- Email: [email protected]
Molecular simulation of carbon nanotube membrane for Li+ and Mg2+ separation
Journal of Membrane Science 444 (2013) 327–331
Contents lists available at SciVerse ScienceDirect
Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
Molecular simulation of carbon nanotube membrane for Li+ and Mg2+ separation Dengfeng Yang a, Qingzhi Liu b, Hongman Li a, Congjie Gao a,n a b
College of Chemistry and Chemical Engineering, Ocean University of China, No.238 SongLing road, Qingdao city 266100, China College of Chemistry and Pharmaceutical Science, Qingdao Agriculture University, No. 700 Changcheng road, Qingdao city 266109, China
art ic l e i nf o
a b s t r a c t
Article history: Received 8 February 2013 Received in revised form 9 May 2013 Accepted 11 May 2013 Available online 20 May 2013
Using molecular dynamics simulations, we study the transport of Li+ and Mg2+ through membranes formed from armchair type (8,8) to (11,11) single wall carbon nanotubes (SWNTs) under hydrostatic pressure, and investigate the potential of mean force(PMF), conductance and axial density distributions of ions in the carbon nanotubes, the results show that under 100 MPa, (9,9) and (10,10) SWNT are capable of extracting Li+; when adjusting pressure to 200 MPa, (8,8), (9,9) and (10,10) tubes can separate Li+ and Mg2+ with higher ion fluxes which also augment with increasing tube diameters. By calculating the potential mean force for ion translocation, we show that Mg2+ face greater energy barrier than Li+ when passing the carbon nanotube with same diameter, and free energy difference of ∼8 kJ/mol seem to be the threshold to effectively separate Li+ and Mg2+. Membrane incorporating narrow carbon nanotubes is found to be promising in Li+ and Mg2+ separation. & 2013 Elsevier B.V. All rights reserved.
Keywords: Carbon nanotube Membrane Lithium Magnesium Ion
1. Introduction With the development of new energy automotive, energy storage technology as well as the flat computer etc., the global demand for Li-ion battery keeps growing. How to effectively explore and utilize lithium resources have increasingly become the focus of research [1]. In the world, about more than 69% of terrestrial lithium resources deposit in Saline Lakes [2], and most with a high magnesium lithium ratio feature range from 40 to 1800 [3]. For the close radius of Li+ and Mg2+, it is difficult to extract lithium from the brine, and the final products like lithium carbonate or lithium hydroxide contaminated with residual magnesium cannot be used to manufacture lithium cell for their drastically weakened performance. Hitherto, many research has been done in lithium recovery methods like precipitation, adsorption, solvent extraction and ion exchange etc [4–7], while the precipitation process is only suitable for brine with low magnesium lithium ratio; the adsorption method needs to find the absorbent easily prepared with high adsorption capacity; solvent extraction consume much organic solvent and is not realistic to operate in large scale. Each method mentioned above has its own shortcomings and how to effectively separate lithium and magnesium is still a challenging task.
n
Corresponding author. Tel.: +86 532 66782481. E-mail address: [email protected] (C. Gao).
0376-7388/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.memsci.2013.05.019
Carbon nanotubes as a kind of hollow, non-polar, smooth inner surface nano material have many special structural, mechanical and electronic properties [8]. Water molecule can permeate through the carbon nanotubes with a massive flow rate in the form of “water chain” [9–11]. The hydrophobic pore similar to biological ion channel also has intrinsic ion selectivity, Song [12] conducted molecular dynamics simulations to show that Na+, K+, and Cl− face different free energy barriers when entering carbon nanotubes; Peter and Hummer [13] have studied that Na+ can pass through (10,10) carbon nanotube spontaneously, while Cl− can not; Liu et al. [14]also showed that a membrane of uncharged nanotubes of the appropriate diameter can select K+ against Na+ without the presence of external electric field or surface charges. As described above, there have been some studies that aimed to determine the selectivity of Na+, K+, and Cl− through CNTs under osmotic or hydrostatic pressures, but still no study of the conductance of Li+ and Mg2+ in a range of differently sized SWNTs has been done to determine the selectivity of them. In this paper, using molecular dynamics simulations, the energetics of Li+ and Mg2+are examined in a range of differently sized carbon nanotubes such that their potential use as ion sieves for Li+ recovery can be investigated. We determine the energy barriers to ion conductance in membranes formed from (8,8), (9,9), (10,10) and (11,11) “armchair” type SWNTs and examine their conduction rate in the carbon nanotubes when 100 and 200 MPa hydrostatic pressure differences are applied separately. These results show that the membranes incorporating narrow carbon nanotube have
328
D. Yang et al. / Journal of Membrane Science 444 (2013) 327–331
distinguishing ion selectivity to Li+ and Mg2+, and maybe a promising way to lithium recovery.
site–site interaction potential used is of the form " 6 # qi qj r m 12 rm U LJ ¼ εij −2 þ r ij r ij r ij
ð1Þ
where εij and rm are Lennard–Jones parameters and rij is the distance between the sites; The Lennard–Jones energy (εij) and diameter (rm) parameters of the cross interactions are calculated on the basis of the Lorentz–Berthelot combining rules.
2. Methods 2.1. Model system Carbon nanotube membranes are created by hexagonally packing 4 SWNTs each ∼13.5 Å-long as shown in Fig. 1. Periodic boundaries are used to form a continuous two-dimensional membrane. There are four separate simulation systems constructed comprising (8,8), (9,9), (10,10) and (11,11) armchair type SWNTs, the diameters of the nanotubes as measured between the carbon centers as well as the effective internal diameter assuming a carbon atom van der Waals radius of 1.7 Å are indicated in Table 1. Each SWNT membrane is solvated by ∼20-Å-thick layers of TIP3P water on either side, the ions are randomly placed to yield a net concentration of 250 mM LiCl or MgCl2 in one unit cell. In our simulations, all of the SWNTs axes are parallel to the z axis, and all of the tubes are located between z¼ 0 and z¼14 Å, with the center at z ¼7 Å.
2.2. MD simulation parameters Simulations are conducted using NAMD 2.8 [15] with the CHARMM27 force field [16]. All carbon atoms are taken to be neutral. A value of 12 Å is used to cut off the nonbonded van der Waals interactions, and the particle mesh Ewald (PME) method is used to calculate the full electrostatic interactions. Lennard–Jones
εij ¼
r m;i þ r m;j pffiffiffiffiffiffiffiffiffi εii εjj ; r m ¼ 2
ð2Þ
The parameters of particles are listed in Table 2. To mimic the friction caused by jostling of solvent and the system perturbation induced by occasional high velocity collision, langevin dynamics is used to control the temperature and pressure. The time step is set to 1 fs. 2.3. Hydrostatic-pressure simulations Before collecting data, all systems are energy minimized and equilibrated for 1.0 ns under a constant pressure of 1 atm and temperature of 300 K. To keep the membrane in place and avoid overall translation of the system, the carbon atoms are fixed. Then the method developed by Zhu et al. [17,18] is employed to introduce a hydrostatic pressure difference in our simulations, a constant force is applied to water molecules greater than 13 Å from either side of the membrane, to generate a hydrostatic pressure difference across the membrane using the fact that ΔP¼ nf/A where n is the number of water molecules in the section to which forces are applied and A is the cross sectional area of the membrane. For each of the four membranes, 10-ns hydrostaticpressure MD simulations are performed under pressures of 100 and 200 MPa. The total simulation time is 80 ns. 2.4. PMF Calculations The potentials of mean force (PMF) for Li+ and Mg2+ ions passing through SWNTs are determined using the Adaptive Biasing Forces (ABF) method which compute the mean force acting on reaction coordinate along z axis and then remove it from the system to get a uniform sampling and a diffusive-like motion along the reaction coordinate [19,20]. In order to improve computing efficiency, five windows are divided along the axial coordinates.
3. Results and discussion 3.1. PMF for ions
Fig. 1. A model of simulation system, which is formed by hexagonally packing 4 nanotubes in a periodic cell and immersing in a solution of LiCl in water.
Table 1 Diameter of CNT pores. Size
In many simulations, ions like Na+, K+, Cl− have been found facing different free energy barriers when entering hydrophobic pores of CNTs, and carbon nanotube membranes can act as ion sieves, with the pore radius and pressure determining which ions will permeate through the membrane [12,21]. To better understand the selectivity of Li+ and Mg2+ through the nanotubes, in this study, the PMF for Li+ and Mg2+ moving along the tube axis are calculated and shown in Fig. 2. Contrary to homogeneity of SWNT's structure and properties, Table 2 Interaction parameters of C and ions. The ε and rm parameters of the cross interactions are calculated on the basis of the Lorentz–Berthelot combining rules.
Diameter (Å) of CNT C–C
Internal
10.85 12.20 13.56 14.92
7.45 8.80 10.15 11.52
parameter (8,8) (9,9) (10,10) (11,11)
−1
ε (kJ mol ) rm (Å) q (e)
Li+
Mg2+
Cl−
C
0.075 2.39 +1
0.062 2.37 +2
0.63 4.54 −1
0.46 4.00 0
D. Yang et al. / Journal of Membrane Science 444 (2013) 327–331
329
Fig. 2. PMF profiles for Li+ (solid lines), Mg2+ (line with bare circles) within SWNTs (a) (8,8), (b) (9,9), (c) (10,10) and (d) (11,11), the ion is moved from the bulk solution (z¼ 20) into the center of the pore (z ¼7) in each case, the SWNTs are located from z ¼0 to z ¼14 Å.
Table 3 Diameter of water and ions. Diameter (Å) of water and ions [21] Water
Coordination number
Li+
Mg2+
IV V VI VIII
1.18
2.90
1.14 1.32 1.44 1.78
1.52 1.84
the PMF curve of two ions exhibit some irregular change in the pores, and that become especially evident in larger diameter SWNTs like (10,10) and (11,11). This phenomenon means that the ions favor some positions over others in the pores, for reasons like the pulse-like transmission of water molecule in the confined environment [10] which present some favorable space for ions randomly or electrostatic and Van der Waals forces from other ions in the system. In larger diameter (10,10) and (11,11), the lower PMF value makes this fluctuations more clearly. Difficulty for separating Li+ and Mg2+ are generally attributed to similar size. As shown in Table 3, the diameters of Li+ and Mg2+ with the same hydration number seem indeed almost equivalent. But Li+ face much smaller free energy barriers than Mg2+ when entering pores of a particular radius, and the discrimination between them become more obvious as the diameter of SWNTs decrease. For example, in the (8,8) tube, the ΔGMg2+−ΔGLi+≈51 kJ/ mol, which means that it is easier for Li+ to enter the (8,8) CNT. As a contrast, for the (9,9), (10,10) and (11,11) tubes, the PMF differences of Li+ and Mg2+ are 29, 9.7 and 4.6 kJ/mol separately. What cause the differences? We note that previous studies have shown that most of the doubly charged Mg2+ in bulk water has about 6 coordinated water molecules in the first hydration shell [22] and is quite different with Li+ with 4 stable water molecules in the inner hydration shell [23], which leads to the corresponding diameters of Li+ and Mg2+ are 1.18 and 1.44 Å separately [24], and thus the water complexes of Li+ ion have
Fig. 3. Coordination numbers of (a) Li+, (b) Mg2+ permeating each of the carbon nanotubes studied. The tube center is at z ¼7.0.
smaller size than that of Mg2+ ion. Fig. 3 shows the coordination numbers of Li+ and Mg2+ when they enter each of the carbon nanotubes studied during ABF simulations (determined by counting the number of water molecules within 3.8 Å for Li+ and 4.3 Å for Mg2+ separately).The hydration shell of Li+ seem intact in simulated carbon nanotubes except for (8,8) in which the coordination number is down a little. Comparatively, the bigger Mg2+ complexes must be stripped of some water molecule or change geometry of the solvation shell to enter the narrower tubes, and only in the wider (11,11) tubes studied that the hydration shell in the pore can approximate that in the bulk, and the PMF differences of two ions become less obvious. The magnitude of PMF differences has evident effects on ion selectivity [12] and the biologically relevant values for free energy differences between different types of ions are ∼8 kJ/mol [25],
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D. Yang et al. / Journal of Membrane Science 444 (2013) 327–331
previous studies have shown that the free energy difference of ∼8 kJ/mol leads to a 10-fold selectivity for K+ over Na+ in potassium channels. From this point, it seems to indicate that the membrane composed of (8,8), (9,9), and (10,10) CNTs with the PMF differences of Li+ and Mg2+ more than 8 kJ/mol are more suitable to separate the two ions.
3.2. Ion transport in CNTs Information about ions conductance through the pores is important to assess the suitability of CNTs for Li+ and Mg2+ separation. To determine this, 100 MPa and 200 MPa hydrostatic pressure differences are created across the CNT membranes and the number of ions that pass across them during our simulations are counted, as shown in Table 4. From the results, under 100 MPa pressure, no ions pass through the (8,8) SWNT membranes in our simulations, one Li+ crosses (9,9) SWNT, while no Mg2+ permeate. In the remaining situations, conductance is greater in the wider tubes than the narrow ones, and not surprisingly, more Li+ permeate through the membrane than Mg2+, but the preferential conduction of Li+ seems to be less distinct in lager diameter (11,11) SWNTs than in (10,10) SWNTs. Table 4 Numbers of ions and water molecules passing through the membranes within 10 ns under different hydrostatic pressures. Size
100 MPa +
(8,8) (9,9) (10,10) (11,11)
200 MPa
Li
Mg
0 1 5 7
0 0 1 5
2+
H2O
Li+
Mg2+
H2O
2366 3442 4492 5790
1 3 23 35
0 1 4 26
4740 6802 8934 11470
As a contrast, under 200 MPa pressure, only one Li+ permeates through the (8,8) SWNTs, and both Li+ and Mg2+ transport through (9,9), (10,10) and (11,11) SWNT membranes. Unlike the conductance of water under 200 MPa which is double that under 100 MPa, and in other studies the conductance is found to increase linearly over the range of pressures from 5 to 400 MPa [26], the conductance of ions increase drastically following no obvious rules. Compared with the water molecules [26], ions face much higher energy barrier when enter sub nanometer CNT pores. The higher hydrostatic pressure help to overcome energy barriers and obtain increasing ion flux. From data in Table 4, the rising ratio of ion flux with pressure change from 100 to 200 MPa is more evident for larger diameter SWNTs, like in (9,9) SWNT, the throughput of Li+ increase from 1 to 3, while in (10,10) and (11,11) SWNTs, the permeation number increases 4.6 and 5 times respectively; the flux of Mg2+ in (10,10) and (11,11) SWNTs with changing pressure also rise 4 and 5.2 times separately. The lower PMF value of ions into wider pores are more easily crossed by promotion of hydrostatic pressure. These results show that, in our 10 ns MD simulation time duration, the SWNT membranes seem to act as ion sieves to thoroughly separate Li+ from Mg2+ by applying 100 MPa pressure across (9,9) SWNT membrane or applying 200 MPa pressure across a (8,8) SWNT membrane, that different hydrostatic pressure conditions need different sized CNTs. The (10,10) tube under 100 Mpa and (9,9), (10,10) tubes under 200 Mpa also can discriminate between Li+ and Mg2+ with higher ion fluxes, which are in accord with that at least ∼8 kJ/mol PMF differences of Li+ and Mg2+ are required to separate the two ions, and the (11,11) SWNTs with 4.2 kJ/mol PMF difference cannot extract Li+ effectively, no matter under 100 or 200 MPa hydrostatic pressure. We also note that Li+/ Mg2+ selectivity of (9,9) SWNT is not better than (10,10) SWNT with lower PMF difference, which may be partly caused by larger energy barrier to Li+ entrance and limited sampling in our 10 ns simulation.
Fig. 4. Axial density profiles of Li+ (line with bare triangle), Mg2+ (line with bare circles) and water (line with solid squares) for SWNTs (a) (8,8), (b) (9,9), (c) (10,10) and (d) (11,11) under 100 MPa hydrostatic pressure as a function of z axis where the SWNTs lies from z ¼ 0 to z ¼ 14 Å.
D. Yang et al. / Journal of Membrane Science 444 (2013) 327–331
The ensemble-averaged axial density profiles plotted in Fig. 4 provide a detailed picture of Li+ and Mg2+ distribution in the proximity of the carbon walls under 100 MPa hydrostatic pressure. From Fig. 4, the ions density is evidently higher in area near the entrance of carbon nanotubes, especially the narrower tubes like (8,8) and (9,9). The asymmetry seems to be caused by hydrostatic pressure across the membranes and the ions accumulate before the opening due to high energy barrier to permeate the tube. Fig. 4 shows that under 100 MPa pressure difference, there are no ions in (8,8) tubes; in (9,9) tubes, the Li+ ion is visible in contrast to Mg2+, which is still zero; in (10,10) tubes, the mean axial density of Li+ is about 4.5 times of Mg2+, while in (11,11) tubes the density distribution of Li+ is only slightly higher than Mg2+. The facts above evidently display the separation process of Li+ and Mg2+. Although most reverse osmosis membranes operate under pressures of less than 10 MPa, our simulations are conducted under a larger 100 or 200 MPa pressure difference as this allows for a greater number of conduction events to be observed in our limited simulations time, and we think it is possible to achieve similar results using much lower pressure by choosing membranes composed of wider SWNTs which are charged or functionalized that have appropriate PMF difference for two ions, or by waiting longer times. In addition, carbon nanotubes are the strongest and stiffest materials yet discovered in terms of tensile strength and elastic modulus respectively, and standard single-walled carbon nanotubes can withstand a pressure up to 24 GPa without deformation, so, if appropriate polymer or inorganic matrix that seal carbon nanotubes is applied and properly fabricated, the membrane can bear much higher pressure than traditional RO membranes. 4. Conclusions Using membranes comprising sub nanometer diameter carbon nanotubes as models, we have performed extensive MD simulations to examine their potential selectivity to Li+ and Mg2+ and their relationships to pore size. We find that narrow carbon nanotubes like (8,8), (9,9) and (10,10) SWNTs with free energy difference more than ∼8 kJ/mol to Li+ and Mg2+ can provide a promising way to separate the two ions which are supported both by free energy calculations and by simulations under hydrostatic pressure. Under 100 MPa, (9,9) and (10,10) SWNTs are capable of extracting Li+; while under 200 MPa, (8,8), (9,9) and (10,10) tubes all can separate Li+ and Mg2+ with higher ion fluxes. Through cautiously choosing the SWNT size and the pressure across the membrane, Li+ can be extracted from the bulk. Therefore carefully fabricated SWNT membranes can act as Li+ and Mg2+ separator that we hope can be tested experimentally in the future. Acknowledgments Financial support of this research was provided by the National Basic Research Program of China (2009CB623402) and the National Natural Science Foundation of China (No.21006100).
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Contents lists available at SciVerse ScienceDirect
Journal of Membrane Science journal homepage: www.elsevier.com/locate/memsci
Molecular simulation of carbon nanotube membrane for Li+ and Mg2+ separation Dengfeng Yang a, Qingzhi Liu b, Hongman Li a, Congjie Gao a,n a b
College of Chemistry and Chemical Engineering, Ocean University of China, No.238 SongLing road, Qingdao city 266100, China College of Chemistry and Pharmaceutical Science, Qingdao Agriculture University, No. 700 Changcheng road, Qingdao city 266109, China
art ic l e i nf o
a b s t r a c t
Article history: Received 8 February 2013 Received in revised form 9 May 2013 Accepted 11 May 2013 Available online 20 May 2013
Using molecular dynamics simulations, we study the transport of Li+ and Mg2+ through membranes formed from armchair type (8,8) to (11,11) single wall carbon nanotubes (SWNTs) under hydrostatic pressure, and investigate the potential of mean force(PMF), conductance and axial density distributions of ions in the carbon nanotubes, the results show that under 100 MPa, (9,9) and (10,10) SWNT are capable of extracting Li+; when adjusting pressure to 200 MPa, (8,8), (9,9) and (10,10) tubes can separate Li+ and Mg2+ with higher ion fluxes which also augment with increasing tube diameters. By calculating the potential mean force for ion translocation, we show that Mg2+ face greater energy barrier than Li+ when passing the carbon nanotube with same diameter, and free energy difference of ∼8 kJ/mol seem to be the threshold to effectively separate Li+ and Mg2+. Membrane incorporating narrow carbon nanotubes is found to be promising in Li+ and Mg2+ separation. & 2013 Elsevier B.V. All rights reserved.
Keywords: Carbon nanotube Membrane Lithium Magnesium Ion
1. Introduction With the development of new energy automotive, energy storage technology as well as the flat computer etc., the global demand for Li-ion battery keeps growing. How to effectively explore and utilize lithium resources have increasingly become the focus of research [1]. In the world, about more than 69% of terrestrial lithium resources deposit in Saline Lakes [2], and most with a high magnesium lithium ratio feature range from 40 to 1800 [3]. For the close radius of Li+ and Mg2+, it is difficult to extract lithium from the brine, and the final products like lithium carbonate or lithium hydroxide contaminated with residual magnesium cannot be used to manufacture lithium cell for their drastically weakened performance. Hitherto, many research has been done in lithium recovery methods like precipitation, adsorption, solvent extraction and ion exchange etc [4–7], while the precipitation process is only suitable for brine with low magnesium lithium ratio; the adsorption method needs to find the absorbent easily prepared with high adsorption capacity; solvent extraction consume much organic solvent and is not realistic to operate in large scale. Each method mentioned above has its own shortcomings and how to effectively separate lithium and magnesium is still a challenging task.
n
Corresponding author. Tel.: +86 532 66782481. E-mail address: [email protected] (C. Gao).
0376-7388/$ - see front matter & 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.memsci.2013.05.019
Carbon nanotubes as a kind of hollow, non-polar, smooth inner surface nano material have many special structural, mechanical and electronic properties [8]. Water molecule can permeate through the carbon nanotubes with a massive flow rate in the form of “water chain” [9–11]. The hydrophobic pore similar to biological ion channel also has intrinsic ion selectivity, Song [12] conducted molecular dynamics simulations to show that Na+, K+, and Cl− face different free energy barriers when entering carbon nanotubes; Peter and Hummer [13] have studied that Na+ can pass through (10,10) carbon nanotube spontaneously, while Cl− can not; Liu et al. [14]also showed that a membrane of uncharged nanotubes of the appropriate diameter can select K+ against Na+ without the presence of external electric field or surface charges. As described above, there have been some studies that aimed to determine the selectivity of Na+, K+, and Cl− through CNTs under osmotic or hydrostatic pressures, but still no study of the conductance of Li+ and Mg2+ in a range of differently sized SWNTs has been done to determine the selectivity of them. In this paper, using molecular dynamics simulations, the energetics of Li+ and Mg2+are examined in a range of differently sized carbon nanotubes such that their potential use as ion sieves for Li+ recovery can be investigated. We determine the energy barriers to ion conductance in membranes formed from (8,8), (9,9), (10,10) and (11,11) “armchair” type SWNTs and examine their conduction rate in the carbon nanotubes when 100 and 200 MPa hydrostatic pressure differences are applied separately. These results show that the membranes incorporating narrow carbon nanotube have
328
D. Yang et al. / Journal of Membrane Science 444 (2013) 327–331
distinguishing ion selectivity to Li+ and Mg2+, and maybe a promising way to lithium recovery.
site–site interaction potential used is of the form " 6 # qi qj r m 12 rm U LJ ¼ εij −2 þ r ij r ij r ij
ð1Þ
where εij and rm are Lennard–Jones parameters and rij is the distance between the sites; The Lennard–Jones energy (εij) and diameter (rm) parameters of the cross interactions are calculated on the basis of the Lorentz–Berthelot combining rules.
2. Methods 2.1. Model system Carbon nanotube membranes are created by hexagonally packing 4 SWNTs each ∼13.5 Å-long as shown in Fig. 1. Periodic boundaries are used to form a continuous two-dimensional membrane. There are four separate simulation systems constructed comprising (8,8), (9,9), (10,10) and (11,11) armchair type SWNTs, the diameters of the nanotubes as measured between the carbon centers as well as the effective internal diameter assuming a carbon atom van der Waals radius of 1.7 Å are indicated in Table 1. Each SWNT membrane is solvated by ∼20-Å-thick layers of TIP3P water on either side, the ions are randomly placed to yield a net concentration of 250 mM LiCl or MgCl2 in one unit cell. In our simulations, all of the SWNTs axes are parallel to the z axis, and all of the tubes are located between z¼ 0 and z¼14 Å, with the center at z ¼7 Å.
2.2. MD simulation parameters Simulations are conducted using NAMD 2.8 [15] with the CHARMM27 force field [16]. All carbon atoms are taken to be neutral. A value of 12 Å is used to cut off the nonbonded van der Waals interactions, and the particle mesh Ewald (PME) method is used to calculate the full electrostatic interactions. Lennard–Jones
εij ¼
r m;i þ r m;j pffiffiffiffiffiffiffiffiffi εii εjj ; r m ¼ 2
ð2Þ
The parameters of particles are listed in Table 2. To mimic the friction caused by jostling of solvent and the system perturbation induced by occasional high velocity collision, langevin dynamics is used to control the temperature and pressure. The time step is set to 1 fs. 2.3. Hydrostatic-pressure simulations Before collecting data, all systems are energy minimized and equilibrated for 1.0 ns under a constant pressure of 1 atm and temperature of 300 K. To keep the membrane in place and avoid overall translation of the system, the carbon atoms are fixed. Then the method developed by Zhu et al. [17,18] is employed to introduce a hydrostatic pressure difference in our simulations, a constant force is applied to water molecules greater than 13 Å from either side of the membrane, to generate a hydrostatic pressure difference across the membrane using the fact that ΔP¼ nf/A where n is the number of water molecules in the section to which forces are applied and A is the cross sectional area of the membrane. For each of the four membranes, 10-ns hydrostaticpressure MD simulations are performed under pressures of 100 and 200 MPa. The total simulation time is 80 ns. 2.4. PMF Calculations The potentials of mean force (PMF) for Li+ and Mg2+ ions passing through SWNTs are determined using the Adaptive Biasing Forces (ABF) method which compute the mean force acting on reaction coordinate along z axis and then remove it from the system to get a uniform sampling and a diffusive-like motion along the reaction coordinate [19,20]. In order to improve computing efficiency, five windows are divided along the axial coordinates.
3. Results and discussion 3.1. PMF for ions
Fig. 1. A model of simulation system, which is formed by hexagonally packing 4 nanotubes in a periodic cell and immersing in a solution of LiCl in water.
Table 1 Diameter of CNT pores. Size
In many simulations, ions like Na+, K+, Cl− have been found facing different free energy barriers when entering hydrophobic pores of CNTs, and carbon nanotube membranes can act as ion sieves, with the pore radius and pressure determining which ions will permeate through the membrane [12,21]. To better understand the selectivity of Li+ and Mg2+ through the nanotubes, in this study, the PMF for Li+ and Mg2+ moving along the tube axis are calculated and shown in Fig. 2. Contrary to homogeneity of SWNT's structure and properties, Table 2 Interaction parameters of C and ions. The ε and rm parameters of the cross interactions are calculated on the basis of the Lorentz–Berthelot combining rules.
Diameter (Å) of CNT C–C
Internal
10.85 12.20 13.56 14.92
7.45 8.80 10.15 11.52
parameter (8,8) (9,9) (10,10) (11,11)
−1
ε (kJ mol ) rm (Å) q (e)
Li+
Mg2+
Cl−
C
0.075 2.39 +1
0.062 2.37 +2
0.63 4.54 −1
0.46 4.00 0
D. Yang et al. / Journal of Membrane Science 444 (2013) 327–331
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Fig. 2. PMF profiles for Li+ (solid lines), Mg2+ (line with bare circles) within SWNTs (a) (8,8), (b) (9,9), (c) (10,10) and (d) (11,11), the ion is moved from the bulk solution (z¼ 20) into the center of the pore (z ¼7) in each case, the SWNTs are located from z ¼0 to z ¼14 Å.
Table 3 Diameter of water and ions. Diameter (Å) of water and ions [21] Water
Coordination number
Li+
Mg2+
IV V VI VIII
1.18
2.90
1.14 1.32 1.44 1.78
1.52 1.84
the PMF curve of two ions exhibit some irregular change in the pores, and that become especially evident in larger diameter SWNTs like (10,10) and (11,11). This phenomenon means that the ions favor some positions over others in the pores, for reasons like the pulse-like transmission of water molecule in the confined environment [10] which present some favorable space for ions randomly or electrostatic and Van der Waals forces from other ions in the system. In larger diameter (10,10) and (11,11), the lower PMF value makes this fluctuations more clearly. Difficulty for separating Li+ and Mg2+ are generally attributed to similar size. As shown in Table 3, the diameters of Li+ and Mg2+ with the same hydration number seem indeed almost equivalent. But Li+ face much smaller free energy barriers than Mg2+ when entering pores of a particular radius, and the discrimination between them become more obvious as the diameter of SWNTs decrease. For example, in the (8,8) tube, the ΔGMg2+−ΔGLi+≈51 kJ/ mol, which means that it is easier for Li+ to enter the (8,8) CNT. As a contrast, for the (9,9), (10,10) and (11,11) tubes, the PMF differences of Li+ and Mg2+ are 29, 9.7 and 4.6 kJ/mol separately. What cause the differences? We note that previous studies have shown that most of the doubly charged Mg2+ in bulk water has about 6 coordinated water molecules in the first hydration shell [22] and is quite different with Li+ with 4 stable water molecules in the inner hydration shell [23], which leads to the corresponding diameters of Li+ and Mg2+ are 1.18 and 1.44 Å separately [24], and thus the water complexes of Li+ ion have
Fig. 3. Coordination numbers of (a) Li+, (b) Mg2+ permeating each of the carbon nanotubes studied. The tube center is at z ¼7.0.
smaller size than that of Mg2+ ion. Fig. 3 shows the coordination numbers of Li+ and Mg2+ when they enter each of the carbon nanotubes studied during ABF simulations (determined by counting the number of water molecules within 3.8 Å for Li+ and 4.3 Å for Mg2+ separately).The hydration shell of Li+ seem intact in simulated carbon nanotubes except for (8,8) in which the coordination number is down a little. Comparatively, the bigger Mg2+ complexes must be stripped of some water molecule or change geometry of the solvation shell to enter the narrower tubes, and only in the wider (11,11) tubes studied that the hydration shell in the pore can approximate that in the bulk, and the PMF differences of two ions become less obvious. The magnitude of PMF differences has evident effects on ion selectivity [12] and the biologically relevant values for free energy differences between different types of ions are ∼8 kJ/mol [25],
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previous studies have shown that the free energy difference of ∼8 kJ/mol leads to a 10-fold selectivity for K+ over Na+ in potassium channels. From this point, it seems to indicate that the membrane composed of (8,8), (9,9), and (10,10) CNTs with the PMF differences of Li+ and Mg2+ more than 8 kJ/mol are more suitable to separate the two ions.
3.2. Ion transport in CNTs Information about ions conductance through the pores is important to assess the suitability of CNTs for Li+ and Mg2+ separation. To determine this, 100 MPa and 200 MPa hydrostatic pressure differences are created across the CNT membranes and the number of ions that pass across them during our simulations are counted, as shown in Table 4. From the results, under 100 MPa pressure, no ions pass through the (8,8) SWNT membranes in our simulations, one Li+ crosses (9,9) SWNT, while no Mg2+ permeate. In the remaining situations, conductance is greater in the wider tubes than the narrow ones, and not surprisingly, more Li+ permeate through the membrane than Mg2+, but the preferential conduction of Li+ seems to be less distinct in lager diameter (11,11) SWNTs than in (10,10) SWNTs. Table 4 Numbers of ions and water molecules passing through the membranes within 10 ns under different hydrostatic pressures. Size
100 MPa +
(8,8) (9,9) (10,10) (11,11)
200 MPa
Li
Mg
0 1 5 7
0 0 1 5
2+
H2O
Li+
Mg2+
H2O
2366 3442 4492 5790
1 3 23 35
0 1 4 26
4740 6802 8934 11470
As a contrast, under 200 MPa pressure, only one Li+ permeates through the (8,8) SWNTs, and both Li+ and Mg2+ transport through (9,9), (10,10) and (11,11) SWNT membranes. Unlike the conductance of water under 200 MPa which is double that under 100 MPa, and in other studies the conductance is found to increase linearly over the range of pressures from 5 to 400 MPa [26], the conductance of ions increase drastically following no obvious rules. Compared with the water molecules [26], ions face much higher energy barrier when enter sub nanometer CNT pores. The higher hydrostatic pressure help to overcome energy barriers and obtain increasing ion flux. From data in Table 4, the rising ratio of ion flux with pressure change from 100 to 200 MPa is more evident for larger diameter SWNTs, like in (9,9) SWNT, the throughput of Li+ increase from 1 to 3, while in (10,10) and (11,11) SWNTs, the permeation number increases 4.6 and 5 times respectively; the flux of Mg2+ in (10,10) and (11,11) SWNTs with changing pressure also rise 4 and 5.2 times separately. The lower PMF value of ions into wider pores are more easily crossed by promotion of hydrostatic pressure. These results show that, in our 10 ns MD simulation time duration, the SWNT membranes seem to act as ion sieves to thoroughly separate Li+ from Mg2+ by applying 100 MPa pressure across (9,9) SWNT membrane or applying 200 MPa pressure across a (8,8) SWNT membrane, that different hydrostatic pressure conditions need different sized CNTs. The (10,10) tube under 100 Mpa and (9,9), (10,10) tubes under 200 Mpa also can discriminate between Li+ and Mg2+ with higher ion fluxes, which are in accord with that at least ∼8 kJ/mol PMF differences of Li+ and Mg2+ are required to separate the two ions, and the (11,11) SWNTs with 4.2 kJ/mol PMF difference cannot extract Li+ effectively, no matter under 100 or 200 MPa hydrostatic pressure. We also note that Li+/ Mg2+ selectivity of (9,9) SWNT is not better than (10,10) SWNT with lower PMF difference, which may be partly caused by larger energy barrier to Li+ entrance and limited sampling in our 10 ns simulation.
Fig. 4. Axial density profiles of Li+ (line with bare triangle), Mg2+ (line with bare circles) and water (line with solid squares) for SWNTs (a) (8,8), (b) (9,9), (c) (10,10) and (d) (11,11) under 100 MPa hydrostatic pressure as a function of z axis where the SWNTs lies from z ¼ 0 to z ¼ 14 Å.
D. Yang et al. / Journal of Membrane Science 444 (2013) 327–331
The ensemble-averaged axial density profiles plotted in Fig. 4 provide a detailed picture of Li+ and Mg2+ distribution in the proximity of the carbon walls under 100 MPa hydrostatic pressure. From Fig. 4, the ions density is evidently higher in area near the entrance of carbon nanotubes, especially the narrower tubes like (8,8) and (9,9). The asymmetry seems to be caused by hydrostatic pressure across the membranes and the ions accumulate before the opening due to high energy barrier to permeate the tube. Fig. 4 shows that under 100 MPa pressure difference, there are no ions in (8,8) tubes; in (9,9) tubes, the Li+ ion is visible in contrast to Mg2+, which is still zero; in (10,10) tubes, the mean axial density of Li+ is about 4.5 times of Mg2+, while in (11,11) tubes the density distribution of Li+ is only slightly higher than Mg2+. The facts above evidently display the separation process of Li+ and Mg2+. Although most reverse osmosis membranes operate under pressures of less than 10 MPa, our simulations are conducted under a larger 100 or 200 MPa pressure difference as this allows for a greater number of conduction events to be observed in our limited simulations time, and we think it is possible to achieve similar results using much lower pressure by choosing membranes composed of wider SWNTs which are charged or functionalized that have appropriate PMF difference for two ions, or by waiting longer times. In addition, carbon nanotubes are the strongest and stiffest materials yet discovered in terms of tensile strength and elastic modulus respectively, and standard single-walled carbon nanotubes can withstand a pressure up to 24 GPa without deformation, so, if appropriate polymer or inorganic matrix that seal carbon nanotubes is applied and properly fabricated, the membrane can bear much higher pressure than traditional RO membranes. 4. Conclusions Using membranes comprising sub nanometer diameter carbon nanotubes as models, we have performed extensive MD simulations to examine their potential selectivity to Li+ and Mg2+ and their relationships to pore size. We find that narrow carbon nanotubes like (8,8), (9,9) and (10,10) SWNTs with free energy difference more than ∼8 kJ/mol to Li+ and Mg2+ can provide a promising way to separate the two ions which are supported both by free energy calculations and by simulations under hydrostatic pressure. Under 100 MPa, (9,9) and (10,10) SWNTs are capable of extracting Li+; while under 200 MPa, (8,8), (9,9) and (10,10) tubes all can separate Li+ and Mg2+ with higher ion fluxes. Through cautiously choosing the SWNT size and the pressure across the membrane, Li+ can be extracted from the bulk. Therefore carefully fabricated SWNT membranes can act as Li+ and Mg2+ separator that we hope can be tested experimentally in the future. Acknowledgments Financial support of this research was provided by the National Basic Research Program of China (2009CB623402) and the National Natural Science Foundation of China (No.21006100).
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