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A network microcapillary model for electrokinetic phenomena through microporous membranes
Colloids and Surfaces A: Physicochemical and Engineering Aspects 145 (1998) 11–24
A network microcapillary model for electrokinetic phenomena through microporous membranes A. Herna´ndez *, R. Lo´pez, J.I. Calvo, P. Pra´danos Dpto. do Termodina´mica y Fı´sica Aplicada, Facultad de Ciencias, Universidad de Valladolid, 47071 Valladolid, Spain Received 7 July 1997; accepted 16 July 1998
Abstract The streaming potential and the electro-osmotic pressure are calculated through a simulation procedure, for a membrane consisting of a sample of equally charged pores, but are arranged differently to approach the real structure of actual membranes. It is presumed that a microporous membrane consists in an ensemble of cylindrical microcapillaries with the same pore charge density. The pores can have two sizes, which can be interconnected in three main ways: (1) parallel, (2) series, and (3) square network. On the other hand, it is assumed that these two pore sizes are, in fact, the central values of two Gaussian distributions, of equal relative standard deviations. In particular, here, mean diameters 0.2 or 0.1 and 0.02 or 0.01 mm, with standard deviations of 0%, 15% and 25%, in parallel, series and network structures, are considered. This is carried out for several charge densities and concentrations of LiCl aqueous solutions. In these conditions, both the studied electro-kinetic parameters are determined by the widest pore in parallel, and its narrowest section. Interconnections of pores are seen to play a significant role, only when narrow and wide pores are almost equally present. Increasing standard deviations affect differently, but only slightly, the overall result for the electro-kinetic parameters, depending on the assumed structure. All these results indicate that high concentration results of streaming potential should be preferred, to obtain the membrane charge versus the concentration profiles. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Electro-kinetic phenomena; Microcapillary model; Microporous membranes; Network structure; Porous materials
1.
Nomenclature
a: a: (1)
mean hydrodynamic radius (m) mean hydrodynamic radius for the small pores (m) a: (2) mean hydrodynamic radius for the large pores (m) a: (i, j, k) mean hydrodynamic radius for the pore P(i, j, k) (m) * Corresponding author. Tel: +34 834 23134; Fax: +34 834 23136; e-mail: [email protected]
a(i, j, k) hydrodynamic radius for the pore P(i, j, k) (m) c bulk concentration (mol/m3) C combination of coefficients given by Eq. (25) (dimensionless) C combination of coefficients given by ab Eqs. (21)–(24) (dimensionless) c concentration of the ith ion, i=+, − i (mol/m3) D ionic diffusivity, i=+, − (m2/s) i F Faraday number (C/mol ) i(i) electric current for the pore i (A)
0927-7757/98/$ – see front matter © 1998 Elsevier Science B.V. All rights reserved. PII S0 9 2 7- 7 7 5 7 ( 9 8 ) 0 06 6 9 - 4
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i (i) p i (i) s i(i, j, k) I :i I9 j (r) i j (i) V j (i) Vp j (i) Vs
A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
electric current for the pair of parallel pores i (A) electric current for the composite series pore i (A) electric current for a pore P(i, j, k) (A) electric current per pore (A) electric current through a unit surface of pores (A/m2) electric current through a unit surface of the membrane (A/m2) ionic flux in the pore at a distance r from the pore centre (mol/m2 s) volume flow for the pore i(m3/s) volume flow for a pair of parallel pores (m3/s) volume flow for a pair of pores arranged in series (m3/s) volume flow for a pore P(i, j, k) (m3/s)
j (i, j, k) V J volume flow per pore (m3/s) V :j volume flow per surface unit of the pores V (m/s) J9 volume flow per surface unit of the memV brane (m/s) :l one of the four phenomenological ab coefficients of the pores (A/V m2, m/Pa s, or m/V s=A/Pa m2) L9 one of the four phenomenological ab coefficients of the membrane (A/V m2, m/Pa s, or m/V s=A/Pa m2) l(i) length of the pore i (m) l (i) one of the four phenomenological ab coefficients for a pore i (A m/V, m4/Pa s, or m4V s=A m/Pa) one of the four phenomenological l (i, j, k) coefficients for a pore P(i, j, k) ab (A m/V, m4/Pa s, or m4/V s=A m/Pa) L one of the four phenomenological ab coefficients per pore (A m/V, m4/Pa s, or m4/V s=A m/Pa) n number of connections between both sides of the membrane (dimensionless) N(i, j) node i, j (dimensionless) p pressure at the interface between pore 1 C and 2 in the series model (Pa) p pressure at the interface between the i membrane and the solution i (Pa)
radial coordinate from the centre of a mean pore (m) R gas constant (J/mol K ) T absolute temperature ( K ) v fluid speed in a mean pore (m/s) x structural characteristic of the network, defined in the text (dimensionless) y longitudinal coordinate along the pore (m) z charge number of the ith ion, i=+, − i (dimensionless) Dp pressure difference across the membrane (Pa) Dp(i) pressure difference for the pore i (Pa) Dp(i, j, k) pressure difference for the pore P(i, j, k) (Pa) Dy membrane thickness (m) Dw electric potential difference across the membrane ( V ) Dw(i) electric potential difference across the pore i ( V ) Dw(i, j, k) electric potential difference across the pore P(i, j, k) ( V ) e permittivity (F/m) f zeta potential, w(a: ) ( V ) g solution viscosity ( Kg/m s) h azimuth angle in the pore (rad) H membrane porosity (dimensionless) l Debye’s length (m) n stoichiometric coefficient, i=+,− i (dimensionless) n streaming potential ( V/Pa) p m electro-osmotic pressure (Pa/V ) p s surface charge density on the shear surface (C/m2) s standard deviation of the pore size distris butions (dimensionless) Q total electric potential somewhere in the pore ( V ) w dynamic electric potential ( V ) w(i, j) dynamic electric potential at the node N(i, j) ( V ) w dynamic electric potential at the interface i between the membrane and the solution i (V) w dynamic electric potential at the interface C between the two pores in the series model (V) r
A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
y
normalized radial portion of the electric potential (dimensionless)
2. Introduction It has been recognized on empirical grounds, that in a system characterized by a single, small force and a single flow, the flow is a linear function of the conjugated driving force. In general, however, when two or more flows are taking place simultaneously, there may be coupling between non-conjugated flows and forces. For a membrane that separates two electrolyte solutions, the gradients of pressure and electric potential play the role of driving forces which are coupled, leading to the conjugated fluxes; i.e. the volume flow and the electric current [1]. This occurs in such a way, that forces and fluxes are related through: J9 =L9 Dp+L 9 Dw V 11 12 , (1) I9=L9 Dp+L9 Dw 21 22 where J9 and I9 are, respectively, the volume and V the electric charge passing through the membrane per unit time and membrane area, and the L9 are ab the phenomenological coefficients. The electro-kinetic phenomena that arise in these conditions lead to parameters that, in any case, can be written as a combination of the phenomenological coefficients. For example, the streaming potential is:
H
A B
Dw L9 =− 12 , n = p Dp 9I=0 L9 22
(2)
while for the electro-osmotic pressure we have:
A B
Dp L9 m = =− 12 . (3) p Dw J9 =0 L9 11 V In order to evaluate an electro-kinetic coefficient, a microscopic model for the solution movement and its interactions with the membrane surfaces, is required. Thirty years ago, a simple, but extraordinarily fruitful, space–charge model, was originally raised by Dresner [2], and developed in the
13
following years by Osterle and co-workers [3–5], and Westerman-Clark and co-workers [6,7]. This model allows one to obtain the phenomenological coefficients for a membrane, by assuming that it consists of a bunch of parallel cylindrical and equal microcapillary tubes, with the same surface charge density uniformly distributed on their surfaces. The question of how the phenomenological coefficients of a composite membrane change, when it can be thought of as consistent in two or more simple membranes, has been addressed in Refs [8–10]. The authors have arrived at equations relating the phenomenological coefficients for the composite membranes, with those for the constituent membrane elements, when they are placed both in parallel and series. Of course, real membranes include pores that are size-distributed, attending to their sections at any plane parallel to the membrane surfaces, but as well as along any direction perpendicular to the external surfaces, and with interconnected pores. A further step to approach a real membrane has been faced by Jin [11,12], by considering that we have a complex membrane that can be imagined as a square network of capillary tubes of two sizes, and solving each of them by a modification of the space–charge model. The authors have obtained some results for the streaming potential in terms of the zeta potential (f) for the pore walls. Our aim here is to study the predictions of this model for streaming potential and electro-osmotic pressure, in terms of the surface charge density and the solute concentration. These parameters have been chosen because concentration is directly controllable, while the surface charge density should be a constant for a given membrane material–solution interface. This simulation is not expected to fit exactly any really complex microporous membrane or porous plug [13,14], but it should be better suited than the pure one-size microcapillary representation. Actually, the series and parallel models should work relatively well for reasonably regular microporous membranes, such as track etched and inorganic, anodically formed ones. In any case, such a study will provide some insights on where and how the simpler one-size model can or cannot be
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A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
applied to real, more complex, microporous membranes [15,16 ]. These apects are relevant for measurements of the electro-kinetic phenomena; the surface charge density can be obtained once the pore sizes are known. This allows one to predict solute–membrane interactions in actual filtration conditions.
3. Microcapillary model The solid–liquid interface corresponding to the solution in contact with the pore walls can be described (see, for example, Ref. [17]), in terms of a charged surface in contact with a diffuse region of mobile co-ions and counter-ions, whose distribution is given by the Poisson–Boltzmann equation. This is known as the Gouy–Chapman picture, which assumes the ions as point charges. According to this model, time mobile solution is filling a pore of an effective radius a: and length l(l&a: ), which slides on a surface characterized by a uniform immobile surface charge density, given by s. The liquid phase is an incompressible Newtonian aqueous solution of an electrolyte whose uniform permittivity is e, with viscosity g, and ionic diffusivities D , i=+,−. i Pressure and electric gradients are assumed to be fully established along the axial direction of the pore, which produces a steady flow through the capillary. In order to benefit from cylindrical symmetry, r, y coordinates can be used, where r denotes the radial distance from the pore axis, and y the distance along it, while the azimuth angle h does not need to be considered as a result of symmetry. The electric potential, Q, inside the pore can be split into a static and a dynamic contribution, according to: RT
y(r)+w( y), (4) z F + where RTy/z F is the electric potential at the + solid–liquid interface, taking the bulk solution as a reference. This should be the only potential drop present at the interface, if no gradient acted through the membrane. While w is the exclusive
Q(r, y)=
longitudinal contribution caused by the pressure or electric potential gradients. The Navier–Stokes equation gives, as shown elsewhere — see for example Herna´ndez and co-workers [18–20] — the velocity profile v(r), at low Reynolds numbers, while those of Nernst–Planck give the ionic fluxes j of the ions i (i=+,−). On the other hand, the volume charge density r (r) and the ith ion molar concentration c c (r) come from the Poisson and Boltzmann equai tions, respectively. The fluxes and the corresponding phenomenological coefficients per unit area pore are related through: :j =l: Dp+l: Dw V 11 12 . (5) :i=l: Dp+l: Dw 22 21 The volume flow per surface unit through the pore results in:
H
2 :j = V a: 2 −
P
a:
v(r)rdr=
0 eRT
A
a: 2 Dp 8gl
2
P
a:
B
(6) y(r)rdr Dw, lgz F a: 2 0 + where RTf/z F is the electric potential on the + pore wall, i.e. f is the dimensionless zeta potential. In the same way, the current density through the pore can be obtained by integration of the Nernst–Planck equations: :i=
2F a: 2
−
P
a:
0 eRT
f−
[z j (r)+z j (r)]rdr= ++ −−
C
f−
2
P
a:
D C
y(r)dr Dp+
2F2c
P
a:
gz Fl a: 2 a: 2RTl 0 0 + ×{n z 2D e−y(r)+n z 2D e−z−y(r)/z+ }rdr + + + − − − 2cRT a: {n z e−y(r)+n z e−z−y(r)/z+ )} − + + − − a: 2z l 0 +
P
D
×{f−y(r)}rdr Dw.
(7)
Eqs. (5)–(7) allow one to evaluate all the phenomenological coefficients :l , for each pair a: /l (l ab being the Debye length [eRT/F2c (~∑~n z )2]1/2 i i
A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
15
and s, once the potential profile inside a pore, including y(a: )=f is known. This radial profile of the electric potential can be calculated by a numerical or variational integration of the Poisson– Boltzmann equation [20]. The bulk solution values, as given by [21], can be taken for g, e, D and D , if the fluid dynamics + − and electrostatics of the solution are not substantially modified by the presence of electrostatic interactions with the solid surfaces. This is certainly a very good approximation for the small electric fields, that can be expected to hold within the pores when containing low concentration solutions. Thus, every combination of phenomenological coefficients, :l , can be related with the surface ab charge density and the bulk concentration (through the Debye’s length), if the structural parameters (the effective mean or hydrodynamic pore radius and the pore length) are known.
4. Structural models
Fig. 1. Scheme of the parallel (a) and the series (b) models.
4.1. Equal pores If all the pores are assumed to be parallel, equal microcapillary tubes placed perpendicularly to the membrane walls, the membrane thickness is equal to the pore length, l=Dy, and the transport characteristics of the membrane are given by those of a single pore. The fluxes through the membrane, J9 and I9, are V related with the volume flow and electric intensity per surface unit for a pore, by: J9 =Hj: V V , I9=Hi:
H
(8)
respectively. This defines H, which is the membrane porosity, as the open surface fraction or the void volume fraction. However, L9 is linked to the mean pore radius 11 up to the shear surface, a: , (see Eqs. (5) and (6), by means of the Hagen–Poisseuille equation: Ha: 2
. L9 =Hl: = 11 11 8gDy
(9)
Hence, a: can be obtained from L9 once poros11 ity, H, and thickness, Dy, are previously measured. Then, Eqs. (5)–(8) can be used to obtain J9 , I9 V and, by comparing with Eq. (1), n and m through p p their definitions, Eqs. (2) and (3). 4.2. Pores in parallel We can assume that there are n(1) pores of radius a: (1) and n(2) with a: (2), as shown in Fig. 1(A). Consequently, the total volume and charge flowing per pore and unit time, can be obtained from the total volume and charge flowing per unit time through each pore of types 1 and 2 according to: J = V
n(1) j (1)+n(2) j (2) V V n(1)+n(2)
I=
n(1)i(1)+n(2)i(2) n(1)+n(2)
H
,
(10)
whose respective phenomenological coefficients
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A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
can be defined through: Dp Dw J =L +L V 11 l 12 l Dw
Dp
+L I=L 22 l 21 l
H
and unit time, are: Dw Dp +L J =L 12 l(1)+l(2) V 11 l(1)+l(2)
,
(11)
Dw
Dp
+l (i ) i(i )=l (i ) 22 21 l l
H
L =(1−x)l (1)+xl (2), ab ab ab
.
(12)
Dw(i ) Dp(i ) +l (i ) j (i )=l (i ) 12 V 11 l(i ) l(i ) Dp(i )
n(1)+n(2)
.
(13)
(14)
(15)
and L is given by Eqs. (13) and (14) as a function ab of x, leading to: n =−L /L p 12 22 . m =−L /L p 12 11
H
(17)
H
,
(18)
where
If the pore radii, a: (1) and a: (2), are experimentally obtained — through scanning electron microscopy (SEM ), liquid displacement, or mercury intrusion techniques, for example — all the :l (i ) ab coefficients can be evaluated by Eqs. (6) and (7). Then: l (i )=lpa: 2(i )l: (i ), ab ab
Dw(i )
i(i )=l (i ) +l (i ) 21 22 l(i ) l(i )
j (1)=j (2)=J V V V i(1)=i(2)=I
where x=
H
,
and, for the volume and charge passed through a pore of type i per unit time:
Then, using Eqs. (10)–(12):
n(2)
Dw
+L I=L 22 l(1)+l(2) 21 l(1)+l(2)
and Dp Dw j (i )=l (i ) +l (i ) V 11 12 l l
Dp
(16)
4.3. Pores in series On the other hand, we can admit that there are two pores in series with different lengths, l(1) for a pore of radius a: (1) and 1(2) for one with a: (2), as shown in Fig. 1(B). The phenomenological equations for the volume and charge flow per opening connecting both sides of the membrane
Dp(1)=p −P , Dw(1)=w −w C 1 C 1 Dp(2)=p −p , Dw(2)=w −w 2 C 2 C
H
.
(19)
Then by using Eqs. (17)–(19), we have: C L = ab , ab C
(20)
where: C =x [l (1)l (1)l (2)−l (1)l (1)l (2)] 11 11 22 11 12 21 11 +(1−x) [l (2)l (2)l (1) 11 22 11 −l (2)l (2)l (1)], (21) 12 21 11 C =x [l (1)l (1)l (2)−l (1)l (1)l (2)] 12 11 22 12 12 21 12 +(1−x) [l (2)l (2)l (1) 11 22 12 −l (2)l (2)l (1)], (22) 12 21 12 C =x [l (1)l (1)l (2)−l (1)l (1)l (2)] 21 11 22 21 12 21 21 +(1−x) [l (2)l (2)l (1) 11 22 21 −l (2)l (2)l (1)], (23) 12 21 21 C =x [l (1)l (1)l (2)−l (1)l (1)l (2)] 22 11 22 22 12 21 22 +(1−x) [l (2)l (2)l (1) 11 22 22 −l (2)l (2)l (1)], (24) 12 21 22
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A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
with: C=
C C −C C 11 22 12 21
2 a [l (i )l (i )−l (i )l (i )] 11 22 12 21 i=1
(25)
and x=
l(2) l(1)+l(2)
.
(26)
If a: (1) and a: (2) are somehow experimentally obtained, the coefficients :l (i) can be evaluated by ab Eqs. (6) and (7). Then l (i) can be calculated ab according to: l (i )=l(i )pa: 2(i )l: (i ), (27) ab ab and L is given by Eqs. (20)–(25) as a function ab of x, leading to n and m by using Eq. (16). p p 4.4. Network model Now, we consider a charged membrane to be modelled as a regular square lattice arrangement of nodes connected by cylindrical tubes, as shown in Fig. 2. Each tube in the network behaves like a single capillary, whose performance is described according to the space–charge model summarized
above. All the tubes are assigned an equal surface charge density, while each of them has one of two possible diameters randomly assigned. The n2 nodes are named as N(i, j), and the (2n2−n) pores as P(i, j, k), with i, j=1,…,n and k=1, 2. The four pores which meet at N(i, j) with i≠n are called: P(i, j, 1), P(i, j, 2), P(i+1, j, 1), P(i, j−1, 2), according to Fig. 3. The pores P(i, 0, 2) are equal to P(i, n, 2), according with periodic boundary conditions at the network edges, that are adopted in order to avoid undesirable effects at the network ends. Note that there are (2n2−n) pores, because there are two pores per node, but the n pores P(n, j, 2) are not defined, because the nodes N(n, j) are at the surface of the membrane which is in contact with solution 2. The conservation laws at each node, for volume flow and electrical current, can be written as:
H
j (i, j, 1)+j (i, j, 2)+j (i+1, j, 1)+j (i, j−1, 2)=0 V V V V , i(i, j, 1)+i(i, j, 2)+i(i+1, j, 1)+i(i, j−1, 2)=0 (28)
with i=1,…,n−1 and j=1,…,n. On the other hand, the linear phenomenological equations, for each pore P(i, j, k), are:
Fig. 2. Scheme of the square network, where N represents a node, while P refers to a pore.
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A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
Fig. 3. Scheme of the pores meeting at node N(i, j).
j (i, j, k)=l (i, j, k) V 11
Dp(i, j, k) l
+l (i, j, k) 12 i(i, j, k)=l (i, j, k) 12
However, for i=1: Dp(1, j, 1)=p(1, j )−p
1 Dp(1, j, 2)=p(1, j )−p(1, j−1)
Dw(i, j, k) l
Dw(i, j, k) l
H
,
(29)
I (i, j, k)=l: (i, j, k)pa: 2(i, j, k)l, ab ab
(30)
Dp(n−1, j, 1)=p(n−1, j )−p
2 Dp(n−1, j, 2)=p(n−1, j )−p(n−1, j−1)
H
(35)
and DW(n−1, j, 1)=W(n−1, j )−W
where a: (i, j, k) is the mean radius of P(i, j, k). The driving forces are:
H
2 . DW(n−1, j, 2)=W(n−1, j )−W(n−1, j−1) (36)
H
(31)
and
H
DW(i, j, 1)=W(i, j )−W(i−1, j ) DW(i, j, 2)=W(i, j )−W(i, j−1)
(34)
Finally, for i=n−1:
where all the parameters are a function of the pore, and the I (i, j, k) can be calculated from ab :l (i, j, k) by: ab
Dp(i, j, 2)=p(i, j )−p(i, j−1)
H
DW(1, j, 1)=W(1, j )−W 1 . DW(1, j, 2)=W(1, j )−W(1, j−1)
l
Dp(i, j, 1)=p(i, j )−p(i−1, j )
(33)
and
Dp(i, j, k)
+l (i, j, k) 22
H
,
(32)
with i=2,…,n-2 and j=1,…,n, where p(i, j) and W(i, j) are the pressure and potential at the node N(i, j).
Then, we could write Eq. (29) for each node by using Eqs. (30)–(36), arriving at a system of (2n2−2n) equations with (2n2−2n) unknowns, which are p(i, j) and W(i, j), with i=1,…,n−1 and j=11,…,n. In order to find the relation between the phenomenological coefficients of the whole membrane and those for the pores, let us start by noting that the volume flow and the charge per opening connecting both sides of the membrane and unit time,
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A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
are: n ∑ j (n, j, 1) V v= j=1 n n ∑ [l (n, j, 1)Dp(n, j )+l (n, j, 1)DW(n, J ) 11 12 = j=1 nl (37) and n ∑ i(n, j, 1) I= j=1 n
(38) according to Eq. (29). Now, if there was no electrical potential gradient, we should have:
Dp
I=L 21 nl with
H
,
(39)
DW=0, W =W ¬0 2 1 . (40) Dp=p −p =p , p ¬0 2 1 2 1 On the other hand, with no pressure gradient, we should have:
H
DW J =L V 12 nl DW
I=L 22 nl with
H
x=
number of pores with a: (2) total number of pores
=
n(2) (2n2−n)
, (43)
n ∑ [l (n, j, 1)Dp(n, j )+l (n, j, 1)DWDW(n, J )] 21 22 , = j=1 nl
Dp J =L V 11 nl
(2n2−2n) equations with the conditions in Eqs. (40) and (42). Therefore, by comparing Eqs. (37) and (38) with Eqs. (39) and (41) and using Eqs. (40) and (42), the coefficients L can ab be obtained from l (i, j, k). ab Given that there are two possible pore radii, let us say that they are a: (1) and a: (2) with a: (1)
,
(41)
Dp=0, p =p ¬0 2 1 . (42) DW=W −W =W , W ¬0 2 1 2 1 Then, the problem of finding Dp(n, j) and DW(n, j) can be simplified by solving the
H
which is 1 when all the pores have a radius of a: (2). Therefore, the process can be followed for different networks or values of x. 4.5. Random models In the models described above, the pores have been assumed as belonging to two groups of different radii. Nevertheless, in an actual membrane, there is always a distribution of pore sizes around a mean one. Thus, in order to simulate a real membrane with two characteristic pore sizes, it should be appropriate to improve the model, by allowing the pore radii to be distributed according to a more or less broad Gaussian around both the admitted pore sizes. Referring to the network model, in principle, the results should be better for a network that is as large as possible, nevertheless, for practical reasons we must limit ourselves to 10×10 nodes (i.e. 190 pores). For each value of x (the fraction of large pores), the mean sizes are randomly assigned to the 180 independent pores. In fact, the random assignation of mean sizes can affect the method in such a way, that the final results can be different even for equal values of x, mainly around x≈0.5. Hence, the results are averaged for several random networks for each x. Then each pore size is slightly modified following a Gaussian distribution, according to the transformation: (44) a(i, j, k)=a: (i, j, k)+s 앀−2lnA cos(2pB), S where A, B≤1 are random numbers, a: (i, j, k) is the mean radius of the pore P(i, j, k), while a(i, j, k)
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A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
is the actual pore radius which is distributed according to a Gaussian of standard deviation s. Thus, for each random network, the pore radii are modified according to Eq. (44), and the results are again averaged. This has been carried out here for s =15% and 25% of the assumed mean. S As far as the series model is concerned, the same trends as for the network model are followed. Once the mean pore sizes are assigned, we build a parallel array of say 200 composite pores [see Fig. 1(B)] with a fixed value of x. Then, they are modified according, again, to Eq. (44), and finally, the model is solved by using the corresponding procedure, along with Eqs. (17) and (18) with: n ∑ j (i ) Vs J = i=1 V n n ∑ i (i ) s I= i=1 n
H
,
(45)
Fig. 4. L for a membrane with a: (1)=0.02 mm and a: =0.2 mm, 11 c=5 mol/m3 and s=−1 mC/m2.
where J is the volume flow through a pair of Vs pores in series, i is the current and n=200. The s same process is followed for several values of x. Finally, for the parallel model, after assigning the mean pore sizes, we build a parallel array of say 200 parallel pores [see Fig. 1(A)], which are modified, again, according to Eq. (44), and finally, the model is solved by using the corresponding method, along with Eqs. (11) and (12), with: n ∑ j (i ) Vp J = i=1 V n n ∑ i (i ) p I= i=1 n
H
,
(46)
where J is the volume flow through a parallel Vp pore, i is the current, and n=n(1)+n(2)=200. p The operation is repeated for different values of x.
5. Results and conclusions In Figs. 4–6, typical L versus x curves are ab shown for the series, parallel and network models,
Fig. 5. L for a membrane with a: (1)=0.02 mm and a: (2)= 12 0.2 mm, c=5 mol/m3 and s=−1 mC/m2.
with s =0. In particular, they are plotted for a s membrane with a: (1)=0.02 mm and a: (2)=0.2 mm, separating two equal aqueous LiCl solutions (electrolyte which will also be used for the rest of
A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
Fig. 6. L for a membrane with a: (1)=0.02 mm and a: (2)= 22 0.2 mm, c=5 mol/m3 and s=−1 mC/m2.
the simulations) with c=5 mol/m3 and s=−1 mC/m2. It is seen that no significant differences are noted for L , L and L . Common 11 12 22 features are: (1) The parallel model leads to L that increases ab linearly with x. Thus, it seems that a pore size distribution of parallel pores should modify significantly the expected values of L , given ab that all the pore radii should influence the phenomenological coefficients to an extent which should be proportional to the fraction of pores with this radius. (2) In the series model, a high fraction, x≥0.9, of large pores is needed, to see their influence in the value of the corresponding phenomenological coefficient per pore and unit area. Only if more than 90% of the membrane thickness corresponds to large pores does L increase ab very steeply. (3) Interconnections of pores are only relevant for fractions of large pores over 50% of the total. For smaller fractions, x≤0.5, the interconnections are irrelevant, the pores behave as if they were placed in series, and transport is ruled by the smallest pores. While for more than 50% of large pores, L increases sharply with ab x until approaching the linear dependence
21
characteristic of parallel pores. Both these dependencies are reasonable as far as when there are many large pores, and it is more likely for the flux to cross the membrane through pathways perpendicular to the external membrane surface; while when there are many small pores, the flux should pass through many diferent wide and narrow pores in a more intrincated pathway. (4) Of course, when all the pores are small or large, it does not matter if they are placed in parallel or in series, therefore, the corresponding curves coincide at x=0 and 1. However, the network curves also coincide with the others at these points, thus, the interconnections of pores play no role at all if all the pores have equal radii. Nevertheless, the phenomenological coefficients per opening connecting both sides of the membrane and unit time, are not directly accessible from experiments. However, the streaming potentials, as well as the electro-osmotic pressures, refer to the whole membrane and can be simply measured. Thus, the predictions of the model can be easily interpreted in terms of both these parameters. Consequently, in Fig. 7, n is shown as p a function of x for c=5 mol/m3 and s=−1 mC/M2, for the parallel, series and network models, with pore size distributions with different standard deviations, and central values of a: (1)=0.02 mm and a: (2)=0.2 mm. In Fig. 8, m p is shown for the same concentration, charge and pore size distributions. For n and m , the following characteristics are p p noted: (1) For a parallel array of pores, the presence of small pores until 90% (for the electro-osmotic pressure) or around 75% (for the streaming potential ), does not affect the result. Therefore, these electro-kinetic parameters seem to be ruled by the largest pores when they are set in parallel. (2) Nonetheless, for a series membrane, it is the presence of large pores until 90% that does not affect n or m . Therefore, both the streamp p ing potential and the electro-osmotic pressure seem to be ruled by the smallest pores if they are placed in series.
22
A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
Fig. 7. Streaming potential for a membrane with a: (1)=0.02 mm and a: (2)=0.2 mm, c=5 mol/m3 and s=−1 mC/M2, for the parallel, series and network models, with different pore size distributions.
Fig. 8. Electro-osmotic pressure for a membrane with a: (1)= 0.02 mm and a: (2)=0.2 mm, c=5 mol/m3 and s=−1 mC/m2, for the parallel, series and network models, with different pore size distributions.
(3) For a parallel membrane, the presence of a non-zero standard deviation does not change appreciably the values of n or m . Whereas, p p
Fig. 9. Streaming potential for a membrane with a: (1)=0.02 pm and a: (2)=0.2 mm, c=5 mol/m3 and several values of s, for the parallel and series models.
for a series membrane, n decreases while m p p increases, when the standard deviation increases. (4) Only for 0.40.6, n and m almost p p coincide with those obtained, assuming parallel pores. While, for x<0.4, they are very similar to those obtained assuming pores in series. (5) The network curves do not coincide with the others at x=0 and 1, if s ≠0 for the streaming s potential. Whereas, the electro-osmotic pressure is very similar to the predicted values for parallel pores (x>0.6). In any case, a network membrane with a pore size distribution leads to decreasing absolute values of n and m for p p increasing standard deviations. Thus, the interconnections of pores play some limited role if pore size distributions are present. In Fig. 9, the streaming potential is shown for a membrane with a: (1)=0.02 mm and a: (2)=0.2 mm, c=5 mol/m3 and several values of s. Whereas, in Fig. 10, n is shown for a membrane with p
A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
Fig. 10. Streaming potential for a membrane with a: (1)= 0.02 mm and a: (2)=0.2 mm, s=−1 mC/m2 and several values of c, for the parallel and series models.
Fig. 11. Electro-osmotic pressure for a membrane with a: (1)= 0.02 mm and a: (2)=0.2 mm, c=5 mol/m3 and several values of s, for the parallel and series models.
a: (1)=0.02 mm and a: (2)=0.2 mm, s=−1 mC/m2, and several values of c. In both cases, only the results for the parallel and series models with s =0 are included. In Figs. 11 and 12, m is plotted s p for the same concentrations and charge densities.
23
Fig. 12. Electro-osmotic pressure for a membrane with a: (1)= 0.02 mm and a: (2)=0.2 mm, s=−1 mC/m2 and several values of c, for the parallel and series models.
in any case, it is known that the presence of interconnections of pores and or standard deviations should lead to values of both the electrokinetic parameters, that should be placed in between those corresponding to the parallel and series models. Thus, the behaviour and characteristics of the plotted parameters should correspond to minima and maxima for a real membrane. From these, some issues can be noted, namely: (1) The streaming potential increases in absolute value with s, increasing the differences between the predicted values for the series and parallel configurations, in such a way that for small charge densities both the models should lead to substantially equal values for n . p (2) The electro-osmotic pressure also increases in absolute value with s, increasing the differences between the predicted values for the series and parallel configurations. However, m is very small for parallel pores, and only p slightly sensitive to increases in s. In any case, it should be necessary to reach smaller values of s to arrive at equal predictions for pores in series and parallel configurations. (3) The absolute value of the streaming potential decreases very steeply with c; decreasing the differences between the predicted values for
24
A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
the series and parallel configurations, that in any case, remain moderate. (4) The absolute value of the electro-osmotic pressure decreases fairly with c. However, now the differences between the predicted values for the series and parallel configurations, decrease for low concentrations, remaining, in any case, extreme. In conclusion, it has been shown, through a simulation of the electro-kinetic properties of more or less intricately structured membranes, that some general patterns can be inferred as follows [22,23]. The streaming potential and the electro-osmotic pressure are determined mainly by the narrower section of each pore and, among all the pores in parallel, by the widest one. The influence of interconnections and pore size distributions is limited, and, anyway, it can be minimized if high concentrations and low-charged membranes are used. Owing to the unavoidable presence of adsorption which leads to an increase of s with c [15–17], it should only be possible to deal with low charges and high concentrations one at a time, if low adsorptive solids form the membrane. On the other hand both n and m are very p p sensitive to changes in concentration and charge density, as confirmed experimentally [15,16 ]. However, if these parameters are measured in order to elucidate the charge versus concentration characteristics of the membrane–solution system, or to study other electrochemical properties of the membrane interfaces, the streaming potential should be preferred, because the low charge and/or high concentration convergence of the predictions of series and parallel pores is faster, and the concentration and charge dependencies are also steeper.
Acknowledgment We would like to thank the Spanish ‘‘Plan Nacional de Investigacio´n y Desarrollo’’ (CICYT ) (project no. QU196-0767) and the J. Castilla y
Leo´n (project no. VA10496), for financial support of this work.
References [1] R.C. Srivastava, M.G. Abraham, J. Non-equilib. Thermodyn. 4 (1979) 107. [2] J.L. Dresner, J. Phys. Chem. 67 (1963) 1635. [3] F.A. Morrison, J.F. Osterle, J. Chem. Phys. 43 (1965) 2111. [4] J.C. Fair, J.F. Osterle, J. Chem. Phys. 49 (1968) 228. [5] R.J. Gross, J.F. Osterle, J. Chem. Phys. 49 (1968) 228. [6 ] G.B. Westerman-Clark, J.L. Anderson, J. Electrochem. Soc. 130 (1983) 839. [7] C.C. Christoforou, G.B. Westerman-Clark, J.L. Anderson, J. Colloid Interface Sci. 106 (1985) 1. [8] O. Kedem, A. Katchalsky, Trans. Faraday Soc. 59 (1963) 1918. [9] O. Kedem, A. Katchalsky, Trans. Faraday Soc. 59 (1963) 1941. [10] O. Kedem, A. Katchalsky, Trans. Faraday Soc. 59 (1963) 1931. [11] M. Jin, A theoretical model for coupled processes in inhomogeneous geological systems, M.Sc. Thesis, University of Texas at Austin, 1989. [12] M. Jin, M.M. Sharma, J. Colloid Interface Sci. 142 (1990) 61. [13] J.F. Kuo, M.M. Sharma, T.F. Yen, J. Colloid Interface Sci. 126 (1987) 537. [14] M.M. Sharma, Chem. Engng Sci. 43 (1988) 1203. [15] A. Herna´ndez, L. Martı´nez, M.V. Go´mez, J. Colloid Interface Sci. 158 (1993) 429. [16 ] J. Benavente, A. Herna´ndez, G. Jonsson, J. Membrane Sci. 80 (1993) 285. [17] R.J. Hunter, Zeta Potential in Colloid Science Principles and Applications, Academic Press, London, UK, 1981. [18] L. Martı´nez, M.A. Gigosos, A. Herna´ndez, F. Tejerina, J. Membrane Sci. 35 (1987) 1. [19] L. Martı´nez, F. Martı´nez, A. Herna´ndez, F. Tejerina, J. Colloid Interface Sci. 132 (1989) 27. [20] L. Martı´nez, A. Herna´ndez, A. Gonza´lez, F. Tejerina, J. Colloid Interface Sci. 152 (1992) 325. [21] R.A. Robinson, R.M. Stokes, Electrolyte Solutions, Butterworths, London, UK, 1959. [22] R. Lo´pez, J.I. Calvo, A. Herna´ndez, Potencial de Flujo para una Red Bidimensional de Poros segu´n el Modelo Microcapilar, Primer Congreso Iberoamericano en Ciencia y Tecnologı´a de Membranas, Murcia, 1992, p. 135. [23] R. Lo´pez, J.I. Calvo, A. Herna´ndez, Electrokinetic phenomena for a microporous network membrane, in: Effect of the Pore Size Distribution, Electrokinetic Phenomena ’93. Theory and Applications, Granada, Spain, 1993, p. 86.
A network microcapillary model for electrokinetic phenomena through microporous membranes A. Herna´ndez *, R. Lo´pez, J.I. Calvo, P. Pra´danos Dpto. do Termodina´mica y Fı´sica Aplicada, Facultad de Ciencias, Universidad de Valladolid, 47071 Valladolid, Spain Received 7 July 1997; accepted 16 July 1998
Abstract The streaming potential and the electro-osmotic pressure are calculated through a simulation procedure, for a membrane consisting of a sample of equally charged pores, but are arranged differently to approach the real structure of actual membranes. It is presumed that a microporous membrane consists in an ensemble of cylindrical microcapillaries with the same pore charge density. The pores can have two sizes, which can be interconnected in three main ways: (1) parallel, (2) series, and (3) square network. On the other hand, it is assumed that these two pore sizes are, in fact, the central values of two Gaussian distributions, of equal relative standard deviations. In particular, here, mean diameters 0.2 or 0.1 and 0.02 or 0.01 mm, with standard deviations of 0%, 15% and 25%, in parallel, series and network structures, are considered. This is carried out for several charge densities and concentrations of LiCl aqueous solutions. In these conditions, both the studied electro-kinetic parameters are determined by the widest pore in parallel, and its narrowest section. Interconnections of pores are seen to play a significant role, only when narrow and wide pores are almost equally present. Increasing standard deviations affect differently, but only slightly, the overall result for the electro-kinetic parameters, depending on the assumed structure. All these results indicate that high concentration results of streaming potential should be preferred, to obtain the membrane charge versus the concentration profiles. © 1998 Elsevier Science B.V. All rights reserved. Keywords: Electro-kinetic phenomena; Microcapillary model; Microporous membranes; Network structure; Porous materials
1.
Nomenclature
a: a: (1)
mean hydrodynamic radius (m) mean hydrodynamic radius for the small pores (m) a: (2) mean hydrodynamic radius for the large pores (m) a: (i, j, k) mean hydrodynamic radius for the pore P(i, j, k) (m) * Corresponding author. Tel: +34 834 23134; Fax: +34 834 23136; e-mail: [email protected]
a(i, j, k) hydrodynamic radius for the pore P(i, j, k) (m) c bulk concentration (mol/m3) C combination of coefficients given by Eq. (25) (dimensionless) C combination of coefficients given by ab Eqs. (21)–(24) (dimensionless) c concentration of the ith ion, i=+, − i (mol/m3) D ionic diffusivity, i=+, − (m2/s) i F Faraday number (C/mol ) i(i) electric current for the pore i (A)
0927-7757/98/$ – see front matter © 1998 Elsevier Science B.V. All rights reserved. PII S0 9 2 7- 7 7 5 7 ( 9 8 ) 0 06 6 9 - 4
12
i (i) p i (i) s i(i, j, k) I :i I9 j (r) i j (i) V j (i) Vp j (i) Vs
A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
electric current for the pair of parallel pores i (A) electric current for the composite series pore i (A) electric current for a pore P(i, j, k) (A) electric current per pore (A) electric current through a unit surface of pores (A/m2) electric current through a unit surface of the membrane (A/m2) ionic flux in the pore at a distance r from the pore centre (mol/m2 s) volume flow for the pore i(m3/s) volume flow for a pair of parallel pores (m3/s) volume flow for a pair of pores arranged in series (m3/s) volume flow for a pore P(i, j, k) (m3/s)
j (i, j, k) V J volume flow per pore (m3/s) V :j volume flow per surface unit of the pores V (m/s) J9 volume flow per surface unit of the memV brane (m/s) :l one of the four phenomenological ab coefficients of the pores (A/V m2, m/Pa s, or m/V s=A/Pa m2) L9 one of the four phenomenological ab coefficients of the membrane (A/V m2, m/Pa s, or m/V s=A/Pa m2) l(i) length of the pore i (m) l (i) one of the four phenomenological ab coefficients for a pore i (A m/V, m4/Pa s, or m4V s=A m/Pa) one of the four phenomenological l (i, j, k) coefficients for a pore P(i, j, k) ab (A m/V, m4/Pa s, or m4/V s=A m/Pa) L one of the four phenomenological ab coefficients per pore (A m/V, m4/Pa s, or m4/V s=A m/Pa) n number of connections between both sides of the membrane (dimensionless) N(i, j) node i, j (dimensionless) p pressure at the interface between pore 1 C and 2 in the series model (Pa) p pressure at the interface between the i membrane and the solution i (Pa)
radial coordinate from the centre of a mean pore (m) R gas constant (J/mol K ) T absolute temperature ( K ) v fluid speed in a mean pore (m/s) x structural characteristic of the network, defined in the text (dimensionless) y longitudinal coordinate along the pore (m) z charge number of the ith ion, i=+, − i (dimensionless) Dp pressure difference across the membrane (Pa) Dp(i) pressure difference for the pore i (Pa) Dp(i, j, k) pressure difference for the pore P(i, j, k) (Pa) Dy membrane thickness (m) Dw electric potential difference across the membrane ( V ) Dw(i) electric potential difference across the pore i ( V ) Dw(i, j, k) electric potential difference across the pore P(i, j, k) ( V ) e permittivity (F/m) f zeta potential, w(a: ) ( V ) g solution viscosity ( Kg/m s) h azimuth angle in the pore (rad) H membrane porosity (dimensionless) l Debye’s length (m) n stoichiometric coefficient, i=+,− i (dimensionless) n streaming potential ( V/Pa) p m electro-osmotic pressure (Pa/V ) p s surface charge density on the shear surface (C/m2) s standard deviation of the pore size distris butions (dimensionless) Q total electric potential somewhere in the pore ( V ) w dynamic electric potential ( V ) w(i, j) dynamic electric potential at the node N(i, j) ( V ) w dynamic electric potential at the interface i between the membrane and the solution i (V) w dynamic electric potential at the interface C between the two pores in the series model (V) r
A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
y
normalized radial portion of the electric potential (dimensionless)
2. Introduction It has been recognized on empirical grounds, that in a system characterized by a single, small force and a single flow, the flow is a linear function of the conjugated driving force. In general, however, when two or more flows are taking place simultaneously, there may be coupling between non-conjugated flows and forces. For a membrane that separates two electrolyte solutions, the gradients of pressure and electric potential play the role of driving forces which are coupled, leading to the conjugated fluxes; i.e. the volume flow and the electric current [1]. This occurs in such a way, that forces and fluxes are related through: J9 =L9 Dp+L 9 Dw V 11 12 , (1) I9=L9 Dp+L9 Dw 21 22 where J9 and I9 are, respectively, the volume and V the electric charge passing through the membrane per unit time and membrane area, and the L9 are ab the phenomenological coefficients. The electro-kinetic phenomena that arise in these conditions lead to parameters that, in any case, can be written as a combination of the phenomenological coefficients. For example, the streaming potential is:
H
A B
Dw L9 =− 12 , n = p Dp 9I=0 L9 22
(2)
while for the electro-osmotic pressure we have:
A B
Dp L9 m = =− 12 . (3) p Dw J9 =0 L9 11 V In order to evaluate an electro-kinetic coefficient, a microscopic model for the solution movement and its interactions with the membrane surfaces, is required. Thirty years ago, a simple, but extraordinarily fruitful, space–charge model, was originally raised by Dresner [2], and developed in the
13
following years by Osterle and co-workers [3–5], and Westerman-Clark and co-workers [6,7]. This model allows one to obtain the phenomenological coefficients for a membrane, by assuming that it consists of a bunch of parallel cylindrical and equal microcapillary tubes, with the same surface charge density uniformly distributed on their surfaces. The question of how the phenomenological coefficients of a composite membrane change, when it can be thought of as consistent in two or more simple membranes, has been addressed in Refs [8–10]. The authors have arrived at equations relating the phenomenological coefficients for the composite membranes, with those for the constituent membrane elements, when they are placed both in parallel and series. Of course, real membranes include pores that are size-distributed, attending to their sections at any plane parallel to the membrane surfaces, but as well as along any direction perpendicular to the external surfaces, and with interconnected pores. A further step to approach a real membrane has been faced by Jin [11,12], by considering that we have a complex membrane that can be imagined as a square network of capillary tubes of two sizes, and solving each of them by a modification of the space–charge model. The authors have obtained some results for the streaming potential in terms of the zeta potential (f) for the pore walls. Our aim here is to study the predictions of this model for streaming potential and electro-osmotic pressure, in terms of the surface charge density and the solute concentration. These parameters have been chosen because concentration is directly controllable, while the surface charge density should be a constant for a given membrane material–solution interface. This simulation is not expected to fit exactly any really complex microporous membrane or porous plug [13,14], but it should be better suited than the pure one-size microcapillary representation. Actually, the series and parallel models should work relatively well for reasonably regular microporous membranes, such as track etched and inorganic, anodically formed ones. In any case, such a study will provide some insights on where and how the simpler one-size model can or cannot be
14
A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
applied to real, more complex, microporous membranes [15,16 ]. These apects are relevant for measurements of the electro-kinetic phenomena; the surface charge density can be obtained once the pore sizes are known. This allows one to predict solute–membrane interactions in actual filtration conditions.
3. Microcapillary model The solid–liquid interface corresponding to the solution in contact with the pore walls can be described (see, for example, Ref. [17]), in terms of a charged surface in contact with a diffuse region of mobile co-ions and counter-ions, whose distribution is given by the Poisson–Boltzmann equation. This is known as the Gouy–Chapman picture, which assumes the ions as point charges. According to this model, time mobile solution is filling a pore of an effective radius a: and length l(l&a: ), which slides on a surface characterized by a uniform immobile surface charge density, given by s. The liquid phase is an incompressible Newtonian aqueous solution of an electrolyte whose uniform permittivity is e, with viscosity g, and ionic diffusivities D , i=+,−. i Pressure and electric gradients are assumed to be fully established along the axial direction of the pore, which produces a steady flow through the capillary. In order to benefit from cylindrical symmetry, r, y coordinates can be used, where r denotes the radial distance from the pore axis, and y the distance along it, while the azimuth angle h does not need to be considered as a result of symmetry. The electric potential, Q, inside the pore can be split into a static and a dynamic contribution, according to: RT
y(r)+w( y), (4) z F + where RTy/z F is the electric potential at the + solid–liquid interface, taking the bulk solution as a reference. This should be the only potential drop present at the interface, if no gradient acted through the membrane. While w is the exclusive
Q(r, y)=
longitudinal contribution caused by the pressure or electric potential gradients. The Navier–Stokes equation gives, as shown elsewhere — see for example Herna´ndez and co-workers [18–20] — the velocity profile v(r), at low Reynolds numbers, while those of Nernst–Planck give the ionic fluxes j of the ions i (i=+,−). On the other hand, the volume charge density r (r) and the ith ion molar concentration c c (r) come from the Poisson and Boltzmann equai tions, respectively. The fluxes and the corresponding phenomenological coefficients per unit area pore are related through: :j =l: Dp+l: Dw V 11 12 . (5) :i=l: Dp+l: Dw 22 21 The volume flow per surface unit through the pore results in:
H
2 :j = V a: 2 −
P
a:
v(r)rdr=
0 eRT
A
a: 2 Dp 8gl
2
P
a:
B
(6) y(r)rdr Dw, lgz F a: 2 0 + where RTf/z F is the electric potential on the + pore wall, i.e. f is the dimensionless zeta potential. In the same way, the current density through the pore can be obtained by integration of the Nernst–Planck equations: :i=
2F a: 2
−
P
a:
0 eRT
f−
[z j (r)+z j (r)]rdr= ++ −−
C
f−
2
P
a:
D C
y(r)dr Dp+
2F2c
P
a:
gz Fl a: 2 a: 2RTl 0 0 + ×{n z 2D e−y(r)+n z 2D e−z−y(r)/z+ }rdr + + + − − − 2cRT a: {n z e−y(r)+n z e−z−y(r)/z+ )} − + + − − a: 2z l 0 +
P
D
×{f−y(r)}rdr Dw.
(7)
Eqs. (5)–(7) allow one to evaluate all the phenomenological coefficients :l , for each pair a: /l (l ab being the Debye length [eRT/F2c (~∑~n z )2]1/2 i i
A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
15
and s, once the potential profile inside a pore, including y(a: )=f is known. This radial profile of the electric potential can be calculated by a numerical or variational integration of the Poisson– Boltzmann equation [20]. The bulk solution values, as given by [21], can be taken for g, e, D and D , if the fluid dynamics + − and electrostatics of the solution are not substantially modified by the presence of electrostatic interactions with the solid surfaces. This is certainly a very good approximation for the small electric fields, that can be expected to hold within the pores when containing low concentration solutions. Thus, every combination of phenomenological coefficients, :l , can be related with the surface ab charge density and the bulk concentration (through the Debye’s length), if the structural parameters (the effective mean or hydrodynamic pore radius and the pore length) are known.
4. Structural models
Fig. 1. Scheme of the parallel (a) and the series (b) models.
4.1. Equal pores If all the pores are assumed to be parallel, equal microcapillary tubes placed perpendicularly to the membrane walls, the membrane thickness is equal to the pore length, l=Dy, and the transport characteristics of the membrane are given by those of a single pore. The fluxes through the membrane, J9 and I9, are V related with the volume flow and electric intensity per surface unit for a pore, by: J9 =Hj: V V , I9=Hi:
H
(8)
respectively. This defines H, which is the membrane porosity, as the open surface fraction or the void volume fraction. However, L9 is linked to the mean pore radius 11 up to the shear surface, a: , (see Eqs. (5) and (6), by means of the Hagen–Poisseuille equation: Ha: 2
. L9 =Hl: = 11 11 8gDy
(9)
Hence, a: can be obtained from L9 once poros11 ity, H, and thickness, Dy, are previously measured. Then, Eqs. (5)–(8) can be used to obtain J9 , I9 V and, by comparing with Eq. (1), n and m through p p their definitions, Eqs. (2) and (3). 4.2. Pores in parallel We can assume that there are n(1) pores of radius a: (1) and n(2) with a: (2), as shown in Fig. 1(A). Consequently, the total volume and charge flowing per pore and unit time, can be obtained from the total volume and charge flowing per unit time through each pore of types 1 and 2 according to: J = V
n(1) j (1)+n(2) j (2) V V n(1)+n(2)
I=
n(1)i(1)+n(2)i(2) n(1)+n(2)
H
,
(10)
whose respective phenomenological coefficients
16
A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
can be defined through: Dp Dw J =L +L V 11 l 12 l Dw
Dp
+L I=L 22 l 21 l
H
and unit time, are: Dw Dp +L J =L 12 l(1)+l(2) V 11 l(1)+l(2)
,
(11)
Dw
Dp
+l (i ) i(i )=l (i ) 22 21 l l
H
L =(1−x)l (1)+xl (2), ab ab ab
.
(12)
Dw(i ) Dp(i ) +l (i ) j (i )=l (i ) 12 V 11 l(i ) l(i ) Dp(i )
n(1)+n(2)
.
(13)
(14)
(15)
and L is given by Eqs. (13) and (14) as a function ab of x, leading to: n =−L /L p 12 22 . m =−L /L p 12 11
H
(17)
H
,
(18)
where
If the pore radii, a: (1) and a: (2), are experimentally obtained — through scanning electron microscopy (SEM ), liquid displacement, or mercury intrusion techniques, for example — all the :l (i ) ab coefficients can be evaluated by Eqs. (6) and (7). Then: l (i )=lpa: 2(i )l: (i ), ab ab
Dw(i )
i(i )=l (i ) +l (i ) 21 22 l(i ) l(i )
j (1)=j (2)=J V V V i(1)=i(2)=I
where x=
H
,
and, for the volume and charge passed through a pore of type i per unit time:
Then, using Eqs. (10)–(12):
n(2)
Dw
+L I=L 22 l(1)+l(2) 21 l(1)+l(2)
and Dp Dw j (i )=l (i ) +l (i ) V 11 12 l l
Dp
(16)
4.3. Pores in series On the other hand, we can admit that there are two pores in series with different lengths, l(1) for a pore of radius a: (1) and 1(2) for one with a: (2), as shown in Fig. 1(B). The phenomenological equations for the volume and charge flow per opening connecting both sides of the membrane
Dp(1)=p −P , Dw(1)=w −w C 1 C 1 Dp(2)=p −p , Dw(2)=w −w 2 C 2 C
H
.
(19)
Then by using Eqs. (17)–(19), we have: C L = ab , ab C
(20)
where: C =x [l (1)l (1)l (2)−l (1)l (1)l (2)] 11 11 22 11 12 21 11 +(1−x) [l (2)l (2)l (1) 11 22 11 −l (2)l (2)l (1)], (21) 12 21 11 C =x [l (1)l (1)l (2)−l (1)l (1)l (2)] 12 11 22 12 12 21 12 +(1−x) [l (2)l (2)l (1) 11 22 12 −l (2)l (2)l (1)], (22) 12 21 12 C =x [l (1)l (1)l (2)−l (1)l (1)l (2)] 21 11 22 21 12 21 21 +(1−x) [l (2)l (2)l (1) 11 22 21 −l (2)l (2)l (1)], (23) 12 21 21 C =x [l (1)l (1)l (2)−l (1)l (1)l (2)] 22 11 22 22 12 21 22 +(1−x) [l (2)l (2)l (1) 11 22 22 −l (2)l (2)l (1)], (24) 12 21 22
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A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
with: C=
C C −C C 11 22 12 21
2 a [l (i )l (i )−l (i )l (i )] 11 22 12 21 i=1
(25)
and x=
l(2) l(1)+l(2)
.
(26)
If a: (1) and a: (2) are somehow experimentally obtained, the coefficients :l (i) can be evaluated by ab Eqs. (6) and (7). Then l (i) can be calculated ab according to: l (i )=l(i )pa: 2(i )l: (i ), (27) ab ab and L is given by Eqs. (20)–(25) as a function ab of x, leading to n and m by using Eq. (16). p p 4.4. Network model Now, we consider a charged membrane to be modelled as a regular square lattice arrangement of nodes connected by cylindrical tubes, as shown in Fig. 2. Each tube in the network behaves like a single capillary, whose performance is described according to the space–charge model summarized
above. All the tubes are assigned an equal surface charge density, while each of them has one of two possible diameters randomly assigned. The n2 nodes are named as N(i, j), and the (2n2−n) pores as P(i, j, k), with i, j=1,…,n and k=1, 2. The four pores which meet at N(i, j) with i≠n are called: P(i, j, 1), P(i, j, 2), P(i+1, j, 1), P(i, j−1, 2), according to Fig. 3. The pores P(i, 0, 2) are equal to P(i, n, 2), according with periodic boundary conditions at the network edges, that are adopted in order to avoid undesirable effects at the network ends. Note that there are (2n2−n) pores, because there are two pores per node, but the n pores P(n, j, 2) are not defined, because the nodes N(n, j) are at the surface of the membrane which is in contact with solution 2. The conservation laws at each node, for volume flow and electrical current, can be written as:
H
j (i, j, 1)+j (i, j, 2)+j (i+1, j, 1)+j (i, j−1, 2)=0 V V V V , i(i, j, 1)+i(i, j, 2)+i(i+1, j, 1)+i(i, j−1, 2)=0 (28)
with i=1,…,n−1 and j=1,…,n. On the other hand, the linear phenomenological equations, for each pore P(i, j, k), are:
Fig. 2. Scheme of the square network, where N represents a node, while P refers to a pore.
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A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
Fig. 3. Scheme of the pores meeting at node N(i, j).
j (i, j, k)=l (i, j, k) V 11
Dp(i, j, k) l
+l (i, j, k) 12 i(i, j, k)=l (i, j, k) 12
However, for i=1: Dp(1, j, 1)=p(1, j )−p
1 Dp(1, j, 2)=p(1, j )−p(1, j−1)
Dw(i, j, k) l
Dw(i, j, k) l
H
,
(29)
I (i, j, k)=l: (i, j, k)pa: 2(i, j, k)l, ab ab
(30)
Dp(n−1, j, 1)=p(n−1, j )−p
2 Dp(n−1, j, 2)=p(n−1, j )−p(n−1, j−1)
H
(35)
and DW(n−1, j, 1)=W(n−1, j )−W
where a: (i, j, k) is the mean radius of P(i, j, k). The driving forces are:
H
2 . DW(n−1, j, 2)=W(n−1, j )−W(n−1, j−1) (36)
H
(31)
and
H
DW(i, j, 1)=W(i, j )−W(i−1, j ) DW(i, j, 2)=W(i, j )−W(i, j−1)
(34)
Finally, for i=n−1:
where all the parameters are a function of the pore, and the I (i, j, k) can be calculated from ab :l (i, j, k) by: ab
Dp(i, j, 2)=p(i, j )−p(i, j−1)
H
DW(1, j, 1)=W(1, j )−W 1 . DW(1, j, 2)=W(1, j )−W(1, j−1)
l
Dp(i, j, 1)=p(i, j )−p(i−1, j )
(33)
and
Dp(i, j, k)
+l (i, j, k) 22
H
,
(32)
with i=2,…,n-2 and j=1,…,n, where p(i, j) and W(i, j) are the pressure and potential at the node N(i, j).
Then, we could write Eq. (29) for each node by using Eqs. (30)–(36), arriving at a system of (2n2−2n) equations with (2n2−2n) unknowns, which are p(i, j) and W(i, j), with i=1,…,n−1 and j=11,…,n. In order to find the relation between the phenomenological coefficients of the whole membrane and those for the pores, let us start by noting that the volume flow and the charge per opening connecting both sides of the membrane and unit time,
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A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
are: n ∑ j (n, j, 1) V v= j=1 n n ∑ [l (n, j, 1)Dp(n, j )+l (n, j, 1)DW(n, J ) 11 12 = j=1 nl (37) and n ∑ i(n, j, 1) I= j=1 n
(38) according to Eq. (29). Now, if there was no electrical potential gradient, we should have:
Dp
I=L 21 nl with
H
,
(39)
DW=0, W =W ¬0 2 1 . (40) Dp=p −p =p , p ¬0 2 1 2 1 On the other hand, with no pressure gradient, we should have:
H
DW J =L V 12 nl DW
I=L 22 nl with
H
x=
number of pores with a: (2) total number of pores
=
n(2) (2n2−n)
, (43)
n ∑ [l (n, j, 1)Dp(n, j )+l (n, j, 1)DWDW(n, J )] 21 22 , = j=1 nl
Dp J =L V 11 nl
(2n2−2n) equations with the conditions in Eqs. (40) and (42). Therefore, by comparing Eqs. (37) and (38) with Eqs. (39) and (41) and using Eqs. (40) and (42), the coefficients L can ab be obtained from l (i, j, k). ab Given that there are two possible pore radii, let us say that they are a: (1) and a: (2) with a: (1)
,
(41)
Dp=0, p =p ¬0 2 1 . (42) DW=W −W =W , W ¬0 2 1 2 1 Then, the problem of finding Dp(n, j) and DW(n, j) can be simplified by solving the
H
which is 1 when all the pores have a radius of a: (2). Therefore, the process can be followed for different networks or values of x. 4.5. Random models In the models described above, the pores have been assumed as belonging to two groups of different radii. Nevertheless, in an actual membrane, there is always a distribution of pore sizes around a mean one. Thus, in order to simulate a real membrane with two characteristic pore sizes, it should be appropriate to improve the model, by allowing the pore radii to be distributed according to a more or less broad Gaussian around both the admitted pore sizes. Referring to the network model, in principle, the results should be better for a network that is as large as possible, nevertheless, for practical reasons we must limit ourselves to 10×10 nodes (i.e. 190 pores). For each value of x (the fraction of large pores), the mean sizes are randomly assigned to the 180 independent pores. In fact, the random assignation of mean sizes can affect the method in such a way, that the final results can be different even for equal values of x, mainly around x≈0.5. Hence, the results are averaged for several random networks for each x. Then each pore size is slightly modified following a Gaussian distribution, according to the transformation: (44) a(i, j, k)=a: (i, j, k)+s 앀−2lnA cos(2pB), S where A, B≤1 are random numbers, a: (i, j, k) is the mean radius of the pore P(i, j, k), while a(i, j, k)
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A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
is the actual pore radius which is distributed according to a Gaussian of standard deviation s. Thus, for each random network, the pore radii are modified according to Eq. (44), and the results are again averaged. This has been carried out here for s =15% and 25% of the assumed mean. S As far as the series model is concerned, the same trends as for the network model are followed. Once the mean pore sizes are assigned, we build a parallel array of say 200 composite pores [see Fig. 1(B)] with a fixed value of x. Then, they are modified according, again, to Eq. (44), and finally, the model is solved by using the corresponding procedure, along with Eqs. (17) and (18) with: n ∑ j (i ) Vs J = i=1 V n n ∑ i (i ) s I= i=1 n
H
,
(45)
Fig. 4. L for a membrane with a: (1)=0.02 mm and a: =0.2 mm, 11 c=5 mol/m3 and s=−1 mC/m2.
where J is the volume flow through a pair of Vs pores in series, i is the current and n=200. The s same process is followed for several values of x. Finally, for the parallel model, after assigning the mean pore sizes, we build a parallel array of say 200 parallel pores [see Fig. 1(A)], which are modified, again, according to Eq. (44), and finally, the model is solved by using the corresponding method, along with Eqs. (11) and (12), with: n ∑ j (i ) Vp J = i=1 V n n ∑ i (i ) p I= i=1 n
H
,
(46)
where J is the volume flow through a parallel Vp pore, i is the current, and n=n(1)+n(2)=200. p The operation is repeated for different values of x.
5. Results and conclusions In Figs. 4–6, typical L versus x curves are ab shown for the series, parallel and network models,
Fig. 5. L for a membrane with a: (1)=0.02 mm and a: (2)= 12 0.2 mm, c=5 mol/m3 and s=−1 mC/m2.
with s =0. In particular, they are plotted for a s membrane with a: (1)=0.02 mm and a: (2)=0.2 mm, separating two equal aqueous LiCl solutions (electrolyte which will also be used for the rest of
A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
Fig. 6. L for a membrane with a: (1)=0.02 mm and a: (2)= 22 0.2 mm, c=5 mol/m3 and s=−1 mC/m2.
the simulations) with c=5 mol/m3 and s=−1 mC/m2. It is seen that no significant differences are noted for L , L and L . Common 11 12 22 features are: (1) The parallel model leads to L that increases ab linearly with x. Thus, it seems that a pore size distribution of parallel pores should modify significantly the expected values of L , given ab that all the pore radii should influence the phenomenological coefficients to an extent which should be proportional to the fraction of pores with this radius. (2) In the series model, a high fraction, x≥0.9, of large pores is needed, to see their influence in the value of the corresponding phenomenological coefficient per pore and unit area. Only if more than 90% of the membrane thickness corresponds to large pores does L increase ab very steeply. (3) Interconnections of pores are only relevant for fractions of large pores over 50% of the total. For smaller fractions, x≤0.5, the interconnections are irrelevant, the pores behave as if they were placed in series, and transport is ruled by the smallest pores. While for more than 50% of large pores, L increases sharply with ab x until approaching the linear dependence
21
characteristic of parallel pores. Both these dependencies are reasonable as far as when there are many large pores, and it is more likely for the flux to cross the membrane through pathways perpendicular to the external membrane surface; while when there are many small pores, the flux should pass through many diferent wide and narrow pores in a more intrincated pathway. (4) Of course, when all the pores are small or large, it does not matter if they are placed in parallel or in series, therefore, the corresponding curves coincide at x=0 and 1. However, the network curves also coincide with the others at these points, thus, the interconnections of pores play no role at all if all the pores have equal radii. Nevertheless, the phenomenological coefficients per opening connecting both sides of the membrane and unit time, are not directly accessible from experiments. However, the streaming potentials, as well as the electro-osmotic pressures, refer to the whole membrane and can be simply measured. Thus, the predictions of the model can be easily interpreted in terms of both these parameters. Consequently, in Fig. 7, n is shown as p a function of x for c=5 mol/m3 and s=−1 mC/M2, for the parallel, series and network models, with pore size distributions with different standard deviations, and central values of a: (1)=0.02 mm and a: (2)=0.2 mm. In Fig. 8, m p is shown for the same concentration, charge and pore size distributions. For n and m , the following characteristics are p p noted: (1) For a parallel array of pores, the presence of small pores until 90% (for the electro-osmotic pressure) or around 75% (for the streaming potential ), does not affect the result. Therefore, these electro-kinetic parameters seem to be ruled by the largest pores when they are set in parallel. (2) Nonetheless, for a series membrane, it is the presence of large pores until 90% that does not affect n or m . Therefore, both the streamp p ing potential and the electro-osmotic pressure seem to be ruled by the smallest pores if they are placed in series.
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A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
Fig. 7. Streaming potential for a membrane with a: (1)=0.02 mm and a: (2)=0.2 mm, c=5 mol/m3 and s=−1 mC/M2, for the parallel, series and network models, with different pore size distributions.
Fig. 8. Electro-osmotic pressure for a membrane with a: (1)= 0.02 mm and a: (2)=0.2 mm, c=5 mol/m3 and s=−1 mC/m2, for the parallel, series and network models, with different pore size distributions.
(3) For a parallel membrane, the presence of a non-zero standard deviation does not change appreciably the values of n or m . Whereas, p p
Fig. 9. Streaming potential for a membrane with a: (1)=0.02 pm and a: (2)=0.2 mm, c=5 mol/m3 and several values of s, for the parallel and series models.
for a series membrane, n decreases while m p p increases, when the standard deviation increases. (4) Only for 0.4
A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
Fig. 10. Streaming potential for a membrane with a: (1)= 0.02 mm and a: (2)=0.2 mm, s=−1 mC/m2 and several values of c, for the parallel and series models.
Fig. 11. Electro-osmotic pressure for a membrane with a: (1)= 0.02 mm and a: (2)=0.2 mm, c=5 mol/m3 and several values of s, for the parallel and series models.
a: (1)=0.02 mm and a: (2)=0.2 mm, s=−1 mC/m2, and several values of c. In both cases, only the results for the parallel and series models with s =0 are included. In Figs. 11 and 12, m is plotted s p for the same concentrations and charge densities.
23
Fig. 12. Electro-osmotic pressure for a membrane with a: (1)= 0.02 mm and a: (2)=0.2 mm, s=−1 mC/m2 and several values of c, for the parallel and series models.
in any case, it is known that the presence of interconnections of pores and or standard deviations should lead to values of both the electrokinetic parameters, that should be placed in between those corresponding to the parallel and series models. Thus, the behaviour and characteristics of the plotted parameters should correspond to minima and maxima for a real membrane. From these, some issues can be noted, namely: (1) The streaming potential increases in absolute value with s, increasing the differences between the predicted values for the series and parallel configurations, in such a way that for small charge densities both the models should lead to substantially equal values for n . p (2) The electro-osmotic pressure also increases in absolute value with s, increasing the differences between the predicted values for the series and parallel configurations. However, m is very small for parallel pores, and only p slightly sensitive to increases in s. In any case, it should be necessary to reach smaller values of s to arrive at equal predictions for pores in series and parallel configurations. (3) The absolute value of the streaming potential decreases very steeply with c; decreasing the differences between the predicted values for
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A. Herna´ndez et al. / Colloids Surfaces A: Physicochem. Eng. Aspects 145 (1998) 11–24
the series and parallel configurations, that in any case, remain moderate. (4) The absolute value of the electro-osmotic pressure decreases fairly with c. However, now the differences between the predicted values for the series and parallel configurations, decrease for low concentrations, remaining, in any case, extreme. In conclusion, it has been shown, through a simulation of the electro-kinetic properties of more or less intricately structured membranes, that some general patterns can be inferred as follows [22,23]. The streaming potential and the electro-osmotic pressure are determined mainly by the narrower section of each pore and, among all the pores in parallel, by the widest one. The influence of interconnections and pore size distributions is limited, and, anyway, it can be minimized if high concentrations and low-charged membranes are used. Owing to the unavoidable presence of adsorption which leads to an increase of s with c [15–17], it should only be possible to deal with low charges and high concentrations one at a time, if low adsorptive solids form the membrane. On the other hand both n and m are very p p sensitive to changes in concentration and charge density, as confirmed experimentally [15,16 ]. However, if these parameters are measured in order to elucidate the charge versus concentration characteristics of the membrane–solution system, or to study other electrochemical properties of the membrane interfaces, the streaming potential should be preferred, because the low charge and/or high concentration convergence of the predictions of series and parallel pores is faster, and the concentration and charge dependencies are also steeper.
Acknowledgment We would like to thank the Spanish ‘‘Plan Nacional de Investigacio´n y Desarrollo’’ (CICYT ) (project no. QU196-0767) and the J. Castilla y
Leo´n (project no. VA10496), for financial support of this work.
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