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Fig. 5. Measurement of the membrane potential (¢,, mV) as function of low and high salt concentration (Cj next to lines; Cpjgn plotted on x-axis). Solid lines: standard TMS theory.

Figure 5 Measurement of the membrane potential (¢,, mV) as function of low and high salt concentration (Cj next to lines; Cpjgn plotted on x-axis). Solid lines: standard TMS theory.

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'Wetsus, European Centre of Excellence for Sustainable Water Technology, Oostergoweg 9, 8911 MA Leeuwarden, The Netherlands ?Sub-Department of Environmental Technology, Wageningen University, Bornse Weilanden 9, 6708 WG Wageningen, The Netherlands 3Membrane Institute, Kuban State University, 149 Stavropolskaya str., 350040 Krasnodar, Russia ‘Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherland:
'Wetsus, European Centre of Excellence for Sustainable Water Technology, Oostergoweg 9, 8911 MA Leeuwarden, The Netherlands ?Sub-Department of Environmental Technology, Wageningen University, Bornse Weilanden 9, 6708 WG Wageningen, The Netherlands 3Membrane Institute, Kuban State University, 149 Stavropolskaya str., 350040 Krasnodar, Russia ‘Laboratory of Physical Chemistry and Colloid Science, Wageningen University, Dreijenplein 6, 6703 HB Wageningen, The Netherland:
Fig. 1. Schematic representation of concentration, c, and potential, ¢, profile in an ion exchange membrane (IEM) between two electrolyte phases of different concentration, bulk L and bulk R, when no current is applied. Here X is the fixed membrane charge, @ui the diffusion potential, 4@p the Donnan potential of side L or R, and db, is the resulting membrane potential. where ¢» is the membrane potential (V), 4¢p the Donnan potential (V) on the left hand (LZ) and right hand (R) side of the membrane, and ¢qiz is the diffusion potential (V). Figure 1 shows these potentials schematically. The Donnan potential is also referred to as the exclusion potential as it can include both electrostatic (repulsion) and steric (size exclusion) effects [49]. For ionic solutions steric effects are much smaller than electrostatic effects and are not further discussed in this work. 2.2.1. Effect of the stagnant diffusion layer
Fig. 1. Schematic representation of concentration, c, and potential, ¢, profile in an ion exchange membrane (IEM) between two electrolyte phases of different concentration, bulk L and bulk R, when no current is applied. Here X is the fixed membrane charge, @ui the diffusion potential, 4@p the Donnan potential of side L or R, and db, is the resulting membrane potential. where ¢» is the membrane potential (V), 4¢p the Donnan potential (V) on the left hand (LZ) and right hand (R) side of the membrane, and ¢qiz is the diffusion potential (V). Figure 1 shows these potentials schematically. The Donnan potential is also referred to as the exclusion potential as it can include both electrostatic (repulsion) and steric (size exclusion) effects [49]. For ionic solutions steric effects are much smaller than electrostatic effects and are not further discussed in this work. 2.2.1. Effect of the stagnant diffusion layer
Fig. 2. Schematic representation of concentration, c, and potential, ¢, profile in the bulk solutions, the stagnant diffusion layers (SDLs), and in an ion exchange membrane (IEM) when no current is applied. Symbols as in Figure 1.
Fig. 2. Schematic representation of concentration, c, and potential, ¢, profile in the bulk solutions, the stagnant diffusion layers (SDLs), and in an ion exchange membrane (IEM) when no current is applied. Symbols as in Figure 1.
Fig. 3. Schematic representation of concentration, c, and potential, ¢, profile in the bulk solution, the stagnant diffusion layer (SDL), and in an ion exchange membrane (IEM) when a current is applied. Driving forces acting on the ions are shown as vectors. Generally, the diffusion potentials due to transport across the SDLs a1 eglected when considering the membrane potential, which is justified at lo oncentration differences over the membrane, very thin SDLs, and densel harged IEMs. However, effects of the SDLs are often considered i rocesses where an electric potential is applied over one or multiple IEM uch as in electrodialysis. In that case, when an electric current is applied ove n IEM (and the accompanying SDLs), concentration polarization takes plac 1 these SDLs [38,63,65,69]. Concentration polarization is then caused by th 1embrane selectivity between counterions and co-ions [24,43,48]. Th iffusional and migrational forces acting on ions when a current is applied a1 chematically shown in Figure 3. So, besides formation of a diffusio otential in the SDL, concentration polarization in the SDL affects the io oncentrations at the membrane-solution interfaces, which also affects th 1embrane potential. The relative influence of the SDL depends on i) the rati f SDL thickness, dsp, over the membrane thickness, 6, (m), ii) the ratio « 1e ion diffusion coefficient in the SDL, D over the ion diffusion coefficie1 1 the membrane, D (m2/s), and iii) the ratio of the external ion concentratio , over the internal ion concentration Cc (mM) [22,25].
Fig. 3. Schematic representation of concentration, c, and potential, ¢, profile in the bulk solution, the stagnant diffusion layer (SDL), and in an ion exchange membrane (IEM) when a current is applied. Driving forces acting on the ions are shown as vectors. Generally, the diffusion potentials due to transport across the SDLs a1 eglected when considering the membrane potential, which is justified at lo oncentration differences over the membrane, very thin SDLs, and densel harged IEMs. However, effects of the SDLs are often considered i rocesses where an electric potential is applied over one or multiple IEM uch as in electrodialysis. In that case, when an electric current is applied ove n IEM (and the accompanying SDLs), concentration polarization takes plac 1 these SDLs [38,63,65,69]. Concentration polarization is then caused by th 1embrane selectivity between counterions and co-ions [24,43,48]. Th iffusional and migrational forces acting on ions when a current is applied a1 chematically shown in Figure 3. So, besides formation of a diffusio otential in the SDL, concentration polarization in the SDL affects the io oncentrations at the membrane-solution interfaces, which also affects th 1embrane potential. The relative influence of the SDL depends on i) the rati f SDL thickness, dsp, over the membrane thickness, 6, (m), ii) the ratio « 1e ion diffusion coefficient in the SDL, D over the ion diffusion coefficie1 1 the membrane, D (m2/s), and iii) the ratio of the external ion concentratio , over the internal ion concentration Cc (mM) [22,25].
Fig. 4. Schematic representation of a membrane pore and the concentration gradients in the radial direction according to the theory of Teorell-Meyer-Sievers (TMS) and the space charge model (SCM). The solid line refer to membranes with a relative low fixed charge density, X, and the dashed lines to a membrane with a high fixed charge density. The external salt concentration, Cey, is indicated by the thin line, pore radius is shown as r, and the Debye length is Ap. For membranes with larger r, (e.g. used in nanofiltration, NF) and a considerable fixed charge density, these gradients are more important However, for membranes with large r, and low fixed charge density, the gradient is also of less importance and predictions of the membrane potential by the TMS theory and the SCM will become closer again. Interesting are the cases with a low external concentration on one side and a high concentration on the other side of the membrane. In this situation, the EDLs can change from (fully) overlapping to (partly) non-overlapping along the axial direction. Proper estimation of the internal ion concentration is in that case required. Figure 4 gives a schematic representation of the concentration gradients in membrane pores according to the TMS theory and the SCM.
Fig. 4. Schematic representation of a membrane pore and the concentration gradients in the radial direction according to the theory of Teorell-Meyer-Sievers (TMS) and the space charge model (SCM). The solid line refer to membranes with a relative low fixed charge density, X, and the dashed lines to a membrane with a high fixed charge density. The external salt concentration, Cey, is indicated by the thin line, pore radius is shown as r, and the Debye length is Ap. For membranes with larger r, (e.g. used in nanofiltration, NF) and a considerable fixed charge density, these gradients are more important However, for membranes with large r, and low fixed charge density, the gradient is also of less importance and predictions of the membrane potential by the TMS theory and the SCM will become closer again. Interesting are the cases with a low external concentration on one side and a high concentration on the other side of the membrane. In this situation, the EDLs can change from (fully) overlapping to (partly) non-overlapping along the axial direction. Proper estimation of the internal ion concentration is in that case required. Figure 4 gives a schematic representation of the concentration gradients in membrane pores according to the TMS theory and the SCM.
Fig. 6. Testing of six modifications of TMS theory. Diamonds: experimental values of membrane potential (¢,,, mV). Low external NaCl solution concentration 0.01 M; high concentration plotted on the x-axis. Solid line: standard TMS theory. Effect of A) Ion activity (activity coefficients in ref. [126]); B) Inhomogeneous charge distribution; C) Stagnant diffusion layers adjacent to the membrane of specified thickness (dsp,); D) Lowering the fixed membrane charge density; E) Osmosis at different water-membrane friction coefficients, f,,6, F) Lowering the diffusion coefficient of the counterion (D,yq,) in the membrane. to the Donnan relation (Eq. 4), which is very close to TMS theory. This implies that the effect of the diffusion potential iz, on the membrane potential, ¢,,, is very small, which is expected because X>>ce. This also implies that in this case, activity coefficients in the membrane can be neglected. For that reason as a first approximation we can simply replace in the TMS theory the ion concentrations in the external solutions by the ion activities which then affects the Donnan potentials [86]. Use of ion activities
Fig. 6. Testing of six modifications of TMS theory. Diamonds: experimental values of membrane potential (¢,,, mV). Low external NaCl solution concentration 0.01 M; high concentration plotted on the x-axis. Solid line: standard TMS theory. Effect of A) Ion activity (activity coefficients in ref. [126]); B) Inhomogeneous charge distribution; C) Stagnant diffusion layers adjacent to the membrane of specified thickness (dsp,); D) Lowering the fixed membrane charge density; E) Osmosis at different water-membrane friction coefficients, f,,6, F) Lowering the diffusion coefficient of the counterion (D,yq,) in the membrane. to the Donnan relation (Eq. 4), which is very close to TMS theory. This implies that the effect of the diffusion potential iz, on the membrane potential, ¢,,, is very small, which is expected because X>>ce. This also implies that in this case, activity coefficients in the membrane can be neglected. For that reason as a first approximation we can simply replace in the TMS theory the ion concentrations in the external solutions by the ion activities which then affects the Donnan potentials [86]. Use of ion activities
Fig. 8. Experimental membrane potential, ¢,,, as a function of the logarithm of the ratio of the activity of the external solution with the high NaCl concentration, Gnign, over the activity of the external solution with the low NaCl concentration, aj,,,.. The dashed line indicates the activity based maximum potential, dru, according to Eqs. (4) and (20); ciow either 0.001 M (circles, crosses) or 0.01 M (squares), and Cpjg, is varied between cj, and 5 M.
Fig. 8. Experimental membrane potential, ¢,,, as a function of the logarithm of the ratio of the activity of the external solution with the high NaCl concentration, Gnign, over the activity of the external solution with the low NaCl concentration, aj,,,.. The dashed line indicates the activity based maximum potential, dru, according to Eqs. (4) and (20); ciow either 0.001 M (circles, crosses) or 0.01 M (squares), and Cpjg, is varied between cj, and 5 M.
Inserting Eq. (A16) into Eq. (A15) leads to the following expression of 1e (dimensionless) diffusion potential At the solution-membrane interfaces on the left hand and right hand side of the membrane, the expressions for the Donnan equilibrium can be used to relate the internal ion concentration to the external concentration and the fixed membrane charge, according to References
Inserting Eq. (A16) into Eq. (A15) leads to the following expression of 1e (dimensionless) diffusion potential At the solution-membrane interfaces on the left hand and right hand side of the membrane, the expressions for the Donnan equilibrium can be used to relate the internal ion concentration to the external concentration and the fixed membrane charge, according to References
[44] A.H. Galama, J.W. Post, M.A. Cohen Stuart, P.M. Biesheuvel, Validity of the Boltzmann equatio to describe Donnan equilibrium at the membrane-solution interface, J Mem Sci, 442 (2013) 131
[44] A.H. Galama, J.W. Post, M.A. Cohen Stuart, P.M. Biesheuvel, Validity of the Boltzmann equatio to describe Donnan equilibrium at the membrane-solution interface, J Mem Sci, 442 (2013) 131
Related topics:
ChemistryMembrane ScienceNernst-Planck equation
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