The Interfacial Barrier Model

In recent years two novel models [122,123] have appeared that were proposed to describe the heterogeneous features of drug dissolution. They are considered here as continuous (in well-stirred media) or discrete (in understirred media) reaction-limited dissolution models. Their derivation and relevance is discussed below. 

---

Fig. 15 Two of the simplest theories for the dissolution of solids (A) the interfacial barrier model, and (B) the diffusion layer model, in the simple form of Nemst [105] and Brunner [106] (dashed trace) and in the more exact form of Levich [104] (solid trace). c is the concentration of the dissolving solid, cs is the solubility, cb is the concentration in the bulk solution, and x is the distance from the solid-liquid interface of thickness h or 8, depending on how it is defined. (Reproduced with permission of the copyright owner, John Wiley and Sons, Inc., from Ref. 1, p. 478.)...

The interfacial barrier theory is illustrated in Fig. 15A. Since transport does not control the dissolution rate, the solute concentration falls precipitously from the surface value, cs, to the bulk value, cb, over an infinitesimal distance. The interfacial barrier model is probably applicable when the dissolution rate is limited by a condensed film absorbed at the solid-liquid interface this gives rise to a high activation energy barrier to the surface reaction, so that kR kj. Reaction-controlled dissolution is somewhat rare for organic compounds. Examples include the dissolution of gallstones, which consist mostly of cholesterol,... 

Two of the simplest theories to explain the dissolution rate of solutes are the interfacial barrier model and the diffusion-layer model (Figures 17.1 and 17.2). Both of these theories make the following two assumptions ... 

Figure 5.2 Schematic representation of the dissolution mechanisms according to (A) the diffusion layer model, and (B) the interfacial barrier model.

In the interfacial barrier model of dissolution it is assumed that the reaction at the solid-liquid interface is not rapid due to the high free energy of activation requirement and therefore the reaction becomes the rate-limiting step for the dissolution process (Figure 5.1), thus, drug dissolution is considered as a reaction-limited process for the interfacial barrier model. Although the diffusion layer model enjoys widespread acceptance since it provides a rather simplistic interpretation of dissolution with a well-defined mathematical description, the interfacial barrier model is not widely used because of the lack of a physically-based mathematical description. 

Stochastic variation may be introduced in other models as well. In this context, Lansky and Weiss [130] have also considered random variation for the parameter k of the interfacial barrier model (5.20). 

Studies based on the Frenkel-Kontorova model reveal that static friction depends on the strength of interactions and structural commensurability between the surfaces in contact. For surfaces in incommensurate contact, there is a critical strength, b, below which the depinning force becomes zero and static friction disappears, i.e., the chain starts to slide if an infinitely small force F is applied (cf. Section 3). This is understandable from the energetic point of view that the interfacial atoms in an incommensurate system can hardly settle in any potential minimum, or the energy barrier, which prevents the object from moving, can be almost zero. 

It has been proposed recently [28] that static friction may result from the molecules of a third medium, such as adsorbed monolayers or liquid lubricant confined between the surfaces. The confined molecules can easily adjust or rearrange themselves to form localized structures that are conformal to both adjacent surfaces, so that they stay at the energy minimum. A finite lateral force is required to initiate motion because the energy barrier created by the substrate-medium system has to be overcome, which gives rise to a static friction depending on the interfacial substances. The model is consistent with the results of computer simulations [29], meanwhile it successfully explains the sensitivity of friction to surface film or contamination. 

The promise of photoelectrochemical devices of both the photovoltaic and chemical producing variety has been discussed and reviewed extensively.Cl,, 3,4) The criteria that these cells must meet with respect to stability, band gap and flatband potential have been modeled effectively and in a systematic fashion. However, it is becomirg clear that though such models accurately describe the general features of the device, as in the case of solid state Schottky barrier solar cells, the detailed nature of the interfacial properties can play an overriding role in determining the device properties. Some of these interface properties and processes and their potential deleterious or beneficial effects on electrode performance will be discussed. 

It is worth repeating the above. T o date, the effects of the presence of interfacial barriers or interface regions have not been explicitly considered in models of charge injection at such interfaces. The... 

It has been shown that the magnitude of the rate constant for crossing the octanol-water interface makes the energy barrier significantly larger than the diffusional barrier. It has also been shown that for compounds with log Pow less than =1.2, the overall rates are faster and the interfacial kinetics term more important. However, detailed development of a model would be needed to understand what the relative importance of diffusion and interfacial terms (such as cuticle or membrane permeation) are in vivo. No clear dependence of interfacial rate constants on log Pow was seen, but the initial emphasis of such a study should be on the intermediate... 

Willig and co-workers used near-infrared spectroscopy to measure excited-state interfacial electron transfer rates after pulsed light excitation of cis-Ru(dcb)2(NCS)2-Ti02 in vacuum from 20 to 295 K [208]. They reported that excited-state electron injection occurred in less than 25 fs, prior to the redistribution of the excited-state vibrational energy, and that the classical Gerischer model for electron injection was inappropriate for this process. They concluded that the injection reaction is controlled by the electronic tunneling barrier and by the escape of the initially prepared wave packet describing the hot electron from the reaction distance of the oxidized dye molecule. It appears that some sensitizer decomposition occurred in these studies as the transient spectrum was reported to be similar to that of the thermal oxidation product of m-Ru(dcb)2(NCS)2. 

A Gf for an anodic reaction is decreased and A Gt increased by applying a positive voltage with respect to the equilibrium condition (Fig. 7.2b). Accordingto this model, the potential dependence of the interfacial current is caused by the potential dependence of the rate constants. As will be shown later, in this aspect metal electrodes behave completely different from semiconductor electrodes. It also becomes clear from Figs. 7.1 and 7.2 that a variation of the concentration ratio leads to the same effect as that caused by an application of a voltage to the cell. This is reasonable because an increase of tVej speeds up the reaction rate, i.e. the barrier height must be smaller as shown in Fig. 7.1a. 

It has been proved that the elimination of lower energetic carriers brings the enhancement of Z and the optimal height of potential barrier within the two band model is given analytically. Such elimination (confinement) of minority carriers has already studied in GaAs/AlAs systems and it has been known that the use of graded alloy composition at interfaces of hetero-junction are important to confine the minority carriers effectively[8] due to decrese in the defect density near interfacial region. 

---