Molecular Dynamics Simulations of Ion Transport in Carbon Nanotube

Jan 9, 2015 - *Phone: (254) 710-2576. ... For the smallest tubes studied, the energy penalty to access the pore interior is too great for most ions, l...
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Molecular Dynamics Simulations of Ion Transport in Carbon Nanotube Channels Olga N. Samoylova, Emvia I. Calixte, and Kevin L. Shuford* Department of Chemistry and Biochemistry, Baylor University, One Bear Place #97348, Waco, Texas 76798-7348, United States S Supporting Information *

ABSTRACT: We present a computational modeling study examining ion transport dynamics of aqueous electrolytes under severely confined conditions. Ionic current and solvent transport through carbon nanotubes in an external electric field are studied using all atom molecular dynamics simulations. Specifically, we have examined the behavior of sodium and chloride ions in nanotubes of different radii to assess the influence of confinement on the ionic current. We find a linear relationship between the current computed and potential applied for the wider nanotubes; however, there is a significant departure from linearity when the tube diameter becomes comparable to the size of the solvated ion. For the smallest tubes studied, the energy penalty to access the pore interior is too great for most ions, leading to minimal current. We provide analyses of the energy barriers associated with ion entry as well as the hydration shell properties, which supports the absence of ionic current in the smallest carbon nanotubes.

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INTRODUCTION The exceptional properties of carbon nanotubes1 (CNTs) have garnered significant attention in recent years. The ability to tune the size and material properties of CNTs makes them attractive components for nanodevices. Broad investigations have led to a number of practical applications ranging from novel subnanometer supercapacitors,2 hydroelectric voltage generation, power converters,3 transformative technologies for chemical separations and desalinations of water,4 to a variety of biomedical applications such as nanoscale pipes,5 devices to mimic fluidic transport in biological channels,6 water pumps,7 carbon nanotube membranes in microfluidics to control electro-osmotic flow,8 electrophoretic transport of RNA9 and DNA,10 drug delivery,11,12 etc. In particular, there have been a number of computational studies on water transport through CNTs that provide fundamental insights into confined water behavior and explore the extent that environment can alter this behavior. Examples include research on water conduction through the hydrophobic CNTs,13 solvent kinetics of filling and emptying of CNTs,14 water flux through modified and unmodified CNTs,15 and water ordering in the nanotubes.16,17 However, in many physical systems, the fluids of interest are not pure water, but solutions whose collective behavior under confinement can be drastically different. A number of factors contribute to the discrepancies, and detailed computational studies are often required to obtain a clear understanding of the observed physical and chemical phenomena.18−20 In this work, we have concentrated on ion transport behavior in a hydrophobic environment, a pristine CNT. We utilize a set of seven different armchair carbon nanotubes 40 Å in length with radii ranging from ∼4.1−8.2 Å to provide varying degrees of confinement.21 The CNTs have been solvated in a water box with a 1 M concentration of NaCl. The simulated system, © 2015 American Chemical Society

comprised of a carbon nanotube, ions, and the water box, is shown as Figure 1. Detailed system information including the carbon nanotube radius as well as the number of carbon atoms, sodium and chloride ions, and water molecules can be found in Table 1. We studied ionic current in these systems in the presence of an external electric field to ascertain when confinement significantly affects ion transport through the nanochannel. We have examined ion permeation events from the bulk fluid area into the nanotube and determined that transport strongly depends on the extent of confinement induced by the CNT and the water structure inside the channel. These findings are substantiated by a free energy analysis for ion permeation and an ion hydration shell analysis, which support our computational results and provide further mechanistic clarity.

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SIMULATION METHOD AND MODELS We used all atom molecular dynamics (MD) to simulate ion transport through CNTs of various radii in the presence of an electric field. All MD production runs have been performed using the program NAMD2.922 with the CHARMM36 force field.23 Armchair carbon nanotubes were constructed using the tcl code6 with the C−C bond equilibrium distance set to 1.42 Å and an equilibrium angle between carbon atoms of Θ0 = 120°. The following parameters for the C−C intramolecular interactions were employed: stretch constant kstretch = 305 kcal/mol·Å2, anglestretching constant kangle = 40 kcal/mol·rad2, dihedral constant kdihedral = 3.1 kcal/mol, equilibrium torsion angle ϕ0 = 180°, and Lennard-Jones parameters for the carbon−carbon interaction σC−C = 3.55 Å and εC−C = 0.07 kcal/mol. Each tube was solvated Received: October 14, 2014 Revised: December 15, 2014 Published: January 9, 2015 1659

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temperature at 300 K, and the extended system pressure method was used to maintain a constant pressure of 1 atm.

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COMPUTING IONIC CURRENT IN THE NANOTUBE We computed the ionic current over the pore by counting every ion inside the carbon nanotube passing the CNT center of mass: I(t + Δt /2) =

1 Δtl

N

∑ qi[ξi(t + Δt ) − ξi(t )] i

(1)

where l = ztop − zbot is a nanotube length, ztop and zbot are the zcoordinates of the top and bottom positions of the CNT, N is the number of ions, qi is an ion charge, zi is the z coordinate of the ion, Δt is the time between trajectory frames, and ⎧ zi(t ), if z bot ≤ zi(t ) ≤ z top ⎪ ⎪ ξi(t ) = ⎨ z bot , if zi(t ) < z bot ⎪ ⎪ z top , if zi(t ) > z top ⎩

(2)

The first 2 ns of the trajectory was not included when calculating the mean value of the current to allow the system time to reach the steady-state regime.

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POTENTIAL OF MEAN FORCE FOR ION ENTRY Steered molecular dynamics (SMD)25 has been employed to define the ion permeation pathway. We then applied the adaptive biasing force (ABF) method26−28 to generate quasi-equilibrium trajectories for ions and quantitatively evaluate the potential barrier for the ions to enter the confinement of the tube from the bulk water area. A number of factors affect the error in the ABF method, as has been discussed in the literature previously.27,28,30 In the ABF method, the force Fξ is accumulated in small windows of finite size δξ along a reaction coordinate ξ to provide an estimate of the free energy derivative. The free energy of a state at a value of ξ along the reaction coordinate is 1 A(ξ) = − ln Pξ + A 0 β (3)

Figure 1. VMD screenshot of the simulated system containing a carbon nanotube solvated in water with sodium (cyan) and chloride (yellow) ions. Water is rendered here with a space-filling representation. An electric field is applied along the z direction of the CNT.

Table 1. Carbon Nanotube Systems of Interesta (n,m)

radius, Å

carbon

ions

water

(6,6) (7,7) (8,8) (9,9) (10,10) (11,11) (12,12)

4.09 4.76 5.44 6.12 6.79 7.47 8.15

480 559 640 720 800 880 960

36 34 34 32 30 30 28

932 898 857 839 796 752 714

where Pξ is the probability density to find the system of interest at reaction coordinate ξ, β = (kBT)−1, kB is Boltzmann’s constant, T is temperature, and A0 is an additive constant. The first derivative of free energy by the reaction coordinate can be written in terms of configurational averages as27,29

a Columns are labeled by CNT chiral vectors (n,m), CNT radius, the number of carbon atoms in the CNT, and the number of ions and water molecules, respectively. The tube length is 40 Å in all cases.

d A (ξ ) = dξ

by a 1 M NaCl solution in an orthorhombic simulation box with dimensions 24 Å × 24 Å × 70 Å. Flexible TIP3P water molecules24 have been used for the simulations. Carbon atoms of the tube were fixed to prevent the motion of the tube. The tube was aligned along the z-axis, corresponding to the long axis of the hexagonal system. The system was minimized and then equilibrated for 5 ns at 300 K using the NPT ensemble and a 1 fs time step. Periodic boundary conditions were applied in three dimensions, and the Particle Mesh Ewald (PME) summation was used with a grid size of 1 Å for long-range electrostatic interactions. The van der Waals interactions were calculated with a cutoff 12 Å. The system was biased by an external electric field in the z-direction, with voltage drops ranging from 0.2 to 1 V. Production MD runs of 50 ns including the biasing electrical field were performed on every system. A Lowe−Anderson thermostat was used to control the

∂U (x) 1 ∂ ln|J | − ∂ξ β ∂ξ

= −⟨Fξ⟩ξ ξ

(4)

where U is potential energy, |J| is the determinant of the Jacobian for the inverse transformation from generalized to Cartesian coordinates, and ⟨Fξ⟩ξ is the average force acting along the reaction coordinate. The ABF force applied along the reaction coordinate to overcome free energy barrier is F ABF = ∇x à = −⟨Fξ⟩ξ ∇x ξ

(5)

where à and ⟨Fξ⟩ξ are the current estimate of the free energy and average of Fξ, respectively.26,27,30 The value of ∇xà is updated as sampling proceeds. First, we applied a SMD force (with force constant k = 5 kcal/ mol·Å) to the sodium or chloride ion in the −z direction to pull it with a constant speed of 10 Å/ns in a 5 ns MD simulation to define the reaction coordinate for the ion pathway along the 1660

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Figure 3 shows the maximum ionic current passing through CNTs of various radii biased by 1 V after 50 ns of MD simulation.

CNT channel. Additional restraints on the ions were added to limit their motion to only inside the nanotube cylindrical confinement. Next, we used the ABF methods in NAMD31 to calculate the potential of mean force (PMF) for the ion entering the nanotube. The reaction coordinate was chosen as the distance between the permeating ion and the center of mass of the CNT. We were interested in the energy barrier between the bulk water and the area strongly confined within the nanotube, so the reaction coordinate in the interval 10 ≤ z ≤ 30 was divided into four windows each 5 Å wide. From the SMD trajectory obtained earlier, we chose a different starting point for every window with the initial position of the ion in the bottom of each window to run unconstrained MD simulations for 5 ns. The results from the four windows were then combined to obtain the PMF for ion entry into the CNT.

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RESULTS AND DISCUSSION All atom MD simulations have been performed for 50 ns with an electric field applied in the z-direction, E = 23.045 (Vz/Lz) kcal/ mol·Å·e. Lz is ∼60 Å, which corresponds to the range of z values of the relaxed system after the equilibration. Figure 2 displays the

Figure 3. Current dependence on nanotube radius for CNT systems: (8,8), (9,9), (10,10), (11,11), (12,12). Values correspond to mean current attained for a 1 V bias after 50 ns of simulation.

A value of I = (0.96 ± 0.02) nA was computed for the (12,12) tube. The current for the (11,11) CNT is almost 50% less, and for the (10,10) case, the current drops to ∼40% of the value achieved in the widest CNT. There is a dramatic drop off in current for CNTs with a radius 23 Å) into the tube (z < 23 Å). We can see that the free energy barrier for the sodium ion in group I is ∼(1.5−2) kcal/mol [(6.3−8.4) kJ/mol], where first value is the barrier at the pore entrance and the second value is the PMF value inside the pore. In the group II CNTs, the barrier for sodium is (1.2−1.7) kcal/mol [(5−7.1) kJ/mol], but it is significantly larger in the group III CNTs. The computed values

Figure 2. Mean value of ionic current through CNTs after 50 ns versus external electric field. The traces are labeled by nanotube chiral vectors.

ionic current for the different CNT systems as a function of electric field. Applied voltages ranged from 0.2−1 V with the step 0.2 V. The I(V) curve for the widest tube (radius 8.15 Å, (12,12) CNT) shows essentially linear behavior consistent with Ohm’s law, indicating that the confinement effects are not substantial for this system. The equivalent current−voltage analysis for the electrolyte in the absence of a CNT is also linear with an effective resistance that is approximately a factor of 4 less (see Supporting Information Figure S1). By decreasing the tube diameter slightly, deviations from linear behavior begin to emerge for the (11,11) CNT with a 7.47 Å radius and with an effective radius 5.97 Å. The effective radius is the radius of nanotube that is geometrically accessible to solvated ions and water molecules (rCNT − rC, where rCNT is the radius of CNT and rC is the radius of carbon atom). This trend continues as the CNT diameter is decreased for the (10,10) case, showing a further decline in current passing through the CNT. The ionic current passing the (9,9) and (8,8) CNTs was almost negligible; tube confinement results in near complete suppression of current for these systems. We did not register any current for the two smallest tubes studied, the (6,6) and (7,7) CNTs with effective radii of ∼2.6 and 3.3 Å. 1661

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Figure 4. Free energy profile obtained from unconstrained MD simulations with adaptive biasing force. Top: PMFs for the ions in group I CNTs, which pass significant ionic current. Middle: PMFs for group II systems with small currents. Bottom: PMFs for group III systems with no current. The left column is for a sodium cation; the right column is the chloride anion.

(8,8) CNT (2.6−3.8) kcal/mol [(10.9−15.9) kJ/mol]; group III, (7,7) CNT (4−6) kcal/mol [(16.7−25.1) kJ/mol], (6,6) CNT (9−23) kcal/mol [(36−96) kJ/mol]. These values are generally consistent with previous studies on the free energy penalty for (6,6), (7,7), and (10,10) CNTs.32,33

are about (2−5) kcal/mol [(8.4−21) kJ/mol] for the CNT (7,7) and (14−21) kcal/mol [(59−88) kJ/mol] in the (6,6) nanotube. For negatively charged Cl− ion, the energy barriers are the following: group I (0.6−1.3) kcal/mol [(2.5−5.4) kJ/mol]; group II, (9,9) CNT (1.4−2) kcal/mol [(5.9−8.4) kJ/mol], 1662

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To explore the effects of molecular interactions and hydration in the wider CNTs, we did a hydration shell analysis for the ions inside the wide channels of group I. Figure 6 presents the radial distribution functions of the oxygen and hydrogen atoms from water around the ions within group I CNTs. g(r) is the probability to find the water oxygen or water hydrogen at a radial distance r from the ion, normalized by the total number of reference ions. The positive sodium ion (left panel) is hydrated primarily by the oxygen atoms in water, with the first oxygen peak centered at ∼2.2 Å and hydrogen peak at about 3 Å. For the negative chloride ion (right panel), the situation is reversed with the anion being hydrated primarily by hydrogen due to water rotations resulting from the electrostatic attraction. Barring the peak ordering, the radial distribution functions do not show any major qualitative difference between structural characteristics of the hydration shells for the anion and cation. For both ions, all of the distribution functions show two well-defined hydration shells, where the inner shell is notably more structured than the outer shell, as indicated by a more intense/narrow distribution peak. A more subtle effect is the slight contraction of hydration in smaller CNTs for sodium, where the peaks in g(r) for wateroxygen and water-hydrogen become more intense and narrower with a decrease of the nanotube radius. To further understand the relationship between ionic current and energy barriers, we examined the hydration behavior of the sodium cation in the widest nanotubes of groups I and II. These systems were chosen to display the effects of confinement, while still allowing the ions to pass (i.e., ionic current was found in the simulations). Also included for size comparison are results for the chloride ion in the widest (12,12) CNT. The analysis is limited to this subgroup because if we partition the current from the production runs by ion, 82% is due to sodium and 18% is from chloride in the (12,12) CNT (See Supporting Information Figure S2), while the contribution of chloride ions to the ionic current in a (9,9) CNT is approximately zero. Coordination numbers were determined by integrating the ion-O radial distribution functions out to the first minimum, approximately 3.0 Å for the sodium and 3.8 Å for the chloride. We found the noteworthy differences in the energy barriers for the wide (12,12) and narrow (in this analysis) (9,9) tube correlated with ion coordination number in the first hydration shell. Very often the ion has to shed water molecules from its hydration shell to be able to penetrate into the narrow tube. This effect is displayed more clearly in Figure 7 and Table 2, which show the coordination number statistics (see Supporting Information Figure S3 for second shell data). For the widest tube, the majority of sodium ions have coordination number 5 or 6, which is similar to cation hydration in the bulk water. However, for smaller tube (9,9), the vast majority of sodium ions have 5 water molecules in the solvation shell. As a comparison, the larger chloride ions in the (12,12) CNT favor a coordination number of 7 or 8, which is similar to to the value found in bulk. We also found a large discrepancy in the amount of time ions spend inside the tube as confinement effects become substantial. For example, every sodium ion sampled the interior of the largest nanotubes but only ∼63% spent time in the smaller (9,9) CNT; the vastly preferred region for the latter was in the bulk water outside of the tube. We can quantify this general statement by reporting the average time ions reside within the nanotube as a percentage of the total simulation time (⟨t⟩ in Table 2). For the (9.9) CNT with radius 6.12 Å, sodium cations spent only 0.86% of the total 50 ns simulation time inside the channel, and the ionic current is almost negligible in this system. For the wider

The general trend observed is an increase in the free energy barrier as the tube diameter decreases, as one would expect. This is more clearly displayed by the chloride (right column) but is followed within method accuracy for both ions. In particular, the PMF to enter pores notably larger than the solvated ions is only about 1.5 kcal/mol, while the energy penalty to migrate deep within the smallest CNT examined is in excess of 21 kcal/mol. These larger energy barriers deter ions from entering the pore and contributing to the current for group III CNTs. We note that the barriers for the sodium ion in group I and group II are similar; however, the ionic current in group II is much lower than that in group I. This highlights the fact that the computed barrier values are sensitive to the interactions and collective effects along the chosen reaction coordinate. Similar studies on ion conductance in larger, biological channels have shown that ion correlation effects can be considerable.34,35 It is likely the case that the barrier for the wider tubes is even lower than predicted, because the value reported represents the mean force distribution from only one target ion that has been placed at various positions along the z axis of the nanotube. In reality, several positive and negative ions are present in the wider tubes at the same moment during the production run. This can clearly be seen in Figure 5, which shows a snapshot of water and ions inside

Figure 5. Snapshot of water and ions (sodium ions are indicated with cyan spheres; chloride ions are yellow spheres) confined within (9,9)left and (12,12)-right nanotubes. Water molecules and ions in the bulk area are not shown.

the (9,9) and (12,12) CNTs. The wider group I CNTs naturally have a noisier PMF because of less restriction of the accessible space in {x,y} plane and numerous interactions with other ions inside the tube, which can both lead to more variability in computed barriers. Also, the extent of hydration of a particular ion varies with the degree of confinement induced by the CNT, which will also affect the PMF, as will be discussed in more detail below. 1663

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Figure 6. Radial distribution functions of the water-oxygen and water-hydrogen atoms around the ions in the wide CNTs.

nanotubes. Here, we present sodium data, as it is the primary contributor to the current in these systems, but a similar analysis could be presented using chloride in slightly larger nanotubes. Figure 8 shows the average time sodium ions with coordination numbers 5 and 6 spend inside the (9,9) and (12,12) CNTs at certain radial positions. Comparing the most frequent radial position of the sodium ion with coordination number 6, there are two clear preferred radial positions in the (12,12) tube (r ≈ 0.5 Å, r ≈ 3 Å), whereas in the smaller (9,9) tube, there is only one peak at r ≈ 1.2 Å. The geometrical confinement induced by the smaller CNT does not support multiple “streams” of ions passing concurrently. So by going from the (12,12) to the (9,9) systems, the exterior distribution located near the tube wall is effectively eliminated, and the ions that were passing in the channel center are redistributed to slightly longer radial positions. We also note that the ions with smaller coordination numbers are strongly favored near the nanotube walls, which is particularly important as the tube diameter and accessible cross-sectional area decrease. For example, the majority of ions with coordination number 6 do not come closer to the nanotube wall than ∼5.2 Å in (12,12) CNT and ∼4.9 Å in (9,9) CNT. This is depicted schematically in Figure 9, where R is a radius of CNT, r is peak position from the ion distribution for the (9,9) system and the second peak position for (12,12) system, and Δr is a critical distance. The shaded areas indicate regions more accessible to sodium cations with coordination number 6 and correspond to ∼4% of CNT

Figure 7. Coordination number statistics of water molecules around sodium and chloride ions confined within different CNTs.

(12,12) CNT with a radius of 8.15 Å, the largest ionic current is obtained. In this case, sodium spent an average 23.03% of the time within the nanotube, and chloride ion spent about 16.76% of the time in the tube. We can further analyze the preferred locations and observed hydration properties of ions to connect energy barriers to current by examining the time spent at a particular radial position within 1664

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Table 2. Percentage of Sodium and Chloride Ions Inside the Tube with Different Coordination Numbers, and Average Time Ions Spend in the Tube as a Percentage of the Total Simulation Time coord. number Na+ (9,9) Na+ (12,12) coord. number Cl− (12,12)

1 0 0.43 ≤4 0.77

2 0 1.55 5 3.76

3 0 5.17 6 13.88

4 5.05 19.45 7 31.81

5 66.5 36.12 8 31.98

6 27.81 37.17 9 14.22

7 0.63 0.55 ≥10 3.59

⟨t⟩ % 0.86 23.03 ⟨t⟩ % 16.76

Figure 8. Average time sodium ions with coordination numbers 5 and 6 spend inside nanotubes at certain radial positions for the (9,9) and (12,12) CNT systems.

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CONCLUSIONS Our simulation results show that the ionic current through a pristine carbon nanotube strongly depends on the radius of the CNT. We showed a direct correlation between CNT channel radius, water structure in the tube, and ion permeation events. The energy barrier for ions to enter the nanotube increases dramatically for smaller CNTs. Ions lose water molecules to be able to permeate from the bulk region into the narrow tubes, until the energy penalty to do so becomes prohibitively large. The wider channels provide more opportunities for ions to leave the bulk region and enter the tube without any change in their hydration shell. As a result, permeation events occur more frequently for the wider tubes with radius 6.8−8.2 Å, where ions only have to overcome a barrier about 1.5−2 kcal/mol. Conversely, the narrow pores with radius 4.1−4.8 Å and energy barriers up to ∼20 kcal/mol deter ionic current from passing through the pore. From this, we can deduce the conditions for successful ion permeation into severely confined environments. These results are of fundamental importance and have broad applicability to a number of scientific areas including electrical energy storage, microfluidics, separations, filtration, and biological transport processes.

Figure 9. Schematic illustration of the allowed cross-sectional area inside the CNT for sodium ions with coordination number 6.

volume for the (9,9) system and ∼14% in the (12,12) system. From simple geometric considerations, this suggests that the sodium ion in the wide CNT has a significantly larger chance of leaving the bulk region and entering the tube without any changes to its hydration shell. Moreover, to access spatial regions closer to the CNT walls, corresponding to most of the tube volume, the ion must decrease its hydration shell. As the tubes continue to get smaller, the only accessible volume becomes this critical region close to the walls, and a decrease in water coordination is required to enter the pore, which can occur until the energy penalty to do so becomes too great. This interpretation is supported by the coordination data in Table 2 and the average times (vertical axis) in Figure 8, which are an order of magnitude greater for the (12,12) system than for (9,9) CNT. Thus, as nanotubes become smaller and confinement is substantial, both the statistics and the energetics work against ion migration, and, as a result, the current decreases dramatically with CNT diameter.

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ASSOCIATED CONTENT

S Supporting Information *

Several additional plots. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*Phone: (254) 710-2576. Fax: (254) 710-4272. E-mail: kevin_ [email protected]. Notes

The authors declare no competing financial interest. 1665

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ACKNOWLEDGMENTS This work is supported by the Chemical Sciences, Geosciences, and Biosciences Division, Office of Basic Energy Sciences, Office of Science, U.S. Department of Energy, under award number DESC0010212. K.L.S. thanks Baylor University for start-up funding.

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