Isotropic shell model

Continuum shell models used to study the CNT properties and showed similarities between MD simulations of macroscopic shell model. Because of the neglecting the discrete nature of the CNT geometry in this method, it has shown that mechanical properties of CNTs were strongly dependent on atomic structure of the tubes and like the curvature and chirality effects, the mechanical behavior of CNTs cannot be calculated in an isotropic shell model. Different from common shell model, which is constmcted as an isotropic continuum shell with constant elastic properties for SWCNTs, the MBASM model can predict the chirality induced anisotropic effects on some mechanical behaviors of CNTs by incorporating molecular and continuum mechanics solutions. One of the other theory is shallow shell theories, this theory are not accurate for CNT analysis because of CNT is a... 

This method requires only a crude structural model as a starting model. In this analysis, the starting model was a homogeneous spherical shell density for the carbon cage. As for the temperature factors of all atoms, an isotropic harmonic model was used an isotropic Gaussian distribution is presumed for a La atom in the starting model. Then, the radius of the C82 sphere was refined as structural parameter in the Rietveld refinement. 

To the extent that the polarization of physical atoms results in dipole moments of finite length, it can be argued that the shell model is more physically realistic (the section on Applications will examine this argument in more detail). Of course, both models include additional approximations that may be even more severe than ignoring the finite electronic displacement upon polarization. Among these approximations are (1) the representation of the electronic charge density with point charges and/or dipoles, (2) the assumption of an isotropic electrostatic polarizability, and (3) the assumption that the electrostatic interactions can be terminated after the dipole-dipole term. 

For the first model, tibia was considered as an elastic isotropic shell and the following data (see Table 1) were used [4] ... 

The most popular model which takes into account both the ionic and electronic polarizabilities is the shell model of DICK and OVERHAUSER [4.12]. It is assumed that each ion consists of a spherical electronic shell which is isotropically coupled to its rigid ion-core by a spring. To begin with we consider a free ion which is polarized by a static field E. The spring constant is k, the displacement of the shell relative to its core is v and the charge of the shell is ye (Fig.4.7). In equilibrium, the electrostatric force yeE is equal to the elastic force kv yeE = kv. The induced dipole moment is d = yev = aE from which we obtain the free ion polarizability... 

Fig,4 7. In the shell model, an ion consists of a spherical electronic shell which is isotropically coupled to its rigid ion-core by a spring with force constant k. xe charge of the core, ye charge of the shell ... 

The two-shell model of Kerner [65] conforms to the conditions of the second group of models. The dilatation of a spherical inclusion surrounded by a homogeneous medium is derived subject to the condition that displacements and tractions at the surface of the inclusion are continuous. The homogeneous medium is supposed to have the elastic properties of the composite as a whole. The model interrelates shear (Gj) and compressive (Kj) moduli (or Poisson s ratios p ) of an arbitrary number of isotropic elements with the macroscopic moduli Gc and Kc. 

Neal and Nader [260] considered diffusion in homogeneous isotropic medium composed of randomly placed impermeable spherical particles. They solved steady-state diffusion problems in a unit cell consisting of a spherical particle placed in a concentric shell and the exterior of the unit cell modeled as a homogeneous media characterized by one parameter, the porosity. By equating the fluxes in the unit cell and at the exterior and applying the definition of porosity, they obtained... 

Quantitative models of solute-solvent systems are often divided into two broad classes, depending upon whether the solvent is treated as being composed of discrete molecules or as a continuum. Molecular dynamics and Monte Carlo simulations are examples of the former 8"11 the interaction of a solute molecule with each of hundreds or sometimes even thousands of solvent molecules is explicitly taken into account, over a lengthy series of steps. This clearly puts a considerable demand upon computer resources. The different continuum models,11"16 which have evolved from the work of Bom,17 Bell,18 Kirkwood,19 and Onsager20 in the pre-computer era, view the solvent as a continuous, polarizable isotropic medium in which the solute molecule is contained within a cavity. The division into discrete and continuum models is of course not a rigorous one there are many variants that combine elements of both. For example, the solute molecule might be surrounded by a first solvation shell with the constituents of which it interacts explicitly, while beyond this is the continuum solvent.16... 

The corresponding flow equation for K obtained in [16] deviates slightly from (23a), which can be traced back to the different RG procedures. In [16] the authors performed the RG at strictly zero temperature and used a symmetric, circular shape of the momentum-shell , i.e., treated the model as, effectively, isotropic in the 1+1-dimensional space-time. 

Continuum solvation models consider the solvent as a homogeneous, isotropic, linear dielectric medium [104], The solute is considered to occupy a cavity in this medium. The ability of a bulk dielectric medium to be polarized and hence to exert an electric field back on the solute (this field is called the reaction field) is determined by the dielectric constant. The dielectric constant depends on the frequency of the applied field, and for equilibrium solvation we use the static dielectric constant that corresponds to a slowly changing field. In order to obtain accurate results, the solute charge distribution should be optimized in the presence of the field (the reaction field) exerted back on the solute by the dielectric medium. This is usually done by a quantum mechanical molecular orbital calculation called a self-consistent reaction field (SCRF) calculation, which is iterative since the reaction field depends on the distortion of the solute wave function and vice versa. While the assumption of linear homogeneous response is adequate for the solvent molecules at distant positions, it is a poor representation for the solute-solvent interaction in the first solvation shell. In this case, the solute sees the atomic-scale charge distribution of the solvent molecules and polarizes nonlinearly and system specifically on an atomic scale (see Figure 3.9). More generally, one could say that the breakdown of the linear response approximation is connected with the fact that the liquid medium is structured [105],... 

The mean absolute deviations for isotropic NMR chemical shifts show quite clearly that the atomic functions must be allowed to depend on the magnetic field if close agreement with experiment is to be obtained at the minimal or split valence shell levels. Within the GIAO framework, the results show a marked improvement as the theoretical model is improved. However, the study presented here is by no means exhaustive and further theoretical work is necessary to document more adequately the performance of these computational methods for magnetic shielding parameters. 

Solid state physicists are familiar with the free- and nearly free-electron models of simple metals [9]. The essence of those models is the fact that the effective potential seen by the conduction electrons in metals like Na, K, etc., is nearly constant through the volume of the metal. This is so because (a) the ion cores occupy only a small fraction of the atomic volume, and (b) the effective ionic potential is weak. Under these circumstances, a constant potential in the interior of the metal is a good approximation—even better if the metal is liquid. However, electrons cannot escape from the metal spontaneously in fact, the energy needed to extract one electron through the surface is called the work function. This means that the potential rises abruptly at the surface of the metal. If the piece of metal has microscopic dimensions and we assume for simplicity its form to be spherical - like a classical liquid drop, then the effective potential confining the valence electrons will be spherically symmetric, with a form intermediate between an isotropic harmonic oscillator and a square well [10]. These simple model potentials can already give an idea of the reason for the magic numbers the formation of electronic shells. 

Theret et al. [1988] analyzed the micropipette experiment with endothelial cell. The cell was interpreted as a linear elastic isotropic half-space, and the pipette was considered as an axisymmetric rigid ptmch. This approach was later extended to a viscoelastic material of the cell and to the model of the cell as a deformable layer. The solutions were obtained both analytically by using the Laplace transform and numerically by using the finite element method. Spector et al. [ 1998] analyzed the application of the micropipette to a cylindrical cochlear outer hair cell. The cell composite membrane (wall) was treated as an orthotropic elastic shell, and the corresponding problem was solved in terms of Fourier series. Recently, Hochmuth [2000] reviewed the micropipette technique applied to the analysis of the cellular properties. 

Extrapolation of the SANS data in [4] to the isotropic state confirms, indirectly, the presence of a diffuse PS-PI transition layer between filler and rubbery matrix with thickness A 0.5 nm around the PS domain with a mean filler radius of about 84 A. Excellent agreement between measured reinforcing factor and corresponding model predictions could be realized within a very recent approach of Huber and Vilgis [5] for the hydrodynamic reinforcement of rubbers filled with spherical fillers of core-shell structures [6]. 

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