Interfacial barrier theory

According to the diffusion layer theory, for which the transport process is rate-limiting, kT kR, so that k = kT. According to the interfacial barrier theory, for which the surface reaction is rate-limiting, kR kT, so that it, = R-... 

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,... 

Figure 7.48 The appearance of indoxole in the oil phase as a function of time. Points, experimental data dashed line, the theoretical rate of transport based on diffusion theory solid line, the theoretical rate based on the interfacial barrier theory. Results for 1 % and 2 % polysorbate 80 are shown. The oil phase is isopropyl myristate. From Goldberg and Higuchi [227] with permission.

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 conclusions we may draw from these results are that, in general, interfacial turbulence will occur, and that it will increase the rate of mass transfer in these otherwise unstirred systems. Monolayers will prevent this turbulence, and theory and experiment are then in good agreement, in spite of spontaneously formed emulsion. There are no interfacial barriers greater than 1000 sec. cm. due to the presence of a mono-layer, though polymolecular films can set up quite considerable barriers. Usually there are no appreciable barriers due to re-solvation however, in the passage of Hg from the liquid metal into water, the change between the metallic state and the Hg2++ (aq) ion reduces the transfer rate by a factor of the order 1000. 

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 ... 

Albery et al. [16] have used Marcus Theory [16] interfacial processes to calculate that an interfacial rate constant of 2p,ms-1 constitutes a free energy barrier of 44 kJ mol-1. The slowest rate here (2,4-D) gives 49kJmol-1, and rates >100jxms-1 have barriers of 34kJmol-1, still rather larger than that of diffusion itself (20 kJ mol-1). However, the RDC method is not suitable for accurate measurement of k > 20 xm s-1. 

With cetyl alcohol, there is the complication that the polarity of the molecule may cause it to reside at the surface of the droplet, imparting additional colloidal stability. Here, the surfactant and costabilizer form an ordered structure at the monomer-water interface, which acts as a barrier to coalescence and mass transfer. Support for this theory lies in the method of preparation of the emulsion as well as experimental interfacial tension measurements [79]. It is well known that preparation of a stable emulsion with fatty alcohol costabilizers requires pre-emulsification of the surfactants within the aqueous phase prior to monomer addition. By mixing the fatty alcohol costabilizer in the water prior to monomer addition, it is believed that an ordered structure forms from the two surfactants. Upon addition of the monomer (oil) phase, the monomer diffuses through the aqueous phase to swell these ordered structures. For long chain alkanes that are strictly oil-soluble, homogenization of the oil phase is required to produce a stable emulsion. Although both costabilizers produce re-... 

It is in this light that one may judge the significance of the theory of electrified interfaces and thus electrochemistry. It is of interest to note how interfacial charge-transfer theories are based on a combination of the electric currents of Maxwell s theory and the quantum-mechanical tunneling of electrons through energy barriers. 

According to the classic nucleation theory, a free-energy barrier must be overcome to form a stable nucleus. The energy needed to form a crystal is proportional to the interfacial tension, y, and the surface area. However, once a nucleus is formed, there is a release of energy (latent heat) associated with the phase change. 

In conclusion, the antifluorescein system provides a reasonable model with which to evaluate interfacial interactions utilizing transition-state theory. Evaluations like those presented herein provide means to develop mechanistic models to describe interfacial interaction from an energetic barrier viewpoint. 

Before a protein molecule can adsorb and exert its influence at a phase boundary or take part in an interfacial reaction, it must arrive at the interface by a diffusion process. If we assume there is no barrier to adsorption other than diffusion, simple diffusion theory may be applied to predict the rate of adsorption. Under these conditions, after formation of a clean interface, all the molecules in the immediate vicinity will be rapidly adsorbed. The protein concentration in a sublayer, adjacent to the interface.and of several molecular diameters in thickness, will thus be depleted to zero. A diffusion process then proceeds from the bulk solution to the sublayer. The rate of adsorption, dn/dt, will be simply equal to the rate of this diffusion step given by classical diffusion theory (Crank, 1956) as... 

Protein monolayers can attain very high interfacial viscosities and elasticities. Moore and Eyring (1938) have developed a theory of interfacial viscosity, based on the theory of absolute reaction rates. In this theory, the flow of a molecule in a monolayer is treated as a movement of flow units, normally molecules, from one equilibrium position to another, passing over an intermediate activation energy barrier. The equation for the interfacial viscosity, tjs, which is derived is... 

To summarize, although rate constants (or, perhaps, apparent rate constants) for IT across the ITIES have been reported for more than 20 years, there is still controversy about the interpretation of this phenomenon, not least because the reported rate constants have increased over the years as experimental measurements became more and more sophisticated [127]. As noted above, a general problem has been that the characteristic timescales of the (apparent) kinetic process are often not markedly lower than the timescales of the experimental technique, a fact that has been remarked upon in the literature [128]. For example, in some of the recent data [94, 96] the time constants of the technique are frequently of the same order of magnitude as the timescales of the process they are purporting to measure. The question therefore becomes whether the rate constants reported for IT using nanopipettes [104, 107, 108] will increase in the future or whether these represent true values. Clearly at this point it is reasonable to ask whether theory predicts that a barrier to interfacial IT should exist and, if so, what the physical origin of such a barrier might be. 

The reliability of this theory has been the subject of a great amount of study. A careful review of the results indicates that departure from concentration equilibrium at the interface must be a rarity (Treybal, 1980). Consequently, in most situations the interfacial concentrations in Figure 3.3 are those corresponding to a point on the equilibrium-distribution curve. Notice that the concentration rise at the interface, from yA to xA., is not a barrier to diffusion in the gas-to-liquid direction. They are equilibrium concentrations and hence correspond to equal chemical potentials of substance A in both phases at the interface. 

The induction of the high supersaturation and the initiation of burst nucleation are explained by the theory of homogeneous nucleation. As discussed in Section 6.3.1, the interfacial free energy acts as an energy barrier for the nucleation reaction. Because the... 

Because (T in Equation (66) pertains to a hypothetical isotropic nucleus, it cannot be measured. Furthermore, true (anisotropic) solid-liquid interfacial energies are measured near the melting temperature [86], whereas what would be needed for an independent confirmation of the theory is Cf(T) in the supercooled region. Consequently, the theory of homogeneous nucleation, as it applies to supercooled liquids, has been used mainly to calculate effective interfacial tensions from measurements of nucleation rates [82,86]. In summary, the application of nucleation theory to supercooled liquids involves two major simplifications the replacement of the true, anisotropic embryo by an "equivalent" spherical object, and the ad-hoc introduction of a diffusion-like activation energy barrier to account for hindered molecular mobility in the dense supercooled liquid. It is therefore not surprising that the resulting theory has been mostly used descriptively rather than predict vely. 

Ciani, S., Laprade, R., Eisenman, G., Szabo, G., 1973. Theory for carrier-mediated zero-current conductance of bilayers extended to allow for nonequilibrium of interfacial reactions, spatially dependent mobilities and barrier shape. J. Membrane Biol. 11 255. 

In such an instrument, the dosing system and the light barrier are linked via an interface to the serial port of a standard computer, which controls the complete measurement sequence and then calculates the interfacial tension values according to the theory. Due to the force balance between the acceleration due to gravity and the interfacial tension, the volume of a detaching drop correlates directly with the interfacial tension y and the density difference Ap of the two adjacent phases, and is given by the following ... 

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