Dielectric Screening in a Spherical Cavity

arXiv:0807.1553v1 [physics.class-ph] 9 Jul 2008 Dielectric screening in a spherical cavity Christopher J Glosser and Roger C Hill Department of Physics, Southern Illinois University, Edwardsville, IL 62026-1654 E-mail: cglosse@siue.edu, rhill@siue.edu Abstract. In this work we examine the electrostatic screening potential due to a point charge located off-centre in a spherical dielectric cavity. This potential is expanded for the case in which the dielectric constant ǫ is large, several methods of finding the terms in the expansion are investigated, and closed-form expressions are found through third order in ǫ along with error bounds. Finally, possible uses of these expressions in molecular dynamics simulations of isolated charged molecules is discussed. PACS numbers: 02.30.–f, 02.70.Ns, 41.20.Cv, 87.10.–e Submitted to: J. Phys. A: Math. Gen. Dielectric screening in a spherical cavity 2 1. Introduction Over the last generation, molecular dynamics simulations have become a vital theoretical tool in the analysis of the physical interaction of proteins. This has been driven in part by the dramatic increase in cheap computer power in the last decade, which has grown at almost an exponential rate. This growth in computer power has resulted in a similarly dramatic increase in the size of the systems studied utilizing this technique. What began as a study of modest proteins such as myoglobin has expanded to include complicated systems such as structures embedded in cellular membranes and even a tobacco mosaic virus [1]. Due to the complexity of these systems, previously ignorable errors due to approximations in the model are likely to accumulate, resulting in inaccurate results and unstable simulations. Therefore, it is of vital importance to implement as accurate a representation as possible, particularly with respect to the long-range interactions in the model. Of these, the most problematic is the electrostatic interaction between charged elements in the simulation. In addition to generating long-range forces, the electrostatic field also polarizes the media in which the simulation is taking place, effectively creating more sources for the field in the simulation. It is this aspect of the electrostatic interaction that is the most troublesome to implement accurately while keeping computational time and expense to a minimum. There have been numerous attempts to circumvent the electrostatics problem in molecular dynamics models. The classic way of doing this is to place the system in a periodic cell and implement Particle Mesh Ewald dynamics [2] to account for the long range fields. This model indeed handles the electrostatic problem while keeping the system size reasonable. However, it artificially imposes a crystalline structure on the system which may not be desirable for some applications. If one wishes to investigate an isolated structure, then the options are fairly limited. A cutoffon the electrostatic interaction is usually imposed, but this effectively isolates portions of the system from one another. These models also suffer from the defect that the system in effect becomes finite in size, ignoring a large portion of the solvent. Since the solvent — which is usually water — has a large dielectric constant (ǫ ≈80), it is quite polarizable. Therefore, the field generated by the solvent is very sensitive to the background field. Multipole methods [3] historically have had some success in dealing with these long range terms. We wish to reformulate the approach to the electrostatic interaction in molecular dynamics simulations to take into account this sensitive dependence of the system on the background field. Our model should have the following properties, • It should accurately represent the field of the solvent. • It should be relatively inexpensive from a computational point of view. • The potential in question should be a solution to Poisson’s equation, so that it represents a physically possible charge distribution. 2. Green’s Function for the Screening Potential If a charge distribution is placed in a dielectric medium that is uniform and infinite in extent, the well known result is that the electric potential is reduced by a factor of ǫ, the relative permittivity of the dielectric. This reduction is caused by an additional “screening potential” due to the polarization induced in the dielectric, which partly Dielectric screening in a spherical cavity 3 cancels the original potential. The problem is more complicated when there are dielectric boundaries involved, as in the case of a charge distribution inside a cavity within a dielectric. The interaction of charges embedded in a dielectric cavity is a surprisingly complicated and rich subject in the study of classical electromagnetic theory. Even simple systems fail to have closed-form solutions for the potential. If one wishes to construct a realistic model in which charge interacts with a dielectric, then some approximation is inevitably necessary. To begin buildi

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