pubs.acs.org/Langmuir © 2010 American Chemical Society
Determining the Dielectric Constant inside Pores of Nanofiltration Membranes from Membrane Potential Measurements Aurelie Escoda,† Yannick Lanteri,† Patrick Fievet,*,† Sebastien Deon,† and Anthony Szymczyk‡,§ †
Institut UTINAM, UMR CNRS 6213, Universit e de Franche-Comt e, 16 Route de Gray, :: Besanc-on Cedex 25030, France, ‡Universit e Europ eenne de Bretagne, 5 Boulevard Laennec, § 35000 Rennes, France, and Universit e de Rennes 1, CNRS, Laboratoire des Sciences Chimiques de Rennes UMR 6226 CNRS/UR1/ENSCR, Chimie et Ing enierie des Proc ed es, 263 Av. du G en eral Leclerc, B^ at. 10 A, CS 74205, F-35042 Rennes, France Received June 11, 2010. Revised Manuscript Received August 10, 2010 The membrane potential technique was applied to a nanofiltration polyamide membrane to determine its mean pore radius and the dielectric constant of electrolyte solutions inside pores. To our knowledge, this is the first attempt to assess these features from membrane potential measurements. Membrane potential data were analyzed by means of the SEDE (steric electric and dielectric exclusion) transport model. Experiments were conducted with single-salt solutions of NaCl and CaCl2 and mixed-salt solutions of NaCl and CaCl2 at various concentrations. It was shown that the pore-size values deduced from the high-concentration limit of the membrane potential measured with the two single-salt solutions are in good agreement. With this parameter being known, the membrane potential measured at high salt concentration with electrolyte mixtures was further used to compute the dielectric constant inside pores. The latter was found to be smaller than its bulk value and to decrease when sodium ions were replaced by calcium ions.
1. Introduction The application potential of nanofiltration (NF) for the separation or purification of liquid mixtures (industrial effluent treatment, production of drinking water, etc.) opened, in the past decade, intensive research directed toward the modeling of mass transfer1-9 with the aim of identifying the mechanisms involved in solute separation and the development of predictive tools allowing the optimization of the existing NF applications. The combination of pore diameters of around a few nanometers with electrically charged materials implies that ion exclusion results from a complex mechanism involving several phenomena. Most recent transport models for NF are based on the extended Nernst-Planck equation, which is modified by hydrodynamic coefficients to take into account the hindered transport through narrow pores comparable with dimensions of the permeating species and an equilibrium partitioning relation to describe the distribution of species at the pore inlet and outlet. In the first models, the Donnan equilibrium was assumed to be the only electrostatic phenomenon involved in ion partitioning at the interfaces between the membrane and liquid phases. Although particularly interesting and innovative, these models suffer from several problems. In particular, they were proven to be unable to describe the high rejection rates observed with some NF membranes in the case of ionic solutions containing divalent counterions. As an example, in the case of negatively charged membranes, CaCl2 rejection was predicted to be well below the *Corresponding author. Tel: þ33 81 66 20 32. Fax: þ33 81 66 62 88. E-mail:
[email protected]. (1) (2) (3) (4) (5) (6) (7) (8) (9)
Wang, X.-L.; Tsuru, T.; Nakao, S.; Kimura, S. J. Membr. Sci. 1997, 135, 19. Bowen, W. R.; Mohammad, A. W.; Hilal, N. J. Membr. Sci. 1997, 126, 91. Palmeri, J.; Blanc, P.; Larbot, A.; David, P. J. Membr. Sci. 1999, 160, 141. Yaroshchuk, A. E. Adv. Colloid Interface Sci. 2000, 85, 193. Bowen, W. R.; Welfoot, J. S. Chem. Eng. Sci. 2002, 57, 1121. Bandini, S.; Vezzani, D. Chem. Eng. Sci. 2003, 58, 3303. Lefebvre, X.; Palmeri, J.; David, P. J. Phys. Chem. B 2004, 108, 16811. Szymczyk, A.; Fievet, P. J. Membr. Sci. 2005, 252, 77. Deon, S.; Dutournie, P.; Bourseau, P. AIChE J. 2007, 53, 1952.
14628 DOI: 10.1021/la1023949
experimental value.10,11 The problem was also put into evidence in the case of magnesium salts by Schaep et al.12 and Szymczyk and Fievet.8 This suggests that the Donnan mechanism is not sufficient to explain the high rejection rates measured for multivalent cations because the Donnan potential acts to pump cations through negatively charged membranes. The free energy of ion transfer from external solution into nanopores is also likely to be affected by structural changes of water in a confined medium as well as by the interaction of ions with the polarization charges that are induced at the interface between media characterized by different dielectric constants (i.e., the membrane matrix and the solution inside the pores).13-15 This latter phenomenon is usually described as the production of image forces.16 It should be noted that, unlike the Donnan exclusion, which excludes co-ions from pores and attracts counterions inside, the dielectric exclusion is always unfavorable to ion transfer, irrespective of the ion sign, provided that the dielectric constant of the solution inside the pores is smaller than that of the outer solution (Born effect) and larger than that of the membrane matrix (image forces production). However, accounting for the dielectric exclusion mechanism inevitably increases the number of fitted parameters (the dielectric constant of the solution inside the pores is used as a fitting parameter), making the use of the models less appropriate for predictive purposes. The transport models including the dielectric exclusion mechanism, in terms of the Born dielectric effect or both the Born dielectric effect and image forces contribution, were found to be (10) Vezzani, D.; Bandini, S. Proceeding of the Third Nanofiltration and Application Workshop; Lappeenranta University of Technology: Finland, 2001,. (11) Vezzani, D.; Bandini, S. Desalination 2002, 149, 477. (12) Schaep, J.; Vandecasteele, C.; Mohammad, A. W.; Bowen, W. R. Sep. Purif. Technol. 2001, 22, 169. (13) Parsegian, A. Nature 1969, 221, 844. (14) Dresner, L. Desalination 1974, 5, 39. (15) Glueckauf, E. Desalination 1976, 18, 155. (16) Dukhin, S. S.; Churaev, N. V.; Shilov, V. N.; Starov, V. M. Russ. Chem. Rev. 1988, 57, 572.
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able to describe the experimental rejection rates of many membranes for various solutions. As an example, the steric, electric, and dielectric exclusion (SEDE) model, which is the latest one developed,8 was shown to provide a rather good description of the rejection properties of (i) a negatively charged organic membrane for single-salt solutions of KCl and MgCl2,8 (ii) an amphoteric ceramic membrane for three-ion solutions,17 (iii) a positively charged organic membrane for three- and four-ion solutions,18 and (iv) a negatively charged organic membrane for a five-ion mixture.19 In these studies, a decrease in the dielectric constant inside the pores with respect to its bulk value was found. Although these results are in qualitative agreement with molecular dynamics simulations showing the effect of confinement on the decrease of the dielectric constant,20 the determination of this parameter is crucial in reducing the number of fitting parameters and thus in performing a better evaluation of transport models. However, the measurement of this parameter is extremely difficult because of the multilayer (thick support layer) and composite (membrane material and confined solution) structure of NF membranes. Only a few reports show that dielectric spectroscopy can be efficient for determining the dielectric constant of filling pores.21-23 Recently, Lanteri et al. investigated the membrane potential arising from NF membranes separating binary electrolytes24 or ternary mixtures (i.e., three different ions coming from two binary electrolytes with a common ion)25 within the scope of the SEDE model. The effect of the pore size, the membrane fixed charge, and the dielectric exclusion on the membrane potential was examined. In the case of binary electrolytes, it was shown that the Donnan and Born dielectric exclusions affect the membrane potential of charged membranes similarly; in other words, a number of pairs of pore dielectric constant and volume charge density values can lead to the same membrane potential value.24,26 As a result, neither the dielectric constant inside the pores nor the membrane fixed charge can be deduced from single membrane potential measurements. It was also found that the diffusion potential (that is, the high-concentration limit of the membrane potential) in solutions of single binary electrolytes depends on neither the dielectric constant inside the pores nor the membrane fixed charge but is affected only by the pore size. In the case of ternary electrolytes, the membrane potential is also influenced by the Donnan exclusion and Born effect similarly, but unlike binary electrolytes, the diffusion potential depends on both the pore size and the dielectric constant inside the pores.25 Consequently, membrane potential measurements performed at high concentration with such electrolyte mixtures could be used to assess the dielectric constant inside the pores, provided the pore size is known. The aim of the present work was to show the possibility of determining the dielectric constant of a solution confined inside nanodimensional pores from membrane potential measurements. To this end, experiments were conducted with a NF polyamide membrane separating single-salt solutions of NaCl and CaCl2 and mixed-salt solutions of NaCl and CaCl2 at various concentrations. The mean pore size of the membrane was first deduced from
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the high-concentration limit of the membrane potential obtained for the two single-salt solutions. With this parameter being known, the membrane potential measured at high salt concentration with electrolyte mixtures was then used to compute the dielectric constant inside the pores. The dependence of this parameter on the mixture composition could be determined. To our knowledge, this is the first attempt to determine the dielectric constant insides pores from membrane potential measurements.
2. Theory When a membrane is put between two solutions of the same electrolyte(s) at the same temperature and hydrostatic pressure but at different concentrations, an electrical potential difference develops between the two solutions. The difference between the electrical potential in the bulk solution of higher concentration and that in the bulk solution of lower concentration is called the membrane potential. There are basically two approaches to investigating the membrane potential phenomenon. The first one is based on the Teorell-Meyer-Sievers (TMS) model,27,28 and the second one is based on the so-called space charge (SC) model.29 The SEDE (steric, electric, and dielectric exclusion) model, which is an improved version of the TMS model, was recently proposed.24,25 It describes the exclusion mechanism of charged solutes as the result of steric effects, the Donnan exclusion, and dielectric effects.8 Because a theoretical description of the SEDE model for the study of the membrane potential has been given in detail in refs 24 and 25, just a brief presentation of the model is given below. The membrane is modeled as a bundle of identical cylindrical pores with the following parameters: the pore radius, rp; the thickness-to-porosity ratio, Δx/Ak; the effective volume charge density X (i.e., the mole number of fixed charges per unit of pore volume); and the dielectric constant of the solution filling pores, εp. The last two parameters are dependent on the physicochemical properties of the surrounding solution. Within the scope of the TMS model and related models such as the SEDE model, the membrane potential (ΔΨm) can be split in two components, namely, the difference in the Donnan potentials developing at 0 both membrane/external solutions interfaces (ΔΨΔx D - ΔΨD) and the diffusion potential (ΔΨdiff) arising from the membrane pores30 0 ΔΨm ¼ ðΔΨΔx D - ΔΨD Þ þ ΔΨdiff
ð1Þ
Superscripts Δx and 0 refer to the interface between the membrane and the most diluted solution and the interface between the membrane and the most concentrated solution, respectively. In the formalism of the SEDE model, the Donnan potential reads as follows ΔΨint D
¼ ψ
int
-ψ
int
kB T κint cint ln i inti ¼ zi e ci
! ð2Þ
with (17) Szymczyk, A.; Sbaı¨ , M.; Fievet, P.; Vidonne, A. Langmuir 2006, 22, 3910. (18) Bouranene, S.; Fievet, P.; Szymczyk, A Chem. Eng. Sci. 2009, 64, 3789. (19) Cavaco Mor~ao, A. I.; Szymczyk, A.; Fievet, P.; Brites Alves, A. M. J. Membr. Sci. 2008, 322, 320. (20) Senapati, S.; Chandra, A. J. Phys. Chem. B 2001, 105, 5106. (21) Asaka, K. J. Membr. Sci. 1990, 50, 71. (22) Coster, H. G. L.; Chilcott, T. C.; Coster, A. C. F. Bioelectrochem. Bioenerg. 1996, 40, 79. (23) Li, Y. H.; Zhao, K. S. J. Colloid Interface Sci. 2004, 276, 68. (24) Lanteri, Y.; Szymczyk, A.; Fievet, P. Langmuir 2008, 24, 7955. (25) Lanteri, Y.; Szymczyk, A.; Fievet, P. J. Phys. Chem. B 2009, 113, 9197. (26) Lanteri, Y.; Fievet, P.; Szymczyk, A. J. Colloid Interface Sci. 2009, 331, 148.
Langmuir 2010, 26(18), 14628–14635
0
0
int int int κint i ¼ φi expð - ΔW i, Born Þ expð - ΔW i, image Þ
ð3Þ
(27) Theorell, T. Proc. Soc. Exp. Biol. 1935, 33, 282. (28) Meyer, K. H.; Sievers, J. F. Helv. Chim. Acta 1936, 19, 649.Meyer, K. H.; Sievers, J. F. Helv. Chim. Acta 1936, 19, 665.Meyer, K. H.; Sievers, J. F. Helv. Chim. Acta 1936, 19, 987. (29) Gross, R. J.; Osterle, J. F. J. Chem. Phys. 1968, 49, 228. (30) Asaka, K. J. Membr. Sci. 1990, 52, 57.
DOI: 10.1021/la1023949
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Superscript int stands for 0 or Δx depending on the interface that is considered, and the bar refers to a magnitude inside the membrane. kB is the Boltzmann constant, T is the absolute temperature, zi is the charge number of ion i, e is the elementary charge, ci is the concentration of ion i, φint i represents the steric partitioning coefficient for ion i (which is defined as the ratio of the available section for ion i to the pore cross section), and ΔW 0 int i,Born and ΔW 0 int i,image denote the increase in the interaction energy due to the Born dielectric effect and image forces, respectively, where the prime symbol indicates that both terms are scaled by kBT. Expres0 int 0 int sion φint i , ΔW i,Born, and ΔW i,image are given in the Appendix. It should be noted that all solutions are assumed to be ideal. Consider that c0i = 2cΔx i in eq 2 yields the following expression for the difference in the Donnan potentials appearing in eq 1: 0 ΔΨΔx D - ΔΨD ¼
kB T ln zi e
0 κΔx i ci Δx 0 2ci κi
!
int electroneutrality condition outside the membrane (i.e., cint þ = c- ), we can write
cint þ ¼
X ci zi ¼ - X
ð4Þ
i
ð5Þ where Di is the diffusion coefficient of ion i at infinite dilution and Ki is the hindrance diffusion factor (which is expressed in the Appendix) accounting for the effect of finite ion and pore sizes. Introducing eqs 4 and 5 into eq 1 gives the following expression for the membrane potential: ΔΨm 0P -
kB T κ Δx c 0 ¼ ln i Δx i 0 zi e 2ci κi
K i Di zi ðciΔx - ci0 Þ
!
1 0P
K i Di zi 2 ci0
Substituting eq 8 into eq 9 yields
f ¥, the By solving eq 10 and considering anion concentration just inside the membrane is written as
i
Now, we will consider only the limiting case of high salt concentration for binary (1-1 and 2-1) and ternary electrolytes, the latter being composed of a symmetric salt possessing a monovalent cation and a monovalent anion mixed with an asymmetric salt possessing a divalent cation and a monovalent anion, with both having a common anion. 2.1. Donnan Potential Difference at High Concentration. It must be kept in mind that at high salt concentration the interaction between ions and their image charges becomes negligible, which leads to ΔW 0 0i,image ≈ ΔW 0 Δx i,image ≈ 0. Moreover, neither the steric partitioning coefficient (φint i ) nor the excess solvation energy due to the Born effect (ΔW 0 int i,Born) depends on the concentration (eqs A1 and A2). They are thus independent of the membrane/solution interface (0 or Δx). According to eq 3, we can then write κ0i ¼ κiΔx
ð7Þ
2.1.1. 1-1 Electrolytes. By writing eq 2 for both cations and anions, identifying the equations thus obtained, and considering the 14630 DOI: 10.1021/la1023949
int int 1=2 int 1=2 cint - ¼ c þ ðκ þ Þ ðκ - Þ
ð11Þ
Substituting eq 11 into eq 4 leads to the following expression of the Donnan contribution: 0 ΔΨΔx D - ΔΨD
0 1=2 0 1=2 0 κΔx kB T - ðκ þ Þ ðκ - Þ c þ ln ¼ 1=2 Δx 1=2 Δx e 2κ0- ðκΔx þ Þ ðκ - Þ c þ
! ð12Þ
0 Δx By considering κ0i = κΔx i (eq 7) and the condition cþ = 2cþ , eq 12 shows that the difference in the Donnan potentials becomes zero as cint þ f ¥. 2.1.2. 2-1 Electrolytes. Again, by writing eq 2 for both cations and anions, identifying the equations thus obtained, and considering the electroneutrality condition outside the membrane int (i.e., 2cint þ = c- ), we obtain
cint þ ¼ ð6Þ
ð10Þ
the limiting case of cint þ
1
kB T B i C B i C @P A ln@P A e K i Di zi 2 ðciΔx - ci0 Þ K i Di zi 2 ciΔx i
ð9Þ
i
2 int int 2 int int ðcint - Þ - X ðc - Þ - ðc þ Þ ðκ - Þκ þ ¼ 0
1 0P 1 0P 0 Ki Di zi 2 c0i Ki Di zi ðcΔx i - ci Þ kB T B i C C B i ¼ ln P A @P 0 A @ e K i Di zi 2 ðcΔx K i Di zi 2 cΔx i - ci Þ i i
ð8Þ
The electroneutrality condition inside the membrane pores is as follows:
In the formalism of the SEDE model, the diffusion potential takes the following form24
ΔΨdiff
int 2 int κint þ ðc þ Þ ðκ - Þ ðc int -Þ
int 3 int 2 4κint þ ðc þ Þ ðκ - Þ 2 ðcint -Þ
ð13Þ
Substituting eq 13 into eq 9 leads to the following cubic equation: 3 int 2 int 3 int 2 int ðcint - Þ - Xðc - Þ - 8ðc þ Þ ðκ - Þ κ þ ¼ 0
ð14Þ
In a previous paper,24 it was shown that by considering eq 14 and the limiting case of cint þ f ¥, the anion concentration just inside the membrane could be given by qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi int 3 int int 2 cint κ þ ðκ - Þ - ¼ 2c þ
ð15Þ
Again, substituting eq 15 into eq 4 leads to the following expression for the difference in Donnan potentials:
0 ΔΨΔx D - ΔΨD
0 !sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi1 0 0 0 2 c 1 @ κΔx 3 κ þ ðκ - Þ - þ A ¼ ln Δx Þ2 z2κ0- cΔx κΔx ðκ þ þ -
ð16Þ
0 Δx By considering κ0i = κΔx i (eq 7) and cþ = 2cþ in eq 16, we can Δx 0 write ΔΨD - ΔΨD = 0. 2.1.3. Ternary Mixtures. For ternary mixtures, let us define R as the ratio of external (i.e., outside the membrane pores)
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By considering eqs 23 and 25, eq 22 can be rewritten as
cation concentrations: R ¼
cint Mþ cint M2þ
ð17Þ
Mþ and M2þ denote monovalent and divalent cations, respectively. Introducing eq 17 into Pthe electroneutrality condition outside the membrane pores ( i zicint = 0) leads to the following i expression for the external anion concentration: int cint A - ¼ ð2 þ RÞcM2þ
R
eΔΨint D ¼ exp kB T
κ þ κ þ 2K M2þ DM2þ þ K Mþ DMþ R M R - 1 - K A - DA - 2 þ R M R - 1 κM2þ κM2þ 4K M2þ DM2þ þ K Mþ DMþ R
i
! ð19Þ
int zi int zi cint i κi ðR Þ ¼ - X
i
ð21Þ
In a previous paper,25 it was shown that the roots of eq 21 at high salt condepend only on R and κi. Because κ0i = κΔx i centration, we thus have R 0 = R Δx. According to eq 19, we 0 obtain ΔΨΔx D = ΔΨD. For both binary electrolytes and electrolyte mixtures, the 0 Donnan contribution (i.e., ΔΨΔx D - ΔΨD) to the membrane potential is then null at high salt concentration and the membrane potential then becomes equal to the diffusion potential:
ΔΨm
1
0P
1
0 K i Di zi ðcΔx K i Di zi 2 c0i i - ci Þ kB T B i C B i C ln P ¼ @P A 0 A @ 2 cΔx e K i Di zi 2 ðcΔx c Þ K D z i i i i i i i
i
ð22Þ 2.2. Expression of Membrane Potential at High Concentration. 2.2.1. Ternary Mixtures. Let us write cint Mþ cint M2þ
¼ R
κMþ - 1 R κM2þ
ð23Þ
Within the scope of the SEDE model, the partitioning coefficient accounts for steric effects, the Donnan exclusion, and dielectric exclusion and can be written as cint zi eΔΨint D i ¼ κint i exp int kB T ci
! ð24Þ
Writing eq 24 for int = 0 and Δx and considering the conditions 0 0 Δx ΔΨΔx D = ΔΨD and κi = κi , we obtain cΔx cΔx 1 i i ¼ ¼ 2 c0i c0i Langmuir 2010, 26(18), 14628–14635
K Mþ KA¼ f ðrp Þ; ¼ f ðrp Þ K M2þ K M2þ
ð27Þ
and
int zi int zi cint i κi ðR Þ 0
0P
! lnð2Þ
Equation 26 shows that the limiting value of the membrane potential at high concentration depends on the mixture composition (through the cation concentration ratio R), the pore radius (rp), and the effective dielectric constant of the solution confined inside the pores (εp). Indeed
ð20Þ
When the ionic strength is high enough, the effect of the membrane fixed charge is totally screened. Considering X/ciint f 0, eq 20 can then be rewritten as X
κMþ - 1 κ þ R þ K A - DA - 2 þ R M R - 1 κM2þ κM2þ
!
ð26Þ
By considering eqs 2 and 19, eq 9 can be rewritten as X
kBT e
ð18Þ
Let us write int
ΔΨm ¼ -
φint κMþ Mþ ¼ int expðΔW 0M2þ , Born - ΔW 0Mþ , Born Þ ¼ f ðrp , εp Þ ð28Þ κM2þ φM2þ 2.2.2. Binary Electrolytes. Let us now consider binary electrolytes. The high-concentration limit of the membrane potential for single MA and MA2 solutions can be easily deduced from eq 26 by setting R f ¥ and R f 0, respectively. For example, setting R f ¥ in eq 26 yields ΔΨm, MA ¼ -
kB T K Mþ DMþ - K A - DA lnð2Þ e K M þ DM þ þ K A - DA -
ð29Þ
Equation 29 shows that for single-salt solutions the highconcentration limit of the membrane potential is no longer affected by the Born dielectric effect and depends only on steric effects. As can be seen, the influence of pore radius on the membrane potential arises through hindrance diffusion factors KMþ and KA-. The membrane potential measured at high salt concentration with single-salt solutions and salt mixtures could be used to determine the pore size and the effective dielectric constant inside the pores, respectively.
3. Experimental Section 3.1. Membrane and Chemicals. Membrane potential measurements were performed with a commercially available membrane, labeled Desal 5DK (GE Osmonics). This membrane is a flat thin-film composite membrane with a polyamide top layer on a polyester support. According to the supplier, its molecular weight cutoff is in the range of 150-300 g mol-1. An isoelectric point of 4.0 is reported by Hagmeyer and Gimbel.31 Single-salt solutions and salt mixtures were prepared from pure analytical grade NaCl and CaCl2 (Fisher Scientific) and Milli-Qquality water (conductivity
