Membrane potential model for an asymmetrical nanofiltration membrane — consideration of noncontinuous concentration at the interface

Study of the surface character as responsible for controlling interfacial forces at membrane–feed interface

“…In the last decades, increasing interest has been turned to design new functional polymeric membranes in order to control interfacial phenomena, traditionally affecting separation and purification processes [1][2][3][4]. It is well known, the first step of each membrane separation process involves chemical interactions at the membrane-feed interface, directing positively or negatively the productivity and efficiency of the overall transport [5][6][7][8].…”

Applicability of Donnan equilibrium theory at nanochannel–reservoir interfaces

“…Note that the standard electrochemical potential is defined at a standard state and is independent of concentration and electric potential for dilute solution, therefore Q 0 will also not change with these parameters. For the study of asymmetrical membrane consisting of a cation-exchange layer and an anionexchange layer, a contact factor n is multiplied with Q 0 to compensate the error of Donnan equilibrium due to different materials' characteristics [18,28]. By choosing an appropriate n, the experimental data showed well consistence with modified Donnan theory.…”

“…(10) is much less popular than Eq. (6), because it is hard or even impossible to calculate l 0 i in practice and Q 0 is a correction factor for non-ideal conditions due to the electrostatic forces unbalance between the fixed charge and the free ions in the membrane phase [18]. Note that the standard electrochemical potential is defined at a standard state and is independent of concentration and electric potential for dilute solution, therefore Q 0 will also not change with these parameters.…”

“…The bulk concentrations and electrical potentials in both reservoirs are known, whereas we need the real boundary conditions to refine the ion transport through the nanochannels. In previous studies, the Donnan equilibrium is commonly used to quantify such an interfacial drop of electrical potential [5,[12][13][14][15][16][17][18][19]. As shown in Section 2, Donnan equilibrium is derived from the classical thermodynamic theories with an equilibrium assumption, and gives a succinct relationship of the quantities between the bulk solution and nanochannel-reservoir interface.…”

“…It is clear that this assumption brings many doubts and challenges. Therefore, after a long time of developments the traditional Donnan equilibrium theory is still the most popular one in the literature [5,15,[17][18][19]22]. However, the nanofluidic system may be in a strongly non-equilibrium state.…”