Electric(al) Double Layer for Unsymmetrical Electrolytes: Tutorials

Abstract

A tutorial of the theory of the electrical double layer for unsymmetrical electrolyte is given. The GCS (Gouy-Chapman-Stern)[1, 2, 3] model based on the Grahame's method[4] is reformulated and the results calculated by the analytical model are compared with the numerical calculation by the use of Poisson-Boltzmann(PB) equation. We also show that the GCS model may be not good approximation for the electrical double layer in multivalent electrolytes solution, because our Monte Calro(MC) simulation showed the potential inversion region is appeared[5], which can not reproduce by the GCS model. GCS PB MC GCS 1 GCS theory of Electric Double Layer: Unsymmetric electrolytes In the previous tutorial (TU1) the basics of the eletric-double-layer was reviewd[6]. In the present tutorial the electric-double-layer of unsymmetrical electrolyte is given and we will show that the formulation by Grahame[4] and Smagala and Fawcett[7] may be not correct in some equations. Our formulation may be justified by comparing the results obtained by the analytical solution in the present study with the numerical solutions by solving the PB equation directly. We will also show GCS theory does not work in the case of the multi-valent electrolytes by our MC simulation. The MC simulation reproduces the Torrie-Valleau results[5] which showed that the potential inversion region is appeared. When the valence of the ion is given by Z, for example, in the case of the unsymmetrical electrolytes such as MgCl 2 , Z(cation) = +2, and Z(anion) =-1, and Na 2 SO 4 , Z(cation) = +1, and Z(anion) =-2. If we define the electrolyte by the valence, we will call MgCl 2 is 2:1 electrolyte and Na 2 SO 4 1:2 electrolyte 1 The electric neutrality condition in bulk solution can be written by ∑ i eZ i n 0 i = 0 (1) Here e is the elementary charge, Z i is the valence of ion speceis i, n 0 i is the bulk number density of species i.