Rapid force-flux transitions in highly porous membranes

Abstract

When a steady electric current is passed through a porous membrane which separates two electrolyte solutions at different concentrations the system can, in a suitable experimental configuration, enter a state of stable oscillations of the trans-membrane pressure and potential. This system, sometimes called the Teorell membrane oscillator, also displays unusual stationary state behaviour when the pressure difference across the membrane is held constant. These phenomena arise because the pressure-driven flow of volume through the membrane is virtually independent of the concentration of the solution in its pores, whereas the electro-osmotic flow decreases as the concentration increases. If the pressure-driven and electro-osmotic flows are opposed and pressure is applied to the concentrated solution then at low currents pressure drives the concentrated solution into the pores and at high currents electro-osmosis drags the dilute solution into the pores. At some intermediate current the transition from concentrated to dilute solution in the pores occurs and is accompanied by a sudden increase in the membrane resistance and potential difference. These observations have been made on various membranes of ill-defined structure, it is shown here that they can be reproduced with Nuclepore filters which have readily characterized uniform circular and parallel pores. This observation has facilitated the development and testing of a quantitative theory of the phenomenon. The theory developed here follows the lines laid down by Kobatake & Fujita (1964) and by Mikulecky & Caplan (1966). The membrane pores are treated as independent capillaries lined by an electrical double layer with a diffuse counter charge in the pore solution. A system of flux equations consistent with non-equilibrium thermodynamics is developed and the minimum assumptions and idealizations needed to obtain well-defined results are identified. Flow in the pores is treated via the Navier—Stokes equation and equations for the membrane fluxes and forces are obtained in terms of the membrane properties and external parameters under the control of an experimenter. Two cases are considered. In the first the surface charge density on the pore walls is independent of the solution concentration and in the second the surface charge density is proportional to the cube root of the concentration. The second case applies to Nuclepore membranes because the surface charge on polycarbonate is probably due mainly to adsorption of anions from solution. In the first case the electroosmotic flow is inversely proportional to the one-half power of the concentration and in the second case to the one-sixth power. In order to convert the barycentric local flux equations to equations describing the macroscopic phenomena account must be taken of the radial variation of parameters over a pore cross-section and to their variation along the pore from one membrane boundary to the other. Radial variation is dealt with first by transferring from the barycentric to the membrane-fixed frame. The flux equations are then averaged over the pore cross-section and finally integrated across the membrane under the assumption of a steady state. This procedure has the advantage of producing manageable flux equations, with no adjustable parameters, which can therefore be given an unequivocal test. It has the disadvantage that no information is obtained concerning the precise mechanisms of the sudden transitions between the high and low resistance states. The flux equations predict the observed types of transitions in fluxes and forces and give a correct general picture of the expected behaviour of the system. In order to devise a quantitative test it is necessary to bear in mind that it is not technically possible to stir bulk solutions very vigorously against the surfaces of a thin porous membrane without exaggerating pore end effects or even pulsing solutions right through the pores. This difficulty has been dealt with by superimposing upon the membrane flux equations, transport equations across Nernst hypothetical diffusion layers at the membrane/solution boundaries. Methods of computation have been developed which permit the observable behaviour of the system to be predicted in terms of the ascertainable concentration of the bulk solutions. It is interesting to note that, whereas the low power of the concentration dependence of electro-osmosis in Nuclepore membranes, as compared with constant charge membranes, lowers the scale of the expected force-flux transitions, the presence of stagnant solution layers at the membrane faces increases the scale of these transitions. The quantitative test of this theory has been based on the comparison of observed and predicted current-voltage curves. A membrane cell has been devised in which the solutions are stirred by paddles and pump circulation to give stable and characterizable stirring conditions. An electric current can be passed through the cell but the products from the electrodes are excluded by ion-exchange membranes. The membrane potential can be measured, and controlled with a potentiostat, via probe holes close to the membrane faces. The pressure difference across the membrane can be maintained at a desired level or allowed to fluctuate in attached vertical tubes. The pore characteristics of the membranes have been measured optically and by studying hydrodynamic and electro-osmotic flows through them. Teorell-type oscillations and steady-state dynatron-type current-voltage curves have been recorded in membranes of 0.5, 0.8, 1.0 and 2.0 nm pore diameter at various currents and pressure differentials in the presence of solutions of NaCl at several concentrations, MgCla and Na2S 0 4. Transitions were observed in current—voltage curves at high enough pressures. The curves showed hysteresis-like loops. They were characterized by the values of the upper or * flip * current and lower or ‘ flop ’ current on the boundaries of the hysteresis loop and by the difference in the cell resistance between the low current and high current states. Theoretical values were calculated for the flip and flop currents and the resistance changes under the conditions studied experimentally and a direct comparison made. Within the limits of the assumptions of ideal solutions and of simple diffuse double-layer theory it was found that agreement was satisfactory on all important matters. In particular, factors that decreased the hydrodynamic permeability or increased the electro-osmotic permeability were correctly found to make the system less sensitive to changes of pressure. This was borne out by the response of the membrane system to changes in the membrane pore size and surface charge, to varying the valence type of the electrolyte and to varying the concentrations of the bulk solutions. Proceeding from this evidence in favour of the correctness of the theory with regard to current-voltage characteristics, it has been possible to make various predictions regarding the behaviour of the volume and ion fluxes through the membrane in the transition region. It has not yet been possible to measure these fluxes and so to test the predictions which emphasize the importance of the volume flux in controlling the nonlinearity of the system.