Ja n 20 07 Ewald sums for Yukawa potentials in quasi - two - dimensional systems

In this note, we derive Ewald sums for Yukawa potential for three dimensional systems with two dimensional periodicity. This sums are derived from the Ewald sums for Yukawa potentials with three dimensional periodicity [G. Salin and J.-M. Caillol, J. Chem. Phys. 113, 10459 (2000)] by using the method proposed by Parry for the Coulomb interactions [D.E. Parry, Surf. Sci. 49, 433 (1975); 54, 195 (1976)]. ∗Electronic mail: [email protected] 1 The Yukawa interaction energy between two particles is given by E(r) = yiyj ǫ exp(−κr) r (1) where ǫ is the dielectric constant, κ the inverse of the screening length and yi the ”Yukawa charges” defined by the properties and the state of the system ; for instance, at the Debye-Hückel approximation for electrolytes or in the Derjaguin-Landau-VerweyOverbeek (DLVO) theory of colloids, κ and yi are related to physical parameters of systems as κ = √ q2ρ kBTǫ and yi = q exp(κσi) (1 + κσi) where σi is the diameter of the hard core of the ions, in the Debye-Hückel approximations, or the radius of macroions, in DLVO theory, and ρ and q are respectively the density of ions or counterions and their charge, kB the Boltzmann constant and T the temperature. Yukawa interactions between particles are used in numerical simulations as effective potentials to simulate systems as plasmas, dusty plasma, colloids, etc. ; on general ground, such potentials may be used as a reasonable approximation, as soon as some microscopic degrees of freedom may be approximated to a continuous background leading to a screening of the direct interaction between particles, while the spherical symmetry of the interaction is preserved. As outlined in ref.[1], if κ is large enough, the screening length can be much smaller than simulation box lengths, then interactions between particles are not long ranged and, in practice, a simple truncation of the potential, with the use of the minimum image convention, could be sufficient. On the contrary, if κ is not large or quite small, then interactions between particles may be long ranged and images of particles introduced by the periodic boundary conditions may contribute significantly to the energy of the system. In these cases, a crude truncation of the potential could lead to strong bias in computations (for Coulomb interactions, see for instance refs.[2-4] for errors introduced by crude truncations of long ranged potentials). To handle these latter cases, an Ewald method for systems with 2 three dimensional periodicity and Yukawa interaction potentials has been exhibited. Many interesting systems which interaction between particles can be approximated by Yukawa potentials are also confined to quasi-two dimensional geometries, therefore an Ewald method is of interest to permit to simulate the properties of these quasi-two dimensional systems for any value of the κ parameter including at low counterions concentration or high temperatures. In this note, we derive Ewald sums for Yukawa potential in quasi-two dimensional systems from results of ref.[1] following the same derivation done by Parry for Coulomb interactions. For Coulomb interaction in quasi-two dimensional systems several methods exist, in particular some methods used the Ewald method for three dimensional systems with a highly asymmetric box and by adding correction terms related to the total dipole of the simulation box ; a general review on Coulomb interaction in quasi two-dimensional systems is done in ref.[10]. In a forthcoming work, some numerical implementations on a test system will be given ; the present work is devoted only to provide a simple derivation of Ewald sums for Yukawa potential in quasi-two dimensional systems. As computed by Salin and Caillol, the Ewald-Yukawa interaction energy is given by E = Er + Ek − ESelf (2) with the short ranged contribution Er = 1 4 ∑