Continuous medium approach

Impurity atoms, cation or anion vacancies are considered as defects placed in a perfect host lattice in the continuous medium approach (see Fig. 4.11). We consider several cases ... 

T, R) is the temperature and size dependent dielectric permittivity of incipient ferroelectric nanoparticles of radius R, (x is the fermion effective mass, is the effective permittivity of the particle environment, 8q is the dielectric permittivity of vacuum (in SI units). Due to the high values of e(r, R) the radius (T, R) > 5 nm is much higher than the lattice constant a = 0.4 nm, proving the validity of the effective mass approximation as well as the self-consistent background for the introduction of dielectric permittivity in the continuous medium approach [60]. 

In the quantum mechanical continuum model, the solute is embedded in a cavity while the solvent, treated as a continuous medium having the same dielectric constant as the bulk liquid, is incorporated in the solute Hamiltonian as a perturbation. In this reaction field approach, which has its origin in Onsager s work, the bulk medium is polarized by the solute molecules and subsequently back-polarizes the solute, etc. The continuum approach has been criticized for its neglect of the molecular structure of the solvent. Also, the higher-order moments of the charge distribution, which in general are not included in the calculations, may have important effects on the results. Another important limitation of the early implementations of this method was the lack of a realistic representation of the cavity form and size in relation to the shape of the solute. 

In the molecular-kinetic theory of molten polymers there is another approach to the problem which does not employ the notion of engagements. It is based on the analysis of dynamics of individual macromolecules in the medium of their like. In this case the real environment of macromolecules is substituted by a certain averaged continuous medium. The major problem of this approach is to determine the character of this me-... 

The total interaction can be determined in two ways, either by a calculation of the contributions from all the terms described above or by making proper approximations. The former way is of course more accurate, but since it leads to unsolvable equations, it is more fruitful to use the second approach. One common assumption in the theory of electrolyte solutions is to replace the solvent by a continuous medium with a dielectric constant grso- This approximation dramatically reduces the number of interacting molecules in the calculations. The only remaining particles are now the ions and the colloids, which they are treated as monopoles interacting according to Coulomb s law reduced by grso- Thus, the expression for the force F is... 

Current efforts in quantum-chemical modeling of the influence of solvents may be divided into two distinct approaches. The first, the supermolecular approximation, involves the explicit consideration of solvent molecules in quantum-chemical calculations. Another possibility for simulating solvent influence is to replace the explicit solvent molecules with a continuous medium having a bulk dielectric constant. Models of this type are usually referred to as polarized continuum models (PCMs). 

An alternative simulation procedure is to replace the explicit solvent molecules with a continuous medium having the bulk dielectric constant. - " Once the solvent has been simplified, it is much easier to employ quantum mechanical techniques for the ENP relaxation of electronic and molecular structure in solution thus this approach is complementary to simulation insofar as it typically focuses on the response of the solute to the solvent. Since the properties of the continuum solvent must represent an average over solvent configurations, such approaches are most accurately described as quantum statistical models. 

Because no general theories exist even for concentrated non-food suspensions of well defined spherical particles (Jeffrey and Acrivos, 1976 Metzner, 1985), approaches to studying the influence of the viscosity of the continuous medium (serum) and the pulp content of PF dispersions, just as for non-food suspensions, have been empirical. In PF dispersions, the two media can be separated by centrifugation and their characteristics studied separately (Mizrahi and Berk, 1970). One model that was proposed for relating the apparent viscosity of food suspensions is (Rao, 1987) ... 

Several combinations of QM approaches with a continuum dielectric model [175, 325, 360-369] have been focused upon. Continuum solvation models have their origin in the work of Onsager [325] for describing ions in solution these models have shown flexibility and accuracy enough to become a popular tool nowadays. Generally in these methods the solute is placed inside a cavity with appropriate shape, made in a continuous medium characterised by a dielectric constant. The electronic distribution of the solute induces a charge density at the surface of the cavity which creates a field that modifies the energy... 

The cyclone, or inertial separation method, is a common industrial approach for segregating a dispersed phase from a continuous medium based upon the difference in density between the phases. The concept takes advantage of the velocity lag which occurs for dense particles with respect to a lower density medium when both phases are subject to an accelerating flow field, such as within a rotating vortex. The larger the acceleration, the smaller the particle which fails to follow the continuous phase streamlines and will migrate to the outer wall of the cyclone for collection. 

In this Eulerian theoretical approach, a number of obstructions is considered as a continuous medium, or even media. They may have their own motion, for example, the waving of leafs in the case of vegetation canopy. In the case of droplets, their motion in the vertical direction and along the wind can be described by the following momentum equation for each size r ... 

The purpose of calculating Henry s Law constants is usually to determine the parameters of the adsorption potential. This was the approach in Ref. [17], where the Henry s Law constant was calculated for a spherically symmetric model of CH4 molecules in a model microporous (specific surface area ca. 800 m /g) silica gel. The porous structure of this silica was taken to be the interstitial space between spherical particles (diameter ca. 2.7 nm ) arranged in two different ways as an equilibrium system that had the structure of a hard sphere fluid, and as a cluster consisting of spheres in contact. The atomic structure of the silica spheres was also modeled in two ways as a continuous medium (CM) and as an amorphous oxide (AO). The CM model considered each microsphere of silica gel to be a continuous density of oxide ions. The interaction of an adsorbed atom with such a sphere was then calculated by integration over the volume of the sphere. The CM model was also employed in Refs. [36] where an analytic expression for the atom - microsphere potential was obtained. In Ref. [37], the Henry s Law constants for spherically symmetric atoms in the CM model of silica gel were calculated for different temperatures and compared with the experimental data for Ar and CH4. This made it possible to determine the well-depth parameter of the LJ-potential e for the adsorbed atom - oxygen ion. This proved to be 339 K for CH4 and 305 K for Ar [37]. On the other hand, the summation over ions in the more realistic AO model yielded efk = 184A" for the CH4 - oxide ion LJ-potential [17]. Thus, the value of e for the CH4 - oxide ion interaction for a continuous model of the adsorbent is 1.8 times larger than for the atomic model. 

The properties of the ions and the solvent which are ignored are similar to those ignored in the Debye-Hiickel treatment. These are very important properties at the microscopic level, but it would be a thankless task to try to incorporate them into the treatment used in the 1957 equation. Furthermore, Stokes Law is used in the equations describing the movement of the ions. This law applies to the motion of a macroscopic sphere through a structureless continuous medium. But the ions are microscopic species and the solvent is not structureless and use of Stokes Law is approximate in the extreme. Likewise, the equations describing the motion also involve the viscosity which is a macroscopic property of the solvent and does not include any of the important microscopic details of the solvent structure. The macroscopic relative permittivity also appears in the equation. This is certainly not valid in the vicinity of an ion because the intense electrical field due to an ion will cause dielectric saturation of the solvent immediately around the ion. In addition, alteration of the solvent stmcture by the ion is an important feature of electrolyte solutions (see Section 13.16). However, solvation is ignored. As in the Debye-Hiickel treatment the physical meaning of the distance of closest approach, i.e. a is also open to debate. 

The use of Coulomb s law should be regarded as a purely empirical procedure because, when two partial charges are not well separated, the solvent molecules and the rest of the solute between and around the two charges do not behave like a continuous medium of constant dielectric constant, and it is also difficult to know where the point dipoles should be located. For two partial charges separated by greater than one width of water layer it has been suggested that the effective dielectric constant approaches that of bulk water, 80 hence electrostatic interactions would be negligible at these distances. 

In his classic investigation of capillarity, Laplace [76] explained the adhesion of liquids to solids in terms of central fields of force between the volume elements of a continuous medium. This approach was illuminating about the origin of surface tension and energy and their relation to the internal pressure, and it resulted in the fundamental differential equation of capillarity which has been the basis of all... 

The theoretical result obtained for a continuous medium sphere confirms the computational results Eloc(x) x/R and is linearly proportional to the applied field Eo. The dispersion of values for Eioc(x) around the x/R line in Fig. 1 is due to the presence of Stone-Wales defects that induce some dispersion of the direction of Eloc which is perpendicular to the surface on the average. Furthermore, the fullerenes studied here are far from being a continuous surface due to their small size, which explains the deviation with the continuous model. The perspective of this approach is to extend these calculations with a frequency-dependent model by including dynamical polarizabilities and kinetic energy for dipoles and charges. 

Meredith and Tobias49 noted that Bruggeman s equation overcorrects in the concentrated ranges and devised another approach called the Distribution Model by considering only two size fractions. As in the Bruggeman equation, the smaller size fraction is added first and then is considered as part of a continuous medium having its own bulk conductivity when the larger size fraction of bubbles is added ... 

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