Electrostatics continuous dielectric media

C) and this is the main reason for its ability to solubilize ionic solids, because the electrostatic attraction between anions and cations decreases 78.4 times in water. Polar liquids have much higher er values than non-polar molecules, and hydrogen-bonded liquids have exceptionally high values. It should be noted that e, is a macroscopic parameter derived for a continuous dielectric medium and cannot be used for interactions over molecular distances. 

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. 

The ions are regarded as rigid balls moving in a liquid bath. It is assumed that the macroscopic laws of motion in a viscous medium hold, and that the electrostatic interaction is determined by the theory of continuous dielectrics. This assumption implies that the moving particles are large compared to the molecular structure of the liquid. The most successful results of continuous theories can be found in any textbook of physical chemistry Stokes , law for viscous motion, Einstein s derivation of the dependence of viscosity on the concentration... 

The charge density p of the solute may be expressed either as some continuous function of r or as discrete point charges, depending on the theoretical model used to represent the solute. The polarization energy, Gp, discussed above, is simply the difference in the work of charging the system in the gas phase and in solution. Thus, in order to compute the polarization free energy, all that is needed is the electrostatic potential in solution and in the gas phase (the latter may be regarded as a dielectric medium characterized by a dielectric constant of 1). 

A number of attempts have been made to predict thermodynamic functions for ionizations on the basis of electrostatic theory (Benson, 1960d Frost and Pearson, 1961a). The simple Born treatment, which considers the solvent as a continuous dielectric, gives for the free energy of separation of a pair of spherical charges, ZK e and ZB e, in a medium of dielectric constant D,... 

Implicit solvation models developed for condensed phases represent the solvent by a continuous electric field, and are based on the Poisson equation, which is valid when a surrounding dielectric medium responds linearly to the charge distribution of the solute. The Poisson equation is actually a special case of the Poisson-Boltzmann (PB) equation PB electrostatics applies when electrolytes are present in solution while the Poisson equation applies when no ions are present. Solving the Poisson equation for an arbitrary equation requires numerical methods, and many researchers have developed an alternative way to approximate the Poisson equation that can be solved analytically, known as the... 

Besides the specific solvent effects such as catalysis and solvation by a protic solvent, the effects related to the properties of a solvent regarded as a continuous dielectric can be important along the reaction path. Until recently, the problem of quantification of such phenomena within the framework of the quantum-meehanical theory received relatively little attention. An early attempt to simulate solvent as a continuous medium in the ab initio geometry optimization was described by Tomasi et al. [139, 163]. They used an electrostatic interaction potential, added as a perturbation to the Hamiltonian of the reactive system, to account for the interaction between... 

The simplest starting point for the quantitative treatment of the electrostatic part of the medium effect is the Born model of rigid spherical ions in a continuous dielectric. For a molecular acid whose ions have a mean radius r, the Born estimate of the electrostatic contribution to the medium effect is... 

The popular Poisson-Boltzmann equation considers the mean electrostatic potential in a continuous dielectric with point charges and is therefore, an approximation of the actual potential. An improved model and mathematical solution resulted in the MPB equation (26). This equation is based on a restricted primitive electrolyte model that considers ions as charged hard spheres with diameter d in a continuous uniform structureless dielectric medium of constant dielectric permittivity s. The sphere representing an ion has the same permittivity e. The model initially was developed for an electrolyte at a hard wall with dielectric permittivity and surface charge density a. The charge is distributed over the surface evenly and continuously. 

Implicit solvation models developed for condensed phases represent the solvent by a continuous electric field, and are based on the Poisson equation, which is valid when a surrounding dielectric medium responds linearly to the charge distribution of the solute. The Poisson equation is actually a special case of the Poisson-Boltzmann (PB) equation PB electrostatics applies when electrolytes are present in solution, while the Poisson equation applies when no ions are present. Solving the Poisson equation for an arbitrary equation requires numerical methods, and many researchers have developed an alternative way to approximate the Poisson equation that can be solved analytically, known as the Generalized Born (GB) approach. The most common implicit models used for small molecules are the Conductor-like Screening Model (COSMO) [96,97], the Dielectric Polarized Continuum Model (DPCM) [98], the Conductor-like modification to the Polarized Continuum Model (CPCM) [99], the Integral Equation Formalism implementation of PCM (lEF-PCM) [100] PB models and the GB SMx models of Cramer and Truhlar [52,57,101,102]. The newest Miimesota solvation models are the SMD (universal Solvation Model based on solute electron Density [57]) and the SMLVE method, which combines the surface and volume polarization for electrostatic interactions model (SVPE) [103-105] with semiempirical terms that account for local electrostatics [106]. Further details on these methods can be found in Chapter 11 of reference 52. 

Although not rigorously correct, the approximation of water as a structureless homogeneous continuum dielectric medium is used by many simulative methodologies. Both Brownian dynamics (see the section entitled Implicit Solvation Brownian Dynamics) and electrodiffusive approaches (see the section on Flux-Based Simulation) include the water in the electrostatic picture as a continuous dielectric background with polarizability appropriately tuned... 

Equation (21) represents electrostatic interactions in vacuum, i.e. the dielectric constant has been given the value D = 1. Some authors have used D >1, arguing that interactions in a polarizable medium are weaker than in vacuum. This argument seems, however, to be based on a misconception. When the charge-carrying atoms constitute the medium, then their interaction must be considered as interaction in vacuum. Thus intramolecular interactions or intermo-lecular interactions in crystals must be represented by Eq. (21) with D = 1. On the other hand intermolecular electrostatic interactions between molecules dissolved in a solvent of dielectric constant D must be reduced by a factor D if (and only if) the solvent is considered as a continuous medium. The problem of the meaning of a dielectric medium at microscopic distances is very complex and could not be discussed here. 

The electrostatic polarization theory is commonly employed to describe ER response. The model assumes that ER fluids are dispersions of nonionic polarizable particles in a low dielectric medium and that free charges and charge-transfer electrochemical processes can be neglected. This model is based on the fact that, due to the permittivity mismatch between the particles 6p and the continuous phase e, the dipolar particles are polarized and aligned with the neighboring particles. When an electric field is superimposed on the point dipole interaction, the orientation of the dipoles in relation to the exter-... 

On the assumption that = 2, the theoretical values of the ion solvation energy were shown to agree well with the experimental values for univalent cations and anions in various solvents (e.g., 1,1- and 1,2-dichloroethane, tetrahydrofuran, 1,2-dimethoxyethane, ammonia, acetone, acetonitrile, nitromethane, 1-propanol, ethanol, methanol, and water). Abraham et al. [16,17] proposed an extended model in which the local solvent layer was further divided into two layers of different dielectric constants. The nonlocal electrostatic theory [9,11,12] was also presented, in which the permittivity of a medium was assumed to change continuously with the electric field around an ion. Combined with the above-mentioned Uhlig formula, it was successfully employed to elucidate the ion transfer energy at the nitrobenzene-water and 1,2-dichloroethane-water interfaces. 

Continuum solvation models are generally focused on purely electrostatic effects the solvent is a homogeneous continuous medium and its response is determined by its dielectric constant. Electrostatic effects usually constitute the dominating part of the solute - solvent interaction but in some cases explicit solute-solvent (or solute-solute)... 

Across real surfaces and interfaces, the dielectric response varies smoothly with location. For a planar interface normal to a direction z, we can speak of a continuously changing s(z). More pertinent to the interaction of bodies in solutions, solutes will distribute nonuniformly in the vicinity of a material interface. If that interface is charged and the medium is a salt solution, then positive and negative ions will be pushed and pulled into the different distributions of an electrostatic double layer. We know that solutes visibly change the index of refraction that determines the optical-frequency contribution to the charge-fluctuation force. The nonuniform distribution of solutes thereby creates a non-uniform e(z) near the interfaces of a solution with suspended colloids or macromolecules. Conversely, the distribution of solutes can be expected to be perturbed by the very charge-fluctuation forces that they perturb through an e(z).5... 

The Poisson Equation From classical electrostatics, the free charge density p(r)—that is, the charge density due to the solute as opposed to the polarization charges in the solvent—in a continuous medium of homogeneous dielectric constant (relative permittivity) e, where r denotes the position in space, is related to the electrostatic potential, ( )(r), by Poisson s equation, which takes the following form, in this case in Gaussian units ... 

The use of Coulomb s law should be regarded as a purely empirical procedure because, when two partial charges are not well separated, the solvent molecules and the rest of the solute between and around the two charges do not behave like a continuous medium of constant dielectric constant, and it is also difficult to know where the point dipoles should be located. For two partial charges separated by greater than one width of water layer it has been suggested that the effective dielectric constant approaches that of bulk water, 80 hence electrostatic interactions would be negligible at these distances. 

In the more realistic discrete-charge electrostatic theory (Tanford and Kirkwood ), the amino acid groups are point charges positioned at fixed sites on the surface of the protein or are buried at a short distance within the interior of the molecule which is assumed to be a continuous medium of low dielectric constant. The theory was successfully tested on a variety of model compounds. However, this calculation was also limited to the mutual effect of two groups only, such as the iron atom and the amino acid in a hemoglobin molecule. 

All theories depend upon the development of expressions for the variation in the relative free energies of reactants and the transition state as a function of the solvation by the medium. At this stage the fully developed theories consider the solvent as a continuous medium and use electrostatic theories to calculate the work required to produce a certain charge in the dielectric and hence changes in solute free energy resulting from changes in the solvent. 

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