Entropy of counterions

When the network chains contain ionic groups, there will be additional forces that affect their swelling properties. Translational entropy of counterions, Coulomb interactions, and ion pair multiplets are forces that lead to interesting phenomena in ion-containing gels. These phenomena were studied in detail by Khokhlov and collaborators [74-77]. The free energy of the networks used by this group is... 

Other molecular thermodynamic models for protein-reverse micelle complexes have also emerged. Bratko et al. [171] presented a model for phase transfer of proteins in RMs. The shell and core model was combined with the Poisson-Boltzmann approximation for the protein-RM complex and for the protein-free RM. The increase in entropy of counterions released from RMs on solubilization of a protein was the main contribution to the decrease in free energy of com-plexation. Good agreement was found with SANS results of Sheu et al. [151] for cytochrome C solubilization and the effect of electrolytes on it. However, this model assumes that filled and empty RMs are of the same size, independent of salt strength and pH, which is not true according to experimental evidence available since then. 

In this paper we would like to discuss the theoretical approaches developed for the description of coil-globule transition of polyelectrolyte macromolecules in these two limiting situations and in the intermediate case when both of the contributions (due to the electrostatic repulsion and due to the translational entropy of counterions) are of the same order. We will discuss in detail mainly the theories constructed or developed by members of the Moscow group. 

T. Hanaoka, R. Tanaka, M. Tasaka, M. Hamada and K. Yoshie, Thermal membrane potential across test anion-exchange membrane Aciplex STA and the transported entropy of counterions, Maku (Membr.), 1993,18, 363-370. 

The MD simulation study of the interaction between PE stars in a salt-free solution has been performed in [39]. The simulation results were fitted using a simplified analytical equation for repulsive force, which arises due to the decrease in translational entropy of counterions entrapped in overlapping star coronae. In this model, the repulsion starts at distances between star centers smaller than 2R, where R is the unperturbed size of an individual star. A similar approach has been applied in [40] to describe repulsion between colloidal PE brushes. 

In the low salt limit, atCp > the coronal contribution to the free energy is dominated by the translational entropy of counterions entrapped inside the corona, int = k TabNAi riCp - 1). In this case, all results of the blob model are recovered both for osmotic starlike and crew-cut spherical micelles (59), (61), and (62). 

Under the so-called salt dominance conditions, the association of block copolymers into micelles does not lead to significant losses in the translational entropy of counterions (whose concentrations inside the corona and in the bulk of the solution are approximately equal). Therefore, within the accuracy of the main term, the CMC is controlled by the hydrophobicity of the block B ... 

When analyzing the behavior of ion-containing gels, the following main physical factors should be taken into account (1) translational entropy of counterions, (2) Coulomb interactions, and (3) formation of ion pairs and multiplets. 

Below, the simplest theory of swelling of polyelectrolyte gels is described, mainly to illustrate that the translational entropy of counterions for this case is much more important than the Coulomb interactions. 

Thus, already a simple theory shows that the swelling of a weakly charged gel in a thermodynamically good solvent is much higher than for uncharged gel, and this is due to the translational entropy of counterions. 

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.

Figure 12). Thus, the discontinuous collapse transition is due to the translational entropy of counterions. 

As to the entropy of counterions that form ion pairs, it is related to the combinatorial entropy of their distribution. In particular, the number of ways of ion pair formation from [N/y]vif/ counterions and (N/y)v ions on polymer chains is determined as... 

In very dilute salt-free solutions the interaction between polyelectrolyte chains and counterions is weaker than the contribution of the configurational entropy of counterions, and counterions are thus distributed almost... 

Basically, the confinement free energy of a polyelectrolyte chain inside a cavity is due to the translational entropy of counterions, electrolyte ions, and solvent molecules. Simple scaling formulas based on the radius of gyration of the polyelectrolyte, analogous to Equations 5.28 and 5.26, are not applicable for the confinement free energy of a polyelectrolyte in spherical cavities, unlike the case of uncharged polymers. 

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