Acta Chim. Slov. 2014, 61, 241-245 241 Scientific paper Relative Permittivity in Stern and Diffuse Layers Ekaterina Gongadze1'2 and Ale{ Igli~1,2 * 1 Laboratory of Biophysics, Faculty of Electrical Engineering, University of Ljubljana, Tržaška cesta 25, SI-1000 Ljubljana, Slovenia 2 Laboratory of Clinical Biophysics, Faculty of Medicine, University of Ljubljana, Zaloška 9, SI-1000 Ljubljana, Slovenia * Corresponding author: E-mail: ales.iglic@fe.uni-lj.si Received: 16-09-2013 Paper based on a presentation at the 4th RSE-SEE 2013 Symposium on Electrochemistry in Ljubljana, Slovenia Abstract Stern layer and outer Helmholtz plane (i.e. the distance of closest approach) are considered within electric double layer models, where the orientational ordering of water dipoles is explicitly taken into account. It is shown that permittivity of the Stern layer is not independent of the surface charge density as it is frequently assumed in different theoretical models and simulations, but strongly depends on the magnitude of the surface charge density. Therefore, to predict the behaviour and realistic values of the surface potential and electric field, in the electrolyte solution near the charged surface, requires a surface charge density dependent permittivity of the Stern layer. Keywords: Outer Helmholtz plane, Stern layer, relative permittivity, water ordering 1. Introduction In the complex interface charged surface - surrounding biosystem, electric double layer (EDL) plays the leading role. It causes the ions and water molecules to rearrange near the charged surface and thus to screen the electric potential.1-9 The electrostatic forces acting there make the counterions (i.e. ions with a charge of the opposite sign than the charged surface) to accumulate close to the surface, while the coions (i.e. the ions with a charge of the same sign as the surface) to deplete near the surface (Fig. 1). Most of the theoretical models, of an electrolyte solution in contact with a charged surface, assume that the relative (dielectric) permittivity (er) is constant everywhere in the solution.5-8 The classical Poisson-Boltzmann (PB)2-3,5 theory treats ions in the electrolyte solution as di-mensionless and does not consider the reduced permittivity of electrolyte solution near the charged surface.10 Thus the PB theory has been upgraded by hydration models, where interplay between solvent polarization and the diffuse double layer takes place.11-18 Study of the orienta-tional ordering of water dipoles at the charged surface has shown that dipoles on average are oriented perpendicularly to the charged surface.8 The spatial decay of solvent polarization for increasing distance from the charged membrane surface was predicted.11-17 a<0 OHP Stem b diffuse layer layer Figure 1. Schematic figure of an electrolyte solution, near a charged surface, consisting of Stern layer (0 < x < b) and diffuse electric double layer (b < x < <»). The outer Helmholtz plane (OHP) is located at the distance of closest approach (x = b) which is approximately equal to the hydrated radius of the counterions involved. Note that the water dipoles are oriented in a close vicinty to the charged surface as well as around cations and anions in the bulk electrolyte solution. Here C stands for the surface charged density and p for the volume charge density of the electrolyte solution. Gongadze and Iglic: Relative Permittivity in Stern and Diffuse Layers 242 Acta Chim. Slov. 2014, 61, 241-245 242 Helmholtz1 treated the double layer as a simple capacitor, assuming that the surface charge density (a) is neutralized by the counterions located at a distance equal to their hydrated radius. Gouy2 and Chapman3 considered the thermal motion of ions within PB approach, while Stern4 combined the Helmholtz1 and Gouy-Chapman models2-3 by introducing the outer Helmholtz plane (OHP), where the hydrated counterions are at the distance of closest approach (x = b) and the diffuse double layer starts (Fig. 1). In this work we combine Helmholtz model and EDL models which takes into account orientational ordering of water, resulting in a spatial variation of permittivity, and the cavity field. Two different EDL models are considered: the modified Langevin Poisson-Boltzmann model8,19-20 for point-like ions and Gongadze-Iglic model8,17 which takes into account the finite size of ions. The permittvity within Stern layer (see Fig. 1) is calculated for different values of the surface charge density using finite element method (FEM) in Comsol Mul-tiphysics 4.3a as described in details in the references the- rein 8,17 2. Results and Discussion Taking into account that the ions in the Stern layer are depleted, the Helmholtz Langevin Poisson-Boltzmann (Helmoltz LPB) equation19-20 for point-like ions can be written in the form: > o 0(>■)) where the space dependent relative permittvity is8,20 £r(x)-n2 (2+n-\L(mAx)ß) 3 J E(x) (2) and 0(x) is the electric potential, E(x) is the magnitude of the electric field strength, e0 is the unit charge, n is the optical refractive index of water, n0 is the number density of ions in the bulk far away from the charged surface, n0w is the constant number density of water molecules, p0 is the magnitude of the water external dipole moment17, e0 is the permittivity of the free space, ¡3 = 1/kT, kT is the thermal energy, L(u) = (coth(u) - 1/u) is the Langevin function and Y = (2 + n2)/2. In the Stern layer (0 < x < b) the electric field strength is constant, therefore also the relative permittivity in this region is constant, i.e. er(x) = er (x = 0). The magnitude of the electric field strength E(x) in the region b < x is spatial dependent and consequently also er(x) varies with the distance from the charged surface.8,17 In the generalized HLPB model, the electronic polarization of water is taken into account by assuming that the point-like rigid (permanent) water dipole is embedded in the center of the sphere with a volume equal to the average volume of a water molecule in the electrolyte solution.8,20 The permittivity of the single molecule's water sphere is n2. Within Helmholtz Gongadze-Iglic (Helmholtz GI) model,8,17 the consideration of Stern layer and outer Helmholtz plane yields the following governing model equations: dx Ü -2eansn0 0 b (see Fig. 1). Therefore, also in the region (b - r) < x < b the water molecules are partially depleted and their number density nw(x) is smaller than the bulk water number density n0w (i.e. partially depleted). The boundary conditions in both models are: dx x-0 £,{x = Q)£û (5) The validity of Gauss's law at x = b is fulfilled by the following equations: dx d0 dx (6) where also =