Charging free energy contributions

The electrostatic free energy contribution in Eq. (14) may be expressed as a thennody-namic integration corresponding to a reversible process between two states of the system no solute-solvent electrostatic interactions (X = 0) and full electrostatic solute-solvent interactions (X = 1). The electrostatic free energy has a particularly simple form if the thermodynamic parameter X corresponds to a scaling of the solute charges, i.e., (X,... 

The different free energy contributions lead, upon minimization with respect to the two length scales h and d, to different behaviors. Let us first consider the weak charging limit, i.e. the situation where the counterions leave the brush, d>h.ln this case, minimization of Fion + Fint with respect to the counterion height d in the limiting case of vanishing brush height (h = 0) and monomer volume = 0) leads to... 

Whereas the charging approach could be applied to any geometry but only at constant surface charge or potential, the Langmuir expression could be employed for any surface conditions but only for parallel planar plates. The addition of electrostatic, entropic, and chemical contributions would allow the calculation of the free energy of interaction for systems of any shape and any surface conditions, if one could derive a general expression for the chemical free energy contribution. 

If the grafted polymer molecules are also adsorbed on the second plate, then the force between the two plates includes both bridging and steric interactions. In this case, the adsorption contributions to the total free energy must be included among the free-energy contributions. When the polymer is charged, eqs... 

Thus finally, the free energy contributions to G show explicitly that the electrostatic charge q interacts with the potential, the electric dipole moment vector m (Fig. 2.10) interacts with the external electric field E, the traceless electric quadrupole moment Q(/ interacts with the external field gradient, and so on ... 

However, we can account for such proximity effects by correcting the electrostatic contribution to the charging free energy. 

The charging free energy, originating from the attractive interaction between solute i and solvent , has three contributions electrostatic, dispersion, and repulsion (Tomasi and Persico, 1994),... 

Figure 12 The dependence of the total free energy (b) and its additive components (a) free energy of non-Coulomb interactions F,d, free energy of elastic deformation Pei, and free energy contribution of translational entropy of counterions firansi on the swelling ratio for uncharged (i) and weakly charged (ii) gels.

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