Transmembrane potential across a charged nanochannel subjected to a salinity gradient

The transmembrane voltage, V_{i=0}, which is the potential drop required to nullify the electrical current (i=0), is a key characteristic of water desalination and energy harvesting systems that utilize macroscopically large nanoporous membranes, as well as for physiological ion channels subjected to asymmetric salt concentrations. To date, existing analytical expressions for V_{i=0} have been limited to simple scenarios under simplifying assumptions. In this work, we derive two expressions for V_{i=0}. First, we consider the much simpler scenario of two species. Then, we can consider an electrolyte composed of an arbitrary number of species. The difference in the models is that the latter solution utilizes an ad hoc assumption of a linear concentration profile, while the former solution does not require such an ad hoc assumption. However, to derive a closed-form solution, another assumption is needed. In both models, we explicitly assume that the system is locally electroneutral. We show that both electroneutral models display remarkable correspondence with the numerical simulations of the one-dimensional Poisson-Nernst-Planck equations that do not assume electroneutrality. We show how the interplay between diffusion coefficients and ionic valencies significantly varies the system response and why it is essential to account for all system parameters. Importantly, we show that the new models can be reduced to several known models. Ultimately, this model can be used to improve experimental interpretation of ion transport measurements.