pubs.acs.org/Langmuir © 2009 American Chemical Society
Pressure-Driven Ionic Transport through Nanochannels with Inhomogenous Charge Distributions Anthony Szymczyk,* Haochen Zhu, and Beatrice Balannec Universit e Europ eenne de Bretagne, France, and Sciences Chimiques de Rennes, UMR 6226 CNRS - Universit e de Rennes 1 - ENSCR, 263 Avenue du G en eral Leclerc, B^ atiment 10 A, CS 74205, 35042 Rennes Cedex, France Received July 1, 2009. Revised Manuscript Received August 20, 2009 The effect of spatially inhomogeneous fixed charge distributions on the pressure-driven transport of ions through cylindrical nanopores have been investigated theoretically by means of an approximate version of the PoissonNernst-Planck model that can be used with confidence for moderately charged nanopores with radius smaller than the Debye screening length of the system. Salt rejection rate has been computed as a function of the applied pressure difference for various one-dimensional (1D) unipolar charge distributions and has been compared with that obtained for a homogeneously charged nanochannel with an identical average volume charge density. The ion rejection capabilities of charged nanopores can be optimized by an appropriate distribution of the fixed charge concentration. When ions are forced to enter the nanopores by the end with the lowest fixed charged concentration, the salt rejection rate exhibits a nonmonotonous variation with the applied pressure. This phenomenon has been attributed to the influence of the inhomogeneous charge distribution on the electric field that arises spontaneously so as to maintain the electroneutrality within the nanopore.
Introduction Nanofluidics is the part of science that focuses on fluid flow in and around nanometer-sized objects with at least one characteristic dimension below 100 nm.1 The high surface-to-volume ratio in nanochannels results in a surface-charge-governed ion transport, and it is now well established that the fixed charges are largely responsible for nanochannel properties such as conductance,2 salt rejection3 or electromechanical energy conversion.4 The ion-selective features of nanoporous media were first described in membrane filtration processes.5-9 Recent advances in the fabrication of nanochannels, by means of track-etching techniques10,11 or e-beam lithography followed by a self-limiting poreshrinking process,12 allow the design of well-controlled sub-10 nm nanopores that are ideal physical modeling systems. The combination of pore diameters around a few nanometers with electrically charged materials enables the occurrence of phenomena that are impossible at bigger length scales. For example, conically shaped nanopores can act as rectifiers of ionic current (i.e., the ratio of currents flowing through the nanopores for the same voltage but opposite polarities is a value other than 1), provided there is an
excess surface charge on the pore walls.13-16 This physical phenomenon has recently been exploited to design nanofluidic diodes that rectify ion current in a similar way as a bipolar semiconductor diode rectifies electron current.17,18 Siwy et al. have shown that electrostatic interactions between ions and the surface charge of conically shaped nanopores can also produce an asymmetry in the rates of diffusive transport of ions, i.e., that diffusion current depends on the direction of the applied concentration gradient.19 Ion transport through nanoporous media is usually described by models that consider homogenously charged nanopores,4,20-22 although nanopores with fixed charge distributions that deviate to some extent from homogeneity are common place and include conical track-etched nanopores,18 ion channels,23 nanofiltration (NF) membranes,24 and so forth. It is worth mentioning that a pH gradient can result in an inhomogeneous charge distribution through a nanochannel with weak-acid surface groups, even if these groups are homogeneously distributed over the nanochannel surface. Indeed, the pH gradient occurring through the nanopore produces a variable concentration of the dissociated weak-acid fixed groups, and so an inhomogeneous fixed charge distribution arises.25-27 Interestingly, nanopores that are symmetric in shape but
*Corresponding author. E-mail:
[email protected]. (1) Schoch, R. B. Rev. Mod. Phys. 2008, 80, 839–883. (2) Stein, D.; Kruithof, M.; Dekker, C. Phys. Rev. Lett. 2004, 93, 035901. (3) Szymczyk, A.; Sbai, M.; Fievet, P.; Vidonne, A. Langmuir 2006, 22, 3910– 3919. (4) Daiguji, H.; Yang, P.; Szeri, A. J.; Majumdar, A. Nano Lett. 2004, 4, 2315– 2321. (5) Morrison, F. A.; Osterle, J. F. J. Chem. Phys. 1965, 43, 2111–2115. (6) Schl€ogl, R. Ber. Bunsen-Ges. Phys. Chem 1966, 70, 400–414. (7) Gross, R. J.; Osterle, J. F. J. Chem. Phys. 1968, 49, 228–234. (8) Fair, J. C.; Osterle, J. F. J. Chem. Phys. 1971, 54, 3307–3316. (9) Dresner, L. J. Phys. Chem. 1972, 76, 2256–2267. (10) Cervera, J.; Schiedt, B.; Neumann, R.; Mafe, S.; Ramirez, P. J. Chem. Phys. 2006, 124, 104706. (11) Constantin, D.; Siwy, Z. Phys. Rev. E 2007, 76, 041202. (12) Nam, S. W.; Rooks, M. J.; Kim, K. B.; Rossnagel, M. Nano Lett. 2009, 9, 2044–2048. (13) Apel, P.; Korchev, Y. E.; Siwy, Z.; Spohr, R.; Yoshida, M. Nucl. Instrum. Methods Phys. Res., Sect. B 2001, 184, 337–346. (14) Siwy, Z.; Fulinski, A. Phys. Rev. Lett. 2002, 89, 198103. (15) Cervera, J.; Schiedt, B.; Ramirez, P. Europhys. Lett. 2005, 71, 35–41.
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(16) Davenport, M.; Rodriguez, A.; Shea, K. J.; Siwy, Z. Nano Lett. 2009, 9, 2125–2128. (17) Karnik, R.; Duan, C.; Castelino, K.; Daiguji, H.; Majumdar, A. Nano Lett. 2007, 7, 547–551. (18) Vlassiouk, I.; Siwy, Z. Nano Lett. 2007, 7, 552–556. (19) Siwy, Z.; Kosinka, I. D.; Fulinski, A.; Martin, C. R. Phys. Rev. Lett. 2005, 94, 048102. (20) Lefebvre, X.; Palmeri, J.; David, P. J. Phys. Chem. B 2004, 108, 16811– 16824. (21) Vlassiouk, I.; Smirnov, S.; Siwy, Z. Nano Lett. 2008, 8, 1978–1985. (22) Lanteri, Y.; Szymczyk, A.; Fievet, P. Langmuir 2008, 24, 7955–7962. (23) Alcaraz, A.; Nestorovich, E. M.; Aguilella-Arzo, M.; Aguilella, V. M.; Bezrukov, S. M. Biophys. J. 2004, 87, 943–957. (24) Freger, V. Langmuir 2003, 19, 4791–4797. (25) Ramirez, P.; Mafe, S.; Aguilella, V. M.; Alcaraz, A. Phys. Rev. E 2003, 68, 011910. (26) Cervera, J.; Aguilella-Arzo, M.; Lopez, M. L.; Alcaraz, A.; Aguilella, V. M. Electrostatic regulation of ionic transport in nanoporous membranes. In Surface Electrical Phenomena in Membranes and Microchannels; Szymczyk, A., Ed.; Transworld Research Network: Kerala, India, 2008; p 173.
Published on Web 09/08/2009
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asymmetric in fixed charge distribution also exhibit asymmetric properties such as ionic current rectification.28-30 The use of continuum models to describe ion transport through nanopores is currently a matter of controversy,25 notably because they cannot describe the layering structures of solvent and ion distributions near the solid surface.31 These models have been found, however, to account fairly well for the experiments for nanopores with diameters of a few nanometers, even with limited number of fitting parameters.15,25,32,33 Therefore, continuous models giving at least a semiquantitative description of ion transport can help to optimize the construction of synthetic nanochannels with desired transport characteristics.28 The pioneer works devoted to the influence of inhomogeneous fixed charge distribution on ion transport focused on diffusive transport through synthetic membranes.34 It was shown that the usual Henderson assumption, which consists in considering constant ion concentration gradients inside pores, breaks down when the fixed charge density exhibits spatial variation over the pore length since this assumption leads to physically unrealistic results, e.g., the existence of a nonzero steady-state value of the ion molar fluxes when the membrane separates two solutions of the same electrolyte at identical hydrostatic pressure and concentration.35 The effect of nonuniform charge distribution on the reversal potential (usually called membrane potential in membrane science) was investigated by Levadny and Aguilella within the scope of the space charge model36 and by Yamamoto et al.37 by means of a modified Theorell-Meyer-Sievers theory including the concept of ion-pair formation in the membrane through the phenomenon of counterion site binding onto the charged surface groups. Very recently, Garcia-Gimenez et al. have shown that biological nanopores with bipolar charge distribution (induced by an imposed pH gradient through the nanopore) can behave as either a cation selective channel or an anion selective one just by changing the direction of the salt concentration gradient.27 Ionic current rectification properties (under an external electric field) of nanopores with inhomogeneous fixed charged concentrations have also been modeled recently by means of various models based on the so-called Poisson-Nernst-Planck (PNP) formalism.10,11,28,30 Pressure-driven transport (i.e., generated by a volume flux) through nanochannels constitutes another field with substantial applications in various areas such as electromechanical energy conversion,4,38 characterization of the electric properties of solid-liquid interfaces,39-41 desalination,3,42,43 and so forth. In (27) Garcia-Gimenez, E.; Alcaraz, A.; Aguilella, V. M.; Ramirez, P. J. Membr. Sci. 2009, 331, 137–142. (28) Fulinski, A.; Kosinska, I. D.; Siwy, Z. Europhys. Lett. 2004, 67, 683–689. (29) Cheng, L. J.; Guo, L. J. Nano Lett. 2007, 7, 3165–3171. (30) Ramirez, P.; Gomez, V.; Cervera, J.; Schiedt, B.; Mafe, S. J. Chem. Phys. 2007, 126, 194703. (31) Chen, Y.; Ni, Z.; Wang, G.; Xu, D.; Li, D. Nano Lett. 2008, 8, 42–48. (32) Szymczyk, A.; Fievet, P. J. Membr. Sci. 2005, 252, 77–88. (33) Lanteri, Y.; Fievet, P.; Szymczyk, A. J. Colloid Interface Sci. 2009, 331, 148–155. (34) Takagi, R.; Nakagaki, M. J. Membr. Sci. 1986, 27, 285–299. (35) Manzanares, J. A.; Mafe, S.; Pellicier, J. J. Phys. Chem. 1991, 95, 5620– 5624. (36) Levadny, V.; Aguilella, V. J. Phys. Chem. B 2001, 105, 9902–9908. (37) Yamamoto, R.; Matsumoto, H.; Tanioka, A. J. Phys. Chem. B 2003, 107, 10615–10622. (38) Van der Heyden, F. H. J.; Bonthuis, D. J.; Stein, D.; Meyer, C.; Dekker, C. Nano Lett. 2007, 7, 1022–1025. (39) Aguilella, V.; Aguilella-Arzo, M.; Ramirez, P. J. Membr. Sci. 1996, 113, 191–204. (40) Szymczyk, A.; Aoubiza, B.; Fievet, P.; Pagetti, J. J. Colloid Interface Sci. 1999, 216, 285–296. (41) Van der Heyden, F. H. J.; Stein, D.; Meyer, C.; Dekker, C. Phys. Rev. Lett. 2005, 95, 116104. (42) Corry, B. J. Phys. Chem. B 2008, 112, 1427–1434. (43) Song, C.; Corry, B. J. Phys. Chem. B 2009, 113, 7642–7649.
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Figure 1. Schematic of the cylindrical nanopore used in the modeling: rp is the pore radius, L is the pore length, 0- and Lþ denote the axial coordinates just outside the nanopore, 0þ and L- denote the axial coordinates just inside the nanopore, ci and Ci are the concentrations inside and outside the nanopore, respectively, P and π are the hydrostatic and osmotic pressures, respectively, and JV is the volume flux of the fluid flowing through the nanopore (by convention, we consider JV > 0 when the solution flows from the left to the right).
this work we present a detailed theoretical account of the effects of inhomogeneous charge distributions on pressure-induced transport of ions through nanochannels and their ion rejection capabilities. Several types of one-dimensional (1D) unipolar distributions (i.e., the fixed charge concentration is spatially changing, but the sign of the surface charge remains the same) have been analyzed on the basis of a simplified version of the PNP formalism that is valid for sufficiently narrow channels (with respect to the Debye screening length) and moderate fixed charge concentrations. Ion rejection rates have been compared with those obtained for the case of a homogeneous nanopore with the same average fixed charge concentration.
Formulation of the Problem We consider a cylindrical nanochannel of length L=2 μm and radius rp=2 nm (see Figure 1) so that transport can be considered as 1D since L . rp. We note that this pore radius value is relevant with the pore size of NF membranes3 as well as the tip of conical nanopores obtained by recent track-etching techniques.18 The external solutions are assumed to be ideal and perfectly stirred. We will restrict our consideration to the case of an isothermal system (with T=298 K) and a millimolar solution of uniunivalent electrolyte with cations and anions having identical diffusion coefficients (Dþ = D- = 10-9 m2 s-1). We will presume that continuum treatment is applicable (at least qualitatively or semiquantitatively) down to the dimensions of our interest, i.e. for pores of 4 nm in diameter. Within the scope of pressure-induced transport, the molar flux (ji) of each ion i (with i=þ or -) of charge zi is provided by the extended Nernst-Planck (NP) equation that includes the contribution of the convection caused by the pressure gradient applied across the pore.6,9 When dealing with nanopores, the extended NP equation is frequently modified by hydrodynamic factors (accounting for the drag force arising due to both the finite ion size and the finite pore size) according to the hinderedtransport theory26,44-46 ji ¼ -Ki, d Di rci -
FKi, d zi Di ci rψþKi, c ci u RT
ð1Þ
where ci is the local ion-concentration inside the nanopore, F is the (44) Anderson, J. L.; Quinn, J. A. Biophys. J. 1974, 14, 130–150. (45) Deen, W. M. AIChE J. 1987, 33, 1409–1425. (46) Han, J.; Fu, J.; Schoch, R. B. Lab Chip 2008, 8, 23–33.
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Faraday constant, R is the ideal gas constant, ψ denotes the local electric potential inside the nanopore, u is the solution flow velocity inside the nanopore, and Ki,d and Ki,c are the hindrance factors for diffusion and convection, respectively. The hydrodynamic factors Ki,d and Ki,c in eq 1 are usually assessed from approximate polynomial functions that depend on the ratio λi=ri,Stokes/rp (where ri,Stokes denotes the Stokes radius of the ion species i; see Supporting Information). All calculations shown in this work were performed with the approximate functions derived by Bungay and Brenner for cylindrical pores (see Supporting Information).45 It must be stressed that these analytical expressions were derived for neutral solutes. Very recently Dechadilok and Deen have investigated the influence of charge interactions on the intrapore diffusivity of charged species.47 According to their findings, there is only a slight effect of charge interactions provided λi is less than ∼0.2. Since λi is only ∼0.1 in our study, we have considered that the expressions of Ki,d given in the Supporting Information can be used, and we have assumed that the same is true for hindered convection factors Ki,c. The molar flux of each ion species is related to the solution volume flux flowing through the nanopore by the filtration condition ji ¼ CiLP JV
ð2Þ
where CLP i is the bulk concentration in the low-pressure compartment, and JV is the solution volume flux (equivalent to the solution flow velocity u). in the extended NP We note that the concentration CLP i equations via eq 2 is unknown, which implies that the model must be solved iteratively. At steady state, the molar flux satisfies the continuity equation, and no net electrical current flows through the nanopore rji ¼ 0 X
ð3Þ
ji zi ¼ 0
ð4Þ
i
The solution flow velocity (u) in eq 1 obeys the Stokes equation, which reads as follows within the scope of the creeping flow approximation (i.e., inertial terms are neglected): -rPþηr2 u -F
X
ci zi rψ ¼ 0
ð5Þ
i
where η is the solution viscosity. The local electric potential is governed by the Poisson equation: r2 ψ ¼ -
F X ci zi ε0 εr i
ð6Þ
where ε0 is the vacuum permittivity and εr is the solution dielectric constant. When the electric potential in the (extended) NP equation is determined from the Poisson equation in a self-consistent manner, the combined system of equations form the so-called Poisson-Nernst-Planck (PNP) theory.15,48 Considering the Stokes equation allows accounting for the electro-osmotic flow generated through the nanopore. The complete set of PNP and (47) Dechadilok, P.; Deen, W. M. J. Membr. Sci. 2009, 336, 7–16. (48) Corry, B.; Kuyucak, S.; Chung, S. H. Biophys. J. 2000, 78, 2364–2381.
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Figure 2. Radial profile of the (normalized) electric potential obtained from the numerical solution of the Poisson equation (eq 6) for various fixed charge densities. Calculations were performed for a millimolar solution of uniunivalent electrolyte with cations and anions having identical diffusion coefficients (Dþ = D- = 10-9 m2 s-1).
Stokes equations is known as the space charge model and was first proposed in membrane science by Osterle and co-workers.5,7,8 It provides a full two-dimensional description of transport through a nanopore, but this model is rather cumbersome and requires significant computational efforts. When dealing with nanochannels, it is, however, possible to use an approximate model thanks to the strong overlap of electrical double layers inside the nanopore. A common approximation consists in neglecting the radial variation of the electric potential inside the nanopore. This approximation is called “homogeneous approximation” and it holds only when the Debye screening length is much larger than the pore radius49 (a condition that is fulfilled with nanopores in many practical situations) and for sufficiently low fixed charge densities.40,50,51 In order to check the relevance of the homogeneous approximation for our system, we solved the Poisson equation for various fixed charge densities (σ), boundary conditions arising from the Gauss law, and the symmetry of the problem: Dψ σ r ¼r ¼ Dr p ε0 εr
ð7Þ
Dψ jr ¼0 ¼ 0 Dr
ð8Þ
Results are shown in Figure 2. Calculations were performed for three different surface charge densities that are relevant to many practical applications including polymeric membranes,52 silica nanochannels,2,29,41 and so forth. For fixed charge densities up to -4.5 10-3 C m-2 and a 0.001 mol/L solution, the local electric potential deviates by no more than 4% with respect to the radially averaged electric potential (additional calculations show that the maximal deviation of the electric potential with respect to the radially averaged value does not exceed 10%, even if the electrolyte concentration is increased up to 0.01 mol/L), and the (49) Plecis, A.; Schoch, R. B.; Renaud, P. Nano Lett. 2005, 5, 1147–1155. (50) Wang, X. L.; Tsuru, T.; Nakao, S. I.; Kimura, S. J. Membr. Sci. 1995, 103, 117–133. (51) Palmeri, J.; Blanc, P.; Larbot, A.; David, P. J. Membr. Sci. 1999, 160, 141– 170. (52) Rohani, M. M.; Zydney, A. L. J. Membr. Sci. 2009, 337, 324–331.
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homogeneous approximation can be used with confidence. In this case, all quantities in the extended NP equations are defined as radially averaged quantities, and the extended NP equations can be rewritten this way (considering the filtration condition given by eq 2): dÆci ðzÞæ JV zi FÆci ðzÞæ dÆψðzÞæ ¼ ð9Þ ðKi, c Æci ðzÞæ -CiLP Þ dz RT dz Ki, d Di where the symbol Ææ indicates radially averaged quantities. When dealing with the homogeneous approximation, eq 9 has to be coupled with an equation describing the local electroneutrality inside the nanopore: X
zi ci ðzÞþXðzÞ ¼ 0 for 0þ ezeL -
ð10Þ
i
where X(z) denotes the local volume charge density of the nanopore (i.e., the fixed charge concentration at a point of coordinate z within the nanopore) that is related to the surface charge density of a cylindrical pore by XðzÞ ¼
2σðzÞ Frp
ð11Þ
Equation 10 is expected to be valid for the 1D transport problem under consideration since we deal with nanochannels much longer than the Debye length.30 Considering eqs 9 and 10, the electric field can be expressed as follows: dÆψðzÞæ dz P zi JV LP dXðzÞ i Ki, d Di, ¥ ðKi, c Æci ðzÞæ -Ci Þ dz P P ¼ F F 2 2 i Æci ðzÞæzi i Æci ðzÞæzi RT RT
ÆEðzÞæ ¼ -
have underlined, however, the significant role of induced polarization charges in polymer NF membranes.32,53 For the sake of simplification, we will disregard induced polarization charges in the present work. This approximation is supported by the recent computer simulations performed by Boda et al., which suggest that the total induced polarization charge should be negligible for nanopores of 2 nm in radius.54 It is then possible to compute the theoretical rejection rate of each ion species for a given solution volume flux (JV) according to its definition: Ri ¼ 1 -
¼ CiHP φi expð -zi ΔΨHP cHP i D Þ
ð13Þ
¼ CiLP φi expð -zi ΔΨLP cLP D Þ i
ð14Þ
where superscripts HP and LP stand for high-pressure and lowpressure sides, respectively, ci is the concentration just inside the nanopore (it corresponds to ci(0þ) or ci(L-) depending on the flow direction; see Figure 1), Ci is the bulk concentration outside the nanopore (it corresponds to Ci(0-) or Ci(Lþ) depending on the flow direction), φi (= (1 - λi)2 for a cylindrical nanopore45) is the steric partitioning coefficient accounting for finite ion-size, and ΔΨD denotes the dimensionless Donnan potential arising at the nanopore/external solution interface. Equations 13 and 14 are modified Donnan equations including steric hindrance at pore ends. The discontinuous Donnan approach is approximately valid here because the nanopore thickness is much greater than the Debye length.30 Equations 13 and 14 are commonly used in NF modeling. Some recent works (53) Yaroshchuk, A. E. Adv. Colloid Interface Sci. 2000, 85, 193–230.
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ð15Þ
We note that Rþ = R- = Rsalt (salt rejection rate) for a uniunivalent electrolyte as a result of the electroneutrality of bulk P solutions (i.e., iziCi=0). In order to compute Rsalt as a function of the applied pressure gradient (and so, to account for electroosmosis), eq 5 has to be considered. Within the scope of the homogeneous model, Lefebvre et al. have proposed a simplified approach based on the average Stokes equation.6,20,55 Sufficiently far from the interfacial regions and assuming that the flow is well developed, it is approximately given for a cylindrical nanochannel by20,56,57 X 8η dÆPðzÞæ dÆΨðzÞæ -F JV ≈ Æci ðzÞæzi 2 rp dz dz i
ð16Þ
Integration of eq 16 (by considering the local electroneutrality condition, i.e., eq 10) over the nanopore length gives the pressure drop inside the nanochannel (ÆΔPmæ): ÆΔPm æ ¼ ÆPð0þ Þæ -ÆPðL - Þæ 8ηL ¼ 2 JV þF rp
ð12Þ Equations 9 and 10 form a system of coupled nonlinear algebraic/ differential equations (considering eq 12) that can be solved iteratively with the following boundary conditions (assuming ideal and perfectly stirred solutions)
CiLP CiHP
Z
z ¼0þ
XðzÞ z ¼L -
dÆΨðzÞæ dz dz
ð17Þ
The total pressure difference (ΔP=P(0-) - P(Lþ)) is obtained by adding the osmotic pressure jumps occurring at both nanopore/solution interfaces, π(0-) - π(0þ) and π(L-) - π(Lþ), due to ion concentration jumps:20,56,57 ΔP ¼ RT Z þF
z ¼0þ
XðzÞ z ¼L -
X 8ηL ðCi ð0 - Þ -Æci ð0þ ÞæÞþ 2 JV rp i X dÆΨðzÞæ dzþRT ðÆci ðL - Þæ -Ci ðLþ ÞÞ dz i ð18Þ
Results and Discussion We have considered the three following inhomogeneous fixed charge distributions, which are shown in Figure 3
(54) Boda, D.; Valisko, M.; Eisenberg, B.; Nonner, W.; Henderson, D.; Gillespie, D. Phys. Rev. Lett. 2007, 98, 168102. (55) Sonin, A. A. Osmosis and ion transport in charged porous membranes: Macroscopic mechanistic model. In Charged Gels and Membranes; Selegny, E., Ed.; Reidel: Dordrecht, 1976; Vol. 1, p 255. (56) Lefebvre, X.; Palmeri, J. J. Phys. Chem. B 2005, 109, 5525–5540. (57) Palmeri, J.; Lefebvre, X. Modeling of neutral solute and ion transport in charged nanofiltration membranes using computer simulation programs. In Handbook of Theoretical and Computational Nanotechnology; Rieth, M.; Schommers, W., Eds.; American Scientific Publishers, 2006; Vol. 5, p 93.
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Figure 3. Fixed charge distributions under consideration. All distributions give the fixed charge concenR same average normalized R tration, ξavg = 10ξ(z)dz = (1/CsaltHP) 10X(z)d(z) = -15.
ξðzÞ ¼
Linear distribution
Xmax z HP Csalt
ð19Þ
Hyperbolic distribution ξðzÞ ¼
Skin distribution
Xmax HP ð3:5125 -2:5125zÞCsalt
ξðzÞ ¼
Xmax 2 2 -z þz HP 3 Csalt
ð20Þ
ð21Þ
where ξ is the normalized volume charge density (defined as the HP ratio X/CHP salt, with Csalt being the salt concentration in the high pressure side), z=z/L is the dimensionless axial coordinate, and Xmax denotes the maximum fixed charge concentration used in the calculations. The first two distributions are asymmetric, while the last one (i.e., skin distribution) is symmetric. As mentioned previously, such fixed charge distributions can result either from an inhomogeneous distribution of surface sites (generated during the fabrication of nanopores) or from a pH gradient arising inside nanopores with homogeneously distributed ionizable groups. We have set Xmax = -30 mmol/L, which corresponds to σ = -2.9 10-3 C m-2. According to Figure 2, Xmax is low enough to apply the homogeneous approximation. It is worth mentioning that 1D approximations of the PNP equations have been shown to provide reasonable agreement with the full PNP equations for systems with wider pores carrying much greater fixed charge density.21,58 The various coefficients in eqs 19-21 have been chosen to give the same average fixed charge concentration (ξavg) Z
1
ξavg ¼ 0
1 ξðzÞdz ¼ HP Csalt
Z
1
XðzÞdz ¼ -15
ð22Þ
0
Figure 4 shows the salt rejection rates versus the pressure difference (ΔP) across the nanopore (computed from eq 18) for the various fixed charge distributions. Positive and negative values of ΔP (= P(0-) - P(Lþ)) in Figure 4 are associated with the direction of the volume flow (ΔP > 0 when the solution flows
(58) Vlassiouk, I.; Smirnov, S.; Siwy, Z. ACS Nano 2008, 2, 1589–1602.
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Figure 4. Salt rejection rate versus the pressure difference across the nanochannel for the various fixed charge distributions (ξavg = -15) shown in Figure 3.
from the left to right in Figures 1 and 3, and ΔP < 0 when the flow direction is reversed). For ΔP < 0, the rejection rates follow the series Linear ≈ Hyperbolic > Skin > Homogeneous. This behavior results from partitioning effects at the pore inlet, i.e., higher rejection rates are obtained when ions enter the nanopore end with higher charge density (keeping in mind that the average fixed charge concentration is identical for all charge distributions). These results suggest that designing nanoporous materials with controlled inhomogeneous charge distributions could be beneficial for desalination purposes. We have performed additional calculations (results not shown) for nanopores with increasing homogeneous fixed charge densities. Results show that a homogeneously charged nanopore with ξ=X/CHP salt=-28 leads to the same limiting salt rejection rate (i.e., the intrinsic rejection rate at infinite pressure difference) as the one obtained in Figure 4 for the linear distribution (for which ξavg is only -15). For ΔP > 0 (i.e., the solution enters the nanopores with asymmetric charge distributions by the end with the lowest fixed charged concentration; see Figure 3), the rejection rates follow the series Skin > Homogeneous > Hyperbolic > Linear, in accordance with the sequence of fixed charge concentrations at the pore entrance (see Figure 3). As expected, reversal of flow direction does not affect the rejection rate for the skin distribution since it is symmetric with respect to the pore center (z=0.5). On the other hand, asymmetric fixed charge distributions lead to asymmetric Rsalt - ΔP curves. A striking result is the nonmonotonous variation of Rsalt with ΔP for both the linear and hyperbolic distributions. These results are evidence that there is an optimum pressure difference beyond which the separation performances of the nanopores with asymmetric charge distributions decrease. It must be stressed that this behavior can be observed (in the case of binary electrolytes) neither with homogeneously charged nanopores59,60 (whatever the volume flow direction) nor for nanopores with inhomogeneous charge distributions when ΔP < 0. A sound physical explanation of this phenomenon can be given by investigating the electric field arising through the nanopores, which can be computed at any given volume flux by means of eq 12. Figure 5 shows the variation of the local electric field inside the nanopores for a relatively low volume flux fixed at JV=10-4 m/s. As can be seen, the electric field is positive (i.e., it has the same direction as the z-axis; see Figure 1) for both the linear and (59) Spiegler, K. S.; Kedem, O. Desalination 1966, 1, 311–326. (60) Yaroshchuk, A. E. J. Membr. Sci. 2000, 167, 163–185.
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Figure 5. Local electric field inside the nanopore for ΔP > 0
corresponding to a volume flux JV = 10-4 m/s for all charge distributions. The various fixed charge distributions correspond to those shown in Figure 3.
hyperbolic distributions. It means that the electric field drives the counterions toward the low pressure side just like the convection does. This peculiar behavior cannot be observed with homogeneously charged nanopores for which the electromigrative flux of counterions is always oriented in the opposite direction of the volume flux so as to maintain the electroneutrality within the nanopore (counterions being in excess with respect to co-ions).20,61 This is illustrated in Figure 5, where it can be seen that the electric field is negative (i.e., it is oriented in the opposite direction of the z-axis; see Figure 1) for the homogeneous charge distribution, whatever the axial coordinate inside the nanopore. In passing, it can be noted that the electric field remains approximately constant throughout the pore for the homogeneous charge distribution, which justifies the frequently used approximation known as the Goldman constant field approximation, which allows linearizing the NP equations.30,62 Figure 5 clearly shows that this approximation does not hold for inhomogeneous charge distributions anymore. The phenomenon observed for the linear and hyperbolic distributions can be explained by considering eq 12, which shows that the electric field can be written as an algebraic sum of two terms. The first term in the right-hand side of eq 12 depends explicitly on the volume flux (JV) and is always negative for negative surface charge densities (it therefore contributes to drive counterions in the opposite direction of the volume flux), while the second term arises as a result of inhomogeneous charge distribution (note that it is also dependent on JV through Æci(z)æ), and it can be either positive or negative depending on the type of charge distribution. For charge distributions characterized by dX(z)/dz < 0 the contribution of charge inhomogeneities can then be dominant and overcompensate the first term in eq 12, leading to a strong positive electric field as shown in Figure 5. Physically, this positive electric field spontaneously arises to ensure the electroneutrality condition since counterions must be “pumped” into the nanopore so as to balance the increase in the negative charge density through the nanopore. The nonmonotonous behavior with ΔP is not observed when the solution enters the nanopores with asymmetric charge distributions by the most highly charged end (ΔP < 0 in Figure 4) because, in that case, both terms in eq 12 have the same sign. Consequently, the inhomogeneous charge distribution just strengthens the electric (61) Szymczyk, A.; Labbez, C.; Fievet, P.; Vidonne, A.; Foissy, A.; Pagetti, J. Adv. Colloid Interface Sci. 2003, 103, 77–94. (62) Ku, J. R.; Lai, S. M.; Ileri, N.; Ramirez, P.; Mafe, S.; Stroeve, P. J. Phys. Chem. C 2007, 111, 2965–2973.
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R1 0ÆE(z)ædz) for various (positive) volume fluxes. The various fixed charge distributions correspond to those shown in Figure 3.
Figure 6. Average electric field inside the nanopore (Eavg =
field through the nanopores, which tends to increase the rejection rate with respect to the homogeneous distribution (the effect of the electric field adds further to pore end effects discussed previously). The positive electric field that develops for charge distributions characterized by dX(z)/dz < 0 has a strong influence on the salt rejection. Indeed, according to Figure 3, the pore entrance is uncharged for the nanopore with a linear charge distribution when ΔP > 0, which means that the ion partitioning at the pore entrance is purely steric in nature. Additional calculations performed by considering the steric hindrance as the sole exclusion mechanism indicate that the salt rejection should not exceed ∼6%, which is far below the actual salt rejection, which can be as high as ∼84%, as shown in Figure 4. This relatively high salt rejection results from the positive electric field, which drives the co-ions in the opposite direction of the volume flux and thus limits the transport of the co-ions toward the low pressure side. When the applied pressure increases (i.e., for greater volume fluxes) the electric field becomes weaker as a result of the competition between the two terms of opposite signs in the right-hand side of eq 12. Thus, the contribution of electromigration decreases progressively with respect to convection and diffusion, which leads to the maximum of the salt rejection observed in Figure 4. At even higher volume fluxes, the first term in eq 12 overcompensates the contribution of the inhomogeneous charge distribution and the electric field becomes negative (see Figure 6) and so the electric field drives the co-ions and counterions toward the low pressure and high pressure sides, respectively. From a qualitative point of view, it corresponds to the standard behavior obtained with a homogeneous charge distribution,61 and the rejection rate therefore tends to an asymptotic limit at infinite pressure difference (the reason is that the diffusive flux of counterions becomes negligible with respect to their convective and electromigrative fluxes that compensate each other63). Another fundamental difference between homogeneously and inhomogeneously charged nanopores is the existence of a nonzero electric field inside inhomogeneously charged nanopores, even in the absence of any volume flow. This appears explicitly in eq 12 and is illustrated in Figure 6, which shows the volume flux dependence R of the electric field averaged over the nanopore length (Eavg = 10ÆE(z)ædz). The reason is that an electric field spontaneously arises to balance the diffusive ion-flux induced by the fixed charge concentration gradient so that no net salt flux flows through the nanopores in the absence of an applied pressure. (63) Yaroshchuk, A. E. Adv. Colloid Interface Sci. 1995, 60, 1–93.
DOI: 10.1021/la902355x
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Finally, it can be noted in Figure 4 that the salt rejection rate computed for the skin distribution varies monotonously with the pressure difference, although the electric field is positive in onehalf of the nanopore (see Figure 5) due to the symmetry of this inhomogeneous charge distribution. The reason is that the average electric field (Eavg) is globally negative, as shown in Figure 6. It can be noted in Figure 6 that both symmetrical distributions, i.e., homogeneous and skin distributions, lead to an identical average electric field whatever the volume flux.
Conclusion In this work, we have theoretically described the effects of spatially inhomogeneous fixed charge distributions on the pressure-induced transport of a single uniunivalent electrolyte through negatively charged cylindrical nanopores. A set of 1D symmetrical and asymmetrical distributions have been investigated. Starting from the PNP formalism, we have focused on systems for which the homogeneous approximation, which consists in neglecting the radial variations of the electrostatic potential, can be applied confidently. Salt rejection rate has been computed as a function of the applied pressure difference for
1220 DOI: 10.1021/la902355x
Szymczyk et al.
various inhomogeneous charge distributions and has been compared with that obtained for a homogeneously charged nanochannel with an identical average volume charge density. It has been shown that the occurrence of spatial inhomogeneities in the fixed charge concentration may lead to an enhancement of the rejection properties of the nanopore with respect to the homogeneously charged nanopores, which can be especially attractive for desalination applications involving NF or reverse osmosis membranes. In the case of nanopores with asymmetric charge distributions, it has been shown that the salt rejection varies nonmonotonously with the applied pressure when the solution enters the nanopore by the end with the lowest fixed charged concentration. This peculiar behavior has been shown to be related to the influence of the inhomogeneous charge distribution on the electric field that arises spontaneously through the nanopore so as to maintain the electroneutrality condition. Supporting Information Available: Expressions of hindrance factors for diffusion and convection. This material is available free of charge via the Internet at http://pubs.acs. org.
Langmuir 2010, 26(2), 1214–1220
