Water fluxes

A typical simulation box consists of a single-layer MoS 2 , a graphene sheet (acting as a rigid piston to apply the external pressure), water and ions (Fig. 1a). Here three pore edge types for MoS 2 are considered to study the effect of terminating atoms and pore chemistry on the rate of water permeation and ion rejection. The first type of pore, which is labelled as mixed in this study, is a combination of molybdenum and sulfur atoms. The other two pore types are labelled as Mo only and S only, as these are terminated by molybdenum and sulfur atoms, respectively (Fig. 1b). Water fluxes through various MoS 2 nanopores as a function of the applied pressure gradient are presented in Fig. 2a. Three MoS 2 pore types (mixed, Mo only and S only) were studied to explore their rejection rate and flux. To investigate the relative performance of MoS 2 over other two-dimensional materials, a graphene nanopore, which has been shown to be promising for water desalination, is also considered11,19. For the sake of comparison, the three MoS 2 pores and the graphene pore have approximately equivalent accessible pore areas (mixed, A=55.45 Å2; Mo only, A=56.42 Å2; S only, A=57.38 Å2; and graphene, A=59.67Å2). Our results indicate that the Mo only pore has the highest rate of water permeation followed by the mixed, S only and the graphene pore for all the applied pressures (Fig. 2a). Water flux through the mixed pore is intermediary between Mo only and S only nanopores. The higher water fluxes through MoS 2 nanopores compared with graphene nanopores imply that for a desired water flux, a smaller applied pressure is needed with MoS 2 nanopores. Later, in this paper, we will explain the physical chemistry and geometrical foundations of MoS 2 pore that give rise to a higher flux.

Figure 1: Simulation box and different pore architectures. (a) Schematic of the simulation box consisting of a MoS 2 sheet (molybdenum in blue and sulfur in yellow), water (transparent blue), ions (in red and green) and a graphene sheet (in gray). (b) Left: Mo only pore type. Right: S only pore type. Bottom: mixed pore type. Full size image

Figure 2: Water permeation and salt rejection. (a) Water flux as a function of the applied pressure for mixed, Mo only, S only and graphene nanopores with similar pore areas. (b) Percentage of ion rejection by various pores as a function of the applied pressure. Pores with different edge chemistries as well as various pore areas (denoted by A) are considered. (c) Number of water molecules (#) filtered through Mo only pores as a function of simulation time for different pore areas at a fixed pressure of 250 MPa. Full size image

Salt rejection efficiency

The other important aspect in water desalination is the ability of the membrane to reject ions. The percentage of total ions rejected by the MoS 2 and graphene pores is plotted as a function of the applied pressure in Fig. 2b. The rejection is calculated after 1,700 water molecules have filtered through the pores for all pressures. Pore sizes ranging from 20 to 60 Å2 are considered for the three types of MoS 2 pores. The ion rejection decreases at higher pressures as high pressures induce higher forces on the ions giving rise to more ion translocation events. The ion rejection of small pores (for example, 18.02 Å2) is found to be 100% for all types of pores. For larger pore sizes, ions escape through the pore reducing the rejection efficiency. For the pores with equivalent areas (mixed, A=55.45 Å2; Mo only, A=56.42 Å2; S only, A=57.38 Å2; and graphene, A=59.67 Å2), the general trend for ion rejection is quite similar regardless of the type of the pore (Fig. 2b). In other words, ion rejection is mainly dependent on the pore area and the type of the pore plays a less important role, for example, for the four pores considered, the difference in rejection is <10% even at a high pressure of 350 MPa.

As shown in Fig. 2c, the water filtration rate increases sharply as the pore area increases from ∼20 to ∼50 Å2. The sharp change in the water flow rate is due to the formation of single-file chain of water in small pores (∼20 Å2). As shown in ref. 11, the water flow rate is considerably reduced because of the weak hydrogen bonding in single-file chains. For efficient water desalination, pore sizes should be chosen such that both the ion rejection and water filtration rate are optimized since very small pores lack high permeation rates and large pores (wider than 60 Å2) fail to effectively reject ions.

As observed by Cohen-Tangui et al.19 for graphene, the polarizability of water also has a little effect on ion rejection in MoS 2 nanopores. To introduce the effect of polarization, the flexible simple point charge (SPC/F) model38 was used. The ion rejection percentages associated with the flexible water model are within 2% of those modelled with the SPC/E water.

Permeation coefficient

To quantify the water permeability through various pores, we compute the permeability coefficient, p, across the pore. For dilute solutions39,

where J w is the flux of water (# ns−1), V w is the molar volume of water (18.91 ml mol−1), ΔC s is the concentration gradient of the solute (1.0 M), N A is the Avogadro number, k B is the Boltzmann constant, T is the temperature (300 K) and ΔP is the applied hydrodynamic pressure (MPa). The permeability coefficients of the mixed, Mo only, S only and graphene pores were calculated to be 71.64, 83.61, 62.69 and 59.32 # ns−1, respectively. These coefficients are expected to also hold true for small applied pressures (<10 MPa), which are normally used in water desalination, since the relationship between the external 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 only>mixed>S only>graphene) 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 ΔE Mo only =8.50 k B T, ΔE mixed =8.84 k B T, ΔE S only =9.01 k B T, ΔE graphene =11.05 k B T, 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 (ρ) inside the pore, velocity (V) of water through the pore and the area of the pore (A), (Q=ρ·V·A). 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 ρ and V as the control parameters to increase water flux through the pore.

As shown above, Mo only pore exhibits the highest rate of water permeation. This can be explained by the higher water density (ρ) and velocity (V) in the Mo only pore compared with those of the S only and mixed pores (Fig. 3a–c). The average density of water follows the order of Mo only>mixed>S only (1.47, 1.37 and 1.31 g cm−3, respectively). The denser packing of water molecules at the Mo only 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).

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=55.45 Å2 ; Mo only, A=56.42 Å2; S only, A=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. Full size image

Next, we explored the velocity profiles in the pore for all the three different pores. The velocities are also higher in Mo only pores compared with mixed and S only 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).

Figure 4: Effect of pore type on water permeation and salt rejection. (a) Axial velocity of water molecules in the radial direction at the location of S and Mo atom layers in the Mo only nanopore of A=56.42 Å2 at 250 MPa. (b) Axial velocity of water molecules in the radial direction at the location of S and Mo atom layers in the S only nanopore of A=57.38 Å2 at 250 MPa. (c) Cartoon representation of the pore architecture for Mo only, S only and graphene nanopore. (d) Performance of various membranes in terms of their ion rejection and water permeation rate. Water permeation rate is expressed per unit area of the membrane and per unit pressure as l cm−2 per day per MPa. Full size image