Conductor models

The generally applicable relations for a two-conductor model are derived in the following section. For simplicity, local potential uniformity is assumed for one of the two conductor phases. Relationships for the potential and current distributions, depending on assumed current density-potential functions, are derived for various applications. 

Figure 24-5 contains the general data for the two-conductor model [12]. The conductor phase II has a locally constant potential based on a very high conductiv-... 

The ideal conductor model does not account for diffuseness of the ionic distribution in the electrolyte and the corresponding spreading of the electric field with a potential drop outside the membrane. To account approximately for these effects we apply Poisson-Boltzmann theory. The results for the modes energies can be summarized as follows [89] ... 

The index of helicity can be used in an alternative formulation of the basic equation for the helical conductor model of optical activity (1) ... 

Fig. 3—15 show four-, five- and six-atom chains (A—C -2—B) in their non-planar staggered conformations (dihedral angles of 60° and 180°). The individual bond conformations are denoted P (positive), M (minus), consistent with the proposals of Cahn, Ingold and Prelog 15>, and T (trans), as shown in 4. Using the familiar properties of triangles and the tetrahedral bond angle (cos r = — 1/3) (see Ref. 12 for the derivation of the equation in Fig. 3) we have derived expressions for the subtended areas (A) as needed for use with the helical conductor model (Eq. (1)). It turns out that all of these expressions contain the term L... 

This equation leads to a reductio ad absurdum that may provide a significant refinement of the helical conductor model. It will be noted that as the step helices (ya = 180°) approach the absolutely planar zigzag chain structure (ya = /b = 180°) they acquire very large cross sections. The index of helieity approaches a constant value of about 0.385 but the expected residue rotations approach infinity. (See Table 12). This result is, at least intuitively, absurd. 

These considerations, thus, lay the groundwork for tests among several semi-empirical approaches to the estimation of optical rotation of bond systems regarded as helices. Should it be necessary to use Eq. (lb) rather than (la), then a sweeping reassessment of the use of the helical conductor model will be required. However that test turns out, a test between that model and the simple conformational dissymmetry model becomes possible on the basis of the material shown in Table 1. At this point it should be said that our calculations on twistane 16> support the helical conductor model but that the results obtained by Pino and his co-workers 17 18> on the chiroptical properties of isotactic polymers prepared from chiral a-olefins support the conformational dissymmetry model. [We are not able, at present anyhow, to account for their results with the helical conductor model]. 

Although quite obvious, it is important to remark that a perfect conductor-like model completely neglects the chemical nature of the metal (i.e., all the metals behave in the same way). This is different to what happens for solvents, where even the simplest models include at least one solvent-specific parameter, the static dielectric constant. However, the chemical nature of the metal is relevant for another aspect of the static dielectric response that is neglected by the conductor model the nonlocal effects. They will be discussed in the following Section, The Response Properties of a Molecule Close to a Metal Specimen Surface Enhanced Phenomena and Related Continuum Models. 

Two hidden assumptions implicit in the model of Sato and Mooney (1960) are (1) that the conductor consists of a single phase, such as graphite or pyrite and (2) that oxidation of the conductor would result in its conversion to a non-conductive phase. Thomber (1975a, 1975b) presents a reactive conductor model in which the conductor itself is the reducing agent, which is in apparent contrast to the model of Sato and Mooney. However, the reactive conductor model is based on the presence of one oxidised phase and at least one reduced phase relative to the first phase, all of which are electronically conductive. Such scenarios have been noted in terrain with deep weathering profiles due to the phase conversion of reduced sulphide minerals to more oxidised forms. 

In both the reactive groundwater and reactive conductor models, the impetus for electronic current flow in mineralisation comes from the redox differential between the oxidised groundwater environment surrounding the upper part of the conductor and reducing agents in contact with its lower part. The upward movement of electrons consumes oxidising agents in basal overburden and results in the development of a... 

An impressive example of the use of the helical conductor model of the optical... 

The helical conductor model is useful for the prediction of optical rotations and configurations of aliphatic helices which do not contain chromophoric skeletons, and for twisted chains of atoms. In studies of the chiroptical properties of isotactic... 

The first of the two recent PCM versions quoted above, namely that we shall call COSMO-PCM, has been implemented by Barone and Cossi [32]. It formally derives from the computational model proposed by Klamt and Schiiiirman [33], which consists in defining apparent charges g by first exploiting a liquid conductor model (i.e. e = oo), and then introducing a correction factor multiplying each ASC in order to make the obtained formulas coherent with the isotropic dielectric model. 

Daniel, E.E., Bardakjian, B.L., Huizinga, J.D., and Diamant, N.E. 1994. Relaxation oscillators and core conductor models are needed for understanding of G1 electrical activities. Am. J. Physiol, 266 G339 349. 

The example network in Figure 23.1, is solving a bioelectric field problem for a dipolar source in a volume conductor model of a head. The domain is discretized with linear tetrahedral finite elements, with five different conductivity types assigned through the volume. The problem is numerically approximated with a linear system, and is solved using the CG method. A set of virtual electrode points are rendered as pseudocolored spheres, to visualize the potentials at those locations on the scalp, and an iso-potential surface and several pseudocolored electric field streamlines are also shown. 

FIGURE 25.4 Simulated currents and extracellular potentials of frog sartorius muscle fiber (radius a = 50 /xm). (a) The net fiber current density is the summation of the current density through the sarcolemma and that passing the tubular mouth, (b) Extracellular action potentials calculated at increasing radial distances (in units of fiber radius) using a bidomain volume conductor model and the net current source in panel (a). The time axes have been expanded and truncated. 

In a cylindricaUy symmetrical volume conductor model, the lead field flow fines are concentric circles and do not cut the discontinuity boundaries. Therefore, the sensitivity distribution in the brain area of the spherical model equals that in an infinite, homogeneous volume conductor. 

One 45-kA conductor model has been constructed and its use for large energy storage magnets and poloidal field coils in fusion reactors is under investigation. 

Fig. 6.2 Three-conductor model of ion-exchange membrane conductivity.

The transport numbers can be calculated using the three-conductor model. The co-ion concentration in gel areas of ion-exchange membrane is far less than in intergel areas, so it would be a valid assumption that co-ions are transported through intergel areas only. As can be seen from the picture of the three[Pg.267]

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