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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms9616 pressure and the rate of water permeation is observed to be quite linear (Fig. 2a). Previous studies40,41 also show that water flux in small nanochannels is linear with respect to external pressure. The permeation rates through various pores (Mo only4mixed4S only4graphene) can also be explained by the energy barrier that a water molecule needs to overcome to enter the pore. These barriers were computed to be DEMo only 1⁄4 8.50 kBT, DEmixed 1⁄4 8.84 kBT, DES only 1⁄4 9.01 kBT, DEgraphene1⁄411.05 kBT, which are consistent with the results in Fig. 2a. The details on the energy barrier calculations are documented in Supplementary Fig. 1. Physical chemistry and geometry of the pore. Water flux (Q) is a function of density (r) inside the pore, velocity (V) of water through the pore and the area of the pore (A), (Q1⁄4rVA). In water desalination, increasing the area of the pore limits the salt rejection capability of the pore. As the area of the pore increases, the efficiency of rejection decreases25, leaving r and V as the control parameters to increase water flux through the pore. As shown above, Moonly pore exhibits the highest rate of water permeation. This can be explained by the higher water density (r) and velocity (V) in the Mo only pore compared with those of the Sonly and mixed pores (Fig. 3a–c). The average density of water follows the order of Mo only4mixed4S only (1.47, 1.37 and 1.31 g cm 3, respectively). The denser packing of water molecules at the Moonly pore can be attributed to the hydrophilic nature of Mo sites42 at the edge of the nanopore, which attracts water molecules to the pore interior. It has been shown that the molybdenum surface has a water contact angle close to 0° (molybdenum is a transition metal with a large atomic diameter)42. Attraction of water molecules towards Mo sites becomes more obvious by comparing the mixed and S only pores densities (Fig. 3a). In the mixed pore, the existence of 50% Mo sites gives rise to higher density in the centre of the pore compared with that of S only pore (Fig. 3a). Next, we explored the velocity profiles in the pore for all the three different pores. The velocities are also higher in Moonly pores compared with mixed and Sonly pores (Fig. 3c). The average velocity of water is 8.26, 7.53 and 7.51 m s 1 for Mo only, mixed and S only pores, respectively. To shed deeper insight into the physical understanding of why the velocity of Mo only pore is higher compared with mixed and S only pores, we computed velocity profiles at the sites of S and Mo for both pore types of Mo only and S only (Fig. 4a,b). This is achieved by binning both pore types at Mo and S sites and averaging velocity at each point for a large number of sets of simulations. We observed that in the Mo only pore, the velocity is higher at Mo site compared with the S sites. Unlike Mo only pore, we did not observe the velocities to be higher in Mo site in the S only pore, (Fig. 4a,b) which implies that the arrangement of Mo and S sites matter for velocity profiles (see Supplementary Fig. 2 for more evidence on geometry dependency of the velocity in the pore). It has been shown that conical nanopores have higher fluxes and permeation rates25,43,44. Many biological nanopores, including aquaporin25,45,46, have an hourglass shape, which facilitates rapid water permeation47. Solid-state nanopores have also been designed for conical/hourglass shape to enhance solute and DNA transport48,49. Here in Moonly pores, due to the fish-bone structure of MoS2 (ref. 9), the pore can be tailored13,27 to an hourglass shape at sub-nanometre length scale (see cartoon representation of comparison between Mo only, S only and graphene pores in Fig. 4c). Moonly pore has a contraction centre with hydrophobic S sites at the entrance and S only pore has an expanding centre (Fig. 4c). Graphene has a flat entrance and exit geometry with a single-atom-type exposure at the pore surface50. Water molecules slip on the hydrophobic edges of S and are attracted by the hydrophilic sites of Mo at the pore centre in Moonly case. This arrangement of hydrophobic and hydrophilic atoms along with the conical shape of the pore enhances the flux of water. Also, the water flux highly correlates with the energy barrier of each pore type. The computed potential of mean force for water molecules in each pore type is the reflection of pore chemistry and geometry. In Mo only pore, the potential of mean force is the lowest because of the conical/ hourglass and the hydrophobic–hydrophilic arrangement of the pore atoms (Supplementary Fig. 1). The fundamental advantage of Mo only pore architecture over other pores is the interplay of geometry and chemistry to produce a higher flux of water. Discussion Ion rejection and water flux are the two important factors defining the effectiveness and performance of a water desalination membrane. In Fig. 4d, ion rejection and water permeation rate are plotted for various nanomembrane materials51 (MFI-type zeolite52, commercial polymeric seawater RO53, brackish RO53, nanofiltration53 and high-flux RO53) including MoS2 and graphene investigated in this work. As shown in Fig. 4d, water permeation rate is theoretically enhanced by five orders of magnitude using MoS2 compared with conventional MFI-type zeolite. Also, there is a 70% improvement in the permeation rate of MoS2 compared with graphene. In the study by Cohen-Tanugi et al.19, the permeation rate for graphene is shown to be higher than the rate we observed for graphene. This is because, in our abc 3.5 3.0 2.5 i 2.0 1.5 1.0 0.5 0.0 0246 Distance from the center of the pore (Å) 0246 Distance from the center of the pore (Å) Mixed Mo only S only 12 10 8 6 4 2 ii 0 Figure 3 | Water density and velocity profiles. (a) Water density distribution in the radial direction in the mixed, Mo only and S only pores with equivalent pore sizes (mixed, A 1⁄4 55.45 Å2 ; Mo only, A 1⁄4 56.42 Å2; S only, A 1⁄4 57.38 Å2 ) at a fixed pressure of 250 MPa. (b) Density map of water distribution in Mo only (i) and S only (ii) pores. Blue denotes a zero probability of finding a water molecule and red indicates the highest probability of observing a water molecule. (c) Axial velocity of water molecules in the radial direction for mixed, Mo only and S only nanopores. 4 NATURE COMMUNICATIONS | 6:8616 | DOI: 10.1038/ncomms9616 | www.nature.com/naturecommunications & 2015 Macmillan Publishers Limited. All rights reserved. Mixed Mo only S only Water density (g cm–3) Velocity (m s–1)PDF Image | Water desalination with a single-layer MoS2 nanopore
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