Charge Renormalization Concept

If the value of the electrostatic potential at the particle surface is low, V (I c) hQ/Rc C 1, the electrostatic potential at r is even lower and the linearized form of the PB equation, often referred to as the Debye Hiickel (DH) approximation  

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A quantitative analysis of counterion localization in a salt-free solution of star-like PEs is carried out on the basis of an exact numerical solution of the corresponding Poisson-Boltzmann (PB) problem (Sect. 5). Here, the conformational degrees of freedom of the flexible branches are accounted for within the Scheutjens-Fleer self-consistent field (SF-SCF) framework. The latter is used to prove and to quantify the applicability of the concept of colloidal charge renormalization to PE stars, that exemplify soft charged colloidal objects. The predictions of analytical and numerical SCF-PB theories are complemented by results of Monte Carlo (MC) and molecular dynamics (MD) simulations. The available experimental data on solution properties of PE star polymers are discussed in the light of theoretical predictions (Sect. 6). 

It turns out that concepts of dielectric media are helpful to describe such a system so the effective interaction between a pair of opposite charges at distance r is not the bare interaction 2rr J ln(r/a), but rather it is screened by a distance-dependent dielectric constant (r) which must be calculated self-consistently. A renormalization group treatment (Kosterlilz, 1974 Young, 1978) shows that the quantity K n r/a) = J/(k Teir)) vanishes above 7kt for r - oo but behaves as... 

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