Three-layer model

In a three-layer model, we represent one part of the system by MM, one part by a semi-empirical Hamiltonian and one part by an ab initio Hamiltonian. 

From the difference of these morphological characteristics, a simplified three-layer model could be set up and the inhomogeneous structure could be classified into these three layers. Figure 23 illustrates the relative thickness of the core, transition, and skin layer for the four sample groups, depending on the processing condition. 

The Three-Layer Model Based on Values of Thermal Expansion.155... 

In order to simplify the procedure of evaluating the extent of mesophase and its mechanical and thermal properties, a simple but effective three-layer model may be used, which is based on measurements of the thermal expansions of the phases and the composite, below and above the transition zone of the composite, lying around its glass transition temperature Tgc. 

Then, the three-layer model provides an easy method for evaluating the characteristics of the mesophase, by introducing a significant flexibility in the study of the physical behaviour of particulates. The drawback of the model is its instability to the values of the thermal expansions and the moduli of the composite, which must be evaluated with very high accuracy, fact which is a difficult task. Small deviations in measuring the a s and the E s may vary considerably the balance of characteristic values of the composite. However, the introduction of the influence of the mesophase to the physical behaviour of the composite, made in this model, is a certain advancement in the knowledge of the behaviour of these complicated substances. 

Thus, in the three-layer model, with the intermediate layer having variable physical properties (and perhaps also chemical), subscripts f, i, m and c denote quantities corresponding to the filler, mesophase, matrix and composite respectively. It is easy to establish for the representative volume element (RVE) of a particulate composite, consisting of a cluster of three concentric spheres, that the following relations hold ... 

A three-layer model for fiber composites may be developed, based on the theory of self-consistent models and adapting this theory to a three-layered cylinder, delineating the representative volume element for the fiber composite. 

The three-layer model, as previously mentioned, as well as the multi-layer model, were previously applied to study the behaviour, especially of fiber composites 3A). The three-layer model, based on the self-consistency of phases, gave relationships between stresses and displacements between phases, which, when solved, may give... 

Thermally spiked 204 Thermal spikes 195 Three-layer models 152, 174, 175 Three-term models 149 Three-term unfolding model 176,185 Tie molecules 124 Tie-taut molecules 141 Time-dependent hardness 121 Tire cord 6... 

In the three layer model shown in Fig. 5-8, the electric capacity C of an interfacial electric double layer is represented by a series connection of three... 

Some results of the simulation experiment are given in Figures 6.5 and 6.6. Figure 6.5 shows the tendency vs. time of the average content of radionuclear pollution on the whole Arctic water area. The distribution with depth is represented by a three-layer model, upper waters (z < 1 km), deep water (z > 1 km), and sediments. Bottom depth is taken as 1.5 km. More realistic depth representations of both shallow seas and the deeper Arctic Basin will be considered in a future refinement of the model. The curves describe the vertical distribution with time of the radionuclide content in two water layers and in sediments. The transfer of radionuclides from upper water to deep water occurs at a speed which results in the reduction of radionuclear pollution in upper water by 43.3% over 20 years. Such distributions for each Arctic sea are given in Table 6.11. 

Fig. 2 A, B. Three layers model of water at the interface with mineral according to Dorst-Hansen164) O = clathrate-like ordering ] = water-dipole. A Vicinal water near non-polar surface. Extensive clathrate-like ordering near interface but minimal disordered region. B Vicinal water near polar surface, disordered region...

With FITEQL numeric procedure Hayes et al. fitted edl parameters to the three models of electric double layer DLM (diffuse layer model), CCM (constant capacity model) and TLM (three layer model) for the following oxides a-FeOOH, AI2O3 and TiC>2 in NaNC>3 solutions [51]. The fitting was performed for surface reaction constants, edl capacity and the densities of the hydroxyl groups on the surface of the oxides. The quality of the fitting was evaluated by the minimization of the function of the sum of the square deviations of the calculated value from the standard error of measured charge. The lower value of the function the better was the fit... 

According to three layer model of site binding of edl (TLM, model SCM) also the adsorption of background electrolyte ions may be considered as a... 

The thickness of the aqueous core h2 is most often determined done by the three-layer model, according Duyvis formula [59]... 

Since, however, each model involves some assumptions, the calculation of h2 always renders certain inaccuracy. The most important problem in the three-layer model concerns the position of the plane that divides the hydrophobic and hydrophilic parts of the adsorbed surfactant molecule. In some cases it seems reasonable to have this plane passing through the middle of the hydrophilic head of the molecule, in others the head does not enter into the aqueous core. That is why it is worth comparing film thicknesses determined by the interferometric technique using the three-layer model, to those estimated by other methods. An attempt for such a comparison is presented in [63]. Discussed are phospholipid foam films the thickness of which was determined by two optical techniques the microinterferometric and FT-IR (see Section 2.2.5). The comparison could be proceeded with the results from the X-ray Reflectivity technique that deals not only with the foam film itself but also with the lamellar structures in the solution bulk, the latter being much better studied. Undoubtedly, this would contribute to a more detailed understanding of the foam film structure. 

Fig. 2.6. A three-layer model of the foam film 1 - organic phase 2 - water phase 3 - air.

Optical measurements of foam films are very complicate. In order to have a clear and precise interpretation of the results, especially for the thinnest black films, it is necessary to make models. The simplest and the most widely used one is the three-layer model (see Section 2.1.3). 

An optical three-layer model has proved superior to a one-layer model for the interpretation of the ellipsometric data. The refractive indices of the film and surface layers are determined and it is found that the index for the surface is higher than that for the film core. A Lorenz-Lorentz type treatment of NBF reveals that there are approximately seven water molecules per molecule of surfactants in both NaDoS and NaDoBS films. The optical data obtained by the three-layer model for NBF from NaDoS indicate that the thickness of the aqueous core is zero while that of the adsorption monolayers of surfactant molecules with refractive index 1.365 is 1.8 nm, i.e. the thickness of NBF is 3.6 nm. 

Corkill et al. [56] have used for the first time the infrared spectroscopy for foam films. The measurement of the adsorption of the infrared light provides information about the water content in the foam films which is of major significance for the black foam films. These studies involved the use of dispersion type instruments. In order to obtain measurable values of adsorption, the infrared light is passed through a series of vertical films (up to 10) formed in a cylindrical tube acting as a frame. Additional information about the film structure the authors derived from the correlation between the optical infrared transmission data and the film reflectance measurements. Here a three-layer model of the film structure consisting of an aqueous core sandwiched between two adsorption layers is assumed (see Section 2.1.3). 

On the contrary the plateau values for the two copolymers are very different. Since the higher copolymer gives thicker films a surface force component of steric origin may be evoked. However, the thickness hK is an effective parameter which is too crude. As a reasonable compromise between physical relevance and tractability, the three-layer model is adopted. Within the three-layer model the foam film is viewed as a symmetric sandwich structure [159] two adsorption layers symmetrically confine an aqueous core (Fig. 3.34). 

Fig. 3.34. Three layers model the brush thickness is assumed constant the thickness decrease with...

At film thickness larger than twice the adsorption layer thickness this type of force vanishes [248], Therefore, such a mechanism is operative only at Ijtot 2Iii = 21.2 nm, i.e. hw < 28.0 nm (Table 3.5). The solid line in Fig. 3.40 is the best fit of Eq. (3.87). The van der Waals component has no practical influence on the numerical procedure. The fitted value h = 11.1 nm is in good agreement with the value of 10.6 nm used in the three layers model. Thus, de Gennes theory [248] gives a satisfactory description of the steric interactions at film thickness where brush-to-brush contact is realised. 

Obviously, the reliability of the above data for Y0 and Q0 depends significantly on the assumptions made to convert hw into d and on the accuracy of the Uvw evaluation. A five-layer film model that comprise the hydrophilic heads of the surfactant as separate layers may be considered as an alternative [283], This yielded d values 0.14 nm higher than those based on the three-layer model. The difference is less than the experimental scatter in the hw data used and is, as it appears from Fig. 3.45, insignificant. This, therefore, gives some supporting evidence for the assumptions made. As far as the evaluation of IIvw is concerned, no indications of considerable errors can be found through the region of the film thicknesses considered [166]. 

Values of the equivalent water-alcohol thickness kw, the thickness of the hydrocarbon layers hly the thickness of the polar core h2 and the thickness h (according to three-layer model) of DMPC foam bilayers at different temperatures. 

Note The results for the thickness of foam films according to the three-layer model depend on the place of the conditional boundary between the hydrophilic and hydrophobic parts of foam bilayer (see Section 3.4.1.2). The above values are calculated under the assumption that these boundaries are situated between the polar head groups and hydrocarbon chains of DMPC molecules. 

The more accurate analysis of the data for foam bilayer thickness obtained by the three-layer model (see Section 2.1.1) enables a better understanding of the structural changes in foam bilayers due to temperature variations. 

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