Analysis of Generalized Debye-H\"uckel Equation from Poisson-Fermi Theory

The Debye-H\"uckel equation is a fundamental physical model in chemical thermodynamics that describes the free energy (chemical potential, activity) of an ion in electrolyte solutions at variable salt concentration, temperature, and pressure. It is based on the linear Poisson-Boltzmann equation that ignores the steric (finite size), correlation, and polarization effects of ions and water (or solvent molecules). The Poisson-Fermi theory developed in recent years takes these effects into account. A generalized Debye-H\"uckel equation is derived from the Poisson-Fermi theory and is shown to consistently reduce to the classical equation when these effects vanish in limiting cases. As a result, a linear fourth-order Poisson-Fermi equation is presented for which unique solutions are shown to exist for spherically symmetric systems. Moreover, a generalized Debye length is proposed to include the size effects of ions and water.