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UNIVERSITÉ D’AIX MARSEILLE ÉCOLE DOCTORAL ED353 SCIENCES POUR L’INGÉNIEUR : MÉCANIQUE, PHYSIQUE, MICRO ET NANOELÉCTRONIQUE THÈSE DE DOCTORAT pour obtenir le grade de Docteur de l’Université d’Aix Marseille Specialté : Mécanique et Énergétique Mustafa HADJ NACER Tangential Momentum Accommodation Coefficient in Microchannels with Different Surface Materials (measurements and simulations) préparé au laboratoire IUSTI, UMR CNRS 7343, Marseille Soutenue publiquement devant le jury composé de: Rapporteurs Stéphane Colin - Université de Toulouse David Newport - University of Limerick Éxaminateurs Lounes Tadrist - Université d’Aix-Marseille Guy Lauriat - Université de Paris-Est Hiroki Yamaguchi - University of Nagoya Martin Wuest - INFICON, Liechtenstein. Directeur de thèse Irina Graur - Université d’Aix-Marseille Co-directeur Pierre Perrier - Université d’Aix-Marseille Invités Amor Bouhdjar - CDER- Alger Gilbert Méolans - Université d’Aix-Marseille 17 Décembre 2012 Tangential Momentum Accommodation Coefficient in Microchannels with Different Surface Materials (measurements and simulations) Mustafa HADJ NACER UNIVERSITY OF AIX MARSEILLE DOCTORAL SCHOOL ED353 THESIS submitted to obtain the degree of Doctor of Philosophy of the University of Aix Marseille Specialty : Mechanical engineering defended on December 17, 2012 to my Mother and Father, to my sister and brothers, and especially to my wife "Soumia". Mustafa Acknowledgments First of all I want to thank "Allah" for giving me the courage and the strength to complete this thesis. Then, I would like to take this opportunity to thank and express my sincere gratitude to my supervisor Professor Irina Graur and my co-supervisor Pierre Perrier, without forgetting J. Gilbert Méolans, for giving me the opportunity to do a research degree in a project where I feel very lucky to be a part of, and for their experts’ guidance and support. I would also thank Doctor Martin Wüest for receiving me within the Company INFICON, Liechtenstein for a period of two months and Professor Julio Croce for receiving within the Universita degli studi di Udine (UNIUD), Italy for a period of six months. On this occasion I do not forget to thank my previous professors Philippe Bournot, Olivier Vauquelin, George Le Palec, Amor Boudjar and Bouzid Benkoussas who were the cause to come to France and to start this thesis. I am also deeply grateful to the team DTF and all the personnel of the IUSTI Laboratory, with whom I developed good professional relationships and friendships. Most of all, I thank Vincent Pavan, Jean Luc Frippo, Tadrist Lounès, Lazhar Houas, Georges Jourdan, Christian Mariani, Yann Jobic, Jeanne Pullino for their kindness and helps. I thank also my office mates: Alice Chauvin, Mariusz Wozniak, Alexey Polikarpov, Charely André, Ali Dinler, Laurent Biamino, Minh Tuan Ho and special thank to my friends Abdelafour Zaabout, Mohammed Drissi, Salim Zeguai. Many thanks go to my housemates over the last three years Hadj Youb Bouras, Mo- hammed Bahachou, Mahfoud Bakli and Brahim Bazamlel for their invaluable friendship and help, and many thanks also to all the community of Tawat of Marseille, especially Seddik Ben Yahia, Bahmed Zaabi, Mohammed and Mustapha Ben Drissou and Bamoune Abderahmane and to all the members of my family in Marseille. I thank all of the various support provided me by my family. To my parents for their love and encouragement, to my brothers and sister for their love and friendship, thank you all. Finally, of all the contributions, my wife’s stands out as the most meaningful. Her understanding and steadfast love have always been a source of profound tranquility during this sometimes hectic journey. Thank you Soumia. The research leading to these results has received funding from the European Community’s Seventh Framework Program (ITN - FP7/2007-2013) under grant agreement n◦ 215504. Abstract This thesis is devoted to the study of rarefied gas flows through micro-channels of various cross sections (circular and rectangular) under isothermal and stationary conditions. The objective of this thesis is to contribute to the study of gas-surface interaction by determining the tangential momentum accommodation coefficient for different surface materials (gold, silica, stainless steel and Sulfinert) and associated to various gases (helium, nitrogen, argon and carbon-dioxide). To achieve this goal three aspects are considered: experimental, theoretical and numerical. The experimental aspect is considered by measuring the mass flow rate through microchannels using the constant volume technique. The theoretical aspect is considered by the development of a new approach based on the Stokes equations. This approach yields to the analytical expression of the mass flow rate in the slip regime, which takes into account the second order effects. The last aspect, numerical, is considered by the numerical simulations of the mass flow rate in the transitional and free molecular flow regimes by solving the linearized BGK kinetic model. The comparison between the measured mass flow rates and the analytically expressions in the slip regime or with the results of numerical simulations in the transitional and free molecular regimes enabled to deduce the tangential momentum accommodation coefficients corresponding to each pair gas-surface in all flow regimes. Keywords: Rarefied flow, mass flow rate, accommodation coefficient, kinetic model, continuum model. Résumé Cette thèse est consacrée à l’étude des écoulements de gaz raréfiés à travers divers micro- conduits de type circulaire et rectangulaire dans des conditions isotherme et stationnaire. L’objectif de la thèse est de contribuer à l’étude de l’interaction gaz-surface notamment en déterminant le coefficient d’accommodation de la quantité de mouvement pour différent matériaux de surface (Or, Silice, Acier inoxydable et Sulfinert) associés à différents types de gaz (hélium, azote, argon et dioxyde-de-carbone). Afin d’atteindre cet objectif, on adopte un triple point de vue : expérimental, théorique et numérique. L’aspect expérimental est réalisé par des mesures de débit massique à travers les micro-conduits, en utilisant la méthode dite « à volume constant ». L’aspect théorique original est développé à travers une nouvelle approche basée sur la résolution de l’équation de Stokes. Cette approche a permis d’écrire une expression analytique de débit massique en régime de glissement, qui prenne en compte les effets bidimensionnels dans une section de conduit rectangulaire. Cette approche complètement explicite, est conduite au deuxième ordre. Enfin l’aspect numérique permet de calculer le débit massique, en régimes transitionnel et moléculaire libre, en résolvant numériquement l’équation cinétique BGK linéarisée. La comparaison des mesures de débit massique avec l’équation analytique, en régime de glissement, ou avec les calculs numériques, en régimes transitionnel et moléculaire libre, nous a permis de déduire des coefficients de glissement et les coefficients d’accommodation correspondant à chaque couple gaz-surface dans tous les régimes de raréfaction. Keywords: Écoulement raréfié, débit massique, coefficient d’accommodation, modèle cinétique, modèle continu. Contents 1 Introduction1 1.1 Flow regimes....................................1 1.2 Tangential momentum accommodation coefficient (TMAC)..........4 1.3 Aims and structure of the thesis.........................5 2 Microchannels fabrication7 2.1 Lithography.....................................8 2.2 RIE etching..................................... 10 2.3 Wafer coating.................................... 11 2.3.1 Oxidation.................................. 11 2.3.2 Physical Vapor Deposition (PVD).................... 11 2.4 Wafer bonding................................... 12 2.5 Some problems encountered with the microchannels.............. 12 2.5.1 Microchannels Fabrication Summary.................. 13 2.6 Technical characteristics of the microchannels.................. 15 2.6.1 Rectangular cross-section microchannels................. 15 2.6.2 Circular cross-section microchannels................... 18 3 Analytical and Numerical Modeling 21 3.1 Statement of the problem............................. 22 3.2 Continuum and slip regimes............................ 23 3.2.1 Different expressions of the velocity slip coefficient........... 25 3.2.2 Conservation equation........................... 30 3.2.3 Flow between two parallel plates..................... 30 3.2.4 First order velocity slip condition for rectangular cross-section channels 31 3.2.5 Bi-dimensional approach developed in this thesis............ 34 3.2.6 The theoretical bases of our approach.................. 35 3.2.7 Analysis of the boundary conditions. Choice of an orthogonal function set...................................... 37 3.2.8 Search for the reduced velocity...................... 38 3.2.9 Expansion method............................. 40 3.2.10 Implementation of the expansion method................ 41 3.2.11 Mass flow rate calculation......................... 43 3.2.12 Comparison with other methods..................... 44 3.3 Transitional and free molecular regimes..................... 45 3.3.1 Problem formulation............................ 47 3.3.2 Linearized BGK model........................... 48 3.3.3 Discrete velocity method.......................... 51 3.3.4 Numerical results.............................. 53 xii Contents 4 Description of the experimental approach 61 4.1 Experiments in TMAC............................... 61 4.1.1 Molecular beam technique......................... 62 4.1.2
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