Diffusion between finite layers

Fig. 4. Schematic diagram of the layered model for a pore (47). The two nuclear spins diffuse in an infinite layer of finite thickness d between two flat surfaces. The M axes are fixed in the layer system. The L axes are fixed in the laboratory frame, with Bq oriented at the angle P from the normal axis n. The cylindrical polar relative coordinates p, (p, and z are based on the M axis. The smallest value of p corresponding to the distance of minimal approach between the two spin bearing molecules is 5.

How must the expressions derived in the sections above be modified to take into account the variation in rj and the finite distance over which it increases The answer is that rj — the viscosity within the double layer —must be written as a function of location. Our objective in discussing this variation is not to examine in detail the efforts that have been directed along these lines. Instead, it is to arrive at a better understanding of the relationship between f and the potential at the inner limit of the diffuse double layer and a better appreciation of the physical significance of the surface of shear. 

Occasionally (e.g., thin-layer electrochemistry, porous-bed electrodes, metal atoms dissolved in a mercury film), diffusion may be further confined by a second barrier. Figure 2.7 illustrates the case of restricted diffusion when the solution is confined between two parallel barrier plates. Once again, the folding technique quickly enables a prediction of the actual result. In this case, complete relaxation of the profile results in a uniform finite concentration across the slab of solution, in distinct contrast to the semi-infinite case. When the slab thickness t is given, the time for the average molecule to diffuse across the slab is calculable from the Einstein equation such that... 

In Fig. 2.10, the boundary between the enzyme-containing layer and the transducer has been considered as having either a zero or a finite flux of chemical species. In this respect, amperometric enzyme sensors, which have a finite flux boundary, stand apart from other types of chemical enzymatic sensors. Although the enzyme kinetics are described by the same Michaelis-Menten scheme and by the same set of partial differential equations, the boundary and the initial conditions are different if one or more of the participating species can cross the enzyme layer/transducer boundary. Otherwise, the general diffusion-reaction equations apply to every species in the same manner as discussed in Section 2.3.1. Many amperometric enzyme sensors in the past have been built by adding an enzyme layer to a macroelectrode. However, the microelectrode geometry is preferable because such biosensors reach steady-state operation. 

This section analyzes the response of a charge transfer process under conditions of finite linear diffusion which corresponds to a thin layer cell. This type of cell can be achieved by miniaturization process for obtaining a very high Area/Volume ratio, i.e., a maximum distance between the working and counter electrodes that is even smaller than the diffusion layer [31], In these cells it is easy to carry out a bulk electrolysis of the electroactive species even with no convection. Two different cell configurations can be described a cell with two working electrodes or a working electrode versus an electro-inactive wall separated at distance / (see Fig. 2.23). 

In regards to the stationarity of the reaction-diffusion process, it should be emphasised that the number of the B atoms diffusing across the ApBq layer is always equal to their number combined by the A surface into the ApBq compound at interface 1, if the growth of this layer is not accompanied by the formation of other compounds or solid solutions. The case under consideration is characterised by a kind of forced stationarity due to (z) the impossibility of any build-up of atoms at interfaces between the solids, (z z) the limited number of diffusion paths in the ApBq layer for the B atoms to travel from interface 2 to interface 1 and (Hi) the finite value of the reactivity of the A surface towards the B atoms. The stationarity is only... 

Compared to small molecules the description of convective diffusion of particles of finite size in a fluid near a solid boundary has to account for both the interaction forces between particles and collector (such as van der Waals and double-layer forces) and for the hydrodynamic interactions between particles and fluid. The effect of the London-van der Waals forces and doublelayer attractive forces is important if the range over which they act is comparable to the thickness over which the convective diffusion affects the transport of the particles. If, however, because of the competition between the double-layer repulsive forces and London attractive forces, a potential barrier is generated, then the effect of the interaction forces is important even when they act over distances much shorter than the thickness of the diffusion boundary layer. For... 

A lattice model for an electrolyte solution is proposed, which assumes that the hydrated ion occupies ti (i = 1, 2) sites on a water lattice. A lattice site is available to an ion i only if it is free (it is occupied by a water molecule, which does not hydrate an ion) and has also at least (i, - 1) first-neighbors free. The model accounts for the correlations between the probabilities of occupancy of adjacent sites and is used to calculate the excluded volume (lattice site exclusion) effect on the double layer interactions. It is shown that at high surface potentials the thickness of the double layer generated near a charged surface is increased, when compared to that predicted by the Poisson-Boltzmann treatment. However, at low surface potentials, the diffuse double layer can be slightly compressed, if the hydrated co-ions are larger than the hydrated counterions. The finite sizes of the ions can lead to either an increase or even a small decrease of the double layer repulsion. The effect can be strongly dependent on the hydration numbers of the two species of ions. 

Here D is the diffusion coefficient, t is the time, t is a dummy integration variable. Using Equation (8), respective T(t) dependencies can be obtained, while the Equations (l)-(7) serve as boundary condition for the diffusion model. This set of equations yield a quasi-equilibrium diffusion model which means that at a given surface pressure the composition of the surface layer under dynamic conditions is equal to that in the equilibrium. Another regime of adsorption kinetics, called kinetic model, can also be described by assuming compositions of the adsorption layer that can differ from the equilibrium state. The deviation of the adsorption layer from the equilibrium composition is the result of the finite rate of the transition process between the adsorption states. In case of two adsorption states we have6... 

---