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January 03, 2014

C 2014 American Chemical Society

Implications of Permeation throughIntrinsic Defects in Graphene on theDesign of Defect-Tolerant Membranesfor Gas SeparationMichael S. H. Boutilier,† Chengzhen Sun,†,‡ Sean C. O’Hern,† Harold Au,† Nicolas G. Hadjiconstantinou,† and

Rohit Karnik†,*

†Department of Mechanical Engineering, Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, Massachusetts 02139, United States, and‡State Key Laboratory of Multiphase Flow in Power Engineering, Xi'an Jiaotong University, Xianning West Road 28, Xi'an 710049, P.R. China

Separating the components of a gas-eous mixture is a critical step in severalimportant industrial processes includ-

ing natural gas purification, hydrogen pro-duction, carbon dioxide sequestration, andoxy-combustion.1�4 Membrane separationsystems offer definite benefits compared toconventional cryogenic and sorption-basedmethods, especially for small-to-mediumscale applications.1 Membrane separationsemploy no moving parts, require no exoticchemicals, and typically exhibit low energyrequirements and flexibility in configu-ration.4 However, the inherent trade-offbetween the permeability and selectivityof conventional membrane materials haslargely limited gas separation membranesto systems requiring relatively low produc-tion rates or to mixtures with high impurityconcentrations in the feed gas.5

In contrast, graphene gas separationmembranes have the potential to signifi-cantly surpass the permeance and selec-tivity limits of conventional membranes.6,7

With this single atom thick material, it is

possible to create subnanometer-scalepores that allow smaller gas molecules topass through but produce a physical barrierseverely limiting the passage of largermolecules.8 Theoretical studies employingvarious computational and analytical toolshave predicted permeance and selectivityvalues for various graphene pore geome-tries that are orders of magnitude higherthan have been achieved with existingmembrane technology.6,7,9�11 The feasibil-ity of constructing such selective pores wasrecently demonstrated by Koenig et al.8

using micrometer-scale graphene mem-branes in which pores were created byUV-induced oxidative etching. They wereable to produce two graphene membraneswith selectivities of approximately 15 000,one being strongly permeable to gas mol-ecules with kinetic diameters smaller than3.4 Å and the other permeable to gases withkinetic diameters smaller than 4.9 Å.However, the imperfect quality of large

areas of graphenepresents a significant chal-lenge in scaling such single-layer graphene

* Address correspondence tokarnik@mit.edu.

Received for review October 23, 2013and accepted December 20, 2013.

Published online10.1021/nn405537u

ABSTRACT Gas transport through intrinsic defects and tears is a critical yet

poorly understood phenomenon in graphene membranes for gas separation. We

report that independent stacking of graphene layers on a porous support

exponentially decreases flow through defects. On the basis of experimental

results, we develop a gas transport model that elucidates the separate contribu-

tions of tears and intrinsic defects on gas leakage through these membranes. The

model shows that the pore size of the porous support and its permeance critically

affect the separation behavior, and reveals the parameter space where gas separation can be achieved regardless of the presence of nonselective defects,

even for single-layer membranes. The results provide a framework for understanding gas transport in graphene membranes and guide the design of

practical, selectively permeable graphene membranes for gas separation.

KEYWORDS: graphene membranes . gas separation . multilayer graphene . nanofiltration . nanofluidics

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membranes to macroscopic sizes. Small areas of pris-tine graphene are impermeable even to helium,12 butdue to the difficulty in isolating large areas of pristinegraphene, macroscopic graphene membranes inher-ently have a nonzero permeance due to the presenceof defects in the graphene.13 These defects includeintrinsic nanometer-scale holes that develop duringthe graphene growth process, and tears or othermicrometer-scale gaps in the graphene that formduring the membrane manufacturing process.13

Gas leakage through micrometer-scale tears andnanometer-scale intrinsic defects in macroscopicgraphene membranes can severely limit their selec-tivity and make separation impossible. Indeed, to dategas separation has been demonstrated only in multi-layered membranes prepared from grapheneoxide,14�16 where permeance is 3�4 orders of magni-tude lower than that predicted by simulations acrosssingle-layer graphene membranes. Even in these mul-tilayer membranes, gas transport through defects canplay a role in addition to interlayer transport.15 It istherefore critical to understand transport through in-trinsic defects and tears in graphene and to developstrategies to mitigate nonselective leakage if the fullpotential of graphenemembranes for gas separation isto be realized.In this paper, we quantify the effects of tears and

intrinsic defects on the gas permeance and separationperformance of graphene composite membranes(GCM) comprising graphene situated on a poroussupport membrane. We show that independent stack-ing of two to five layers of graphene is a promisingmethod for reducing the effects of defects whilemaintaining the nanometer-scale thickness of thegraphene layer. The inherent permeance of thesemultilayer macroscopic graphene membranes is mea-sured and a gas transport model is developed that canaccurately explain the experimentally observed flowrates. From this model, the separate contributions oftears and intrinsic defects to the inherent permeanceof macroscopic graphene membranes are estimated.This model is further used to identify design con-straints for realizing selectively permeable graphenemembranes for gas separation.

RESULTS AND DISCUSSION

Graphene Composite Memrbanes. Graphene compositemembranes were fabricated by transferring graphenegrown on copper foil by chemical vapor deposition(CVD) to a polycarbonate track-etched membrane(PCTEM) support using a direct transfer method devel-oped previously (Figure 1a).13 PCTEMs were chosen asthe porous support because they contain a highdensity of straight, well-defined pores with uniformsizes17 that can be tuned from 10 nm to over 10 μm.The porous support allows the graphene layer to behandled without damage.13 The series of parallel pore

channels in the PCTEM also isolate small areas ofgraphene, such that the graphene composite mem-brane is effectively formed from numerous smallergraphene membranes arrayed in parallel. Flowthrough any defect in the graphene is restricted bythe narrow channels in the PCTEM, which can poten-tially prevent leakage from dominating over the flowthrough intentionally created selective pores.

In addition, we hypothesized that the effects ofleakage may be further mitigated by independentlystacking multiple layers of graphene on the poroussupport, so that defects in one layer are covered byanother layer. To achieve this goal, the transfer processwas repeated to create GCMs with multiple indepen-dently stacked layers of graphene (Figure 1a). In a mul-tilayer graphene composite membrane (Figure 1b),individual layers of graphene can be distinguished bythe successively darker areas over the white PCTEM.Scanning electron microscope (SEM) images of GCMswith a single layer of graphene (Figure 1c) clearly showseveral polycarbonate pores with micrometer-scaletears in their graphene coverage. In contrast, SEMimages of a five layer graphene membrane (Figure 1d)show a much smaller fraction of tears as compared tothe single layer membrane. The images clearly showthat independently stacked graphene layers improvegraphene coverage on the PCTEM.

Independent Stacking of Multiple Layers of Graphene Re-duces Leakage. Gas transport through the GCMs wasmeasured by supplying the desired gas on the gra-phene side of the membrane and monitoring the

Figure 1. Graphene composite membranes. (a) Membranefabrication process by direct transfer of graphene fromcopper foil to a polycarbonate track-etched membrane(PCTEM). The copper foil was etched using ammoniumpersulfate (APS). Multilayer graphene composite mem-branes were fabricated by repeated application of the onelayer transfer process. (b) Photograph of a four-layer gra-phene compositemembrane. Scale bar, 5mm. (c andd) SEMimages of a (c) one-layer and (d) five-layer graphene mem-brane and tears. Scale bars are 5 and 1 μm.

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pressure rise in a downstream reservoir (Figure 2a,inset). Transferring a single layer of graphene onto aPCTEM reduced the flow rate of helium through thePCTEM by approximately 60%, as seen in Figure 2a.Since no selective pores were created in graphene, weexpect that the residual 40% leakage represents he-lium flow through intrinsic defects and tears in thegraphene. Next, we examined helium flow ratesthrough GCMs with one to five layers of graphene,and observed an exponential reduction in the heliumflow rate as the number of graphene layers wasincreased (Figure 2). By stacking multiple layers ofgraphene on a PCTEM, it was possible to produce amacroscopic graphene membrane with a 99% smallerpermeance (i.e., only 1% leakage) compared to thebare support membrane. Furthermore, the apparentexponential decay of helium flow rates with increasingnumber of layers of graphene suggests that approxi-mately the same fraction of polycarbonate pores arecovered by graphene in each layer, and that thelocations of the micrometer-scale tears in each layerare independent of those in the other layers.

Modeling Transport through Graphene Composite Mem-branes. We developed amodel to quantitatively under-stand the effects of defects and independent stackingof graphene layers on the gas transport behavior of themembranes. Since pristine graphene is impermeable

to gases,12 we assume that the net permeance is aresult of flow through micrometer-scale tears over thePCTEM pores (see Figure 1c,d), intrinsic nanometer-scale defects dispersed across areas of continuousgraphene,13 and any intentionally created selectivenanopores (Figure 3).

We account for any micrometer-scale tears byassuming that each layer of graphene covers a fractionγ of the pores in the PCTEM support, with subsequentlayers covering the same fraction of pores indepen-dent of the other layers (Figure 3a,b). As a result, a two-layer graphene membrane has polycarbonate poreswith zero, one, or two layers of graphene (Figure 3c).For a general N-layer graphene composite membrane,the fraction of polycarbonate pores with n layers ofgraphene coverage, γn, is given by,

γn ¼ N!n!(N � n)!

(1 � γ)N � nγn (1)

When multiple layers of graphene are stacked, weassume that intrinsic defects in one layer are coveredby graphene in another layer, except where intrinsic

Figure 2. Measurements of gas flow rates through gra-phene composite membranes with multiple layers of gra-phene. (a) Effect of the number of layers of graphene onheliumflow rate, normalizedby theflow rate through a barePCTEM. Data points represent flow rates measured throughdifferent GCMs that were manufactured in parallel. Insetshows a sketch of the gas flowmeasurement setup. (b) Timetraces of pressure rise in the downstream reservoir duringhelium flow rate measurements for the membranes.

Figure 3. Transport pathways considered in the gas flowrate model. (a�c) Illustration of the change in tears andintrinsic defects over polycarbonate track-etched poresupon transferring (b) one and (c) two layers of grapheneonto the PCTEM. (d) Equivalent resistance network forflow through GCMs. Note that RPC = 4(πDPC

2 PPCA )�1, RS =

4(πDPC2 PS

A)�1, and RIP(n) = (LIP

2 PIP,nA )�1.

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defects in all layers randomly align. The average dis-tance between intrinsic defects in one-layer graphene,LIP(1), was obtained from the defect size distribution datain ref 13 (where we reported characterization of gra-phene from the same source), combined with theintrinsic porosity of graphene, η, i.e., the fraction ofgraphene area occupied by intrinsic defects. The aver-age spacing between aligned intrinsic defects in n

stacked layers of graphene, LIP(n), was computed by

Monte Carlo simulations (see Supporting InformationSection 1 for details). Given the spacing betweenaligned intrinsic defects, the fraction of polycarbonatepores (of diameter DPC) with n layers of graphene thathave k aligned intrinsic defects can then be calculatedfrom the Poisson distribution,

fn, k ¼ 1k!

π

4DPC

L(n)IP

!224

35k

exp � π

4DPC

L(n)IP

!224

35 (2)

To limit complexity, we assumed identical per-meance for all intrinsic defects regardless of the size,an assumption that was independently validated (seeSupporting Information Section 1.3). The average per-meance due to intrinsic defects in n-layer graphene(PIP,n

A ) was estimated as the equilibrium ideal gas fluxthrough the fraction of graphene area occupied byaligned intrinsic defects, assuming that the intrinsicpores are much larger than the gas molecules. Finally,any selective nanopores intentionally introduced intothe graphene will provide a flow path in parallel to theintrinsic defects. We assume that their density is suffi-ciently high to represent flow through these nano-pores by an average selective pore permeance, PS

A,defined as the molar flow rate per unit graphene areaper unit pressure difference.

Gas transport through the graphene membranewas computed using an equivalent resistance net-work18 (Figure 3d). Each branch of this networkrepresents polycarbonate pores with a certain num-ber of layers of graphene coverage and a certainnumber of intrinsic defects. The graphene composite

membrane permeance for arbitrary gas A is thencomputed as,

PA ¼ γ0PAPC þ ∑

N

n¼ 1∑¥

k¼ 0

γnfn, k1PAPC

þ 4kπPAIP, n

L(n)IP

DPC

!2

þ PAS

0@

1A

�1264

375

�1

(3)

The polycarbonate permeance (PPCA ) for each gas

was measured on a bare PCTEMwithout any graphene(see Supporting Information Section 2 and Figure S9).This leaves three unspecified parameters for thismodel: (1) the fraction of polycarbonate pores coveredby a single layer of graphene, γ, (2) the fraction ofgraphene area occupied by intrinsic defects, η, and ifapplicable, (3) the intentionally created selective nano-pore permeance, PS

A.

Model Predicts Transport Characteristics of IndependentlyStacked Graphene Membranes. The number density anddistance between intrinsic defects is an importantparameter governing the membrane performance.Power-law fits to the Monte Carlo simulation resultsfor a range of intrinsic porosities (Figure 4a) show ascaling (LIP

(n))�2∼ ηn, i.e., the average number density ofaligned intrinsic defects in independently stackedlayers of graphene decreases as the nth power ofthe intrinsic porosity of graphene. This scaling be-havior is explained by the fact that when a layer ofgraphene with porosity η is placed on a previouslayer, only a fraction η of the existing defects alignwith defects in the new layer. Therefore, each suc-cessive layer reduces the number of defects by afraction η. The exact coefficients of the scaling rela-tions depend on the intrinsic defect size distribution.

Next, we compared model predictions (SupportingInformation Section 1.3) of gas transport through themultilayer membranes with measured flow rates ofgases with a range of molecule sizes (He, N2, andSF6, with kinetic diameters of 2.6, 3.64, and 5.5 Å,respectively19), without any intentionally createdselective nanopores (PS

A = 0, Figure 4b,c). Withleast-squares fitting, the graphene composite

Figure 4. Multiple layer graphene composite membrane flow rate measurements and model fit. (a) Computed averagedistance between intrinsic defects in multilayer graphene membranes. (b) Flow rate model fit, giving a coverage of 69.8%(γ = 0.698) and an intrinsic porosity of 0.687% (η = 0.00687). Flow rates are normalized by the value for a bare PCTEM. (c)Measured flow rate ratios compared to model fit for flow rates.

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membrane characteristics were well described by γ =0.698 andη= 6.87� 10�3. Themodel results accuratelymatched the exponential decrease of measured flowrates as the number of layers of graphene was in-creased from one to four (Figure 4b). The model alsocaptures the gradual rise in flow rate ratios of the gaseswith increasing number of graphene layers. Since theintrinsic defects were measured to be larger than thekinetic diameter of the gases, the intrinsic defect selec-tivity is approximately PIP,1

A /PIP,1B = (MB/MA)

1/2 from thekinetic theory of gases, which is somewhat higherthan that for flow through the polycarbonate pores(Figure S10b). As a result, increasing the contribution ofintrinsic defects compared to tears upon adding morelayers of graphene causes a slight increase in the flowrate ratio.

Quantitative agreement between the experimen-tally measured and theoretical flow rate and flow rateratio with increasing number of graphene layers pro-vides validation for the model developed here. Themodel was further verified by comparing its predic-tions to those of a more detailed model, discussed inSupporting Information Section 1. We considered thepossibility that the deviation of the model trends fromthe measured flow rates for five layer graphene com-posite membranes (Figure 4b) is in-plane gas trans-port between layers of graphene, which scales weakly(inversely) with the number of layers and will eventuallygovern transport as the number of layers increases. If thedeviation for the five-layer membrane is attributedentirely to in-plane transport, we can estimate a con-servative lower bound on in-plane transport resistanceof 1.2 � 104 Pa 3 s 3m

2/mol for helium, corresponding toan interlayer diffusivity of 1.2� 10�15m3/s (see Support-ing Information Section 3). However, such a low inter-layer transport resistance would cause significantdeviation between measured flow rates and modelpredictions for two, three, and four layer membranes,which we do not observe. Therefore, it is concluded thatinterlayer transport is not responsible for the deviationbetween model and measured flow rates for the fivelayer membranes in Figure 4b, and the computed in-plane transport resistance represents a very conservativelower bound. The discrepancy between the measure-ments and model predictions for the five layer mem-brane in Figure 4b is thus likely due to damage inducedduring the repeated mechanical pressing steps, whichcould become a significant factor at low flow rates.

A recent report in which transport across stackedgraphene layers was measured speculated that theincrease in flow rate ratio is due to interlayer trans-port.15 Our experiments and modeling results indicatethat the permeability for interlayer transport in gra-phene is low and that intrinsic defects alone canaccount for the increased selectivity observed fromstacking successive layers. This result is consistentwith measurements of in-plane gas diffusion along a

graphene�copper interface, which show significantlylower transport than has been observed betweenlayers of graphene oxide.20

Feasibility of Selective Transport in the Presence of Non-selective Defects. Wenowuse the gas transportmodel toexplore whether gas separation may be accomplishedin macroscopic, single- or few-layer graphene mem-branes regardless of the presence of defects for twomodel gases, He and SF6. To do this, we estimatedintentionally created selective nanopore permeances,and used these in the model to predict overall mem-brane selectivities in the presence of intentionallycreated selective nanopores. On the basis of moleculardynamics simulations9 and gas flux measurements byKoenig et al.8 (assuming that the reported flux corre-sponds to one measured pore), the achievable resis-tance of a single selective nanopore to He is estimatedto be in the range of 9� 1019 to 2� 1022 Pa 3 s/mol. Wefurther assume a selective pore density of 1012 cm�2

(one nanopore per 10 � 10 nm2 area) to calculate thepermeance due to selective pores, PS

He. On the basis ofthe measurements of Koenig et al.,8 the selective poreselectivity is set to PS

He/PSSF6 = 15 000.

The available graphene quality (specified byη = 6.87� 10�3 and LIP

(n) in Figure 4a) and the graphenepermeance due to selective pores are limited by thegraphene synthesis and selective pore creation proce-dures. However, it is possible to precisely control thesupport membrane pore size and permeance, and toenhance the graphene coverage through the gra-phene transfer process and the number of layers used.The model predicts that for the 1 μm diameter poly-carbonate membranes with∼70% graphene coveragefabricated in this study, it is unlikely that any selectivitywill be obtained even with five layers of graphene forthe expected range of selective pore permeance(Figure 5, top left plot). This lack of selectivity resultsprimarily from the high permeance of polycarbonatepores, causing leakage throughmicrometer-scale tearsto dominate. By increasing the polycarbonate poreresistance to mitigate the effect of leaks through tears,it is possible to achieve modest selectivity with fivelayers of graphene (top row of plots in Figure 5).Further increase in polycarbonate pore resistancereduces selectivity because flow is then limited bythe nonselective polycarbonate pores.

Using polycarbonate membranes with the smallestcommercially available pore size of 10 nm shows thepotential for much better selectivity, using the experi-mentally measured resistance of those polycarbonatemembranes (center row of plots in Figure 5). Since thesupport membrane pore diameter is much smallerthan the average distance between intrinsic defects(see Figure 4a), it tends to isolate the defects to only asmall fraction of graphene-covered polycarbonatepores, leaving the majority of the pores covered withdefect-free graphene. Three layers of graphene and

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a factor of 1000 increase of the polycarbonate poreresistance further mitigates leaks, yielding selectivitieson the order of 100. With a modest increase in thefractional area coverage of graphene from 70% to 85%,selectivities on the order of 1000 are feasible (bottomrow of plots in Figure 5). These results suggest thefeasibility of gas separation using graphene with typicalquality obtained by chemical vapor deposition, pro-vided that an appropriate porous support is chosen.

Dimensionless Parameters Defining Regimes of SelectiveTransport. We now specify dimensionless parametersto define gas permeation regimes through graphenemembranes for the purpose of guiding their designand optimization. For simplicity, we focus on mem-branes with a single layer of graphene and omitnotations specifying the number of layers. Given twogases A and B, under the assumptions that intrinsicdefects are nonselective and that the polycarbonatepore resistances for both gases are equal to thoseunder rarefied gas flow, we can express themembraneselectivity PA/PB in terms of dimensionless parametersas follows (see Supporting Information Section 1.3):

PA

PB¼ f γ,

LIPDPC

,PPCPAS

,PIPPAS

,PBSPAS

!(4)

The number of intrinsic defects per polycarbonatepore is proportional to (DPC/LIP)

2. PPC/PSA and PIP/PS

A

define the permeance of the support membrane andintrinsic defects, respectively, relative to that due toselective pores, while the selectivity of the selective

pores is defined by PSA/PS

B (assumed equal to 15 000here, see Supporting Information Section 1).

Analytical expressions for the GCM selectivity(PA/PB) function in eq 4 and corresponding GCM per-meance (PA/PS

A) are derived in Supporting InformationSection 1.3. Given a certain intrinsic defect permeance(PIP/PS

A), both the GCM selectivity (PA/PB) and per-meance (PA/PS

A) are defined by the choice of supportpermeance (PPC/PS

A) and intrinsic defect spacing rela-tive to the support pore size (LIP/DPC). This resultsin a trade-off between the selectivity (PA/PB) andpermeance (PA/PS

A), illustrated for PIP/PSA = 0.01 and

PIP/PSA = 1 for γ = 0.99 (Figure 6a,b). When the

permeance of intrinsic defects is small (Figure 6a), highGCM selectivity can be achieved for any value of LIP/DPC. However, for PIP/PS

A = 1 (Figure 6b), high selectivityis only achieved when the spacing between intrinsicdefects is large compared to the support pore diameter(LIP/DPC J 1). This illustrates that, when the selectivepore permeance dominates over the intrinsic defectpermeance, intrinsic defects in the graphene over thesupport membrane pore will not compromise selec-tivity. However, when the intrinsic defect permeance issignificant, it is important that the support membranepore diameter be sufficiently smaller than the intrinsicdefect spacing, so that the intrinsic defects will beisolated to a small fraction of the support pores, leavinga large number of support pores with high selectivity.In addition to the dependence on LIP/DPC, the mem-brane is selective only for a certain range of permeance

Figure 5. Model predictions of helium (He) to sulfur hexafluoride (SF6) flow rate ratios for various values of graphenefractional area coverage (γ), support membrane pore diameter (DPC), and support membrane pore resistance (RPC).

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PA/PSA. Nonselective transport through tears result in

poor selectivity at high PA/PSA, corresponding to high

PPC/PSA, whereas the dominant nonselective backing

membrane resistance prevents selective transport atlow PA/PS

A, corresponding to low PPC/PSA. Selectivity is

observed at intermediate PA/PSA values, when the sup-

port resistance approaches the selective pore per-meance PPC/PS

A ∼ 1. Further decrease in the supportpermeance decreases the total permeance by severalorders ofmagnitude for onlymodest gain in selectivity.

When the support membrane resistance matchesthat of the selective pores (PPC/PS

A = 1), high permeanceoccurs for large PIP/PS

A and small LIP/DPC (Figure 6c), forwhich all support pores have highly permeable intrin-sic defects in the graphene over them resulting in poorselectivity (Figure 6d). Under these conditions, PA/PS

A =PPC/PS

A = 1, because the resistance of the graphenebecomes negligible compared to that of the supportmembrane. For low intrinsic defect permeance PIP/PS

A

or large spacing between intrinsic defects LIP/DPC, trans-port through intrinsic defects becomes negligible andthe permeance approaches PA/PS

A = (1 � γ)PPC/PSA þ

γ/(1 þ PSA/PPC), as shown in Supporting Information

Section 1.4. It is immediately seen from Figure 6d that,even for a properly balanced membrane where thesupport membrane resistance matches that of theselective pores, selective transport is possible only ifeither of the two criteria are met: (a) the permeancedue to intrinsic defects is small compared to thepermeance due to selective pores (PIP/PS

A j 1), OR (b)the spacing between intrinsic defects is large

compared to the support pore diameter (LIP/DPC J 1).The first criterion implies insignificant leakage throughintrinsic defects. We find that this is practically unlikelyfor the currently available quality of graphene, basedon the estimated η and permeance due to selectivepores, which suggest 2j PIP/PS

A j 2� 104. However, ifleakage due to intrinsic defects is negligible withadvances in graphene quality and selective pore crea-tion, the selectivity is governed by micrometer-scaletears. For PPC/PS

A = 1, the selectivity approaches PA/PB =[1 � (1/2)γ]/(1 � γ) for infinite selectivity (PS

A/PSB f ¥),

and is only slightly lower for PSA/PS

B= 15 000 (SupportingInformation Section 1.4). The second criterion results inisolation of intrinsic defects. Under these conditions,most of the support membrane pores have no intrinsicdefects in the graphene suspended over them andflow through intrinsic defects can be restricted bycontrolling the support membrane pore resistance,enabling high overall membrane selectivity. Leakageis then limited to the remaining pores that havegraphene with intrinsic defects, approaching thesame selectivity limit, PA/PB = [1 � (1/2)γ]/(1 � γ).The nonselective regime occurs when these two criter-ia are not satisfied, and it is not possible to achieveselectivity through control of the support membraneresistance. In this regime, the support membrane porediameter is larger than the average distance betweenintrinsic defects (LIP/DPC < 1), resulting in most of thesupport membrane pores having intrinsic defects inthe graphene over them. Consequently, the gas flowinto the support membrane pores is dominated by the

Figure 6. Model predictions of the general GCM permeance characteristics. (a and b) Dependence of GCM selectivity (PA/PB)on GCMpermeance (PA/PS

A) and intrinsic defect spacing (LIP/DPC) for 99% coverage (γ= 0.99) and intrinsic defect permeance of(a) PIP /PS

A = 0.01 and (b) PIP/PSA = 1. (c and d) Dependence of (c) GCM permeance (PA/PS

A) and (b) GCM selectivity (PA/PB) on theintrinsic defect permeance (PIP/PS

A) and intrinsic defect spacing (LIP/DPC) for 99% coverage (γ = 0.99) when the supportmembrane is matched to that of the selective pores (PPC/PS

A = 1). (e) Support membrane permeance (PPC/PSA) that maximizes

GCM selectivity (PA/PB) for various graphene area coverage fractions (γ), intrinsic defect permeances (PIP/PSA), and intrinsic

defect spacings (LIP/DPC), and (f) corresponding maximum GCM selectivities (PA/PB).

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lower resistance path through the intrinsic defects,bypassing the selective nanopores and producing nodetectable overall membrane selectivity.

We now address the question of how to select anappropriate support membrane to achieve gas separa-tion for graphene of a given quality, as characterized bythe distance between intrinsic defects and the per-meance due to intrinsic defects. The support mem-brane permeance that maximizes selectivity and thecorresponding maximum possible selectivity dependon the quality of graphene (Figure 6e,f). As in Figure 6d,selectivity requires either low PIP/PS

A or high LIP/DPC.Under these conditions, theGCMapproaches a limitingselectivity of PA/PB = 1/(1� γ) for PS

A/PSBf¥ rather than

the selectivity of PA/PB = [1 � (1/2)γ]/(1 � γ) obtainedfor the nonoptimal value of PPC/PS

A = 1 shown inFigure 6d. For finite PS

A/PSB, the GCM selectivity is

maximized at low PIP/PSA or high LIP/DPC to PA/PB =

{{[(PSB/PS

A)(1 � γ)]1/2 þ 1}/{(PSB/PS

A)1/2 þ (1 � γ)1/2}}2

(Figure 6f, Supporting Information Section 1.4) with asupport membrane permeance of PPC/PS

A = [(PSB/PS

A)/(1 � γ)]1/2 (Figure 6e). However, as seen by comparingthe plot in Figure 6f to Figure 6d for γ = 0.99, selectivityis not significantly compromised with a membranepermeance of PPC/PS

A = 1 instead of the optimal valueof PPC/PS

A = 0.082, due to the relatively wide peakobserved for high LIP/DPC in Figure 6a,b. However,choosing a support permeance PPC/PS

A = 0.082 drasti-cally compromises the overall membrane permeance,which shows that approximatelymatching the supportpermeance to that of selective pores (PPC/PS

A ∼ 1) maybe preferable to choosing a support permeance thatoptimizes selectivity.

These results demonstrate that by choosing a sup-port membrane with appropriate resistance and with asmaller pore size than the average distance betweenintrinsic defects, it is possible to achieve selective gastransport despite the presence of defects. While wehave discussed the design of a porous support withisolated cylindrical pores, the results can also informthe design of membranes where the support com-prises a thin isotropic layer with controlled permeance.In this case, the thickness of the layer is the

characteristic dimension that replaces the supportpore diameter DPC in the above analysis, i.e., it isimportant to choose a thickness that is smaller thanthe spacing between intrinsic defects. The results inFigure 6 can then be used to identify the appropriatesupport membrane pore size (or thickness) and per-meance once the graphene coverage, intrinsic defectcharacteristics, and achievable selective pore perfor-mance have been estimated.

CONCLUSIONS

Realizing the full potential of graphene for use in gasseparation membranes on a macroscopic scale willrequire understanding and controlling the adverseeffects of micrometer-scale tears and nanometer-scaleintrinsic defects in the graphene. We quantified theseeffects by measuring the permeance of macroscopicgraphene composite membranes to different gases. Itwas shown that, by stacking individual layers of gra-phene, it is possible to exponentially reduce leakagethrough defects in the membrane. A model for gaspermeance through macroscopic graphene mem-branes accounting for micrometer-scale tears andnanometer-scale intrinsic defects was developed andshown to accurately explain measured flow rates. Thecontributions of tears and intrinsic defects to theoverall permeance of the membrane were quantifiedby fitting this model to the measured permeancevalues. The results indicate that interlayer transport isnegligible in graphene membranes, and intrinsic de-fects can give rise to amodest selectivity. The graphenepermeance model was applied to estimate selectivitiesof graphene composite membranes with intentionallycreated selective nanopores. Although tears and intrin-sic defects have a detrimental effect on the overallmembrane selectivity, the model shows that by con-trolling supportmembranepore size and resistance, it ispossible to design a selectively permeable graphenecomposite membrane. Model predictions of the opti-mal support membrane specifications are provided fora wide range of graphene characteristics to guide in thedesign of defect-tolerant selectively permeable gra-phene composite membranes for gas separation.

METHODS

Membrane Fabrication. Multiple layers of graphene werestacked on a polycarbonate track-etched membrane (PCTEM)support by repeated application of the process for transferring asingle layer of graphene onto PCTEMs developed in ref 13. Thisprocedure is outlined in Figure 1a. Graphene on copper foil (ACSMaterials) grown by chemical vapor deposition (CVD) was firstcut into approximately 5 mm squares and the back sides ofthese pieces were etched for 5 min in ammonium persulfate(APS-100, Transene) to expose the copper and reduce the foilthickness. After a rinsing step, the piece of graphene on copperfoil was mechanically pressed onto a larger PCTEM with 1 μmpores and without a polyvinylpyrrolidone coating (Sterlitech).

The remaining copper foil was then completely removed byfloating the stack on an ammonium persulfate solution. Theresulting composite membrane, composed of a single layergraphene on a PCTEM, was then rinsed in ethanol before airdrying. Subsequent layers of graphene were transferred one-by-one by mechanically pressing graphene on copper foil ontoa PCTEM onto which graphene had already been transferred,etching the copper away, rinsing, and drying.

Membrane Characterization. SEM images were acquired withthe JEOL 6320FV Field-Emission High-Resolution SEM at theMITCenter for Materials Science and Engineering. All images wereobtained at an acceleration voltage of 5 kV in secondary elec-tron imaging mode. Carbon tape was used to create a currentpath between the graphene on the GCM and the microscope

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stage to reduce charging of the polycarbonate supportmaterial.The CVD graphene was characterized by Raman spectroscopyand scanning transmission electronmicroscopy with the resultsreported in ref 13.

Gas Permeance Measurements. Gas flow rates through thegraphene composite membrane were measured using theapparatus sketched in the inset of Figure 2a. The membranewas sealed between an upstream pressure line and a down-stream reservoir, both initially evacuated. During measure-ments, the upstream line was continuously supplied with asingle gas species at an absolute pressure of 1 atm by a reg-ulated gas cylinder. A pressure transducer was used to measurethe rate of pressure rise in the downstream reservoir, resultingin time traces such as those presented in Figure 2b. The slope ofthe pressure-time line was used in an ideal gas law relation todetermine the flow rate through the graphene membrane (seeSupporting Information Section 2 for additional details on themeasurement equipment and procedure).

Conflict of Interest: The authors declare no competingfinancial interest.

Acknowledgment. The authors thank J. Kong and Y. Song forhelpful discussions and suggestions. Funding for this work wasprovided by the MIT Energy Initiative Seed Fund and the NSERCPGS program (to MSHB). This work made use of the MRSECShared Experimental Facilities atMIT, supported by the NationalScience Foundation under award number DMR-08-19762.

Supporting Information Available: Additional details on gaspermeation models, gas flow rate measurement, and estima-tion of interlayer flux. This material is available free of charge viathe Internet at http://pubs.acs.org.

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