- Email: [email protected]
Filling carbon nanotubes with liquid acetonitrile
Chemical Physics Letters 496 (2010) 50–55
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Filling carbon nanotubes with liquid acetonitrile Vitaly Chaban School of Chemistry, V.N. Karazin Kharkiv National University, Svoboda Square 4, 61077 Kharkiv, Ukraine
a r t i c l e
i n f o
Article history: Received 4 June 2010 In final form 3 July 2010 Available online 8 July 2010
a b s t r a c t Carbon nanotubes and acetonitrile are of interest for modern electrochemistry since they are used to make supercapacitors more efficient. In order to assess the feasibility of this setup, molecular dynamics simulations were performed to investigate the hydrophobic degasified single-walled nanotubes filling with liquid acetonitrile. The simulation shows that nanotubes with 10 nm of length can be completely filled with acetonitrile during less than 100 ps. Surprisingly, the filling process is not significantly affected by nanotube diameter and ambient conditions. In general, the ability of small hydrophobic carbon nanotubes to be completely filled with acetonitrile is an important feature for supercapacitors. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The static and dynamic behavior of fluids inside the nanosized confinements is now an important and broad field [1–7] of investigation generally called nanofluidics. With a widespread availability of carbon nanotubes, interest in nanofluidics has greatly increased mainly due to the revolutionary possibilities of exploring the fluids inside the extremely narrow channels of about 1 nm or even less. The use of nanotubes to study the confined fluids also raises the interest to the specific atomic interactions between nanotube and confined particles of different chemical nature that basically defines the nature of the confined liquid flow [1,2,7,8]. The recent studies of nanofluidics comprise various aspects: both the fluids entering the tube and the overall flow inside the tube. By now, experimental studies dealt with very different nanotube diameters (from 1 to 100 nm) and lengths [9–15], whereas computer simulations were concentrated only on the rather smaller CNTs (with diameters of a few nanometers) [1,5,16–19]. Various substances were recently used to fill the hollow carbon nanotubes including molten metals, salts [9,20], oxides, aqueous and nonaqueous solutions [35] through metal evaporation [10], capillary filling [21] and other techniques. These works raised the production of a large number of hybrid materials [22,23]. An empirical law for single-walled CNTs filling was suggested by Dujardin et al. [11] stating that no liquid with a surface tension of more than 180–200 mN/m could enter the tube inner cavity. This law works well for the nanotubes with diameters of up to 4 nm. Although the surface tension of many polar liquids, including water (72 mN/m), is significantly below the threshold, they are generally believed not to wet pristine carbon nanotubes (in analogy with basal planes of graphite).
E-mail addresses: [email protected], [email protected] 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.07.003
The present work focuses on acetonitrile (ACN), an important organic dipolar liquid both for science and chemical technology, which is also often used as a solvent for modern supercapacitors [24–29]. Among various possible setups, such devices are often designed with their electrodes containing carbon nanotubes or porous carbon to increase the electrolyte adsorption. Obviously, the performance of supercapacitors critically depends on the ability of the particular electrolyte solution to enter, and subsequently fill, the CNTs of the certain sizes. Since the transport properties of electrolyte solutions tightly depend on a particular solvent, direct investigation of the latter is important. Moreover, the pristine carbon nanotube can be considered as an ideal model of a porous carbon with nanosized channels. The another important question arising in this context is how far the solvent and its solution can go inside carbon nanotube. In spite of the considerable interest to supercapacitors in modern electrochemistry and general nanofluidics, the mentioned questions still remain open. For this kind of investigations, computer simulation appears to be the most productive tool because of the principal experimental limitations on the nanoscale. Here, the molecular dynamics (MD) study of filling CNTs with acetonitrile was carried out with a series of narrow armchair single-walled nanotubes ranging from (5, 5) to (11, 11) and the constant length of 10 nm for all species. The influence of the different ambient conditions, temperature and elevated pressure, has been considered. The adsorption processes of ACN molecules on the internal and external surfaces of the CNTs were separated in time by means of the specific consequent MD setups. 2. Simulation details 2.1. Force fields The GROMACS [30] program package was used to perform MD simulations on several systems containing a few carbon nanotubes
V. Chaban / Chemical Physics Letters 496 (2010) 50–55
and liquid acetonitrile. The exact composition of all MD systems is presented in Table 1. To represent all bonded and non-bonded interactions within the carbon nanotubes, AMBER force field [31] was applied. All simulated nanotubes were treated as flexible non-polarizable (hydrophobic) particles with 1–4 carbon–carbon interactions switched on. The acetonitrile molecule was represented using the six-site model of Nikitin et al. [32] which appears to be the best force field for this liquid by now. It reproduces thermodynamics and structure properties of bulk ACN very well and also gives a pretty good viscosity (0.4 cP at 298 K and under 1 bar, according to our own calculations). Surprisingly, the self-diffusion constant is a bit lower than experimental value (3.5 vs 4.3 10-9 m2/s at 298 K and 1 bar). Unfortunately, no other acetonitrile force field gives better result for diffusion. The electrostatics was treated by means of the Ewald summation (rcut = 1.4 nm) and Lennard–Jones (12, 6) interactions were accounted with shifted force method (switch region between 1.2 nm and 1.3 nm). The cross-term Lennard–Jones (12, 6) parameters between carbon nanotubes and ACN were obtained using the standard Lorenz– Berthelot combination rules. The molecular dynamics time-step was 0.001 ps in conjunction with a leap-frog integration algorithm. The linear velocity of the nanotube, initially centered in the box, was gradually removed to simplify the subsequent analysis. 2.2. Simulation setup The multi-stage MD simulations were carried out to investigate different aspects of the dynamic behavior of the CNT-ACN systems. Five narrow single-walled armchair carbon nanotubes were considered: (5, 5), (6, 6), (7, 7), (9, 9) and (11, 11) with the constant length of about 10 nm. The study of the system dynamics was started with the CNT centered in the box and surrounded by 0.6 nm of vacuum. Initially, no solvent molecules were present in the inner cavity of the nanotube corresponding to the degasified case. The liquid acetonitrile (see Table 1 for details) was located as far as 0.6 nm from the outer nanotube wall and filled all the available volume in the 5 5 14 nm parallelepipedic MD box. Because of this somewhat exotic setup, the initial densities of the simulated systems were 20–30% lower than equilibrium ones (Table 1). Firstly, in order to
Table 1 Diameters of CNTs, dCNT, numbers of acetonitrile molecules in the MD system, numbers of confined solvent molecules, times of adsorption on the external surfaces of CNTs, times of filling and system densities, d. CNT
dCNT (nm)
N (ACN)
Nconfined (ACN)
texter.ads, (ps)
tfilling (ps)
d (kg/m3)
(5, 5) (6, 6) (7, 7) (9, 9) (11, 11)
0.68 0.82 0.95 1.22 1.49
2463 2397 2292 2217 2137
0 19 28 69 130
20 20 15 20 20
– 50 55 60 100
828 841 854 878 904
51
study the adsorption of ACN on the external surface of the nanotubes only, the tubes were capped preventing the solvent to penetrate inside. Secondly, when the systems were equilibrated in their previous configurations, the caps were removed and the solvent was allowed to enter the open-ended tubes (Fig. 1). The rate of the filling process was investigated at temperatures of 278, 298 and 323 K and pressures of 1, 100 and 2000 bar (T = 323 K). Finally, equilibrium MD runs were performed to explore the structure patterns of acetonitrile in confinements and estimate its dielectric constant in the specific single-file chains. All MD simulations were carried out in the constant temperature and constant pressure (NPT) ensemble. V-rescale thermostat [33] with a time constant of 0.5 ps and Parrinello–Rahman barostat [34] with a time constant of 1.0 ps were applied to maintain the temperature and pressure values, respectively. Importantly, the pressure coupling was applied only in the radial direction of the CNT to minimize its possible artificial effect on the filling process of acetonitrile. 2.3. Data analysis The rate of carbon nanotubes filling was determined by the interaction energy between nanotube and ACN. The corresponding energy values were dumped every 2 ps (2000 time-steps) and plotted vs simulation time. The structure patterns of acetonitrile confined inside CNTs were analyzed in terms of atomic densities in axial and radial directions of the nanotubes separately. The nitrogen atom of ACN was selected for that as the heaviest one in the solvent molecule. The dielectric constants of the confined liquid were estimated from the fluctuations of the molecule dipole moment. 3. Results and discussion 3.1. Effect of CNT diameter Fig. 2 illustrates the evolution of interaction energy between the nanotube and acetonitrile for CNT (5, 5), CNT (6, 6), CNT (7, 7), CNT (9, 9) and CNT (11, 11). Interestingly, both internal and external surfaces of these CNTs are wetted unexpectedly rapidly (during less than 100 ps). For external surface, the equilibration time is not more than 20 ps (Table 1). It does not also depend on the particular tube diameter. The general shape of the curves is also very similar indicating that the solvent approaches the tube external walls very rapidly and wets them within 20 ps. The penetration of ACN into the internal cavity of the nanotube is more complicated. While CNT (6, 6), CNT (7, 7) and CNT (9, 9) are filled rapidly and similarly during 50–70 ps (Table 1), the filling of CNT (11, 11) is longer and exhibits somewhat different mechanism. The first stage occurs during the first 50 ps and corresponds to the intense penetration of solvent molecules into the nanotube. The second stage (50 ps) can be understood as repacking of ACN inside the spatial confinements tending to the minimal potential en-
Fig. 1. The simulated system containing centered CNT (11, 11) surrounded by 2137 acetonitrile molecules.
52
V. Chaban / Chemical Physics Letters 496 (2010) 50–55
number of carbon atoms in each CNT (Fig. 3). The energy of about 3 kJ/mol per one carbon atom corresponds to the interaction of the CNTs with the outside solvent molecules. The total interaction energy slightly increases as the tube curvature increases. Filling the inner surface of the CNT corresponds to 1.2–2 kJ/mol per one carbon atom of the nanotube. The energy value also correlates well with a tube diameter. This is rather trivial since bigger tubes are able to encapsulate a larger number of solvent molecules (Table 1). All these molecules interact with CNT sidewalls directly through van der Waals forces. The ability of liquid acetonitrile to fill carbon nanotubes of sub-molecular sizes favors its use as a solvent in supercapacitors together with carbon nanotubes [24–29]. The simulation results suggest that the geometric size of the proper carbon pore size can be as small as 0.82 nm.
3.2. Effect of temperature
Fig. 2. The evolution of the interaction energy between CNT and (a) inside ACN, (b) outside ACN derived from MD simulations at 298 K and 1 bar. Horizontal solid lines of the same color show the respective equilibrium energies. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
ergy of the simulated system. This repacking stage is absent for smaller nanotubes because their diameters allow only one molecular layer to be located near each sidewall. Thus, subsequent repacking is not further necessary. Rapid filling of nanotubes was recently demonstrated for the case of water [35] which is attributed to the tight hydrogen bonds. Although acetonitrile is even more polar (l = 3.9 D), it is known as rather unstructured liquid with low maxima on radial distribution functions [32] and high diffusion constant (4.3 10 9 m2/s). There is also no hydrogen bonds in liquid ACN as opposed to water [35]. In spite of no tight intermolecular bonding, the filling process is unexpectedly fast and complete, even at atmospheric pressure and low temperature. Importantly, the internal cavity of the CNT (5, 5) can not be filled with acetonitrile probably because of its too small diameter (0.68 nm). The van der Waals radius of carbon atom is 0.17 nm decreasing the available space to 0.34 nm. As current MD study shows, this value is too little for methyl group (the largest group of ACN) to enter the CNT (5, 5). The independency of the nanotube filling rate from its diameter is rather curious provided that recently self-diffusion of the confined ACN was shown to correlate well with CNT diameter [36]. It proves that the equilibrium diffusion constants of the confined liquid cannot be directly used to predict the dynamics of filling. The main reason for this is a considerable difference between diffusion in the bulk phase and at the liquid–vapor interface. For small nanotubes, this effect is pronounced even more than at the ordinary liquid–vapor interface because the tube geometry drastically limits the number of acetonitrile molecule neighbors. In fact, during the first filling stage (Fig. 2) we observe quasi–gas phase of the confined acetonitrile. The molecular mobility in such a phase is found to be high and weakly dependent on the nanotube size. To compare the interaction energies with ACN for different nanotubes, the corresponding energy values were divided by the
Three temperatures (278, 298, 323 K) at which ACN is a liquid were considered in this study. Their influence on the filling dynamics is depicted in Fig. 4. The role of temperature is not crucial, although some insignificant differences can be observed for CNT (11, 11) whereas the filling rate for CNT (6, 6) is not affected at all. The first stage of the overall filling process becomes shorter with the system temperature increase (at 298 and 323 K) but the repacking stage remains nevertheless rather long. At 278 K, the first stage is a bit longer but the second one disappears. Overall, the system simulated at 278 K comes to equilibrium faster (within 70 ps). This is not correlated with the self-diffusion constants of bulk ACN which are 2.3, 3.4, 4.3 (10 9 m2/s) at 278, 298 and 323 K, respectively. The difference in the equilibrium interaction energies can be attributed to the density decrease when tempera-
Fig. 3. The evolution of the interaction energy per one carbon atom between CNT and (a) inside ACN, (b) outside ACN derived from MD simulations at 298 K and 1 bar: CNT (6, 6), red solid line; CNT (7, 7), green dash-dotted line; CNT (9, 9), blue dashed line; CNT (11, 11), pink dash-dot-dotted line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
V. Chaban / Chemical Physics Letters 496 (2010) 50–55
53
ture rises. Bigger system density always corresponds to more compact molecular packing resulting in larger potential energies. Generally, the weak effect of temperature suggests that ACN flow inside the partially filled small hydrophobic CNTs is even faster than in its bulk phase. One can conclude that the filling rate is limited only by geometry of confinements and does not considerably depend on media temperature. 3.3. Effect of pressure Three pressures (1, 100, 2000 bar) at T = 323 K were considered to investigate their influence on the nanotube filling process. CNT (6, 6) and CNT (11, 11), as the biggest and the smallest ones, were selected for this investigation (Fig. 5). The influence of high and very high pressures is more pronounced than that of temperature and affects both nanotubes similarly. The increase of pressure to 2000 bar leads to the increase of the equilibrium interaction energy between nanotube and solvent by about 20%. Also the filling process becomes somewhat faster as the pressure grows. Note, the degasified nanotubes are very readily filled with ACN both at atmospheric and elevated pressure. Thus, the elevated pressure is not necessary for practical applications (supercapacitors, nanofluidic devices) where the carbon nanotube filling is required. 3.4. Confined acetonitrile structure Fig. 4. The evolution of the interaction energy between CNTs and confined ACN, derived from MD simulations at 278 K (red solid line), 298 K (green dash-dotted line), 323 K (blue dashed line) and 1 bar: (a) CNT (6, 6); (b) CNT (11, 11). Horizontal solid lines of the same color show the respective equilibrium energies. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. The evolution of the interaction energy between CNTs and confined ACN derived from MD simulations at 1 bar (red solid line), 100 bar (green dash-dotted line) and 2000 bar (blue dashed line) and 323 K: (a) CNT (6, 6); (b) CNT (11, 11). Horizontal solid lines of the same color show the respective equilibrium energies. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In order to understand the above specificity of CNT filling with acetonitrile, the partial atomic densities along axial and radial
Fig. 6. Partial atomic (nitrogen, ACN) densities along (a) radial and (b) axial directions of the nanotubes derived from the equilibrium MD simulations at 298 K and 1 bar. The straight vertical lines correspond to the CNT sidewalls. The shown coordinates are the absolute coordinates of the respective nitrogen atoms in the MD box.
54
V. Chaban / Chemical Physics Letters 496 (2010) 50–55
Fig. 7. The instantaneous configurations of acetonitrile confined by (a) CNT (6, 6), (b) CNT (7, 7), (c) CNT (11, 11) derived from the equilibrium MD simulations at 298 K and 1 bar.
directions of the nanotube were calculated (Fig. 6). The number of maxima on Fig. 6a corresponds to the number of ACN molecules located along the radial direction inside the CNT. So, CNT (6, 6) is able to encapsulate only one solvent layer, CNT (7, 7) and CNT (9, 9) – two layers each and CNT (11, 11) – three layers. Inside CNT (6, 6), the confined acetonitrile molecules are quite free to move because of this nanotube diameter (0.82 nm). The effective (available to solvent) CNT (6, 6) diameter is 0.48 nm which is about 1.4 times bigger than the van der Waals diameter of the ACN carbon atom. This is bigger than intermolecular distances in bulk liquid ACN leading to the anomalous mobility of the confined solvent molecules. The same conclusion is derived from Fig. 6b which demonstrates almost no oscillations of acetonitrile molecules along the nanotube axial direction. The effect is well pronounced for the CNT filling process resulting in extremely fast slipping of the ACN molecules from bulk liquid into confinement. Analyzing other systems, we get the following relations between the diameters of the carbon atom and the nanotube: for CNT (7, 7) – 1.8, for CNT (9, 9) – 2.6, for CNT (11, 11) – 3.4. It is interesting that Fig. 6a indicates two maxima for ACN inside CNT (7, 7) although two full-grown layers of solvent cannot exist inside this tube, since the CNT diameter is evidently smaller than two diameters of the solvent molecule. This fact is also confirmed by the total number of ACN molecules inside the CNT (7, 7) – 28. If two full-grown solvent layers existed in CNT (7,7), the number of the confined solvent molecules should have been two times more than inside CNT (6, 6), e.g. 38. The answer is given by the instantaneous configurations of the confined ACN inside CNT (6, 6) and CNT (7, 7) (Fig. 7). Inside CNT (6, 6), ACN molecules are aligned, so that their dipole moments coincide with the nanotube axis. In turn, inside CNT (7, 7), the solvent dipole moments are perpendicular to the CNT axis. Such configuration of the confined acetonitrile allows a bigger number of molecules to be encapsulated inside the tube of the same size. Thus, two maxima for the nitrogen atoms (Fig. 6a) correspond to the neighboring ACN molecules located along the nanotube axial direction. Their antiparallel dipole moments are oriented towards the CNT sidewall. Both CNT (6, 6) and CNT (7, 7) show single-file chains of liquid but with different orientations in reference to the nanotube sidewalls. Importantly, Fig. 6b proves that for all nanotubes the complete filling with acetonitrile (10 nm) occurs. For all investigated nanotubes it is performed within the first 100 ps of run. The repacking stage for the CNT (11, 11) (Fig. 7) case is presumably connected with the central solvent layer which can be formed only after the two parietal layers of the confined liquid are equilibrated. For smaller tubes this stage (Fig. 2) is not observed since
they have no central liquid layer due to their particular sizes (Fig. 6a). 3.5. Dielectric constant of confined acetonitrile The molecular single-file patterns of the confined polar liquid should result in a specific dielectric constant as was recently shown for liquid water [37]. For acetonitrile, the dielectric constants, e, of 1.01, 1.02, 1.08 and 1.22 were obtained for CNT (6, 6), CNT (7, 7), CNT (9, 9) and CNT (11, 11), respectively. These values are in good agreement with [37] for the same narrow tube lengths exhibiting single-file liquid structures. Note, the experimental bulk values for water and acetonitrile are e = 80 and e = 36, respectively. The present ACN force field model slightly underestimates e giving 26 instead of 36 for pure bulk liquid [32]. According to [37], at much bigger nanotube lengths (hundreds of lm), the dielectric constants of the confined dielectric media is expected to grow significantly (up to 10 000). This fact favors using narrow carbon nanotubes to make supercapacitors even more efficient provided that the penetration of the electrolyte solution is complete. 4. Conclusions Molecular dynamics simulations were performed to investigate the filling process of the degasified narrow single-walled carbon nanotubes with liquid acetonitrile at different temperatures and pressures. It is found that the filling rates are unexpectedly fast and weakly dependent from the nanotube diameter. Moreover, the changes of ambient temperature and pressure do not significantly affect the filling rate. This is unexpected since the diffusion of the bulk ACN increases significantly as temperature grows. The ability of small hydrophobic carbon nanotubes to be completely filled with acetonitrile is an important feature for efficient supercapacitors [24–29]. It can be also found useful for the modern nanofluidic devices that are actively developed nowadays. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
M.C. Gordillo, J. Marti, Chem. Phys. Lett. 341 (2001) 250. M.C. Gordillo, J. Marti, Chem. Phys. Lett. 329 (2000) 341. Y.C. Liu, Q. Wang, T. Wu, L. Zhang, J. Chem. Phys. 123 (2005) 234701. Y.C. Liu, Q. Wang, L. Zhang, T. Wu, Langmuir 21 (2005) 12025. R.J. Mashl, S. Joseph, N.R. Aluru, E. Jakobsson, Nano Lett. 3 (2003) 589. D. Mattia, Y. Gogotsi, Microfluidics Nanofluidics 5 (2008) 289. T. Werder, J.H. Walther, R.L. Jaffe, et al., Nano Lett. 1 (2001) 697. G. Garberoglio, Eur. Phys. J. E 31 (2010) 73. D. Ugarte, T. Stockli, J.M. Bonard, et al., Appl. Phys. A – Mater. Sci. Process. 67 (1998) 101.
V. Chaban / Chemical Physics Letters 496 (2010) 50–55 [10] P.M. Ajayan, S. Iijima, Nature 361 (1993) 333. [11] E. Dujardin, T.W. Ebbesen, H. Hiura, K. Tanigaki, Science 265 (1994) 1850. [12] Y. Gogotsi, J.A. Libera, A. Guvenc-Yazicioglu, C.M. Megaridis, Appl. Phys. Lett. 79 (2001) 1021. [13] H.W. Yu, B. Fugetsu, J. Hazard. Mater. 177 (2010) 138. [14] J. Meng, M. Yang, F.M. Jia, et al., Nanotechnology 21 (2010) 145104. [15] B. Tawabini, S. Al-Khaldi, M. Atieh, M. Khaled, Water Sci. Technol. 61 (2010) 591. [16] R.E. Tuzun, D.W. Noid, B.G. Sumpter, R.C. Merkle, Nanotechnology 7 (1996) 241. [17] S. Banerjee, S. Murad, I.K. Puri, Chem. Phys. Lett. 434 (2007) 292. [18] S. Joseph, N.R. Aluru, Nano Lett. 8 (2008) 452. [19] J. Lee, N.R. Aluru, Appl. Phys. Lett. 96 (2010) 133108. [20] C.G. Xu, J. Sloan, G. Brown, et al., Chem. Commun. 24 (2000) 2427. [21] J. Mittal, M. Monthioux, H. Allouche, O. Stephan, Chem. Phys. Lett. 339 (2001) 311. [22] J. Sloan, A.I. Kirkland, J.L. Hutchison, M.L.H. Green, C. R. Phys. 4 (2003) 1063.
[23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
55
M. Monthioux, Carbon 40 (2002) 1809. Q.F. Xiao, X. Zhou, Electrochim. Acta 48 (2003) 575. P.L. Taberna, G. Chevallier, P. Simon, et al., Mater. Res. Bull. 41 (2006) 478. P. Azais, L. Duclaux, P. Florian, et al., J. Power Sources 171 (2007) 1046. J.S. Huang, B.G. Sumpter, V. Meunier, Chem. – A Eur. J. 14 (2008) 6614. Y. Korenblit, M. Rose, E. Kockrick, et al., ACS Nano 4 (2010) 1337. S. Jagannathan, T. Liu, S. Kumar, Compos. Sci. Technol. 70 (2010) 593. B. Hess, C. Kutzner, D. van der Spoel, E. Lindahl, J. Chem. Theor. Comput. 4 (2008) 435. R. Luo, J.M. Wang, P.A. Kollman, Abstr. Pap. Am. Chem. Soc. 224 (2002) U470. A.M. Nikitin, A.P. Lyubartsev, J. Comput. Chem. 28 (2007) 2020. G. Bussi, D. Donadio, M. Parrinello, J. Chem. Phys. 126 (2007) 014101. M. Parrinello, A. Rahman, J. Appl. Phys. 52 (1981) 7182. G. Hummer, J.C. Rasaiah, J.P. Noworyta, Nature 414 (2001) 188–190. O.N. Kalugin, V.V. Chaban, V.V. Loskutov, O.V. Prezhdo, Nano Lett. 8 (2008) 2126. J. Kofinger, C. Dellago, Rev. Lett. 103 (2009) 080601.
Contents lists available at ScienceDirect
Chemical Physics Letters journal homepage: www.elsevier.com/locate/cplett
Filling carbon nanotubes with liquid acetonitrile Vitaly Chaban School of Chemistry, V.N. Karazin Kharkiv National University, Svoboda Square 4, 61077 Kharkiv, Ukraine
a r t i c l e
i n f o
Article history: Received 4 June 2010 In final form 3 July 2010 Available online 8 July 2010
a b s t r a c t Carbon nanotubes and acetonitrile are of interest for modern electrochemistry since they are used to make supercapacitors more efficient. In order to assess the feasibility of this setup, molecular dynamics simulations were performed to investigate the hydrophobic degasified single-walled nanotubes filling with liquid acetonitrile. The simulation shows that nanotubes with 10 nm of length can be completely filled with acetonitrile during less than 100 ps. Surprisingly, the filling process is not significantly affected by nanotube diameter and ambient conditions. In general, the ability of small hydrophobic carbon nanotubes to be completely filled with acetonitrile is an important feature for supercapacitors. Ó 2010 Elsevier B.V. All rights reserved.
1. Introduction The static and dynamic behavior of fluids inside the nanosized confinements is now an important and broad field [1–7] of investigation generally called nanofluidics. With a widespread availability of carbon nanotubes, interest in nanofluidics has greatly increased mainly due to the revolutionary possibilities of exploring the fluids inside the extremely narrow channels of about 1 nm or even less. The use of nanotubes to study the confined fluids also raises the interest to the specific atomic interactions between nanotube and confined particles of different chemical nature that basically defines the nature of the confined liquid flow [1,2,7,8]. The recent studies of nanofluidics comprise various aspects: both the fluids entering the tube and the overall flow inside the tube. By now, experimental studies dealt with very different nanotube diameters (from 1 to 100 nm) and lengths [9–15], whereas computer simulations were concentrated only on the rather smaller CNTs (with diameters of a few nanometers) [1,5,16–19]. Various substances were recently used to fill the hollow carbon nanotubes including molten metals, salts [9,20], oxides, aqueous and nonaqueous solutions [35] through metal evaporation [10], capillary filling [21] and other techniques. These works raised the production of a large number of hybrid materials [22,23]. An empirical law for single-walled CNTs filling was suggested by Dujardin et al. [11] stating that no liquid with a surface tension of more than 180–200 mN/m could enter the tube inner cavity. This law works well for the nanotubes with diameters of up to 4 nm. Although the surface tension of many polar liquids, including water (72 mN/m), is significantly below the threshold, they are generally believed not to wet pristine carbon nanotubes (in analogy with basal planes of graphite).
E-mail addresses: [email protected], [email protected] 0009-2614/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.cplett.2010.07.003
The present work focuses on acetonitrile (ACN), an important organic dipolar liquid both for science and chemical technology, which is also often used as a solvent for modern supercapacitors [24–29]. Among various possible setups, such devices are often designed with their electrodes containing carbon nanotubes or porous carbon to increase the electrolyte adsorption. Obviously, the performance of supercapacitors critically depends on the ability of the particular electrolyte solution to enter, and subsequently fill, the CNTs of the certain sizes. Since the transport properties of electrolyte solutions tightly depend on a particular solvent, direct investigation of the latter is important. Moreover, the pristine carbon nanotube can be considered as an ideal model of a porous carbon with nanosized channels. The another important question arising in this context is how far the solvent and its solution can go inside carbon nanotube. In spite of the considerable interest to supercapacitors in modern electrochemistry and general nanofluidics, the mentioned questions still remain open. For this kind of investigations, computer simulation appears to be the most productive tool because of the principal experimental limitations on the nanoscale. Here, the molecular dynamics (MD) study of filling CNTs with acetonitrile was carried out with a series of narrow armchair single-walled nanotubes ranging from (5, 5) to (11, 11) and the constant length of 10 nm for all species. The influence of the different ambient conditions, temperature and elevated pressure, has been considered. The adsorption processes of ACN molecules on the internal and external surfaces of the CNTs were separated in time by means of the specific consequent MD setups. 2. Simulation details 2.1. Force fields The GROMACS [30] program package was used to perform MD simulations on several systems containing a few carbon nanotubes
V. Chaban / Chemical Physics Letters 496 (2010) 50–55
and liquid acetonitrile. The exact composition of all MD systems is presented in Table 1. To represent all bonded and non-bonded interactions within the carbon nanotubes, AMBER force field [31] was applied. All simulated nanotubes were treated as flexible non-polarizable (hydrophobic) particles with 1–4 carbon–carbon interactions switched on. The acetonitrile molecule was represented using the six-site model of Nikitin et al. [32] which appears to be the best force field for this liquid by now. It reproduces thermodynamics and structure properties of bulk ACN very well and also gives a pretty good viscosity (0.4 cP at 298 K and under 1 bar, according to our own calculations). Surprisingly, the self-diffusion constant is a bit lower than experimental value (3.5 vs 4.3 10-9 m2/s at 298 K and 1 bar). Unfortunately, no other acetonitrile force field gives better result for diffusion. The electrostatics was treated by means of the Ewald summation (rcut = 1.4 nm) and Lennard–Jones (12, 6) interactions were accounted with shifted force method (switch region between 1.2 nm and 1.3 nm). The cross-term Lennard–Jones (12, 6) parameters between carbon nanotubes and ACN were obtained using the standard Lorenz– Berthelot combination rules. The molecular dynamics time-step was 0.001 ps in conjunction with a leap-frog integration algorithm. The linear velocity of the nanotube, initially centered in the box, was gradually removed to simplify the subsequent analysis. 2.2. Simulation setup The multi-stage MD simulations were carried out to investigate different aspects of the dynamic behavior of the CNT-ACN systems. Five narrow single-walled armchair carbon nanotubes were considered: (5, 5), (6, 6), (7, 7), (9, 9) and (11, 11) with the constant length of about 10 nm. The study of the system dynamics was started with the CNT centered in the box and surrounded by 0.6 nm of vacuum. Initially, no solvent molecules were present in the inner cavity of the nanotube corresponding to the degasified case. The liquid acetonitrile (see Table 1 for details) was located as far as 0.6 nm from the outer nanotube wall and filled all the available volume in the 5 5 14 nm parallelepipedic MD box. Because of this somewhat exotic setup, the initial densities of the simulated systems were 20–30% lower than equilibrium ones (Table 1). Firstly, in order to
Table 1 Diameters of CNTs, dCNT, numbers of acetonitrile molecules in the MD system, numbers of confined solvent molecules, times of adsorption on the external surfaces of CNTs, times of filling and system densities, d. CNT
dCNT (nm)
N (ACN)
Nconfined (ACN)
texter.ads, (ps)
tfilling (ps)
d (kg/m3)
(5, 5) (6, 6) (7, 7) (9, 9) (11, 11)
0.68 0.82 0.95 1.22 1.49
2463 2397 2292 2217 2137
0 19 28 69 130
20 20 15 20 20
– 50 55 60 100
828 841 854 878 904
51
study the adsorption of ACN on the external surface of the nanotubes only, the tubes were capped preventing the solvent to penetrate inside. Secondly, when the systems were equilibrated in their previous configurations, the caps were removed and the solvent was allowed to enter the open-ended tubes (Fig. 1). The rate of the filling process was investigated at temperatures of 278, 298 and 323 K and pressures of 1, 100 and 2000 bar (T = 323 K). Finally, equilibrium MD runs were performed to explore the structure patterns of acetonitrile in confinements and estimate its dielectric constant in the specific single-file chains. All MD simulations were carried out in the constant temperature and constant pressure (NPT) ensemble. V-rescale thermostat [33] with a time constant of 0.5 ps and Parrinello–Rahman barostat [34] with a time constant of 1.0 ps were applied to maintain the temperature and pressure values, respectively. Importantly, the pressure coupling was applied only in the radial direction of the CNT to minimize its possible artificial effect on the filling process of acetonitrile. 2.3. Data analysis The rate of carbon nanotubes filling was determined by the interaction energy between nanotube and ACN. The corresponding energy values were dumped every 2 ps (2000 time-steps) and plotted vs simulation time. The structure patterns of acetonitrile confined inside CNTs were analyzed in terms of atomic densities in axial and radial directions of the nanotubes separately. The nitrogen atom of ACN was selected for that as the heaviest one in the solvent molecule. The dielectric constants of the confined liquid were estimated from the fluctuations of the molecule dipole moment. 3. Results and discussion 3.1. Effect of CNT diameter Fig. 2 illustrates the evolution of interaction energy between the nanotube and acetonitrile for CNT (5, 5), CNT (6, 6), CNT (7, 7), CNT (9, 9) and CNT (11, 11). Interestingly, both internal and external surfaces of these CNTs are wetted unexpectedly rapidly (during less than 100 ps). For external surface, the equilibration time is not more than 20 ps (Table 1). It does not also depend on the particular tube diameter. The general shape of the curves is also very similar indicating that the solvent approaches the tube external walls very rapidly and wets them within 20 ps. The penetration of ACN into the internal cavity of the nanotube is more complicated. While CNT (6, 6), CNT (7, 7) and CNT (9, 9) are filled rapidly and similarly during 50–70 ps (Table 1), the filling of CNT (11, 11) is longer and exhibits somewhat different mechanism. The first stage occurs during the first 50 ps and corresponds to the intense penetration of solvent molecules into the nanotube. The second stage (50 ps) can be understood as repacking of ACN inside the spatial confinements tending to the minimal potential en-
Fig. 1. The simulated system containing centered CNT (11, 11) surrounded by 2137 acetonitrile molecules.
52
V. Chaban / Chemical Physics Letters 496 (2010) 50–55
number of carbon atoms in each CNT (Fig. 3). The energy of about 3 kJ/mol per one carbon atom corresponds to the interaction of the CNTs with the outside solvent molecules. The total interaction energy slightly increases as the tube curvature increases. Filling the inner surface of the CNT corresponds to 1.2–2 kJ/mol per one carbon atom of the nanotube. The energy value also correlates well with a tube diameter. This is rather trivial since bigger tubes are able to encapsulate a larger number of solvent molecules (Table 1). All these molecules interact with CNT sidewalls directly through van der Waals forces. The ability of liquid acetonitrile to fill carbon nanotubes of sub-molecular sizes favors its use as a solvent in supercapacitors together with carbon nanotubes [24–29]. The simulation results suggest that the geometric size of the proper carbon pore size can be as small as 0.82 nm.
3.2. Effect of temperature
Fig. 2. The evolution of the interaction energy between CNT and (a) inside ACN, (b) outside ACN derived from MD simulations at 298 K and 1 bar. Horizontal solid lines of the same color show the respective equilibrium energies. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
ergy of the simulated system. This repacking stage is absent for smaller nanotubes because their diameters allow only one molecular layer to be located near each sidewall. Thus, subsequent repacking is not further necessary. Rapid filling of nanotubes was recently demonstrated for the case of water [35] which is attributed to the tight hydrogen bonds. Although acetonitrile is even more polar (l = 3.9 D), it is known as rather unstructured liquid with low maxima on radial distribution functions [32] and high diffusion constant (4.3 10 9 m2/s). There is also no hydrogen bonds in liquid ACN as opposed to water [35]. In spite of no tight intermolecular bonding, the filling process is unexpectedly fast and complete, even at atmospheric pressure and low temperature. Importantly, the internal cavity of the CNT (5, 5) can not be filled with acetonitrile probably because of its too small diameter (0.68 nm). The van der Waals radius of carbon atom is 0.17 nm decreasing the available space to 0.34 nm. As current MD study shows, this value is too little for methyl group (the largest group of ACN) to enter the CNT (5, 5). The independency of the nanotube filling rate from its diameter is rather curious provided that recently self-diffusion of the confined ACN was shown to correlate well with CNT diameter [36]. It proves that the equilibrium diffusion constants of the confined liquid cannot be directly used to predict the dynamics of filling. The main reason for this is a considerable difference between diffusion in the bulk phase and at the liquid–vapor interface. For small nanotubes, this effect is pronounced even more than at the ordinary liquid–vapor interface because the tube geometry drastically limits the number of acetonitrile molecule neighbors. In fact, during the first filling stage (Fig. 2) we observe quasi–gas phase of the confined acetonitrile. The molecular mobility in such a phase is found to be high and weakly dependent on the nanotube size. To compare the interaction energies with ACN for different nanotubes, the corresponding energy values were divided by the
Three temperatures (278, 298, 323 K) at which ACN is a liquid were considered in this study. Their influence on the filling dynamics is depicted in Fig. 4. The role of temperature is not crucial, although some insignificant differences can be observed for CNT (11, 11) whereas the filling rate for CNT (6, 6) is not affected at all. The first stage of the overall filling process becomes shorter with the system temperature increase (at 298 and 323 K) but the repacking stage remains nevertheless rather long. At 278 K, the first stage is a bit longer but the second one disappears. Overall, the system simulated at 278 K comes to equilibrium faster (within 70 ps). This is not correlated with the self-diffusion constants of bulk ACN which are 2.3, 3.4, 4.3 (10 9 m2/s) at 278, 298 and 323 K, respectively. The difference in the equilibrium interaction energies can be attributed to the density decrease when tempera-
Fig. 3. The evolution of the interaction energy per one carbon atom between CNT and (a) inside ACN, (b) outside ACN derived from MD simulations at 298 K and 1 bar: CNT (6, 6), red solid line; CNT (7, 7), green dash-dotted line; CNT (9, 9), blue dashed line; CNT (11, 11), pink dash-dot-dotted line. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
V. Chaban / Chemical Physics Letters 496 (2010) 50–55
53
ture rises. Bigger system density always corresponds to more compact molecular packing resulting in larger potential energies. Generally, the weak effect of temperature suggests that ACN flow inside the partially filled small hydrophobic CNTs is even faster than in its bulk phase. One can conclude that the filling rate is limited only by geometry of confinements and does not considerably depend on media temperature. 3.3. Effect of pressure Three pressures (1, 100, 2000 bar) at T = 323 K were considered to investigate their influence on the nanotube filling process. CNT (6, 6) and CNT (11, 11), as the biggest and the smallest ones, were selected for this investigation (Fig. 5). The influence of high and very high pressures is more pronounced than that of temperature and affects both nanotubes similarly. The increase of pressure to 2000 bar leads to the increase of the equilibrium interaction energy between nanotube and solvent by about 20%. Also the filling process becomes somewhat faster as the pressure grows. Note, the degasified nanotubes are very readily filled with ACN both at atmospheric and elevated pressure. Thus, the elevated pressure is not necessary for practical applications (supercapacitors, nanofluidic devices) where the carbon nanotube filling is required. 3.4. Confined acetonitrile structure Fig. 4. The evolution of the interaction energy between CNTs and confined ACN, derived from MD simulations at 278 K (red solid line), 298 K (green dash-dotted line), 323 K (blue dashed line) and 1 bar: (a) CNT (6, 6); (b) CNT (11, 11). Horizontal solid lines of the same color show the respective equilibrium energies. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
Fig. 5. The evolution of the interaction energy between CNTs and confined ACN derived from MD simulations at 1 bar (red solid line), 100 bar (green dash-dotted line) and 2000 bar (blue dashed line) and 323 K: (a) CNT (6, 6); (b) CNT (11, 11). Horizontal solid lines of the same color show the respective equilibrium energies. (For interpretation of the references to color in this figure legend, the reader is referred to the web version of this article.)
In order to understand the above specificity of CNT filling with acetonitrile, the partial atomic densities along axial and radial
Fig. 6. Partial atomic (nitrogen, ACN) densities along (a) radial and (b) axial directions of the nanotubes derived from the equilibrium MD simulations at 298 K and 1 bar. The straight vertical lines correspond to the CNT sidewalls. The shown coordinates are the absolute coordinates of the respective nitrogen atoms in the MD box.
54
V. Chaban / Chemical Physics Letters 496 (2010) 50–55
Fig. 7. The instantaneous configurations of acetonitrile confined by (a) CNT (6, 6), (b) CNT (7, 7), (c) CNT (11, 11) derived from the equilibrium MD simulations at 298 K and 1 bar.
directions of the nanotube were calculated (Fig. 6). The number of maxima on Fig. 6a corresponds to the number of ACN molecules located along the radial direction inside the CNT. So, CNT (6, 6) is able to encapsulate only one solvent layer, CNT (7, 7) and CNT (9, 9) – two layers each and CNT (11, 11) – three layers. Inside CNT (6, 6), the confined acetonitrile molecules are quite free to move because of this nanotube diameter (0.82 nm). The effective (available to solvent) CNT (6, 6) diameter is 0.48 nm which is about 1.4 times bigger than the van der Waals diameter of the ACN carbon atom. This is bigger than intermolecular distances in bulk liquid ACN leading to the anomalous mobility of the confined solvent molecules. The same conclusion is derived from Fig. 6b which demonstrates almost no oscillations of acetonitrile molecules along the nanotube axial direction. The effect is well pronounced for the CNT filling process resulting in extremely fast slipping of the ACN molecules from bulk liquid into confinement. Analyzing other systems, we get the following relations between the diameters of the carbon atom and the nanotube: for CNT (7, 7) – 1.8, for CNT (9, 9) – 2.6, for CNT (11, 11) – 3.4. It is interesting that Fig. 6a indicates two maxima for ACN inside CNT (7, 7) although two full-grown layers of solvent cannot exist inside this tube, since the CNT diameter is evidently smaller than two diameters of the solvent molecule. This fact is also confirmed by the total number of ACN molecules inside the CNT (7, 7) – 28. If two full-grown solvent layers existed in CNT (7,7), the number of the confined solvent molecules should have been two times more than inside CNT (6, 6), e.g. 38. The answer is given by the instantaneous configurations of the confined ACN inside CNT (6, 6) and CNT (7, 7) (Fig. 7). Inside CNT (6, 6), ACN molecules are aligned, so that their dipole moments coincide with the nanotube axis. In turn, inside CNT (7, 7), the solvent dipole moments are perpendicular to the CNT axis. Such configuration of the confined acetonitrile allows a bigger number of molecules to be encapsulated inside the tube of the same size. Thus, two maxima for the nitrogen atoms (Fig. 6a) correspond to the neighboring ACN molecules located along the nanotube axial direction. Their antiparallel dipole moments are oriented towards the CNT sidewall. Both CNT (6, 6) and CNT (7, 7) show single-file chains of liquid but with different orientations in reference to the nanotube sidewalls. Importantly, Fig. 6b proves that for all nanotubes the complete filling with acetonitrile (10 nm) occurs. For all investigated nanotubes it is performed within the first 100 ps of run. The repacking stage for the CNT (11, 11) (Fig. 7) case is presumably connected with the central solvent layer which can be formed only after the two parietal layers of the confined liquid are equilibrated. For smaller tubes this stage (Fig. 2) is not observed since
they have no central liquid layer due to their particular sizes (Fig. 6a). 3.5. Dielectric constant of confined acetonitrile The molecular single-file patterns of the confined polar liquid should result in a specific dielectric constant as was recently shown for liquid water [37]. For acetonitrile, the dielectric constants, e, of 1.01, 1.02, 1.08 and 1.22 were obtained for CNT (6, 6), CNT (7, 7), CNT (9, 9) and CNT (11, 11), respectively. These values are in good agreement with [37] for the same narrow tube lengths exhibiting single-file liquid structures. Note, the experimental bulk values for water and acetonitrile are e = 80 and e = 36, respectively. The present ACN force field model slightly underestimates e giving 26 instead of 36 for pure bulk liquid [32]. According to [37], at much bigger nanotube lengths (hundreds of lm), the dielectric constants of the confined dielectric media is expected to grow significantly (up to 10 000). This fact favors using narrow carbon nanotubes to make supercapacitors even more efficient provided that the penetration of the electrolyte solution is complete. 4. Conclusions Molecular dynamics simulations were performed to investigate the filling process of the degasified narrow single-walled carbon nanotubes with liquid acetonitrile at different temperatures and pressures. It is found that the filling rates are unexpectedly fast and weakly dependent from the nanotube diameter. Moreover, the changes of ambient temperature and pressure do not significantly affect the filling rate. This is unexpected since the diffusion of the bulk ACN increases significantly as temperature grows. The ability of small hydrophobic carbon nanotubes to be completely filled with acetonitrile is an important feature for efficient supercapacitors [24–29]. It can be also found useful for the modern nanofluidic devices that are actively developed nowadays. References [1] [2] [3] [4] [5] [6] [7] [8] [9]
M.C. Gordillo, J. Marti, Chem. Phys. Lett. 341 (2001) 250. M.C. Gordillo, J. Marti, Chem. Phys. Lett. 329 (2000) 341. Y.C. Liu, Q. Wang, T. Wu, L. Zhang, J. Chem. Phys. 123 (2005) 234701. Y.C. Liu, Q. Wang, L. Zhang, T. Wu, Langmuir 21 (2005) 12025. R.J. Mashl, S. Joseph, N.R. Aluru, E. Jakobsson, Nano Lett. 3 (2003) 589. D. Mattia, Y. Gogotsi, Microfluidics Nanofluidics 5 (2008) 289. T. Werder, J.H. Walther, R.L. Jaffe, et al., Nano Lett. 1 (2001) 697. G. Garberoglio, Eur. Phys. J. E 31 (2010) 73. D. Ugarte, T. Stockli, J.M. Bonard, et al., Appl. Phys. A – Mater. Sci. Process. 67 (1998) 101.
V. Chaban / Chemical Physics Letters 496 (2010) 50–55 [10] P.M. Ajayan, S. Iijima, Nature 361 (1993) 333. [11] E. Dujardin, T.W. Ebbesen, H. Hiura, K. Tanigaki, Science 265 (1994) 1850. [12] Y. Gogotsi, J.A. Libera, A. Guvenc-Yazicioglu, C.M. Megaridis, Appl. Phys. Lett. 79 (2001) 1021. [13] H.W. Yu, B. Fugetsu, J. Hazard. Mater. 177 (2010) 138. [14] J. Meng, M. Yang, F.M. Jia, et al., Nanotechnology 21 (2010) 145104. [15] B. Tawabini, S. Al-Khaldi, M. Atieh, M. Khaled, Water Sci. Technol. 61 (2010) 591. [16] R.E. Tuzun, D.W. Noid, B.G. Sumpter, R.C. Merkle, Nanotechnology 7 (1996) 241. [17] S. Banerjee, S. Murad, I.K. Puri, Chem. Phys. Lett. 434 (2007) 292. [18] S. Joseph, N.R. Aluru, Nano Lett. 8 (2008) 452. [19] J. Lee, N.R. Aluru, Appl. Phys. Lett. 96 (2010) 133108. [20] C.G. Xu, J. Sloan, G. Brown, et al., Chem. Commun. 24 (2000) 2427. [21] J. Mittal, M. Monthioux, H. Allouche, O. Stephan, Chem. Phys. Lett. 339 (2001) 311. [22] J. Sloan, A.I. Kirkland, J.L. Hutchison, M.L.H. Green, C. R. Phys. 4 (2003) 1063.
[23] [24] [25] [26] [27] [28] [29] [30] [31] [32] [33] [34] [35] [36] [37]
55
M. Monthioux, Carbon 40 (2002) 1809. Q.F. Xiao, X. Zhou, Electrochim. Acta 48 (2003) 575. P.L. Taberna, G. Chevallier, P. Simon, et al., Mater. Res. Bull. 41 (2006) 478. P. Azais, L. Duclaux, P. Florian, et al., J. Power Sources 171 (2007) 1046. J.S. Huang, B.G. Sumpter, V. Meunier, Chem. – A Eur. J. 14 (2008) 6614. Y. Korenblit, M. Rose, E. Kockrick, et al., ACS Nano 4 (2010) 1337. S. Jagannathan, T. Liu, S. Kumar, Compos. Sci. Technol. 70 (2010) 593. B. Hess, C. Kutzner, D. van der Spoel, E. Lindahl, J. Chem. Theor. Comput. 4 (2008) 435. R. Luo, J.M. Wang, P.A. Kollman, Abstr. Pap. Am. Chem. Soc. 224 (2002) U470. A.M. Nikitin, A.P. Lyubartsev, J. Comput. Chem. 28 (2007) 2020. G. Bussi, D. Donadio, M. Parrinello, J. Chem. Phys. 126 (2007) 014101. M. Parrinello, A. Rahman, J. Appl. Phys. 52 (1981) 7182. G. Hummer, J.C. Rasaiah, J.P. Noworyta, Nature 414 (2001) 188–190. O.N. Kalugin, V.V. Chaban, V.V. Loskutov, O.V. Prezhdo, Nano Lett. 8 (2008) 2126. J. Kofinger, C. Dellago, Rev. Lett. 103 (2009) 080601.