Constant-thickness model

The interfacial curvature probability densities are displayed in Figs. 29a and 29b for the G morphology in the SIS copolymer and the constant-thickness model surface, respectively. To facilitate comparison, Ph(H) and Pk K) have been scaled with respect to the interfacial area per unit volume (Z) in the same fashion as that described earlier (Eq. 33). P(H,K) was scaled in the following way ... 

Here, H = and K = KL, with Z=0.070 and 0.074 nm for the SIS copolymer and constant-thickness model, respectively. Close examination of the scaled probability densities in Fig. 29a reveals that a part of the scaled joint probability density for the SIS G morphology possesses H < 0 and K>0, implying that the interface is an elliptic surface curved inward relative to the I microphase. Such interfacial concavity is not evident from P(H,K) derived from the constant-thickness model of the G morphology in Fig. 29b, in which nearly all (just under 100%) of the measured points possess k <0. Moreover, P(H,k) of the constant-thickness model exhibits two interesting characteristics. The first is that the measured data are distributed along H = Cok, where the constant Cq is related to the displacement used to construct the constant-thickness model in Fig. 27b from the Schoen G sur-... 

According to the capacitor model of the double layer, assuming constant thickness and electric permittivity, the dependence of AG° on <7m should be linear. " Deviations from linearity can be viewed as resulting from changes of X2 and/or e in the inner part of the double layer. A linear plot ofAG° vs. is observed for adsorption of ions and thiourea. ... 

Why such a difference between the real EDLC and the model proposed here In other words, what are the specific features the equivalent circuit does not take into account The oversimplified equivalent circuit presented in Figure 1.22 considers two planar electrodes face to face, with a constant thickness of dielectric material between them. The reality is much more complex since EDLCs use three-dimensional porous electrodes, and the porous electrodes are responsible for the particular shape of the Nyquist plot presented in Figure 1.23. 

A physical explanation of the process where the injection rate is changed can be described with help from the mass balance of the liquid within the fluidized bed. For simplicity, the suspension is regarded as a pure liquid. The particle movement follows the model of the ideal mixing, and the liquid on the particles is a thin film of constant thickness. Under these stationary conditions, the degree of wetting can be formulated from the moisture balance around the entire apparatus... 

The second model assumes an interfacial layer of finite and constant thickness T, so that K = T/lj is the volume of the interfacial layer, which has to be defined on the basis of some appropriate model of gas adsorption. For most practical purposes the two models are equivalent. The first model is easier to apply, but most of the authors in the early development of statistical mechanical theories of adsorption have expressed the problem in terms of an interfacial layer. For completeness, the appropriate definitions are given in relation to both formulations. 

A number of studies have been reported in the literature for the development and testing of mathematical descriptions for solute transport through hquid membranes. The existing models can be broadly classified into (i) the membrane film model in which the entire resistance to mass transfer is assumed to be concentrated in a membrane film of constant thickness and (ii) distributed resistance model which considers the mass transfer resistance to be distributed throughout the emulsion drop. 

Calandra et al. [44] adapted Muller s model to potentiodynamic conditions. Mac Donald [45] corrected a typographical error found in the mathematical expressions in the article. Devilliers et al. [46] developed a general model for the formation of low-conductivity films, considering a process controlled by the solution resistance in the pores of the film. The authors simulated the potentiodynamic curves for the following particular cases constant film thickness (bidimensional growth), three-dimensional growth, and a decomposition/dissolution process coupled to the electrochemical reaction. The potentiodynamic curves simulated for constant thickness are identical to those obtained by Calandra et al. [44]. 

Plotted in Figures 8 and 9 is equation 2, the thickness calculated for uniform layers of silica on spherical alumina particles. The XPS- and TEM-derived coating thicknesses are in fair agreement with the assumption of complete coatings of constant thickness. The TEMs are consistent with the notion that the thick coatings are uniform. A comparison of model... 

What happens with increasing ED current can be presented mathematically with the aid of a simplified model called the Nernst idealization. This model, which is illustrated in Figure 8.17, is based on the simplifying assumptions of stagnant boundary layers of constant thickness and a well-mixed region in the center of the solution compartment. These idealized boundary-layer conditions are obviously unrealistic, especially in the presence of the spacer screens used in an ED stack, but they allow a simplified approach to an otherwise complex problem. 

It can be shown that the total amount of surfactant adsorbed at the bubble surface under stationary conditions is equal or even exceeds the respective equilibrium value Analfg. Using Levich s model of a diffusion layer of constant thickness (Levich 1962), the surfactant flux density to the bubble surface can be evaluated by... 

Three interface layers occur within the electrical or the diffuse double layer (DDL) of a clay particle the inner Helmholtz plane (IHP) the outer Helmholtz plane (OHP) with constant thicknesses of Xi and X2, respectively and third is the plane of shear where the electro kinetic potential is measured (Rg. 2.10). This plane of shear is sometimes assumed to coincide with the OHP plane. The IHP is the outer limit of the specifically adsorbed water, molecules with dipoles, and other species (anions or cations) on the clay solid surface. The OHP is the plane that defines the outer limit of the Stem layer, the layer of positively charged ions that are condensed on the clay particle surface. In this model, known as the Gouy-Chapman-Stera-Grahame (GCSG) model, the diffuse part of the double layer starts at the location of the shear plane or the OHP plane (Hunter, 1981). The electric potential drop is linear across the Stem layer that encompasses the three planes (IHP, OHP, and shear planes) and it is exponential from the shear plane to the bulk solution, designated as the reference zero potential. 

Emulsion Liquid Membranes. Emulsion liquid membranes have been modeled by numerous researchers. Chan and Lee (77) reviewed the various models. The simplest representation characterizes the emulsion globule (membrane phase) as a spherical shell of constant thickness surrounding a single Internal phase droplet. This representation Is equivalent to assuming that the membrane and internal phase are well mixed. In practice, this Is usually a poor assumption. 

The result, Eq. (43), can also be used to calculate the elastic constants of interfaces in ternary diblock-copolymer systems [100]. The saddle-splay modulus is found to be always positive, which favors the formation of ordered bicontinuous structures, as observed experimentally [9] and theoretically [77,80] in diblock-copolymer systems. In contrast, molecular models for diblock-copolymer monolayers [68,69], which are applicable to the strong-segregation limit, always give a negative value of k. This result can be understood intuitively [68], as the volume of a saddle-shaped film of constant thickness is smaller than... 

This EVB model was employed to the systematic study of model pores or channels. Taking a simphstic view, the polymer was regarded as a rigid framework in which slab or cylinder pores of constant thickness or radius, respectively, are formed. Within this approach, proton transport in pores has been studied as a function of a variety of generic structural and dynamical features of the polymer and operational parameters of the working fuel cell (such as temperature and humidity). These studies revealed a number of factors determining the proton mobihty, such as the width of the channel, distance between the sidechains, and their flexibility. The main lessons of the simulations and the theoretical analysis were ... 

Figure 2.16 Models of the gyroid phase, (a) A constant thickness gyroid model surface, (b] A model for the cubic G phase block copolymer phase unit cell. The matrix phase is removed and the two network phases colored blue and red.

To determine the flux, Nemst model of stationary diffusion with linear distribution of the concentration across the layers of the constant thickness is used, and the concentration of the intermediate is assumed to be zero, i.e. the anode discharge proceeds at limit current condition. Then... 

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