Polyelectrolytes in multivalent salt solutions: Monomolecular versus multimolecular aggregation

We analyze the segregation of strongly charged chains of N monomers of size a in the presence of multivalent salts with valence z as a function of the concentration of monomers φ in the solutions. The multivalent ions of opposite charge condense along the monomers and induce monomer attractions that lead to the formation of dense finite size aggregates (micelles) of chains and multivalent ions when their valence z > 1. We compute the number density of chains in aggregates with p = 1, 2, 3, 4, ... chains by equating the chemical potential of all the p-aggregated chains and of the free and condensed ions in the aggregates. At low concentration of multivalent salts m, we observe monomolecular (p = 1) precipitated chains when φ < φ*, where φ* ≅ mz. When φ increases above φ*, the chains redissolve in the solution and adopt stretched conformations. As m increases above a critical value m**, the number density of aggregates with p > 1 chains increases such that there are more aggregates with p* > 1 chains when φ** < φ < φ*. If we only include the surface free energy in the analysis, we find p* = ∞. However, if we include the chain entropy of confining p chains in a region R(p) = (pN)^1/3a in the free energy, p* can be finite and greater than or equal to one. This situation arises when the chains are constrained to stretched conformations, such as in metastable toroidal and in spherical coil aggregates observed in long double-stranded DNA in multivalent ions.