Journal of Colloid and Interface Science 216, 285–296 (1999) Article ID jcis.1999.6321, available online at http://www.idealibrary.com on
Electrokinetic Phenomena in Homogeneous Cylindrical Pores Anthony Szymczyk,* ,1 Boujemaa Aoubiza,† Patrick Fievet,* and Jacques Pagetti* *Laboratoire de Corrosion et Traitements de Surface, 32 rue Me´gevand, 25030 Besanc¸on cedex, France; and †Laboratoire de Calcul Scientifique, 16 route de Gray, 25030 Besanc¸on cedex, France Received December 8, 1998; accepted May 19, 1999
The electrokinetic phenomena occurring in homogeneous cylindrical pores containing symmetric electrolytes are studied. The local relations for flow in the pores (Nernst–Planck and Navier–Stokes equations) are developed. The analysis includes a numerical solution of the nonlinear Poisson–Boltzmann equation. The integral expressions of the phenomenological coefficients coupling the solvent flow and the electrical current with the hydrostatic pressure and the electrical potential gradients are established and calculated numerically. The mobilities of anions and cations are individually specified and the electroviscous effects as well as the surface conductance are taken into account. Streaming potentials obtained from numerical calculations are compared with results given by classical relations in a range of zeta potentials and electrokinetic radii that may commonly occur in experimental investigation of micro- and ultrafiltration membranes. In this work, it is shown that classical approximated relations can give rise to very misleading conclusions and that the determination of the true zeta potential requires a full analysis (including numerical calculations) of the basic relations for flow and potential distribution in charged pores. © 1999 Academic Press Key Words: electrokinetic phenomena; zeta potential; surface conductance; electroviscous effects; streaming potential; space charge model.
where e 0 is the vacuum permittivity, e r the relative dielectric constant of the solvent, m the viscosity of the solution, l 0 the conductivity of the electrolyte in the bulk, G s the surface conductance, and a the pore radius. The term ( l 0 1 (2G S / a)) represents the conductivity of the electrolyte inside pores (l pore ) which can be substantially different from the bulk conductivity. The well-known Helmholtz–Smoluchowski relationship is obtained by neglecting the surface conductance contribution and is often used by experimental investigators to deduce the zeta potential from streaming potential measurements:
SP 5
It is now recognized that electric charges on the pore walls of filtration membranes play an important role in their separation performances and fouling behavior (1–3). The streaming potential is among the most convenient experimental techniques for studying the electrical potential properties of porous membranes. These properties are frequently characterized in terms of zeta potential (z) (4, 5). A usual relation linking the zeta potential and the streaming potential (SP) for cylindrical pores is given by
e 0 e rz , 2G S m l0 1 a
S
D
[2]
The above equation considers that the streaming potential is proportional to the zeta potential and independent of the pore size. The use of this equation is grossly incorrect in many situations and can lead to substantial underestimation of the true zeta potential, particularly when the Debye length of the solution is comparable to the pore radius of the membrane. It can also lead to possible serious error in the interpretation of relative changes in membrane properties (6). Rice and Whitehead (7) have calculated analytically the correction factors that must be applied to the Helmholtz– Smoluchowski equation. However, their approach is subject to the restriction of low zeta potential (,25 mV for a 1–1 electrolyte) because of the use of the Debye–Hu¨ckel approximation. Levine et al. (8) have extended the Rice and Whitehead theory to higher potentials by developing analytical approximations to the Poisson–Boltzmann equation but their method is confined to symmetric monovalent electrolytes having equal ionic mobilities. More recently, Bowen and Cao (6) have presented calculations based on an extension of the approach of Levine et al. that includes a numerical solution of the nonlinear Poisson–Boltzmann equation for symmetric electrolytes in which mobilities of anions and cations are individually specified.
1. INTRODUCTION
SP 5
e 0e rz . ml 0
[1]
To whom correspondence should be addressed. Fax: 133.3.81.66.56.52; E-mail:
[email protected]. 1
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0021-9797/99 $30.00 Copyright © 1999 by Academic Press All rights of reproduction in any form reserved.
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Electrokinetic phenomena occurring within a membrane can lead to severe misinterpretations of the membrane electrokinetic properties. The aim of this work is to present and discuss the main electrokinetic phenomena that are likely to make difficult the interpretation of experimental data. This paper presents calculations based on the thermodynamics of irreversible processes and the space charge model developed by Osterle et al. (9 –11). The nonlinear Poisson– Boltzmann equation is solved numerically and calculations allow for a proper account of the individual diffusivities of the cations and anions in symmetrical electrolytes. Electroviscous effects and surface conduction are studied in a range of zeta potentials and electrokinetic radii that are frequently encountered in practical situations. The zones of predominant influence of primary and secondary electroviscous effects are located by studying the variation of the electroviscous effects as a function of the streaming potential. It is also shown that the determination of the zeta potential of a membrane filled with an electrolyte solution at a given pH cannot be made from a single streaming potential measurement but requires a full study of the streaming potential over a range of pH. Streaming potentials calculated numerically are compared to those given by the approximated relations usually used in experimental investigations. The results show that is important to account properly for the individual ionic diffusivities in calculating the correction factor that should be applied to the Helmholtz– Smoluchowski equation. 2. SPACE CHARGE MODEL
A porous membrane (having identical cylindrical pores of length l and radius a, with l @ a) separating two aqueous solutions at the same temperature but different pressures and electrical potentials and containing slightly different concentrations of the same electrolyte is first considered. The steady state transport phenomena in such a membrane can be characterized by three flows (solvent flow q, differential flow J d , and electrical current I) and three driving forces (transmembrane pressure P, osmotic pressure P, and electrical potential w) (10). The space charge model assumes ions as point charges, so the steric effects of the size of ions are neglected (12). Calculations presented further in this work have been carried out for pore radii which are at least an order of magnitude larger than the averaged size of ions. The study of pores with radii comparable to the size of ions and water molecules (nanofiltration) would require the inclusion of hindrance factors for diffusion and convection in the Nernst–Planck equation (13, 14) which is needed to describe the ions flux within pores.
2.1. Phenomenological Relations For sufficiently small forces that are near the thermodynamic equilibrium, the flows are linear functions of the generalized forces
S D S D S D
S D S D S D
S D S D S D
q 5 K 11 2
dP dP dw 1 K 12 2 1 K 13 2 dx dx dx
[3a]
J d 5 K 21 2
dP dP dw 1 K 22 2 1 K 23 2 dx dx dx
[3b]
dP dP dw 1 K 32 2 1 K 33 2 , dx dx dx
[3c]
I 5 K 31 2
where the terms K ij denote the coupling coefficients between the different flows and forces. It should be noted that these global linear relationships fail if the thermodynamic forces are not sufficiently small. In such a case, Yaroshchuk and Dukhin (15) have proposed a locally linear approach. They consider the phenomenological relations as written down for sufficiently thin membrane layers in which the linearity may be restored. In order to study the electrokinetic phenomena, the solutions on each side of the membrane are taken at the same concentration. In such a case, the steady state transport phenomena can be completely characterized by the coupling of only two flows and their conjugated forces, and the set of Eqs. [3] reduces to
S D S D
S D S D
q 5 K 11 2
dP dw 1 K 13 2 dx dx
[4a]
I 5 K 31 2
dP dw 1 K 33 2 . dx dx
[4b]
Taking into account Eqs. [4] the streaming potential can be written as SP 5
S D Dw DP
52 I50
K 31 , K 33
[5]
where Dw and DP are the electrical potential difference and the hydrostatic pressure difference across the pores. 2.2. Local Flow Relations In the following section we introduce the cylindrical coordinates (r, x, u ); where r denotes the distance measured radially from the pore axis, x denotes the distance along the axis, and u is the azimuthal angle. Due to axial symmetry, all quantities are independent of u.
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ELECTROKINETIC PHENOMENA
The total electrical potential, f, can be split into two parts (9),
f ~ x, r! 5 w ~ x! 1 C~ x, r!,
[6]
where w is the component due to the streaming potential in the axial direction and C is the electrostatic potential due to the electrochemical double layer. The flow of ions in the pore is described by the Nernst– Planck equation. Considering that there is no radial solute flow, the ion densities ( j i ) in the x direction are given by j i 5 2K i z i c i F
S D
f 1 c iu x
i 5 1, 2,
[7]
where c i is the ion concentration in the pore, K i is the ion mobility, z i is the ion valence, u is the fluid velocity, and where i 5 1 indicates the cations and i 5 2 the anions. The above equation describes the transport of ions across the pore in terms of migration and convection resulting from electrical potential and pressure gradients, respectively. The diffusion term does not appear in Eq. [7] since the establishment of Eqs. [4] assumes no axial concentration gradient. FIG. 2. Distribution of the electrostatic potential as a function of r/a for various zeta potentials; a 5 2 nm; 0.001 M KCl.
The fluid flow in the pore is governed by the Navier–Stokes equation which can be written (at the steady state):
O z c ¹f 1 m¹ u 5 0. 2
2¹P 2 F
2
i i
[8]
i51
Applying the condition of no radial flow to the radial component of the Navier–Stokes equation and since it can be shown that the axial derivatives of velocity can be neglected (16, 17), Eq. [8] can be written in cylindrical coordinates as 2
S F S DGD
P f 2 zF~c 1 2 c 2 ! 1 m r 21 x x r
r
u r
5 0. [9]
Integrating the above equation with respect to r and to the following boundary conditions, u ~r5a! 5 0 FIG. 1. Distribution of the electrostatic potential across the pore as a function of dimensionless radial distance (r/a) for various values of a; 0.001 M KCl.
S D u r
5 0, r50
[10] [11]
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we obtain u5
S D
dP a2 2 r2 2 4m dx zF m
1
E E a
1 r
r
S D
r
~c 1 2 c 2 !rdrdr 2
0
dw . dx
[12]
2.3. Expression of the Coupling Coefficients The general expressions of the solvent flow and electrical current in a cylindrical pore are
q 5 2p
E
a
urdr
[13]
0
I 5 2 p zF
E
a
~ j 1 2 j 2 !rdr.
[14]
0
Substituting Eqs. [7] and [12] into Eqs. [13] and [14], the four coupling coefficients expressed in Eqs. [4] can be written as follows: K 11 5
pa 4 8m
K 13 5
2 p zF m
K 31 5
[15a]
p zF 2m
E E E E E E E a
a
r
0
FIG. 3. Dependence of streaming potential (SP) on electrokinetic radius (a/ k 21 ) for various values of the normalized zeta potential (j); 0.001 M KCl.
r
1 r
~c 1 2 c 2 !rdrdrdr
r
[15b]
with the following boundary conditions:
0
C ~r5a! 5 z
a
~c 1 2 c 2 !~a 2 2 r 2 !rdr
and
0
a
K 33 5 2 p z F 2
2
0
2 p z 2F 2 ~K 1 c 1 1 K 2 c 2 !rdr 1 m a
3 r~c 1 2 c 2 !
r
1 r
E
S D dC dr
a
0
r
~c 1 2 c 2 !rdrdrdr.
[15d]
0
The local concentrations of cations and anions (c 1 and c 2 , respectively) are given by the Boltzmann equation,
S
c i 5 C exp 2
z i FC RT
D
S F S DGD
[19]
The resolution of Eq. [17] allows the calculation of the local concentrations c i which are needed to integrate numerically the expressions of the coupling coefficients expressed in Eqs. [15]. The Poisson–Boltzmann equation is solved numerically using the substitution described by Bowen and Jenner (18).
[16]
S D
5 sinh
5 0. r50
3. RESULTS AND DISCUSSION
i 5 1, 2,
where C is solution of the Poisson–Boltzmann equation
e 0 e r RT 21 zFC r 2 r 2C~ zF! r r RT
[18]
[15c]
zFC , [17] RT
All numerical simulations presented in this work have been carried out for a range of electrokinetic radii and zeta potentials which may commonly occur in experimental investigations with micro- and ultrafiltration membranes. Figure 1 shows the distribution of the electrostatic poten-
ELECTROKINETIC PHENOMENA
289
FIG. 4. Streaming potential vs zeta potential (z) for various values of the electrokinetic radius; 0.001 M KCl. Semi-interrupted line, SP calculated from Eq. [2]; interrupted lines, SP calculated from Eq. [1]; continuous lines, SP calculated from space charge model.
tial (C) across pores of various sizes obtained using the nonlinear Poisson–Boltzmann equation. For the smallest pore radius (a 5 2 nm), the radial variation of C is small and the assumption of a constant potential is reasonable. In such a case, the assumption of a Donnan partitioning at the entrance of the pores may be justified. However, cautions must be taken before making such an assumption. Indeed, the electrostatic potential distribution across the pore also depends on the zeta potential level ( z 5 C (r/a51) ) as exposed in Fig. 2. In this example, the assumption of a constant potential is justified for not too high zeta potentials but it appears that the radial variation of C becomes significant as z increases. Then, the use of the Poisson–Boltzmann equation may be required even in the case of small pores if the zeta potential is sufficiently high. Figure 3 shows the streaming potential calculated from Eq. [5] (space charge model) as a function of the electrokinetic radius (a/ k 21 ) for various values of normalized zeta
potential ( j 5 zF z /RT) in a 0.001 M KCl solution. When the pore radius is comparable to the Debye length of the solution (k 21 ), the streaming potential shows a dependence on pore radius. The decrease in streaming potential with decreasing pore radius is a direct consequence of the surface conduction and electroviscous effects which will be discussed later in this paper. For a low charged surface (e.g., j 5 0.5), Fig. 3 shows that streaming potential becomes almost constant when the pore radius is about ten times the Debye length (k 21 ). This value depends on the zeta potential level and it appears for higher electrokinetic potentials (j 5 2) that the streaming potential is still dependent on the pore size for pore radii much larger than the Debye length of the solution. Figure 4 shows the zeta potential dependence of the streaming potential for various electrokinetic radii in 0.001 M KCl. Streaming potential determined from the Helmholtz–Smoluchowski equation (Eq. [2]) is directly proportional to the zeta
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potential and is independent of the electrokinetic radius. Equations [1] and [5] give different results and show that the streaming potential reaches a maximum value before decreasing even if the zeta potential still increases (it can be noted that the same behavior was obtained by O’Brien and White for electrophoretic mobility (19)). Thus, it appears that a particular streaming potential value can be associated with two significantly different values of zeta potential. The experimental investigator will be able to remove this ambiguity by plotting the streaming potential as a function of the pH of the solution. Indeed, if the above ambiguity occurs, the plot SP 5 f(pH) will show a particular streaming potential value associated with two different pH. The H 1 and OH 2 ions being potential determining ions for many membranes, the closer the pH is to the isoelectric point (that is, the pH for which SP 5 0), the lower the zeta potential is. Then, the streaming potential value corresponding with the pH closest to the isoelectric point will be associated with the smallest zeta potential value obtained from the plot SP 5 f( z ). Experimentally, the use of the Helmholtz–Smoluchowski equation to deduce the zeta potential from streaming potential measurements is convenient but can give rise to very misleading conclusions because it only allows the determination of an apparent zeta potential. A correct relation FIG. 6. Conductivity in the pore (l pore) versus pore radius for various value of j; 0.001 M KCl; interrupted line, conductivity of the bulk solution (l0).
between the streaming potential and the zeta potential can be expressed as SP 5
e 0e rz F, ml 0
[20]
where F is the correction factor that should be applied to the classic Helmholtz–Smoluchowski equation. By considering Eqs. [5] and [20], F can be written as F52
FIG. 5. Zeta potential dependence of the corrective factor (F) to apply to the Helmholtz–Smoluchowski equation for various values of the electrokinetic radius; 0.001 M KCl.
K 31 ml 0 . K 33 e 0 e r z
[21]
The correction factor F depends on the electrokinetic radius as well as the zeta potential level as exposed in Fig. 5. The determination of the true zeta potential from experimental streaming potential measurements therefore requires a full analysis of the electrokinetic phenomena occurring within pores as presented in this work. Figure 5 shows that the use of the Helmholtz–Smoluchowski equation can lead to a significant underestimation of the true zeta potential, not only when the pore size is comparable to the double layer thickness (ultrafiltration) but also when the surface is highly charged.
ELECTROKINETIC PHENOMENA
FIG. 7. KCl.
291
Dukhin number (Du) vs z for various values of a/ k 21 ; 0.001 M
The Helmholtz–Smoluchowski relation assumes that the conductivity in the pores is the same as the bulk conductivity (l 0) whereas the space charge model as well as Eq. [1] take into account the true conductivity of the electrolyte inside the membrane pores (l pore). This latter can be written as
l pore 5 l 0 1
2G s K 33 5 , a S
FIG. 8. Streaming potential (SP) vs surface conductance (G s ) for various values of a/ k 21 ; 0.001 M KCl.
Figure 7 shows the variation of the Dukhin number versus the zeta potential for various electrokinetic radii. The Dukhin number increases with zeta potential because the concentration
[22]
where S is the cross-section of the pore. Figure 6 shows the l pore variations as a function of pore radius for a 0.001 M KCl solution. It is clearly shown that l pore can be substantially greater than the bulk conductivity, especially for small pores at high zeta potentials. During the streaming potential process, the electric field created by the pressure gradient produces a backflow of ions by electro-osmotic effect. This backflow is enhanced by the surface conduction. This latter is expressed in terms of surface conductance (G s ) which is an excess conduction that takes place in the pore due to the presence of electrical double layers. The surface conductance is expressed in V 21. The bulk conductivity being expressed in V 21.m 21, the introduction of a parameter of dimension length is then required to express the surface contribution with respect to the bulk one. To this end, we introduce the Dukhin number as defined by Lyklema and Minor in a recent work (20): Du 5
Gs . l 0a
[23]
FIG. 9.
q p /q vs z for various values of a/ k 21 ; 0.001 M KCl.
292
FIG. 10.
FIG. 11.
SZYMCZYK ET AL.
q p /q vs streaming potential for a/ k 21 5 1; 0.001 M KCl.
q p /q vs streaming potential for a/ k 21 5 10; 0.001 M KCl.
FIG. 12.
q p /q vs streaming potential for a/ k 21 5 100; 0.001 M KCl.
of counterions in the double layer increases exponentially with z as shown by Eq. [16]. For small pores and low electrolyte concentration (that is for small a/ k 21 ), Du is rapidly much larger than 1. In such a case, the conduction takes place mainly along the charged surface. It clearly appears from Fig. 7 and Eq. [23] that the surface conductance G s varies exponentially with z at fixed electrokinetic radius. Figure 8 is a plot of the streaming potential against the surface conductance for various values of a/ k 21 in a 0.001 M KCl solution. The streaming potential has a maximum with respect to G s at fixed electrokinetic radius. The increase in surface conductance leads to a greater backflow of counterions during the streaming potential process. When the conductance of the region near the pore wall reaches a critical value, the backflow of counterions becomes so important that the axial polarization of the pore decreases. As a result, the streaming potential decreases even if G s still increases (that is, even if more counterions are present in the diffuse layer). That explains the global shape of curves presented in Fig. 4 (SP 5 f( z )) since an increasing zeta potential corresponds to an increasing surface conductance. It can be also noted from Fig. 4 that the use of Eq. [1], which takes into account the surface conductance, leads to higher streaming potential values than Eq. [5] (space charge model). The discrepancy between streaming potentials given by both equations increases as the electrokinetic radius becomes
ELECTROKINETIC PHENOMENA
293
FIG. 13. Zeta potential dependence of the SP for various electrolytes; C 5 0.0149 M; a/ k 21 5 10; Diffusivities used in calculations (in m 2.s 21): Li 1 5 1.03 3 10 29; Na 1 5 1.33 3 10 29; K 1 5 1.96 3 10 29; H 1 5 9.30 3 10 29; Cl 2 5 2.03 3 10 29.
smaller. That can be explained by the presence of some electroviscous effects which are not taken into account by Eq. [1]. Indeed, the presence of the electrical double layer may exert a profound influence on the behavior of the fluid flowing through the membrane pores. The solvent flow through an uncharged pore is given by the Poiseuille equation:
qp 5
S D
S D
pa 4 dP dP 2 5 K 11 2 . 8m dx dx
[24]
In the presence of an electrical double layer the solvent flow q is given by Eq. [4a]. The electroviscous effects can be described in terms of the ratio of the Poiseuille flow q p to the observed solvent flow q. Using Eqs. [4a] and [24], we obtain qp 5 q
1 . K 13 K 31 12 K 11 K 33
[25]
The electroviscous effects can be split in two parts: the primary and the secondary effect. During the streaming potential process, the backflow of liquid due to the electro-osmotic phenomenon leads to a diminished flow in the forward direction. The liquid appears to exhibit a greater viscosity (usually called apparent viscosity) if its flow is compared with the Poiseuille flow. This is called the primary effect. The secondary electroviscous effect occurs when the double layers within the pore cannot be properly developed. It leads to a decrease of the solvent flow as a result of the double layers interactions. When this effect occurs in the pore, it cannot be completely separated from the primary effect (21). Figure 9 shows the variation of q p /q due to both electroviscous effects versus the normalized zeta potential for various electrokinetic radii. The ratio q p /q presents a maximum with respect to j at fixed a/ k 21 . It can be noted that the electroviscous effects can have a substantial influence on the liquid flow in the pore. Figures 10 –12 present q p /q as a function of the streaming potential for three electrokinetic radii (a/ k 21 5 1, 10 and 100)
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SZYMCZYK ET AL.
FIG. 14. Dependence of F on negative zeta potential for various electrolytes; a/ k 21 5 10; Diffusivities as in Fig. 12.
in 0.001 M KCl. These curves give information about the dominant electroviscous effect occurring in the pore. In these figures, each point corresponds to a given normalized zeta potential value. Figure 10 refers to a/ k 21 5 1. At first, q p /q increases with the streaming potential which means that the primary electroviscous effect is greater than the secondary effect. Indeed, the primary effect results from the polarization between the entrance and the exit of the pore. Thus, an increase in streaming potential leads to a more substantial backflow of ions, that is, a greater primary electroviscous effect. For higher normalized zeta potentials (from j 5 2 to ;6.5), q p /q still increases whereas the streaming potential decreases. This negative slope shows the presence of the secondary electroviscous effect which is more significant than the primary effect in this range of j. The interactions between the double layers act to diminish the fluid flow in the streaming direction. Consequently, less counterions are carried toward the end of the pore and then the streaming potential is smaller. For the highest j values, both electroviscous effects decrease. The slope of the curve becomes positive again which indicates that the secondary effect
is negligible with respect to the primary effect. This latter also decreases because the streaming potential becomes smaller at high zeta potential (due to the surface conduction). Figure 11 shows the same plot as Fig. 10 for a/ k 21 5 10. It appears that the secondary electroviscous effect still exists (negative slope from j 5 3 to 5) but it is less important than for a/ k 21 5 1 which is not unexpected because the overlap of the double layers is less significant. When the pore radius is a hundred times larger than the Debye length (see Fig. 12) the interactions between the double layers are negligible and the secondary effect virtually vanishes. The primary electroviscous effect still exists but it can be noted that its influence on the solvent flow is quite negligible (the solvent flow is very close to the Poiseuille flow for all zeta potentials). Figure 13 shows the zeta potential dependence of the streaming potential obtained from the space charge model for various electrolytes having a common anion. The electrolyte concentration and the electrokinetic radius are fixed at 0.0149 M and 10, respectively. For a membrane with a negative zeta potential, the streaming potential is lowest (in absolute value) for
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ELECTROKINETIC PHENOMENA
charged pores F is greatest for cations with the lowest diffusivity (or mobility) whereas the sequence of F is reversed for pores with a positive zeta potential. These results agree with those obtained by Bowen and Cao (6). 4. CONCLUSION
FIG. 15. Dependence of F on positive zeta potential for various electrolytes; a/ k 21 5 10; Diffusivities as in Fig. 12.
counterions with the highest mobility (or diffusivity). Indeed, the backflow of counterions induced by the electric field (electro-osmotic effect) is more significant for ions with a higher mobility. Consequently, the end of the pore is less polarized and the streaming potential is therefore smaller. For positive zeta potentials, the discrepancies between the electrolytes still exist but are less significant because the counterions (Cl 2) are the same for all electrolytes. It can be seen that the streaming potential is lowest for cations with the highest mobility. The space charge model uses the Boltzmann equation to describe the radial distribution of the ion concentrations. Thus, it considers coions to be present in the diffuse layer even if their concentration is much lower than that of the counterions. The electro-osmotic effect occurring during the streaming potential process produces a backflow of counterions but it induces a flow of coions in the streaming direction. This flow of coions acts to reduce the polarization between the pore ends. The higher the coion mobility, the less the pore ends are polarized. It can be noted that the streaming potential tends to reach the same value when the zeta potential becomes sufficiently high, whatever the electrolyte. Indeed, at high zeta potential, the concentration of coions becomes negligible with respect to that of counterions which leads to a quite identical streaming potential for all electrolytes. Figures 14 and 15 show the corrective factor F that must be applied to the Helmholtz–Smoluchowski equation as a function of z for various electrolytes in the same conditions of concentration and electrokinetic radius as in Fig. 13. For negatively
The electrokinetic phenomena occurring in homogeneous cylindrical pores containing symmetric electrolytes have been studied in the framework of the space charge model. The analysis includes a numerical solution of the nonlinear Poisson–Boltzmann equation and accounts for the individual diffusivities of the cations and anions in the electrolyte used for measurements. The zones of predominance of primary and secondary electroviscous effects are located by studying the variation of the electroviscous effects as a function of the streaming potential. It is also shown that the determination of the zeta potential at a given pH from experimental streaming potential measurements requires an additional study of the pH dependence of the streaming potential. Streaming potentials determined numerically are compared with results given by approximated relations in a range of zeta potentials and electrokinetic radii that may be commonly encountered in experimental investigations. It is shown that the approximated relations can give rise to very misleading conclusions and that the determination of the true value of the zeta potential from experimental measurements requires numerical calculations allowing for the ionic diffusivities to be individually specified. 5. APPENDIX: NOMENCLATURE
a C ci Di Du F Gs I ji Ki K ij P q qp r R SP T u x zi z
Pore radius (m) Bulk electrolyte concentration (mol.m 23) Ion concentration in the pore (mol.m 23) Ion diffusivity (m 2.s 21) (5RTK i) Dukhin number Faraday constant (596485 C.mol 21) Surface conductance (V 21) Electrical current (A) Ionic flux density (mol.m 22.s 21) Ion mobility (m.s 21.N 21.mol) Coupling coefficients Hydrostatic pressure (N.m 22) Solvent flow (m 3.s 21) Poiseuille’s flow (m 3.s 21) Radial coordinate (m) Universal gas constant (8.31 J.mol 21.K 21) Streaming potential (V.N 21.m 2) Temperature (K) Fluid velocity (m.s 21) Axial coordinate (m) Charge number of the ionic species i Absolute value of z i
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SZYMCZYK ET AL.
Greek Letters
e0 er k 21 l0 l pore m p P f w C z j
Vacuum permittivity (8.854.10 212 F.m 21) Relative dielectric constant of the solvent (79.8) Debye length (m) Conductivity of bulk electrolyte (V 21.m 21) Conductivity of the electrolyte in the pore (V 21.m 21) Viscosity of the electrolyte (0.001 kg.m 21.s 21) Standard dimensionless constant (3.1415927) Osmotic pressure (N.m 22) Total electrical potential (V) Axial electrical potential (V) Electrostatic potential (V) Zeta potential (V) Normalised zeta potential ( j 5 zF z /RT) REFERENCES
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