Dimensionless reduced surface parameter

The concentration profile of fixed oxidized and reduced sites within the film depends on the dimensionless parameter Dcjr/d2, where r is the experimental timescale, i.e. RT/Fv in cyclic voltammetry, and d is the polymer layer thickness. When Dcix/d2 1, all electroactive sites within the film are in equilibrium with the electrode potential, and the surface-type behavior described previously is observed. In contrast, Dcjx/d2 <3C 1 when the oxidizing scan direction is switched before the reduced sites at the film s outer boundary are completely oxidized. The wave will exhibit distinctive diffusional tailing where these conditions prevail. At intermediate values of Dcjr/d2, an intermediate ip versus v dependence occurs, and a less pronounced diffusional tail appears. 

On a RDE, in the absence of a surface layer, the EHD impedance is a function of a single dimensionless frequency, pSc1/3. This means that if the viscosity of the medium directly above the surface of the electrode and the diffusion coefficient of the species of interest are independent of position away from the electrode, then the EHD impedance measured at different rotation frequencies reduces to a common curve when plotted as a function of p. In other words, there is a characteristic dimensionless diffusional relaxation time for the system, pD, strictly (pSc1/3)D, which is independent of the disc rotation frequency. However, if v or D vary with position (for example, as a consequence of the formation of a viscous boundary layer or the presence of a surface film), then, except under particular circumstances described below, reduction of the measured parameters to a common curve is not possible. Under these conditions pD is dependent upon the disc rotation frequency. The variation of the EHD impedance with as a function of p is therefore the diagnostic for... 

Here H is the surface pressure, F is the adsorption, c is the concentration, b is the adsorption equilibrium constant, co is the area per molecule in the surface layer, and F and G are some functions dependent on F, co and other model parameters denoted here as a, 02,. .. a . For the simplest models considered, namely Langmuir and Frumkin models, co is the model parameter, while for more advanced models this is a property which is defined via model equations. It is essential that in each case F enters the equations via the surface coverage coefficient, 8 = Fco. Also, for each model there exists a parameter, say co, which has the dimension of the area per molecule, and, being introduced into Eq. (7.1), enables one to reduce this equation to a dimensionless form... 

To obtain Eqs. 5-10, it was assumed that the concentration of solute within the adsorption boundary layer is related to the solute-surface interaction energy by a Boltzmann distribution. The essence of the thin-layer polarization approach is that a thin diffuse layer can still transport a significant amoimt of solute molecules so as to affect the solute transport outside the diffuse layer. For a strongly adsorbing solute (e.g., a surfactant), the dimensionless relaxation parameter fila (or Kid) can be much greater than imity. If all the adsorbed solute were stuck to the surface of the particle (the diffuseness of the adsorption layer disappears), then L = 0 and there would be no diffusiophoretic migration of the particle. In the limit of [l/a 0 (very weak adsorption), the polarization of the diffuse solute in the interfacial layer vanishes and Eq. 5 reduces to Eq. 1. 

I is the groove depth (normal to the surface), and Z ei is the double-layer impedance per unit of the true surface area. Equation (9.14) reduces to the impedance of a perfectly flat surface for ji = 90° and to the impedance of cylindrical porous electrode for p = 0°. Gunning [414] obtained an exact solution of the de Levie grooved surface not restricted to a pseudo-one-dimensional problem in the form of an infinite series. Comparison with de Levie s equation (9.14) shows that the deviations arise at higher frequencies or, more precisely, at high values of the dimensionless parameter Q. = coC ialp, where a is half of the distance of the groove opening, a = Itan p (Fig. 9.8). 

Veriable surface excess concentration. In general case F const of a falling film of weak volatile surfactant solution in which the surfactant mass transfer is governed by the diffusion, evaporation and adsorption-desorption processes in the near-surface layer the development of instability depends on nine dimensionless external parameters. We can take 7, 6, Cq, Pe, G, Bi, T, Di, and Ma as these independent parameters. If the parameters are given, the problem reduces to the numerical solution of the dispersion relation for various of the wave number a and the spectral analysis of cj = u a). 

A typical, unsealed plot of versus the nonisothermal Thiele modulus is shown in Figure 9.10. Two additional parameters that contain the thermal factors make their appearance here the Arrhenius number EJRT which contains the important activation energy E and the dimensionless parameter P, which reflects the effect due to the heat of reaction and the transport resistances. For p = 0 (i.e., for a vanishing heat of reaction or infinite thermal conductivity), the effectiveness factor reduces to that of the isothermal case. P > 0 denotes an exothermic reaction, and here the rise in temperature in the interior of the pellet is seen to have a significant impact on E which may rise above unity and reach values as high as 100. This means that the overall reaction rate in the pellet is up to 100 times faster than would be the case at the prevailing surface conditions. This is due to the strong exponential dependence of reaction rate on temperature, as expressed by the Arrhenius relation... 

In addition, Figs. 2-4a, 2-4b and 2-4c indicate that the reduced droplet velocity ut increases with the packing size d. The material of the packing elements also has an influence on the parameter ut. It follows from this that the resistance coefficient i[fo in Eq. (2-22) is a function of the size and the surface properties of the packing. The first influencing factor can be expressed dimensionless by the quotient f2(dh/dT). The second factor is linked to the resistance coefficient lr of the dry packing. These two effects are reflected in the general correlation (2-24) ... 

The dimensionless parameter S (also called slope or screening parameter) varies between zero and unity. If the density of surface states is low, S = 1 and Eq. (26) reduces to the MS limit. The reverse case, 5 = 0, corresponds to the Bardeen limit. 

Since the solubility of lipids in water is very low, the number of lipid molecules in a membrane is essentially constant over typical experimental time scales. Also, the osmotic pressure generated by a small number of ions or macromolecules in solution, which cannot penetrate the lipid bUayer, keeps the internal volume essentially constant. The shape of fluid vesicles [176] is therefore determined by the competition of the curvature elasticity of the membrane, and the constraints of constant volume V and constant surface area S. In the simplest case of vanishing spontaneous curvature, the curvature elasticity is given by (98). In this case, the vesicle shape in the absence of thermal fluctuations depends on a single dimensionless parameter, the reduced volume V = V/Vo, where Vb = (47t/3)1 o nd Ro = (5/4 r) are the volume and radius of a sphere of the same surface area S, respectively. The calculated vesicle shapes are shown in Fig. 23. There are three phases. For reduced volumes not too far from the sphere, elongated prolate shapes are stable. In a small range of reduced volumes of V e [0.592,0.651], oblate discocyte shapes have the lowest curvature energy. Finally, at very low reduced volumes, cup-like stomatocyte shapes are found. 

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