Water transport inside carbon nanotubes mediated by phonon-induced oscillating friction

Transcription

1 SUPPLEMENTARY INFORMATION DOI: /NNANO Water transport inside carbon nanotubes mediated by phonon-induced oscillating friction Ming Ma, Francois Grey, Luming Shen, Michael Urbakh, Shuai Wu, Jefferson Zhe Liu, Yilun Liu, Quanshui Zheng In this supplementary information, we provide additional details on certain aspects of the study reported in the manuscript. Mainly these are concerned with the accuracy and reliability of the simulations performed. The following issues are discussed: 1. MD simulation methodology and data processing 2. Choice of boundary conditions 3. Estimation of computational power needed 4. Choosing the initial conditions among different samples 5. Method for estimating diffusion coefficient and slip length 6. Mechanism for the enhanced diffusion 1. MD simulation methodology and data processing a. Simulation System In the simulations, double-walled CNTs, are employed with flexible innertubes and rigid outertubes. Since the moduli of the membrane matrix materials are much higher than the effective modulus between the inner and outertubes, it is reasonable to assume the outertubes to be rigid 1. The TIP4P/2005 model 2 is used for water molecules. The carbon atoms in innertubes are modeled by the Dreiding force field potential 3 and subject to van der Waal s (vdw s) interaction with the atoms in outertubes. Both ends of the innertube are fixed by fixing two carbon atom rings on each end. We note that the flexibility of the solid surface is crucial to account for the momentum transfer between liquid and solid 4, and is physically more realistic than a rigid one 5. The long-range NATURE NANOTECHNOLOGY 1

2 Coulomb interactions are described by using the particle-particle particle-mesh technique with a root-mean-square accuracy of The 12-6 Lennard-Jones potential, V r = 4ε σ/r!" σ/r!, r being the distance between the two atoms, is used to describe the vdw interaction with a 1 nm cutoff distance. The two parameters, ε and σ, are chosen as ε=6.012 mev and σ=3.296 Å, which correspond to a contact angle θ of 54.4 o ±0.1 o. Standard procedures 6 are used to estimate θ. We notice that this contact angle is smaller than the value of ~79 o reported recently 7. However, as discussed below, this difference does not affect the conclusions made in this paper as both contact angles result in slip length on the order of tens of nm. All the MD simulations are carried out using the LAMMPS 8 code with a time step of 1 fs. b. Simulation Procedure and data processing The simulation is composed of the following steps: 1) Obtaining the correct water density for water confined inside CNTs using NPT ensemble. 2) Equilibrating the system Using NVT for 10 ps. 3) Setting the flow rate to be 0.0 m/s before applying the driving force. 4) Applying the driving force and collecting data for 10 ps. 5) Statistical analysis of the data. Steps b.2-b.4 are included in the World Community Grid work units and described in more detail below. b.1 Obtaining the correct water density for water confined inside CNTs. To obtain the correct water density, we immersed the CNTs (2.7 nm diameter, with different lengths of 15 nm, 20 nm and 30 nm) in a water bath of nm. The systems is then equilibrated at 298 K, 1 bar for 100 ps using Nosé-Hoover chain thermostats and barostats. This equilibration is long enough to get a stable number of water molecules inside CNTs. After that, we remove the water molecules outside CNTs, obtaining the 2

3 configuration for the following steps. This procedure has been shown to produce a reasonable density of water confined in carbon nanotubes 9. b.2 Equilibrating the system Using NVT for 10 ps. In order to obtain a large number of independent systems, we combine two methods, namely (i) generating the velocity using different random number seeds and (ii) running the simulation for 10 ps under NVT ensemble using Nosé-Hoover chain thermostats to reduce their correlations. The first method will apply different velocities for a given molecules in different simulations. The second method, which is based on Lyapunov instability, improves the independence between the coordinates of different simulations. Using these two methods we get partially correlated distributions of initial conditions as discussed in section 4. b.3 Setting the flow rate to be 0.0 m/s before applying driven force. The flow rate is set to be 0.0 m/s before applying the driving force. This is to make sure that no net flow exists when there is no pressure drop. b.4 Applying driving force and collecting data for 10 ps. After the initial equilibration, a constant force of pn is applied to each water molecule to accelerate the flow inside the tube. During this process, the temperature of the innertube is maintained at 298 K using Nosé-Hoover chain thermostats, while the temperature of the water is not controlled. With this thermostat method, the dynamics of water molecules is governed by Newton s Laws only. According to a classic paper 10 titled Thermostating highly confined fluids, this thermostating approach is the most appropriate for studying nanofluidics systems, as it shows results in agreement with hydrodynamic predictions. The negligible spurious effects of this thermostating method on our results are also confirmed by the following two observations. First, the oscillation angular frequency from MD (~3.05 THz) is in good agreement with that estimated from continuum mechanics theory to describe free vibrations of a bar with hinged ends (3.26 THz), which shows that the thermostat has 3

4 negligible effects on the lowest longitudinal mode the key mode for the coupling between phonons and water flow. Second, the diffusion coefficients for water confined in DWNTs have been calculated under either NVE or NVT (see section S5 below), and we find negligible difference between them (<10% difference). As it is the coupling between water and the longitudinal modes of phonons that accounts for the enhanced diffusion, we conclude that any possible spurious effects of using NVT on our results are negligible. The flow rate, v(t), at time t is defined as the mean value of the axial velocities of all water molecules over a properly chosen time interval, t, ending at t. In other words, v(t) is the velocity of the center-of-mass of the water group concerned. Given the noise level in v and the number of steps available for each sample, the time interval t is chosen to be 0.1 ps here. This is also a reasonable value to be able to observe the oscillation in τ(t) of which the period is about 1-2 ps, as shown in Fig. 1c. The acceleration, a(t), is obtained using v v + t v v t /(2 t). According to Newton s second law, the instantaneous interfacial frictional stress τ(t) is thus equal to F Ma t /(πdl), where F is the total force applied onto all the water molecules, M is the total mass of the water, and D (=2.7 nm) and L (=15, 20 and 30 nm) are the innertube diameter and cell length, respectively. We name the relation between τ and v the t-averaged τ-v relation. The above results have been obtained under accelerating conditions. The equivalence of these results with the case of steady state flow has a solid physical basis which has been well explained thanks to the time-scale separation 11. As noted in the reference 11, the time scales concerned are the relaxation time corresponds to liquid/solid friction τ relax and the momentum diffusion transfer time scale associated with the building up of the flow profile τ h = R 2 /(2v), where v is the kinematic viscosity of water and R is the radius of innertube. Both time scales serve as the main physical parameters that should be considered to justify the methodology used here. The time, τ relax, is of the order of η 1, where η is the damping coefficient at the liquid-solid interface, which can be estimated as η = 2k/ρR, where k = τ/v, τ and v are the stress and velocity, and ρ is the water 4

5 density and R. This gives τ relax ~110 ps, that is about two orders higher than τ h ~2.8 ps. Thus, these two time scales are well separated. In addition, the relaxation time τ relax (~110 ps) is two orders higher than the temporal period of the lowest mode with the largest oscillation period studied here (~ 2 ps), which is not affected by the water flux. This again shows a good separation of time scales. Using periodic boundary conditions like those employed in the original paper 11 will only reduce the period of the mode from 2 ps to 1 ps. The periodicity of water flow through CNTs is also in the nanosecond range rather than in the picosecond range. Thus, careful consideration of all relevant time scales justifies the method used here. It should be noted that the consistency between the results obtained under steady-state and under acceleration (non-steady state) has been validated directly using MD simulations 1. b.5 Statistics of the data To check the statistics of our samples, we calculate the standard deviation in v (Δv) and in τ (Δτ) as a function of number of samples (N) for L=20 nm and the results are shown in Fig. S1. This method is similar to the one used in our previous work 1. For each N at a given value, at least 50 samples are used, e.g. for N=10 5, samples are used. From central limit theorem, we know that for independent samples, both Δv and Δ τ should scale with N 1/2, i.e. Δv = Δv! N!!/! and Δτ = Δτ! N!!/!, which is exactly what we show in Fig. S1. The parameters fitted are Δv! = ± m/s and Δτ! = ± MPa. For N=5 10 6, the standard deviations (errors) for v and τ are then estimated to be Δv ~ m/s and Δτ ~ 0.6 kpa. 5

6 Figure S1 Standard deviation in (a) v and (b) τ for L=20 nm. The dots are MD results and dashed lines are one-parameter fitting to the central limit theorem. 2. Choice of boundary conditions In any practical system such as a CNT-based water filter 12, CNTs are always of finite length and include various kinds of defects 13. For multi-walled CNTs studied here, one of common defects is the covalent bond between the shells of a MWCNT 14. Another type of defects is the kink formed in non-perfectly straight CNTs 15. Both types of defects have been observed experimentally. These defects apply constraints on the innertube, which prevent its axially translational motion. To describe the dynamics of the innertube, these constraints can be modeled by introducing springs connecting some of the carbon atoms of the innertube, which are associated with the defects, to their equilibrium position. The stiffness of the springs depends on the nature of the constraints. For the softest ones which are due to the van der Waals interactions between the layers, which is the case for the kinks, the corresponding spring constant is about 0.02 ev/å 2 estimated based on the L-J potential used here. For the strongest ones, which are the interlayer carbon-carbon bonds, the spring constant is about 47 ev/å 2 estimated based on the Dreiding force field used here. Clamped conditions correspond to springs with much larger spring constant. To check the effects of spring constant on our results, we calculated the peak frequencies of the vibrations of the center of mass of the innertube in a 20 nm-long DWNTs for different stiffnesses (0.02, 0.1, 1.0, 10.0, and 50.0 ev/å 2 used 6

7 here). In all cases we found a single sharp peak (similar to that in Fig. 2b) with the peak frequency that increases from 0.478±0.01 to 0.484±0.01 THz with an increase of stiffness. These peak frequencies are close to that obtained for the clamped conditions (0.52 THz). Thus, we see that the effect of spring constants on the vibration of the innertube is practically negligible The effects of the defects on the friction between water and CNTs studied here are also negligible. Using MD simulations, Nicholls et al. 16 found that the effects of defects on flow rate only become noticeable with a concentration of defects larger than an extremely high value (10%). For an innertube with diameter larger than 1 nm, 10% defects corresponds to at least one defect per 0.3 nm along the tube. Obviously this is neither the case for the systems studied in our MD simulations nor that in experiments 13. The periodic boundary conditions along the axial direction for confined water do not affect the main conclusion of the present paper either. This is because the interaction among water molecules across the periodic boundaries has no effect on either the friction force or diffusion coefficient. The primary difference between the systems studied here and real finite systems is that the simulated system does not include end effects 17, for example a meniscus formed near the ends or an additional dissipation caused by the ends. However, these effects do not influence the shear stress at the interface and diffusion of the center of mass of the water. It has been previously shown that the end effects do not affect qualitatively diffusion of confined nanoparticles Estimation of computational power needed For water transport in 20 nm-long DWNTs, based on central limit theory 19, we first established a quantitative relation between the standard deviation of flow rate Δv and the number of samples N as Δv = Δv! N!!.! with Δv! =4.3±0.2 m/s as shown in section 1. The flow rate studied in experiments 12,20-25 is usually about to 0.04 m/s (Fig. 1b), which corresponds to N= Therefore, we simulated samples. 7

8 Such a huge computational task was accomplished by >150,000 volunteers participating in the Computing for Clean Water project, donating in total 37,535 CPU-years of processing time 26. This allowed us to obtain data points with small relative errors (Δv ~ m/s and Δτ ~ 0.6 kpa, see section 1) over a flow rate range which covers the entire gap between experiments and previous MD simulations studies (Fig. 1b). As Δτ equals to Δτ 0 N 0.5, where Δτ 0 =1.34 MPa, such a huge computational resource is also needed to observe the oscillation in friction force. For example for Δτ ~ 4 kpa, N= samples are needed, correspond to 2.4 µs simulation in total. 4. Choosing initial conditions for different samples The fraction of water confined in CNTs and responding to phonon modes can be estimated as β = δ/r! + 2 δ/r, where δ = 2ν/ω!/! is the decay length 27,28 corresponding to the mode with angular frequency ω, and ν is the kinematic viscosity of water. In our case δ is about 0.8 nm for ω = 3.05 THz, a large portion of the water (~80%) couples with the lowest mode. As β decreases sharply as ω increases, only a negligible fraction of water is coupled with higher-frequency modes (see also Sect.5). It has been proven by tribological experiments that coupling between the liquid flow and surface oscillations strongly influences the interfacial dynamics and energy dissipation 29. To study the effects of the low-frequency modes on the τ-v relation, one has to reduce significantly the effect of thermal noise. This has been achieved by averaging results over a large amount of samples, as shown in Fig. S1. In the first order approximation, for an individual CNT the coupling between the shear stress and the water flow can be presented as τ = k v V! cos ωt + φ! + τ! where V 1 can be estimated from the kinetic energy of the mode (~0.5 kbt) that in our case is about 3.4 m/s. This is a linear relation, and in order to observe the coupling between the dynamics of water and the vibration modes in CNTs, the phases of oscillations in different samples have to be partially correlated, otherwise the effect will be washed out under averaging. The partial 8

9 synchronization has been achieved by shifting the mean value of the distribution of V!"# 0 = V! cos (φ! ) to a non-zero value, as shown in Fig. S2. According to the central limit theorem, the quantity V cnt (0) is a random number which follows the Gaussian distribution since it is an average of N independent quantities V i (0) (see Fig. S2 for the statistics of V cnt (0)). The standard deviation σ = 5.00±0.04 m/s for V cnt (0) is in agreement with that predicted by the Maxwell Boltzmann distribution for empty tubes (~5.71 m/s) given that the coupling to the confined water will cause a frequency shift, i.e. reduce the peak frequency. Together with the huge amount of samples (5 million for each DWNT), the described procedure enabled us to reduce significantly the effects of thermal noise on τ and v (e.g. Δτ ~ 0.6 kpa and Δv ~ m/s), and to obtain the intrinsic τ-v relation at low flow rate. In experiments, a direct observation of the physically intrinsic coupling of water flow with the phonon modes in CNTs is challenging since it requires a partial synchronization of the phases for different samples similarly to that done in the simulations. However, as we found here, this coupling is important as it leads to interesting and novel phenomena, for example to the enhanced diffusion. It should also be emphasized that it is easier to observe the enhanced diffusion than to measure oscillations of shear force. The enhanced diffusion can be observed by measuring the diffusion coefficient of either water molecules or the center of mass of water confined in CNTs with different lengths. The frequency of the phonon modes depends on the length of the tube, and for very short tubes the frequency will be larger than the intrinsic frequency of water molecules, and the water flow will not respond to the oscillations. However, for a longer tube the coupling between the water flow and oscillations becomes important and enhancement of diffusion should be observed. 9

10 Figure S2 Statistics of V cnt at t=0, i.e. when the driving force is applied. The dots are MD data and the curve corresponds to a exp!!"#!!!!, where b = 2.89±0.03 m/s is the mean value of V cnt (0) and σ = 5.00±0.04 m/s is the standard deviation of V cnt (0).! 5. Method for estimating diffusion coefficient and slip length To check the effects of the coupling between water and the collective modes in CNTs on diffusion, we first estimate the diffusion of water molecules in nanotubes without collective modes. The mw water model 30 is used as it is a computing-efficient water model (~one order higher efficiency compared with an all-atomistic model) while retaining many of the water properties. For example, for water confined in nanotubes, the mw water model has an entropy profile which closely tracks the all atomistic water model (e.g. SPC-E) except for the subnanometer CNTs 31. The viscosity of the mw model is estimated to be mpa s following the procedure proposed by Gonzalez et al 32. We use the same interaction between mw water and carbon atoms as that between TIP4P/2005 and carbon atoms used in our main manuscript. This corresponds to a slip length b 14 nm for water with the mw model in (20,20)@(25,25) carbon nanotubes, which is estimated using the method proposed by Bocquet et al 33,34. Nanotubes without collective modes are achieved by connecting the carbon atoms to their equilibrium 10

11 positions using springs with spring constant being 1.0 ev/å 2 and removing all the interactions among them, an approach which has been used before 18. The spring constant is strong enough for the Lindemann criterion to be satisfied 35. For convenience, in the following we name DWNTs with no collective modes phonon-free DWNTs. Langevin thermostats are used with a damping coefficient chosen to assure critical damping for the carbon atoms. A time step of 2 fs is used. We tested the parameter settings using an NVE ensemble and found that the energy drift is less than mev per ns per degree-of-freedom. The systems are first equilibrated with a water bath in an NPT ensemble at 298 K using Nosé-Hoover chain thermostats and 1 bar using Nosé-Hoover chain barostats for 1 ns to get the correct number of water molecules in the tube. Then the mw water molecules outside the tube are removed, and the systems remain with correct periodic boundary conditions for use as the starting configurations. This method has been shown to generate reasonable water density in nanotubes 9. The systems generated are further equilibrated at 298 K with an NVT ensemble using Langevin thermostats with critical damping for 1 ns. Then production runs ~ 130 ns under NVT with Langevin thermostat of critical damping are used to estimate the diffusion coefficient 18. For this phonon-free DWNT, D is found to be ~0.81 nm 2 /ns with error less than 1% (the error remains < 1% for all the diffusion coefficients reported later). We have also calculated D under NVT using Nosé-Hoover chain thermostats or NVE and found a difference < 30%. This diffusion coefficient (D~0.81 nm 2 /ns) is in reasonable agreement with D estimated theoretically (D~1.06 nm 2 /ns) using the results based on fluctuating hydrodynamics 18. We then replace the spring models for CNTs with a Dreiding force field 3 to estimate the diffusion coefficient for water in normal DWNTs under NVT ensemble using Nosé-Hoover chain thermostats. We found D to be ~3.8 nm 2 /ns for normal DWNTs. Compared with D for phonon-free DWNTs (~0.81 nm 2 /ns), there is an enhancement of ~370% for normal DWNTs. Further tests either under NVE ensemble or with thermostat applied to CNTs and for water molecules along directions perpendicular to the axis only 11

12 are also carried out and less than 10% difference in D is found. 6. Mechanisms for the enhanced diffusion To understand the origin of the enhanced diffusion, we first calculate D either based on the friction coefficient or direct MD simulations. The friction coefficient k for water in flexible DWNTs with two ends fixed is estimated to be 5.5± MPa s/m using methods based on Green-Kubo relation 33,34, which is in agreement with that calculated based on a non-equilibrium approach in the manuscript (k=6.0± MPa s/m). The value of k for rigid DWNTs calculated independently using an equilibrium ensemble based on Green-Kubo relation 33 is ~ MPa s/m, which is in agreement with the result calculated with non-equilibrium methods, given that the flexibility of the tube should lower k by up to 20% 11. Using a Green-Kubo based approach 33,34, we also estimated k for phonon-free DWNT to be 5.7± MPa s/m. The corresponding diffusion coefficient is thus ~1.06 nm 2 /ns, which generally agrees with D calculated directly using MD for water in phonon-free DWNTs (0.81 nm 2 /ns). And both diffusion coefficients are significantly smaller than D for water in normal DWNTs(3.8 nm 2 /ns). Taking the scaling law between slip length and contact angle into consideration 36, the slip length predicted here for a contact angle of 79 o (~60 nm for water on graphene, ~260 nm for water in CNTs, see SI for more details) is in good agreement with the experiments ( nm) 37. In order to further investigate the effects of vibration modes of the innertube on the diffusion of the center of mass of confined water, using MD simulation, we explicitly actuate the modes individually and calculate the corresponding diffusion coefficient of the center of mass of water (D ac ). For water molecules, the thermostat is applied along directions perpendicular to the tube axis only. We expect that application of this thermostat produces negligible artificial effects on D ac, since the use of it for calculations of D for water confined in CNTs equilibrated at 298 K leads to less than 10% difference 12

13 compared to other calculations. For the nth phonon mode, this was done by setting the displacement of carbon atoms in the innertube as u! (x, t) = C! sin nπx/l cos(ω! t + φ! ) with n=1, 2, 3 The vibration frequency ω n is predicted by continuum mechanics theory for hinged end bar as ω! = nπc/l with c being the speed of sound along the tube. The vibration amplitude C n can be estimated from the vibration energy. The average kinetic energy for the n th mode during one vibration period is!! E!! =!! ρ!!!!dxu!! (x, t)dt, where T!!! = 2π/ω! is the temporal period and ρ! = 2πRhρ! is the line mass density with R being the radius and h=0.34 nm is the thickness of the tube. Then the kinetic energy can be calculated as E!! = ρ! c!! ω!! L/8, and based on the equipartition theorem, E!! = k! T/2. Thus, the amplitude of the oscillation is c! =!!!!!!!!!! c! = α/n with α =!!!!!!!!!!/!, given that ω! = nπc/l, c n can be expressed as!/! and E=1 TPa being the Young s modulus of a CNT. In our case, α is 3.4 pm. Considering that the overall longitudinal displacement caused by the nth mode is d! t =!!! u! (x, t) dx =!! 1 cos nπ cos ω!!"!t + φ!, we obtain that the amplitude of d n (t) is inversely proportional to n 2 (d! t = α/n! π) for n=1, 3, 5 In other words the main contribution to vibrations of the center of mass of the innertube comes from the lowest longitudinal mode (~2.2 pm), which is in good agreement with our direct MD results (~2 pm). The results for the diffusion coefficient calculated for the nth mode of the innertube are shown in Fig. 3a. The frequency of modes is normalized by the intrinsic frequency of a water molecule in the potential well of the innertube, which is f! =!!! k/m with k = u! 2π/a! for u = u! cos 2πx/a. For the L-J potential used here, u 0 =9.42 mev and a=0.246 nm, that gives f THz. For modes with f/f! 1 (n=1, 5 shown here as example), we found a significant increase in D ac (~100% larger) compared to that calculated for water confined in a phonon-free CNT (D 0 ). For modes with f/f! 1 (n=9, 21, 33 shown here as example), however, we found a negligible enhancement. 13

14 These results validate our conclusions based on the normal mode analysis of the innertube that it is the lowest mode that contributes to the enhancement of the diffusion of center of mass of confined water. These results also bring to mind a similar phenomenon in tribology, where it was found that a giant enhancement of surface diffusion coefficient of particles can be achieved by actuating the surface with mechanical oscillations at frequencies near f The amplitude A used there, however, was much larger than the one in our case. In order to show a relation between the two phenomena we carried out simulations for the system shown in the inset to Fig. 3b. We considered a set of particles interacting with the surface through the L-J potential, and it is assumed that there is no interaction between particles. The parameters of the potential are chosen in such a way that they reproduce the 1 st term of the Fourier transform of the potential experienced by water molecules located in the first layer in contact with the CNT surface. The amplitude of this potential is U 1 and period is a sub. The particles are equilibrated at T=0.01U 1 /k B using a Langevin thermostat. The substrate oscillates as a rigid body with amplitudes A/a sub ranging from 0.01 to 0.2 and with frequencies f/f 0 ranging from 0.1 to 5.0, where f 0 is the intrinsic frequency of small oscillations of particles on the surface. As shown in Fig. 3b, for all systems we found oscillation-induced enhancement of self-diffusion at f/f! 1, compared to the self-diffusion for particles on immovable surfaces, where the diffusion coefficient is D eq. For the oscillation amplitudes studied, the enhancement (D osc /D eq ) increases linearly with A/a sub. For the 1 st mode of CNTs studied, the oscillation amplitude A/a sub is ~0.014, and lies within the considered range. Of course, a comparison between these results and the effect of oscillations on the diffusion of the center of mass of particles is not straightforward, as the modes of oscillation are different, and it is hard to introduce hydrodynamic interactions within a simple one-dimensional model considered here. However, we can still establish a useful qualitative relation between the direct MD simulations and this theoretical model, as discussed below. 14

15 For water in CNTs, it is the modes with long spatial periods that contribute to the enhanced diffusion of the center of mass Taking the 1 st mode as an example, the displacement of the carbon atoms can be written as u! (x, t) = C! sin πx/l cos(ω! t + φ! ). It is evident that only some of the CNT s atoms are oscillating with a non-negligible amplitude (those near x/l~0.5), and this section of the tube can be mimicked by a rigid oscillating surface as described by the theoretical model discussed above. Thus, the self-diffusion of water molecules confined in the oscillating section of the tube will be enhanced. Based on the central limit theorem, D ac for the center of mass should show the same dependence on the parameters of the oscillations as D for water molecules, since the coordinate of the center of mass is the average of coordinates of the water molecules. Thus the enhancement of self-diffusion of water molecules results in the enhancement of the diffusion of the center of mass of confined water. Based on this analysis we can provide a semi-quantitative comparison between the model results and MD simulations. For oscillations of a rigid surface with A/a!"# = 0.01, we found approximately a 10 times enhancement of diffusion. So taking into account that the fraction of tube experiencing large amplitude oscillations is about 0.1 of its total length, we obtain approximately 100% enhancement of D for the center of mass of water, which is in good agreement with the direct predictions of our MD simulations (~130% enhancement for A/a!"# = 0.014). While we are able to explain the resonance-like mechanisms for the enhanced diffusion with the help of numerical simulations and a simple model, it is difficult to derive analytical equations describing this effect. This is because the discussed phenomenon results from nonlinear dynamics of a stochastic system The effect of higher-frequency phonon modes on diffusion and the coupling between phonon modes and water can be rationalized by considering the decay length of fluid velocity δ = 2ν/ω!/! induced by surface oscillations 27,28, here ν is kinematic viscosity, ω is the angular frequency of oscillations. The decay length characterizes the 15

16 depth within which the liquid responds to the surface oscillations. The fraction of confined water which responds to the oscillation could be estimated as β = δ/r! + 2 δ/r. In our case where δ is about 0.8 nm for ω = 3.05 THz, we obtain that a large fraction of the water (~80%) is coupled to the lowest mode. As β decreases sharply with increase in ω, only a negligible fraction of water is coupled with the higher-frequency modes. Based on this analysis we can conclude that for a longer tube, more low-frequency modes should be coupled with the water flow. This is exactly what we found for water confined in 30 nm-long DWNTs that shows two peaks (Fig. 2b), where the 1 st peak corresponds to the lowest mode (0.3 THz) and the 2 nd peak corresponds to the 3 rd mode (0.9 THz). Considering that β decreases sharply with ω and bearing in mind that the resonance-like mechanism accounts for the enhanced diffusion, it is reasonable to conclude that the higher-frequency modes will contribute negligibly to the enhancement of the center of mass diffusion. This is validated by our direct MD simulations for higher-frequency longitudinal modes (Fig. 3a). References 1 Ma, M.D. et al. Friction of water slipping in carbon nanotubes. Phys. Rev. E 83, (2011). 2 Abascal, J.L.F. & Vega, C. A general purpose model for the condensed phases of water: TIP4P/2005. J. Chem. Phys. 123, (2005). 3 Mayo, S.L., Olafson, B.D. & Goddard III, W.A. Dreiding - a generic force-field for molecular simulations. J. Phys. Chem. 94, (1990). 4 Martini, A., Hsu, H.Y., Patankar, N.A. & Lichter, S. Slip at high shear rates. Phys. Rev. Lett. 100, 4 (2008). 5 Thomas, J.A. & McGaughey, A.J.H. Reassessing fast water transport through carbon nanotubes. Nano Lett. 8, (2008). 6 Werder, T., Walther, J.H., Jaffe, R.L., Halicioglu, T. & Koumoutsakos, P. On the water-carbon interaction for use in molecular dynamics simulations of graphite and carbon nanotubes. J. Phys. Chem. B 107, (2003). 7 Li, Z. et al. Effect of airborne contaminants on the wettability of supported graphene and graphite. Nature Mater. 12, (2013). 8 Plimpton, S. Fast parallel algorithms for short-range molecular-dynamics. J. 16

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