Spherical dielectric boundary

The simplest simulated system is a Stockmayer fluid structureless particles characterized by dipole-dipole and Lennard-Jones interactions, moving in a box (size L) with periodic botmdary conditions. The results described below were obtained using 400 such particles and in addition a solute atom A which can become an ion of charge q embedded in this solvent. The long range nature of the electrostatic interactions is handled within the effective dielectric environment seheme. In this approach the simulated system is taken to be surrounded by a continuum dielectric environment whose dielectric constant e is to be chosen self consistently with that eomputed from the simulation. Accordingly, the electrostatic potential between any two partieles is supplemented by the image interaction associated with a spherical dielectric boundary of radius (taken equal to L/2) placed so that one of these... 

Boundary effects on the electrophoretic mobility of spherical particles have been studied extensively over the past two decades. Keh and Anderson [8] applied a method of reflections to investigate the boundary effects on electrophoresis of a spherical dielectric particle. Considered cases include particle motions normal to a conducting wall, parallel to a dielectric plane, along the centerline in a slit (two parallel nonconducting plates), and along the axis of a long cylindrical pore. The double layer is assumed to be infinitely thin... 

A dielectric sphere of dielectric coefficient e embedded in an infinite dielectric of permittivity 82 is an important case from many points of view. The idea of a cavity formed in a dielectric is routinely used in the classical theories of the dielectric constant [67-69], Such cavities are used in the studies of solvation of molecules in the framework of PCM [1-7] although the shape of the cavities mimic that of the molecule and are usually not spherical. Dielectric spheres are important in models of colloid particles, electrorheological fluids, and macromolecules just to mention a few. Of course, the ICC method is not restricted to a spherical sample, but, for this study, the main advantage of this geometry lies just in its spherical symmetry. This is one of the simplest examples where the dielectric boundary is curved and an analytic solution is available for this geometry in the form of Legendre polynomials [60], In the previous subsection, we showed an example where the SC approximation is important while the boundaries are not curved. As mentioned before, using the SC approximation is especially important if we consider curved dielectric boundaries. The dielectric sphere is an excellent example to demonstrate the importance of curvature corrections . 

Let us consider a spherical dielectric particle (phase 1), which is immersed in a nonpolar medium (phase 2), near its boundary with a third dielectric medium (phase 3) see the inset in Figure 4.26. The interaction is due to electric charges at the particle surface. The theoretical problem has been solved exactly, in terms of Legendre polynomials, for arbitrary values of the dielectric constants of the three phases, and expressions for calculating the interaction force, F, and energy, W, have been derived [350] ... 

Slightly different sets of values for the model compound pK s in water, P amodel,/ ave been tabulated. The protein dielectric boundary is usually taken to be a Richards probe-accessible surface,computed with a spherical probe of radius of 1.4 A, and an initial dot density of 500 per atom. ... 

Since our treatment of the ionic atmosphere around a dipolar molecule makes use of the Onsager model, it becomes necessary to adopt a similar model for the ion. Consequently we are going to assume that the ion is also represented by a spherical cavity in the surrounding dielectric with a point charge at its center. Then the constants by the ordinary boundary conditions become... 

In 1920, Max Born, a Nobel Prize winner, published some work on the free energy of solvation of ions, AGgon [21]. He conceived the idea of approximating the solvent surrounding the ion as a dielectric continuum. Defining a spherical boundary between the ion and the continuum by an effective ion-radius, f lon, he got the simple result... 

Lorentz calculated EL in the following way. A spherical region within the dielectric, centred on the point X at which EL is required, is selected. The radius is chosen so that, as viewed from X, the region external to the spherical boundary can be regarded as a continuum, whereas within the boundary the discontinuous atomic nature of the dielectric must be taken into consideration. EL can then be written... 

As shown by Mie " and Debye,the electromagnetic field of the light scattered by a sphere can be presented as an infinite series over associated Legendre polynomials, P (cos0), multiphed by spherical Bessel functions, (InrlX). The coefficients in this series must be determined from the boundary conditions and afterward can be used to calculate the angular dependence of the amplitude and polarization of the scattered field. Different boundary conditions were imposed in the case of conducting or dielectric materials of the sphere and of the medium. 

Water and matrix are characterized by their respective dielectric constants, e and t. Typical value are e n. 80, e 2, so that e e. We first write the self energy of one cation in the vesicle. It expresses the repulsion by the electrical image due to the spherical boundary. As is well known (15), there is no exact analytical expression for it in the spherical geometry, but if we neglect terms in 1/e2 it can be written as... 

Equation 3 is valid for any volume. For a limited volume boundary conditions may be considered. Thus for a spherical volume surrounded by its own medium (spherical cavity in a dielectric), the field E0 must be replaced by... 

These difficulties have been avoided by Frdhlich20 whose Reasoning is very similar to Kirkwood s but who has chosen his model in such a manner that he, need consider no boundary effect. He has treated the deformation polarization as a macroscopic phenomenon. Molecules are replaced by a set of nondeformable point dipoles, having a moment p and placed in a continuous medium of dielectric constant n2(n — refractive index), accounting for deformation effects. The moment of a spherical molecule is given by... 

It is the right-hand side that is to be identified with (2.37) and the left-hand side with (2.30c). The latter equation can be thought of as the equation relevant to a sample of material of macroscopic radius R embedded in an infinite system of the same material (i.e., an infinite system of dielectric constant e). Moreover, on the macroscale determined by the length unit R, the system external to the sphere can be regarded dielectrically simply as a continuum of dielectric constant e. Thus (2.37) and (2.30c) are the relevant equations for the same macroscopic spherical sample embedded in continua of dielectric constant 1 and e, respectively. These results can be generalized to a sample embedded in a continuum of arbitrary dielectric constant e, as discussed by de Leeuw, Perram, and Smith, who use the generalization to illuminate the status of Ewald summation in systems with periodic boundary conditions. We review their work in Section III.C. 

Two other methods, both of which explicitly avoid the use of periodic boundary conditions, have been proposed for the simulation of dipolar systems. Friedman has proposed that a sample of N particles be enclosed in a spherical cavity within a dielectric continuum and has shown how the energy of such a system can be obtained. This method, however, has been applied in MC calculations by Valleau and Minns and does not appear to work well in practice. 

It is a simple matter to rework the standard electrostatic problem of a sphere in a slightly divergent field using the requirement that the normal component of D (rather than ) ) is continuous across the boundary. The resulting relation between E inside the sphere and 0, the field in the absence of the spherical body, as shown in standard texts for ideal dielectrics, but with complex dielectric constants, and as justifiable by the procedure in Ref. 1, is... 

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