Gatekeeper model for ion selective membranes

Abstract

A two-parameter model describes the resistance of a membrane challenged with different salt concentrations on either side of the membrane. This model treats the membrane as a circuit consisting of a resistor located between two switches. One switch is on the diluate side of the membrane. The second switch is on the concentrated side. The probability of the diluate side switch being active (loaded with a counter-ion) is β (dimensionless) and the probability of the switch being active (loaded with a counter-ion) on the concentrated side is γ (dimensionless). The probability of the circuit being active is the product of the probabilities βγ in which case the circuit is actively transporting a counter-ion across the membrane. The probability functions are calculated from the conductivity of the diluate λD (mS/cm), the concentrate λC (mS/cm) and a fitted parameter {Ksat} (mS/cm). The functions are β = λD/(λD +{Ksat}) and γ = λD/(λD + {Ksat}). These functions derived from enzyme kinetics where {Ksat} is analogous to the Michaelis-Menton saturation coefficient. The second fitted parameter is the minimum resistance rmin (Ωˑcm2) representing the case where the product βγ approaches unity (λD and λC ≫ {Ksat}). The product of the minimum resistance rmin and the applied voltage is analogous to the reciprocal of the maximum rate of an enzyme reaction. The resistance of the membrane rmem (Ωˑcm2) is rmem = rmin/(βγ). Measurements made with equal salt solutions on both sides of the membrane are used to calculate the resistance under any salt conditions. The probability functions β and γ represent the relative activity of the counter-ions on the surfaces of the membrane. This suggests that the Nernst potential across the membrane vmem (Volts) is expressed as vmem= (ɌT/Ƒ)ˑln (γ/β) where Ɍ (8.314 J/°K⋅mol) is the gas law constant, T (°K) is the temperature, and Ƒ (96,485 C/mol) is the Faraday constant. The current density (A/cm2) generated across the membrane under reverse electrodialysis conditions is i = vmem/rmem and the power density P (W/cm2) is P = i2rmem. It is, therefore, possible to estimate the performance of a membrane from a limited data set of resistance data such as may be found in the literature.