Designing an All-Carbon Membrane for Water Desalination

TL;DR

This paper presents the design of an all-carbon membrane combining layered carbon nanostructures for efficient water desalination, offering enhanced durability, chemical resistance, and ion rejection capabilities.

Contribution

It introduces a novel layered all-carbon membrane structure and provides atomic-level insights into water permeation and ion rejection mechanisms.

Findings

High mechanical strength and chemical resilience of the membrane.

Selective rejection of Na+ ions demonstrated through atomic simulations.

Structural stability of the membrane in desalination conditions.

Abstract

We design an all-carbon membrane for the filtration and desalination of water. A unique layered assembly of carbon nanostructures including graphite oxide (GO), buckypaper consisting of carbon nanotubes, and a strong carbon fabric provides high mechanical strength and thermal stability, resilience to harsh chemical cleaning agents and electrical conductivity, thus addressing major shortcomings of commercial reverse osmosis membranes. We use ab initio density functional theory calculations to obtain atomic-level insight into the permeation of water molecules in-between GO layers and across in-layer vacancy defects. Our calculations elucidate the reason for selective rejection of solvated Na$^+$ ions in an optimized GO membrane that is structurally stabilized in a sandwich arrangement in-between layers of buckypaper, which are protected on both sides by strong carbon fabric layers.

Tables

Table 1. Table 1: Adsorption energy E a ​ d subscript 𝐸 𝑎 𝑑 E_{ad} of an isolated H 2 O molecule at graphene edges. Reported results are for bare as well as H-, O- and OH-terminated armchair and zigzag edges.

Edge termination	$E_{ad}$(armchair)	$E_{ad}$(zigzag)
Bare (H2O$\to$Had+OH)	$-4.75$ eV	$-5.27$ eV
Bare	$-0.02$ eV	$-0.21$ eV
H-terminated	$-0.10$ eV	$-0.03$ eV
O-terminated	$-0.14$ eV	$-0.10$ eV
OH-terminated	$-0.08$ eV	$-0.03$ eV

Equations

$$<|\mathbf{r}(t)-\mathbf{r}(0){|}^2>=2\tilde{d}Dt.$$

Full text

Graphite oxide, GO, water, flow, DFT, ab initio

I Introduction

Availability of potable water is one of the most pressing needs of humankind Elimelech and Phillip (2011). Even though water is plentiful on earth, most of it is not suited for human consumption and must be treated beforehand Elimelech and Phillip (2011); Werber et al. (2016a). The key to processing contaminated or brine water is filtration. Micro- and nano-porous carbon has been used successfully for filtration due to its small pore size and large surface area, combined with its high chemical, thermal and mechanical stability as well as low cost Boretti et al. (2018); Liu et al. (2015). Also salty seawater can be made potable using, among others, ultra-fine membranes. We focus here on desalination based on reverse osmosis Cohen et al. (2017), which is more challenging than filtration due to the difficulty to separate solvated ions from water on the molecular level. State-of-the-art nano-porous membranes for water desalination achieve selective rejection of specific ions, typically utilizing polymers such as polysulfone. Such membranes do not operate well at high temperatures approaching that of boiling water, pressures of several hundred bar, and show limited resilience to chemicals used for cleaning of bio-fouling debris Warsinger et al. (2018). Carbon nanostructures have been considered in search of suitable alternatives, but found to be rather brittle and unsuited for membrane applications Sun et al. (2016). There have been calls for atomic-level theoretical investigations Elimelech and Phillip (2011) and for a paradigm shift Elimelech and Phillip (2011); Shahzad et al. (2019) to achieve serious progress in water desalination.

Here we present a design of an all-carbon membrane for the filtration and desalination of water that should not suffer from previously reported materials limitations. The outermost layer of the proposed sandwich structure of the membrane would be a strong, yet flexible fabric of carbon fibers or carbon nanotubes (CNTs), capable of withstanding extremely high pressures used in the reverse osmosis process. The innermost layer, which should separate water molecules from solvated ions including Na+, would consist of layered, highly oriented graphite oxide (GO). This core layer would be sandwiched in-between layers of buckypaper consisting of CNTs, which would also provide cushioning against the outermost layers. Using different polymorphs of graphitic carbon as membrane components should provide superior thermal and mechanical stability as well as resilience to harsh chemicals used to clean off bio-fouling debris. To investigate the viability of our concept, we use ab initio density functional theory calculations to obtain atomic-level insight into the permeation of water molecules in-between GO layers and across in-layer vacancy defects. Our results elucidate the reason for the reported selective rejection of solvated Na+ ions in an optimized GO membrane.

II Structure and functionality of the proposed all-carbon

membrane

A schematic view of the proposed all-carbon membrane, along with a likely trajectory of a water molecule permeating the membrane, is presented in Fig. 1(a). In the following, we will first discuss the properties of the outermost layers B and C of the membrane, which are well known in the carbon community.

II.1 Outermost layers of bucky paper and carbon fabric

The two outermost layers B and C of the membrane in Fig. 1(a) consist of quasi-1D graphitic carbon nanostructures including CNTs and carbon fibers. Both benefit from extraordinary properties of graphite or graphene, including its flexibility, in-plane tensile strength hundred times that of steel, melting temperature close to $4,000$ K, and resilience to harsh cleaning chemicals containing chlorine Tománek (2014).

The buckypaper layer B consists of entangled single-wall CNTs with diameters of $1-2$ nm, with its structure resembling a brillo pad Tománek (2014); Rinzler et al. (1998); Sweetman et al. (2012). The typical pore size, representing the inter-nanotube spacing, ranges between $20-50$ nm. Buckypaper does not impede passage of water molecules or ions, but blocks larger debris Sweetman et al. (2012); Rashid et al. (2014). The main purpose of layer B is to contain the layered, but brittle GO with typical flake sizes exceeding $1$ $\mu$m in the core layer A.

The outermost carbon fabric layer C plays its most important role in structurally supporting and containing material in layers A and B. The pore size in layer C typically exceeds $1$ $\mu$m and does not really matter as long as it contains the buckypaper layer. Since the pore sizes exceed those of buckypaper, this layer also does not impede passage of water molecules or ions. The strength and flexibility of the carbon fabric, which also protects the membrane from stone impact, is provided by CNTs, which can be harvested, spun into yarns Zhang et al. (2004) and woven to a fabric Luo et al. (2019). Presence of CNTs has been shown to reduce bio-fouling of membranes Rashid and Ralph (2017). Since layers B and C are electrically conductive, applying a voltage may furthermore assist in rejecting ions Ho et al. (2018); Pérez-Roa et al. (2009); Park et al. (2006).

II.2 Layered GO

Key to the selective rejection of ions is the core layer A of the membrane consisting of GO. The structure of GO consists of graphene layers that have been chemically functionalized by epoxy-O and OH groups Boehm et al. (1961). A simplified schematic image of a realistic GO structure is presented in Fig. 1(b). Unlike hydrophobic graphite and graphene, GO is hydrophilic and requires water for stabilization. In comparison to graphite, GO also has a lower bio-fouling tendency Hu et al. (2016).

The two commonly used approaches to synthesize GO are the Hummer Hummers and Offeman (1958) and the Brodie Brodie (1859) process. Both techniques yield an ordered material containing a substantial fraction of in-layer pores, seen in Fig. 1(b). For a long time, GO has been considered a viable material to form membranes, especially suitable for water desalination based on reverse osmosis Boehm et al. (1961). GO has been found very useful for the filtration and purification of water Akhavan and Ghaderi (2010) and other substances, which have to cross many pores in many layers. GO membranes have come back into the focus of research due to progress in this direction. Due to rather small grain size, as-produced GO is typically regarded a powder. Related multilayer graphene oxide (MLGO), a new term coined for layered GO with large grains, has indeed proven useful for water purification and desalination Jung et al. (2009). A significant improvement of the nanofiltration capability of GO has been achieved by shear alignment Akbari et al. (2016). We believe that homogenization of the GO sample by ball milling followed by shear alignment may optimize the performance of GO for water desalination.

Obviously, controlling the microscopic structure and layer alignment in GO and understanding the flow mechanism of water molecules in-between layers of graphite Cicero et al. (2008) and GO Korobov et al. (2016); Gogoi et al. (2018) are crucial for further progress. Molecular dynamics simulations with model potentials Lerf et al. (2006); Dai et al. (2016) and neutron diffractions studies Buchsteiner et al. (2006); Lerf et al. (2006) have provided initial insight into the system. Still, atomic-scale information about the mechanism of water permeation and selective ion rejection by GO membranes is either absent or extremely limited.

II.3 Known facts about the permeation of H2O molecules

and salt ions in GO

Presence of in-layer pores in the layered structure of GO is a necessary prerequisite for water permeation since no atoms could penetrate through the atomic lattice of an ideal membrane consisting of infinitely large, defect-free layers. We distinguish in-layer pores, consisting of extended vacancies and defects in a layered structure, from slit pores associated with the interlayer region that can accommodate external atoms and molecules. To allow water to permeate, pore sizes in the membrane must exceed the van der Waals diameter $d_{vdW}$(H2O)$=2.8$ Å of a H2O molecule Jorgensen et al. (1983). The key challenge is to allow only water and not salt ions to pass through the membrane by limiting the size and chemical termination of in-layer pores.

Description of ion transport across in-layer pores in GO must necessarily consider the fact that ions are hydrated and that the ionic hydration shell may change during the passage. It would therefore be misleading to compare the observed bare radii of salt ions Volkov et al. (1997) $r{\rm{(Na}}^{+}{\rm{)}}=1.17$ Å, $r{\rm{(K}}^{+}{\rm{)}}=1.49$ Å, and $r{\rm{(Cl}}^{-}{\rm{)}}=1.64$ Å to the van der Waals radius $r$(H2O)$=1.4$ Å of a water molecule Jorgensen et al. (1983).

In aqueous environment, all ions are surrounded by water molecules, which form a tightly bound hydration shell. Neutron diffraction studies Mancinelli et al. (2007) and DFT calculations Bankura et al. (2013) agree on the hydration shell radii that are slightly larger than the ion-oxygen distances $d$(Na${}^{+}-$O)$=2.3-2.4$ Å, $d$(K${}^{+}-$O)$=2.7-2.8$ Å, and $d$(Cl${}^{-}-$O)$=3.2$ Å. Unlike the cations, hydrated Cl- ions form hydrogen-bonded bridges with water molecules Mancinelli et al. (2007). The hydration shell of Na+ and K+ cations contains about $6$ water molecules, that of larger Cl- anions about $7$ water molecules. The values of the sequential detachment energies of water molecules from the hydration shells of many ions are now well established Meot-Ner (Mautner). For ions of interest here, the detachment of all water molecules requires $E_{hyd}$(Na*+)$=4.03$ eV, $E_{hyd}$(K+)$=3.46$ eV, and $E_{hyd}$(Cl-*)$=3.27$ eV. These hydration energy values are significantly higher than the hydration energy of a water molecule Pang (2014) $E_{hyd}{\rm{(H}}_{2}{\rm{O)}}=0.43$ eV and have to be considered seriously in any desalination study.

Steric and energetic considerations suggest that with their largest hydration shell, Cl- anions should pass with much less ease than the Na*+* and K*+* cations across a membrane containing nanopores. This expectation has been confirmed by the experimental finding Rollings et al. (2016) that K*+* and Na*+* behave in a similar way, and that K*+* ions pass 100 times faster than Cl*-* ions through a membrane. With their even larger hydration shells, divalent cations Ca*++* and Mg*++* have been observed to pass still more slowly than monovalent cations Rollings et al. (2016). Therefore, we will focus on the selective passage of water molecules and monovalent cations in this study.

Passage of ions through a membrane may be suppressed by limiting the pore size to that of the ionic hydration shell, with $6$ Å for hydrated Na+ representing a minimum value Mähler and Persson (2012); Bankura et al. (2013). The major challenge in the desalination field is production of membranes with a very narrow range of pore sizes between $0.3-0.8$ nm that enable passage of water, but not of hydrated ions.

In this study, we combine reported observations with original atomic-scale calculation results that should elucidate the permeation of water and specific rejection of ions in a GO membrane, which have eluded a consistent microscopic description so far.

II.4 Cleaning of bio-fouled all-carbon membrane

Bio-fouling of membranes can never be prevented and poses a significant problem in the RO process Warsinger et al. (2018); Rachman et al. (2013); Zhang et al. (2013); Hu et al. (2016). Conventional cleaning agents, which contain chlorine, not only remove bio-fouling agents, but also attack currently used polymer membranes, limiting their useful lifetime. One major benefit of an all-carbon membrane is the known resilience of carbon materials to cleaning agents containing chlorine and to other chemicals.

Stability of carbon membrane materials under temperatures well beyond $1,000^{\circ}$C suggests an unconventional alternative to chemical cleaning. For the purpose of cleaning, an all-carbon membrane could be separated from its holder, suspended in an oxygen-free nitrogen atmosphere, and heated resistively to high temperatures, when all bio-fouling deposits would decompose. The most stable product of the decomposition process should be non-specific carbon nanostructures, which should provide additional structural reinforcement to the membrane material and potentially improve its performance without clogging it.

III Computational approach

Our computational approach to study liquid water and Na+ ions interacting with chemically functionalized, defective graphite and graphite oxide is based on ab initio density functional theory (DFT) as implemented in the SIESTA Artacho et al. (2008) and VASP Kresse and Furthmüller (1996); Kresse and Joubert (1999) codes. While computationally rather demanding, DFT calculations are free of adjustable parameters and provide bias-free predictions for molecular and solid systems. In this, our approach differs from parameterized force fields that are computationally faster, but offer limited transferability among different systems. We described the effect of electron exchange and correlation using the nonlocal Perdew-Burke-Ernzerhof (PBE) Perdew et al. (1996) functional and compared selected results to those based on the Local Density Approximation (LDA) Ceperley and Alder (1980); Perdew and Zunger (1981). The DFT-PBE total energy functional has been used extensively to provide an unbiased description of water and its interaction with solids Cicero et al. (2008); Ambrosetti and Silvestrelli (2011). We have used periodic boundary conditions throughout the study. Our SIESTA calculations used norm-conserving Troullier-Martins pseudopotentials Troullier and Martins (1991), a double-$\zeta$ basis including polarization orbitals, and a mesh cutoff energy of $180$ Ry to determine the self-consistent charge density, which provided us with a precision in total energy of ${\lesssim}2$ meV/atom. The VASP calculations were performed using the projector augmented wave (PAW) method Kresse and Joubert (1999) and $400$ eV as energy cutoff. The reciprocal space has been sampled by a uniform $k$-point grid Monkhorst and Pack (1976), as noted specifically. Systems with very large unit cells have been represented by the $\Gamma$ point only. Geometries have been optimized using the conjugate gradient (CG) method Hestenes and Stiefel (1952), until none of the residual Hellmann-Feynman forces exceeded $10^{-2}$ eV/Å. Equilibrium structures and energies based on SIESTA have been checked against values based on the VASP code. Microcanonical (NVE) and canonical (NVT) molecular dynamics (MD) calculations were performed using short $0.3$ fs time steps, which were sufficiently short guarantee energy conservation in microcanonical ensembles.

To represent the effect of pressure on the permeation of water and Na+ ions, we modified the SIESTA source code to subject selected atoms to a constant force. To preserve interatomic distances and obtain a constant acceleration within a group of selected atoms, we subject each atom in the group to the force ${\bf{F}}={\bf{F_{0}}}{\times}m_{a}$, where $m_{a}$ is the atomic mass in a.m.u. These driving forces, which typically amount to $|{\bf{F_{0}}}|{\lesssim}2.5{\times}10^{-2}$ eV/Å, induce water flow along and across defective graphene and graphite oxide layers in our MD simulations.

IV Results

All our results reported in the following are based on ab initio total energy calculations and corresponding MD simulations. We use DFT to obtain static equilibrium structures including their energies and DFT-based MD simulations to describe the evolution of a system in time. Such numerical simulations are often referred to as computational experiments as they produce vast quantities of data, which need to be further analyzed. Both in real and computational experiments, error bars represent the uncertainty in the reported values. In computer simulations, error bars may be obtained from ensemble averages or estimated based on finite-size fluctuations in finite-time MD simulations.

IV.1 Properties of liquid water

Whereas isolated H2O molecules are well understood, correct description of liquid water has remained a challenging problem for decades Chen et al. (2017). Very many energy functionals have been developed to describe the delicate interplay among the strong and weak forces that determine the behavior of liquid water, but none has been able to satisfactorily reproduce all aspects of its behavior and interaction with solids in an unbiased manner. In spite of their relatively high requirement on computational resources, we use here ab initio DFT calculations to provide an unbiased description of water interacting with carbon-based membranes.

Our calculated hydration energy of a water molecule $E_{hyd,theo}{\rm{(H}}_{2}{\rm{O)}}=0.408$ eV agrees well with the observed value Pang (2014) $E_{hyd,expt}{\rm{(H}}_{2}{\rm{O)}}=0.430$ eV. The calculated gravimetric density of water $\rho_{theo}{\rm{(H}}_{2}{\rm{O)}}=0.9$ g/cm3 agrees with previously reported DFT-PBE values Miceli et al. (2015); Gaiduk et al. (2015); Chen et al. (2017) of $0.81-0.87$ g/cm3, but underestimates the observed value at room temperature Franks (2000) $\rho_{expt}{\rm{(H}}_{2}{\rm{O)}}=1.0$ g/cm3.

To obtain insight into the dynamics of liquid water, we first considered its viscous behavior. To do so, we used the Einstein-Smoluchowsky equation, which relates the self-diffusion coefficient $D$ of a Brownian particle in a ${\tilde{d}}$-dimensional space to its trajectory ${\bf r}(t)$ by

[TABLE]

The dimensionality of bulk water in this equation is ${\tilde{d}}=3$. We performed a limited-size MD simulation of 30 H2O molecules per unit cell and traced the positions of water molecules by their center of mass as a function of time. Our simulation results shown in Fig. 3(e) suggest $D_{theo}=(1.0{\pm}0.8){\times}10^{-5}$ cm2/s for the self-diffusion coefficient. In spite of the large uncertainty caused by limited simulation time and the significant finite-size fluctuations, the estimated value of $D$ is somewhat lower than the observed value Holz et al. (2000) $D_{expt}=2.3{\times}10^{-5}$ cm2/s$=2.3{\times}10^{-9}$ m2/s in bulk liquid H2O at $25^{\circ}$C. The fact that DFT-PBE MD simulations typically underestimate the value of $D$, with results ranging from $0.1{\times}10^{-5}-0.6{\times}10^{-5}$ cm2/s, has been reported previously Chen et al. (2017); Gillan et al. (2016). Also, the higher initial slope of the mean square displacement in our MD simulation, caused by initial ballistic motion of H2O, agrees with previously reported behavior Youssef et al. (2011).

IV.2 Representation of graphite oxide (GO)

Whereas the honeycomb lattice of graphene, with atoms separated by $1.42$ Å, has been known for a long time Debye and Scherrer (1916), its oxidized counterpart – GO – is not as clearly defined and contains covalently attached epoxy-O and OH-groups that may vary in their spatial distribution. GO may contain isolated non-oxidized graphitic regions amounting to less than $15\%$ by area Erickson et al. (2010). The most commonly used representation of fully oxidized GO is the Lerf-Klinowski model Lerf et al. (1998); Buchsteiner et al. (2006) shown in Fig. 2.

IV.3 Interaction of H2O molecules with graphite and GO layers

As a water molecule adsorbs on graphene or graphite, it gains the adsorption energy $E_{ad,theo}{\rm{(H}}_{2}{\rm{O)/MLG}}=0.14$ eV according to our DFT-PBE results. This is similar to the observed value Avgul and Kiselev (1970) $E_{ad,expt}{\rm{(H}}_{2}{\rm{O)/MLG}}{\lesssim}0.20$ eV, which is likely overestimated Ambrosetti and Silvestrelli (2011). The interaction energy of a 2D water monolayer with graphene, amounting to $0.02$ eV per H2O molecule, is one order of magnitude smaller. Adsorption of initially free water molecules and their subsequent condensation to a 2D monolayer on graphene provides a larger energy gain of $0.39$ eV per water molecule. All these values are significantly lower than the hydration energy $E_{hyd,theo}{\rm{(H}}_{2}{\rm{O)}}=0.408$ eV of a water molecule in water. Energetic preference of H2O to be surrounded by other water molecules rather than graphitic carbon makes graphene and graphite hydrophobic.

We could not confirm the reported formation of 2D square ice on a graphene monolayerAlgara-Siller et al. (2015), which have been questioned subsequently Zhou et al. (2015), based on our total energy results, which suggest energy differences of only few meV/molecule due to molecular ordering. Such small energy differences are consistent with previously reported theoretical results Boukhvalov et al. (2013) indicating easy sliding of H2O molecules in-between graphene layers.

We find the adsorption energy of a water molecule on GO to be significantly higher than on graphene and graphite. Depending on the adsorption site, typical values range near $E_{ad,theo}{\rm{(H}}_{2}{\rm{O)/GO}}=0.73$ eV, exceeding the hydration energy of H2O significantly. This indicates that water molecules prefer energetically to be near GO rather than surrounded by water only, making GO hydrophilic on the molecular level. Our findings for isolated water molecules are only indicators that should not be taken as substitutes for the behavior of liquid water.

In a more realistic description, we considered the permeation of water in a carbon-based layered system, namely GO or graphite, using the periodic geometry shown in Fig. 3(a). A snap shot of water molecules arranged in-between GO layers is shown in Fig. 3(b). Per unit cell with in-layer area $A$ and height $d$, the water permeation energy ${\Delta}E_{perm}$ associated with the transfer of $m$ water molecules from liquid water to the interlayer region can be estimated by obtaining the energy difference between the water-permeated system, bulk water and the bulk dry carbon-based system. The carbon-based system will be hydrophobic if ${\Delta}E_{perm}>0$ and hydrophilic if ${\Delta}E_{perm}<0$, indicating that water molecules prefer energetically the interlayer space in the carbon-based system to bulk water. Beyond a critical content of water, amounting to many H2O molecules per unit cell and corresponding to several layers of water in the inter-layer space, ${\Delta}E_{perm}$ should approach a constant value describing non-interacting GO layers solved in bulk water.

We found the energy balance in the calculation of ${\Delta}E_{perm}$ to depend delicately on the arrangement of water molecules in the interlayer region. To compensate for this uncertainty to some degree, we first calculated ${\Delta}\tilde{E}_{perm}$ not with respect to bulk H2O, but rather free-standing layers of H2O in exactly the same configuration as in the interlayer space of the carbon-based layered system. Then, we determined ${\Delta}E_{perm}={\Delta}\tilde{E}_{perm}+2{\gamma}A$, where $A$ is the area of the unit cell in the plane of the layers and $\gamma$ is the surface tension of liquid water, which in turn depends on its surface structure in calculations. We used $A=41.9$ Å2 for optimized graphene with a 16-atom supercell, $A=43.1$ Å2 for GO terminated with O-epoxy groups, $A=41.0$ Å2 for GO terminated with OH-groups, and $A=32.3$ Å2 for the Lerf-Klinowski model Lerf et al. (1998); Buchsteiner et al. (2006) of GO shown in Fig. 2. Based on the density distribution of water molecules and structural snap shots such as Fig. 3(b), we find that distinguishable water layers contain typically 5 molecules per unit cell in our calculation, which helps in correlating the number $m$ of water molecules with the number of water layers.

Theoretical estimates of $\gamma$ range from $6.6{\times}10^{-2}$ J/m2 based on DFT-PBE MD simulations Ohto et al. (2019) to the estimated value $\gamma{\lesssim}65.9{\times}10^{-2}$ J/m2 based on our DFT-PBE structure optimization studies for different geometries. The observed value H2O (1986) $\gamma=7.3{\times}10^{-2}$ J/m2 at $20^{\circ}$C falls into this value range and so does the value $\gamma=14.0{\times}10^{-2}$ J/m2 we selected to estimate the water permeation energy.

Our results for ${\Delta}E_{perm}$ as a function of the number $m$ of the water molecules per unit cell are presented in Fig. 3(c). Based on the sign of ${\Delta}E_{perm}$, we conclude that the H2O/GO system is hydrophilic and the H2O/graphite system is hydrophobic in agreement with observation. The simulation data points are subject to variations caused by the amorphous structure of liquid water that is continuously changing. Among the GO systems, we find GO-OH to be most and GO-epoxy to be least hydrophilic, with the Lerf-Klinowski model containing both OH- and O-epoxy groups to lie in-between. There even appears to be preference for about two water layers in GO according to the Lerf-Klinowski model, in agreement with observation Korobov et al. (2016), and about three water layers in GO-OH, probably due to favorable registry of water molecules with the functionalized carbon layers. We find that energy changes in the system are largest during the initial permeation of water in-between the layers, eventually reaching a saturation value. With $m{\approx}5$ water molecules per water layer in the unit cell, we may consider systems with $m{\gtrsim}15$ to represent a fully hydrated system.

We present the optimized interlayer distance $d$ as a function of water filling in Fig. 3(d). The increase of $d$ due to water permeation, starting from $d{\approx}6$ Å in dry GO, is called swelling. In an RO system containing a GO-based membrane, swelling will be limited by the pressure difference between the two sides of the membrane and by inter-layer bonds that hold a defective GO membrane together.

In agreement with previously reported theoretical Bankura et al. (2013) and experimental data Mähler and Persson (2012), we find the optimum size of the Na+ hydration shell to be close to $6$ Å. Since the interlayer distance $d$ in GO permeated with water is significantly larger, hydrated Na+ ions are likely to permeate in and propagate within the interlayer region in a similar way as bulk water.

To learn more about water flow within slit pores in GO, we first studied the diffusion of a water molecule along the armchair direction in the GO layer, aligned with the $y-$axis in Fig. 2(a). To obtain bias-free energy information along the diffusion path, we performed a set of global optimizations while constraining the $y-$coordinate of the O atom of the water molecule and allowing this atom to move only within the $x-z$ plane normal to GO layers. The energy change ${\Delta}E$ along the trajectory is shown in Fig. 4(a) and a superposition of corresponding snapshots of the molecule is shown in Fig. 4(b). Due the activation barriers for water propagation along the slit pore, the self-diffusion coefficient is reduced significantly with respect to the bulk water value, as seen in Fig. 3(e). Our estimated value of the self-diffusion coefficient of water in GO is $D_{theo}=8{\times}10^{-7}$ cm2/s. The significant reduction in the calculated value of $D$ from bulk water to water in GO is consistent with the observation Raviv et al. (2001); Major et al. (2006) that $D$ is several times lower for water under both hydrophilic and hydrophobic confinement than in bulk water.

We should note that self-diffusion under ambient conditions is unrelated to the pressure-driven propagation of water molecules along the slit pores. Still, the related reduction of the mobility of water molecules adjacent to the GO layers is another validation of the assumptions used in the Hagen-Poisseuille description of pressure-driven flow of liquids through pores and pipes Hunt et al. (2014).

To study pressure-induced water permeation along slit pores, we filled the space in-between GO layers, defined in Fig. 3(a), partly with water. To observe the formation of a meniscus and the dynamics of the propagating water front, we separated segments completely filled with water by water-free segments. All atoms in the water molecules were subject to the force $F=F_{0}{\times}m_{a}$, where $m_{a}$ is the atomic mass and $F_{0}=6.25{\times}10^{-3}$ eV/Å, acting along the slit pores. We performed MD simulations of the evolution of the system lasting about $1$ ps in total and followed the acceleration of the water molecules. Unlike in unconstrained bulk water, water molecules in-between GO layers were slowed down by a drag force $F_{0,drag}$, acting on both H and O atoms and originating in the interaction between the moving molecules and the GO layers. Movies of the propagation of water molecules along the slit pores are presented in Section A of the Supporting Information (SI) des .

For one monolayer of water, the distance between GO layers was fixed at $d=9.03$ Å and the drag force was found to be $F_{0,drag}=5.0{\times}10^{-3}$ eV/Å, almost $80\%$ of the driving force. For three monolayers of water in-between GO layers separated by $d=12.9$ Å, the drag force dropped to $F_{0,drag}=2.5{\times}10^{-3}$ eV/Å, about half the value for a water monolayer. For the sake of comparison, we also considered a monolayer of water contained in-between graphene layers separated by $d=6.57$ Å. Applying the same accelerating force as in GO, we found a drag force of only $F_{0,drag}=0.3{\times}10^{-3}$ eV/Å, one order of magnitude smaller than for water in GO.

We interpret these results in the following way. Whereas the water layers next to GO are nearly immobile, other water molecules in the middle of the slit pore are not affected much by the constraints. This is different in (hydrophobic) graphite, where none of the water layers would be slowed down by contact with graphene.

IV.4 Interaction of H2O molecules with defects

in graphite and GO

As mentioned above, presence of defects and pores in graphite and GO is essential for the permeation of water through corresponding membranes. The two characteristic edges of extended in-layer pores of interest here are the armchair and zigzag edge, obtained by cleaving graphene along the corresponding direction. Under ambient conditions, graphitic edges are not bare, but chemically terminated. The interaction of water molecules with these edges depends to a significant degree on the termination with hydrogen, hydroxy, and oxygen groups. Presence of these groups in the edge termination of GO has been established He et al. (1998) and their functionality should be similar to that in a graphene edge. The optimum geometries of water molecules interacting with graphitic edges is depicted in Fig. 5 and our numerical results for the adsorption energies of water molecules are summarized in Table 1.

Energetically most favorable for the water molecule near a bare graphene edge is dissociative adsorption shown in Figs. 5(a) and 5(f), which splits H2O and binds the resulting OH-radical and hydrogen to the edge covalently with a net energy gain of $4-6$ eV according to Table 1. The metastable physisorbed structures, shown in Fig. 5(b) and 5(g), are bound much more weakly according to Table 1, similar to the adsorption on H-, O- and OH-terminated edges. Among these, there is a small energetic preference for the functionalized armchair edges.

IV.5 Representation of in-layer pores in graphite and GO

In our numerical study, we represent in-layer pores in GO by an infinite array of graphene nanoribbons with chemically functionalized armchair and zigzag edges and changing the in-layer separation width $W$. The structure of such pores is shown in Fig. 6.

IV.6 Permeation of H2O and ions through

in-layer pores in GO

The key functionality of GO membranes in the desalination process is in allowing water molecules to pass through in-layer pores, while suppressing passage of hydrated salt ions such as Na+. Since the interlayer distance in GO permeated with water is significantly larger than the size of hydrated Na+ ions, we must consider the presence of both Na+ ions and H2O molecules in the interlayer region in GO. Then, the passage of hydrated Na+ ions across in-layer pores needs to be investigated in the same fashion as that of water molecules.

Most important for the passage of molecules and ions through an in-layer pore is its geometry, including the edge termination. As mentioned before, the molecular sieving process is not affected by the layer geometry beyond the atomically close vicinity of the pore. Thus, we represent in-layer pores in defective GO layers by infinite arrays of graphene nanoribbons with armchair and zigzag edges, shown in Fig. 6, which are better defined and easier to treat computationally. We considered pores of different width $W$, defined by the separation of closest carbon atoms at pore edges, terminated with hydrogen, hydroxy, and oxygen groups He et al. (1998).

Our results for the passage of isolated water molecules and hydrated Na+ ions through different in-layer pores are presented in Fig. 7. Corresponding movies of permeating water molecules are presented in Section B of the SI and those of permeating hydrated ions in Sections D and E of the SI des . To provide a realistic description of the activated process, we performed global optimization of the system while only fixing in-layer carbon atoms and the height $h$ of the molecule or ion above the in-layer pore. As seen in Figs. 7(a) and 7(b), passage of both H2O and Na+ through most narrow pores with $W=7$ Å is energetically activated, independent of edge geometry and termination. Activation-free passage is only possible through $W=7$ Å pores with H-terminated armchair edges for H2O and O-terminated armchair edges for Na+. While passing an in-layer pore, water molecules lie preferentially in a plane normal to the carbon layer that passes through the pore.

{video}

[t]

\setfloatlinkvideo1.mp4 Concerted motion of water molecules permeating across an in-layer pore, represented by a pair of hydrogen-terminated armchair graphene nanoribbons separated by $W=5$ Å.

According to Figs. 7(c) and 7(d), the activation barrier for water passage drops down to zero in wider pores with $W=9$ Å, whereas the activation barrier for hydrated Na+ ions remains substantial with the exception of O-terminated zigzag edges. As seen in the snapshots from the optimized trajectory in Figs. 7(c) and 7(f), Na+ ions lose most of their hydration shell during the passage through the pore, which accounts for much of the energy cost. As expected, the energy difference ${\Delta}E(h)$ is symmetric around $h=0$.

Taking into the account the van der Waals radius of hydrogen Bondi (1964) of $1.2$ Å, the effective size of an in-layer armchair pore, shown in Fig. 6(a), is reduced significantly in presence of terminating hydrogen atoms. Thus, terminating hydrogens reduce the effective opening of a $W=7$ Å wide pore to $4.6$ Å and that of the even narrower $W=5$ Å pore to merely $2.6$ Å, less than the van der Waals diameter $d_{vdW}$(H2O)$=2.8$ Å of a H2O molecule Jorgensen et al. (1983). In the latter case, the passage of a single H2O molecule requires local deformation of the pore at the energy cost ${\Delta}E=1.1$ eV. Most favorable deformations at the pore edge will likely involve bending and not stretching or compression of C-H bonds, as indicated by observed IR-frequencies in aromatic hydrocarbons, which suggest that bending a $C_{ar}-H$ bond requires three times less energy than stretching this bond Silverstein et al. (2014). The same rule applies for other terminating groups as well. Therefore, in order to allow a water molecule subject to an external force to pass through an in-layer pore, the terminating atoms at the two sides of pore will preferably deflect in the same direction, as indicated in Fig. 8(a), in much the same way as a saloon door swings open when a drunk patron is thrown out.

In a process benefiting water, but not solved ions, energetically activated passage of H2O molecules through narrow in-layer pores may be further promoted by concerted motion involving many molecules connected by hydrogen bonds as discussed in the following Subsection.

IV.7 Concerted motion of H2O molecules through

in-layer pores in GO

Understanding the passage of a single water molecule across the pore is of limited value in view of the fact that water molecules are connected in a network of hydrogen bonds. A more realistic scenario considers many water molecules passing through the in-layer pore at the same time, like a waterfall, depicted schematically in Fig. 8(a). Instead of one water molecule, we have considered a 2D array of molecules confined in a plane normal to and containing the pore in GO. It is appealing to consider water molecules assisting each other energetically in the passage through an in-layer pore by preventing the relaxation of the terminating edges to the ‘closed’ position, same as saloon patrons may keep the swing door open for each other when entering. Our results, shown in Fig. 8(b) for a hydrogen-terminated and in Fig. 8(c) for an oxygen-terminated edge, indicate that concerted motion of water molecules may indeed reduce the energy barrier for their passage from $1.1$ eV to $0.6$ eV in case of hydrogen termination and from $2.3$ eV to $1.3$ eV for oxygen termination, almost to half the original value. Movies of the concerted motion of H2O molecules across in-layer pore are shown in Video 8 and in Section C of the SI.

IV.8 Permeation of different ions through

in-layer pores in GO

As mentioned above, both isolated and hydrated K+ ions are significantly larger than Na+ ions. Nevertheless, K+ ions are known to be transported across protein membranes in preference to Na+ ions Doyle et al. (1998); Zhou et al. (2001). Even though protein membranes are significantly more complex that in-layer pores in GO, the postulated reason reason for this behavior being the stronger hydration shell of Na+ as compared to K+ ions Sadhu et al. (2015) applies in both cases.

We compared permeation of Na+ and K+ ions in water across hydrogen-terminated in-layer armchair pores in graphene with $W=5$ Å and $W=7$ Å. To simulate this process, we performed microcanonical MD simulations of the system that had been initially equilibrated at $T=300$ K. We monitored changes in the potential energy ${\Delta}E_{pot}(t)$ of the system as a function of time. During the run, all water molecules and ions have been subject to an external force characterized by $|{\bf{F_{0}}}|=6.25{\times}10^{-3}$ eV/Å, applied normal to the graphene layers. No constraints have been applied during the $0.6$ ps-long simulations except for fixing all carbon atoms in their positions. Results of our single-trajectory simulations are presented in Fig. 9. We should note that these results should be considered only illustrative of the process and that ensemble averages over many trajectories would be required for any quantitative conclusions.

As expected, the action of the external force gradually increases the total energy and, in view of the virial theorem, also the potential energy of the system. Geometries corresponding to relative minima in ${\Delta}E_{pot}(t)$ graphs are to be considered more stable. We found it interesting that the point of ion penetration across the layer, defined by $h=0$, is not prominent in any of the graphs. This indicates that the relative position of water molecules with respect to the pore plays as important role as that of the ions. In view of the uncertainty caused by the finite size and complex structure of the system, ions pass faster through wider pores than through narrower pores. We found relative maxima in ${\Delta}E_{pot}(t)$ in all trajectories at $t{\lesssim}200$ fs from the beginning and associate these peaks with the necessity of structural rearrangements in water molecules adjacent to the ions and the pore that would subsequently enable the passes of the ion across the pore. With respect to this configuration prior to passage through the pore, the one corresponding to $h=0$ may be either more stable as in Fig. 9(c), or less stable, as in the other panels. Based on these preliminary results, we can not find any conclusive evidence for preferential passage of Na+ or K+ ions through in-layer pores in graphene or GO.

V Discussion

The major objective of our study was to find whether GO is a viable material for water desalination Boehm et al. (1961) as postulated by many since the 1960’s. To mitigate the brittle nature and related shortcomings of GO, we postulated a sandwich structure containing ultra-strong carbon nanotubes and graphitic carbon fabric in the outer layers enclosing the GO core layer. One of the key motivations for our study has been the urgent call for a paradigm shift in the desalination membrane community, largely unaware of recent progress in ultra-strong carbon nanotubes, and the desire of the carbon nanotube community to locate a ‘killer application’. Based on what has been known about each component of the system, and based on our microscopic ab initio calculations detailing atomic-scale processes occurring in the GO layer, we find the proposed all-carbon membrane well worth considering seriously for water desalination and encourage its experimental realization.

The proposed membrane design addresses the most urgent problems in water desalination, namely selective salt rejection falling short of its objective Werber et al. (2016b) and irreversible bio-fouling. We have identified structural parameters that should maximize salt rejection by GO without sacrificing water permeability. We believe that mechanical treatment of GO by ball milling and shear alignment may provide a uniform system well suited for its purpose. Graphitic carbon nanostructures forming the layers surrounding GO are now well established for their mechanical strength, flexibility, electrical and thermal conductivity, and resilience to extreme heat and to chemicals. Very important is the fact that all components of the all-carbon membrane are commercially available at very low cost.

Selective rejection of ions and reduction of its bio-fouling propensity may result from charging the conductive membrane possibly by a pulsed current Ho et al. (2018). Additional resistance against bio-fouling may be provided by continuously changing the morphology of the outermost layers of the proposed all-carbon membrane Pocivavsek et al. (2018). This should be possible, because individual components are graphitic structures that, under compression, may kink or wrinkle, but not uniformly shrink due to the high in-plane rigidity or graphene. Since reverse osmosis membranes are not static objects, but rather vibrate under operating conditions, the membrane surface will continuously change its morphology and thus detach debris and bio-fouling agents Pocivavsek et al. (2018).

Key to selective rejection of ions and permeation of water molecules is the morphology and chemical treatment of the central GO layer. The hydrophilic nature of GO can be regulated by modifying the density of OH terminating groups. The size and nature of in-layer pores, which we find critical for the desalination process, can initially be optimized during the ball milling and shear alignment process. Membranes with fine pores are suitable for deionizing water, whereas coarse-pored membranes will find their use in multi-stage RO systems or in decreasing the salinity of brines. Besides their smaller size compared to hydrated ions, water molecules benefit from their ability to pass through even very narrow pores by concerted motion with quite low activation barriers. Even $5$ Å wide pores are still permeable while providing high salt rejection.

According to our results, $7$ Å pores provide the best permeability ratio between H2O an Na+, whereas still wider $9$ Å pores provide lower selectivity. The mechanical strength and flexibility of graphitic carbon allows adjusting the operating pressure, which should modify the pore size in order to adjust salt rejection and water flux as needed. Measuring and feeding these data continuously to a control system should allow on-the-fly tuning of operating pressure in response to changing feed water parameters Lior et al. (2012). Pores should open at higher and close at lower pressures, modifying the performance of the membrane. The range of operating parameters, in particular pressure, which optimize specific aspects of the membrane performance, can be studied in pilot plants and provided to the consumer by the manufacturer.

According to our simulations and common chemical intuition, presence of OH groups is largely responsible for turning hydrophobic graphite into hydrophilic GO. As shown in our study, hydrophilic interlayer slit pores in GO attract ions and reduce their mobility, thus further decreasing permeation of salt in a similar way as in-layer pores do Dai et al. (2016). For this reason, increasing the number of OH groups at the expense of epoxy-groups is highly desirable from the point of view of salt rejection.

As mentioned in the text, edges of in-layer pores are unlikely to remain bare and will acquire terminating groups under ambient conditions. We studied three likely candidates for chemical termination, namely H, epoxy-O and OH groups, but did not observe significant differences from the viewpoint of water and salt permeation except a minor preference for hydrogen termination. Given a fixed in-layer pore width $W$, hydrogen termination hinders entering water molecules least due to its small size. Also, the activation barriers for Na+ passage through H-terminated pores are not the highest. However, hydrogen-terminated pores provide the highest ${\Delta}E$(H2O)/${\Delta}E$(Na+) ratio, which means they maximize the permeability difference – the quantity we strive to increase. Hydrogen termination is also favored, because C-H groups interact only weakly with dissolved ions.

As we can infer from Fig. 7, Na+ ions appear to be stabilized inside some O- and OH-terminated pores, resulting in a “negative barrier height”. The reason for this behavior is the arrangement of O atoms in these pores, which provides an excellent coordination sphere for sodium cations, resembling the structure the ion channels selectivity filter Doyle et al. (1998). If the cation binding is too strong, the first cation that enters the pore would clog it. If the binding is weaker, small energetic preference for the in-pore site may promote cation passage by the adsorption-desorption mechanism. Unlike O- and OH- terminating groups, terminating hydrogens act mostly like a mechanical brush without the potential negative side effects.

VI Summary and Conclusions

We have designed an all-carbon membrane for the purpose of filtering and desalinating water. The membrane contains graphite oxide (GO) sandwiched in-between layers of buckypaper consisting of carbon nanotubes, and the entire structure is contained in-between layers of strong carbon fabric from both sides. The structure combines high mechanical strength with thermal stability, resilience to harsh chemical cleaning agents with electrical conductivity, thus addressing major shortcomings of commercial reverse osmosis membranes. We have used ab initio DFT calculations to provide atomic-level insight into the permeation of water molecules in-between GO layers and across in-layer vacancy defects. Our calculations elucidate optimum conditions for permeation of water and selective rejection of solvated Na+ ions by the membrane. We anticipate that the viability of the designed structure will be confirmed experimentally in the near future.

Note added in proof. We have recently learned of a different all-carbon desalination membrane, which has demonstrated superior performance over current technology Yang et al. (2019).

Acknowledgements.

D.T. acknowledges financial support by the NSF/AFOSR EFRI 2-DARE grant number EFMA-1433459. A.K. acknowledges financial support by the Fulbright program. We thank Igor Baburin, Morinobu Endo, Katsumi Kaneko, Takeyuki Kawaguchi, James Linnemann, Dan Liu, Alexander Quandt, Gotthard Seifert, Alexandr Talyzin, Akihiko Tanioka and Volodymyr Tarabara for valuable discussions. Computational resources have been provided by the Michigan State University High Performance Computing Center.