Diffusion across a barrier layer

The relative displacement rates of the interfaces AIAB and ABIB in any particular system will, of course, depend on the relative migration velocities of all mobile participants across the barrier layer and reaction will continue while appropriate reactant constituents remain available. Such migrant entities travel by the most efficient route and therefore overall rates of such reactions are frequently sensitive to the concentration of imperfections in the product crystal lattice [1171]. One possible 

A detailed account of transport phenomena in crystals is outside the scope of the present review, though it is relevant to point out that factors which determine the rate at which reactants penetrate a barrier layer include the numbers, distributions and mobilities of vacancies. Oleinikov et al. [1173] conclude that Arrhenius parameters are devoid of any physical significance if due allowance is not made for imperfection concentration, which may vary with temperature (and a [77]). 

In reactions of the type discussed, the more bulky anions are often regarded as an immobile framework within the solid through which the cations diffuse. Some experiments have been directed towards investigation of the significance of anion migration but results are restricted in scope and more work is clearly desirable. It is possible that anion migrations may occur along dislocation lines, at surfaces, or even by desorption-adsorption, and may be sensitive to the presence of specific impurities. 

In a study of the spinel formation reaction CoO + A1203 = C0AI2O4 

The kinetic expressions applicable to diffusion-controlled reactions have been discussed in Chap. 3, Sect. 3.3. Mass transport in ionic solids has been reviewed by Steele and Dudley [1182], 

---

The properties of barrier layers, oxides in particular, and the kinetic characteristics of diffusion-controlled reactions have been extensively investigated, notably in the field of metal oxidation [31,38]. The concepts developed in these studies are undoubtedly capable of modification and application to kinetic studies of reactions between solids where the rate is determined by reactant diffusion across a barrier layer. 

If diffusion across a barrier layer is rapid compared with the rate of reaction at the advancing interface, then the overall rate is determined by the interface step (Sect. 3.1). 

This account of the kinetics of reactions between (inorganic) solids commences with a consideration of the reactant mixture (Sect. 1), since composition, particle sizes, method of mixing and other pretreatments exert important influences on rate characteristics. Some comments on experimental methods are included here. Section 2 is concerned with reaction mechanisms formulated to account for observed behaviour, including references to rate processes which involve diffusion across a barrier layer. This section also includes a consideration of the application of mechanistic criteria to the classification of the kinetic characteristics of solid-solid reactions. Section 3 surveys rate processes identified as the decomposition of a solid catalyzed by a solid. Section 4 reviews other types of solid + solid reactions, which may be conveniently subdivided further into the classes... 

While it is inherently probable that product formation will be most readily initiated at sites of effective contact between reactants (A IB), it is improbable that this process alone is capable of permitting continued product formation at low temperature for two related reasons. Firstly (as discussed in detail in Sect. 2.1.1) the area available for chemical contact in a mixture of particles is a very small fraction of the total surface (and, indeed, this total surface constitutes only a small proportion of the reactant present). Secondly, bulk diffusion across a barrier layer is usually an activated process, so that interposition of product between the points of initial contact reduces the ease, and therefore the rate, of interaction. On completion of the first step in the reaction, the restricted zones of direct contact have undergone chemical modification and the continuation of reaction necessitates a transport process to maintain the migration of material from one solid to a reactive surface of the other. On increasing the temperature, surface migration usually becomes appreciable at temperatures significantly below those required for the onset of bulk diffusion within a product phase. It is to be expected that components of the less refractory constituent will migrate onto the surfaces of the other solid present. These ions are chemisorbed as the first step in product formation and, in a subsequent process, penetrate the outer layers of the... 

Further investigations of spinel formation reactions are to be found in the literature [1], but the above representative selection illustrates a number of typical features of these rate processes. Following migration of cations from one constituent onto the surfaces of the other, the process is limited by the rate of diffusion across a barrier layer. While obedience to a particular kinetic expression is sometimes reported, the data available are not always sufficiently precise to enable the fit found to be positively... 

The method makes use of the tendency of a metal foil to bend when it is oxidised unilaterally, i.e. on one side only, and has been developed from the method used by Stoney to measure stresses in electrodeposits, as long ago as 1909. There are important problems in the translation of the technique to oxidation studies. Firstly, it is difficult to completely arrest oxidation on the Inert side of the testpiece. The problems of diffusion across a barrier layer of electrodeposited or vapour-deposited metal for a long time restricted its use to low temperatures, e.g. Ta at 350-550 C Nb at 425 C" and Cu at 200-400 C, but recently Pawel and Cathcart have reported the use of Al-3Au evaporated layers which, on uranium alloys, will enable temperatures of up to 800°C to be used" . Secondly, there is an interpretive difficulty since although there are two stress systems, acting at right angles in the plane of the oxide, a strip (or even a disc) generally bends in a plane perpendicular to only one of them . 

The successful application of in vitro models of intestinal drug absorption depends on the ability of the in vitro model to mimic the relevant characteristics of the in vivo biological barrier. Most compounds are absorbed by passive transcellular diffusion. To undergo tran-scellular transport a molecule must cross the lipid bilayer of the apical and basolateral cell membranes. In recent years, there has been a widespread acceptance of a technique, artificial membrane permeation assay (PAMPA), to estimate intestinal permeability.117118 The principle of the PAMPA is that, diffusion across a lipid layer, mimics transepithelial permeation. Experiments are conducted by applying a drug solution on top of a lipid layer covering a filter that separates top (donor) and bottom (receiver) chambers. The rate of drug appearance in the bottom wells should reflect the diffusion across the lipid layer, and by extrapolation, across the epithelial cell layer. 

Radiochemical studies indicate that the pore base is the actual site of formation of aluminium oxide, presumably by transport of aluminium ions across the barrier-layer, although transport of oxygen ions in the opposite direction has been postulated by some authorities. The downward extension of the pore takes place by chemical solution, which may be enhanced by the heating effect of the current and the greater solution rate of the freshly formed oxide, but will also be limited by diffusion. It has been shown that the freshly formed oxide, y -AljOj, is amorphous and becomes slowly converted into a more nearly crystalline modifipation of y-AljO . 

In reviewing reported values of E for calcite decompositions, Beruto and Searcy [121] find that most are close to the dissociation enthalpy. They suggest, as a possible explanation, that if product gas removal is not rapid and complete, readsorption of C02 on CaO may establish dissociation equilibria within the pores and channels of the layer of residual phase. The rate of gas diffusion across this barrier is modified accordingly and is not characteristic of the dissociation step at the interface. 

Fig. 20. Schematic representation of the solid + solid reaction A + B -> AB in which constituents of the relatively mobile reactant (A) are transported to the outer surfaces of the product phase (AB) and rate is controlled by diffusion of constituents of A and/ or B across the barrier layer AB.

Since, in both these reactions (i.e. KI or Rbl and Agl), product formation occurs on both sides of the original contact interface, it is believed that there is migration of both alkali metal and silver ions across the barrier layer. Alkali metal movement is identified as rate limiting and the relatively slower reaction of the rubidium salt is ascribed to the larger size and correspondingly slower movement of Rb+. The measured values of E are not those for cation diffusion alone, but include a contribution from... 

The meaning of this term is shown by Figure 2.5 and it is essentially the time required to attain steady state flux across a barrier. When the resistance in the boundary layer is negligible, the lag-time equation provides a convenient means of calculating membrane or polymer-diffusion coefficients. 

Although forming a protective barrier on the skin is important, some cosmetic products also contain physiologically active ingredients that will improve skin conditions only if they penetrate the skin [431], The active substance(s) can be encapsulated in the internal aqueous phase or the internal oleic phase depending on the type of emulsion and whether the active ingredient is lipophilic or hydrophilic. If the protective film is not to be broken then the active substance has to diffuse across the oleic layer that has been deposited on the skin surface [910], This diffusion will be approximately described by Fick s law (Section 5.5), but is complicated by the fact... 

From Figure 1.3 it can be seen that in order to reach the underlying blood capillaries to be absorbed, the drug must pass through at least two epithelial membrane barriers (the apical and basolateral epithelial cell membranes) and also the endothelial membrane of the capillaries. In some cases, for example in stratified epithelia such as that found in the skin and buccal mucosa, the epithelial barrier comprises a number of cell layers rather than a single epithelial cell. Thus the effective barrier to drag absorption is not diffusion across a single membrane as described above, but diffusion across the entire epithelial and endothelial barrier, which may comprise several membranes and cells in series. 

The resistance to diffusion of a molecular species across a barrier equals the reciprocal of its permeability coefficient (Chapter 1, Section 1.4B). In this regard, we will let f COi be the permeability coefficient for CO2 diffusion across barrier j. To express the resistance of a particular mesophyll or chlo-roplast component on a leaf area basis, we must also incorporate Am sIA to allow for the actual area available for diffusion—the large internal leaf area acts like more pathways in parallel and thus reduces the effective resistance (Fig. 8-4). Because the area of the plasma membrane is about the same as that of the cell wall, and the chloroplasts generally occupy a single layer around the periphery of the cytosol (Figs. 1-1 and 8-11), the factor AmesIA applies to all of the diffusion steps of CO2 in mesophyll cells (all five individual resistances in Eq. 8.21). In other words, we are imagining for simplicity that the cell wall, the plasma membrane, the cytosol, and the chloroplasts are all in layers having essentially equal areas (Fig. 8-11). Thus, the resistance of any of the mesophyll or chloroplast components for CO2 diffusion,, is reduced from 1 /P co, by the reciprocal of the same factor, Ames/A ... 

Diffusion processes participate in, and may even control the rates of many types of chemical reactions involving sohds. For example, (i) in decompositions of solids the rate may be controlled by the diffusive escape of a volatile product, and (ii) in reactions between solids a solid product may be formed between the reactants [5], so that progress is maintained by transport across the barrier layer (Figure 3.3.). 

A drug is absorbed through diffusion across a series of separate barriers where the single layer of epithelial cells is the most significant barrier to absorption. Many in vitro methods have been developed for the study of this phenomenon. These methods include small animal gut studies, cell culture (i.e., Caco-2 cell culture model), octanol-water partition coefficients, measures of hydrogen bonding and desolvation energies, immobilized artificial membranes, and retention time on reversed-phase HPLC columns. 

The characteristic feature of solid—solid reactions which controls, to some extent, the methods which can be applied to the investigation of their kinetics, is that the continuation of product formation requires the transportation of one or both reactants to a zone of interaction, perhaps through a coherent barrier layer of the product phase or as a monomolec-ular layer across surfaces. Since diffusion at phase boundaries may occur at temperatures appreciably below those required for bulk diffusion, the initial step in product formation may be rapidly completed on the attainment of reaction temperature. In such systems, there is no initial delay during nucleation and the initial processes, perhaps involving monomolec-ular films, are not readily identified. The subsequent growth of the product phase, the main reaction, is thereafter controlled by the diffusion of one or more species through the barrier layer. Microscopic observation is of little value where the phases present cannot be unambiguously identified and X-ray diffraction techniques are more fruitful. More recently, the considerable potential of electron microprobe analyses has been developed and exploited. 

The maintenance of product formation, after loss of direct contact between reactants by the interposition of a layer of product, requires the mobility of at least one component and rates are often controlled by diffusion of one or more reactant across the barrier constituted by the product layer. Reaction rates of such processes are characteristically strongly deceleratory since nucleation is effectively instantaneous and the rate of product formation is determined by bulk diffusion from one interface to another across a product zone of progressively increasing thickness. Rate measurements can be simplified by preparation of the reactant in a controlled geometric shape, such as pressing together flat discs at a common planar surface that then constitutes the initial reaction interface. Control by diffusion in one dimension results in obedience to the... 

The sole purpose of the filter support and any applied extracellular matrix is simply to provide a surface for cell attachment and thus to provide mechanical support to the monolayer. However, the filter and matrix also can act as serial barriers to solute movement after diffusion through the cell monolayer. The important variables are the chemical composition of the filter, porosity, pore size, and overall thickness. In some cases, pore tortuosity also can be important. It is desired that the filter, with or without an added matrix, provide a favorable surface to which the cells can attach. However, in some cases these properties can also result in an attractive surface for nonspecific adsorption of the transported solute. In these instances, the appearance of the solute in the receiver compartment of the diffusion cell will not be a true reflection of its movement across the mono-layer. Such problems must be examined on a case-by-case basis. 

---