Chemical Engineering Science 66 (2011) 2898–2911

Contents lists available at ScienceDirect

Chemical Engineering Science

0009-25

doi:10.1

n Corr

E-m

journal homepage: www.elsevier.com/locate/ces

Electrical characterization of NF membranes. A modified model with chargevariation along the pores

Veronica Silva a, Alvaro Martın a, Fernando Martınez a, Jorge Malfeito b, Pedro Pradanos a, Laura Palacio a,Antonio Hernandez a,n

a Grupo de Superficies y Materiales Porosos, Dpto. Fısica Aplicada, Facultad de Ciencias, Universidad de Valladolid, 47071 Valladolid, Spainb R&D Department, Acciona Agua, Parc de Negocis Mas Blau II, Avda. de Les Garrigues, 22-2a, 08820 El Prat de Llobregat, Barcelona, Spain

a r t i c l e i n f o

Article history:

Received 17 August 2010

Received in revised form

2 February 2011

Accepted 14 March 2011Available online 30 March 2011

Keywords:

Nanofiltration

Modified SEDE model

Dielectric exclusion

Variable charge density

Membrane potential

Active and support layer

09/$ - see front matter & 2011 Elsevier Ltd. A

016/j.ces.2011.03.025

esponding author. Tel.: þ34 983423134; fax

ail address: membrana@termo.uva.es (A. Her

a b s t r a c t

Nanofiltration rejection measurements have been performed for KCl solutions to characterize a non-

commercial NF membrane. Membrane Potential (MP) and Tangential Streaming Potential (TSP)

measurements have also been carried out for a wide range of concentrations. The charge of the

membrane and the pore dielectric constant has been obtained as functions of the salt concentration

from these three ensembles of experiments, showing a fair agreement.

For the data treatment, two modifications of the Steric Electric and Dielectric Exclusion (SEDE)

model have been proposed to allow the electrical characterization of the NF membrane: (1) the

volumetric charge density is considered to change along the pores by means of the inclusion of a charge

isotherm (which correlates the membrane charge with the ion concentration) in the transport

equations, and (2) the porous support is considered by the inclusion of the adequate transport

equations similarly to how the active layer is considered.

& 2011 Elsevier Ltd. All rights reserved.

1. Introduction

The use of nanofiltration (NF) membranes is steeply increasingin the separation industrial processes, which leads to a consider-able effort in research. Apart from the economical advantages dueto the decrease of operational costs when using nanofiltrationinstead of more traditional procedures, the improvement ofperformances makes them particularly useful. Their specificfeatures are due to the combination of size and electrical effectsbecause their pores are typically near 1 nm in diameter and havefixed charges. Due to these characteristics, nanofiltration mem-branes retain multivalent complex ions and transmit smalluncharged solutes and low charged ions. The electric charge of aNF membrane plays an important role in the charge separationduring a filtration process due to the formation of electricaldouble-layers that are comparable or bigger than the pore size.The charge density can be obtained by different techniques liketangential streaming potential (TSP) (Martınez et al., 2002; Silvaet al., submitted for publication), pore conductivity (PC) (Fievetet al., 2001) and Membrane Potential (MP) (Fievet et al., 1999).These three possible ways to characterize the electric condition ofthe membrane were applied and compared by Sbaı et al. (2003).

ll rights reserved.

: þ34 983423136.

nandez).

In this work TSP and MP measurements are conducted andanalyzed for a non-commercial membrane. It is important toremember that the TSP technique provides information on the topsurface of the membrane active layer while the MP method givesinformation on the inner part of the membrane, which includesthe active layer and also the support layer. Of course thenanofiltration models describe the transport of ionic speciesthrough the membrane and its rejection considering the innercharge, thus MP should, in principle, lead to a more accuraterepresentation than TSP of the charge properties that are relevantfor NF. In any case both these ways give complimentary portraitsof the electrical properties of nanofiltration membranes, whichare crucial to understand NF experiments.

The Space Charge Model (SCM) is one of the most usedapproaches to investigate the membrane potential phenomenon(Dresner, 1963; Gross and Osterle, 1968; Sbaı et al., 2003;Westermannclark and Anderson, 1983). The Teorell–Meyer–Sie-vers (TMS) model is also used due to its simplicity. It has beenproved that in the NF range the TMS and SCM modeling givesubstantially equivalent results (Shang et al., 2006). It was alsodemonstrated that the two models agree with each other forslightly charged membranes with pore radii of less than 5.0 nm.While the TMS model seems to overestimate the membranepotential for pore sizes over 5 nm. The work of Shang and co-workers was based on a previous work of Lefebvre et al. (2004),where a method slightly simpler than SCM was proposed. Theweakness of using the SCM or the TMS models is that they

V. Silva et al. / Chemical Engineering Science 66 (2011) 2898–2911 2899

consider neither the hindered nature of the transport inside thepores nor the dielectric effects at the interfaces. Recently, theSteric Electric and Dielectric Exclusion (SEDE) model has beenused to obtain the charge volume density X using also MPmeasurements (Lanteri et al., 2008,2009). Here we will followthis way using the same model used for nanofiltration experi-ments in order to study the membrane potential results.

Actually, NF will be analyzed here using the SEDE model withtwo very relevant modifications that will be proven to be realimprovements. These modifications are

�

Firstly, the volumetric charge density inside the pores, X, willbe considered as a variable along the pore by the inclusion of acharge isotherm in the transport equations. This isothermcorrelates the membrane charge with the ion concentrationand consequently with the axial coordinate along the pores. � Moreover, the porous support will be taken into account by

the inclusion of a specific term in the transport equations.

The results of the enhanced model that we will call SEDE-VChwill be compared with that without these modifications.

The charge isotherm will be obtained from the TSP results andsubsequently used in the SEDE-VCh model in order to fit thedielectric constant inside the pores. It has been shown that thesetwo parameters (charge X and dielectric constant ep) cannot besimultaneously fitted to experimental results on retention bynanofiltration of univalent salts (Lanteri et al., 2008). This is whyan independent method to get the volumetric charge densityshould help. Actually, ep is the most ambiguous parameter in theSEDE model and it has a very strong dependency on the geometryof the pores (Silva et al., submitted for publication) and onconcentration. There is a certain controversy in common litera-ture on how to model the dielectric effects inside nanofiltrationpores because this parameter is not easy to obtain independently.

On the other hand, the SEDE model, with the mentionedadditions of the charge variation and the transport throughoutthe support (SEDE-VCh model), can be used to fit the bestFreundlich isotherm to reproduce the experimental results. More-over measurements of membrane potential can be modeled onthe same basis. All these results on the charge isotherm and thedielectric constant dependence on concentration can be com-pared and the advantages of the SEDE-VCh model revealed.

2. Theory

2.1. Nanofiltration model

In this section we will describe the basic features of the SEDEmodel and the modifications that will be proposed and used hereto study, as an example, a non-commercial nanofiltration mem-brane. Fig. 1 shows schematically the concentration profile acrossboth the active and support layers of the membrane.

The assumption of equal volumetric charge density all alongthe pores has been considered from the very beginning of NFmodeling (Bowen et al., 1997) and retained in the SEDE model.However, such an assumption is actually unreasonable becauseevidently a concentration gradient appears through the mem-brane and charge and concentration are known to be correlated(Adamson and Gast, 1997; Bowen et al., 2002). Thus, conse-quently, a charge gradient has to be considered as well. Here,these variations are taken into account by means of the inclusionof a Freundlich isotherm (Eq. (1)) in the transport equations(Calvo et al., 1996; Bowen et al., 2002):

X ¼ X0þacb2 ð1Þ

The subscript 2 refers to the anion (here Cl�), which is usuallylarger than the cation and easily delocalizes its own chargeleading to a preferential attachment on the membrane walls ifno or low proper charge is present (X0E0). For this reasonthe isotherm is customarily correlated only with the anionconcentration.

Here we will include the charge isotherm, as given by Eq. (1),in the SEDE model to correlate the charge volume density,through a and b, with concentration. Thus leading to a substitu-tion of X by a and b as fitting parameters and letting X to changewith the x coordinate along the pores.

The influence of the support layer has not been commonlytaken into account. For the sake of a complete description of thenanofiltration transport, the possible relevance of the transportthrough the support layer will be also included in the model here.

2.1.1. Transport equations in the active layer

The concentration variation through the membrane, obtainedfrom the Nernst–Planck equation for the transport, can becalculated with the following equation:

dci

dx¼

Jv

Di,pAkðK 0i,cci�ci,pÞ�

ziF

RTci

dcdx

ð2Þ

Di,p¼Ki,dDi,N is the diffusivity inside the pores corrected with thediffusion hindrance factors, Ki,d, for both the ions i¼1 and 2. TheK 0i,c parameter is the convective hindrance parameter.

The electroneutrality condition inside the membrane isXi

ciziþX ¼ 0 ð3Þ

Substituting Eq. (1) into Eq. (3) and after derivation:

Xi

zidci

dxþbacðb�1Þ

2

dc2

dx¼ 0 ð4Þ

Then, the electric potential gradient can be written, accordingto Eqs. (2) and (4), as

dcdx¼

Pi

ziJv

Di,pAk

� �ðK 0i,cci�ci,pÞþ

Jvbacðb�1Þ2

D2,pAk

� �ðK 02,cc2�c2,pÞ

FRT ðP

iciz2i þz2bacb

2Þð5Þ

The hindrance factors: Kid and K0ic are taken as functions of theratio l¼ri/rp (for slit-like pores rp¼0.5h, h being the thickness ofthe slit while for cylindrical geometry rp being the pore radius).These hindrance factors will be calculated here using the correla-tions proposed (Dechadilok and Deen, 2006; Silva et al., 2009) inTable 1.

Eqs. (2) and (5) can be numerically solved to obtain thepotential and concentration profiles across the membrane withthe following boundary conditions:

ciðx¼ 0Þ ¼ ci,0

ciðx¼DxÞ ¼ ci,Dx ð6Þ

The 0 and Dx subscripts refer to the feed and permeateinterfaces inside the membrane, respectively.

2.1.2. Partitioning equations

At thermodynamic equilibrium, a difference of solute concen-tration is generated from one side to the other at each interface.This concentration discontinuity can be described, for the x¼0interface, as

ci,0

ci,m¼fi

gi,m

gi,0

expð�ziDCÞexpð�DW 0i,BÞexpð�DW 0

i,imÞ ð7Þ

A similar equation holds for the x¼Dx interface. In theseequations all partitioning effects are included: fi takes intoaccount the steric effect, DC represents the Donnan potential,

Fig. 1. Schematic representation of both active and support layers for NF measurements.

Table 1Hindrance factor for diffusion and convection.

In a recent work, Dechadilok and Deen (2006) showed different correlation based on averaged sections for diffusion and convection that are valid for values of lup to 0.95.

Hindrance factors for slit geometry

Kd ¼1þð9=16Þl lnl�1:19358lþ0:4285l3

�0:3192l4þ0:08428l5

1�l

Kc ¼1�3:02l2

þ5:776l3�12:3675l4

þ18:9775l5�15:2185l6

þ4:8525l7

1�l

K 0i,c ¼ Ki,cþð2�fÞKi,d2l2

3

!

Hindrance factors for cylindrical geometry

Kc ¼1þ3:867l�1:907l2

�0:834l3

1þ1:867l�0:741l2ðA1:17Þ

Kd ¼1þð9=8Þl lnl�1:5603lþ0:52815l2

þ1:9152l3�2:8190l4

þ0:27078l5þ1:10115l6

�0:43593l7� �

ð1�lÞ2ðA1:18Þ

K 0i,c ¼ Ki,cþð2�fÞKi,d16l2

9

!

V. Silva et al. / Chemical Engineering Science 66 (2011) 2898–29112900

and the dielectric effects are considered as separable in a Borncontribution, DW0i,B, and a contribution corresponding to theeffect of the image forces DW0i,im (Szymczyk and Fievet, 2005;Yaroshchuk, 2000). The expressions customarily used for thesedielectric effects are shown in Table 2. The deviation from idealityof the bulk solutions at both sides of the membrane is also takeninto account in Eq. (7) through the activity coefficients, gi,m andgi,0, that can be calculated according to the Debye–Huckel theory(Robinson and Stokes, 2002) for diluted solutions up to 0.1 M ofionic strength.

The electroneutrality conditions state thatXi

ciðxÞziþX0þacb2ðxÞ ¼ 0 0rxrDx ð8Þ

Xi

cmizi ¼ 0 andX

i

cpizi ¼ 0 ð9Þ

From Eqs. (7) to (9), it is possible to calculate numerically: DC,ci,0 (the concentration just inside the membrane) and ci,Dx.

2.1.3. Transport equation through the support layer

In the support layer the electric charge density is considered tobe zero, which seems reasonable when the pores are wide enough

to give minimum effects of the charged walls. This can beassumed when concentrations are high enough as to give com-paratively small double layer thickness and/or when the charge isactually very low as it happens for low concentrations. TheNernst–Planck equations and electroneutrality equations describ-ing the transport through the support layer can be written as

dcs,i

dxs¼

Jv

Ds,ipAs,kðK 0s,i,ccs,i�ci,pÞ�

ziF

RTcs,i

dcs

dxsð10Þ

Xi

cs,izi ¼ 0 ð11Þ

The s-subscript refers to the support layer.Moreover, the electric potential gradient can be derived from

Eqs. (10) and (11):

dcs

dxs¼

Pi

ziJv

Ds,ipAs,k

� �ðK 0s,i,ccs,i�ci,pÞ

FRT

Pics,iz

2i

ð12Þ

cs,iðx¼ 0sÞ ¼ cs,i,0

cs,iðx¼DxsÞ ¼ ci,p ð13Þ

Table 2Summary of the equations used to evaluate the dielectric exclusion effects.

The dielectric Born energy (DWi,B)

DW 0i,B ¼

ðzieÞ2

8pkBTe0as

1

ep�

1

eb

� �ðII:1Þ

where as is the cavity radius defined by Rashin and Honig (1985) as the distance from the center of the ion to the point where the dielectric constant becomes different

than the vacuum one, e0.

The dielectric image forces energy (DWi,im).

This effect depends on the geometry of the pores:

DW 0i,im ¼

2ai

p

Z 10

K0ðkÞK1ðnÞ�bK0ðnÞK1ðkÞ

I1ðnÞK0ðkÞþbI0ðnÞK1ðkÞdk ðCylindrical geometryÞ ðII:2Þ

DW 0i,im ¼�ai ln 1�

ep�em

epþem

� �expð�2mÞ

� �ðSlit-like geometryÞ ðII:3Þ

where,

ai ¼ðziFÞ

2

8pe0epRTNArpðII:4Þ

n¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffik2þm2

qðII:5Þ

m¼ Frp

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiXi

z2i ci,mfiðgi,m=gi,0Þexpð�ziDC�DW 0

i,B�DW 0i,imÞ

RTe0eb

vuut ðII:6Þ

b¼kffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

k2þm2p em

ep

� �ðII:7Þ

The K0, K1, I0 and I1 are the modified Bessel functions, and em is the dielectric constant of the dry polymer of the membrane, which is usually a constant value obtained

from bibliography.

V. Silva et al. / Chemical Engineering Science 66 (2011) 2898–2911 2901

In these equations the hindrance factors for diffusion andconvection in the support layer have been included, but they alsocould be considered as being 1 since the pore size in this layer ismuch higher than the solute size and the ratio li¼ri,s/rpffi0.

2.1.4. System resolution

The model described above will be called here SEDE-VCh withVCh meaning that it includes the variation of the volumetric chargedensity along the pores. The model is totally defined in terms ofthree independent parameters: the dielectric constant inside thepores (ep) and the two parameters of the Freundlich isotherm(a and b). The model resolution algorithm is shown in Fig. 2.Although the inclusion of these parameters adds some complexityto the mathematical resolution of the system, it is clear that a and b

parameters are unique for a given solute and membrane system, sothey can be easily calculated. When X0 is included as a fittingparameter it results to be so small that it can be considered as zerowithin the error range, thus it has been fixed as zero. This does nothave any effects on the other parameters.

Finally, the theoretical intrinsic rejection of the ion is calcu-lated as

Ri,T ¼ 1�ci,p,T

ci,mð14Þ

where ci,p,T is the theoretical permeate concentration.

2.2. Concentration polarization

The effects of concentration polarization can be calculatedusing the model developed by Geraldes and Afonso (2007). Theyproposed the inclusion of suction effects and an additional termthat accounts for the ion migration due to the electric potentialappearing in the polarization layer on the membrane. If weassume that the concentration and the electric potential profilesare both linear without any convection or friction, we can use theNernst–Planck extended relationship (Eq. (2)) without hindrance

for neither convection nor diffusion (K0 i,c¼1 and Ki,d¼1) and withAk¼1. If we assume that both the gradients are linear and wedefine x¼Dc/d being the thickness of the concentration polariza-tion layer, then

Jvci,mR¼ Ki,mðci,m�ci,bÞþzici,mDi,1F

RTx ð15Þ

The mass transfer coefficient is Ki,m¼Di,N/d. The electroneu-trality conditions are those shown in Eq. (9).

Ki,m can be obtained using the correlation of dimensionlessnumbers:

Shi ¼ 2:24dh

L

� �1=3

ðReUSciÞ1=3

ð16Þ

This correlation of the dimensionless numbers of Sherwood,Shi¼Ki,mdh/Di,N, Reynolds, Re¼rvdh/Z, and Schmidt, Sci¼Z/rDi,N

(Z and r are the solution viscosity and density, respectively) can beused because the regime is laminar (Re¼1180) and the velocity andconcentration limit layers develop at Lv¼0.029dhRe¼0.034 m andLc¼0.6vdh3/D¼635 m, being thus LvoL and Lc4L (Cheryan, 1998).

This Ki,m is applicable to the case of an impermeable wallmembrane, but when the effect of permeation flux is considered,Ki,m can be corrected by taking into account the suction effect asdescribed in the work of Geraldes and Afonso (2006).

2.3. Membrane potential

The membrane potential technique is based on the diffusivetransport of ions through the membrane induced by an electro-lyte concentration gradient without any pressure gradientapplied. When a membrane is placed between two solutions withdifferent concentrations of the same electrolyte, the electricalpotential difference generated is the Membrane Potential (MP).

When there is not any applied pressure, the ions pass throughthe membrane from the higher to the lower concentrationregion and water moves in the opposite direction due to the

Optimization Parameters:

εp, a, b

Independent equations

Transport equations

Transport equations

in the feed interface

∗ Born Energy* Viscocity

∗ Δx/Ak

Partitioning effects in the feed interface

Partitioning effects in the permeate

interface

Transport equationsresolution

Input constant values:* Membrane: rp, εm, Lp

* Ions: rs, rc, z, Di* Operation: T, Jv, cm

ε

ΔWim, ΔWD,0Ci,0

ΔWim, ΔWD,ΔxCi,Δx

Ci,0Relative error

Less than TOL?

R = 1- Cm/Cp

Function errorLess than TOL?

SOLUTIONOF THE

OPTIMIZATION

Input experimentalvalues of rejection

and permeate fluxes

YES

YES

NO

NO

pGuess

Fig. 2. Nanofiltration resolution algorithm.

V. Silva et al. / Chemical Engineering Science 66 (2011) 2898–29112902

osmotic effects until a steady state is reached. Under theseconditions the transport in the support layer of the membranehas to be considered although the charge, as mentioned, could betaken as negligible.

To perform the MP measurements, a membrane is placed in asuitable cell and an electrode is positioned at each side of themembrane in order to measure the cell potential (Ec), which isrelated to membrane potential (Em) by

Em ¼ Ec�Es ð17Þ

where Es is the concentration potential, which is the electricalpotential difference between the high and the low concentratedsolutions measured separately against a reference electrode:

Es ¼ Ehigh�Elow ð18Þ

The membrane potential, Em, is used to calculate the surface, s,or volumetric charge density, X.

According to the TMS theory, the membrane potential (Em)consists of the difference of the two Donnan potentials at theinterfaces (DCD) and the diffusion potentials inside the active

layer (DcA) and within the support layer (DcS):

Em ¼DCDxD �DC

0DþDcAþDcS ð19Þ

2.3.1. SEDE-VCh model for membrane potential calculation

The electric potential terms shown in Eq. (19) are calculatedusing a rearrangement of the SEDE-VCh model. This reorganiza-tion is done in the transport equations through the membraneand in the concentration polarization equations.

In order to calculate the diffusion potential in the active layer, theextended Nernst–Planck equation of transport must be rearrangedconsidering that, in the case of membrane potential, the ion fluxes(Ji) of all the ions in presence are unknown variables (instead of ci,p

for the NF case that now are known) and they cannot be correlatedwith Jv because now Jv is an osmotic flux (of solvent) going from thepermeate side to the feed side and there is a fix concentration in thepermeate side, thus (ci,pJv)/AkaJi. So, with these considerations andfollowing a similar procedure as in Section 2.1.1— i.e. assuming thatthe volumetric charge density is correlated with concentration alongthe pore by the Freundlich isotherms—the expressions for the

V. Silva et al. / Chemical Engineering Science 66 (2011) 2898–2911 2903

concentration and electric potential profiles are

dci

dx¼ðK 0i,cJvci�AkJiÞ

Di,pAk�

ziF

RTci

dcdx

ð20Þ

dcdx¼

Pizi

JvK 0i,c

ci�AkJi

Di,pAk

� �þ

bacðb�1Þ2

D2,pAk

� �ðJvK 02,cc2�AkJ2Þ

FRT ðP

iciz2i þz2bacb

2Þð21Þ

Nevertheless in membrane potential measurements the Jv

value is that corresponding to the osmotic flux, which dependson the osmotic pressure difference between the pore ends. Inmost cases, the osmotic pressure is calculated by following theVan’t Hoff theory (Robinson and Stokes, 2002), which is valid fordilute solutions:

P¼ ðn1þn2ÞcRT ð22Þ

where n1 and n2 are the stoichiometric numbers for the ions.Then, the osmotic flux is calculated using the Spiegler–Kedem

Fig. 3. Membrane potentia

equation:

Jv ¼ LwðDp�sDPÞ ð23Þ

According to Eq. (22) and (23) we have

Jv

Lw¼Dp�sRTðn1þn2ÞDc ð24Þ

Then a plot of Jv/Lw as a function of Dc for a constant Dp wouldlead to s. When this is done for the data obtained in thenanofiltration experiments, we obtain s values that are very closeto 1 (Garcia-Aleman and Dickson, 2004; Mohammad et al., 2007).Thus,

Jv ��LwDP ð25Þ

Eqs. (20), (21) and (25) can be solved numerically to obtainthe potential and concentration profiles across the membraneactive layer with the same boundary conditions given by Eq. (6).The diffusion potential in the active layer is calculated from the

l resolution algorithm.

0.200

nm

2

4

6

8

nm

V. Silva et al. / Chemical Engineering Science 66 (2011) 2898–29112904

electric potential values in the pore ends:

DcA ¼cx ¼ 0�cx ¼ Dx ð26Þ

The terms of this equation are easily obtained from the evaluatedelectric potential profile.

For the support layer, the diffusion potential Dcs is obtainedby a procedure similar to that used for the active layer but withzero electric charge density. Thus, the equations that describe thetransport through the support layer in a MP system are

dcs,i

dxs¼ðK 0s,i,ccs,iJv�AkJiÞ

Ds:iAks�

zics,iF

RT

dcs

dxsð27Þ

dcs

dxs¼

Pi

ziDs,i,pAks

� �K 0s,i,ccs,iJv�AkJi

� �F

RT ðP

ics,iz2i Þ

ð28Þ

The corresponding equations can be solved numerically toobtain the potential and concentration profiles across the mem-brane support, with the same boundary conditions of Eq. (13). Thediffusion potential in the membrane support layer is

Dcs ¼cxs ¼ 0�cxs ¼ Dxsð29Þ

The Donnan potentials at the interfaces (DCD) – appearing inEq. (19) – are calculated using the partitioning equations. Since theconcentrations in both sides are known, these equations are wellspecified and the Donnan potentials and concentrations inside themembrane can be calculated numerically. In the feed–membraneinterface the membrane concentration (ci,m) just outside the poresis obtained by taking into account the concentration polarization inthe surface. To accomplish this, Eq. (15) is rearranged as

Ji ¼�Ki,mðci,m�ci,bÞþ Jvci,m�zici,mDi,1F

RTx ð30Þ

Finally the electroneutrality conditions, to be used, areXici,mzi ¼ 0 F

XiJizi ¼ 0 ð31Þ

Eqs. (30) and (31) must be solved along with Eqs. (20)–(22),(25), (27) and (28) to get the unknown Ji. Although, the electricalpotential gradient in the film layer is small compared to the otherpotentials, it can be included in Eq. (19) as

Em ¼DCDxD �DC

0DþDcAþDcSþxd ð32Þ

Finally, using the measurements of the membrane potential(Em) and Eq. (32), it is possible to solve the parameters: ep, a and b

according to the resolution algorithm sketched in Fig. 3.

35

30 0.98 ± 0.08 nm

25

20

gaussian15

10

5

00.7 0.8 0.9 1.0 1.1 1.2 1.3 1.4

dp (nm)

coun

ts

Fig. 4. (left) AFM image and (right) the corresponding pore size distribution and

the Gaussian fitted.

3. Experimental methods

3.1. Membrane characterization

For this study a non-commercial polyamide NF membrane hasbeen used. This membrane was prepared by interfacial polymer-ization of an aromatic polyamide over a polysulfone membranemade by phase inversion in the R&D department of AccionaAgua. The membrane also has a non-woven polyester backinglayer that has been removed only for the membrane potentialmeasurements.

The pore size distribution has been obtained using AtomicForce Microscopy (AFM). Atomic Force Microscopy has beenperformed with a Nanoscope Multimode IIIa scanning probemicroscope from Digital Instruments (Veeco Metrology Inc., SantaBarbara, CA). The images have been captured in contact mode.Nanosensors silicon tips of the CONTR#19809 type have beenused. Their force constant is 0.12 N/m with a curvature radiusbelow 10 nm. A scanner A type with a maximum lateral (x, y) scanof 1 m has been used. All images have been acquired with 512

points per scan line and with 512 lines. Images of 10�10 and40�40 nm have been analyzed using the command Spectrum 2D

function, which transforms images by applying a 2D fast Fouriertransform (FFT), to pass or remove specific frequencies from theimages. This modification is necessary to eliminate the electricalnoise and some other parasite vibrations (that affect specially thehigh resolution images). The image acquisition and treatment hasbeen done with Nanoscope Software, version 5.12 rev.B.

An example of the AFM analyzed pictures is shown in Fig. 4along with the corresponding pore size distribution obtained froma big number of such images. The mean pore radius according tothese images is 0.4970.04 nm. The pore radii obtained by AFMrefer to the pore entrances as appearing on the surface of themembrane, nevertheless it has been shown to agree withthose determined by solute retention tests and liquid–liquiddisplacement porosimetry for typical nanofiltration membranes(Otero et al., 2008).

An environmental scanning electron microscopy (ESEM)Quanta 200FEG apparatus provided by FEI has been used toobtain cross section images of the membrane to determine thethickness of the support layer, which resulted to be approxi-mately 4475 mm, as shown in Fig. 5. Fig. 5b is a magnification ofthe Fig. 5a where it can be easily seen it is not possible todetermine the active layer thickness.

Fig. 5. ESEM images of transversal sections of the studied membrane at two magnifications. (a) The thickness of the membrane without the non-woven support

(previously peeled off) is shown. (b) Magnification of the active side of the membrane where it is seen that the active layer is not well defined.

p (Pa)

5

1.0

4

3

20.5

v Int

r. (m

L/g

)

dvin

tr./d

d (1

0-2m

L/g

)

1

00.0

d (nm)d (nm)

105 104 103 102 101 100 104 103 102 101 100

108107106105104

p (Pa)

104 105 106 107 108

Pore size populationsof the support layer

Fig. 6. Mercury porosimetry results (porograms) including the cumulative intrusion (a) and the differential one (b). The arrows in the differential porograms correspond to

8 and 200 nm.

V. Silva et al. / Chemical Engineering Science 66 (2011) 2898–2911 2905

In order to obtain the porosity of the support layer (Aks),mercury porosimetry measurements have been performed formembranes where their non-woven sheet was peeled off. A valueof 50% for the porosity was obtained. This porosity can beattributed to the support layer because the contribution of theactive layer to the measured porosity should be negligible. Ineffect the pores in the active layer are not detected by thistechnique because at the maximum working pressure, Hg cannotpenetrate into pores with diameters below 3 nm. In Fig. 6, thecorresponding porograms are shown. These results also provideinformation on the pore size distribution in the support layer,which results to be bimodal containing pores around 8 nm andothers with around 200 nm of diameter.

Fig. 7. Schematic representation of the experimental set-up for: (a) nanofiltration

and (b) tangential streaming potentials measurements.

3.1.1. Salt rejection measurements

The experimental rig is described in detail in Fig. 7a. Themembrane holder included in such a device is a flat sheet crossflow module with a single channel of length L¼110 mm, widthW¼60 mm and height H¼0.5 mm, resulting in a membrane areaof 0.0066 m2 (a hydraulic diameter of dh¼1 mm). The pressuredifference has ranged from 11 to 17 bar. Total recirculation hasbeen used and the permeate and concentrate are returned to thefeed reservoir. The concentrate has been recirculated with a crossflow of 120 L/h at a temperature of 25 1C (which corresponds to aspeed v¼1 m/s and a Reynolds number Re¼1180).

V. Silva et al. / Chemical Engineering Science 66 (2011) 2898–29112906

The salt rejection measurements have been made for aqueousKCl solutions with concentrations of 5�10�5, 1�10�4, 5�10�4,1�10�3, 2�10�3, 5�10�3 and 1�10�2 M, prepared with ion-exchange and reverse osmosis treated demineralized water. Feedand permeate concentrations have been obtained by conductivitymeasurements.

3.1.2. Tangential streaming potential measurements

The tangential streaming potential on the membrane is mea-sured using a membrane holder where two membranes areplaced in such a way that their active layers face each other andno permeation is allowed through them. They leave a singlechannel of length L¼75 mm, width W¼20 mm and heightH¼160 mm with a transversal area of 3.2�10�6 m2. The heightof such a channel has been evaluated from flow versus pressuremeasurements according to the Hagen–Poiseuille law.

Two Ag/AgCl Cl�-selective electrodes are placed at both thechannel entrance and exit. The electrical potentials have beenmeasured by using a high impedance (41010 O) voltmeter fromHewlett-Packard (HP3456A) with an accuracy of 1 mV. The systemis shown in Fig. 7b. The applied pressure differences in the systemwent from 1 to 5.5 bars, which correspond to tangential fluxesfrom 1 to 2.5 L/min.

3.1.3. Membrane potential measurements

The membrane potential has been measured using a cylind-rical holder where the membrane is placed leaving a circulareffective transversal area A¼3�10�5 m2. In order to reduce theconcentration polarization, it is very important to keep both sidesproperly stirred; this has been done by the recirculation of thesolution at both sides of the membrane with a centrifugal pumpwith a flux of Q¼0.6 L/min.

Membrane Cl� selective electrodes (ISE 9652 of Crisons) areplaced at both sides of the cell and connected to the same highimpedance voltmeter used for the streaming potential experi-ments. The electrodes were immersed in the solution with thelowest concentration for more than 2 h in order to stabilize thembefore each measurement.

Hydrostatic pressure is kept equal on both sides of the cell byplacing the solution reservoirs containing the adequately concen-trated salt solutions at equal height on both sides of themembrane. Temperature is controlled at 2571 1C using a ther-mostatic bath. The system is shown schematically in Fig. 8.

The concentration values used in each MP experiment wereselected from the results of the NF rejection measurements, in sucha way that the low concentration side has the same concentration

Fig. 8. Scheme of the MP measurement system.

than the permeate in the nanofiltration experiments and the highconcentration side corresponds to the concentration on the mem-brane surface in the nanofiltration experiments.

The potentials in Eq. (18) have been measured against acommercial reference electrode (Ag/ClAg Ref. 5044 of Crisons).This procedure has been followed with each electrode and amean value of them is used in the calculation in order to avoidasymmetry effects.

4. Results

4.1. Salt rejection measurements

The observed and intrinsic rejection for the different KClconcentrations versus permeate flux (Jv) is shown in Fig. 9. Thedotted lines in the figure are only an eye guide. These resultsshow the normal behavior for NF membranes with an importantcontribution of concentration polarization. Note that the trueretention increases with Jv while the observed one is much lowerand decreases with Jv.

4.2. Tangential streaming potential and membrane potential results

The TSP measurements have been analyzed using thesame calculation procedure described in a previous work (Silvaet al., submitted for publication) according to the Helmholtz–Smoluchowski equation and the Gouy-Chapman theory (Hunter,1988).

The resulting XTSP values can be correlated with concentrationaccording to a Freundlich isotherm (Eq. (1)), which representsproperly the experimental results as shown in Fig. 10. Thisequation includes the X0 value that represents the proper chargeof the membrane; the fitted Freundlich parameters are shown inthe figure. For the X0 value, the standard error has the samemagnitude than the mean value, and thus it can be assumed asnegligible.

The results for the membrane potential, Em, obtained experi-mentally using Eqs. (17) and (18), are summarized in Fig. 11 as afunction of the mean concentration at both sides of the mem-brane. The error of the potential measurements is about 3% that,considering the error propagation, leads to a mean error inmembrane potential of 9%.

1.0

0.5

Ro

(bla

ck) a

nd R

(whi

te)

0.01.5 2.0 2.5 3.0 3.5 4.0 4.5

Jv (µm/s)

Fig. 9. Intrinsic and observed rejection for the studied concentrations.

-Em

(mV)

Cmean (mol/m3)

Fig. 11. Membrane Potential as a function of the mean concentration in the

membrane (arithmetic mean of the concentration at both sides of the membrane).

Table 3Modeling fixed parameters.

Parameter Slit pore

Water permeability (m/s Pa) 3.34�10�11

Active layer pore radius (m)a 0.493�10�9

Support layer pore radius (m)b 10�10�9

Dx/Ak (m)c 7.16�10�7 Cylindrical pores

3.82�10�6 Slit pores

Dxs/Aks (m)d 8.8�10�5

eme 3

X0 0

a From AFM.b Mercury porosimetry.c Using Hagen–Poiseuille law (with viscosity correction (Wesolowska et al.,

2004)).d Using Dxs from ESEM images and Aks from mercury porosimetry.e From literature (Lide, 1994).

100

10

1

0.1

0.010.001 0.01 0.1 1 10

c (mol/m3)

- XTS

P (m

ol/m

3 )

XTSP = -25.2 c20.41

Fig. 10. Isotherm correlation of X with concentration as obtained from TSP.

V. Silva et al. / Chemical Engineering Science 66 (2011) 2898–2911 2907

4.3. Nanofiltration modeling results

The modeling parameters for SEDE-VCh and SEDE models arepresented in Table 3. These values will be used in all modelingsections. The parameters a and b can be obtained by fitting therejection results or can also be kept equal to those obtained fromthe TSP results. Note that all the parameters are equal for both thegeometries, with the sole exception of Dx/Ak.

When the charge is taken as known by assuming the isothermobtained from TSP results, it has to be taken into account thatwhen two surfaces are close to each other (as inside nanofiltrationpores) the charge density decreases due to the charge regulationprocess (Israelachvili, 1991), and consequently the actual X insidethe pore should be less than XTSP. For that reason, Szymczyk et al.(2007) suggested a tentative correction for the charge volumetricdensities:

X ¼ XTSP

ffiffiffiffiffiep

eb

rð33Þ

This correction takes into account the difference of thedielectric constants outside (eb) and inside the pore (ep). Notethat, by writing this equation, it has been assumed that the

surface potential inside the pores equals that on the externalmembrane surfaces. In particular, a possible effect of a change inep on the degree of dissociation of the ionogenic groups inside thepores has been neglected.

The experimental results have been treated by means of thefollowing different procedures:

(1)

From the TSP results, the Freundlich isotherm (Fig. 10 cor-rected according with Eq. (33)) is introduced in the SEDE andSEDE-VCh models and ep is kept the only fitting parameter forboth slit-like and cylindrical pores. In this way the effect ofconsidering the volumetric charge as changing along the porecan be studied.

(2)

The support effect is analyzed via the SEDE-VCh model as inpoint 1 in order to compare the support layer relevance in theNF transport phenomena. Here the pores are assumed to beslit-shaped.

(3)

The SEDE-VCh model is used to fit the three parameters ep, a

and b from the NF measurements at each solution concentra-tion and for both the geometries.

(4)

The dielectric constants, ep, obtained in point 3 is used to fitthe MP results to get the a and b Freundlich’s parameters. Thisis done assuming slit-shaped pores.

4.3.1. Modeling using TSP measurements.

The results for ep after fitting the experimental nanofiltrationdata, assuming cylindrical and slit-like pore geometries, aredepicted in Fig. 12. In this case the SEDE and SEDE-VCh modelshave been applied with ep as the only parameter to fit and usingthe isotherm parameters (a and b) obtained from TSP to evaluatethe volumetric charge density. It is important to remark that theconcentrations used to evaluate the charge (X) in the SEDE-VChmodel are those inside the membrane, which are much lowerthan those on the surface of the membrane (that have been usedwithin the SEDE model).

The effect of considering the volumetric charge density as avariable along the pore can also be analyzed in Fig. 12. For slitgeometry, the SEDE-VCh gives ep values below those using theSEDE model in all the range of concentrations. The situation is justthe opposite for cylindrical geometry. The SEDE-VCh model leadsto ep values that change much less with concentration than in thecase of the SEDE model.

The high values of ep obtained for cylindrical pores could bedue to the appearance of new water structures inside nanotubularpores, which could lead to very high dielectric constants or to newdielectric effects (Silva et al., submitted for publication). Of course

100

10

0.01 0.1 1 10 100cf (mol/m3)

Fig. 12. ep versus feed concentration for different geometries and according to

different fitting procedures.

V. Silva et al. / Chemical Engineering Science 66 (2011) 2898–29112908

it could also be an artefact as far as this prediction should becaused by trying to model the pores as cylinders when they aremore slit-shaped than cylindrical.

Actually the concentration inside the membrane is alwaysbelow that outside the pores, which means that when using theSEDE-VCh model the evaluated charges should be lower thanthose corresponding to the external concentrations. Then accord-ing to Eq. (7), lower charges would mean higher free energy dueto the dielectric effects (Born plus Image forces). The image forcesenergy is totally different for the cylindrical pores than for the slitpores (see Eqs. (II.2) and (II.3)) and it increases more forcylindrical pores when the charge inside the pore decreases, dueto their more restrictive geometry. In the case of cylindrical poresit increases so strongly that the Born energy must change its sign(see Eq. (II.1)) to keep the balance leading to dielectric constantsover that of bulk water. This should mean that the desolvation ofthe ions should be a spontaneous phenomenon attracting theminto the pores, which is clearly erroneous. For slit like pores, boththe contributions to the free energy due to the dielectric effectsincrease leading to decrease in dielectric constant within thepores according to Eqs. (II.1) and (II.3).

4.3.2. Effect of the support layer on the rejection modeling

The effect of the support layer on the transport has beenanalyzed using the data on porosity, thickness and pore radius(obtained from ESEM and Hg-Porosimetry) and testing how thedielectric constant changes when the support is or not taken intoaccount. Given that these experimental characteristics of thesupport layer could be affected by big errors, quite differentDxs/Aks and rps are tested too. In effect, they have been decreasedand increased 4 times and the corresponding changes on thedielectric constant tested.

This test has been made using the a and b parameters obtainedfrom the TSP experiments and fitting the ep for slit-like pores(calculations are by far faster for such slit-shaped pores) for a KClconcentration of 5�10�3 M. This study shows that the supporthas no relevance in the transport because there are no anyrelevant differences in values obtained for ep (less than 1%).Another way to get the same conclusion is to compare thepermeate concentration, cp, with the concentration just outsidethe permeate side, csup

o (see Fig. 1) at a concentration of5�10�3 M of KCl and Jv¼1.72�10�6 m/s. Now, the differencesare irrelevant too. In effect, in the worst case (4 times for Dxs/Aks

and 1/4 times for rp) the differences in these concentrations arebelow 1.28%.

Referring to the effect of the charges of the porous support itcan be taken into account that in the porous support there aretwo pore populations. Some of them have (the least) pore sizesaround 8 nm and others (the majority) have pore sizes around200 nm (see Fig. 6). These pore sizes can be compared with thedouble layer thickness that for the concentrations used would gofrom 3 nm (for 10 mol/m3) to 42 nm (for 0.05 mol/m3); thismeans that there should be double layer overlapping only forlow concentrations for the tiny fraction of small pores in thesupport layer. But, actually for low concentrations the chargeshould be very low, giving a negligible effect on the transport.

Thus, it can be concluded that the effect of the support canbe neglected for the study of rejection experiments. In any casethe presence of the support will be taken into account in thefollowing.

4.3.3. Results of SEDE-VCh model with the three free parameters:

ep, a and b

As has already been explained, the NF rejection values havebeen calculated using the SEDE-VCh model to fit the threeparameters ep, a and b. Although the a and b parameters cannotdepend on concentration (Navarro et al., 2008), each concentra-tion is fitted independently in order to obtain the correspondingep for each concentration. The average of the a and b couplesobtained is used as the initial value for the fitting of the results forall the concentration together and ep fixed at the previously fittedvalue for each concentration in order to obtain concentrationindependent a and b values.

The fitted values of ep for both geometries are plotted againstthe NF feed concentration value, in Fig. 12, and compared withthose values obtained using the a and b parameters obtained fromTSP measurements in the previous section with the same model.

The isotherms obtained for both geometries are expressed, forX and c in mol/m3, as

X ¼�2:73c0:542 For slit pores ð34Þ

X ¼�1:85c0:652 For cylindrical pores ð35Þ

For both these pore geometries, the ep values are quite similarto those obtained when a and b were taken from the TSP results(Fig. 10).

It is worth noting that, the SEDE-VCh model with a, b and ep

fitted, allows a modeling of the transport from simple nanofiltra-tion retention experiments. As mentioned, the simple SEDE modelcannot be solved by fitting the charge and the dielectric constantsimultaneously (Lanteri et al., 2008).

The question on the actual dielectric constant inside the poresshould be solved by direct measurements of such parameter. Inthis direction interesting contributions have been done by Li andZhao (Li and Zhao, 2004; Zhao and Li, 2006), who use impedancespectroscopy to obtain the dielectric overall dielectric constant ofwhat could be called wet membrane. Some attempts have beendone to adequately take into account the specificities of electricand dielectric interactions inside nanofiltration pores to interpretimpedance spectroscopy results that seem to advance in thecorrect direction (Montalvillo et al., 2011).

4.3.4. Results of SEDE-VCh model with MP systems

The aim of this section is to prove the ability of the SEDE-VChmodel to describe a different kind of system like that correspond-ing to the membrane potential.

In the membrane potential modeling the Jv values correspond-ing to osmotic flux has gone from 8�10�8 to 1.6�10�6 m/s for

V. Silva et al. / Chemical Engineering Science 66 (2011) 2898–2911 2909

the lowest and the highest concentration experiences, respec-tively (Eq. (22) and (25)). These fluxes are small as compared withthose appearing in the NF experiments; in the worst case (at thehighest concentration) the osmotic flux value is 15% lower thanthe convective flux under pressure.

As was already said, in order to treat the MP results the ep

values obtained in Section 4.3.3 have been used and the a and b

parameters have been fitted for slit-like pores using the whole setof MP data. The SEDE-VCh model has been solved including thesupport layer. The fixed parameters for both layers are thosealready shown in Table 3.

The theoretical membrane potential versus concentrationagrees relatively well (see Fig. 11) with the experimental results.The discrepancies are attributable to the experimental errorswhen measuring the membrane potential and retention, as wellas to the approximations done in the modeling (see Eq. (25) forexample). The corresponding isotherm, with X and c in mol/m3, is

X ¼�0:58c0:152 ð36Þ

4.4. Comparison of electric characterization techniques

The Freundlich isotherm that results from the TSP measure-ments and that obtained by fitting the SEDE-VCh model iscompared in Fig. 13a for slit geometry. The isotherm obtainedfrom MP measurements (Eq. (36) is also shown in Fig. 13a. Notethat the MP isotherm is quite similar to those obtained from TSPand NF experiments. In Fig. 13b, the isotherms obtained from theTSP and NF results are shown for cylindrical pores. In this case the

Fig. 13. Isotherm comparison of X with concentration: (a) MP, NF and TSP results are

pores.ger than for slit-like pores.

Fig. 14. Active and support layer Cl� concentration (mol/m3) profiles

differences between both approaches are bigger than for slit-like pores.

These dissimilarities can be due to an inadequacy of Eq. (33) totake into account the differences in charge volumetric densityinside and outside the membrane due to the diverse dielectricconstants. Alternatively the bigger differences found in theresulting isotherms when cylindrical pores are assumed agreewith the assumption on the better modeling of the actual poresby slit-like ones.

Small differences can also appear because the concentrationdeveloped inside the membrane for NF measurements is lowerthan outside the membranes, and even lower for MP measure-ments. The MP and NF concentration profiles in both layers areshown in Fig. 14 for 5�10�3 M of KCl concentration consideringslit-like pores.

5. Conclusions

An important modification has been introduced in the SEDEmodel for nanofiltration, which consists of including the volu-metric charge density variation inside the membrane pores. Thismodification leads to what could be called the SEDE-VCh model. Ifthe charge density was considered constant along the pore, thisshould be clearly inconsistent because it is known that a con-centration gradient is established along the pores for membranepotential measurements and it appears as a consequence ofretention during nanofiltration experiments. This inconsiste-ncy should obviously lead to incorrect solutions of the systemparameters.

compared for slit-like pores and (b) TSP and NF results are shown for cylindrical

for MP (solid) and NF (dashed) experiments at KCl of 5�10�3 M.

V. Silva et al. / Chemical Engineering Science 66 (2011) 2898–29112910

When the charge is allowed to change with concentration andthus along the pore, it is necessary to assume a certain isothermlinking charge and concentration explicitly. Here the isothermchosen has been a Freundlich’s one including two parameters a

and b. Other way that has been tested is to use the a and b asobtained from the tangential streaming potential results thatactually fit well to a Freundlich’s isotherm, assuming that thesituation for the charges inside the pores is well described by theexternal surface behavior.

The MP measurements provide additional information thathelps to compare the results of using different techniques. TheSEDE-VCh model shows acceptable results for both the mem-brane potential and nanofiltration systems, especially if the poresare assumed as slit shaped. The similarity between the ep para-meter predicted by the different approaches that give quitedissimilar charges indicates that the differences in charge arenot probably very significant.

The use of the three fitting parameters ep, a and b can beadvantageous because it opens some interesting algorithmicpossibilities that could be beneficial as time of calculus could bereduced substantially. In effect, a and b do not depend on theconcentration but also ep can be considered as constant atrelatively high concentrations. Therefore, the three parameterscould be solved simultaneously for several high feed concentra-tion experiments and a and b fixed to obtain ep for the lowerconcentrations (leaving a single fitting parameter).

The effect of the support layer has been included in the model.When NF rejection results are studied the support layer has norelevance in the transport, mainly because the constant permeateflux is continuously dragging the ions in the solution to thepermeate side, thus avoiding the possible concentration polariza-tion effect. On the other hand, for membrane potential measure-ments where there is no applied pressure, the support layer andthe osmotic pressure are relevant and must be considered. Theosmotic water flux carries the ions from the permeate to the feedside and by diffusion the ions tend to go from high to lowconcentrations until a steady state is reached.

Nomenclature

a Freundlich parameteras Cavity radius (M)Ak Porosity (dimensionless)b Freundlich parameterc Concentration (mol/m3)Di,p Solute/ion diffusivity in the pore (m2/s)Di,N Solute/ion diffusivity at infinite dilution (m2/s)dh Hydraulic diameter (M)e Elementary charge (C)Em Membrane potential (V)Ec Cell potential (V)Es Solutions potential (V)Ehigh High concentration solution potential (V)Elow Low concentration solution potential (V)F Faraday constant (C/mol)I0, I1 Modified Bessel functions (dimensionless)Ji Ionic molar flux per unit of membrane area (mol/s m2)Jv Volumetric flux per unit of membrane area (m/s)K0, K1 Modified Bessel functions (dimensionless)K0i,c Hindrance factor for convection with pressure gradient

effect (dimensionless)Ki,d Hindrance factor for diffusion (dimensionless)Ki,m Mass transfer coefficientLw Water permeability (m/s Pa)P Pressure (Pa)

R Universal constant of gases (J/mol K)r Radial coordinate (M)Ri Intrinsic solute rejection (dimensionless)Ro Observed rejection (dimensionless)rp Pore radius (M)ri Stokes solute radius (M)S Objective functionT Temperature (1C)X Volumetric charge density (mol/m3)X0 Volumetric charge density of the membrane material

(mol/m3)x Axial coordinate (M)z Ion valence (dimensionless)

Subscripts

b Bulkm membrane0 just inside de membrane in the feed sideDx just inside de membrane in the permeate sidep permeatei ions relative to the support layer

Greek letters

r Solute density (kg/m3)l Ratio of ion radius to pore radius (dimensionless)Dc Diffusion potential in the active layer (V)DW0i,B Free energy difference due to Born effects

(dimensionless)DW0i,im Free energy difference due to images forces effects

(dimensionless)Dx Pore length (M)DC Donnan potential (V)d Thickness of the polarization film layer (M)e0 Vacuum permittivity (F/m)eb Dielectric constant of the solution in the bulk

(dimensionless)em Dielectric constant of the membrane material dry

(dimensionless)ep Dielectric constant inside the pores (dimensionless)e Steric coefficient (dimensionless)gi Solute/ion activity coefficient (dimensionless)Zp Dynamic pore viscosity (Pa s)kB Boltzmann constant (J/K)u Stechiometric coefficientx Potential difference in the polarization layer (V)

Acknowledgements

Authors want to thank the Spanish ‘‘Ministerio de Ciencia eInnovacion (MCINN)’’ for financing this work within the frame ofthe ‘‘Plan Nacional de I+D+I’’ and through the projects CTQ2009-07666 and MAT2008-00619. Also the ‘‘Junta de Castilla y Leon’’has contributed through the project Grupos de Excelencia-GR18.

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