Water Flow in Carbon Nanotubes: Transition to Subcontinuum Transport

John A. Thomas and Alan J. H. McGaughey*

Department of Mechanical Engineering, Carnegie Mellon University, Pittsburgh, Pennsylvania 15213, USA(Received 23 December 2008; published 8 May 2009)

The structure and flow of water inside 75 and 150 nm-long carbon nanotubes with diameters ranging

from 0.83 to 1.66 nm are examined using molecular dynamics simulation. The flow rate enhancement,

defined as the ratio of the observed flow rate to that predicted from the no-slip Poiseuille relation, is

calculated for each tube and the liquid structure is examined using an axial distribution function. The

relationship between the intermolecular water structure and water flow is quantified and differences

between continuum and subcontinuum flow are discussed.

DOI: 10.1103/PhysRevLett.102.184502 PACS numbers: 47.56.+r, 47.61.�k, 61.20.Ja

An important step towards understanding liquid flow innanoscale systems is to predict the transition from contin-uum to subcontinuum transport as the flow area decreases.In a continuum system, the behavior of a liquid can bedescribed in terms of infinitesimal volumetric elementsthat are small compared to the flow domain but have well-defined thermophysical properties. Applying Newton’ssecond law to a system of volumetric elements gives riseto the Cauchy and Navier-Stokes equations, which can beused to derive the Poiseuille and other continuum-levelflow relations [1]. In a system where the size of a liquidmolecule is comparable to the size of the flow domain,however, the notion of a representative volumetric ele-ment is invalid and the applicability of continuum-basedrelations is questionable. Within such ‘‘subcontinuum’’systems, the movement of individual molecules mustbe considered when predicting mass and momentum trans-port [2].

In this Letter, we use molecular dynamics (MD) simu-lation to examine pressure-driven water flow through car-bon nanotubes (CNTs) with diameters ranging from 0.83 to1.66 nm, where a transition from continuum to subcontin-uum flow with decreasing CNT diameter is expected [3,4].We examine flow through 75 and 150 nm-long CNTs,which are much longer than the 1–3 nm-long CNTs exam-ined in a previous investigation [4]. We begin by quantify-ing the relationship between pressure gradient and averageflow velocity for each CNT and then identify the transitionto subcontinuum water flow. Next, we predict the liquidstructure inside each CNT and correlate the structure tomolecular transport. For subcontinuum systems we findthat the liquid structure relaxation time, which we defineas the time required for a water molecule to becomeuncorrelated from an initial structural arrangement, ex-ceeds a time scale characteristic of the flow. This behaviorleads to coordinated molecular transport inside the CNTand a nonmonotonic relationship between CNT diameterand average flow velocity. We find that the water structureis long-ranged, and discuss how the relationship between

pressure gradient and flow velocity will depend on tubelength for CNT fragments shorter than 10 nm.The average flow velocity �v of an incompressible, creep-

ing liquid (Reynolds number much less than one) inside atube with a uniform cross-sectional area is given by the

Darcy law, �v ¼ �ð�PL Þ, where�P is the pressure difference

across the tube, L is its length, and � is the hydraulicconductivity [1]. Although the Darcy law is an empiricalexpression, the hydraulic conductivity of a Newtonianliquid in a circular tube subject to the no-slip boundarycondition, �no-slip, can be found directly from the no-slip

Poiseuille relation, �no-slip ¼ D2

32� , where D is the tube

diameter and � is the liquid viscosity [3].Liquid slip at the solid-liquid boundary, confinement-

induced reductions in the liquid viscosity, and subcontin-uum changes to the liquid structure can all cause the actualhydraulic conductivity (�actual, as measured from experi-ment or predicted from MD simulation) to exceed thatcalculated from the Poiseuille relation [2]. This increasein � has led to the definition of a flow enhancement factor,", given by " ¼ �actual=�no-slip [3]. For CNTs with diam-

eters larger than 1.6 nm, the variation in " with CNTdiameter can be understood in terms of slip at the water-CNT boundary and diameter-related changes to the waterviscosity [3]. In CNTs with smaller diameters, however,water molecules have been shown to assemble intodiameter-dependent one-dimensional structures for whichneither the slip length nor the effective viscosity is welldefined [2]. This confinement-induced change to the liquidstructure necessitates a subcontinuum description of theliquid [5,6].We simulate pressure-driven water flow through 0.83,

0.97, 1.10, 1.25, 1.39, and 1.66 nm-diameter single-walledarmchair CNTs. A snapshot from a typical simulation,which consists of a water-filled CNT fragment connectedto two water-filled reservoirs, is presented in Fig. 1. Allsimulations are performed in the NVT ensemble (constantnumber of particles, volume, and temperature) at a tem-perature of 298 K and flow is driven by a reflecting particle

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membrane placed between the reservoirs [7]. We use theTIP5P potential to model interactions between water mole-cules [8], the Lennard-Jones potential of Werder et al. tomodel the interactions between water molecules and car-bon atoms [9], and keep the carbon atoms fixed. We main-tain the temperature of the system by applying a Berendsenthermostat to the velocity components transverse to theflow direction [10]. We note that applying the thermostat toall three directions does not affect the flow predictions.

We find that fixing the carbon atoms has no effect on thestructure of water molecules inside the CNT. The numberof molecules in the system is chosen such that the waterdensity in the reservoirs is 1000 kg=m3 and the averagenumber of water molecules in the CNT fragment is con-stant. As presented in Fig. 2(a), the CNT diameters exam-ined here have water structures corresponding to single-filemolecular chains (0.83 nm), tilted pentagonal rings(0.97 nm), stacked pentagonal rings (1.10 nm), stackedhexagonal rings (1.25 nm), and disordered bulklike water(1.39 and 1.66 nm). The water structure and density wepredict for each CNTare unaffected by flow and consistentwith previous MD simulation results [11].

In Fig. 3(a), we plot �v vs �P=L for the 75 nm-longCNTs. Each point in Fig. 3(a) is the average of at least fourindependent 1 ns simulations. Consistent with the Darcylaw, the average flow velocity for each CNT increases withincreasing pressure gradient. For a fixed value of �P=L,however, the average flow velocity does not increasemonotonically with CNT diameter, as would be predictedfrom the Poiseuille relation. Instead, when subject to thesame pressure gradient, the average velocity decreasesfrom the 0.83 nm CNT to the 1.10 nm CNT, is similar inthe 1.10 and 1.25 nm CNTs, and then increases from the1.25 nm CNT to the 1.66 nm CNT.

The nonlinear relationship between �v and �P=L is theresult of inertial losses (i.e., minor losses) at the two CNT-reservior boundaries. Inertial losses are velocity-dependent

and caused by sudden expansions, contractions, and otherobstacles in the flow field. The penalty associated withsuch losses, �Pmð �vÞ, can be incorporated into the Darcy

law: �v ¼ �½�P��PmðvÞL � [1]. Given that the inertial losses

are unknown, the hydraulic conductivity can be extractedfrom this modified form of the Darcy law as follows:Consider a long CNT with length Ll, and a shorter CNTwith length Ls, with identical diameters and connected toidentical reservoirs. Let the ratio of length to diameter forboth tubes be large such that the entrance and exit effectsare decoupled. If both systems have the same average flowvelocity, the inertial losses in each system will be the same.However, due to additional frictional flow resistance in thelonger CNT, the total pressure drop across it, �Pl, willexceed that across the short CNT, �Ps. Applying themodified Darcy law to both systems and eliminating�Pmð �vÞ gives � ¼ �v½ðLl � LsÞ=ð�Pl � �PsÞ�. For the1.66 nm-diameter CNT, the hydraulic conductivity calcu-lated for the systems investigated here is within 17% of thevalue we predicted in our previous work, which was free of

FIG. 2 (color online). (a) Water structures inside the 0.83–1.66 nm-diameter CNTs. Carbon atoms are removed for clarityand the chirality vector for each CNT is provided. (b) Axialdistribution function (ADF), structure relaxation time �, andextent of the ADF, Lt, for each CNT. (c) Pðn; tcÞ, the probabilitythat n molecules cross the system midplane over the character-istic flow time tc. The CNT permeability � is provided for�P=L ¼ 4� 1014 Pa=m.

FIG. 1 (color online). Snapshot from a typical flow simulationfor the 0.83 nm-diameter CNT. A constant pressure difference isestablished between the reservoirs using a reflecting particlemembrane. After a 0.25 ns initialization period, the mean pres-sure in each reservoir remains constant over the duration of theensuing data collection period and flow through the tube issteady. Periodic boundary conditions are imposed in the flowdirection. The number of molecules inside the CNT ranges from310� 5 for the 75 nm-long, 0.83 nm-diameter CNT to 3676�20 for the 150 nm-long, 1.66 nm-diameter CNT.

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reservoirs and had no inertial losses. We estimate theuncertainty in � (and thus �) to be �25% from the mea-sured variances in �v and �Pl ��Ps. We find that, withinthis prediction uncertainly, � is invariant over the flowvelocities considered for a given CNT diameter.

In Fig. 3(b), we present the variation of " with CNTdiameter. We also include the enhancement values for the1.66–to 3.33 nm-diameter CNTs we reported previously[3]. In CNTs with diameters larger than 1.66 nm, where acontinuum description of water flow is valid, we found that" increased monotonically with decreasing CNT diameter.Here, we find that this trend extends to the 1.39 nm-diameter CNT. The abrupt reduction in flow enhancementbetween the 1.39 and 1.25 nm-diameter CNTs suggests atransition to subcontinuum transport, which is consistentwith the change from disordered bulklike water to one-dimensional structures with decreasing CNT diameter [seeFig. 2(a)]. The variation in flow enhancement within CNTswith diameters smaller than 1.25 nm cannot be describedusing continuum relations and, as we will show, is relatedto the water structure.

In Fig. 2(c), we provide the molecular permeability � ofeach CNT corresponding to a pressure gradient of 4�1014 Pa=m. Despite the decrease in flow velocity fromthe 0.83 to the 1.10 nm-diameter CNT, the permeability(which incorporates the effects of diameter-related changesto the flow area, water density, and velocity) increasesmonotonically with CNT diameter for all tubes. This trendarises from the fact that the cross-sectional flow areaincreases faster with diameter than the structure-relatedreductions in the flow velocity.

To quantify the variation in liquid structure with tubediameter, in Fig. 2(b) we present the axial distribution

function (ADF) for water molecules inside each CNT.

We define the ADF, aðzÞ, by Rz¼�Lz¼0 aðzÞ �D2

4 dz ¼ Nt � 1 ’Nt, where Nt is the number of water molecules betweenz ¼ �L. In CNTs with diameters greater than 1.66 nm, theADF has a small peak at z ¼ 0:3 nm and is invariant withCNT diameter. The position of this peak is similar to theposition of the first peak in the radial distribution functionof bulk water. The ADF inside the 1.39 nm-diameter CNTis comparable to that of larger tubes, suggesting thatconfinement-induced changes to the liquid structure areinsignificant in this CNT [12]. Inside the 1.25–0.83 nm-diameter CNTs, where the layered and single-file molecu-lar structures are present, the axial positions of the watermolecules are strongly correlated and oscillations in theADF extend 4–10 nm from the origin molecule. Such long-range positional correlation, which is not present in bulkwater, is a distinguishing feature of subcontinuum liquids[2]. We find no difference between the ADFs in the 75 and150 nm-long CNTs.In CNTs with diameters less than 1.39 nm and lengths

shorter than about 10 nm, all the water molecules inside thetube will have correlated positions. Although the distinc-tion between solid phase water and liquid phase water isunclear for such systems, we expect that the solidlikemolecular structure will limit the mobility of individualmolecules and lower the average flow velocity of waterinside such short CNTs. This hypothesis is supported bythe MD simulation data of Corry [4], who investigatedpressure-driven water flow through CNT membranes withdiameters similar to those investigated here, but withlengths of 1.3 and 2.6 nm. The positions and movementof water molecules inside the short CNTs were indeedcorrelated, and transport through the tubes occurred viacollective bursts. Moreover, the flow enhancement factorswe calculate from the Corry data are 1–10, a range lowerthan the 100–1000 range we predict here for 75 and150 nm-long CNTs. This result suggests that as the CNTlength decreases below 10 nm and the molecules at thetube inlet become increasingly coupled to those at the tubeoutlet, mass transport will become less like flow through apipe and more like coordinated diffusion through a two-dimensional pore. This transition will reduce the flow ratethrough the tube and cause a reduction in the flow enhance-ment factor.Although the CNTs investigated here are longer than the

oscillations in their ADFs, the local liquid structure in eachtube still governs the molecular transport through it. Thiscoupling between structure and flow can be elucidated byexamining the cumulative midplane mass flux nðtÞ. InFig. 2(c), we present the probability Pðn; tcÞ that n mole-cules cross the CNT midplane over a characteristic flow

time tc. We define tc from tc ¼ Lc

�v , where Lc is a character-

istic length scale. We set Lc equal to the location of the firstpeak in the ADF and calculate Pðn; tcÞ for each flowsimulation. Since the liquid structure does not vary withflow velocity (over the range considered here), Pðn; tcÞ for

FIG. 3 (color online). (a) Relationship between average flowvelocity �v and applied pressure gradient �P=L for the 75 nm-long CNTs. A similar relationship exists for the 150 nm-longCNTs. We note that �v is well below the molecular thermalvelocity (340 m=s at T ¼ 298 K). For each data point, thestandard deviation in �v and �P=L is 0:5 m=s and 2�1013 Pa=m, leading to a �25% uncertainty in ". Guide linesare added to highlight the trends. (b) Variation in flow enhance-ment factor " with CNT diameter D. The dashed line betweenD ¼ 1:25 nm and D ¼ 1:39 nm delineates continuum and sub-continuum flow regimes. The line in the continuum regime is amodel we developed previously [3].

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each CNT collapses onto a velocity-independent distribu-tion. We note that with Lc � 0:17–0:3 nm and �v�1–7 m=s, tc is Oð100 psÞ.

Inside the 1.66 nm-diameter CNT (where flow can bemodeled using continuum-based relations) Pðn; tcÞ is non-zero over a large range of n. This large range is accessiblebecause the structure relaxation time, which we estimate tobe 0.3 ps from the decay of the velocity autocorrelationfunction, is very small compared to tc. The flow is there-fore independent of liquid structure and a random numberof molecules, subject to the distribution of Pðn; tcÞ, cancross the midplane during tc. Although the accessiblerange of n increases with CNT diameter, we find that theshape of the Pðn; tcÞ distribution in larger CNTs is similarto what we report here for the 1.66 nm diameter tube andfind no appreciable change in the structure relaxation time.This smooth distribution of Pðn; tcÞ is also present in the1.39 nm-diameter CNT, further suggesting that the flow inthis tube is independent of structure and a continuumdescription of mass flow will be appropriate.

Inside the 1.25 nm-diameter CNT, Pðn; tcÞ exhibitspeaks at n ¼ 6 and 12. A similar distribution is presentinside the 1.10 nm-diameter CNT, where Pðn; tcÞ exhibitspeaks at n ¼ 5 and 10. These peaks, which are related tothe stacked hexagonal and pentagonal rings in the tubes,indicate a coupling between structure and flow. The re-laxation time of the layered water structure inside theseCNTs, as estimated fromMD simulation, isOð10 nsÞ [5,6],indicating that water molecules may travel from the tubeinlet to the tube outlet as members of a single layer. Unlikebulklike systems, where water molecules can move inde-pendently, the transport of molecules in these structuredsubcontinuum systems is conditional upon the movementof nearby layers. A layer of water molecules that fills anenergetically stable CNT surface site can therefore impedethe movement of other layers. Such localized limitationson transport will reduce �v and lower the overall flowenhancement (see Fig. 3). This hypothesis is consistentwith the findings of Mamontov et al., who demonstratefrom neutron scattering experiments and MD simulationthat the water molecules in layered structures have, onaverage, less mobility than the molecules in bulklikewater [12].

Between the 1.10 and 0.83 nm-diameter CNTs, theliquid structure transitions from stacked pentagonalrings to tilted pentagonal rings and finally to a single-filemolecular chain. With decreasing CNT diameter, we findthat the effects of layer-by-layer transport become lessprominent, the spatial extent of the ADF decreases [seeFig. 2(b)], and the average flow velocity increases for agiven pressure gradient [see Fig. 3(a)]. Although the masstransport mechanisms within these tubes are not yet clear,in a previous work we found that water molecules becomeless coupled to the CNT surface with decreasing CNT

diameter [3]. This effect will reduce flow friction in smallerCNTs and may contribute to the increase in flow velocitywith decreasing diameter. In the 0.83 nm-diameter CNT,where the single-file water structure relaxation time hasbeen estimated to be Oð0:1 sÞ [13], the probability of twoor more molecules crossing the midplane in either thepositive or negative direction is 0.6. This behavior is theresult of coordinated molecular motion inside the CNT. Itconfirms that, although the tube inlet and tube outlet aredecoupled in our system, localized bursting still governsmass transport inside long-length, small-diameter CNTs.Although this work is focused on water flow through

CNTs, several trends can be extrapolated to the moregeneral field of subcontinuum liquid transport. First, unlikepredictions from continuum mechanics, the flow enhance-ment in subcontinuum systems may not increase mono-tonically with decreasing flow area. Instead, when the flowarea is comparable to the size of the liquid molecules,confinement-induced changes to the liquid structure mayreduce the flow enhancement and must be considered.Second, if the system length is comparable to the correla-tion length, the liquid inlet and outlet are not independentand the hydraulic conductivity may depend on systemlength. Within short systems, transport may therefore beless like flow through a pipe and more like coordinateddiffusion through a two-dimensional pore. Third, liquidstructure and liquid flow are independent in systems wherethe characteristic flow time scale is much longer than thestructure relaxation time. These scales must be consideredwhen comparing results from different flow investigationsand examining flow-structure interactions.

*mcgaughey@cmu.edu[1] J. Bear, Dynamics of Fluids in Porous Media (American

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(2008).[4] B. Corry, J. Phys. Chem. B 112, 1427 (2008).[5] S. Joseph and N. R. Aluru, Phys. Rev. Lett. 101, 064502

(2008).[6] A. Striolo, Nano Lett. 6, 633 (2006).[7] J. Li et al., Phys. Rev. E 57, 7259 (1998).[8] M.W. Mahoney and W.L. Jorgensen, J. Chem. Phys. 112,

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