Water transport confined in graphene oxide channels through the rarefied effect

Bo Chen , Haifeng Jiang *, Xiang Liu and Xuejiao Hu *
Key Laboratory of Hydraulic Machinery Transients (Wuhan University), Ministry of Education, School of Power and Mechanical Engineering, Wuhan University, Wuhan, Hubei 430072, China. E-mail: hfjiang@whu.edu.cn; xjhu@whu.edu.cn

Received 11th December 2017 , Accepted 7th February 2018

First published on 8th February 2018


Understanding the mechanism of water transport inside an interlayer between graphene-based plates has tremendous value for theoretical studies and industrial applications. The fluid flow confined in nano-scaled spaces experiences a slip velocity near the wall, which is significantly different to that of bulk water. Here we propose a model combining classic hydrodynamics with kinetic theory to depict the dependency of the slip effect on the oxide concentration of valley plates. The influence of oxidized graphene on water flow is a comprehensive result of a slipped boundary, and depends on both the diffuse reflection coefficient of the wall, and the shrunken effective passageway caused by the electrostatic interactions between the oxidized surface and the water molecules. The former effect enhances the water flow, which reduces with increasing oxide concentration, while the latter effect inhibits water flow. We examine the diffuse reflection coefficient and the shrunken effective passageway at different oxide concentrations of the GO sheets by molecular dynamics simulations, and we quantitively predict the flux relationship at various concentrations. This work provides a molecular insight into transport processes of confined water and a useful guideline for the design of perfect graphene-derived membranes for desalination.


1. Introduction

With the increasing human population and fast development of modern industry, the problem of water shortage is a severe challenge. Recently, nanoporous materials such as graphene-derived membranes have gained much attention for nanofiltration1–6 and solar desalination7–12 because of their high membrane flux13–17 and salt rejection,18–24 which offer potential for purification and energy conservation. Understanding the mechanism of water transport confined in graphene-based sheets is crucial for resolving many theoretical and industrial problems and it would help to improve membrane performance. Present research has revealed that water confined in graphene nanochannels is transported at exceptionally high rates, which deviate distinctly from the predictions of the classical continuum hydrodynamics model.13,25,26 The observation is mainly explained by the velocity slippage at the boundary of the flow.27–29 This is similar for water in graphite and carbon nanotube (CNT) membranes, in which water molecules move freely due to the low friction with hydrophobic surfaces.30 In contrast, for graphene oxide (GO) laminates, the oxidized regions with varied functional groups prohibit the flow enhancement on account of the attractive interactions between the oxidized graphene sheets and the water molecules.27,28,31–33 Additionally, hydrogen-bond networks and structures of liquid water, which are correlated with functionalization and nanoconfinement, have effects on water flow.27,33–35 Although current research has clarified the qualitative relationship between oxide concentrations and slip effects, it is still unclear how the slipped velocity occurs, and the quantitative relationship between oxide concentration and water flux has yet to be determined.

Under usual circumstances the continuum model is adopted for fluid flows, and the properties of a fluid observed by ordinary measuring apparatus are continuous and smooth. Actually, in reality, fluid is not continuous and consists of numerous discrete molecules. For large volumes, the number of molecules is big enough to ensure that the average property of the fluid is not influenced by the number of molecules in the volume. However, under the special condition of extremely tiny dimensions (like the fluid flow interlaminated in GO sheets), the discrete particle effect becomes significant and we are obliged to give up the continuum hypothesis. Thus, we depict the process of nanoconfined water flow at the atomic level and propose a model combining classic hydrodynamics and kinetic theory. The dynamics of wall-sided water molecules have been deeply investigated using molecular dynamics (MD) simulations. The collisions between water molecules and GO sheets are critical to the slip effect, and frequent specular reflection is able to enhance the slip velocity. With the increase of the oxide concentration, the electrostatic interactions between functional groups and water molecules prohibit the slip effect and moreover trap adjacent molecules, resulting in shrinking the effective flow domain. We examine the diffuse reflection coefficient and the shrunken effective passageway at different oxide concentrations of the sheets by numerous simulations and quantitively predict the flux relationship with oxide concentration.

2. Theory

For a classical Poiseuille flow confined between two flat plates with a separated distance d (see Table 1 for the definitions of other symbols), the velocity can be estimated as,
 
image file: c7cp08281a-t1.tif(1)
The volumetric flux is calculated by the integral of the velocity,
 
image file: c7cp08281a-t2.tif(2)
Table 1 Nomenclature used in formulas
Nomenclature
u Velocity of Poiseuille flow K n Knudsen number
u s Slipped velocity λ Mean free path
Q 0 Volumetric flux of Poiseuille flow L Characteristic length
Q s1 Volumetric flux considering slip effect n s Number density of fluid
Q s2 Volumetric flux considering effective passageway shrinkage k Boltzmann constant
η Viscosity of water T s Fluid temperature near surface
P Applied pressure m Mass of fluid
W Width of the plate ψ Incident molecular flux
H Height between two plates f Diffuse reflection coefficient
ΔH Effective passageway shrinkage


In a continuum system, the fluid can be described in terms of infinitesimal volumetric elements when these are small compared to the flow domain, and have well-defined thermophysical properties. This is because the molecules of the fluid collide with each other frequently, and successfully transmit the state from one volumetric element to another. However, in a system where the motion scope of the molecules is comparable to the scale of the flow domain, the applicability of the continuum-based model is uncertain because of the discrete effect of the molecules; the kinetics of the molecules near the boundary no longer have a negligible effect on the fluid flow. This is also known as the rarefied effect. To characterize the degree of rarefaction, the Knudsen number is introduced and is defined as the ratio of the molecular mean free path λ to the characteristic length of the flow L,

 
Kn = λ/L,(3)

Rarefied fluid flows can be divided into three regimes according to the degree of rarefaction, i.e., the slip flow regime, the transitional regime and the free molecular regime. According to the range of the appropriate Knudsen number Kn, the three regimes are:

0.01 < Kn < 0.1 Slip flow regime,

0.1 < Kn < 10 Transitional flow regime,

Kn > 10 Free molecular regime.
Under the circumstance that water flows through GO channels of several nanometers, Kn is around 0.1 and the flow in these GO channels is in the slip flow regime. In this regime, the fluid is slightly rarefied, and the discrete molecular effects do not manifest themselves sharply. The fluid motion can also be investigated from the point of view of the continuum model, but it is necessary to introduce some modifications into the boundary conditions. A diagram of plate flow is shown in Fig. 1. The mainstream flow is in the x direction with a macro-scaled velocity ux and there is a velocity gradient image file: c7cp08281a-t3.tif near the boundary. We consider the wall-sided slip velocity as us. Near the wall surface, some of the fluid molecules come from the external flow and the others are reflected from the surface. The momentum exchange of reflected molecules inside the Knudsen layer corresponds to the viscous stressimage file: c7cp08281a-t4.tif, where η represents the viscosity of the fluid. The velocity distribution of incident molecules obeys the Maxwellian distribution,
 
image file: c7cp08281a-t5.tif(4)
The tangential momentum of these molecules can be derived by the integral,
 
image file: c7cp08281a-t6.tif(5)
There are two scattering types after molecule reflection from wall surface, which give rise to different momentum accommodation. In diffuse reflection, the molecules leaving the surface also scatter with a Maxwellian distribution,
 
image file: c7cp08281a-t7.tif(6)


image file: c7cp08281a-f1.tif
Fig. 1 The diagram of laminar flow between GO sheets. The lower left panel shows the continuum model of water flow, and the lower right panel gives the molecular insight into reflection dynamics on the GO surface.

The incident molecular flux Ψ can be calculated by the integral,

 
image file: c7cp08281a-t8.tif(7)
The tangential momentum of diffuse-reflected molecules can be derived by the integral,
 
image file: c7cp08281a-t9.tif(8)
For the momentum conservation in specular reflection, there is a zero contribution in momentum exchange. We consider the diffusion reflection coefficient f to denote the proportion of diffuse-reflected molecules, and the net momentum exchange of molecules corresponds to the shear stress,
 
image file: c7cp08281a-t10.tif(9)
Finally, we can deduce the slipped velocity according to eqn (3), (6) and (7),
 
image file: c7cp08281a-t11.tif(10)
We define the former term of eqn (10) as the slipped coefficient ζ1, which is related to the diffuse reflection coefficient,
 
image file: c7cp08281a-t12.tif(11)
Considering the slipped boundary, the volumetric flux of Poiseuille flow is enhanced as,
 
image file: c7cp08281a-t13.tif(12)

Many researchers have shown that the flow behavior of water confined between GO plates is mainly dominated by an interaction between water and hydroxyl groups.27,28,33 GO surfaces with high oxide concentration break down the fast interlaminar transport because that the diffuse reflection coefficient increases for larger wall–molecule interactions. A further increase in the hydrogen-bonding interactions leads to the locking of water molecules in the vicinity of the wall. Hence, the viscosity of water near the wall is higher than that of bulk water, which causes a “negative slip effect”.36 We consider this to result in a shrunken flow passageway compared with the real d-spacing of GO laminates. A shrunken parameter ΔH, which is defined as the shrinkage of the effective flow path, is introduced to depict the flux decrease, and eqn (12) is transformed to eqn (13),

 
image file: c7cp08281a-t14.tif(13)
 
image file: c7cp08281a-t15.tif(14)
Thus, the flux in the GO channels should be the comprehensive result of both the enhancement effect and the inhibition effect,
 
image file: c7cp08281a-t16.tif(15)
 
image file: c7cp08281a-t17.tif(16)
All symbols used in the formulas above are summarized in Table 1.

According to eqn (15) and (16), we can conclude that both the slipped boundary and shrunken effective passageway have significant effects on confined water flow between GO sheets. The former effect enhances the water flux, depending on the diffuse reflection coefficient of the surface wall. On the contrary, the latter, caused by the electrostatic interaction between surface oxidation and water molecules, inhibits the flow. To examine the dependency of the diffuse reflection coefficient f and the shrunken effective passageway ΔH on different oxide concentrations c, we performed MD simulations and quantitively predict the flux relationship at various oxide concentrations.

3. Simulation model and method

We carried out two MD simulations to address the diffuse reflection coefficient and the shrunken effective passageway at different oxide concentrations of GO sheets.

For investigation of the diffuse reflection coefficient, we computed the collision dynamics of a water molecule and GO sheet, made of pristine graphene and functional groups. Considering quantity and chemical stability, the hydroxyl group was chosen as the representative functional group on the GO sheet. The initial configuration is depicted in Fig. 2(a). Variable oxide concentrations were monitored ranging from c = nOH/nC = 0.1 to 0.5, where nOH and nC are respectively the number of hydroxyl groups and carbon atoms. The GO surfaces at low oxide concentrations were constructed by deleting the hydroxyl groups, as shown in Fig. 2(c). A random distribution of hydroxyl groups was sampled in the oxidized region; the flow characteristics of different arrays of functional groups are presented in the Supplementary Information. Simulations were performed in the microcanonical (NVE) ensemble. After 50 ps relaxation, the water molecule was sent towards the GO laminates assigned with the velocity of the Maxwellian distribution at 298.15 K. Diverse velocity vectors in the xy plane were set by initializing the velocity of (vx, vy), (−vx, vy), (vx, −vy) and (−vx, −vy) with random values to avoid bias from the same incident direction. The water molecule was initially placed at 20 nm away from the GO laminate to allow a sufficient moving time. A total of 190 MD simulations were performed at each concentration. The trajectories and energy of the molecule were recorded every 1 ps to depict the whole process of reflection dynamics.


image file: c7cp08281a-f2.tif
Fig. 2 The initial configuration of the simulation system. (a) The investigation of the diffuse reflection coefficient. (b) The investigation of effective passageway shrinkage. (c) The top view of GO surfaces at oxide concentrations from 0.05 to 0.5, with the interval of 0.05 between two adjacent panels. Gray, red and white spheres represent carbon atoms, oxygen atoms and hydrogen atoms, respectively.

For investigation of the shrunken effective passageway, we created a system of one dimensional (1D) GO channel flow as shown in Fig. 2(b). Both sides were pure water reservoirs and were bounded by rigid pistons to control the transmembrane pressure. A GO channel with a d-spacing of 1.4 nm, measured from the basal planes of GO, was set between the two reservoirs. The molecular structure of the GO channel was the same as that used in the former simulation, with oxide concentrations from 0.1 to 0.5. It should be noted that the GO channel was held rigid to avoid any spurious influence of a layer spacing change caused by thermal fluctuations. Before the simulation, 1 atm pressure was exerted on the two side pistons for 1 ns to reach an equilibrium state. Then the pressure of the upper piston was increased to force the water through the 1D GO channel. The canonical (NVT) ensemble was employed for the whole MD simulation. A sufficient simulation time (20 ns) was used at 298 K for data collection.

Period boundary conditions were utilized on all simulation dimensions. We chose the rigid simple point charge effective pair (SPC/E) model32 to describe the potential of water molecules. The interactions between water molecules and the functional groups (–C–OH) on GO sheets were determined by the all-atom optimized potentials for liquid simulations (OPLS-AA).37,38 Both of these models include van der Waals and electrostatic terms. To ensure that the interaction between water molecules and hydroxyl groups was correctly applied, we performed contact angle simulations of a water droplet on graphene and a GO covered copper surface (Fig. S1–S4, ESI). The MD results were in agreement with previous research.39–43 The potential parameters are given in Table 2. The characteristic length σ and energy parameter ε between different atoms were determined by the common Lorentz–Berthelot combination rule.44 The van der Waals interactions were truncated at 1.0 nm, and the long-range Coulomb interactions were computed by utilizing the particle–particle particle-mesh (PPPM) algorithm.45 All the MD simulations were implemented using the LAMMPS package.46 The post-processing was conducted by Visual Molecular Dynamics (VMD)47 and The Open Visualization Tool (OVITO).48

Table 2 Potential parameters of atoms
Atom σ (Å) ε (kcal mol−1) Charge (q)
C(C–C) 3.851 0.1050 0
C(C–OH) 3.550 0.0700 +0.15
O(C–OH) 3.070 0.1700 −0.585
H(C–OH) 0.000 0.0000 +0.435
O(H2O) 3.166 0.1553 −0.82
H(H2O) 0.000 0.0000 +0.41


4. Results and discussion

4.1 Slip effect

Three types of collision dynamics were observed on GO surfaces. As seen in the left panel of Fig. 3, the water molecule could undergo immediate reflection upon contact, which is considered to be specular reflection. The energy profile and coordinate evolution in the z direction of the reflected molecule are portrayed in Fig. 4. The immediate collision results in a transient change of potential energy and kinetic energy of the molecule in the specular reflection regime. The middle panel of Fig. 3 shows that the molecule could be trapped on the GO surface, and the admolecule then undergoes surface diffusion for ∼50 ps. During this time, the admolecule continuously exchanges energy with the GO plate and eventually regains enough energy to desorb from the surface as shown in the middle panel of Fig. 4. This collision type is regarded as diffusion reflection. A collision dynamic that “permanently” traps the molecule within the simulation time period is shown in the right panel of Fig. 3. This is often observed with highly oxidized GO sheets because the large binding energy prevents the admolecule from desorption. We considered 7 kinds of GO sheets with oxide concentrations ranging from c = 0.1 to 0.5, and 190 MD simulations were performed at each concentration. The trajectories and energy evolutions were tracked to obtain the frequency of collisions in the diffuse reflection regime, which is considered as the diffuse reflection coefficient f. As depicted in Fig. 5, the coefficient f is plotted at different oxide concentrations, and it shows that f increases with the increase of oxide concentration. The logarithmic curve is used to fit the relation between oxide concentration c and diffuse reflection coefficient f,
 
f = 0.7996 + 0.20273[thin space (1/6-em)]ln(c + 0.02004).(17)

image file: c7cp08281a-f3.tif
Fig. 3 The diagram showing three kinds of collisions of a water molecule with a GO surface.

image file: c7cp08281a-f4.tif
Fig. 4 The energy profiles (blue lines) and z-coordinate trajectories (black lines) of water molecules upon three kinds of collisions with a GO surface.

image file: c7cp08281a-f5.tif
Fig. 5 The diffuse reflection coefficient f at different oxide concentrations. The black squares denote the MD results and the red dashed line denotes the logarithmic fitting curve.

4.2 Shrunken effective passageway

During the pressure driven flow, we recorded the water density distribution perpendicular to the GO channels with different oxide concentrations, as shown in Fig. 6. At low oxide concentrations (c < 0.2), there are density peaks near to the GO surface, because the low friction enables the water molecules to flow freely and offers enough flowing space for the molecules. With the increase in oxide concentration, the outermost density peaks diminish gradually, while the inside peak rises. Water molecules near the GO surface experience larger energy barriers at high oxide concentrations according to the free energy calculation G(x) = −kBT[thin space (1/6-em)]ln(ρ(x)) (Fig. S5, ESI). This is because the hydroxyl groups attach water molecules around the GO surface, which occupy the flowing space of other molecules. Thus, the water flow path through GO sheets with high oxide concentrations is actually narrower than their geometrical structures. To quantitively calculate ΔH at different c, we used the ratio of the outermost peak value and the sub exterior one p1/p2 as a yardstick to evaluate the extent of the flowing passageway shrinkage. The velocity and density distribution at c = 0.45 are shown in Fig. 7. In the vicinity of the GO surface, the velocity of water molecules is around zero, which also locates at the outermost peak of the density distribution. This region of highly trapped molecules is regarded as the flowing passageway shrinkage. The reduction of the flowing path at other concentrations can be deduced by the peak ratios as displayed in Fig. 8; it shows that ΔH rises with the increase of oxide concentration. The logarithmic curve is used to fit the relation between oxide concentration c and effective passageway shrinkage ΔH,
 
ΔH = 3.5582 + 1.14127[thin space (1/6-em)]ln(c − 0.1551)(18)

image file: c7cp08281a-f6.tif
Fig. 6 The density distribution of water perpendicular to GO channels with different oxide concentrations.

image file: c7cp08281a-f7.tif
Fig. 7 The density and velocity distribution of water perpendicular to the GO channel with oxide concentration c = 0.45.

image file: c7cp08281a-f8.tif
Fig. 8 The effective passageway shrinkage ΔH for GO channels with different oxide concentrations. The black squares denote the MD results and the red dashed line denotes the logarithmic fitting curve.

5. Conclusions

In summary, we have investigated the mechanism of water flow confined in GO laminations with different oxide concentrations. Both a slipped boundary and a shrunken effective passageway have significant effects on water flow between the GO sheets. The slip effect positively enhances the flux, and the increase can be calculated by eqn (12), which is controlled by the slip coefficient ζ1. A further understanding of the slip effect was achieved through the rarefied effect for Kn close to 0.1. Our theoretical analysis demonstrated that the slip coefficient depends on the reflection dynamics of water molecules on the GO surface, and through the MD simulations, it was observed that the diffuse reflection coefficient f shows a logarithmic increase with the change of oxide concentration c, as shown in eqn (17). We also detected that the water molecules tend to be absorbed on GO surfaces with high oxide concentration, which results in the shrinkage of the effective passageway of water flow. This reverses the enhancement of water transport and acts like a “negative slip effect” with the coefficient ζ2. With the increase of oxide concentration, greater shrinkage of the effective passageway (ΔH) occurs and leads to more blockage in water transport. All this work further demonstrates the limitation of the continuum fluid mechanics model and provides a modified and valuable model for water flow confined in different oxidic GO channels.

Conflicts of interest

There are no conflicts to declare.

Acknowledgements

The authors acknowledge the support of the National Natural Science Foundation of China (No. 51706157), the China Postdoctoral Science Foundation (No. 2017M612498), and the Fundamental Research Funds for the Central Universities (No. 2042016kf0023). The numerical calculations in this paper have been done on the supercomputing system in the Supercomputing Center of Wuhan University.

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Footnote

Electronic supplementary information (ESI) available. See DOI: 10.1039/c7cp08281a

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