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Electrochemical and structural characterizations of an experimental track-etched membrane in KCl solutions
Separation and Purification Technology 20 (2000) 169 – 175 www.elsevier.com/locate/seppur
Electrochemical and structural characterizations of an experimental track-etched membrane in KCl solutions A. Can˜as, J. Benavente * Departamento de Fı´sica Aplicada, Grupo de Caracterizacio´n Ele´ctrica y de Transporte en membranas e Interfases, Facultad de Ciencias, Uni6ersidad de Ma´laga, E-29071 Ma´laga, Spain Received 8 January 2000; received in revised form 6 March 2000; accepted 6 March 2000
Abstract Characteristic electrochemical transport parameters for an experimental poly(ethylene)terephtalate (PET) tracketched membrane with well-defined structure and low porosity (U = 0.13%) were determined with the membrane in contact with KCl solutions at different concentrations. Membrane potential, Df m, measurements were performed to investigate the effective fixed charge concentration, Xf, and transport number of the ions, ti, in the membrane using two different procedures: keeping the concentration ratio constant, or keeping one concentration constant and changing the other one. Results show the membrane presents a weak cation-exchanger character, since the following values were obtained: Xf = −(2.590.2)×10 − 2 M, tK = (0.569 0.06), tCl− = (0.449 0.05); taking into account these values, concentration dependence of membrane potential was predicted. Membrane electrical resistance, Rm, was obtained from Impedance Spectroscopy (IS) measurements using equivalent circuits as models, and the membrane porosity U=(0.11 9 0.02)% was also obtained from resistance values, which agrees very well with the value determined from geometrical parameters. From Rm, Dfm and U values, the diffusion coefficient of the ions in the membrane pores can be calculated, and the following average values were obtained: DK+ =(1.99 0.4)× 10 − 9 m2/s and DCl−=(0.8 9 0.2)×10 − 9 m2/s, but for an average concentration higher than 0.06 M, their values do not differ practically from solution in agreement with the small negative charge previously indicated. © 2000 Elsevier Science B.V. All rights reserved. +
Keywords: Impedance spectroscopy; Membrane potential; Salt permeability; Ionic diffusion coefficients
1. Introduction
* Corresponding author. Tel.: +34-95-2131929; fax: + 3495-2132000. E-mail address: j – [email protected] (J. Benavente).
The use of porous membranes for separation of differently sized molecules has many industrial applications, mainly in those involved in the treatment of food, beverage biological materials and waste treatment. The predition of the applicability of such membranes depends on the development of effective methods for their characterization,
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which are mainly carried out on the base of morphological parameters such as pore structure, pore size distribution and porosity [1]. However, other parameters can also affect the separation ability of porous membranes, for instance the electrochemical properties of membranes have a significant influence on the nature and magnitude of the interactions between the membrane and the substances to be processed [2]. Membrane characteristic electrochemical parameters, such as fixed charge concentration, Xf, and ion transport numbers, ti, are usually obtained from membrane potential measurements [3 – 5], while ionic diffusion coefficients in the membrane pores, Di, can be obtained, if the electrical resistance of the membrane, Rm, is also known. On the other hand, Impedance Spectroscopy (IS) is a non-destructive technique, which is being used as a successful tool to determine the electrical properties of layered systems such as membrane/electrolyte systems, since it allows the separate evaluation of the electrical contribution of both the membrane and the electrolyte solution [6 – 9]. Moreover, according to Coster et al. [7] membrane geometrical parameters (porosity and thickness) can be estimated from IS results. In this work, electrochemical and structural characterizations of a poly(ethylene)terephtalate (PET) track-etched membrane in contact with KCl solutions at different concentrations were carried out. Fixed charge concentration and ionic transport numbers in the membrane were obtained from membrane potential, which was measured following two different procedures in order to see the influence of the external conditions on transport numbers. The membrane electrical resistance was determined from IS data using an equivalent circuit consisting of a parallel association of a resistance and a non-ideal capacitor, but membrane porosity was also estimated from electrical results in order to compare it with that calculated from geometrical parameters. The diffusion coefficient of the ions into the membrane pores was also determined from electrical resistance, membrane potential and porosity. Concentration dependence of all these parameters was also studied.
2. Experimental
2.1. Membranes The membrane studied was an experimental nuclear track filter produced in the Laboratoty of Nuclear Filters, Shubnikov Institute of Cristallography, (Russian Academy of Sciences), Moscow, by etching the track in a poly(ethylene) terephtalate (PET) film irradiated by heavy ions [10]. The membrane geometrical parameters (given by suppliers) are: thickness d= (1091) mm, pore radii rp = (4.59 0.5) nm and number of pores by membrane area n= 2×1013 m − 2, from these values a porosity U= (0.139 0.03)% was calculated. Electrochemical parameters were measured with aqueous KCl solution at different concentrations, at a constant temperature t= (25.09 0.3)°C and standard pH = (5.79 0.3); before use, the membranes were immersed for at least 6 h, in a solution of the appropriate salt concentration.
2.2. Water flux, impedance spectroscopy, membrane potential and salt diffusion measurements Measurements of water flux due to an applied pressure difference were carried out in a crossflow filtration cell for pressure ranging between 80 and 200 kPa. The dead-end cell used for electrochemical characterization is similar to that indicated in the literature [11]. The membrane was tightly clamped between two glass half-cells by using silicone rubber rings; each half-cell has a reversible Ag–AgCl electrode and a magnetic stirrer (stirring rate around 500 rpm) to minimize concentration–polarization at the membrane surfaces: Impedance spectroscopy measurements were carried out with the electrodes connected to an Impedance Analyzer (Solartron 1260), from which data can be sent to a computer for further treatment and storage. The experimental data were corrected by the software, taking into account the influence of connecting cables and other parasite capacitances. Measurements were carried out with 100 different frequencies in the range 10–106 Hz at a maximum voltage
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of 0.01 V, with both half-cells filled with solutions of identical concentration. Measurements were carried out for concentration ranging between 10 − 3 M and 5×10 − 2 M. Electromotive force (DE) between both sides of the membrane caused by a concentration gradient was measured by connecting the reversible electrodes to a digital voltmeter (Yokogawa 7552, 1GV imput resistance). Membrane potential measurements were carried out using two different procedures: (i) by keeping the concentration ratio of the solutions at both sides of the membrane constant, c2/c1 =g =2, for concentration c1 ranging between 5 × 10 − 3 and 10 − 1 M; ii) keeping the concentration c1 constant (c1 = 0.01 M) and changing gradually c2 from 10 − 2 to 2 ×10 − 1 M. Membrane potential, Dfm, was determined from measured DE values by subtracting the electrode potential contribution. For each experiment, three series of measurements were carried out, the experimental values were obtained as the average of these measurements; good reproducibility was found and relative errors lower than 10% were obtained.
Fig. 1. Water flux, Jv, versus applied pressure difference, DP.
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3. Results and discussion Due to the well-defined geometrical structure of track-etched membranes, the uniform capillary physical model, which consists of straight pores without tortuosity or constriction, is usually adopted. In these cases, the Haagen–Poiseuille expression can be used to determine the water flow through the membrane, Jv, due to the applied pressure difference, DP: Jv = Lp DP = (r 2pu/8hd)DP
(1)
where Lp is the membrane hydraulic permeability and h is the viscosity of water. Fig. 1 shows the experimental Jv –DP linear relationship and, according to Eq. (1), the membrane hydraulic permeability can be calculated from the slope of this straight line; it was obtained that Lp = (2.09 0.3)×10 − 13 m/s Pa. Taking into account Eq. (1) and the value of the geometrical parameters previously reported, the theoretical value of the hydraulic permeability was also determined: L tp = (3.29 0.6)× 10 − 13 m/s Pa; the good agreement found between experimental and theoretical values indicates the adequacy of the capillary model used. Fig. 2 shows the typical impedance plots: Nyquist plot (− Zimg vs Zreal) and Bode (− Zimg vs f ) plot, obtained by impedance data. Two dielectric relaxations can be observed: the semicircle obtained at high frequencies in Fig. 2(a) (2×104 5 f(Hz)5 106) is assigned to the electrolyte solution, while the depressed semi-circle for frequency ranging between 102 and 2× 103 Hz corresponds to the membrane [9,12]; this assignation is made on the basis of the Bode plot (Fig. 2(b)) where a comparison of the data obtained with two different concentrations (10 − 3 and 2 × 10 − 3 ?M) is shown. The analysis of impedance data was carried out by the complex plane Z* method, which involves plotting the impedance imaginary part (− Zimg) versus the real part (Zreal) as is shown in Fig. 2(a). A parallel R–C circuit gives rise to a semi-circle in the Z* plane, which has intercepts on the Zreal axis at R (v ) and R0 (v0), being (R0 − R ) the resistance of the system. The maximum of the semi-circle equals 0.5(R0 − R ) and occurs
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Experimental data were fitted to a circuit formed by a series association of: (i) a resistance in parallel with a capatior (ReCe), for the electrolyte part; (ii) a parallel association of a resistance and a non-ideal capacitor (RmQm), for the membrane itself. The fitting of the experimental points by means of a non-linear program [13] allows the determination of the different circuit parameters (relative errors lower that 7%), although only the membrane contribution will be discussed. Dependence of Rm values with salt concentration is shown in Fig. 3. The decrease of membrane resistance when KCl concentration increases is attributed to the concentration dependence of the electrolyte filling the membrane pores [8,9]. According to the uniform capillary physical model previously indicated, the membrane electrical resistance might be calculated as that corresponding to the pores filled by the electrolyte solution, this means: R cm = Rp/N=(1/l0)(d/USm)
(3)
where l0 is the solution conductivity and Sm is the membrane area. A comparison between experimental (Rm) values and those calculated using Eq. (3) (R cm) is also shown in Fig. 3, where very good agreement between both Rm and R cm values can be
Fig. 2. Membrane (m) and electrolyte (e) contributions to the impedance plots. (a) Nyquist plot ( − Zimg vs Zreal) for a given KCl concentration c=0.001 M; (b) A comparison of Bode plot ( − Zimg vs f ) for two KCl solutions: () c = 0.001 M; () c=0.002 M.
at frequency such that vRC =1, RC being the relaxation time [12]. Complex systems may present different relaxation times and the resulting plot is a depressed semi-circle. In such cases a non-ideal capacitor, which is called a constant phase element (CPE), is considered [12]. The impedance for the CPE is expressed by: Q(v) = Y0( jv) − n
(2) −n
where the admittance Y0 (V s ) and n are two empirical parameters (05 n 51).
Fig. 3. Concentration dependence for membrane electrical resistance. ( ) experimental points; () theoretical values calculated by Eq. (3).
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+ wU ln[(1+4y 21)1/2 − wU]/(1 + 4y 22)1/2 − wU]}
Fig. 4. Membrane potential, Dfm, as a function of the external concentration 1/c2.
(4)
where w is + 1 or − 1, for anionic or cationic membranes respectively; yj = zjkscj/wXf, ks being the salt partition coefficient (ks : 1 for porous membranes), Xf is the fixed charge concentration in the membrane, zj is the valence of the ions, cj represents the concentration of the solutions separated by the membrane. The parameter U is related to the transpor number of the anion (t − ) and cation (t + ) in the membrane (U= [t + / z + ]+ [t − / z − ]), R and F are the gas and Faraday constants and T is the temperature of the system. For external concentrations higher than the fixed charge concentration in the membrane, this means, when the Donnan potential can be neglected, a linear dependence between the membrane potential and 1/c was obtained by Aizawa et al. [17]. Dfm = (RT/F)[(1− 2t + )
seen. This result confirms the possibility of using electrical parameters for geometrical determinations as was previously indicated by different authors [6,7]; it is worth indicating that the average porosity from IS results is U =(0.11 9 0.02)%, while that obtained by geometrical parameters was (0.1390.03%). The good agreement obtained is directly related to the welldefined structure of track-etched membranes due to their narrow pore distribution and known pore length. According to the TMS theory [14,15], the membrane potential, Dfm, or electrical potential difference at both sides of a membrane due to a concentration gradient, can be taken as the sum of two different contributions: (i) a Donnan potential on each membrane – solution interface due to a possible exclusion of co-ions in the membrane; and (ii) a diffusion potential caused by the concentration gradient in the membrane; for symmetrical electrolytes (z + =z − =z), the membrane potential can be expressed as [16]: Dfm =(RT/wzF) {ln(c1[(1+ 4y 21)1/2 +1]/c2[(1 + 4y 22)1/2 +1]
+ (2g − 1)(t + t − /g)(Xf/c2)]
(5)
Fig. 4 shows the Dfm versus 1/c2 linear relationship obtained. From the slope of these experimental points, the fixed charge concentration and transport number of the ions in the membrane were determined by Eq. (5). Results were: t + = (0.569 0.06), t − = (0.4490.04) and Xf = − (2.59 0.2)×10 − 2 M, which indicate the membrane behaves as a weak cation-exchanger. Concentration dependence of membrane potentials measured keeping one concentration constant, this means, Dfm versus ln(c2/0.01), are shown in Fig. 5; for comparison, calculated membrane potential by Eq. (4) using the values previously obtained for the fixed charge concentration and the transport number of the ions in the membrane, Df cm, is also shown in Fig. 5. As can be seen, experimental values hardly differ from calculated ones, which confirms the reliability of the estimated electrochemical parameters. From membrane potential and conductance, lm, (or resistance, Rm = 1/lm) values the ionic permeabilities into the membrane pores, PK+ and PCl−, can be determined by Eqs. (6) and (7) [18]:
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PK+?/PCl − =exp[(F/RT)Dfm] −g/(1 − g exp[(F/RT)Dfm) PK+ +PCl − = lmRT/F 2%j(z 2j cj)
(6) (7)
Since ion permeability is related to ion diffusion coefficient, Dj, by: Pj =DjU/d, cation and anion diffusion coefficients into the membrane pores were obtained. Conductance values at the average
concentration of the external solutions, cavg, were obtained by interpolation of the results presented in Fig. 3, and membrane porosity determined by IS measurements was used for DK+ and DCl− calculations. Variation of the ionic diffusion coefficients with KCl concentration is shown in Fig. 6; the type of concentration dependence obtained for each ion, the decrease of DK+ values and the increase of DCl− ones, when cavg increases, agrees very well with the cation-exchange character found for this membrane, moreover at high concentrations (cavg \ Xf) ion diffusion coefficients do not differ very much from their values in solution.
4. Conclusions
Fig. 5. Comparison of experimental ( ) and calculated (---) membrane potential values.
Fig. 6. Concentration dependence of ionic diffusion coefficients: () DK+, () DCl−.
Electrochemical and structural characterizations of a track-etched membrane were carried out by hydraulic permeability, membrane potential and impedance spectroscopy measurements with the membrane in contact with KCl solutions at different concentrations. Good agreement among the different parameters determined, using different techniques, was obtained. Membrane porosity was determined from geometrical and electrical parameters assuming a membrane capillary model and very good agreement between both types of results was obtained. Concentration dependence of membrane electrical resistance is attributed to the electrolyte filling the membrane pores. Fixed charge concentration and ion transport numbers in the membrane were obtained, and these results show its slight cation-exchanger character. These values clearly predict the concentration dependence of membrane potential measured under different experimental conditions. Diffusion coefficients of ions in the pores of the membrane were determined and they also confirm the small negative charge in the membrane; however, values similar to those in solution are obtained for an average concentration of higher than 0.06 M, when the effect of the fixed charge can be neglected.
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Acknowledgements The authors would like to thank Dr A. Nechaev and the membrane group in the Laboratoty of Nuclear Filters, Shubnikov Institute of Cristallography, Moscow, for membrane preparation, and the European Comission (Project no. IC 15 CT 96-0826) and CICYT, Spain, (Proyect no. QUI97-1851-CE) for financial support.
References [1] J.I. Calvo, A. Hernandez, G. Caruana, L. Martinez, J. Coll. Interface Sci. 175 (1995) 138. [2] W.R. Bowen, D.T. Huges, J. Coll. Interface Sci. 143 (1991) 252. [3] S.G. Wong, J.C.T. Kwak, Desalination 15 (1974) 213. [4] H.-U. Demish, W. Pusch, J. Coll. Interface Sci. 69 (1979) 247. [5] J. Benavente, C. Ferna´ndez-Pine´da, J. Membr. Sci. 23 (1985) 121.
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[6] T. Hanai, K. Zhao, K. Asaka, K. Asami, J. Membr. Sci. 64 (1991) 153. [7] H.G.L. Coster, K.J. Kim, K. Dahlan, J.R. Smith, C.J.D. Fell, J. Membr. Sci. 66 (1992) 19. [8] J. Benavente, J.M. Garcı`a, J.G. de la Campa, J. de Abajo, J. Membr. Sci. 114 (1996) 51. [9] J. Benavente, J.R. Ramos-Barrado, A. Heredia, Coll. Surf. 140 (1998) 333. [10] V.V. Berezkin, A.N. Nechaev, S.V. Fomichev, B.V. Mchedlishvili, Kolloidn. Zh. 53 (1991) 339. [11] J. Benavente, A. Mun˜oz, A. Heredia, J. Membr. Sci. 139 (1998) 147. [12] J.R. MacDonald, Impedance Spectroscopy, Wiley, New York, 1987. [13] B.A. Boukamp, Solid State Ionics 18, 19 (1986) 136. [14] T. Teorell, Discuss. Faraday Soc. 21 (1956) 9. [15] K.H. Meyer, J.F. Sievers, Helv. Chim. Acta 19 (1936) 649. [16] N. Laksminarayanaiah, Transport Phenomena in Membranes, Academic Press, New York, 1969. [17] M. Aizawa, S. Tomoto, S. Suzuki, J. Membr. Sci. 6 (1980) 235. [18] S.G. Schultz, Basic Principles of Membrane Transport, Cambridge University Press, Cambridge, 1980.
.
Electrochemical and structural characterizations of an experimental track-etched membrane in KCl solutions A. Can˜as, J. Benavente * Departamento de Fı´sica Aplicada, Grupo de Caracterizacio´n Ele´ctrica y de Transporte en membranas e Interfases, Facultad de Ciencias, Uni6ersidad de Ma´laga, E-29071 Ma´laga, Spain Received 8 January 2000; received in revised form 6 March 2000; accepted 6 March 2000
Abstract Characteristic electrochemical transport parameters for an experimental poly(ethylene)terephtalate (PET) tracketched membrane with well-defined structure and low porosity (U = 0.13%) were determined with the membrane in contact with KCl solutions at different concentrations. Membrane potential, Df m, measurements were performed to investigate the effective fixed charge concentration, Xf, and transport number of the ions, ti, in the membrane using two different procedures: keeping the concentration ratio constant, or keeping one concentration constant and changing the other one. Results show the membrane presents a weak cation-exchanger character, since the following values were obtained: Xf = −(2.590.2)×10 − 2 M, tK = (0.569 0.06), tCl− = (0.449 0.05); taking into account these values, concentration dependence of membrane potential was predicted. Membrane electrical resistance, Rm, was obtained from Impedance Spectroscopy (IS) measurements using equivalent circuits as models, and the membrane porosity U=(0.11 9 0.02)% was also obtained from resistance values, which agrees very well with the value determined from geometrical parameters. From Rm, Dfm and U values, the diffusion coefficient of the ions in the membrane pores can be calculated, and the following average values were obtained: DK+ =(1.99 0.4)× 10 − 9 m2/s and DCl−=(0.8 9 0.2)×10 − 9 m2/s, but for an average concentration higher than 0.06 M, their values do not differ practically from solution in agreement with the small negative charge previously indicated. © 2000 Elsevier Science B.V. All rights reserved. +
Keywords: Impedance spectroscopy; Membrane potential; Salt permeability; Ionic diffusion coefficients
1. Introduction
* Corresponding author. Tel.: +34-95-2131929; fax: + 3495-2132000. E-mail address: j – [email protected] (J. Benavente).
The use of porous membranes for separation of differently sized molecules has many industrial applications, mainly in those involved in the treatment of food, beverage biological materials and waste treatment. The predition of the applicability of such membranes depends on the development of effective methods for their characterization,
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which are mainly carried out on the base of morphological parameters such as pore structure, pore size distribution and porosity [1]. However, other parameters can also affect the separation ability of porous membranes, for instance the electrochemical properties of membranes have a significant influence on the nature and magnitude of the interactions between the membrane and the substances to be processed [2]. Membrane characteristic electrochemical parameters, such as fixed charge concentration, Xf, and ion transport numbers, ti, are usually obtained from membrane potential measurements [3 – 5], while ionic diffusion coefficients in the membrane pores, Di, can be obtained, if the electrical resistance of the membrane, Rm, is also known. On the other hand, Impedance Spectroscopy (IS) is a non-destructive technique, which is being used as a successful tool to determine the electrical properties of layered systems such as membrane/electrolyte systems, since it allows the separate evaluation of the electrical contribution of both the membrane and the electrolyte solution [6 – 9]. Moreover, according to Coster et al. [7] membrane geometrical parameters (porosity and thickness) can be estimated from IS results. In this work, electrochemical and structural characterizations of a poly(ethylene)terephtalate (PET) track-etched membrane in contact with KCl solutions at different concentrations were carried out. Fixed charge concentration and ionic transport numbers in the membrane were obtained from membrane potential, which was measured following two different procedures in order to see the influence of the external conditions on transport numbers. The membrane electrical resistance was determined from IS data using an equivalent circuit consisting of a parallel association of a resistance and a non-ideal capacitor, but membrane porosity was also estimated from electrical results in order to compare it with that calculated from geometrical parameters. The diffusion coefficient of the ions into the membrane pores was also determined from electrical resistance, membrane potential and porosity. Concentration dependence of all these parameters was also studied.
2. Experimental
2.1. Membranes The membrane studied was an experimental nuclear track filter produced in the Laboratoty of Nuclear Filters, Shubnikov Institute of Cristallography, (Russian Academy of Sciences), Moscow, by etching the track in a poly(ethylene) terephtalate (PET) film irradiated by heavy ions [10]. The membrane geometrical parameters (given by suppliers) are: thickness d= (1091) mm, pore radii rp = (4.59 0.5) nm and number of pores by membrane area n= 2×1013 m − 2, from these values a porosity U= (0.139 0.03)% was calculated. Electrochemical parameters were measured with aqueous KCl solution at different concentrations, at a constant temperature t= (25.09 0.3)°C and standard pH = (5.79 0.3); before use, the membranes were immersed for at least 6 h, in a solution of the appropriate salt concentration.
2.2. Water flux, impedance spectroscopy, membrane potential and salt diffusion measurements Measurements of water flux due to an applied pressure difference were carried out in a crossflow filtration cell for pressure ranging between 80 and 200 kPa. The dead-end cell used for electrochemical characterization is similar to that indicated in the literature [11]. The membrane was tightly clamped between two glass half-cells by using silicone rubber rings; each half-cell has a reversible Ag–AgCl electrode and a magnetic stirrer (stirring rate around 500 rpm) to minimize concentration–polarization at the membrane surfaces: Impedance spectroscopy measurements were carried out with the electrodes connected to an Impedance Analyzer (Solartron 1260), from which data can be sent to a computer for further treatment and storage. The experimental data were corrected by the software, taking into account the influence of connecting cables and other parasite capacitances. Measurements were carried out with 100 different frequencies in the range 10–106 Hz at a maximum voltage
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of 0.01 V, with both half-cells filled with solutions of identical concentration. Measurements were carried out for concentration ranging between 10 − 3 M and 5×10 − 2 M. Electromotive force (DE) between both sides of the membrane caused by a concentration gradient was measured by connecting the reversible electrodes to a digital voltmeter (Yokogawa 7552, 1GV imput resistance). Membrane potential measurements were carried out using two different procedures: (i) by keeping the concentration ratio of the solutions at both sides of the membrane constant, c2/c1 =g =2, for concentration c1 ranging between 5 × 10 − 3 and 10 − 1 M; ii) keeping the concentration c1 constant (c1 = 0.01 M) and changing gradually c2 from 10 − 2 to 2 ×10 − 1 M. Membrane potential, Dfm, was determined from measured DE values by subtracting the electrode potential contribution. For each experiment, three series of measurements were carried out, the experimental values were obtained as the average of these measurements; good reproducibility was found and relative errors lower than 10% were obtained.
Fig. 1. Water flux, Jv, versus applied pressure difference, DP.
171
3. Results and discussion Due to the well-defined geometrical structure of track-etched membranes, the uniform capillary physical model, which consists of straight pores without tortuosity or constriction, is usually adopted. In these cases, the Haagen–Poiseuille expression can be used to determine the water flow through the membrane, Jv, due to the applied pressure difference, DP: Jv = Lp DP = (r 2pu/8hd)DP
(1)
where Lp is the membrane hydraulic permeability and h is the viscosity of water. Fig. 1 shows the experimental Jv –DP linear relationship and, according to Eq. (1), the membrane hydraulic permeability can be calculated from the slope of this straight line; it was obtained that Lp = (2.09 0.3)×10 − 13 m/s Pa. Taking into account Eq. (1) and the value of the geometrical parameters previously reported, the theoretical value of the hydraulic permeability was also determined: L tp = (3.29 0.6)× 10 − 13 m/s Pa; the good agreement found between experimental and theoretical values indicates the adequacy of the capillary model used. Fig. 2 shows the typical impedance plots: Nyquist plot (− Zimg vs Zreal) and Bode (− Zimg vs f ) plot, obtained by impedance data. Two dielectric relaxations can be observed: the semicircle obtained at high frequencies in Fig. 2(a) (2×104 5 f(Hz)5 106) is assigned to the electrolyte solution, while the depressed semi-circle for frequency ranging between 102 and 2× 103 Hz corresponds to the membrane [9,12]; this assignation is made on the basis of the Bode plot (Fig. 2(b)) where a comparison of the data obtained with two different concentrations (10 − 3 and 2 × 10 − 3 ?M) is shown. The analysis of impedance data was carried out by the complex plane Z* method, which involves plotting the impedance imaginary part (− Zimg) versus the real part (Zreal) as is shown in Fig. 2(a). A parallel R–C circuit gives rise to a semi-circle in the Z* plane, which has intercepts on the Zreal axis at R (v ) and R0 (v0), being (R0 − R ) the resistance of the system. The maximum of the semi-circle equals 0.5(R0 − R ) and occurs
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Experimental data were fitted to a circuit formed by a series association of: (i) a resistance in parallel with a capatior (ReCe), for the electrolyte part; (ii) a parallel association of a resistance and a non-ideal capacitor (RmQm), for the membrane itself. The fitting of the experimental points by means of a non-linear program [13] allows the determination of the different circuit parameters (relative errors lower that 7%), although only the membrane contribution will be discussed. Dependence of Rm values with salt concentration is shown in Fig. 3. The decrease of membrane resistance when KCl concentration increases is attributed to the concentration dependence of the electrolyte filling the membrane pores [8,9]. According to the uniform capillary physical model previously indicated, the membrane electrical resistance might be calculated as that corresponding to the pores filled by the electrolyte solution, this means: R cm = Rp/N=(1/l0)(d/USm)
(3)
where l0 is the solution conductivity and Sm is the membrane area. A comparison between experimental (Rm) values and those calculated using Eq. (3) (R cm) is also shown in Fig. 3, where very good agreement between both Rm and R cm values can be
Fig. 2. Membrane (m) and electrolyte (e) contributions to the impedance plots. (a) Nyquist plot ( − Zimg vs Zreal) for a given KCl concentration c=0.001 M; (b) A comparison of Bode plot ( − Zimg vs f ) for two KCl solutions: () c = 0.001 M; () c=0.002 M.
at frequency such that vRC =1, RC being the relaxation time [12]. Complex systems may present different relaxation times and the resulting plot is a depressed semi-circle. In such cases a non-ideal capacitor, which is called a constant phase element (CPE), is considered [12]. The impedance for the CPE is expressed by: Q(v) = Y0( jv) − n
(2) −n
where the admittance Y0 (V s ) and n are two empirical parameters (05 n 51).
Fig. 3. Concentration dependence for membrane electrical resistance. ( ) experimental points; () theoretical values calculated by Eq. (3).
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+ wU ln[(1+4y 21)1/2 − wU]/(1 + 4y 22)1/2 − wU]}
Fig. 4. Membrane potential, Dfm, as a function of the external concentration 1/c2.
(4)
where w is + 1 or − 1, for anionic or cationic membranes respectively; yj = zjkscj/wXf, ks being the salt partition coefficient (ks : 1 for porous membranes), Xf is the fixed charge concentration in the membrane, zj is the valence of the ions, cj represents the concentration of the solutions separated by the membrane. The parameter U is related to the transpor number of the anion (t − ) and cation (t + ) in the membrane (U= [t + / z + ]+ [t − / z − ]), R and F are the gas and Faraday constants and T is the temperature of the system. For external concentrations higher than the fixed charge concentration in the membrane, this means, when the Donnan potential can be neglected, a linear dependence between the membrane potential and 1/c was obtained by Aizawa et al. [17]. Dfm = (RT/F)[(1− 2t + )
seen. This result confirms the possibility of using electrical parameters for geometrical determinations as was previously indicated by different authors [6,7]; it is worth indicating that the average porosity from IS results is U =(0.11 9 0.02)%, while that obtained by geometrical parameters was (0.1390.03%). The good agreement obtained is directly related to the welldefined structure of track-etched membranes due to their narrow pore distribution and known pore length. According to the TMS theory [14,15], the membrane potential, Dfm, or electrical potential difference at both sides of a membrane due to a concentration gradient, can be taken as the sum of two different contributions: (i) a Donnan potential on each membrane – solution interface due to a possible exclusion of co-ions in the membrane; and (ii) a diffusion potential caused by the concentration gradient in the membrane; for symmetrical electrolytes (z + =z − =z), the membrane potential can be expressed as [16]: Dfm =(RT/wzF) {ln(c1[(1+ 4y 21)1/2 +1]/c2[(1 + 4y 22)1/2 +1]
+ (2g − 1)(t + t − /g)(Xf/c2)]
(5)
Fig. 4 shows the Dfm versus 1/c2 linear relationship obtained. From the slope of these experimental points, the fixed charge concentration and transport number of the ions in the membrane were determined by Eq. (5). Results were: t + = (0.569 0.06), t − = (0.4490.04) and Xf = − (2.59 0.2)×10 − 2 M, which indicate the membrane behaves as a weak cation-exchanger. Concentration dependence of membrane potentials measured keeping one concentration constant, this means, Dfm versus ln(c2/0.01), are shown in Fig. 5; for comparison, calculated membrane potential by Eq. (4) using the values previously obtained for the fixed charge concentration and the transport number of the ions in the membrane, Df cm, is also shown in Fig. 5. As can be seen, experimental values hardly differ from calculated ones, which confirms the reliability of the estimated electrochemical parameters. From membrane potential and conductance, lm, (or resistance, Rm = 1/lm) values the ionic permeabilities into the membrane pores, PK+ and PCl−, can be determined by Eqs. (6) and (7) [18]:
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PK+?/PCl − =exp[(F/RT)Dfm] −g/(1 − g exp[(F/RT)Dfm) PK+ +PCl − = lmRT/F 2%j(z 2j cj)
(6) (7)
Since ion permeability is related to ion diffusion coefficient, Dj, by: Pj =DjU/d, cation and anion diffusion coefficients into the membrane pores were obtained. Conductance values at the average
concentration of the external solutions, cavg, were obtained by interpolation of the results presented in Fig. 3, and membrane porosity determined by IS measurements was used for DK+ and DCl− calculations. Variation of the ionic diffusion coefficients with KCl concentration is shown in Fig. 6; the type of concentration dependence obtained for each ion, the decrease of DK+ values and the increase of DCl− ones, when cavg increases, agrees very well with the cation-exchange character found for this membrane, moreover at high concentrations (cavg \ Xf) ion diffusion coefficients do not differ very much from their values in solution.
4. Conclusions
Fig. 5. Comparison of experimental ( ) and calculated (---) membrane potential values.
Fig. 6. Concentration dependence of ionic diffusion coefficients: () DK+, () DCl−.
Electrochemical and structural characterizations of a track-etched membrane were carried out by hydraulic permeability, membrane potential and impedance spectroscopy measurements with the membrane in contact with KCl solutions at different concentrations. Good agreement among the different parameters determined, using different techniques, was obtained. Membrane porosity was determined from geometrical and electrical parameters assuming a membrane capillary model and very good agreement between both types of results was obtained. Concentration dependence of membrane electrical resistance is attributed to the electrolyte filling the membrane pores. Fixed charge concentration and ion transport numbers in the membrane were obtained, and these results show its slight cation-exchanger character. These values clearly predict the concentration dependence of membrane potential measured under different experimental conditions. Diffusion coefficients of ions in the pores of the membrane were determined and they also confirm the small negative charge in the membrane; however, values similar to those in solution are obtained for an average concentration of higher than 0.06 M, when the effect of the fixed charge can be neglected.
A. Can˜as, J. Bena6ente / Separation/Purification Technology 20 (2000) 169–175
Acknowledgements The authors would like to thank Dr A. Nechaev and the membrane group in the Laboratoty of Nuclear Filters, Shubnikov Institute of Cristallography, Moscow, for membrane preparation, and the European Comission (Project no. IC 15 CT 96-0826) and CICYT, Spain, (Proyect no. QUI97-1851-CE) for financial support.
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