Multilayered model

Comparison of the Surface Areas from the Various Multilayer Models... 

From Ref. 22) the values of the composite moduli in the direction of the fibers were taken for various fiber-volume contents, whereas the mesophase volume fractions Uj were determined from the experimental values of these quantities for certain values of uf s, by applying the approximate multilayer model introduced by Ref. 4). These values fitted excellently a second degree curve expressed by ... 

Meyers TP, Finkelstein P, Clarke J, Ellestad T, Sims PF. 1998. A multilayer model for inferring dry deposition using standard meteorological measurements. J Geophys Res 103 22645-22661. 

The purpose of this section is to describe the general results of models that contain more than one of the layers described above. It is beyond the scope of this article to analyze every model and its results in detail, especially since they have already been discussed to a certain extent in section 2. Many of the models make tradeoffs between complexity, dimensionality, and what effects are emphasized and modeled in detail. It is worth noting that those models that employ a CFD approach seem to be the best suited for considering multidimensional effects. In this section, the ways in which the multilayer models are solved and connected are discussed first. Next, some general trends and results are presented. 

The theory of Frank and Wen (54) for ion hydration involves the notion that some water molecules intimately contact the ion under consideration and subject to the strong centrosymmetric force field, are highly ordered. Beyond this area is a region of disorder, beyond which, in dilute solutions, unaffected water prevails. Typical literature values for primary hydration numbers range from 2-8 water molecules. For divalent ions, primary hydration numbers range from 10-20 water molecules while some authors have suggested hydration numbers for trivalent ions (based on compressibility data) between 20 and 30 water molecules per ion. Many attempts have been made to extend theories of this type to account better for the hydration of ions. Thus, Azzam (7, 8) and Horne and Birkett (80) have proposed a multilayer model of ion hydration. 

A somewhat intermediate view has also been adopted by Horne and Birkett (SO), who also propose a multilayer model of hydration where both the firmly bonded, first hydration layer and the disordered zone of the Frank-Wen model are accepted. However, they suggest the existence of a second layer of water molecules (separating the primary hydration shell and the disordered zone) around the ion, consisting of rarified or extended clusters of water molecules with density less than waters but definitely not of Ice-I like structure. We return to this aspect later. In this connection, compare also our discussion of the studies by Vaslow (150), Griffith and Scheraga (67), and Luz and Yagil (103). 

Figure 20 Scheme of the multilayer model of the Li-solution interphase, the division of the various layers, and the corresponding equivalent circuit analog, which can be fitted very well to the experimental data [49]. (With copyrights from The American Chemical Society, 1998.)... 

Single- and double-chained lipid derivatives of purines and purine nucleosides, such as 1-7, have been prepared. Ultrasonication of such compounds leads to either nucellar (single-chain compounds) or liposomal aggregates (double-chained compounds). Their surface behavior in monolayer, bilayer, and multilayer model membrane systems has been studied. 

In the case of a multilayered model, this function represents a combination of the bandlimited delta-functions with the positions of extremum at the reflecting... 

Fig. 3. Calculated squared reflectance by multilayer model according to Tausch U) in a wide-range calibration...

Laviron, E. 1980. A multilayer model for the study of space-distributed redox modified electrodes Part 1. Description and discussion of the model. Journal of Electroanaytical Chemistry 112, 1-9. 

The interface between the measuring solution and the membrane corresponds to x = 0, the sensor side of the membrane to x = d. In multilayer models an additional index, i, is introduced. In all cases the modeling starts at t = 0. 

The linear approach described here is expandable to multienzyme electrodes as well as multilayer electrodes. At least for the stationary case, multilayer models of bienzyme electrodes may be easily treated, too. The whole system is readily adaptable to potentiometric electrodes (Carr and Bowers, 1980). It must be noted, however, that the superiority over purely numerical solution procedures decreases with increasing number of enzyme species and in the multilayer model. The advantage in calculation speed using the sum formulas described (e.g., in Section 2.5.2) amounts to about two orders of magnitude. With multilayer electrodes and formulas containing double and triple sums it is reduced to one order of magnitude. 

Eischens and Selwood (5) explained the susceptibihty data shown on Fig. 6 in terms of a monolayer-multilayer model of surface chromia. They used the following expression for the variation of the Weiss constant A with the number of nearest neighbors ... 

As the polymerization proceeds, the initial catalyst support fragments and is dispersed within the growing polymer matrix. In the case of clay-supported catalysts, one needs to include another level of mass transfer to account for monomer diffusion and polymer formation between the clay platelets. A model to describe this process, called the multilayer model (MLM), is depicted in Figure 3.11. 

Figure 3.11 A schematic of the multilayer model (reproduced from Ref [61]).

Rahmoun K et al (2009) A multilayer model for describing hardness variations of aged porous silicon low-dielectric constant thin films. Thin Solid Films 518 213-221 Rice RW (ed) (1998) Porosity of ceramics. Marcel Dekker, New York... 

Model Calculations Based on XPS Results. This section concerns a multilayer model for the SAM-covered surface and corresponding calculations of XPS intensities based on standard equations and depending on the molecular orientation and packing density within the adlayer. The calculations are first applied to the less complex, smooth surfaces and then to the rough (SLA) case. 

A Variable Angle Spectroscopic EUipsometer of the type M200-F (J.A. Woollam Co. Inc., Lincoln, USA) with a spectral range from 245 to 995 nm was used to determine the thickness of the adsorbed polymer layers. Measurements were performed in ambient at three different angles (65, 70, and 75° with respect to the surface normal). For each polymer adlayer, i.e. Sil-PEG (from toluene), Sil-PEG (from acidic aqueous solution), and PLL-g-PEG (from aqueous HEPES buffer), five samples were prepared to obtain statistical data. The measurements were fitted with multilayer models using WVASE32 analysis software. The analysis of optical constants was based on a bulk silicon/ SiOj, layer, fitted in accordance with the Jellison model. After adsorption of the molecules, the adlayer thickness was determined using a Cauchy model A = 1.45, B = 0.01, C = 0). 

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