American Institute of Mathematical Sciences

Advanced
March  2011, 4(1): 53-85. doi: 10.3934/krm.2011.4.53

A hierarchy of models related to nanoflows and surface diffusion

Kazuo Aoki 1, , Pierre Charrier 2, and  Pierre Degond 3,

1. 

Department of Mechanical Engineering and Science, Graduate School of Engineering, Kyoto University, Kyoto 606-8501
2. 

Institut de Mathméatiques de Bordeaux, ENSEIRB-MATMECA, IPB, Université de Bordeaux, F-33405 Talence cedex, France
3. 

1-Université de Toulouse; UPS, INSA, UT1, UTM, Institut de Mathématiques de Toulouse, F-31062 Toulouse, France

Received  September 2010 Revised  October 2010 Published  January 2011

In last years a great interest was brought to molecular transport problems at nanoscales, such as surface diffusion or molecular flows in nano or sub-nano-channels. In a series of papers V. D. Borman, S. Y. Krylov, A. V. Prosyanov and J. J. M. Beenakker proposed to use kinetic theory in order to analyze the mechanisms that determine mobility of molecules in nanoscale channels. This approach proved to be remarkably useful to give new insight on these issues, such as density dependence of the diffusion coefficient. In this paper we revisit these works to derive the kinetic and diffusion models introduced by V. D. Borman, S. Y. Krylov, A. V. Prosyanov and J. J. M. Beenakker by using classical tools of kinetic theory such as scaling and systematic asymptotic analysis. Some results are extended to less restrictive hypothesis.
Keywords: surface diffusion., nanoflows, Kinetic theory.
Mathematics Subject Classification: Primary: 82C40, 76P05, 41A60, 82D05; Secondary: 74A2.
Citation: Kazuo Aoki, Pierre Charrier, Pierre Degond. A hierarchy of models related to nanoflows and surface diffusion. Kinetic & Related Models, 2011, 4 (1) : 53-85. doi: 10.3934/krm.2011.4.53
References:
[1]

K. Aoki and P. Degond, Homogenization of a flow in a periodic channel of small section, Multiscale Model. Simul., 1 (2003), 304-334. doi: 10.1137/S1540345902409931.  Google Scholar

[2]

K. Aoki, P. Degond, S. Takata and H. Yoshida, Diffusion models for Knudsen compressors, Phys. Fluids, 19 (2007), 117103: 1-20. Google Scholar

[3]

J. J. M. Beenakker, Reduced dimensionality in gases in nanopores, Phys. Low-Dim. Struct., 10/11 (1995), 115-124. Google Scholar

[4]

J. J. M. Beenakker and S. Yu. Krylov, One-dimensional surface diffusion: density dependence in a smooth potential, J. Chem. Phys., 107 (1997), 4015-4023. doi: 10.1063/1.474757.  Google Scholar

[5]

J. J. M. Beenakker, V. D. Borman and S. Yu. Krylov, Molecular transport in the nanometer regime, Phys. Rev. Lett., 72 (1994), 514-517. doi: 10.1103/PhysRevLett.72.514.  Google Scholar

[6]

J. J. M. Beenakker, V. D. Borman and S. U. Krylov, Molecular transport in subnanometer pores: Zero-point energy, reduced dimensionality and quantum sieving, Chem. Phys. Letters, 232 (1995), 379-382. doi: 10.1016/0009-2614(94)01372-3.  Google Scholar

[7]

V. D. Borman, S. Yu. Krylov and A. V. Prosyanov, Theory of nonequilibrium phenomena at a gas-solid interface, Sov. Phys. JETP, 67 (1988), 2110-2121. Google Scholar

[8]

V. D. Borman, S. Yu. Krylov and A. V. Prosyanov, Fundamental role of unbound surface particles in transport phenomena along a gas-solid interface, Sov. Phys. JETP, 70 (1990), 1013-1022. Google Scholar

[9]

P. Charrier and B. Dubroca, Asymptotic transport models for heat and mass transfer in reactive porous media, Multiscale Model. Simul., 2 (2003), 124-157. doi: 10.1137/S1540345902411736.  Google Scholar

[10]

P. Degond, Transport of trapped particles in a surface potential, in "Nonlinear Partial Differential Equations and Their Applications," Collège de France Seminar, North Holland, XIV (2002), 273-296. Google Scholar

[11]

P. Degond, C. Parzani and M.-H. Vignal, A Boltzmann model for trapped particles in a surface potential, Multiscale Model. Simul., 5 (2006), 364-392. doi: 10.1137/050642897.  Google Scholar

[12]

E. Frenod and K. Hamdache, Homogenisation of transport kinetic equations with oscillating potentials, Proceedings of the Royal Society of Edinburgh A, 126 (1996), 1247-1275.  Google Scholar

[13]

J. K. Holt, H. G. Park, Y. Wang, M. Stadermann, A. B. Artyukhin, C. P. Grigoropoulos, A. Noy and O. Bakajin, Fast mass transport sub-2-nanometer carbon nanotubes, Science, 312 (2006), 1034-1037. doi: 10.1126/science.1126298.  Google Scholar

[14]

G. Karniadakis, A. Beskok and N. Aluru, "Microflows and Nanoflows," Springer-Verlag, 2005. Google Scholar

[15]

S. Yu. Krylov, A. V. Prosyanov and J. J. M. Beenakker, One dimensional surface diffusion. II. Density dependence in a corrugated potential, J. Chem. Phys., 107 (1997), 6970-6979. doi: 10.1063/1.474937.  Google Scholar

[16]

S. Yu. Krylov, Molecular transport in sub-nano-scale systems, in "Rarefied Gas Dynamics,'' A. D. Ketsdever and E. P. Muntz (eds.), Am. Inst. Phys., (2002). Google Scholar

[17]

Y. Sone, "Kinetic Theory and Fluid Dynamics," Birkäuser, 2002. Google Scholar

[18]

Y. Sone, "Molecular Gas Dynamics: Theory, Techniques, and Applications," Birkäuser, 2007. Google Scholar

show all references

References:
[1]

K. Aoki and P. Degond, Homogenization of a flow in a periodic channel of small section, Multiscale Model. Simul., 1 (2003), 304-334. doi: 10.1137/S1540345902409931.  Google Scholar

[2]

K. Aoki, P. Degond, S. Takata and H. Yoshida, Diffusion models for Knudsen compressors, Phys. Fluids, 19 (2007), 117103: 1-20. Google Scholar

[3]

J. J. M. Beenakker, Reduced dimensionality in gases in nanopores, Phys. Low-Dim. Struct., 10/11 (1995), 115-124. Google Scholar

[4]

J. J. M. Beenakker and S. Yu. Krylov, One-dimensional surface diffusion: density dependence in a smooth potential, J. Chem. Phys., 107 (1997), 4015-4023. doi: 10.1063/1.474757.  Google Scholar

[5]

J. J. M. Beenakker, V. D. Borman and S. Yu. Krylov, Molecular transport in the nanometer regime, Phys. Rev. Lett., 72 (1994), 514-517. doi: 10.1103/PhysRevLett.72.514.  Google Scholar

[6]

J. J. M. Beenakker, V. D. Borman and S. U. Krylov, Molecular transport in subnanometer pores: Zero-point energy, reduced dimensionality and quantum sieving, Chem. Phys. Letters, 232 (1995), 379-382. doi: 10.1016/0009-2614(94)01372-3.  Google Scholar

[7]

V. D. Borman, S. Yu. Krylov and A. V. Prosyanov, Theory of nonequilibrium phenomena at a gas-solid interface, Sov. Phys. JETP, 67 (1988), 2110-2121. Google Scholar

[8]

V. D. Borman, S. Yu. Krylov and A. V. Prosyanov, Fundamental role of unbound surface particles in transport phenomena along a gas-solid interface, Sov. Phys. JETP, 70 (1990), 1013-1022. Google Scholar

[9]

P. Charrier and B. Dubroca, Asymptotic transport models for heat and mass transfer in reactive porous media, Multiscale Model. Simul., 2 (2003), 124-157. doi: 10.1137/S1540345902411736.  Google Scholar

[10]

P. Degond, Transport of trapped particles in a surface potential, in "Nonlinear Partial Differential Equations and Their Applications," Collège de France Seminar, North Holland, XIV (2002), 273-296. Google Scholar

[11]

P. Degond, C. Parzani and M.-H. Vignal, A Boltzmann model for trapped particles in a surface potential, Multiscale Model. Simul., 5 (2006), 364-392. doi: 10.1137/050642897.  Google Scholar

[12]

E. Frenod and K. Hamdache, Homogenisation of transport kinetic equations with oscillating potentials, Proceedings of the Royal Society of Edinburgh A, 126 (1996), 1247-1275.  Google Scholar

[13]

J. K. Holt, H. G. Park, Y. Wang, M. Stadermann, A. B. Artyukhin, C. P. Grigoropoulos, A. Noy and O. Bakajin, Fast mass transport sub-2-nanometer carbon nanotubes, Science, 312 (2006), 1034-1037. doi: 10.1126/science.1126298.  Google Scholar

[14]

G. Karniadakis, A. Beskok and N. Aluru, "Microflows and Nanoflows," Springer-Verlag, 2005. Google Scholar

[15]

S. Yu. Krylov, A. V. Prosyanov and J. J. M. Beenakker, One dimensional surface diffusion. II. Density dependence in a corrugated potential, J. Chem. Phys., 107 (1997), 6970-6979. doi: 10.1063/1.474937.  Google Scholar

[16]

S. Yu. Krylov, Molecular transport in sub-nano-scale systems, in "Rarefied Gas Dynamics,'' A. D. Ketsdever and E. P. Muntz (eds.), Am. Inst. Phys., (2002). Google Scholar

[17]

Y. Sone, "Kinetic Theory and Fluid Dynamics," Birkäuser, 2002. Google Scholar

[18]

Y. Sone, "Molecular Gas Dynamics: Theory, Techniques, and Applications," Birkäuser, 2007. Google Scholar

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