Finite-N correction to the electrostatic correlation free energy

The electrostatic correlation free energy due to composition fluctuations is needed to model polyelectrolyte (PE) complexation. This contribution has been treated in previous studies using the random phase approximation (RPA) for infinitely long chains, which neglects the effect of chain ends. We show that long-range electrostatic interactions result in a strong finite-N correction to the free energy that originates from the miscounting of linked charges near chain ends. Without this correction, the standard RPA expression fails to predict the finite charge density needed for PE complexation. The nature of the correction depends on PE architecture, leading to opposite contributions for linear and ring topologies. As an example, we demonstrate that this contribution is necessary to explain experimentally observed partitioning in disperse PE solutions. We provide a compact expression for the finite-N correction and apply it to analyze related solution properties, including interfacial tension and ion-binding equilibrium.