, 7km is a kinetically controlled current which is independent of the diffusion coefficient and of the geometry of the diffusion field, i.e., it is a pure kinetic current. The other two currents have a diffusive character, and, therefore, depend on the geometry of the diffusion field. I((((s corresponds to the maximum current achieved for very negative potentials and I N is a current controlled by diffusion and by the applied potential which has no physical meaning since it exceeds the limiting diffusion current 7 ss when the applied potential is lower than the formal potential (E < Ef"). This behavior is indicated by Oldham in the case of spherical microelectrodes [15, 20, 25]. 

When any of the electron transfer is slow the situation is much more complex and the geometry of the diffusive field plays a relevant role in the electrochemical response [4, 10]. 

When the electrochemical properties of some materials are analyzed, the timescale of the phenomena involved requires the use of ultrafast voltammetry. Microelectrodes play an essential role for recording voltammograms at scan rates of megavolts-per-seconds, reaching nanoseconds timescales for which the perturbation is short enough, so it propagates only over a very small zone close to the electrode and the diffusion field can be considered almost planar. In these conditions, the current and the interfacial capacitance are proportional to the electrode area, whereas the ohmic drop and the cell time constant decrease linearly with the electrode characteristic dimension. For Cyclic Voltammetry, these can be written in terms of the dimensionless parameters yu and 6 given by... 

As a result, a stationary voltammogram cannot be expected under these conditions since it shows a behavior similar to that of a macrointerface with respect to the egress of the ion, and features of radial diffusion for the ingress process, reaching a time-independent response [73, 74]. Both are consequences of the markedly different diffusion fields inside and outside the capillaries which give rise to very different concentration profiles (see Fig. 5.21). A similar voltammetric behavior has been reported for electron transfer processes at electrode I solution interfaces where the diffusion fields of the reactant and product species differ greatly. 

In summary, although the construction of micro-ITIES is, in general, simpler than that of microelectrodes, their mathematical treatment is always more complicated for two reasons. First, in micro-ITIES the participating species always move from one phase to the other, while in microelecrodes they remain in the same phase. This leads to complications because in the case of micro-ITIES the diffusion coefficients in both phases are different, which complicates the solution when nonlinear diffusion is considered. Second, the diffusion fields of a microelectrode are identical for oxidized and reduced species, while in micro-ITIES the diffusion fields for the ions in the aqueous and organic phases are not usually symmetrical. Moreover, as a stationary response requires fDt / o (where D is the diffusion coefficient, r0 is the critical dimension of the microinterface, and t is the experiment time), even in L/L interfaces with symmetrical diffusion field it may occur that the stationary state has been reached in one phase (aqueous) and not in the other (organic) at a given time, so a transient behavior must be considered. 

The effects of the catalytic reaction on the CV curve are related to the value of dimensionless parameter A in whose expressions appear variables related to the chemical reaction and also to the geometry of the diffusion field. For small values of A, the surface concentration of species C is scarcely affected by the catalysis for any value of the electrode radius, such that r)7,> —> c c and the current becomes identical to that corresponding to a pseudo-first-order catalytic mechanism (see Eq. (6.203)). In contrast, for high values of A and f —> 1 (cathodic limit), the rate-determining step of the process is the mass transport. In this case, the catalytic limiting current coincides with that obtained for a simple charge transfer process. 

The advantages derived from the use of microscopic liquid-liquid interfaces have been highlighted in Sect. 5.5.3, and different approaches to support such small liquidlliquid interfaces in pores, pipettes, and capillaries have been addressed. The theoretical treatment of ion transfer through these interfaces needs to consider the asymmetry of the diffusion fields inside and outside the pore or pipette (i.e., diffusion can be approximated as linear in the inner phase, whereas radial diffusion is significant in the outer phase, especially for small sizes) [36, 40, 42-44]. 

Provided that the diffusion field Ax is approximated by r in Fig. 10(b) and 1-dimensional growth is assumed for simplicity of the argument, the following relationship is expected ... 

We proceed from (R,Z) space, where it is easy to define an envelope that encloses the diffusion field to a good approximation. This is the length A, defined, for times T other than very large (see below), by... 

When the dimensions of the crystallite are small compared to those of the diffusion field, then the crystallite appears as a point source. As the diffusion fields expand, they coalesce (as the current-density increases)... 

The fraction of the surface that is active is given by (d/L), where d is the diameter of each electrode and L is the distance between their centers. Designing such an ensemble of ultramicro electrodes, one must compromise between the desire to make the ratio dlL as small as possible (to decrease the overlap between the diffusion fields of the individual electrodes), and the desire to make d L as large as possible (to increase the total active area). Values of d L in the range of 0.03-0.1, corresponding to 0.1-1% of active area, seem to be a reasonable choice, as we shall see. 

Fig. 20L Development of the diffusion field near the surface of an ensemble of micro electrodes, (a) planar diffusion-, (b) spherical diffusion with no overlap-, (c) spherical diffusion with substantial overlap-, (d) total overlap, equivalent to planar diffusion to the whole surface.

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