  We study the reentrant condensation of polyelectrolytes in dilute solutions
of small multivalent salts, whose phase-transition mechanism remains poorly
understood. Motivated by recent full atomic simulation results reported by the
Caltech group on phase behaviors of polyelectrolytes in presence of multivalent
salts (DOIs: 10.1021/acs.macromol.3c02437 and 10.1021/acs.langmuir.3c03640), in
this work we construct a simple but effective mean-field model which can
rationalize the essential features of the reentrant condensation including the
phase diagram of polyelectrolyte. The model unveils that the strong adsorption
between the ionic monomers and multivalent ions can be at the origin of the
peculiar phenomenon that rather low concentrations of multivalent salts trigger
both collapse and re-entry transitions. For the first time, the analytical
solution of the model indicates that a minimum of coupling energy due to
sharing multivalent salt ions between ionic monomers is essential for a phase
transition to occur, which can explain the enigmatic observation that
polyelectrolytes can only show phase transition in a dilute solution of salts
with selective multivalency. Our analytical calculations also show that the
incompatibility of the uncharged moieties of the polyelectrolytes with water is
critical to regulate phase behaviors of polyelectrolytes in aqueous solutions.
This is in agreement with recent experimental investigations on solution
properties of amphiphilic proteins. The obtained results will contribute to the
understanding of liquid-liquid phase separation in biological systems where
multivalent ions bound to bio-polyelectrolytes play an essential role.
