Modelling of the Electrical Membrane Potential for Concentration Polarization Conditions

Based on Kedem–Katchalsky formalism, the model equation of the membrane potential (∆ψ_s ) generated in a membrane system was derived for the conditions of concentration polariza-tion. In this system, a horizontally oriented electro-neutral biomembrane separates solutions of the same electrolytes at different concentrations. The consequence of concentration polarization is the creation, on both sides of the membrane, of concentration boundary layers. The basic equation of this model includes the unknown ratio of solution concentrations (C_i /C_e ) at the membrane/concentration boundary layers. We present the calculation procedure (C_i /C_e ) based on novel equations derived in the paper containing the transport parameters of the membrane (L_p, σ, and ω), solutions (ρ, ν), concentration boundary layer thicknesses (δ_l, δ_h ), concentration Raileigh number (R_C ), concentration polarization factor (ζ_s ), volume flux (J_v ), mechanical pressure difference (∆P), and ratio of known solution concentrations (C_h /C_l ). From the resulting equation, ∆ψ_s was calculated for various com-binations of the solution concentration ratio (C_h /C_l ), the Rayleigh concentration number (R_C ), the concentration polarization coefficient (ζ_s ), and the hydrostatic pressure difference (∆P). Calculations were performed for a case where an aqueous NaCl solution with a fixed concentration of 1 mol m⁻³ (C_l ) was on one side of the membrane and on the other side an aqueous NaCl solution with a concentration between 1 and 15 mol m⁻³ (C_h ). It is shown that (∆ψ_s ) depends on the value of one of the factors (i.e., ∆P, C_h /C_l, R_C and ζ_s ) at a fixed value of the other three.