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June  2015, 8(2): 235-254. doi: 10.3934/krm.2015.8.235

A kinetic theory description of liquid menisci at the microscale

Paolo Barbante 1, , Aldo Frezzotti 2, and  Livio Gibelli 2,

1. 

Politecnico di Milano, MOX, Dipartimento di Matematica, Piazza Leonardo da Vinci, 32, 20133 Milan, Italy
2. 

Politecnico di Milano, Dipartimento di Scienze e Tecnologie Aerospaziali, Via La Masa 34, 20156 Milan, Italy, Italy

Received  July 2014 Revised  December 2014 Published  March 2015

A kinetic model for the study of capillary flows in devices with microscale geometry is presented. The model is based on the Enskog-Vlasov kinetic equation and provides a reasonable description of both fluid-fluid and fluid-wall interactions. Numerical solutions are obtained by an extension of the classical Direct Simulation Monte Carlo (DSMC) to dense fluids. The equilibrium properties of liquid menisci between two hydrophilic walls are investigated and the validity of the Laplace-Kelvin equation at the microscale is assessed. The dynamical process which leads to the meniscus breakage is clarified.
Keywords: two-phase flow., Enskog-Vlasov equation, microchannels, surface tension, liquid meniscus.
Mathematics Subject Classification: Primary: 82C40, 82C26; Secondary: 82C8.
Citation: Paolo Barbante, Aldo Frezzotti, Livio Gibelli. A kinetic theory description of liquid menisci at the microscale. Kinetic and Related Models, 2015, 8 (2) : 235-254. doi: 10.3934/krm.2015.8.235
References:
[1]

M. Allen and D. Tildesley, Computer Simulation of Liquids, Clarendon Press, 1989.

[2]

R. Ardito, A. Corigliano and A. Frangi, Multiscale finite element models for predicting spontaneous adhesion in MEMS, Mecanique Industries, 11 (2010), 177-182. doi: 10.1051/meca/2010028.

[3]

P. Barbante, A. Frezzotti, L. Gibelli and D. Giordano, A kinetic model for collisional effects in dense adsorbed gas layers, in Proceedings of the 27th International Symposium on Rarefied Gas Dynamics (eds. I. Wysong and A. Garcia), AIP Conference Proceedings, Vol. 1333, 2011, 458-463. doi: 10.1063/1.3562690.

[4]

P. Barbante, A. Frezzotti, L. Gibelli, P. Legrenzi, A. Corigliano and A. Frangi, A kinetic model for capillary flows in MEMS, in Proceedings of the 28th International Symposium on Rarefied Gas Dynamics (eds. M. Mareschal and A. Santos), AIP Conference Proceedings, Vol. 1501, 2012, 713-719. doi: 10.1063/1.4769612.

[5]

G. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1995.

[6]

N. Carnahan and K. Starling, Equation of state for nonattracting rigid spheres, J. Chem. Phys., 51 (1969), 635-636. doi: 10.1063/1.1672048.

[7]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer, Berlin, 1988. doi: 10.1007/978-1-4612-1039-9.

[8]

S. Cheng and M. Robbins, Capillary adhesion at the nanometer scale, Phys. Rev. E, 89 (2014), 062402. doi: 10.1103/PhysRevE.89.062402.

[9]

J. Eggers, Nonlinear dynamics and breakup of free-surface flows, Reviews of Modern Physics, 69 (1997), 865-929. doi: 10.1103/RevModPhys.69.865.

[10]

D. Enskog, Kinetische theorie der wärmeleitung, reibung und selbstdiffusion in gewissen verdichteten gasen und flüssigkeiten, K. Svensk. Vet. Akad. Handl., 63 (1922), 5-44.

[11]

J. Fischer and M. Methfessel, Born-Green-Yvon approach to the local densities of a fluid at interfaces, Phys. Rev. A, 22 (1980), 2836-2843. doi: 10.1103/PhysRevA.22.2836.

[12]

A. Frezzotti, A particle scheme for the numerical solution of the Enskog equation, Phys. Fluids, 9 (1997), 1329-1335. doi: 10.1063/1.869247.

[13]

A. Frezzotti and L. Gibelli, A kinetic model for equilibrium and non-equilibrium structure of the vapor-liquid interface, in Proceedings of the 23rd International Symposium on Rarefied Gas Dynamics (eds. A. Ketsdever and E. Muntz), AIP Conference Proceedings, Vol. 663, 2003, 980-987. doi: 10.1063/1.1581646.

[14]

A. Frezzotti and L. Gibelli, A kinetic model for fluid wall interaction, Proc. IMechE, Part C: J. Mech. Eng. Science, 222 (2008), 787-795. doi: 10.1243/09544062JMES718.

[15]

A. Frezzotti, L. Gibelli and S. Lorenzani, Mean field kinetic theory description of evaporation of a fluid into vacuum, Phys. Fluids, 17 (2005), 012102. doi: 10.1063/1.1824111.

[16]

A. Frezzotti, S. Nedea, A. Markvoort, P. Spijker and L. Gibelli, Comparison of molecular dynamics and kinetic modeling of gas-surface interaction, in Proceedings of the 26th International Symposium on Rarefied Gas Dynamics (ed. T. Abe), AIP Conference Proceedings, Vol. 1084, 2008, 635-640. doi: 10.1063/1.3076554.

[17]

M. Grmela, Kinetic equation approach to phase transitions, J. Stat. Phys., 3 (1971), 347-364. doi: 10.1007/BF01011389.

[18]

Z. Guo, T. Zhao and Y. Shi, Simple kinetic model for fluid flows in the nanometer scale, Phys. Rev. E, 71 (2005), 035301(R). doi: 10.1103/PhysRevE.71.035301.

[19]

J. Hansen and I. McDonald, Theory of Simple Liquids, Academic Press, 2006.

[20]

A. Hariri, J. Zu, J. Zu and R. B. Mrad, Modeling of wet stiction in microelectromechanical systems MEMS, J. Microelectromech. Syst., 16 (2007), 1276-1285. doi: 10.1109/JMEMS.2007.904349.

[21]

J. Hirschfelder, C. Curtiss and R. Bird, The Molecular Theory of Gases and Liquids, Wiley-Interscience, 1964.

[22]

W. Kang and U. Landman, Universality crossover of the pinch-off shape profiles of collapsing liquid nanobridges in vacuum and gaseous environments, Physical Review Letters, 98 (2007), 064504. doi: 10.1103/PhysRevLett.98.064504.

[23]

J. Karkheck and G. Stell, Mean field kinetic theories, J. Chem. Phys., 75 (1981), 1475-1487. doi: 10.1063/1.442154.

[24]

G. Karniadakis, A. Beskok and A. Narayan, Microflows and Nanoflows: Fundamentals and Simulation, Springer, Berlin, 2005.

[25]

R. Maboudian and R. Howe, Critical review: Stiction in surface micromechanical structures, J. Vac. Sci. Technol. B, 15 (1997), 1-19.

[26]

J. Rowlinson and B. Widom, Molecular Theory of Capillarity, Dover Pubns, 2003.

[27]

H. van Beijeren and M. Ernst, The modified Enskog equation, Physica, 68 (1973), 437-456.

show all references

References:
[1]

M. Allen and D. Tildesley, Computer Simulation of Liquids, Clarendon Press, 1989.

[2]

R. Ardito, A. Corigliano and A. Frangi, Multiscale finite element models for predicting spontaneous adhesion in MEMS, Mecanique Industries, 11 (2010), 177-182. doi: 10.1051/meca/2010028.

[3]

P. Barbante, A. Frezzotti, L. Gibelli and D. Giordano, A kinetic model for collisional effects in dense adsorbed gas layers, in Proceedings of the 27th International Symposium on Rarefied Gas Dynamics (eds. I. Wysong and A. Garcia), AIP Conference Proceedings, Vol. 1333, 2011, 458-463. doi: 10.1063/1.3562690.

[4]

P. Barbante, A. Frezzotti, L. Gibelli, P. Legrenzi, A. Corigliano and A. Frangi, A kinetic model for capillary flows in MEMS, in Proceedings of the 28th International Symposium on Rarefied Gas Dynamics (eds. M. Mareschal and A. Santos), AIP Conference Proceedings, Vol. 1501, 2012, 713-719. doi: 10.1063/1.4769612.

[5]

G. Bird, Molecular Gas Dynamics and the Direct Simulation of Gas Flows, Clarendon Press, Oxford, 1995.

[6]

N. Carnahan and K. Starling, Equation of state for nonattracting rigid spheres, J. Chem. Phys., 51 (1969), 635-636. doi: 10.1063/1.1672048.

[7]

C. Cercignani, The Boltzmann Equation and Its Applications, Springer, Berlin, 1988. doi: 10.1007/978-1-4612-1039-9.

[8]

S. Cheng and M. Robbins, Capillary adhesion at the nanometer scale, Phys. Rev. E, 89 (2014), 062402. doi: 10.1103/PhysRevE.89.062402.

[9]

J. Eggers, Nonlinear dynamics and breakup of free-surface flows, Reviews of Modern Physics, 69 (1997), 865-929. doi: 10.1103/RevModPhys.69.865.

[10]

D. Enskog, Kinetische theorie der wärmeleitung, reibung und selbstdiffusion in gewissen verdichteten gasen und flüssigkeiten, K. Svensk. Vet. Akad. Handl., 63 (1922), 5-44.

[11]

J. Fischer and M. Methfessel, Born-Green-Yvon approach to the local densities of a fluid at interfaces, Phys. Rev. A, 22 (1980), 2836-2843. doi: 10.1103/PhysRevA.22.2836.

[12]

A. Frezzotti, A particle scheme for the numerical solution of the Enskog equation, Phys. Fluids, 9 (1997), 1329-1335. doi: 10.1063/1.869247.

[13]

A. Frezzotti and L. Gibelli, A kinetic model for equilibrium and non-equilibrium structure of the vapor-liquid interface, in Proceedings of the 23rd International Symposium on Rarefied Gas Dynamics (eds. A. Ketsdever and E. Muntz), AIP Conference Proceedings, Vol. 663, 2003, 980-987. doi: 10.1063/1.1581646.

[14]

A. Frezzotti and L. Gibelli, A kinetic model for fluid wall interaction, Proc. IMechE, Part C: J. Mech. Eng. Science, 222 (2008), 787-795. doi: 10.1243/09544062JMES718.

[15]

A. Frezzotti, L. Gibelli and S. Lorenzani, Mean field kinetic theory description of evaporation of a fluid into vacuum, Phys. Fluids, 17 (2005), 012102. doi: 10.1063/1.1824111.

[16]

A. Frezzotti, S. Nedea, A. Markvoort, P. Spijker and L. Gibelli, Comparison of molecular dynamics and kinetic modeling of gas-surface interaction, in Proceedings of the 26th International Symposium on Rarefied Gas Dynamics (ed. T. Abe), AIP Conference Proceedings, Vol. 1084, 2008, 635-640. doi: 10.1063/1.3076554.

[17]

M. Grmela, Kinetic equation approach to phase transitions, J. Stat. Phys., 3 (1971), 347-364. doi: 10.1007/BF01011389.

[18]

Z. Guo, T. Zhao and Y. Shi, Simple kinetic model for fluid flows in the nanometer scale, Phys. Rev. E, 71 (2005), 035301(R). doi: 10.1103/PhysRevE.71.035301.

[19]

J. Hansen and I. McDonald, Theory of Simple Liquids, Academic Press, 2006.

[20]

A. Hariri, J. Zu, J. Zu and R. B. Mrad, Modeling of wet stiction in microelectromechanical systems MEMS, J. Microelectromech. Syst., 16 (2007), 1276-1285. doi: 10.1109/JMEMS.2007.904349.

[21]

J. Hirschfelder, C. Curtiss and R. Bird, The Molecular Theory of Gases and Liquids, Wiley-Interscience, 1964.

[22]

W. Kang and U. Landman, Universality crossover of the pinch-off shape profiles of collapsing liquid nanobridges in vacuum and gaseous environments, Physical Review Letters, 98 (2007), 064504. doi: 10.1103/PhysRevLett.98.064504.

[23]

J. Karkheck and G. Stell, Mean field kinetic theories, J. Chem. Phys., 75 (1981), 1475-1487. doi: 10.1063/1.442154.

[24]

G. Karniadakis, A. Beskok and A. Narayan, Microflows and Nanoflows: Fundamentals and Simulation, Springer, Berlin, 2005.

[25]

R. Maboudian and R. Howe, Critical review: Stiction in surface micromechanical structures, J. Vac. Sci. Technol. B, 15 (1997), 1-19.

[26]

J. Rowlinson and B. Widom, Molecular Theory of Capillarity, Dover Pubns, 2003.

[27]

H. van Beijeren and M. Ernst, The modified Enskog equation, Physica, 68 (1973), 437-456.

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