  Interplay of Coulomb interaction energy, free ion entropy, and conformational
elasticity is a fascinating aspect in polyelectrolytes (PEs). We develop a
theory for complexation of two oppositely charged PEs, a process known to be
the precursor to the formation of complex coacervates in PE solutions, to
explore the underlying thermodynamics of complex formation, at low salts.
Explicit calculation of the free energy of complexation and its components
indicates that the entropy of free counterions and salt ions and the Coulomb
enthalpy of bound ion-pairs dictate the equilibrium of PE complexation. This
helps decouple the self-consistent dependency of charge and size of the
uncomplexed parts of the polyions, derive an analytical expression for charge,
and evaluate the free energy components as functions of chain overlap.
Complexation is observed to be driven by enthalpy gain at low Coulomb
strengths, driven by entropy gain of released counterions but opposed by
enthalpy loss due to reduction of ion-pairs at moderate Coulomb strengths, and
finally prohibited by enthalpy loss at higher Coulomb strengths. Phase diagrams
are constructed which identify the stability of fully-, partially- and
un-complexed states as functions of electrostatic strength. Thermodynamic
predictions from our model are in good quantitative agreement with simulations
in literature, and may motivate simulations and experiments at higher
electrostatic strengths at which complexation is found to be unfavourable.
