Electrokinetic characterization of hollow fibers by streaming current, streaming potential and electric conductance

Journal of Membrane Science 411–412 (2012) 193–200

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Electrokinetic characterization of hollow fibers by streaming current, streaming potential and electric conductance Y. Lanteri a , P. Fievet a,∗ , S. Déon a , P. Sauvade b , W. Ballout a , A. Szymczyk c,d a

Institut UTINAM, UMR CNRS 6213, Université de Franche-Comté, 16 route de Gray, Besanc¸on Cedex 25030, France Degremont Technologie – AQUASOURCE, 20 Avenue Didier Daurat, 31029 Toulouse, France Université Européenne de Bretagne, France d Université de Rennes 1, Institut des Sciences Chimiques de Rennes, UMR CNRS 6226, 263 Av. du Général Leclerc, CS 74205, 35042 Rennes, France b c

a r t i c l e

i n f o

Article history: Received 31 January 2012 Received in revised form 10 April 2012 Accepted 20 April 2012 Available online 27 April 2012 Keywords: Electrokinetic measurements Streaming current/potential Zeta potential Hollow fibers

a b s t r a c t The electrokinetic properties of hollow fiber polymer membranes were investigated from tangential streaming current/streaming potential and electric conductance measurements. The experiments were conducted with a number of fibers n between 1 and 10 and for three fiber lengths l. The quite good linearity of (i) streaming current/potential data versus pressure difference and (ii) streaming current coefficient and “SP × G” (SP: streaming potential coefficient; G: cell electric conductance) data versus n/l shows that expressions of the streaming current and streaming potential derived in laminar flow are also valid for turbulent flux conditions (provided the electrical double layer lies within the laminar sublayer near the surface). The high experimental conductance, the nonlinear dependence of electric conductance on the number of fibers and the variation of streaming potential coefficient with n and l suggest that the solution in which fibers are immersed makes contribution to the cell electric conductance. A non negligible part of the total streaming current is likely to flow through the macroporous body of fibers. Unlike flat membranes, the contributions of the skin surface and the porous body of the fibers to the streaming current cannot be separated for this type of material due to the impossibility of varying channel cross section. The conversion of tangential electrokinetic measurements into zeta-potential of lumen surface is then no more possible. In such cases, it is advisable to carry out streaming current measurements (or to combine streaming potential measurements with electric conductance measurements) because the streaming current (or the product SP × G) is not affected by the cell electric conductance and can then be considered a property of membrane surface. © 2012 Elsevier B.V. All rights reserved.

1. Introduction The zeta potential originates from the accumulation of charges at a solid–liquid interface where an electrical double layer is formed. It is defined as the electrostatic potential at the hydrodynamic plane of shear. It is an important and reliable indicator of the membrane surface charge that interacts with its surroundings and its knowledge is essential (i) to characterize new membrane materials as well as modified membranes, to study the effect of solution properties (type of ions, ionic strength, pH. . .) on membrane charge properties, (ii) to control the efficiency of cleaning treatments, (iii) to better understand the rejection mechanisms of charged solutes as well as interactions between the membrane surface and various charged foulants such as macromolecules or colloids. . . The zeta potential can be determined from the measurement of

∗ Corresponding author. Tel.: +33 81 66 20 32; fax: +33 81 66 62 88. E-mail address: patrick.fi[email protected] (P. Fievet). 0376-7388/$ – see front matter © 2012 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.memsci.2012.04.031

the streaming potential or streaming current. Streaming potential measurements can be performed in two different ways: by flow through the membrane pores (transversal streaming potential) [1–6] or by flow along the top surface of the membrane (tangential streaming potential) [6–14]. In the case of asymmetric/composite membranes or fine-porous membranes, it is advisable to use the second procedure because it allows avoiding undesirable effects such as the contribution of both supporting layer(s) to the measured signal [15–17] and the membrane potential induced by the concentration difference across the selective layer of the membrane [18–20]. These contributions make the interpretation of experimental data difficult. Unlike streaming potential, streaming current measurements are seldom carried out through membranes due to their unknown pore structure (the calculation of the zeta potential from streaming current requires the knowledge of both the pore length and the membrane porosity). This drawback is eliminated by measuring the streaming current in a channel, the geometry of which is precisely known (typically slit-like channels of 50–500 ␮m in height), formed by two identical flat membranes facing each

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other. With so large channels, it could have been expected that the Helmholtz–Smoluchowski (H–S) relation is applicable to the interpretation of tangential streaming potential measurements in terms of zeta potential of membrane surface. However, ten years ago, Yaroshchuk and Ribitsch [21] have underlined that in the case of channels whose walls are formed by porous materials soaked with electrolyte solution, a non negligible share of the conduction current involved in the streaming potential phenomenon is likely to flow through the channel walls (i.e. the substrate body). They have shown theoretically that in such cases neither the H–S equation nor related relations accounting for the surface conductivity are suitable to compute the zeta potential because these equations are derived with the implicit assumption that both streaming and conduction currents flow through identical paths. In the case of channels with conducting walls, it was theoretically shown that the correct value of the zeta potential can be inferred either from a series of streaming potential measurements performed at various channel heights (extrapolation method) [14] or directly from coupled streaming potential and electric conductance measurements (coupling method) [9,22]. The influence of porous body conductance was investigated experimentally in refs. [9–12,14] for organic and ceramic membranes. It was found that the magnitude of this effect was quite different from a study to another. Indeed, the ratio of the zeta potential determined from the abovementioned extrapolation or coupling method (i.e. by taking into account the effect of the porous body conductance) to that calculated via the H–S equation ( corr / H–S ) was found to be in the range 1–10. Since about five years, the availability of streaming current measurement (in addition to the streaming potential) with the new commercial electrokinetic analysers has enabled a deeper insight into the tangential electrokinetic phenomena with porous substrates. Indeed, Yaroshchuk and Luxbacher [14] have recently shown that the porous structure could make contribution not only to the cell electric conductance (as demonstrated previously) but also to the streaming current and these contributions are considerable especially in the case of membranes with large pores like MF membranes. They have also highlighted that the type of cell used may have a significant influence on measurements of the streaming current or streaming potential. Indeed, the contribution of the support layer of membrane to the measured streaming current (in addition to the measured conductance), and consequently to the streaming potential, may be very different depending on whether the support layer of membrane is or not directly exposed to the pressure drop occurring along the cell. The contribution of the porous body, to a greater or lesser extent, to the streaming current and the type of cell used for electrokinetic measurements could then explain the different ratios  corr / H–S reported in literature. The influence of the measuring cell on the electrokinetic measurements was recently demonstrated by Buksek et al. [13] by comparing the results of two differently designed measuring cells but operating on the same principle. Up to now, the tangential technique was very little applied to the characterization of channels of tubular membranes or membrane hollow fibers probably due to the fact that no tangential measuring cell for this type of membranes has been marketed yet. Another possible explanation could be the large hydraulic diameter of channels that prevents the establishment of a laminar flow and the use of standard equations (e.g. the H–S equation) derived from the Hagen–Poiseuille equation (which is used to derive the expression of the streaming current). However, it was demonstrated in ref. [11] that the expression of streaming current usually used in the electrokinetics theory is still applicable even if the flow is not wholly laminar provided the electrical double layer lies within the laminar sublayer near the channel surface. In this paper, the electrokinetic behavior of a bundle of hollow fibers is studied by streaming current, streaming potential and cell electric conductance under conditions for which the flux is

Hollow fiber Resin Tube

St

Fiber lumen (channel) St - Sc Sc

Porous fiber body Lumen surface Fig. 1. Schematic cross section view of a bundle of fibers.

not wholly laminar. To the best of our knowledge, this is the first time that tangential electrokinetic measurements are performed with hollow fiber membranes. It will be shown that, in addition to the contribution of the membrane body to the overall electric conductance, the electrolyte solution around fibers (indeed, in the measuring cell, the permeate compartment is filled up with the measuring solution and the permeate outlet is closed in order to avoid any permeation through fibers during tangential measurements) also makes contribution. 2. Theory Lets us consider a set of n identical fibers immersed in an electrolyte solution, the porous structure of which is hydraulically exposed (Fig. 1 and Table 1). When a solution is forced to flow through the lumen of the fibers, a tangential gradient of hydrostatic pressure occurs inside their porous structure. As recently explained by Yaroshchuk and Luxbacher [14], when the pores are not large enough and/or the electrokinetic properties of their surface are not the same as those of external surface of the porous substrate (in this work, the lumen of the fiber) forming the channel, one has to explicitly account for the streaming current (Is ) occurring inside the pores of the fiber and in the fiber lumen (i.e. the channel). The contribution of the two media to the electric conductance of the system has also to be taken into account. Unlike the streaming current which has a convective nature and arises only where the liquid flow is possible (i.e. inside the channel and membrane pores), the conduction current flows wherever the electric conductivity is non zero. Consequently, if the fibers are immersed in a conducting liquid, it could also contribute to the conductance of the system. The equivalent electrical circuit describing this system consists of three conductances in parallel due to the channels, pores of the fibers and external solution (i.e. the solution around the fibers). The streaming potential (ϕs ) is given by the ratio of the streaming current (Is ) to the total electric conductance (G) of the system. The expressions for the streaming current, electric conductance, streaming potential and parameter SP × G (where SP denotes the streaming potential coefficient) are collected in Table 1 for various situations. All symbols are defined in the nomenclature section. The situation A corresponds to the case where the streaming current flows in both the channel (i.e. the lumen of the fiber) and the membrane pores, and conduction current exists in the channel, the membrane pores and also in the external solution around the fibers. That is why the electric conductance of the system is expressed as the sum of three components. However, the expression of the external solution conductance is not derived because its representation is not simple. The situation B considers that the streaming and conduction currents take the same path, namely the channel and membrane pores. The situation C corresponds to the situation where the contribution of the porous structure to the streaming current is negligible but not its contribution to the system electric conductance. Finally, in

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195

Table 1 Expressions for the streaming current (Is ), cell electric conductance (G), streaming potential (ϕs ) and parameter SP × G for various situations. Situation A εo εr P Is = −n (Sc e + (St − Sc )Am p )  l  Sc c + Am (St − Sc )p G=n + Ge (n, l) l ϕs = −

SP G = n



ε0 εr (Sc e + (St − Sc )Am p ) l

Sc c + Am (St − Sc )p

SP G = n



G=n

ε0 εr (Sc e + (St − Sc )Am p ) l

Sc c + Am (St − Sc )p

(2a)

(2b)

ε0 εr P(Sc e + (St − Sc )Am p ) Is = G (Sc c + Am (St − Sc )p )

Situation C ε0 εr P Sc e Is = −n l

(1d)



l

ϕs = −

(1b)

ε0 εr P(Sc e + (St − Sc )Am p ) Is (1c) = G (Sc c + Am (St − Sc )p + (l/n)Ge (n, l))

Situation B ε0 εr P Is = −n (Sc e + (St − Sc )Am p ) l G=n

(1a)

(2c)

(2d)

(3a)



l

(3b)

ϕs = −

ε0 εr P Sc e Is = G (Sc c + Am p (St − Sc ))

(3c)

SP G = n

ε0 εr Sc e l

(3d)

Situation D ε0 εr P Sc e Is = −n l G=

n (Sc c ) l

(4b)

ε0 εr Pe Is = G c

(4c)

ε0 εr Sc e l

(4d)

ϕs = − SP G = n

(4a)

situation D, the contribution of porous structure to the streaming current and electric conductance is negligible as compared to that of the channel (e.g. due to fibers too thin as compared to their inner diameter or fibers with too small porosity). In these conditions, the well-known H–S relation (Eq. (4c)) is then obtained provided that the channel is large enough (so that surface conductivity can be neglected). This relation is also obtained for large channels formed by non porous substrates (Am = 0). It should be noted that the porosity of the fiber (Am ) includes the pore tortuosity and  p is the apparent zeta potential of pore surface. The apparent zeta potential is the zeta potential calculated from H–S relation regardless of the pore size. It can be smaller than the true one due to the overlap of electrical double layers or pore radii not sufficiently large as compared with the thickness of the electric double layer (contribution of the surface conductivity). As explained in ref. [14], the porosity appearing into the expressions of the streaming current and electric conductance are the same only in the limiting case of sufficiently large pores.

It should be noted that the expression of the streaming current flowing through a single channel, i.e.: Is =

−(ε0 εr PSc e ) l

(5)

is valid for laminar flow conditions and also for turbulent flow conditions provided the electrical double layer lies within the laminar sublayer near the channel surface (i.e. the thickness of the laminar sublayer near the channel surface is larger than the Debye length) [11]. The hydrodynamic theory [23] gives tls , the thickness of this laminar sublayer, as: tls ≈ 116ri (Re)−7/8

(6)

where Re is the Reynolds number, given by: Re =

¯vdh 

(7)

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Fig. 2. SEM micrographs of a hollow fiber cross section view.

v¯ represents the mean velocity of the liquid, , its density, , its dynamic viscosity and dh , the hydraulic diameter of the channel. 3. Materials and methods 3.1. Membrane and chemicals Hydrophilic polysulfone AlteonTM hollow fiber membranes manufactured by Degrémont Technologies-Aquasource were used. They are hollow fibers filtering from the inside to the outside. They have an asymmetric structure with a gradual change of pore size, which varies from 0.020 ␮m (MWCO of 200 kDa) in the inner active skin up to 5–10 ␮m in the depth of porous structure (Fig. 2). The outer skin has pores of ∼0.1 ␮m. The thickness of inner and outer skins is some ␮m. The membrane porosity is estimated as 80–85%. The fibers have internal and external diameters of 0.8 and 1.3 mm, respectively, which gives cross sections of 0.50 mm2 for the channel and 0.82 mm2 for the porous body. Fibers with length of 10, 20 and 30 cm were studied. Electrokinetic measurements were performed in 10−3 M KCl solutions at natural pH, i.e. 5.6 ± 0.1. KCl was of pure analytical grade (Fisher Scientific) and solutions were prepared from Milli-Q quality water (conductivity < 1 ␮S cm−1 ). 3.2. Streaming potential, streaming current and electric conductance measurements A ZETACAD (CAD Inst., France) zeta meter was used for electrokinetic measurements. This instrument measures the streaming potential and the streaming current generated by the imposed movement of an electrolyte solution through a cell containing hollow fibers. The liquid is forced through the cell using the nitrogen gas whose the pressure is measured by means of a differential pressure sensor (pressure range: 0–1 bar; sensitivity: 0.25 mbar) (Fig. 3a). The cell used for electrokinetic measurements is composed of two PTFE cylinders (total length, 74 mm; external diameters, 50 and 32 mm) inside which the ends of a PMMA tube (external diameter, 20 mm) containing hollow fibers are inserted. The ends of hollow fibers are potted with epoxy resin over ∼25 mm in length. The tube is maintained inside the two cylinders with two screwcaps. Each cylinder is equipped with a rod-type Ag/AgCl electrode (length, ∼15 mm; diameter, 2 mm) positioned at 20 mm from the end of the fibers and with two PTFE tubes, one being linked to the feed circuit and the other to the pressure sensor. The Ag/AgCl electrodes are linked to a Keithley multimeter to measure either the electrical potential difference developed in the solution along the fibers or the streaming current flowing through the cell. The advantage of this cell lies in the possibility to perform measurements for a variable number of fibers with different dimensions.

For conductance measurements, two other PTFE cylinders are used (total length, 95 mm; external diameters, 50 and 32 mm). As shown in Fig. 3b, a PMMA tube (external diameter, 20 mm) having a disk-shaped Ag/AgCl electrode at one end (diameter, 14 mm) is inserted at the back of each cylinder and maintained inside with a screwcap. The electrode is embedded in an epoxy resin and only one side is exposed to the solution. The distance between the electrode and the inlet (or outlet) of the tube containing hollow fibers is ∼5 mm. The electrode area exposed to the electrolyte is 1.5 cm2 which represents about 75% of the internal cross-sectional area of the tube. Conductance measurements are performed by using the ac potentiostatic mode. The disk-shaped Ag/AgCl electrodes are used to apply the voltage and also to measure the resulting current. The equipment used is an electrochemical impedance spectrometer (composed of a Solarton 1286 electrochemical interface linked to a solartron 1255 frequency response analyser). Electrical resistance measurements are carried out with frequencies ranging from 0.1 to 105 Hz. Prior to the measurements, milli-Q quality water (conductivity < 1 ␮S cm−1 ) was filtered at ∼1 bar (600 L per square meter of membrane surface) in order to remove the wetting agent. The measuring solution was then filtered at the same pressure during ∼1 h so as to equilibrate the fibers with the measuring solution. It was checked this time was long enough to allow the fibers equilibration. Indeed, about 45 min of solution permeation was needed for both pH and conductivity of the permeate to reach constant values. Next, the permeate compartment (volume around the fibers) was fully filled up with the measuring solution and the permeate outlet was closed in order to avoid any permeation through fibers during tangential measurements. The electric potential difference (ϕs ) and the streaming current (Is ) were measured alternatively in the two flow directions for continuously increasing pressure values (from 0 to ∼750 mbar). The repeated alteration of the flow direction allows the reduction of the electrode polarization in the measurements of streaming current. Temperature and conductivity were continuously monitored online. Measurements were made at 20 ± 1 ◦ C. The streaming potential and streaming current coefficients were determined from the slope of the plots of ϕs and Is versus P, respectively. The measurement of these coefficients was immediately followed by the conductance measurement of the system. 4. Results Fig. 4 shows an example of the volume flow rate variation as a function of the applied pressure difference (P) for a set of 5 fibers of 20 cm in length. As can be seen, the flow rate does not vary linearly on the pressure range studied. Volume flow rates between 72.5 and 476 mL mn−1 (measured for pressure values

Y. Lanteri et al. / Journal of Membrane Science 411–412 (2012) 193–200

197

Fig. 3. Schematic view of (a) the experimental set-up used for the tangential streaming current/potential measurements of hollow fibers, and (b) the cell used for conductance measurements.

comprised between 45 and 730 mbar) correspond to mean velocities v¯ between 0.481 and 3.155 m s−1 . The Reynolds numbers Re (Eq. (7)) calculated at the lowest and highest applied pressures for the three fiber lengths studied are collected in Table 2. It appears

Volume flow rate (mL/mn)

500 400 300 200

that the flow cannot be regarded as laminar on the whole pressure range. When Re = 2840 (the highest value), the purely laminar sublayer extends to ∼45 ␮m from the channel surface. The electrical double layer whose thickness is ∼10 nm is therefore included within this laminar sublayer. Consequently, Eq. (5) can be used to express the streaming current arising in a single channel. The linear variations of streaming current and streaming potential with the pressure difference (examples are shown in Fig. 5) are the experimental proof that Eqs. (1a), (2a), (3a) and (4a) and (1c), (2c), (3c) and (4c), which are derived using Eq. (5), are valid even if the flow is not wholly laminar. Fig. 6a and b presents the dependence of streaming current coefficient (Is /P) on the number of fibers (n) and reciprocal fiber length (1/l), respectively. The dependences are linear as predicted by Eqs. (1a), (2a), (3a) and (4a). The degree of linearity is very high, the

100 0 0

100

200

300

400

500

600

700

800

Δ P (mbar) Fig. 4. Volume flow rate as a function of pressure difference for 5 fibers of 20 cm in length.

Table 2 Values of the Reynolds number (Re) at 45 and 730 mbar for three fiber lengths (l). l (cm)

10

20

30

Re (P = 45 mbar) Re (P = 730 mbar)

590 2840

385 2520

315 2200

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80

Δϕ s (mV) or Is (nA)

R2 = 0.9997

60 40

R2 = 0.9999

20 0

-800

-600

-400

-200

-20

0

200

400

600

800

-40 -60 -80 Δ P (mbar)

Fig. 5. Streaming potential (full symbols) and streaming current (empty symbols) measurements in alternating flow directions as a function of the pressure difference.

correlation coefficient ranging from 0.9967 to 0.9996 for the data as a function of n and from 0.9975 to 0.9996 for those as a function of 1/l. The strict linearity of the data makes possible reliable determination of the slopes. However, as shown by Eqs. (1a), (2a), (3a) and (4a), the variation of Is /P with n or l does not enable to know if the streaming current occurring within the porous structure is or not negligible and thus, if Eq. (3a) (or Eq. (4a)) can be applied to determine the zeta potential of lumen surface. To this end, fibers of the same material with the same cross section of membrane body (i.e. the difference between the outer and inner cross sections, St − Sc , is kept constant) but with different channel cross sections (Sc ) would be needed. As to the streaming potential coefficient (SP), Fig. 7a shows that this parameter depends on both the number and length of fibers. As established in the previous theoretical section, the dependence of

Fig. 6. Streaming current coefficient (Is /P) as a function of the (a) number of fibers and (b) reciprocal fiber length (1/l).

Fig. 7. (a) Streaming potential coefficient (SP) as a function of the number of fibers for various lengths (l) and (b) Electric conductance (G) as a function of the number of fibers for various lengths (l). Symbols: experimental data; Straight lines: theoretical conductance due to both channels and layers of electrolyte solution having the same dimensions as the porous body of fibers. (——) l = 10 cm; (- -) l = 20 cm; (––) l = 30 cm.

SP with n and l can be explained only by the additional conduction outside the fibers (Eq. (1c)). Thus, the streaming and conduction currents involved in the streaming potential process do not flow through identical paths. The streaming current flows only through the channels (Eq. (3a)) or through both the channels and fiber pores (Eq. (1a)) whereas a part of the conduction current is likely to flow inside the external solution (in addition to the channels and pores). The high values of the measured electric conductance and its nonlinear variation with the number of fibers (Fig. 7b) confirms that the external solution contributes to the conduction phenomenon. The straight lines in Fig. 7b represent the theoretical conductance due to both channels and layers of electrolyte solution having the same dimensions as the porous body of fibers. As can be seen from Eqs. (1a) and (1c), no information about the electrokinetic properties of lumen and pore surfaces can be obtained from the dependences of Is /P and SP on the number or length of fibers. In a recent paper devoted to the interpretation of electrokinetic measurements with membranes forming a slit channel, Yaroshchuk and Luxbacher [14] showed that the contribution of porous structure to the streaming current decreases with pore size, the contribution becoming negligible for a channel height of ∼120 ␮m and membranes having an average pore size of 0.025 ␮m, porosity of 0.7 and thickness of 105 ␮m. On the contrary, the contribution of porous body was found to become dominant for membranes with an average pore size of 5 ␮m, porosity of 0.85 and thickness of 135 ␮m. As shown by Eq. (1a), the contribution of streaming current flowing through the membrane pores depends on both the structural characteristics of the porous body through the parameters St –Sc and Am , and electric properties of pore surface through the parameter  p . In the study by Yaroshchuk and Luxbacher [14], it should be noted that for a channel height of 120 ␮m the ratio of the membrane cross section

Y. Lanteri et al. / Journal of Membrane Science 411–412 (2012) 193–200

(a)

0 -50

0

20

40

60

80

100

120

Is /Δ P (nA/bar)

-100 -150 -200 -250 -300 -350

l = 30 cm l = 20 cm

2

R = 0.997

l = 10 cm

-400

-1

n/l (m )

-1

SP G (mV bar μ S)

(b)

20

40

60

80

100

120

-100 -150 -200 -250 -300 -350 -400

streaming current or the streaming potential together with electric conductance. Indeed, measuring the streaming potential only is not beneficial in this case because the contribution of external electrolyte solution to the cell electric conductance (Eq. (1b)) is poorly controllable. Unlike streaming potential, the streaming current (or the product SP × G) is not affected by the cell conductance (as shown by Eq. (1a)) and can be considered rigorously as a property of membrane surface (including lumen and pore surfaces). Therefore, when the effects of solution (pH, species in solution, ionic strength. . .) on the surface charge properties of hollows fiber membranes are investigated, it is advisable to consider the streaming current coefficient instead of the streaming potential coefficient (otherwise the interpretation of the electrokinetic properties could be distorted). 5. Conclusion

0 -50 0

199

l = 30 cm l = 20 cm l = 10 cm

2

R = 0.996

-1

n/l (m ) Fig. 8. (a) Streaming current coefficient (Is /P) versus the ratio of the number of fibers to their length (n/l) and (b) Product of the streaming potential coefficient by the electric conductance (SP G) versus n/l.

to the channel cross section is 1.75. This value is close to 1.64 which is obtained for fibers studied in this work. Moreover, their membranes have porosities similar to our fibers. Given that the thickness (some ␮m) of inner and outer skins is negligible with respect to that of the central part of fiber body (∼250 ␮m), it results that the mean pore size is a few micrometers. Consequently, according to their results concerning the effect of pore size on the contribution of membrane porous body to the streaming current, it would be unreasonable to consider that this contribution is negligible in our case. By the way, if the contribution was assumed to be negligible, zeta potentials of −81.6 and −79.8 mV would be obtained for the lumen surface of fibers ( e ) from the slope of the plots of Is /P (streaming current coefficient) and the product SP × G (streaming potential coefficient by electric conductance) versus n/l, by using Eqs. (3a) and (3d), respectively (Fig. 8a and b). As shown by Eq. (1a), the total streaming current is the sum of a component due to the fiber porous body and another component due to the channel. If the porous body brings about an increase in the absolute value of total streaming current, then ascribing this latter to the channel surface only leads to an overestimation of the actual zeta potential of the channel surface. Nevertheless, it should be stressed that although the zeta potentials given above are overestimated, the fact that very close values are obtained from independent measurements (i.e. Is /P and SP × G) gives evidence of the reliability of electrokinetic and electric measurements carried out in this work. Unlike flat membranes for which the channel height between two identical membrane samples can be varied, the contributions of active and porous body surfaces to the streaming current cannot be separated for this type of material (due to the impossibility of variation of channel cross section). Nonetheless, a combined information on the electrokinetic properties of active surface and pore surface of hollow fibers remains very useful. This study demonstrates that for obtaining this information, one needs to measure either the

It was shown that tangential electrokinetic measurements usually performed with flat membranes are also possible with hollow fiber membranes even if the flow is not wholly laminar. This is demonstrated experimentally by the quality of linear fits of (i) streaming current and streaming potential with pressure difference, (ii) streaming current coefficient with fiber number to fiber length ratio (n/l) and (iii) product “streaming potential coefficient × electric conductance” (SP × G) with n/l. The equality between SP × G and Is /P (obtained from independent measurements) is also a proof of the reliability of measurements. The nonlinear dependence of electric conductance and streaming potential coefficient on the number of fibers reveals that, in addition to the contribution of the porous structure to the overall electric conductance, the solution around fibers also makes contribution. Because the major part of the porous body of fibers considered in this work is macroporous it is very unlikely that the porous structure does not contribute to the streaming current. In this case (i.e. when the streaming current flowing through the porous structure is not negligible), it is advisable to perform measurements of streaming current or combined measurements of streaming potential and electric conductance because the parameters Is and SP × G are not complicated by the cell electric conductance, which is poorly controllable. Finally, the tangential electrokinetic characterization of hollow fibers (in terms of zeta potential of both lumen and pore surfaces) is found to be more complex than for flat membranes due to the impossibility of variation of channel cross section. Acknowledgments This research was carried out under funding from the FiltrACOPPE program (ANR). L’Agence Nationale pour la Recherche is kindly acknowledged for its financial support. References [1] M. Nyström, M. Lindström, E. Matthiasson, Streaming potential as a tool in the characterization of ultrafiltration membranes, Colloids Surf. 36 (1989) 297–312. [2] E. Staude, D. Duputell, F. Malejka, D. Wyszynski, Determination of surface properties of membranes based on polysulfone derivatives by electrokinetic measurements, J. Dispersion Sci. Technol. 12 (1991) 113–127. [3] C. Werner, H.J. Jacobasch, G. Reichelt, Surface characterization of hemodialysis membranes based on streaming potential measurements, J. Biomater. Sci. 7 (1995) 61–76. [4] A. Szymczyk, A. Pierre, J.C. Reggiani, J. Pagetti, Characterisation of the electrokinetic properties of plane inorganic membranes using streaming potential measurements, J. Membr. Sci. 134 (1997) 59–66. [5] J. Benavente, G. Jonsson, Effects of adsorbed protein on the hydraulic permeability, membrane and streaming potential values measured across a microporous membrane, Colloids Surf. A 138 (1998) 255–264. [6] C. Lettmann, D. Möckel, E. Staude, Permeation and tangential flow streaming potential measurements for electrokinetic characterization of track-etched microfiltration membranes, J. Membr. Sci. 145 (1999) 243–251.

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Nomenclature Roman letters Am fiber porosity including the pore tortuosity dh hydraulic diameter (m) total electric conductance of the system ( −1 ) G Ge electric conductance due to the solution around fibers ( −1 ) streaming current (A) IS l fiber length (m) number of fibers n ri inner radius of a fiber or channel radius (m) outer radius of a fiber (m) re Re Reynolds number Sc channel cross section; Sc = ri 2 (m2 ) total fiber cross section; St = re 2 (m2 ) St tls thickness of the laminar sublayer v¯ mean velocity of the liquid (m s−1 )  density (kg m−3 ) Greek letters ϕs streaming potential (V) P hydrostatic pressure difference (Pa) ε0 vacuum permittivity (8.854 × 10−12 F m−1 ) εr relative dielectric constant of the solvent electric conductivity of electrolyte solution in the c channel ( −1 m−1 ) electric conductivity of electrolyte solution in the p pores ( −1 m−1 )  dynamic viscosity of the solution (kg m−1 s−1 ) zeta potential of lumen surface of the fiber (V) e p zeta potential of pore surface (V)

[9]

[10]

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