Continuum models theory

Computational methods have accompanied the development of the Polarizable Continuum Model theory throughout its history. In the building of the molecular cavity and its sampling together with the resolution of the BEM equations we nowadays have a large choice of alternative algorithms, suitable for all kinds of molecular calculations. Linear scaling both in time and space is achieved in both fields. 

R. Cammi, L. Frediani, B. Mennucci, K. Ruud, Multiconfigurational self-consistent field linear response for the polarizable continuum model Theory and application to ground and excited-state polarizabilities of para-nitroanUine in solution. J. Chem. Phys. 119, 5818 (2003)... 

Continuum Model Theory in Order to Predict Macroscopic Data. 257... 

The advantage of the continuum model theories [24] is that they provide simple expressions for the orientational correlation functions that can be tested against experiments. This is especially tme for complex liquids where a microscopic theory is bound to be complicated. The main drawback of the continuum model theories is that they ignore the intermolecular correlations that are present in a dense dipolar liquid. If these correlations are important in a relaxation process, then the continuum model is obviously inadequate. 

Constanciel R and R Contreras 1984. Self-Consistent Field Theory of Solvent Effects Representation by Continuum Models - Introduction of Desolvation Contribution. Theoretica Chimica Acta 65 1-11. 

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 quantitative theory of ionic reactions, within the limitations of a continuum model of the solvent, is based on the Bom equation for the electrostatic free energy of transfer of an ion from a medium of e = 1 to the solvent of dielectric constant... 

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. 

The subject of this chapter is single-phase heat transfer in micro-channels. Several aspects of the problem are considered in the frame of a continuum model, corresponding to small Knudsen number. A number of special problems of the theory of heat transfer in micro-channels, such as the effect of viscous energy dissipation, axial heat conduction, heat transfer characteristics of gaseous flows in microchannels, and electro-osmotic heat transfer in micro-channels, are also discussed in this chapter. 

When the length scale approaches molecular dimensions, the inner spinning" of molecules will contribute to the lubrication performance. It should be borne in mind that it is not considered in the conventional theory of lubrication. The continuum fluid theories with microstructure were studied in the early 1960s by Stokes [22]. Two concepts were introduced couple stress and microstructure. The notion of couple stress stems from the assumption that the mechanical interaction between two parts of one body is composed of a force distribution and a moment distribution. And the microstructure is a kinematic one. The velocity field is no longer sufficient to determine the kinematic parameters the spin tensor and vorticity will appear. One simplified model of polar fluids is the micropolar theory, which assumes that the fluid particles are rigid and randomly ordered in viscous media. Thus, the viscous action, the effect of couple stress, and... 

Marten, B., K. Kim, C. Cortis, R. A. Friesner, R. B. Murphy, M. N. Ringnalda, D. Sitkoff, and B. Honig. 1996. New Model for Calculation of Solvation Free Energies Correction to Self-consistent Reaction Field Continuum Dielectric Theory for Short-Range Hydrogen-Bonding Effects. J. Phys. Chem. 100, 11775. 

Fortunelli, A. and J. Tomasi. 1994. The implementation of density functional theory within the polarizable continuum model for solvation. Chem. Phys. Lett. 231, 34. 

Theories of solvated electrons may be divided as follows (Jortner, 1970 Webster and Howat, 1972 Kevan, 1974 Kestner, 1976) (1) molecular orbital models, (2) structural models, (3) continuum models, and (4) semicontinuum models. We will consider these models a little in detail. 

Certain difficulties remain, however, with this approach. First, such an important feature as a secondary structure did not find its place in this theory. Second, the techniques of sequence design ensuring exact reproduction of the given conformation are well developed only for lattice models of polymers. The existing techniques for continuum models are complex, intricate, and inefficient. Yet another aspect of the problem is the necessity of reaching in some cases beyond the mean field approximation. The first steps in this direction were made in paper [84], where an analog of the Ginzburg number for the theory of heteropolymers was established. 

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. 

Bokkers, G. A., Van Sint Annaland, M., and Kuipers, J. A. M., Comparison of continuum models using kinetic theory of granular flow with discrete particle models and experiments extent of particle mixing induced by bubbles. Proceedings of Fluidization XI, May 9-14, 2004, 187-194, Naples, Italy (2004). 

R. Constanciel and R. Contreras, Self-consistent field theory of solvent effects representation by continuum models-introduction of desolvation contribution, Theor. Chim. Acta 65 1 (1984). 

Continuum models have a long and honorable tradition in solvation modeling they ultimately have their roots in the classical formulas of Mossotti (1850), Clausius (1879), Lorentz (1880), and Lorenz (1881), based on the polarization fields in condensed media [32, 57], Chemical thermodynamics is based on free energies [58], and the modem theory of free energies in solution is traceable to Bom s derivation (1920) of the electrostatic free energy of insertion of a monatomic ion in a continuum dielectric [59], and Kirkwood and Onsager s... 

Section 2 presents a review of the theory underlying self-consistent continuum models, with section 2.1 devoted to electrostatics and section 2.2 devoted to the incorporation of non-electrostatic effects into continuum solvation... 

Finally we address the issue of contributions. In our view it is unbalanced to concentrate on a converged treatment of electrostatics but to ignore other effects. As discussed in section 2.2, first-solvation-shell effects may be included in continuum models in terms of surface tensions. An alternative way to try to include some of them is by scaled particle theory and/or by some ab initio theory... 

We have given some highlights of a theory which combines the familiar multistate VB picture of a molecular system with a dielectric continuum model for the solvent which accounts for the solute s boundary effects — due to the presence of a van der Waals cavity which displays the solute s shape — and includes a quantum model for the electronic solvent polarization. 

Continuum model has been applied for the first time to predict the Fukui functions of formaldehyde, methanol, acetone, and formamide in water medium [54], The results reveal that the potential for electrophilic and nucleophilic attack increases when passing from the gas phase to an aqueous medium. The calculated Fukui functions for formaldehyde at Hartree-Fock (HF) level of theory are presented in Table 26.2. 

Even here, specific solvent effects which are not considered in the continuum theory begin to appear, most notably for hydrox-ylic solvents. However, recent work of our own suggests that it may be possible to make straightforward empirical corrections to the dielectric continuum model for such cases. 

Marten B, Kim K, Cortis C, Friesner RA, Murphy RB, Ringnalda MN, Sitkoff D, Honig B. New model for calculation of solvation free energies correction of self-consistent reaction field continuum dielectric theory for short-range hydrogen-bonding effects, / Phys. Chem. 1996, 100, 11775-11788. 

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