Operando study of water vapor transport through ultra-thin graphene oxide membranes

Today, the mainstream in membrane technology is captured by 2D layered materials represented by graphene and graphene oxide [1], transition metal dichalcogenides [2] or MXenes [3] finding their application in ultrafiltration [4], desalination [5], pervaporation [6] and gas separation [7]. Among the listed, graphene oxide (GO) have received most of attention due to its unique ability of adaptive structural changes in solvents, impressive water transport characteristics and ease of fabrication [8]. Besides exceptional permeance for liquid water, GO membranes exhibits high permeability to water vapors exceeding 10⁵ Barrer with H₂O/N₂ selectivity over 10⁴ at ~80% relative humidity. These characteristics make GO extremely attractive for air and natural gas dehumidification [9, 10]. These properties are mainly dictated by week interlayer interaction in GO layers and its strong hydrophilicity provided by a huge number of oxygen-containing functional groups in the layers [10, 11].

Unusual properties of graphite oxide membranes—high permeation rate of water vapor and barrier properties for the permanent gases were demonstrated in 1961 by Boehm et al for the first time [12]. However, the huge interest to the study of water vapor permeation trough graphene oxide membranes had arisen fifty years later after the work by Nair et al [1]. At the same time the mechanism of water molecules diffusion in GO structure remains unclear. Initially it was attributed to a high capillary pressure in a network of graphene capillaries. Since then, different approaches involving molecular dynamic simulation, combination of classic hydrodynamics with the kinetic theory [13] and classical continuum prediction [14, 15] have been adopted to explain water diffusion in GO layers. However a lack of well-defined relation between water diffusion constants with interlayer spacing in GO impedes model description, while pure theoretical studies do not cover all the features of real GO structure leading to controversial results and divergence with the experimental data.

It is currently accepted that the performance of GO-based membranes is governed by d-spacing of graphene oxide, like the pore diameter determines the transport characteristics of conventional membranes. Generally, interlayer distance of solid GO varies in the range from ~0.7 to ~1.1 nm depending on environment humidity [16], and can be further expanded to 1.5–6 nm in liquid water [17]. Thus, the variation of d-spacing at certain conditions (humidity, pressure, temperature) gains a decisive role in the membrane performance. However, only a few studies report in situ measurement of d-spacing at various conditions. However neither of those reports simultaneously the permeance of GO or diffusivities of water in the interlayer space. Water adsorption in 5 µm-thick GO papers was studied using x-ray diffraction under relative humidity from 0 to 100% during 12 h [18]. It was established, that after exposing in dry atmosphere the d-spacing was equal to 0.77 nm and did not change, whereas the elevation of the relative humidity forced the d-spacing to expand up to 0.96 nm. The same trend was observed in [19] by x-ray scattering and quasi-elastic neutron scattering techniques. It was concluded the hydration of GO occurs by continuous filling of the d-spacing with water molecules. Moreover it was reported the interlayer spacing of GO can be filled with several layers depending on GO oxidation degree [16, 20]. It was suggested that the first monolayer of water is tightly attached to GO adsorption sites, but an extension of interlayer distance over 1 nm results in formation of the second layer exhibiting transitional motion along the interlayer space [21, 22]. In situ neutron reflectivity studies of water, ethanol and water-ethanol mixtures intercalation into d-spacing of 25 nm-thick graphene oxide film were performed in [23]. It was shown that water and ethanol molecules intercalate slowly into d-spacing achieving complete saturation in 4–5 h, whereas deintercalation occurs faster: within an hour. In situ scanning force microscopy was applied to investigate the hydration of bi-layered graphene oxide in the humidity range of 2%–80% [24]. It was revealed, that being exposed to water vapors, the d-spacing of the graphene oxide film expands gradually up to ~1 nm with increasing of humidity. While immersing of the GO film in liquid water enlarges the d-spacing up to 3 nm. It was also established, that the liquid water intercalates directly as monolayers into GO d-spacing, whereas in vapor phase single water molecules incorporate continuously in between GO sheets.

Thus, the studies published to date consider structural changes in graphene oxide under various conditions, but provide only fractional results on the mass-transport through GO. Despite the interrelation between the d-spacing and permeance of GO membranes is of great scientific and practical demand, we have not found any comprehensive and complete studies in this field. To bridge this gap, here we present operando results on simultaneous evaluation of d-spacing and water vapor permeance in GO membranes as well as the evolution of these characteristics under various operational conditions. We show that the water vapor permeance strongly correlates with the d-spacing in GO-based membranes being strongly governed by partial water vapor pressures on membrane interfaces. The obtained results provide a deep insight into the fundamental structure-permeance relation in graphene oxide and open the way to accurate design of 2D membranes for a wide range of separation processes.

Colloid solutions of graphene oxide (GO) with typical flake sizes of 500–1000 nm were synthesized from medium flake graphite by a modified Hummer's method [25]. After the synthesis, graphene oxide suspensions were subjected to a multiple centrifugation and washing cycles with distilled water to pH 4 and purified by dialysis for 30 d. The procedure is reported in details in [10].

The substrates for deposition of selective layers were prepared by anodic oxidation of high purity aluminum foils (99.999%) in oxalic acid at stepwise voltage sweep from 120 to 10 V. To enhance permeability of the supports, anodization was performed by two-stage technique [26] and anodization regime was adjusted to form asymmetric AAO films with layered structure having 120 nm pores in the supporting layer (90 µm thick), 40 nm pores in the intermediate layer (10 µm thick) and 10 nm pores in covering layer (2 µm thick) [27]. This allows to improve AAO permeance for N₂ up to 500 000 l/(m² · atm · h) resulting in minimal support flow resistance to transport of water vapors [27–29]. The detailed description of the AAO preparation is given in [27].

Composite membranes were prepared by spin-coating of AAO substrates with GO suspensions diluted with methanol under vacuum suction. Concentration of water-methanol suspension of 1 mg ml⁻¹ and substrate rotation speed of 2800 rpm were used. To confirm integrity of the membrane and the absence of defects in GO layer dry nitrogen permeance test were performed prior to diffraction experiments. The resulting N₂ permeance of 8 l/(m² · barh) coincides well to earlier reported data and illustrates strong barrier-type of MFGO/AAO membrane under evaluation [10]. The test was repeated after operando experiments giving very close result of 6.7 l/(m² · bar · h). Characterization of initial GO was performed according to the protocol described in [30] with SEM analysis of average flake size of GO and XPS data revealing it's chemical composition [10] (see supplementary information SI 1 (stacks.iop.org/TDM/6/035039/mmedia)). IR and Raman spectroscopy data is also provided in SI 1. The results of SEM and Raman spectroscopy mapping of GO layers on AAO substrate as well as permanent and condensable gas transport characteristics of membranes are reported in [10].

Due to highly disordered structure and ultralow thickness of GO layers XRD experiments were carried out in Bragg-Brentano geometry using surface scattering setup of ID10 beamline at ESRF (Grenoble, France). X-ray reflectivity (XRR) and grazing incidence scattering (GIS) signal from the sample surface was measured using a new diffractometer-deflector assembly, specifically designed for studies on fluid surfaces and interfaces based on a double crystal deflector (DCD) beam steering mechanism. X-ray beam, tilted by the DCD, hit the horizontal surface at a grazing angle µ. Monochromatic incident beam with energy of 8 keV (wavelength 0.155 nm) was focused at the sample position by a set of 2D parabolic compound refractive lenses (CRLs) to the cross-sectional size of 250  ×  54 µm² in horizontal and vertical directions respectively. Sample was mounted on an active antivibration system. Scattering patterns were recorded by 2D detector Pilatus 300 K (487  ×  619 pixels with the pixel size 172  ×  172 µm²) at the reflection angle θ  =  5.5°, and incidence angle µ  =  5.5°, with the sample-detector distance of 38.5 mm. XRR data was acquired by the linear detector Mythen 1 K (1280 pixels with the pixel size 8000  ×  50 µm²) at the distance 970 mm with variable θ and µ. Reflected beam was collimated by the two slits S1 (1  ×  2.3 mm²) and S2 (1.5  ×  4.5 mm²) positioned at 465 mm and 950 mm from the sample, respectively.

Due to highly textured nature of GO layer prior to operando GIS experiments the membrane placed in a humidity cell was accurately aligned respect to the x-ray beam to achieve the grazing angle corresponding the diffraction maximum found at XRR experiment. 2D detector used for GIS was positioned so to cover scattering from the direct beam position to the diffraction pick of the interest. Alignment was performed at constant humidity level of 55%. Post-experiment data treatment was carried our using Fit2D software.

Water vapor permeance in operando experiments was measured using a self-developed two-compartment membrane cell (see SI 2) with a top compartment having x-rays transparent windows (12.7 µm Kapton foil) to measure scattering signal. The gaseous scheme of the setup is also given in supplementary information SI 2. The humidity in the top compartment was controlled by two SLA5850 mass-flow controller delivering dry nitrogen (from a gas cylinder) and ~100% humid air from a bottle-type humidifier. Bottom compartment was vacuum sealed and equipped with He sweep gas blow-off. Helium flow was also controlled using SLA5850 mass-flow controller (Brooks, UK). Bottom compartment exit was connected to vacuum port through a regulating needle valve and a full-section bypass for permeate side pressure control. Pressure in a bottom compartment was registered by Carel SPKT00E3R pressure transducer. Humidity levels at both feed and permeate sides of the membrane was monitored with KIP-20 (Teplopribor, Russia) temperature-humidity sensors. Humidity sensor of top compartment was positioned right over the membrane. Both humidity sensors were recalibrated just prior to experiment. The measurements were performed at ambient temperature (22 °C  ±  1 °C) and continuous flow of feed gas at the atmospheric pressure. At certain points of experiment dew point of permeate flux was measured with a self-developed chilled mirror hygrometer.

Water vapor flux through the membrane was determined from the partial pressure of water vapors in permeate:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle F={{J}_{He}}\cdot R{{H}_{out}}\frac{{{P}_{0,water}}}{{{P}_{out}}}\nonumber \end{align} \tag{ 1 }$$

where J_He –helium flux (sweep gas), RH_out –outlet relative humidity, which was determined from dew point temperature and humidity sensors, P_out –permeate side pressure, P_0,water–saturated water vapor pressure at the measurement temperature (calculation was performed by Antoinie equation with coefficients determined by Stull [31]).

Membrane permeance for water vapors was calculated as:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle P=\frac{F}{S\cdot \left( R{{H}_{in}}-R{{H}_{out}} \right)\cdot {{P}_{0,water}}}.\nonumber \end{align} \tag{ 2 }$$

RH_in—feed stream humidity and S—membrane area.

Equilibrium interlayer distance in water containing GO and barrier energies for H₂O molecules transport were determined with unrestricted Hartree–Fock method with semiempirical PM7 Hamiltonian [32] in MOPAC2016 [33]. The computations were performed in several steps. Initially, C:O ratio was determined from XPS analysis and hydroxyl-/epoxy-group ratio by deconvolution of C1s an O1s peaks in XPS spectra (see SI 1). Then a model of GO was built with a maximum possible stable oxidation degree and a given hydroxyl-to-epoxy groups ratio. The choice of utmost oxidation degree was justified by expected maximum water permeance of the structure. We also suspected that in case of strong overoxidation or hydration the modeled layer, GO will release water molecules or CO/CO₂ in course of semiemperical optimization of a layer, which has occurred at low C:O ratios. Numerous attempts have been made to obtain a layer with minimal C:O ratio. Finally we have proceeded with a reported model with 54 hydroxy and 8 epoxy groups per 100 carbon atoms, which slightly exceed an oxidation degree of the samples as obtained from XPS. Next, a conformational analysis of GO layers with a unit cell containing 100 carbon atoms, with randomly connected 54 hydroxy- and 8 epoxy- groups in the structure was carried out using periodic boundary conditions. Neither atomic coordinates no cell parameters of GO structure were fixed. Ten attempts have been made to distribute oxygen-containing groups in the structure and a model having the lowest energy after geometry optimization was further used. At the third stage, the interlayer space of GO was statistically populated with water molecules with a corresponding extension of cell c-parameter, and geometry optimization step was repeated. Computations of structures containing from 1 to 60 water molecules per unit cell were performed. The resulting cell c-parameter was extracted as an interlayer distance in water-containing GO. At the final stage, the barrier energies for H₂O molecule transport in the interlayer space of GO were extracted from the potential energy profile calculations. Refined models containing different amount of water molecules in the interlayer space were used as initial geometries. Geometric and energy profile for transport of water molecule through the structure were built by the same geometry optimization with y -coordinate of a single travelling water molecule and (x, y ) of 4 peripheral carbon atoms in the layer of GO being fixed. Neither cell parameter nor cell tilting angles was fixed in profile analysis. All profiles were built with 0.1 Å Δy  step and full geometry optimization on each step. Transport barrier energies were extracted from the potential energy profiles as maximal variation of the total system energies on ~3 Å length scale (corresponding to a single molecule jump).

Due to the necessity of destructive specimen preparation for determination of GO layer thickness in MFGO/AAO membrane by electron microscopy, SEM studies were conducted after operando experiments. Cross-sectional SEM image of membrane are given in figure 1. Statistically evaluated thickness of GO layer measured at 30 distant points of membrane equals 55  ±  15 nm. The value corresponds to ~60 layers of GO.

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Typical 2D diffraction patterns for MFGO/AAO acquired in GIS experiment are represented on figure 2(a). The analysis of GIS data involved 2D pattern integration, fit with mixed Gaussian/Lorentzian function, extracting peak position and FWHM. To establish a reference point for further operando experiments a dependence of d-spacing of graphene oxide on water vapor humidity was first acquired. At stationary humidity conditions an equilibrium d-spacing in membrane was achieved in a few minutes allowing to collect rather a detailed dataset appropriate for further analysis. Figure 3(a) reveals a strong dependency of d-spacing on humidity levels, which coincides principally to the earlier published data [16, 22, 34–36] Notably the obtained curve pattern at humidity levels over 10% recalls in mind Kelvin equation. Thus we take an effort to fit the dependency with a modified function:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle d={{d}_{0}}+\frac{-2\sigma M\cos \theta }{\rho RT\cdot {\rm In}\left( \frac{P}{{{P}_{0}}+{{P}_{b}}} \right)}\approx {{d}_{0}}-\frac{const}{{\rm In}\left( \frac{P}{{{P}_{0}}+{{P}_{b}}} \right)}\nonumber \end{align} \tag{ 3 }$$

where d₀—corresponds to the interlayer distance in dry graphene oxide, σ—is surface tension of water, θ—is contact angle of water with GO flakes, M—is water molecular weight, P₀—is water condensation pressure at the temperature (T) of experiment and P_b represents an equilibrium pressure shift due to binding of water molecules to graphene oxide layers.

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Notably, equation (3) involves both surface tension and contact angle of water deviating significantly from a bulk values in shale nanopores [37]. Thus fitting of experimental data was performed with unfixed $\frac{-2\sigma M\cos \theta }{\rho RT}$ coefficient. Three-parameter fit gives d₀  ≈  8.0 Å and P_b  ≈  800 Pa with χ²  <  0.0015, illustrating conformity of the proposed concept. Despite experimental d(P) follows principally Kelvin equation, the presence of positive P_b term implies water molecules in GO are bound more tightly than compared to bulk water. Extracted P_b allows to estimate an immersion energy of water in graphene oxide of 0.6 kJ mol⁻¹. The value is significantly smaller than compared to condensation enthalpy of bulk water (44.0 kJ mol⁻¹) and can be regarded as energy for separation of GO layers from an equilibrium state at 100% humidity to an infinite distance in liquid water. The calculated value of immersion energy is smaller than that early reported in [39] for GO containing a monolayer of water at 0.05–0.4 P₀, which can be ascribed to different number of water layers in the initial state. However it stays in an agreement with theoretical predictions of interlayer binding energy in GO with similar interlayer distances [37, 38]. Thus condensation of water vapors between the layers of graphene oxide can be accounted as capillary condensation in an elastic slit with boundary plates bound by electrostatic interactions.

FWHM of GO interlayer distance predictably increases with air humidity, revealing some inhomogeneity of both the distribution of oxygen containing groups in GO and water condensation between the layers (see SI 3), revealing the role of intra- and interstratification effects [22, 40]. Nevertheless RSD value of d-spacing distribution close to condensation point does not exceeds ~0.15, indicating the uniformity of GO nanosheet spreading.

To estimate water quantity in GO interlayer space, structural modeling with semi-empirical Hartree–Fock method was performed (figures 3(b)–(d)). We should stress here that the model, containing rather a limited number of atoms, as restrained by the computing resources of semiemperical method, is more suitable for describing of ordered Brodie GO, rather than disordered Hummers GO. The model does not include vacancies, defects or peripheral edge groups; it also cannot address interstratification phenomena and limitedly addresses intrastratification, specific for Hummers GO. Thus it should be considered as a rather simplified and idealized model of the real material. Nevertheless, it is still valuable for deducing general dependences of the interlayer spacing in GO with an absorbed water quantity and calculation of energy barriers for water molecules movement between the layers.

Minimal experimentally observed value of d(0 0 1) of ~7.2 Å does not coincide with simulations results for flat GO sheet containing no interstitial water those reveal minimal d-spacing of water-free relaxed GO of ~5.4 Å. This interlayer distance is close to an interlayer distance in dry Brodie GO [41]. Experimentally derived vale fits rather well to a sum of GO layer thickness and the length of O···H–O hydrogen bond of 2.8 Å tilted at 70° (dihedral angle in tetrahedron). One can explain the difference by a diversity of structures of Brodie and Hummers GO. Graphene oxide formed by Brodie technique has a planar layers, while layers in Hummers GO are known to deviate planar shape due to intrastratification phenomena and overall disruptions of the carbon–carbon bond network [20]. According to MOPAC modeling, GO with an interlayer distance of 7.2 Å, corresponds to H₂O content of ~8% mol and contains a single layer of water molecules between GO nanosheets. It agrees earlier published results of molecular dynamic simulation [35] reporting the formation of a single layer of water molecules closely packed around GO functional groups at relative water vapour pressure of 0.15 P₀, corresponding to d-spacing of 6.9 Å and H₂O:C ratio of 0.13. Thus, one can conclude that dry graphene oxide in operando experiment (at  <30 Pa H₂O partial pressure) contain likely some interstitial water with interlayer distance expanded due to intrastratification (SI 4, figure 3(c)). Maximum value of d(0 0 1) of ~11.2 Å is ~4 Å larger as compared to dry GO structure, and can fit two additional water layers in between GO nanosheets. This is confirmed by the simulation results, revealing a water content in GO for this structure of ~50%–60% mol (figure 3(d)).

An obvious quantization of d-spacing has been obtained with increasing absorbed water quantity in the model (figure 3(b)), corresponding to layer-by-layer growth of sandwiched water film between GO planes. These steps are not typically observed in the humidity dependences of interlayer distance of Hummers GO. However the layer-by-layer filling of absorbed water quantity was earlier retrieved by molecular dynamic simulation [35] reporting also the formation of water moving layer and described in occasional experimental reports [18]. Similar steps were found in the experimental dependences of interlayer spacing on humidity levels in the absorption and operando (continuous desorption) experiments (figure 3(e)): stable d-spacing steps are well distinguished at d(0 0 1)  ≈  8 Å on the absorption curve and at d(0 0 1)  ≈  10 Å on the desorption curve. According to modeling results those states correspond likely to GO containing two and three water layers. Notably these states can contain lower water quantity with the interlayer diastance expended due to intrastratification. Nevertheless the difference of d-spacing for adsorption and desorption isotherms in the range of relative humidity from ~5% to ~40% can be ascribed to the presence of the additional water between GO planes. Consolidation of theoretical and experimental results allows to reveal absorption capacity dependence on the humidity (figure 3(e)). Extracted absorption capacities coincide principally with data reported recently by Liu et al [18] and by Korobov et al [42]. Also it should be noted that absorption capacities values are more close to the values for GO powder, than for GO membrane formed by vacuum assisted filtration reported in [42]. We associate this with more packless structure of membrane formed by spin-coating in comparison with membrane formed by vacuum assisted filtration.

The observed phenomenon stands for structuring of water molecules in GO interlayer space. Indeed intralayer distribution of oxygen atomic coordinates reveals well defined maxima on z-profiles (see supplementary information SI3). Those illustrate maximum molecular density at the centre of GO interlayer space at the beginning of each absorption step and minimum density at the step ending. Moreover the distance from the centre of GO plane to nearest oxygen atoms decrease from ~2.6 Å at the beginning of each step to ~2.3 Å at the step ending. Structuring of matter in a confined subnanometer space stays in line with recent reports [43]. As soon as each step in a humidity dependence of water content appears at certain humidity level (figure 3(e)), the extension of GO can be considered as local corrugation of GO by water molecule introduced to the center of interflake gap followed by reorganization of water molecules to provide energetically favored state with fully hydrated GO planes.

Besides the main diffraction maximum corresponding to an average d-spacing of graphene oxide in a selective layer, we have also detected an additional reflection corresponding to ~14.9 Å which is about ~3.6 Å larger as compared to the maximum interlayer spacing observed in series (see figure 1(b)). The maximum persists in all the diffraction patters acquired at a feed pressure over 0.97 P₀, and can correspond to the structure of swelled GO reported in [44]. Accounting to experimental setup geometry the reflection appears likely from the outer layer of graphene oxide contacting with a feed stream. Accounting to an average intermolecular distance in bulk water, this structure can be attributed to graphene oxide containing 4–5 interstitial water layers between GO nanosheets. Emergence of such structures with d-pacing attaining few nanometers were reported recently for graphene oxide soaked in water [17]. Nevertheless, an appearance of this particular reflection and the stability of some special structure at high humidity levels is not well understood to the moment. The most probable explanation involves local ordering of water molecules in the interlayer space. Indeed, the difference of the observed interlayer distance of 14.9 Å with GO structure containing a single layer of water molecules (~7.2 Å) fits well to Ih ice cell parameter of 7.82 Å. I.e. the structure can correspond to the disordered fragments of 2D hexagonal ice layers separated by GO nanosheets. Such structure appears to be predictably stable as supported by semi-empirical calculations (see supplementary information SI3). Its stability does not change much while adding additional water molecules into the ice cages. Despite being very surprising, similar structures with the same cell parameter have already received experimental evidence [45]. Notably the above hypothesis presumes ordering of water molecules in 2D layer which is detectable in wide angle diffraction experiments. Despite we failed to find out any obvious ordering of water molecules between the layers of Hummers GO with wide-angle XRD, it still requires detailed PDF analysis, which can be a challenge for further studies.

Operando experiments were started with varying permeate side partial pressure of water vapors while keeping feed side humidity close to 100%. Despite we have expected here a serious widening of the diffraction reflections it has not been observed experimentally until low partial pressures were reached (figure 2(d), compare curves 1–3 and 4–5). Conversely, rather a sharp maximum shifting with permeate humidity and having FWHM exceeding slightly that of equilibrium graphene oxide with the same interlayer distance was detected (see supplementary SI2). The plot of d(0 0 1) value as a function of the permeate partial water vapor pressure indicates a classical behavior of type H2 adsorption-desorption isotherms characteristic for microporous solids with bottleneck constrictions (figure 3(e)). Early hysteresis on the absorption–desorption isotherm for graphene oxide was found in [46, 47], however the hysteresis loop width for GO powder is lower than for GO membrane. These facts points out a limitation of water transport by a few graphene oxide layers at graphene/AAO interface. Due to a very limited number and high degree of disorder these layers provide quite a small wide-angle shoulder of the diffraction reflection (see curves 4–5 on figure 2(d)). However these particular layers strongly restrict transport characteristics of membranes. The interlayer distance in those layers can be expected to meet an equilibrium condition with permeate partial water pressure.

Water vapors permeance in operando experiment was calculated from the humidity of feed and permeate streams (see experimental section for details). At high vapor pressures MFGO/AAO membrane exhibits huge water permeance of ~90 000 l/(m² · atm · h), which equals or even exceeds most of liquid water permeance values through GO reported in literature [11]. This confirms capillary condensation mechanism of water transport through GO layers and allows application of corresponding models for theoretical description. An analysis of transmembrane water transport as a function of an average interlayer distance in graphene oxide illustrates continuous reduction of permeance with decreasing interlayer distance between GO sheets to ~20 000 l/(m² · atm · h) at d~9.2 Å. A sharp decrease of GO permeance is observed for smaller d-spacings.

An estimate of membrane permeance with Poiseuille flow accounting for capillary pressure produced in GO was done assuming diffusion through slit-like pores:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J=\frac{D{{(d-{{d}_{g}})}^{3}}}{12\,\mu {\rm l}}\Delta P=\frac{D{{(d-{{d}_{g}})}^{3}}}{12\,\mu {\rm l}}\frac{4\sigma }{d}\nonumber \end{align} \tag{ 4 }$$

where J is water flux through a slit, D is a lateral slit size (slit width), µ is viscosity of water, σ is water surface tension, l is the length of diffusion path, d is the interlayer spacing in graphene oxide and d_g is the interlayer distance in graphene oxide containing no interstitial water molecules.

To evaluate the length of diffusion path we assumed statistical distribution of nanosheets, resulting in an effective overlap of graphene layers of 1/4 of their average size. Total slit width was estimated as an average nanosheet diameter multiplied by GO flakes density in a single layer. The thickness of GO selective layer of 55 nm and an average flake size of 750 nm was used in evaluations. Notably the capillary pressure in GO was evaluated as hundreds MPa [48].

Simple calculations result in some overestimation of the membrane permeance, slightly exceeding experimental values. However theoretical tendency reproduces rather well experimental results at d  >  9.2 Å with closely the same slope value. Most likely, an excess in theoretical flow originates from an underevaluation of the average travelling distance from GO particle size, requiring more detailed mathematical description. Nevertheless, rather a good agreement of the results allows the use of continuous flow approximation at least for rough estimation of condensate flux through nanoslits and comparative assessment of membranes permeability at the slit sizes twice exceeding intermolecular distances in a liquid. With reducing interlayer distance in GO below ~9.2 Å the model of continuous medium becomes inappropriate.

To describe the permeance changes at low interflake distances and reveal the nature of a well-defined kink on the dependence of membrane permeance on GO d-spacing we have further engaged statistical mechanics approximation to describe water transport through GO layers. First we have estimated water molecule migration energies in GO by semi-empirical calculations those were performed for GO with an interlayer space varying from 7.2 to 11.2 Å (table 1). Optimized GO geometries with an appropriate quantity of interstitial water located between the layers were used as starting geometries. Then, a single water molecule was allowed to find an optimal path between GO layers by changing its single coordinate in y  direction. Neither interstitial water nor GO atoms were fixed in computations and full geometry optimization was performed at each step. Energy profiles and corresponding activation energies for this travelling process were extracted.

Expectedly, computation results (table 1) reveal substantial diminishing of the activation barrier height with increasing d-spacing from 7.2 to 9.2 Å, while at larger d-spacing activation energy stays nearly constant. This pattern is rationally explained by perturbations introduced to the conformation of GO layer by travelling water molecule. In case of GO containing two or more interstitial water layers and central line transport all the disturbances are effectively damped by hydrogen bonds of marginal water layers, while any movements in the proximity of rigid GO necessary result in total energy rise.

To compare the theoretical barrier energies with experimental results we have further described GO water permeance as hopping diffusion of water molecules between GO nanosheets. Applying an activation-based approach one can write an expression for water flux (J) in GO:

$$\begin{align} \newcommand{\e}{{\rm e}} \displaystyle J=\frac{{{L}_{jump}}}{L}N\cdot {{f}_{0}}\cdot {{e}^{-\frac{{{E}_{a}}}{kT}}}\nonumber \end{align} \tag{ 5 }$$

where f ₀ represents an attempt frequency for water molecules jump with corresponding activation energies E_a; N—is a number of water molecules in graphene oxide layer, k is Boltzmann constant, and T is absolute temperature, L_jump is an average jump length and L is total travelling distance in GO.

As soon as travelling of water molecule in graphene oxide is strongly associated with changes in hydrogen bond network, a typical microwave absorption frequency of 2.54 GHz, was exploited in calculations as f ₀. Some doubt on the choice of the attempt frequency can be placed here by the presence of low-frequency translational vibrations of water at ~1 THz as detected by Raman spectroscopy and ultrafast dynamics experiments. Those doubts are however disposed as such processes do not involve molecular reorientations and relaxations in the coordination sphere appearing on much longer timescale [49–51]. Barrier energies extracted with equation (5) fit well theoretical values and exhibit the same kink at d  ≈  9.2 Å (table 1). At large interlayer distances, corresponding to  >3 interstitial water layers in GO, jump activation energies are nearly identical (~0.25 eV) while the permeance changes are mostly dictated by the quantity of the absorbed water. With diminishing interlayer distance below 9.2 Å barrier rises significantly, attaining ~0.75 eV at 7.2 Å. According to theoretical evaluations this corresponds to direct interaction of travelling water molecule with GO functional groups.

To summarize, our studies represent a full dataset for water permeance through graphene oxide and methods for calculation of GO membranes permeance depending on external humidity conditions and GO interlayer spacing. It is quantitatively shown that the absorbance of water molecules into the interlayer space of GO plays a crucial role for GO transport characteristics. In common conditions (1%–100% humidity range at s.c.) GO can contain 1–4 water layers in the interlayer space unequally distributed in the interlayer space due to intrastratification. Until the accumulation of two water layers in GO, water transport is strongly hindered due to direct interaction of travelling water molecules with GO functional groups. Higher water content allows quick transport of water due to effective damping of structure distortions by hydrogen bonds of marginal water layers. Theoretically, water transport in GO containing over two interstitial water layers (d  >  9.2 Å) is well described by Poiseuille flow in a slit pore, while lower d-spacing necessary requires statistical mechanics approximation to predict permeance of the membrane.

Our study reveals both advantages and drawbacks of utilizing graphene oxide membranes for water transport. The major advantage of GO is that the permeance of GO membranes does not generally depend on a selective layer thickness due to chemical potential gradient at the outer surface at least for thin GO selective layers. This allows a wide flexibility for GO membrane technologies, allowing utilization of both free-standing and supported GO membranes for selective water transport with virtually any thickness. However, this also entails a major drawback of GO membranes: partial pressures in permeate stream affect strongly the water permeance of outer GO layers. This hardly restricts the possibility of attaining low dew point values in dehumidification processing. I.e. in spite of ultimate water permeances reported for GO, in real technological tasks the values of GO permeance, characteristic for graphene oxide equilibrium to permeate side partial water pressure have to be applied. This seriously diminishes favour of GO membranes. On the other hand, our findings provide the waymarks for further development of GO membranes: an enhancement of GO permeance can certainly be achieved by restricting minimal interlayer distance in GO with incorporating some particles or ions between the layers. Such studies are already known in literature and now receive theoretical background. We believe our findings have potential impact on both the technological assessment of future GO dehumidification membranes and better understanding of fundamental reasons of GO transport characteristics.

The work is supported by the Russian Foundation for Basic Research grant No. 18-29-19105. The authors acknowledge Kapitanova OO for providing medium flake graphene oxide suspensions for membrane preparation and Lebedev V for the help during synchrotron experiment. We are grateful to ESRF for providing beamtime at the beamline ID10. Authors acknowledge Karim Lhoste, Pierre Lloria, Diego Pontoni, Harald Muller, Lea Bourcet, Katty Paermentier (ESRF) for the technical support of the experiments. The authors are also thankful to the M.V. Lomonosov Moscow State University Program of Development for the partial support of instrumental studies.

Results of graphene oxide characterization chemical composition using Raman, IR and XPS spectroscopy and analysis of flake size is given in SI 1. Scheme of experimental cell is given in SI 2. The dependencies of FWHM of d(0 0 1) distribution on water vapor humidity is given in SI 3. Intralayer distribution of water molecules (oxygen atom coordinates) in graphene oxide depending on absorbed water quantity is given in SI 4. The dependence of d-spacing on transmembrane pressure is given in SI 5.