Polyelectrolytes such as single and double stranded DNA and many synthetic polymers undergo two structural transitions upon increasing the concentration of multivalent salt or molecules. First, the expanded-stretched chains in low monovalent salt solutions collapse into nearly neutral compact structures when the density of multivalent salt approaches that of the monomers. With further addition of multivalent salt the chains redissolve acquiring expanded-coiled conformations. We study the redissolution transition using a two state model [F. Solis and M. Olvera de la Cruz, {it J. Chem. Phys.} {bf 112} (2000) 2030]. The redissolution occurs when there is a high degree of screening of the electrostatic interactions between monomers, thus reducing the energy of the expanded state. The transition is determined by the chemical potential of the multivalent ions in the solution $mu$ and the inverse screening length $kappa$. The transition point also depends on the charge distribution along the chain but is almost independent of the molecular weight and degree of flexibility of the polyelectrolytes. We generate a diagram of $mu$ versus $kappa^2$ where we find two regions of expanded conformations, one with charged chains and other with overcharged (inverted charge) chains, separated by a collapsed nearly neutral conformation region. The collapse and redissolution transitions occur when the trajectory of the properties of the salt crosses the boundaries between these regions. We find that in most cases the redissolution occurs within the same expanded branch from which the chain precipitates.
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