Spherical cavity

The first term on the right-hand side of Equation 5.10 is due to the fact that one end of the chain can be placed anywhere inside the volume of the cavity. The second term, called the confinement free energy, arises from all allowed conformations due to chain connectivity and confinement effects, provided that one end is already placed somewhere inside the cavity. For large values of N such that Rg R, the leading term in Equation 5.11 dominates the sum. The 

is the free energy of interaction of the chain end with the pore. This term also includes a term ksTlaz associated with the loss of one segment in the extensive part of Equation 2.12. 

R) for a Gaussian tail of N segments is (Hiergeist and Lipowsky 1996, Park and Sung 1998a) 

Since only one segment is interacting with the hole, the corresponding free energy is 

We shall return to these results when we discuss the kinetics of polymer translocation under an electric field in later chapters. 

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Four volumetric defects are also included a spherical cavity, a sphere of a different material, a spheroidal cavity and a cylinderical cavity (a side-drilled hole). Except for the spheroid, the scattering problems are solved exactly by separation-of-variables. The spheroid (a cigar- or oblate-shaped defect) is solved by the null field approach and this limits the radio between the two axes to be smaller than five. 

Microscopic analyses of the van der Waals interaction have been made for many geometries, including, a spherical colloid in a cylindrical pore [14] and in a spherical cavity [15] and for flat plates with conical or spherical asperities [16,17]. 

Kirkwood generalized the Onsager reaction field method to arbitrary charge distributions and, for a spherical cavity, obtained the Gibbs free energy of solvation in tenns of a miiltipole expansion of the electrostatic field generated by the charge distribution [12, 1 3]... 

Two important contributions to the study of solvation effects were made by Bom (in 192( and Onsager (in 1936). Bom derived the electrostatic component of the free energ) c solvation for placing a charge within a spherical solvent cavity [Bom 1920], and Onsagi extended this to a dipole in a spherical cavity (Figure 11.21) [Onsager 1936]. In the Bor... 

A drawback of the SCRF method is its use of a spherical cavity molecules are rarely exac spherical in shape. However, a spherical representation can be a reasonable first apprc mation to the shape of many molecules. It is also possible to use an ellipsoidal cavity t may be a more appropriate shape for some molecules. For both the spherical and ellipsoi cavities analytical expressions for the first and second derivatives of the energy can derived, so enabling geometry optimisations to be performed efficiently. For these cavil it is necessary to define their size. In the case of a spherical cavity a value for the rad can be calculated from the molecular volume ... 

The Onsager model describes the system as a molecule with a multipole moment inside of a spherical cavity surrounded by a continuum dielectric. In some programs, only a dipole moment is used so the calculation fails for molecules with a zero dipole moment. Results with the Onsager model and HF calculations are usually qualitatively correct. The accuracy increases significantly with the use of MP2 or hybrid DFT functionals. This is not the most accurate method available, but it is stable and fast. This makes the Onsager model a viable alternative when PCM calculations fail. 

The self-consistent reaction held (SCRF) method is an adaptation of the Poisson method for ah initio calculations. There are quite a number of variations on this method. One point of difference is the shape of the solvent cavity. Various models use spherical cavities, spheres for each atom, or an isosurface... 

When diree spherical particles are sintered together, die volume between them decreases as the necks increase until a spherical cavity is left. The source of material to promote further neck growth is now removed by die coalescence of... 

A variety of methodologies have been implemented for the reaction field. The basic equation for the dielectric continuum model is the Poisson-Laplace equation, by which the electrostatic field in a cavity with an arbitrary shape and size is calculated, although some methods do not satisfy the equation. Because the solute s electronic strucmre and the reaction field depend on each other, a nonlinear equation (modified Schrddinger equation) has to be solved in an iterative manner. In practice this is achieved by modifying the electronic Hamiltonian or Fock operator, which is defined through the shape and size of the cavity and the description of the solute s electronic distribution. If one takes a dipole moment approximation for the solute s electronic distribution and a spherical cavity (Onsager s reaction field), the interaction can be derived rather easily and an analytical expression of theFock operator is obtained. However, such an expression is not feasible for an arbitrary electronic distribution in an arbitrary cavity fitted to the molecular shape. In this case the Fock operator is very complicated and has to be prepared by a numerical procedure. 

FIG. 11 Particles inside a spherical cavity. This is the generic case for a strongly confining situation. 

The simplest SCRF model is the Onsager reaction field model. In this method, the solute occupies a fixed spherical cavity of radius Oq within the solvent field. A dipole in the molecule will induce a dipole in the medium, and the electric field applied by the solvent dipole will in turn interact with the molecular dipole, leading to net stabilization. 

Continuum models of solvation treat the solute microscopically, and the surrounding solvent macroscopically, according to the above principles. The simplest treatment is the Onsager (1936) model, where aspirin in solution would be modelled according to Figure 15.4. The solute is embedded in a spherical cavity, whose radius can be estimated by calculating the molecular volume. A dipole in the solute molecule induces polarization in the solvent continuum, which in turn interacts with the solute dipole, leading to stabilization. 

The simplest reaction field model is a spherical cavity, where only the net charge and dipole moment of the molecule are taken into account, and cavity/dispersion effects are neglected. For a net charge in a cavity of radius a, the difference in energy between vacuum and a medium with a dielectric constant of e is given by the Bom model. ... 

The spherical cavity, dipole only, SCRF model is known as the OnMger model.The Kirkwood model s refers to a general multipole expansion, if the cavity is ellipsoidal the Kirkwood—Westheimer model arise." A fixed dipole moment of yr in the Onsager model gives rise to an energy stabilization. 

Let us imagine that the liquid cage is a spherical cavity in a continuous medium. When the molecule is in its centre, the orienting field is equal to zero. At this point the anisotropic part of the rotator-neighbourhood interaction appears only in the case of asymmetrical breathing of the... 

Quiben JM, Thome JR (2007b) Flow pattern based two-phase pressure drop model for horizontal tubes. Part II. New phenomenological model. Int. J. Heat and Fluid Flow. 28(5) 1060-1072 Rayleigh JWS (1917) On the pressure developed in a liquid during the collapse of a spherical cavity. Phil Mag 34 94-98... 

The tricyclic cryptand 20 has 10 binding sites and a spherical cavity.Another molecule with a spherical cavity (though not a cryptand) is 21, which complexes... 

Rayleigh L (1917) On the pressure developed in a liquid during the collapse of a spherical cavity. Philanthropic Mag 34 94-98... 

Fig. 5. Probabilities pn of observing n water-oxygen atoms in spherical cavity volumes v. Results from Monte Carlo simulations of SPC water are shown as symbols. The parabolas are predictions using the flat default model in Eq. (11). The center-to-center exclusion distance d (in nanometers) is noted next to the curves. The solute exclusion volume is defined by the distance d of closest approach of water-oxygen atoms to the center of the sphere. (Hummer et al., 1998a)... 

A fascinating insight into the impact that modelling can make in polymer science is provided in an article by Miiller-Plathe and co-workers [136]. They summarise work in two areas of experimental study, the first involves positron annihilation studies as a technique for the measurement of free volume in polymers, and the second is the use of MD as a tool for aiding the interpretation of NMR data. In the first example they show how the previous assumptions about spherical cavities representing free volume must be questioned. Indeed, they show that the assumptions of a spherical cavity lead to a systematic underestimate of the volume for a given lifetime, and that it is unable to account for the distribution of lifetimes observed for a given volume of cavity. The NMR example is a wonderful illustration of the impact of a simple model with the correct physics. 

Fig. 2.2 Self-Consistent Reaction Field (SCRF) model for the inclusion of solvent effects in semi-empirical calculations. The solvent is represented as an isotropic, polarizable continuum of macroscopic dielectric e. The solute occupies a spherical cavity of radius ru, and has a dipole moment of p,o. The molecular dipole induces an opposing dipole in the solvent medium, the magnitude of which is dependent on e.

The macrotricyclic ligands (240) and (241) may be synthesized by multistep high-dilution procedures (Graf Lehn, 1975). They contain spherical cavities which are able to accommodate suitable guests whether they be cationic, neutral, or anionic. 

Parchment et al25 also found reasonable agreement between the PCM and MD-FEP methods. Simpler SCRF approaches however differ widely. For example spherical cavity ab initio SCRF calculations predict a solvation free energy of the keto-N2H tautomer of 3-hydroxypyrole (see Sec. 3.1) of -93.5 kJ/mol in comparison to the PCM and FEP values of -9.0 kJ/mol and -12.5 kJ/mol respectively. 

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