Structureless continuum

If we now transfer our two interacting particles from the vacuum (whose dielectric constant is unity by definition) to a hypothetical continuous isotropic medium of dielectric constant e > 1, the electrostatic attractive forces will be attenuated because of the medium s capability of separating charge. Quantitative theories of this effect tend to be approximate, in part because the medium is not a structureless continuum and also because the bulk dielectric constant may be an inappropriate measure on the molecular scale. Eurther discussion of the influence of dielectric constant is given in Section 8.3. 

For a structureless continuum (i.e., in the absence of resonances), assuming that the scattering projection of the potential can only induce elastic scattering, the channel phase vanishes. The simplest model of this scenario is depicted schematically in Fig. 5a. Here we consider direct dissociation of a diatomic molecule, assuming that there are no nonadiabatic couplings, hence no inelastic scattering. This limit was observed experimentally (e.g., in ionization of H2S). 

Considering again the case of a structureless continuum, we have that 8j3 arises from excitation of a superposition of continuum states, hence from coupling within PHmP [69]. The simplest model of this class of problems, depicted schematically in Fig. 5b, is that of dissociation of a diatomic molecule subject to two coupled electronic dissociative potential energy curves. Here the channel phase can be expressed as... 

Uncoupled continuum, two-pathway excitation, coherence spectroscopy isolated resonances, 167-168 structureless continuum, 166... 

This effective medium or mean field assumption is easy to understand if there is a very large size difference between the newly added particles and any there previously, for example if we think of adding particles to a molecular liquid, we merely treat the liquid as a structureless continuum. However when the dimensions become comparable, the finite volume of the particles present prior to each addition must be considered, i.e. new particles can only replace medium and not particles. The consequence of this crowding is that the concentration change is greater than expressed in Equation (3.52) and it must be corrected to the volume available ... 

The potential energy curves of excited electronic states need not have potential energy minima, such as those shown in Fig. 3.6. Thus Fig. 3.7 shows two hypothetical cases of repulsive states where no minima are present. Dissociation occurs immediately following light absorption, giving rise to a spectrum with a structureless continuum. Transition a represents the case where dissociation of the molecule AB produces the atoms A and B in their ground states, and transition b the situation where dissociation produces one of the atoms in an electronically excited state, designated A. ... 

This equation is approximate since it assumes that the solvent is a structureless continuum, and so cannot interact with the reactants and the activated complex. Electrolyte solution studies demonstrate conclusively that this is not the case. In... 

In this section, we shall focus on the use of CMs to study molecules at the interface between a solid and a fluid (gas or liquid). In particular, we reserve the term continuum models to approaches that consider both the solid and the fluid as structureless continuum bodies characterized by their dielectric response, and treat the molecule at some microscopic level. 

In the diffusion region the reorientational motion of the molecules is impeded by a frictional force exerted by a medium considered structureless (continuum). For a spherical molecule, the rotational diffusion coefficient, D, is given by the Stokes-Einstein-Debye equation42... 

By contrast, the description given by a continuum description does not require any knowledge of the solvent configuration around the solute as a structureless continuum dielectric is introduced instead. The response of such a dielectric to the presence of the solute is determined by its macroscopic properties (namely the dielectric constant and the refractive index) and thus it will be implicitly averaged. Contrary to what happens in a QM/MM approach, here a single calculation on a given solute contained within the continuum dielectric will be sufficient to get the correct picture of the solvated system. 

The Polarizable Continuum Model (PCM)[18] describes the solvent as a structureless continuum, characterized by its dielectric permittivity e, in which a molecular-shaped empty cavity hosts the solute fully described by its QM charge distribution. The dielectric medium polarized by the solute charge distribution acts as source of a reaction field which in turn polarizes back the solute. The effects of the mutual polarization is evaluated by solving, in a self-consistent way, an electrostatic Poisson equation, with the proper boundary conditions at the cavity surface, coupled to a QM Schrodinger equation for the solute. 

To go further than Eq. (4.108), one has to examine the factors that govern the mean jump distance / and the jump frequency k. For this, the picture of a liquid (in which diffusion is occurring) as a structureless continuum is inadequate. In reality, the liquid has a structure—ions and molecules in definite arrangements at any one instant... 

We recall sec. 1.2.22, and in particular fig. 1.2.13 where the consequences of basing the Gibbs plane on a major or a minor component are illustrated. Statistically, adsorption from dilute solution is easy when the solvent may be interpreted primitively, i.e. as a structureless continuum. Then, much of chapter 1 may be applied after minor modification. For binary mixtures this becomes more problematic. In practice, adsorption from (dilute) solution is more frequently met than that from binarj mixtures. 

In principle all kind of interactions are contained In (3.6.11. In the present section we shall consider a solid-liquid interface although the treatment is also valid for liquid-liquid Interfaces. Solid and liquid su e taken as primittve, l.e. as structureless continuums with dielectric permittivities and = , respectively. In this model the surface is hard, planar and uniformly charged. Considering the surface charge a° as discrete would mean a further improvement. The... 

In a melt or in a concentrated solution each polymer chain is basically unperturbed and shielded from hydrodynamic interaction by the surrounding chains. Accordingly, if the medium embedding each chain could be assimilated to a structureless continuum, polymer dynamics should be well described by the so-called Rouse model. Actually, this is true only if the... 

Q.14.2 Describe some of the atomic properties of water. Is it a structureless continuum ... 

A.14.2 Water is not a structureless continuum. Water is a polar molecule with a dipole moment of 1.85 debye. Water is ideal for forming hydrogen bonds, and in bulk water these bonds are extensively formed with other water molecules... 

The study of a particular adsorption process requires the knowledge of equilibrium data and adsorption kinetics [4]. Equilibrium data are obtained firom adsorption isotherms and are used to evaluate the capacity of activated carbons to adsorb a particular molecule. They constitute the first experimental information that is generally used as a tool to discriminate among different activated carbons and thereby choose the most appropriate one for a particular application. Statistically, adsorption from dilute solutions is simple because the solvent can be interpreted as primitive, that is to say as a structureless continuum [3]. Therefore, all equations derived firom monolayer gas adsorption remain vafid. Some of these equations, such as the Langmuir and Dubinin—Astakhov, are widely used to determine the adsorption capacity of activated carbons. Batch equilibrium tests are often complemented by kinetics studies, to determine the external mass transfer resistance and the effective diffusion coefficient, and by dynamic column studies. These column studies are used to determine system size requirements, contact time, and carbon usage rates. These parameters can be obtained from the breakthrough curves. In this chapter, I shall deal mainly with equilibrium data in the adsorption of organic solutes. 

Dielectric continuum models such as the Bom model consider the solvent to be a structureless continuum of relative permittivity s. The Gibbs energy of solvation of an ion, AsoivG, is calculated by the difference of the charging process in a vacuum (s = 1) and in the solvent (e) ... 

The theories of van der Waals and double-layer forces are both continuum theories wherein the intervening solvent is characterized solely by its bulk properties such as refractive index, dielectric constant, and density. When a liquid is confined within a restricted space, it ceases to behave as a structureless continuum. At small surface separations, the van der Waals force between two surfaces is no longer a smoothly varying attraction instead, there arises an additional solvation force that generally oscillates between attraction and repulsion with distance, with a periodicity equal to some mean dimension of the liquid molecules. 

Statistically, adsorption from dilute solutions is simple because the solvent can be interpreted as primitive, that is to say as a structureless continuum. Therefore, all the equations derived from monolayer gas adsorption remain valid after replacing pressure by concentration and modifying the dimensions of some parameters. 

Since the advent of sensitive techniques for the measurement of surface forces in liquids in the 1970s, it has been apparent that a liquid medium in close proximity to a solid surface is not a structureless continuum as found in bulk, but rather, the discrete molecular nature of the solid and the liquid at the interface leads... 

As shown above the size of the explicit water simulations can be rather large, even for a medium sized protein as in the case of the sea raven antifreeze protein (113 amino acid residues and 5391 water). Simulations of that size can require a large amount of computer memory and disk space. If one is interested in the stability of a particular antifreeze protein or in general any protein and not concerned with the protein-solvent interactions, then an alternative method is available. In this case the simulation of a protein in which the explicit waters are represent by a structureless continuum. In this continuum picture the solvent is represented by a dielectric constant. This replacement of the explicit solvent model by a continuum is due to Bom and was initially used to calculate the solvation free energy of ions. For complex systems like proteins one uses the Poisson-Boltzmann equation to solve the continuum electrostatic problem. In... 

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