Planar models

Figure 8.16. Planar model of part of a screw pump...

Sundararajan, S. andXhakurta, D., Two-Dimensional Wafer-Scale Chemical-Mechanical Planarization Models Based on Lubrication Theory and Mass Transport, Journal of the Electrochemical Society, Vol. 146, No. 2, 1999, pp. 761-766. 

Rainer, D. R., Vesecky, S. M., Koranne, M. et al. (1997) The CO + NO reaction over Pd A combined study using single-crystal, planar-model-supported, and high-surface-area Pd/Al203 catalysts , J. Catal., 167, 234. 

In this paper we examined quantum aspects of special classical configurations of two-electron atoms. In the doubly excited regime, we found quantum states of helium that are localized along ID periodic orbits of the classical system. A comparison of the decay rates of such states obtained in one, two and three dimensional ab initio calculations allows us to conclude that the dimension of the accessible configuration space does matter for the quantitative description of the autoionization process of doubly excited Rydberg states of helium. Whilst ID models can lead to dramatically false predictions for the decay rates, the planar model allows for a quantitatively reliable reproduction of the exact life times. 

While the development of planar model catalysts has largely led to the closing of the material gap, the pressure gap still remains. Rupprechter et al.109 nicely cover some of the concerns of the pressure... 

N. L. Thompson, A. G. Palmer, L. L. Wright, and P. E. Scarborough, Fluorescence techniques for supported planar model membranes, Commun. Mol. Cell. Biophys. 5, 109-113 (1988). 

K. H. Pearce, R. G. Hiskey, and N. L. Thompson, Binding kinetics of fluorescently labeled bovine prothrombin fragment 1 at planar model membranes measured by total internal reflection fluorescence microscopy, Biophys. J. 59, 622a (1991). 

Using assumed molecular models and force constants based on the force constants derived from the paraffin series, normal co-ordinate calculations for the simple alkylcarbonium ions were carried out. These calculations were made in order to predict the vibrational spectra. Comparison with the experimentally obtained infra-red spectra show that the main observed features can indeed be reasonably explained in terms of the modes calculated for the planar models of the ions and allowed an assignment of the fundamentals (Table 11). 

To investigate whether computational methods could be used for an accurate description of the conformation of helicenes a tt-SCF-force field method was applied to hexahelicene, its 1-methyl- and 1,16-dimethyl derivatives and heptahelicene ll2). To avoid predestinate outcome of the calculations minimization of the energy was started from a nearly planar model, and the minima obtained were compared with those of a procedure starting with the geometry found by X-ray analyses. 

Using planar tetrakaidecahedra as the model, on the other hand, causes the square faces to shrink to zero area at the yield point. The unit cell therefore resembles a true rhombic dodecahedron. The elastic response was found to be anisotropic (i.e. dependent on initial cell orientation) for the planar model, up to the elastic limit. This is in contrast to the monodisperse 2D case, which is... 

Through, for reasons of mathematical simplicity, planar electrodes are assumed in most voltammetric derivations, they are seldom used in practice because of the difficulties of their fabrication. The favorite, and most convenient, electrode of experimental voltammetrists is the static mercury drop. This spherical electrode cannot always be adequately approximated by a planar model. In particular, eqn. (139) does not hold for a spherical electrode and hence semi-integration cannot be used to generate accurate values of surface concentrations. Indeed, even eqn. (137) is invalid, the concentrations of O and R not being linearly related at the surface of a spherical electrode. 

In contrast to aromatic molecules which have An + 2 n electrons, cyclobutadiene and cyclooctatetraene do not have An + 2 7r electrons and are not aromatic. In fact, diese molecules, which contain An jt electrons (n is an integer), are less stable than die planar model compounds and are termed antiaromatic. Bodi of these molecules adopt shapes that minimize interactions of die n orbitals. 

If 1 1 complexes are formed (1) in the ground state, then two types of photoreactions will occur parallel to each other, that of the bare and that of the complexed solute, each in its own type of cage. Mixed alcohol/ alkane systems, for example, show an indication of preformed solute-solvent complexes as evidenced by the picosecond experiments of Wang and Eisenthal80 81 on DMABN. Planar model systems like the indolines 3 and 5 (Sec. II.A.l) indicate that an additional channel opens for the B state in alcoholic solvents which increases the nonradiative decay path. This can explain the observed reduction of the fluorescence quantum yield in jirotic solvents about 0.1 in the aprotic polar solvent n-butyl chloride, about 0.01 in 1,2-propanediol, and about 0.001 in water.228... 

In addition to out-of-plane bending, significant in-plane splaying-out of the exocyclic valency bonds was found in the above acids. Thus, the angle between exocyclic carbon-carbon and carbon-halogen vectors is increased beyond the value of 60° appropriate to a regular planar model to 67 4° in o-chlorobenzoic acid, to 68 3° in o-bromobenzoic acid, and to 67 2° in 2-chloro-5-nitrobenzoic acid. 

The crystal structure analysis (by two-dimensional Fourier methods) was facilitated by the fact that the crystal space group requires the molecule to have symmetry 222, the asymmetric crystal unit consisting of one-quarter of the chemical molecule. If there were no distortions from a regular planar model with a trigonal arrangement of bonds... 

These displacements show up the fact that the atoms in ring D are well out of the anthracene plane. Ring D itself is fairly planar and is inclined at 18-5° to the plane of the anthracene nucleus. The bond C9—C16 is inclined at 15° to the anthracene plane, atom 15 being displaced 0-565 A, and atom 9, 0-172 A, from this plane. These distortions allow the C15 C17 separation to increase beyond the distance (2-5 A) appropriate to a regular planar model to 2-97 A and so help in... 

The molecular orbital bond length calculations, although based on a planar model, show no marked differences from those obtained experimentally, probably because the molecular distortion is not severe. No significant extensions of the bonds BC (1-48 0-03 A) and GH (1-46 0-03 A) were detected, but the accuracy of the analysis does not exclude the possibility that they may occur. 

In 3,4-benzophenanthrene, the two overcrowded carbon atoms Cx and C12 would be only 2-4 A apart in a regular planar model. This distance, while much less than the sum of the van der Waals radii, is considerably greater than a covalent bond distance. The effect of linking Cx and C12 by a covalent bond is to convert 3,4-benzophen-anthrene into 2,13-benzofluoranthene (82). The crystal structure of this... 

In dianthronylidene (98) pairs of overcrowded carbon atoms would be only 1-4 A apart in a planar model. A two-dimensional analysis by Harnik and Schmidt (1954) using Fourier methods shows that the molecule approximates to symmetry 2/m although the only crystallographic requirement is a centre. Distortion, as shown in Fig. 19, occurs to increase the separation between the overcrowded carbon atoms to... 

The model Mr+1 contains the r-th degree term in the mixture components only along with the product of this term with the first degree terms in the Zj s. For example, a planar or first-degree model in the mixture components, and a main effects only model in the process variables, is y=M1+i+e. A planar model in the Xj s, combined with a main effect plus first-order interaction effects model in the Zj s, would be y=Mi+i+Mi+2+ . The model containing up to quadratic blending terms by main effects in the Zfs is defined as y=Mi+i+M2+l+H. This continues, up to the complete 2q+n-2n term model that is defined as ... 

P. Piecuch and J. Paldus, Phys. Rev. A, 49, 3479 (1994). Application of Hilbert-Space Coupled-Cluster Theory to Simple (H2 )2 Model Systems. II. Non-Planar Models. 

The planar model which introduces covalent bonds must be excluded. Within the limits of the separation of the n and a energies, Fig. 12 gives the two possible curves, Q(U). There are... 

Rainer, D.R., Xu, C. and Goodman, D.W. (1997) Characterization and catalysis studies of small metal particles on planar model oxide supports. /. Mol. Catal. A Chem., 119, 307. 

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