Non-equilibrium molecular dynamics simulations are performed to investigate water transport through (7,7) CNTs and to examine how changing the CNT length affects the flow dynamics. We show that fluid flow rates are well in advance of continuum expectations and that this flow enhancement increases with increasing CNT length. This enhancement is related to the internal fluid structure. Water molecules form a tightly packed cylindrical shell inside (7,7) CNTs, with densities nearly 3.5 times that of the water reservoir

Borg, M.K.

Nicholls, W.D.

Reese, J.M.

English

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryNicholls, William and Borg, Matthew Karl and Reese, Jason (2011) Simulating the fast transport ofwater through carbon nanotubes. In: 14th NSTL Nanotech Conference, 2011-06-13 - 2011-06-16,Boston, Massachusetts.Strathprints is designed to allow users to access the research output of the University of Strathclyde.Copyright c© and Moral Rights for the papers on this site are retained by the individual authorsand/or other copyright owners. You may not engage in further distribution of the material for anyprofitmaking activities or any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or study, educational, ornot-for-profit purposes without prior permission or charge.Any correspondence concerning this service should be sent to Strathprints administrator:mailto:strathprints@strath.ac.ukhttp://strathprints.strath.ac.uk/Simulating the Fast Transport of Water Through Carbon NanotubesW.D. Nicholls*, M.K. Borg and J.M. ReeseDepartment of Mechanical Engineering,University of Strathclyde, Glasgow G1 1XJ, UK*william.nicholls@strath.ac.ukABSTRACTNon-equilibrium molecular dynamics simulations areperformed to investigate water transport through (7,7)CNTs and to examine how changing the CNT lengthaffects the flow dynamics. We show that fluid flow ratesare well in advance of continuum expectations and thatthis flow enhancement increases with increasing CNTlength. This enhancement is related to the internal fluidstructure. Water molecules form a tightly packed cylin-drical shell inside (7,7) CNTs, with densities nearly 3.5times that of the water reservoir.Keywords: carbon nanotubes, molecular dynamics,water flow1 INTRODUCTIONRecent experiments [1–3] and molecular dynamics(MD) simulations [4–7] have shown that water is trans-ported through carbon nanotubes (CNTs) at unexpect-edly high flow rates. The contained fluid structure hasalso been shown to be dependent upon the CNT diame-ter: single-file molecule chains at the smallest diametersand high-density structures at larger diameters. Theflow of water inside CNTs of diameters below 1.66 nm isnon-continuum: the problem cannot be accurately de-scribed using conventional continuum fluid mechanicswith its associated linear constitutive relations and no-slip boundary conditions [8]. This truly atomistic prob-lem requires a molecular dynamics simulation method.Very few studies have investigated the effect thatvarying the CNT length has on the resulting flow rateand fluid structure. In this paper, we use MD simula-tions to investigate water transport through (7,7) CNTs,which have a diameter of 0.96 nm, with lengths rangingfrom 2.5 to 25 nm and examine how the change of lengthaffects: fluid flow rate, radial density distributions andflow enhancements over hydrodynamic expectations.2 SIMULATION METHODOLOGYOur MD simulations are performed usingmdFoam [9–12], a parallelised non-equilibrium molecular dynamicssolver, that is open-source and available to downloadfrom the OpenFOAMwebsite [13]. The motion of molec-ules is governed by Newton’s second law, and the equa-tions of motion are integrated using the Verlet leapfrogscheme. A time-step of 1 fs is used in all the followingsimulations.The rigid TIP4P water model is used, which consistsof a Lennard-Jones (LJ) interaction potential at the oxy-gen atom site (O), positive Coulomb charges at the twohydrogen sites (H) and a negative charge at a site M,located a small distance away from O. The O-O LJ in-teractions use the following parameters: σOO = 3.154A˚ and ǫOO = 0.6502 kJ mol−1. The electrostatic pointcharge values for water are -0.8476e and +0.4238e forthe M and hydrogen sites, respectively. The carbon-water interaction is modelled using the LJ potential ofWerder et al. [14]. Electrostatic and Lennard-Jones in-teractions are smoothly truncated at 1.0 nm.The configuration of our pressure-driven flow simu-lation domain is shown in Fig. 1. Two graphene sheetsare positioned at the inlet and outlet of the CNT toform a simplified CNT membrane representation. TheCNT and graphene sheets are modelled as rigid struc-tures to speed up the MD runs: this has been reportedto be a fair approximation in a previous study [6]. Pe-riodic boundary conditions are employed in the y- andz-directions, while non-periodic boundary conditions areapplied in the x-direction: the left-hand boundary is aspecular-reflective wall, while the right-hand boundarydeletes molecules upon collision. The wall helps con-trol the fluid pressure and density upstream while thedeletion patch creates an “open” system [15]. A pres-sure difference of 200 MPa is applied (see below) acrossthe membrane in all our simulations; such a large pres-sure difference is required to develop the dynamics ofthe simulation over a shorter time period due to thelarge computational cost associated with MD. Berend-sen thermostats are applied to both fluid reservoirs tomaintain a constant temperature of 298 K and elim-inate the contribution of any temperature gradients tothe fluid transport. The fluid is not controlled inside theCNT so as not to disturb the dynamics of the containedwater molecules.The upstream pressure is controlled using a proporti-onal-integral-derivative (PID) control feedback loop al-gorithm, similar to that used in [12], in addition to adap-Control Zone 1CNT MembranexyzSampling Zone 1 Sampling Zone 2Control Zone 2Figure 1: Simulation Domain.tive control of mass-flux at the inlet. An external forceis distributed over all molecules which reside in controlzone 1 in order to create the required pressure in theneighbouring sampling region, see Fig. 1. The requiredexternal force is calculated using three separate com-ponents: a proportional term fp, an integral term fi,and a derivative term fd. This technique is commonlyused in modern feedback control systems. The propor-tional force term is calculated from the pressure errorbetween the measured pressure in the sampling regionand the target pressure. The integral force term is cal-culated using the accumulation of past pressure errors.The derivative force term is calculated using the rate ofchange of the pressure error. The equation of motionfor molecule i located in control zone 1 is then:ai = fi/mi + fext/mi, (1)where ai is the acceleration of molecule i, fi is the in-teraction force, mi is the molecule mass and fext is theexternal force given by the sum of all three PID compo-nents:fext = fp + fi + fd. (2)A mass flux of water molecules is imposed at the inletof the system in order to compensate for those moleculesthat leave the system, and to keep the upstream reser-voir in a steady thermodynamic state. Our numericalimplementation controls density adaptively in the inletcontrol zone: the target mass density in the control zoneis set to the measured fluid density in the sampling re-gion, because the pressure and temperature of the fluidin this region are set at the desired values. The pressurecontrol process helps in establishing a steady and homo-geneous density distribution because it forces moleculesin and out of the control zone. To improve the stabilityof our algorithm, we use relaxation, so that the targetdensity within the control zone is given by:ρt = β〈ρs〉+ (1− β)〈ρc〉, (3)where 〈ρs〉 is the measured pressure in the samplingzone, 〈ρc〉 is the measured pressure in the control zone,and β is a relaxation parameter (e.g. ∼0.5). The num-ber of molecules to insert/delete in the control zone isthen:∆N =(ρt − 〈ρc〉)Vcm, (4)where Vc is the volume of the control zone and m isthe mass of a molecule. For molecule insertions, theUSHER algorithm [16] is used, which searches for a sitewithin the potential energy landscape via a steepest-descent iteration scheme. A molecule is inserted if thepotential energy is equal to the cell-averaged poten-tial energy. In this way, the algorithm ensures thatthe inserted molecule does not overlap with existingwater molecules, nor does the local potential energyor temperature change after insertions/deletions. Fordeletion, the candidate molecule is chosen with its po-tential energy closest to the cell-averaged potential en-ergy. Momentum and energy conservation steps are alsoperformed after a molecule is inserted/deleted by dis-tributing the added/removed momentum to neighbour-ing molecules, and updating the pair-forces. The down-stream pressure is controlled using a pressure-flux tech-nique, which applies to all molecules in control zone 2an external force that corresponds to the target. Bycontrolling pressure in this way, it creates an “open”boundary allowing a fluid flux to pass through the sys-tem. A key advantage of our pressure control techniquesis that the required reservoir pressures can be specifiedexplicitly. 0 50 100 150 200 2  3  4  5  6  7  8  9  10Pressure (MPa)x-Position (nm)CNTMembraneMembrane Interaction RegionsFigure 2: Pressure profile across domain showing the applicationof a 200 MPa pressure difference.Four different CNT lengths are investigated: 2.5,5.0, 12.5, and 25 nm. Initially the CNT is closed whilethe reservoirs are filled with water molecules and equi-librated to the correct conditions, see Fig. 2. After thisinitial equilibration, the CNT is opened and allowed tofill naturally. The system is then allowed to run until thenumber of water molecules inside the CNT remains con-stant before averaging of properties is performed. Thesame reservoir conditions are adopted in each simulationrun. The maximum variation in the applied pressuredifference was measured to be 1.76 MPa.3 RESULTS AND DISCUSSIONAs the fluid transport is non-continuum, the net flowrate is measured by averaging the number of moleculeswhich cross the centre plane inside each CNT over aprescribed time period; molecules which cross in thepositive x-direction are counted as positive and thosewhich cross in the opposite direction are counted as neg-ative. The measured mass flow rates are presented inFig. 3. For all CNT lengths investigated in this studythe fluid flow rate remained fairly constant under thesame applied pressure difference. This phenomenon hasbeen reported for small changes in nanotube length pre-viously [5], but not for large extensions as presentedin this paper. The measured flow rates are comparedto predicted, hydrodynamic flow rates via the no-slipPoiseuille relation for flow in a cylindrical pipe:m˙ =πr4ρ∆P8µL, (5)where r is the radius of the CNT, ∆P is the pressure dif-ference, ρ is the density, µ is the dynamic viscosity, andL is the CNT length. The radius used here is related tothe volume which the water can occupy inside a CNTand is half of the carbon to carbon diameter, less σCO. 0 20 40 60 80 100 120 140 160 0  5  10  15  20  25  30Molecule Transport Rate (#/ns)CNT Length (nm)Figure 3: Relationship between flow rate and CNT length under200 MPa pressure difference. 0 200 400 600 800 1000 1200 1400 0  5  10  15  20  25  30Flow Enhancement FactorCNT Length (nm)Figure 4: Flow enhancement values for different CNT lengths.Bulk properties for ρ and µ for water at 298 K are used.The flow enhancement factor, i.e. the ratio of the mea-sured mass flow rate to the hydrodynamic prediction, foreach CNT length is shown in Fig. 4. This shows a lin-ear increase with length because the measured mass flowrate stays constant with increasing CNT length, whereashydrodynamics predicts a linear decrease in mass flowrate with increasing pipe length (if the applied pressuredifference is constant).The average fluid streaming velocity for the differ-ent nanotube lengths under the same applied pressuredifference are shown in Fig. 5. The fluid velocity ismeasured at the longitudinal centre of each CNT. Asthe CNT length increases and the pressure gradient de-creases, there is only a slight reduction in the fluid ve-locity. Thomas and McGaughey [8] measured fluid ve-locities of 4 nm/ns for the same diameter of CNT undermuch smaller pressure gradients and found the relation-ship between the velocity and applied pressure gradientto be non-linear. This suggests that at higher pressuregradients the fluid velocity may tend towards a maxi-mum value for a given CNT diameter; 15 nm/ns for this 0 5 10 15 20 0  5  10  15  20  25  30Flow Velocity (nm/ns)CNT Length (nm)Figure 5: Variation of flow velocity with CNT length under 200MPa pressure difference. 0 0.5 1 1.5 2 2.5 3 3.5 0  0.1  0.2  0.3  0.4  0.5Normalised DensityRadius (nm)2.5 nm Long CNT5 nm Long CNT12.5 nm Long CNT25 nm Long CNTBulkFigure 6: Radial density distributions normalised with the reser-voir density for CNTs with different lengths. The vertical dashedline indicates the position of the CNT surface.case. This extraordinary non-continuum transport be-haviour may be related to the internal fluid structure,which we now describe.The fluid structure within the CNTs is investigatedby measuring the mean radial density at the longitu-dinal centre of each CNT. Figure 6 shows that watercontained within each CNT forms tightly-packed cylin-drical shells, whose densities are dependent upon thenanotube length. The maximum density of the water inthese shells in the longer CNTs is greater than that inthe shorter CNTs, with the 25 nm long CNT having amaximum of almost 3.5 times that of the water reservoirdensity. The distance between where the peak densityof the cylindrical shells occurs and the CNT surface cor-responds to the interaction length of the Lennard-Jonespotential between the carbon and oxygen atoms. Al-though the fluid velocity decreases with increasing CNTlength, the fluid density at the centre increases, whichmeans the flow rate stays constant with increasing CNTlength. The density profiles measured at the longitudi-nal centre of each CNT vary with length: the maximumdensity in a 25 nm CNT is higher than that in a 2.5nm CNT. This suggests that the structure of the trans-ported water is able to develop in longer CNTs i.e. CNTswith greater lengths optimise the fluid molecular pack-ing for significant flow rate enhancements.4 CONCLUSIONSThe results of non-equilibrium MD simulations havebeen reported on water transport through (7,7) CNTs,in particular, how changing the CNT length affects theinternal flow dynamics. Using new fluid pressure MDcontrol techniques we have shown that (i) under thesame applied pressure difference an increase in CNTlength can result in increased flow enhancements overcontinuum expectations, (ii) water molecules form a tigh-tly packed cylindrical shell inside (7,7) CNTs with den-sities reaching nearly 3.5 times that of the water reser-voirs for certain CNT lengths, and (iii) the internal fluidstructure is dependent upon the CNT length: longerCNTs form water structures with a greater density.5 ACKNOWLEDGEMENTSThe authors thank Rafael Delgado Buscalioni of theUniversidad Autonoma de Madrid for useful discussions.This work is funded in the UK by the Engineering andPhysical Sciences Research Council under grant EP/F00-2467/1, and by the Institution of Mechanical Engineers.JMR would like to thank the Royal Academy of Engi-neering and the Leverhulme Trust for support througha Senior Research Fellowship.REFERENCES[1] M. Majumder, N. Chopra, R. Andrew, and B. J.Hinds. Nanoscale hydrodynamics: enhanced fluowin carbon nanotubes. Nature, 438:44, 2005.[2] J. K. Holt, H. G. Park, Y. Wang, M. Stader-mann, A. B. Artyukhin, C. P. Grigoropoulos,A. Noy, and O. Bakajin. Fast mass transportthrough sub-2-nanometer carbon nanotubes. Sci-ence, 312(5776):1034–1037, 2006.[3] M. Whitby, L. Cagnon, M. Thanou, and N. Quirke.Enhanced fluid flow through nanoscale carbonpipes. Nano Letters, 8(9):2632–2637, 2008.[4] J. A. Thomas, A. J. H. McGaughey, and O. Kuter-Arnebeck. Pressure-driven water flow through car-bon nanotubes: insights from molecular dynamicssimulation. International Journal of Thermal Sci-ences, 49(2):281–289, 2010.[5] B. Corry. Designing carbon nanotube membranesfor efficient water desalination. Journal of PhysicalChemistry B, 112(5):1427–1434, 2008.[6] S. Joseph and N. R. Aluru. Why are carbon nan-otubes fast transporters of water? Nano Letters,8(2):452–458, 2008.[7] M. E. Suk, A. V. Raghunathan, and N. R. 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Controllers for imposing continuum-to-molecular boundary conditions in arbitrary ﬂuid ﬂow geometries.

Designing carbon nanotube membranes for eﬃcient water desalination.

Energy controlled insertion of polar molecules in dense ﬂuids.

Enhanced ﬂuid ﬂow through nanoscale carbon pipes.

Fast mass transport through sub-2-nanometer carbon nanotubes.

Fast reverse osmosis using boron nitride and carbon nanotubes.

Flux boundary conditions in particle simulations. Physical Review E,

Generation of initial molecular dynamics conﬁgurations in arbitrary geometries and in parallel.

Molecular dynamics in arbitrary geometries: Parallel evaluation of pair forces.

Nanoscale hydrodynamics: enhanced ﬂuow in carbon nanotubes.

On the water-carbon interaction for use in molecular dynamics simulations of graphite and carbon nanotubes.

Particle tracking in unstructured, arbitrary polyhedral meshes for use in cfd and molecular dynamics.

Pressure-driven water ﬂow through carbon nanotubes: insights from molecular dynamics simulation.

Water ﬂow in carbon nanotubes: transition to subcontinuum transport.

Why are carbon nanotubes fast transporters of water?

Simulating the fast transport of water through carbon nanotubes

Nicholls, William

Borg, Matthew Karl

Reese, Jason

Name not available

http://strathprints.strath.ac.uk/36522/5/Reese_JM_Pure_Simulating_the_fast_transport_of_water_through_carbon_nanotubes_Apr_2011.pdf

Simulating the fast transport of water through carbon nanotubes

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