IMPOSING DIRICHLET BOUNDARY CONDITION ON CLOSED IONIC SYSTEMS CONTAINING INHOMOGENEOUS DIELECTRICS. 
APPLICATION IN MONTE CARLO SIMULATIONS OF ELECTRICAL DOUBLE LAYERS
                         
			 Dezso Boda and Monika Valisko
                       Department of Physical Chemistry
                           University of Veszprem
                               H-8201 Hungary
  
                              Douglas Henderson
                  Department of Chemistry and Biochemistry
                           Brigham Young University
                                Provo UT 84602
  
 
                        Bob Eisenberg and Dirk Gillespie
              Department of Molecular Biophysics and Physiology
                             Rush Medical School
                              Chicago IL 60612                  
			      
                             Wolfgang Nonner
                     Department of Physiology and Biophysics
                       University of Miami Medical School
                               Miami FL 33101
		
It has been a central problem of computer simulations of ionic systems to correctly handle electrostatics for decades.
Problems usually arise from the unavoidable facts that the simulation cell is necessarily final and that the electrostatic interactions are long ranged.
Several methods (such as various summation techniques) to take into account
long range corrections of charges outside the central simulation cell have been proposed using periodic boundary conditions, which are generally used to mimic a macroscopic sample.

We propose an alternative route which assumes a finite simulation cell confined by surfaces for wich various boundary conditions can be applied.
The Dirichlet boundary condition imposes a prescribed potetial on the confining surfaces (which we call electrodes). 
The boundary conditions are satisfied by placing an appropriate amount of surface charge (the electrode charge) on the electrodes.
We propose a numerical procedure with which this electrode charge can be calculated in every simulation step by solving a matrix equation.
The matrix equation is obtained from an integral equation after discretizing the surface of the electrode.
In this approach, the effect of the charges outside the simulation cell is no longer a consideration.
The region outside the central cell is isolated from the cell by the Dirichlet boundary; the boundary condition implicitely contains the effect of outer world.
The method can be coupled to calculations of polarization charges induced on dielectric boundaries (Boda et al. Phys. Rev. E, 69, 046702, 2004).
		
The advantages of this approach, wich is commonly used in simulations of semiconductor devices, are manyfold:
(1) The electrostatics is correctly treated in a framework that is the natural choice for small finite systems.
(2) If dielectric boundaries are present, the dielectric polarization charges induced by ions harm the condition of electroneutrality.
In this approach, the appropriate amount of electrode charge is induced on the Dirichlet boundaries and overall charge neutrality is automatically maintained.
(3) In some situations, this geometry is closer to experimental situations. The best example is the electrical double layer, which is formed at the interface of an electrolyte and an electrode. 
The model that underlies classical theories (such as the Gouy-Chapman theory) of the electrochemical double layer as well as earlier simulation studies assumes that the electrolyte (which is modelled as hard sphere ions solvated in a continuum solvent) is in contact with a hard wall carrying uniform surface charge and that the dielectric constant of the electrode is the same as that of the solution.  
In reality, the electrode is a metal and the surface charge charge in nonuniform.
Furthermore, usually, it is the voltage rather than the charge that is fixed.

In this contribution, we present our methodology, report Monte Carlo results on the electrochemical double layer in this new approach and compare our results to those obtained in the traditional approach.