Anomalous water transport in narrow-diameter carbon nanotubes

Significance

Typically, as the channel length increases, the flow rate decreases due to the increase in flow resistance. Based on molecular dynamics simulations, we show that the pressure-driven flow rate of water through narrow-diameter carbon nanotubes (CNTs) exhibits anomalous transport behavior, whereby the flow rate increases markedly first and then either slowly decreases or changes slightly as the CNT length increases. This anomalous transport behavior of water cannot be simply explained by the tube-length dependence of the free energy barrier for a single water molecule passing through the CNTs; rather, it can be attributed to the tube-length–dependent stability of the cross–CNT orifice hydrogen bonds that formed between water molecules inside and outside the CNT.

Abstract

Carbon nanotubes (CNTs) mimicking the structure of aquaporins support fast water transport, making them strong candidates for building next-generation high-performance membranes for water treatment. The diffusion and transport behavior of water through CNTs or nanoporous graphene can be fundamentally different from those of bulk water through a macroscopic tube. To date, the nanotube-length–dependent physical transport behavior of water is still largely unexplored. Herein, on the basis of molecular dynamics simulations, we show that the flow rate of water through 0.83-nm-diameter (6,6) and 0.96-nm-diameter (7,7) CNTs exhibits anomalous transport behavior, whereby the flow rate increases markedly first and then either slowly decreases or changes slightly as the CNT length l increases. The critical range of l for the flow-rate transition is 0.37 to 0.5 nm. This anomalous water transport behavior is attributed to the l-dependent mechanical stability of the transient hydrogen-bonding chain that connects water molecules inside and outside the CNTs and bypasses the CNT orifice. The results unveil a microscopic mechanism governing water transport through subnanometer tubes, which has important implications for nanofluidic manipulation.

Water transport through membranes incorporating carbon nanotubes (CNTs) or nanoporous graphene has attracted considerable interest over the past two decades (1–18), largely due to the striking physics of flow enhancement by pressure. In turn, such a dramatic enhancement of flow rate renders CNT-based membranes promising nanofluidic media for drug delivery (1), biomimetic selective transport of ions (2), and desalination (11, 19, 20). It is known that the diffusion and transport behavior of water through a nanotube can be fundamentally different from those of bulk water through a macroscopic tube (3, 4, 16–18). As an example, single-file water transport through a (6,6) CNT occurs in a burst due to collective water motion (4, 15). Many molecular dynamics (MD) simulations have been employed to investigate the filling mechanism of water into CNTs and the self-diffusion of water molecules within the CNTs (3, 8, 14–18). On the other hand, the zero-dimensional (0D) nanopore can be viewed as a lower limit for the length of CNTs. The transport behavior of water molecules through 0D nanopores, such as nanoporous graphene and graphyne, has also been studied (21, 22). To date, the nanotube-length–dependent physical transport behavior of water is still largely unexplored.

In this article, we report results of MD simulations of pressure-driven water transport through CNTs as the length l varies from atomic length (a 0D nanopore) to finite length. Contrary to water flow through a (8,8) CNT with a wider diameter, the flow rate of water through subnanometer (7,7) or (6,6) CNTs exhibits anomalous behavior; that is, the rate increases first and then either decreases slowly or varies slightly with increase of the length l. The critical range for the length l to observe the flow-rate transition is 0.37 to 0.5 nm. We find that the free energy barrier for a single water molecule across the orifice of a (6,6) or (7,7) CNT is extremely low, essentially independent of the length l. Hence, this free energy barrier cannot account for the anomalous flow rate of single-file water passing through the CNT. It turns out that the underlying physics of the anomalous flow-rate behavior can be attributed to the dynamic hydrogen-bonding chain (HBC) that connects water molecules inside and outside the subnanometer CNTs and bypasses the CNT orifice. As a result, the length l can markedly affect the stability of the cross–CNT orifice (CCO) HBC, and a relatively weak HBC can facilitate disruption of the concerted motion of the water molecules through subnanometer CNTs, thereby affecting the single-file water transport.

Results and Discussion

A schematic of a typical system is shown in SI Appendix, Fig. S1, where a CNT membrane divides the water cell into two parts. External pressure of 600 MPa is applied to the piston to push the water toward the CNT membrane. SI Appendix, Fig. S2 shows the number of water molecules passing through the (6,6), (7,7) and (8,8) CNTs (all with a length of 0.74 nm) versus time. The number of filtered water molecules increases linearly with the simulation time, and the slope of the corresponding plot gives the flow rate. In a continuum fluid system, the average flow rate is given by the Darcy law in Eq. 1:

$$\begin{array}{c}\upsilon =\gamma \left(\frac{\mathrm{\Delta }P}{l}\right),\end{array}$$ [1]

where υ, γ, l, and $\mathrm{\Delta }P$ are the flow rate, the hydraulic conductivity, the length of the tube, and the pressure difference between the two ends of the tube, respectively (18). Note that the Darcy law is generally applicable for an incompressible, creeping liquid inside a macroscopic tube with a uniform cross-sectional area, and the hydraulic conductivity γ of a Newtonian liquid in a circular tube, subject to the no-slip boundary condition, is given by the no-slip Poiseuille relation,

$$\begin{array}{c}{\text{γ}}_{no-\mathit{slip}}=\frac{D^2}{32\mu },\end{array}$$ [2]

where D and μ are the tube diameter and the liquid viscosity, respectively (18). Although the applicability of the Darcy law is questionable in a system where the size of a liquid molecule is comparable to the size of the flow domain, here we simply viewed the Darcy law as a reference without trying to justifying its applicability on the nanoscale. The rate increases with enlargement of the tube diameter, indicating that the Darcy law could be somehow applicable to the nanoscale dimension heuristically. However, water flow rate depends on the length l in a complex way. As shown in Fig. 1A, for (8) CNTs, the water flow rate decreases from 390 to 347 ns⁻¹ as the length increases from 0 to 0.12 nm (the zero-length CNT is defined as having a thickness of only one layer of carbon atoms). The water flow rate seems not correlate strongly with l in the range of 0.25 to 0.74 nm. At l = 2.58 nm, the flow rate drops to 311 ns⁻¹. On the other hand, in (6,6) CNT, the water flow rate increases from 68.7 to 75.0 ns⁻¹ as l increases from 0 to 0.37 nm and then increases slightly for l from 0.49 to 2.58 nm. In contrast, for the (7,7) CNT, the water flow rate decreases from 190.4 to 174.8 ns⁻¹ with l from 0 to 0.12 nm and then increases from 174.8 to 192.0 ns⁻¹ with $\text{l}$ l from 0.12 to 0.25 nm before decreasing again from 192.0 to 187.5 ns⁻¹ with l from 0.25 to 2.58 nm. The different behaviors of water flow through these CNTs become more apparent when l increases from 0 to 0.37 nm. For (8,8) CNT, the water flow rate is higher for l = 0 nm, whereas the water flow rate through (6,6) and (7,7) CNTs is higher for l of ∼0.25 to 0.37 nm. Similar anomalous behavior was also observed for a system under a lower external pressure of 120 or 20 MPa (SI Appendix, Fig. S3).

Fig. 1.

Water flow and structure in CNTs. (A) Water flow rate versus length of (6,6), (7,7), and (8,8) CNTs. Three simulations with the same conditions but different randomly given initial velocities. Standard deviation (SD) error bars were used. (B) Side and top views of water structures inside the (6,6), (7,7), and (8,8) CNTs. C (gray), O (red), H (white). Arrows show the lengths of the flow-rate transition for (6,6) and (7,7) CNTs.

To gain deeper insight into the anomalous behavior for the water system, we also performed MD simulations of a Lennard-Jones (LJ) fluid system. Interestingly, the flow rates of the LJ fluid decrease as the lengths of both (6,6) CNT and (8,8) CNT increase from 0 to 0.25 nm, contrary to the anomalous behavior observed for water (SI Appendix, Fig. S4). Hence, the existence of hydrogen bonds (HBs) in a water system (not in the LJ system) seems critical to the anomalous flow-rate behaviors. Moreover, independent MD simulations were performed using the four-point transferable intermolecular potential (TIP4P) water model (SI Appendix, Fig. S5) or with CNTs that have similar diameters but different CNT indexes (SI Appendix, Fig. S6). Again, the anomalous flow-rate behaviors were observed, indicating that the qualitative results and conclusions are generic, insensitive to the water model or atomic details of the CNT walls used in the MD simulations.

Because the confined structure of water molecules inside CNTs plays a key role in the water flow (23, 24), we compute the water distribution inside CNTs. Fig. 1B presents the front and top views of water molecules inside (6,6), (7,7) and (8,8) CNTs, all with l = 0.74 nm and at the external pressure of 600 MPa. The narrowest (6,6) CNT entails a single-file water chain, consistent with previous MD simulations (18). For the (7,7) CNT, the water molecules pass through the subnanotube one or two at a time. The cross-sectional area of the widest (8,8) CNT is large enough for three or more water molecules to pass through at a time.

To better understand the process by which water enters a CNT, Fig. 2 and SI Appendix, Fig. S7 show two-dimensional (2D) concentration contours for oxygen atoms of water molecules at several axial positions for (6,6) and (7,7) CNTs, respectively. The 2D profiles were plotted by counting oxygen atoms in a thickness Δz = 0.01 nm during a 10-ns simulation after the steady state is reached. Consider a 0.74-nm-long (6,6) CNT as an example (Fig. 2): six representative 2D profiles show the distribution of oxygen atoms of water molecules as the orifice is approached from the bulk water. Fig. 2A shows the density map of oxygen atoms at the water-graphene interface (z = −0.30 nm). Approaching the orifice of the CNT (−0.25 nm ≤ z ≤ −0.15 nm), the oxygen atoms are increasingly distributed in a hexagonal pattern (Fig. 2 B–D) due to the interaction between the carbon atoms of the CNT rim and the water molecules. The oxygen atoms are arranged in one-to-two correspondence with the rim carbon atoms. Near the CNT orifice (z ≥ −0.10 nm), the oxygen atoms are distributed in a disk pattern (Fig. 2 E and F), indicating that only one water molecule can exist near the orifice (within a 0.1-nm axial distance). As a comparison, the oxygen atoms near the orifice of the (7,7) CNT (z ≥ −0.10 nm) are distributed in a doughnut pattern, suggesting that more than one water molecule can exist near the orifice.

Fig. 2.

Oxygen density maps of a (6,6) CNT. Oxygen density maps at different axial positions (or z values) of a 0.74-nm-long (6,6) CNT. Dark-blue regions indicate the highest probability of finding an oxygen atom. The red and pink plus signs represent the carbon atoms in the membrane. The left orifice of the CNT is fixed at z = 0 nm, and a piston is allowed to push water in the +z direction at an external pressure of 600 MPa. A–F correspond to z = −0.30, −0.25, −0.20, −0.15, −0.10, and 0 nm, respectively.

For l = 0 to 0.25 nm, the enhancement of the water flow rate through (6,6) CNT is more notable than that through (7,7) CNT, which may be due to the formation of a single-file chain structure in (6,6) CNTs. Hereafter, we focus on water transport through the (6,6) CNT only to better understand the underlying physical mechanism. First, we calculate the potential mean forces (PMFs) for passing a single water molecule through the (6,6) CNT with different l (SI Appendix). As shown in SI Appendix, Fig. S8, the PMF profile entails two peaks, located at z = ∼−0.2 and 0 nm, respectively. The first peak is due to the water molecules breaking their in-plane HB network in the interfacial water layer (Fig. 2C). The second peak corresponds to the water molecules passing through the orifice of the CNT (Fig. 2F). The former peak is higher than the latter peak, implying breaking the in-plane HB network plays a more important role for water entering the CNT.

Fig. 3 shows the free energy barrier E_b for a water molecule to overcome when passing through a (6,6) CNT with a different length. Mainly, the free energy barrier is due to the penetration of the in-plane HB network in the interfacial water layer—specifically, for the (6,6) CNT with l = 0 nm, E_b = 0.54 kcal/mol and for CNTs with l = 0.12, 0.25, 0.37, 0.49, 0.61, or 0.74 nm, E_b = 0.33, 0.71, 0.28, 0.41, 0.32, and 0.50 kcal/mol, respectively. Compared to (6,6) CNTs, the free energy barriers that a water molecule need to overcome to pass through (7,7) CNTs with a different l are similar (SI Appendix, Figs. S8 and S9). These free energy barrier values are very low compared to the typical energy associated with thermal fluctuations at 300 K (kT = 0.6 kcal/mol) and do not vary monotonically with l. We could assume that water transport is an activated process, following the Arrhenius model

$$\begin{array}{c}\upsilon =A_0{e}^{(PV-E_b)/kT},\end{array}$$ [3]

where A₀ is the attempt rates for water passage, the PV term represents the possibility of modulating activation energy by pressure, and V is the effective volume of a water molecule. Taking the effective volume V as ∼3.0 × 10⁻²⁹ m³, the PV term is ∼2.6 kcal/mol, which is 4 to 10 times higher than the E_b term. As a result, water transport through CNTs is not an activated process, and apparently, the strongly enhanced water flow rate over the critical range of l seems surprising.

Fig. 3.

Free energy barriers encountered for a single water molecule passing through the (6,6) CNT versus the CNT length l from 0 to 2.58 nm. SD error bars were used.

Previous studies show that water transport through CNTs occurs in bursts with collective water motion (15), and in this case, the water flow rate is not dominated by E_b. In (6,6) CNTs, water molecules, including two nearest neighbor water molecules outside the CNT, form a one-dimensional single-file chain while moving along the nanotube in a concerted fashion (Fig. 1B). The entry, exit, and transport of water molecules is thus highly correlated. A continuous-time random-walk model is used to describe the motion of the single-file water chain inside the CNT (23, 25). To quantify the variation in the local liquid structure as water molecules approach the CNT orifice, the axial distribution function (ADF) a(z) of water molecules, given by counting the number of water molecules in each z interval of 0.02 nm, is presented in Fig. 4A and SI Appendix, Fig. S10. In addition to the conventional interfacial water layer (region II), a low peak (region I) (inset in Fig. 4A) arises in a(z) near the CNT orifice (cf. Fig. 2D). Following previous work (26), we define water molecules in region I as active water. The active water molecules can form HBs with water molecules inside the CNT. These CCO HBs play a key role for the entry of water molecules into the (6) CNT.

Fig. 4.

Axial distribution of water molecules and probability distribution of oxygen atoms. (A) ADF for oxygen atoms along the z axis in a 0.74-nm-long (6,6) CNT. Regions I and II (blue and green bars) are explained in the main text. The inset shows an expanded view of the ADF in region I. (B) Probability distributions of the distance between the two oxygen atoms d_OO in the two water molecules that form a CCO HB for (6,6) CNTs of various lengths l.

Fig. 4B shows the probability distributions of the distance between two oxygen atoms d_OO in the two water molecules that form a CCO HB in (6,6) CNTs at various lengths. For all CNTs, a peak in the d_OO distribution appears ∼0.27 nm. Compared with CNTs with l > 0, the peak is much lower and broadened for a CNT with l = 0 nm. Furthermore, the peak becomes wider and higher as l increases from 0 to 0.49 nm. Thus, the water chain and the associated concerted motion can be easily disrupted.

As l increases from 0 to 0.49 nm, the increase in the strength of CCO HBs is further confirmed through computation. As shown in Fig. 5, the strength of the COO HB increases from 5.31 to 5.48 kcal/mol first as l increases from 0 to 0.49 nm and then only changes slightly as the CNT length l increases from 0.49 to 2.58 nm. The weaker the COO HB, the more easily the water chain can break; therefore, their concerted motion can disrupted more easily. A similar trend is observed for (7,7) CNTs. As shown in SI Appendix, Fig. S11, the strength of the COO HB increases from 5.02 to 5.16 kcal/mol first as l increases from 0 to 0.37 nm and then only changes slightly as the CNT length l increases from 0.37 to 2.58 nm. The transition point coincides well with the critical value of the l-dependent flow-rate profile.

Fig. 5.

Strength of CCO HBs. The strength of a CCO HB near the orifice of (6,6) CNTs at various lengths l. Three simulations with the same conditions but different randomly given initial velocities. SD error bars were used.

Conclusion

In summary, we systematically investigate the physical transport of water wires through CNT with its length l varying from 0 nm (i.e., an atom-length 0D nanopore) to a nonzero length. The MD simulation results indicate that the water flow rate depends on l in a complex manner. Contrary to the macroscopic theory that flow rate across a membrane decreases with membrane thickness, the present results show that the water flow rate through the (6,6) CNT increases as l increases from 0 to 0.37 nm. Because the free energy barrier is quite small for passing a single water molecule through a (6,6) CNT, regardless of its length, the free energy barrier cannot account for the flow-rate transition. Further analysis shows that this anomalous behavior is mainly due to variations in the stability of the hydrogen-bonded water chain as it crosses the orifice of a CNT. The strength of the COO HB increases first as l increases from 0 to 0.49 nm and then only changes slightly as the CNT length l increases from 0.49 to 2.58 nm. The weaker the COO HB, the more easily the water chain can break; therefore, their concerted motion is easier to disrupt. This anomalous behavior may have biological implications, because the pore diameter of the transmembrane protein aquaporin-1 (a biological water channel) is comparable to that of a (6,6) CNT (27, 28).

Methods

Classical MD simulations were performed to simulate water flow through two subnanometer-wide CNTs, (6,6) and (7,7), as well as a wider (8,8) CNT. The diameters of the three CNTs are 0.83, 0.96, and 1.10 nm, respectively. A typical simulation box consists of a CNT membrane, a graphene layer acting as a rigid piston, and water (SI Appendix, Fig. S1). In addition, a rigid graphene layer is anchored on the permeating side of the CNT membrane to prevent water molecules from leaving the permeating side. The initial number of water molecules on the feeding side and the permeating side are about 2,900 and 1,030, respectively. The simulation stops before water molecules reach the rigid graphene layer on the permeating side, so there is no pressure on the water molecules in the direction parallel to the flow direction on the permeating side. The size of the simulation box is 5.0 $\times$ 5.1 $\times$ 11.0 nm (x $\times$ y $\times$ z). Periodic boundary conditions are applied in all directions. The movable graphene on one side of the CNT membrane is subjected to an external pressure of 600 MPa and is thus used for pressing the water through the CNT (SI Appendix shows detailed pressure calculations). The CNT length l ranges from atom long to 2.58 nm. During the MD simulations, all the carbon atoms of the CNT membrane are fixed. In the movable graphene, the carbon-carbon interactions are described by a second-generation reactive empirical bond order potential (29). Water molecules are modeled using the extended simple point charge (SPC/E) potential (30). The interactions between water molecules and carbon atoms are described by the 12-6 LJ potential. The LJ parameters for carbon atoms are σ = 0.355 nm and ɛ = 0.293 kJ/mol. The long-range electrostatic interaction is computed by using a particle-particle particle-mesh solver (31). The cutoff distances for long-range electrostatic and LJ interactions are set at 1.0 nm. All MD simulations are performed using the Large-scale Atomic/Molecular Massively Parallel Simulator (LAMMPS) package (32). The time step of the MD simulation is 2.0 fs, and each simulation lasts about 10 ns. Data are collected every 5,000 steps. The temperature of the system is controlled at 300 K by using a Nóse-Hoover thermostat, which is applied only to the degrees of freedom orthogonal to the flow direction (33). For computing the velocity, data are averaged over three independent sets of MD simulations with different initial conditions.

Data, Materials, and Software Availability

All study data are included in the article and/or SI Appendix.