E LS E V I E R Journal of Membrane Science 113 (1996) 191-204
Electrokinetic phenomena in microporous membranes with a fixed transverse charge distribution
V. Aguilella *, M. Aguilella-Arzo, P. Ramfrez Dep. Ciencias Experimentales, Universidad Jaume 1. Ap. 224, 12080 Castell6n, Spain
Accepted 6 February 1995
Abstract
The charged capillary model for microporous membranes has been extended to account for a distribution of fixed charge transversal to the pore wall. The non-linearized Poisson-Boltzmann equation has been solved numerically and results obtained for an exponential charge distribution function have been analyzed. The expressions used for computing the streaming potential and the pore conductivity arise from Poisson-Boltzmann equation together with Nernst-Planck and Navier-Stokes equations. The effect of charge displacement on streaming potential and wall potential is significant, especially for high bulk electrolyte concentration, unlike Pore Conductivity which slightly changes with respect to the case of charge smeared at the pore wall. The model is expected to be useful for microporous membranes whose pore dimensions are very small, and also for some membrane- like biological systems. In particular, this transverse-charge model could be used in the study of electrokinetic phenomena through human Stratum Corneum, provided fixed charge density and "pore" dimensions are known.
Ionic-transport through microporous membranes has been often described in terms of the so-called space charge (SC) model, which assumes that the membrane (or any membrane-like system) is composed of an array of charged cylindrical capillaries (see [ 1 ] and references therein). In addition, the classical version of the SC model includes the Gouy-Chapman theory lbr the electrical double layer, and the assumption that charge is spread uniformly on the hydrodynamic sur- face, i.e. the pore wall. Despite its apparent simplicity, this model has been applied successfully to a wide variety of porous media, whose resemblance to a micro- porous membrane would be at first sight rather doubt-
ful, though most of the systems described in terms of the SC model, have more or less well defined pores [2,3]. The theory of electrokinetic effects in porous membranes appears now to be well established [2 ,4- 9] in terms of the classical SC model and the coupling between charge and mass transport seems adequately explained, not only qualitatively but quite often also quantitatively. However, two weak points are often recognized in this theory, namely the assumptions that charge is: (a) uniformly spread; and (b) distributed along the pore surface. Concerning the former, the effect of the discreteness of charge is still under debate, though it seems to be only relevant when charge spac- ing is much greater than the characteristic Debye length of the problem [ 10]. We restrict this preliminary anal- ysis to systems in which this condition is not met, and avoid taking into consideration other lhctors which can
192 ~L Aguilella et al. / Journal of Membrane Science 113 (1996) 191-204
modify the validity of the continuum hypothesis: the charge number of lipids (for the case of a lipid bilayer), the type of ions in solution, the adsorption of ions to membranes, etc. Our aim in this paper is relaxing the second assumption and discussing the consequences of a transverse distribution of fixed charges (i.e. perpen- dicular to the membrane-solution interface) on elec- trokinetic phenomena within the context of the above mentioned SC model. Moreover, we intend to apply this model, after some changes, to a particular biolog- ical system which can be regarded at first approxima- tion as a microporous ion selective membrane.
Cevc et al. [ 11,12] have studied the effect on the electrostatic potential of the smearing of the structural charge perpendicular to the surface of lipid bilayer membranes. In addition, they have provided some argu- ments which support this picture for some biological charged surfaces: thermal motion of the charge carrying groups, molecular stretchings and rotations, and other related phenomena, cause the fixed charges to lie out of the plane determined by the membrane-solution interface, and to extend over a small part of the solution. Their analysis was made for a planar geometry and described a static problem.
Tsui and co-workers [ 13] assumed also a distribu- tion of charge on photoreceptor disc membranes to interpret the surface potentials they measured, and sug- gested that protein charges could be distributed over some distance normal to the membrane surface.
Other similar approaches, always dealing with a pla- nar geometry, have been reported by Donath and Pas- tushenko [ 14], Wunderlich [ 15], Levine [ 16] and Ohshima and Ohki [ 17] among others. A thorough study by Vorotyntsev and co-workers [ 18] deserves special mention. With some differences, a common feature is allowing the fixed charge region to be per- meable to electrolyte ions. While only indirect infor- mation about this charge displacement is usually available [ I 1 ], its existence appears to be a very rea- sonable hypothesis. In some biological systems, such as human erythrocytes, there is experimental evidence that the dissociable chemical groups which give rise to the cell surface charge extend out of this surface [ 16].
We have recently reported [ 19 ] some measurements of streaming potential and fixed charge density on human Stratum Corneum (the uppermost skin layer, which exerts the main control over passive transport) in vitro, together with a simple model for ionic transport
through this layer, based on the so-called "brick and mortar" picture for Stratum Corneum [20]: Keratin- ocytes represent the bricks and the lipid mixture between them constitute the mortar. This skin layer exhibits ion selective properties [21] and specifically a cation-exchange behavior at physiological electrolyte concentration. The structure of Stratum Corneum has been extensively studied, particularly in what concerns passive transport through it, due to the wide scope of therapeutic and cosmetic applications which demand a detailed knowledge of its permeability to neutral and charged substances and a better use of a relatively novel technique known as iontophoresis (basically, the appli- cation of an electric field to enhance the transport of charged solutes). Some similarities between ionto- phoretic transport through skin and electrically driven transport across a porous charged membrane have been recognized [ 22,23]. Although there is not yet a com- plete picture of the Stratum Corneum intercellular domain, some studies (see [24] and references therein) agree on the existence of lipid bilayers within the intercellular space between corneocytes. This means that two phases can be distinguished, namely, the lipophilic phase through which non-polar material can penetrate, and the hydrophilic or aqueous phase in contact with the polar head groups of the lipid bilayers.
In a previous paper [ 19], the standard SC model was used to estimate the size of the aqueous pores which form the "intercellular route" through the lipid phase from streaming potential and charge density measure- ments. However, this attempt to characterizing Stratum Corneum structure from electrokinetic measurements still needs a lot of refinement and a more detailed micro- scopic model if one expects to get useful information about its structure. Lipid bilayers have been extensively studied for a long time and their electrical properties and basic structure have been characterized, no matter that their precise chemical composition which, even in the simplest system, is not known with certainty [ 25 ]. Since our study is focused on transport of inorganic ions in aqueous solutions, it is the polar region rather than the hydrocarbon core that matters. The charged groups can be more or less extended out of the hydro- carbon phase (reported estimates of the polar region are, e.g. 5 .& for compact head groups and 14 .& for a fully extended configuration [25]). Consequently, charge is likely to be distributed perpendicularly to the bilayer surface.
V. Aguilella et al. /Journal of Membrane Science 113 (1996) 191-204 193
The nature of these idealized pores in the lipid phase provides some basis to extend the SC model and con- sider a fixed charge distribution along the radial direc- tion for a cylindrical geometry (no assumption is made at this stage concerning the lamellar or non-lamellar structure of the bilayers). Our interest lies in analyzing the influence of the mean displacement of fixed charge on the predicted values for streaming potential, con- ductivity and electric potential at the pore "wal l " . Unfortunately, to our knowledge there is no evidence in support of a specific type of distribution, therefore we have used an exponential one and taken the average fixed charge displacement d as the distance at which charge density decreases by a factor 1/e from its value in the pore wall.
As is known, the exact Poisson-Boltzmann (P-B) equation cannot be solved in closed form for a cylin- drical geometry. Usual approximations consider either the assumption that the electric potential is very small (this allows tbr linearization of P-B equation, as in Debye-Htickel 's theory) or the assumption that the electric potential is sufficiently high (this leads to total co-ion exclusion in the pore). When the electric double layer characteristic dimensions are similar to pore radius, we have to resort to a numerical solution of P- B equation if we want to obtain reliable results. In our case, pore dimensions lie around 15-20 nm and the Debye length for 1 mM solution is ca. 10 nm, so that an analytical solution cannot be safely used. This is also true of other biomembranes. We use a simple numerical procedure to solve the P-B equation, which uses the powerful capabilities of Mathematica®.
2. Pore model
Fig. 1. Sketch of the pore model with charged groups extended out of the pore surface.
l >> a, so that end effects can be neglected; and (b) all pores are equal in size, so that transport across a single pore can represent transport through the entire membrane (however, generalization for a distribution of pore sizes is possible [ 19] ).
The theoretical foundations of the standard charged capillary model can be found elsewhere [2,8,9] and are based on: (a) Gouy-Chapman 's theory for the elec- tric double layer at the pore surface (which combines Poisson's equation with Boltzmann's equation); (b) Nernst-Planck equation for ion fluxes; and (c) Navier- Stokes' equation (low Reynolds number) for describing the solution flow through the pore. For a 1 : 1 electrolyte, these equations - in cylindrical coordinates - read as follows:
Let us consider a membrane-like porous media made of an array of identical cylindrical pores of length 1, and radius a, so that our study can be focused on one of them. We assume that negatively charged groups are extended over some average distance d, and both sol- vent (water) and small mobile ions are able to penetrate the structural charge region (see Fig. 1 ). r is the radial coordinate having its origin in the pore axis. Pores are filled with an aqueous electrolyte of dielectric constant e and viscosity /x, and the total charge density at a distance r fi-om the center of the pore is denoted by p(r). Other usual assumptions are: (a) considering
194 V. Aguilella et al. / Journal of Membrane Science 113 (1996) 191-204
(a)
DELTA (nm)
0.8
0.4 0.2
-50
STR. POT. (nV/Pa)
-100
100
80
20
60
40
CONE (raM)
Fig. 2. Numerical results for the streaming potential as a function of: (a) bulk electrolyte concentration and average charge displacement (a = 17 nm and o-~ = 50 mC/m 2 are assigned fixed values); (b) equivalent surface charge density and average charge displacement (a = 17 nm and ~= 100 mM are assigned fixed values).
where ~ ( r ) denotes the electric potential due to pore charge and q~(r,x) = ~ ( r ) + V(x) is the total electric potential, including the contribution V(x) from the externally applied electric field, p r ( r ) is the space charge density; a;~, D i, zi and ci denote the flux, diffusion
coefficient, charge number and concentration, respec- tively, of the ionic species i; ~ is the electrolyte bulk concentration; tT(r) is the solvent velocity in a
membrane-fixed frame of reference, p(r,x) is the hydrodynamic pressure, F is Faraday ' s constant, T the temperature and R the universal gas constant. Boundary
conditions for the P-B equation arise from cylindrical symmetry:
--~-- = o = 0 (4)
and from Gauss ' theorem in the vicinity of the pore wall
dq~dl~ a= t = F~r_.__._aa=RT~ So (5)
V. A guile lla et al. / Journal of Membrane Science 113 (1996) 191-204 195
-11
STR. POT. (nV/Pa)
DELTA(nm)
where ty is the surface charge density at the pore wall and So the corresponding dimensionless magnitude. ck = F ~ b / R T and R = r / a are the dimensionless electric potential and radial coordinate, respectively.
Now, if we assume a structural charge density pf(r) out of the pore wall instead of a surface charge density o-, P-B equation, in dimensionless form, becomes for a symmetric electrolyte (z+ = - z_ = z) :
1
2 F 2 (z2 a 2 F a 2 _ - sinh(z~b(/~) ) - -----~--,of(e) (6)
R T e e_Kl "
There is no experimental evidence supporting a specific functional form for pf(r). On the one hand, a constant pf seems unrealistic and, on the other hand, the number of charged groups near the surface is likely to be much greater than the same density at a certain distance. Fol- lowing the suggestion of Cevc et al., we have chosen
tRGE DENS. (mC/m2)
A v v
0
an exponential distribution of fixed charge density. The same authors found negligible differences when choos- ing a gaussian distribution, provided the mean charge displacement t5 was the same [ 11 ], and our preliminary calculations showed no appreciable differences when using a Gaussian or a linear distribution. In most cases a >> d, so that we have taken:
r - a p r ( r ) = p ° exp(---ff--) (7)
Once we know, e.g. from measurements, the mean vol- ume charge density in the pore tim or the equivalent surface charge density in the pore wall o's, the electro- neutrality condition allows the determination of the parameter pO:
f p f d v = f p m d o = f o ~ s d S (8) Vcyl /J2yl S
Upon integration, we obtain
196
20
V. Aguilella et al. / Journal of Membrane Science 113 (1996) 191-204
100
CONC (mM)
6 ~ ~
40
o I
-50
STR. POT. (nV/Pa)
-100
-150
-2oo I
-100
-80
-60
CHARGE DENS. (mC/m2) -40
-20
Fig. 3. Numerical results for the streaming potential as a function of bulk electrolyte concentration and equivalent surface charge density in the case of no charge displacement (a = 17 nm and 6= 0). (Note that the local maximum at low o-~ is due to the interpolating function and the small number of points used in the plot.)
a c t s
P " - 82 (9) a 6 - 6 2 + e,i),s
By introducing the Debye length, defined as
_ / RTe A-V~ (lo)
the Poisson-Boltzmann equation becomes
l d 1 (a12 ~ - ~ \ ~ ] = ~ - ~ ] sinh(z~b(/~))
Fa3o's (.(/~ 1)a) eRT(82+ea/a(aS_82) ) exp ~ (11)
with the boundary conditions:
d~b a=o = 0 =_~.R~ (12) dR a= 1
V. Aguilella et al. / Journal of Membrane Science 113 (1996) 191-204 197
Table 1 Comparison of numerical results for streaming potential (u), pore conductivity (K) and wall electric potential (gt(a) ) at several bulk electrolyte concentrations (6). a = 17 nm and o-, = 50 mC/m'-
~: (mM) 6 (nm) v (V/Pa) K (,(2-~ a/'(a) (mY) m - l )
in terms of 4,(/~), a, & o- s and 8 are obtained in the form of definite integrals which need to be solved
numerical ly . P denotes the pressure di f ference be tween
both sides o f the membrane.
3 . R e s u l t s
Eq. ( 1 1) relates the electrostat ic part o f the electric
potential in the pore with the structural charge para-
meters (equ iva len t surface charge densi ty and average
charge d i sp lacement ) . Its integration would al low
obtaining 4,(15,) p rovided we know o- s and 6. However ,
this solution must be obtained numerical ly , since the
equat ions cannot be so lved in c losed form (even within
the limits o f D e b y e - H t i c k e l l inear izat ion) . In addition,
we are interested in finding express ions for measurable
e lectrokinet ic parameters which may serve as a test for
the model . F rom Eqs. (2 ) , (3) and (11) , one can
obtain express ions for some macroscopic magni tudes
defined as fol lows:
2 F a [=-zf (z+J+x(r) +Z J-x(r))rdr 0
(e lectr ic current densi ty) (13)
a
]=- -~ f (J+x(r) + J_x(r ) )rdr 0
(average e lec t ro ly te f lux) (14)
(t
: vx r rdr 0
(average f low ve loc i ty ) (15)
and, after some algebra, express ions for the s t reaming
potential
Numer ica l solution o f the differential Eq. ( 11 ) and
further computa t ion of r a n d K was carried out by means
o f Mathemat ica® (a software package which a l lows
for both numerical and symbol ic computa t ion) . 3D
Graphic outputs were also g iven by the same program.
A final accuracy of 10 -6 was imposed to the i terative
procedure.
The main structural characterist ics o f the system
under study have been descr ibed above. Mos t para-
meters involved in the calculat ions have been assigned
typical values, since we are not interested, at this first
stage, in precise quanti tat ive results. Thus, e lec t ro lyte
bulk concentrat ion varies be tween 1 and 100 m M (i.e.
the Debye length varies f rom about 10 to 1 n m ) , equiv-
alent surface charge (nega t ive ) ranges from 10 -3 C /
m 2 up to 0.1 C / m 2, and average charge d isp lacements
are considered up to 1 nm. However , the equiva len t
Table 2 Comparison of numerical results for streaming potential (v), pole conductivity (K) and wall electric potential (qt(a) ) at several equiv- alent surface charge densities (o-~). a = 17 nm and £:= 100 mM
o-~ (mC/m 2) 6 (nm) ~, (V/Pa) K (,t"2 I qr(a) (mY) m - I )
198 V. Aguilella et al. / Journal of Membrane Science 113 (1996) 191-204
-0.5
-0.75
-1 COND (1/Ohm m)
-1.25
-1.5 f (a)
60 '~
CONC (mM)
0.4 ~ 0 " 6
DELTA (nm)
Fig. 4. Numerical results for pore conductivity as a function of: (a) bulk electrolyte concentration and average charge displacement (a = 17 nm and o-~ = 50 mC/m 2 are assigned fixed values) ; (b) equivalent surface charge density and average charge displacement (a = 17 nm and F= 100 mM are assigned fixed values).
pore radius a was taken as 17 nm, according to a pre- vious rough estimation [19], just to compare these results with those obtained for 6 = 0 in Stratum Cor-
neum. We chose KC1 for our calculations, and the ionic diffusion coefficients were taken as constant within the
pore and assigned their infinite dilution values. In addi- tion, the remaining parameters were: ~ = 78,/z = 10-3 kg/m s and R T / F = 2 6 . 7 1 mV.
3.1. Streaming potential
The well known dependence of streaming potential, v, on bulk electrolyte concentration, F, and equivalent
surface charge density, cr S, remains qualitatively the
same whether 6 = 0 or 6 ¢ 0 (see Figs. 2a and 2b): the
higher ~, the lower v (hereafter we refer to its absolute
value), since the Debye length decreases and the mass
transfer-charge coupling becomes weaker; on the other
hand, as shown in Fig. 3, the influence of cr~ on v is different at " l ow " and "h igh" bulk concentration: at
"h igh" & v increases with O's, while at " l ow " F(when
A becomes comparable with the pore radius a and co-
ions - anions in our case - are mostly excluded from
the pore) the trend is just the contrary and v decreases
when o's increases (in fact, when Fis small, the counter
ion concentration near the pore wall is governed by o-~
V. Aguilella et al. / Journal of Membrane Science 113 (1996) 191-204 199
1.4
-l.
COND (1/Ohm m)
- through the electroneutrality condition - so that such concentration is much greater than 6).
Fig. 2a shows the most interesting prediction for streaming potential: a significant dependence on the average charge displacement. For fixed values of 6and o-s, u decreases as 6 gets higher. For instance, for a typical case in which 6 = 100 mM, o'~ = 1 m C / m 2 and 6 = 1 nm, the predicted value for v changes by nearly 50% with respect to the calculated value for 6 = 0. The relative change of u with 8 becomes greater as 6 increases (see Table 1 ). On the other hand, that relative change becomes smaller as o'2 increases (see Table 2).
3.2. ConductiviO~
Figs. 4a and 4b show a slight dependence of con- ductivity on the average charge displacement, although for large surface charge densities it cannot be neglected. From Tables 1 and 2, it follows that differences between K ( 6 = 0 ) and K ( 6 = lnm) become up to 14%.
3E DENS. (mC/m2)
0 -,vv
3.3. Pore wall electric potential ( ~ (a ) )
The maximum electric potential, i.e. the electric potential at the pore "wa l l " in the classical model, is not a directly measurable parameter but, particularly for biological membranes, it influences their properties [ 11 ] and it is worth considering how it changes with 6. As expected, ~ ( a ) changes significantly with bulk concentration and surface charge density (see Fig. 5) specially in the region of low 6 and low o-~. The effect of charge displacement on ~ ( a ) is decreasing its abso- lute value, as appears in Figs. 6a and 6b. Tables 1 and 2 show that the differences between ~ ( a ) (6 = 0) and q~(a) ( 6 = 1 nm) range between 15% (for ~ = 1 mM) and 45% (for 6 = 100 mM).
4 . D i s c u s s i o n
The above results suggest that the extension of ion- izable groups out of the membrane surface gives rise
200 V. Aguilella et al. / Journal of Membrane Science 113 (1996) 191-204
100 80
CONC (raM)
60 ,....._ 40
20
-50
WALL POT. (mV) -100
-150
-20
-100
-80
-60
CHARGE DENS. (mC/m2)
Fig. 5. Numerical results for the wall potential as a function of bulk electrolyte concentration and equivalent surface charge density in the case of no charge displacement ( a = 17 nm and 8=0) .
to important changes in some electrokinetic parame- ters, and it should not be overlooked when pore dimen- sions are sufficiently small or, more specifically, when the ratio a/8 is not large. In our case, a/8> 17. The fact that ions in solution are able to penetrate into the fixed charge region explains the screening of the elec- tric field in this zone and the decrease in the electric potential at the "pore wall" predicted by the theory both for a planar geometry [ 11 ] and for the present case of cylindrical geometry. This leads to remarkable changes in the electric potential profile near the pore wall and, consequently, in the diffuse part of the double
layer. These changes are reflected in the Streaming Potential because space charge density becomes lower and so does the electric potential build-up due to the applied pressure. The influence of charge displacement on the Streaming Potential and Wall Potential is very sensitive to bulk concentration: high differences are predicted for bulk concentration in the physiological range (~3= 100 mM). However, the equivalent surface charge density does not contribute significantly to enhance or weaken this transverse charge effect. Just the opposite happens to pore conductivity: the effect of 8=# 0 is mainly modulated by surface charge density
V. A guilella et al. / Journal r¢ Membrane Science 113 (1996) 191-204 201
DELTA (nm)
0.8 0.6
0.4 0.2
(a) -50
-100 WALL POT. (mV)
-150
100
80
20
60
40
CONC (mM)
Fig. 6. Numerical results for the wall potential as a function of: (a) bulk electrolyte concentration and average charge displacement (a = 17 nm and tr~ = 50 mC/m 2 are assigned fixed values) ; (b) equivalent surface charge density and average charge displacement (a = 17 nm and ~= 100 mM are assigned fixed values). (Note that the apparent local minimum in Fig. 6(a) is due to the interpolating function and the small number of points used in the plot,)
rather than ionic concentrat ion. Thus, for o-~ ~ 1 m C /
m 2, fixed charge d isp lacement is not re levant at all.
This analysis shows that depending on the electro-
kinetic parameter we intend to measure, the particular
membrane system chosen and other exper imental con-
ditions, the effect of charge d isp lacement should be taken into account i f we want to relate the measured
magni tudes (e.g. v or K) with microscopic quantit ies (o-~, a, e tc . ) . A quest ion arises now of whether it is
possible or not to measure 6 indirectly f rom electroki-
netic measurements . It would require an accurate
exper imental determinat ion of a and o-~, apart f rom
measur ing u. a can be obtained f rom hydraul ic per-
meabil i ty measurements and o-~ from e lec t romot ive
force ( E M F - m e t h o d ) measurements (Despi te this, o'~
is frequently regarded as an adjustable parameter ) . So,
provided other effects as specific adsorption, variat ion
of e, etc., can be neglected, 6 could be es t imated by
indirect methods. W e fo l lowed the inverse path and
checked our earl ier est imations o f aqueous pore size a
202 V. Aguilella et al. / Journal of Membrane Science 113 (1996) 191-204
(b)
0.8 DELTA (nm)
0.6 0.4
0.2 0
I0
-20
-4O WALL POT. (mY)
-60
-80
-100
-80
-40
CHARGE DENS. (mC/m2)
-20
Fig. 6 (continued).
in Stra tum C o r n e u m [19] by using the approximate value of 6 = 0.25 nm reported in the literature for lipid bilayers [ 11 ]. Significant differences were found with respect to the case of 6 = 0 in the calculated values for a which ranged from 10 to 50% (depending on bulk concentration and consequently on surface charge den- sity). In addition, the dispersion of those values was much smaller than before. The new mean value for a was ca. 13 rim. Increasing 6 did not yield more reliable results for a since their dispersion was greater.
Since it is the ratio 8 / a that determines to a great extent the influence of the transversely distributed fixed
charge on some electrokinetic parameters, the same predictions would be obtained by assuming either that ionizable groups protrude from the pore wall or that fixed charges are "bu r i ed" in the membrane along a finite region in which mobile ions can penetrate. Although in view of the key factor 8 /a , the effect of a radial distribution of structural charge is expected to be relevant mainly in biological systems, it could also be taken into account for some synthetic microporous membranes with nominal pore radii about 10 nm (e.g. Nuclepore® N0015 or similar track-etched membranes having very narrow pores). Further experimental work
V. Aguilella et al. / Journal of Membrane Science 113 (1996) 191-204 203
is being carried out in this field to confirm this predic- tion in synthetic membranes.
5. List o f symbols
6 Di
E
e~ F ¢ @(r,x) J, K
A l
1]
u
p(r,x) P
R r
p ( r )
par)
Pill
,O o
O"
O'~
So
T v(r)
pore radius (m) electrolyte bulk concentration (mol m-3) concentration of the ionic species i (mol m - 3 )
average fixed charge displacement (m) diffusion coefficient of the ionic species i (m 2 s - l )
dielectric permittivity (C V- t m - ~) dielectric constant Faraday's constant (96487 C equiv -1) dimensionless electric potential (05 = qt/RT) total electric potential (V) flux of the ionic species i (mol m -2 s-~) conductivity (S2 l m- 1) Debye's length (m) length of cylindrical pores (m) viscosity (kg m -j s -~) streaming potential (V Pa - j ) volume (m 3) hydrodynamic pressure (Pa) pressure difference between both sides of the membrane (Pa) dimensionless radial coordinate universal gas constant (8.31 J moi-l K - ~) radial coordinate (m) total charge density at a distance r from the center of the pore (C m-3) structural charge density out of the pore wall (C m 3) mean volume charge density in the pore (C m 3)
parameter for the structural charge density ( C m - 3 )
surface charge density at the pore wall (C m 2)
equivalent surface charge density in the pore wall (C m -2) dimensionless surface charge density at the pore wall temperature (K) solvent velocity in a membrane- fixed frame of reference (m s - l )
V(x)
x
q~(r) Zi £
contribution to the electric potential from the externally applied electric field (V) axial coordinate (m) electric potential due to pore charge (V) Charge number of the ionic species i absolute value of zi
Acknowledgement s
Financial support from DGICYT (project no. PB92- 0516) and Universidad Jaume I-Fundaci6 Caixa Cas- tell6 (B-04-CE) is gratefully acknowledged.
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