Theoretical continuum modelling

Continuum models go one step frirtlier and drop the notion of particles altogether. Two classes of models shall be discussed field theoretical models that describe the equilibrium properties in temis of spatially varying fields of mesoscopic quantities (e.g., density or composition of a mixture) and effective interface models that describe the state of the system only in temis of the position of mterfaces. Sometimes these models can be derived from a mesoscopic model (e.g., the Edwards Hamiltonian for polymeric systems) but often the Hamiltonians are based on general symmetry considerations (e.g., Landau-Ginzburg models). These models are well suited to examine the generic universal features of mesoscopic behaviour. 

The theoretical methods can be divided into two fundamental groups. The so-called continuum models are characterized by assuming that the medium is a structureless and polarizable dielectricum described only by macroscopic physical constants. On the other hand there are the so-called discrete models. The main advantage of... 

For applications in the field of micro reaction engineering, the conclusion may be drawn that the Navier-Stokes equation and other continuum models are valid in many cases, as Knudsen numbers greater than 10 are rarely obtained. However, it might be necessary to use slip boimdaty conditions. The first theoretical investigations on slip flow of gases were carried out in the 19th century by Maxwell and von Smoluchowski. The basic concept relies on a so-called slip length L, which relates the local shear strain to the relative flow velocity at the wall ... 

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. 

Although the trajectory and convective diffusion techniques are conceptually simple, certain mechanisms, in particular, the exact role of the intermolecular force between the particle and the electrode remains an element of debate. Most of these problems arise because continuum models about short-range interactions break down at very short distances, where other factors, much less defined come into play. A complete understanding of the coelectrodeposition process requires a synergy between theoretical models and thorough experimental work. 

Another aspect that has been theoretically studied109,124,129 is experimental evidence that Diels-Alder reactions are quite sensitive to solvent effects in aqueous media. Several models have been developed to account for the solvent in quantum chemical calculations. They may be divided into two large classes discrete models, where solvent molecules are explicitly considered and continuum models, where the solvent is represented by its macroscopic magnitudes. Within the first group noteworthy is the Monte Carlo study... 

Abstract This chapter reviews the theoretical background for continuum models of solvation, recent advances in their implementation, and illustrative examples of their use. Continuum models are the most efficient way to include condensed-phase effects into quantum mechanical calculations, and this is typically accomplished by the using self-consistent reaction field (SCRF) approach for the electrostatic component. This approach does not automatically include the non-electrostatic component of solvation, and we review various approaches for including that aspect. The performance of various models is compared for a number of applications, with emphasis on heterocyclic tautomeric equilibria because they have been the subject of the widest variety of studies. For nonequilibrium applications, e.g., dynamics and spectroscopy, one must consider the various time scales of the solvation process and the dynamical process under consideration, and the final section of the review discusses these issues. 

In addition to heterocycles, other molecular systems have attracted theoretical attention with respect to prediction of tautomeric equilibria and solvation effects thereon. The most commonly studied example in this class is the equilibrium between formamide and formamidic acid, discussed in the next section. In addition, some continuum modeling of solvation effects on keto/enol equilibria have appeared these are presented in section 4.2.2.2. We note that the equilibrium... 

Tomasi, J., Bonaccorsi, R., Cammi, R. and Olivares del Valle, F. J. Theoretical chemistry in solution. Some results and perspectives of the continuum methods and in particular of the polarizable continuum model,. J.Mol.Struct., 234 (1991), 401-424... 

Most theoretical studies of outer-sphere (nonbond-breaking) electron transfer reactions at the metal-solution interface involve major simplifying assumptions regarding the molecular and electronic structure of the solvents and the metal. Although the importance of molecular structure and the dynamics of the solvent has been recognized, most of the theoretical work in this area has been based on a highly simplified continuum model. ... 

Analysis of the experimental measurements of transient solvation (primarily C(r)) in terms of contemporary theoretical models has led to several conclusions [15,22-26,30-33,41], which are reviewed in detail in Section II. Continuum treatments are seen to fail in several cases, but are remarkably predictive considering the simplicity of the model. Qualitative features predicted by theories that go beyond the simple continuum model are borne out in experiment, although the agreement is qualitative at best. 

A self-consistent field theory (SCFT) for micelle formation of block copolymers in selective solvents was developed by Yuan el at. (1992). They emphasized the importance of treating the isolated chain at the same level of theoretical approximation at the micelle, in contrast to earlier approaches. This was achieved by modifying the Edwards diffusion equation for the excluded volume of polymers in solution to the case of block copolymers, with one block in a poor solvent. The results of the continuum model were compared to experimental results for PS-PI diblocks in hexadecane, which is a selective solvent for PI and satisfactory agreement was obtained. 

In more recent work, Johnston and co-workers (17,18,20,27,32) showed quantitatively that the local fluid density about the solute is greater than the bulk density. In these papers, results were presented for CQ2, C2H4, CF3H, and CF3C1. Local densities were recovered by comparison of the observed spectral shift (or position) to that expected for a homogeneous polarizable dielectric medium. Clustering manifests itself in deviation from the expected linear McRae continuum model (17,18,20,27,32,56,57). These data were subsequently interpreted using an expression derived from Kirkwood-Buff solution theory (20). Detailed theoretical... 

The molecular mechanism of the Hoffmann elimination involving (iV-Cl)-N-methyl-ethanolamine has been theoretically characterized by using DFT at the B3LYP/ 6-31++G computing level.49 The role of water as a solvent has been analysed by using both discrete and hybrid discrete-continuum models. The rearrangement proceeds by a water-assisted asynchronous concerted mechanism. 

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