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A Comparison of Electrochemical and Electrokinetic Parameters Determined for Cellophane Membranes in Contact with NaCl and NaNO3 Solutions
Journal of Colloid and Interface Science 246, 150–156 (2002) doi:10.1006/jcis.2001.8004, available online at http://www.idealibrary.com on
A Comparison of Electrochemical and Electrokinetic Parameters Determined for Cellophane Membranes in Contact with NaCl and NaNO3 Solutions A. Ca˜nas, M. J. Ariza, and J. Benavente1 Grupo de Caracterizaci´on Electrocin´etica y de Transporte en Membranas e Interfases, Departamento de F´ısica Aplicada, Facultad de Ciencias, Universidad de M´alaga, E-29071 M´alaga, Spain Received April 30, 2001; accepted October 1, 2001
Electrochemical and electrokinetic characterizations of cellophane membrane samples have been carried out by measuring membrane potential, salt diffusion, and tangential streaming potential, which allow the determination of different characteristic membrane parameters. Experiments were made with the membrane samples in contact with NaCl and NaNO3 solutions at different concentrations and under different external conditions (concentration gradients), in order to obtain differences in transport and membrane characteristic parameters, depending on the electrolyte considered. Salt permeability across the membrane, which was obtained from diffusion measurements, is about twice as high for NaCl solutions as for NaNO3 solutions, which is attributed to the different sizes of the electrolytes. Membrane potential measurements keeping the concentration ratio constant (C1 /C2 = 2) were used to determine both the effective fixed charge concentration in the membrane, Xf , and the average value of transport numbers, hti i; taking into account these values, concentration dependence of membrane potential under a different external condition (C1 = cte = 0.01 M, 5 × 10−3 ≤ C(M) ≤ 5 × 10−2 ) was predicted. Results show that cellophane membrane behaves as a weak cation-exchange membrane and its permselectivity to cations is practically independent of the electrolyte considered. From electrokinetic results, assuming a Langmuir-type adsorption of anions on the cellophane surface, the number of accessible sites per surface unit was obtained, which is higher for Cl− than for NO− 3 , in agreement with the small radii of chlorine ions; however, no significant differences in the specific adsorption free energy were found (1GNacl = −22.0 × 103 J/mol) and (1GNaNO3 = −23.2 × 103 J/mol). °C 2002 Elsevier Science Key Words: membrane potential; salt diffusion; tangential streaming potential; electrochemical and adsorption parameters; NaCl and NaNO3 solutions.
INTRODUCTION
Cellulose and its derivatives provide a very important class of basic materials for membranes, mainly due to the fact that cellulose is a very hydrophilic polymer but is not water solu-
1
To whom correspondence should be addressed.
0021-9797/02 $35.00
° C 2002 Elsevier Science
All rights reserved.
ble, which is related to its crystallinity and the intermolecular hydrogen bonding between its hydroxyl groups (1). Cellophane and other types of regenerated cellulose are mainly used as materials for dialysis membranes, since they allow the diffusion of ions and low molecular weight solutes but they do not permit the diffusion of proteins or macromolecules of high molecular weight (1, 2), while cellulose acetate and nitrate are used for microfiltration/ultrafiltration processes and cellulose triacetate is used in reverse osmosis membranes for desalting applications. However, due to medical applications of dialysis membranes (hemodialysis), they represent one of the major membranes marketed worldwide. Transport of solutions across dialysis membranes is usually characterized by determining the solute’s permeability or diffusion coefficients in the membrane, but in the case of regenerated cellulose membranes some of the –CH2 OH groups are oxidized to –COOH in air, giving a weak negative charge to these membranes. Then, if electrolyte solutions are studied, other characteristic electrochemical parameters such as membrane charge density (surface and bulk fixed charge), salt permeability, and ionic transport numbers must be determined, since they can also provide information on electrical “membrane-solution” interactions, which can affect the mobility of charged solutes across the membranes (2–7). Particularly, when the membranes are used in hemodialysis, the negative charge can reduce the passage of some anions even if their size is lower than that of the membrane pores (i.e., the hydrogen phosphate ions, which must be removed from the patient). For these reasons, an electrochemical characterization of a cellophane membrane, which can be used in dialysis systems, in contact with NaCl and NaNO3 solutions at different concentrations was carried out. Membrane potential, salt diffusion, and tangential streaming potential were measured. Concentration dependence for the different electrochemical and electrokinetic parameters was determined and a comparison of their values across a cellophane membrane was also made for both electrolytes. The influence of external conditions (concentration gradients) on salt diffusion and ion transport numbers was also determined, as well as the Donnan potential contribution to membrane potential at
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CELLOPHANE MEMBRANES IN NaCl AND NaNO3
each membrane–solution interface. Differences found in transport parameters for both electrolytes agree with the lower size and higher mobility of Cl− ions with respect to NO− 3 ones. In order to obtain some fundamental properties of the solid phase (the membrane) from streaming potential measurements, it is necessary to conjecture a particular origin and mechanism of adsorption of charge on the membrane. Assuming that the surface charge density on the Stern layer (electrokinetic charge density) comes from adsorbed anions according to Langmuir-type adsorption (8), electrokinetic results allow the determination of some characteristic adsorption parameters such as the number of accessible sites per surface unit and the Gibb’s free energy; results show that the density of accessible sites is about three times higher for Cl− than for NO− 3 , which also agrees with the small radii of chlorine ions, but only slight differences in the specific adsorption free energy were obtained. 2. EXPERIMENTAL
2.1. Material Different samples of a sheet of cellophane kindly submitted by Cellophane Espa˜nola, S.A. (Burgos, Spain), were studied. Membrane structural parameters are (7) wet membrane thickness, 1X w = (60 ± 3) µm; degree of swelling, sw = (50 ± 2)%; and fractional void volume, ε = (55 ± 3)%. Measurements were carried out with the membrane samples in contact with aqueous NaCl and NaNO3 solutions at different concentrations (5 × 10−4 ≤ C(M) ≤ 10−1 ), at room temperature t = (25.0 ± 0.3)◦ C and standard pH (5.8 ± 0.3). Before use, the membranes were immersed for at least 4 h in a solution of the appropriate concentration. 2.2. Membrane Potential and Salt Diffusion Measurements The test cell used for membrane potential and salt diffusion measurements is similar to that described elsewhere (9). The membrane was tightly clamped between two glass half-cells by using silicone rubber rings; a magnetic stirrer was placed at the bottom of each half-cell and, to minimize concentrationpolarization at the membrane surfaces, measurements were carried out at a stirring rate of 525 rpm. The electromotive force (1E) between the two sides of the membranes caused by a concentration gradient was measured by two reversible Ag/AgCl electrodes connected to a digital voltmeter (Data Precision 3500, 100 MÄ imput impedance). Measurements were carried out using two different procedures: (i) keeping the concentration C1 constant (C1 = 0.01 M) and changing C2 gradually from 5 × 10−3 to 5 × 10−2 M. (ii) keeping the concentration ratio of the solutions on both sides of the membrane constant, C1 /C2 = g = 2, but changing concentration C2 between 0.005 and 0.1 M. Membrane potential, 1øm , was determined from measured 1E values by subtracting the electrode potential contribution.
151
In salt diffusion measurements the membrane was initially separating a concentrated solution (C1 ) from a diluted one (initially distilled water, this means, C2 = 0). Changes in the solution C2 versus time were recorded by means of a conductivity cell connected to a digital conductivity meter (Radimeter CDM 83). Four concentrated solutions (C1 = 10−3 , 5 × 10−3 , 10−2 , and 5 × 10−2 M) were used to see the possible effect of concentration gradients on salt permeability. 2.3. Streaming Potential Measurements Tangential streaming potential (TSP) measurements were carried out with a system which basically consists of (i) a rectangular cell (15.2 cm2 section) with a channel of well-defined dimensions created by the use of two Teflon spacers between two membrane samples; (ii) a mechanical drive unit to produce and measure the pressure (80 ≤ 1P(mB) ≤ 200), which drives the electrolyte solution from a reservoir into the measuring cell. Characteristic hydrodynamic parameters are the Reynolds number Re = (390 + 60) and the hydrodynamic height of the channel h h = (238 ± 7) µm. Black platinum electrodes, one at each end of the channel, connected to a high-impedance voltmeter (Yokogawa 7552, 1 GÄ imput impedance) were used to measure the streaming potential, taking the electrode at lower pressure as reference. Streaming potential coefficient (φ = 1V /1P) was determined from the average of at least six measurements with the solution always flowing in the same direction; 1V values were corrected by electrode potential asymmetry. RESULTS AND DISCUSSION
The electrical potential difference at both sides of a membrane when it is separating two solutions of the same electrolyte but different concentrations (C1 and C2 ) is called “membrane potential” (1øm ). Figure 1 shows membrane potential versus salt concentration for the cellophane membrane with NaCl and NaNO3 solutions under both different external conditions: C1 /C2 = 2 and C1 = 0.01 M, respectively. According to the Teorell–Meyer–Sievers or TMS theory (10, 11), the membrane potential can be considered as the sum of two Donnan potentials (one at each membrane/solution interface) plus a diffusion potential in the membrane; this means that 1øm = 1øDon(I) + 1ødif + 1øDon(II) . The expression for these potentials, when diluted solutions are considered (concentrations are used instead of activities), are (12) £ £ ¤¤ [1] 1øDon = (RT /F) ln (w X f /2C) + (w X f /2C)2 + 1)1/2 1ødif = (RT /F)[(t− − t+ ) ln(C1 /C2 )],
[2]
where X f is the membrane fixed charge concentration, w = −1 or +1 for negatively or positively charged membranes, respectively, and ti is the transport number of the ion in the membrane (i = + for cation, − for anions); t1 represents the amount of current transported for one ion with respect to the total current crossing the membrane, ti = Ii /IT , this means t+ + t− = 1.
˜ CANAS, ARIZA, AND BENAVENTE
152
FIG. 1. Membrane potential vs salt concentration: (a) C1 = cte = 10−2 M; (b) C1 /C2 = cte = 2. Experimental values: (s) NaCl and (n) NaNO3 solutions; calculated values by Eqs. [1] and [2]: dotted line.
However, for slightly charged membranes, or when the external salt concentration is higher than the membrane fixed charge (X f ), Aizawa et al. (13) derived the approximation 1øm = (RT /F)[(1 − 2t+ ) + (2g − 1)(t+ t− /g)(X f /C2 )], [3] where g is the concentration ratio (g = C1 /C2 ). If the parameter g is constant, Eq. [3] represents a linear relationship between 1øm and 1/C2 . Figure 1b shows membrane potential versus salt concentration (1øm vs 1/C1 ) for the cellophane membrane with NaCl and NaNO3 solutions and keeping the concentration ratio constant: C1 /C2 = 2, (10−2 ≤ C2 (M) ≤ 10−1 ). The fitting of the linear relationships obtained allows the determination of the fixed charge concentration and the average ion transport numbers in the membrane by means of Eq. [3]. X f and ht+ i values determined for the cellophane membrane with both electrolytes are indicated in Table 1. As can be seen, the cellophane membrane behaves as a weak cation exchanger, which agrees with results reported in the literature for different cellophane membranes (14, 15), and is more relevant when the membrane is
TABLE 1 Characteristic Electrochemical Parameters Determined for a Cellophane Membrane in Contact with NaCl and NaNO3 Solutions Electrolyte NaCl NaNO3
X f (M)
ht+ i
Psm (+) (%)
Ds (m2 /s)
−(1.5 ± 0.2)×10−2 (0.53 ± 0.2) (23.7 ± 0.9) (3.6 ± 0.3)×10−10 −(0.8 ± 0.1)×10−2 (0.55 ± 0.3) (22.8 ± 1.2) (5.6 ± 0.7)×10−11
Note. Fixed charge concentration, X f , average cation transport number, ht+ i, cation permselectivity, Ps (Na+ ), and salt diffusion coefficient, Ds .
in contact with NaCl solutions than when it is in contact with NaNO3 ones. Figure 1a shows the concentration dependence of membrane potential when C1 = cte = 0.01 M (1φm vs ln(C2 /C1 )). Theoretical values for membrane potential calculated taking into account both Donnan and diffusion contributions by using X f and ht+ i values obtained from C1 /C2 = cte measurements are seen in Fig. 1a as a dotted line. As can be seen, very good agreement between experimental and theoretical membrane potential was obtained, which can be considered a test of the validity of X f and ht+ i values determined from measurements at constant concentration ratios. Membrane permselectivity, Psm (i), is a measure of the selectivity of counterions over co-ions in a membrane; in the case of negatively charged membranes, permselectivity to cations can be obtained by (16) Psm (+) = (t+ − t+o )/(1 − t+o ),
[4]
where t+o represents the cation transport number in solution. Average permselectivity, hPsm (+)i, for both electrolyte solutions was determined taking the ht+ i values previously obtained, and its values are also indicated in Table 1. Results show that membrane permselectivity to cations is practically independent of the electrolyte considered. Salt permeability across a membrane can be obtained from diffusion measurements. According to Fick’s first law, the salt flux (Js ) through a membrane (for a quasi-steady state) can be written Js = Ps (C1 − C2 ), where Ps is the salt permeability in the membrane. On the other hand, the molar salt flux through the membrane at any time is given by Js = (1/Sm )(dn/dt) = (Vo /Sm )(dC2 /dt),
[5]
CELLOPHANE MEMBRANES IN NaCl AND NaNO3
153
FIG. 2. Time dependence for conductivity in compartment 2: (a) at different concentration gradients (1c) and NaNO3 solutions; (b) comparison between NaCl and NaNO3 solutions at a given concentration gradient 1c = 10−2 M.
where Vo and Sm are the volume of the solution on the side with concentration C2 , and the membrane area, respectively. Then, the following expression can be obtained, dC2 /[C1 − C2 (t)] = (Sm /Vo )Ps dt,
[6]
which can be also written as a function of salt conductivity, dλ2 /[λ1 − λ2 (t)] = (Sm /Vo )(dλ/dC)e Ps dt,
[7]
where (dλ/dC)e is characteristic for each electrolyte. Variation in the conductivity of solution C2 as a function of time for NaCl and NaNO3 solutions is shown in Fig. 2b for a given concentration gradient (1C = C1 − C2 = 0.01 M), while Figure 2a shows a comparison of salt diffusion for NaNO3 solutions at different concentration gradients. From the slopes of these straight lines, salt permeability across the cellophane membrane was obtained for the different concentration gradients and electrolytes studied. Ps concentration dependence for both electrolytes is shown in Fig. 3. As can be observed, at low concentrations Ps increases when concentration gradient increases, but an almost constant value is reached at high 1C values, as also indicated by Kimura et al. (17). It should be pointed out that a higher permeability was obtained with NaCl solutions than with NaNO3 , which is attributed to its small size. Salt diffusion coefficients in the membrane, Ds , can be obtained from Ps values by (1) Ps = Ds /1xm , where 1xm , is the membrane thickness. Ds values for both electrolytes at a given concentration gradient (1C = 0.01 M) are shown in Table 1. It is worth noting that salt diffusion coefficients in the membrane are much lower than those in solution, Dso ([Ds /Dso ](NaCl) = 0.23 and [Ds /Dso ](NaNO3 ) = 0.004), which is indicative of the interactions between the solutes and the cross-linked chains of the cellophane matrix. For negatively charged membranes and 1 : 1 electrolytes, the following expression for the salt permeability on the basis of the
TMS theory was obtained by Kimura et al. (17), ©¡ ¢1/2 Ps = [D+ D− RT X f /1xm (D+ + D− )1C] 1 + 4y12 ¡ £¡ ¤± ¢1/2 ¢1/2 − 1 + 4y22 − U ln 1 + 4y12 +U ¤ª ¢1/2 £¡ +U , [8] × 1 + 4y22 where yj = K ± Cj / X f (K ± is the salt partition coefficient), Ci is the concentration of the high and low solutions (i = 1 and 2), respectively, D+ and D− are the diffusion coefficient of cation and anion in the membrane, and U = (D+ − D− )/(D+ + D− ) = t+ − t− . In Fig. 3 the dotted lines represent the concentration dependence of Ps values obtained from Eq. [8] by using the membrane fixed charge concentration and transport numbers determined from membrane potential measurements at different external concentrations (K ± = 1). As can be seen, rather good
FIG. 3. Salt permeability vs concentration gradient. (s) NaCl solutions and (1) NaNO3 solutions. Dotted line: calculated values by Eq. [8].
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agreement between calculated and experimental values determined from different kinds of experiments was obtained. On the other hand, the solid/liquid electrical interface is usually characterized by streaming potential measurements. When an electrolyte solution is forced to move, by means of an external pressure (1P) along a charged solid wall, an electrical potential difference or streaming potential (1φst ) between the ends (at steady state) can be measured. The hydraulic pressure causes movement of liquid and thus ions are stripped off along the shear plane, and a streaming current is formed; due to a charge accumulation at the downstream side, and electrical field is then generated that causes a backflow of ions until steady state is reached. The measurable electrical potential difference between the two ends of the solid/liquid system, 1φst , gives direct information about the electrical potential at the shear plane (zeta potential or ζ ) and the electrokinetic charge density (σs ). Figure 4a shows streaming potential versus applied pressure for NaCl and NaNO3 solutions at different concentrations the
FIG. 5. Electrokinetic parameters vs salt concentration: (s) NaCl solutions and (1) NaNO3 solutions. (a) Zeta potential (ζ ); (b) surface charge density (σs ).
streaming potential coefficient, φst , was obtained from the slope of these straight lines, φst = 1V /1P, and its dependence on salt concentration for both electrolytes is shown in Fig. 4b. Streaming potential is related to the zeta potential by the Helmholtz– Smoluchowski equation (18) ζ = (λη/εo εr )(1φst /1P) = (λη/ε)φ,
FIG. 4. (a) Streaming potential vs applied pressure for both electrolytes at different concentrations. (b) Streaming potential coefficient (Φst = 1V /1P) as a function of electrolyte concentration: (s) NaCl solutions and (1) NaNO3 solutions.
[9]
where λ, η, and ε are the liquid conductivity, viscosity, and dielectric constant (ε = εo εr ). Surface conductivity was not taken into account, since the height of the channel (about 300 µm) was large enough to allow us to ignore the electrical conductance effect at the diffuse part of the double layer. Zeta potential versus salt concentration is shown in Fig. 5a. As can be seen, zeta potential clearly depends on the electrolyte considered, but it does not show any clear dependence on salt concentration, and the following average values for each electrolyte were obtained: hζ i(NaCl) = −(5.6 ± 0.9) and hζ i(NaNO3 ) = −(2.9 ± 0.5). Surface charge density at the shear plane can be determined from zeta potential values by the following equation (18), σS = (2RT ek/z F) sinh(Fζ /2RT ),
[10]
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CELLOPHANE MEMBRANES IN NaCl AND NaNO3
where k and e are the Debye length and the elemental charge, respectively. Concentration dependence for the electrokinetic charge density is shown in Fig. 5b for both electrolytes. As can be seen, more negative values for σS are obtained when the cellophane membrane was in contact with NaCl solutions than with NaNO3 ones, which agrees with the slightly higher values also found for the fixed charge concentration from membrane potential measurements. The slight increase of the negative electrokinetic charge when salt concentration increases, found for both electrolytes, can be attributed to the adsorption of anions from solution on the membrane surface (18, 19). According to the Stern model, the electroneutrality condition for the membrane/solution interface can be written as a surface density charge balance (18), σe = σs + σo ,
[11]
where σe , σs , and σo , are the surface charge densities at the shear, Stern, and membrane surface planes, respectively. If a kind of ion adsorption is assumed (8, 20), this equality allows us to obtain the membrane charge density and some adsorption characteristic parameters such as the molar free energy and the density of accessible sites on the membrane surface. If the surface charge density on the Stern plane comes from adsorbed anions according to a Langmuir model, the following expression can be obtained (8, 18), σS = z − eNs χ− exp[(−z − ζ ∗ \z + ) − (1G c /RT )]/ × {1 + χ− exp[(−z − ζ ∗ \z + ) − (1G c /RT )]}, [12] where ζ ∗ is the dimentionless zeta potential (ζ ∗ = ζ e/RT ),
TABLE 2 Total Number of Adsorption Sites Accesible to the Anions per Unit of Membrane Surface, Ns , and Chemical Part of the Molar Gibbs Free energy of Adsorption, ∆G c Salt
Ns (sites m−2 )
1G c (J/mol)
NaCl NaNO3
(2.3 ± 0.4) × 1016 (0.8 ± 0.1) × 1016
−(22 ± 3) × 103 −(23 ± 4) × 103
1G c is the chemical (or nonelectrostatic) part of the molar Gibbs free energy of adsorption, and Ns is the total number of adsorption sites accessible to the anions per unit of membrane surface; χ− is the molar fraction of anions in the solution at the solid surface, which is given by (18) ¡ ¢±£ ¡ ¢¤ χ− = Cν− MH2 O ρ − C ν− M− + ν+ M+ (ν− + ν+ )MH2 O , [13] with Mi and νi the molecular weight and the stochiometric coefficient of the i species (i = ± for cation/anion), and ρ is the solution density. Solid characteristic parameters (1G c and Ns ) can be obtained by fitting the electrokinetic potential and charge as a function of the ionic molar fraction according to (8, 18) −[A1 /(σe − σo )] − A2 = exp(−ζ ∗ )/χ− ,
[14]
where the constants A1 and A2 are related with 1G c and Ns as follows (18): 1G c = −RT ln A2 and Ns = A1 /e A2 . In the case of zero proper charge, Eq. [14] represents a linear relationship between 1/σs and exp(−ζ ∗ )/χ− , such as that obtained with both electrolytes and shown in Fig. 6. By the fitting of the experimental data, A1 and A2 were determined; values of the specific adsorption free energy (1G c ) and the density of accessible sites (Ns ) are shown in Table 2. A comparison of results obtained for both electrolytes shows that the number of accessible sites is three times higher for Cl− than for NO− 3 , in agreement with the small radii of chlorine ions, but practically no difference in the molar adsorption free energy was obtained. The negative sign of 1G c implies a spontaneous process of adsorption of the anions on the membrane surface; moreover, the calculated density of anion sites is relatively low, in accord with the hydrophilic character of cellophane.
CONCLUSIONS
FIG. 6. Plot of exp(−ζ ∗ )/χ− as a function of 1/σs for both electrolytes: (s) NaCl and (1) NaNO3 solutions.
Transport of electrolyte solutions across a cellophane membrane has been studied by measuring membrane potential, salt diffusion, and streaming potential with NaCl and NaNO3 solutions at different concentrations, which allows the
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˜ CANAS, ARIZA, AND BENAVENTE
comparison of different transport parameters (ion transport numbers, permselectivity, and salt permeability) as well as characteristic adsorption parameters on the membrane surface (molar Gibbs free energy and density of accessible sites) for both electrolytes. —Membrane and streaming potential measurements show the negative character of the membrane, but results seems to indicate that the effective membrane charge is mainly due to the adsorption of anions from solution. —Very good agreement was obtained by comparing experimental membrane potential values and those calculated by using the TMS theory, which can be considered as a probe of the reliability of determined parameters (membrane fixed charge and ion transport numbers). It should be pointed out that Donnan equilibrium affects the ion distribution for concentration lower than 0.02 M, but at higher concentrations the membrane potential is practically a diffusion potential. —The lower values for permeability and diffusion coefficients across the cellophane membrane obtained with NaNO3 solutions at the different concentration gradients studied are attributed to its higher size and strong interaction with the membrane matrix. On the other hand, good concordance has been found between salt permeability values determined from diffusion measurements and those calculated taking into account the electrochemical parameters previously obtained. —Electrokinetic parameters permit us to obtain a representation of the membrane/solution interface, but they have also allowed the determination of membrane parameters assuming a Langmuir adsorption of anions on the surface of the cellophane membrane. The higher density of accessible sites obtained when NaCl solutions are used agrees with the small radii of Cl− ions in comparison with NO− 3 ones ions.
ACKNOWLEDGMENTS We thank the Comision Interministerial de Ciencia y Tecnolog´ıa (Project MAT2000-1140) and Junta de Andaluc´ıa (Grupo FQM 258) for financial support.
REFERENCES 1. Mulder, M., “Basic Principles of Membrane Technology.” Kluwer Academic, Dordrecht, The Netherlands, 1992. 2. Sakai, K., J. Membr. Sci. 96, 91 (1994). 3. van Oss, C. D., Science 139, 1123 (1963). 4. Haase, R., de Greiff, H. J., and Buchner, H. J., Z. Naturforch. 25, 1080 (1970). 5. Wong, S. G., and Kwak, J. C. T., Desalination 15, 213 (1974). 6. Kimura, Y., Lim, H.-J., and Iijima, T., J. Membr. Sci. 18, 285 (1984). 7. Benavente, J., V´azquez, M. I., and de Abajo, J., Sept. Sci. Technol. 31, 189 (1996). 8. Benavente, J., Hernandez, A., and Jonsson, G., J. Membr. Sci. 80, 285 (1993). 9. Benavente, J., Mu˜noz, A., and Heredia, A., J. Membr. Sci. 139, 147 (1998). 10. Teorell, T., Discuss. Faraday Soc. 21, 9 (1956). 11. Meyer, K. H., and Sievers, J. F., Helv. Chim. Acta 19, 649 (1936). 12. Laksminarayanaian, N., “Transport Phenomena in Membranes. Academic Press, New York, 1969. 13. Aizawa, M., Tomono, S., and Suzuki, S., J. Membr. Sci. 6, 235 (1980). 14. Benavente, J., Sep. Sci. Technol. 26, 189 (1991). 15. Kimura, Y., Lim, H.-J., and Iijima, T., J. Membr. Sci. 18, 285 (1984). 16. Schultz, S. G., “Basic Principles of Membrane Transport.” Cambridge Univ. Press, Cambridge, 1980. 17. Kimura, Y., Lim, H.-J., and Iijima, T., Angew. Makromol. Chem. 138, 151 (1986). 18. Hunter, R. J., “Zeta Potential in Colloid Science: Principles and Applications.” Academic Press, London, 1981. 19. Lyklema, J., “Fundamentals of Interface and Colloid Science.” Academic Press, London, 1993. 20. Hernandez, A., Mart´ınez, F., Mart´ın, A., and Pr´adanos, P., J. Colloid Interface Sci. 173, 284 (1995).
A Comparison of Electrochemical and Electrokinetic Parameters Determined for Cellophane Membranes in Contact with NaCl and NaNO3 Solutions A. Ca˜nas, M. J. Ariza, and J. Benavente1 Grupo de Caracterizaci´on Electrocin´etica y de Transporte en Membranas e Interfases, Departamento de F´ısica Aplicada, Facultad de Ciencias, Universidad de M´alaga, E-29071 M´alaga, Spain Received April 30, 2001; accepted October 1, 2001
Electrochemical and electrokinetic characterizations of cellophane membrane samples have been carried out by measuring membrane potential, salt diffusion, and tangential streaming potential, which allow the determination of different characteristic membrane parameters. Experiments were made with the membrane samples in contact with NaCl and NaNO3 solutions at different concentrations and under different external conditions (concentration gradients), in order to obtain differences in transport and membrane characteristic parameters, depending on the electrolyte considered. Salt permeability across the membrane, which was obtained from diffusion measurements, is about twice as high for NaCl solutions as for NaNO3 solutions, which is attributed to the different sizes of the electrolytes. Membrane potential measurements keeping the concentration ratio constant (C1 /C2 = 2) were used to determine both the effective fixed charge concentration in the membrane, Xf , and the average value of transport numbers, hti i; taking into account these values, concentration dependence of membrane potential under a different external condition (C1 = cte = 0.01 M, 5 × 10−3 ≤ C(M) ≤ 5 × 10−2 ) was predicted. Results show that cellophane membrane behaves as a weak cation-exchange membrane and its permselectivity to cations is practically independent of the electrolyte considered. From electrokinetic results, assuming a Langmuir-type adsorption of anions on the cellophane surface, the number of accessible sites per surface unit was obtained, which is higher for Cl− than for NO− 3 , in agreement with the small radii of chlorine ions; however, no significant differences in the specific adsorption free energy were found (1GNacl = −22.0 × 103 J/mol) and (1GNaNO3 = −23.2 × 103 J/mol). °C 2002 Elsevier Science Key Words: membrane potential; salt diffusion; tangential streaming potential; electrochemical and adsorption parameters; NaCl and NaNO3 solutions.
INTRODUCTION
Cellulose and its derivatives provide a very important class of basic materials for membranes, mainly due to the fact that cellulose is a very hydrophilic polymer but is not water solu-
1
To whom correspondence should be addressed.
0021-9797/02 $35.00
° C 2002 Elsevier Science
All rights reserved.
ble, which is related to its crystallinity and the intermolecular hydrogen bonding between its hydroxyl groups (1). Cellophane and other types of regenerated cellulose are mainly used as materials for dialysis membranes, since they allow the diffusion of ions and low molecular weight solutes but they do not permit the diffusion of proteins or macromolecules of high molecular weight (1, 2), while cellulose acetate and nitrate are used for microfiltration/ultrafiltration processes and cellulose triacetate is used in reverse osmosis membranes for desalting applications. However, due to medical applications of dialysis membranes (hemodialysis), they represent one of the major membranes marketed worldwide. Transport of solutions across dialysis membranes is usually characterized by determining the solute’s permeability or diffusion coefficients in the membrane, but in the case of regenerated cellulose membranes some of the –CH2 OH groups are oxidized to –COOH in air, giving a weak negative charge to these membranes. Then, if electrolyte solutions are studied, other characteristic electrochemical parameters such as membrane charge density (surface and bulk fixed charge), salt permeability, and ionic transport numbers must be determined, since they can also provide information on electrical “membrane-solution” interactions, which can affect the mobility of charged solutes across the membranes (2–7). Particularly, when the membranes are used in hemodialysis, the negative charge can reduce the passage of some anions even if their size is lower than that of the membrane pores (i.e., the hydrogen phosphate ions, which must be removed from the patient). For these reasons, an electrochemical characterization of a cellophane membrane, which can be used in dialysis systems, in contact with NaCl and NaNO3 solutions at different concentrations was carried out. Membrane potential, salt diffusion, and tangential streaming potential were measured. Concentration dependence for the different electrochemical and electrokinetic parameters was determined and a comparison of their values across a cellophane membrane was also made for both electrolytes. The influence of external conditions (concentration gradients) on salt diffusion and ion transport numbers was also determined, as well as the Donnan potential contribution to membrane potential at
150
CELLOPHANE MEMBRANES IN NaCl AND NaNO3
each membrane–solution interface. Differences found in transport parameters for both electrolytes agree with the lower size and higher mobility of Cl− ions with respect to NO− 3 ones. In order to obtain some fundamental properties of the solid phase (the membrane) from streaming potential measurements, it is necessary to conjecture a particular origin and mechanism of adsorption of charge on the membrane. Assuming that the surface charge density on the Stern layer (electrokinetic charge density) comes from adsorbed anions according to Langmuir-type adsorption (8), electrokinetic results allow the determination of some characteristic adsorption parameters such as the number of accessible sites per surface unit and the Gibb’s free energy; results show that the density of accessible sites is about three times higher for Cl− than for NO− 3 , which also agrees with the small radii of chlorine ions, but only slight differences in the specific adsorption free energy were obtained. 2. EXPERIMENTAL
2.1. Material Different samples of a sheet of cellophane kindly submitted by Cellophane Espa˜nola, S.A. (Burgos, Spain), were studied. Membrane structural parameters are (7) wet membrane thickness, 1X w = (60 ± 3) µm; degree of swelling, sw = (50 ± 2)%; and fractional void volume, ε = (55 ± 3)%. Measurements were carried out with the membrane samples in contact with aqueous NaCl and NaNO3 solutions at different concentrations (5 × 10−4 ≤ C(M) ≤ 10−1 ), at room temperature t = (25.0 ± 0.3)◦ C and standard pH (5.8 ± 0.3). Before use, the membranes were immersed for at least 4 h in a solution of the appropriate concentration. 2.2. Membrane Potential and Salt Diffusion Measurements The test cell used for membrane potential and salt diffusion measurements is similar to that described elsewhere (9). The membrane was tightly clamped between two glass half-cells by using silicone rubber rings; a magnetic stirrer was placed at the bottom of each half-cell and, to minimize concentrationpolarization at the membrane surfaces, measurements were carried out at a stirring rate of 525 rpm. The electromotive force (1E) between the two sides of the membranes caused by a concentration gradient was measured by two reversible Ag/AgCl electrodes connected to a digital voltmeter (Data Precision 3500, 100 MÄ imput impedance). Measurements were carried out using two different procedures: (i) keeping the concentration C1 constant (C1 = 0.01 M) and changing C2 gradually from 5 × 10−3 to 5 × 10−2 M. (ii) keeping the concentration ratio of the solutions on both sides of the membrane constant, C1 /C2 = g = 2, but changing concentration C2 between 0.005 and 0.1 M. Membrane potential, 1øm , was determined from measured 1E values by subtracting the electrode potential contribution.
151
In salt diffusion measurements the membrane was initially separating a concentrated solution (C1 ) from a diluted one (initially distilled water, this means, C2 = 0). Changes in the solution C2 versus time were recorded by means of a conductivity cell connected to a digital conductivity meter (Radimeter CDM 83). Four concentrated solutions (C1 = 10−3 , 5 × 10−3 , 10−2 , and 5 × 10−2 M) were used to see the possible effect of concentration gradients on salt permeability. 2.3. Streaming Potential Measurements Tangential streaming potential (TSP) measurements were carried out with a system which basically consists of (i) a rectangular cell (15.2 cm2 section) with a channel of well-defined dimensions created by the use of two Teflon spacers between two membrane samples; (ii) a mechanical drive unit to produce and measure the pressure (80 ≤ 1P(mB) ≤ 200), which drives the electrolyte solution from a reservoir into the measuring cell. Characteristic hydrodynamic parameters are the Reynolds number Re = (390 + 60) and the hydrodynamic height of the channel h h = (238 ± 7) µm. Black platinum electrodes, one at each end of the channel, connected to a high-impedance voltmeter (Yokogawa 7552, 1 GÄ imput impedance) were used to measure the streaming potential, taking the electrode at lower pressure as reference. Streaming potential coefficient (φ = 1V /1P) was determined from the average of at least six measurements with the solution always flowing in the same direction; 1V values were corrected by electrode potential asymmetry. RESULTS AND DISCUSSION
The electrical potential difference at both sides of a membrane when it is separating two solutions of the same electrolyte but different concentrations (C1 and C2 ) is called “membrane potential” (1øm ). Figure 1 shows membrane potential versus salt concentration for the cellophane membrane with NaCl and NaNO3 solutions under both different external conditions: C1 /C2 = 2 and C1 = 0.01 M, respectively. According to the Teorell–Meyer–Sievers or TMS theory (10, 11), the membrane potential can be considered as the sum of two Donnan potentials (one at each membrane/solution interface) plus a diffusion potential in the membrane; this means that 1øm = 1øDon(I) + 1ødif + 1øDon(II) . The expression for these potentials, when diluted solutions are considered (concentrations are used instead of activities), are (12) £ £ ¤¤ [1] 1øDon = (RT /F) ln (w X f /2C) + (w X f /2C)2 + 1)1/2 1ødif = (RT /F)[(t− − t+ ) ln(C1 /C2 )],
[2]
where X f is the membrane fixed charge concentration, w = −1 or +1 for negatively or positively charged membranes, respectively, and ti is the transport number of the ion in the membrane (i = + for cation, − for anions); t1 represents the amount of current transported for one ion with respect to the total current crossing the membrane, ti = Ii /IT , this means t+ + t− = 1.
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FIG. 1. Membrane potential vs salt concentration: (a) C1 = cte = 10−2 M; (b) C1 /C2 = cte = 2. Experimental values: (s) NaCl and (n) NaNO3 solutions; calculated values by Eqs. [1] and [2]: dotted line.
However, for slightly charged membranes, or when the external salt concentration is higher than the membrane fixed charge (X f ), Aizawa et al. (13) derived the approximation 1øm = (RT /F)[(1 − 2t+ ) + (2g − 1)(t+ t− /g)(X f /C2 )], [3] where g is the concentration ratio (g = C1 /C2 ). If the parameter g is constant, Eq. [3] represents a linear relationship between 1øm and 1/C2 . Figure 1b shows membrane potential versus salt concentration (1øm vs 1/C1 ) for the cellophane membrane with NaCl and NaNO3 solutions and keeping the concentration ratio constant: C1 /C2 = 2, (10−2 ≤ C2 (M) ≤ 10−1 ). The fitting of the linear relationships obtained allows the determination of the fixed charge concentration and the average ion transport numbers in the membrane by means of Eq. [3]. X f and ht+ i values determined for the cellophane membrane with both electrolytes are indicated in Table 1. As can be seen, the cellophane membrane behaves as a weak cation exchanger, which agrees with results reported in the literature for different cellophane membranes (14, 15), and is more relevant when the membrane is
TABLE 1 Characteristic Electrochemical Parameters Determined for a Cellophane Membrane in Contact with NaCl and NaNO3 Solutions Electrolyte NaCl NaNO3
X f (M)
ht+ i
Psm (+) (%)
Ds (m2 /s)
−(1.5 ± 0.2)×10−2 (0.53 ± 0.2) (23.7 ± 0.9) (3.6 ± 0.3)×10−10 −(0.8 ± 0.1)×10−2 (0.55 ± 0.3) (22.8 ± 1.2) (5.6 ± 0.7)×10−11
Note. Fixed charge concentration, X f , average cation transport number, ht+ i, cation permselectivity, Ps (Na+ ), and salt diffusion coefficient, Ds .
in contact with NaCl solutions than when it is in contact with NaNO3 ones. Figure 1a shows the concentration dependence of membrane potential when C1 = cte = 0.01 M (1φm vs ln(C2 /C1 )). Theoretical values for membrane potential calculated taking into account both Donnan and diffusion contributions by using X f and ht+ i values obtained from C1 /C2 = cte measurements are seen in Fig. 1a as a dotted line. As can be seen, very good agreement between experimental and theoretical membrane potential was obtained, which can be considered a test of the validity of X f and ht+ i values determined from measurements at constant concentration ratios. Membrane permselectivity, Psm (i), is a measure of the selectivity of counterions over co-ions in a membrane; in the case of negatively charged membranes, permselectivity to cations can be obtained by (16) Psm (+) = (t+ − t+o )/(1 − t+o ),
[4]
where t+o represents the cation transport number in solution. Average permselectivity, hPsm (+)i, for both electrolyte solutions was determined taking the ht+ i values previously obtained, and its values are also indicated in Table 1. Results show that membrane permselectivity to cations is practically independent of the electrolyte considered. Salt permeability across a membrane can be obtained from diffusion measurements. According to Fick’s first law, the salt flux (Js ) through a membrane (for a quasi-steady state) can be written Js = Ps (C1 − C2 ), where Ps is the salt permeability in the membrane. On the other hand, the molar salt flux through the membrane at any time is given by Js = (1/Sm )(dn/dt) = (Vo /Sm )(dC2 /dt),
[5]
CELLOPHANE MEMBRANES IN NaCl AND NaNO3
153
FIG. 2. Time dependence for conductivity in compartment 2: (a) at different concentration gradients (1c) and NaNO3 solutions; (b) comparison between NaCl and NaNO3 solutions at a given concentration gradient 1c = 10−2 M.
where Vo and Sm are the volume of the solution on the side with concentration C2 , and the membrane area, respectively. Then, the following expression can be obtained, dC2 /[C1 − C2 (t)] = (Sm /Vo )Ps dt,
[6]
which can be also written as a function of salt conductivity, dλ2 /[λ1 − λ2 (t)] = (Sm /Vo )(dλ/dC)e Ps dt,
[7]
where (dλ/dC)e is characteristic for each electrolyte. Variation in the conductivity of solution C2 as a function of time for NaCl and NaNO3 solutions is shown in Fig. 2b for a given concentration gradient (1C = C1 − C2 = 0.01 M), while Figure 2a shows a comparison of salt diffusion for NaNO3 solutions at different concentration gradients. From the slopes of these straight lines, salt permeability across the cellophane membrane was obtained for the different concentration gradients and electrolytes studied. Ps concentration dependence for both electrolytes is shown in Fig. 3. As can be observed, at low concentrations Ps increases when concentration gradient increases, but an almost constant value is reached at high 1C values, as also indicated by Kimura et al. (17). It should be pointed out that a higher permeability was obtained with NaCl solutions than with NaNO3 , which is attributed to its small size. Salt diffusion coefficients in the membrane, Ds , can be obtained from Ps values by (1) Ps = Ds /1xm , where 1xm , is the membrane thickness. Ds values for both electrolytes at a given concentration gradient (1C = 0.01 M) are shown in Table 1. It is worth noting that salt diffusion coefficients in the membrane are much lower than those in solution, Dso ([Ds /Dso ](NaCl) = 0.23 and [Ds /Dso ](NaNO3 ) = 0.004), which is indicative of the interactions between the solutes and the cross-linked chains of the cellophane matrix. For negatively charged membranes and 1 : 1 electrolytes, the following expression for the salt permeability on the basis of the
TMS theory was obtained by Kimura et al. (17), ©¡ ¢1/2 Ps = [D+ D− RT X f /1xm (D+ + D− )1C] 1 + 4y12 ¡ £¡ ¤± ¢1/2 ¢1/2 − 1 + 4y22 − U ln 1 + 4y12 +U ¤ª ¢1/2 £¡ +U , [8] × 1 + 4y22 where yj = K ± Cj / X f (K ± is the salt partition coefficient), Ci is the concentration of the high and low solutions (i = 1 and 2), respectively, D+ and D− are the diffusion coefficient of cation and anion in the membrane, and U = (D+ − D− )/(D+ + D− ) = t+ − t− . In Fig. 3 the dotted lines represent the concentration dependence of Ps values obtained from Eq. [8] by using the membrane fixed charge concentration and transport numbers determined from membrane potential measurements at different external concentrations (K ± = 1). As can be seen, rather good
FIG. 3. Salt permeability vs concentration gradient. (s) NaCl solutions and (1) NaNO3 solutions. Dotted line: calculated values by Eq. [8].
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agreement between calculated and experimental values determined from different kinds of experiments was obtained. On the other hand, the solid/liquid electrical interface is usually characterized by streaming potential measurements. When an electrolyte solution is forced to move, by means of an external pressure (1P) along a charged solid wall, an electrical potential difference or streaming potential (1φst ) between the ends (at steady state) can be measured. The hydraulic pressure causes movement of liquid and thus ions are stripped off along the shear plane, and a streaming current is formed; due to a charge accumulation at the downstream side, and electrical field is then generated that causes a backflow of ions until steady state is reached. The measurable electrical potential difference between the two ends of the solid/liquid system, 1φst , gives direct information about the electrical potential at the shear plane (zeta potential or ζ ) and the electrokinetic charge density (σs ). Figure 4a shows streaming potential versus applied pressure for NaCl and NaNO3 solutions at different concentrations the
FIG. 5. Electrokinetic parameters vs salt concentration: (s) NaCl solutions and (1) NaNO3 solutions. (a) Zeta potential (ζ ); (b) surface charge density (σs ).
streaming potential coefficient, φst , was obtained from the slope of these straight lines, φst = 1V /1P, and its dependence on salt concentration for both electrolytes is shown in Fig. 4b. Streaming potential is related to the zeta potential by the Helmholtz– Smoluchowski equation (18) ζ = (λη/εo εr )(1φst /1P) = (λη/ε)φ,
FIG. 4. (a) Streaming potential vs applied pressure for both electrolytes at different concentrations. (b) Streaming potential coefficient (Φst = 1V /1P) as a function of electrolyte concentration: (s) NaCl solutions and (1) NaNO3 solutions.
[9]
where λ, η, and ε are the liquid conductivity, viscosity, and dielectric constant (ε = εo εr ). Surface conductivity was not taken into account, since the height of the channel (about 300 µm) was large enough to allow us to ignore the electrical conductance effect at the diffuse part of the double layer. Zeta potential versus salt concentration is shown in Fig. 5a. As can be seen, zeta potential clearly depends on the electrolyte considered, but it does not show any clear dependence on salt concentration, and the following average values for each electrolyte were obtained: hζ i(NaCl) = −(5.6 ± 0.9) and hζ i(NaNO3 ) = −(2.9 ± 0.5). Surface charge density at the shear plane can be determined from zeta potential values by the following equation (18), σS = (2RT ek/z F) sinh(Fζ /2RT ),
[10]
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CELLOPHANE MEMBRANES IN NaCl AND NaNO3
where k and e are the Debye length and the elemental charge, respectively. Concentration dependence for the electrokinetic charge density is shown in Fig. 5b for both electrolytes. As can be seen, more negative values for σS are obtained when the cellophane membrane was in contact with NaCl solutions than with NaNO3 ones, which agrees with the slightly higher values also found for the fixed charge concentration from membrane potential measurements. The slight increase of the negative electrokinetic charge when salt concentration increases, found for both electrolytes, can be attributed to the adsorption of anions from solution on the membrane surface (18, 19). According to the Stern model, the electroneutrality condition for the membrane/solution interface can be written as a surface density charge balance (18), σe = σs + σo ,
[11]
where σe , σs , and σo , are the surface charge densities at the shear, Stern, and membrane surface planes, respectively. If a kind of ion adsorption is assumed (8, 20), this equality allows us to obtain the membrane charge density and some adsorption characteristic parameters such as the molar free energy and the density of accessible sites on the membrane surface. If the surface charge density on the Stern plane comes from adsorbed anions according to a Langmuir model, the following expression can be obtained (8, 18), σS = z − eNs χ− exp[(−z − ζ ∗ \z + ) − (1G c /RT )]/ × {1 + χ− exp[(−z − ζ ∗ \z + ) − (1G c /RT )]}, [12] where ζ ∗ is the dimentionless zeta potential (ζ ∗ = ζ e/RT ),
TABLE 2 Total Number of Adsorption Sites Accesible to the Anions per Unit of Membrane Surface, Ns , and Chemical Part of the Molar Gibbs Free energy of Adsorption, ∆G c Salt
Ns (sites m−2 )
1G c (J/mol)
NaCl NaNO3
(2.3 ± 0.4) × 1016 (0.8 ± 0.1) × 1016
−(22 ± 3) × 103 −(23 ± 4) × 103
1G c is the chemical (or nonelectrostatic) part of the molar Gibbs free energy of adsorption, and Ns is the total number of adsorption sites accessible to the anions per unit of membrane surface; χ− is the molar fraction of anions in the solution at the solid surface, which is given by (18) ¡ ¢±£ ¡ ¢¤ χ− = Cν− MH2 O ρ − C ν− M− + ν+ M+ (ν− + ν+ )MH2 O , [13] with Mi and νi the molecular weight and the stochiometric coefficient of the i species (i = ± for cation/anion), and ρ is the solution density. Solid characteristic parameters (1G c and Ns ) can be obtained by fitting the electrokinetic potential and charge as a function of the ionic molar fraction according to (8, 18) −[A1 /(σe − σo )] − A2 = exp(−ζ ∗ )/χ− ,
[14]
where the constants A1 and A2 are related with 1G c and Ns as follows (18): 1G c = −RT ln A2 and Ns = A1 /e A2 . In the case of zero proper charge, Eq. [14] represents a linear relationship between 1/σs and exp(−ζ ∗ )/χ− , such as that obtained with both electrolytes and shown in Fig. 6. By the fitting of the experimental data, A1 and A2 were determined; values of the specific adsorption free energy (1G c ) and the density of accessible sites (Ns ) are shown in Table 2. A comparison of results obtained for both electrolytes shows that the number of accessible sites is three times higher for Cl− than for NO− 3 , in agreement with the small radii of chlorine ions, but practically no difference in the molar adsorption free energy was obtained. The negative sign of 1G c implies a spontaneous process of adsorption of the anions on the membrane surface; moreover, the calculated density of anion sites is relatively low, in accord with the hydrophilic character of cellophane.
CONCLUSIONS
FIG. 6. Plot of exp(−ζ ∗ )/χ− as a function of 1/σs for both electrolytes: (s) NaCl and (1) NaNO3 solutions.
Transport of electrolyte solutions across a cellophane membrane has been studied by measuring membrane potential, salt diffusion, and streaming potential with NaCl and NaNO3 solutions at different concentrations, which allows the
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comparison of different transport parameters (ion transport numbers, permselectivity, and salt permeability) as well as characteristic adsorption parameters on the membrane surface (molar Gibbs free energy and density of accessible sites) for both electrolytes. —Membrane and streaming potential measurements show the negative character of the membrane, but results seems to indicate that the effective membrane charge is mainly due to the adsorption of anions from solution. —Very good agreement was obtained by comparing experimental membrane potential values and those calculated by using the TMS theory, which can be considered as a probe of the reliability of determined parameters (membrane fixed charge and ion transport numbers). It should be pointed out that Donnan equilibrium affects the ion distribution for concentration lower than 0.02 M, but at higher concentrations the membrane potential is practically a diffusion potential. —The lower values for permeability and diffusion coefficients across the cellophane membrane obtained with NaNO3 solutions at the different concentration gradients studied are attributed to its higher size and strong interaction with the membrane matrix. On the other hand, good concordance has been found between salt permeability values determined from diffusion measurements and those calculated taking into account the electrochemical parameters previously obtained. —Electrokinetic parameters permit us to obtain a representation of the membrane/solution interface, but they have also allowed the determination of membrane parameters assuming a Langmuir adsorption of anions on the surface of the cellophane membrane. The higher density of accessible sites obtained when NaCl solutions are used agrees with the small radii of Cl− ions in comparison with NO− 3 ones ions.
ACKNOWLEDGMENTS We thank the Comision Interministerial de Ciencia y Tecnolog´ıa (Project MAT2000-1140) and Junta de Andaluc´ıa (Grupo FQM 258) for financial support.
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