A Comparison of Electrochemical and Electrokinetic Parameters
Determined for Cellophane Membranes in Contact
with NaCl and NaNO
3
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 cello-phane membrane samples have been carried out by measuring mem-brane 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 NaNO3solutions at different
concentra-tions and under different external condiconcentra-tions (concentration gra-dients), 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,htii; taking into
ac-count these values, concentration dependence of membrane po-tential 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 mem-brane 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= −23.2×10
3
J/mol). °C2002 Elsevier Science
Key Words: membrane potential; salt diffusion; tangential streaming potential; electrochemical and adsorption parameters; NaCl and NaNO3solutions.
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-1To whom correspondence should be addressed.
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 ma-terials 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 mi-crofiltration/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 mar-keted worldwide.
Transport of solutions across dialysis membranes is usually characterized by determining the solute’s permeability or diffu-sion coefficients in the membrane, but in the case of regenerated cellulose membranes some of the –CH2OH groups are oxidized to –COOH in air, giving a weak negative charge to these mem-branes. Then, if electrolyte solutions are studied, other charac-teristic 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” interac-tions, 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 mem-brane 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 NaNO3solutions at different concen-trations was carried out. Membrane potential, salt diffusion, and tangential streaming potential were measured. Concentration de-pendence for the different electrochemical and electrokinetic pa-rameters 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 0021-9797/02 $35.00
C
°2002 Elsevier Science All rights reserved.
each membrane–solution interface. Differences found in trans-port 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 sur-face charge density on the Stern layer (electrokinetic charge den-sity) 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 thick-ness,1Xw=(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 aque-ous NaCl and NaNO3solutions 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 con-centration.
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 concentration-polarization at the membrane surfaces, measurements were car-ried 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 MÄimput impedance). Measurements were carried out using two different procedures:
(i) keeping the concentration C1constant (C1=0.01 M) and changing C2gradually from 5×10−3to 5×10−2M.
(ii) keeping the concentration ratio of the solutions on both sides of the membrane constant, C1/C2=g=2, but changing concentration C2between 0.005 and 0.1 M.
Membrane potential, 1øm, was determined from measured
1E values by subtracting the electrode potential contribution.
In salt diffusion measurements the membrane was initially separating a concentrated solution (C1) from a diluted one (ini-tially distilled water, this means, C2=0). Changes in the solu-tion C2versus 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 5×10−2M) were used to see the possible effect of concen-tration gradients on salt permeability.
2.3. Streaming Potential Measurements
Tangential streaming potential (TSP) measurements were car-ried out with a system which basically consists of (i) a rectan-gular 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 num-ber Re=(390+60) and the hydrodynamic height of the chan-nel hh=(238±7)µm. Black platinum electrodes, one at each end of the channel, connected to a high-impedance voltmeter (Yokogawa 7552, 1 GÄimput 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 mem-brane when it is separating two solutions of the same elec-trolyte but different concentrations (C1and C2) is called “mem-brane potential” (1øm). Figure 1 shows membrane potential versus salt concentration for the cellophane membrane with NaCl and NaNO3 solutions under both different external con-ditions: C1/C2 =2 and C1=0.01 M, respectively. Accord-ing 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 (concentra-tions are used instead of activities), are (12)
1øDon=(RT/F ) ln £
(wXf/2C)+ £
(wXf/2C)2+1)1/2 ¤¤
[1]
1ødif=(RT/F )[(t−−t+) ln(C1/C2)], [2]
where Xfis the membrane fixed charge concentration,w= −1 or+1 for negatively or positively charged membranes, respec-tively, and tiis 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 cur-rent crossing the membrane, ti=Ii/IT, this means t++t−=1.
FIG. 1. Membrane potential vs salt concentration: (a) C1=cte=10−2M; (b) C1/C2=cte=2. Experimental values: (s) NaCl and (n) NaNO3solutions; calculated values by Eqs. [1] and [2]: dotted line.
However, for slightly charged membranes, or when the exter-nal salt concentration is higher than the membrane fixed charge (Xf), Aizawa et al. (13) derived the approximation
1øm=(RT/F )[(1−2t+)+(2g−1)(t+t−/g)(Xf/C2)], [3]
where g is the concentration ratio (g=C1/C2). If the parameter g is constant, Eq. [3] represents a linear relationship between
1ømand 1/C2.
Figure 1b shows membrane potential versus salt concentra-tion (1øm vs 1/C1) for the cellophane membrane with NaCl and NaNO3solutions and keeping the concentration ratio con-stant: 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 num-bers in the membrane by means of Eq. [3]. Xfandht+ivalues determined for the cellophane membrane with both electrolytes are indicated in Table 1. As can be seen, the cellophane mem-brane behaves as a weak cation exchanger, which agrees with results reported in the literature for different cellophane mem-branes (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 NaNO3Solutions
Electrolyte Xf(M) ht+i Psm(+) (%) Ds(m2/s) NaCl −(1.5±0.2)×10−2 (0.53±0.2) (23.7±0.9) (3.6±0.3)×10−10 NaNO3 −(0.8±0.1)×10−2 (0.55±0.3) (22.8±1.2) (5.6±0.7)×10−11 Note. Fixed charge concentration, Xf,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 NaNO3ones.
Figure 1a shows the concentration dependence of membrane potential when C1=cte=0.01 M (1φm vs ln(C2/C1)). Theo-retical values for membrane potential calculated taking into ac-count both Donnan and diffusion contributions by using Xfand
ht+ivalues obtained from C1/C2=cte measurements are seen in Fig. 1a as a dotted line. As can be seen, very good agree-ment between experiagree-mental and theoretical membrane potential was obtained, which can be considered a test of the validity of Xfandht+ivalues determined from measurements at constant concentration ratios.
Membrane permselectivity, Psm(i ), is a measure of the selec-tivity 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. Av-erage permselectivity,hPsm(+)i, for both electrolyte solutions was determined taking theht+ivalues previously obtained, and its values are also indicated in Table 1. Results show that mem-brane 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 Psis the salt permeability in the membrane. On the other hand, the molar salt flux through the membrane at any time is given by
FIG. 2. Time dependence for conductivity in compartment 2: (a) at different concentration gradients (1c) and NaNO3solutions; (b) comparison between NaCl and NaNO3solutions at a given concentration gradient1c=10−2M.
where Voand Smare 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)Psdt, [6]
which can be also written as a function of salt conductivity,
dλ2/[λ1−λ2(t)]=(Sm/Vo)(dλ/dC)ePsdt, [7]
where (dλ/dC)e is characteristic for each electrolyte. Varia-tion in the conductivity of soluVaria-tion 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 so-lutions at different concentration gradients. From the slopes of these straight lines, salt permeability across the cellophane mem-brane was obtained for the different concentration gradients and electrolytes studied. Psconcentration dependence for both elec-trolytes is shown in Fig. 3. As can be observed, at low concen-trations Psincreases when concentration gradient increases, but an almost constant value is reached at high1C 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 diffu-sion coefficients in the membrane, Ds, can be obtained from Psvalues by (1) Ps=Ds/1xm, where1xm, is the membrane thickness. Ds values for both electrolytes at a given concen-tration 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, Do
s([Ds/Dos]
(NaCl)=0.23
and [Ds/Dso]
(NaNO3)=0.004), which is indicative of the
inter-actions 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),
Ps=[D+D−RT Xf/1xm(D++D−)1C] ©¡
1+4y12¢1/2
−¡1+4y22¢1/2−U ln£¡1+4y12¢1/2+U¤±
×£¡1+4y22¢1/2+U¤ª, [8]
where yj=K±Cj/Xf(K±is the salt partition coefficient), Ciis
the concentration of the high and low solutions (i=1 and 2), re-spectively, 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) NaNO3solutions. Dotted line: calculated values by Eq. [8].
agreement between calculated and experimental values deter-mined from different kinds of experiments was obtained.
On the other hand, the solid/liquid electrical interface is usu-ally characterized by streaming potential measurements. When an electrolyte solution is forced to move, by means of an ex-ternal 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 di-rect information about the electrical potential at the shear plane (zeta potential orζ) and the electrokinetic charge density (σs).
Figure 4a shows streaming potential versus applied pressure for NaCl and NaNO3 solutions at different concentrations the
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.
FIG. 5. Electrokinetic parameters vs salt concentration: (s) NaCl solutions and (1) NaNO3solutions. (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. Stream-ing potential is related to the zeta potential by the Helmholtz– Smoluchowski equation (18)
ζ =(λη/εoεr)(1φst/1P)=(λη/ε)φ, [9]
whereλ, η, andεare the liquid conductivity, viscosity, and di-electric constant (ε=εoεr). Surface conductivity was not taken into account, since the height of the channel (about 300µm) was large enough to allow us to ignore the electrical conduc-tance 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: hζi(NaCl)= −(5.6±0.9) andhζi(NaNO3) = −(2.9±0.5).
Surface charge density at the shear plane can be determined from zeta potential values by the following equation (18),
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 solu-tions 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]
whereσe, σs, andσo, 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 char-acteristic parameters such as the molar free energy and the den-sity 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−eNsχ−exp[(−z−ζ∗\z+)−(1Gc/RT )]/
× {1+χ−exp[(−z−ζ∗\z+)−(1Gc/RT )]}, [12]
where ζ∗ is the dimentionless zeta potential (ζ∗=ζe/RT ),
FIG. 6. Plot of exp(−ζ∗)/χ−as a function of 1/σsfor both electrolytes: (s) NaCl and (1) NaNO3solutions.
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,∆Gc
Salt Ns(sites m−2) 1Gc(J/mol)
NaCl (2.3±0.4)×1016 −(22±3)×103 NaNO3 (0.8±0.1)×1016 −(23±4)×103
1Gcis the chemical (or nonelectrostatic) part of the molar Gibbs free energy of adsorption, and Nsis the total number of adsorp-tion 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−MH2O
¢±£
ρ−C¡ν−M−+ν+M+(ν−+ν+)MH2O
¢¤
,
[13]
with Miandνithe molecular weight and the stochiometric co-efficient of the i species (i= ±for cation/anion), andρ is the solution density.
Solid characteristic parameters (1Gcand N
s) 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 A1and A2are related with1Gcand Nsas follows (18):1Gc= −RT ln A2and 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 ob-tained with both electrolytes and shown in Fig. 6. By the fitting of the experimental data, A1and A2were determined; values of the specific adsorption free energy (1Gc) and the density of accessible sites (Ns) are shown in Table 2. A compari-son 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 en-ergy was obtained. The negative sign of1Gc implies a spon-taneous process of adsorption of the anions on the mem-brane surface; moreover, the calculated density of anion sites is relatively low, in accord with the hydrophilic character of cellophane.
CONCLUSIONS
Transport of electrolyte solutions across a cellophane membrane has been studied by measuring membrane poten-tial, salt diffusion, and streaming potential with NaCl and NaNO3solutions at different concentrations, which allows the
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 experi-mental membrane potential values and those calculated by using the TMS theory, which can be considered as a probe of the relia-bility of determined parameters (membrane fixed charge and ion transport numbers). It should be pointed out that Donnan equi-librium 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 coeffi-cients across the cellophane membrane obtained with NaNO3 solutions at the different concentration gradients studied are at-tributed to its higher size and strong interaction with the mem-brane matrix. On the other hand, good concordance has been found between salt permeability values determined from diffu-sion measurements and those calculated taking into account the electrochemical parameters previously obtained.
—Electrokinetic parameters permit us to obtain a represen-tation 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 Applica-tions.” 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).
