Determining the Zeta Potential of Porous Membranes Using Electrolyte Conductivity inside Pores

Journal of Colloid and Interface Science 235, 383–390 (2001) doi:10.1006/jcis.2000.7331, available online at http://www.idealibrary.com on

Determining the Zeta Potential of Porous Membranes Using Electrolyte Conductivity inside Pores P. Fievet,∗,1 A. Szymczyk,∗ C. Labbez,∗ B. Aoubiza,† C. Simon,‡ A. Foissy,∗ and J. Pagetti∗ ∗ Laboratoire de Chimie des Mat´eriaux et Interfaces, 16 route de Gray, 25030 Besanc¸on cedex, France; †Laboratoire de Calcul Scientifique, 16 route de Gray, 25030 Besanc¸on cedex, France; and ‡SINTEF Materials and Technology, P.O. Box 124, Blindern 0314, Oslo, Norway Received July 26, 2000; accepted November 6, 2000

The zeta potential is an important and reliable indicator of the surface charge of membranes, and knowledge of it is essential for the design and operation of membrane processes. The zeta potential cannot be measured directly, but must be deduced from experiments by means of a model. The possibility of determining the zeta potential of porous membranes from measurements of the electrolyte conductivity inside pores (λpore ) is investigated in the case of a ceramic microfiltration membrane. To this end, experimental measurements of the electrical resistance in pores are performed with the membrane filled with KCl solutions of various pHs and concentrations. λpore is deduced from these experiments. The farther the pH is from the isoelectric point and/or the lower the salt concentration is, the higher the ratio of the electrolyte conductivity inside pores to the bulk conductivity is, due to a more important contribution of the surface conduction. Zeta potentials are calculated from λpore values by means of a space charge model and compared to those calculated from streaming potential measurements. It is found that the isoelectric points are very close and that zeta potential values for both methods are in quite good agreement. The differences observed in zeta potentials could be due to the fact that the space charge model does not consider the surface conductivity in the inner part of the double layer. Measurements of the electrolyte conductivity within the membrane pores are proved to be a well-adapted procedure for the determination of the zeta potential in situations where the contribution of the surface conduction is significant, i.e., for small and charged pores. °C 2001 Academic Press Key Words: zeta potential; electrolyte conductivity; streaming potential; ceramic membrane.

1. INTRODUCTION

Surface charges of a membrane are known to greatly influence its filtration properties such as permeability and permselectivity. All recent works dealing with membrane retention (1–6) or fouling phenomena (7–9) underline the role of surface interactions that modify the transport of species that would be given by simple sieving effects. From then on, the determination of parameters representing membrane–solution interactions, such as 1 To whom correspondence should be addressed. Fax: +33.3.81.66.20.33. E-mail: [email protected].

the zeta potential (or electrokinetic potential), is very important in understanding and predicting the filtration performance of a membrane. Zeta potential cannot be measured directly but must be deduced from experiments by means of a model. Literature shows that a variety of methods are available for assessing the zeta potential of membranes. The main features of these methods are given in Table 1. The zeta potential can be determined by standard electrokinetic methods such as streaming potential (SP) (10–14) and electroosmosis measurements (EO) (10, 12, 15–19). Between these two methods, the more widely used procedure is the streaming potential, which has the advantage of being determined during the filtration process. Moreover, the streaming potential method is less time-consuming than the electroosmosis method (20). Recently, Huisman et al. (21) have developed a new method for assessing the zeta potential of membranes based on the electroviscous effect. This involves measuring the volume flux through the membrane at various salt concentrations. The variations in water flux with salt concentration can be then related to zeta potential through the electrokinetic flow theory (22–24). According to these authors, the main advantage of this new method lies in its experimental simplicity. It only uses measurements of pressure and flux at various salt concentrations. However, a disadvantage of this method is its low sensitivity at low values of zeta potential (< ∼12 mV). Very recently, Huisman et al. (25) presented an alternative method of determining the zeta potential of membranes that can even be used for low values of zeta potential. These values are calculated from measured values of salt retention of membranes with the help of a model based on the electrokinetic flow theory. However, a disadvantage of the retention-based method is that concentration polarization phenomena may cause observed retention values to be quite different from real ones. Another way to access to the charge state of a membrane is to measure the membrane potential E m induced by a concentration gradient through the membrane (26). In a previous paper, Fievet et al. (27) showed that the membrane potential can be related to the zeta potential by applying the theory of nonequilibrium thermodynamics to the membrane process and considering a space charge model. In an additional paper (28), it was shown that the membrane potential method is a well-adapted procedure

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TABLE 1 Characteristics of Various Methods for the Determination of the Zeta Potential of Membranes Driving force/ measured parameter/ determined parameter a

Methods

Range of applicability

Membrane geometry

Information obtained from a single measurement

Advantages/disadvantages

Streaming potential (10–14)

Pressure difference 1P/ electrical potential difference1ϕ/ SP = (1ϕ/1P) I =0

MF and UF

Plane or tubular

• Sign of ζ • Two different values of ζ are possible (28)

• Measurements can be performed using the filtration equipment itself (not necessary to remove the membrane from the filtration plant) • Experimental simplicity

Electroosmosis (10, 12, 15–19)

Electrical current I (or electrical potential difference)/fluid flow J / (J/I )1P=0

MF and UF

Plane

• Sign of ζ • Two different values of ζ are possible

• Measurements cannot be performed using the filtration equipment itself • Experimental difficulties (20) • Relatively time-consuming (20)

Electroviscous effect-based method (21)

Pressure difference 1P/ fluid flow J / apparent viscosity µa = 1P/J Rh

MF and UF

Plane or tubular

• Neither sign nor value of ζ

• Measurements can be performed using the filtration equipment itself • Low sensitivity at low values of ζ

Salt retention-based method (25)

Pressure difference/ fluid flow J / salt retention R = 1–Cfeed /Cp

UF and NF

Plane or tubular

• Neither sign nor value of ζ

• Measurements can be performed using the filtration equipment itself • Experimental simplicity • Concentration polarization phenomena may cause observed retention values to be different from real retention values

Membrane potential (26, 27)

Concentration difference/ electrical potential difference (1ϕ) I =0 / E m = (1ϕ) I =0

UF and NF

Plane

• Sign and value of ζ

• Measurements cannot be performed using the filtration equipment itself • Concentration polarization phenomena may cause observed membrane potential values to be different from real membrane potential values

aR

h,

hydraulic resistance of the membrane; Cfeed and Cp , salt concentration in feed and permeate.

for characterizing small and slightly charged pores as well as larger and strongly charged pores. This electrokinetic model also permits one to relate the electrolyte conductivity inside pores (λpore ) to the membrane zeta potential. Indeed, the conductivity within the pores reflects both the mobilities of ions and their concentrations, which are dependent on the zeta potential. The present study aims at evaluating the pore conductivitybased method for the determination of the zeta potential. To this end, we have compared the zeta potentials deduced from measurements of λpore with those deduced from streaming potential measurements (which represent the best-established method) in the case of a ceramic microfiltration membrane. The effects of pH and ionic strength are investigated. Zeta potential values are numerically calculated from experimental data of SP and λpore by considering a model based on irreversible thermodynamics and a space charge model. 2. THEORETICAL BACKGROUND

Membranes acquire a characteristic electric surface charge when in contact with an aqueous solution. To maintain the electroneutrality of the system, charges (ions) in the adjacent solu-

tion reorganize. Consequently, the potential and the ionic concentration vary progressively from the charged surface to the bulk solution, leading to the formation of an “electrical double layer” (29). Due to the excess of counterions in the adjacent solution, the conductivity inside pores (λpore ) may significantly exceed the conductivity of the bulk solution (λ0 ). The difference between λpore and λ0 is referred to as the surface conductivity (λs ) (30). This is a quantity representing the surface conduction, that is, the excess conduction that occurs in charged pores due to the presence of electrical double layers. Thus, surface conductivity (λs ) reflects the distribution of charges in the double layer as well as their mobility. It results that the conductivity inside charged pores (λpore ) depends on the charge on the pore walls. In a previous paper (32), the electrokinetic phenomena occurring in homogeneous cylindrical pores were studied in the framework of the linear thermodynamics of irreversible processes and the space charge model outlined by Osterle and coworkers (33–35). The local relations for transport through pores (Nernst–Planck and Navier–Stokes equations) and the nonlinear Poisson–Boltzmann equation for the electrostatic condition of the pore fluid were developed. This model considers a charged cylindrical capillary of radius a, with a ¿ l, where l is the pore length. The pore radius was

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THE ZETA POTENTIAL OF POROUS MEMBRANES

set as representing the distance from the capillary axis to the slip plane separating the compact and diffuse parts of the double layer. The compact layer was then not considered in this model. The influence of the zeta potential on both the conductivity inside pores (λpore ) and the streaming potential was examined. To this end, the integral expressions of the phenomenological coefficients (K i j ) coupling the solvent flow and the electrical current (I ) with both the hydrostatic pressure gradient (1P) and the electrical potential gradient (1ϕ) were established and calculated numerically. The conductivity of the electrolyte inside a charged pore (λpore ) could then be expressed under isobaric conditions as µ ¶ I l K 33 λpore = , [1] = S 1ϕ 1P=0 S where l is the length of the pore and S is its cross section. The ratio of the conductivity of the electrolyte inside a charged pore (λpore ) to the conductivity of the bulk electrolyte (λ0 ) can be written as λpore (K 33 )1ϕ6=0 = . λ0 (K 33 )1ϕ=0

[2]

Also, the streaming potential (SP) can then be expressed as µ SP =

1ϕ 1P

¶ I =0

=−

K 31 . K 33

[3]

The calculations presented in this work are carried out using the expressions for K i j listed in Appendix A. The membrane is assumed to be composed of identical and parallel cylindrical pores so that the electric and electrokinetic behavior of a single pore can represent the behavior of the entire membrane. 3. EXPERIMENTAL

3.1. Chemicals and Membranes Electrolyte solutions are prepared from potassium chloride of pure analytical grade and milli-Q quality water (conductivity 8. The solution is placed in the module at a pressure difference across the membrane of 0.2 bar. Permeate and retentate are continuously recycled. The streaming potential is found by applying, at a given time, an over-pressure (1P) of 1 bar on the retentate side and measuring the instantaneous resulting potential difference (1ϕ) on both sides of the membrane (“pulse method”). A preliminary study was performed to check the linear variation of 1ϕ versus 1P (37). The 1ϕ and 1P values are measured at ±0.1 mV and ±0.01 bar, respectively. 4. RESULTS AND DISCUSSION

Influence of pH The aim is to measure streaming potentials and electrolyte conductivities inside pores over a range of pH values at fixed ionic strength to study the variation of the zeta potential with pH due to ionization of charge-determining hydroxyl groups. Figure 1 shows the influence of pH on the ratio λpore /λ0 for a salt concentration of 0.006 M. The curve obtained presents a minimum with respect to the pH of the solution. The pH value corresponding to this minimum is the isoelectric point (IEP, pH for which the zeta potential ζ is zero). Indeed, when ζ = 0 mV/bar, the pore walls are globally uncharged. The distribution of ions in the region near the surface is the same as in the bulk and λpore = λ0 .

FIG. 2. Variation of the experimental streaming potential (SP) versus pH: KCl, 0.006 M.

A value of about 9.6 is obtained for the IEP. When the pH moves away from the IEP, the zeta potential increases (in absolute value) and it results a higher pore conductivity. This is expected behavior since the concentration of counterions near the pore wall becomes much higher than the bulk concentration. The excess concentration of ions in the diffuse layer makes the region near the pore wall more conductive. As for the streaming potential method, the pore conductivitybased method allows one to determine the IEP of a porous membrane without the help of a model. Figure 2 presents the pH dependence of the streaming potential for the same KCl concentration as that in Fig. 1. The streaming potential has a maximum with respect to pH. The downward trend of the streaming potential for pH sufficiently far from IEP (pH for which SP = 0 V) results from the excess conductance in the region near the pore walls. Indeed, it is known that the surface conductance increases exponentially with the zeta potential (32). Consequently, the higher the surface conductance, the greater the counterion backflow induced by the electroosmotic effect during the streaming potential process is, which in turn causes the streaming potential to decrease as the zeta potential increases in magnitude, i.e., as pH moves away from the IEP. As can be seen, a value of about 9.3 is obtained for the IEP. This value is close to that determined by the λpore method. The determination of the IEP on the SP–pH curve is essential to determine from the model the correct zeta potential corresponding to one streaming potential value obtained at a known pH (32). Thus, the streaming potential value corresponding to the pH closest to the isoelectric point will be associated with the smaller of the two possible zeta potential values of the theoretical curve SP = f (ζ ) (32). Influence of Salt Concentration

FIG. 1. Variation of the ratio λpore /λ0 versus pH: KCl, 0.006 M.

The charge of the membrane is controlled by the pH and also the ionic strength. Thus, now, the aim is to measure streaming potentials and electrolyte conductivities inside pores over a

THE ZETA POTENTIAL OF POROUS MEMBRANES

387

expected, the streaming potential decreases as the salt concentration in the solution increases, due to a compression of the diffuse part of the double layer (decrease of the Debye length). It can be noted that SP is zero for the 1 M KCl solution, which justifies the use of this concentration to determine the electrolyte conductivity inside pores from electrical resistance measurements (as specified in Section 3.2). Comparison of Zeta Potentials Determined from λpore and SP Measurements

FIG. 3. Variation of the ratio λpore /λ0 versus concentration (C): KCl, pH 5.60 ± 0.05.

range of concentrations at fixed pH to study the variation of the zeta potential with concentration due to the screening effect of the surface charges. Figure 3 shows the concentration dependence of λpore /λ0 at fixed pH (5.6 ± 0.1). As expected, the influence of the charged pore walls on the pore conductivity decreases as the electrolyte concentration (C) increases. Indeed, when C increases, the diffuse layer occupies a relatively smaller and smaller part of the pore and the bulk conductivity increases. Consequently, the contribution of the surface conduction becomes less important compared with the bulk electrolyte (λ0 ), and the electrolyte conductivity inside pores (λpore ) approaches the conductivity of the bulk solution (λ0 ). λpore becomes equal to λ0 for a 1 M KCl solution. Figure 4 presents the variation of the streaming potential (SP) versus the electrolyte concentration (C) at pH = 5.6 ± 0.05. As

FIG. 4. Variation of the experimental streaming potential (SP) versus concentration (C); KCl, pH 5.60 ± 0.05.

The pH-dependence of the zeta potential (ζ ) determined from pore conductivity and streaming potential is given in Fig. 5. For both methods, the profiles of curves are similar: the zeta potential increases (in absolute value) as the pH moves away from the IEP, as would be expected for a surface with acid–base groups. However, the numerical values of zeta potential differ from one method to another. As can be seen, the reproducibility of the results for both methods is better than the differences observed, confirming that the precision of the measurement is not the cause of these differences. It appears that the pore conductivity method gives absolute values of the zeta potential lower than those calculated from SP measurements for ζ > 80 mV and inversely for lower ζ values. It is interesting to note that, at pH 9.3, the reproducibility of the calculated zeta potential values by the pore conductivitybased method is small in contrast to that of the measured pore conductivity values at the same pH (Fig. 1). This is due to the fact that conductivity measurements are not very sensitive at low zeta potentials. This low sensitivity at low zeta potentials is caused by the nonlinear dependence of λpore on the zeta potential (28). Figure 6 presents the concentration dependence of the zeta potential (ζ ) calculated from the pore conductivity and streaming potential measurements. For both techniques, the zeta potential

FIG. 5. pH dependence of zeta potentials (ζ ) calculated according to the space charge model from streaming potential (◦) and pore conductivity experiments (•); KCl, 0.006 M.

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FIG. 6. Concentration dependence of zeta potentials (ζ ) calculated according to the space charge model from streaming potential (◦) and pore conductivity experiments (•).

curves are similar. As expected, the zeta potential decreases with increasing salt concentration since the addition of electrolyte compresses the double layer. It can be noted that the membrane is strongly charged in a wide range of salt concentrations. Zeta potentials determined by the two methods are identical for C ≥ 0.05 M and differ for lower values; zeta potentials calculated from streaming potential values are higher than those obtained by pore conductivity method. However, the ratio of the zeta potentials does not exceed 1.4. An explanation for the differences observed in zeta potentials calculated by both methods could be that the model used does not take into account the conduction behind the shear plane (conduction within the Stern layer). The ionic fluxes are only assumed to occur in the diffuse part of the double layer (beyond the shear plane), so this theory only considers the surface conductivity in the diffuse layer. Recently, Lyklema et al. (30) and Minor et al. (31) have shown that under certain conditions the conduction behind the slip plane might be of the same order of magnitude as that beyond it. It is well known that surface conduction affects electrokinetic phenomena. For example, if computed by Eq. [3], the streaming potential as a function of zeta potential passes through a maximum, unlike the streaming potential calculated by the wellknown Helmholtz–Smoluchowski relationship (Fig. 7) (32). The decrease in streaming potential with increasing zeta potential is a direct consequence of the surface conduction. It should be noted in Fig. 7 that the zeta potential value corresponding to this maximum (ζSPmax ) is 95 mV. If we suggest that the surface conduction in the inner part is not negligible, i.e., that the total surface conductivity (in both diffuse and inner parts) exceeds that computed (surface conductivity in the diffuse layer), the dashed line plotted in Fig. 7 can then represent a possible variation of the streaming potential as a function of the zeta potential. For a given value of the streaming potential, the zeta potential will then be higher than that obtained by the model used for ζ < ζSPmax

FIG. 7. Streaming potential (SP) versus zeta potential (ζ ): (—) SP calculated from the Helmholtz–Smoluchowski relation, SP = ε0 εr ζ /µλ0 ; (—) SP calculated from Eq. [3], 0.006 M KCl, a = 75 nm; (- - -) hypothetical curve that could be obtained by taking into account the contribution of the inner part of the double layer to the surface conduction.

and, in contrast, lower for ζ > ζSPmax . Consequently, the zeta potentials calculated from streaming potentials in Fig. 5 will be underestimated for ζ < ζSPmax and overestimated for ζ > ζSPmax . Moreover, the zeta potential value from which the streaming potential will decrease with increasing zeta potential is expected to occur at ζ < ζSPmax = 95 mV since the surface will be more conductive. Of course, the same analysis must be also considered for the pore conductivity method. Figure 8 shows the influence of zeta potential on the conductivity within pores (λpore ) calculated from Eq. [2] (continuous line). As can be seen, λpore increases exponentially with ζ . It is important to keep in mind that λpore

FIG. 8. Pore conductivity (λpore ) versus zeta potential (ζ ): (—) λpore calculated from Eq. [3], 0.006 M KCl, a = 70 nm; (- - -) hypothetical curve obtained with contribution of the inner part of the double layer to the surface conduction.

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THE ZETA POTENTIAL OF POROUS MEMBRANES

method. It was found that the isoelectric points obtained from the two methods are similar (9.3 ± 0.1 and 9.6 ± 0.1 from the SP method and the λpore -based method, respectively) and that the zeta potential values are in quite good agreement. The differences observed in zeta potentials could be due to the fact that the model does not consider the surface conductivity in the inner part of the double layer. The conductivity-based method has the advantage of being a simple and fast procedure for determining zeta potentials for porous membranes. Moreover, it appears to be an attractive method for characterizing small pores (NF) or large pores with strongly charged surfaces (UF) (28). Thus, it is complementary to the streaming potential method, which is an adapted procedure for characterizing larger pores (MF, UF). APPENDIX A FIG. 9. Concentration dependence of zeta potentials (ζ ): ζSP , ζ calculated according to the space charge model from streaming potential experiments; ζSPmax , zeta potential value corresponding to the maximum of the SP-ζ curves obtained according to the space charge model (Eq. [3]).

calculated from the model (Eq. [2]) takes into account the conductivity in the diffuse layer and the bulk conductivity. If the surface conductivity in the inner part is not negligible, then the measured conductivity exceeds the conductivity in “diffuse layer + bulk.” It would result that the zeta potentials inferred from conductivity experiments with the help of the model would be overestimated on the whole range of zeta potentials. This is illustrated in Fig. 8. The dashed line in this figure could then represent a possible variation of the pore conductivity versus zeta potential. However, due to the exponential shape of the curve λpore = f (ζ ), the model used would lead to an overestimation of the true zeta potential, this overestimation being lower as zeta potential increases. The zeta potential corresponding to the maximum of the SP– ζ curves such as in Fig. 7 increases with the concentration, as shown in Fig. 9 (curve ζSPmax ). It appears that the zeta potentials calculated from the streaming potentials would be overestimated for concentrations less than ∼0.02 M if the inner part of the double layer would contribute significantly to the total surface conductivity. For higher concentrations, the total surface conductivity becomes negligible with respect to the bulk conductivity λ0 (λpore /λ0 ∼ 1; see Fig. 3) and then both methods lead to similar values of the zeta potential as shown in Fig. 6.

The coupling coefficients used in Eqs. [1] and [3] are given by the explicit expressions (28, 32) K 31 =

πzF 2µ

Za (c1 − c2 )(a 2 − r 2 )r dr 0

Za K 33 = 2π z F 2

(K 1 c1 + K 2 c2 )r dr +

2 0

Za ×

Za r (c1 − c2 )

1 r

r

0

2π z 2 F 2 µ

Zr (c1 − c2 )r dr dr dr. 0

The concentrations of cations and anions (c1 and c2 , respectively) are linked to the electrostatic potential 9 by the Boltzmann equation ¶ µ z i F9 ci = C exp − RT

i = 1, 2

and the Poisson–Boltzmann equation provides the relationship between 9 and r needed to evaluate the integral expressions for the above coefficients: µ ¶¸¶ µ ¶ µ · ∂ z F9 z F9 ε0 εr RT −1 ∂ r = sin h . r 2C(z F)2 ∂r ∂r RT RT The boundary conditions that this equation must satisfy are

5. CONCLUSION

The present paper clearly shows that pore conductivity measurements can be used to determine the zeta potentials of porous membranes. The pore conductivity is related to the membrane zeta potential via a space charge model. The zeta potential of a ceramic microfiltration membrane was determined at different values of pH and salt concentration both by the conductivity-based method and by the streaming potential

9(r =a) = ζ and µ

d9 dr

¶ r =0

= 0,

with ζ the zeta potential and a the effective pore radius.

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APPENDIX B: NOMENCLATURE

Roman Letters a C ci F I Ki Ki j l P r R Rcell Rm h Rm

Rsol S SP T z

Effective pore radius (m) Electrolyte concentration in the membrane (mol m−3 ) Ion concentration in the pore (mol m−3 ) Faraday constant (=96,485 C mol−1 ) Electrical current (A) Ion mobility (m s−1 N−1 mol) Coupling coefficients Length of pores (m) Hydrostatic pressure (N m−2 ) Radial coordinate (m) Universal gas constant (8.31 J mol−1 K−1 ) Electrical resistance of the measuring cell (Ä) Electrical resistance of the electrolyte inside pores (Ä) Electrical resistance of the electrolyte inside pores with high salt concentration solution (Ä) Electrical resistance of the electrolyte between the membrane and the two voltage electrodes (Ä) Cross section of the pore (m2 ) Streaming potential (V N−1 m2 ) Temperature (K) Charge number of the ionic species (in absolute value)

Greek letters ε0 εr κ −1 λ0 λpore λs λh µ π ϕ 9 ζ

Vacuum permittivity (8.854 × 10−12 F m−1 ) Relative dielectric constant of the solvent Debye length (m) (κ: Debye parameter (m−1 )) Conductivity of bulk electrolyte (Ä−1 m−1 ) Conductivity of the electrolyte in the pore (Ä−1 m−1 ) Surface conductivity (Ä−1 m−1 ) Conductivity of the electrolyte at high salt concentration (Ä−1 m−1 ) Viscosity of the electrolyte (0.001 kg m−1 s−1 ) Standard dimensionless constant (3.1415927) Electrical potential (V) Electrostatic potential (V) Zeta potential (V) REFERENCES

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