ARTICLE pubs.acs.org/Langmuir
Free Energies of the Ion Equilibrium Partition of KCl into Nanofiltration Membranes Based on Transmembrane Electrical Potential and Rejection Cong-Hui Tu,† Yan-Yan Fang,† Jie Zhu,† Bart Van der Bruggen,‡ and Xiao-Lin Wang* † ‡
State Key Laboratory of Chemical Engineering, Department of Chemical Engineering, Tsinghua University, Beijing 100084, P. R. China Division Applied Physical Chemistry and Environmental Technology, Department of Chemical Engineering, K. U. Leuven, Belgium ABSTRACT:
The free energies of ion equilibrium partition between an aqueous KCl solution and nanofiltration (NF) membranes were investigated on the basis of the relationship of the transmembrane electrical potential (TMEP) and rejection. The measurements of TMEP and rejection were performed for Filmtec NF membranes in KCl solutions over a wide range of salt concentrations (160 mol 3 m3) and pH values (310) at the feed side, with pressure differences in the range 0.10.6 MPa. The reflection coefficient and transport number, which were used to obtain the distribution coefficients on basis of irreversible thermodynamics, were fitted by the two-layer model with consideration of the activity coefficient. Evidence for dielectric exclusion under the experimental conditions was obtained by analyzing the rejection of KCl at the isoelectric point. The free energies were calculated, and the contribution of the electrostatic effect, dielectric exclusion, steric hindrance, and activity coefficient on the ion partitioning is elucidated. It is clearly demonstrated that the dielectric exclusion plays a central role.
1. INTRODUCTION Nanofiltration (NF) is a promising pressure-driven membrane separation technique that uses membranes with pores of nanoscale (∼1 nm) dimensions and electric charges in aqueous media. These special properties imply that solute exclusion results from a complex mechanism. The key points to clarify the mechanism can be identified both in the characterization of the membrane, which can be seen as a homogeneous or as a porous medium, and in the understanding of the phenomena that give rise to ion partitioning, which involves size, electrical, and dielectric mechanisms. Several models have been established according to different interpretations of the transport mechanism through NF membranes. The ions were considered point charges in the space charge (SC) model13 and the TeorellMeyerSievers (TMS) model,4,5 in which ion partitioning is thought to depend only on electrical interactions. However, because the sizes of the pores of NF membranes and ions are comparable, some researchers have proposed that the steric-hindrance effect should also be taken r 2011 American Chemical Society
into account. In the electrostatic and steric-hindrance (ES) model,6 the ion flux inside the charged capillaries of NF membranes was expressed by the modified NernstPlanck equation by considering the steric-hindrance effect. Meanwhile, Bowen7 proposed the Donnan and steric partitioning model (DSPM), which is also based on the extended NernstPlanck equation and which takes into account the partitioning of ions between external and internal solutions of the membrane through the Donnan equation, where both equations have been modified to include the sieving effects. Nevertheless, some negative results have been obtained with this model, especially for NF membranes with small pores. In some cases, the rejection was underestimated,810 and unrealistically high values for the volume charge (sometimes even with the wrong sign) were predicted for some NF membranes11,12 with consideration of the Received: January 18, 2011 Revised: June 23, 2011 Published: July 05, 2011 10274
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Langmuir Donnan exclusion and steric-hindrance effect. The dielectric exclusion, which included two mechanisms, was proposed. First, by considering a greater degree of spatial and orientational order of water molecules in the confined environments, the water dielectric constant will decrease, which creates an energy barrier to the solvation of ions into pores and repulses co-ions and counterions.13 However, some controversial opinions about the dielectric constant of water in fine pores still exist.14 Second, the “image force” that arises from the difference between the dielectric constant of an aqueous solution and the corresponding value of the polymeric matrix also allows the prediction of ion exclusion.15 The combination of these two mechanisms with both Donnan exclusion and the steric-hindrance effect was shown to be valid in many NF systems.9,12,1620 However, methods for the direct measurement of the dielectric constant of water in NF membranes or the charge density are not available. Whether Born energy or image force dominates the dielectric exclusion is still a subject of discussion. Some researchers believe that the porewater dielectric constant approaches that of the polymeric matrix because of the small radii of the NF pores. The image charges are screened in electrolyte solutions because of electrical double layers; consequently, the Born energy would dominate the dielectric exclusion mechanism under most NF conditions.12,16 Other authors have assumed that the difference between the dielectric constant of water and the polymeric matrix is dominant.9 Both of these mechanisms were also considered by Szymczyk and Fievet.19 Moreover, the value of charge density varied between the different methods. Currently, two primary methods exist for obtaining the charge density of NF membranes. One method is based on the fitting of experimental data by models, such as DSPM and the dielectric exclusion model (DSPM&DE), by measuring the rejection of salt solutions of different concentrations. Nevertheless, the problem arises that both the dielectric constant of water in NF pores and the effective volume charge density are unknown. Accurately obtaining these two parameters by a single fitting program is therefore difficult.9,12,16 An alternative approach is based on the tangential streaming potential. Szymmczyk and Fievet obtained the charge density to calculate the dielectric constant of water molecules in NF pores.19 The value of charge density from tangential streaming potential was considered the maximal value for the possible differences in the chemistries of inner and outer membrane surfaces.21 Therefore, the values of volume charge density were still undetermined by this method. In this work, evidence for the existence of dielectric exclusion was obtained, and a new method to obtain free energies of the equilibrium partition of KCl at the entrance of NF pores was developed. The process was divided into four parts. First, the rejection, transmembrane electric potential (TMEP),2228 and permeate volume flux were measured during the filtration process. TMEP was chosen because it could reflect the actual electric characteristics of NF membranes in the filtration process. Second, transport numbers were fitted by the two-layer model from our previous work.29 Reflection coefficients were obtained from the relationship between rejection and the permeate volume flux using a combination of the extended NernstPlanck equation and irreversible thermodynamics. Third, these two coefficients were combined to calculate the distribution coefficient on the basis of the irreversible thermodynamics. Fourth, the rejection of KCl solutions with different concentrations at the isoelectric point, determined by the zero TMEP,29 was measured. The dielectric exclusion mechanism was studied without the influence of the electrostatic effect. On the basis of these findings,
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Table 1. Governing Equations zero electric current condition (steady state) 2
F
∑ zi Ji i¼1
¼ 0
ð1Þ
transport equations d ln γi, act ci, act zi F Ji dj Dip ci, act ¼ Dip ci, act Rg T dx ðAk Þact dx þ Kic ci, act
Jv ðactivelayerÞ ðAk Þact
Ji zi F dj Jv þ ci, sup Di ci, sup ¼ ðsupportlayerÞ Rg T dx ðAk Þsup ðAk Þsup
ð2Þ ð3Þ
with6 Kid ¼ SDi HDi H ¼ 1 Di SDi ¼ ð1 ηi Þ2 Kic ¼ SFi HFi , 16 2 , 2 2 , η SFi ¼ ð1 ηi Þ ½2 ð1 ηi Þ HFi ¼ 1 þ Dip ¼ Kid Di 9 i ηi ¼ rsi =rp ð4Þ membrane parameters zi 2 Dip ci, act , i ¼ 1, 2 ti, act ¼ 2 zi 2 Dip ci, act
∑ i¼1
ð5Þ
1σ ¼
K1c c1, act t2, act K2c c2, act t1, act þ c1, f c2, f
ð6Þ
Rg Tω ¼
v1 D2p K1c þ v2 D1p K2c ð1 σÞ ðv1 þ v2 ÞD1p D2p
ð7Þ
free energies of the equilibrium partition of KCl at the entrance of the NF pores were obtained from the distribution coefficient, and the image force contribution and Born energy were then calculated. It is clearly demonstrated that the dielectric exclusion was the important parameter of the resulting ion partition, and it was dependent regarding both concentration and pH.
2. THEORY NF membranes are considered to be composed of two layers: an active layer and a support layer. The active layer is thought to dominate the separation performance. Because of the comparable sizes of NF membrane pores and most ions, the ion flux inside the charged capillaries of the active layer obeys the extended Nernst Planck equation, including the steric hindrance effects. The support layer is considered incapable of separating the ions, and the concentration gradient in this layer is therefore almost zero. Thus, the concentration in the support layer equals that at the permeate side. The steric hindrance effect is considered to be absent in the support layer. As a result, the ion flux in the support layer can be described by the NernstPlanck equation. A Boltzmann distribution of ions is assumed at the two interfaces: the interface of the feed solution and active layer, and the interface of the active layer and support layer. The concentration and the potential are continuous at the interface of the support layer and permeate side, which means ci,sup = ci,p and jT,sup = jT,p at this interface (subscripts act and sup refer to the active layer and the support layer; f is the feed, and p is the permeate; i refers to ion i). The governing equations, including the NernstPlanck equation in the active layer and support layer and the Boltzmann distribution 10275
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at the interface, are listed in Table 1. To simplify the expression of TMEP, the membrane parameters, including the electric transport number ti, the reflection coefficient σ, and the solute permeability coefficient ω in the active layer, are also listed in Table 1. These parameters can be obtained by combining the extended Nernst Planck equation and irreversible thermodynamics.6,30 Here, the activity coefficient γi is taken into account using the extended DebyeH€uckel law pffiffi Azi 2 I pffiffi log γi ¼ 1 þ Bδi I
"
γp cp ti, act ln þ zi γf cf i¼1 2
∑
ti, act Kic Δx Jv Ak act i ¼ 1 zi Dip 2
∑
3 þ
Jv S 2
∑ zi
i¼1
2D
i ci, p
7 Δx 7 S ¼ ðX Þ 7 w sup 5 Ak sup
ð9Þ
For KCl, when the membranes are negatively charged, the transport number of K+ is larger than that of Cl; therefore, the first two terms in the right bracket of eq 9 are positive, which means that the TMEP in the active layer would be positive. Conversely, when the membranes are positively charged, the TMEP in the active layer would be negative. In steady-state nanofiltration, the electrical current is zero. Thus, the integration of eq 2 for the active layer gives the relationship between rejection and flux for the KCl solution R ¼ 1
1σ γp Jv ð1 σÞ þ ln 1 σ exp P γf
¼ 1
!
1σ ! γp Jv ð1 σÞ Δx þ ln 1 σ exp Rg Tω Ak act γf ð10Þ
ð11Þ
Thus, the reflection coefficient can be obtained from the relationship between the rejection and permeate flux. The combination of eq 5 and eq 6 gives the distribution coefficients as follows:
ð8Þ
where I is the ion strength, and its units are mol 3 kg1. A and B are constants that depend on the temperature and dielectric constant of the solvent, and δi is a parameter that depends on the sizes of the ions. For a KCl solution at 20 °C, A is 0.5070, B is 0.3282. and both δCl- and δK+ are 3.31 Thus, under these conditions, the activity coefficients of K+ and Cl are the same. By considering the effect of the activity coefficient, the transport number of K+ in the active layer can be fitted using the following equation by multivariate linear regression with the experimental data of ΔjT and Jv in KCl solution.29 For other solutes, the activity coefficients of the cation and anion may be different, which would render this equation invalid. Rg T ΔjT ¼ F
The term (Δx/Ak)act can be calculated as follows: ðv2 þ v1 ÞD1p D2p ð1 σÞ Δx ¼ Ak act P v2 D1p K2c þ v1 D2p K1c
k1, f ¼
z2 D2p ð1 σÞ t2 z2 D2p K1c þ z1 D1p K2c
ð12Þ
k2, f ¼
z1 D1p ð1 σÞ t1 z2 D2p K1c þ z1 D1p K2c
ð13Þ
Equations 913 are the approximate results of combining the extended NernstPlanck equation and irreversible thermodynamics for the change in the parameters, such as ci,act, ti,act, σ, and ω, in the x-direction in the membranes. A difference between the accurate and approximate results would exist, especially in the intermediate flux regime. This difference, however, depends on which average values have been selected for these parameters; when proper average parameters are chosen, the difference will be small. According to the Boltzmann distribution, which gives the equilibrium concentration of a particular type of molecule in two regions/phases, if the molecular free energy has different values μi1 and μi2, then the difference in the molecular free energy Δμi,f = μi,act μi,f in the bulk phase and in the active layer of the NF membranes includes four parts.19 The first part is the electrostatic interaction between the ions and charges on the NF membranes, which causes an electrostatic potential, called the Donnan ΔjD potential, at the interface. For ion i with electrovalence zi, the difference in the molecular free energy is written as ΔμiD,f = zi eΔjD,f. The second part is the steric hindrance effect, and this free energy can be written as ΔμiS,f = kT ln(SDi). The third part is the dielectric exclusion, which can be divided into two mechanisms: the dielectric exclusion is caused by the interactions between the ions and solvent and between the ions and polymer membranes. The ionsolvent interaction can be expressed by the Born energy32 ΔμiB,f, which excludes the ions from a solvent with a low dielectric constant. However, the original Born equation leads to an overestimation of the experimental solvation free energy when the crystal radius for an ion is used, because it assumes a uniform dielectric constant of the solvent. Adjustments to either the ion radius or the solvent dielectric constant have been carried out to improve the Born model.3340 Because the characteristics of water in a nanopore are different from those in the bulk phase, the characteristics of ions, especially the hydration number, may be different; therefore, all the adjustments were applied to the bulk. Without knowledge of the sizes of the ions in the pores, some researchers used the Stokes radius to provide consistency between the transport and distribution equations.12 Here, we have used three kinds of radii to compare the differences. Table 2 lists the three radii of K+ and Cl. The Born equation was followed:
where P is the solute permeability. According to eq 7, (1σ)/ RgTω only depends on the diffusion coefficient and the steric hindrance effect of the solute, and not on the concentration of the solution. Therefore, when the data of the rejection and flux are fitted for different concentrations, (1σ)/P remains constant.
ΔμiB, f ðpÞ 10276
ðzi eÞ2 1 1 ¼ 8πε0 ri εw, act εw, f
! ð14Þ
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Table 2. Three Kinds of Radii of K+ and Cl Pauling crysallographic
3. EXPERIMENTS
cavity radius by Rashin and Stokes radius
ions
radius31 (nm)
Honig40 (nm)
(nm)
K+
0.133
0.2172
0.124
Cl
0.181
0.1937
0.12
The ionpolymer membrane interaction can be defined as the image interaction in confined charged pores ΔμiI,f,. This interaction has been recently reviewed by Yaroshchuk,15 who derived approximate expressions for the interaction energy due to image forces. The followed equations are the simplified versions by Szymczyk that neglect the radial position:19 Z 2Ri kT ∞ K0 ðxÞK1 ðvÞ βðxÞK0 ðvÞK1 ðxÞ dx ΔμiI, f ¼ π 0 K0 ðxÞI1 ðvÞ þ βðkÞI0 ðvÞK1 ðxÞ ð15Þ Ri ¼
ðzi FÞ2 8πε0 εw, act RTNA rp
ð16Þ
pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi x 2 þ μ2
ð17Þ
sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi z2i ci, f ki, f μ ¼ kf rp 2I i
ð18Þ
v¼
∑
kf ¼ I ¼
1 2
ε0 εw, f RT 2F 2 I
1=2
ð19Þ
∑i ci, f z2i
x εm β¼ v εw, act
ð20Þ
3.2. Rejection, Permeate Volume Flux, and TMEP Measurements. The rejection, permeate volume flux, TMEP, and pressure difference were measured synchronously in a laboratory setup. The experimental setup was the same as previously reported.29 After each experiment, the pure-water permeability was measured to ensure that the properties of the membranes did not change. The transmembrane pressures ranged from 0.1 to 0.6 MPa. The cross-flow was fixed at 6.0 L 3 min1. Theoretically, the influence of concentration polarization on the rejection and TMEP could be ignored under these conditions. A heat exchanger was used to maintain the temperature at 20 ( 0.2 °C. The permeate volume flux Jv was determined by weighing the amount of permeate that flowed through the membrane. The rejection was obtained from measurements of the solution concentrations, which were determined by conductivity measurements performed in both the feed and permeate compartments. The electrical potential difference was measured by two Ag/AgCl electrodes. The electrical potential measurements include the Nernst (electrode) potential for Cl reversible electrodes. The electrode potential is given by the following expression:
!
Δμi, f ¼ ΔμiD, f þ ΔμiS, f þ ΔμiA, f þ ΔμiB, f þ ΔμiI, f ð22Þ Therefore, the energies causing the dielectric exclusion in the KCl solutions can be expressed by the distribution coefficients:
þ ΔμiI, f
Δjelectrode ¼
ð21Þ
The last part of molecular free energy in the bulk phase and in the active layer of the NF membranes can be defined as the electrostatic interaction between ions in solution, which can be expressed by the activity coefficient. For different ion strengths in the bulk phase and in the membranes, the difference in the molecular free energy should be ΔμiA,f = kT ln(γi,act/γi,f). Thus, according to the Boltzmann distribution, the distribution coefficient can be written as follows: ! Δμi, f ci, act ¼ exp ki, f ¼ ci, f kT
∑i ΔμiB, f
3.1. Membranes and Chemicals. The membrane used in this work was Filmtec NF (Dow). The membrane is made of aromatic polyamide and is negatively charged in the pH range of 510. The pore size of two NF membranes was determined by the separation experiments of neutral molecules (ethanol, isopropyl alcohol, and t-butyl alcohol) on the basis of the steric hindrance pore model.41 In our primary work,29 the mean pore radii of Filmtec NF was determined as 1 nm. Experiments were performed with reagent-grade potassium chloride (Beijing Modern Eastern Fine Chemical Corp.). Millimolar solutions at different concentrations were prepared in deionized water. The salt concentrations ranged from 1.0 to 60.0 mol 3 m3. The pH was adjusted from 3.0 to 10.0 by the addition of 1000 mol 3 m3 KOH or HCl solution to 10 mol 3 m3 KCl solution. Therefore, the ratios of the concentrations of H+ to K+ or OH to Cl were less than 0.1, and the influence of the rejection of H+ and OH and the change in concentration of the KCl solutions could be ignored.
k1, f k2, f γ1, f γ2, f ¼ kT ln SD1 SD2 γ1, act γ2, act
! ð23Þ
R g T af ln ap F
af ðpÞ ¼ γf ðpÞ cf ðpÞ
ð24Þ
4. RESULTS AND DISCUSSION 4.1. Distribution Coefficients. The relationship between rejection, TMEP, and permeate volume flux in KCl solutions with different concentrations has already been reported.29 However, in the previous work, only the qualitative relationship of the TMEP and permeate volume flux was discussed. In this work, we combine the fitting results of the TMEP and the rejection, and we also take the activity coefficients into account. The fitting procedure follows our previous work.29 First, the pore radius was obtained by the separation experiments of neutral solutes in the membranes, then Ak and the reflection coefficient were fitted by eq 9 on the basis of the relationship of the rejection and permeate volume flux in KCl solutions. With the experimental data of the TMEP and permeate volume flux, ti,act was obtained by multivariate linear regression according to eq 10. The reflection coefficient and transport number of K+ determined by fitting are listed in Table 3. The activity coefficient of KCl in the considered concentration range is greater than 0.85, and the rejection of KCl in the considered pressure range is between 0.1 and 0.7. When the concentration is small, the activity coefficient is close to 1.0. When concentration is increased, the 10277
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Table 3. Reflection Coefficients and Transport Numbers Calculated by eq 9 and eq 10 in KCl Solutions with Different Concentrations concentration
reflection
transport
solutes
(mol 3 m3)
coefficient (-)
number of K+ (-)
2
0.8669
0.8701
KCl
10
0.8506
0.8019
22
0.7886
0.762
52
0.6978
0.7349
Figure 2. Relationship between permeate volume flux and effective pressure for a KCl solution in a Filmtec NF membrane. (9, pH = 6.5, cf = 2 mol 3 m3; 0, pH = 4.3, cf = 4 mol 3 m3; b, pH = 6.5, cf = 10 mol 3 m3; O, pH = 4.3, cf = 8 mol 3 m3; 2, pH = 6.5, cf = 22 mol 3 m3; Δ, pH = 4.3, cf = 13 mol 3 m3).
Figure 1. Relationship between the distribution coefficient and concentration for a KCl solution in a Filmtec NF membrane (subscript 1 indicates K+, and 2 indicates Cl).
rejection decreases, and the differences between γp and γf become smaller. Therefore, the value of γp/γf is close to 1.0; in this system, it is no greater than 1.05. The activity coefficient exerted a small influence on the fitting results. From the table, the reflection coefficients and transport numbers of K+ decrease with increasing concentration. From these two parameters, distribution coefficients can be calculated as described in section 2. The relationship between the distribution coefficients and concentration is shown in Figure 1. As evident in Figure 1, the distribution coefficient of K+ first decreases and then increases with concentration, whereas the distribution coefficient of Cl increases monotonically The distribution coefficients are all below 1.0, which leads to the prediction that the forces between the Filmtec NF membranes and both of the ions are repulsive. Moreover, given the electroneutrality inside the membranes, the effective volume charge density of the membrane would be smaller than the salt concentration in the feed side. The attraction between a negative charge on the membrane and K+ is not sufficiently strong to counteract the repulsion induced by other interactions. The screening effect caused by the increase in the concentration may weaken the electrostatic effect between the membranes and ions, which can explain why the distribution coefficient of Cl increases monotonously. In addition to the electrostatic effect, another effect, i.e., the concentration dependence, exerts an opposite influence on K+, which results in the inflection of the distribution coefficient of K+. 4.2. Dielectric Exclusion. The isoelectric point of the Filmtec NF membrane was 4.3 according to the experimental and theoretical results from our previous work.29 The relationship of permeate volume and effective pressure is shown in Figure 2. As shown in Figure 2, the solvent permeation coefficient is larger when the pH is 4.3. This effect has also been reported by others.42,43
Figure 3. Comparison of the relationship between TMEP and permeate volume flux at different feed pH levels and concentrations of KCl solutions in a Filmtec NF membrane. (9, pH = 6.5, cf = 2 mol 3 m3; 0, pH = 4.3, cf = 4 mol 3 m3; b, pH = 6.5, cf = 10 mol 3 m3; O, pH = 4.3, cf = 8 mol 3 m3; 2, pH = 6.5, cf = 22 mol 3 m3; Δ, pH = 4.3, cf = 13 mol 3 m3).
The TMEP and the rejection for KCl solutions with different concentrations are compared at pH = 6.5 and pH = 4.3 in Figure 3 and Figure 4. At pH = 4.3, the TMEP is almost zero, and the rejection appears to be almost independent of concentration. Among the four previously described effects, the electrostatic effect and the image charge contribution depend on the feed concentration, which plays an important role in the charge formation and the screening effect on the membranes. At the isoelectric point, the electrostatic effect is ignored. Without consideration of dielectric exclusion, the relationship of rejection and permeate volume flux was predicted with the sizes of the solutes and membrane pores; the activity coefficient was also taken into account. As evident in Figure 4, the prediction underestimates the rejection. The reflection coefficient of KCl at pH = 4.3 is approximately 0.5, whereas the predicted value is 0.25. Moreover, the reflection coefficient of ethanol is 0.333, and an ethanol molecule is larger than K+ and Cl. All the phenomena indicate that the dielectric exclusion should not be neglected in the system of ions and NF membranes. Moreover, under the considered conditions, the image charge contribution seems to be weakly independent of concentration. 4.3. Free Energy. The free energies that lead to ion partitioning at pH = 6.5 and pH = 4.3 are shown in Figure 5. The dielectric 10278
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Figure 4. Comparison of the relationship between rejection and permeate volume flux at different feed pH levels and concentrations of KCl solutions in a Filmtec NF membrane. The broken lines were simulated by consideration of the steric hindrance effect and activity coefficient only. (9, pH = 6.5, cf = 2 mol 3 m3; 0, pH = 4.3, cf = 4 mol 3 m3; b, pH = 6.5, cf = 10 mol 3 m3; O, pH = 4.3, cf = 8 mol 3 m3; 2, pH = 6.5, cf = 22 mol 3 m3; Δ, pH = 4.3, cf = 13 mol 3 m3).
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Figure 6. Free energy of dielectric exclusion of Cl and the prediction of image charge contribution by eq 15 in a Filmtec NF membrane at pH = 6.5 and pH = 4.3.
Figure 7. Dielectric constant of water molecules in nanopores calculated with consideration of only the Born effect in a Filmtec NF membrane at pH = 6.5 and pH = 4.3. Figure 5. Relationship between the free energies of Cl and the feed concentration of KCl solutions in a Filmtec NF membrane at pH = 6.5 and pH = 4.3. (Solid symbols: pH = 6.5; hollow symbols: pH = 4.3).
exclusion effect is the dominant factor; it first increases and then decreases with increasing concentration. This effect coincides with the curve of k1, and this association may be caused by the change in the dielectric constant of water molecules in nanopores. The change in the dielectric results in a change of the Born effect or the screening effect, which leads to a change in the image charge contribution. When the membranes are neutral (pH = 4.3) and therefore do not exhibit an electrostatic effect, the dielectric exclusion is still the most significant contributor to the ion partitioning. Therefore, when the concentration is sufficiently large to result in the total screening effect (electrostatic effect could be ignored), the dielectric exclusion will still be the most important factor. The Donnan effect was the second most important factor when the concentration was not sufficiently large. From Figure 5, the Donnan effect decreases with increasing concentration for the screening effect. The steric hindrance effect in this system is not particularly important; the change in the activity is sufficiently small to be neglected. Calculations were performed to clarify the contribution of the image charge and the Born effect on the dielectric exclusion. The image charge contribution is calculated by eq 15. Figure 6 represents the prediction of the image charge contribution with the assumption of the same dielectric constant of water molecules in the bulk phase and in nanopores. If the dielectric
constant in the nanopores is smaller than in the bulk, which would mean that the Born effect is positive and that the image charge contribution is larger according to the weaker screening effect,19 then the prediction would be larger than that calculated in Figure 6. Therefore, even when the dielectric constant of water molecules is assumed to be the same as in the bulk phase, which means that the image charge contribution is the smallest in the charged pores, eq 15 still overestimated the image charge contribution. This situation would be exacerbated if the variational formalism for Coulombic systems in the presence of dielectric discontinuities is introduced.44 According to the prediction, image charge contribution decreases with increasing concentration when Xw is zero or negative. However, the energy calculated by the experimental results of TMEP and rejection seems uniform when Xw equals zero. Furthermore, when Xw is negative, although it also decreases as the concentration increases, the rate of decrease is much smaller. Some researchers have assumed that only the Born effect leads to the dielectric exclusion.12,16 Under this assumption, Figure 7 represents the dielectric constant calculated by considering three different kinds of radii. The values of the dielectric constant are all smaller than those in the bulk phase in this system and range from 45 to 70. The values of εw,act calculated from Stokes radii are larger. These values are in good agreement with those reported in the literature that only took the Born effect into account.16 According to some researchers, the lowering of the dielectric constant inside the nanopores depends on the nature of the 10279
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Langmuir membrane, the pore radius, and the electrolyte type and concentration.45 In our system, we find that, when the pH is 6.5, the dielectric constant of water in the nanopores is concentration dependent; when the pH equals 4.3, the values become larger and appear to remain constant, irrespective of the concentration. This behavior may be caused by the confinement and orientation of the water. The electric field generated by the volume charge density may disrupt the orientation of the water molecules, which leads to a decrease of εw,act with increasing concentration.46 When the concentration is increased, the electric double layer might become thinner; the confinement of the water molecules would therefore become weaker, which would result in an increase of εw,act. At pH = 4.3, the volume charge density was almost zero, and the effect of the electric field was small; the dielectric constant therefore increased.45 However, the decrease in εw,act is based on the assumption that only the Born effect leads to the dielectric exclusion, and this assumption still lacks evidence.
5. CONCLUSIONS The evidence for dielectric exclusion and the free energies contributing to ion partitioning were discussed using results of experiments on the TMEP and rejection in the two-layer model. The rejection when the NF membranes were neutral indicated that the dielectric exclusion should not be neglected in the system of ions and NF membranes. On the basis of the free energies, the dielectric exclusion plays a crucial role in the separation of ions in NF membranes. If the dielectric constant in nanopores is assumed to be smaller than in the bulk phase, the prediction of the image charge made by eq 15 appears to overestimate the energy in this system in low concentration ranges. When only the Born effect was considered, the dielectric constant of the water molecules in nanopores was smaller than that of molecules in the bulk phase and was also concentration and pH dependent. Future research should focus on obtaining more evidence about the composition of dielectric exclusion. ’ AUTHOR INFORMATION Corresponding Author
*E-mail:
[email protected]. Telephone: +86-10-62794741. Fax: +86-10-62794742.
’ ACKNOWLEDGMENT This work was supported by the National High Technology Research and Development Program of China (2009AA062901), the National Basic Research Program of China (2009CB623404), and Beijing Natural Science Foundation (2100001). The authors also thank the project of bilateral scientific cooperation between Tsinghua University and the University of Leuven.
’ NOMENCLATURE A Activity (mol 3 m3) Concentration of ion i (mol 3 m3) ci Diffusion coefficient of ions i (m2 3 s1) Di Effective diffusion coefficient of ions i (m2 3 s1) Dip
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Elementary charge (=1.6 1019) (C) Faraday constant (=96487) (C 3 mol1) Steric-hindrance parameters related to the wall correction factors of ions i under diffusion condition (-) Steric-hindrance parameters related to the wall correcHFi tion factors of ions i under convection condition (-) I Ion strength (mol 3 m3) Flux of ion i over the membrane surface Ji (mol 3 m2 3 s1) Solution volume flux over the membrane surface Jv (m 3 s1) K Boltzmann constant (=1.38 1023) (J/K) Local distribution coefficient of ion (-) ki Convection hindrance factor of ions i (-) Kic Diffusion hindrance factor of ions i (-) Kid Amedeo Avogadro Constant (=6.02 1023) NA Stokes radius of ions i (m) rsi P Solute permeability (m 3 s1) Membrane pore radius (m) rp R Rejection (-) Gas constant (=8.314) (J 3 mol1 3 K1) Rg Contribution to the averaged distribution coefficients SDi caused by the steric-hindrance effects of ions under diffusion condition (-) Contribution to the averaged distribution coefficients SFi caused by the steric-hindrance effects of ions under convection condition (-) T Temperature (K) Transport number of ions i (-) ti Effective volume charge density (mol 3 m3) Xw Δx/Ak The ratio of effective membrane thickness over porosity (m) Electrochemical valence of ion (-) zi Greek Letters Difference of molecular free energy between phases (J) Δμi Difference of molecular free energy caused by electroΔμiA static interaction of ions in solution between phases (J) Difference of molecular free energy caused by Born ΔμiB energy between phases (J) Difference of free energy caused by steric-hindrance ΔμiD effect in membranes between phases (J) Difference of molecular free energy caused by image ΔμiI charge between phases (J) Difference of molecular free energy caused by electroΔμiS static interaction of ions and charged membranes between phases (J) δ The parameter with respect to the size of ions (-) Vacuum permittivity (C2 3 N1 3 m2) ε0 The dielectric constant of membranes (-) εm The dielectric constant of water molecule in nanopores (-) εw.act The dielectric constant of water molecule in feed solution εw.f (-) Convection potential (V) Δjc Diffusion potential (V) Δjd Donnan potential (V) ΔjD Membrane potential (V) Δjm Transmembrane electrical potential (V) ΔjT Δjelectrode The electrode potential (V) Ratio of Stokes radius of ions to pore radius (-) ηih σ Reflection coefficient (-) Activity coefficient (-) γi E F HDi
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Langmuir ω Solute permeability coefficient (mol 3 m2 3 J1 3 s1) Subscript act Active-layer sup Support-layer f Feed side p Permeate side i i-th ion (=1 cation; =2 anion)
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