Three plane model

Madrid, L. et al., Use of a three-plane model to describe charge properties of some iron oxides and soil clays, J. Soil Sci., 34, 57, 1983. 

Figure 1. The metal oxide/aqueous solution interface in the presence of anions forming inner sphere complexes according to the Three Plane Model.

Equations (l)-(4) are the foundations of electrical double layer theory and are often used in modeling the adsorption of metal ions at interfaces of charged solid and electrolyte solutions. In a typieal TLM, the outer layer capacitance is often fixed at a lower value (i.e., C2 = 0.2 F/m ), whereas iimer layer capacitance (Ci) can be adjusted to between 1.0 and 1.4 F/m [25]. It should be noted that the three-plane model (TPM) is a variation of the classical triple-layer model, in which the outer layer eapaeitanee is not fixed. Although the physical presentations of the TLM and TPM are identical as shown in Fig. 2, i.e., both involve a surface layer (0), an inner Helmholtz plane (p), and an outer Helmholtz plane d) where the diffuse double layer starts, a one-step protonation process (i.e., 1 piC approach) is, in general, assumed in the TPM, in eontrast to a two-step protonation process (i.e., 2 p/C approach) in the TLM. Another distinct difference is that pair-forming ions are assumed to be on the outer Helmholtz plane in the TPM but on the inner Helmholtz plane in the TLM. In our study, the outer layer capacitance is allowed to vary while the pair-forming ions are placed on the iimer Helmholtz plane with a complete set of surface eomplexation reactions being considered. Therefore, our approach represents a hybrid of the TPM and TLM. 

In the previous sections, we have demonstrated that a hybrid approach integrating the useful features of the triple-layer model and the three-plane model allows accurate description of the zeta potential (pH) for a range of electrolyte concentrations and excellent prediction of the surface charge density vs. pH relationship. As noted, the effect of replacing v]/ by in the TLM needs to be further examined by... 

Additional surface planes The three-plane model allows eharges to be plaeed in the surfaee plane, in the plane where the head end of the diffuse layer is situated, and in an intermediate plane. In the typieal applieation of the model, the eleetrolyte ions are plaeed in the d plane with respeet to the triple-layer models, where no eharges are plaeed in this plane, the eleetrolyte ions bound as ion pairs are in eontaet with the diffuse layer. 

Aeid-base parameters for a three-plane model can be evaluated by the equivalent Stem-layer model this is eomparable to the relation between the constant-capacitance model (model a) and the extended eonstant-capacitance model (model b). The relation between the capacitance values allows eonstraints on the optimum couples of C, and C2. Problems will arise when eharges of adsorbed solutes may reach further toward the solution than what is allowed by... 

For meehanistie deseriptions of ion adsorption, the three-plane model with charge distribution offers extensive flexibility. This ean, of eourse, be used to fit aU involved parameters like the eomposition of surfaee eomplexes, eapaeitanee values, eharge distribution coefficients,... 

Fig. 6.3. Three-dimensional model of calibration, analytical evaluation and recovery spatial model (A) the three relevant planes are given separately in (B) as the calibration function with confidence interval, in (C) as the recovery function with confidence interval, and in (C) as the evaluation function with prediction interval (D)... 

In contrast to the common calibration procedure measuring y — f xstandard) as shown in the front plane of Fig. (6.3a,b), the glucose reference calibration takes place in the base area of the three-dimensional model, xsampie = f ixstandard)> see Sect. 6.1.2. 

A section in the full-bed models was isolated that was comparable to the WS model. The layout of these different sections was identical, except that the WS model had a two-layer periodicity and the full-bed models had a six-layer periodicity. To be able to make direct comparisons of velocity profiles, several sample-points needed to be defined. In the three different models seven tangential planes were defined and on each plane three axial positions were defined. This reduced the data to single radial velocity profiles at corresponding positions in all three models, as shown in Fig. 10, for the WS model. Identical planes were defined in the full-bed models. Some spheres and sample planes 4 and 5 are not displayed to improve the visibility of the sample planes and lines. In the right-hand part of the figure, plane 4 is shown with the axial positions at which data were taken and compared. 

A further point is that for a multiply-twinned particle of diameter 1 nm, for example, the constituent single crystal regions are half of this size or less and so contain only two or three planes of atoms. One can not expect, under these circumstances, that the diffraction pattern will be made up merely by addition of the intensities of the single crystal regions. Coherence interference effects from atoms in adjacent regions will become important. It is then necessary to compare the experimental patterns with patterns calculated for various model structures. 

A coupled analysis need not be all encompassing. For example, a two dimensional plane frame analysis of a building employing two or more degrees of freedom is considered a coupled analysis approach. Separate plane frames for each orthogonal horizontal direction can be used in lieu of a single comprehensive three dimensional model. Refer to Section 6.6.2 for a discussion on modeling considerations for this type of structure. 

To obtain coordinates for a three-dimensional model the two-dimensional (2D) projection was used to locate roughly the x, y coordinates of the caffeine molecule. The z coordinate of each atom in caffeine was calculated by assuming the planar molecule was positioned in the 1,2,2 plane. (The 1,2,2 was selected because it was the strongest reflection observed.) The orientation of pyrogallol was assumed to be as in I. (There was no justification for this assumption other than it appeared to give the least steric hindrance.) The z coordinates of pyrogallol were calculated by assuming it to be parallel to caffeine and unit cell above the plane of caffeine. The water molecule uncovered in the 2D projec-... 

For TEE molecules the dipolar terms in the three level model are of special interest as they contain the difference dipole moment Ap which vanish perpendicular to a mirror plane. The dipolar contribution yD to the overall nonlinearity... 

Figure 3.5. Diagram showing the change of the resonance energy of the (001) planes in the transition from the surface plane to a bulk plane. In the model adopted, the resonance energy shifts upwards by 200,10 and about 2cm 1 for the first three planes S S2, and S3, respectively.

Using this model, one cannot forecast the adsorption of the background electrolyte ions because this model do not consider the reactions responsible for such a process. Zeta potential values, calculated on the basis of this model, are usually too high, nevertheless, because of its simplicity the model is applied very often. In a more complicated model of edl, the three plate model (see Fig. 3), besides the mentioned surface plate and the diffusion layer, in Stern layer there are some specifically adsorbed ions. The surface charge is formed by = SOHJ and = SO- groups, also by other groups formed by complexation or pair formation with background electrolyte ions = SOHj An- and = SO Ct+. It is assumed that both, cation (Ct+) and anion (A-), are located in the same distance from the surface of the oxide and form the inner Helmholtz plane (IHP). In this case, beside mentioned parameters for two layer model, the additional parameters should be added, i.e., surface complex formation constants (with cation pKct or anion pKAn) and compact and diffuse layer capacities. 

Beginning with Pasteur s work in 1860 [4] the fields of stereochemistry and biology were dominated for almost nine decades by the phenomenon now called chirality. Chiral molecules are those for which a three-dimensional model of the molecule is not superimposable on the mirror image of the model. Since the operation determining the existence of chirality is reflection in a plane mirror, this... 

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. 

Despite some different results between the one- and three-dimensional models that might be due to the fact that simple models (such as the one-dimensional MRTM) provide averaged values, results from both models seem to be consistent. The concentrations estimated by the three-dimensional MRTM seem to better reflect the actual phenomena since this MRTM shows that immediate saturation of the top soil layers does not occur (this would occur only if flow and dispersion in the top horizontal planes would have unrealistic high values). Furthermore, even after ten time steps the topsoil layers are only partially saturated. 

The results provided by three-dimensional MRTM are consistent with the numerical output of one-dimensional MRTM. The concentration-depth curves are shown to be similar for a nominal test case that is independent of temporal and spatial scales. Besides the numerical output that the model generates, the visualization component of the model gives an almost instantaneous look into the spatial distribution of the contaminant. This visualization is made by sliding three planes (horizontal, longitudinal, and transversal) across the entire simulation domain. Concentrations are scaled from 0.0 to the maximum values so that the trace concentrations can be easily visualized. The numerical value of the maximum concentration is also output in the visualization window, together with the current position of the visualization plane. When the trace compound is hazardous (e.g., a heavy metal such as mercury), it is also necessary to monitor the spatial distribution of very low concentrations. The current three-dimensional, MRTM visualization method provides the means to track these types of trace concentrations. 

In a left-handed (cf. 22.1) coordinate system with the z axis vertical, let x=F, y=Sy z=E be the total volume, entropy, and energy of a substance in its different states, the aggregate of which is represented by a surface, which is Gibbs s thermodynamic model. Three planes perpendicular to the axes of Vi Sy and J5", respectively, are the loci of all states of given volume, entropy, and energy. Of these, the plane V=0 is fixed, but the planes S=0 and =0 are subject to arbitrary choice of the initial states of zero entropy and energy, and hence the origin may be chosen anywhere in the plane of zero volume. 

Figure 4. Hypothetical three-state model for binding of in the lateral plane of anionic...

Figure 3.39. Water concentration in an interdigitated PEM fuel cell structure, for three planes at x-values corresponding to the flow channel exit (a top left), the middle (b top right) and the entrance (c bottom left). In each pair of pictures, the y-z plots depict the hydrogen side at the top (GDL is lower bar) and the oxygen side at the bottom (GDL is upper bar). The cell current is at its maximum (about 0.8 A cm" ). (From M. Hu et al, (2004). Three dimensional, two phase flow mathematical model for PEM fuel ceU Part II. Analysis and discussion of the internal transport mechanism. Energy Conversion Management. 45,1883-1916. Used with permission from Elsevier.)...

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