Molecular Dynamics Simulation of Salt Rejection in Model Surface

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Molecular Dynamics Simulation of Salt Rejection in Model Surface-Modified Nanopores Jacob Goldsmith and Craig C. Martens* Department of Chemistry, University of California, Irvine, Irvine, California 92697-2025

ABSTRACT This Letter describes molecular dynamics simulations of pressureinduced flow of water and aqueous salt solutions through model nanopores. The systems studied are comprised of (n,n) carbon nanotubes (CNT) that span a membrane constructed of parallel graphene walls separating two solution reservoirs. We employ this system as an idealized model of surface-modified nanoporous membranes, and thus, both native hydrophobic CNT and nanotubes with artificial surface partial charge patterns are considered. The dependence of the fluxes of water and ions on the nanopore size, nanopore charge patterns, and pressure difference are explored using nonequilibrium molecular dynamics simulation. We demonstrate size- and structure-dependent salt rejection and show evidence of salt flux rectification for our asymmetric nanopore model. SECTION Statistical Mechanics, Thermodynamics, Medium Effects

T

he transport of water and aqueous solutions through nanoporous materials is a process with relevance to a range of applications, including nanofluidic pumps, reactors, sensors, lab on a chip systems, membrane protein channels, reverse osmosis, and water purification. Nanoporous membranes incorporating carbon nanotubes have received considerable attention, both theoretically and experimentally, as model systems with well-defined structures that exhibit novel and desirable properties, including high tensile strength, well-defined nanoscale structure, controllable electronic conductivity, and potential applications in separations science and medicine.1-3 Fluid flow in carbon nanotubes and nanopipes was recently reviewed by Whitby and Quirke.4 Hummer and coworkers have considered water flow in (6,6) single-walled carbon nanotubes (SWCNT).5 These systems exhibit rapid concerted transport of aligned water in a single-file fashion, a mechanism that is similar to flow in biological pores. Experiments have also probed the properties of water inside larger diameter carbon nanotubes.6,7 Giovambattista et al. investigated hydration of confined water by nanoscale surfaces with patterned hydrophobicity and hydrophilicity.8 Maniwa et al. observed the formation of pentagonal to octagonal ice structures in carbon nanotubes,9 and as an extension of that work, a phase diagram for water in carbon nanotubes has been proposed.2 It is believed that the structure of carbon nanotubes closely mimics some biological pores due to their similarity in size and hydrophobic nature.10 Of these, SWCNT have been suggested as analogues to biological pores through lipid bilayers,11 aquaporins,12 and other transmembrane channels.13 Several groups have investigated the dynamics of water permeation through the membrane channel protein aquaporin.14-17 Here, a narrow hydrophobic passage allows facile single-file transport of water molecules through the core

r 2009 American Chemical Society

of the protein. The filling of a cavity by water has been shown to be highly sensitive to the size of the cavity and the strength of the interaction between water and the cavity walls.18 Wettability has been shown to be highly correlated with the physical nature of microstructures and nanostructures on a surface.19,20 These investigations highlight the profound effects tha.t nanoscale size and structure can have on the properties of liquid water. Ionic solute dynamics inside of nanotubes have also been studied. For example, Majumder et al. prepared aligned SWCNT, measured the flux of a large cation, and stressed that tip functionalization can affect the occupancy of a pore.21 It was suggested that a strong correlation exists between ions and water inside of a confined pore and that possible charge density variations22 and hydrophobic gating mechanisms23 of the channel mediate or impede water and ion transport. Salt solutions in nanopores have also been investigated theoretically by Hansen and co-workers.24 In these studies, the ability of ions to enter water-filled pores can be affected by external electric fields, suggesting nanoscale control of solutes via electrical field modulations.25 Yang and Garde modeled the selective partitioning of cations into negatively charged nanopores in water.22 In pores, the nanoscale functionality is often characterized as hydrophilic or hydrophobic, and this functionality influences the flow profile of solutions through a pore.26 The Hagen-Poiseuille (HP) equation of laminar incompressible flow was shown to give a poor representation of Ar atom transport in a cylindrical pore.27 It is believed that liquids do

Received Date: October 22, 2009 Accepted Date: December 18, 2009 Published on Web Date: December 29, 2009

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DOI: 10.1021/jz900173w |J. Phys. Chem. Lett. 2010, 1, 528–535

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not follow the HP equation in hydrophobic pores on the nanoscale.4 Other work has shown that water in a hydrophobic carbon nanotube exhibits “plug”-like flow.28 In the work of Holt et al., transport of water through carbon nanotube pores was shown to be greater than continuum hydrodynamic model predictions,29 which is an advantageous property for large flux nanofluidic applications. Studies related to desalination attempt to classify nanofiltration technologies in terms of variations in solute and solvent viscosity, dielectric constant, and radius of the pore.30,31 These studies, however, are based on commercial membranes and do not attempt to modify salt flows through material design at the molecular scale but rather to classify existing technology for ion rejection. It would be of interest to actively control pore surfaces to alter occupancies and flux rates of ionic species for separations. An important paradigm for nanoscale separation and transport is the Brownian ratchet, which refers to a class of systems where asymmetric structure, driving force, and outof-equilibrium fluctuations result in net directed transport. Generally, transport can occur only if the fluctuations, driving forces, or nonequilibrium spontaneous fluctuations have broken symmetry. These systems were conceived from the description first presented by Feynman.32 A critique of Feynman's ratchet is discussed by Parrondo,33 but the idea has remained as inspiration for various forms of separations and mechanisms of transport. A recent review discusses theoretical, experimental, and practical aspects of rectification via Brownian ratchets and motors.34 Parrondo analyzes the efficiency of such systems.35 Other asymmetric systems have been studied in the context of device applications.21,25,36-38 Further work in this field is needed to realize the technological promise of this paradigm. In this Letter, we investigate nonequilibrium solution flow in nanopores. We perform molecular dynamics simulations39,40 of water and salt solution transport through surface-modified nanopores under externally applied pressure. The pores act as models to study the effects of asymmetric charge patterns on transport through nanopores driven by an externally imposed pressure difference across a membrane. Simulations are performed using NAMD and VMD for molecular dynamics and analysis, respectively.41,42 We use the CHARMM 22 force field with a rigid TIP3P water and a slight modification of aromatic carbon “CA” for the nanotube and graphene in our simulations (see ref 10 for a DFT study and parametrization). Molecular species are treated as rigid, and a time step of 2 fs is used in all simulations. Periodic boundary conditions (PBC) and particle mesh Ewald (PME) electrostatics calculations are used to take into account finite system size and the electrostatic interactions. The systems are first equilibrated under a constant pressure of 1 atm and then used as input for nonequilibrium simulations under a pressure difference across the membrane. Nonequilibrium simulations are then carried out for constant particle number and system volume. A Langevin thermostat is used to control temperature. Typical simulation times are 2 ns, with data output every 500 time steps for subsequent analysis. The gradient in pressure across the membrane is quite large in the


Figure 1. Typical charge patterns employed for the surfacemodified model nanopores. Partial charges are added to selected carbon atoms, as indicated in the figure by large radius spheres in light and dark blue for positive and negative partial charges, respectively. The graph above each pore diagram indicates the total charge on the corresponding ring of charged carbon atoms below. The left case consists of rings of charge with equal magnitude but an unequal spacing along the pore (denoted as the AR system). On the right, the case with equally spaced rings of charge with increasing magnitude along the pore is depicted (denoted as the AC system).

simulations reported here, reflecting the limitations in length and time scales typical of molecular dynamics simulations.39,40 Our model systems are constructed from two graphene sheets comprising a membrane spanned by a carbon nanotube of variable size, all of which are held rigid during the simulations. The size of the graphene wall is 24 37 carbon atoms or about 5.0 4.4 nm, which is then repeated periodically in the (x,y) plane by the PBC. The opening of the slab is modified slightly to accept the nanotube by moving graphene carbon atoms slightly in relation to the pore opening. We treat nonchiral “armchair” nanotubes characterized by chiral vector (n,m) with n = m.1 Nanotubes are considered with (n,n) = (8,8), (10,10), (12,12), (14,14), and (16,16). Each nanotube is 30 carbon atoms long, and the width of the simulation box in the z direction parallel to the pore is approximately 6 nm. The pore radii range from approximately 0.3 to 0.9 nm, which we consider here to be model pores with controllable surface structure rather than literal carbon nanotubes. Charge patterns are added to the nanotube walls in two asymmetric patterns, as shown in Figure 1. Two classes of surface-modified systems are considered. The first consists of rings of charge with alternating sign located on carbon atoms, incorporated by artificially modifying the partial charges of the atoms on the nanotube wall. The magnitude of the charges on the ring remains the same along the tube, but the spacing between the rings increases with increasing z (Figure 1, left). The second class of systems consists of nanotubes with equally spaced charged rings but with the magnitude of the charge increasing with z (Figure 1, right). The charge magnitude of each ring is the same for each system, but the charge density decreases for larger tubes due to an increase in interior surface area. The total charge of the pore surface is zero, while each individual charge is small (a fraction of e) and is comparable to the polarization of a carbon atom in various molecular environments or in a nanotube.10 The number of water molecules in each simulation varies according to the size of the tube. We equilibrate at constant

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Table 1. Summary of System Properties (see text for discussion) (n,n)

waters

NaCl

diameter [nm]

(8,8)

1593

16

0.69

(10,10)

1638

16

0.96

(12,12)

1689

17

1.2

(14,14) (16,16)

1765 1846

17 18

1.5 1.8

Figure 3. Water flux versus pressure difference ΔP for unmodified nanopores, for a range of diameters. See text for discussion.

Figure 2. A snapshot from a nonequilibrium molecular dynamics simulation of pressure-induced flow of a NaCl solution though a model nanopore constructed from a (16,16) carbon nanotube spanning a membrane formed by two graphene sheets. Carbon atoms are shown in blue, oxygen atoms are red, hydrogens are gray, Naþ ions are yellow, and Cl- ions are green.

(T,P). Each equilibrium simulation has a similar number of waters in the reservoir (to within 1%). The time-averaged density for simulations with water only does not vary considerably from 1.0 g mL-1 from tube to tube, and the density on either side of the membrane is also approximately 1.0 g mL-1. Salt water solutions are constructed with an approximately 100:1 H2O to NaCl ratio, which is similar to seawater concentrations. The density for these simulations is 1.02 g mL-1 (similar to seawater) but varies slightly from tube to tube. The exact number density is determined and is used in the calculations shown below. A brief summary of the systems, number of waters, and number of ions is shown in Table 1, while a typical simulation snapshot is shown in Figure 2. Steady-state solution flow in the z direction is induced by adding a force proportional to the desired pressure to the system. By definition, pressure is applied force divided by area, and this total force is the sum of the external forces acting on molecules in a subset of the system. We calculate the total force F of this subset as X fi ¼ f FAl ð1Þ F ¼

Figure 4. Total ion flux versus pressure difference ΔP for unmodified nanopores, for a range of diameters. See text for discussion.

set to T = 300 K. Flux rates are determined for water molecules and ions passing through a pore by counting as they pass from one end to the other. Back fluxes are negligible at pressures greater than 100 MPa. We compute the flux of water and ions through the nanopore models under pressure-induced steady-state flow as a function of size and pressure for both the unmodified pores and the pores with charge patterns on the pore walls. Water flux versus pressure difference ΔP for the NaCl solution in unmodified nanotube pores is shown in Figure 3. (The results for pure water flow under external pressure are qualitatively similar and not shown.) The flux shows a linear dependence on the pressure difference, and the flux for a given pressure difference increases with pore diameter, as expected. In Figure 4, the corresponding total ion flux is shown as a function of pressure difference for unmodified pores of varying diameter. The concentration of ions is much lower than the water concentration; therefore, the magnitude of the flux is less than that for water. The dependence of the salt flux on the pressure and pore diameter nonetheless qualitatively parallels that of the water flux shown in Figure 3. Recent experiments on water flow through membranes spanned by hydrophobic carbon nanotubes exhibit fluxes

i

where the sum is over the affected atoms within the desired area A, l is the length of the area, and fi f is the identical force applied to each active atom. The force applied in the simulations acts on a slab at the (z boundary with thickness l = 0.4 nm, resulting in a small fraction of the overall molecules feeling the force. The force is not applied to hydrogen atoms to avoid spurious rotational excitation, and therefore, the total force on a water molecule is felt by the oxygen atom. We apply pressure in the þz (forward) and -z (reverse) directions under identical (V,T) conditions, with a Langevin thermostat


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Figure 5. Water flux versus pressure difference ΔP. Simulation results (blue lines) are compared with the predictions of the continuum model, the Hagen-Poiseuille equation, eq 2 (dashed black lines). Four diameters are considered, indicated by the corresponding (n,n) designation.

flow through nanotubes occurs in a manner not consistent with the continuum no-slip boundary condition.28 Our results show velocities an order of magnitude smaller for a given pressure difference than those reported by Hinds et al., who estimate velocities of approximately 20 cm s-1 for pores of 7 nm diameter, but our results are comparable when the difference in diameters is taken into account. We now describe the corresponding behavior of surfacemodified nanopores. We focus on the AR system, characterized by unequally spaced rings of charge of constant magnitude (see Figure 1); the other system exhibits similar behavior. In Figure 6, we show the forward and reverse water fluxes for this family of asymmetric systems as a function of the magnitude of the pressure difference |ΔP|, indicated by blue and red lines, respectively. Also shown are the corresponding (symmetric) fluxes for the unmodified pores (solid black lines) and the continuum predictions of eq 2 (dashed black lines). Despite the asymmetry of the nanopore, the water fluxes are nearly symmetric. The fluxes are systematically less than the flux in the unmodified systems but still significantly higher than the continuum prediction. In Figure 7, we show the total ion fluxes versus |ΔP| for the asymmetric modified pore systems. Forward ion flux is shown in blue, while reverse flux is shown in red. The unmodified pore results, which are symmetric, are shown in black. Error bars displayed are estimated as (1/N1/2 of the flux value, where N is the number of events (e.g., the number of ions transported during the simulation) that determine the flux. For all cases, the salt flux for the modified pores is less than that for the corresponding unmodified nanotube pores.

that are significantly larger than the predictions of the Hagen-Poiseuille (HP) equation of continuum mechanics.3,29 The HP equation for the flux Q is πD4 F ΔP ð2Þ Q ¼ 128μL where ΔP is the pressure difference across the membrane, D is the pore diameter, L is the length of the pore, μ is the dynamic viscosity of water (10-3 Pa s), and F is the density. We compare our results with the predictions of eq 2, keeping in mind that this bulk continuum result will not be rigorously applicable to our nanoscale pore models due to expected deviations from bulk viscosity in these atomistic simulations, water model-dependent effects, boundary effects due to finite pore length, and other factors. Still, this provides a useful qualitative comparison. Water flux results for the unmodified carbon nanotube pore systems from Figure 3 are separately compared with the predictions of eq 2 in Figure 5. For all cases, the simulated flux is several times larger than the continuum prediction. The results here are in qualitative agreement with the results reported experimentally3,29 and with previous simulations of water flux through hydrophobic and hydrophilic pores from this group.43 Radial velocity profiles of the water oxygens and the ionic species are calculated and appear nonparabolic, and in particular, the averaged velocities of the species do not vanish at the wall of the uncharged carbon nanotube (results not shown). These detailed results are qualitatively in agreement with our previous work43 and also support reports that water


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Figure 6. Forward and reverse water flux versus the magnitude of the pressure difference |ΔP| for the AR system (asymmetrically placed rings of equal charge magnitude). Forward flux is indicated by blue lines, and reverse flux is indicated by red lines. The results are compared with the fluxes of unmodified pores (solid black lines) and the predictions of the continuum Hagen-Poiseuille equation, eq 2 (dashed black lines).

Figure 7. Forward and reverse total ion flux versus the magnitude of the pressure difference |ΔP| for the AR system (asymmetrically placed rings of equal charge magnitude). Forward flux is indicated by blue lines, and reverse flux is indicated by red lines. The results are compared with the fluxes of ions for unmodified pores (solid black lines).


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pressure, there is a small but significant water flux of ∼60 molecules per nanosecond, as indicated in Figure 6. When asymmetric charges are added to the tubes, the dynamics are altered in two distinct ways. First, our observed flux rates become more comparable to the predictions of the HP equation. This is due to the stronger interactions between the partial charges on the pore wall and the water molecules, leading to slower water flow velocities at the wall and better conformity of the dynamical behavior with the assumptions underlying the HP equation. Second, the pressure-induced flux for salt becomes smaller, regardless of the applied pressure's direction and magnitude. This can be seen by comparing Figure 4 to Figure 7. This is most simply understood as resulting from electrostatic interactions between the ions and wall charges. The dynamical and thermodynamic effects underlying the overall process evidently also involve enhanced water structuring within the pore due to the charge pattern on the walls and the resulting interference of the passage of solvated ions traveling through the pore with solvation shells that are not easily compatible with this structure. The details of these structural and dynamical effects are currently under investigation. Surface patterning with partial charges leads to a decreased overall flux of water but enhances the selectivity of the system. Importantly, surface patterning allows ion rejection to be achieved for larger pores than is possible by simple size exclusion. Surface modification of nanopores holds promise for the design of systems that allow higher water flux while still exhibiting significant ion rejection. Such systems would allow, for instance, desalination of water with lower energy requirements. In this Letter, we have presented the results of nonequilibrium MD simulations of pressure-induced flow of water and the ionic solutes Naþ and Cl- through model nanopores. The systems studied consist of idealized nanopores composed of (n,n) carbon nanotubes spanning a membrane of parallel graphene walls separating two solution reservoirs. We considered both native hydrophobic nanotubes and systems with artificial asymmetric surface charge patterns. The dependence of the fluxes of water and ions on the nanopore size, nanopore charge pattern, and pressure difference were explored using nonequilibrium molecular dynamics simulation. We demonstrated size- and structure-dependent salt rejection and evidence of salt flux rectification for our asymmetric nanopore model. The hydrophobic nature of the pore produces fast flux for salt water solutions. This is most pronounced for the unmodified systems, but the modified pores continue to exhibit significantly higher fluxes than predicted by the HagenPoiseuille equation of continuum mechanics. It was observed that salt rejection can result from simple size exclusion for very small (8,8) nanopores but that rejection is enhanced for model asymmetric charge patterns. We conclude that rejection can be achieved for structured pores that are significantly larger in diameter than unmodified carbon nanotubes and note the potential implications for the design of new materials for separation. We believe that the underlying mechanism for the observed enhancement of ion rejection is due to both direct electrostatic interactions

Figure 8. Ratio of ion flux to total ion plus water flux for the (12,12) AR surface-modified system. The forward simulations (ΔP > 0) are shown in blue, while the reverse simulations (ΔP < 0) are shown in black. The dashed horizontal line indicates the relative flux expected for transport of solution with the bulk salt concentration. See text for discussion.

The smallest (8,8) and (10,10) systems exhibit nearly complete ion rejection; here, we show only results for the (10,10) system. This is to be contrasted with the unmodified pore case, shown for the (10,10) pore as the yellow curve in Figure 4, which shows a low but significant ion flux over the pressure range considered. For the unmodified CNT system, the very small (8,8) system, with a diameter of under 0.7 nm, was necessary to achieve effective ion exclusion. It is important to note that the water flux, although small, is nonzero for the (10,10) modified pore system. Thus, this case exhibits an enhanced separation of water and salt over the unmodified carbon nanotube of the same size. The (12,12) system shown in Figure 7 also exhibits significant, although incomplete, salt rejection, at least for the lower pressures considered. Evidence for salt rectificationasymmetric transport of salt in the forward versus reverse direction under the same magnitude of pressure difference is also observed. For practical application to separations technology, an important parameter is the ratio of salt flux to the total (salt and water) flux through the pore. Clearly, for a membrane to be useful for reverse osmosis, it must reject salt while allowing water to pass. In Figure 8, the ratio of salt flux to total flux is shown for the (12,12) asymmetric AR-modified system. The water flux is nonzero for all pressures considered, as indicated in Figure 6. The forward (ΔP > 0) results are shown in blue, while the reverse (ΔP < 0) results are in black, and these show the relative flux ratio as a function of the magnitude of the pressure difference |ΔP|. The horizontal dashed line indicates the flux expected if the bulk ionic solution were transported through the pore with no ion rejection. It is noted that the relative ion flux can exceed this limit due presumably to the ability of the charged pore to concentrate ions within the channel. Although the effect is not much larger than the statistical uncertainty in the data, the figure suggests an asymmetry in flux exhibited by the modified pore. In particular, the complete salt rejection is achieved in the forward direction at the finite pressure of ΔP = 50 MPa. At this


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between the surface charges and solution ions and the indirect effect of charge-pattern-induced water structuring, which changes the thermodynamic properties of the ions in the solution within the pore. These effects will be quantified in future publications. There is some evidence of rectification of ions for the asymmetric charge distributions in the (12,12) system. The presence of asymmetry in the flux versus the pressure difference behavior of the membrane suggests separation strategies beyond simple unidirectional reverse osmosis, based on the paradigm of the Brownian ratchet.21,25,32-38 Continuing work will focus on the role of water structuring and other effects to reveal the mechanisms for transport and separation at the molecular scale, with the goal of understanding the rational connection between the nanoarchitecture and the technological performance of new membranes for separation.

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AUTHOR INFORMATION

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Corresponding Author: (18)

* To whom correspondence should be addressed. E-mail: cmartens@ uci.edu.

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ACKNOWLEDGMENT We thank Zuzanna Siwy and Gerhard Hummer for insightful comments. This work was supported in part by the National Science Foundation.

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Molecular Dynamics Simulation of Salt Rejection in Model Surface

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