Counterion condensation inverted forces

A recent attempt to extend the scope of counterion condensation theory to the calculation of counterion-polyion, coion-polyion, and polyion-polyion pair potentials retains structural idealization but may nonetheless be capable of generating useful new information about the real molecular structure of polyelectrolyte solutions [57-59], The validity of the first and second stages of the theory, as discussed above, has been heavily documented, including the physical reality of the condensed layer and the onset of condensation at a critical charge density [55,56,60,61], In contrast, the inverted forces predicted by our extended theory have yet to be confirmed by experiment or simulation. We will argue, however, that their presence is at least suggested by current experimental knowledge. 

J. Ray and G. S. Manning, in Physical Chemistry of Polyelectrolytes, T. Radeva, Ed., Marcel Dekker, New York, 2001, pp. 111-133. Inverted Forces in Counterion Condensation Theory. 

The cause of the inverted attraction in the polyion-polyion case is the strong distance dependence of the condensed layer partition function Q(r) revealed in Figure 4. The entropic tendency toward attraction mentioned above in connection with Figure 4 turns out to dominate the balance of forces. Staying with a free volume interpretation of Q(r), we can observe that when two polyions approach, their electric fields merge, and the common field is relatively flat in the space between the polyions. The barrier to entropic expansion of the condensed layers having been removed, the condensed counterions flood into the large volume of space between the poly-... 

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