| We study the phase diagram of polyelectrolyte solutions in salt and salt-free environments. We examine the phase behavior of polyelectrolyte solutions, in the semidilute regime, using different physical models, namely the Random Phase Approximation (RPA) and the cross-linked model. In the RPA, we calculate the electrostatic free energy by summing all the fluctuations of the chains and all present ionic species. Within this approximation, the phase diagrams of salt-free polyelectrolyte solutions show phase separation even without including short-range attractions or ion condensation. We find that the phase behavior of large chains resembles the phase diagram of polymer network solutions. That is, the equilibrium is established between a network phase and a chain-free phase. Upon the addition of salt, the dissociated ions increase the entropy of the system and overcome the energy from the electrostatic fluctuations. When the short-range attraction between monomers is included in the model, the free energy predicts phase segregation for all salt valences at high salt concentrations (1 mol/l and higher). The phenomenon is called salting-out and occurs simply because the addition of salt reduces the quality of the solvent and induces precipitation. However, phase segregation in the presence of multivalent ions in polyelectrolyte solutions occurs at low salt concentrations (less than 1 mol/l). We propose that this phase separation is due to polyions cross-linked by multivalent ions. We constructed a phenomenological two-state model to examine this phenomenon. The two phases coexisting in the solution are a network-like phase and a polymer-free phase. The polymer-free phase is modeled using Debye-Hückel theory. In the cross-linked phase, each condensed multivalent ion attracts an equal number of monomers creating a neutral cluster. The energy of the cluster is evaluated by a simple Coulombic energy. The bare monomer charges between the linkages are treated as line of charges. The cross-linked model solves self-consistently for the fraction of multivalent ions and counterions condensed along the polyions. The calculated phase diagram predicts the precipitation found experimentally; however, it fails to predict the redissolution transition at higher salt concentrations. |
