Excluded volume forces mean field potential

It is clear that a perfect agreement with experiment cannot be provided by a theory which ignores the additional interactions between ions, and ions and surfaces, not included in the mean field potential (such as image forces,14 excluded volume effects,15 and ion-dispersion16 or ion-hydration forces17). However, it will be shown that the experimental results reported by Lopez-Leon et al.1 can be more than qualitatively reproduced for uniunivalent electrolytes by the present polarization model for hydration/double layer forces, if one accounts for the association equilibria with the surface sites for all the ions present in the electrolyte (H+, OH , anions, and cations).11 Some additional reasons for the quantitative disagreements, involving the structural modifications of the adsorbed protein layer and the nonuniformity of the colloidal particles, will be also noted. 

The van der Waals interactions between the polyelectrolyte segments and plates are included into the mean-field approach together with the excluded volume and electrostatic interactions. A repulsive contribution of the force between plates is generated by the van der Waals interactions between the segments and the plates. With increasing van der Waals interactions between the segments and the plates, the force between the plates at constant surface potential becomes more repulsive at small distances between the plates and more attractive at large... 

This is an empirical equation of the mean-field type, based on the assumption that should be proportional to the local density of monomers. The magnitude of the excluded volume interactions is described by the volume-like parameter Ve, with typical values in the order of 0.01-1 nm. The factor kT is explicitly included, not only for dimensional reasons, but also in order to stress that excluded volume energies, like hard core interactions in general, are of entropic nature (entropic forces are always proportional to T, as is exemplified by the pressure exerted by an ideal gas, or the restoring force in an ideal rubber, to be discussed in a later chapter). If the local potential experienced by a monomer is given by Eq. (2.78), then forces arise for all non-uniform density distributions. For the coil under discussion, forces in radial direction result since everywhere, with the exception of the center at x = 0, we have dcm/d x < 0. The obvious consequence is an expansion of the chain. 

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