Theoretical development of pressure-induced electrical potential: Consideration of the concentration polarization for membranes with narrow pores

The pressure-induced electrical potential (Δφ) considering the concentration polarization for membranes with narrow pores is presented theoretically in this paper. The theoretical calculations are mainly based on the Nernst–Planck equation, Donnan equilibrium and the law of mass conservation. The relevant parameters including fixed ion concentration (X), hydrodynamic permeability (DH) and thickness (d) of the membrane itself, boundary layer thicknesses (δ) and ratio of diffusion coefficient of cation to anion (D+/D− in the solution phase and D ¯ + / D ¯ − in the membrane phase) are considered. The effect of each parameter on Δφ is discussed one by one with others unchanged. Results show that X and DH play more important roles than others (including d, δ, D+/D− or D ¯ + / D ¯ − ) especially at low bulk solution concentrations (C0). Further, Δφ remarkably increases when X and DH rise, and it slightly enhances with d, δ and D+/D− (or D ¯ + / D ¯ − ). The theoretical simulation at specific conditions was compared with experiment at identical conditions and the results showed the consistency.