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Modelling the transport of asymmetric electrolytes throughnanofiltratio n membranes
Anthony Szymczyka,*, Yannick Lanterib, Patrick Fievetb
aUniversité Rennes 1, UMR 6226 CNRS-UR1-ENSCR, équipe Chimie et Ingénierie des Procédés, 263 Avenuedu Général Leclerc, Bâtiment 10 A, CS 74205, 35042 Rennes Cedex, France
Email: [email protected]é de Franche-Comté, Institut UTINAM, UMR CNRS 6213, 16 route de Gray, 25030 Besançon Cedex,
France
Received 27 June 2008; revised 06 January 2009; accepted 09 February 2009
Abstract
In this work, we used the SEDE (Steric, Electric and Dielectric Exclusion) model to investigate the rejectionrate of asymmetric electrolytes by nanofiltration membranes. The SEDE model predicts that the rejection rate ofasymmetric electrolytes with divalent counter-ions is a non-monotonous function of the volume charge density.Because the Donnan exclusion screens the interaction between ions and their images, it was shown that there is arange of membrane volume charge densities for which the increase in the Donnan exclusion is overcompensatedby the strong decrease in the interaction between the ions and their images. As a result, the rejection rate decreaseseven if the membrane volume charge density increases. At very high membrane charge densities, the image chargeinteraction vanishes and the rejection rate tends to the value predicted by the Donnan theory. The separation per-formances of two NF polyamide membranes were investigated. It was shown that they cannot be described withouttaking into account the dielectric exclusion.
*Corresponding author.
Presented at the conference Engineering with Membranes 2008; Membrane Processes: Development, Monitoring andModelling – From the Nano to the Macro Scale – (EWM 2008), May 25–28, 2008, Vale do Lobo, Algarve, Portugal.
1. Introduction
Nanofiltration (NF) membranes are made of
organic or ceramic materials the most of which
develop a surface electric charge when brought
into contact with a polar medium. The generation
of a fixed charge onto the membrane surface
results from various phenomena including sur-
face site dissociation and/or adsorption of
charged species [1,2].
NF membranes have a molecular weight cut-
off ranging from a few hundreds to a few thou-
sands Dalton, i.e. intermediate between reverse
osmosis and ultrafiltration membranes. The
Desalination 245 (2009) 3 –96 407
doi:10.1016/j.desal.2009.02.003
0011-9164/09/$– See front matter © 2009 Elsevier B.V. All rights reserved.
combination of pore sizes around a few nanome-
ters with electrically charged materials makes
the prediction of NF membrane performances
extremely difficult since the separation of
solutes results from a complex mechanism
including steric hindrance as well as Donnan
and dielectric effects. These dielectric effects
can be split in two contributions. The first one is
connected to the lowering of the dielectric con-
stant of a fluid trapped in nanodimensional cav-
ities [3] and is referred as the Born effect. It
corresponds to the variation of the solvation
energy of an ion when it is transferred from the
bulk solution to the pore inside. If the effective
dielectric constant of the solution confined
inside pores is lower than the dielectric constant
of the bulk solution, the excess solvation energy
is positive and ions are rejected by the mem-
brane pores. The second kind of dielectric effect
arises because of the difference between the
effective dielectric constant of the solution
inside pores and the dielectric constant of the
membrane itself. Because of the dielectric dis-
continuity at the pore surface, the electric field
generated by an ion close to the surface polar-
izes the surface, and the induced polarization
charge interacts with the ion. This phenomenon
is usually referred as the interaction between
ions and their images [4].
The standard theory of NF is based on a
macroscopic description of transport which is
actually a simplified version of the so-called
Space Charge model [5]. The solute separation
process is described in three steps. The first one
is the solute partitioning at the interface between
the feed solution and the pore inlet. Next, the
transport of solutes through the membrane pores
is considered. Finally, the last step is the parti-
tioning of solutes at the interface between the
membrane and the permeate. During the last
decade, several transport models have been
developed which mainly differ from each other
by the equation used to describe the partitioning
of solutes at the membrane/solution interfaces.
We recently proposed the SEDE (Steric, Electric
and Dielectric Exclusion) model, as a tool of
investigation of transport properties of NF mem-
branes [6,7]. Within the scope of the SEDE
model, the partitioning coefficient of an ion is
expressed by means of a modified Donnan equa-
tion that takes into account the steric hindrance,
the Donnan exclusion, i.e. the electrostatic inter-
action between ions and the membrane fixed
charge, and the so-called dielectric exclusion
(DE) which is expressed in terms of Born effect
and image charges. In this work, we used the
SEDE model with the aim of investigating the
contribution of Donnan and non-Donnan exclu-
sion mechanisms in the nanofiltration of an asym-
metric electrolyte (calcium chloride) in aqueous
solution.
2. Theoretical background
The SEDE model has been described in
details in previous works [6,7]. The governing
equations of the model are collected in Tables 1
and 2. The distribution of ions at the feed solu-
tion/membrane (0+|0–) and the membrane/perme-
ate (Δx–|Δx+) interfaces is described by Eqs. (1a)
and (1b), respectively, which are modified Don-
nan relations including steric hindrance and DE.
Within the scope of the SEDE model, the Born
effect is described by Eq. (4) which are modified
Born equations that consider the radius of the
cavity formed by the ion i in the solvent (ri,cav).
This latter was estimated according to the proce-
dure proposed by Rashin and Honig [8]. The
interaction between the ions and the induced
polarization charges is described by Eq. (5) that
were first derived by Yaroshchuk [9]. The solute
transport through the membrane is described by
means of the extended Nernst–Planck equation
that considers diffusion, electromigration and
convection (see Table 2, Eq. (10)). This equation
can be rewritten so as to establish the expression
of the concentration gradient inside pores, i.e. for
0+ ≤ x ≤ Δx– (see Eq. (11)). The expression of the
A. Szymczyk et al. / Desalination 24 (2009) –5 396 407 397
A. Szymczyk et al. / Desalination 245 (2009) 396–407
Table 1Partitioning equations used in the SEDE model
Partitioning equations at the membrane/solution interfaces
(1a)
(1b)
with,
(slit-like pores) (2)
(3a)
(3b)
(4)
(slit-like pores) (5a)
(slit-like pores) (5b)
(6)
(7a)
c
cz Wi
ii
i
ii i
0
0
0
0 0 0
+
−
−
+
( )( ) =
( )( ) −( ) −( )+ −φ
γ
γexp exp
( ) ,'Δ ΔΨ Born eexp ', ( )
−( )+ −ΔWi im 0 0
μ κ
φγγ
0 0
2
0 0
0
00
0+ −
+ −
( )−
−−
+ ( )= ( )
( ) ( )( ) − −
rz c z Wi i i
i
ii i
p
exp ' ,Δ ΔΨ BBorn im−⎛
⎝⎜⎞⎠⎟
( )+ −( )
−∑ΔW
I
i
i
', 0 0
2 0
απε εi
iz FRTN r
= ( )2
08 p A p
ΔΔ Δ Δ Δ
Wi x x i x x' ln exp
,im
p m
p m
− + − +( ) ( )= − −−+
⎛
⎝⎜⎜
⎞
⎠⎟⎟ −⎛
⎝⎜α
ε εε ε
μ1 2⎞⎞⎠⎟
⎡
⎣⎢⎢
⎤
⎦⎥⎥
ΔWi i' ln exp,im
p m
p m0 0 0 0
1 2+ − + −( ) ( )= − −−+
⎛
⎝⎜⎜
⎞
⎠⎟⎟ −⎛
⎝⎜⎞⎠⎟
⎡α
ε εε ε
μ⎣⎣⎢⎢
⎤
⎦⎥⎥
ΔWz ekTrii
i
' ,
,
Born
cav p b
= ( ) −⎛
⎝⎜⎜
⎞
⎠⎟⎟
2
08
1 1
πε ε ε
Δ ΔΔ Δ Δ Δ
Ψx x D x x
ekT− + − +( ) ( )= ψ
Δ ΔΨ0 0 0 0+ − + −( ) ( )= e
kT Dψ
φiir
r= −
⎛
⎝⎜⎜
⎞
⎠⎟⎟1
,Stokes
p
c x
c x
x
xz Wi
ii
i
ii x x i
Δ
Δ
Δ
ΔΔ Δ
Δ Δ
−
+
+
−
( )( ) =
( )( ) −( ) −− +φ
γ
γexp exp '
( ) ,Ψ BBorn im( ) −( )− +exp '
, ( )Δ
Δ ΔW
i x x
398
electric potential gradient (Eq. (12)) is derived
from Eq. (10) and the condition of zero electric
current flowing through the membrane at the
steady state (Eq. (9)). The electroneutrality con-
dition in the external bulk solutions is given by
Eqs. (8a) and (8b). The electroneutrality inside
pores is expressed by Eq. (8c), where X is the
membrane volume charge density which can be
approximately estimated from tangential stream-
ing potential measurements. Solving transport
equations and partitioning equations allows com-
puting the rejection rate of a solute i (Ri), which
is defined as:
(15)
3. Results and discussion
Fig. 1 shows the variation of the intrinsic
rejection rate of a calcium chloride solution ver-
sus the membrane volume charge density. Rejec-
tion rates were computed out by considering
(solid line) or not considering (dashed curve) the
contribution of image charges. The membrane
was characterized by a dielectric constant εm of 3
which is representative of NF organic mem-
branes. As can be seen, the rejection rates com-
puted from the two approaches differ
significantly over a large range of charge densi-
ties for negatively charged membranes. This
makes electrolytes with divalent counter-ions
very good candidates to put in evidence dielectric
effects in NF. Fig. 2 focuses on these electrolytes
with divalent counter-ions, keeping the example
of calcium chloride. At very low charge densities,
the rejection rate first decreases as more fixed
charges are brought onto the pore walls. This first
decrease is not related to any dielectric effect
since it also occurs when no image charge is
included in the calculations (see dashed line in
Fig. 2). This phenomenon occurs when co-ions
are more mobile than counter-ions. It occurs at
low charge densities because of the interplay
between the co-ion exclusion and the acceleration
of these co-ions by the electric field arising
through the pores to ensure the electroneutrality
R xi = −
+
−10
cci
i
( )
( )
Δ
A. Szymczyk et al. / Desalination 24 (2009) –
Table 1 (continued)
(7b)
Electroneutrality conditions
(8a)
(8b)
for 0+ ≤ x ≤ Δx– (8c)
μ κ
φγγ
Δ Δ
Δ Δ
Δ
ΔΔ
ΔΔ
x x
i i ii
ii x x
x rz c x
xx
z
− +
− +
( )+
++
−
= ( )( ) ( )
( ) −
p
2 exp Ψ(( ) ( )+
− −⎛⎝⎜
⎞⎠⎟
( )− +
∑Δ Δ
Δ
Δ ΔW W
I x
i i x x
i
' ', ,Born im
2
z c x Xi ii
( ) + =∑ 0
z c xi ii
Δ +( ) =∑ 0
z ci ii
0 0−( ) =∑
5 396 407 399
[11,12]. The phenomenon disappears as the mem-
brane charge increases and the rejection rate
increases with the membrane volume charge den-
sity. For higher charge densities the rejection rate
decreases again. This behaviour is brought about
by the combination between the Donnan exclu-
sion and the interaction between ions and their
images. This can be approximately shown in Fig.
3 which presents the variation of the co-ion par-
titioning coefficient due to image charges versus
the co-ion partitioning coefficient due to the Don-
nan exclusion. In this figure, the membrane vol-
ume charge density increases from the right to the
left. It can be seen that the interaction with image
charges decreases (i.e. the co-ion partitioning
coefficient due to image charges increases) as the
Donnan exclusion increases (i.e. the co-ion parti-
tioning coefficient due to the Donnan exclusion
decreases). Otherwise stated, the Donnan exclu-
sion screens the image forces. In Fig. 3, the
dashed line corresponds to the hypothetic case for
which the increase in the Donnan exclusion
would be exactly balanced by the decrease in the
DE, so that the rejection rate would be constant
whatever the membrane charge. Let us compare
this hypothetic case with the real one which is
A. Szymczyk et al. / Desalination 24 (2009) –
Table 2Transport equations used in the SEDE model
Zero electric current condition (steady state)
(9)
Transport equations
(10)
Concentration gradients inside pores
(11)
Electrical potential gradient inside pores
(12)
with [10],
(slit-like pores) (13)
(slit-like pores) (14)
F z ji ii
∑ = 0
Ki ii i i
i,
. . .
. .c =
− + −
+ −−φ
λ λ λ
λ λ1
2 3 4
5
1 3 02 5 776 12 3675
18 9775 15 2185 ii i6 74 8525+
⎛
⎝⎜⎜
⎞
⎠⎟⎟. λ
Ki ii i i
i i
,
ln .
. . .d =
+ −
+ − +
−φλ λ λ
λ λ
1
3 4
19
161 19358
0 4285 0 3192 0 08428λλi5
⎛
⎝
⎜⎜⎜
⎞
⎠
⎟⎟⎟
ddx
z K D dcdx
JA
z K c
FRT
z c K D
i i d ii V
ki
ii c i
i
i i i d ii
ψ =− +∞
∞
∑∑
∑
, , ,
, ,
2
dcdx
JK D A
K c c x z FcRT
ddx
i V
i d i ki c i i
i i= −( ) −∞
+
, ,
, ( )Δ ψ
j K D dcdx
z c K D FRT
ddx
K cVJ c x
Ai i d ii i i i d i
i c iV i
k
= − − + =( )
∞∞
+
, ,
, ,
,
ψ Δ
5 396 407400
A. Szymczyk et al. / Desalination 24 (2009) –
0.0
0.2
0.4
0.6
0.8
1.0
−500 −375 −250 −125 0 125 250
X (eq/m3)
R intWith images
Without images
Fig. 1. Intrinsic rejection rate (Rint
) vs. membrane volume charge density (X); 0.0005 M CaCl2
solution; rp
= 0.5 nm;
Δx/Ak = 5 μm; εm = 3; JV = 10–5 m/s; no Born effect.
0.0
0.2
0.4
0.6
0.8
1.0
1 × 10–3 1 × 10–2 1 × 10–1 0 1 × 101 1 × 102 1 × 103
−X (eq/m3)
Rin
t
With image charges - no BorneffectNo dielectric exclusion
Fig. 2. Variation of the intrinsic rejection rate (Rint
) of negatively charged membranes with the membrane volume charge
density (X); 0.0005 M CaCl2
solution; rp
= 0.5 nm; Δx/Ak = 5 μm; εm = 3; JV = 10–5 m/s.
5 396 407 401
A. Szymczyk et al. / Desalination 245 (2009) 396–407
0
1 × 10–1
4 × 10–1
6 × 10–1
8 × 10–1
1 × 100
0 5 × 10–2 1 × 10–1 2 × 10–1
exp(−zCl-ΔΨD)
exp(
–ΔW
’ im, C
l–)
Increasing Donnan exclusion
Uncharged membraneDecreasing
image exclusion
X (eq/m3) −1000 −60 −0.6 −0.03 0
Fig. 3. Image charge partitioning coefficient vs. Donnan partitioning coefficient for co-ions at the feed/membrane inter-
face; rp
= 0.5 nm; εm = 3; no Born effect; solid line: 0.0005 M CaCl2
solution; dashed line: hypothetic case for which the
increase in the Donnan exclusion would be exactly balanced by the decrease in the image charge contribution.
represented by the solid line. For low charge den-
sities, the solid line is under the dashed line,
which means that the increase in the Donnan
exclusion is not balanced by the decrease in the
DE. This partitioning effect therefore tends to
increase the membrane rejection rate. It must be
stressed that Fig. 3 cannot explain the first
decrease of the rejection rate observed in Fig. 2 at
very low membrane charge densities. The reason
is that the argument based on the results shown
in Fig. 3 takes into account only the partitioning
effect at the membrane/solution interface and it
totally disregards the transport phenomena
through the membrane (i.e. the influence of the
induced transmembrane electric field on the
rejection of charged solutes). Nevertheless, Fig. 3
is helpful to explain the behaviour of more
strongly charged membranes. Indeed, for higher
charge densities, it can be seen that there is a
sharp increase of the solid line which goes over
the dashed line. It means that there is a range of
charge densities for which the increase in the
Donnan exclusion is overcompensated by the
decrease in the DE and so, the rejection rate
decreases even if the membrane charge increases
(as can be seen in Fig. 2). For even more strongly
charged membranes, the situation is reversed (i.e.
the increase in the Donnan exclusion is no more
balanced by the decrease in the DE) and the rejec-
tion rate is expected to be an increasing function
of the membrane charge. This is in agreement
with the results shown in Fig. 2. It is worth men-
tioning that this situation is characterized by par-
titioning coefficients due to image charges close
402
A. Szymczyk et al. / Desalination 245 (2009) 396–407
0
0.2
0.4
0.6
0.8
1.0
−X (eq/m3)
Rin
t
1 × 10–3 1 × 10–2 1 × 10–1 1 × 100 1 × 101 1 × 102 1 × 103
Fig. 4. Influence of the Born effect on the intrinsic rejection rate (Rint
); 0.0005 M CaCl2
solution; rp
= 1 nm; Δx/Ak = 5
μm; εm = 3; JV = 10–5 m/s; (–): εp = 40; (×): εp = 50; (○): εp
= 60; (▲): εp = 70; (�♦): no Born effect.
to 1 (see Fig. 3). Otherwise stated, the interaction
with image charges is totally screened by the
Donnan exclusion and so, the rejection rate is
expected to tend to the value that is predicted by
the standard Donnan theory (see Fig. 2).
According to Eq. (4), the Born effect (i.e. the
work required to transfer an ion from the bulk
solution to the pore inside) is stronger as the
effective dielectric constant inside pores (εp)
decreases. This leads to higher rejection rates (see
Fig. 4) because of the stronger exclusion of both
co-ions and counter-ions from the membrane
pores (as shown by Eq. (4), the Born effect varies
with the square of the ion charge and so, it expels
both cations and anions from the membrane pores
unlike the Donnan exclusion).
Fig. 5 shows the rejection rate of a 0.0005 M
calcium chloride solution by a polyamide mem-
brane, labelled AFC 40 (PCI Membrane Sys-
tems). The mean pore size of the membrane as
well as its thickness to porosity ratio were esti-
mated by fitting the experimental rejection of
neutral solutes according to the conventional pro-
cedure described in the literature [13,14]. Con-
sidering slit-like pores, the pore half-width (rp)
and the thickness to porosity ratio (Δx/Ak) was
found to be 0.54 nm and 7.8 μm, respectively.
The membrane volume charge density, X, was
inferred from tangential streaming potential
measurements by Szymczyk et al. [15]. This elec-
trokinetic characterization indicated that the
membrane is negatively charged in the calcium
chloride solution (X = –41.3 eq/m3). Both struc-
tural and electrical properties of the membrane
were first used to compute the theoretical rejec-
tion rate by neglecting dielectric effects (i.e. using
403
A. Szymczyk et al. / Desalination 245 (2009) 396–407
0
0.25
0.5
0.75
1
0 5 × 10–6 1 × 10–5 1.5 × 10–5 2 × 10–5 2.5 × 10–5 3 × 10–5
Jv (m s−1)
Rin
tt Experimental
SEDE model
Model withoutDE
Model: best fitwith X < 0 andwithout DE
Model: best fitwith X > 0 andwithout DE
Xfit = + 2 eq/m3
Xfit = − 650 eq/m3
Fig. 5. Experimental and theoretical intrinsic rejection rates of the AFC 40 membrane vs. permeate volume flux (JV);
0.0005 M CaCl2
solution; rp
= 0.54 nm; Δx/Ak = 7.8 = μm; εm = 3; X = –41.3 eq/m3 [15].
the standard Donnan theory). As can be seen in
Fig. 5, the rejection rate of calcium chloride is
highly underestimated when dielectric effects are
disregarded whereas a satisfying description of
the experimental data is obtained from the SEDE
model (i.e. by taking into account both image
charges and Born effect). The results of the SEDE
model shown in Fig. 5 were obtained by decreas-
ing the dielectric constant inside pores down to a
value of 69. We also tried to fit the experimental
data by neglecting dielectric effects but using the
volume charge density as a fitting parameter
(instead of using the approximate value inferred
from electrokinetic measurements). It can be
noted from Fig. 5 that a fair description of the
experimental data can be obtained by using either
a positive or a negative membrane charge density.
Of course, the positive value of X is not relevant
since it is in contradiction with the experimental
measurements. The best fit obtained by consider-
ing a negatively charged membrane led to X = –
650 eq/m3. Tangential streaming potential
measurements led to a much smaller value (X = –
41.3 eq/m3). It is true that tangential streaming
potential measurements can only provide a rough
estimate of the real membrane volume charge
density inside pores. But because of phenomena
like the so-called charge regulation process [16],
the true charge density inside pores is expected
to be lower than the value inferred from tangen-
tial streaming potential measurements, which can
be viewed as the upper limit of the real membrane
charge density.
Similar qualitative results were obtained with
another polyamide membrane (AFC 30 from PCI
Membrane Systems) in a more concentrate (0.01
M) calcium chloride solution. At this concentra-
tion, the concentration polarization was found to
404
be non-negligible as shown in Fig. 6. It was taken
into account within the scope of the standard film
theory according to the procedure described in
[17]. Structural features (i.e. rp
and Δx/Ak) of the
membrane sample were determined by Bouranene
et al. [18] and a negative volume charge density (X= –50 eq/m3) was determined from tangential
streaming potential measurements. The results of
the SEDE model shown in Fig. 6 were obtained
from the experimental membrane volume charge
density and a fitted value of the dielectric constant
inside pores equal to 70. It can be noted that the
description of the experimental data without tak-
ing into account the DE requires a huge and phys-
ically unrealistic (fitted) membrane charge density
(around –9000 eq/m3) unless a positive charge is
considered (but this latter is in contradiction with
the negative charge density obtained from electro-
kinetic measurements).
Consequently, it can be concluded that the
DE plays a significant role in the separation
performances of both the AFC 30 and AFC 40
membranes.
4. Conclusion
The SEDE (Steric, Electric and Dielectric
Exclusion) model was used to investigate the
contribution of Donnan and non-Donnan exclu-
sion mechanisms in the nanofiltration of asym-
metric electrolytes. The SEDE model is a 1-D
transport model that considers the separation of
solutes as being the result of both interfacial phe-
nomena (including steric hindrance, Donnan
exclusion and dielectric exclusion through both
the Born effect and the image charges) and trans-
port effects described by means of the so-called
extended Nernst–Planck equations. In the case of
asymmetric electrolytes with divalent counter-
ions, the SEDE model predicts that the rejection
rate is a non-monotonous function of the volume
charge density. It was shown that there is a range
A. Szymczyk et al. / Desalination 245 (2009) 396–407
0
0.25
0.5
0.75
1
0 5 × 10–6 1 × 10–5 1.5 × 10–5 2 × 10–5 2.5 × 10–5 3 × 10–5
Jv (m s-1)
Rej
ecti
on
Experimental: observed rejection rate
Experimental: intrinsic rejection rate
SEDE model
Model: best fit with X < 0 and without DE
Model: best fit with X > 0 and without DEXfit = +34 eq/m3
Xfit = -9020 eq/m3
Fig. 6. Experimental and theoretical intrinsic rejection rates of the AFC 30 membrane vs. permeate volume flux (JV);
0.01 M CaCl2
solution; rp
= 0.43 nm [17]; Δx/Ak
= 4.2 μm [17]; εm
= 3; X = –50 eq/m3.
405
of membrane volume charge densities for which
the increase in the Donnan exclusion is overcom-
pensated by the strong decrease in the interaction
between ions and their images, which leads to a
decrease in the rejection rate although more fixed
charges are brought onto the pore walls. It was
also shown that the separation performances of
two NF polyamide membranes (AFC 30 and AFC
40) cannot be described by neglecting the dielec-
tric exclusion.
Nomenclature
Ak porosity of the membrane active layer
ci concentration of ion i
Di,∞ bulk diffusion coefficient of ion i at
infinite dilution
e elementary charge
F Faraday constant
I ionic strength
ji molar flux density of ion i
JV permeate volume flux
k Boltzmann constant
Ki,c hydrodynamic coefficient accounting
for the effect of pore walls on convec-
tive transport
Ki,d hydrodynamic coefficient for hindered
diffusion inside pores
NA
Avogadro number
ri, cavcavity radius of ion i
ri, StokesStokes radius of ion i
rp
pore half-width
R ideal gas constant
Ri rejection rate of ion i
T temperature
V fluid velocity inside pores
x axial coordinate
X effective volume charge density inside
the membrane pores
zi charge number of ion i
Greek symbols
ΔΨ dimensionless Donnan potential
Δx effective thickness of the active layer
ΔW’i,Borndimensionless excess solvation energy
due to Born effect for ion i
ΔW’i,im dimensionless excess solvation energy
due to “image charges” for ion i
ε0
vacuum permittivity
εb dielectric constant of the bulk solution
outside pores
εm dielectric constant of the membrane
active layer
εp effective dielectric constant inside
pores
φi steric partitioning coefficient for ion i
γi activity coefficient for ion i
�κ: Debye parameter
μ: effective dimensionless reciprocal
screening length for interaction
between ions and induced polarization
charges
ψ local electrical potential inside pore
References
[1] A.E. Childress and M. Elimelech, Effect of solutionchemistry on the surface charge of polymericreverse osmosis and nanofiltration membranes, J. Membr. Sci., 119 (1996) 253–268.
[2] S. Bandini, Modelling the mechanism of charge for-mation in NF membranes: theory and application,J. Membr. Sci., 264 (2005) 75–86.
[3] S. Senapati and A. Chandra, Dielectric constant ofwater confined in a nanocavity, J. Phys. Chem. B,105 (2001) 5106.
[4] A.E. Yaroshchuk, Dielectric exclusion of ions frommembranes, Adv. Colloid Interface Sci., 85 (2000)193.
[5] R.J. Gross and J.F. Osterle, Membrane transportcharacteristics of ultrafine capillaries, J. Chem.Phys., 49 (1968) 228–234.
[6] A. Szymczyk and P. Fievet, Investigating transportproperties of nanofiltration membranes by means ofa Steric, Electric and Dielectric Exclusion model, J. Membr. Sci., 252 (2005) 77.
[7] A. Szymczyk, M. Sbaï and P. Fievet, Transportproperties and electrokinetic characterization of anamphoteric nanofilter, Langmuir, 22 (2006) 3910.
[8] A.A. Rashin and B. Honig, Reevaluation of theBorn model of ion hydration, J. Phys. Chem., 89(1985) 5588.
A. Szymczyk et al. / Desalination 245 (2009) 396–407406
[9] A.E. Yaroshchuk, Non-steric mechanisms ofnanofiltration: superposition of Donnan and dielectric exclusion, Sep. Purif. Technol., 22–23(2001) 143.
[10] P. Dechadilok and W.M. Deen, Hindrance factorsfor diffusion and convection in pores, Ind. Eng.Chem. Res., 45 (2006) 6953.
[11] A. Szymczyk, C. Labbez, P. Fievet, A. Vidonne, A.Foissy and J. Pagetti, Contribution of convection,diffusion and migration to electrolyte transportthrough nanofiltration membranes, Adv. ColloidInterface Sci., 103 (2003) 77.
[12] A.E. Yaroshchuk, Negative rejection of ions in pres-sure-driven membrane processes, Adv. ColloidInterface Sci., 139 (2008) 150.
[13] J. Schaep, C. Vandecasteele, A.W. Mohammad andW.R. Bowen, Analysis of the salt retention ofnanofiltration membranes using the Donnan-StericPartitioning Pore, Model, Sep. Sci. Tech., 34 (1999)3009.
[14] C. Labbez, P. Fievet, A. Szymczyk, A. Vidonne, A.Foissy and J. Pagetti, Analysis of the salt retentionof a titania membrane using the “DSPM” model:effect of pH, salt concentration and nature, J. Membr. Sci., 208 (2002) 315.
[15] A. Szymczyk, N. Fatin-Rouge, P. Fievet, C. Ram-seyer and A. Vidonne, Identification of dielectriceffects in nanofiltration of metallic salts, J. Membr.Sci., 287 (2007) 102.
[16] J.N. Israelachvili, Intermolecular and SurfaceForces, Academic Press, 1985.
[17] S. Bouranene, P. Fievet, A. Szymczyk, M. El-HadiSamar and A. Vidonne, Influence of operating condi-tions on the rejection of cobalt and lead ions in aque-ous solutions by a nanofiltration polyamidemembrane, J. Membr. Sci., 325 (2008) 150.
[18] S. Bouranene, P. Fievet, A. Szymczyk and A.Vidonne, Investigating nanofiltration of multi-ionicsolutions using the SEDE model, Chem. Eng. Sci.,submitted for publication.
A. Szymczyk et al. / Desalination 245 (2009) 396–407 407