Articleshttps://doi.org/10.1038/s41565-017-0031-9

Fast water transport in graphene nanofluidic channelsQuan Xie   1, Mohammad Amin Alibakhshi1, Shuping Jiao2, Zhiping Xu2, Marek Hempel3, Jing Kong3, Hyung Gyu Park   4 and Chuanhua Duan   1*

1Department of Mechanical Engineering, Boston University, Boston, MA, USA. 2Department of Engineering Mechanics and Center for Nano and Micro Mechanics, Tsinghua University, Beijing, China. 3Department of Electrical Engineering and Computer Science, Massachusetts Institute of Technology, Cambridge, MA, USA. 4Department of Mechanical and Process Engineering, Eidgenössische Technische Hochschule (ETH) Zürich, Zürich, Switzerland. *e-mail: duan@bu.edu

© 2017 Macmillan Publishers Limited, part of Springer Nature. All rights reserved.

SUPPLEMENTARY INFORMATION

In the format provided by the authors and unedited.

NAture NANoteCHNoloGy | www.nature.com/naturenanotechnology

Fast Water Transport in Graphene

Nanofluidic Channels

Quan Xie, Mohammad Amin Alibakhshi, Shuping Jiao,

Zhiping Xu, Marek Hempel, Jing Kong, Hyung Gyu Park and

Chuanhua Duan

1

1 Washburn Equation

In the capillary filling process, the mass flow rate Q in the nanochannel can be written as:

Q =∆P

R(S1)

Here, ∆P is the capillary pressure applied at the meniscus. R is the mass flow resistance

from the entrance to the meniscus. For a channel with uniform cross section, the capillary

pressure is a constant along the channel length direction and the mass flow resistance is

proportional to the length of liquid column. Assuming that X represents the meniscus

position from the entrance, the mass flow resistance can be written as:

R = RGX (S2)

Here, RG is the mass flow resistance per unit length for graphene channel.

The mass flow rate can also be correlated with the moving speed of meniscus:

Q = ρwhdX

dt(S3)

Here ρ is the water density, w and h represent the width and height of the channel,

respectively.

Combining equation (S1)-(S3), we have:

ρwhdX1(t)

dt=

∆P

RGX1(t)(S4)

After rearrangement and integration:

X21 (t) =

2

ρwh

∆P

RG

t (S5)

As mentioned in main manuscript, the capillary constant A is first determined by fitting

the first capillary filling experimental data X1(t) with equation (1):

A =1

ρwh

∆PGRG

(S6)

For the capillary filling process in our hybrid channel, the mass flow rate Q in the

nanochannel can be written as:

Q =∆P

R=

∆P

RSL+RGX2(t)(S7)

Combining with equation (S2), after rearrangement and integration:

X22 (t) + 2

RSL

RG

X2(t) =2

ρwh

∆P

RG

t (S8)

2

2 Error Analysis

As presented in main manuscript, capillary flow constant A is extracted from first filling

experiment and used as a known parameter for extraction of β from second filling experiment.

Here we perform the error (accuracy) analysis of our hybrid channel technique by calculating

the total error in this two-step extraction.

First, A is determined from a set of (T1i, X1i), i = 1..m, measured in the graphene

channel when the water is introduce from the graphene channel side (Fig. 1a in the main

manuscript). The total error is minimized using following relations:

ε2 =m∑i=1

(X21i − 2AT1i)

2 (S9)

∂ε2

∂A= 0 (S10)

A =

m∑i=1

X21iT1i

2m∑i=1

T 21i

(S11)

Similarly, β can be found from a set of (T2i, X2i), i = 1..n, measured in the graphene

channel when the water is introduced from the silica channel side (Fig. 1b in the main

manuscript):

ε2 =n∑i=1

(X22i + 2LsβX2i − 2AT2i)

2 (S12)

∂ε2

∂β= 0 (S13)

β =

2An∑i=1

X2iT2i −n∑i=1

X32i

2Lsn∑i=1

X22i

(S14)

Here we assume that there is no temporal error (δτ) for the snapshots taken with high

speed camera. Thus, β is only a function of X2i and A (β = f(X2i, A)). The total error

involved in this two-step extraction, Eβ, is associated with the spatial error (Ex), and the

error associated with A (EA):

Eβ =√E2A + E2

x (S15)

3

The derivations of EA and Ex are presented here.

Error associated with A: EA can be written as:

EA =1

β

∣∣∣∣ ∂β∂AδA∣∣∣∣ (S16)

in which both ∂β/∂A and δA must be determined. δA can be expressed as:

δA =

√√√√ m∑i=1

(∂A

∂X1i

δX)2 + (∂A

∂T1iδT )2 (S17)

Here, δX is the spatial resolution (of the microscope)(1µm), and δT is the time interval

between two consecutive frames (2ms). From equation (S11) we can get ∂A∂X1i

=√

2AT

3/21i

m∑j=1

T 21j

.

Ignoring contribution of the temporal error, δA can be written as:

δA =√

2AδX

√m∑i=1

T 31i

m∑i=1

T 21i

(S18)

The total number of data points is calculated as: m = Tmax/δT = L2G/2AδT , with LG

being length of the graphene channel. Moreover, one can write T1i = iδT , i = 1..m. If we

assume m� 1, δA can be written as:

δA ∼=√

2AδX

√14m4√δT

13m3

=3√

2AδT A δX

L2G

(S19)

As of ∂β/∂A, it can be calculated based on equation (S14):

∂β

∂A=

1

LS

n∑i=1

X2iT2i

n∑i=1

X22i

(S20)

Hence, EA can be calculated by plugging equation (S18) and equation (S20) into equation

(S16).

Spatial error (Ex): A similar approach is adopted for determining the spatial error. Ex

can be written as:

Ex =

√√√√ n∑j=1

(1

β

∣∣∣∣ ∂β∂X2j

δX2j

∣∣∣∣)2

(S21)

4

∂β/∂X2j can be found from equation (S14):

∂β

∂X2j

=1

2Ls

(n∑i=1

X22i)(2AT2j − 3X2

2j)− 2X2j(2An∑i=1

X2iT2i −n∑i=1

X32i)

(n∑i=1

X22i)

2

(S22)

Given that δX2j = δX, j = 1..n, Ex can be calculated by plugging equation (S22) into

equation (S21).

The total error in this two-step method can be calculated by equation (S15) and typ-

ical results were presented in Fig. S1. The estimation was done in 25 nm and 120 nm

nanochannels. For each device, while the total length of hybrid channel is fixed, the length

of graphene channel and silica channel can have many combinations. For example, the total

length is 300µm. The graphene/silica channel length could be 50µm/250µm, 150µm/150µm,

250µm/50µm for different channels.

It is obvious that longer channel can provide more snapshots and can reduce the ex-

perimental error. Since we always track the meniscus movement in graphene nanochannel,

experiments from longer graphene channel should have better accuracies (Fig. S1). It is

worth noting that the total error is still lower than 10% even in the worst scenario (deep

nanochannel, long silica channel) when β > 1. This shows feasibility of our hybrid channel

scheme and the resistance ratio extracted from this method is trustworthy.

Figure S1: Estimated experiment error for a) 300µm and b) 500µm long hybrid channels.

5

3 Meniscus Movement Detection

The capturing process of meniscus movement inside nanochannels is described in the manuscript.

However, the meniscus is not always clear during the capillary filling process, especially in

shallow nanochannels (∼25 nm). As shown in Fig. 2c,d in the main manuscript, the con-

trast between liquid phase and gas phase is extremely low in microscope snapshots and the

meniscus movement can only be seen in spatiotemporal diagrams with special colormaps.

For some other shallow channels, the meniscus movement is untraceable even with special

colormaps. A typical filling result of 25 nm nanochannel is exhibited in Fig. S2. The starting

point of capillary filling process is pointed out in Fig. S2, indicating the timestamp when

microchannel is filled with water. However, no clear boundaries of liquid/gas phase in channel

area can be seen in either colormaps. Therefore, it is impossible to extract resistance ratio

from this channel. Corresponding filling results are abandoned. A typical unsuccessful filling

process can be found in Supplementary Video S1 (plays at half speed). For comparison,

we’ve also added a video (Supplementary Video S2, plays at half speed) where meniscus

can be traced with special colormaps.

Figure S2: No clear boundaries between liquid phase and gas phase can be seen in both colormaps. a) Gray

colormap, b) Lines colormap.

For relatively deep nanochannels, the meniscus can be clearly determined because of the

contrast between liquid phase and gas phase (See Supplementary Video S3, plays at half

speed). However, the meniscus movement may not follow the Washburn equations (equa-

6

tion (1) and equation (2)) if the graphene surface is heterogeneous. As shown in earlier

sections, residues and defects could be introduced to graphene surface during the fabrica-

tion process. The heterogeneous graphene surface can cause distorted meniscus movement

inside nanochannel and makes it difficult to extract the resistance ratio. The spatiotemporal

diagrams presented in Fig. S3 shows typical distorted meniscus movement inside a 102

nm channel. According to the filling results from both ends of the channel, the meniscus

movement is impeded by two marked residues in nanochannel. The positions of impeded

movement stay the same in both filling results. Therefore, for some nanochannel, although

the meniscus can be clearly located, the resistance ratio cannot be extracted due to the

meniscus distortion. Fortunately, the meniscus distortion was only observed in small portion

of channels. Corresponding filling results in these channels are also abandoned.

Figure S3: Meniscus movement distortion observed in a 102 nm channel. a) Filling start from the lower end

of channel. b) Filling start from the upper end of channel.

It is worth noting that we observed air bubble trapping in some of the hybrid nanochannels

with heights between 30 nm and 90 nm during the capillary filling process (in both silica

channels and graphene channels). However, based on several previous works (including one

of our own work),[1, 2] this gas trapping phenomenon would only affect the capillary driving

pressure, but not the hydraulic resistance of the nanochannel. We therefore did not consider

the corresponding nanochannels as faulty channels in our measurements.

7

4 Capillary Flow Resistance

The capillary flow inside our nanochannel is considered as steady state fully developed in-

compressible laminar flow. The shape of cross section is rectangle and the width of our

nanochannel is much larger than the height. Therefore, the flow is two-dimensional and

Navier-Stokes equation can be written as:

η(∂2u

∂z2) =

∂P

∂x(S23)

The integration of Eq S23 will give us:

∂u

∂z= −1

η

(− ∂P

∂x

)z + c1 (S24)

u = − 1

2η

(− ∂P

∂x

)z2 + c1z + c2 (S25)

c1 and c2 are two constants, which can be solved with specific boundary conditions.

For our silica channel, the boundaries are non-slip.

u|z=0 = 0 (S26)

u|z=h = 0 (S27)

In this channel the velocity profile would be:

u = − 1

2η

(− ∂P

∂x

)(z − h

2)2 +

h2

8η

(− ∂P

∂x

)(S28)

The integration of equation (S28) will give us the mass flowrate Q:

Q = ρ

∫ h

0

udz · w =ρwh3

12η

(− ∂P

∂x

)(S29)

The theoretical mass flow resistance per unit length for silica channel is:

RS =12ηS

ρSwSh3S(S30)

For our graphene channel, while the channel ceiling is still no-slip silica surface, the

channel floor is slip graphene surface. If the slip length is defined as LSlip, the boundary

condition can be written as:∂u

∂z

∣∣∣∣z=0

=u

LSlip(S31)

8

u|z=h = 0 (S32)

In this channel the velocity profile would be:

u = − 1

2η

(− ∂P

∂x

)(z2 − z + LSlip

h+ LSliph2) (S33)

The integration of equation (S33) will give us the mass flowrate Q′:

Q′ = ρ

∫ h

0

udz · w =ρwh3

12η

(− ∂P

∂x

)(h+ 4LSliph+ LSlip

) (S34)

The theoretical mass flow resistance per unit length for silica channel is:

RG =12ηG

ρGwGh3G

hG + LSliphG + 4LSlip

(S35)

The mass flow resistance ratio can be written as :

β =RS

RG

=ηSηG

ρGρS

wGwS

(hGhS

)31 +

4LSlip,G

hG

1 +LSlip,G

hG

(S36)

9

5 Defects and Residues on Graphene Surface

During the capillary filling experiments, we expect that all graphene channels are uniform

and smooth, as shown in Fig. S4 (a). According to the captured snapshots, there exists

clear optical contrast between silica channel (lighter) and graphene channel (darker). More

importantly, uniform graphene coverage with no cracks can be seen in graphene channel area.

However, not all graphene channels are uniform and smooth. During the device fabri-

cation, defects and residues can be introduced to the graphene surface and the graphene

channel becomes no longer uniform. As exhibited in Fig. S4 (b), lighter contrast appears

in the graphene channel area, indicating the crack of graphene. This will expose the silica

surface beneath, which can no longer hold a slip boundary for water flow inside nanochannels.

Therefore, the corresponding mass flow resistance of the channel with discontinous graphene

coverage could be increased. In addition, residues such as folded graphene and polymer left-

overs can also be seen in Fig. S4 (b). These high roughness structures on graphene surface

can impede the water flow if the drag force is significant. This explains why β < 1 could

happen inside these channels.

It is also worth noting that the uniform contrast within the graphene channel area does

not always indicate a perfect graphene coverage. Therefore, we did not simply rely on the

microscope images to quantify the coverage and quality of graphene. Instead, the capillary

pressure inside graphene nanochannels were extracted and used as a measure of graphene

coverage and quality, which is explained in the main manuscript.

Figure S4: Microscope images of (a) uniform graphene channels and (b) non-uniform graphene nanochannels.

Residues (folded graphene and polymer residues) and defects (graphene cracks) can be seen from non-uniform

graphene nanochannel areas.

10

6 Raman Spectroscopy of Bonded Graphene

Nanochannel

Generally, the transferred CVD graphene can be visually seen under optical microscope with

the help of 280-nm-thick silicon dioxide underneath graphene and the graphene coverage

could be easily characterized with optical microscopic images. However, in our enclosed

graphene nanochannel area, the optical contrast of graphene becomes extremely low because

of the presence of the 500-µm-thick glass capping layer. Thus, the graphene coverage charac-

terization within the channel region becomes challenging. Here we use Raman spectroscopy

mapping to overcome this challenge. This non-invasive technique allows graphene character-

ization even with 500-µm-thick glass substrate above the graphene surface.

Figure S5: Raman intensity mapping images for different channels. (a) 25 nm nanochannel, (b) 47 nm

nanochannel, (c) 119 nm nanochannel. Three intensity maps (Si band: ∼ 520cm−1, G band: ∼ 1580cm−1,

2D band: ∼ 2700cm−1) are presented for each channel area.

Figure S5 shows the intensity mapping images of the Raman signals (Si band: ∼

520cm−1, G band: ∼ 1580cm−1, 2D band: ∼ 2700cm−1) for three representative graphene

11

nanochannels with different channel heights (25 nm, 47 nm, and 119 nm) and flow resistance

ratios. The silicon band intensity map can help us locate the graphene channel area: the

center brighter strip represent the channel area where additional interference could increase

the intensity; the darker strips represent the patterned-graphene-covered areas where the

intensity is weakened. [3] The G band and 2D band intensity maps can be used to quantify

the graphene coverage after anodic bonding. As it can be seen in Fig. S5 (a), the graphene

coverage in the 25 nm nanochannel is almost 100% inside the channel area, indicating that

patterned graphene survives both inside and outside of the channel area. The measured flow

resistance ratio (β = 3.2) in this channel is also high, suggesting that the water flow inside

graphene channel is greatly enhanced if we have good graphene coverage.

For deeper channels (47 nm and 119 nm), the G band and 2D band intensity maps show

that the graphene inside the channel area is damaged in anodic bonding process although

good graphene coverage outside channel area is still observed. While there still exists partial

graphene coverage in 47 nm nanochannel (Fig. S5 (b)), the graphene in 119 nm nanochannel

is almost gone completely (Fig. S5 (c)). This resonates well with our hypothesis that

deeper channels are more prone to have fatigue-cracks in graphene and have higher chance

of graphene damage during fabrication process.

The Raman intensity maps also echo the measured flow resistance ratio (β) in the cor-

responding nanochannels. For 47 nm nanochannel, the low flow resistance ratio (β = 0.7)

agrees with the partial coverage within the channel region which can significantly increase

the hydraulic resistance of the channel due to bad coverage/quality induced changes of flow

boundary conditions (e.g. from smooth hydrophobic surface to rough hydrophilic surface)

and channel heights (e.g. folded graphene in lower half of the channel).[4] For 119 nm

nanochannel, the flow resistance ratio (β = 0.95) is close to 1, indicating that the water

flow in the graphene channel is close to that in the silica channel. This is the result of the

almost-zero-coverage graphene nanochannel, which is equivalent to silica nanochannel.

12

7 Correlation of Raman Intensity Maps with β

As discussed in the manuscript, we observed flow impediment (β < 1) in some of our graphene

nanochannels. We attribute this impediment to the partial graphene coverage along with

consequent flow resistance increase in those graphene nanochannels.[4] To further confirm

this, we conducted Raman spectroscopy mapping to some of our devices where both high β

channel and low β channel were observed. The results are presented in Fig. S6. For better

illustration, only the G band intensity maps, the fingerprint of graphene, are presented. For

each device, two channels are scanned where one channel shows high β and the other channel

shows low β. Corresponding intensity maps are shown in the same column in Fig. S6.

Figure S6: Raman intensity maps comparison for high β channel and low β channel on the same device. Two

maps from each column are collected from two channels on the same device and five different devices are

scanned.

It is worth noting that to rule out device to device variations, including channel geometries

and fabrication imperfections, only the channels from the same device (in each column) are

compared. Based on the Raman intensity maps comparison, the Raman signals of the high

β channels are stronger and more uniform while the low β channels always show some issues

such as bad coverage/quality and many graphene/silica interfaces. Thus, we believe that

13

the low β (β < 1) collected from capillary filling experiments are associated with the bad

coverage/quality of the graphene nanochannels.

14

8 Capillary Pressure inside Graphene Nanochannel

As discussed in the main manuscript, it is likely that the graphene coverage inside our

channel area is not 100% and the graphene channel properties (hydraulic resistance, capillary

pressure) are affected by this incomplete graphene coverage.

As shown in Fig. 3b, the maximum effective capillary pressure ∆PG extracted from our

experiment can be as high as 0.9∆P0, which is very close to the value of silica nanochannels

[1]. This corresponds to the least coverage of graphene and/or worst quality of graphene. On

the other hand, the minimum ∆PG value of our graphene nanochannels is around 0.1∆P0,

which was mainly observed in the graphene nanochannels with heights below 30 nm. Given

that ∆PSiO2 is close to 0.9∆P0 and graphene in these shallow channels have complete coverage

and best quality [3], the limit of capillary pressure ∆PGraphene is thus close to −0.7∆P0. This

value is greater than the predication based on the Young-Laplace equation if bulk liquid-

vapor surface tension and advancing contact angle measured on plain graphene surface were

used [5], indicating that the nanoscale confinement, fabrication-induced stress [6] and the

rapid capillary filling process itself [7] may significantly affect the surface energy and the

contact angle, which is worthy of further investigation.

Nevertheless, our results show that the extracted capillary pressure ∆PG can be considered

as a good measure of the graphene coverage/quality inside the graphene channel. If we assume

that the ∆PGraphene has a constant value of −0.7∆P0 for all graphene nanochannels regardless

of the actual graphene coverage and quality, we can estimate the graphene coverage in the

graphene nanochannel from ∆PG based on equation (4).

15

9 Coverage Analysis

Intuitively, Raman spectroscopy mapping can provide the graphene coverage (%) inside

nanochannel. For instance, the G band and 2D band intensity maps of the 47 nm nanochan-

nel mentioned in earlier section can be used to calculate the graphene coverage (%) inside

the channel with certain signal threshold settings. However, due to the low signal-to-noise

ratio, the graphene coverage (%) is dependent on the signal threshold settings and can vary

in the range of ∼ 40% to ∼ 60%, as seen in Fig. S7.

Figure S7: Graphene coverage calculation based on Raman intensity mapping images of the 47 nm nanochan-

nel (a) G band intensity map (b) 2D band intensity map.

In comparison, we can also estimate the effective graphene coverage (%) based on capillary

pressure analysis, as discussed in the manuscript. According to our model (equation (4)), the

effective coverage of this 47 nm nanochannel is 67.5%. In addition, the effective coverage of

the 25 nm and 119 nm channel mentioned in earlier section are 100% and 7.2%, respectively.

This shows that the graphene coverage calculated from two different approaches are in close

agreement with each other.

It is worth noting that the low signal-to-noise ratio of the Raman signal is a result of the

16

thick glass substrate above graphene surface and it is very difficult to resolve with parameter

adjustment such as laser power and integration time. In addition, the spatial resolution of

the mapping is limited by the laser spot size, which is in the order of hundreds of nanometers

in our setup. Therefore, although Raman intensity mapping could yield an approximate cov-

erage analysis, the accurate estimation of the graphene coverage (%) in our bonded graphene

nanochannel remains challenging. Furthermore, to guarantee the Raman peak strength and

the spatial resolution, the mapping is very time consuming and it is unrealistic to scan all our

bonded graphene nanochannels. Considering the close agreement of the coverage calculated

from Raman mapping and the coverage based on capillary pressure analysis, the effective

coverage estimated with capillary pressure can reflect the actual graphene coverage inside

the bonded nanochannel to a great extent. Given the decent accuracies and time effective-

ness of this method, we employed the capillary pressure analysis to estimate the effective

graphene coverage in our bonded nanochannels.

17

10 High Coverage Graphene Channel Data

Table S1: Data of Graphene Nanochannels with coverage higher than 90%

Channel No. Channel Height h(nm) Coverage Extracted slip length LSlip,G(nm)

1 25 100.0% 198.34

2 25 100.0% 192.41

3 25 100.0% 5.70

4 25 98.3% 3.69

5 25 100.0% 9.60

6 25 100.0% 3.56

7 25 100.0% 27.26

8 70 95.1% 23.17

9 37 94.3% 0.47

10 37 91.7% 5.94

11 37 92.2% 7.56

12 25 100.0% 18.86

13 25 100.0% 6.90

14 25 100.0% 45.18

15 34 99.0% 159.01

18

11 Surface Charge of Graphene inside Nanochannels

According to our previous studies,[3, 8] the ionic conductance of nanochannel did not change

much at 10−7 ∼ 10−5M HCl, KCl and NaCl solutions. This indicates that at very low ionic

concentrations where electrical double layers overlap in the nanochannels, the surface charge

density of the nanochannels will be a very weak function of the pH and concentration of the

bulk solution. This phenomenon has also been reported and explained in details by Jensen

et al.[9]

The fundamental reason for such weak dependence of surface charge density on pH and

salt concentration at such low salt concentrations is that electro-neutrality and surface chem-

istry are tightly coupled in such situations. Taking surface charge of graphene nanochannel

as an example, if we assume the surface charge comes from the function group −COOH,

proton concentration inside the nanochannel will be determined by the surface charge den-

sity ([H+] = 2σ/eh). However, proton is also involved in the reversible surface dissociation

reaction (−COOH −COO− + H+) which provides the surface charge (−COO−). This

tight coupling makes the surface charge density only a function of the channel height and the

density of the −COOH functional group.

19

12 Molecular Dynamics Model and Methods

We modeled nanoconfined water between graphene and/or silica substrates as shown in Fig.

S8. The channel length along the Y direction is 14.1 nm, and width along the X direction is

2.7 nm. Periodic boundary conditions (PBCs) are applied in both X and Y directions. The

Si, O, C atoms in silica and graphene coatings are fixed in the simulations. In order to ensure

that the fluid in channel is fully equilibrated, we adjusted the inter-wall distance h at 300 K

till the wall pressure reaches 1 atm. In our model, there are 1816 water molecules within the

channel, and the channel height h is ∼1.84 nm with minor variations for different types of

walls under consideration. Here the value of h is measured as the distance between surface

atoms in the two opposite walls. Although this height is one order of magnitude smaller than

the nanochannel height used in our experiment, it is large enough to exclude the appearance

of layered water structures that occurs at h < 1.4 nm. A quantitative understanding is still

expected, since the slip length would not change with the channel height beyond h = 1.4

nm.[10] Two models (case a and b) were explored in this study, with asymmetric walls, i.e.,

silica on one side and monolayer graphene (or graphene-coated silica) on the other.

Figure S8: Atomic structures of carboxyl-functionalized graphene that is considered in our MD simulations.

The carbon, oxygen and hydrogen atoms are plotted in blue, red and white colors, respectively.

During the preparation and transfer processes of graphene-coated silica surfaces, defects

are unavoidable. The silica substrate may be oxidized as exposed to the environment. To

consider this factor that may modify water transport in the asymmetric graphene nanochan-

nel, we construct two additional models (case c and d) with graphene oxidized by carboxyl

functional groups (Fig. S8). The concentration of -COOH groups under consideration are

20

cCOOH = 0.135% and 0.5%. Here cCOOH is defined as the ratio between the number of

carboxyl groups and the total number of carbon atoms in pristine graphene.

Our molecular dynamics (MD) simulations are performed by using the large-scale atomic/

molecular massively parallel simulation (LAMMPS).[11] The all-atom optimized potentials

for liquid simulations (OPLS-AA) force field is used to construct atomic models of graphene

and graphene oxide.[12] We constructed carboxyl-functionalized graphene (on both sides of

the sheet) with various concentrations. The SPC/E model[13, 14] is used for water with the

SHAKE algorithm, which predicts the density and viscosity of bulk water as 0.9913 kg/L

and 0.729 mPa·s. The van der Waals interaction between water and graphene is described

following the Lennard-Jones 12-6 form, that is V = 4ε[(σ/r)12 − (σ/r)6], where r is the

interatomic distance between oxygen and carbon atoms. Parameters ε = 0.09365 kcal/mol

and σ = 0.3190 nm are chosen, yielding a water contact angle of 95◦ on graphene.[15] The

silica surfaces are created by cutting the β-cristobalite by its (111) surface that consists of

low-density (4.54 -OH/nm2) hydroxyl groups. The CLAYFF force field is used for SiO2, with

parameters listed in Table S2 [16]. This set of parameters predicts a water contact angle

of 95◦.[16] The Lorentz-Berthelot mixing rules are used for the van der Waals interactions

between SiO2 and water atoms described in the Lennard-Jones 12-6 form. The van der

Waals forces are truncated at 1.2 nm and long-range Columbic interactions are computed

using the particle-particle particle-mesh (PPPM) algorithm.[17] A time step of 1.0 fs is used

to integrate the equations of motion. The total time of simulation is a few nanoseconds.

Water molecules are equilibrated at 300 K using the Berendsen thermostat.

Using MD simulations, we explored the interfacial friction between the nanoconfined water

and the channel walls of graphene or silica, which can be related to the interfacial slippage

for water flow in the channel. The coefficient of liquid-solid friction, λ, is calculated from

the autocorrelation function of interfacial forces in equilibrium molecular dynamics (EMD)

runs, which can be expressed in terms of the Green-Kubo formulation,

λ =1

SkBT

∫ ∞0

⟨Fα(t)Fα(t)

⟩dt (S37)

Here Fα(t) is the time-dependent interfacial force along the α direction, acting on the

surface with area S. The value of λ, averaged in the in-plane (x and y) directions, can be

21

related to the slip length Lslip through Lslip = η/λ, where η is the shear viscosity.[13, 18] The

value of η = 0.729mPa · s predicted from the SPC/E water model at 300 K is used for the

evaluation using equation (S37).[19]

Table S2: Parameters for the Lennard-Jones parameters of SiO2

Atom σ (kcal/mol) ε(A) Atomic charge (e)

Si 0 3.302 2.1

BO1 (in SiO2) 0.15533 3.166 -1.05

SO2 (in SiO2) 0.15533 3.166 -0.95

H (in SiO2) 0 0 0.425

To validate the values of slip lengths calculated using our EMD approach, we conducted

non-equilibrium MD simulations (NEMD) as well. In the NEMD simulations, flow is driven

by applying a specific acceleration, from 0.0175 to 0.1225 nm/ps2, to set up the steady state

of water flow with the peak velocity along the across section profile up to 200 m/s. It takes a

few hundred picoseconds to reach the steady state usually, and we continue the simulations

for additional 1-3 ns to collect data for analysis. We fit the data in Fig. S9 using an

inverse hyperbolic sine (IHS) relationship between the shear stress τ = F/A and velocity v

at the graphene/water interface with area A, that is τ/τ0 = arcsinh(v/v0),[20] where F is

the interfacial force between water and graphene, v is averaged over the steady state and

along the interface directions, τ0 and v0 are fitted parameters. The friction coefficient λ and

slip length LSlip are then evaluated by λ = τ0/v0, LSlip = η/λ and compared to the values

obtained from EMD simulations.[20] The results summarized in Table S3 clearly indicate

the consistence between EMD and NEMD predictions on the slip lengths.

22

Figure S9: (a)-(d) Velocity profiles measured during the NEMD simulations, where flow is driven at different

pressure gradients in cases a (Graphene/H2O/SiO2), b (cCOOH = 0%), c (cCOOH = 0.135%), d (cCOOH =

0.5%); (e) Interfacial shear stress τ plotted as a function of the slip velocity v at water/graphene interfaces.

23

Table S3: The slip lengths calculated by EMD and NEMD simulations for cases a, b, c, d (Fig. 5). For

NEMD simulations, we include the errors in fitting the shear stress (τ) - velocity (v) relation using τ/τ0 =

arcsinh(v/v0).

Graphene/ SiO2/GO/ SiO2/GO/ SiO2/GO/

H2O/SiO2 H2O/SiO2 H2O/SiO2 H2O/SiO2

cCOOH = 0% cCOOH = 0.135% cCOOH = 0.5%

LSlip,G (nm)

EMD

90 75.2 10.8 3.3

LSlip,G (nm)

NEMD

127±12.8 72.1±15.8 8.24±1.1 2.3±0.38

24

13 Apparent Step-height of Graphene/Silica Surface

Figure S10 presents the AFM characterization of the graphene/silica channel connection

areas from different 25 nm channel devices. Upper subfigures show the surface profiles and

lower subfigures show the section profiles of graphene channel area (blue) and silica channel

area (red). Although different devices share the same channel height, there exist variations

of apparent step-height of graphene/silica surface among devices.

Wrinkles and high roughness surface can also be seen in upper subfigures of Fig. S10(b)(c).

Figure S10: Apparent step-height of graphene/silica surface for different 25 nm nanochannels

25

References

1. Alibakhshi, M. A., Xie, Q., Li, Y. & Duan, C. Accurate measurement of liquid transport

through nanoscale conduits. Scientific Reports 6, 24936 (2016).

2. Chauvet, F., Geoffroy, S., Hamoumi, A., Prat, M. & Joseph, P. Roles of gas in capillary

filling of nanoslits. Soft Matter 8, 10738–10749 (2012).

3. Xie, Q., Xin, F., Park, H. G. & Duan, C. Ion transport in graphene nanofluidic channels.

Nanoscale 8, 19527–19535 (2016).

4. Wei, N., Peng, X. & Xu, Z. Understanding Water Permeation in Graphene Oxide Mem-

branes. ACS Applied Materials & Interfaces 6. PMID: 24669772, 5877–5883 (2014).

5. Raj, R., Maroo, S. C. & Wang, E. N. Wettability of Graphene. Nano Letters 13. PMID:

23458704, 1509–1515 (2013).

6. Xiong, W., Liu, J. Z., Zhang, Z.-L. & Zhen, Q.-S. Control of surface wettability via

strain engineering. Acta Mechanica Sinica 29, 543–549 (2013).

7. Deng, Y., Chen, L., Liu, Q., Yu, J. & Wang, H. Nanoscale View of Dewetting and Coat-

ing on Partially Wetted Solids. The Journal of Physical Chemistry Letters 7. PMID:

27115464, 1763–1768 (2016).

8. Duan, C. & Majumdar, A. Anomalous ion transport in 2-nm hydrophilic nanochannels.

Nature Nanotechnology 5, 848–852 (2010).

9. Jensen, K. L. et al. Hydronium-dominated ion transport in carbon-dioxide-saturated

electrolytes at low salt concentrations in nanochannels. Phys. Rev. E 83, 056307 (5

2011).

10. Wei, N., Peng, X. & Xu, Z. Breakdown of fast water transport in graphene oxides. Phys.

Rev. E 89, 012113 (1 2014).

11. Plimpton, S. Fast Parallel Algorithms for Short-Range Molecular Dynamics. Journal of

Computational Physics 117, 1–19 (1995).

26

12. Shih, C.-J., Lin, S., Sharma, R., Strano, M. S. & Blankschtein, D. Understanding the

pH-Dependent Behavior of Graphene Oxide Aqueous Solutions: A Comparative Exper-

imental and Molecular Dynamics Simulation Study. Langmuir 28, 235–241 (2012).

13. Falk, K., Sedlmeier, F., Joly, L., Netz, R. R. & Bocquet, L. Molecular Origin of Fast

Water Transport in Carbon Nanotube Membranes: Superlubricity versus Curvature

Dependent Friction. Nano Letters 10, 4067–4073 (2010).

14. Wei, N., Lv, C. & Xu, Z. Wetting of Graphene Oxide: A Molecular Dynamics Study.

Langmuir 30, 3572–3578 (2014).

15. Rafiee, J. et al. Wetting transparency of graphene. Nat. Mater. 11, 217–222 (2012).

16. Ho, T. A. et al. Interfacial water on crystalline silica: a comparative molecular dynamics

simulation study. Molecular Simulation 37, 172–195 (2011).

17. Hockney, R. W. & Eastwood, J. W. Computer Simulation Using Particles (Taylor &

Francis, Inc., Bristol, PA, USA, 1988).

18. Bocquet, L. & Barrat, J.-L. On the Green-Kubo relationship for the liquid-solid friction

coefficient. The Journal of Chemical Physics 139, 044704 (2013).

19. Gonzalez, M. A. & Abascal, J. L. F. The shear viscosity of rigid water models. The

Journal of Chemical Physics 132, 096101 (2010).

20. Xiong, W. et al. Strain engineering water transport in graphene nanochannels. Phys.

Rev. E 84, 056329 (2011).

27