Representative layer theory

There are situations where the assumptions of proportionality of the absorption coefficients and constancy of remission coefficient are reasonable. If the fraction of light absorbed by a single particle is small, the assumptions are good. In order to describe other situations, it is desirable to use an approach that does not suffer from the complexities of coefficients referred to above. In the following examples, we will illustrate the use of the approach of discontinuum theory as embodied in the representative layer theory. 

From the preceding example, it can be seen that the representative layer theory predicts that the unique matrix referencing technique would not work. In this example, we will show that the shape of the experimental curves is predicted by the representative layer theory. In this study, log(l//i) and K-M matrix referenced data were obtained from three series of infinitely thick samples ... 

FIGURES.9 A unique matrix referencing scheme can yield a linear K-M plot only with a nonabsorbing matrix and when plotted against a volume based measure of concentration. The log(l/i oo) curves are in general not linear. From top to bottom, the lines in each chart represent 0, 1, and 5% absorption by matrix particles. The lines represent values calculated by the representative layer theory. (Reproduced from D. J. Dahm, NIR News, 15 6-10 (2007), by permission of NIR Publications Copyright 2007.)... 

Additional experimental confirmation for the usefulness of the representative layer theory comes from the work of DeVaux et al. [71,72]. This study was concerned with the fact that, in a mixture of two components with different particle sizes, the smaller particles will be over-represented in an absorption metric compared to the weight fractions of the components. This effect was systematically examined with a combination of image analysis and NIR spectroscopy on mixtures of fine and coarse fractions of wheat and rape seed meal. ... 

Focusing on the line where the absorption metric is reflectance (the fraction of the incident beam that is remitted), the black rape seed particles have high absorption, while the wheat particles absorb far less. However, the line is curved because the layer is not opaque as discussed previously. Further, because the effective sample thickness is higher for wheat than for rape seed, the line is super-linear as a function of wheat fraction. As in Example 2, the representative layer theory can reasonably define the shape of these curves. 

These assigned values may be quite unreasonable for remission from a layer that contained nothing but those particles. This is a curve fitting exercise. We are manipulating four parameters to fit a line. There are many combinations of these four parameters that give the same line. Presumably, none of the sets will be the properties of a single layer of particles, because we have not included voids in the model, and a layer of particles, one particle thick, would certainly contain voids. Nonetheless, this exercise illustrates the capability of the Dahm equation and the representative layer theory to describe particulate systems, even when there is very high absorption, a place where there... 

FIGURE 3.13 Variation of log(l /Roo) vs. the fraction of wheat in a sample composed of rape seed and wheat. The markers show the experimental values of log(l/f oo) vs. the fraction of wheat in the mixture while the lines show the shape predicted by the representative layer theory. The top curve is for wheat and rape seed of the same particle size. In the bottom curve, the wheat particles are twice the size of the particles of rape seed. (Reproduced from D. J. Dahm and K. D. Dahm, Near-Infrared Technology in the Agriculture and Food Industries, 2nd edn. (R Williams and K. Norris, eds.), St. Paul, MN, pp. 1-17 (2001) by permission of American Association of Cereal Chemists, Inc. Copyright 2001.)... 

We believe these examples show both the usefiilness of the representative layer theory in describing particulate samples. Further we have illustrated the applicability of the Dahm equation to situations where continuous theories do not work well. Finally, we have tried to make clear that, however useful the theory may be in explaining what we observe, we have not created an easy way to determine particle properties, or to correct simply absorption data so it will be linear with analyte concentration. 

Beginning with the properties of the individual particles in a mixture, the representative layer theory gives a way of calculating the properties of a layer of particles. The merging of the continuous and discontinuous approaches, as embodied in the Dahm equation, gives a way to calculate the spectroscopic properties of a sample from that of such a layer. 

Clearly, then, the chemical and physical properties of liquid interfaces represent a significant interdisciplinary research area for a broad range of investigators, such as those who have contributed to this book. The chapters are organized into three parts. The first deals with the chemical and physical structure of oil-water interfaces and membrane surfaces. Eighteen chapters present discussion of interfacial potentials, ion solvation, electrostatic instabilities in double layers, theory of adsorption, nonlinear optics, interfacial kinetics, microstructure effects, ultramicroelectrode techniques, catalysis, and extraction. 

When transient techniques are employed for fundamental research on these and other subjects, the effect of double-layer charging has to be accounted for in the analysis procedures. It has been observed frequently that at solid—solution interfaces, this process does not obey the capacitive behaviour predicted by double-layer theories. For example, the doublelayer admittance, Fc, cannot be represented by Yc = jciCd, but rather follows the relation [118]... 

The analysis of co-current flame spread is very similar to that of opposed flame spread. However, it is further complicated because the flame covers the fuel thus, the flame length is a further parameter that needs to be analyzed. The flame length can be represented empirically as being proportional to the pyrolysis length or can be calculated using boundary layer theory and the assumption of infinitely fast gas-phase chemistry [22], Despite the added complexity, co-current flame spread is controlled by the same physical parameters as ignition or opposed flame spread. 

Friedel oscillations — Oscillations of the electronic density caused by a disturbance such as a surface or an excess charge. At surfaces, they decay asymptotically with 1/z2, where z is the distance from the surface. Within electrochemistry they play a role in - double-layer theories that represent the metal as -+ jellium. 

Any transfer of electrons giving rise to changes of semiconductivity during chemisorption must be controlled, inter alia, by the concentration of electrons or holes available in the semi-conductor. The boundary-layer theory of chemisorption 65) is built within the framework of this entirely physical model of the chemisorption act. The gas being adsorbed is represented solely as a donor or acceptor of electrons the adsorbent is represented as a conventional semiconductor with a given concentration of ionized donor or acceptor centers and whose ability to participate in chemisorption is otherwise uniquely determined by the height of the Fermi level. 

Problem 10-12. Higher-Order Approximations for the Blasius Problem. The classical boundary-layer theory represents only the first term in an asymptotic approximation for Re 1. However, in cases involving separation, we do not seek additional corrections because the existence of a separation point signals the breakdown of the whole theory. When the flow does not separate, we can calculate higher-order corrections, and these provide useful insight and results. In this problem, we reconsider the familiar Blasius problem of streaming flow past a semi-infinite flat plate that is oriented parallel to a uniform flow. 

FIG. 20 Current-distance curves obtained with different concentrations of TEAC1 in the bottom aqueous layer and constant composition of the organic phase (0.4 mM TEATPBC1 + 10 mM BTPPATPB in DCE). The filling aqueous solution contained 10 mM LiCl. Concentration of TEAC1 in the bottom aqueous phase was (1) 0, (2) 0.4, and (3) 10 mM. Aqueous phase also contained 10 mM LiCl. Solid lines represent SECM theory for an insulating substrate (curve 1) and pure positive feedback (curve 4). The radius of the pipet orifice was (1) 18, (2) 10, and (3) 12 /rm. The tip was scanned at 1 /rm/s. (From Ref. 53.)... 

Tn recent years, the influence of counterions on the properties of A ionized monolayers has received much attention. Even though Davies (I) application of the Gouy-Chapman double layer theory to ionized monolayers represented a major advance in the understanding of the properties of these systems, it has been increasingly recognized that we must account for the different effects (i.e., specific counterion effects) that counterions of the same net charge may have on the charged mono-layer. Because of counterion sequence inversions which have been ob-... 

The functional dependence of ju on Reynolds number has been the subject of study by many investigators, and a variety of equations have been proposed for correlation of the available data for fixed-bed (79, 89) and fluidized-bed reactors (85-87). Boundary layer theory indicates that the Chilton-Colbum analogy, = jjj, represents an asymp-... 

The adsorption behavior is depicted in Figs. 13 and 14. Our scaling results are in agreement with numerical solutions of discrete lattice models (the multi-Stern layer theory) [61, 62, 111, 117-120]. In Fig. 13, r is plotted as function of / (Fig. 13a) and the pH (Fig. 13b) for different salt concentrations. The behavior as seen in Fig. 13(b) represents annealed PFs where the nominal charge fraction is... 

Stern 1 has therefore altered the model underlying the double layer theory for a solid wall by dividing the liquid charge into two parts. One part is thought of as a layer of ions adsorbed to the wall, and is represented in the theory by a surface charge concentrated in a plane at a small distance 5... 

The potential at the shear plane is termed the electrokinetic or (zeta) potential and represents the actual value determined in the procedures discussed in the next section. It is generally assumed in tests of double-layer theory that the potential and ips are the same, since any error introduced will be small under ordinary circumstances. More significant errors may be introduced at high potentials, high electrolyte concentrations, or in the presence of adsorbed bulky nonionic species that force the shear plane further away from the surface, reducing the potential relative to ips-... 

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 potential flow, the stream function and the potential function are used to represent the flow in the main body of the fluid. These ideal fluid solutions do not satisfy the condition that = Vy = 0 on the wall surface. Near the wall we have viscous drag and we use boundary-layer theory where we obtain approximate solutions for the velocity profiles in this thin. boundary layer taking into account viscosity. This is discussed in Section 3.10. Then we splice this solution onto the ideal flow solution that describes flow outside the boundary layer. 

Fig. 6 Logarithm of applied pressure (log P) plotted versus the distance between bilayers (if) for gel phase bilayers composed of mixtures of the zwitterionic lipid DPPE and the negatively charged lipid DPPA. The circle on the x-axis represents the fluid spacing of gel phase DPPE bilayers in water and the arrow indicates the fluid spacing for liquid-crystalline phase BPE bilayers. The line represents the repulsive electrostatic pressure calculated from double-layer theory for 80 20 DPPE DPPA bilayers. Data were taken from [19,51]...

Recently, the discontinuous approach has been applied by Dahm and Dahm [62,63] to the two-flux results obtained from the radiation transfer equation. This has resulted in being able to recast the results obtained from the continuous approach in terms of a layer of particles. This has been termed the representative tayer theory. In the discussion of deviations from the K-M equation, we gave without proof an analogous mathematical expression reached by the discontinuous treatment (Equation (3.65)). In this section we give the derivation of some of the more important formulas in the discontinuous treatment. 

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