Solid Friction in Carbon Nanotubes: Comprehensive

Article pubs.acs.org/Langmuir

Ultralow Liquid/Solid Friction in Carbon Nanotubes: Comprehensive Theory for Alcohols, Alkanes, OMCTS, and Water Kerstin Falk,† Felix Sedlmeier,‡ Laurent Joly,*,† Roland R. Netz,§ and Lydéric Bocquet† †

LPMCN, Université de Lyon, UMR 5586 Université Lyon 1 et CNRS, F-69622 Villeurbanne, France Physik Department, Technische Universität München, 85748 Garching, Germany § Fachbereich Physik, Freie Universität Berlin, 14195 Berlin, Germany ‡

S Supporting Information *

ABSTRACT: In this work, we perform a theoretical study of liquid flow in graphitic nanopores of different sizes and geometries. Molecular dynamics flow simulations of different liquids (water, decane, ethanol, and OMCTS) in carbon nanotubes (CNT) are shown to exhibit flow velocities 1−3 orders of magnitude higher than those predicted from the continuum hydrodynamics framework and the no-slip boundary condition. These results support previous experimental findings obtained by several groups that reported exceptionally high liquid flow rates in CNT membranes. The liquid/graphite friction coefficient is identified as the crucial parameter for this fast mass transport in CNT. The friction coefficient is found to be very sensitive to wall curvature: friction is independent of confinement for liquids between flat graphene walls with zero curvature, whereas it decreases with increasing positive curvature (liquid inside CNT), and it increases with increasing negative curvature (liquid outside CNT). Furthermore, we present a theoretical approximate expression for the friction coefficient, which predicts qualitatively and semiquantitatively its curvature dependent behavior. The proposed theoretical description, which works well for different kinds of liquids (alcohols, alkanes, and water), sheds light on the physical mechanisms at the origin of the ultra low liquid/solid friction in CNT. In fact, it is due to their perfectly ordered molecular structure and their atomically smooth surface that carbon nanotubes are quasiperfect liquid conductors compared to other membrane pores like nanochannels in amorphous silica.

I. INTRODUCTION In the last few years, the concept of “nanofluidics” as its own research area developed by downsizing further and further the scales at which fluid transport can be studied.1,2 Motivations to strive for ever smaller realizations of fluid technology arise from different fields, including, e.g., biochemistry, with the aim to develop highly sensitive analytic techniques to isolate and study individual macromolecules,3 or biomedicine technology for precision drug delivery.4 Continuum descriptions, like hydrodynamics, are expected to break down if the key assumption, scale-separation between typical time and length scales of the macroscopic fluid flow and the underlying molecular dynamics, fails. Indeed, fluid transport at the smallest scales follows unforeseen principles, as the incredible efficiency of functional biomolecules like aquaporine, as channels, filters, etc., demonstrates.5,6 Specific nanofluidic phenomena would thus offer new promising possibilities, if they could be exploited for technological applications, like water desalination7 or energy conversion.8 Carbon nanotubes (CNT)9 are seen as promising candidates for the realization of such applications. Several experimental studies feature exceptionally high permeability for liquid flow through CNT membranes.10−12 These membranes were claimed to exhibit flow rates up to 4 orders of magnitude higher than predicted by hydrodynamics, assuming a no-slip boundary condition (BC). Furthermore, several independent © 2012 American Chemical Society

investigations with molecular dynamics (MD) simulations showed fast fluid flow in CNT, corresponding to flow enhancements up to a thousand.13−17 However, due to large quantitative differences between the various results and the lack of a conclusive explanation, the existence of fast liquid transport in CNT remained questionable. Here, we present MD simulation results of flow through graphitic channels for different liquids, namely water, decane, ethanol, and octamethylcyclotetrasiloxane (OMCTS). Furthermore, we also investigated the influence of wall structure and channel geometry by using CNT of armchair and zigzag chiralities (corresponding to graphene sheets rolled along a different orientation,18 see Figure 1), as well as a slab geometry, and by varying the size of each channel type. Some results concerning water have already been published elsewhere.19 This initial focus on water was motivated by the high hopes CNT had generated with respect to possible applications and by the central role of aqueous solutions in the aforementioned applications as well as in many biological and chemical processes on the micro- or nanoscale. The results from this preliminary study of water flow in CNT were in accordance Received: July 24, 2012 Revised: September 11, 2012 Published: September 13, 2012 14261

dx.doi.org/10.1021/la3029403 | Langmuir 2012, 28, 14261−14272

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For a given driving force, e.g., the pressure gradient ∂zP, the shape of the velocity profile depends only on the shear viscosity η of the fluid and the slip length b. However, this approach becomes inefficient when b is substantially larger than the confinement (here the tube radius R). In this case the velocity profile tends to a plug which leads to large uncertainties for the parabolic fit. In this work, we have thus taken a different approach and focused on the liquid/solid friction coefficient. The friction coefficient is the physically relevant property characterizing the interfacial dynamics, a large slip length being associated with a low friction. The partial slip BC stems physically from the identification of the “bulk” viscous stress σf = η ∂nv with a surface liquid/solid friction force F /( = − λvslip = − σf , with λ the liquid/solid friction coefficient and ( the contact area. The slip length is accordingly deduced by the relation b = η/λ. For large slip lengths b ≫ R, the velocity profile eq 2 tends to a plug flow

Figure 1. Upper: Snapshots of all considered liquids inside a CNT with radius 1.36 nm. For better visibility of the liquids, parts of the CNT are not displayed. Sketches of the different liquid molecules give an idea of their structure, but are not to scale. Lower: Sketch of the various configurations explored in this study: liquid outside CNT (a), inside armchair (b) or zigzag (c) CNT, and confined between graphene sheets (d).

v(r ) ≡ vslip =

II. MEASUREMENT OF LIQUID−SOLID FRICTION COEFFICIENT A. Measurement Methods and System Geometries. The friction of liquids on solid surfaces is usually discussed in terms of the partial slip hydrodynamic BC that relates the fluid velocity at the solid surface vslip to its gradient ∂nv in the direction normal to it as vslip = b∂nv (1) where b is the slip length.20 Previous numerical studies have quantified slippage of water in CNT by measuring directly the velocity profile and eventually comparing it to the expected Poiseuille flow, e.g., in a CNT of radius R 2b ⎞ R r v (r ) = ⎟( −∂zP) ⎜1 − 2 + 4η ⎝ R⎠ R

(3)

and depends only on the friction coefficient. Under the condition b ≫ R, the fluids’ dynamics are completely determined by the interfacial friction, and not by the viscosity: Due to the absence of shear (plug flow), no dissipation occurs in the liquid. In this study, we measured the liquid/solid friction coefficient λ for water, decane, ethanol, and OMCTS in graphitic channels of different geometries and sizes (Figure 1). For decane, ethanol, and OMCTS, armchair CNT with radii R = 0.54, 0.68, 1.36, and 2.71 nm as well as graphene slabs of width between roughly 1 and 7 nm were considered. OMCTS did not enter in CNT with radii smaller than ∼0.6 nm, due to its large molecular diameter. For water, we extended the range of considered CNT radii up to about 5 nm and down to 0.34 nm. Comparing the results for liquids inside CNT and in graphene slabs enables to distinguish between the influence of confinement and curvature. In addition to the classic setup of a water filled nanotube, we also looked at empty nanotubes surrounded by water. This expands the observation range to negative wall curvature and makes it possible to look at the influence of curvature in the absence of confinement. Like other groups that have already performed MD simulations of liquid flow in CNT,13,17,21 we focused mainly on armchair CNT (i.e., with an armchair-shaped rim). Additionally, some simulations were performed with water in zigzag CNT (with a zigzag-shaped rim), to check for an influence of the tube chirality18 on the friction force. To measure the friction coefficient, two independent methods were used, implying equilibrium and nonequilibrium MD simulations, respectively. First, we performed measurements of the liquid−solid friction in flow simulations: The liquid/solid friction force F at the surface and the slip velocity vslip were measured to determine the friction coefficient according to

with the flow rates measured by Holt et al.11 and Whitby et al.12 The question arises if the observed high permeability is a unique property of water/CNT systems. In the following, we show that this is not the case. We find for all considered systems that liquid/graphite friction is low and that the friction coefficient depends crucially on wall curvature and interface structure. We show that this dependence can be understood within a simplified theoretical picture connecting the friction coefficient to static microscopic properties of the liquid/ graphite interface, in particular the wall roughness and the commensurability between liquid and solid structures. This article is organized as follows. First, we give a detailed description of the MD simulations measuring the friction coefficient. Then, we discuss system properties that are, against the expectation, not responsible for the curvature/structure dependence of the friction coefficient. In a second part, a theoretical discussion follows, where the aforementioned approximate expression for the friction coefficient is derived. Finally, we demonstrate that this expression indeed explains the observed behavior of the friction coefficient for all considered liquids, pore geometries, and wall structures.

2⎛

R ( −∂zP) 2λ

λ=−

F (vslip

(4)

For this, fluid flow was driven by a constant acceleration on the order of 10−4 nm/ps2. The resulting velocity profile was pluglike: Within the limit of statistical fluctuations, the velocity was constant over the whole channel width (Figure S1, Supporting

2

(2) 14262


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Figure 2. (a) Linear relationship between the friction force F and the slip velocity vslip measured in nonequilibrium simulations (pressure driven flow) of water inside CNT of different radii. F is the total tangential force of the carbon wall on the water molecules. (b) Integral of the force auto correlation function measured in equilibrium simulations of water inside CNT for various radii. The friction coefficient was obtained from the plateau value. Note that on larger time scales, the integral decreases to zero due to finite size effects. (c) Friction coefficient versus confinement for water inside and outside armchair CNT, as well as in the graphene slab geometry. R denotes the radius of the CNT, or for the slab geometry the half distance between the graphene planes. Results of both measurement methods match quantitatively. Plots a−c show results obtained for the TIP3P water model and AMBER interaction parameters. (d) Same as in (c) but with SPC/E water model and Werder et al. parameters. This plot combines results obtained with two different simulation packages (LAMMPS and GROMACS), yielding the same behavior of the friction for SPC/E water as for TIP3P water (water/graphene contact angle θ = 95° and θ = 57° for SPC/E and TIP3P, respectively).

= (1/2)Kθ(θ − θ0)2 and Vb = (1/2)Kb(l − l0)2. Spring constants were chosen to be Kθ = 520 kJ/mol rad2 (ref 21) and Kb = 25 kJ/mol nm2. An intramolecular rotational potential (dihedral potential) for groups of four directly linked (pseudo)atoms limited torsion of the middle bond. Atoms of different molecules interacted as Lennard-Jones particles, and for ethanol also via electrostatic potentials. For OMCTS, we used a model introduced in 2010 by Matsubara et al.29 that takes into consideration the intramolecular structure. It is modeled as a rigid body of 16 point masses: the inner ring of 4 silicon and 4 oxygen atoms and the outer shell of 8 methylene groups. To reduce the computational cost, molecules interact with each other only via the outer methylene groups. The optimized Lennard-Jones parameters for the CH3 pseudoatoms therefore implicitly include interactions with the inner oxygen and silicon atoms. The resulting effective potential is different from the CH3 potential in the OPLS model. To reduce the necessary computational effort, the positions of the carbon atoms building the wall were fixed at all times. For water flow in CNT, this simplification was found by others to give somewhat lower flow rates, but the difference is acceptable since it is systematic and relatively small.16−18,26,30 Simulations were performed with the LAMMPS molecular dynamics package.31 Electrostatic interactions were computed with the particle−particle particle−mesh (PPPM) method. Water molecules were held rigid with the SHAKE algorithm. All simulations were performed at constant number of particles, volume and temperature (NVT). During equilibrium as well as flow simulations, the temperature was held constant at 300 K by a Nosé-Hoover thermostat (relaxation time 0.2 ps), which was coupled on the velocity components perpendicular to the flow direction. The number of liquid molecules was chosen to ensure equal pressure 1 atm in all systems. It was previously determined in additional equilibrium simulations with tubes of finite length connected to particle reservoirs, which were kept at 1 atm by a piston. After the systems reached equilibrium, the reservoirs were removed and periodic BC were applied. For the larger slab simulations, where a bulk water region could be defined in the middle of the slab, the number of molecules was fixed arbitrarily, and the volume was adjusted instead to control the pressure: One graphene plane was used as a piston while the second plane was fixed. For the subsequent measurements,

Information). Hence, the slip velocity could be measured as the mean velocity of all liquid particles. Furthermore, due to the extremely low friction at the smooth graphitic wall, the system needed a long time to reach a stationary state (∼1 ns). Compared to this, the build-up of the plug flow happened almost instantly, on the much shorter time scale of the momentum diffusion transfer (Figure S1, Supporting Information). Due to this time scale separation, F and vslip could be measured for different mean flow velocities in one single simulation by treating different states during acceleration as quasi-stationary. For flow velocities up to 50 m/s, we observed the expected proportional relationship between F and vslip (Figure 2a). Second, we used the Green−Kubo (GK) relation for the friction coefficient22 λ=

1 (kBT

∫0

∞

⟨F(t )F(0)⟩equ dt

(5)

which relates the friction coefficient to the equilibrium force fluctuations, in order to extract λ from equilibrium simulations, without any external driving force. As a side remark, it should be mentioned that integrals such as ∫ t01⟨F(t)F(0)⟩equ dt are expected to decrease to zero for very long times, t1 → ∞. This is a known numerical effect, which takes its origin in the finite size (N) of the simulated systems.19,23,24 In principle, the thermodynamic limit (N → ∞) should be taken before the infinite time limit t1 → ∞. This numerical difficulty can be circumvented by identifying λ with the plateau value of the integrated force auto correlation function at intermediate times (Figure 2b), which mimics the proper order of limits. Altogether, we found that both measurements give results which are in very good agreement (Figure 2c). B. Simulation Details. For water, we used the TIP3P water model in combination with the AMBER96 force field.25 For comparison, we also used the SPC/E water model in combination with the oxygen−carbon interaction parameters given by Werder et al.26 For the Lennard-Jones interactions, a cutoff distance of 1.0 nm was applied. Decane and ethanol were modeled with the united atom model OPLS (optimized potentials for liquid simulations).27,28 In order to constrain bond lengths l and angles θ, we applied harmonic potentials Vθ 14263


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the positions of both planes were fixed at the equilibrium distance that was found. This process requires less particles and therefore less computing time than an equilibration with particle reservoirs. As a final note, we quote that we also crosschecked our results with equilibrium simulations performed with a different MD simulation package, namely GROMACS,32 in order to discard any possible error in the implementation.33 Results were compared for the TIP3P water model inside a CNT with diameter 1.36 nm and for the SPC/E model outside CNT with diameters 2.04, 4.06, and 6.78 nm. The values for the friction coefficient agree within 20%. C. Parameters That Do/Do Not Affect Liquid-Graphite Friction. Curvature Dependent Friction. Results for the liquid/solid friction coefficient for different tube and slab sizes are shown in Figures 2c,d (for water) and 3 (for decane,

is about 40% smaller. This shows that the key parameter influencing friction is indeed curvature and not confinement. Parameters That Do Not Affect Liquid/Graphite Friction. We have then explored the influence of various factors on friction: curvature-induced change of the normal pressure at the liquid/wall interface, wettability, depletion effects at the surfaces, and structure changes at the interface. We focused on water in order to investigate this question. As we now show, these factors are not responsible for the curvature dependence of solid−liquid friction in graphitic pores. In numerical studies of simple liquids, variations of the pressure lead to changing properties of a liquid confined between two hydrophobic walls (slab geometry)34 and lower pressure was found to result in lower friction.22 Furthermore, an empirical correlation between slip length and contact angle b ∝ (1 + cos θ)2, or alternatively between slip and depletion length δ at the solid−liquid interface, b ∝ δ4, has been shown to hold for hydrophobic planar surfaces.35,36 It has been suggested that water inside CNT has actually a nonwetting behavior and that a depletion area between the carbon wall and the first water layer is responsible for the low friction.16 However, in the following we show that neither variations of the normal pressure nor the depletion length or contact angle correlate with the curvature dependence of the friction coefficient. Surface tension is expected to change the normal pressure at the curved liquid/wall interface, for a given tangential pressure (which remains equal to the bulk liquid pressure/reservoir pressure). For a fixed reservoir/tangential pressure pT = 1 atm, we measured the normal pressure pN of water on the CNT walls, as a function of the CNT radius R. We found that the normal pressure behaved like pN(R) = pT − (γls/R) with γls ≈ 0.07 N/m (Figure S2, Supporting Information). This corresponds to a strong decrease of pN(R) with the CNT radius, from 1 atm (zero curvature) to about −1000 atm (R = 0.34 nm). However, we checked independently that the friction coefficient λ was merely independent of pressure in the range from −1000 to +500 atm for water in a graphene slab (zero curvature), see Figure S2 (Supporting Information). Thus the change in normal pressure due to curvature cannot be at the origin of the friction change. Second, for water as a liquid, we also simulated alternative models of water, using the SPC/E water model and the water/ carbon force field introduced by Werder et al.,37 using the same number of particles as for the TIP3P simulations. A noticeable difference between both models is the water-graphene contact angle: simulations of a water droplet on a graphene sheet gave a water−carbon contact angle of θ = 95° for SPC/E and θ = 57° for TIP3P. However, we found that the results for SPC/E water (shown in Figure 2d) exhibit the same behavior as found before with the TIP3P model. This shows that wettability is not a key parameter at the origin of the observed phenomenon. Going more into details of the liquid−CNT interface, we also compared the depletion lengths at the interface between the liquid and the graphite system.35,36 We define the depletion lengths as δplanar = (1/2)∫ (H/2) −(H/2) dz (1 − (ρ(z)/ρbulk)) for a graphene slab with carbon positions z = ± H/2 and δcylindrical = (1/R)∫ R0 dr r(1 − (ρ(r)/ρbulk)) for a CNT with radius R. We have measured these lengths for water/graphite interfaces and could not find any correlation between the friction coefficient and the depletion length. The depletion length decreases slightly with the CNT radius, but variations are less than 10% for R ranging from 5 to 1 nm. Note also that the depletion length is not a good parameter to characterize the water/CNT

Figure 3. Friction coefficient for decane (left), ethanol (middle), and OMCTS (right) confined inside armchair CNT of diameter 2R (○ equilibrium, ★ flow measurement) and in graphene slabs (□ equ. measurement) of width 2R. For all liquids, including water (compare Figure 2), the friction coefficient exhibits the same curvature dependent behavior: It decreases with the CNT radius, while it rests independent of confinement in the absence of curvature. Note one exception concerning decane: the graphene/decane friction coefficient is significantly reduced for slab sizes smaller than 2 nm.

ethanol, and OMCTS). Almost all considered liquids show the same qualitative behavior: The friction coefficient is independent of confinement in the slab geometry (only for decane, we find a certain dependence on confinement which becomes eminent in the graphene slab simulations with slab size H < 3 nm, Figure 3); but in contrast, the measured friction is very sensitive to the wall curvature. For liquids flowing inside CNT, the friction coefficient decreases strongly with the radius. Furthermore, even for tube diameters as large as 6 − 10 nm, we found significant discrepancies between results for the friction on graphene and in CNT. This finding is further confirmed by the results for water outside CNT (negative wall curvature), where the friction coefficient increases for smaller radii. And for very large CNT diameters >10 nm the results for the friction coefficient for both water inside and outside of the CNT approach the one for zero curvature. We further checked the conclusion that confinement does not influence the friction coefficient by simulating the flow of water in a block-shaped carbon nanotube with a square cross section of 4.26 × 4.26 nm2. In this system water is confined in two dimensions (as for CNT) but with flat interfaces, as for the graphene slab geometry. We found that the friction coefficient in this square tube λ = 1.3 × 104 Ns/m2 is in reasonable agreement with the results for the slab geometry, whereas in a CNT with comparable diameter ∼4 nm, the friction coefficient 14264


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interaction for high confinement R < 1 nm, because no bulk density can be defined. Last but not least, we considered that the increasing wall curvature might induce a change of the spatial arrangement of the molecules in the first water layer, in contact with the wall (thereafter referred to as the “contact layer”). A restructuring could in turn influence the liquid/carbon friction due to a changing commensurability between the 2-dimensional (2D) structure of liquid molecules in the contact layer and the 2D structure of the CNT wall. We quantify local, tangential ordering in the contact layer with the 2D static structure factor N

S(q) =

Figure 5. Coordinates of the “unrolled” contact layer, here for a water/ CNT interface.

N

1 ⟨∑ ∑ exp(iq· (x l − x j))⟩ N j=1 l=1

(6)

where x is tangential to the liquid/carbon interface: x = (r1θ;z) with r1 the radial position of the contact layer (position of density maximum in Figure 4). This can be imagined as

Figure 6. 2D structure factor (SF) of the water contact layer (definition in eq 6). (a) S(q) for the water contact layer on a plane graphene sheet. We observed the same isotropic structure for the first water layer in CNT of radii R > 1 nm. (b) S(q) for the graphene slab and several armchair CNT with radii between 0.81 and 3.39 nm. For low confinement the SF is independent of the tube radius. For smaller tube radius (not shown here), S(q) gets anisotropic and water in the two smallest tubes (R = 0.34 nm and R = 0.41 nm) is arranged in single file.19 The arrows mark the principal wave vectors q+ of the periodic potential V felt by a water molecule in the contact layer next to a graphene wall (green), the inside (orange) or the outside (blue) of a CNT with radius R = 1.36 nm, which enters the friction via the corresponding SF, see eq 16: For water inside CNT, S(q∥) decreases and for water outside CNT S(q∥) increases with respect to the value for zero curvature. Figure 4. Density profile and potential for TIP3P water in armchair CNT. Upper plot: density ρ(r) of oxygen and potential energy V(r) of one oxygen atom positioned at (r;θ = 0, z = 0) for R = 2.03 nm. The contact density increases along with the curvature (lower left) due to the deepening of the potential well (lower right). One exception is the (7,7) CNT (R = 0.48 nm), where the maximum density is strongly reduced due to size exclusion: two water molecules barely fit in the cross section, inhibiting the formation of a true interfacial layer. However, the tube is still too large to impose a real single file. In this intermediate situation, water molecules are distributed almost equally over the tube cross section which leads to a smaller maximum density.

S(q) = S(q) (q = |q|) and it is not affected by confinement for CNT radii above 0.8 nm. This is exhibited in the right plot of Figure 6, where the structure factor S(q) of oxygen atoms in the first water layer close to the surface is shown for various CNT radii. Consequently, the curvature dependence of the friction coefficient which is eminent even for tube radii as large as 3 nm can not stem from a rearrangement of the water molecules in the CNT due to confinement. A significant change in the structure factor was only observed for CNT diameter smaller than 1 nm. For the highest confinements, water molecules are forced to arrange in single file on the tube axis.19 Note however that this behavior measured above for water does not generalize systematically to more complex liquid molecules. In Figure 7, with decane as a liquid, we measured a far more complex structure factor of the interfacial decane molecule at graphitic interfaces, with furthermore a marked difference between decane/graphene and decane/CNT interfaces. The relatively stiff rod-like decane molecules orientate preferentially parallel to the wall, building the first liquid layer as demonstrated in Figure 7. In CNT, the tube geometry introduces a second preferential direction − decane molecules tend to orientate parallel to the tube axis. For higher curvature, more and more molecules reorientate themselves to evade bending that the wall curvature imposes in angular direction.

calculating the 2D structure factor of the “unrolled” liquid layer. “Unrolled” means that the positions of all atoms in the contact layer are mapped according to (Figure 5) ⎛r ⎞ ⎛ s = r1θ ⎞ r = ⎜θ ⎟ → x = ⎜ ⎟ ⎜ ⎟ ⎝ z ⎠ ⎝z⎠ (7) The structure factor S(q) for the water contact layer at a plane graphene sheet is shown in the left part of Figure 6. It is found to be isotropic and independent of confinement for slab sizes larger than three molecular diameters. Surprisingly, we found that the structure factor of the interfacial water in CNT matched its planar counterpart: the structure factor is isotropic 14265


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Figure 8. Friction coefficient versus static rms force for water (left): measured static rms force ⟨F2⟩/( ; right: theoretical estimate ⟨F2⟩theo/ ( = ρ1 f2q∥S(q∥)), for all considered water/graphite systems: water inside and outside armchair CNT, as well as inside zigzag CNT and water in a graphene slab, modeled with different scales for the carbon honeycomb lattice.

Figure 7. Structural properties of the decane/graphite interface for different curvatures. (a) Left: Snapshot of the decane interface layer on graphene. The rod-like decane molecules adhere parallel to the wall and build a liquid layer of molecular thickness. (a) Right: 2D static structure factor of the interface decane layer on graphene. (b) Same as in (a) for decane in a CNT with radius 1.36 nm. For decreasing tube diameter, the decane molecules align increasingly in the direction of the tube axis, and the structure factor gets anisotropic. (c) Static structure factor of all considered decane/graphite systems in direction of the wave vector q∥ determining the periodic liquid/wall potential, see text. For the cases shown in panels a and b (graphene/R = 1.36 nm), q∥ is drawn as an example (green/orange). Due to the developing anisotropy, S(q) changes drastically with the radius. Red dots mark the values of q∥ for various curvatures (graphene and R = 2.71, 1.36, 0.68, and 0.54 nm).

This observation is in agreement with simulation results for decane structure in CNT from Supple and Quirke.14

III. THEORY In this part, we develop a theoretical model of the liquid/ graphitic friction, with the aim to reveal the physical mechanisms controlling friction in the considered systems. A. Microscopic Picture of Liquid/CNT Friction. In order to identify the origin of the curvature dependent friction coefficient, we take a closer look at the GK expression and rewrite eq 5 as λ = βτF

⟨F 2⟩ (

with β = 1/(kBT), defining the decorrelation time ∞ 1 τF = 2 dt ⟨F(t )F(0)⟩ ⟨F ⟩ 0

∫

Figure 9. Friction coefficient (upper) and theoretical prediction eq 18 (lower) versus the measured static rms force, for decane (left), ethanol (middle), and OMCTS (right); ○ CNT, □ graphene. The curvature dependence of the friction coefficient is a static effect: Variations of λ with the tube radius are proportional to variations of the rms force ⟨F2⟩, with a predicted value ⟨F2⟩theory ∝ ρ1 fq∥2S(q∥). The deviation of the ethanol/graphene results is due to qualitative differences in the layering of the ethanol molecules next to CNT and graphene (see text).

(8)

(9)

different topology in low and high confinement - bulk water and layering in the former, single file structure in the latter case. We concentrate on the low confinement case, which is closer to the plane geometry .considered in reference 22. All four liquids exhibit a pronounced layering on the graphitic surfaces with a prominent density peak about one atomic diameter from the carbon wall, as can be seen in Figure 4 for water, and in Figure S3 of the Supporting Information for all four liquids. This “contact layer” is followed by a zone of depletion, then by further “layers”, and finally by the bulk liquid. Now keeping in mind these density profiles, we assume that the main contribution to the force stems from the atoms in the contact layer and so we write

that characterizes the decay of the force−force autocorrelation function. It turned out that changes of the decorrelation time remained small in low confinement. In contrast, curvature had a strong influence on the static rms force. The variation of the friction coefficient λ on CNT curvature was found to directly correlate with the static rms force ⟨F2⟩ for all considered liquids (Figure 8 and 9). This relation holds for positive and negative curvature (water inside and outside CNT, Figure 8). In the following, we focus on the analysis of the static rms force ⟨F2⟩. The water/carbon force can be written in the general form F=

∫

d3r ρ(r)f(r) with f(r) = −∇V (r)

(10)

F≈−

where V(r) denotes the potential energy of one oxygen atom at position r due to interactions with all wall atoms. In order to simplify eq 10 for the force, we have to distinguish between the

∫

dz

∫

ds ∂zV (r1 , s , z)

(11)

where r1 = R ± σ is the mean radial position of the atoms in the contact layer (± for liquid outside/inside CNT) and s = r1θ is 14266


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the angular coordinate (see Figure 5). We further note that the periodicity of the potential V(s,z) at the radial position of the contact layer is given by the projection of the carbon structure on the cylinder with radius r1. The graphene structure is given by a honeycomb lattice with a carbon−carbon distance d = 0.142 nm. Carbon positions are n+a+ + n−a− with n± ∈  and the lattice vectors a+ = (l0/2)(√3; 1) and a− = (l0/2)(√3; −1), where l0 = √3d. Carbon nanotubes are ’prepared’ by rolling rectangular graphene sheets to a cylinder. Depending on the orientation of the tube axis na+ +ma− with respect to the graphene structure, CNT are called “armchair”, “zigzag” or “chiral”. For armchair CNT the cylinder axis is in direction a+ −a−; for zigzag CNT it is in direction a+ +a−. For armchair CNT, the lattice vectors for the projected structure are therefore r1 ⎞ ⎛ l0 ⎜ 3 ⎟ R a± = ⎜ ⎟ 2 ⎝ ±1⎠

3 the 2-dimensional structure factor S(q) of the liquid contact layer, evaluated at the main reciprocal lattice vector q∥ of the liquid/wall potential at the position of the contact layer, which measures the commensurability between the 2D tangential structures of the liquid contact layer and the graphitic wall. All quantities in the expression above are (i) characteristics of the interface and (ii) easily accessible in MD simulations. One more complication enters the theoretical analysis for liquid molecules where several distinct atom types contribute to the liquid/wall force. Resolving the force into the contributions of the various atom types F = ΣiF(i) (i = O, CH2, and CH3 for ethanol and i = CH2 and CH3 for decane) the rms force reads ⟨F 2⟩ =

i

(12)

λ = βτF ∑

(13)

(14)

∫ d2x ∫ d2x′f (x)f (x′)⟨ρ(x)ρ(x′)⟩

≈ fq 2

∫ d2x ∫ d2x′(sin(q +·x) − sin(q −·x))

(sin(q +·x′) − sin(q −·x′))⟨ρ(x)ρ(x′)⟩ =

1 N1(S(q +) + S(q −)) 2

i

N1(i) ((i)

f (θ , z) = −∑ ∂zULJ(R i − r) (15)

i

(f q(i) )2 S(i)(q )

(18)

(19)

where ULJ is the 12-6-Lennard-Jones potential. This force field is calculated for a discrete sample of positions rn which is then used in the next step to calculate the Fourier coefficient of the wave vector q± according to

The last equality holds due to the relation ⟨ρ̃(q)ρ̃(−q)⟩ = N1S(q) where N1 is the number of molecules in the contact layer. Finally, using the isotropy of the water structure factor and the length of the reciprocal lattice vectors q+ = q− ≡ q∥, we find

fq =

2

N ⟨F ⟩ λ = ≈ 1 fq2 S(q ) βτF ( (

((i)

≈ βτF ∑

All equilibrium simulations were evaluated according to eq 18. Comparison to Simulations. The contact density is given by the ratio of mean number of atoms in the contact layer N1 and the contact area ( . We get N1 from a time average of the number of atoms in the interface area defined in Figure 4. The distance to the wall of the various first “atomic” layers was found to be very similar for all atom types and liquids. Furthermore, the position of the contact layer is curvature independent in low confinement (Figure S3, Supporting Information). Note ethanol as an exception, where the density profile on graphene is clearly distinct from the one in the R = 2.7 nm CNT. The contact area is calculated with the effective tube radius R ± σ/2. The structure factor is calculated according to eq 6 and averaged over a few hundred configurations. Finally, the force field at the position of the contact layer is the sum of the pair interactions between one atom at position r = (r1 cos θ;r1 sin θ;z) and all carbon atoms Ri = (R cos θi;R sin θi;zi)

the mean force can in first approximation be written as ⟨F 2⟩ =

(F (i))2

i

where q0 = 2π/l0 ≈ 25.6 nm−1. The resulting force field f(s,z) = −∂zV for an oxygen atom in the contact layer has the same periodicity. Expressing the force as a Fourier series and omitting higher orders (see Figure S4, Supporting Information) f (x = (s , z)) ≈ fq (sin(q +·x) − sin(q −·x))

(17)

i≠j

It turns out that the mixed parts ⟨F(i)F(j)⟩ are negligible compared to the first sum. We can therefore conveniently write the friction coefficient as the sum

where l0 = √3d is the length of the lattice vectors for the graphene structure which is recovered in the limit r1/R → 1. The corresponding reciprocal lattice vectors are ⎛ R ⎞ ⎜ ⎟ q ± = q0⎜ 3 r1 ⎟ ⎜ ⎟ ⎝±1⎠

∑ ⟨(F (i))2 ⟩ + ∑ ⟨F (i)F (j)⟩

∑ f (θn , zn) exp(iq ±·{r1θn; zn}) n

(20)

Comparing left- and right-hand sides of eq 18 for the considered liquids, we find a proportional relationship: the estimation catches all effects that lead to the curvature dependence of the rms force and thus of the friction coefficient (Figure 8 and 9). We mention that the slight underestimation (i) 2 (i) of the rms force ⟨F2⟩ by the approximation ΣiN(i) 1 ( fq∥ ) S (q∥) (a constant prefactor of 10 or less, depending on the liquid) is caused by the projection of the contact layer on its mean radial position eq 7. We found that the lateral force amplitude fq∥ increases exponentially for decreasing distance to the wall

(16)

This expression constitutes a microscopic description of the liquid/carbon friction. It relates the friction coefficient λ via the mean squared force ⟨F2⟩ to: 1 the average number of atoms in the liquid contact layer per interface area; in other words the contact density (N1/( ); 2 the force field amplitude fq∥, which characterizes the roughness of the lateral interaction potential; 14267


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Figure 10. Contributions to the predicted static rms force, ⟨F2⟩ ∝ (N1/( )fq∥2S(q∥): Contact density (N1/( ), squared force amplitude fq∥2, and commensurability factor S(q∥): for both water (left) and decane (right), normalized by the values for zero curvature, for the graphene slab geometry. Findings for ethanol and OMCTS (not shown) are qualitatively the same as those of decane, namely N1/( and fq∥2 decrease monotonously, and S(q∥) varies in a nonsystematic way.

So for constant ρ(r1), the contact density N1/( decreases with the tube radius R. This is a purely geometric effect, that contributes nevertheless to the curvature dependence of the rms force. Interfacial Structure. For water, the structure factor S(q) (eq 6) of the first water layer was found to be isotropic and independent of the tube radius in low confinement. Still, the structure contribution S(q∥) to the rms force plays a crucial part for the curvature dependence of ⟨F2⟩/( . Indeed, while the water structure remains the same, the periodicity of the lateral oxygen/wall potential changes with the wall curvature and induces a monotonously varying commensurability between both structures. The wall structure is characterized by the radius-dependent wave vector q∥. Technically, this means that the water structure factor in Figure 6 has to be evaluated at another q-vector for each CNT radius. For other liquids, the situation may be actually more complex. Contrarily to water, the structure of decane, ethanol and OMCTS inside the contact layer was found to change considerably with curvature, due to spatial constraints on the larger molecules of these liquids, as we discussed previously, see Figure 7 for decane. Consequently, two effects add to the curvature dependence of the structure contribution S(q∥). First, the wave vector q∥ changes in length and direction. And second, the form of the structure factor itself is influenced by the CNT size, becoming more and more anisotropic for decreasing diameter. The interplay of both effects, changing decane structure and changing wave vector q∥, results in a nonsystematic curvature dependence of S(q∥). Here, we focused on decane because its simple shape makes its structural rearrangement at curved walls easily understandable. But anisotropic and curvature dependent structure factors were also found for OMCTS and ethanol. Finally, we mention that we observed an influence of confinement on the friction for decane in a graphene slab, contrary to what was found for the other liquids. In fact, the friction coefficient measured for slab sizes 1.15 and 1.97 nm is smaller compared to lower confinements. The effect is well captured with the approximate rms force (see Figure 9). The reason for this friction decrease is that decane arranges in layers throughout the whole slab (2 layers for H = 1.15 nm; 4 for H = 1.97 nm). These layers influence each other and build stable domains of parallel aligned decane “blocks” that impact the structure factor at the interface and thus the decane/graphene commensurability S(q∥).

(Figure S5, Supporting Information). Atoms at a distance smaller than σ feel a considerably rougher potential as estimated in eq 16 and thus their contribution to the friction force is underestimated. The exact form of the density distribution at the interface is therefore responsible for the quantitative discrepancy between the rms force and the approximate expression. This also explains a discrepancy between graphene and CNT results for ethanol, since the density profile of ethanol in proximity of graphene was qualitatively different from the one in CNT. Consequently, the quality of the approximation eq 18 deteriorates. This leads to the deviation of the data points for graphene from the line through origin defined by the CNT data points which can be seen in Figure 9. The other applied approximations, namely the neglect of contributions to the friction force by the second/third liquid layer and the truncation of the liquid/wall force field Fourier expansion, are very accurate. Deviations due to these simplifications are smaller than statistical errors of the measurement. To conclude, we checked that all three static quantities contributing to eq 16 were left unchanged in flowing conditions. This is a necessary condition for the model to work. B. Curvature Dependence of Different Contributions. In the previous part, we demonstrated that the curvature dependence of the friction coefficient λ is very well reproduced by eq 16. We can now make use of this effective theoretical description to get further insights at a microscopic level for the curvature dependence of the friction coefficient. To this end, we take a closer look at each contribution: contact density, roughness and commensurability (Figure 10). Contact Density. The curvature dependence of the contact densities is similar for all atom types figuring in the considered liquids. Differences of the mass density ρ of the contact layer in CNT of various radii remain rather small. But the surface density at the interface decreases anyway, due to a changing ratio between the effective interface area ( and the area defined by the position of the first liquid layer A1 = 2πr1Lz. The effective interface area is given by ( = 2πreffLz, with reff = (R + r1)/2 ≈ R − σ/2. Estimating the number of atoms in the contact layer as N1 ∝ ρ(r1)A1σ, one can write ( N1 R−σ ∝ ρ(r1)σ 1 ≈ ρ(r1)σ ( ( R − σ /2

(21) 14268


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Potential Roughness. The roughness of the lateral force field inside armchair CNT decreases with increasing curvature, because maxima and minima of the force field lie closer together and smooth each other out. The Fourier coefficient fq∥, which is a good measure for the roughness of the force field, depends only on the Lennard-Jones parameters of the interaction between liquid and wall atoms (for a given curvature). Given that σ for the various liquid atom types are all similar, it is not surprising to find qualitatively the same radius dependence of fq∥ for all of them. C. Other Wall Structures (Zigzag/Artificial Graphene). The friction coefficient depends sensibly on the exact form of the liquid/wall potential at the position of the first liquid layer next to the carbon wall, via its roughness and the commensurability between liquid and carbon structure λ ∝ S(q∥)fq∥2. This was demonstrated in the previous part for armchair CNT of varying radii where the potential changed along with the wall curvature. Now, for other CNT chiralities18 the form of the potential evolves differently with increasing curvature. Consequently, we expect a different friction coefficient for two CNT of comparable radius but different chirality. To test this prediction, we performed equilibrium and flow simulations of water in zigzag CNT with radii between 0.35 and 9.8 nm (Figure 11). There is indeed a measurable influence of

fz (x = (s , z)) ≈ fq (sin(q +·x) − sin(q −·x)) + fq sin((q + + q −)·x) z

(22)

Furthermore, the definitions of q± are also subtly different because switching from armchair to zigzag chirality means that the axial (z) and angular (s = rθ) directions are inverted. Consequently, keeping the notation that the first vector component denotes the radius dependent s-coordinate, we write ⎛ R⎞ ⎜± ⎟ r1 ⎟ q ± = q0 ⎜ ⎜ 1 ⎟ ⎜ ⎟ ⎝ 3 ⎠

(23)

Calculating the modulus of the Fourier coefficients for force fields on graphene and in CNT, we find that the main contribution stems from the linear combination q+ + q− instead of the individual vectors. For graphene, fqz2 is about twice the magnitude of fq∥2 and for CNT the proportions shift more and more in favor of fqz2 with decreasing tube radius. So, contrary to the armchair case, the main q vector for the periodicity of the potential landscape in zigzag tubes ⎛ 0 ⎞ ⎜ ⎟ q + + q − = q0 ⎜ 2 ⎟ ⎜ ⎟ ⎝ 3⎠

(24)

is radius independent. The length of this vector is qz = 29.5 nm−1. With the above representation of the force f(s,z), the rms force can, in complete analogy to the derivation for armchair CNT, be estimated to ⟨F 2⟩ ≈ N1(fq 2 S(q ) + 1/2fq 2 S(qz)) z

Figure 11. Friction coefficient versus tube radius for water inside CNT of armchair and zigzag chiralities. The different wall structure causes a measurable variation of the friction coefficient for radii smaller than 2− 3 nm. We did not find a significant change of friction for different flow directions on graphene (see text for a discussion).

(25)

where N1 is the number of molecules in the contact layer and S(q) is the static structure factor of this layer (we specialized here to the contribution of a single atom inside the liquid molecule). The length of the wave vectors |q±| = q∥ is again radius dependent, but qz and consequently S(qz) are not affected by curvature. Friction results for water as a liquid superimpose reasonably well for armchair and zigzag CNT when displayed as λ versus (N1/( )fq∥2S(q∥) and (N1/( )( fq∥2S(q∥) + 1/2fqz2S(qz)), respectively, see Figure 8. This good agreement further emphasizes that these theoretical expressions for the friction coefficient capture the relevant effects: contact density, potential roughness, and commensurability. Whereas the contact density is unaffected by the CNT chirality, the structure factor and especially the potential roughness behave very differently in the two considered cases. In armchair CNT, the changing structure contribution is the main effect in low confinement. Additionally, the potential roughness decreases continuously and is, in particular, responsible for the frictionless motion of single file water in (5,5) and (6,6) CNT. In zigzag CNT, the behavior of S(q∥) and fq∥ is similar to the one in armchair CNT, but the most important contribution to the friction coefficient stems not from q∥ but from qz: For the zigzag flow direction on graphene, fqz is about the double that of

the CNT chirality: In zigzag CNT of R ≲ 2 nm, we find a significantly higher friction coefficient than in armchair CNT of the same size. In particular, although λ decreases again with increasing curvature, it does not disappear completely for single file motion in zigzag CNT, contrary to armchair CNT. These differences can be explained within the presented theory. In the low confinement regime R > 1 nm, the friction coefficient for zigzag CNT is again proportional to the rms force (Figure 8). In this case, the rms force can be expressed similarly as in eq 15. The only difference to the derivation for armchair CNT is the Fourier expansion of the force field. In fact, more parts of the Fourier series have to be included to get a decent approximation (see Figure S4, Supporting Information). We write the force fz on one oxygen atom in the first water layer for the zigzag chirality as 14269


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in zigzag CNT, a much higher friction is expected compared to an armchair CNT of similar (small) radius. A test for decane in a (0,14) CNT (R = 0.55 nm) resulted indeed in a friction coefficient of λ ≈ 1.2 × 104 Ns/m3, a value that is 300 times higher than in the (8,8) CNT (R = 0.54 nm).

fq∥ and is increasing further for decreasing radius, leading to an increasing roughness of the potential energy landscape at the position of the first water layer. Furthermore, S(qz) is curvature independent in weak confinement, because neither the water structure factor nor the qz vector change with the radius. That means no incommensurability like in armchair CNT occurs. The friction decrease for radii larger than ∼1 nm is only due to the decreasing contact density. This explains why the effect is smaller in zigzag than in armchair CNT in weak confinement. Finally, note that for water on graphene the approximate expressions for the rms force in the two different flow directions predict indeed the same value, although they are composed of different contributions. In short, for both CNT types increasing curvature leads to decreasing friction, but according to our understanding, it is caused by different effects. Finally, we want to stress again the importance of commensurability between liquid and solid interfacial structures for the friction coefficient (λ ∝ S(q∥)). In fact, this understanding permits us to “tune” the amount of interface friction by changing either the liquid or the solid structure. This can be demonstrated very easily for water on graphene by varying the carbon−carbon distance of the graphene honeycomb: We performed equilibrium simulations of the graphene slab system exactly as before, but with a fixed carbon−carbon distance that was either larger (d = 0.173 nm) or smaller (d = 0.118 nm) than the real value (d = 0.142 nm). The values for the carbon−carbon distances of the “artificial” graphene were chosen to result in q∥ values that mark the position of the first −1 min maximum (qmax ∥ = 24.2 nm ) and the following minimum (q∥ −1 = 35.5 nm ) of the interface water structure factor (see Figure 6). As expected we find a smaller friction coefficient (λ = 0.18 × 104 Ns/m3) for the smaller C−C distance and a larger value (λ = 4.9 × 104 Ns/m3) for the larger C−C distance. We note that, apart from the structure contribution, the contact density N1/ ( and especially the roughness fq∥ also change due to the different surface density of the carbon atoms. Recently, Xiong et al. performed the same kind of investigation with another water model: They measured the water/graphene friction coefficient for different C−C distances d ∈ [1.28; 1.56] nm using SPC/E+Werder parameters.38 They observed variations of λ from 0.4 × 104 to 3.4 × 104 Ns/m3 that are consistent with our values given above for TIP3P+Amber parameters. Overall, these results for water/graphene systems with different carbon lattice sizes, from ref 38 as well as from this work (Figure 8), are again consistent with the approximation for the rms force eq 15. For the other liquids, no incommensurability between the contact layer and the carbon structure was observed. (On the contrary, we found S(q∥) > 1 in most cases.) In other words, for decane, ethanol, and OMCTS, the structure factor does not play a role for the curvature dependence of the friction coefficient. For decane, ethanol, and OMCTS, a systematic decrease with curvature is only observed for the contact density and the potential roughness. Although this still results in the same overall trend for the friction coefficient in armchair CNT, it might well be very different in zigzag CNT, since, contrarily to armchair CNT, the roughness of the force field inside zigzag CNT is increasing for small radii. If in parallel to a higher roughness the structure contribution changes unpredictably because the liquid adapts to the wall curvature, the behavior of the friction coefficient is equally unforeseeable. If no incommensurability occurs to counter the higher roughness

IV. CONCLUSION We used a combination of numerical simulations and an analytic approach to establish the special properties of carbon nanotubes for liquid transport: Molecular dynamics flow simulations of different liquids in carbon nanotubes exhibited flow velocities 1−3 orders of magnitude higher than predicted from the continuum hydrodynamics framework and the no-slip boundary condition. These results support previous experiments reporting exceptionally high flow rates for different liquids in carbon nanotube membranes. Our simulation results suggest that a variety of fluids exhibit a very similar flow behavior in graphitic nanoscale systems like graphene slabs or carbon nanotubes: Flow rates lie far above what could be expected from the hydrodynamic no-slip boundary condition, due to extremely low friction on the carbon wall. This leads to plug flows instead of the parabolic velocity profiles expected for small liquid/solid slippage. In the CNT systems studied so far (armchair CNT for decane, ethanol, OMCTS, and water; zigzag CNT for water), the friction coefficient is decreasing for increasing curvature, favoring fast fluid transport even for CNT with very small cross sections (diameter comparable to the molecular size). Furthermore, we presented a theoretical approximate expression for the friction coefficient, which predicts qualitatively and semiquantitatively its curvature dependent behavior. Moreover, the proposed microscopic description sheds light on the physical mechanisms at the origin of the ultra low liquid/solid friction in carbon nanotubes. Changes in the friction coefficient are directly related to changes in the equilibrium rms force between the liquid and carbon wall. The rms force, in turn, is determined by three interfacial properties: liquid contact density, wall roughness, and (in)commensurability of wall structure and the structure of the liquid contact layer. This description works very well for all four considered liquids. In fact, for many liquids carbon nanotubes seem to be quasi-perfect conductors compared to other membrane pores like, for instance, nanochannels in amorphous silica, which is due to their perfectly ordered molecular structure and their atomically smooth surface. Based on the presented theory it is now possible to make predictions for flow rates in different systems. Since our results suggest that the structure of the wall and the liquid are a crucial factor, further investigations of the influence of the CNT chirality might permit us to find liquid/nanotube combinations with particularly low friction. For this, the high confinement regime is especially interesting since it favors strong ordering of the liquid molecules, like single file water for example. In principle, such crystalline structures offer the possibility of frictionless sliding due to complete incommensurability (zero liquid structure factor at the characteristic wavelength of the potential). Therefore, it would also be very interesting to investigate the transport of larger molecules, like DNA or other biopolymers, where the breakdown of the hydrodynamic theory, which marks the onset of high confinement, occurs at larger system sizes. Finally, to predict flow rates through real CNT membranes for a given pressure difference, entrance effects at the tube 14270


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openings have to be taken into consideration.39,40 These entrance effects can even contribute much more to the pressure drop than friction inside the CNT. In this case, the membrane permeability will effectively be limited by the resistance to liquid in- and output. In fact, MD simulations of water uptake in CNT with radii R ∈ [0.51; 1.87] nm have shown that the resistance to capillary filling is determined by viscous dissipation at the tube entrance.41 Consequently, flow measurements with different CNT lengths would be needed to distinguish between the effects of dissipation inside and at the end of the tubes.

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ASSOCIATED CONTENT

S Supporting Information *

Additional information presented in Figures 1−5. This material is available free of charge via the Internet at http://pubs.acs.org.

■

AUTHOR INFORMATION

Corresponding Author

*E-mail: [email protected]. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS L.B. thanks support from ERC AG program, project Micromega. REFERENCES

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