Bio-Medical Materials and Engineering 24 (2014) 383–390 DOI 10.3233/BME-130822 IOS Press
383
The effect of counter-ions on the ion selectivity of potassium and sodium ions in nanopores Dai Tang and Daejoong Kim*
Department of Mechanical Engineering, Sogang University, Seoul, Republic of Korea
Abstract. The ion selective transport plays an important role in the function of cell membranes, and promotes the application of artificial permeable membranes. This phenomenon has been studied in case for different diameters and functional groups of nanopores. In this work, we focus on the effect of anions on cation selectivity, in particular the influence of various halide ions on K+ and Na+ selectivity. We adopted molecular dynamics simulations with non-charged nanopores under constant temperature and uniform concentration. The results show K+-selectivity in the solution with Cl- and Na+-selectivity in the solution with Br- and I-. This selectivity, on the contrary, disappears in the solution with F-. We also investigate the change of the hydration shell of ions and cation-anion interactions between in the bulk region and in the nanopores, which could explain this selective phenomenon. Keywords: ion selectivity; halide ions; potassium and sodium; hydration shell of ion
1. Introduction The ion selectivity refers to the ability of a nano-scale channel which prefers the permeation of specific ions. Biological ion channels and protein channels play a critical role in the metabolic process of animals and they are also essential in the mass transport and the consumption process of plants [1-5]. Understanding the mechanism of ion selectivity is the foundation of many applications, such as drug delivery technique in pharmaceutical industry and a membrane-based technology in desalination of seawater and brackish ground water [6, 7]. It has been found that the ion selective transport is influenced by physical and chemical environments, including the structure of a channel [8, 9], temperature [10-12], pressure [13, 14] and the properties of solutions [15, 16]. The computer simulations have greatly extended our understanding of such a transport of water molecules and ions in nano-scale channels. By using molecular dynamics (MD) simulation, researchers have observed that the transport properties of water inside nano-scale channels are significantly different from that in bulk solution [17]. Water can pass through a short 8.1 Å diameter single-walled nanotube, forming a single file water structure within the nanotube [18]. As the nanopore diameter increases, more water molecules can enter *
Corresponding author. E-mail: [email protected], Phone:+82-2-705-8644, Fax:+82-2-712-0799
0959-2989/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved
384
D. Tang and D. Kim / The effect of counter-ions on the ion selectivity
and phase transition temperature is affected by the diameter, which proves that the physical properties of water are vastly different from the bulk solution [19]. Once ions appear inside a nanopore, the space of nanopore more or less squeezes the ion hydration shell, which is closely related to the ion selectivity of nano-scale channels [20]. The first shell of a cation hydration shell is fairly disordered inside a nanopore than in the bulk solution, but a minor change occurs for an anion hydration shell [10]. The K+ ions are more sensitive to the spatial or orientation distributions of water molecules in the first hydration shell than Na+ ions and Li+ ions [21]. The first shell of ion hydration is also correspondent to the energy of a system, found from the calculation of the potential of mean force of ion-pair [11]. A molecular-level study on the effect of halide ions in aqueous solutions on the ion selectivity of nano-scale channels in vitro is interesting. We applied a MD simulation to a non-charged nanopore model to study the spontaneous ion transport at equilibrium to various halide ions solutions. In our previous study, we showed that potassium and sodium, which carry the same charge and similar size, can be separated by choice of nanopore diameter and temperature [31]. Here we demonstrate that as the system temperature, concentration of ions and the geometry properties of nanopore fixed, the ion selectivity is tend to depend on the property of chemical solution, specifically on the type of halide ions in this study. Our work could provide a possible reference for the pretreatment of solution in a permeable membrane. 2. Simulation details We considered a 1:1 Na-anion and K-anion solution for each kind of halide ions at constant temperature (300 K). This system contains 810 water molecules, 21 Na+ ions, 21 K+ ions, and 42 halide ions with a constant bulk concentration (~1.29 M). Figure 1a shows a schematic of the system.
(a)
(b) Fig. 1 Schematic of the simulation system.
Two walls of carbon atoms 30 Å apart represent a membrane and they are connected with an 18.02 Å-diameter nanopore. The carbon atoms forming the walls and the nanopore are all frozen and they have zero charge. The simulation box is 30 Å * 30 Å * 60 Å. In this study, we performed a molecular dynamics simulation with DL_Poly 2.20 software [22]. Periodic boundary conditions were set for all three Cartesian directions in the NVT ensemble systems. We used the SPC/E water model [23, 24] and we have a time step of 2 fs to minimize the computational load while maintaining accuracy. The interactions between the ions and the water molecules are described by the Lennard-Jones poten-
D. Tang and D. Kim / The effect of counter-ions on the ion selectivity
385
tial [25], which is expressed as, U( ri , r j ) = 4ε ij [(
ε ij = ε iiε jj ,
ij rij
) 12 − (
ij rij
)6 ] +
qiq j , rij
(1)
1 σ ij = (σ ii + σ jj ) , ( i ≠ j ),
(2)
2
where Tij, Fij and qi are the atomic hard-sphere distance, the depth of the potential well and the atomic charge, respectively. The Lorentz-Berthelot mixing rules [26] expressed in Eq. (2) were used to obtain heterogeneous Lennard-Jones parameters [27]. The halide-water potential parameters are collected from Koneshan’s work [28]. The Nosé-Hoover thermostat was applied to keep the system temperature at 300 K [29] and the length of the van der Waals cutoff radius is 10 Å, which coincides with the length of short range Coulomb cutoff. Initially, all the ions and the water molecules are randomly located inside pool region. On average, 6 ns simulations were employed for each system for the purpose of achieving an equilibrium state while all the solid parts were fully solvated. In the following production simulations for over 36 ns, the particles spontaneously access the single nanopore. We checked the system equilibration state by performing the blocking average method [30]. The velocity and trajectory of particles were collected during the production simulations.
(a)
(b)
(c)
(d)
Fig. 2. Full trajectory of ions inside the nanopore. (a) F-; (b) Cl-; (c) Br-; (d) I-.
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D. Tang and D. Kim / The effect of counter-ions on the ion selectivity
Fig. 3. The average number of ions inside the nanopore.
3. Results and discussion We first investigate the filling process in the 18.02 Å diameter nanopore with various halide ions solutions. Figure 2 shows the full trajectory of ion density inside the nanopore over the production simulation. It shows that the ion density fluctuation in the solution with F- is significantly weaker than the other tested cases. We calculated the average number of ions which quantitatively describes the ion selectivity of the nanopores. We compare the ion selectivity of Na+ and K+ using the average number of ions, as shown in Figure 3. The entries of Na+ ions increase in the systems with the halide ions having lower affinity. No ion selectivity occurs within the nanopore in the solution with F-, which can be observed with approximate zero value of number of ions. The magnitude of the average number of Na+ and K+ in the solution with Cl- is consistent with our previous results [31]. The average number of K+ is larger than Na+, which indicates that K+-selectivity occurs. We note that more Na+ ions are attracted anions inside the nanopore than K+ in the solution with Br- or I-, which suggests that Na+-selectivity occurs. We thus find that the ion selectivity can be changed by different anions (counter-ions) in solution as well as by adjusting the geometry of nanopores, solution concentration, and temperature. By collecting equilibrium trajectories of ions and using an evaluation of the volume of intersection of a sphere with a cylinder by elliptic integrals [32], we calculated the radial distribution function (RDF) from sufficiently long production simulations [30]. Here, the RDF analysis includes the calculation of cation-anion interactions and the hydration shell of cation, as shown in Figure 4. The RDF curves exhibit the nonlinear density distribution surrounding a center particle and approach to unity as the distance between center particles and surrounding particles goes to infinity. In Figure 4a and 4b, the hydration shells of Na+ and K+ inside the nanopore are the same as that in the pool. The density distribution of Na+-F- and K+-F- are similar and they are more concentrated than the hydration shell of cations in both of the nanopore, and the pool. This demonstrates that almost all of cation ions are detained by F- ions in the pool. It implies that neither cations nor F- ions enter the 18.02 Å diameter nanopore at 300 K. The corresponding RDF analysis in the solution with Cl- (Figure 4c and 4d) shows that the hydration shell are the same in the pool and in the nanopore for Na+ and K+, respectively. Na+ ions attract a little more water molecule than K+ ions. Cations hardly enter the nanopore because of stably bonded ion pair of NaF (KF). In contrast to the solution with F-, the interactions among particles are moderate and ions move freely inside the whole system. K+ ions attract more Cl- ions than Na+ ions in the pool and the nanopore. However, both of K+ and Na+ attract less chloride ions when moving inside the nanopore. The Cl- ions are mainly locate at about 3 Å apart from a cation, which can be from that the first peak in cation-anion RDF is in a dominant position than the
D. Tang and D. Kim / The effect of counter-ions on the ion selectivity
387
second peak. It indicates that the tendency of forming ion pair K+-Cl- is higher than Na+-Cl- inside the nanopre. We conclude that under the same condition (system volume, temperature and concentration) the 18.02 Å diameter nanopore exhibits switching for cation ions that it closes in fluorine solution but opens in the solution with Cl-.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 4. The radial distribution function of cation-anion interactions and the hydration shell of cations. (a & b) F-; (c & d) Cl-; (e & f) Br-; (g & h) I-.
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D. Tang and D. Kim / The effect of counter-ions on the ion selectivity
A similar relationship of hydration shell of cation is observed in the solution with larger-size halide
(a)
(b)
(c)
Fig. 5. The potential of mean force of ions inside the nanopore.
ions, where the hydration shell of Na+ and K+ in the pool is same as in the nanopore. The cation-anion RDF decreases as ions enter the nanopore. Here, the first peak of Na+-Br- appears to be significantly lower than that of K+-Br- and has equal density distribution at a distance of 5 Å far from Na+ where the second peak of Na+-Br- is located. The results show that the weaker bond between Na+ and Br- relative to the strong activity anion F- and Cl- can be attributed to the less concentrated density distribution surrounding Na+. We therefore conclude that the bond between cation and anion restrict them to enter the nanopore alone but this restriction relaxes in less active halide ions solution. In particular, Na+ ions are more sensitive to this change than K+ ions. In the solution with I-, the drop of the hydration shell of cation is pronounced inside the nanopore. The reduction of ionic bond strength may partially account for ion selectivity. In particular, not only the values of first peak and second peak of Na+-I- RDF appear to be comparable but also the conversion of primary and secondary peaks would largely determine the type of ion selectivity. Our results here thus suggest that the geometric factor of the nanopore and other physical parameters, each of them alone would not give rise to the variable and complexity of ion selectivity. When only altering the anion in solution, the ion selectivity can be affected. To explore the ion selectivity further, we calculated the potential of mean force (PMF) [33] which shows the free energy along the reaction coordinate. We employ a simplified Fokker-Planck equation of PMF, as expressed F ( z ) = −k BT ln ρ ( z )
(3)
where F(z) is the potential of mean force, kB is Boltzmann constant, T is temperature, S(z) is the equilibrium particle probability density and z is the reaction coordinate. In this study, z is the axis of the nanopore. Figure 5 shows the PMF profiles, which is symmetric with respect to the center of the nanopore. At the entrances, the PMF of ions is relatively low and fluctuates near 2.0 kBT, which allows cation and anion to access the nanopore. At the center of the nanopore, there is a free energy barrier of Na+ with respect to K+ in the solution with Cl-. However, the trend is reversed in the other two cases. We believe that this can explain the phenomenon of ion selectivity shown in Figure 3. The free energy barrier of K+ in the solution with I- is considerably lower than that of Br-, while the free energy of Na+ remains at the same level for these two cases. This explains the increase of the average number of K+.
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4. Conclusion In this work, we observe that solution with Br- and I- provides a chemical environment where selects Na+ more effectively than the solution with F- or Cl-. The investigation of the liquid state inside the nanopore shows that the hydration shell of a cation inside the pool and the nanopore is independent of halide ions [30]. We also investigate the potential of mean force of ions inside the nanopore and the results show that there is a free energy barrier of Na+ (K+) in the solution of Cl- (Br- and I-) [33]. Simulation results show that the chemical nature of halide ions solutions is an important factor beside the nanopore diameter and temperature in the separation of Na+ and K+ ions. On the other hand, the range of possible selective nanopore and corresponding physical condition is strongly restricted by the type of positive and negative solutes. We think that this work would contribute to the understanding of ion separation processes and the development of permeable membranes. 5. Acknowledgement This work was supported by Basic Science Research Program (Grant No. NRF-2011-0009993), the Pioneer Research Center Program (Grant No. NRF-2012-0009578) through the National Research Foundation of Korea, and the Multiphenomena CFD Engineering Research Center (Grant No. 2009-0093136) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
J. Aqvist, and V. Luzhkov, Ion permeation mechanism of the potassium channel, Nature 404 (2000), 881-881. S. Berneche, and B. Roux, Energetics of ion conduction through the K+ channel, Nature 414 (2001), 73-77. A. Alam, and Y. X. Jiang, Structural studies of ion selectivity in tetrameric cation channels, J. Gen. Pysiol. 137 (2011), 397-403. B. Egwolf, and B. Roux, Ion selectivity of the KcsA channel: A perspective from multi-ion free energy landscapes, J. Mol. Bioi. 401 (2010), 831-842. D. P. Schachtman, Molecular insights into the structure and function of plant K+ transport mechanisms, Biochimica et Biophysica Acta, 1465 (2000), 127-139. C. Y. Tang, Y. Zhao, R. Wang, C. Helix-Nielsen, and A. G. Fane, Desalination by biomimetic aquaporin membranes: Review of status and prospects, Desalination. 308 (2013), 34-40. V. G. Gude, N. Nirmalakhandan, S. Deng, and A. Maganti, Feasibility study of a new two-stage low temperature desalination process, Energ. Convers. Manage 56 (2012), 192-198. W. Sparreboom, A. van den Berg, and J. C. T. Eijkel, Transport in nanofluidic systems: a review of theory and applications, New Journal of Physics 12 (2010), 015004. J. H. Park, S. B. Sinnott, and N. R. Aluru, Ion separation using a Y-junction carbon nanotube, Nanotechnology 17 (2006), 895-900. Q. Shao, L. Huang, J. Zhou, L. Lu, L. Zhang, X. Lu, S. Jiang, K. E. Gubbins, and W. Shen, Molecular simulation study of temperature effect on ionic hydration in carbon nanotubes, Phys. Chem. Chem. Phys. 10 (2008), 1896-1906 K. Yui, M. Sakuma, and T. Funazukuri, Molecular dynamics simulation on ion-pair association of NaCl from ambient to supercritical water, Fluid Phase Equilibria 297 (2010), 227-235. E. M. Zahran, V. Gavalas, M. Valiente, and L. G. Bachas, Can temperature be used to tune the selectivity of membrane ion-selective electrodes, Anal. Chem. 82 (2011), 3622-3628. L. L. Huang, L. Z. Zhang, Q. Shao, L. H. Lu, X. H. Lu, S. Y. Jiang, and W. F. Shen, Simulations of binary mixture adsorption of carbon dioxide and methane in carbon nanotubes: temperature, pressure, and pore size effects, J. Phys. Chem. C 111 (2007), 11912-11920.
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[14] M. V. Fedotova, Effect of temperature and pressre on structural self-organization of aqueous sodium chloride solutions, J. Mol. Liq. 153 (2010), 9-14. [15] V. A. Kolombet, and A. I. Frolov, Na+/K+ selectivity in the formation of ion pairs in aqueous solutions, Russ. J. Phys. Chem. B 4 (2011), 875-882. [16] A. Criscuoli, M. C. Carnevale, H. Mahmoudi, S. Gaeta, F. Lentini, and E. Drioli, Membrane contactors for the oxygen and pH control in desalination, J. Membrane. Sci. 376 (2011), 207-213. [17] J. Behler, D. W. Price, and M. G. B. Drew, Water structuring properties of carbohydrates, molecular dynamics studies on 1,5-anhydro-D-fructose, Phys. Chem. Chem. Phys. 3 (2011), 588-601. [18] G. Hummer, J. C. Rasaiah, and J. Noworyta, Water conduction through the hydrophobic channel of a carbon nanotube, Nature 414 (2001),188-190 . [19] J. Shiomi, T. Kimura, and S. Maruyama, Molecular dynamics of ice-nanotube formation inside carbon nanotubes, J. Phys. Chem. C 111 (2007), 12188-12193. [20] S. B. Remper, and L. R. Pratt, The hydration number of Na+ in liquid water, Fluid Phase Equilibria 183 (2001), 121-132. [21] Y. Zhu, X. Guo, Q. Shao, M. Wei, X. W. L. Lu, and X. Lu, Molecular simulation study of the effect of inner wall modified groups on ionic hydration confined in carbon nanotube, Fluid. Phase. Equilibr 297 (2010), 215-220. [22] W. Smith, and T. R. Forester, DL_POLY2.0: A general-purpose parallel molecular dynamics simulation package, J. Mol. Graph. Model. 14 (1996), 136-141. [23] H. J. C. Berendsen, J. R. Grigera, and T. P. Straatsma, The missing term in effective pair potentials, J. Phys. Chem. 91 (1987), 6269-6271. [24] S. Miyamoto, and P. A. Kollman, SETTLE: An analytical verstion of the SHAKE and RATTLE algorithm for rigid water models, J. Comput. Chem. 13 (1992), 952-962. [25] A. V. Raghunathan, J. H. Park, and N. R. Aluru, Interatomic potential-based semiclassical theory for Lennard-Jones fluids, J. Chem. Phys. 127 (2007), 174701. [26] T. Werder, J. H. Walther, R. L. Jaffe, T. Halicioglu, and P. Koumoutsakos, On the water-carbon interaction for use in molecular dynamics simulations of graphite and carbon nanotubes, J. Phys. Chem. B 107 (2003), 1345-1352. [27] L. X. Dang, and P. A. Kollman, Free energy of association of the K+: 18-Crown-6 Complex in water: a new molecular dynamics study, J. Phys. Chem. 99, 55 (1995), 55-58. [28] S. Koneshan, J. C. Rasaiah, R. M. Lynden-Bell, and S. H. Lee, Solvent structure, dynamics, and ion mobility in aqueous solutions at 25 °C, J. Phys. Chem. B 102 (1998), 4193-4204. [29] W. G. Hoover, Canonical dynamics: Equilibrium phase-space distributions, Phys. Rev. A 31 (1985), 1695-1697. [30] D. Frenkel, and B. Smit, Understanding Molecular Simulation (2 edn), in: San Diego, Academic Press, California, USA, 2001, pp. 529-532. [31] J. J. Cannon, D. Tang, N. Hur, and D. Kim, Competitive entry of sodium and potassium into nanoscale pores, J. Phys. Chem. B 114 (2010), 12252-12256. [32] F. Lamarche, and C. Leroy, Evaluation of the volume of intersection of a sphere with a cylinder by elliptic integrals, Comput. Phys. Commun. 59 (1990), 359-369. [33] H. Lu, X. Zhou, F. Wu, and Y. Xu, Monte Carlo simulation of adsorption of binary and quaternary alkane isomers mixtures in zeolites: Effect of pore size and structure, J. Phys. Chem. B 112 (2008), 135-145.
383
The effect of counter-ions on the ion selectivity of potassium and sodium ions in nanopores Dai Tang and Daejoong Kim*
Department of Mechanical Engineering, Sogang University, Seoul, Republic of Korea
Abstract. The ion selective transport plays an important role in the function of cell membranes, and promotes the application of artificial permeable membranes. This phenomenon has been studied in case for different diameters and functional groups of nanopores. In this work, we focus on the effect of anions on cation selectivity, in particular the influence of various halide ions on K+ and Na+ selectivity. We adopted molecular dynamics simulations with non-charged nanopores under constant temperature and uniform concentration. The results show K+-selectivity in the solution with Cl- and Na+-selectivity in the solution with Br- and I-. This selectivity, on the contrary, disappears in the solution with F-. We also investigate the change of the hydration shell of ions and cation-anion interactions between in the bulk region and in the nanopores, which could explain this selective phenomenon. Keywords: ion selectivity; halide ions; potassium and sodium; hydration shell of ion
1. Introduction The ion selectivity refers to the ability of a nano-scale channel which prefers the permeation of specific ions. Biological ion channels and protein channels play a critical role in the metabolic process of animals and they are also essential in the mass transport and the consumption process of plants [1-5]. Understanding the mechanism of ion selectivity is the foundation of many applications, such as drug delivery technique in pharmaceutical industry and a membrane-based technology in desalination of seawater and brackish ground water [6, 7]. It has been found that the ion selective transport is influenced by physical and chemical environments, including the structure of a channel [8, 9], temperature [10-12], pressure [13, 14] and the properties of solutions [15, 16]. The computer simulations have greatly extended our understanding of such a transport of water molecules and ions in nano-scale channels. By using molecular dynamics (MD) simulation, researchers have observed that the transport properties of water inside nano-scale channels are significantly different from that in bulk solution [17]. Water can pass through a short 8.1 Å diameter single-walled nanotube, forming a single file water structure within the nanotube [18]. As the nanopore diameter increases, more water molecules can enter *
Corresponding author. E-mail: [email protected], Phone:+82-2-705-8644, Fax:+82-2-712-0799
0959-2989/14/$27.50 © 2014 – IOS Press and the authors. All rights reserved
384
D. Tang and D. Kim / The effect of counter-ions on the ion selectivity
and phase transition temperature is affected by the diameter, which proves that the physical properties of water are vastly different from the bulk solution [19]. Once ions appear inside a nanopore, the space of nanopore more or less squeezes the ion hydration shell, which is closely related to the ion selectivity of nano-scale channels [20]. The first shell of a cation hydration shell is fairly disordered inside a nanopore than in the bulk solution, but a minor change occurs for an anion hydration shell [10]. The K+ ions are more sensitive to the spatial or orientation distributions of water molecules in the first hydration shell than Na+ ions and Li+ ions [21]. The first shell of ion hydration is also correspondent to the energy of a system, found from the calculation of the potential of mean force of ion-pair [11]. A molecular-level study on the effect of halide ions in aqueous solutions on the ion selectivity of nano-scale channels in vitro is interesting. We applied a MD simulation to a non-charged nanopore model to study the spontaneous ion transport at equilibrium to various halide ions solutions. In our previous study, we showed that potassium and sodium, which carry the same charge and similar size, can be separated by choice of nanopore diameter and temperature [31]. Here we demonstrate that as the system temperature, concentration of ions and the geometry properties of nanopore fixed, the ion selectivity is tend to depend on the property of chemical solution, specifically on the type of halide ions in this study. Our work could provide a possible reference for the pretreatment of solution in a permeable membrane. 2. Simulation details We considered a 1:1 Na-anion and K-anion solution for each kind of halide ions at constant temperature (300 K). This system contains 810 water molecules, 21 Na+ ions, 21 K+ ions, and 42 halide ions with a constant bulk concentration (~1.29 M). Figure 1a shows a schematic of the system.
(a)
(b) Fig. 1 Schematic of the simulation system.
Two walls of carbon atoms 30 Å apart represent a membrane and they are connected with an 18.02 Å-diameter nanopore. The carbon atoms forming the walls and the nanopore are all frozen and they have zero charge. The simulation box is 30 Å * 30 Å * 60 Å. In this study, we performed a molecular dynamics simulation with DL_Poly 2.20 software [22]. Periodic boundary conditions were set for all three Cartesian directions in the NVT ensemble systems. We used the SPC/E water model [23, 24] and we have a time step of 2 fs to minimize the computational load while maintaining accuracy. The interactions between the ions and the water molecules are described by the Lennard-Jones poten-
D. Tang and D. Kim / The effect of counter-ions on the ion selectivity
385
tial [25], which is expressed as, U( ri , r j ) = 4ε ij [(
ε ij = ε iiε jj ,
ij rij
) 12 − (
ij rij
)6 ] +
qiq j , rij
(1)
1 σ ij = (σ ii + σ jj ) , ( i ≠ j ),
(2)
2
where Tij, Fij and qi are the atomic hard-sphere distance, the depth of the potential well and the atomic charge, respectively. The Lorentz-Berthelot mixing rules [26] expressed in Eq. (2) were used to obtain heterogeneous Lennard-Jones parameters [27]. The halide-water potential parameters are collected from Koneshan’s work [28]. The Nosé-Hoover thermostat was applied to keep the system temperature at 300 K [29] and the length of the van der Waals cutoff radius is 10 Å, which coincides with the length of short range Coulomb cutoff. Initially, all the ions and the water molecules are randomly located inside pool region. On average, 6 ns simulations were employed for each system for the purpose of achieving an equilibrium state while all the solid parts were fully solvated. In the following production simulations for over 36 ns, the particles spontaneously access the single nanopore. We checked the system equilibration state by performing the blocking average method [30]. The velocity and trajectory of particles were collected during the production simulations.
(a)
(b)
(c)
(d)
Fig. 2. Full trajectory of ions inside the nanopore. (a) F-; (b) Cl-; (c) Br-; (d) I-.
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D. Tang and D. Kim / The effect of counter-ions on the ion selectivity
Fig. 3. The average number of ions inside the nanopore.
3. Results and discussion We first investigate the filling process in the 18.02 Å diameter nanopore with various halide ions solutions. Figure 2 shows the full trajectory of ion density inside the nanopore over the production simulation. It shows that the ion density fluctuation in the solution with F- is significantly weaker than the other tested cases. We calculated the average number of ions which quantitatively describes the ion selectivity of the nanopores. We compare the ion selectivity of Na+ and K+ using the average number of ions, as shown in Figure 3. The entries of Na+ ions increase in the systems with the halide ions having lower affinity. No ion selectivity occurs within the nanopore in the solution with F-, which can be observed with approximate zero value of number of ions. The magnitude of the average number of Na+ and K+ in the solution with Cl- is consistent with our previous results [31]. The average number of K+ is larger than Na+, which indicates that K+-selectivity occurs. We note that more Na+ ions are attracted anions inside the nanopore than K+ in the solution with Br- or I-, which suggests that Na+-selectivity occurs. We thus find that the ion selectivity can be changed by different anions (counter-ions) in solution as well as by adjusting the geometry of nanopores, solution concentration, and temperature. By collecting equilibrium trajectories of ions and using an evaluation of the volume of intersection of a sphere with a cylinder by elliptic integrals [32], we calculated the radial distribution function (RDF) from sufficiently long production simulations [30]. Here, the RDF analysis includes the calculation of cation-anion interactions and the hydration shell of cation, as shown in Figure 4. The RDF curves exhibit the nonlinear density distribution surrounding a center particle and approach to unity as the distance between center particles and surrounding particles goes to infinity. In Figure 4a and 4b, the hydration shells of Na+ and K+ inside the nanopore are the same as that in the pool. The density distribution of Na+-F- and K+-F- are similar and they are more concentrated than the hydration shell of cations in both of the nanopore, and the pool. This demonstrates that almost all of cation ions are detained by F- ions in the pool. It implies that neither cations nor F- ions enter the 18.02 Å diameter nanopore at 300 K. The corresponding RDF analysis in the solution with Cl- (Figure 4c and 4d) shows that the hydration shell are the same in the pool and in the nanopore for Na+ and K+, respectively. Na+ ions attract a little more water molecule than K+ ions. Cations hardly enter the nanopore because of stably bonded ion pair of NaF (KF). In contrast to the solution with F-, the interactions among particles are moderate and ions move freely inside the whole system. K+ ions attract more Cl- ions than Na+ ions in the pool and the nanopore. However, both of K+ and Na+ attract less chloride ions when moving inside the nanopore. The Cl- ions are mainly locate at about 3 Å apart from a cation, which can be from that the first peak in cation-anion RDF is in a dominant position than the
D. Tang and D. Kim / The effect of counter-ions on the ion selectivity
387
second peak. It indicates that the tendency of forming ion pair K+-Cl- is higher than Na+-Cl- inside the nanopre. We conclude that under the same condition (system volume, temperature and concentration) the 18.02 Å diameter nanopore exhibits switching for cation ions that it closes in fluorine solution but opens in the solution with Cl-.
(a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
Fig. 4. The radial distribution function of cation-anion interactions and the hydration shell of cations. (a & b) F-; (c & d) Cl-; (e & f) Br-; (g & h) I-.
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D. Tang and D. Kim / The effect of counter-ions on the ion selectivity
A similar relationship of hydration shell of cation is observed in the solution with larger-size halide
(a)
(b)
(c)
Fig. 5. The potential of mean force of ions inside the nanopore.
ions, where the hydration shell of Na+ and K+ in the pool is same as in the nanopore. The cation-anion RDF decreases as ions enter the nanopore. Here, the first peak of Na+-Br- appears to be significantly lower than that of K+-Br- and has equal density distribution at a distance of 5 Å far from Na+ where the second peak of Na+-Br- is located. The results show that the weaker bond between Na+ and Br- relative to the strong activity anion F- and Cl- can be attributed to the less concentrated density distribution surrounding Na+. We therefore conclude that the bond between cation and anion restrict them to enter the nanopore alone but this restriction relaxes in less active halide ions solution. In particular, Na+ ions are more sensitive to this change than K+ ions. In the solution with I-, the drop of the hydration shell of cation is pronounced inside the nanopore. The reduction of ionic bond strength may partially account for ion selectivity. In particular, not only the values of first peak and second peak of Na+-I- RDF appear to be comparable but also the conversion of primary and secondary peaks would largely determine the type of ion selectivity. Our results here thus suggest that the geometric factor of the nanopore and other physical parameters, each of them alone would not give rise to the variable and complexity of ion selectivity. When only altering the anion in solution, the ion selectivity can be affected. To explore the ion selectivity further, we calculated the potential of mean force (PMF) [33] which shows the free energy along the reaction coordinate. We employ a simplified Fokker-Planck equation of PMF, as expressed F ( z ) = −k BT ln ρ ( z )
(3)
where F(z) is the potential of mean force, kB is Boltzmann constant, T is temperature, S(z) is the equilibrium particle probability density and z is the reaction coordinate. In this study, z is the axis of the nanopore. Figure 5 shows the PMF profiles, which is symmetric with respect to the center of the nanopore. At the entrances, the PMF of ions is relatively low and fluctuates near 2.0 kBT, which allows cation and anion to access the nanopore. At the center of the nanopore, there is a free energy barrier of Na+ with respect to K+ in the solution with Cl-. However, the trend is reversed in the other two cases. We believe that this can explain the phenomenon of ion selectivity shown in Figure 3. The free energy barrier of K+ in the solution with I- is considerably lower than that of Br-, while the free energy of Na+ remains at the same level for these two cases. This explains the increase of the average number of K+.
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4. Conclusion In this work, we observe that solution with Br- and I- provides a chemical environment where selects Na+ more effectively than the solution with F- or Cl-. The investigation of the liquid state inside the nanopore shows that the hydration shell of a cation inside the pool and the nanopore is independent of halide ions [30]. We also investigate the potential of mean force of ions inside the nanopore and the results show that there is a free energy barrier of Na+ (K+) in the solution of Cl- (Br- and I-) [33]. Simulation results show that the chemical nature of halide ions solutions is an important factor beside the nanopore diameter and temperature in the separation of Na+ and K+ ions. On the other hand, the range of possible selective nanopore and corresponding physical condition is strongly restricted by the type of positive and negative solutes. We think that this work would contribute to the understanding of ion separation processes and the development of permeable membranes. 5. Acknowledgement This work was supported by Basic Science Research Program (Grant No. NRF-2011-0009993), the Pioneer Research Center Program (Grant No. NRF-2012-0009578) through the National Research Foundation of Korea, and the Multiphenomena CFD Engineering Research Center (Grant No. 2009-0093136) through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13]
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