We consider an incompressible fluid in a rectangular nanochannel. We solve numerically the three
dimensional Fourier heat equation to get the steady solution for the temperature. Then we set and
solve the Langevin equation for the temperature. We have developed equations in order to determine
relaxation time of the temperature fluctuations, τT = 4.62 × 10-10s. We have performed a
spectral analysis of the thermal fluctuations, with the result that temporal correlations are in the
one-digit ps range, and the thermal noise excites the thermal modes in the two-digit GHz range.
Also we observe long-range spatial correlation up to more than half the size of the cell, 600 nm;
the wave number, q, is in the 106 m-1 range. We have also determined two thermal relaxation
lengths in the z direction: l1 = 1.18 nm and l2 = 9.86 nm.
References
[1]
Soong, R.K., Bachand, G.D., Neves, H.P., Olkhovets, A.G., Craihead, H.G. and Montemagno, C.D. (2000) Powering an Inorganic Nanodevice with a Biomolecular Motor. Science, 290, 1555. http://dx.doi.org/10.1126/science.290.5496.1555
[2]
Tsong, T.Y. (2002) Na, K-ATPase as a Brownian Motor: Electric Field-Induced Conformational Fluctuation Leads to Up-Hill Pumping of Cation in the Absence of ATP. Journal of Biological Physics, 28, 309-325. http://dx.doi.org/10.1023/A:1019991918315
[3]
Reisner, W., Pedersen, J.N. and Austin, R.H. (2012) DNA Confinement in Nanochannels: Physics and Biological Applications. Reports on Progress in Physics, 75, Article ID: 106601. http://dx.doi.org/10.1088/0034-4885/75/10/106601
[4]
Hou, X., Guoa, W. and Jiang, L. (2011) Biomimetic Smart Nanopores and Nanochannels. Chemical Society Reviews, 40, 2385-2401. http://dx.doi.org/10.1039/c0cs00053a
[5]
Lappala, A., Zaccone, A. and Terentjev, E.M. (2013) Ratcheted Diffusion Transport through Crowded Nanochannels. Scientific Reports, 3, 3103. http://dx.doi.org/10.1038/srep03103
[6]
Cheng, L.-J. (2008) Ion and Molecule Transport in Nanochannels. Dissertation of the Requirements for the Degree of Doctor of Philosophy, Electrical Engineering and Computer Science in the University of Michigan.
Yang, J. and Kwok, D.Y. (2003) Microfluid Flow in Circular Microchannel with Electrokinetic Effect and Naviers Slip Condition. Langmuir, 19, 1047-1053. http://dx.doi.org/10.1021/la026201t
[9]
Eigen, M. and Rigler, R. (1994) Sorting Single Molecules: Application to Diagnostics and Evolutionary Biotechnology. Proceedings of the National Academy of Sciences of the United States of America, 91, 5740-5747. http://dx.doi.org/10.1073/pnas.91.13.5740
[10]
Goodwin, P.M., Johnson, M.E., Martin, J.C., Ambrose, W.P., Marrone, B.L., Jett, J.H. and Keller, R.A. (1993) Rapid Sizing of Individual Fluorescently Stained DNA Fragments by Flow Cytometry. Nucleic Acids Research, 21, 803-806. http://dx.doi.org/10.1093/nar/21.4.803
[11]
Sauer, M., Angerer, B., Ankenbauer, W., Foldes-Papp, Z., Gobel, F., Han, K.T., Rigler, R., Schulz, A., Wolfrum, J. and Zander, C. (2001) Single Molecule DNA Sequencing in Submicrometer Channels: State of the Art and Future Prospects. Journal of Biotechnology, 86, 181-201. http://dx.doi.org/10.1016/S0168-1656(00)00413-2
[12]
Kuo, T.-C., Sloan, L.A., Sweedler, J.V. and Bohn, P.W. (2001) Manipulating Molecular Transport through Nanoporous Membranes by Control of Electrokinetic Flow: Effect of Surface Charge Density and Debye Length. Langmuir, 17, 6298-6303.
[13]
Liu, Y., Liu, M., Lau, W.M. and Yang, J. (2008) Ion Size and Image Effect on Electrokinetic Flows. Langmuir, 24, 2884-2891.
[14]
Stepisnik, J. and Callaghan, P.T. (2000) The Long Time Tail of Molecular Velocity Correlation in a Confined Fluid: Observation by Modulated Gradient Spin-Echo NMR. Physica B: Condensed Matter, 292, 296-301. http://dx.doi.org/10.1016/S0921-4526(00)00469-5
[15]
Stepisnik, J. and Callaghan, P.T. (2001) Low-Frequency Velocity Correlation Spectrum of Fluid in a Porous Media by Modulated Gradient Spin Echo. Magnetic Resonance Imaging, 19, 469-472. http://dx.doi.org/10.1016/S0730-725X(01)00269-7
[16]
Callaghan, P.T. and Stepisnik, J. (1995) Frequency-Domain Analysis of Spin Motion Using Modulated Gradient NMR. Journal of Magnetic Resonance, Series A, 117, 118-122. http://dx.doi.org/10.1006/jmra.1995.9959
[17]
Detcheverry, F. and Bocquet, L. (2013) Thermal Fluctuations of Hydrodynamic Flows in Nanochannels. Physical Review E, 88, Article ID: 012106.
[18]
Lifshitz, E.M. and Pitaevskii, L.P. (2003) Landau and Lifshitz Course of Theoretical Physics. Volume 9: Statistical Physics, Part 2, Elsevier Butterworth-Heinemann, Oxford.
[19]
Landau, L.D. and Lifshitz, E.M. (2004) Landau and Lifshitz Course of Theoretical Physics. Volume 6: Fluid Mechanics, Elsevier Butterworth-Heinemann, Oxford.
[20]
Gardiner, C.W. (1985) Handbook of Stochastic Methods for Physics, Chemistry and the Natural Sciences. Springer-Verlag, Berlin. http://dx.doi.org/10.1007/978-3-662-02452-2
[21]
Scherer, C. (2005) Métodos Computacionais da Fsica. Editora Livraria da Fsica, USP, Sao Paulo. (In Portuguese)
[22]
Dorfman, J.R., Kirkpatrick, T.R. and Sengers, J.V. (1994) Generic Long-Range Correlations in Molecular Fluids. Annual Review of Physical Chemistry, 45, 213-219. http://dx.doi.org/10.1146/annurev.pc.45.100194.001241 Kirkpatrick, T.R., Belitz, D. and Sengers, J.V. (2002) Long Time Tails, Weak Localizations, and Classical and Quantum Critical Behavior. Journal of Statistical Physics, 109, 373-405. http://dx.doi.org/10.1023/A:1020485809093
[23]
Grinstein, G., Lee, D.H. and Sachdev, S. (1990) Conservation Laws, Anisotropy, and “Self-Organized Criticality” in Noisy Nonequilibrium Systems. Physical Review Letters, 64, 1927-1930. http://dx.doi.org/10.1103/PhysRevLett.64.1927
[24]
J. Dufty, in Spectral Line Shapes, p. 1143, edited by B. Wende (W. de Gruyter, NY, 1981); in Long Range Correlations, p. 1, edited by J-R. Buchler et al. (Annals NY Acad. Sci., vol. 848,1998).
[25]
Lutsko, J.F. and Dufty, J.W. (2002) Long-Ranged Correlations in Sheared Fluids. Physical Review E, 66, Article ID: 041206. http://dx.doi.org/10.1103/PhysRevE.66.041206
[26]
Kirkpatrick, T.R., Cohen, E.G.D. and Dorfman, J.R. (1986) Light Scattering by a Fluid in a Nonequilibrium Steady State. I. Small Gradients. Physical Review A, 26, 972-994. http://dx.doi.org/10.1103/PhysRevA.26.972
[27]
Schmitz, R. (1988) Fluctuations in Nonequilibrium Fluids. Physics Reports, 171, 1-58. http://dx.doi.org/10.1016/0370-1573(88)90052-X
[28]
Alder, B.J. and Wainwright, T.E. (1970) Decay of the Velocity Autocorrelation Function. Physical Review A, 1, 18-21. http://dx.doi.org/10.1103/PhysRevA.1.18
[29]
Hagen, M.H.J., Pagonabarraga, I., Lowe, C.P. and Frenkel, D. (1997) Algebraic Decay of Velocity Fluctuations in a Confined Fluid. Physical Review Letters, 78, 3785-3788. http://dx.doi.org/10.1103/PhysRevLett.78.3785
[30]
Fornés, J.A. and de Zárate, J.M.O. (2007) Low-Frequency Velocity Correlation Spectrum of Fluid in a Rectangular Microcapillary. Langmuir, 23, 11917-11923. http://dx.doi.org/10.1021/la702502q
