# American Institute of Mathematical Sciences

2014, 7(2): 219-251. doi: 10.3934/krm.2014.7.219

## Gas-surface interaction and boundary conditions for the Boltzmann equation

 1 Institut de Mathématiques de Bordeaux, ENSEIRB-MATMECA, IPB, Université de Bordeaux, F-33405 Talence cedex, France, France 2 Institut de Mathméatiques de Bordeaux, ENSEIRB-MATMECA, IPB, Université de Bordeaux, F-33405 Talence cedex

Received  May 2013 Revised  February 2014 Published  March 2014

In this paper we revisit the derivation of boundary conditions for the Boltzmann Equation. The interaction between the wall atoms and the gas molecules within a thin surface layer is described by a kinetic equation introduced in [10] and used in [1]. This equation includes a Vlasov term and a linear molecule-phonon collision term and is coupled with the Boltzmann equation describing the evolution of the gas in the bulk flow. Boundary conditions are formally derived from this model by using classical tools of kinetic theory such as scaling and systematic asymptotic expansion. In a first step this method is applied to the simplified case of a flat wall. Then it is extented to walls with nanoscale roughness allowing to obtain more complex scattering patterns related to the morphology of the wall. It is proved that the obtained scattering kernels satisfy the classical imposed properties of non-negativeness, normalization and reciprocity introduced by Cercignani [13].
Citation: Stéphane Brull, Pierre Charrier, Luc Mieussens. Gas-surface interaction and boundary conditions for the Boltzmann equation. Kinetic & Related Models, 2014, 7 (2) : 219-251. doi: 10.3934/krm.2014.7.219
##### References:
 [1] K. Aoki, P. Charrier and P. Degond, A hierarchy of models related to nanoflows and surface diffusion,, Kinetic and Related Models, 4 (2011), 53. doi: 10.3934/krm.2011.4.53. [2] K. Aoki and P. Degond, Homogenization of a flow in a periodic channel of small section,, Multiscale Model. Simul., 1 (2003). doi: 10.1137/S1540345902409931. [3] K. Aoki, P. Degond, S. Takata and H. Yoshida, Diffusion models for Knudsen compressors,, Phys. Fluids, 19 (2007). doi: 10.1063/1.2798748. [4] G. Arya, H.-C. Chang and E. Magin, Knudsen Diffusivity of a Hard Sphere in a Rough Slit Pore,, Phys. Rev. Lett., 91 (2003). doi: 10.1103/PhysRevLett.91.026102. [5] H. Babovsky, Derivation of stochastic reflection laws from specular reflection,, Trans. Th. and Stat. Phys., 16 (1987), 113. doi: 10.1080/00411458708204299. [6] J. J. M. Beenakker, Reduced Dimensionality in Gases in Nanopores,, Phys. Low-Dim. Struct., (1995), 115. [7] J. J. M. Beenakker, V. D. Borman and S. Yu Krylov, Molecular Transport in the Nanometer Regime,, Phys. Rev. Lett., 72 (1994). doi: 10.1103/PhysRevLett.72.514. [8] J. J. M. Beenakker, V. D. Borman and S. Yu. Krylov1, Molecular transport in subnanometer pores: Zero-point energy, reduced dimensionality and quantum sieving,, Chem. Phys. Letters, 232 (1995), 379. doi: 10.1016/0009-2614(94)01372-3. [9] J. J. M. Beenakker and S. Yu. Krylov, One-dimensional surface diffusion: Density dependence in a smooth potential,, J. Chem. Phys., 107 (1997). doi: 10.1063/1.474757. [10] V. D. Borman, S. Yu. Krylov and A. V. Prosyanov, Theory of nonequilibrium phenomena at a gas-solid interface,, Sov. Phys. JETP, 67 (1988). [11] 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). [12] F. Celestini and F. Mortessagne, The cosine law at the atomic scale: Toward realistic simulations of Knudsen diffusion,, Phys.Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.021202. [13] C. Cercignani, The Boltzman Equation and Its Applications,, Springer, (1988). doi: 10.1007/978-1-4612-1039-9. [14] C. Cercignani, Scattering kernels for gas-surface interactions,, Transp. Th. and Stat. Phys., 2 (1972), 27. [15] C. Cercignani, Scattering kernels for gas-surface interaction,, Proceedings of the Workshop on Hypersonic Flows for Reentry Problems, I (1990), 9. [16] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Springer: New York, (1994), 133. [17] C. Cercignani and M. Lampis, Kinetic models for gas-surface interactions,, Transp. Th. and Stat. Phys., 1 (1971), 101. doi: 10.1080/00411457108231440. [18] C. Cercignani, M. Lampis and A. Lentati, A new scattering kernel in kinetic theory of gases,, Trans. Theory Statist. Phys., 24 (1995), 1319. doi: 10.1080/00411459508206026. [19] S. Chandrasekhar, Radiative Transfer,, Dover Publications, (1960). [20] P. Charrier and B. Dubroca, Asymptotic transport models for heat and mass transfer in reactive porous media,, Multiscale Model. Simul., 2 (2003). doi: 10.1137/S1540345902411736. [21] F. Coron, F. Golse and C. Sulem, A classification of well-posed kinetic layer problems,, Commun. Pure Appl. Math., 41 (1988). doi: 10.1002/cpa.3160410403. [22] P. Degond, Transport of trapped particles in a surface potential,, in Nonlinear Partial Differential Equations and Their Applications, XIV (2002), 273. doi: 10.1016/S0168-2024(02)80014-5. [23] P. Degond and S. Mas-Gallic, Existence of solutions and diffusion approximation for a model Fokker-Planck equation,, Transport Theory Statist. Phys., 16 (1987), 589. doi: 10.1080/00411458708204307. [24] P. Degond, C. Parzani and M.-H. Vignal, A Boltzmann Model for Trapped Particles in a Surface Potential,, SIAM J. Multiscale model. Simul., 5 (2006), 364. doi: 10.1137/050642897. [25] L. Falk, Existence of solutions to the stationary linear boltzmann equation,, Transport Theory and Statistical Physics, 32 (2003), 37. doi: 10.1081/TT-120018651. [26] F. Golse, Knudsen layers from a computational viewpoint,, Transport Theory Statist. Phys., 21 (1992), 211. doi: 10.1080/00411459208203921. [27] F. Golse and A. Klar, A numerical method for computing asymptotic states and outgoing distributions for kinetic linear Half-Space problems,, J. of Sta. Physics, 80 (1995). doi: 10.1007/BF02179863. [28] G. Karniadakis, A. Beskok and N. Aluru, Microflows and Nanoflows,, Springer, (2005). [29] A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift-diffusion equations,, SIAM J. Sci. Comput., 19 (1998), 2032. doi: 10.1137/S1064827595286177. [30] 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). doi: 10.1063/1.474937. [31] S. Yu Krylov, Molecular transport in Sub-Nano-Scale systems,, RGD, 663 (2003). doi: 10.1063/1.1581616. [32] J. C. Maxwell, On stresses in rarefied gases arising from inequalities of temperature,, Phil. Trans. Royal Soc., (1879), 231. [33] F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers,, Asymptotic Anal., 4 (1991), 293. [34] Y. Sone, Kinetic Theory and Fluid Dynamics,, Birkäuser, (2002). doi: 10.1007/978-1-4612-0061-1. [35] Y. Sone, Molecular Gas Dynamics,, Birkäuser, (2007). doi: 10.1007/978-0-8176-4573-1. [36] H. Struchtrup, Maxwell boundary condition and velocity dependent accommodation coefficients,, Phys. Fluids, 25 (2013). doi: 10.1063/1.4829907.

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##### References:
 [1] K. Aoki, P. Charrier and P. Degond, A hierarchy of models related to nanoflows and surface diffusion,, Kinetic and Related Models, 4 (2011), 53. doi: 10.3934/krm.2011.4.53. [2] K. Aoki and P. Degond, Homogenization of a flow in a periodic channel of small section,, Multiscale Model. Simul., 1 (2003). doi: 10.1137/S1540345902409931. [3] K. Aoki, P. Degond, S. Takata and H. Yoshida, Diffusion models for Knudsen compressors,, Phys. Fluids, 19 (2007). doi: 10.1063/1.2798748. [4] G. Arya, H.-C. Chang and E. Magin, Knudsen Diffusivity of a Hard Sphere in a Rough Slit Pore,, Phys. Rev. Lett., 91 (2003). doi: 10.1103/PhysRevLett.91.026102. [5] H. Babovsky, Derivation of stochastic reflection laws from specular reflection,, Trans. Th. and Stat. Phys., 16 (1987), 113. doi: 10.1080/00411458708204299. [6] J. J. M. Beenakker, Reduced Dimensionality in Gases in Nanopores,, Phys. Low-Dim. Struct., (1995), 115. [7] J. J. M. Beenakker, V. D. Borman and S. Yu Krylov, Molecular Transport in the Nanometer Regime,, Phys. Rev. Lett., 72 (1994). doi: 10.1103/PhysRevLett.72.514. [8] J. J. M. Beenakker, V. D. Borman and S. Yu. Krylov1, Molecular transport in subnanometer pores: Zero-point energy, reduced dimensionality and quantum sieving,, Chem. Phys. Letters, 232 (1995), 379. doi: 10.1016/0009-2614(94)01372-3. [9] J. J. M. Beenakker and S. Yu. Krylov, One-dimensional surface diffusion: Density dependence in a smooth potential,, J. Chem. Phys., 107 (1997). doi: 10.1063/1.474757. [10] V. D. Borman, S. Yu. Krylov and A. V. Prosyanov, Theory of nonequilibrium phenomena at a gas-solid interface,, Sov. Phys. JETP, 67 (1988). [11] 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). [12] F. Celestini and F. Mortessagne, The cosine law at the atomic scale: Toward realistic simulations of Knudsen diffusion,, Phys.Rev. E, 77 (2008). doi: 10.1103/PhysRevE.77.021202. [13] C. Cercignani, The Boltzman Equation and Its Applications,, Springer, (1988). doi: 10.1007/978-1-4612-1039-9. [14] C. Cercignani, Scattering kernels for gas-surface interactions,, Transp. Th. and Stat. Phys., 2 (1972), 27. [15] C. Cercignani, Scattering kernels for gas-surface interaction,, Proceedings of the Workshop on Hypersonic Flows for Reentry Problems, I (1990), 9. [16] C. Cercignani, R. Illner and M. Pulvirenti, The Mathematical Theory of Dilute Gases,, Springer: New York, (1994), 133. [17] C. Cercignani and M. Lampis, Kinetic models for gas-surface interactions,, Transp. Th. and Stat. Phys., 1 (1971), 101. doi: 10.1080/00411457108231440. [18] C. Cercignani, M. Lampis and A. Lentati, A new scattering kernel in kinetic theory of gases,, Trans. Theory Statist. Phys., 24 (1995), 1319. doi: 10.1080/00411459508206026. [19] S. Chandrasekhar, Radiative Transfer,, Dover Publications, (1960). [20] P. Charrier and B. Dubroca, Asymptotic transport models for heat and mass transfer in reactive porous media,, Multiscale Model. Simul., 2 (2003). doi: 10.1137/S1540345902411736. [21] F. Coron, F. Golse and C. Sulem, A classification of well-posed kinetic layer problems,, Commun. Pure Appl. Math., 41 (1988). doi: 10.1002/cpa.3160410403. [22] P. Degond, Transport of trapped particles in a surface potential,, in Nonlinear Partial Differential Equations and Their Applications, XIV (2002), 273. doi: 10.1016/S0168-2024(02)80014-5. [23] P. Degond and S. Mas-Gallic, Existence of solutions and diffusion approximation for a model Fokker-Planck equation,, Transport Theory Statist. Phys., 16 (1987), 589. doi: 10.1080/00411458708204307. [24] P. Degond, C. Parzani and M.-H. Vignal, A Boltzmann Model for Trapped Particles in a Surface Potential,, SIAM J. Multiscale model. Simul., 5 (2006), 364. doi: 10.1137/050642897. [25] L. Falk, Existence of solutions to the stationary linear boltzmann equation,, Transport Theory and Statistical Physics, 32 (2003), 37. doi: 10.1081/TT-120018651. [26] F. Golse, Knudsen layers from a computational viewpoint,, Transport Theory Statist. Phys., 21 (1992), 211. doi: 10.1080/00411459208203921. [27] F. Golse and A. Klar, A numerical method for computing asymptotic states and outgoing distributions for kinetic linear Half-Space problems,, J. of Sta. Physics, 80 (1995). doi: 10.1007/BF02179863. [28] G. Karniadakis, A. Beskok and N. Aluru, Microflows and Nanoflows,, Springer, (2005). [29] A. Klar, Asymptotic-induced domain decomposition methods for kinetic and drift-diffusion equations,, SIAM J. Sci. Comput., 19 (1998), 2032. doi: 10.1137/S1064827595286177. [30] 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). doi: 10.1063/1.474937. [31] S. Yu Krylov, Molecular transport in Sub-Nano-Scale systems,, RGD, 663 (2003). doi: 10.1063/1.1581616. [32] J. C. Maxwell, On stresses in rarefied gases arising from inequalities of temperature,, Phil. Trans. Royal Soc., (1879), 231. [33] F. Poupaud, Diffusion approximation of the linear semiconductor Boltzmann equation: Analysis of boundary layers,, Asymptotic Anal., 4 (1991), 293. [34] Y. Sone, Kinetic Theory and Fluid Dynamics,, Birkäuser, (2002). doi: 10.1007/978-1-4612-0061-1. [35] Y. Sone, Molecular Gas Dynamics,, Birkäuser, (2007). doi: 10.1007/978-0-8176-4573-1. [36] H. Struchtrup, Maxwell boundary condition and velocity dependent accommodation coefficients,, Phys. Fluids, 25 (2013). doi: 10.1063/1.4829907.
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