Size and temperature effects on the viscosity of water inside carbon nanotubes
 Hongfei Ye^{1},
 Hongwu Zhang^{1}Email author,
 Zhongqiang Zhang^{1, 2} and
 Yonggang Zheng^{1}
DOI: 10.1186/1556276X687
© Ye et al; licensee Springer. 2011
Received: 3 August 2010
Accepted: 17 January 2011
Published: 17 January 2011
Abstract
The influences of the diameter (size) of singlewalled carbon nanotubes (SWCNTs) and the temperature on the viscosity of water confined in SWCNTs are investigated by an "EyringMD" (molecular dynamics) method. The results suggest that the relative viscosity of the confined water increases with increasing diameter and temperature, whereas the sizedependent trend of the relative viscosity is almost independent of the temperature. Based on the computational results, a fitting formula is proposed to calculate the size and temperature dependent water viscosity, which is useful for the computation on the nanoflow. To demonstrate the rationality of the calculated relative viscosity, the relative amount of the hydrogen bonds of water confined in SWCNTs is also computed. The results of the relative amount of the hydrogen bonds exhibit similar profiles with the curves of the relative viscosity. The present results should be instructive for understanding the coupling effect of the size and the temperature at the nanoscale.
Introduction
Water conduction through singlewalled carbon nanotubes (SWCNTs) has been paid much attention in recent years [1–5]. It is a significant topic for studying and designing the nanodevices such as the nanochannel for drug delivery and the membrane for water desalination [6–8]. The previous studies have revealed that the flow behavior of water at the nanoscale strongly depends on the characteristic length of nanochannel [9–12], which implies that the classical continuum theory for the macroscopic fluid may be no longer applicable for the fluid confined in nanochannels. Hence, many researches focused on the unique feature of the confined fluid and its relationship with the continuum fluid [9–13]. In classical continuum theory, the viscosity is an essential transport property and thereby has been extensively measured and computed [14, 15]. The previous results have identified that the water viscosity relies on the temperature and the characteristic length of the nanochannel [9, 12–15]. So far, the viscosity of fluids in nanoconfinement on a scale comparable to the molecular diameter is seldom explored owing to the extremely small scale on which the transport properties are difficult to be captured by experiments and the intrinsic limitations of the existing computational methods in the MD simulations [16–18]. This restricts the application of the classical continuum theory to the nanoflows.
Recently, an "EyringMD" method was proposed to calculate the viscosity of water by using the MD simulations [18]. In this article, we redetermine the coefficients in the "EyringMD" method through more numerical experiments and evaluate the viscosity of water inside SWCNTs at 298, 325, and 350 K. The objective of this study is to examine the size and the temperature effects on the water viscosity. Here, the size effect on the viscosity of the confined water implies the influence of the diameter of SWCNTs.
The computational method
in which U _{coul} and U _{van} are the coulomb energy and the van der Waals energy extracted from the MD simulations. The coefficients f _{1} = 2.062576 and f _{2} = 8.984223 kcal mol^{1} at 298 K, f _{1} = 2.058061 and f _{ 2 }= 8.742694 kcal mol^{1} at 325 K, and f _{1} = 2.065280 and f _{2} = 8.502127 kcal mol^{1} at 350 K. Thus, by using Equations 1, 2, and 3, the viscosity of water can be predicted by the MD simulations. The correlation coefficient between the viscosity calculated by the "EyringMD" method and that obtained from the numerical experiments (StokesEinstein relation) is about 0.99.
Results and discussion
where p _{1} = 0.00285 mPa s, p _{2} = 1632 K, p _{11} = 0.000225 1/K, p _{12} = 0.055547, p _{13} = 1197.417113 K, p _{21} = 0.007639 1/K, p _{22} = 4.910991, p _{31} = 0.011533 1/K, and p _{32} = 7.240463. The computational results of Equation 4 are also displayed in Figure 2 (lines). The correlation coefficient between the fitting results (lines in Figure 2) and the relative viscosity (symbols in Figure 2) is about 0.96. Furthermore, it should be noted that the η _{bulk} in Equation 5 calculates the temperaturedependent viscosity of the bulk water, which is fitted according to the widely accepted exponential relationship [23] and the viscosities of bulk water within the temperature range from 275 to 400 K from the MD simulations. This term will become dominant when the size (d) gradually tends to infinite, which is consistent with the physical role of the confinement. Equation 4 describes the size and the temperature effects on the water viscosity and should be significant for the research on the flow behavior at the nanoscale.
Conclusions
In summary, we have studied the influences of the diameter of SWCNTs and the temperature on the viscosity of the confined water by using the "EyringMD" method whose coefficients are redetermined through considering new numerical experiments. For a specified temperature, the relative viscosity nonlinearly increases with enlarging diameter of SWCNTs. For a given diameter, the relative viscosity of water inside the SWCNTs increases with increasing temperature. An approximate formula of the relative viscosity with consideration of the size and the temperature effects is proposed, which can avoid the timeconsuming MD simulations and should be significant for the research on the water flow inside the nanochannels. Furthermore, the amount of the hydrogen bonds of water confined in SWCNTs is also computed. The results suggest that the relative amount of the hydrogen bonds has similar profile with the relative viscosity, which demonstrates the present predictions of the relative viscosity. The computations in this study reveal that the trend of the size dependence is almost insensitive to the temperature, whereas the sizedependent extent could vary with the temperature. This finding provides an insight into the researches on the nanoflows and is instructive for understanding the coupling effect of the size and the temperature at the nanoscale.
Authors contributions
HZ and HY conceived and designed this work. HY and ZZ performed the MD simulations. HY, YZ and ZZ collected and analyzed the data. All authors discussed the results and edited the manuscript. All authors read and approved the final manuscript.
Abbreviations
 LJ:

LennardJones
 MD:

molecular dynamics
 SWCNTs:

singlewalled carbon nanotubes.
Declarations
Acknowledgements
The supports of the National Natural Science Foundation of China (11072051, 90715037, 10902021, 91015003, 10728205, 10721062), the 111 Project (No.B08014), the National Key Basic Research Special Foundation of China (2010CB832704), and the Program for Changjiang Scholars and Innovative Research Team in University of China (PCSIRT) are gratefully acknowledged.
Authors’ Affiliations
References
 Hummer G, Rasaiah JC, Noworyta JP: Water Conduction through the Hydrophobic Channel of a Carbon Nanotube. Nature 2001, 414: 188. 10.1038/35102535View ArticleGoogle Scholar
 Holt JK: Carbon Nanotubes and Nanofluidic Transport. Adv Mater 2009, 21: 3542. 10.1002/adma.200900867View ArticleGoogle Scholar
 Hanaski I, Yonebayashi T, Kawano S: Molecular dynamics of a water jet from a carbon nanotube. Phys Rev E 2009, 79: 046307. 10.1103/PhysRevE.79.046307View ArticleGoogle Scholar
 Liu L, Qiao Y, Chen X: Pressuredriven water infiltration into carbon nanotube: The effect of applied charges. Appl Phys Lett 2008, 92: 101927. 10.1063/1.2857474View ArticleGoogle Scholar
 Zuo GC, Shen R, Ma SJ, Guo WL: Transport Properties of SingleFile Water Molecules inside a Carbon Nanotube Biomimicking Water Channel. ACS Nano 2010, 4: 205. 10.1021/nn901334wView ArticleGoogle Scholar
 Bianco A, Kostarelos K, Prato M: Applications of Carbon Nanotubes in Drug Delivery. Curr Opin Chem Biol 2005, 9: 674. 10.1016/j.cbpa.2005.10.005View ArticleGoogle Scholar
 Corry B: Designing Carbon Nanotube Membranes for Efficient Water Desalination. J Phys Chem B 2008, 112: 1427. 10.1021/jp709845uView ArticleGoogle Scholar
 Zhu FQ, Schulten K: Water and Proton Conduction through Carbon Nanotubes as Models for Biological Channels. Biophys J 2003, 85: 236. 10.1016/S00063495(03)744695View ArticleGoogle Scholar
 Thomas JA, McGaughey AJH: Reassessing Fast Water Transport through Carbon Nanotubes. Nano Lett 2008, 8: 2788. 10.1021/nl8013617View ArticleGoogle Scholar
 Thomas JA, McGaughey AJH: Water Flow in Carbon Nanotubes: Transition to Subcontinuum Transport. Phys Rev Lett 2009, 102: 184502. 10.1103/PhysRevLett.102.184502View ArticleGoogle Scholar
 Wang LQ, Fan J: Nanofluids Research: Key Issues. Nanoscale Res Lett 2010, 5: 1241–1252. 10.1007/s1167101096386View ArticleGoogle Scholar
 Chen X, Cao GX, Han AJ, Punyamurtula VK, Liu L, Culligan PJ, Kim T, Qiao Y: Nanoscale Fluid Transport: Size and Rate Effects. Nano Lett 2008, 8: 2988. 10.1021/nl802046bView ArticleGoogle Scholar
 Zhang ZQ, Zhang HW, Ye HF: Pressuredriven flow in parallelplate nanochannels. Appl Phys Lett 2009, 95: 154101. 10.1063/1.3247892View ArticleGoogle Scholar
 David RL: CRC Handbook of Chemistry and Physics. 84th edition. New York: CRC press; 2004.Google Scholar
 Powell RE, Roseveare WE, Eyring H: Diffusion, Thermal Conductivity, and Viscous Flow of Liquids. Ind Eng Chem 1941, 33: 430. 10.1021/ie50376a003View ArticleGoogle Scholar
 Bertolini D, Tani A: Stress Tensor and Viscosity of Water: Molecular Dynamics and Generalized Hydrodynamics Results. Phys Rev E 1995, 52: 1699. 10.1103/PhysRevE.52.1699View ArticleGoogle Scholar
 Mallamace F, Branca C, Corsaro C, Leone N, Spooren J, Stanley HE, Chen SH: Dynamical Crossover and Breakdown of the StokesEinstein Relation in Confined Water and in MethanolDiluted Bulk Water. J Phys Chem B 2010, 114: 1870. 10.1021/jp910038jView ArticleGoogle Scholar
 Zhang HW, Ye HF, Zheng YG, Zhang ZQ: Prediction of the viscosity of water confined in carbon nanotubes. Microfluid Nanofluid 2010. Online First ArticlesGoogle Scholar
 Steve P: Fast Parallel Algorithms for Shortrange Molecular Dynamics. J Comput Phys 1995, 117: 1. 10.1006/jcph.1995.1039View ArticleGoogle Scholar
 Hans WH, William CS, Jed WP, Jeffry DM, Thomas JD, Greg LH, Teresa HG: Development of an Improved Foursite Water Model for Biomolecular Simulations: TIP4PEW. J Chem Phys 2004, 120: 665.Google Scholar
 Mashl RJ, Joseph S, Aluru NR, Jakobsson E: Anomalously Immobilized Water: A New Water Phase Induced by Confinement in Nanotubes. Nano Lett 2003, 3: 589. 10.1021/nl0340226View ArticleGoogle Scholar
 Giovambattista N, Rossky PJ, Debenedetti PG: Phase Transitions Induced by Nanoconfinement in Liquid Water. Phys Rev Lett 2009, 102: 050603. 10.1103/PhysRevLett.102.050603View ArticleGoogle Scholar
 Poling BE, Prausnitz JM, O'Connell JP: The Properties of Gases and Liquids. 5th edition. New York: McGrawHill; 2001.Google Scholar
 Alenka L, David C: Hydrogenbond Kinetics in Liquid Water. Nature 1996, 379: 55. 10.1038/379055a0View ArticleGoogle Scholar
 Martí J: Analysis of the Hydrogen Bonding and Vibrational Spectra of Supercritical Model Water by Molecular Dynamics Simulations. J Chem Phys 1999, 110: 6876.View ArticleGoogle Scholar
Copyright
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.