Explicit interrelationship between Donnan and surface potentials and explicit quantification of capacitance of charged soft interfaces with pH-dependent charge density

A charged soft interface is characterized by a charged polyelectrolyte layer (PEL) sandwiched between an uncharged rigid surface and an electrolyte solution. The PEL harbors or consists of a particular kind of ion (henceforth denoted as PEL ion), different from the electrolyte ions. The PEL ions remain excluded from locations outside the PEL. This gives rise to a Donnan potential (ψD) deep within the PEL, with the PEL-electrolyte interface (having an electrostatic potential, also called the surface potential, ψ0) behaving as a possible ion-selective permeable membrane. In this paper, we provide closed-form explicit analytical expressions for the interrelationship between ψD and ψ0 for both positively and negatively charged PEL with pH-dependent charge densities. We demonstrate that for a given magnitude of ψD, magnitude of ψ0 is identical for both positively and negatively charged PEL, dictated only by a parameter α, which depends on the difference between pH and pKa values for negatively charged PEL or the difference between pKb and pOH values for positively charged PEL. Most importantly, derivation of explicit form of interrelationship between ψD and ψ0, hitherto missing in the existing literature, allows us to provide substantially new insights into the relative variation of ψD and ψ0 and at the same time helps us to rectify several earlier misconceptions. Further, using this explicit relationship, it becomes possible to quantify the capacitance of a soft interface as explicit functions of α and the Donnan potential, with relevance for development of possible soft-interface-based energy storage system.