Journal of Electroanalytical Chemistry 661 (2011) 226–238

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Journal of Electroanalytical Chemistry

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Dielectric analysis of nanofiltration membrane in electrolyte solutions: Influencesof permittivity of wet membrane and volume charge density on ion permeability

Kongshuang Zhao ⇑, Guizhi NiCollege of Chemistry, Beijing Normal University, Beijing 100875, China

a r t i c l e i n f o

Article history:Received 10 April 2011Received in revised form 4 August 2011Accepted 8 August 2011Available online 25 August 2011

Keywords:Dielectric spectroscopyPermittivityVolume charge densityIon solvation energy barrierTransport parameters

1572-6657/$ - see front matter � 2011 Elsevier B.V. Adoi:10.1016/j.jelechem.2011.08.005

⇑ Corresponding author. Tel.: +86 1058808283; faxE-mail address: zhaoks@bnu.edu.cn (K. Zhao).

a b s t r a c t

Dielectric measurements were carried out for the systems composed of nanofiltration membranes andelectrolyte solutions over a frequency range of 40 Hz to 10 MHz, and double dielectric relaxations wereobserved at about 102–106 Hz. Based on the interfacial polarization theory, the dielectric relaxation athigh frequency was analyzed theoretically, and the permittivity and conductivity of the membraneand electrolyte solutions, were obtained. By using the permittivity of wet membrane, the ion solvationenergy barrier in eight electrolyte solutions was calculated. The results showed that the ion solvationenergy barrier of 2-2 type of electrolyte solution was the biggest among all types of electrolyte. The influ-ence of solvation energy barrier on separation performance was also evaluated. Because the permittivityof wet membrane takes the place of that of dry membrane, the calculations can ensure more accurateresults close to practical separation process. Furthermore, the variation of the ratio of membrane/solutionconductivity, jm/jw, with the electrolyte concentration was explained by Donnan equilibrium theory,and the ratio together with the TMS model was used to calculate the volume charge density of mem-brane. On the basis of this, by using the volume charge density of membrane and the existing transportmodels, such as TMS, TMS-DE, DSPM model, four transport parameters were calculated. It was concludedthat in the process of ion permeation through the membrane, Donnan exclusion is the most importantinfluencing factor and steric effect is secondary in low electrolyte concentration, whereas dielectricexclusion plays a dominant role in high electrolyte concentration.

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1. Introduction

Nanofiltration (NF) membranes have found application recentlyin several areas such as water softening and water treatment be-cause of the effective multi-valent ion rejection [1]. In order tounderstand how the membrane structures affect ion permeationinside membrane and ion rejection, a number of studies on theseparation mechanism of membrane have been reported. In otherwords, the study on membrane structures and ion permeation inmembrane has become the highlight of the basic researches onNF membranes [1,2].

The steric effect, Donnan exclusion and dielectric exclusion haveplayed a vital role in the study on separation mechanisms of NFmembranes. In other words, there is more than one mechanismto determine separation processes. This results in an increased dif-ficulty in the prediction of separation performances of membrane.To solve the problem, many transport models have been developed.These models include the fixed-charge model (TMS) [3], Donnansteric pore model (DSPM) [4], Donnan-Steric-Pore-Dielectric-Exclusionmodel (DSPM-DE) [1] and fixed-charge model taking into

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account of Donnan exclusion (TMS-DE) [5]. In the previous studies,the focus was on the following two aspects: the first is to calculatethe electrical and structural parameters of membranes, such as vol-ume charge density of membrane and pore radius, by means ofexperiment data (rejection rate and membrane potential, etc.);and the second is to find out the factors that effect ion transportin membrane by theoretical simulation with assumed membraneparameters. The former can gain information on separation perfor-mance directly, and the latter can investigate the influences ofmembrane parameters on the separation performance, but the bothcannot predict the separating result of NF membrane. However, ifthe electrical and structural parameters can be obtained by someotherways, after substituting these parameters into those transportmodels, some transport parameters can be calculated. As a conse-quence, the separation performance can be predicted without alot of experiments. Nevertheless, the permittivity of membraneused in computation should be the permittivity of wet membrane,i.e. permittivity of membrane in electrolyte solutionwhich is differ-ent completely from that of dry membrane.

The dielectric spectroscopy method is a good solution to theabove questions, because it is a non-invasive way which can detectelectrical properties of the whole system in situ, especially that ofthe heterogeneous systems consisting of polymer membrane and

K. Zhao, G. Ni / Journal of Electroanalytical Chemistry 661 (2011) 226–238 227

electrolyte solutions. Therefore, with the aid of certain model themethod can obtain permittivity of wet membrane which corre-sponds to permittivity of the membrane in working conditions.This method has been applied successfully in many areas in recentyears[6]. Moreover, the dielectric spectroscopy method (refers to‘‘dielectric analyzing” in this work, to calculate the parameters ofthe constituent phases of heterogeneous system from dielectricspectra data) coupled with transport models can also be used toestimate the factors that effect the ion transport in membrane.That is, what the electrical and transport parameters of membranewhich can be obtained by dielectric analysis combined with thetransport models are much closer to the real separation process.

The studies on electrical properties of NF membrane have beenreported, such as tangential streaming potential studied bySzymczyk et al. [7,8], membrane potential studied by Wang andXu et al. [9,10], streaming potential [11,12] and Zeta-potential[13]. Besides, in recent years, the studies by using dielectric spec-troscopy or electrical impedance spectroscopy (EIS) have also beenreported: Coster et al. investigated the electrical properties of thefouled reverse osmosis membranes and BSA-fouling of ion-ex-change membrane by EIS [14,15]; Benavente et al. discussed theinfluence of modification of polysulfone membranes with polyeth-ylene glycol and lignosulfate on the electrical properties of mem-brane[16]; Moon Seung Hyeon et al. examined the foulingmechanism of a fouling layer formed by sodium dodecylbenzene-sulfonate to ion exchange membranes[17]; Freger Viatcheslavet al. characterized ion transport in thin film [18]. In our previousstudies [19], an effective method to calculate the inner parametersin multi-membrane was proposed, and the method has beenadapted to detect the structural and electrical properties of NFmembrane [20].

In the present study, dielectric measurements were carried outon a NTR7450 NF membrane immersed in eight individual electro-lyte solutions at various concentrations. In order to obtain the per-mittivity of whole membrane in solution which is an indispensablevalue for calculating the fixed charge density, solvation energy bar-rier and transport parameters of the membrane, the dielectricrelaxation observed at high frequency was analyzed theoreticallybased on the interfacial polarization theory. The ion permeationand ion selectivity of the NF membrane was discussed, and ion sol-vation energy barrier was calculated by using phase parameters.Final, volume charge density of membrane was obtained by com-bining dielectric analysis with TMS transport model, and by meansof the volume charge density and TMS, TMS-DE, DSPM model[3,5,4], the transport parameters, such as reflection coefficientand transport number of ion, were calculated.

Fig. 1. Dielectric model for the membrane-solution system. (a) The measurementsystem; (b) the equivalent circuit model of (a); (c) equivalent capacitance C(f) and

2. Theory background

2.1. Applicability of M-W theory to membrane/solution systems

2.1.1. Current continuity in membrane/solution systemsAccording to Maxwell equations in the quasi-stationary approx-

imation and MWS approach it is valid for describing interfacepolarization in higher frequencies, when an ac field (E) with anangular frequency x is applied to the heterogeneous medium con-sisting of homogeneous phases which are all isotropic, the mag-netic field H, effective conductivity of the whole system (j�

eff ðxÞ)and the total current density of Maxwell (Jtot) satisfy followingcondition:

r� H ¼ fjþ jxe�ðxÞgE � j�eff ðxÞE ¼ Jtot ð1Þ

The complex effective permittivity e�eff ðxÞ is defined as

e�eff ðxÞ � e�ðxÞ � jj=x ð2Þ

As opposed to the complex permittivity e*(x) which can be usu-ally is measured in experiment, the Eq. (1) can be written in thealternative form [21]:

Jtot ¼ jxe�eff ðxÞE ð3ÞAnd at the interface between two different media, a and b, the

current density satisfies the following condition:

ðJtot;a � Jtot;bÞ � n ¼ 0 ð4Þwhere n is the unit normal to the interface in the direction a? b.Eq. (4) expresses that the total current is conserved during a transi-tion from one medium to another medium and neither accumula-tion nor sources in the various bulk phases. Because the totalcurrent is measurable, the effective, complex conductivity and com-plex permittivity are all measurable, as can be seen from Eqs. (1)and (3).

When an ac electric field is applied to a heterogeneous system,as our system, the charges will accumulate on the interface and aninductive field will be generated to keep the continuity of the elec-tric current. Under such conditions, the permittivity and conduc-tivity of membrane phase and solution phase (em, jm and ew, jw)meet the following relation: em/jm – ew/jw, leading to interfacialrelaxation occurs, the relaxation can be observed by measuring fre-quency dependence of the complex permittivity for the wholesystem.

2.1.2. Method for dielectric analysisIn order to obtain the electrical properties of membrane and

solution from dielectric spectra, it is necessary to analyze thedielectric data of the system on the basis of appropriate model.In this work, the membrane is sandwiched between two solutionphases to form a solution/membrane/solution system as shownin Fig. 1a. From the viewpoint of Maxwell theory, when the solu-tions at both sides of the membrane are the same, the systemcan be represented by a series combination of a membrane withcomplex capacitance C�

m and a solution with complex capacitanceC�w (subscript m and w denote the membrane phase and solution

phase, respectively). This is a two-phase system and its equivalentcircuit is shown in Fig. 1b, which consists of a resistor (of conduc-tance G) in parallel with a capacitor (of capacitance C). In fact, whatis observed experimentally is the frequency dependence of appar-ent capacitance C(f) and apparent conductance G(f) of whole sys-tem represented by equivalent circuit Fig. 1c.

Here, it should be noted that the circuit model used in this workis different that in our previous paper [20], this is because that theaim of present work is to obtain the permittivity of whole mem-brane and regardless of inner structure of the membrane. There-fore, in this case, we need to consider only one interface

conductance G(f) for the whole cell system.

228 K. Zhao, G. Ni / Journal of Electroanalytical Chemistry 661 (2011) 226–238

polarization between the membrane and solution occurring in thehigher frequency range. We therefore used the two electric circuitmodel.

According to M-W theory, the complex capacitance C* of thewhole measuring system is expressed by Eq. (5),

1C� ¼

1C�mþ 1C�w

ð5Þ

where the complex capacitance can be defined as:

C� � G�

jx¼ C þ G

jxð6Þ

Eq. (6) can also be written in the alternative form:

e� ¼ eþ jjxe0

ð7Þ

where e and j are the permittivity and conductivity of the wholesystem, respectively, e0 (=8.8541 � 10–12 F/m) is vacuum permit-tivity. Substituting Eq. (6) into Eq. (5), the links between capaci-tance and conductance of the system and each of phases areestablished. The analysis method for such two phases system hasbeen developed by Hanai et al. [22]. A series of equations whichcan give the relations between the phase parameters (C and G ofmembrane phase and solution phase (see Fig. 1b) and the dielectricparameters(the limiting values of C and G at high frequency, sub-script h, and low frequency, subscript l) have also been given.Through programmable computation, the solution of the expres-sions can calculate the values of Cm, Cw, Gm, Gw (or em, jm, ew, jw)[23].

2.2. Transport models of charged membrane

The membrane is usually characterized by three adjustableparameters, namely the pore radius (rp), volume charge density(Xm) and the effective membrane thickness. Xm is a crucial param-eter for understanding ion permeation of membrane. It can be cal-culated by using phase parameters obtained from dielectricspectra. Therefore, the dielectric analysis coupled with transportmodels can provides a new approach to explore the ion permselec-tivity in membrane. The dielectric analyzing method of membranesystem has been introduced in Section 2.1, here we give a basicdescription of transport models of charged membrane.

In this work, three kinds of models (TMS, TMS-DE and DSPMmodel) were used. The basic equations of TMS model contain Don-nan equation, Nernst–Planck equation. The TMS model gives theexpression of ion partitioning coefficients as follows:

ki ¼ cmici

¼ exp � ziFD/RT

� �ð8Þ

where cmi and zi are the concentration and the valence of the ion i inmembrane, respectively, ci is ionic concentration in electrolyte solu-tions; Du is Donnan potential difference (V).

The TMS-DE model is based on Nernst–Planck equation,Donnan equation and Born solvation energy barrier, this model in-cludes not only Donnan exclusion effect but also the dielectricexclusion effect. Due to the difference of permittivity in membranepore and in electrolyte solution, when ions transfer from solutionphase to pores, they will subject to hindrance of energy barrier.Thus, the ion partitioning coefficients is modified as follows:

ki ¼ cmici

¼ exp � ziFD/RT

� �exp �DWi

kbT

� �ð9Þ

where kb and DWi are the Boltzmann constant and Born solvationenergy barrier, respectively. To calculate DWi, the permittivity ofmembrane pore and the parameters that reflect the structural char-

acteristics of membrane are needed, the formulas for calculatingthese parameters are given in Section 4.3.

The DSPM model assumes the membrane is a charged porouslayer; the partitioning effects at the interfaces between the mem-brane and the external phases are described through steric hin-drance and Donnan equilibrium; mass transfer through themembrane is based on the extended Nernst–Planck equation. Soin DSPMmodel the hindered diffusion and convection are also con-sidered. The ion partitioning coefficients is expressed by:

ki ¼ cmici

¼ /i exp � ziFD/RT

� �ð10Þ

where ui is steric partitioning coefficient of ion i.In all the three models, the electroneutrality condition in mem-

brane defined below must be taken into account when the ion par-titioning coefficients is calculated.

Xni¼1

zici þ Xm ¼ 0 ð11Þ

Eq. (11) implies that the volume charge density Xm is a criticalparameter for investigating ion transport through membrane. Inthis work, Xm are estimated using phase parameters obtained fromdielectric analysis. Consequently, the information about ion perm-selectivity in membrane also be obtained by means of Xm coupledwith the three transport models.

3. Experimental section

3.1. Nanofiltration membrane

The membrane used in this work is a composite NF membrane,NTR7450, supplied by Nitto Denko Co. Japan. The membrane has adouble-layer structure with a supporting fabric and an effectivelayer composed of sulfonated polyether sulfone. The membraneis negatively charged in solution because of partial dissociationof the sulfonic groups (–SO3H) in the membrane, which can beunderstood from following chemical structure of NTR7450 mem-brane materials. The pore radius of the membrane is 0.8 nm [24],and its thickness (t) fully swollen in solution is 120 lm whichwas measured with a vernier caliper. To remove organic impuritiesand salt in the membrane, the membrane was rinsed with enoughdeionized water, and then immersed in deionized water for at least2 days. The whole membrane was divided into eight pieces of thesame size as specimen membrane for dielectric measurements.

3.2. Dielectric measurements

The dielectric measurements were carried out over a frequencyrange of 40 Hz–107 Hz using 4294A Precision Impedance Analyzer(Agilent Technologies). The measuring cell consists of a pair of plat-inum disk electrodes whose areas S is 3.14 cm2, and the cell con-stant (S/L) is 2.415 cm (L is the distance between two electrodes).A specimen membrane separated the cylindrical compartmentsbetween the electrodes, and the two compartments were filledwith the same electrolyte solutions of identical concentration toform a solution/membrane/solution configuration as shown in

K. Zhao, G. Ni / Journal of Electroanalytical Chemistry 661 (2011) 226–238 229

Fig. 1a. Measuring systems were composed of a piece of the mem-brane and eight respective electrolyte solutions (NaCl, KCl, MgCl2,CuCl2, K2SO4, Na2SO4, MgSO4 and CuSO4) with varying concentra-tions from 0.005 to 10 mol m�3. Before the measurements, the cellwas kept still for 15 min to allow the membrane to equilibrate withthe solution at the applied concentration; after the measurement,the solution in the cell was changed to a higher concentration toperform next measurement, until the last one. All dielectric mea-surements were performed at a temperature range of 18 ± 1 �Cand pH = 6.5.

The stray capacitance Cf and cell constant Cl were determinedbeing 0.217 and 0.79 pF, respectively, by which the conductanceG and capacitance C of the whole system at each frequency weretransferred to corresponding conductivity j and permittivity eaccording to the following equations:

e ¼ ðC � Cf Þ=Cl ð12Þand

j ¼ Ge0=Cl ð13ÞPrior to the transformation, all the raw data were subjected to cer-tain corrections for the errors arising from Cf and residual induc-tance Lr due to the terminal leads[25]. The value of Lr,1.93 � 10�7, was determined by use of standard KCl solutions withdifferent concentrations (the relation C = LrG

2 was considered). Andthen the measured values of C and G were subjected to correctionby the following equations:

Cs ¼ Cxð1þx2LrCxÞ þ LrG2x

ð1þx2LrCxÞ2 þ ðxLrGxÞ2� Cf ð14Þ

Gs ¼ Gx

ð1þx2LrCxÞ2 þ ðxLrGxÞ2ð15Þ

where the subscript x and s denote the directly measured and cor-rected values, respectively.

4. Results and discussions

4.1. Dielectric analysis of membrane/solution systems

4.1.1. Dielectric spectrumFig. 2 shows the three-dimensional representations for the con-

centration dependence of dielectric spectra for the NTR7450membrane-KCl solution system. It is obvious that there are tworemarkable dielectric relaxations at the frequency range of

200

400

600

800

1e+21e+31e+41e+51e+61e+72

46

810

ε

f/(Hz)c/(

mol/m3 )

(a)Lo

Diele

Di

Fig. 2. Three-dimensional representations of concentration dependence of (a) the permimembrane and KCl solution.

102–104 and 104–106 Hz, respectively, as indicated in Fig. 2a andb. The two relaxations are characterized with the following eightdielectric parameters: eh, emid, el, jh, jmid, jl, fl and fh, which arethe permittivity and the conductivity at high-, middle-, and low-frequency, and the characteristic relaxation frequencies of thelow-, and high-frequency relaxations, respectively.

4.1.2. Estimation of dielectric parametersTo obtain the electric properties of the constituent phases of the

system, it is necessary to estimate the values of high-frequencydielectric parameters. For clarity, we took the system of mem-brane-KCl solution as an example to introduce how these parame-ters were determined from the dielectric spectrum. The dielectricdata of membrane-0.005 mol/m3 KCl system that cut from theFig. 2a were plotted on complex planes of permittivity and conduc-tivity respectively as shown in Fig. 3a and b. In the two figures,three parameters, eh,el and jh were determined with accuracy byextrapolation of the respective plots to low and higher frequencies.The conductivity of low frequency jl was determined directlythrough the curve of conductivity spectrum j/f as shown inFig. 3c. emid and jmid were estimated according to the literature[19] as below: the plot of permittivity e(f) against conductivity j(f) gives two straight lines with different slopes, as shown inFig. 3d, and the intersection of the two lines gave the value of emid

on abscissa and the value of jmid on ordinate. fl and fh, were deter-mined from the plot in Fig. 3e. Such, the eight dielectric parame-ters, eh, emid, el, jl, jmid, jh, fl and fh, were obtained for all systemsstudied in this work. Here, only the high-frequency dielectricparameters for membrane-KCl solution at various concentrationsare listed in Table 1.

It can be seen from Table 1 that, for the high-frequency relaxa-tion, De is nearly invariable with the increment in external concen-tration, whereas fh and Dj depended linearly on the concentration,as illustrated in Fig. 4. This indicated a distinctive feature of inter-facial polarization [26], so, the high-frequency relaxation may beconsidered to be caused by the interfacial polarization mechanism.

4.1.3. Calculation of phase parameters and comments on the resultsBy the analyzing method described in Section 2.1.2, the phase

parameters, Cm, Gm, Cw, Gw, were calculated from dielectric param-eters, fh, Cmid, Ch, Gmid and Gh which were gotten from eh, emid, jh andjmid with the Eqs. (12) and (13). Subsequently, they were trans-ferred to the corresponding permittivity e and conductivity j, thatis, em, jm, ew, jw, by means of membrane thickness (t) and the dis-tance between electrodes (L) and the results were listed in Table 2.

100

200

300

1e+21e+31e+41e+51e+624

68

10

ε"

f/Hz

c/(mol/m

3)

(b)

w-Frequency ctric Relaxation

High-Frequency electric Relaxation

ttivity spectrum and (b) the dielectric loss spectrum for the system composed of NF

0 100 200 300 400 500 6000

75

150

225

εmid

ε lεh

0.005mol/m3

ε"ε

(a)

0.0 1.0x10-4 2.0x10-4 3.0x10-40.00000

0.00005

0.00010

0.00015 0.005mol/m3

κh

(b)

κ"

κ/(S/m)101 102 103 104 105 106

1E-5

1E-4

κmid

κ/(S

/m)

f/(Hz)

0.005mol/m3

κ l

(c)

200 400 600 800

0.00

0.01

0.02

0.03

0.04

0.05

10mol/m3

κ/(S

/m)

ε

κmid

εmid

(d)

103 104 105 106 10710

100 10mol/m3

fh

ε"

f/(Hz)

f l

(e)

Fig. 3. The example of estimation of dielectric parameters for the KCl-membrane system. (a) Complex plane plots of permittivity, (b) Complex plane plots of conductivity, (c)Frequency dependence of conductivity, (d) Plots of conductivity j against permittivity e, (e)Frequency dependence of dielectric loss.

Table 1High-frequency Dielectric Parameters for NTR7450 Membrane in KCl Solutions at Various Concentrations.

c (mol/m3) emid eh jmid (lS/m) jh (lS/m) fh (kHz) De Dj (lS/m)

0.005 510.4 62.8 4.27 307.9 10.08 447.6 303.60.01 495.4 62.0 5.48 461.0 15.54 433.4 455.50.05 459.4 61.1 8.48 735.9 26.08 398.3 727.40.1 469.8 60.7 11.58 1161.0 44.65 409.2 1149.40.2 450.4 61.6 20.88 2234.5 87.52 388.8 2213.60.4 459.7 60.1 29.50 3759.2 146.83 399.6 3729.70.7 454.0 57.3 36.58 6112.6 236.54 396.6 6076.01 453.0 55.3 50.54 9195.3 364.67 397.6 9144.82 455.7 48.8 92.97 17506.9 793.79 406.9 17414.04 446.9 46.1 138.70 31910.8 1454.15 400.8 31772.17 457.1 45.2 198.24 55587.3 2334.71 411.9 55389.110 443.4 46.0 245.40 80713.6 3456.26 397.3 80468.2

Notes: De and Dj are the dielectric and conductivity increments for high-frequency, respectively, defined as De = emid � eh and Dj = jh � jmid, c is the concentration ofelectrolyte solution.

230 K. Zhao, G. Ni / Journal of Electroanalytical Chemistry 661 (2011) 226–238

Permittivity of the solution ew From Table 2, it can be seen thatthe permittivity of electrolyte solutions ew are nearly invariablewith electrolyte concentration and the mean permittivity ew isabout 81.7 which can be thought of apparently reasonable valuewhen compared with the permittivity of water (80.59–81.33 at18 ± 1 �C) [27].

This can be verified by an individual experiment: by measuringthe dielectric data of KCl solution, the permittivity e0w and conduc-tivity j0

w of KCl solution at various concentrations were obtained.The ratio of ew/e0w is close to 1 as illustrated in Fig. 5, showing thepermittivity is independence of electrolyte concentration. That is,the permittivity of KCl solution ew obtained by analyzing dielectric

0 2 4 6 8 100.8

0.9

1.0

1.1

1.2

ε w/ε w

'

C/(mol/m3)

Fig. 5. Plots of the ratio of ew to e0w against electrolyte concentration c. ew isdielectrically analyzed permittivity; e0w is permittivity obtained by measuring theKCl solution without a membrane.

Table 3Mean values of permittivity of different electrolyte solutions.

Solutions KCl NaCl MgCl2 CuCl2 K2SO4 Na2SO4 MgSO4 CuSO4

ew 81.7 81.3 81.1 80.8 80.5 80.6 80.2 80.1

0 2 4 6 8 10

0.0

2.0x104

4.0x104

6.0x104

8.0x104

0

1000

2000

3000

4000Δκ

/(μS/

m)

C/(mol/m3)

f h/(K

Hz)

Fig. 4. Conductivity increment, Dj, and characteristic relaxation frequency, fh, forhigh-frequency relaxation vs external electrolyte concentration for systems of KClsolutions-membrane.

K. Zhao, G. Ni / Journal of Electroanalytical Chemistry 661 (2011) 226–238 231

spectrum is in good agreement with that measured alone e0w. Themean value of ew of the seven other electrolyte solutions (NaCl,MgCl2, CuCl2, K2SO4, Na2SO4, MgSO4, CuSO4) were also obtainedin the same way and listed in Table 3.

Conductivity of the solution and membrane jw, jm It also be seenfrom Table 2 that the conductivity of membrane phase jm andelectrolyte solution phase jw increase with the increment of elec-trolyte concentration, and the value of jm is much smaller thanthat of jw at any concentration. The change of the ratio of jm/jw

with concentration c, shown in Fig. 6a, showed a typical Donnanexclusion effect. The Fig. 6b shows a linear relation between the

Table 2Phase parameters for membrane-KCl solution system.

c Cm Cw Gm Gw

(mol/m3) pF pF S � 10�9 S � 1

0.005 104.82 17.52 105.98 10.30.01 110.86 17.43 135.70 15.10.05 103.13 17.64 210.07 24.70.1 104.93 17.58 286.50 38.80.2 101.05 17.70 516.19 75.70.4 102.87 17.86 728.55 127.0.7 100.86 16.83 902.17 204.1 101.21 17.50 1245.98 310.2 101.28 17.48 2291.85 591.4 99.55 17.44 3417.34 10267 101.46 17.26 4881.40 177210 100.46 17.36 6040.69 2478Mean value

conductivity jw of KCl solution obtained by analyzing dielectricspectra and the conductivity j0

w of the KCl solution without mem-brane. This figure shows that jw are in good agreement with j0

w,this means the reliability and accuracy of the theoretical modeland analyzing method used in this work.

Permittivity of the membrane em Permittivity of membrane em isthe most important one among all phase parameters, and it is vitalto calculate ion solvation energy barrier and transport parametersusing DSPM model in subsequent sections. As shown in Table 2, emis nearly invariable with the concentration and the mean value ofem is 4.43, slightly larger than that of dry polymers (permittivityof polysulfones is between 3 and 3.5). This is not hard to under-stand because of permeation of water into the pore passage ofmembrane. In this case, em can be simply expressed as [28]:

em ¼ ewfw þ ð1� fwÞe0m ð16Þ

where fw is the volume fraction of water in whole membrane phase,and e0m is permittivity of dry polymer matrix.

Taking the permittivity e0m of dry membrane as 3 and substitut-ing e0m, em and ew into Eq. (16), the values of fw at various concentra-tion were calculated. The results showed that fw decreases with theincrease of concentration and finally tends to remain unchanged asillustrated in Fig. 7a. The volume fraction of water in membranephase fw is around 0.017, indicating that the water content inmembrane is very low. Therefore, we can speculate that the innerstructure of membrane is compact and it is hard for ions to perme-ate membrane, leading that ions can be effectively retained inmembrane. It should be note here that the concentration depen-dency of em and fw change in a similar trend as seen in Fig. 7, shar-ply near the concentration of 0.5 mol/m3. The reason for thisphenomenon may be interpreted as that because the membraneconsists of polymer materials with electrical charge and the poly-mer matrix are with empty space, i.e. pore passage, as illustratedin Fig. 7, when counterions in solutions enter the interspaces, elec-trostatic repulsion between polymer backbones will be shieldedgradually, giving rise to aggregate and reunite between polymerchains in a relatively compact state. Accompanied with the change,fw and em will decrease. With the increase of counter-ion concen-tration, the fixed charge will be screened completely and the elec-trolyte concentration will no longer impact on the aggregationstate of polymer backbones. Therefore, fw and em have no change

em ew jm jw

0�6 S/m � 10�9 S/m

0 4.52 81.9 40.50 0.000436 4.78 81.5 51.86 0.000637 4.45 82.5 80.28 0.001036 4.53 82.2 109.49 0.001616 4.36 82.8 197.27 0.0031413 4.44 83.5 278.43 0.0052633 4.35 78.7 344.78 0.0084660 4.37 81.8 476.17 0.0128607 4.37 81.8 875.87 0.02447.86 4.3 81.6 1305.99 0.04481.93 4.38 80.7 1865.50 0.07739.49 4.33 81.2 2308.54 0.1289

4.43 81.7

0.0

1.0x10-3

2.0x10-3

3.0x10-3

c/mol m-3

(a)

0 2 4 6 8 10 0.00 0.03 0.06 0.09 0.12 0.15

0.00

0.03

0.06

0.09

0.12

0.15

κ w/(S

/m)

κ mκ w

/

κw'/(S/m)

(b)

Fig. 6. (a) Plots of membrane/solution conductivity ratio jm/jw against the KCl concentration c. (b) Plots of dielectrically analyzed conductivity jw of KCl solutions against j0w

from directly measuring the KCl solution without a membrane.

0.016

0.018

0.020

0.022

f w

c/(mol/m3)

(a)

0 2 4 6 8 10 0 2 4 6 8 104.2

4.4

4.6

4.8

ε m

c/(mol/m3)

(b)

Fig. 7. Plots of (a) volume fraction of water in membrane phase fw and (b) permittivity of membrane against KCl concentratin.

104 105 106

3.0x10-11

6.0x10-11

9.0x10-11

1.2x10-10

0.0

1.0x10-5

2.0x10-5

3.0x10-5

C/F

f/(Hz)

0.1mol/m3

G/S

CG

Fig. 8. Frequency dependence of capacitance and conductance of membrane at0.1 mol/m3 KCl solutions. Dots are the experimental points; the curves aretheoretical values.

232 K. Zhao, G. Ni / Journal of Electroanalytical Chemistry 661 (2011) 226–238

over the concentration range of 1–10 mol/m3. We take the meanvalue of em as permittivity of membrane in solution in this work.

Table 4 gives the mean values of permittivity of membrane ineight electrolyte solutions, which is also essential to evaluate theion solvation energy barrier, which will be seen in Section 4.3.

4.1.4. Comparison of theoretically calculated curves with experimentalones

In order to verify the validity of dielectric analysis theoretically,the phase parameters obtained from dielectric analysis are substi-tuted into the theoretical formulas to calculate theoretical relaxa-tion curves. The theoretical formulas were about the relationbetween capacitance and conductance of the whole system andthat of each constituent phases, derived by Eq. (5) and Eq. (6),and given in literature 23. Through the method, the high-frequencyrelaxation theoretical curves are calculated and compared withexperimental ones. The result of membrane-0.1 mol/m3 KCl solu-tion is shown in Fig. 8. It can be seen that the theoretical curvesare in good accordance with the experimental data besides thecapacitance at low frequency which is affected by low-frequencyrelaxation. This shows that the dielectric analysis method used inthis work is valid.

4.2. Volume charge density

Volume charge density Xm plays a crucial role in the evaluationof separation performance of membrane. According to the discus-

Table 4The mean value of permittivity of membrane in eight electrolyte solutions.

Solutions KCl NaCl MgCl2 CuCl2 K2SO4 Na2SO4 MgSO4 CuSO4

em 4.35 4.44 4.05 6.26 4.59 5.52 4.82 5.92

sions in Section 4.1.3, it can be deduced that when ions enter mem-brane, the ion can interact with the fixed charge in the membraneowing to Donnan potential. With increasing the concentration ofelectrolyte, the electrostatic interaction decreases because thefixed charge was neutralized gradually. As a result, jm/jw reachesa steady value. In order to calculate Xm, it is necessary to get therelation between Xm and ion concentration in membrane cmi .According to the foregoing analysis, the relation for different elec-trolyte solutions between ionic distribution coefficient ki and Xm

were established using the expression of ionic distribution coeffi-cient in TMS model (Eq. (8)), electroneutrality condition in mem-brane (Eq. (11)), and Eq. (17). Further, the expressions for cmi andXm were derived through Eq. (8), and the corresponding formulasare listed in Appendix A.

z1v1 þ z2v2 ¼ 0 ð17Þ

Table 5Membrane volumetric charge density Xm in different electrolyte solutions obtained by fitting the curves of jm/jw � c.

Solutions KCl NaCl MgCl2 CuCl2 K2SO4 Na2SO4 MgSO4 CuSO4

Xm (mol/m3) 0.716 1.122 1.262 1.232 1.256 1.256 1.226 1.282Xm 1.169

0.0

2.0x10-4

4.0x10-4

KCl NaCl MgCl2 CuCl2

c/mol m-3

(a)

0 2 4 6 8 10 0 2 4 6 8 100.0

2.0x10-4

4.0x10-4

6.0x10-4

Na2SO4

K2SO4

MgSO4

CuSO4

c/mol m-3

(b)

κ mκ w

/

κ mκ w

/

Fig. 9. Plots of jm/jw against c for the membrane in eight individual electrolyte solutions. (a) for mono–monovalent and bi–monovalent electrolyte solutions; (b) for mono–divalent and bi–divalent electrolyte solutions. Symbols are the result of dielectric analysis, and the curves are calculated from expressions derived by Eq. (19) and formulas ofion concentration in membrane in Appendix A.

Table 6Stokes radius of ions rs.

Ions K+ Na+ Mg2+ Cu2+SO2�

4Cl�

rs (nm) 0.125 0.184 0.347 0.325 0.23 0.121

Table 7pore dielectric constant ep in different solutions calculated using Eq. (20).

Solutinos KCl NaCl MgCl2 CuCl2 K2SO4 Na2SO4 MgSO4 CuSO4

ep 37.97 37.8 37.71 37.62 37.47 37.53 37.37 37.29

K. Zhao, G. Ni / Journal of Electroanalytical Chemistry 661 (2011) 226–238 233

where z is ion valence, subscripts 1 and 2 denote counter-ions andcoions, respectively, v is stoichiometric coefficient.

There is a certain relation between concentration of cation, an-ion in membrane and the ratio of conductivity in membrane andbulk solution jm/jw. For ByAx type electrolyte, the relation canbe expressed in Eq. (18) [23].

jm

jw¼ p

tmþcmBþ þ tm�c

mA�

twþcwBþ þ tw�c

wA�

ð18Þ

where p ¼ Umþ t

wþ=U

wþt

mþ , U

mþ and Uw

þ are the mobility of cation inmembrane and solution, respectively; tmþ , t

m� , t

wþ and tw� are defined

as tiþ ¼ Uiþ=ðUi

þ þ Ui�Þ and ti� ¼ 1� tiþ (where i denotes the phase

of membrane or solution), twþ and tw� are the transport numbers ofcation and anion in solution, respectively, but tmþ and tm� are notthe transport numbers of cation and anion in membrane, becausethe concentration of cation and anion in membrane are different;cmBþ , c

mA� , c

wBþ and cwA� are the concentration of cation and anion in

membrane and solution, respectively. With cmi and Eq. (18), theexpressions of jm/jw and Xm for different electrolyte solutions were

Table 8Ion solvation energy barrier for each species of ions in different solutions calculated using

Solutions KCl NaCl MgCl2 CuCl2

K+ Cl� Na+ Cl- Mg2+ Cl- Cu2+ C

4Wi (�10�20J) 2.3 2.34 1.88 2.35 6.34 2.47 4.43 1

derived. And then the values of Xm in different electrolyte solutionswere obtained by fitting the derived expression and listed in Table5. Fig. 9 shows the fitting results for the eight systems. It can beseen from Fig. 9 that the calculated curves are in good agreementwith the result of dielectric analysis. The mean value of Xm is1.169 mol/m3, which represents the electric charge carried by themembrane itself, and Xm is independent of electrolyteconcentration.

4.3. Influence of solvation energy barrier on ion permeation ofmembrane

The Born solvation energy barrier represents the dielectricexclusion effect. Dielectric exclusion is caused by the interactionsof ions with the polarization electric charges induced by ions atinterfaces between media of different permittivity. Thus, owingto the difference of permittivity in membrane pores and bulk solu-tion, when ions transfer from bulk solution of high permittivity tomembrane pores of low permittivity, dielectric exclusion effect andthere is an energy barrier to ions into pores (which will clearly in-crease salt rejection). The magnitude of the energy barrier stronglyinfluences the ions to enter the membrane pore. The permeabilityof ion in membrane can be evaluated from ion solvation energybarrier of different kinds of electrolytes.

Considering that the change of the electrostatic free energy ofan ion between the bulk solution and the membrane pore (or thechange of the permittivity), based on the Born equation in whichonly the permittivity of membrane was considered [29], an expres-sions of ion solvation energy barrier for ion i was proposed [5]:

DWi ¼ z2i e2

8peors1ep

þ 0:393rp=rsem

1� emep

� �2 !

� 1ew

!ð19Þ

Eq. (19).

K2SO4 Na2SO4 MgSO4 CuSO4

l� K+SO2�

4Na+ SO2�

4Mg2+ SO2�

4Cu2+

SO2�4

.99 2.26 6.66 1.64 5.84 5.47 6.43 4.63 5.58

-0.4

-0.2

0.0

0.2

0.4

0.6

0.8

1.0TMS

z1v1C/(mol/m3) z1v1C/(mol/m3)

z1v1C/(mol/m3) z1v1C/(mol/m3)

z1v1C/(mol/m3) z1v1C/(mol/m3)

KCl NaCl MgCl2 CuCl2 Na2SO4 K2SO4

MgSO4 CuSO4σ σ

(a)

0.994

0.996

0.998

1.000

TMS-DE

KCl NaCl MgCl2 CuCl2 Na2SO4

K2SO4

MgSO4

CuSO4

(b)

-0.4-0.20.00.20.40.60.81.0

KCl NaCl MgCl2 CuCl2 Na2SO4 K2SO4 MgSO4 CuSO4

σ

DSPM

(c)

0.0

0.2

0.4

0.6

0.8

1.0

1.2

TMS KCl NaCl MgCl2 CuCl2 Na2SO4

K2SO4

MgSO4

CuSO4

(d)

0.000

0.002

0.004

0.006TMS-DE

KCl NaCl MgCl2 CuCl2 Na2SO4

K2SO4

MgSO4

CuSO4

RTω

/Ds

RTω

/Ds

RTω

/Ds

(e)

1E-3 0.01 0.1 1 10 100 1000 1E-3 0.1 10 1000 100000

1E-3 0.01 0.1 1 10 100 1E-3 0.01 0.1 1 10 100 1000

1E-3 0.1 10 1000 100000 1E-3 0.01 0.1 1 10 100

0.0

0.2

0.4

0.6

0.8

1.0 KCl NaCl MgCl2

CuCl2

Na2SO4

K2SO4

MgSO4

CuSO4

DSPM(f)

Fig. 10. Variation of reflection coefficient (a–c) and solute permeability coefficient (d–f) of NTR7450 membrane with equivalent concentration of eight electrolyte solutions inthree transport models.

234 K. Zhao, G. Ni / Journal of Electroanalytical Chemistry 661 (2011) 226–238

where zi is valence for ion i, rs is Stokes radius of ion and the valuesfor eight kinds of ions studied in this work were listed in Table 6[30]; rp is pore radius; ew, em and ep are the permittivity of bulk,membrane and membrane pore, respectively. The ew and em forthe eight electrolyte solutions have been obtained in Section 4.1.3by analyzing dielectric spectra; ep can be calculated by the followingformula [29]. (Here, it should be noted that in Eq. (19), because theem is the permittivity of the membrane matrix without pore, usingthe value of em estimated by dielectric analysis should be anapproximation.):

ep ¼ ew � 2ðew � edÞ drp

� �þ ew � edð Þ d

rp

� �2

ð20Þ

This formula assumed that the solvent within the pore have thesame permittivity with bulk solution, and there is one thin layer ofwater molecular at pore walls. The permittivity of the water layernear the pore wall is different from that of inside pore and bulksolution because the oriented water molecules is limited. The per-mittivity and the thickness of this layer, ed and d, are about 6 and0.28 nm, respectively[29].

By substituting ew of in Table 3 into Eq. (20), the permittivity ofpore ep in different solutions was calculated and the results werelisted in Table 7. Further, by using Stokes radius of ions rs in Table6, ep in Table 7 and ew, em in Tables 3 and 4, the solvation energybarrier for each kind of ions in different solutions were calculatedfrom Eq. (19) and are listed in Table 8. It can be seen from Table 8that, the sequence of solvation energy barrier DWi for electrolytesolutions is 2-2 > 1-2, 2-1 > 1-1. Obviously, DWi for 2-2 type ofelectrolyte solution is much larger than that for 1-1 type, 1-2 and2-1 type of solutions, suggesting that the higher ion valence resultsin the larger solvation energy barrier thereby causing more diffi-culty for ions to permeate membrane. Therefore, for 2-2 type ofsolution, ions permeation of membrane is the most difficult interms of energy barrier. However, because DWi is not the only fac-tor that affect ion permeation of membrane, the sequence of DWi

cannot be accepted as the salt retention sequence.According to the literature[24], the membrane follows the salt

retention sequence obtained by analyzing experimental rejections:1-2 > 2-2 > 1-1 > 2-1 when the concentration of electrolyte solu-tion was below a certain value(0.5 eq/l) and 2-2 > 1-2 > 1-1 > 2-1when the concentration was above the certain value. The differentsequence shows that the main influencing factors on the ion

K. Zhao, G. Ni / Journal of Electroanalytical Chemistry 661 (2011) 226–238 235

permeability are different at low and high concentrations. Bycomparison with retention sequences gotten using Donnan anddielectric exclusion mechanisms, respectively, we find that for anegatively charged membrane, when the concentration of electro-lyte solutions is below a certain value, the salt retention sequencefrom literature 32 is close to that deduced by Donnan exclusion(1-2 > 1-1 = 2-2 > 2-1). Therefore, we can conclude that in low con-centration range Donnan exclusion is the dominant influencingfactor, the solvation energy barrier playing a secondary role;Whereas, when the concentration is above a certain value, theeffect of solvation energy barrier becomes the dominant role, andthe salt retention sequence is similar to that obtained by the abovecalculations of solvation energy barrier. This suggests that in highconcentration range, solvation energy barrier plays a predominaterole, Donnan exclusion effect is secondary and bringing about thelowest rejection of 2-1 type of the solution.

4.4. Transport parameters

Transport parameters describing the transport phenomena ofcharged membrane include reflection coefficient, solute perme-ability coefficient, ion conductivity in membrane, transport num-ber of ion, permeability of pure water and electroosmoticcoefficient [31]. In this section, by means of volume charge densityin Section 4.2 (Table 5), permittivity of membrane in Section 4.1.3(Table 4), the expressions of ion partitioning coefficient derived inAppendix A and three transport models TMS, TMS-DE and DSPM,we calculated the four transport parameters: reflection coefficientr, solute permeability coefficient -, ion conductivity j in mem-brane, transport number of ion ti.

4.4.1. Reflection coefficient and solute permeability coefficientReflection coefficient r and solute permeability coefficient �x

characterize the largest rejection rate of ion and the ionpermeability, respectively. The relation between r and �x can bedescribed by r = 1 � RT �x/Ds (diffusive coefficient Ds = (|z1| + |z2|)

1000 1000000

4000

8000

12000

16000

20000 TMS

z1v1C/(mol/m3)

z

KCl NaCl MgCl2 CuCl2 Na2SO4

K2SO4

MgSO4

CuSO4

(a)

103

0

500

1000

1500

2000

2500

κ/(S

/m)

κ/(S

/m)

KC Na Mg Cu Na K2S Mg Cu

(c)

Fig. 11. Variation of conductivity of NTR7450 membrane with equivalent c

D1D2/(|z1|D1 + |z2|D2)). From the both parameters, the relation be-tween rejection rate and the volume flux can be obtained by usingthe Spiegler–Kedem equation. We derived two equations describ-ing the r and �x by TMS, TMS-DE and DSPM models:

TMS, TMS-DE model:

r ¼ 1� k1k2ðD1 þ D2 þ v 0D1 þ 1v 0 D2Þ

ð1þ v 0Þðk1D1 þ 1v 0 k2D2Þ

ð21Þ

TMS, TMS-DE model:

- ¼ k1k2D1D2ð2þ v 0 þ 1v 0Þ

RTð1þ v 0Þðk1D1 þ 1v 0 k2D2Þ

ð22Þ

where the expressions of ion partitioning coefficient k are listed inAppendix A, v0 is the ratio of v2 to v1, subscripts 1 and 2 denotecounter-ion and co-ion respectively as have been defined.

By using the parameters (em, Xm and ew) which are obtainedfrom dielectric analysis and three models (TMS, TMS-DE andDSPM), the change of r for the NTR7450 membrane with equiva-lent concentration of eight electrolyte solutions were calculatedas shown in Fig. 10. It can be seen from Fig. 10 that, in the threemodels, the value of r for the same electrolyte have the samechanging trend: For 1-1, 2-2 and 1-2 type electrolytes, r decreaseswith the increment of concentration, and when the concentrationreaches a certain value, r tends to remain unchanged; For 2-1 typeelectrolytes, r decreases to a minimum and then increases, finallyreaches a steady value. However, r for the same electrolyte withthe same concentration are different in the three models (seeFig. 10). In low concentration, r of three models have only minorchanges; whereas in high concentration, r of TMS-DE model varieswidely compared with the other two models, but the difference be-tween TMS and DSPM is very little. That is, the concentration rangein which r changes rapidly is different in the three models. ForTMS, DSPM models, the concentration range is from 0.01 to1 mol/m3, while it is from 1 to 1000 mol/m3 for TMS-DE model.All these characteristics shows Donnan exclusion, steric effectand dielectric exclusion play different role in ion permeability. In

z1v1C/(mol/m3)

1v1C/(mol/m3)

1000 1000000

20

40

60

80 TMS-DE

KCl NaCl MgCl2 CuCl2 Na2SO4

K2SO4

MgSO4

CuSO4

(b)

105

κ/(S

/m)

lClCl2Cl22SO4

O4

SO4

SO4

DSPM

oncentration of eight electrolyte solutions for three transport models.

10-3 10-2 10-1 100 101 102 103

0.4

0.6

0.8

1.0TMS KCl

NaCl MgCl2 CuCl2 Na2SO4

K2SO4

MgSO4

CuSO4

t 1 t 1

t 1

(a)

10-3 10-1 101 103 105 107

0.4

0.6

0.8

1.0(b)TMS-DE

KCl NaCl MgCl2 CuCl2 Na2SO4

K2SO4

MgSO4

CuSO4

10-3 10-2 10-1 100 101 102 103

0.2

0.4

0.6

0.8

1.0(c) KCl NaCl MgCl2 CuCl2 Na2SO4

K2SO4

MgSO4

CuSO4

DSPM

10-3 10-2 10-1 100 101 102 103

0.0

0.2

0.4

0.6(d) TMS KCl

NaCl MgCl2 CuCl2 Na2SO4

K2SO4

MgSO4

CuSO4

10-3 10-1 101 103 105 107

0.0

0.2

0.4

0.6(e) TMS-DE KCl

NaCl MgCl2 CuCl2 Na2SO4

K2SO4

MgSO4

CuSO4

t 2

t 2t 2

10-3 10-2 10-1 100 101 102 103

0.0

0.2

0.4

0.6

0.8

(f) KCl NaCl MgCl2 CuCl2 Na2SO4

K2SO4

MgSO4

CuSO4

DSPM

z1v1C/(mol/m3) z1v1C/(mol/m3)

z1v1C/(mol/m3) z1v1C/(mol/m3)

z1v1C/(mol/m3) z1v1C/(mol/m3)

Fig. 12. Variation of counterions (a–c) and coions (d–f) transport number in eight electrolyte solutions with equivalent concentration calculated using three transportmodels.

Table 9Surface charge density of NTR7450 membrane in different electrolyte solutions

Solutions KCl NaCl MgCl2 CuCl2 K2SO4 Na2SO4 MgSO4 CuSO4

qw � 105 (C/m2) 2.76333 4.33025 4.87056 4.75478 4.84741 4.84741 4.73162 4.94775

0.0 3.0x10-5 6.0x10-5 9.0x10-5

-150

-120

-90

-60

-30

KCl NaCl K2SO4

Na2SO4

MgCl2 CuCl2 MgSO4

CuSO4

ζ(m

v)

c/(mol/m3)

Fig. 13. Variation of zeta potential of NTR7450 membrane with concentration foreight electrolyte solutions (pH = 6.5).

236 K. Zhao, G. Ni / Journal of Electroanalytical Chemistry 661 (2011) 226–238

low concentration, Donnan exclusion is the most important factorsand in high concentration, dielectric exclusion is the most impor-tant. This conclusion is the same as that described in Section 4.3.

4.4.2. Ion conductivity in membraneMembrane conductivity j is an important parameter to demon-

strate the difficulty of ion transport inside membrane. Fig. 11shows the relations between j and z1v1c, which were calculatedwith the three model and following formulas:

TMS model, TMS-DE model:

j ¼ F2

RTz21v1k1D1 þ z22v2k2D2� �

c ð23Þ

DSPM model

K. Zhao, G. Ni / Journal of Electroanalytical Chemistry 661 (2011) 226–238 237

j ¼ F2

RTðz21v1k1D1p þ z22v2k2D2pÞc ð24Þ

where Dip is pore diffusivity (m2 s�1) of the ion i, the calculatingformulas of Dip are given in Appendix B. Fig. 11 shows the equiv-alent concentration z1v1c dependency of membrane conductivity jfor eight electrolyte solutions and three models. From Fig. 11 it isobvious that in low concentration, j is almost unchanged withz1v1c, whereas in high concentration j increases sharply withthe concentration. However, for TMS model, the z1v1c dependencesof j are nearly identical for the eight kinds of electrolyte solutionsin the whole range of concentration, while for TMS-DE and DSPMmodels, the values of j for different electrolyte solutions at thesame concentration shows very big difference when the concen-tration is above a certain value which varies from model to model.These results suggest that the three factors, Donnan exclusion, ste-ric effect and dielectric exclusion, play different roles in wholeconcentration range. Moreover, the membrane conductivity inthe TMS model is much larger than that of the other two modelsin high concentration, suggesting the importance of steric effectand dielectric exclusion to ion rejection in high concentration. Thisis consistent with the results in Section 4.3 in which it was con-cluded that in high concentration range, solvation energy barrier(i.e. dielectric exclusion) plays a predominate role in ionpermeability.

4.4.3. Transport number of ion in membraneThe transport number of ion i in membrane ti for eight kinds of

electrolyte solutions were calculated using Eqs. (25)–(28)[32], andthe equivalent concentration z1v1c dependence of ti for the threemodels are shown in Fig. 12.

TMS model, TMS-DE model:

t1 ¼ z21v1k1D1

z21v1k1D1 þ z22v2k2D2ð25Þ

t2 ¼ z22v2k2D2

z21v1k1D1 þ z22v2k2D2ð26Þ

DSPM model:

t1 ¼ z21v1k1D1p

z21v1k1D1p þ z22v2k2D2pð27Þ

t2 ¼ z22v2k2D2p

z21v1k1D1p þ z22v2k2D2pð28Þ

where t1 and t2 are the transport number of counter-ion and co-ionin membrane, respectively.

As depicted in Fig. 12, in the low concentration range, ti wasnearly unchanged, but the transport number of counter-ion inmembrane t1 is very large and transport number of co-ion in mem-brane t2 is very small. Nevertheless, as the concentration increases,t1 decreases rapidly and t2 increases sharply. Finally both of themtend to a stable value in higher concentration. This indicates theion transport number in membrane ti can quickly reach equilib-rium in the membrane. The smaller the ion transport number,the slower the ion migration in membrane, resulting in mast ionswere rejected in membrane. The results of Fig. 12 also showed thatthe concentration of counter-ion in membrane is more than that ofco-ion in membrane in low concentration range. However, the con-centration of counter-ion is not very high because it was limited bythe electroneutrality condition. This leads to the high rejection. Asthe concentration increases, still t1 is larger than t2 even though t2increases and t1 declines, implying that the co-ion still plays a sig-nificant role in retention in higher concentration range. On theother hand, with the increment of t2 (i.e. concentration increase),

the rejection descends. When the transport of ion in membranereaches an equilibrium state, from Fig. 12 ion transport numberwere gotten. It should be noted that the ion transport number insolutions are the same as that in membrane gotten by TMS andTMS-DE models, and are different from that obtained using DSPMmodel except that in KCl and MgSO4 solutions. This is because thatthe equivalent concentration of the eight electrolyte solutions atwhich the ion transport number reach equilibrium are differentin the three models. For TMS and DSPM model the equivalent con-centration is about 10 mol/m3, while for TMS-DE is more than 103

mol/m3. The difference shows the importance of dielectric exclu-sion to ion rejection.

4.5. Surface charge density and zeta potential

The volume charge density Xm obtained in Section 4.3 can beconverted to surface charge density on pore walls qw by Eq. (29)which is based on the assumption that there are straight cylindri-cal pores with uniform radius rp in membrane[33]:

Xm ¼ 2qw

Frpð29Þ

where F is Faraday constant. Through the Eq. (29), the value of qwfor eight systems composed of membrane and eight individual elec-trolytes were calculated and the results are listed in Table 9.

According to Gouy–Chapman electrical double layer theory,surface charge density qw and zeta potential f satisfy the followingrelation [33]:

qw ¼ ðsignfÞ

� ð2ceRTÞ12 vþ exp � zþFfRT

� �þ v� exp � z�Ff

RT

� �� vþ � v�

� �12

ð30ÞThe symbols in Eq. (30) have the same meaning as before. From

the Eq. (30) the zeta potential f can be calculated using surfacecharge density. Fig. 13 shows the electrolyte concentration depen-dency of f calculated by the Eq. (30). It is obvious from the Fig. 13that the zeta-potential in membrane surface in eight electrolytesolutions increases with the increment of the electrolyte concen-tration. This changing trend will be helpful to investigate the pene-trating quality of NF membrane, and provide information on thechange of charge in membrane surface. Sequentially, this resultcan give us an important reference for analyzing the salt rejection.

5. Conclusions

Thedielectricmeasurementswere carriedout onNTR7450mem-brane in eight individual electrolyte solutions. On the basis of inter-facial polarization theory, the high frequency dielectric spectrawereanalyzed theoretically, giving important parameters such as permit-tivityandconductivity inmembraneandsolutions,which is inacces-sible by other any individual experimental methods. Permittivity ofthe membrane em varied in different solutions, but was less affectedby the electrolyte concentration. Conductivity of membrane jmwasmuch smaller than the conductivity of the solution jw, indicatingthat membrane had a retention effect to ions.

Making use of the permittivity of membrane em, ion solvationenergy barrier DW was calculated and its influence on separationperformances of membrane was evaluated. The results showedthat DW is very significant for estimating the difficulty of ionstransport in the membrane. By combining the ratio of conductivityof membrane to solution jm/jw with TMS model, volumetriccharge density Xm of membrane was calculated. By using Xm andem coupled with the transport models, transport parameters were

Table A.1Expressions of ki for TMS model.

Types of solutions 1-1 type and 2-2 type 1-2 type 2-1 type

(1) ð27þ 4n3f Þ > 0 时, K ¼ 108þ 12ffiffiffi3

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið27� 4n3f Þ

qK ¼ 108þ 12

ffiffiffi3

p ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffið27þ 4n3f Þ

qki ¼ ðK1=3

6 � 2nfK1=3 Þzi

Expressions of kiki ¼

�sgnðXmzi Þnf þffiffiffiffiffiffiffiffin2f þ4

p2

ki ¼ ½K1=3

6 � 2nfK1=3 ��zi (2) ð27þ 4n3f Þ 6 0 时, h ¼ ar cosð12

ffiffiffiffi27n3f

rÞ

ki ¼ ð2ffiffiffiffinf3

qcos h

3 Þzi

238 K. Zhao, G. Ni / Journal of Electroanalytical Chemistry 661 (2011) 226–238

calculated. The following conclusions were drawn: in the processof ion permeation inside membrane, Donnan exclusion and stericeffect play a dominant role in low concentration, whereas dielec-tric exclusion is dominant influencing factor in high concentration.

In summary, in this paper, the membrane systems immersed inelectrolyte solutions were systematically studied, and some uniqueinformation on ion permeation through membrane and someimportant membrane parameters were obtained by analyzingdielectric spectra coupled with the transport models. The phaseparameters characterizing the electric properties of membranethat is close to virtual separation condition are obtained, and thetransport parameters obtained by calculation based on phaseparameters provide us a guidance to evaluate the separation per-formance of membrane and design new membrane. Additionally,dielectric spectroscopy method also provides a new idea for study-ing ion permeability in membrane.

Acknowledgement

Financial support of this work by the National Natural ScienceFoundation of China (21173025 and 20976015) is gratefullyacknowledged.

Appendix A

The expression of ion partitioning coefficients for TMS model(Eq. (8)), electroneutrality condition in the membrane (Eq. (11))and Eq. (18) give the following relation:

Cz1 � Cz2 � n ¼ 0 ðA:1Þ

where

C ¼ expð� FD/RT

Þ ðA:2Þ

n ¼ jXmjjz1jv1c

ðA:3Þ

Substituting Eq. (A.2) into Eq. (8), the expression of ion parti-tioning coefficients is described by:

ki ¼ Czi ðA:4ÞThe solution of Eq. (A.1) gives the expression of C, and the com-

bination of C and Eq. (A.4) gets the expressions of ion partitioningcoefficients ki as listed in Appendix Table A.1.

Rearranging Eq. (8), ion concentration in membrane can be gi-ven by:

cmi ¼ kici ðA:5ÞCombing Eq. (A.5) with the expressions of ion partitioning coef-

ficients in Appendix Table A.1, the expressions of ion concentrationin membrane for different electrolyte solutions can be derived.

Appendix B

Pore diffusivity Dip of ion i can be calculated using the followingformulas:

Dip ¼ KidDigo

g0 ðB:1Þ

where

g0

go¼ 1:0þ 18ðd

rpÞ � 9ðd

rpÞ2 ðB:2Þ

where d is thickness of the oriented solvent layer, Kip is diffusivehindrance factor of ion i and described by:

Kid ¼ 1:0� 2:30ki þ 1:154k2i þ 0:224k3i ðB:3Þwhere

ki ¼ rsrp

ðB:4Þ

rs and rp are Stokes radius of ions i and pore radius, respectively ashave been defined above.

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