Conformation and Energetics in Asymmetric Polyelectrolyte Complexation

Asymmetry in chain lengths between polycations and polyanions plays a pivotal role in determining the structure and dynamics of polyelectrolyte complex coacervates, yet how such asymmetry modulates the energetics underlying the complexation between polycation and polyanion chains remains poorly understood. Here we develop an efficient Monte Carlo technique to calculate the Helmholtz free energy as a function of the electrostatic coupling parameter Γ for polyelectrolyte solutions in an implicit dielectric continuum. We apply this method to investigate the conformation and energetics in both single-chain systems with their monovalent counterions and mixtures composed of one polycation and one polyanion chain along with their respective counterions, all under the salt-free conditions. Comparison between the two-chain systems and the corresponding single-chain systems allows us to quantify the complexation free energy Δ F between two oppositely charged polyions. We further examine the energetic and entropic contributions of Δ F by either neglecting or incorporating the temperature dependence of dielectric constant. We find that only the latter can reproduce the experimentally observed entropy-dominated driving force over a wide range of Γ. At weak to moderate Γ, the entropy gain mainly arises from the orientation degrees of freedom of solvent molecules. As Γ increases, the contribution of counterion-release entropy gradually becomes dominant. In addition, Δ F of asymmetric pairs is mainly governed by the shorter chain and varies weakly with increasing the chain length of the longer polyion. Our study offers valuable insights into the thermodynamic nature of polyelectrolyte complexation with asymmetric chain lengths, which is crucial for understanding the structure and stability of biomolecular condensates and the rational design of coacervate-based polymeric materials.