Role of entropy and enthalpy in the complexation of a pair of oppositely charged asymmetric and partially ionized polyelectrolytes

Coulomb interaction energy, free ion entropy, and conformational elasticity form a fascinating trio in the thermodynamics of polyelectrolytes (PEs). We present a framework for the complexation of two oppositely charged PEs of asymmetric lengths and partially ionized backbones, considering maximal ion pairing and general degrees of solvent polarity and dielectricity within an implicit solvent model, to explore the underlying thermodynamics. The dominance of free ion entropy and ion pairing enthalpy helps decouple the self-consistent dependency of charge and size of the PEs, thus allowing for an analytical estimate of the free energies of complexation. For a pair of asymmetric PEs with partially ionized backbones, the effective charge and size of the complex, larger than sub-Gaussian globules as for symmetric chains, is found to increase with asymmetry in PE length and charge density. The thermodynamic drive for complexation grows with the ionizability of symmetric PEs and, with a decrease in asymmetry in length for equally ionizable PEs. The crossover Coulomb strength demarcating the ion pair enthalpy-driven and counterion release entropy-driven complexation, depends marginally on the charge density of PEs because so does the degree of counterion condensation, and strongly depends on the dielectric environment and salt. The model, treating counterion condensation explicitly, provides a direct way to calculate thermodynamic dependencies of complexation on experimental parameters such as electrostatic strength and salt, thus laying the foundation for a better understanding of the thermodynamics of the complexation of a general pair of charged biopolymers.