Debye-Hiickel screening parameter

The GB model is a modification of the Coulomb equation to include the Born radius of the particle or atom which estimates the degree of the particle s burial within the molecule. Equation 5 relates AGdec to the solvent/solute dielectric (e), the separation between the partial atomic charges r, the effective Born radii R(i and /), and the smoothing function fGR. A Debye-Hiickel screening parameter (k) similar to that used in the PB equation is used to account for the monovalent ions. 

This is the screened image interaction between a point charge and an uncharged plate, both immersed in an electrolyte solution of Debye-Hiickel parameter k. Further, in the absence of electrolytes (k 0), Eq. (14.78) becomes... 

Equation (14.101) is the screened image interaction between a line charge and an uncharged plate, both immersed in an electrolyte solution of Debye-Hiickel parameter K. We see that in the former case (Cpj = 0), the interaction force is repulsion and the latter case (epi = cxd) attraction. Eurther, in the absence of electrolytes (k 0), we can show from Eq. (14.101) that the interaction force —dV IdH per unit length between plate 1 and cylinder 2 with 02 0 is given by... 

This is exactly the form expected for gyir) on the basis of the Debye-Hiickel model if 2F replaces k as the screening parameter. 

The MSA is fundamentally connected to the Debye-Hiickel (DH) theory [7, 8], in which the linearized Poisson-Boltzmann equation is solved for a central ion surrounded by a neutralizing ionic cloud. In the DH framework, the main simplifying assumption is that the ions in the cloud are point ions. These ions are supposed to be able to approach the central ion to some minimum distance, the distance of closest approach. The MSA is the solution of the same linearized Poisson-Boltzmann equation but with finite size for all ions. The mathematical solution of the proper boundary conditions of this problem is more complex than for the DH theory. However, it is tractable and the MSA leads to analytical expressions. The latter shares with the DH theory the remarkable simplicity of being a function of a single screening parameter, generally denoted by r. For an arbitrary (neutral) mixture of ions, this parameter satisfies a simple equation which can be easily solved numerically by iterations. Its expression is explicit in the case of equisized ions (restricted case) [12]. One has... 

The equilibrium pdfs can be computed nowadays very accurately by theories such as HNC [22] and some of its improved versions [23], The MSA [24, 25] is the simplest theory that satisfies all of the above conditions. It is the Debye-Hiickel theory, but solved with the condition (5.11) for all pairs of ions. The final result introduces a new screening parameter F (intead of the Debye screening parameter /c) which is calculated from an algebraic equation [26, 27, 28]. It was found that the MSA is sufficiently accurate, in all of the studied cases. 

FIGURE 2.6 Parameter space spanned by the surface density of electric charge on the particle (o, in units of e/rmd-) and the contact line radius (Tq = R sin 9y, in i,m) of a system of spherical particles at an air/water interface, assuming the Debye-Hiickel theory for water with a screening length of 1 pm. The solid line is the locus of values of Ep such that above that line the capillary attraction given by Equation 2.32 dominates over the electric repulsion. The dashed line is the locus Ep = 1 (the small-deformation approach corresponds formally to Ep < 1). (Reproduced from Dominguez, A. et al. /. Chem. Phys. 127, 204706, 2007. With permission.)... 

We begin by a calculation of the (Helmholtz) free energy in the Debye-Hiickel approximation. The Debye screening parameter k is defined by... 

---