  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.
