The Continuum Models

Although the continuum model of the ion could be analyzed by Gauss law together with spherical symmetry, in order to treat more general continuum models of electrostatics such as solvated proteins we need to consider media that have a position-specific permittivity e(r). For these a more general variant of Poisson s equation holds ... 

The continuum model, in which solvent is regarded as a continuum dielectric, has been used to study solvent effects for a long time [2,3]. Because the electrostatic interaction in a polar system dominates over other forces such as van der Waals interactions, solvation energies can be approximated by a reaction field due to polarization of the dielectric continuum as solvent. Other contributions such as dispersion interactions, which must be explicitly considered for nonpolar solvent systems, have usually been treated with empirical quantity such as macroscopic surface tension of solvent. 

The integral equation method is free of the disadvantages of the continuum model and simulation techniques mentioned in the foregoing, and it gives a microscopic picture of the solvent effect within a reasonable computational time. Since details of the RISM-SCF/ MCSCF method are discussed in the following section we here briefly sketch the reference interaction site model (RISM) theory. 

We recently proposed a new method referred to as RISM-SCF/MCSCF based on the ab initio electronic structure theory and the integral equation theory of molecular liquids (RISM). Ten-no et al. [12,13] proposed the original RISM-SCF method in 1993. The basic idea of the method is to replace the reaction field in the continuum models with a microscopic expression in terms of the site-site radial distribution functions between solute and solvent, which can be calculated from the RISM theory. Exploiting the microscopic reaction field, the Fock operator of a molecule in solution can be expressed by... 

The third group is the continuum, models, and these are based on simple concepts from classical electromagnetism. It is convenient to divide materials into two classes, electrical conductors and dielectrics. In a conductor such as metallic copper, the conduction electrons are free to move under the influence of an applied electric field. In a dielectric material such as glass, paraffin wax or paper, all the electrons are bound to the molecules as shown schematically in Figure 15.2. The black circles represent nuclei, and the electron clouds are represented as open circles. 

The continuum model with the Hamiltonian equal to the sum of Eq. (3.10) and Eq. (3.12), describing the interaction of electrons close to the Fermi surface with the optical phonons, is called the Takayama-Lin-Liu-Maki (TLM) model [5, 6], The Hamiltonian of the continuum model retains the important symmetries of the discrete Hamiltonian Eq. (3.2). In particular, the spectrum of the single-particle states of the TLM model is a symmetric function of energy. 

The continuum models represent a real alternative to the supermolecule approach. In this cases the solvation energy Esolv is assumed to be a sum of individual terms which can be calculated separately (see Eq. (6)). 

In Eq. (6) Ecav represents the energy necessary to create a cavity in the solvent continuum. Eel and Eydw depict the electrostatic and van-der-Waals interactions between solute and the solvent after the solute is brought into the cavity, respectively. The van-der-Waals interactions divide themselves into dispersion and repulsion interactions (Ed sp, Erep). Specific interactions between solute and solvent such as H-bridges and association can only be considered by additional assumptions because the solvent is characterized as a structureless and polarizable medium by macroscopic constants such as dielectric constant, surface tension and volume extension coefficient. The use of macroscopic physical constants in microscopic processes in progress is an approximation. Additional approximations are inherent to the continuum models since the choice of shape and size of the cavity is arbitrary. Entropic effects are considered neither in the continuum models nor in the supermolecule approximation. Despite these numerous approximations, continuum models were developed which produce suitabel estimations of solvation energies and effects (see Refs. 10-30 in 68)). 

The continuum model introduced by Huron and Claverie 69,70) was used in the article presented here. That model is distinguished by some special qualities, for example, the problem of the shape and size of the cavity is solved by fitting the shape to the concrete form of the molecule. The size of the cavity is chosen according to... 

The model process Eq. (15) has been studied by means of the MINDO/3 method to clarify the energetic conditions during the formation of cyclic reactive intermediates in cationic propagation of alkoxy-substituted monomers. The enthalpies of formation in the gas phase AH°g of both the alternative structures e and /were supplemented by the solvation energies Eso]v for transition into solvent CH2C12 with the assistance of the continuum model of Huron and Claverie which leads to heats of formation in solution AH° s. Table 13 contains the calculated results. 

We consider the problem of liquid and gas flow in micro-channels under the conditions of small Knudsen and Mach numbers that correspond to the continuum model. Data from the literature on pressure drop in micro-channels of circular, rectangular, triangular and trapezoidal cross-sections are analyzed, whereas the hydraulic diameter ranges from 1.01 to 4,010 pm. The Reynolds number at the transition from laminar to turbulent flow is considered. Attention is paid to a comparison between predictions of the conventional theory and experimental data, obtained during the last decade, as well as to a discussion of possible sources of unexpected effects which were revealed by a number of previous investigations. 

Consider the mass, thermal and momentum balance equations. The key assumption of the present analysis is that the Knudsen number of the flow in the capillary is sufficiently small. This allows one to use the continuum model for each phase. Due to the moderate flow velocity, the effects of compressibility of the phases, as well as mechanical energy, dissipation in the phases are negligible. Assuming that thermal conductivity and viscosity of vapor and liquid are independent of temperature and pressure, we arrive at the following equations ... 

It should be pointed out that the flow rate in the case of the Couette flow is independent of the inverse Knudsen number, and is the same as the prediction of the continuum model, although the velocity profiles predicted by the different flow models are different as shown in Fig. 4. The flow velocity in the case of the plane Couette flow is given as follows (i) Continuum model ... 

The flat interface model employed by Marcus does not seem to be in agreement with the rough picture obtained from molecular dynamics simulations [19,21,64-66]. Benjamin examined the main assumptions of work terms [Eq. (19)] and the reorganization energy [Eq. (18)] by MD simulations of the water-DCE junction [8,19]. It was found that the electric field induced by both liquids underestimates the effect of water molecules and overestimates the effect of DCE molecules in the case of the continuum approach. However, the total field as a function of the charge of the reactants is consistent in both analyses. In conclusion, the continuum model remains as a good approximation despite the crude description of the liquid-liquid boundary. 

The theoretical aspects highlighted in this section show that describing heterogeneous ET dynamics by the continuum model can be regarded as reasonable. In the next section, we shall discuss in detail the experimental advances and how some results cast doubts in the picture outlined so far. 

Marcus model for flat interfaces [Eqs. (17)-(19)], it was realized that the continuum model for a mixed solvent layer also approaches these experimental values [3-6]. 

The equations of motion can either be formulated for individual particles and the surrounding fluid, or the fluid and the particulate phases can each be considered a continuum. Both approaches yield identical results, see Glicksman et al. (1994) for a complete derivation. For our purposes, we will base the derivation on the continuum model formulated by Jackson. 

Clearly, there are two quite different types of models for a gas flow the continuum models and the molecular models. Although the molecular models can, in principle, be used to any length scale, it has been almost exclusively applied to the microscale because of the limitation of computing capacity at present. The continuum models present the main stream of engineering applications and are more flexible when applying to different macroscale gas flows however, they are not suited for microscale flows. The gap between the continuum and molecular models can be bridged by the kinetic theory that is based on the Boltzmann equation. 

The methods used for modeling pure granular flow are essentially borrowed from that of a molecular gas. Similarly, there are two main types of models the continuous (Eulerian) models (Dufty, 2000) and discrete particle (Lagrangian) models (Herrmann and Luding, 1998 Luding, 1998 Walton, 2004). The continuum models are developed for large-scale simulations, where the controlling equations resemble the Navier-Stokes equations for an ordinary gas flow. The discrete particle models (DPMs) are typically used in small-scale simulations or... 

From a computational view point, chemical reactions in solution present a yet not solved challenge. On one hand, some of the solvent effects can be approximated as if the solute molecule would be in a continuum with a given dielectric characterization of the liquid, and this view point has been pioneered by Bom [1], later by Kirkwood [2] and Onsager [3] and even later by many computational quantum chemists [4-9], On the other hand, the continuum model fails totally when one is interested in the specific... 

Meanwhile, computational methods have reached a level of sophistication that makes them an important complement to experimental work. These methods take into account the inhomogeneities of the bilayer, and present molecular details contrary to the continuum models like the classical solubility-diffusion model. The first solutes for which permeation through (polymeric) membranes was described using MD simulations were small molecules like methane and helium [128]. Soon after this, the passage of biologically more interesting molecules like water and protons [129,130] and sodium and chloride ions [131] over lipid membranes was considered. We will come back to this later in this section. 

Various components of the interactions are calculated using different formalisms. In fact, the shape and size of the cavity are defined differently in various versions of the continuum models. It is generally accepted that the cavity shape should reproduce that of the molecule. The simplest cavity is spherical or ellipsoidal. Computations are simpler and faster when simple molecular shapes are used. In Bom model, with simplest spherical reaction held, the free energy of difference between vacuum and a medium with a dielectric constant is given as [16]... 

In addition to the continuum models, the explicit solvation has also been used to quantify the reactivity [48]. In this study, the effect of solvent on the... 

Coarse-grained polymer models neglect the chemical detail of a specific polymer chain and include only excluded volume and topology (chain connectivity) as the properties determining universal behavior of polymers. They can be formulated for the continuum (off-lattice) as well as for a lattice. For all coarse-grained models, the repeat unit or monomer unit represents a section of a chemically realistic chain. MD techniques are employed to study dynamics with off-lattice models, whereas MC techniques are used for the lattice models and for efficient equilibration of the continuum models.36 2 A tutorial on coarse-grained modeling can be found in this book series.43... 

The continuum model of solvation has evolved from these beginnings. The solvent is treated as a continuous polarizable medium, usually assumed to be homogeneous and isotropic, with a uniform dielectric constant e.11-16 The solute molecule creates and occupies a cavity within this medium. The free energy of solvation is usually considered to be composed of three primary components ... 

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