Non-Continuum Intercalated Water Diffusion Explains Fast Permeation

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Non-Continuum Intercalated Water Diffusion Explains Fast Permeation through Graphene Oxide Membranes Shuping Jiao, and Zhiping Xu ACS Nano, Just Accepted Manuscript • DOI: 10.1021/acsnano.7b05419 • Publication Date (Web): 25 Oct 2017 Downloaded from http://pubs.acs.org on October 25, 2017

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Non-Continuum Intercalated Water Diffusion Explains Fast Permeation through Graphene Oxide Membranes Shuping Jiao and Zhiping Xu* Applied Mechanics Laboratory, Department of Engineering Mechanics, and Center for Nano and Micro Mechanics, Tsinghua University, Beijing 100084, China *

Email: [email protected]

ABSTRACT: Recent experimental studies have revealed unconventional phase and transport behaviors of water confined within lamellar graphene oxide membranes, which hold great promises not only in improving our current understanding of nanoconfined water, but also in developing high-performance filtration and separation applications. In this work, we explore molecular structures and diffusive dynamics of water intercalated between graphene or graphene oxide sheets. We identify the monolayer structured water between graphene sheets at temperature T below Tc = ~315 K and an interlayer distance d = 0.65 nm, which is absent as the sheets are oxidized. The non-continuum collective diffusion of water intercalation between graphene layers facilitates fast molecular transport due to reduced wall friction. This solid-like structural order of intercalated water is disturbed as T or d increases to a critical value, with abnormal declines in the coefficients of collective diffusion. Based on a patched model of graphene oxide sheets consisting of spatially distributed pristine and oxidized regions, we conclude that the non-continuum collective diffusion of intercalated water can explain fast water permeation through graphene oxide membranes as reported in recent experimental studies, in stark contrast to the conventional picture of pressure-driven continuum flow with boundary slip, which has been widely adopted in the literature but may apply only at high humidity or in the fully hydrated condition.

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The presence of water intercalation in nanoconfined channels and slits has called upon much attention because of its relevant applications in filtration, separation and energy storage. Nanofluidic applications based on lamellar membranes have been proposed for highly selective transport of gas, liquid molecules and ions, as well as efficient heat exchange.1-7 With molecular structures distinctly different from that in the bulk,8-10 water intercalated within lamellae such as graphene oxide (GO) membranes is expected to exhibit unique diffusive, flow behaviors and unconventional mechanisms for selective mass transport. Recent experimental and theoretical studies have demonstrated the existence of two-dimensional (2D) ordered water structures while intercalated within graphitic, MoS2 layers or between graphene and mica/MoS2 substrates, as a result of the nanoconfinement and/or pressure.11-16 Although the nature of ‘ice’ formation under nanoconfinement is still not clear from experimental evidences, and distinction between ice and metastable water needs further clarification,12,

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computational studies have

demonstrated the existence of ordered water structures as intercalated between paralleled hydrophobic plates, in forms of square, hexagon, pentagon or mixed lattices.8-10 More interestingly, the phase behaviors and transitions of nanoconfined water shows high contrast with those of bulk water, and are controlled by a number of parameters such as pressure, temperature, interlayer spacings and the interfacial commensurability between the water lattices and solid walls.10, 11 The solid nature of intercalated water implicates interesting kinetic and dynamical behaviors that differ from the conventional picture of molecular liquids, as demonstrated in recent studies for water on solid surfaces or within nanopores.18-23 Ultrahigh permeation rates of water vapor through GO membranes were reported recently, with an enhancement factor of ~107 compared to the non-slip viscous-flow prediction.24 This unexpected enhancement was attributed to the slippery nature of water flow between pristine graphene regions in the GO sheets, which offer smooth potential energy surfaces, highly-reduced wall friction, and the capillary pressure acting as an additional driven force for the flow. However, either the form of water intercalation or the microstructures of GO membranes has not been resolved with direct experimental evidences. At high humidity or a fully hydrated condition, the interlayer gallery of the membrane may be filled up by water and thus a pressure-driven flow mechanism can be 2


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manifested in a percolated network. However, for vapor- or liquid-phase water transport in a partially hydrated membrane, the continuum flow picture may fail to capture the underlying physics. In this situation, as the interlayer distance in GO membranes is much shorter than the mean free paths of gas molecules at ambient condition, the hydrodynamic pressure between water intercalations cannot be transmitted, and their collective diffusion instead of advection may contribute dominantly to the mass transport through the membrane as a percolated flow network cannot be attained. However, this diffusioncontrolled mechanism of water transport through GO membranes has not been discussed yet. In this work, we explore the molecular structures and collective diffusion of finite-size water clusters intercalated between two parallel graphene or GO layers, by performing molecular dynamics (MD) simulations. The results are discussed to elucidate the underlying mechanism of efficient mass transport through GO membranes. We calculate the coefficients of collective diffusion and wall friction, which demonstrate abnormal dependence on temperature and the interlayer distance, conforming to the undergoing solid-liquid structural transition. The contribution of collective water diffusion to mass transport through the GO membrane is estimated, which explains the fast water permeation measured in experiments, offering an alternative picture to the continuum flow with boundary slip.

RESULTS AND DISCUSSION Molecular Structures and Solid-Liquid Phase Transition of Intercalated Water. To model the intercalated water, we place a droplet of NW water molecules between two graphene sheets at a specific interlayer distance d. The top and bottom layers are constrained in plane. Experimental evidences show that the d in GO membranes increases the relative humidity (RH), due to the formation of water capillary in hydrophilic regions with oxygen-rich functional groups.4,

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Monolayer structured water with a 2D quasi-

square lattice are identified at d = 0.65 nm and T = 300 K (Figure 1a), which aligns with recent simulation and experimental results.9, 12 This value of d corresponds to a local energy minimum of graphene layers containing the intercalation with NW = 58-986 water molecules (Figure S1). However, this regular 2D lattice structure no longer exists as T or

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d increases, which can be quantified from the changes in the structure factors (Figures 1b and S2). Specifically, the order-to-disorder structural transition is first-order,25 occurring as T increases to ~310-325 K for intercalated water with d = 0.65 nm and NW = 58-986. The transition temperature Tc slightly decreases with NW. At T = 300 K the structural transition also occurs by tuning d, at dc = 0.77 nm for NW = 221, 486 and 986 (Figure S2). These results indicate that the ordered 2D quasi-square structure is sensitive to temperature and the geometric confinement. No ordered water structure can be identified for the intercalation thicker than a bilayer by increasing d in our MD simulations. Comparative simulations are carried out for water intercalated between GO layers, where no ordered lattice structure form because of the surface roughness due to the presence of hydroxyl functionalization, as well as the hydrogen-bond (H-bond) interaction between the hydroxyl groups and water molecules that perturbs the H-bond network within intercalation (Figure 1a). Diffusive Behaviors of Intercalated Water. We measure the coefficients of collective diffusion D for intercalated water droplets from the MD-simulated trajectories, by using the Einstein’s relation. The results show distinct dependence of D on the size of droplet at d = 0.65 nm and T = 300 K, which decreases rapidly with NW (Figure 2a). With NW changes from 58 to 986, D decreases from 30.87×10-5 cm2/s to 1.49×10-5 cm2/s (Figure 2a). This size effect was previously reported for water droplets diffusing on the top of graphene,18 which indicates that the collective diffusion is driven by thermal fluctuation, and the value of D would then scale inversely with the mass. For comparison, the molecular self-diffusion coefficient of water in the bulk phase is D0 = 2.5×10-5 cm2/s at ambient condition.18 On the other hand, increasing d from 0.65 to 0.78 nm enhances D by 1-3 times, to D = ~10-4 cm2/s for NW = 221-486 (Figure 2b), resulted from the weakened confinement. These results suggest that the intercalated water could diffuse in a collective way that is much faster than the self-diffusion of water molecules in the bulk phase, which may lead to rapid water transport in graphitic channels. This enhanced transport could be attributed to the fact that the collective diffusion of intercalated water, which is absent in the bulk, is critically modulated by the friction at the solid-liquid interface, while the pristine graphene provides an atomistically smooth surface with ultralow friction that facilitates fast collective mass transport over it. The coefficient of self-

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diffusion in the intercalated water, however, is much lower (by two orders) than the value in the bulk within the temperature range we explored. To gain more insights into the nature of collective diffusion, we then explore the Tdependence of D for water intercalation with NW = 986. As shown in Figure 3a, D increases with T in general, but exhibits a sudden drop in its value at T = 305-315 K, which corresponds to the critical temperature for the order-to-disorder transition in intercalated water (Figure 1b). This finding reveals that the collective diffusion of intercalated water is closely associated with its molecular structures, and the change in phases of water leads to a shift in the diffusion mechanisms. To further illustrate this fact, we carry out MD simulations for droplet sizes with NW = 221 and 486 (Figure S3). The simulation results demonstrate abnormal reduction of D as well, at T = 320-325 K, 315320 K, respectively, which is consistent with the critical temperature Tc predicted for the solid-liquid structural transition (Figure S4). Moreover, we find that the statistical error in D near Tc (Tc = 300-315 K for NW = 986) is significant and the values of D along the x (armchair) and y (zigzag) directions (Dxx, Dyy) are anisotropic below Tc, but isotropic above Tc. This contrast can also be seen from the distribution of diffusive distance ld (extracted for each 30 ps interval), which demonstrates this anisotropy-isotropy transition at Tc (Figure 4b). This is a direct consequence from the structural change with T, where water intercalation features a crystalline lattice below Tc. The relative orientation of ice lattice aligned to the graphene sheet shows discrete jumps of π/3 along the path of diffusion, which matches the lattice symmetry of graphene (Figures 4c and 4d). Along with these findings, the T-dependence of D can also be attributed to the interfacial friction between intercalated water and the graphene wall. We calculate the interfacial friction coefficient λ for water intercalation with NW = 986 and d = 0.65 nm, and find a rapid increase in λ near Tc (T = 310-315 K) that indicates a solid-solid interface is less resistant to the collective diffusion than the solid-liquid interface (Figure 3b). This rapid jump of λ conforms to the abnormal decline of D as we have seen in Figure 3a, along with the structural transition. The order-disorder structural transition could also be triggered by tuning the interlayer distance d. From the MD simulation results, we find that D for the collective diffusion increases with d in general as the nanoconfinement enforced by the graphene or GO 5


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sheets is weakened, and there is a sudden drop at d = ~0.77 nm where the monolayer solid-like order is lost, as shown in Figure 2b specifically for NW = 221 and 486. This critical value of d aligns with the solid-liquid transition as indicated by the structure factors of intercalated water (Figure S2). These results further confirm our conclusion made on the correlation between collective diffusion of intercalated water and its molecular structures. As there is no in-plane structural order in the intercalated water between oxidized graphene sheets, the efficient mass transport by collective diffusion could be much reduced. We consider oxidized graphene sheets functionalized with hydroxyl groups (O:C = 20%) and d = 0.65 nm and 0.9 nm. Our MD simulations show that, the narrow channel with d = 0.65 nm limits the collective motion of water intercalation, and the value of D is ~10-8 cm2/s. However, at a larger interlayer distance d = 0.9 nm, clustered water intercalation could form. The coefficient of its collective motion increases with NW, d (Figures 2c and 2d). These results indicate that the intercalated water in hydrated GO membranes could diffuse 3-4 orders lower than that between graphene sheets at the same interlayer distance, and 2-3 order lower than that for molecular diffusion in bulk water. It should be noticed here that the structure of water intercalation between oxidized graphene sheets is highly sensitive to the spatial distribution of chemical groups, and its effect awaits further investigation. In this work, to exclude the effects of static and thermal corrugations in the highly defective GO sheets, we consider both full and partial planar constraints in the MD simulations, which shows that this effect can be neglected. With the remarkable dependence of D on the molecular structures of intercalated water identified from MD simulations, we then probe the dynamical behaviors of collective diffusion. We track the center-of-mass trajectory of the intercalated droplet, with NW = 986 for example, and find a distinct change in the pattern of diffusive motion near Tc (Figure S5). The distributions of stepwise displacements dx and dy have longer tails at temperature below Tc than that above Tc (Figure 4b), demonstrating deviation from the normal Gaussian behavior before the structural phase transition takes place, which is recovered as T increases beyond Tc. The deviation from Gaussian behavior can be captured by the non-Gaussian parameter α2(t) = 3 /(52) - 1.26, 27

Our MD simulation results suggest that the parameter α2 reaches a peak and then 6


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decreases at a characteristic time scale of td = ~20 ps for T > Tc (liquid-like), and ~100 ps for T > Tc (solid-like) (Figure 4a). Consequently, the heavy-tail nature of diffusion observed here could also be responsible for the faster diffusion of ordered intercalation compared to the normal diffusion, and our total simulation time for the calculation of D, 10 ns, is sufficient to extract the coefficient of collective diffusion using the Einstein’s relation. Mass Transport through Lamellar GO Membranes. Ultrafast mass transport of water vapor through GO membranes was reported recently, and the measured water evaporation rate with the presence of a GO film is almost the same as that in the absence of the film.24 This much-enhanced mass transport was attributed to the highly slippery nature of capillary force driven viscous flow within the percolated 2D channels. However, even considering the finite slip length of water flow between graphene sheets (~60 nm), the factor of flow enhancement estimated theoretically for the slip flow is ~102, which is much lower than the factor reported from experiments.20 Moreover, the formation of a continuous percolation network across the whole GO membrane has not yet been validated by any experimental evidence, which could be difficult considering the irregular patterns of oxidized and pristine graphene regions in the GO sheets and the complicate brick-and-motar microstructures, especially at low humidity.28 An alternative view of the advective mass transport process is that water transported may exist in forms of 2D intercalated droplets, which could diffuse in both the oxidized and pristine graphene regions but with contrastive diffusion constants. Recently, an experimental study of nanoscale graphitic channels shows that liquid transport inside the channels is the rate-limiting process, compared to the diffusion and evaporation outside.29 We could thus assume the rates for water vapor to condense into, or evaporate from the membrane are high enough so that the diffusive mass transport within the membrane is the rate-limiting process. Specifically, for diffusion between graphene sheets with an average lateral size of L, quantitative relation could be derived between the microstructural characteristics and permeability based on our simulation results for the diffusivity. By assuming that the water intercalation is randomly distributed in the interlayer gallery, diffusing through the membrane through the shortest path in the complicate network (Figure 5), the characteristic distance for an intercalation to diffusive 7


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across one layer of the stacks is l = 0.25L. In this analysis, we neglect the cross-layer barrier for the collective diffusion of intercalated water droplets, which is reasonable for nanopores or nanoslits with size larger than or comparable with the droplets, or for GO sheets with large lateral span and low density of nanopores, where lateral diffusion dominates the mass transport process. One can then extract the value of l from the permeability Pr measured from experiments, that is (mW/A)/(l2/4D) = Pr or l2 = 4DmW/(APr)

(1)

Here mW/A is the areal mass density of water intercalation in the interlayer space available for the collective diffusion. The area of the intercalation we studied in this work is ~10-100 nm2, which is compared to the values reported from experimental studies.28, 3032

Our MD simulation results predict that the value of D is in the range of 1.49 to

30.87×10-5 cm2/s for water intercalation with NW = 986-58 and d = 0.65 nm at 300 K (Figure 2a). The measured vapor permeation through GO membranes shows that the values of d and Pr increase with the RH, which reach ~1.1 nm, 0.48 mgh-1mm-2

at 100% RH,

respectively.24 The permeation rate is Pr = ~0.024 mgh-1mm-2 at the RH corresponding to the condition d = 0.65 nm.24 Although at high humidity, continuous percolated networks of water in the GO membrane could form and the pressure-driven flow may dominate the mass transport process,30 collective diffusion is expect to control at low RH conditions,28 where intercalated water cannot fill the whole transport pathway in the GO membrane. As there are no experimental measurements for the amount of water intercalated in GO, we consider a range of areal number density N/A of water molecules from 1 µm-2 to 103 µm-2 in estimating l, which corresponding to the mass density mW/Ad = 0.088-88 g/m3. The values of l estimated from Eq. 1 for specific size of the intercalation NW are summarized in Figure 5a. The results show that l decreases with NW and decreases with mW/A for a certain permeation rate Pr measured from experiments. Specifically, the length of diffusing path l ranges from a few nanometers to ~0.7 µm for NW = 58-986, and for the diffusion of a single water molecule (NW = 1), l is ~0.17-5.4 µm. These estimated values of l is in a reasonable range considering the typical GO sheet size in the membrane is ~1 µm,24 and the smaller value of l than L could be attributed to the presence of trans-

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sheet nanopores that cut the diffusive paths within the 2D interlayer gallery and the irregular shape of GO sheet. Intercalated water could also diffuse over the hydrophilic regions of GO sheets that are functionalized with oxygen-rich groups. We extend our estimation of l by considering the collective diffusion of intercalated water droplets within this pathway to the overall mass transport across GO membranes. With the experimentally measured value Pr = 0.024 mgh-1mm-2 at d = 0.65 nm, and using the values of D calculated from our MD simulations (Figure 2c), we estimate that l = ~0.062-4.55 nm for NW = 986-58, N/A = 11000 µm-2. l could be interpreted here as the averaged inter-pore distance in the oxidized region considering its highly defective nature.33,

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We also make estimations for the

measured permeability Pr = 0.21 mgh-1mm-2 at d = 0.9 nm, which yield l = ~0.060-7.97 nm for NW = 986-58, N/A = 1-1000 µm-2 (Figure 5b). The largest value of l estimated here is comparable to the value ~1-10 nm characterized from GO samples, although one should be noted that these values highly depend on the preparing processes of GO membranes.35-38 This agreement leads to the conclusion that both fast diffusion in the interlayer gallery between pristine graphene regions and slow diffusion a cross the GO sheets contribute to the high permeability rate of water vapor measured in experiments.24 Consequently, our results can be used to explain the ultra-high rate of water mass transport, through an alternative mechanism manifested by efficient collective diffusion of intercalated water droplets through percolated graphene channels, or nanopores in the oxidized region, as illustrated in Figures 5c and 5d. This picture differs from the slipflow view that has been widely adopted in recent studies.20, 24, 39 A clarification of the underlying molecular processes calls for more detailed experimental characterization of the membrane microstructures, as well as the distribution of defects and intercalated water content and size. More detailed investigation of RH-dependent mass transport could also offer some clues that may identify a crossover from the diffusion-dominated mechanism to slip flow. Additional Remarks. It should be noted that our discussion on diffusion-controlled mass transport in GO membranes is limited by excluding a few factors that are related to their complex microstructures. First, both pristine and oxidized regions in the GO sheets have

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a limited areal span. Consequently, in a pristine channel with size comparable to the intercalated droplet, the collective diffusion could be retarded by the side-pinning effect resulted from the capillary force, and the droplet could even attract the droplet into the oxidized region.39 Secondly, wide channels could form in the wrinkled regions of GO sheets, the nanodroplets would then need to diffuse over the surface of GO sheets.18 Thirdly, the layer-by-layer microstructure of GO membranes indicates that the crosslayer transport through nanopores or nanoslits may become the rate-limiting process due to the presence of a cross-layer energy barrier, especially in the situation of very thin membranes with high densities of nanopores or nanoslits.40, 41 Although the intercalated droplets could diffuse collectively over pores or slits with size larger than or comparable with them, evaporation-condensation process may be critical for diffusion through small pores. Conclusive arguments on the consideration on these complexities requires further clarification. The understanding of collective diffusion explored in this work could also shed light on the selective gas and ion transport processes through GO membranes that are not fully wetted with water continuum inside. The water intercalation could enhance the contrast in permeability of gas molecules by blocking non-solvable molecules, while carrying solvable ones, although the diffusive coefficient of solvable gases such as CO2 is reduced once immersed in water.42 Ions that are immobile in dry GO membranes could be encapsulated in the droplet and diffuse with the water shell.43,

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The mechanism of

collective diffusion identified for the intercalated water and the calculated coefficients of diffusion enable quantitative estimation of the contributions for mass transport through both the pristine and oxidized regions, provided with their areal fraction and spatial patterns identified for the GO membranes.

CONCLUSION In this work, we explored diffusion of intercalated water between pristine and oxidized graphene sheets in order to understand water vapor permeation through GO membranes. Our MD simulation results show that the diffusive behaviors of droplets are defined by the structural ordering in it, which demonstrate anomalous deviation from normal Gaussian diffusion and reduced interfacial friction below the critical temperature for

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solid-fluid structural transition. The calculated coefficients of diffusion are implemented in a patchy structural model of GO sheets to estimate the characteristics of diffusive pathway within the lamellar GO membrane, which explains the high permeation rate identified in recent experimental studies, indicating that the non-continuum diffusive processes play a dominant role in mass transport at low humidity and offers an alternative picture to the continuum slip flow of water across GO membranes at high humidity or in a fully hydrated condition.

Models and Methods Molecular Structures. Models with NW = 58-986 water molecules are constructed, which are intercalated between two 20×20 nm graphene or GO sheets (Figure 1a). The number of water molecules, or the lateral span of water droplets considered here covers the typical range of pristine/oxidized region sizes in the graphene oxide sheets.33,

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Consequently, the structural and dynamical properties of water droplets investigated here could capture the characteristic mass transport processes of water intercalation in graphene oxide membranes. An open boundary is used in the z direction and periodic boundary conditions (PBCs) are applied in the in-plane x and y directions. The proposed molecular models of GO sheets in the literature consist of hydroxyl and epoxy groups on the basal plane, and carbonyl groups at defective sites and open edges.45 For the surface functional groups on graphene, hydroxyl groups are reported to be able to stay enriched in the long-living quasi-equilibrium state,45 and a typical fraction of hydroxyl species relative to the amount of carbon atoms in GO is ~20%.46 Considering these experimental evidences, we constructed in this work hydroxyl-functionalized graphene with c = nOH/nC = 20%. Here nOH and nC are the numbers of hydroxyl groups and carbon atoms, respectively. The distribution of hydroxyl groups was sampled randomly on both sides of the sheet. Molecule Dynamics Simulations. We perform molecular dynamics simulations using the large-scale atomic/molecular massively parallel simulator (LAMMPS).47 The allatom optimized potentials for liquid simulations (OPLS-AA)48 are used for graphene and GO sheets. The SPC/E model49 of water is used in our study, which is widely adopted for MD simulations of water transport as it predicts reasonable density, diffusivity and

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viscosity compared to other models.8, 49-52 The effect of water models on the intercalated water phases were discussed in our previous work.25 The conclusions in this work are cross-checked by comparing with results simulated using the TIP4P model. We find that, although the critical temperature and interlayer distance for structural order transition are different in these two different models, the characteristics of observations agree well, demonstrating the robustness of the conclusion we made. Long-range Coulomb interactions are computed by using the particle-particle particle-mesh algorithm (PPPM).53 The interaction between carbon atoms in graphene and oxygen atoms in water is modelled with parameters εC-O = 4.063 meV and σC-O = 0.319 nm, which predict a water contact angle (WCA) of θc,G = 98.4° for graphene, in consistency with the value measured experimentally.20, 54 The WCA θc,GO for GO is lower than θc,G, and decreases with the concentration c of oxygen-rich functional groups. For a typical value of c = 20% for GO, our simulation results is θc,GO = 26.8°, which is also close to recent experimental reports.39 The time step to integrate the equation-of-motion is 1 fs, with the SHAKE algorithm applied for the stretching terms between oxygen and hydrogen atoms of water to avoid the need of integration with very short time steps for hydrogen-atom-related high-frequency vibrations. Product simulations are carried out after 1 ns equilibration at specific temperature using the Nosé -Hoover thermostat with a damping time constant of 100 fs. In the simulations, we consider both full and partial (by fixing a few carbon atoms randomly) planar constraints of carbon atoms in the GO sheets that do not lead to notable difference for the conclusions we draw in this work (Figure S6). Key parameters used in our model are listed in Table 1.55 Structure Factor Analysis. To assess the structural order of intercalated water between graphene and GO layers, the in-plane structure factor is calculated as8, 56 2 2  N   NW  1  W S(q) = ∑ cos(q.ri )  + ∑sin(q.ri )   N W  i =1   i =1  

(2)

Here rj is position of the j-th oxygen atom in the plane of intercalation, NW is the number of water molecules within the intercalation, q is a planar wave vector, and denotes the time average in thermal equilibrium.

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Analysis of the Diffusive Behaviors. To elucidate the nature of diffusion, we analyze the distribution of diffusion distance ld in a constant interval of simulation time td. We consider the value of td ranging from 20 to 100 ps. We find that the distribution profile of ld or the stepwise displacements dx and dy depends on the choice of td, but the function that can be used to well fit the data is not affected for td < 100 ps, and thus we choose td = 30 ps for our discussion. Characterization of Molecular and Collective Diffusion. In the Gaussian regime, the molecular diffusion constant D is calculated from the trajectories of water molecules by using the Einstein’s definition relating the correlation function of atomic positions ri, or the mean-square distance (MSD), to the value of D = limt->∞/2dit. Here di is the dimension of space one explores, t is the simulation time, and is the ensemble average. The collective motion of intercalated water in the 2D interlayer gallery (di = 2) is calculated from its center-of-mass trajectories. In our MD simulations with a time span of a few nanoseconds, the MSD of water molecules is calculated based on the time-series of atomic (oxygen) positions. Time averaging and statistical error analysis are performed for 5 independent MD runs, with 3000 or more time-series in each run that are broken down into segments of 100 ps duration that starts at different simulation time. Molecular structures of the whole system are equilibrated using the Nosé-hoover thermostat at 290-350 K for 10 ns, to assure the convergence of calculated values of D as a function of the simulation time (Figure S7). We extract MD-simulated trajectories for the calculation of diffusion constant after the first 1 ns record for equilibration. Calculation of the Interfacial Friction Coefficients. As a parameter that quantifies energy dissipation at the water/wall interface, the friction coefficient λ can be calculated via the Green-Kubo (GK) relation through the autocorrelation function (ACF), , of fluctuating pairwise forces at equilibrium

λ=

1 AkΒT

∫

∞

0

F ( t ) F ( 0 ) dt

(3)

Here F(t) is the total forces exerted on the water intercalation from its interaction with the graphene sheets, kB is the Boltzmann constant, A is the contact area between intercalated water and graphene or GO sheets, and T is the temperature. We calculate the ACF of F(t)

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in our 5 ns-long equilibrium MD simulations. It should be pointed out that there is a welldocumented difficulty to obtain the GK relation via equilibrium MD simulations due to the finite-size effect, which often leads to vanishing of friction coefficients for a very long time simulation.57, 58 As a result, the integration of ACF in Eq. 3 should be done within a reasonable cutoff time tc to resolve this issue. A widely adopted recipe is to use the time corresponding to the onset of a plateau in the integration as tc.57, analysis, tc is estimated to be in the range of 1-2 ps.

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58

In our

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ASSOCIATED CONTENT Supporting Information The Supporting Information is available free of charge on the ACS Publications website at DOI: 10.1021/acsnano.XX. Figures S1−S7 offer more details on the simulation results and analysis (PDF).

AUTHOR INFORMATION Corresponding Author *E-mail: [email protected] Author Contributions Z. X. designed the project. Both authors performed the study and wrote the manuscript. Notes The authors declare no competing financial interest.

ACKNOWLEDGMENTS This work was supported by the National Natural Science Foundation of China through Grants

11472150,

the

National

Science

&

Technology

Major

Project

(2016ZX05011003), the Tsinghua University Initiative Scientific Research Program through Grant 2014z22074, and the State Key Laboratory of Mechanics and Control of Mechanical Structures (Nanjing University of Aeronautics and Astronautics) through Grant No. MCMS-0414G01. The computation was performed on the Explorer 100 cluster system at Tsinghua National Laboratory for Information Science and Technology.

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Tables, Figures and Captions

Table 1. The 12-6 Lennard-Jones (L-J) potential parameters and atomic charges used in the MD models of graphene and GO. The L-J parameters for atomic pairs are evaluated using the Lorentz-Berthelot mixing rules.

models

sites

ε (Kcal/mol)

σ (nm)

q (e)

bond length Ref. (nm)

graphene

C

0.06831

0.33997 0

lC-C = 0.142

38

GO

C

0.06831

0.33997 +0.1966

lC-O = 0.141

38

O

0.15535

0.3166

-0.5260

lO-H = 0.0945

H

0

0

+0.3294

O

0.15535

0.3166

-0.8472

H

0

0

+0.4236

water

16


lO-H = 0.1

39

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Figure 1. (a) Molecular structures and the hydrogen-bonding network of intercalated water confined between two graphene sheets. (b) 2D structure factors of intercalated water between graphene, calculated at T = 300, 310, 315 K at d = 0.65 nm and NW = 986.

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Figure 2. The dependence of collective diffusion coefficients D of intercalated water on (a) the number of water molecule NW at d = 0.65 nm and T = 300 K, and (b) the interlayer distance d between graphene sheets with NW = 221,486 and T =300 K. (c) The dependence of D on NW at d = 0.65 nm, d = 0.9 nm and T = 300 K, and (d) on d for GO sheets with NW = 986 and T = 300 K. Here d for GO is defined as the distance between two basal carbon sheets here.

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Figure 3. (a) The dependence of collective diffusion coefficients D of intercalated water on the temperature T at d = 0.65 nm and NW = 986. (b) Friction coefficients calculated for the intercalated water at d = 0.65 nm and NW = 986, which demonstrate distinct dependence on T. λ is friction coefficient and A is area of the 2D water droplet.

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Figure 4. (a) Non-Gaussian parameter α2 calculated for d = 0.65 nm and NW = 986 at temperature close to Tc, the transition temperature. (b) Distribution of discretized molecular displacements within a time step of td = 30 ps, dx and dy are the in-plane displacements along the x and y directions. For T < Tc = 315 K, subplots show heavytailed distributions that deviate significantly from Gaussian diffusion, while for T ≥ Tc, the Gaussian behavior is recovered. (c) The distribution and (d) time evolution of the angle between the water lattice and graphene lattice (see the inset in panel c) at d = 0.65 nm and NW = 986.

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Figure 5. (a, b) The length l of collective diffusive paths in lamellar GO membranes, estimated by Eq. 1, for water intercalated between (a) pristine (d = 0.65 nm, T = 300 K) and (b) oxidized (d = 0.9 nm, T = 300 K) graphene sheets. (c,d) Illustration of the diffusive transport processes for water vapor permeation through GO membranes. The dash lines highlight the shortest paths of transport. Note that for illustration, a single water molecule is plotted instead of the intercalated clusters, and only one side of the graphene sheet is oxidized in GO, with hydroxyl groups only.

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Non-Continuum Intercalated Water Diffusion Explains Fast Permeation

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