Comprehensive analysis of open and isolated confined electrolyte systems.

In this paper, various aspects of the Poisson-Boltzmann (PB) equation, focusing on the roles of structural confinement and ionic conservation have been addressed. Specifically, this paper provides an analytical and well-behaved solution of the PB equation in finite and infinite structures for open and isolated electrolytic environments. A structure-dependent Debye length for confined systems and a generalized Debye length appropriate for isolated electrolytes are introduced. For an isolated symmetric (1:1) electrolyte, a generalized PB equation has been derived. In the literature, the solution to this equation within a finite region is often presented as complex mathematical expressions, typically constrained to particular scenarios. In this paper, a general solution to the PB equation is derived using a novel mathematical function yielding a simpler closed-form solution than existing analytical approaches. This solution is expressed explicitly in a fully analytical, closed form, and approximation free manner. The results presented yield a simple and general algorithm for treating the PB model under various conditions. Finally, using this framework, we examine the validity range of the Debye model, determine, through pressure analysis, when two immersed electrodes become effectively non-interacting and analyze the stored electrostatic energy in different geometries, highlighting the effect of system isolation on the stored energy.