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 Spinning rotor gauge technique...... 62 4.1.3 Flow through microchannels technique...... 63 4.1.4 Influence of the surface roughness...... 65 4.1.5 Influence of the surface contamination...... 66 4.1.6 Influence of the temperature...... 66 4.2 Mass flow measurement techniques...... 67 4.2.1 Description of the rise-of-pressure technique...... 69 4.2.2 Procedure...... 72 4.2.3 Leak and outgassing controls...... 73 4.3 Mass flow rate calculation...... 76 4.4 Volume measurement...... 78 4.4.1 Uncertainty on the volume measurement...... 79 4.5 Total uncertainty on mass flow...... 80

5 Results in Microtubes 83 5.1 Continuum and slip regimes...... 83 5.1.1 Extraction of the slip and accommodation coefficients...... 87 5.1.2 Comparison between the HS and the VHS models...... 89 5.1.3 Comparison with other works...... 90 5.2 Transitional and near free molecular regimes...... 92

6 Results in Rectangular Microchannels 97 6.1 Continuum and slip regimes...... 98 6.1.1 Validation of the continuum approach with practical example...... 99 6.1.2 Microchannels A ...... 101 6.1.3 Microchannels E ...... 114 6.1.4 Microchannels S ...... 127 6.2 Transitional and free molecular regimes...... 138 6.2.1 Microchannels A ...... 140 6.2.2 Microchannels E ...... 144 6.2.3 Microchannels S ...... 149

7 General Conclusion and Perspectives 153

A Publications 157

B Unsteady technique 159

Bibliography 161 List of Figures

1 Les régimes d’écoulement suivant le nombre de Knudsen...... xxvi 2 Réflexion spéculaire ou diffuse d’une molécule sur une surface...... xxix

1.1 Flow regimes categorized by Knudsen numbers...... 2 1.2 Operation regimes for MEMS and nano-devices under standard conditions of pressure and temperature [13]...... 3 1.3 Gas/surface interaction...... 5

2.1 Schematic representation of the microchannels published by Colin et al. 2004 [30] (left side) and Hsieh et al. 2004 [52] (right side) used for the mass flow rate experiments...... 8 2.2 Lithography steps: (a) resin deposition, (b) spinning, (c) resin layer...... 9 2.3 Photoresist exposition to U.V. rays...... 9 2.4 Lithography process: (1) Silicon wafer;(2) Photoresist deposit;(3) Exposure to U.V. beams;(4) Development...... 10 2.5 DRIE technique (Bosch process)...... 11 2.6 Microchannel bonding with a gold intermediate layer...... 12 2.7 Pictures showing the misalignment (a) and the gold layer takeoff (b) problems. 13 2.8 Rectangular microchannels fabrication process summary...... 14 2.9 Entrance section of microchannels A1 and A2. Left picture A1, right picture A2. 16 2.10 Image processing. Left side: zoom inside the microchannel. Right side: defi- nition of the image edges using ImageJ software...... 16 2.11 Graph of the roughness for the microchannel S3...... 17

3.1 Rectangular and circular microchannels diagram...... 22 3.2 Scheme of finite difference method...... 51 3.3 Comparison between the mass flow rate obtained using the BGK kinetic model and the continuum model for α = 1 and for the channel aspect ratio h/w = 0.5. (BGK) BGK kinetic model, (1)continuum model (first order, σp = 1.016) and (2), (3), (4) continuum model (second order, σ2p = 0.184 [28], σ2p = 0.243 [48], σ2p = 0.766 [23], respectively)...... 55 3.4 Reduced mass flow rate Qch for α = 1 and for the channel aspect ratio h/w that varies from 0.01 to 1...... 60 3.5 Reduced mass flow rate Qch for h/w = 0.5 and for the accommodation coeffi- cient α that varies from 0.6 to 1...... 60

4.1 Schematic representative of the experimental setup...... 71 4.2 Pictures of the fixation system for the rectangular microchannels (a) and mi- crotubes (b)...... 72 4.3 Comparison between the temperature in the outlet tank and the room temper- ature...... 74 xiv List of Figures

4.4 Graph of the pressure rise during the outgassing check. The points represent the measured pressure values and the line is a representative fitting curve using the fourth order polynomial form...... 76 4.5 Simplified sketch of the experimental data used for the volumes measurement. 78 4.6 Schematic representation of data from volume measurement...... 79

exp 5.1 Dimensionless mass flow rate Stu as function of the mean Knudsen number for microtube T 1...... 86 exp 5.2 Dimensionless mass flow rate Stu as function of the mean Knudsen number for microtube T 2...... 86 5.3 Dimensionless mass flow rate G as function of the Knudsen number for micro- tube T 1...... 94 5.4 Dimensionless mass flow rate G as function of the Knudsen number for micro- tube T 2...... 94

6.1 Tangential momentum accommodation coefficients (TMAC) plotted as func- tion of gas obtained with the first and second order approximations in mi- crochannels of the group A...... 107 6.2 Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expres- sions for the microchannels A1...... 110 6.3 Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expres- sions for the microchannels A2...... 111 6.4 Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expres- sions for the microchannels A3...... 112 6.5 Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expres- sions for the microchannels A4...... 113 6.6 Dimensionless mass flow rate Sexp (6.1) as function of the mean Knudsen num- ber obtained using the steady and unsteady methods in the microchannels E3. The upper graph is for helium and the bottom graph is for argon...... 119 6.7 Tangential momentum accommodation coefficients (TMAC) plotted as func- tion of gas obtained with the first and second order approximations in mi- crochannels of the group E...... 122 6.8 Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expres- sions for the microchannels E1...... 123 6.9 Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expres- sions for the microchannels E2...... 124 List of Figures xv

6.10 Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expres- sions for the microchannels E3...... 125 6.11 Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expres- sions for the microchannels E4...... 126 6.12 Tangential momentum accommodation coefficients (TMAC) plotted as func- tion of gas obtained with the first and second order approximations in mi- crochannels of the group S...... 133 6.13 Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expres- sions for the microchannels S1...... 134 6.14 Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expres- sions for the microchannels S2...... 135 6.15 Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expres- sions for the microchannels S3...... 136 6.16 Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expres- sions for the microchannels S4...... 137 6.17 Graph illustrating the error bars on the points of the experimental dimension- less mass flow rate Gexp of helium through the microchannels A3...... 139 6.18 Experimental and numerical dimensionless mass flow rates as function of the mean rarefaction parameter δm, for the microchannels A1 (upper) and A2 (bottom)...... 142 6.19 Experimental and numerical dimensionless mass flow rates as function of the mean rarefaction parameter δm, for the microchannels A3 (upper) and A4 (bottom)...... 143 6.20 Experimental and numerical dimensionless mass flow rates as function of the mean rarefaction parameter δm, for the microchannels E1 (upper) and E2 (bottom)...... 146 6.21 Experimental and numerical dimensionless mass flow rates as function of the mean rarefaction parameter δm, for the microchannels E3 (upper) and E4 (bottom)...... 147 6.22 Experimental and numerical dimensionless mass flow rates as function of the mean rarefaction parameter δm, for the microchannels E1 (upper) and E2 (bottom), obtained by assimilating the rectangular microchannels cross-section to a circular one...... 148 6.23 Experimental and numerical dimensionless mass flow rates as function of the mean rarefaction parameter δm for the microchannels S1 (upper) and S2 (bot- tom)...... 151 xvi List of Figures

6.24 Experimental and numerical dimensionless mass flow rates as function of the mean rarefaction parameter δm for the microchannels S3 (upper) and S4 (bot- tom)...... 152 List of Tables

2.1 Dimensions of the rectangular microchannels...... 17 2.2 Dimension of the circular microchannels. ∗ The length of the microtubes T 1 and T 2 were reduced to 9.96 cm and 10.33 cm, respectively, for the experiments yield with CO2 gas...... 18

3.1 Gas parameters used in the experiments and simulations...... 23

3.2 Theoretical values for the second order coefficient A2 and the velocity slip coef- ficient σ2p as function of the tangential momentum accommodation coefficients given by different authors...... 28

3.3 Theoretical value for the coefficients A1 and A2 given by different authors for fully-diffusive boundary...... 29

3.4 Value of the slip correction Qs(w/h) 3.33 calculated for the aspect ratios (w/h) of the microchannels of the group A...... 33

3.5 Comparison of the slip correction Qs(w/h) [101] with corresponding coefficient 4 of the present approach 96Tn/π ...... 44 3.6 Influence of lateral walls on the mass flow rate through a rectangular channel M/˙ M˙ ∞...... 45 3.7 Parameter of the numerical grid used for the simulations...... 53 3.8 Comparison of the reduced mass flow rate Qch obtained in this thesis with the results of Loyalka et al. 1976 [72] and Sharipov, 1999 [101] for α = 1 and h/w = 1...... 54 3.9 Reduced mass flow rate Qch for accommodation coefficient α = 1...... 57 3.10 Reduced mass flow rate Qch for accommodation coefficient α = 0.95...... 57 3.11 Reduced mass flow rate Qch for accommodation coefficient α = 0.90...... 58 3.12 Reduced mass flow rate Qch for accommodation coefficient α = 0.85...... 58 3.13 Reduced mass flow rate Qch for accommodation coefficient α = 0.80...... 59 3.14 Reduced mass flow rate Qch for accommodation coefficient α = 0.60...... 59

4.1 TMAC value found in literature, measured with the molecular beam technique. 62 4.2 TMAC value found in literature, measured with the Spinning Rotor Gauge (SRG)...... 63 4.3 TMAC value found in literature, calculated from the measurement of the flow through microchannels of various cross-section...... 64 4.4 Technical data for the pressure transducers (CDG, Capacitance Diaphragm Gauge) [53]...... 70 4.5 Results of the inlet and outlet volumes measurements for the experimental setup built in INFICON...... 80

5.1 Experimental conditions for microtubes T 1 and T 2...... 87 xviii List of Tables

exp exp 5.2 Experimental coefficients B0 and B1 obtained from the first order polyno- mial approximation for the microtubes T 1 and T 2...... 88 5.3 Slip and tangential momentum accommodation coefficients obtained for mi- crotubes T 1 and T 2...... 89 5.4 Experimental results obtained from the first order polynomial approximation for the microtube T 1 using the HS model...... 89 5.5 The tangential momentum accommodation coefficient (TMAC) obtained from the present experiments and in Refs. [39, 88, 92]...... 91

T 6.1 The influence of the lateral walls on B1 ...... 100 6.2 Slip and tangential momentum accommodation coefficients extracted from the experimental data [44]. The first two lines ∗ correspond to results [44] using the parallel plate expression (3.20). The second two lines ∗∗ represent the implementation of expression (3.82), which takes the influence of the lateral walls into account...... 100 6.3 Experimental conditions for the microchannels A1 and A2...... 101 6.4 Experimental conditions for the microchannels A3 and A4...... 101 6.5 Fitting coefficients of the dimensionless mass flow rate Sexp (6.1) obtained with the first (first line) and second order (second line) approximations (6.2) and (6.3) in the mean Knudsen number ranges [0, 0.1] and [0, 0.3], respectively, for microchannels of the group A...... 102 6.6 Slip and tangential momentum accommodation coefficients found experimen- tally from the first (first line) and second order (second line) approximations of the dimensionless mass flow rate Sexp (6.2) and (6.3), respectively, for mi- crochannels of the group A...... 104 6.7 Experimental conditions for the microchannels E1 and E2...... 114 6.8 Experimental conditions for the microchannels E3 and E4...... 114 6.9 Fitting coefficients of the dimensionless mass flow rate Sexp (6.1) obtained with the first (first line) and second order (second line) approximations (6.2) and (6.3) in the mean Knudsen number ranges [0.0, 0.1] and [0.0, 0.3], respectively, for microchannels of the group E...... 115 6.10 Slip and tangential momentum accommodation coefficients found experimen- tally from the first (first line) and second order (second line) approximations of the dimensionless mass flow rate Sexp (6.2) and (6.3), respectively, for mi- crochannels of the group E...... 117 6.11 Experimental conditions for the microchannels S1 and S2...... 127 6.12 Experimental conditions for the microchannels S3 and S4...... 127 6.13 Fitting coefficients of the dimensionless mass flow rate Sexp (6.1) obtained with the first (first line) and second order (second line) approximations (6.2) and (6.3) in the mean Knudsen number ranges [0.0, 0.1] and [0.0, 0.3], respectively, for microchannels of the group S...... 129 List of Tables xix

6.14 Slip and tangential momentum accommodation coefficients found experimen- tally from the first (first line) and second order (second line) approximations of the dimensionless mass flow rate Sexp (6.2) and (6.3), respectively, for mi- crochannels of the group S...... 131

Nomenclature

Minuscule a Coefficient of first order polynomial approximation of the pressure [P a/s] b Coefficient of zero order polynomial approximation of the pressure [P a] f Velocity distribution function f 0 Absolute Maxwellian distribution function f M Local Maxwellian distribution function fn(y), gn(y) Expansion expression g Perturbation function h Height of the rectangular microchannels [m] k Boltzmann constant 1.3806503 ×1023 [m2kgs−2K−1] kλ Dimensionless coefficient m Molecular mass [kg] p Pressure [P a] p(ti) First order polynomial approximation of the pressure during the experiment [P a] r Radial coordinate r2 Determination coefficient s Relative variation of the temperature sr Square residual sum t Time [s] ti Time at instant I [s] u Streamwise velocity [m/s] uH hydrodynamic solution us Slip correction w Width of the rectangular microchannels [m] x, y, z Cartesian coordinates

Majuscule

A1 Reduced first order Coefficient A1 = σp/kλ 2 A2 Reduced second order Coefficient A2 = σ2p/kλ B0,1,2 Coefficients of the polynomial approximation D Diameter of the microtubes [m] Es Standard error xxii Nomenclature

G Dimensionless mass flow rate obtained from the kinetic theory K Factor taking into account the lateral walls influence K(v− → v+, r) Scattering kernel for the diffusive-specular model in kinetic theory Kn Knudsen number Knm Mean Knudsen number calculated from the mean pressure between the inlet and outlet tanks L Length of the microdevices [m] L2 Functional vector space Lc Characteristic length of the micro-devices [m] Ls Length of the syringe[pixel] M exp Experimental mass flow rate [kg/s] Mp Poiseuille Mass flow rate Q(f, f) Boltzmann collision integral Qm Mass flow rate [kg/s] Qs Slip correction term 3 Qv Volume flow rate [m /s] R Radius of the microtubes [m] S Dimensionless mass flow rate obtained from the continuum theory ch Sn, Tn and Vn Coefficients involved in expression of mass flow rate M (3.82) T Temperature [K] Tref Reference Temperature [K] Tw Wall temperature U Constant speed in the direction z relative to the laboratory Frame of reference V Tank volume [m3] 3 Vs Syringe volume [m ] W (cp) Weight function X Distance traveled by the droplet interface in the constant pressure technique

Other symbols

~n Normal vector to the wall P Pressure ratio pin/pout R Gas specific constant [J/kgK] v Molecular velocity u Bulk velocity Nomenclature xxiii

Subscript

eq Equilibrium in Inlet m Mean value out Outlet ref Reference s Tangential component of the velocity at the channel surface sh Sharipov model [101] ∞ Parallel plates

Superscript

ch Rectangular microchannel exp Experimental tu Microtube + Reflected particles − Impinging particles

Acronyms

BE Boltzmann Equation BGK Bhatnagar-Gross-Krook CAD Computer-Aided Design CDG Capacitance Diaphragm Gauge CVD Chemical Vapor Deposition DRIE Deep reactive-ion etching DSMC Direct Simulation Monte Carlo DVM Discrete Velocity Method MEMS Micro-Electro-Mechanical-Systems MST Micro Systems Technology NS Navier-Stokes PVD Physical Vapor Deposition RIE Reactive-ion etching RMS Root Mean Square RSS Root Square Sum TMAC Tangential Momentum Accommodation Coefficient U. V. Ultra-Violet xxiv Nomenclature

Greek symbols

α Tangential momentum accommodation coefficient δ Rarefaction parameter δ Mean rarefaction parameter calculated from the mean pressure between the inlet and outlet tanks εDL εDL = D/L εhL εhL = h/L ζi Incident tangential momentum ζr Reflected tangential momentum λ Mean free path [m] µ Viscosity coefficient [kg/sm] µref Reference Viscosity coefficient [kg/sm] ξp Local pressure gradient νn Eigenvalues of the partial Laplace operator ρ Density [kg/m3] σ2p Second order velocity slip coefficient σp First order velocity slip coefficient τ Relaxation time used in the kinetic theory ω Viscosity index Résumé en français

Les progrès réalisés au cours de ces dernières années dans l’étude des écoulements de gaz raréfiés et le développement des techniques de miniaturisation ont permis la mise en œuvre de nouvelles applications dans le domaine des MEMS. Le développement de ces microsystèmes remonte au début des années 70 et leur commercialisation commence en 1980. Ces systèmes englobent à la fois des technologies relevant de la mécanique des fluides ou de l’électronique. Ces microsystèmes peuvent aussi être classés comme micro machines (au Japon) ou comme micro systèmes technologiques (en Europe). Les MEMS sont généralement des systèmes comportant un ou plusieurs éléments mécaniques, utilisant l’électricité comme source d’énergie en vue de remplir un rôle de détecteur : de plus un élément au moins est de dimension micrométrique. Ils sont utilisés dans de nombreux domaines tels que l’automation, la médecine, les activités aérospatiales, les télécommunications et aussi dans des applications concernant la vie quotidienne comme les projecteurs, la télévision de haute définition, la sécurité des véhicules (air-bag). Les MEMS sont souvent fabriqués en silicium, mais d’autres matériaux peuvent être employés en fonction de l’adéquation de leurs propriétés physico-chimiques avec l’application recherchée, on trouve ainsi : des métaux, du matériel piézo-électrique, différents polymères, etc. Les professionnels travaillant dans ces domaines estiment que ces systèmes auront un très fort impact économique dans les prochaines années.

Cette thèse fait partie du projet européen GASMEMS. Ce projet est un ITN financé dans le cadre du 7em programme de travail (FPT) de la Commission Européenne Marie Curie. L’idée qui sous-tend ce projet est de dynamiser la recherche dans le champ des micro-écoulements gazeux et de structurer cette recherche au niveau européen pour améliorer l’ensemble des connaissances fondamentales et rendre possibles des applications technologiques jusqu’au niveau industriel et commercial.

Ce travail est destiné à l’étude de micro écoulement gazeux, stationnaires, isothermes à travers des conduits de différentes sections, pour différents matériaux avec des parois internes de rugosités différentes. Cette étude s’étend sur l’ensemble des régimes de raréfaction. Le travail a été réalisé d’un triple point de vue : expérimental, théorique et numérique. Avant de décrire les principaux résultats obtenus dans cette thèse il nous a semblé nécessaire de rappeler quelques concepts fondamentaux qui- pensons nous- faciliteront la compréhension de la problématique ainsi que l’évaluation des avancées apportées par cette étude.

Régimes de raréfaction de l’écoulement

L’écoulement de gaz raréfié peut être caractérisé par le nombre de Knudsen Kn. Le nombre de Knudsen se définit comme un rapport : le rapport entre le libre parcours moyen λ des molécules du gaz et une distance, longueur caractéristique LC de l’écoulement au regard de son échelle spatiale : cette distance est généralement la plus petite dimension du dispositif xxvi Résumé en français spatial considéré : ici le rayon du micro tube, la hauteur de la section rectangulaire du micro canal, ou encore parfois le diamètre hydraulique du conduit. On écrit

λ Kn = , (1) Lc où Kn porte le nom du physicien Danois Martin Knudsen [1870-1949].

On rappelle enfin que le libre parcours moyen correspond à la distance moyenne parcourue par une molécule entre deux collisions intermoléculaires successives.

C’est donc ce nombre de Knudsen, paramètre adimensionnel qui décrit et mesure le degré de raréfaction. Quand ce nombre augmente on dit que le gaz devient plus raréfié. On voit donc que cette augmentation peut se produire pour deux raisons différentes : a) lorsque le libre parcours moyen augmente (ce qui, à température constante, correspond à une diminution de la pression et de la masse volumique) b) ou quand la longueur caractéristique LC de l’écoulement diminue (ce qui est le cas lorsqu’on passe à des conduits de sections micrométriques). On voit donc qu’une forte raréfaction n’est pas caractérisée d’abord par un faible nombre de molécules par unité de volume, mais plutôt par un faible nombre de molécules dans un domaine spatial défini par la longueur caractéristique.

Suivant la suggestion de Schaaf et Chambre datant de 1961 [96], on peut schématiser le classement des différents régimes de raréfaction en fonction du nombre de Knudsen par le diagramme ci-dessous (Fig.1) où les valeurs des bornes limitant chaque régime ne représentent qu’un ordre de grandeur, car la transition entre deux régimes n’est pas brutale mais progressive. On distingue donc habituellement:

Équation de Boltzmann

Équation de Navier-Stokes

Non-glissement Glissement

0 ← Kn 10−3 10−1 10 Kn → ∞

Régime Régime de Régime Régime hydrodynamique glissement transitionnel moléculaire libre

Figure 1: Les régimes d’écoulement suivant le nombre de Knudsen.

Résumé en français xxvii

• Le régime hydrodynamique (défini par Kn ≤ 10−3), où le modèle continue (traduit par les équations de Navier-Stokes) est valide, associé à des conditions limites de paroi, classiques : conditions limites d’adhérence pour la vitesse et de continuité pour la température.

• Le régime de glissement (10−3 ≤ Kn ≤ 10−1), où le modèle continue est encore convenable mais ils doivent alors être associé à des conditions limites de glissement (de vitesse) et de saut (pour la température).

• Le régime transitionnel (10−1 ≤ Kn ≤ 10), où le modèle continu n’est plus valide. Pour ce régime la simulation numérique est basée sur la résolution de l’équation de Boltzmann, en utilisant généralement la méthode de discrétisation de l’espace des vitesses. L’équation de Boltzmann (EB) est souvent remplacée par des équations cinétiques dites « modèles » comme le modèle BGK: ces équations dont la structure est plus simple vérifient les propriétés fondamentales de l’EB et donnent souvent des résultats acceptables. Enfin on utilise aussi des méthodes numériques dites de simulation directe comme la méthode de Monte-Carlo (DSMC).

• Le régime moléculaire libre (Kn ≥ 10), où l’écoulement de gaz est fortement raréfié. Dans ce régime les collisions intermoléculaires sont nettement moins nombreuses que les collisions du gaz avec la surface solide. L’écoulement est donc « piloté » par l’interaction gaz/paroi. L’écoulement peut-être modélisé par l’utilisation de l’équation de Boltzmann sans second membre ou par la méthode DSMC.

Coefficient d’accommodation de la composante tangentielle de la quantité de mouvement (TMAC).

Les micro-dispositifs auxquels on s’intéresse se caractérisent par un rapport [Surface/Volume] (de l’ordre de l’inverse de la longueur caractéristique) dont l’ordre de grandeur est bien plus grand que celui des dispositifs classiques correspondants. C’est pourquoi les collisions molécules/paroi ont une influence souvent prépondérante (par rapport à celle des collisions intermoléculaires).

L’étude de ces collisions et de l’interaction gaz paroi, qu’elles supportent, est donc ici très importante. C’est pourquoi nous allons présenter ici quelques notions fondamentales relatives à ce processus et notamment la notion de TMAC qui est un coefficient fondamental permettant de traduire au niveau macroscopique l’effet des collisions molécules-paroi dont la définition physique sera donnée plus loin. Maxwell introduisit ce concept au milieu du XIXe siècle. D’abord sur plan phénoménologique Maxwell postula que lorsqu’une molécule heurte la paroi deux possibilités peuvent advenir: xxviii Résumé en français

i) soit la molécule est réfléchie de façon spéculaire par la surface, sans aucun transfert de quantité de mouvement (seule la composante normale de sa vitesse est changée en son opposée) ; ii) soit la molécule est réfléchie « de façon diffuse » : la molécule quitte alors la surface en « oubliant » la dynamique de la collision et « s’accommode » aux propriétés de la surface. En d’autres termes la molécule est alors réfléchie «aléatoirement » suivant une distri- bution de vitesse égale à « la Maxwellienne de la paroi » : la température figurant dans cette Maxwellienne est la température de la paroi ; cette Maxwellienne est centrée autour de la vitesse de la paroi, ce qui signifie que la vitesse moyenne des molécules réfléchies de façon diffuse est la vitesse de la paroi. Maxwell refusa l’idée d’une réflexion totalement spéculaire comme celle d’une réflexion complètement diffuse ; il opta pour une réflexion mixte spéculaire diffuse. Il posa au départ que sur un flux donné de particules réfléchies, avec une vitesse réfléchie donnée, la proportion de particules réfléchies de façon diffuse était égale au coefficient α (par définition <1) et donc la proportion de molécules réfléchies de façon spéculaire égale à 1 − α.

D’un point de vue cinétique pour écrire le nombre de molécules (réfléchies) affectées d’une vitesse moléculaire v = (vx, vy, vz) nous devons adopter une écriture statistique et utiliser les fonctions de distribution. La loi de Maxwell s’écrira donc pour une espèce donnée : Nombre (probable) de molécules quittant la paroi = Nombre de molécules issues de la réflexion spéculaire + Nombre de molécules issues de la réflexion diffuse, soit encore :

vx · f(vx, vy, vz) = (1 − α)vx · f(−vx, vy, vz) + αvx · fw, (2) où fw est la Maxwellienne de la paroi. Cette loi a été ensuite écrite dans la formulation de l’opérateur Kernel par Kuscer [61] et Cercignani [24].

En utilisant (2) on peut montrer que la signification physique de α peut notamment être assimilée au coefficient d’accommodation de la composante tangentielle de la quantité de mouvement (TMAC) défini comme un rapport de flux de quantité de mouvement (à travers la surface de la paroi) en tout point de la paroi. Si l’on pose

• ζi = flux de la composante tangentielle de la quantité de mouvement des molécules incidentes.

• ζr = flux de la composante tangentielle de la quantité de mouvement des molécules réfléchies.

• ζracc serait le flux de la composante tangentielle de la quantité de mouvement des molécules réfléchies si elles étaient toutes accommodées avec la paroi. On montre sans difficulté que cette quantité est nulle. Résumé en français xxix

On montre donc que formellement :

ζ − ζ ζ − ζ α = i r = i r . (3) ζi − ζracc ζi C’est en utilisant cette définition que Maxwell établit [77] son expression du coefficient de slip de vitesse, pour un écoulement isotherme et en négligeant l’influence de la couche de Knudsen. On voit sur la relation (3) que α caractérise l’interaction gaz surface à travers le rapport entre le flux de quantité de mouvement réellement transféré à la paroi sur la quantité maximale transférable que l’on enregistre lorsque les molécules sont accommodées avec la surface.

Selon sa définition première, ce coefficient d’accommodation figurant dans la loi de réflexion de Maxwell, varie entre 0 et 1. Dans le cas d’une réflexion purement spéculaire la composante tangentielle de la quantité de mouvement des molécules est conservée et α est alors nul. Dans le cas d’une réflexion purement diffuse, les molécules abandonnent toute leur impulsion tangentielle et quittent la surface avec la distribution Maxwellienne de la paroi : α est alors égal à 1 (cf. Fig.2).

α=0 α=1

Figure 2: Réflexion spéculaire ou diffuse d’une molécule sur une surface.

La détermination expérimentale du TMAC constitue depuis longtemps un domaine de recherche important. De nombreux chercheurs ont conduit différentes expériences, utilisant diverses techniques pour déterminer la valeur du TMAC relatif à différents couples gaz-surface. Nous citerons ici : la méthode du cylindre tournant [80]; la méthode «spinning rotor gauges» [10], [12], [46]; les méthodes utilisant l’écoulement du gaz dans un micro-conduit [8], [7], [30], [76], [134]. Les principales techniques expérimentales utilisées pour mesurer le TMAC d’un «couple gaz-matériau de paroi» sont examinées dans la référence [2]. Cette étude montre que le TMAC dépend d’un certains nombre de paramètre tels que la nature du gaz, le matériau de xxx Résumé en français la surface, sa «propreté », et sa rugosité.

La détermination expérimentale de la valeur du TMAC permet d’améliorer la précision des calculs des écoulements de fluides en régimes raréfiés. De plus notre approche à la fois théorique et expérimentale permet aussi une meilleure compréhension de ce concept, de son domaine de validité de ses limites de son amélioration éventuelle. On pourra parvenir ainsi à une meilleure connaissance des écoulements gazeux dans les conditions des microsystèmes : on sera ainsi conduit finalement à une meilleure conception de ces dispositifs MEMS eux-mêmes.

Objectifs de la thèse et méthodes utilisées

L’un des principaux objectifs de cette thèse est de contribuer à l’étude de l’interaction gaz-surface en déterminant les valeurs expérimentales du TMAC dans des micro-écoulements isothermes à partir de la mesure d’une faible variation de pression du réservoir et du débit stationnaire. Cette étude a été conduite pour différents matériaux de surface (Or, Silice, Acier inoxydable et Sulfinert) et pour différents gaz (hélium, azote, argon et dioxyde-de-carbone). Différentes géométries de micro-conduits ont été explorées. L’influence de la rugosité de surface sur l’échange de moment entre le gaz et la paroi a également été étudiée. Pour atteindre ces objectifs, l’étude a été développée sous un triple point de vue expérimental, théorique et numérique.

L’étude expérimentale a été réalisée au moyen de deux installations et basée sur la technique de mesure à volume constant, afin de mesurer le débit massique stationnaire et isotherme à travers les microcanaux. La première installation a été réalisée dans le laboratoire IUSTI pour la mesure du débit massique à travers des microcanaux de section transversale rectangulaire, avec différents rapports d’aspect (hauteur/largeur), constitués de différents matériaux (Au et SiO2) présentant différentes rugosités de surface. La deuxième installation a été réalisée lors d’un stage de deux mois dans la société INFICON, pour mesurer le débit massique à travers de longs microtubes industriels, fabriqués avec différents revêtements internes (Acier-inoxydable et Sulfinert) et destinés à la chromatographie. L’utilisation de ces deux installations a permis d’examiner le débit massique dans tous les régimes d’écoulement : du régime hydrodynamique au régime moléculaire libre.

Les problèmes liés aux fuites, au dégazage et à l’adsorption de l’eau ont été traités dans le but d’atteindre et mesurer de très faibles valeurs du débit massique de l’ordre de 10−13 kg/s. La technique à la fois simple et précise, utilisée pour mesurer le volume des réservoirs a permis de diminuer l’incertitude sur le débit massique par rapport à celle constatée dans d’autres études (par exemple Ewart et al. [38]).

L’étude théorique a porté sur la modélisation de l’écoulement de gaz, généralement du second ordre suivant le nombre de Knudsen, à travers des microcanaux longs dans le régime Résumé en français xxxi de glissement. Pour les microtubes de section circulaire nous avons utilisé une expression classique connue [44]. Pour les microcanaux, de section rectangulaire, une approche originale a été développée. Cette approche théorique est destinée à prendre en compte, dans l’étude de l’écoulement (vitesse, débit), l’effet « bidimensionnel » du à la limitation de la largeur de la section transversale (souvent considérée comme infinie devant sa hauteur). Cette démarche, basée sur les propriétés spectrales de l’opérateur Laplacien figurant dans l’équation de Stokes, permet d’obtenir une expression de la vitesse sous la forme d’une série construite sur la suite des fonctions propres de cet opérateur. Une forme explicite et nouvelle, de second ordre, en fonction du nombre de Knudsen, est ainsi obtenue pour l’expression du débit massique en régime de glissement. Dans cette expression figurent des fonctions du rapport d’aspect de la section transversale rectangulaire du conduit.

La comparaison de ces expressions avec les mesures du débit massique et avec les courbes « expérimentales» déduites de ces mesures (également obtenues sous une forme polynomiale du second ordre suivant le nombre de Knudsen) a permis l’extraction des caractéristiques de l’écoulement de gaz telles que les coefficients de glissement de vitesse et le coefficient d’accommodation de la quantité de mouvement tangentielle, sans avoir recours aux calculs numériques, comme c’est le cas dans l’approche développée dans la Réf. [101]. Dans le cas des canaux de section rectangulaire la comparaison entre les expressions théoriques et expérimentales du débit permet, en outre, de valider le calcul de l’influence du rapport d’aspect de la section sur les coefficients de l’expression polynomiale du débit (suivant le nombre de Knudsen).

L’étude numérique est menée, notamment, en raison de l’absence de données numériques, sur le débit massique à travers des microcanaux rectangulaires, dans les régimes transi- tionnel et moléculaire libre, où comme on sait, le modèle développé en slip régime n’est plus valable. On utilise ici le modèle cinétique BGK pour des valeurs du coefficient d’accommodation α différentes de 1. Les simulations numériques sont réalisées en ré- solvant l’équation cinétique BGK linéarisée, en utilisant la méthode des vitesses discrétes (DVM) avec l’hypothèse d’une loi de réflexion diffuse-spéculaire des molécules sur les surfaces.

Ces simulations numériques sont réalisées pour différents rapports d’aspect de section rectangulaire des microcanaux et pour différents coefficients d’accommodation α. La comparaison visuelle entre les courbes numériques de débit massique obtenues pour ces différentes valeurs de α et les graphes du débit massique mesuré nous a permis de déduire les valeurs «expérimentales» de α dans le régime transitionnel et le régime moléculaire libre.

Structure de la thèse

La thèse est structurée en six chapitres, en commençant par l’introduction générale où les définitions des régimes de raréfaction, du nombre de Knudsen et celle du TMAC sont données (chapitre 1). Dans le chapitre 2 une brève présentation des techniques de micro-fabrication xxxii Résumé en français mises en œuvre pour fabriquer les microcanaux à section rectangulaire est donnée. Le chapitre 3 est consacré à la description des approches théoriques et numériques utilisées dans la thèse. Dans le chapitre 4, est introduite une brève revue générale des techniques expérimentales utilisées pour la mesure du TMAC. D’autre part, la description de la technique mise en œuvre dans cette thèse pour mesurer le débit massique y est détaillée. Dans les chapitres 5 et 6, on présente les résultats des mesures du débit massique et les calculs des coefficients de glissement de vitesse et du coefficient d’accommodation de la quantité de mouvement tangentielle, dans les microtubes et les microcanaux rectangulaires respectivement. Enfin, une conclusion générale constitue le chapitre 7 où quelques perspectives sont également tracées.

Les principaux résultats obtenus

Les résultats obtenus pour le TMAC ont montré que les surfaces des matériaux constituant les parois (Or, Silice, Acier-inoxydable et Sulfinert) des microcanaux analysés dans cette thèse ne sont pas caractérisées par une réflexion complètement diffuse tout du moins quand on les associe au gaz utilisés ici (en excluant provisoirement les résultats du groupe E dont la validité est discutée plus loin). En outre, les valeurs de TMAC déduites de la comparaison entre les données expérimentales de débit massique et les courbes numériques en régime transitionnel et en régime moléculaire libre sont voisines ou très voisines de celles obtenues en régime de glissement en utilisant l’approche continue. Ainsi, une valeur unique de TMAC peut être utilisée pour tous les régimes d’écoulement.

En régime de glissement, les résultats obtenus pour le coefficient de glissement de vitesse σp et le TMAC avec les approximations de premier ordre sont très proches de ceux obtenus avec le second ordre : les écarts sont inclus dans l’intervalle de l’incertitude expérimentale. Les deux approximations sont pertinentes dans leurs intervalles de validité respectifs, à savoir [0, 0.1] et [0, 0.3], définis en terme de nombre de Knudsen, à l’exception de certains cas (les microcanaux A1, S1 et les microtubes T 1 et T 2) où l’approximation de second ordre n’est pas pertinente. Cette situation d’exception se produit soit parce que l’effet de second ordre est trop faible pour être mesuré correctement soit en raison d’un du manque de données expérimentales dans une partie [0.1, 0.3] de l’intervalle de nombres de Knudsen à considérer pour l’étude de second ordre (ici, [0, 0.3]). Dans le régime transitionnel et le régime moléculaire libre l’allure des courbes numériques de débit massique obtenues pour différentes valeurs de TMAC, à l’aide du modèle BGK nous a permis de retrouver l’allure des résultats expérimentaux et d’identifier la valeur «expérimentale» du coefficient d’accommodation.

Les conclusions suivantes peuvent être tirées de cette étude:

Influence de la rugosité de surface: Les résultats ont montré que la rugosité de surface à une influence importante sur les coefficients de glissement du premier ordre σp et second ordre σ2p et donc aussi sur le TMAC. Cette influence modifie la dépendance de ces coefficients à l’égard de la masse moléculaire du gaz considéré. Lorsque la rugosité de Résumé en français xxxiii surface des microcanaux est très faible (de l’ordre de 1 nm dans notre cas) le TMAC et le coefficient de glissement ne dépendent pratiquement pas de la masse moléculaire des gaz analysés dans cette thèse. Lorsque la rugosité de surface augmente, le TMAC se différencie suivant la masse moléculaire; et pour les gaz monoatomiques considérés ici, on constate qu’une plus forte accommodation se produit pour le gaz plus léger. Toutefois, pour les gaz poly-atomiques cette tendance n’est pas confirmée, de sorte que les valeurs de TMAC obtenues pour l’azote et l’argon sont proches les uns les autres, et ceux obtenus pour le dioxyde-de-carbone et pour l’hélium sont également proches. Par ailleurs, il semble que la rugosité joue un rôle de différenciation, analogue, mais moins prononcé pour le coefficient de glissement de second ordre : il semble qu’une surface encore plus rugueuse soit nécessaire pour différencier clairement σ2p suivant la masse moléculaire du gaz considéré.

Influence du matériau de surface: L’analyse des valeurs de TMAC obtenues pour les microcanaux des groupes A (Au) et S (SiO2), qui ont la même rugosité de surface, nous conduit à conclure que la surface d’or (Au) est plus diffuse que celle de silice (SiO2). Par ailleurs, les résultats de TMAC obtenus respectivement dans les microtubes T 1 (Sulfinert) et T 2 (Acier-inoxydable) restent similaires pour les gaz N2 et Ar, alors qu’ils différent sensiblement d’un tube à l’autre pour He et CO2. Il semble que le revêtement de Sulfinert des microtubes T 1 modifie différemment la valeur du TMAC en fonction du gaz considéré. Néanmoins on constate que si nous considérons les gaz monoatomiques (He, Ar), la hiérarchie, évoquée plus haut, des valeurs du TMAC en fonction de la masse moléculaire est conservée lorsqu’on passe d’une catégorie de tube à l’autre : le gaz le plus léger demeure celui qui s’accommode le plus. On pourra donc noter que même si malheureusement, la rugosité du microtube T 1 n’est pas connue il semble probable, d’après nos observations antérieures, que cette rugosité est suffisante pour rendre discernables les coefficients relatifs aux différents gaz de masse moléculaire différentes.

D’autre part lorsqu’on compare les résultats obtenus pour la silice (groupe S) et le Sulfinert (microtube T 1), qui est un matériau à base de silice, on observe que les valeurs de TMAC sont similaires pour les gaz les plus lourds (N2 et Ar).

Influence des parois latérales des microcanaux: L’investigation et l’analyse de l’influence des parois latérales sur le TMAC et σ2p dans les microcanaux rectangulaires est délicate à réaliser, du fait que les parois latérales des microcanaux ne sont pas véritablement verticales et leur rugosité n’est pas toujours identique à celle des surfaces supérieure et inférieure.

En effet les valeurs de TMAC obtenues dans les microcanaux avec les plus petits rapports d’aspect (w/h) sont différents des ceux obtenus pour les autres microcanaux qui ont des valeurs de TMAC très proches. Or, en principe σp et α ne dépendent pas (où peu) de la géométrie et σ2p non plus, au moins de remettre en cause l’expression (3.36) de la vitesse de xxxiv Résumé en français glissement. Par conséquent : - si les hypothèses qui président à l’établissement de la relation (3.36) de la thèse sont correctes (section parfaitement constante et rectangulaire, rugosité identique sur les parois horizontales et verticales).

-et si la modélisation bidimensionnelle est correcte, alors σp et α ne devraient varier que très peu lorsqu’on change de rapport d’aspect, si les autres propriétés du conduit et de l’écoulement restent égales par ailleurs. C’est pourquoi nous avons mis en cause l’imperfection de la géométrie et l’inhomogénéité de la rugosité des parois.

De même le détail des résultats obtenus pour le coefficient de glissement du second ordre σ2p sont intrigants pour les mêmes raisons. En effet les résultats ont montré que les valeurs de σ2p obtenues sont similaires, pour les deux microcanaux de chaque groupe ayant les plus petits rapports d’aspect (w/h) (par exemple A1, A2), même si ces rapports sont un peu différents entre eux. De même pour les deux autres microcanaux d’un même groupe avec les rapports d’aspect les plus élevés, même différents (par exemple A3, A4), les valeurs de σ2p sont également similaires. Mais par contre, une différence significative entre les valeurs de σ2p est observée si l’on compare celles de la paire des microcanaux (A1, A2) avec celles de la paire des microcanaux (A3, A4). La encore, cette différence est probable- ment due à l’influence de la rugosité inhomogène et de la forme imparfaite des parois latérales.

Il serait souhaitable d’effectuer d’autres investigations sur des microcanaux de différents rapports d’aspect, du même matériau et dont la géométrie de section et la rugosité des parois seraient mieux contrôlées. On pourrait ainsi mieux préciser les raisons des variations observées.

Autres résultats: Quelques résultats surprenants ont été obtenus avec les microcanaux du groupe E où les valeurs de TMAC étaient généralement supérieures à 1 (parfois un peu au-delà de l’erreur expérimentale estimée). Nous nous sommes posé d’abord la question de la pertinence du modèle de réflexion de Maxwell. En effet selon certains auteurs les valeurs de TMAC supérieures à 1 pourraient être dues à un phénomène appelé « backscattering » dont le modèle de Maxwell ne rend pas compte. D’autres modèles comme le modèle Cercignani-Lampis pourraient prendre en compte un tel phénomène. Mais l’utilisation de ce modèle (CL) est plus compliquée car il im- plique l’introduction et la détermination de deux coefficients d’accommodation décrivant les échanges de moment et d’énergie des molécules avec les parois. En conséquence, deux types de mesures seraient nécessaires pour obtenir ces deux coefficients. C’est pourquoi cette autre modélisation n’a pas été développée dans le cadre de la thèse. Néan- moins cette analyse pouvait sembler un peu superflue car il est bien connu que le modèle de Maxwell fonctionne bien lorsque les valeurs du coefficient d’accommodation sont proches de 1.

Finalement l’analyse du problème, et des observations indirectes faites sur des conduits cassés, nous font fortement penser que la cause probable de ces anomalies peut être le Résumé en français xxxv décollage de la couche d’or à l’intérieur des microcanaux, mais cela reste une hypothèse, car il n’était pas possible de vérifier ce fait avec une technique expérimentale directe.

Perspectives

Quelques suggestions sont proposées ici comme perspectives pour d’autres investigations:

• Examiner l’influence du degré de liberté du gaz sur le TMAC et les coefficients de glissement du premier et du second ordre en réalisant des expériences avec des gaz poly-atomiques.

• Mesurer le TMAC pour des mélanges de gaz en utilisant deux gaz ayant des valeurs de TMAC connues. L’effet de la concentration de gaz sur TMAC peut être aussi étudié.

• Etudier plus systématiquement diverses rugosités de la surface.

• Reprendre l’étude de l’influence du rapport d’aspect avec une série de microcanaux dont la géométrie et l’homogénéité de la rugosité seraient rigoureusement contrôlées : répon- dre ainsi entièrement aux questions posées dans le paragraphe consacré à l’influence des parois latérales.

Chapter 1 Introduction

The progress made in recent years in the study of rarefied gas flow and the development of new miniaturization techniques have enabled the development of new applications in the field of MEMS (MicroElectroMechanicalSystems). MEMS have been developed in the early 1970 and initial marketing began in 1980. These systems envelop both tech- nological drilled in the field of mechanics and of electronics. MEMS are also referred to as micromachines (in Japan), or Micro Systems Technology - MST (in Europe). MEMS are systems containing one or more mechanical elements, using electricity as source of energy, in order to achieve a function of sensor and/or actuator, with at least one element having micrometric dimensions. They are used in different fields like automotive, aerospace, medicine, biology, telecommunications, and in some daily applications as some projectors, high-definition TVs or vehicle security (airbag). MEMS are most often made of silicon, but other materials may be used in some applications, according to their physical properties adequacy, such as metals, piezoelectric materials, various polymers...etc. Professionals in the field believe that these systems will have a very important economic impact in next year’s.

This thesis is a part of GASMEMS project, which is an Initial Training Network (ITN), financed by the 7th Framework program (FP7) of the European Commission (Marie Curie Actions). The idea behind this project is to boost the research in the field of gas microflow and to structure research in Europe in the field of micro gas flows to improve global funda- mental knowledge and enable technological applications to an industrial and commercial level.

This thesis is devoted to the study of steady isothermal gas microflows through microchannels of various cross-sections, under rarefaction condition. Triple aspects, exper- imental, theoretical and numerical will be considered in this thesis to study the problem. Before describing the results obtained in this thesis, it seems necessary to review some basic concepts, which, we believe, will simplify the understanding of the problem and the evaluation of the improvements brought by this work.

1.1 Flow regimes

The gas flow in micro size devices or at low pressure may be characterized by the Knudsen number, which is defined as the ratio of the mean distance traveled by molecules between two successive collisions (known as mean free path λ [17]) to a representative physical length scale Lc. This length scale is generally the smallest dimension of a device; it could be a radius of tube, a height of rectangular cross-section or the hydraulic diameter...etc. The Knudsen 2 Chapter 1. Introduction number (Kn) is named after Danish physicist Martin Knudsen (1871 − 1949) and is defined as:

λ Kn = . (1.1) Lc The Knudsen number is the parameter used to describe the degree of gas rarefaction. When the Knudsen number increases the gas becomes more rarefied. That happens when the mean free path λ increases (i.e. in case of pressure decreasing) or when the characteristic length Lc decreases (i.e. in micro or nano-channels).

Boltzmann Equation

Navier-Stokes Equation

No-Slip Slip

0 ← Kn 10−3 10−1 10 Kn → ∞

Continuum Slip regime Transitional Free-Molecular regime regime regime

Figure 1.1: Flow regimes categorized by Knudsen numbers.

Schaaf and Chambre [96] suggested to use the Knudsen number as guidelines for identifying the flow regimes (see Figure 1.1). Usually, the following classification is used:

• The continuum regime (Kn ≤ 10−3), where the classical continuum model (Navier- Stokes equation) with conditions of non-slip and continuity of temperature on the wall

is enough precise to describe the flow. • The slip regime (10−3 ≤ Kn ≤ 10−1), where the continuum model is still appropriate, but it should be subjected to the conditions of velocity slip and temperature jump on the wall. • The transitional regime (10−1 ≤ Kn ≤ 10), where the continuum model is no longer valid. For the flow simulation in this regime, the Boltzmann equation should be resolved using the discrete velocity method or the Direct Simulation Monte Carlo (DSMC) tech- nique. • The free molecular regime (Kn ≥ 10), where the gas flow is highly rarefied. In this regime the number of molecule-molecule collisions are smaller than the numbers of 1.1. Flow regimes 3

molecule-surface collisions. Therefore, the flow is driven by the interaction between gas and wall. The flow in this case is modeled using numerical solution of collisionless Boltzmann equation or DSMC method.

We have to remind that this classification is not strict. The limits between the regimes shown in Figure 1.1 have to be taken as an order of magnitude, because the transition between regimes is not brutal but progressive.

Kn Free Molecular Hard Disk Drive Micro Channels Micro Pumps 10. Micro Gyroscope Accelerometer Micro Valves 1.0 Transitional Micro Nozzles Flow Flow Sensors 0.1 (Helium) Slip Flow 0.01 Continuum Flow (Air)

0.001 Nano Technology MEMS h (µm) 0.01 µm 0.1 µm 1.0 µm 10.0 µm

FIGURE 8.1 The Operation range for typical MEMS and nanotechnology applications under standard conditions Figurespans 1.2: the entire Operation Knudsen regime regimes (continuum, for MEMS slip, transition and nano-devicesand free molecularunder flow regimes). standard conditions of pressure and temperature [13].

conservation equations with more advanced constitutive laws than the Navier–Stokes equations. One Figureexample 1.2 of thisillustrates is the Burnett the equations. operation The regimes second misconception for some MEMS is that, applicationsin the “slip flow” developed regime, in the boundary conditions suddenly change from no slip to slip. This is also misleading, as the no-slip- recentboundary years undercondition standard is just an conditions empirical finding of pressure and the andNavier–Stokes temperature. equations In are MEMS valid both technology for the continuum“slip” and “continuum” approach flow is validregimes. for Hence, many the applications, slip effects become especially important those gradually have with a dimensionincreased of severalKn. hundredNevertheless, of the micrometer identification and of flow working regimes under was made atmospheric for rarefied conditions gas flows almost where a century the mean free pathago. For is Kn approximately ≤ 0.1 flows, the Navier–Stokes 70 nm for equations air (see subject Figure to 1.2 the). velocity-slip However, and devices temperature-jump such as hard diskboundary drives, microchannelsconditions should or be micropumpsused. The slip conditions can be in are the [Kennard, slip or 1938; transitional Schaaf and regimes Chambre, under 1961]: the standard conditions, which makes necessary the study of mass, momentum and energy transport in the whole Knudsen number range. 2 – συ 1 3Pr(γ – 1) = ------τ + ------us – uw 1/2 s (–qs) (8.1) συ ρ /π 4 γρRT (2RTw ) w

2 – σ 2(γ – 1) 1 T – T = ------T------(–q ) (8.2) s w σ γ + 1 ρ /π 1/2 n T R (2RTw )

τ where qn, qs are the normal and tangential heat-flux components; s is the viscous stress component corresponding to the skin friction; R is the specific gas constant; γ is the ratio of specific heats, ρ is the

density; Pr is the Prandtl number; and Tw and uw are the wall temperature and velocity, respectively. The

gas slip velocity and temperature near the wall (jump) are given by us and Ts, respectively. The term in

the above equation proportional to (–qs) is associated with the phenomenon of thermal creep, which can cause variations of pressure along tubes in the presence of tangential temperature gradients [Beskok et al., 1995; Sone, 2000; Vargo and Muntz, 1996; Vargo et al., 1998].

© 2002 by CRC Press LLC 4 Chapter 1. Introduction

1.2 Tangential momentum accommodation coefficient (TMAC)

The microdevices are characterized by the increasing of the surface-to-volume ratio in comparison to the conventional devices. That is why the molecules-surface collisions dominate the molecules-molecules collisions and the study of this type of collision becomes important.

In the middle of nineteenth century Maxwell introduced the concept of the accommodation coefficient [33]. He postulated that when the molecule collide with a surface there are two possibilities: (i) the molecules can be reflected specularly without transferring any of their momentum to the surface (changing only the normal to the wall component of their velocity to an opposite one), (ii) the molecules can be reflected diffusely: a molecule leaving the surface "forgets" all information about upon collision and it leaves accommodating the surface properties (i.e., their average bulk velocity is equal to the surface velocity and the temperature is equal to the temperature of the surface). Maxwell rejected the possibility of full specular reflection and he did not see any reason to assume complete accommodation of the molecule to the surface. So this concept of reflection can be related to the tangential momentum of the incident (ζi) and reflected (ζr) molecules in this form ζ − ζ α = i r . (1.2) ζi This ratio α is known as the Tangential Momentum Accommodation Coefficient (TMAC). TMAC denotes a momentum flux ratio characterizing the gas-surface interaction: namely, the ratio between the tangential momentum flux transferred to the wall and the maximal tangential momentum transfer, occurring when the reflected particles are completely accommodated with the surface. The value of TMAC varies in the range from 0 to 1. In case of a specular reflection, the tangential momentum of gas particles is maintained, and then TMAC value is equal to 0. In case of diffuse reflection, the gas particles lose all their tangential momentum, and leave the surface with the Maxwellian distribution then TMAC value is equal to 1, see Figure 1.3.

We mentioned above that TMAC would normally falls within the range of 0 to 1. However, many experimental studies [10], [92], [55] and some numerical simulations [41] reported TMAC values superior to 1. This phenomenon is called "Backscattering", defined as the reflexion of the particles back to the direction they came from.

Determining TMAC value experimentally was for a long time a very important domain of study. Many researchers carried out different experiments using various techniques to determine the value of TMAC for various couple gas-material. We cite for example, the rotating cylinder method [80], the spinning rotor gauges [10], [12], [46] and flow through a microchannel [8], [7], [30], [76], [134]. The main experimental techniques used to measure TMAC for different couple gas-surface material were reviewed in Ref. [2]. This review shows that TMAC depends on a number of parameters, such as the type of the gas, the surface 1.3. Aims and structure of the thesis 5

α=0 α=1

Figure 1.3: Gas/surface interaction. material, its cleanliness and its roughness. A short review of the techniques used to measure the value of TMAC will be given in Chapter4.

Tangential momentum accommodation coefficient is used to improve the accuracy of fluid flow calculations in rarefied flow regimes. A better understanding of TMAC will improve the understanding of gas flow behaviour under rarefaction condition and will lead to a better design of MEMS devices.

1.3 Aims and structure of the thesis

The aim of this thesis is to contribute in the study of gas/surface interaction by measuring TMAC for various surface materials (Gold, Silica, Stainless Steel and Sulfinert) and different gas types (helium, nitrogen, argon and carbon-dioxide). The influence of the surface roughness on the momentum exchange between gas and surface is also studied. To achieve these goals we have conducted our study from triple points of view, experimental, theoretical and numerical. First, the experimental aspect is focused on the measurements of mass flow rates through microchannels having different shapes of the cross-section (rectangular and circular), different aspect ratios and different materials and roughness of surfaces. Second, the theoretical aspect involves especially the derivation of an analytical expression for the mass flow rate through rectangular microchannels of various cross-section aspect ratios in the slip regime. Finally, the numerical aspect of the thesis is present through the numerical simulations of the gas flow through a microchannel of a rectangular cross-section in transitional and free molecular regimes by the use of the kinetic theory (BGK model kinetic equation). The comparison between the experimental and theoretical results will allow us to extract the values of the velocity slip and tangential momentum accommodation coefficients. 6 Chapter 1. Introduction

The thesis is structured into six chapters, starting by general introduction given above (Chapter 1). In Chapter 2 a brief presentation is given for the microfabrication techniques implemented to manufacture the rectangular cross-section microchannels used in the present work. Chapter 3 is devoted to the description of the theoretical and numerical approaches used. In Chapter 4, a brief review of the experimental techniques employed to measure TMAC is introduced, then the detailed description of the technique used in this thesis to measure the mass flow rate is given. Chapters 5 and 6 provide the results of the mass flow rate measurements and the calculations of the velocity slip and tangential momentum accom- modation coefficients in microtubes and rectangular microchannels, respectively. Finally, a general conclusion is made in Chapter 7. Chapter 2 Microchannels fabrication

Microfluidics is directly outcome from the MEMS (Micro Electro Mechanical Systems), which was developed from the microelectronic fabrication techniques. Microfabrication is a set of manufacturing techniques to produce devices with dimensions in the micrometer range and below. The main characteristics of microsystem technologies, compared to microelectronics, are related to the performance of moving parts, thus relatively detached from the substrate, which is generally obtained by using a sacrificial layer [16]. Mainly based on the silicon, MEMS application domain was extended to microfluidics through the development of deep engravings and bonding methods.

We believe that the information on the fabrication and coating techniques of the microchannels will allow us a better understanding of gas/surface interaction, especially which concerns the used materials nature, the surfaces roughness and the microchannels cross-section shape.

For meaningful extraction of the flow features such as velocity slip coefficients and TMAC from the mass flow rate measurements, it is important that all microchannel internal surfaces have the same characteristics (material and roughness). Several previous studies [132], [30], [52], [49], [91], [84] on rarefied gas flow have used microchannels without the same structure of the upper and bottom surfaces. For example, Colin et al. 2004 [30] and Hsieh et al. 2004 [52] have measured the mass flow rate through rectangular microchannels with no identical structure of the upper and bottom surfaces (see Figure 2.1). The microchannels were etched in silicon wafer and covered with Pyrex plate by anodic bonding. The roughness of the upper and bottom surfaces were not the same, they reported values for the bottom surface (Si) between 5 and 8 nm in Colin et al.[30] and 0.5% of the channel height in Hsieh et al.[52].

In order to avoid this problem the microchannels used in this thesis were etched in two silicon wafers with the same width (w), the same surface roughness and half of depth (h/2), then the two wafers are bonded together using the thermo-compression technique.

A lot of microfabrication techniques were developed in recent years, but in this chapter we will focus only on those used to manufacture the microchannels analyzed in this thesis. No details of the microtube fabrication techniques will be given, although, they can be found on the provider (RESTEK company) website: www.restek.com.

The microfabrication techniques are widely explained in literature and many books are published in this area. The books with references [16],[74], [99] are used, together with the Figure 1 Experimental setup for the measurement of gaseous microflows.

MICROCHANNELS 1–9%. It has been shown that for an outlet Knudsen number Kno = 0.1, a plane flow model overestimates The microchannels have been etched by DRIE (Deep the rate flow, typically by about 1% for a = 0.01 and Reactive Ion Etching) in a silicon wafer and8 closed about 6% for a = 0.1 [34]. For this reason, it is neces- Chapter 2. Microchannels fabrication with Pyrex by anodic bounding. The cross-sections are sary to use the rectangular flow model for comparison rectangular, and several identical microchannels are ar- with our experimental data. S.-S. Hsieh et al. / International Journal of Heat and Mass Transfer 47 (2004) 3877–3887 3879 ranged in parallel in order to obtain a sufficient flow rate to be measured with a good precision (see Figure 3). Outlet surface roughness less than 0.5% wrt channel height The characteristics of the wafers are summarized in were measured. The relevant geometry dimensions of the

Downloaded At: 09:32 2 September 2010 resultant microchannel are presented in Table 2. Table 1. The widths of the microchannels have been measured with an optical microscope, and their depths have been measured with a Tencor P1 profilometer. They 3. Experimental apparatus and procedure range from 4.5 to 0.5 µm, with aspect ratios a from Inlet Pyrex 7740 A schematic diagram of the experiment setup is shown in Fig. 3 in which the device is clamped in a fixture with an internal O-ring to seal the inlet and (a) outlet. Fig. 4 is a photograph of the present micro- channel and test loop. Nitrogen (N2) gas was supplied and controlled from a pressurized cylinder with a regu- lator (Tescom Co.), and flowed through a mass flow controller (MKS Instruments Inc.), two particle filters (TEM Co.; one is 0.5 lm and one is 0.003 lm), passed two pressure transducers (Granville-Phillips 354; y À2 À9 x Micro-channel 5 · 10 –1 · 10 Torr) and two thermistors through the test section at inlet pressure of up to 3.45 bars. The test section inlet was connected to a mass flow meter (MKS z Outlet plenum Instruments Inc.) having a flow rate range of 0.006– 0.321 mg/s for the present flow rate measurements. The Oxidized silicon wafer accuracy of the mass flow rate measurements is within Inlet plenum 3.5%. Noting that prior to N2 gas flow in, the loop was vacuumed by an vacuum pump (Varian EO50/60) and (b) Fig. 1. Close-up view of the present microchannel. maintained at 10À3 Torr. Figure 2.1:Figure Schematic 3 (a) Top view representation of the microchannels etched of the in a silicon microchannels published by Colin et al. 2004 [30Due] to the unavailability of appropriate sensors for Figure 2 Typical example of volume flow rate measurement. wafer (b) and front view of the wafer with its outer connections. inside measurements, both pressures and temperatures (left side) and Hsieh et al. 2004 [52] (right side) usedand for finally, the deionized mass water. flow After rate rinsing, experiments. the wafer was were measured outside the test section. Pressure drop heat transfer engineering vol. 25 no. 3 2004 25 treated with a sulfuric and hydrogen peroxide solution between the upstream and downstream pressure mea- to remove any metal or other inorganic contamination surements was measured with an accuracy less than from the wafer surface. The wafer was then spinned a ±1%. Pressure losses in the fittings and feeding lines technical information provided by the manufacturerthin Femto-ST positive photon to resistant write and this AZ4210PR thesis was part. other than the microchannel are estimated to be a small formed on top of the wafer with a thickness of 1.15 lm. fraction of the total pressure drop. The temperatures The PR pattering conditions is also listed in Table 1. A were measured with miniature thermistors at both the soft bake process was done after PR layer quality inlet and outlet of the test section. The gas entering the The rectangular microchannels were produced byexamination Femto-ST was made. laboratory Once the above using process the had DRIEtest section were at thermal equilibrium with surround- done, an UV light (intensity: 40mW/cm 2) about 365 nm ing, and their temperature (300 K) was determined etching technique with BOSCH process. To be manufacturedto the photon resistant the layermicrochannels was exposed and then, the undergowith an accuracy of ±0.05 °C. The flow provided by the three stages: First, lithography: a thin photosensiblemask pattern resin translated layer tois the deposed photon resistant on thelayer. siliconupstream reservoir was found very stable flowing into After developing, the oxide was removed from the the microchannel under continuous operation for up to wafer and exposed to U.V. rays. Second, DRIE etching:channel using the HF for silicon 15 min, and wafer cleaned is with etched deion- usingfour hours. The microfilter used was to prevent fluctu- ized water. Then, the wafer was subject to a KOH ations and channel blockage caused by trapped parti- DRIE technique. Finally, wafer bonding: the two siliconetching solution wafers (30wt.%) manufactured for varied time with at a tem- the samecles. The uncertainties for the relevant flow parameters perature of 95 °C for different channel depths with an and variables are tabulated in Table 3. All the mea- characteristics are bonded together. These three stagesetching rate are up described to 1.33 lm/min with anisotropically. more Thedetailssurement in devices were connected to an IBM PC for data the following paragraphs. microchannel was fabricated and, then the oxide was acquisition and processing. removed again by HF solution, and the structure as well The present experiments were performed under as surface condition was inspected by a microscope and steady state channel in a clean room, University Mi- a surface profilemeter (Alpha-Step). crosystem Research Laboratory, where the ambient Finally, a 525 lm thick clear Pyrex cover slip was temperature is controlled at 300 K. The flow was con- 2.1 Lithography bonded on the top of the wafer to form the closed sidered to be steady state when the pressure drop mea- channel. The geometry configuration of the channel sured cannot be changed within 5 min or the change rate allowing an accuracy less than ±1% and the roughness is constant for at least 2 min. Each case was repeated at Lithography is a photographic process for printingof the images channel along onto its center a layer with the of surface photosensitive profile- least two times. It shows that it took a longer time for a meter with a relative (wrt channel hydraulics diameter) slower flow rate to reach a steady state. resin (photoresist) used as protective mask against etching. The photolithography (the lithography using the U.V. light source) is so far the most common lithography technique in microfabrication [99].

Three steps are necessary to realize the lithography:

• resin coating;

• exposure;

• development. 2.1. Lithography 9

(a) (b) (c)

Figure 2.2: Lithography steps: (a) resin deposition, (b) spinning, (c) resin layer.

Resin coating: A drop of the resin solution ‘SPR 220’ (developed specially for BOSCH process) is deposed in the center of the silicon wafer then the wafer is spun at high speed between 500 to 5000 rpm for 30 to 60 seconds [16]. Spinning the wafer at certain velocity ω, taking into account the viscosity µ of the resin and the surface tension, spreads the resin solution to a uniform thickness (see Figure 2.2). After the spinning step, the silicon wafer is baked to evaporate the solvent in order to form a solid layer.

Exposure: The obtained resin layer (photoresist) is exposed to U.V. lights through a mask to print on it the desired shape of the microchannel (see Figure 2.3). It results into two regions: exposed and unexposed regions.

U. V. beams Mask Photoresist Silicon Wafer

Non exposed region Exposed region

Figure 2.3: Photoresist exposition to U.V. rays.

The mask consists of an optical opaque chrome pattern generated by a CAD (Computer- Aided Design) tool using laser beam writing. U.V. beams Development: This step consistsMask in removing the exposed or unexposed area of Photoresist the photoresist by immersing the wafer in a chemical solution. There are two kinds of Silicon Wafer photoresists [5]: positive and negative one. The positive photoresist prevents the dissolution of the unexposed area in the developer solution. The opposite process happens with the negative photoresist: the unexposed area dissolves in the developer solution and remains only the exposed area. Figure 2.4 summarizes the three steps of the lithography process. 10 Chapter 2. Microchannels fabrication

(2) (1)

Mask

(3) (4)

Figure 2.4: Lithography process: (1) Silicon wafer;(2) Photoresist deposit;(3)Gold Exposure or Silica to U.V. beams;(4) Development.

5 2.2 RIE etching 6

RIE etching, also called ion-assisted etching, is a combination of physicalGold and chemical processes. Using this technique, the silicon (Si) material is etched by the collision of incident ions from the plasma. The ions are accelerated in perpendicular direction to the wafer surface following the generated electric field, removing the material from the surface at pressure ≈ 10−4 − 10−3 Torr [16]. This process makes tendency to anisotropic etching. To further increase the anisotropic etching, the Deep Reactive Ion Etching technique (DRIE) is used. DRIE technique is a highly anisotropic etching process used to create deep, steep-sided 7 8 holes and trenches in wafers. Etch depths of hundreds of microns can be achieved with aspect ratios of 30 to 1 or more and silicon etching rate higher 6 µm/min [118]. The primary technology is based on the so-called "Bosch process", named after the German company Robert Bosch [75]. This process can fabricate 90◦ (truly vertical) walls, but often the walls are slightly tapered, e.g. 88◦ or 92◦ ("retrograde").

In this process, the passivation deposition and etching steps are performed alternatively in two steps cycle (see Figure 2.5):

• First, the wafer surface is attacked with argon ions (Ar+) in the vertical direction;

• Second, a layer of photoresist9 (SPR220) is deposed on the surface to passivate the side walls.

2.3. Wafer coating 11

Figure 2.5: DRIE technique (Bosch process).

Each phase lasts for several seconds. The passivation layer protects the entire substrate from further ions attack and prevents further etching. However, during the etching phase, the directional ions that bombard the substrate attack the passivation layer at the bottom of the trench (but not along the sides). They collide with it and sputter it off, exposing the substrate to the etchant ions.

These etch/deposit steps are repeated many times, resulting in a large number of very small isotropic etch steps taking place only at the bottom of the etched channel.

2.3 Wafer coating

After the etching step the wafers are coated either with a layer of gold (Au) or silica (SiO2) in the etched part of the wafer. To realize this step, the oxidation technique was used to generate the layer of silicon dioxide and the physical vapor deposition technique to generate the gold layer.

2.3.1 Oxidation

Thermal silicon oxidation is a process to obtain a thin film of SiO2 with a good quality and thickness homogeneity. The oxidation is performed inside a furnace at temperature of 900 ◦C ◦ to 1200 C in the presence of oxygen (O2). This process is called the dry oxidation technique. The reaction for oxide formation is:

Si + O2 =⇒ SiO2. (2.1) The desired thickness of the silicon dioxide layer can be achieved by controlling the timing, temperature and rate of the oxygen entering the furnace.

2.3.2 Physical Vapor Deposition (PVD) This technique permits to depose on the wafer surface different materials using evaporation or sputtering process. To coat the wafer with gold layer the wafer of silicon is deposed in the same vacuum chamber (at pressure of 0.1 − 10 P a) with the gold material and this last is bombarded with high energy argon ions. The ions of argon remove atoms from the gold 12 Chapter 2. Microchannels fabrication

(1)

(2)

Figure 2.6: Microchannel bonding with a gold intermediate layer. material and transfer them on the wafer surface. To improve the adhesion of the gold to the silicon wafer, the titanium (T i) metal is added.

2.4 Wafer bonding

Wafer bonding is a method to join two wafers to create a stuck wafer. This method can be realized by three main processes: direct bonding, anodic bonding and bonding with an intermediate layer.

The bonding with an intermediate layer process was used to make the microchannels analyzed in this thesis. As its name indicates, this method uses an intermediate layer to join two silicon wafers. The intermediate layer can be Glass frits, polymeric adhesives (such as polyimides, silicones, or epoxy resins) or eutectic. In the eutectic bonding process, gold-coated silicon wafers are bonded together at temperature greater than the silicon-gold eutectic1 point (636K, 2.85 % Si and 97.15 % Au). This method is called thermocompression bonding. A layer of gold of about 100 nm is deposed on the silicon wafer (step 1, see Figure 2.6), then the two wafers are aligned and a moderate pressure is applied at a temperature of about 573K (step 2). A removal of silicon dioxide (SiO2) from the wafer surface before the deposing of the gold layer and the cleaning of the gold from any organic contaminant is necessary to realize a good bonding.

2.5 Some problems encountered with the microchannels

As the method chosen to create the rectangular microchannels was to join two silicon wafers etched at half of depth (h/2) with the same width w, the alignment of the wafers have to be done with a good precision to obtain the desired final shape. A misalignment of the wafers can result in non suitable shapes, especially for the microchannels with small width, where

1Composition formed at the lowest possible temperature of solidification for any mixture of specified con- stituents. Used especially of an alloy whose melting point is lower than that of any other alloy composed of the same constituents in different proportions. 2.5. Some problems encountered with the microchannels 13

(a) (b)

Figure 2.7: Pictures showing the misalignment (a) and the gold layer takeoff (b) problems. the lateral walls surface is relatively important comparing to the total surface. Figure 2.7(a) shows a misalignment made in a microchannel with an aspect ratio of about 0.5. One can see that this misalignment have changed the microchannel shape.

Another problem was encountered with the gold coated microchannels, which is the gold layer takeoff, resulting in partial or complete blockage of the microchannel cross-section. Fig- ure 2.7 (b) shows a microchannel where the layer of gold was partially taken-off. This problem can also happen inside the microchannel, which was observed with some microchannels coated with a layer of gold.

2.5.1 Microchannels Fabrication Summary

The summary of the technique used to manufacture the rectangular microchannels is the following. The rectangular microchannels are etched in two identical silicon wafers according to a cavity of h/2 depth using the DRIE technique. A layer of gold (Au) or SiO2 is generated on the wafer surfaces then the two wafers are assembled using a thermo- compression gold-gold bonding. The details of the fabrication process are shown in Figure 2.8.

First, the silicon wafer is prepared and cleaned to remove any metal or organic contami- nation from the surface, and then a drop of photoresist (resin solution ‘SPR 220’) is deposed in the center of the silicon wafer, after that the wafer is then spun at high speed between 500 to 5000 rpm for 30 to 60 seconds. The rotation speed and the viscosity of the resin solution determine the thickness of the photoresist. After the spinning, the silicon wafer is baked to evaporate the solvent in order to form a solid layer (step a). Second, the photoresist is exposed to U.V. light through a mask to define the shape of the channels, which results in two regions: exposed and unexposed regions. Depending on the chemical solution used to develop the photoresist the exposed or unexposed regions are removed (step b). The next step (c) is the DRIE etching, where the silicon wafer is attacked with argon ions (Ar+) that 14 Chapter 2. Microchannels fabrication

(a) Resin coating. (b) Exposure and development.

(c) DRIE etching. (d) Au or SiO2 deposit.

(e) Resin stripping and ions attack. (f) Resin coating.

(g) Exposure and development. (h) Gold (Au) deposit (100 nm).

(i) Resin stripping. (j) Wafers bonding.

Figure 2.8: Rectangular microchannels fabrication process summary.

strike perpendicularly to the surface, removing the material until achieving the desired half depth (h/2) of the microchannel. After the etching step a layer of gold (Au) or silica (SiO2) is generated on the wafer surfaces by cathodic sputtering for gold deposition and by oxidation for silicon dioxide (step d). The resin remains from the first development is striped and the layer of gold or silica is attacked with ions in order to obtain the desired roughness of the surface (step e). Another layer of resin (photoresist) is deposed again on the wafer by sput- tering technique (step f). The photoresist is exposed to the U.V. light and the non-etched regions are developed (step g). Finally, a layer of gold of 100 nm is deposed on the wafer to be uses in the bonding step (step h) and the remaining photoresist is stripped away (step i). The last step is the wafer bonding, the wafers fabricated with the same width (w) and depth (h/2) are assembled together using the thermocompression technique at the eutectic point (636K, 2.85 % Si and 97.15 % Au) by applying a moderate pressure on the wafers at a temperature of 573 K (step j). After the fabrication steps the assembled wafers are cut by a precise saw in order to obtain the microchannels. 2.6. Technical characteristics of the microchannels 15

2.6 Technical characteristics of the microchannels

We will give below the technical characteristics of the rectangular and circular cross-section microchannels used in this thesis for TMAC measurements.

2.6.1 Rectangular cross-section microchannels The total number of the microchannels manufactured is thirty-four; twelve (12) of them were tested. The other microchannels were either blocked with a layer of gold or broken. The microchannels tested have different aspect ratios, surface materials and/or roughness. These microchannels were divided into three groups noted by letters A, E and S (Table 2.1):

• The first group is referenced with the letter A corresponding to the microchannels coated with a gold layer (Au) on the internal surfaces with a mean roughness of 1 nm (RMS, Root Mean Square).

• The second group is referenced with the letter E corresponding to the microchannels coated also with a layer of gold on the internal surfaces, but with a mean roughness of 12 nm.

• The last group (third) is referenced with the letter S corresponding to the microchannels coated with silica (SiO2) layer on the internal surfaces with a mean roughness of 1.12 nm.

The measurement of the cross-section dimensions (height h and width w) for the rectangular microchannels were made after the bonding step of the upper and bottom wafers using optical microscope (Leika DMI3000 M). The microchannels height is measured at ten positions along the channel width and the average value is considered. The microchannels width is taken as the mean value of the max and min measured dimensions. The mi- crochannels length is measured with a numerical caliper. The dimensions of the rectangular microchannels used in the experiments of the mass flow rate measurements are listed in Table 2.1. Each group of microchannels (see Table 2.1) has 4 microchannels with the same characteristics (surface material and roughness) but different aspect ratios (h/w). Four averaged widths are considered: 50, 100, 500 and 1000 µm.

The uncertainty on the microchannels height and width measurements are related essentially to the micro-ruler used to measure the cross-section dimensions and to some eventual other error related to the quality and sharpness of the pictures taken with the microscope. The uncertainty on the microchannels height is estimated as ±0.5 µm. The uncertainty on the microchannels width depends on the average width of the microchannel: ±0.5 µm for the microchannels with average width of 50 and 100 µm, ±1.5 µm for the microchannels with average width of 500 µm and finally, ±3 µm for the microchannels with average width of 1000 µm. This dependence is related to the zoom used to take the pictures of the microchannels cross-section. In another hand, the uncertainty on the microchannels length is only related to the caliper uncertainty and it is ±0.01 mm. 16 Chapter 2. Microchannels fabrication

Figure 2.9: Entrance section of microchannels A1 and A2. Left picture A1, right picture A2.

Figure 2.10: Image processing. Left side: zoom inside the microchannel. Right side: definition of the image edges using ImageJ software.

2.6. Technical characteristics of the microchannels 17

Channel Material Roughness [109 m] h [106 m] w [106 m] L [103 m] A1 27.84 52.23 15.07 A2 0.87 27.60 107.65 15.00 Au A3 27.91 504.00 15.06 A4 1.08 25.80 1005.52 14.87 E1 33.49 55.50 15.02 E2 35.20 103.85 15.19 Au 12.00 E3 34.93 505.00 15.22 E4 34.25 1001.34 15.06 S1 24.30 50.10 13.68 S2 SiO2 1.12 42.30 100.00 15.06 S3 42.00 500.00 15.06 S4 41.50 1000.00 15.06

Table 2.1: Dimensions of the rectangular microchannels.

Some difficulties were encountered with the measurements of the cross-section dimensions of some microchannels, especially with those of the group A. Figure 2.9 shows some of these difficulties. The cut of the microchannels results in some undesirable form of the cross-section (see Figure 2.9). The entrance section of some microchannels was broken because of the cutting technique used, which makes difficult the measurements of microchannel dimensions. To solve this problem we tried to zoom inside the microchannels and with the help of the image processing (see Figure 2.10Alpha-Step) it was possible to have IQ a better view of the microchannels cross-section edges.

Leveling: 2 zones Filter settings Waviness profile Roughness profile

Zoom: none µm Roughness profile, Gaussian Filter, cut-off 0.08 mm

0.012 Ra 0.001114 µm 0.011 Rq 0.001443 µm 0.01 0.009 0.008 0.007 0.006 0.005 0.004 0.003 0.002 0.001 0 -0.001 -0.002 -0.003 -0.004 -0.005 -0.006 -0.007 0 0.05 0.1 0.15 0.2 0.25 0.3 0.35 0.4 0.45 0.5 mm

Figure 2.11: Graph of the roughness for the microchannel S3.

Page 3 / 3 18 Chapter 2. Microchannels fabrication

The roughness of the microchannels was measured using a profilometer (Alpha-Step IQ) before the bonding step of the two wafers. The measurements are made 3 times in the beginning, the center and the end of the microchannel. Figure 2.11 shows an example of roughness measurement made for the microchannel S3. The typical RMS roughness is given in Table 2.1 for each group of microchannels. In Table 2.1 one can see that the roughness of the microchannel A4 is not equal to that of the other microchannels of its group. This is due to the fact that the microchannel A4 was fabricated separately, not at the same fabrication conditions of the other microchannels. However, it was put in the group A only because its roughness is close to those of the other microchannels (A1 − 3) of this group.

2.6.2 Circular cross-section microchannels

Two microtubes T 1 and T 2 (see Table 2.2) used in this study are made from stainless steel and the internal surface of the microtube T 1 is coated with Sulfinert. The Sulfinert is a new fused silica passivation coating used in gas chromatography technique for the storage and transfer of low-level organo-sulfur containing samples. The Sulfinert layer is applied to the stainless steel surface by CVD (Chemical Vapor Deposition) and is completely inert to organo-sulfur compounds. Typically stainless steel adsorbs or reacts with sulfur compounds such as hydrogen sulfide. The Sulfinert layer prevents the sulfur compounds from contacting the reactive stainless steel surface. Both tubes were provided by RESTEK Company. The details of the fabrication process can be found on the company website (www.restek.com).

Channel Material Roughness [109 m] D [106 m] L [103 m] T 1 Sulfinert 287. 2340.∗ not known T 2 Stainless steel 239. 2013.∗

Table 2.2: Dimension of the circular microchannels. ∗ The length of the microtubes T 1 and T 2 were reduced to 9.96 cm and 10.33 cm, respectively, for the experiments yield with CO2 gas.

The indicative diameter of the microtubes given by the provider is 250 µm and its value was corrected (third column of Table 2.2) after comparing the simulated and measured flow rate in the continuum regime when Kn < 0.001 (see Section 5.1 for more details). The measurements of the flow rate were made at least five times for each microtube. The maximum deviation from the mean value of the diameter considered here (see Table 2.2) was around 0.1%. The uncertainty on the diameter measurements is estimated to be ±2 µm.

This kind of microtubes is used in gas chromatography issues to inject samples into the column where the surface treatments of the column give the species separation desired. Their length was set around 2 m and it was measured with a simple metric ruler. The uncertainty on the microtube length measurements is ±5 mm. 2.6. Technical characteristics of the microchannels 19

The microtubes length was reduced when the experiments were carried out with carbon- dioxide (CO2). The molecule of CO2 is the heaviest one so at Knudsen number of order of 1 the mass flow rate becomes too small to be measured with a good precision, that’s why the length of microtubes T 1 and T 2 was reduced to 9.96 cm and 10.33 cm, respectively, in order to increase the mass flow rate.

Chapter 3 Analytical and Numerical Modeling

Due to the small scales involved and therefore the high surface-to-volume ratio, the phenom- ena observed inside micro and nanoscale devices present some special characteristics and differences compared to large scale ones. Thence, the understanding of the phenomena in micro and nanoscale flows, as well as efficient and realistic modeling of those, is of paramount importance for optimal designs of devices operating at these scales.

At microscale, the gas flow can be in a regime different from the classical continuum regime. Using the Knudsen number as a criterion we can distinguish four flow regimes: the hydrodynamic, the slip, the transitional and the free molecular regimes. As previously explained (see Chapter1), all these regimes can be modeled using the kinetic theory, by solving the Boltzmann equation. Nevertheless, it is inefficient to implement this equation or other kinetic equations for gas flow simulation in the hydrodynamic and slip regimes because of the large computational efforts needed for their solution. In the hydrodynamic regime, when the number of collision between molecules is high, the local thermodynamic equilibrium and the continuity of the macroscopic parameters (tangential velocity and temperature) at the wall is achieved, then, the well know continuum model (Navier-Stokes equation) can describe with a very high accuracy the movement of the gas flow in this regime. In the slip regime the Navier-Stokes equation can be still used, but it should be subjected to the velocity slip and temperature jump boundary conditions. In the transitional and free molecular flow regimes, when the continuum approach is no longer valid, the use of the model based on the resolution of the Boltzmann equation such as DSMC (Direct Simulation Monte Carlo), BGK or other kinetic models is paramount.

In this chapter our intention is focused on the study of gas flow through microchannels of rectangular cross-section in all flow regimes. We start by exposing the problem and stating the assumptions used. Then, we divide our description into two parts: the first part concerns the flow in the continuum and slip regimes where the continuum approach based on the Navier-Stokes equation is used. The second part concerns the flow in transitional and free molecular regimes where the linearized kinetic BGK model is used.

In the first part we start by overview some approaches, based on the Navier-Stokes equation, and some slip models given in literature. Then, the expression for mass flow rate through two parallel plates is exposed. After that, the first order model developed by Sharipov [103] for flow through rectangular microchannel with finite width is described. Finally, the details of our original approach, developed in this thesis, leading to completely explicit second order expression for mass flow rate through rectangular microchannel with 22 Chapter 3. Analytical and Numerical Modeling

finite width is given.

In the second part also a brief overview of the kinetic models used in the literature for flow through microchannel is given. Then the description of the numerical calculation of mass flow rate through rectangular microchannels fulfilled in this thesis, based on the linearized BGK kinetic model for flow in the transitional and free molecular regimes is exposed.

3.1 Statement of the problem

Let us consider a channel of rectangular cross-section fixed between two tanks maintained at constant pressure pin and pout, respectively, and at same temperature T (see Figure 3.1). The channel axis coincides with the z axis. The transversal cross-section of the channel, normal to the z axis, is a rectangle whose height h is the smallest critical dimension of the system. The rectangle width w verifies the relation h ≤ w. The channel height to length ratio h/L verifies: h/L = εhL, where εhL  1.

y h/2

-w/2 w/2 x

-h/2

y x pin pout z T T L

Figure 3.1: Rectangular and circular microchannels diagram.

The flow along this channel is engendered by a pressure difference obtained by keeping the inlet and outlet reservoirs at pressure pin and pout, respectively.

The Knudsen number Kn (1.1) is defined as the ratio between the mean free path λ and a characteristic length of the system Lc, where λ can be written as µ√ λ = k 2RT, (3.1) λ p where p is the pressure, R is the specific gas constant and µ is the viscosity coefficient defined 3.2. Continuum and slip regimes 23 as following: !ω T µ = µref . (3.2) Tref

The reference temperature Tref is equal to 273.15K, while the reference values of the viscosity µref for each gas used in this thesis are given in Table 3.1.

kλ is the coefficient that depends on the molecular interaction model. Two models are retained in this thesis:

• The Hard Sphere (HS) model in the Chapman and Cowling version [29], where √ π k = . (3.3) λ 2

• The Variable Hard Sphere (VHS) model [18], where

(7 − 2ω)(5 − 2ω) k = √ . (3.4) λ 15 π

The coefficient ω (viscosity index, see Eq. (3.2)) is equal to 1/2 for the HS model and it depends on the nature of the gas for the VHS model, see Table 3.1. The values of the coefficient kλ when applying the VHS model and for the gases considered in this thesis are given in Table 3.1.

−5 −1 Gas µref ×10 [P a · s] ω kλ R [J/kgK] MolecularMass[g · mol ] He 1.865 0.66 0.786 2077. 4.00 N2 1.656 0.74 0.731 297. 28.00 Ar 2.117 0.81 0.684 208. 39.95 CO2 1.380 0.93 0.607 188.9 44.01

Table 3.1: Gas parameters used in the experiments and simulations.

3.2 Continuum and slip regimes

Continuum theory and the Navier-Stokes equations have formed the basis of studies in fluid and gas dynamics since their derivation by Claude-Louis Navier and George Gabriel Stokes in 1822. For problems of macroscopic dimensions, a solution of the Navier-Stokes equations subjected to the no-slip boundary condition provides a complete and accurate representation of the flow field.

When the dimensions of the system or the pressure decrease, the Knudsen number becomes larger than 10−3 and the continuum regime gives way to the slip regime and the non-slip condition fails to describe the flow in this regime. 24 Chapter 3. Analytical and Numerical Modeling

During the last decades, considerable attention has been given to the study of slip flow in microsystems. Analytical and semi-analytical models of flow in rectangular microchannels derived from the Navier-Stokes equations or from other continuum models require the use of the velocity slip boundary conditions of first or second order.

The two-dimensional flow behaviour in a channel cross-section and the influence of the lateral walls on the main characteristic parameters of the flow (e.g. the mass flow rate) has been studied by several authors. If we disregard some specific cases, like the flow through an equilateral triangular duct [123], the main two-dimensional approaches can be found in the previous works [9, 23, 35, 72, 101, 109].

The solving method proposed by Ebert and Sparrow [35] for rectangular channels in slip regime shows some rough similarities with the present study. These authors used a velocity expansion, following basis functions of functional space, but using a separation variable technique to solve the Stokes equation. Thus the authors could not prove, a priori, the validity of their expansion. Moreover, only the case of a constant pressure gradient in the flow direction seems to have been considered. Finally, the local velocity and the mean velocity on the cross-section were given through very general expressions: the integration constants were not explicitly related to the boundary conditions, where no second order term was present. Furthermore, no expansion of the velocity was considered following Knudsen number orders: thus the slip coefficients were not evaluated.

Later, using an integral transform technique Yu and Amel [133] studied the energy and momentum equations for a rectangular channel and obtained a very complicated formal expression for the velocity. The aim of these authors was to calculate the mean Nusselt numbers. Again, no second order boundary conditions and no expansion according to Knudsen number orders was given. The same limits can be pointed out in the work of Morini and Spiga [82] who also used an integral transform method to obtain the velocity profile in rectangular channels. The author’s purpose was to obtain "two-dimensional correcting factors" to be applied to the shear stress, the momentum flux and the kinetic energy.

Closer to our objectives was a study by Aubert and Colin [9]. These authors undertook a slip regime study of the pressure-gradient-driven flows in rectangular microchannels using a second order boundary condition. They also obtained a velocity expansion on basis function of the L2 functional vector space. Two calculation methods were tested. The first one was related to a separable variable method and did not completely satisfy the authors. The second one, related to a variational approach, required a numerical calculation to determine the basis functions. But, in any case Aubert and Colin [9] used a second order slip condition given by Deissler [34] and based on a phenomenological approach. This condition differs from the second order condition used in this thesis, which was established on the basis of the kinetic theory [23, 109, 111].

Sharipov [101] solved the two-dimensional Stokes equation in the hydrodynamic flow regime with non-slip boundary condition and obtained the explicit expression for the 3.2. Continuum and slip regimes 25 longitudinal component of the bulk velocity and the mass flow rate through a rectangular cross-section channel. However, in the slip flow regime, when the Stokes equation was subjected to the first order slip boundary condition with respect to the Knudsen number, the bulk velocity and the mass flow rate was obtained only numerically.

The following three works [23, 72, 112] have made a fundamental contribution to theory involved in the theme of this thesis: Loyalka et al.[72] presented a general theoretical kinetic study of Poiseuille and thermal creep flows in long rectangular channels, especially in molecular and transitional flow regimes. Thus, their results do not concern the macroscopic formulation to be used in slip flow regime. In contrast, in Cercignani [23], the purpose of the calculation was to obtain a high order slip boundary condition usable at macroscopic level to describe isothermal microflows. Furthermore, the author of Ref [23] focused his interest on the case where two components of the velocity gradient exist in the plane normal to the flow axis. On this basis, the author of Ref. [23] shows that, in the second order term of the velocity slip expansion the second derivative of the velocity according to the normal coordinate is changed into a Laplacian operator reduced to the coordinates describing the points of the channel cross-section. The assumptions used in Ref. [23] and the criticisms of the results, notably formulated in Ref. [112], will be found in Section 3.2.5.

Some of the previous approaches basically allow an evaluation of the influence of a two-dimensional geometry on the procedure for extracting the gas-surface accommodation coefficient or on other features, such as the second order effect in slip regime. Nevertheless, up to now for flows from slip to free molecular regime, the existing two-dimensional approaches required a numerical treatment of the two-dimensional contribution.

3.2.1 Different expressions of the velocity slip coefficient

At macroscopic scale, the conventional fluid mechanics with no-slip velocity condition is appropriate for flow modeling up to approximately a Knudsen number equal to 0.001. When the Knudsen number increases over this value, the non-slip velocity condition should be replaced with the velocity slip condition at wall to describe accurately the flow field.

The first scientists that proposed the presence of the slip of the molecules at wall were Knudt and Warburg [6] in 1875. They observed the fact that when a disk oscillates in vacuum chamber at constant temperature, the decay of oscillation amplitude is not the same as predicted by the continuum viscous model. They explained that the dependence of the oscillation amplitude decrease is due to the slip of the fluid at the interface gas-surface. Later Maxwell, 1878 [78] developed the theoretical basis of the velocity slip at the wall. The author proved, on the basis of the kinetic theory, that when there is no normal to the wall gradient of temperature, the velocity near the wall for planar geometries can be expressed as

2 − α du 3 µ dT u = λ + , (3.5) s α dy s 4 ρT dz s 26 Chapter 3. Analytical and Numerical Modeling where ρ is the density of the fluid, T is the temperature and u is the streamwise velocity and y, z are the streamwise and normal-wall directions, respectively. The subscript s characterizes the values of the tangential component of the velocity at the channel surface. Here α is the so-called tangential momentum accommodation coefficient (TMAC). In this thesis only the isothermal flow conditions are considered, therefore the last term in equation (3.5) is equal to zero.

Later, many authors [23, 57,3, 29, 71, 48, 34] proposed different expressions for the velocity slip at the wall in isothermal gas flow conditions. Albertoni et al. 1963 [3] proposed the following expression for the velocity slip obtained for flow between two parallel plates, in the case of a constant streamwise velocity:

µ√ ∂u u = ±σp 2RT , (3.6) s p ∂y s where σp is the first order velocity slip coefficient. The sign ” + ” and ” − ” in front of the coefficient σp in equation (3.6) refers to the direction of the normal vector to the wall for two parallel plates. Therefore, for the bottom wall the sign is positive and it is the opposite for the upper wall. The authors [3] gave the value of σp = 1.016, which is obtained from the method of elementary solution of the linearized BGK kinetic equation by assuming complete accommodation of the molecules on the wall.

One year later Cercignani [23], on the basis of the linearized BGK equation and also for flow between two parallel plates, proposed a new expression for the velocity slip at the wall, which takes into account the second order term. He obtained the following expression [23]: √ √ µ ∂u µ 2 u = ±σp 2RT − σ2p( 2RT ) ∆u , (3.7) s p ∂y s p s

2 2 2 2 where the operator nabla ∆ is equal to ∂ /∂x + ∂ /∂y and σ2p is the second order velocity slip coefficient that will be discussed bellow. Cercignani, 1964 [23] proposed the value of σp = 1.016 for complete accommodation of the molecules on the wall. This expression (3.7) allows us the extension of the applicability of the Navier-Stokes equation into the beginning of the transitional regime (up to Kn = 0.3).

Many years later in 1989, Loyalka & Hickey [70] and Ohwada et al.[85] solved the Kre- mer’s problem using the linearized full Boltzmann equation and they proposed the following values for first order velocity slip coefficient:

σp = 0.9845 and σp = 0.9849, (3.8) respectively, for full accommodation of the molecules at wall.

By solving numerically the linearized BGK kinetic equation, using the Maxwellian diffuse- specular scattering kernel for various values of the accommodation coefficient α, Loyalka et 3.2. Continuum and slip regimes 27 al.[71] have proposed the following relation between the first order velocity slip coefficient (σp) and the accommodation coefficient α: √ π 2 − α σ (α) = (1 + 0.1621α). (3.9) p 2 α In the case of diffuse scattering (α = 1), the value of the first order velocity slip coefficient found by Loyalka et al.[71] is σp(1) = 1.0299. Later, this expression (3.9) was improved by Sharipov, 1998 [103] and it becomes

2 − α σ (α) = (σ (1) − 0.1211(1 − α)), (3.10) p α p where σp(1) = 1.016 is the velocity slip coefficient for α = 1 calculated by Cercignani & Daneri, 1963 [25] using the BGK kinetic model with assumption of full accommodation of the molecule at the wall.

The value of second order velocity slip coefficient in the case of flow between two parallel plates was calculated by Deissler, 1964 [34](σ2p = 0.884) and Cercignani, 1964 [23](σ2p = 0.766). But these coefficients do not take into account the influence of the Knudsen layer. Recently, in order to take into account the Knudsen layer, Hadjiconstantinou, 2003 [48] proposed a correction to the second order slip coefficient given by Cercignani [23] and the "effective" second order velocity slip coefficient becomes σ2p = 0.310, which is calculated for the case of diffuse scattering.

More recently, Cecignani & Lorenzani, 2010 [28] implemented the variational method to the solution of the full Boltzmann equation with HS molecular model and they calculated the value of the "effective" second order coefficient taking then into account the Knudsen layer influence. They found that this coefficient depends on the gas wall interaction and they obtained this second order coefficient for different values of the tangential momentum accommodation coefficient (see Table 3.2). These values will be compared to the experimental values in Section 6.1.

In the last two decades, many publications appeared where different expressions for the velocity slip were proposed. The authors of these publications introduced the expression for the mean free path in equation (3.7), therefore it becomes

∂u 2 u = ±A1λ − A2λ ∆u . (3.11) s ∂y s s

Expression (3.7) is more general because the molecular interaction model appears only through the viscosity coefficient. However, in expression (3.11) this model must be deter- mined to define the mean free path. Using both expressions (3.7) and (3.11), the coefficients A1 and A2 may be expressed as a function of the first and second order velocity slip coefficients 28 Chapter 3. Analytical and Numerical Modeling

Reference α A2 (HS) σ2p Deissler, 1964 [34] 1 1.125 0.884 Cercignani,1964 [23] 1 0.976 0.766 Hadjiconstantinou, 2003 [48] 1 0.310 0.243 Cecignani & Lorenzani, 2010 [28] 1 0.235 0.184 0.96 0.214 0.168 0.92 0.194 0.152 0.87 0.168 0.132

Table 3.2: Theoretical values for the second order coefficient A2 and the velocity slip coefficient σ2p as function of the tangential momentum accommodation coefficients given by different authors.

(σp and σ2p) as following:

σp σ2p A1 = ,A2 = 2 . (3.12) kλ kλ

The expressions for the coefficient kλ, which depends on the molecular interaction model, are given in Section 3.1.

Table 3.3 lists most of the theoretical values for coefficients A1 and A2 given by the researcher during the last decades. The first order coefficients in Table 3.3 are close to unity, while, there is no consistent between the coefficients of second order, its value ranges from −0.5 to 5π/2. It should be noted that all the values given in Table 3.3 are calculated for the tangential momentum accommodation coefficient α = 1. For an arbitrary value of the accommodation coefficient many expressions are recently proposed in the literature in the frame of the Maxwellian model.

Mitsuya, 1993 [64], [83] developed slip model, known as 1.5 order model, for thin-film gas lubrication problem derived from the kinetic theory by considering the momentum transfer rate of gas molecules impinging on the wall. He propose the following expression:

2 − α ∂u 2 ∂u2 u = λ − λ2 . (3.13) s α ∂y s 9 ∂2y s

From the analysis of the mass flow rate Mitsuya [89] found intermediate characteristic between those corresponding to the velocity slip of first order of Maxwell and of second order of Hsia and Domoto (Table 3.3), hence the name of 1.5 order.

Beskok and Karniadakis, 1999 [14] proposed a "high order" velocity slip conditions of the form 2 − α Kn ∂u u = , (3.14) s α 1 − bKn ∂y s 3.2. Continuum and slip regimes 29

Reference A1 A2 Maxwell, 1878 [79] 1 0 Schamberg, 1947 [97] 1 5π/2 Welander, 1954 [125] 1.21 0 Willis, 1962 [126] 1.121 0 Albertoni et al. 1963 [3] 1.1466 0 Cercignani,1964 [23] 1.1466 0.9756 Deissler, 1964 [34] 1 9/8 Chapman & Cowling, 1970 [29] ≈ 1 ≈ 0.5 Loyalka et al. 1975 [71] 1.0299 0 Hsia & Domoto, 1983 [51] 1 0.5 Mitsuya, 1993 [81] 1 2/9 Beskok & Karniadakis, 1999 [14] 1 −0.5 Wu & Bogy, 2001 [128] 1 1 Hadjiconstantinou, 2003 [48] 1.1466 0.310 Lockerby et al. 2004 [68] 1 0.145 − 0.19 Shen & Chen, 2007 [106] 1 2 Wu, 2008 (Kn < 1)[127] 4/3 1/4

Table 3.3: Theoretical value for the coefficients A1 and A2 given by different authors for fully-diffusive boundary.

where b is an empirical parameter whose value can be determined by DSMC simulations for various Knudsen number regimes. They claimed that they obtained excellent results by utilizing this model as boundary conditions with Navier-Stokes equations in transition region and even in free molecular region.

Using a similar technique, Xue and Fan, 2000 [129] developed a "high order" solution based on the concept of the continuum approach using the Chapman-Enskog method. They introduced a hyperbolic tangent function of Knudsen number in the power series of the distribution function and the expression for the slip boundary condition takes the form

2 − α ∂u u = tanh(Kn) . (3.15) s α ∂y s

Another form of higher order slip condition was suggested by Lockerby et al.[68]. The authors reformulate the generalized full slip condition, but instead of using the linear con- stitutive stress/strain-rate relationship, the authors use a higher order linearized Burnett approximation of the stress tensor, therefore the slip condition for stationary flat surface is given as

2 − α ∂u 9 P r(γ − 1) ∂u2 u = λ − , (3.16) s α ∂y s 4π γ ∂2y s where P r and γ are the Prandtl number and heat capacity ratio. 30 Chapter 3. Analytical and Numerical Modeling

3.2.2 Conservation equation As is well known, in long microchannels under an isothermal flow condition, the Navier-Stokes (NS) equation for the tangential component uz of the bulk velocity ~u = (ux, uy, uz) may be approximated in the form [8, 45]

∂2u ∂2u 1 dp z + z = . (3.17) ∂x2 ∂y2 µ dz Equation (3.17) is obtained using the following assumptions:

• The channel characteristic height-to-length ratio is small enough, εhL = h/L  1.

• The two components of bulk velocity ux and uy are of the order of εhL. • Following on the two previous assumptions the flow Mach number is small, of the order of εhL.

• (pin − pout)/pin ∼ 1, which means that the pressure change between the tanks is of the same order as the pressure itself. In addition, the pressure remains of the same order all along the channel.

As it was mentioned previously, in this section we will proceed in three steps to describe flow through rectangular channels in the continuum and slip regimes:

• First the one-dimensional solution of the Navier-Stokes equation for flow between two parallel plates is given. As this solution is widely presented in the literature (see Graur, 2006 [45] and Ewart, 2007 [38]) only expression for second order for mass flow rate is introduced.

• Second, the two-dimensional approach developed by Sharipov [101] for flow through rectangular microchannel is described. Sharipov [101] followed a methodology based on solving the Stokes equation with the first order slip boundary condition to obtain semi-analytic expression for the mass flow rate through a long rectangular cross-section microchannel.

• Finally, the original two-dimensional approach performed in this thesis to study the flow through rectangular microchannel with a finite width is presented in details. This approach leads to a completely explicit second order expression for the mass flow rate.

3.2.3 Flow between two parallel plates Considering flow between two parallel plates with (h  w), the Navier-Stokes equation (3.17) can be reduced to one-dimensional equation as

∂2u 1 dp z = . (3.18) ∂y2 µ dz 3.2. Continuum and slip regimes 31

Using the second order boundary condition (3.11) in rectangular cross-section microchan- nel with h  w this boundary condition (3.11) becomes ! ∂u ∂2u z 2 z uz∞ = ±A1λ − A2λ , (3.19) s ∂y s ∂y2 s where y is the Cartesian coordinate in the normal direction to the bottom wall.

The mass flow rate between two parallel plates reads

 P + 1  M˙ ∞ = M˙ ∞ 1 + 6A Kn + 12A ln PKn2 , (3.20) P 1 m 2 P − 1 m

˙ ∞ where MP is the classical Poiseuille mass flow rates between two parallel plates calculated per unit of width for a given value of width w, defined as

h3w∆pp M˙ ∞ = m . (3.21) P 12µRTL

Here, ∆p = pin − pout, P = pin/pout and Knm is the mean Knudsen number based on the mean pressure pm = (pin + pout)/2.

Equation (3.20) is often used to calculate the mass flow rate through rectangular microchannels of width w. We will demonstrate later that the use of equation (3.20) for the estimation of the mass flow rate through rectangular channels with aspect ratio w/h less than 50 is no longer accurate. The influence of the lateral walls in such channels must be taken into account.

3.2.4 First order velocity slip condition for rectangular cross-section chan- nels

Some of the microchannels used in this thesis cannot be considered as two parallel plates because their ratios h/w are close to 1, therefore the influence of the lateral walls must be taken into account. For this reason, the use of two-dimensional approach is necessary. The first two-dimensional approach we used in this thesis is developed by Sharipov, 1999 [101]. This approach allows us to calculate the mass flow rate through a rectangular cross-section microchannel. The disadvantage of this method is that the mass flow can be calculated only numerically. Below we present briefly this approach.

In long microchannel the flow may be described by the Stokes equation (3.17). Sharipov [101] postulated that the mass flow rate through a long rectangular microchannel, which satisfy the condition given on page 23, can be divided into two terms as

˙ ch ˙ ch ˙ M = MP + Ms, (3.22) 32 Chapter 3. Analytical and Numerical Modeling

˙ ch where MP is the Poiseuille mass flow rate through a channel of rectangular cross-section and M˙ s is the slip flow correction term.

In order to obtain the first term of expression (3.22) the Stokes equation (3.17) must be solved, subjected to the non-slip boundary condition uz(x, y) = 0 on the walls. This solution has the following form [101]: " # h2 dp 1 y2 ∞ (−1)i cosh(nx/h) cos(ny/h) u (x, y) = − − − 8 X , n = π(2i + 1). zH 2µ dz 4 h2 n3 cosh(nw/2h) i=0 (3.23) Integrating this velocity expression (3.23) over the channel cross-section

Z w/2 Z h/2 ˙ ch p MP = uzH (x, y)dxdy, (3.24) RT −w/2 −h/2

˙ ch the Poiseuille mass flow rate MP (3.22) can be obtained as

h3wp dp M˙ ch = − (1 − K) (3.25) P 12µRT dz with h ∞ 1 nw K = 192 X tanh . (3.26) w n5 2h i=0 Here K is the factor which takes into account the lateral walls influence. For rectangular channel with aspect ratio h/w = 1 the factor K is equal to 0.63, while when the aspect ratio tends to zero (parallel plates), the factor K tends also to zero.

Integrating this expression over the channel length L

Z L ˙ ch 1 ˙ ch MP = MP dz, (3.27) L 0 and using the mass conservation propriety the following expression for the Poiseuille mass flow rate related to the inlet and outlet pressures is obtained

h3w∆pp M˙ ch = m (1 − K) = M˙ ∞(1 − K). (3.28) P 12µRTL P

˙ ∞ Here MP is the Poiseuille mass flow rate between two parallel plates (see Eq. (3.21)). The factor (1 − K) allows taking into account the lateral wall influence.

Let us now represent the bulk velocity as [101]

uz(x, y) = uzH (x, y) + uzs (x, y), (3.29)

where uzH (x, y) is the hydrodynamic solution (3.23) and uzs (x, y) is the slip correction. Since 3.2. Continuum and slip regimes 33

Microchannel(w/h) A1(1.9) A2(3.9) A3(18.1) A4(39.0) Qs(w/h) 0.7339 0.8701 0.9721 0.9872

Table 3.4: Value of the slip correction Qs(w/h) 3.33 calculated for the aspect ratios (w/h) of the microchannels of the group A.

both uzH and uz satisfy equation (3.17) the slip correction uzs must satisfy the following equation: ∂2u ∂2u zs + zs = 0. (3.30) ∂x2 ∂y2

In this case the boundary conditions for uzs reads √ √ " ∞ # µ ∂uzH σp h dp X 1 cosh(nx/h) uz (x, ±h/2) = ∓σp 2RT = 2RT 1 − 8 , s p ∂y 2 p dz n2 cosh(nw/2h) y=±h/2 i=0 √ √ ∞ i   µ ∂uzH h dp X (−1) nw uz (±w/2, y) = ∓σp 2RT = 4σp 2RT tanh cos(ny/h), s p ∂x p dz n2 2h x=±w/2 i=0 (3.31) with n = π(2i + 1). The slip flow correction term M˙ s in Eq. (3.22) is calculated as following:

Z w/2 Z h/2 ˙ p Ms = uzs (x, y)dxdy, (3.32) RT −w/2 −h/2

where the velocity uzs (x, y) is obtained by solving numerically equation (3.30) subjected to the boundary conditions (3.31) using the Gauss method for each aspect ratio h/w. It has to be noted that the duration of calculation depends on the number of points taken in the directions of x and y, respectively. To reach the desired accuracy on the term M˙ s we used 120 points in both direction x and y. The duration of the calculation is around 2 hours for each aspect ratio h/w with computer 2 cores, 2.4 GHZ.

The values of the dimensionless slip flow correction Qs normalized as following: √ 2RT Q = − M˙ , (3.33) s h2wdp/dz s and calculated for various aspect ratios (w/h). Table 3.4 shows the slip flow correction Qs calculated for the aspect ratios of the microchannels of the group A.

The mass flow rate due to the slip flow correction M˙ s can be integrated along the channel length L using the mass conservation propriety and it is found as

h3w∆pp M˙ = m 6A Q (w/h)Kn , (3.34) s 12µRTL 1 s m where Knm is the mean Knudsen number, based on the mean pressure pm = 0.5(pin + pout) 34 Chapter 3. Analytical and Numerical Modeling and on the critical dimension h.

˙ ch ˙ Replacing expressions of MP (3.28) and Ms (3.34) in equation (3.22), the mass flow rate through rectangular microchannel M˙ ch can be obtain as

 Q (w/h)  M˙ ch = M˙ ∞(1 − K) 1 + 6A s Kn . (3.35) P 1 1 − K m

This latter equation (3.35) of the mass flow rate will be compared with the analogous expres- sion developed coming from our approach which is developed in the next section.

3.2.5 Bi-dimensional approach developed in this thesis We showed previously that the Sharipov’s approach [101] has some disadvantages. The term Qs in expression for mass flow rate (3.35) has to be calculated numerically for each microchannel aspect ratio. In addition, this approach leads to expression for the first order for the mass flow rate, useful for Kn ≤ 0.1.

In the frame of this thesis we developed a new approach leads to a completely analytical expression for second order according to the Knudsen number for the mass flow rate. This approach is detailed below.

From expressions of mean free path (3.1) and Knudsen number (1.1), it is clear that the Knudsen number is a local quantity which depends on the z coordinate. Moreover, it is important to remember that, since we have an axis (or a plane) of symmetry, the second order terms with respect to the Knudsen number are negligible in the conservation equation system and are important only when they result from the slip boundary condition [23, 24]. Therefore, the expression for the velocity slip boundary conditions (3.11) in bi-dimensional cross-section can be rewritten in the following form:

∂uz 2 uz = ±A1λ − A2λ ∆uz . (3.36) s ∂~n s s The vector ~n identifies the direction normal to the wall, i.e., successively either x or y. As usual, at any point of the solid surface, ~n is oriented by the entering normal. Consequently in front of A1, the sign + is chosen when relation (3.36) is for a solid surface located at a positive value of the x (or y) coordinate.

Some additional comments are needed regarding equation (3.36). In Cercignani [23], the results were obtained using some assumptions notably a BGK equation model and a gas of Maxwell molecules model. Moreover, the velocity gradient in the flow direction was neglected. Finally, for simplification, the author considered first a flow over a plane wall. Nevertheless, the author applied his result to calculate the mass flow rate in a cylindrical tube. Cercignani’s [23] calculation generated critical comments: the authors of Refs. [111], [112] claimed that the curvature of the cylindrical tube was not correctly taken into account. The same criticism 3.2. Continuum and slip regimes 35 is to be found in Hadjiconstantinou [48]. However, Hadjiconstantinou [48] considered that the assumption of constant velocity in the flow direction led to rather correct results. In any case, no criticisms were voiced against the Laplacian functional form of the second order term.

Furthermore, in the rectangular channel, the wall surface materials, the surface roughness and the surface curvatures are the same on the vertical and horizontal walls and we can thus consider the same coefficient A2 for all these walls. Consequently, Cercignani’s [23] analysis leads directly to equation (3.36). We can add at this point that, as underlined in Section 3.2.1, when referring to the theoretical value of the coefficient A2 we can now take into account the recent Cercignani & Loranzani [28] analysis (see Table 3.2).

Nevertheless, a more fundamental question could be investigated experimentally: is the A2 coefficient dependent on the w/h ratio? Comparing the mass flow rate measurements obtained using the same gas, in the same physical conditions in various rectangular channels differing only by their width, it would be possible to obtain information on this point.

Now we will resolve equation (3.17) subject to boundary condition (3.36). It results from equation (3.17) that ∆uz does not depend on the x and y coordinates whatever the point considered in the channel cross-section. Thus ∆uz in Eq. (3.36) will be constant at any s point of a section perimeter. The same comment may be applied to the molecular mean free path λ, owing to the pressure property. Then it is convenient to use the function change

∗ 2 u = uz + A2λ ∆uz . (3.37) s Consequently equation (3.36) is changed into:

∗ ∗ ∂u u = ±A1λ . (3.38) s ∂n s Naturally, the form of equation (3.17) does not change and it may be rewritten using the new function u∗

1 dp ∆u∗ = . (3.39) µ dz

In the following section, we will present the general approach that will be used to solve equation (3.39) with the boundary condition (3.38) to determine the unknown function u∗ and, further uz, by using relation (3.37).

3.2.6 The theoretical bases of our approach

The present approach is mainly based on the spectral properties of the Laplace oper- ator included in the Stokes equation (3.39). When applied to the vector space of L2 36 Chapter 3. Analytical and Numerical Modeling functions defined on a finite spatial domain D, the Laplacian is characterized by an eigenfunction discrete spectrum. Now on the border of this spatial domain we define a "condition of cancellation" for the L2 functions, i.e. either the nullity of the function, or the nullity of its normal derivative, or a constant ratio between the function and its derivative.

If we require the Laplacian eigenfunctions to verify this "condition of cancellation" on the border of the spatial domain, then their countable set provides a basis of the vector space of L2 functions defined on the spatial domain D and subjected to the same "cancellation condition". Moreover, each function of the set is orthogonal to the other set functions in the sense of the Hermitian product defined on the functional space. Finally the eigenvalues associated with these eigenfunctions form a discrete countable series of real negative numbers. Then, in the finite spatial domain of a microchannel cross-section, it appears convenient to search for the solution of the Stokes equation (3.39), i.e. the velocity u∗ satisfying cancellation conditions (3.38), in the form of an expansion following the Laplacian eigenfunctions which verify equation (3.38) since:

• The basis property of these eigenfunctions guarantees the uniqueness of the solution thus found.

• Generally, the boundary condition verified by the eigenfunctions insures the velocity boundary conditions, because of their respective similarity. Here, for example, u∗ will verify the cancellation condition (3.38).

• The use of the Laplacian eigenfunction basis allows significant simplifications in the Stokes equation: the Laplace operator vanishes and the partial derivative equation reduces to a simple differential equation, generally easy to solve.

But the general process described above concerns an expansion following the eigen- functions of the complete Laplace operator. Thus, this process would rather be suitable for treating non-stationary flows governed by a conservation equation corresponding to a modified equation (3.39) also including time derivatives.

In the present case of a flow through a rectangular channel cross-section, under steady and isothermal conditions, the momentum conservation equation reduces to the Poisson equation (3.39), where the (x, y) Laplace operator is applied to the u∗ function. Thus, the knowledge of the spectrum of the complete operator is not of practical interest here. To simplify the resolution of Eq. (3.39), it is convenient to split the operator into different parts respectively regarding the different spatial variables. Then it is easily shown that all the properties quoted above, concerning the spectral elements of the complete Laplace operator acting on 2 2 2 2 the functional space L2 (x, y), may be transposed on each partial operator ∂ /∂x or ∂ /∂y . Of course, each partial operator acts on the restricted functional space that depends on the spatial variable considered in each partial derivation. Therefore, the most convenient partial operator will be chosen for investigation in regard to its spectral elements. Thus, the solution will again be sought in the form of a series of functions. But for the present stationary study 3.2. Continuum and slip regimes 37 the velocity will be expanded according to the eigenfunctions of the chosen partial Laplace operator.

3.2.7 Analysis of the boundary conditions. Choice of an orthogonal func- tion set

We now have to choose the partial space function depending on the most convenient space parameter leading to the easiest resolution of the problem. This choice requires us to analyze the boundary conditions verified by the velocity u∗. It is obvious that the slip boundary condition (3.38), which relates u∗ and its normal derivative at the wall, is a "cancellation condition" as discussed previously in section 3.2.6. This boundary condition may be rewritten on the wall, at the y limits (±h/2) and at the x limits (±w/2), see Figure 3.1. But it is more convenient, using the symmetry planes (x = 0 and y = 0) of the system to consider only the reduced spatial domain defined by: 0 ≤ x ≤ w/2 and 0 ≤ y ≤ h/2. Then the boundary conditions in a quarter of the channel cross-section read:

∂u∗ (0, y) = 0, ∂x ∂u∗ u∗(w/2, y) = −A λ (w/2, y), (3.40) 1 ∂x and ∂u∗ (x, 0) = 0, ∂y ∂u∗ u∗(x, h/2) = −A λ (x, h/2). (3.41) 1 ∂y

Thus the functional space L2(x) (associated with the boundary condition at x = w/2 and with 2 2 the partial Laplacian ∂ /∂x ) or the functional space L2(y) (associated with the boundary condition at y = h/2 and with the partial Laplacian ∂2/∂y2) are both convenient construct either an orthogonal normal basis of x functions or an orthogonal normal basis of y functions. Thus we may choose to seek the u∗ function in the series form

∞ ∗ X u = gn(x)fn(y), (3.42) n=0 where the summation index n is a natural integer and the set of functions fn(y) is provided by the eigenfunctions of the ∂2/∂y2 operator verifying, at the y limits, conditions similar to (3.40), and where gn(x) are functions of L2(x) space, which will be found by solving equation (3.39) subject to conditions (3.41).

We will see in the next section that gn(x) and fn(y) functions are parametrically dependent on z due notably to the boundary "cancellation conditions". This dependence will appear through the "constants" introduced in the integration process. 38 Chapter 3. Analytical and Numerical Modeling

3.2.8 Search for the reduced velocity

Now we will obtain the explicit expressions for the fn(y) and gn(x) functions involved in expansion (3.42).

The fn(y) basis will be obtained by solving the system

d2f (y) n = −ν2f (y), (3.43) dy2 n n

df f (h/2) = −A λ n (h/2), (3.44) n 1 dy

df n (0) = 0, (3.45) dy

2 where −νn are the eigenvalues of the partial Laplace operator on L2(y) space.

Taking into account (3.45), the general solution of Eq. (3.43) reduces to

fn(y) = an cos(νny), (3.46) where an is a set of constants to be defined. Then using equation (3.44), we obtain the following equation

cotan(βn) = 2A1Knβn, (3.47) with

βn = νnh/2. (3.48)

In expression (3.47), the local Knudsen number Kn, defined by relation (1.1), depends on z via the molecular mean free path (3.1). The solution of equation (3.47) gives the eigenvalues νn (or βn with notation (3.48)) of the partial Laplace operator on L2(y) space.

Using the Hermitian scalar product we normalize the fn(y) functions, thus obtaining the set of corresponding constants an: p 2 2βn/h an = p . (3.49) 2βn + sin(2βn)

Next, we will obtain the explicit expressions for the gn(x) functions. Let us remember that the second member of Eq. (3.39) depends only on the z space variable and so may be 3.2. Continuum and slip regimes 39 noted as 1 dp P (z) = . (3.50) µ dz

Using the u∗ velocity expansion (3.42) in Eq. (3.39) and taking into account the basis property of the fn(y) functions, we obtain the differential equation governing the gn(x) functions

d2g (x) n − ν2g (x) − s = 0, (3.51) dx2 n n n valid for all natural n, where sn are the coefficients of the expansion of P (z) (see Eqs. (3.39),(3.50)) on the fn(y) basis functions

∞ X P (z) = snfn(y). (3.52) n=0

The solution of the homogeneous equation associated with Eq. (3.51) involves the sum of two exponential functions. Moreover, the complete solution of the non-homogeneous Eq. (3.51) includes a constant obvious solution. Finally, the gn(x) expression leads to

νnx −νnx 2 gn(x) = Ane + Bne − sn/νn. (3.53)

Using expression (3.42) we obtain the velocity u∗ in the following form

∞ ∗ X  νnx −νnx 2 u = Ane + Bne − sn/νn an cos(vny). (3.54) n=0

The constants An and Bn are determined using the boundary conditions (3.41) on the x space variable at 0 and w/2. Thus, we find first that An = Bn, and hence we obtain

2 snh −1 An = 2 (cosh(βnw/h) + 2A1Knβn sinh(βnw/h)) , (3.55) 8βn where the infinite set of βn (or νn) values is obtained by solving Eq. (3.47). Finally, expanding P (z) we find

h/2 Z h an sn = P (z) fn(y)dy = P (z) √ sin(βn). (3.56) 0 2 βn

Then, from Eq. (3.54), using expressions (3.48)-(3.50) and (3.56), we find

h2 dp ∞ u∗ = X φ (x, y), (3.57) µ dz n n=0 where the following new notations are used 40 Chapter 3. Analytical and Numerical Modeling

  sin(βn) cos(2βny/h) cosh(2βnx/h) φn(x, y) = 2 × − 1 , (3.58) βn(2βn + sin(2βn)) cosh(βnw/h) + βnε sinh(βnw/h)

ε = 2A1Kn, ε << 1. (3.59)

In contrast to the one-dimensional case, where the flow through the channel of rectangular cross-section is approximated by a one-dimensional flow between two parallel plates [8, 45], the present two-dimensional method leads neither to a reduced velocity u∗, under a linear form of the local Knudsen number Kn(z), nor to a velocity uz under a quadratic form of Kn(z). Then, solution (3.57) of the Stokes equation (3.39) implicitly contains the orders of the Knudsen number, higher than order 1 or 2, which means too small to be pertinent in the framework of the Navier-Stokes approach, which is used here in slip regime conditions. Therefore, to remain in the usual field of the continuum fluid dynamic equations, it is now necessary to develop the velocity u∗ according to the Knudsen number and to limit these expansions in a suitable manner.

This explicit method is all the more useful and essential as, in slip regime, the experimental results are also fitted according to a linear or a quadratic mean Knudsen number functions and are presented in this form. Thus it is very convenient to make comparison with a theoretical expression for a similar form in order.

3.2.9 Expansion method

It is now necessary to describe and justify this expansion process of the velocity functions obtained previously. Let us consider a "point" C in the gas, in a cross-section of the channel. At this point, in the usual laboratory frame of reference, the fluid particle verifies the Navier-Stokes system, especially Eq. (3.17). Moreover, in this section on contact with the wall, the gas particle obeys the boundary condition given in Eq. (3.36). Now consider moving frame of reference characterized by a constant speed U in the z direction relative 2 to the laboratory frame of reference, such that: U = −A2λ ∆uz . From a formal point of s view, in this new Galilean frame of reference, the gas particle located at point C verifies a ∗ Navier-Stokes system similar to the previous one, only changing the velocity uz into u and Eq. (3.17) into Eq. (3.39). In addition, for this gas particle, the boundary condition at the wall, governed by Eq. (3.36), is changed into Eq. (3.38). In the new frame of reference the ∗ particles move with the new velocity u different from uz, and thus the local thermodynamic parameters (density, pressure), calculated in the new frame of reference, undergo the corresponding changes. However, the velocity change introduces terms of second order according to the Knudsen number. Therefore, the changes induced on the thermodynamic parameters are not significant in the Navier-Stokes equations system. Then, from a physics point of view, it becomes obvious that the velocity u∗, which represents the velocity of the gas under consideration with respect to the new moving frame of reference, is to be 3.2. Continuum and slip regimes 41 developed at first order of Knudsen number, because, in this frame of reference, the second or- der terms are negligible or vanishing, so the u∗ expansion will be fulfilled up to first order only.

Finally, coming back to the fluid equations in the laboratory frame of reference, from the ∗ u expansion we obtain uz directly using Eq. (3.37) without any other calculation. Finally to consolidate the consistency of the present approach, let us recall that, as noted in Section 3.2.2, it is pertinent here to associate a first order Navier-Stokes model, governing the fluid particles inside the fluids, with a second order boundary condition describing the gas-wall interaction.

The implementation of the expansion described above is given in the next section, where ∗ the explicit expressions for velocities u and uz, and then the mass flow rate M˙ are obtained.

3.2.10 Implementation of the expansion method

Expression (3.57) of u∗ is expanded up to the first order according to the small parameter ε defined in Eq. (3.59), which is proportional to the local Knudsen number. Since u∗ depends on ε not only explicitly through (3.59), but also implicitly via βn which is related with the local Knudsen number in Eq. (3.47). Using the ε parameter, Eq. (3.47) is now rewritten as follows:

cotan(βn) = βnε. (3.60)

Then the basic form of the expansion u∗ will be:

 ∗  ∗ ∗ du u = u0 + ε , (3.61) dε 0 where du∗  du∗ ∗ ∗ u0 = u , = , (3.62) ε=0 dε 0 dε ε=0 and

du∗  ∞  ∂u∗ dβ  ∂u∗  = X n + . (3.63) dε ∂β dε ∂ε 0 n=0 n 0 0 To make the calculation easier to handle, it is important to note now that, at zero order following ε, Eq. (3.60) reduces to cotan(βn) = 0, so that:

βn = (βn)0 = π/2(2n + 1). (3.64) ε=0 42 Chapter 3. Analytical and Numerical Modeling

We start with the calculation of the first term in Eq. (3.61), which is easily obtained, using Eqs. (3.58) and (3.64) equal to

h2 dp ∞ u∗ = X(φ ) , (3.65) 0 µ dz n 0 n=0 with

n 4(−1) cosh(knx/h) − cosh(ωn) (φn)0 = 3 cos(kny/h) , (3.66) kn cosh(ωn) where kn = π(2n + 1) and ωn = knw/(2h).

Now we calculate the derivatives in Eq. (3.63). First, this equation may be written in the following form using the function φn (3.58)

du∗  h2 dp ∞ ∂φ dβ  ∂u∗  = X n n + . (3.67) dε µ dz ∂β dε ∂ε 0 n=0 n 0 0   The derivative ∂φn is easily calculated from Eq. (3.58). Otherwise, the derivative ∂βn 0 (dβn/dε)0 is calculated from the implicit function defined by Eq. (3.60), so, taking into account (3.64), we obtain:

dβ  n = −π/2(2n + 1). (3.68) dε 0 Then, we finally obtain from the calculation required in Eq. (3.67):   ∂φn dβn = (ϕn)0, (3.69) ∂βn dε 0 where 2(−1)n+1  4 2y  (ϕn)0 = (cosh(ωn) − cosh(knx/h)) cos(kny/h) + sin(kny/h) kn cosh(ωn) kn h 2x w  − cos(k y/h) sinh(k x/h) − sinh(ω ) n h n h n w − tanh(ω ) cos(k y/h) (cosh(k x/h) − cosh(ω )) . (3.70) h n n n n The second term on the right hand side of Eq. (3.67) has the following form deduced from (3.57) et (3.58)

∂u∗  h2 dp ∞ = X(χ ) , (3.71) ∂ε µ dz n 0 0 n=0 3.2. Continuum and slip regimes 43 with 2(−1)n+1 (χn)0 = 2 tanh(ωn) cos(kny/h) cosh(knx/h). (3.72) kn cosh(ωn) Finally, from Eqs. (3.61) and (3.65) and using Eqs. (3.70) and (3.71), we obtain

! h2 dp ∞ ∞ u∗ = X(φ ) + 2A Kn X(ψ ) , (3.73) µ dz n 0 1 n 0 n=0 n=0 with

(ψn)0 = (ϕn)0 + (χn)0.

Then uz is deduced from (3.73), recalling Eq. (3.37) rewritten as

h2 dp u = u∗ − A Kn2 . (3.74) z 2 µ dz

3.2.11 Mass flow rate calculation

The mass flow rate through a rectangular microchannel with cross-section of width w and height h is calculated from expression (3.73). Using expression (3.74) of the velocity uz, the mass flow rate M˙ ch reads " 8h3w dp ∞ tanh(0.5π(2n + 1)w/h)/(0.5π(2n + 1)w/h) − 1 M˙ ch = p X π4µRT dz (2n + 1)4 n=0

∞ 4(1 − tanh(0.5π(2n + 1)w/h)/(0.5π(2n + 1)w/h)) −2A pKn X 1 (2n + 1)4 n=0 ! #  h  ∞ tanh2(0.5π(2n + 1)w/h) π4A p − 1 − X − 2 Kn2 . (3.75) w (2n + 1)4 8 n=0

The local Knudsen number Kn may be expressed through the mean Knudsen number Knm based on the mean pressure pm = (pin + pout)/2, by using

p Kn = m Kn . (3.76) p m

In order to obtain the expression for the mass flow rate as a function of the inlet and outlet pressures we can integrate expression (3.75) along the channel by using the mass conservation property. Finally, the mass flow rate reads

3 4 ˙ ch h w∆ppm Tn A2π P + 1 2 M = Vn(1 + 6A1 Knm + ln PKnm), (3.77) 12µRTL Sn 16Sn P − 1 where ∆p = pin − pout, P = pin/pout, 44 Chapter 3. Analytical and Numerical Modeling

w/h 1 2 5 10 20 50 100 ∞ Qs(w/h) [101] 0.5623 0.7492 0.8985 0.9493 0.9746 0.9902 0.9941 1.000 4 96Tn/π 0.5623 0.7493 0.8986 0.9493 0.9747 0.9899 0.9949 1.000

Table 3.5: Comparison of the slip correction Qs(w/h) [101] with corresponding coefficient of 4 the present approach 96Tn/π .

π4 2h ∞ tanh(0.5π(2n + 1)w/h) S = − X , (3.78) n 96 πw (2n + 1)5 n=0

96 V = S , (3.79) n π4 n and

4 1  h  ∞ tanh2(0.5π(2n + 1)w/h) T = S − 1 − X . (3.80) n 3 n 3 w (2n + 1)4 n=0 To put mass flow rate (3.75) in form (3.77), the value of the following sum is calculated analytically

∞ 1 π4 X = . (3.81) (2n + 1)4 96 n=0

Expression (3.77) can be rewritten using the Poiseuille mass flow rate between two parallel ∞ plates MP (3.21) as

4 ˙ ch ∞ Tn A2π P + 1 2 M = MP Vn(1 + 6A1 Knm + ln PKnm), (3.82) Sn 16Sn P − 1

3.2.12 Comparison with other methods

Now, we will compare the expression obtained for the mass flow rate (3.82) through a channel of rectangular cross-section with the analogous expression (3.35) obtained by Sharipov, 1999 [101].

As we mentioned before, the expression obtained with Sharipov’s approach [101] is of first order, while that with our approach is of the second order. Neglecting the second order term in expression (3.82) one can see that the factors (1 − K) in equation (3.35) and Vn in Eq. (3.82) are equal. By comparing the coefficients in front of the first order Knm term in expressions (3.82) and (3.35), it is possible to observe that the value of the coefficient 4 96Tn/π are equal to the value of the coefficient Qs(w/h) (see Table 3.5).

From this comparison, it is clear that both approaches give similar results. The evident 3.3. Transitional and free molecular regimes 45 advantage of the proposed approach is that the mass flow rate through a rectangular channel in slip regime using the first or second order boundary conditions may be obtained directly by using the simple relation (3.82) without the necessity to solve equation (3.30) with the boundary conditions (3.31) numerically, when w/h changes.

Now, the mass flow rate through a channel of rectangular cross-section (3.82) will be compared with the mass flow rate obtained between two parallel plates (3.20).

We can easily compare the expressions (3.82) and (3.20) in the limit case when the channel width w tends to infinity for a fixed value of the channel height. The factors Vn before the brackets in Eq. (3.82) tend to 1 (Vn → 1). Comparing the coefficients before the second terms in the brackets of Eqs. (3.82) and (3.20), we find that Tn/Sn → 1, when w → ∞, so the coefficients before the second terms are the same. Finally, comparing the coefficients before 4 the third terms we find that π → 6 for w → ∞. Therefore, we find expression (3.20) from 16Sn Eq. (3.82) when w → ∞ for a fixed channel height.

w/h 1 2 5 10 20 50 100 ∞ M˙ ch/M˙ ∞ 0.479 0.712 0.884 0.942 0.971 0.988 0.994 1.000

Table 3.6: Influence of lateral walls on the mass flow rate through a rectangular channel M/˙ M˙ ∞.

In Table 3.6, the ratio between the mass flow rate M˙ ch (3.82) through a channel of rectan- gular cross-section is compared to the mass flow rate M˙ ∞ between two parallel plates (3.20) referred to the same width w. To calculate the value of coefficient A1, see (3.12), the Hard Sphere molecular interaction model [29] is used: this model gives the value of k coefficient √ λ equal to π/2. From the results of Table 3.6, we can conclude that the approximation made using the expression for a flow between two parallel plates overestimates the mass flow rate by approximately 29%, in the case were the aspect ratio is w/h = 2. However, when the width to height ratio of the channel is larger then 50, expression (3.20) may be used with ∼ 1% accuracy.

3.3 Transitional and free molecular regimes

The transitional regime was for a long time the more complicated regime for simulation. In this regime the continuum approach based on the Navier-Stokes equation breaks down and the uses of the molecular approach based on the Boltzmann equation (BE) becomes necessary. Due to the complexity in resolving the Boltzmann equation, especially the collisional term, many models (e.g. BGK, S-model and ES-model) were developed in order to simplify the collisional term and to reduce the computational efforts needed to resolve the BE.

One of the first attempts to use the BGK model was the work of by Cercignani & Daneri 46 Chapter 3. Analytical and Numerical Modeling in 1963 [25]. The authors [25] simulated the gas flow between two parallel plates for a large Knudsen number range. They obtained the reduced mass flow rate by assuming full accommodation of the molecules at the wall. Later in 1975, Loyalka et al.[71] used the BGK model to obtain the mass flow rate through two parallel plates for various accommodation coefficients. One year after Loyalka et al.[72] resolved numerically the BGK equation for flow through a rectangular channel with diffuse assumption for the reflexion of the molecule from the wall.

Cercignani & Sernagiotto, 1966 [27], then Cercignani and Pagani, 1967 [26] solved the BGK equation for flow through microtube using the Discrete ordinate and variational methods, respectively. In the same period, Ferziger [40] studied analytically and numerically the flow through cylindrical tube and obtained analytically the expression for flow rate near the free molecular regime. For the same geometry (circular) the BGK model was applied again by Porodnov & Tuchvetov, 1979 [94] and Lo et al. 1984 [67] for diffuse-specular reflexion at wall and by Lo & Loyalka, 1982 [66] and by Porodnov & Tuchvetov, 1978 [93] for diffuse reflexion. Subsequent works of Lang & Loyalka, 1984 [63] and Valougeorgis & Thomas, 1985 [121] reported numerical results of the Poiseuille and thermal creep flows using the BGK model in a cylindrical tube.

In the early ninety, Ohwada et al.[85] and Hickey & Loyalka [50] developed a kinetic approach based on the linearization of the Boltzmann equation for modeling the flow through parallel plates with the assumption of full accommodation of the molecules at the wall. In 1999 Sharipov [101] used the linearized BGK model equation for flow through rectangular channel with a diffuse assumption. Various aspect ratios of the channel cross-section and large rarefaction parameter range were studied.

Later Sharipov, 2003 [102] have used the S-model of Shakhov [100] to simulate numer- ically the flow of gas through long tube using the discrete velocity method. The author [102] used the Cercignani-Lampis scattering kernel for gas-surface interaction. The reduced mass flow rate was presented in a table as the function of the rarefaction parameter and accommodation coefficient α. Breyiannis et al. 2008 [19] simulated the flow through circular ducts of concentric annular section by the use of the BGK equation subjected Maxwell diffuse-specular boundary condition. The results on the flow rate as function of the accommodation coefficient were given for all flow regimes.

Finally, Sharipov & Selznev in 1998 [103] presented a large review of numerical results on the flow through circular and rectangular channels based on various computational approaches. The authors [103] compared the numerical results of the linearized BGK model with the numerical solution of the linearized Boltzmann equation; they showed that the numerical error made by using the linearized BGK model instead of BE is within 2%, which is considered reasonable.

Due to the lack in the literature of the numerical results of the mass flow rate through rectangular microchannels for diffuse-specular reflexion (0 < α < 1) at wall, in this thesis 3.3. Transitional and free molecular regimes 47 we developed a numerical code based on the linearized BGK approach for the simulations of flow through rectangular channels with the assumption of diffuse-specular reflexion of the molecules at the wall. A large rarefaction parameter ranging from 0.001 (free molecular regime) to 20 (slip regime) and various aspect ratios (h/w = 1, 0.5, 0.2, 0.1, 0.05, 0.02, 0.01 and 0.001) are considered. Different TMAC values are also taken into account (α = 1, 0.95, 0.90, 0.85, 0.80, 0.6). A comparison of the dimensionless mass flow rate obtained in this thesis using the assumption of full accommodation (α = 1) with the results of Loyalka [72] and Sharipov [101] is presented in Table (3.8).

In the following the description of the approach based on the linearized kinetic BGK model used to obtain the mass flow rate through the rectangular channels is given.

3.3.1 Problem formulation

The mass flow rate through a rectangular microchannel fixed between two tanks maintained at constant pressure pin and pout, respectively, and at same temperature T (see Figure 3.1) is calculated for transitional and free molecular regimes. Two assumptions are made:

• The channel width is larger than the channel height (w ≥ h) and the length L is essentially larger than its width (L >> w), in this case the end effects can be neglected.

• The pressure depends only on the longitudinal coordinate z0 and the local pressure gradient is small, i.e.

h dp ξ = , |ξ | << 1. (3.83) p p dz0 p This assumption allows us to linearize the kinetic model equation (3.88). It should be noted also that in a long microchannel the local pressure gradient is always small for any pressure ratio pin/pout [103], [101], because the ratio h/L is always small.

Then, the dimensionless mass flow rate √ 2RT Qch = − M˙ ch. (3.84) hwpξp

M˙ ch here is the mass flow rate through the cross-section of the microchannel, which depends mainly on the rarefaction parameter δ defined as

ph √ δ = , νm = 2RT. (3.85) µνm The rarefaction parameter δ can also be defined as the inverse of the Knudsen number as √ π 1 δ = . (3.86) 2 Kn 48 Chapter 3. Analytical and Numerical Modeling

We consider below the following dimensionless parameters:

x0 y0 z0 1 r m x = , y = , z = , u = u0, (3.87) h h h ξp 2kT where u0 is the bulk velocity having the z0 component only.

3.3.2 Linearized BGK model

To simulate the gas flow in the transitional flow regime the BE must be solved. This integro-differential equation (BE) describes the evolution of the velocity distribution function f(r0, v, t) in time and in space. The complexity of the Boltzmann equation, especially the nature of the collision term, makes direct analysis quite a challenge, so a variety of kinetic models are proposed to get it simplified. To some extent, the BGK operator [15] is the simplest one. It was proposed by Bhatnagar, Gross and Krook (1954), and it has come to be known as the BGK approximation of the Boltzmann collision integral.

For steady flow the kinetic BGK model equation reads:

∂f f M − f v = , (3.88) ∂r0 τ where v is the molecular velocity vector defining the molecular velocity space, r0 is the position vector in the physical space and τ is the relaxation time. The following relaxation time expression: µ τ 0 = , (3.89) p provides the best agreement with the exact Boltzmann equation for isothermal rarefied gas flows.

f M in Eq. (3.88) is the local Maxwellian distribution function given by:

 m 3/2  m(v − u0) f M (n, T, u0) = n exp − . (3.90) 2πkT 2kT

The bulk velocity u0 can be calculated from the distribution function f(r0, v) as following:

1 Z +∞ u0(r0) = vf(r0, v)dv. (3.91) n −∞

Under the assumption of small pressure gradient (see paragraph 3.3.1, page 47) the BGK equation (3.88) may be linearized. 3.3. Transitional and free molecular regimes 49

M Now let us express the absolute Maxwellian distribution function f0 as

 3/2 M m m 2 f0 = n0 exp(− v ), (3.92) 2πkT0 2kT0 where n0 is the equilibrium number density and T0 is the equilibrium temperature. For week non-equilibrium case the local Maxwellian distribution function f M (3.90) can be related to the absolute Maxwellian (3.92) as

 0 0  M 0 M n − n0 mvu (r ) f (n, T, u ) = f0 1 + + . (3.93) n0 kT0 Eq. (3.93) is valid if n − n r m 0 0  1, u  1. (3.94) n0 kT0 The first condition in (3.94) is satisfied since the pressure gradient in the channel is small (3.83)(|ξp|  1), so it is possible to express the number density in the form n = n0(1 + ξpz), therefore the first condition of (3.94) is satisfy. The second condition gives the restriction for the velocity: the bulk velocity must be small compared to the most probable velocity which is the case for the flow under consideration.

Under the assumptions of the small local pressure gradients (3.83) the distribution function f can be linearized as M f(r, c) = f0 {1 + ξp [z + g(x, y, c)]} , (3.95) where g is the perturbation function and c is the molecular velocity defined as

r m c = v. (3.96) 2kT0 Substitute equations (3.95) and (3.93) into (3.88) with the help of equations (3.89) and (3.96) we obtain, after some simplifications, the following expression:

∂ p mvu0  v 0 [ξp + g(x, y, c)] = − ξpg(x, y, c) . (3.97) ∂r µ kT0 Using the expression for the gradient in Cartesian coordinates and the scalar product property,

∂g ∂g ∂g ∂g = ~i + ~j + ~k, (3.98) ∂r ∂x ∂y ∂z we obtain this expression:

 ∂g ∂g  hp mvu0  ξp vz + vx + vy = − ξpg(x, y, c) . (3.99) ∂x ∂y µ kT0 50 Chapter 3. Analytical and Numerical Modeling

We assume that macroscopic velocity u0 (3.91) has z coordinate only so

Z ∞ 0 1 uz(r) = vzf(r, v)dv. (3.100) n ∞ Using again the linearization form (3.95) of the distribution function we obtain s ∞ 1 2kT Z 2 u0 (x, y) = ξ 0 c e−c g(x, y, c)dc. (3.101) z p 3/2 z π m ∞

From equation (3.99) and notation (3.96) with the help of expression for rarefaction parameter (3.85) we obtain the linearized BGK equation as following

∂g ∂g c + c = δ (2c u − g(x, y, c)) − c , (3.102) x ∂x y ∂y z z z where equation of the bulk velocity is given by:

1 Z 2 u (x, y) = c e−c g(x, y, c)dc. (3.103) z π3/2 z

2 √cz −cz Multiplying equation (3.102) by π e and integrating with respect to dcz we obtain

∂φ ∂φ 1 c + c = δ [u − φ(x, y, c , c )] − , (3.104) x ∂x y ∂y z x y 2 where Z +∞ 1 −c2 φ(x, y, cx, cy) = √ cze z g(x, y, cx, cy)dcz. (3.105) π −∞ The bulk velocity is expressed via φ as

Z +∞ 1 −(c2 +c2 ) uz = e x y φ(x, y, cx, cy)dcxdcy. (3.106) π −∞

For the boundary condition we assume diffuse-specular gas/surface interaction, i.e. the frac- tion α of the molecules are reflected diffusely while the fraction (1−α) are reflected specularly. As example, for the vertical channel walls, this condition reads

g(±w/2, y, cx, cy) = (1 − α)g(±w/2, y, −cx, cy). (3.107)

The components cx and cy in Eq. (3.106) may be written as function of polar coordinates (cp and ϕ) ( cx = cp · cos ϕ . 0 ≤ ϕ ≤ 2 π 0 ≤ cp < +∞ (3.108) cy = cp · sin ϕ Equation (3.104) in polar coordinate becomes

∂φ 1 − c = δ [u (x, y) − φ(x, y, c , ϕ)] − , (3.109) p ∂s z p 2 3.3. Transitional and free molecular regimes 51 where s is the characteristic , i.e. the distance from the point (x, y) in the direction opposite to the velocity vectors (cx, cy). The final equation that is used to calculate numerically the bulk velocity is Z π/2 Z ∞ 1 −c2 uz(x, y) = φ(x, y, cp, ϕ)e p cpdcpdϕ. (3.110) π 0 0 ch The dimensionless mass flow rate Q may be found from the bulk velocity uz(x, y) (3.110) as Z w/2h Z 1/2 ch h Q = −2 uz(x, y)dxdy. (3.111) w −w/2h −1/2

3.3.3 Discrete velocity method

In order to calculate the reduced mass flow rate Qch in the transitional and free molecular regimes equation (3.109) subjected to the diffuse-specular boundary conditions is solved numerically using the discrete velocity method. The idea of "discrete velocities" was proposed by Broadwell, 1964 [20] then developed by Gatignol, 1975 [43]. It consist in replacing the velocity space by a set of discrete velocities vk (k = 1, ..., N) and the BGK (3.109) equation is replaced by a set of equations equal to the number of discrete velocities. The systems of the BGK equations are approximated by a finite difference scheme and are solved with an iterative technique.

We consider fixed point on uniform grid in the physical space, where cells are rectangular with size ∆xi and ∆yj in the direction of x and y axis, respectively. Let us note βij the angle characterizing the grid cells (βij = ∆yj/∆xi) as it is shown in Figure 3.2.

cp φ i-1 j Δxi i j

φij

Δs

φB Δyj

βij

i-1 j-1 i j-1

Figure 3.2: Scheme of finite difference method.

Introducing also a grid in the velocity space defined by the coordinate cp and ϕ. cp is distributed according to the Gaussian quadratic rule, which is characterized by the Gaussian abscissas Ncp corresponding to the weight function W (cp)

cp 2 W (c ) = e−cp . (3.112) p π 52 Chapter 3. Analytical and Numerical Modeling

ϕ is the orientation of the molecular velocity vector.

The uniform grid for the variable ϕ is introduced with the number of Nϕ node, where ∆ϕ = 2π/Nϕ. The approximation of Eq. (3.109) at the fixed point (cp, ϕ) of velocity space has the following form:

φ − φ (c , ϕ) 1 ij B p = δ (u + φ (c , ϕ) − ). (3.113) ∆s ij zij ij p 2

From this finite difference equation the function φij in a given node (i, j) may be found as

δuzij − 0.5 + φB(cp, ϕ)cp/∆s φij(cp, ϕ) = , (3.114) δ + cp/∆s where φij(cp, ϕ) = φ(xi, yj, cp, ϕ), and ( ∆x / cos ϕ for ϕ ≤ β , ∆s = i ij (3.115) ∆yj/ sin ϕ for ϕ > βij.

The value of the function φ at the point B can be found as a linear interpolation between two nearest points of the grid as ( (1 − tan ϕ/ tan βij)φi−1j + tan ϕ/ tan βij · φi−1j−1 for ϕ ≤ βij, φB = (3.116) (1 − tan βij/ tan ϕ)φij−1 + tan βij/ tan ϕ · φi−1j−1 for ϕ > βij.

Due to the flow symmetry, the calculation are carried out only on quarter of the physical space (0 ≤ x ≤ w/2 and 0 ≤ y ≤ h/2).

The calculation started by setting the function uz to zero. Then the function φ (3.114) is calculated for all points of the physical space and for all discrete value of the molecular velocities. After that the new value of the bulk velocity is obtained using the Gaussian quadrature rule in following form:

Ncp Nf X X uzij = φij(cpk, ϕm)Wk∆ϕ, (3.117) k=1 m=1 where Ncp and Nϕ are number of points for the variable cp and ϕ, respectively.

When the rarefaction parameter δ is small, typically near the free molecular regime and in the transitional regime (δ ≤ 2 ) this method have a good convergence, whereas in the slip and near hydrodynamic regimes this method have a low convergence. To reduce the computational effort in the slip and near hydrodynamic regimes we used the optimization proposed by Sharipov et al. 1993 [105]. They propose to decompose the perturbation function g(x, y, c) into two parts: g(x, y, c) =g ˜(x, y, c) + gH (x, y, c), (3.118) 3.3. Transitional and free molecular regimes 53

Range of δ Ncp Nϕ nx × ny 0.001 to 2 25 200 0 0t 5 12 200 1000 × 10004to 8 to 20 12 104

Table 3.7: Parameter of the numerical grid used for the simulations.

where gH (x, y, c) = lim g(x, y, c), (3.119) δ→∞ and g˜(x, y, c) tends to zero when δ tends to infinity. The function gH (x, y, c) is the contin- uum solution found using the well known Chapman-Enskog method [105] retaining only the dominate term

gH (x, y, c) = 2czuzH , (3.120) where uH is defined by the following expression: " # δ 1 ∞ (−1)i cosh(nx) cos(ny) u (x, y) = − y2 − 8 X , n = π(2i + 1), (3.121) zH 2 4 n3 cosh(nw/2h) i=0 which is the dimensionless form of equation (3.23) when the non-dimensionalization (3.87) is used.

Substituting (3.118) into (3.102) and using (3.120) the following expression for g˜ can be obtained as: ∂g˜ ∂g˜  ∂u ∂u  c + c = δ (2c u˜ − g˜) − c − 2c c zH + c zH , (3.122) x ∂x y ∂y z z z z x ∂x y ∂y where 1 Z 2 u˜ = c ge˜ −c dc, (3.123) z π3/2 z and ∂u ∞ (−1)i sinh(nx) cos(ny) zH = −4δ X , ∂x n2 cosh(nw/2h) i=0 " # (3.124) ∂u ∞ (−1)i cosh(nx) sin(ny) zH = δ −y + 4δ X , ∂y n2 cosh(nw/2h) i=0 with n = π(2i + 1). The same discrete velocity method described above is applied to Eq. (3.122).

3.3.4 Numerical results The numerical calculation of the reduced mass flow rate Qch is fulfilled for a large range (0.01 − 20) of the rarefaction parameter δ (3.85), for accommodation coefficient α equal to 0.6, 0.8, 0.85, 0.90, 0.95 and 1.0 and for various channel aspect ratios w/h in the range 54 Chapter 3. Analytical and Numerical Modeling

Reduced mass flow rate Qch δ Ref [72] Ref [101] present work 0.001 0.8385 0.8373 0.8375 0.01 0.8281 0.8315 0.8290 0.05 0.8076 0.8124 0.8075 0.1 0.7934 0.7958 0.7933 0.2 - 0.7766 0.7774 0.5 0.7622 0.7607 0.7619 0.8 - 0.7614 0.7629 1. 0.7678 0.7660 0.7674 2. - 0.8076 0.8090 4. 0.9252 0.9209 0.9232 5. 0.9885 0.9846 0.9873 8. 1.1890 1.1790 1.1836 10. 1.3290 1.3140 1.3179 15. 1.7050 1.6530 1.6592 20. - 2.0000 2.0045

Table 3.8: Comparison of the reduced mass flow rate Qch obtained in this thesis with the results of Loyalka et al. 1976 [72] and Sharipov, 1999 [101] for α = 1 and h/w = 1.

from 1 to 100. The error on the reduced mass flow rate Qch is estimated less than 0.1%. This estimation was obtained by comparing the results obtained for different number of the numerical grid points (Ncp , Nϕ, Nx and Ny). The analysis showed that the accuracy of 0.1% can be reached with Nx = Ny = 1000 points in the physical space for the hydrodynamic regime. In the transitional and free molecular regimes the number of points needed to reach the same accuracy is smaller than this value, but we decide to carry out all the series of calculations with the same number of points in the physical space. The number of points in the molecular velocity space Ncp and Nϕ depends also on the rarefaction parameter δ (3.85). Table 3.7 summarizes the parameters of the numerical grid used for the numerical simulations.

A comparison of the reduced mass flow rate Qch fulfilled in this thesis with the results of Loyalka et al. 1976 [72] and Sharipov, 1999 [101] for the accommodation coefficient α = 1 and for the aspect ratio h/w = 1 is shown in Table (3.8). A good agreement with those results is observed. The difference between our results and the results of both authors is less than 0.5% for δ < 10, however, when δ is superior to 10 the results of Loyalka et al.[72] show a little divergence from our results and the results of Sharipov [101], the difference is then higher than 2.5%.

In order to illustrate the validity of the continuum approach developed in this thesis (See Section 3.2.5), the mass flow rate M ch (3.77) calculated using this approach and normalized according to equation (3.84) is compared to the numerical mass flow rate Qch obtained with the BGK kinetic model for the channel aspect ratio (h/w = 0.5) and α = 1 (see Figure 3.3). 3.3. Transitional and free molecular regimes 55

In this Figure 3.3 the curve (BGK) represents the BGK model, the curve (1) represents the continuum model with first order boundary condition (σp=1.016 [25]) and the curves (2), (3), (4) represent the continuum model with the second order boundary conditions of Cercignani & Lorenzani [28], Hadjiconstantinou [48] and Cercignani [23], respectively. It is clear from this comparison that the first order continuum model (curve (1)) is valid for δ down to 10 and that the second order continuum model is valid for δ down to 3 using the second order boundary condition of Ref. [28](σ2p = 0.184). We will see in Chapter6 that the "experimental" value of σ2p obtained with the continuum model are of the same order of the theoretical value obtained in Ref. [28]. We will see also that the values of the coefficient α obtained using the BGK kinetic model in the transitional regime are closer to the value obtained using the second order continuum approach than the first order continuum approach in the slip regime.

3 . 0 ( 4 ) ( 3 ) ( 2 ) 2 . 5

2 . 0 h c

Q 1 . 5

1 . 0 ( B G K ) ( 1 )

0 . 1 1 1 0 δ

Figure 3.3: Comparison between the mass flow rate obtained using the BGK kinetic model and the continuum model for α = 1 and for the channel aspect ratio h/w = 0.5. (BGK) BGK kinetic model, (1)continuum model (first order, σp = 1.016) and (2), (3), (4) continuum model (second order, σ2p = 0.184 [28], σ2p = 0.243 [48], σ2p = 0.766 [23], respectively)

The results of the reduced mass flow rate calculations are given in Tables (3.9, 3.10, 3.11, 3.12, 3.13 and 3.14) for the accommodation coefficients α = 1, 0.95, 0.90, 0.85, 0.80, 0.60, respectively. For a better illustration of the results two examples of the reduced mass flow rate Qch as function of δ are given in Figures 3.4 and 3.5.

In Figure 3.4 the dependence of the reduced mass flow rate Qch on the rarefaction parameter and on the aspect ratio of channel h/w for the accommodation coefficient α = 1 56 Chapter 3. Analytical and Numerical Modeling

(diffuse reflexion) is shown, while in Figure 3.5 the dependence of Qch on the accommodation coefficient for the aspect ratio h/w = 0.5 is given. From Figures 3.4 and 3.5 one can see that the minimum mass flow rate is attended for the rarefaction parameter δ ∼ 1 and that it tends to an asymptote for δ = 0 (free molecular regime).

The increase of the mass flow rate with the decrease of the aspect ratio h/w is observed in Figure 3.4. However, when δ > 0.1 and h/w ≤ 0.02 it seems that the variation of the reduced mass flow rate with the aspect ratio becomes insignificant (less than 2%), then the channel can be considered as a parallel plates.

In Figure 3.5 it is well shown that when the surface of the channel tends to be specular (α decrease) the mass flow rate through the microchannel increases.

As it was mentioned above the numerical simulations of the reduced mass flow rate were carried for some specific values of the aspect ratios (h/w). For an arbitrary value of the aspect ratio the linear interpolation can be used as an approximation of the exact value with maximum error of ±0.5%. The numerical results of the reduced mass flow rate Qch will be used in the Section 6.1 for a comparison with experimental mass flow rate to deduce tangential momentum accommodation coefficient in the transitional and near free molecular regimes. 3.3. Transitional and free molecular regimes 57

δ h/w = 1 0.5 0.2 0.1 0.05 0.02 0.01 0.001 0.8375 1.1500 1.6113 1.9773 2.3457 2.8073 3.1593 0.005 0.8334 1.1419 1.5912 1.9387 2.2754 2.6674 2.9232 0.01 0.8290 1.1336 1.5724 1.9044 2.2151 2.5547 2.7511 0.05 0.8075 1.0968 1.4912 1.7588 1.9719 2.1520 2.2235 0.1 0.7933 1.0733 1.4389 1.6667 1.8285 1.9470 1.9888 0.2 0.7774 1.0483 1.3817 1.5668 1.6820 1.7568 1.7820 0.5 0.7619 1.0273 1.3207 1.4555 1.5279 1.5716 1.5862 0.8 0.7629 1.0326 1.3112 1.4275 1.4872 1.5231 1.5351 1. 0.7674 1.0420 1.3161 1.4258 1.4816 1.5150 1.5262 2. 0.8090 1.1166 1.3925 1.4926 1.5427 1.5728 1.5828 4. 0.9232 1.3093 1.6244 1.7333 1.7878 1.8204 1.8313 5. 0.9873 1.4153 1.7560 1.8728 1.9312 1.9663 1.9779 8. 1.1836 1.7384 2.1623 2.3061 2.3780 2.4211 2.4355 10. 1.3179 1.9585 2.4408 2.6040 2.6856 2.7345 2.7508 15. 1.6592 2.5164 3.1492 3.3626 3.4692 3.5333 3.5546 20. 2.0045 3.0798 3.8660 4.1306 4.2629 4.3423 4.3688

Table 3.9: Reduced mass flow rate Qch for accommodation coefficient α = 1.

δ h/w = 1 0.5 0.2 0.1 0.05 0.02 0.01 0.001 0.9037 1.2419 1.7450 2.1462 2.5502 3.06180 3.4579 0.005 0.8986 1.2320 1.7208 2.1004 2.4677 2.8968 3.1776 0.01 0.8932 1.2221 1.6986 2.0602 2.3977 2.7666 2.9792 0.05 0.8677 1.1783 1.6037 1.8922 2.1210 2.3134 2.3895 0.1 0.8507 1.1504 1.5432 1.7877 1.9609 2.0872 2.1318 0.2 0.8315 1.1204 1.4773 1.6756 1.7989 1.8789 1.9059 0.5 0.8116 1.0934 1.4066 1.5513 1.6290 1.6760 1.6917 0.8 0.8107 1.0964 1.3938 1.5186 1.5829 1.6215 1.6344 1. 0.8145 1.105 1.3975 1.5154 1.5753 1.6113 1.6233 2. 0.8549 1.1789 1.4724 1.5795 1.6332 1.6654 1.6761 4. 0.9697 1.3731 1.7054 1.8206 1.8783 1.9128 1.9244 5. 1.0344 1.4801 1.8380 1.9609 2.0224 2.0593 2.0716 8. 1.2321 1.8053 2.2460 2.3957 2.4705 2.5154 2.5304 10. 1.3672 2.0264 2.5254 2.6944 2.7788 2.8295 2.8464 15. 1.7100 2.5860 3.2352 3.4541 3.5636 3.6293 3.6512 20. 2.0563 3.1505 3.9527 4.2228 4.3579 4.4390 4.4660

Table 3.10: Reduced mass flow rate Qch for accommodation coefficient α = 0.95. 58 Chapter 3. Analytical and Numerical Modeling

δ h/w = 1 0.5 0.2 0.1 0.05 0.02 0.01 0.001 0.9781 1.3451 1.8946 2.3348 2.7782 3.3445 3.7875 0.005 0.9719 1.3330 1.8656 2.2805 2.6812 3.1504 3.4569 0.01 0.9654 1.3212 1.8393 2.2334 2.6000 3.0000 3.2293 0.05 0.9350 1.2694 1.7285 2.0396 2.2851 2.4903 2.5712 0.1 0.9147 1.2363 1.6585 1.9211 2.1063 2.2410 2.2885 0.2 0.8918 1.2007 1.5831 1.7955 1.9275 2.0130 2.0418 0.5 0.8669 1.1668 1.5017 1.6570 1.7406 1.7912 1.8081 0.8 0.8639 1.1673 1.4852 1.6195 1.6887 1.7304 1.7442 1. 0.8670 1.1751 1.4878 1.6146 1.6792 1.7179 1.7309 2. 0.9061 1.2481 1.5611 1.6760 1.7336 1.7682 1.7797 4. 1.0213 1.4439 1.7953 1.9176 1.9788 2.0154 2.0277 5. 1.0867 1.5519 1.9288 2.0586 2.1236 2.1625 2.1755 8. 1.2859 1.8793 2.3387 2.4950 2.5731 2.620 2.6357 10. 1.4219 2.1015 2.6190 2.7944 2.8821 2.9348 2.9523 15. 1.7662 2.6630 3.3302 3.5554 3.6680 3.7356 3.7582 20. 2.1136 3.2286 4.0486 4.3248 4.4629 4.5458 4.5735

Table 3.11: Reduced mass flow rate Qch for accommodation coefficient α = 0.90.

δ h/w = 1 0.5 0.2 0.1 0.05 0.02 0.01 0.001 1.0622 1.4616 2.0628 2.5464 3.0336 3.6603 4.1532 0.005 1.0546 1.4469 2.0281 2.4819 2.9194 3.4320 3.7651 0.01 1.0467 1.4327 1.9970 2.4267 2.8251 3.2585 3.5049 0.05 1.0107 1.3715 1.8676 2.2032 2.4665 2.6852 2.7711 0.1 0.9866 1.3324 1.7869 2.0689 2.2670 2.4104 2.4610 0.2 0.9593 1.2904 1.7006 1.9283 2.0696 2.1611 2.1919 0.5 0.9288 1.2489 1.6075 1.7745 1.8645 1.9190 1.9372 0.8 0.9236 1.2467 1.5871 1.7318 1.8066 1.8515 1.8664 1. 0.9257 1.2535 1.5884 1.7251 1.7949 1.8367 1.8507 2. 0.9634 1.3255 1.6601 1.7837 1.8457 1.8829 1.8953 4. 1.0791 1.5229 1.8957 2.0259 2.0910 2.1300 2.1431 5. 1.1451 1.6320 2.0301 2.1677 2.2365 2.2778 2.2916 8. 1.3459 1.9618 2.4421 2.6057 2.6875 2.7366 2.7530 10. 1.4827 2.1851 2.7233 2.9059 2.9972 3.0520 3.0703 15. 1.8287 2.7486 3.4360 3.6682 3.7843 3.8539 3.8772 20. 2.1772 3.3154 4.1552 4.4382 4.5798 4.6648 4.6931

Table 3.12: Reduced mass flow rate Qch for accommodation coefficient α = 0.85. 3.3. Transitional and free molecular regimes 59

δ h/w = 1 0.5 0.2 0.1 0.05 0.02 0.01 0.001 1.1578 1.5938 2.2532 2.7853 3.3217 4.0152 4.5617 0.005 1.1484 1.5759 2.2115 2.7087 3.1870 3.7465 4.1074 0.01 1.1390 1.5589 2.1748 2.6438 3.07710 3.5464 3.8103 0.05 1.0962 1.4866 2.0236 2.3858 2.6682 2.9012 2.9924 0.1 1.0677 1.4407 1.9306 2.2336 2.4455 2.5983 2.6521 0.2 1.0354 1.3912 1.8320 2.0764 2.2278 2.3257 2.3587 0.5 0.9986 1.3412 1.7260 1.9058 2.0030 2.0618 2.0814 0.8 0.9907 1.3359 1.7013 1.8577 1.9385 1.9871 2.0033 1. 0.9920 1.3418 1.7012 1.8491 1.9246 1.9700 1.9851 2. 1.0280 1.4126 1.7713 1.9048 1.9718 2.0120 2.0254 4. 1.1459 1.6141 2.0114 2.1507 2.2203 2.2621 2.2761 5. 1.2108 1.7219 2.1439 2.2902 2.3634 2.4073 2.4220 8. 1.4132 2.0542 2.5580 2.7299 2.8159 2.8675 2.8847 10. 1.5510 2.2788 2.8402 3.0309 3.1263 3.1836 3.2027 15. 1.8987 2.8443 3.5544 3.7945 3.9145 3.9866 4.0106 20. 2.2483 3.4124 4.2745 4.5653 4.7107 4.7980 4.8271

Table 3.13: Reduced mass flow rate Qch for accommodation coefficient α = 0.80.

δ h/w = 1 0.5 0.2 0.1 0.05 0.02 0.01 0.001 1.7113 2.3576 3.3462 4.1493 4.9603 6.0132 6.8282 0.005 1.6896 2.3175 3.2571 3.9906 4.6878 5.4860 5.9733 0.01 1.6694 2.2822 3.1828 3.8630 4.4790 5.1242 5.4658 0.05 1.5831 2.1385 2.8952 3.3939 3.7711 4.0719 4.1869 0.1 1.5274 2.0509 2.7299 3.1402 3.4204 3.6188 3.6883 0.2 1.4655 1.9589 2.5628 2.8928 3.0956 3.2263 3.2703 0.5 1.3932 1.8619 2.3883 2.6371 2.7726 2.8549 2.8823 0.8 1.3717 1.8406 2.3426 2.5628 2.6777 2.7468 2.7699 1. 1.3679 1.8413 2.3361 2.5457 2.6536 2.7184 2.7401 2. 1.3954 1.9060 2.3995 2.5892 2.6846 2.7419 2.7610 4. 1.5153 2.1157 2.6483 2.8392 2.9348 2.9921 3.0113 5. 1.5826 2.2283 2.7853 2.9819 3.0803 3.1393 3.1590 8. 1.7922 2.5724 3.2094 3.4292 3.5391 3.6051 3.6271 10. 1.9341 2.8030 3.4962 3.7338 3.8526 3.9239 3.9477 15. 2.2900 3.3790 4.2179 4.5030 4.6455 4.7311 4.7597 20. 2.6453 3.9536 4.9422 5.2769 5.4442 5.5447 5.5783

Table 3.14: Reduced mass flow rate Qch for accommodation coefficient α = 0.60. 60 Chapter 3. Analytical and Numerical Modeling

4 . 5 1 4 . 0 0 . 5 0 . 2 3 . 5 0 . 1 0 . 0 5 3 . 0 0 . 0 2 0 . 0 1 h

2 . 5 c

Q 2 . 0

1 . 5

1 . 0

1 E - 3 0 . 0 1 0 . 1 1 1 0 ؄׀

Figure 3.4: Reduced mass flow rate Qch for α = 1 and for the channel aspect ratio h/w that varies from 0.01 to 1.

4 . 5 1 . 0 0 4 . 0 0 . 9 5 0 . 9 0 3 . 5 0 . 8 5 0 . 8 0 3 . 0 0 . 6 0

h 2 . 5 c

Q 2 . 0

1 . 5

1 . 0

1 E - 3 0 . 0 1 0 . 1 1 1 0 ؄׀

Figure 3.5: Reduced mass flow rate Qch for h/w = 0.5 and for the accommodation coefficient α that varies from 0.6 to 1. Chapter 4 Description of the experimental approach

The information about the properties of the gas-surface interaction such as condensation, adsorption, adhesion and accommodation is important in aerodynamic high altitude appli- cations and vacuum technology. One of the most important parameter to characterize the gas-surface interaction is the tangential momentum accommodation coefficient (TMAC).

According to the review of Agrawal and Prabhu, 2008 [2], the firsts who measured the TMAC experimentally were Knudt and Warburg in 1875 on the damping of a vibrating disk by a surrounding gas. They observed that the damping force decreased at low pressures due to slippage of the gas over the disk surface [4]. Then, several techniques were performed to measure the TMAC. There are two types of TMAC measurement technique: direct and indirect. The direct techniques, such as the molecular beam technique, record random reflected molecular angles from one incident molecular angle. Contrarily, in the indirect techniques (e.g. spinning rotor gauge and flow through microchannel techniques) the incident angles are created randomly. Therefore, both measurement techniques can yield to different TMAC value for the same pair gas-surface.

In this Chapter a brief review of TMAC values measured using three different techniques is first introduced. Then, a description of the techniques used in literature to measure the mass flow rate through microchannels is given. It is followed by the description of the experimental apparatus performed in this thesis to measure the mass flow rate through various cross-section microchannels. Finally, the uncertainty evaluation for the mass flow rate measurements is given.

4.1 Experiments in TMAC

The research on TMAC shows it to be sensitive to a number of parameters related to the surface condition and gas type. According to Collins and Knox experiments [31], the TMAC value is a function of the incoming molecule, the energy of the incoming molecule, the temperature and the condition of the surface. The TMAC value can be different if the surface is smooth or rough [119][92] and if it is clean or contaminated engineering surface [69].

Two reviews of the TMAC measurements were made in last decades, are proposed by Agrawal et al. 2008 [2] and Cao et al. 2009 [22]. The different TMAC values measured by 62 Chapter 4. Description of the experimental approach the three widely used techniques [22] (Molecular beam, SRG and flow through microchannels techniques) are reviewed below.

4.1.1 Molecular beam technique In molecular beam experiments, a beam of gas molecules strike against the surface at a fixed incident energy and fixed incident angle. The distributions of the reflected molecules from various angles of attack are gathered and the TMAC is calculated. It is to be noted that the molecular beam technique requires a high vacuum conditions. Several researchers have used the molecular beam technique to measure the TMAC for aerospace applications. The results from this technique can be directly applied to the calculation of forces and torques on a satellite [58].

Molecular beam technique Ref’s Gas Material Kn TMAC (α) Knechtel and Pitts [59][22] Ar+ Au - 0.50-0.95 Seidl and Steinheil [98] He Au - 0.68-0.87 Glass - 0.71-0.79 Omelik [86] N2 Glass 1000 0.30-0.90 Glass cloth 0.80-1.15 Rettner [95] N2 Glass - 0.80-0.88

Table 4.1: TMAC value found in literature, measured with the molecular beam technique.

Knechtel and Pitts [59] obtained the value of TMAC for argon ions (Ar+) colliding on gold (Au) and aluminum (Al) surfaces, in the range 0.42 to 0.95 for both surfaces. While, Dougthy and Schaetzle [22] found that the TMAC value for argon on aluminum surface ranges from 0.7 to a value superior to unity (equal to 1.4), which the authors defined it as the backscattering phenomena. Seidl and Steinheil [98] reported also a value more than unity on rough and contaminated cooper (Cu) surface. Other authors [65], [95] used the same technique and reported values less than unity for helium and nitrogen on aluminum, platinum and glass surfaces. Table 4.1 gives some TMAC values found in literature for He, N2 and Ar on gold and glass surfaces measured using the molecular beam technique.

4.1.2 Spinning rotor gauge technique Using the spinning rotor gauge technique the TMAC is obtained from the measurements of the angular velocity of a spherical rotor exposed to a gas.

In 1974, Thomas and Lord [119] investigated TMAC using the spinning rotor gauge (SRG) technique. They found that the TMAC value is higher when the surface of steel ball 4.1. Experiments in TMAC 63 is rough. The reported TMAC values vary from 0.824 to 1.075 for He, Ne, Ar and Xe with a large value for the heaviest gas (xenon).

Spinning rotor gauge technique Ref’s Gas Material Kn range TMAC (α) Thomas and Lord [119] He Polished steel - 0.824 Rough steel 1.040 Ar Polished steel 0.931 Rough steel - 1.049 Cosma et al.[32] He Steel > 1 1.010-1.031 N2 1.001-1.023 Ar 0.995-1.020 CO2 1.007-1.019 Gabis et al.[42] N2, Ar Steel 0.01 − 1 0.830-1.010 Bentz et al.[10][11] He Steel < 0.1 0.813-0.841 N2 0.830-0.890 Ar 0.783-0.801 Jousten [55] N2 Stainless Steel - 1.158-1.166

Table 4.2: TMAC value found in literature, measured with the Spinning Rotor Gauge (SRG).

Two years later Lord, 1976 [69] investigated the degree of contamination of various surfaces using the SRG technique. The author measured a TMAC value as low as 0.2 for clean surfaces and 0.9 for contaminated surfaces. Other authors, Cosma et al.[32], Gabis et al.[42], Tekasakul et al.[117] and Bentz et al.[10], [11] have used the same technique (SRG) to measure the TMAC. The values reported by these authors are in the range [0.78, 1.11] for various gases (He, Ne, Ar, Kr, Xe,H2, O2, CO, CO2 and CH4) on steel surface. Recently, Jousten, 2003 [55] measured the TMAC of nitrogen (N2) on stainless steel surface with the SRG technique and found backscattering reflection from the surface (its value is between 1.158-1.166).

Summary of the values reported by researchers for the steel surface and the working gases considered in this thesis (He, N2 and Ar) are presented in Table 4.2.

4.1.3 Flow through microchannels technique Experimental investigations on TMAC by measuring the flow through microchannels and micro ducts have been conducted in recent years by several authors.

Veijola et al.[122] presented measurements of the air flow through a gap between silicon (Si) and aluminum (Al) surfaces. The author extracted the TMAC value in the range 0.621−0.661 for surface with 1 nm roughness and in the range 0.749−0.803 for surface with 30 64 Chapter 4. Description of the experimental approach

Flow through channels Ref’s Gas Material Kn TMAC (α) Porodnov et al.[92] He Glass capillaries < 0.1 0.895 > 10 0.944 N2 < 0.1 0.925 > 10 0.977 Ar < 0.1 0.927 > 10 0.982 CO2 < 0.1 0.993 > 10 0.998 Arkilic et al.[7] N2 Si [0.1 − 0.4] 0.830 Ar 0.800 CO2 0.890 Maurer et al.[76] He Glass/Si < 0.8 0.910 N2 0.870 Colin et al.[30] He, N2 Glass/Si < 0.25 0.930 Hsieh et al.[52] N2 Glass/Si < 0.02 0.300-0.700 Ewart et al.[39] He Fused Silica < 0.3 0.914 N2 0.908 Ar 0.871 Jang and Wereley [54] N2 Glass/Silica 0.0137 0.960 Ewart et al.[38] He Si < 0.3 0.910 Graur et al.[44] N2 Glass/Si < 0.3 0.956 Ar 0.910 Pitakarnnop et al.[90] He, Ar Si 0.02 − 1.5 1.000 Perrier et al.[88] He Fused Silica < 0.3 0.879-1.000 N2 0.826-0.981 Ar 0.778-0.954 Yamaguchi et al.[130] N2 Fused Silica < 0.3 0.851 Ar 0.872 Yamaguchi et al.[131] N2 Stainless Steel < 0.3 0.950 Ar 0.890

Table 4.3: TMAC value found in literature, calculated from the measurement of the flow through microchannels of various cross-section. 4.1. Experiments in TMAC 65 nm roughness. Arkilic et al.[8,7] have measured flow of argon, nitrogen and carbon-dioxide through planar micro-fabricated silicon channel in the Knudsen number range [0 − 0.45]. The surface of the silicon channel was smooth with roughness of 0.65 nm (RMS). The au- thors used the first order boundary condition and extracted a TMAC in the range [0.75−0.85].

Maurer et al.[76] conducted experiments on helium and nitrogen flow through mi- crochannel etched in glass wafer and covered with silicon wafer. The glass surface has 20 nm roughness. He used the liquid drop technique to measure the mass flow rate and the TMAC was calculated using the second order boundary conditions in the Knudsen number range from 0 to 0.80. The obtained TMAC value found were 0.91 and 0.87 for helium and nitrogen gases, respectively.

Colin et al.[30] and Hsieh et al.[52] investigated experimentally the flow of helium and nitrogen through silicon microchannel covered with Pyrex plate. Colin et al.[30] used the second order boundary conditions proposed by Deissler (1964) [34] and suggested that the second order model is valid for Knudsen number up to 0.25. The authors [30] reported the same TMAC value of 0.93 for helium and nitrogen gases. Hsieh et al.[52] used the first order boundary condition for Knudsen smaller than 0.02 and re- ported a value of TMAC for nitrogen in the range 0.3−0.7 with channel roughness of 1.47 µm.

Ewart et al.[38] and Perrier et al.[88] presented experimental results for He, N2 and Ar gases in fused silica microtubes with diameter of 25, 50 and 75 µm and relative roughness of 0.04% of the tube diameter. They suggested that TMAC depends on the molecular mass of gas, such as higher value for the lighter gas. The obtained TMAC values were smaller than unity, which reflect the non-complete accommodation. Recently, Pitakarnnop, 2010 et al. [90] reported a TMAC value equal to unity in the Knudsen range 0.02 to 1.5 for both helium and argon gases on glass/silicon surface.

Finally, Yamaguchi et al.[130, 131] measured the TMAC of N2, Ar and O2 on fused silica and stainless steel (SS) surfaces. All the TMAC values reported were in the range [0.85, 0.95].

In Table 4.3 a summary of the TMAC values obtained for the gases (He, N2, Ar and CO2), in microchannels of various surfaces, is presented.

4.1.4 Influence of the surface roughness

One of the most important parameter affecting the TMAC is the surface roughness. Generally, a higher value of TMAC is obtained for the rough surface. Porodnov et al.[92] used unsteady technique to measure the gas flow in rectangular glass channels for various surface roughness. A large Knudsen number range was investigated, from 0.001 to 1000. The measurements for gases He, Ne, Ar indicated an increasing in TMAC values from 0.847 to 0.960 when the surface roughness increased from 0.05 µm (RMS) to 1.5 µm (RMS). 66 Chapter 4. Description of the experimental approach

Thomas and Lord [119] showed an increase in the TMAC value by a ratio of about 1.18 with the increase of the surface roughness of a spinning metal sphere (see Table 4.2). Seidl and Steinheil [98] reported a large TMAC value (over unity) for cooper surface with large roughness, 5 µm.

Mageley [73] investigated the gas flow through annuli over a large Knudsen number range [0.01, 450]. The author compared the measured flow rate with the theoretical one and estimated that 6 % of backscattering reflexion would explain better the finding results for He and Ar. The author pointed that backscattering occurs because of the surface roughness.

However, other authors reported an inverse behaviour with a smaller value of TMAC for the roughest surface [2]. Stacy [113] measured a value of TMAC that changes from 0.809 to 1 for a surface coated with a thin shellac layer. While the shellac is dried by an air blast, the surface condition changes from rough to smooth.

4.1.5 Influence of the surface contamination

Another important parameter affecting the TMAC is the degree of surface contamination. Generally, a smaller value of TMAC is found for a clean surface. Lord [69] measured the TMAC for noble gases and deposited metal on mica surfaces under stringent gettered vacuum conditions. The author found the TMAC values as low as 0.25.

Seidl and Steinheil [98] measured the influence of various surface preparation techniques on TMAC with a monoenergic molecular He beam. Their measurements showed that a slight oxidation of the surface raised the TMAC value from 0.25 to 0.80. Shields [107, 108] showed a change of the TMAC value from 0.375 to 0.75 for He on non-oxide and oxide tungsten surface measured by an acoustical technique.

Using the molecular beam technique Steinheil et al.[114, 120] presented an impressive result were the TMAC for He − Au pair reached a minimum value of 0.1 on a clean surface of gold.

4.1.6 Influence of the temperature

The temperature is another factor which can have effects on the TMAC. Unfortunately, most of the experiments presented in the literatures are conducted at a temperature between 290 to 315 K, while no detailed information on the TMAC dependence on temperature are given. Jousten [55] showed a slight dependence of TMAC on the temperature using the SRG technique. The temperature range investigated was in the range 289 − 313 K. The author [55] reported a linear relative decrease in TMAC of about 1×10−4/K for nitrogen.

However, the numerical investigation of Cao et al.[21] using the MD simulation 4.2. Mass flow measurement techniques 67 reported a decrease in TMAC from 0.28 at 119.8 K to 0.18 at 349.5 K for pair Ar − P t. The reason for a higher TMAC value at low temperature reported by Cao et al.[21] is the trapping-desorption behaviour, which leads to multiple collisions, therefore more momentum exchange between the gas and the surface when temperature decreases. When the temperature increases the behaviour of trapping-desorption becomes a kind of direct scattering, which decreases the TMAC value.

In conclusion, we may underline that a large number of experiments were carried out to obtain the TMAC values, but still a disagreement exists between researchers concerning the dependence of TMAC on the parameters related to the surface conditions and gas nature. In addition, the technique employed can have also some impact on the TMAC value.

4.2 Mass flow measurement techniques

A number of techniques are currently used to measure a low value of the gas mass flow rate. These techniques, which are accurate and derive their accuracy from the constants of nature or from the measurements of mass and time, include the constant pressure techniques and the constant volume techniques.

The first technique, constant pressure measurement, includes two main variants:

• Liquid droplet measurement [30], [49], [62], [76], [134], [37], which involves measuring of the distance traveled by a droplet of liquid (water, oil ...etc.) in a calibrated tube by a speed camera [37] or by a piezoelectric sensors [30]. The liquid droplet technique permits a direct determination of the volume flow rate Qv but suffers from many problems like the measurements of low flow rates, the determination of the droplet interfaces position ...etc. The volumetric flow rate is given by expression [36]:

Vs dX Qv = , (4.1) dt Ls

where Vs is the syringe volume, dX and Ls are the distance traveled by the droplet interface during the time length dt and the syringe length, respectively.

• Variable volume measurement, [56], [87], [124], which uses a variable volume chamber to keep gas pressure constant. The mass flow rate is measured based on gas state equation with mass flow rate depending on the change in volume over time. This technique can work well over large range of mass flow rate, but it needs an elaborate system for controlling the volume in order to keep the pressure constant. Thus, this technique suffers from the limitation in constructing a constant pressure chamber (i.e. a variable volume chamber), which may requires several moving parts and may cause a number of mechanical complications. When using this technique, the expression for the mass flow rate is given by: p dV Q = , (4.2) m RT dt 68 Chapter 4. Description of the experimental approach

where dV/dt is the variation of the chamber volume during the time dt, p is the pressure and R and T are the specific gas constant and temperature, respectively.

The second technique, constant volume measurement, includes also two main variants:

• Dual tank accumulation technique, which was developed by Arkilic et al.[7]. This technique is based also on the equation of state for ideal gases, but unlike the constant pressure techniques, the volume of the chamber is constant while the pressure changes over time. The dual tank accumulation technique consists on measuring a relative pressure variation between a reference and an accumulation tanks. The expression for the mass flow rate is given by [6]:

V d(∆P ) ∆PV dT Q = − , (4.3) m RT dt RT 2 dt where ∆P is pressure difference between the reference and accumulate tanks.

• The rise-of-pressure technique [30], [37], [38], [90], [47], which is very similar to the dual tank accumulation technique, measures the absolute pressure variation in one tank instead of measuring the variation of the pressure difference between two tanks. The rise-of-pressure technique requires a high thermal stability that can be achieved by maintaining a constant temperature [30] or by controlling the temperature of the system [37]. The expression for the mass flow rate is obtained from the law of state for perfect gases and it is explained in details in Section 4.3.

Alternative techniques can be employed [1], such as the gravimetric technique, sonic nozzles (also known as critical flow venturis or critical flow nozzles), laminar flow technique, coriolis flow technique, thermal mass flow technique and others. These techniques can be used over a large range of flow rate but must be continually calibrated.

All these techniques present advantages and disadvantages. From our point of view the simplest and the most accurate one is the rise-of-pressure technique, which was used in this thesis.

Two experimental setups were realized during this thesis. The first setup was developed to measure the gas mass flow rate through microchannels of various rectangular cross-sections in the IUSTI laboratory in Marseille, France. The second setup was developed to measure the gas mass flow rate of gas through microtubes of circular cross-section in the INFICON laboratory company in Blazers’, Liechtenstein.

Both experimental setups are very similar and work exactly with the same principle, however, some of the components making the two setups are not necessary the same. In the setup realized in the IUSTI laboratory most of the components were Swagelok system, while 4.2. Mass flow measurement techniques 69 in INFICON Company all the components were provided from the company itself (INFICON).

In the next section the description of the experimental setup is concentrated on that built in the IUSTI laboratory in Marseille, because the majority of the results obtained in this thesis come from this experimental setup. Nevertheless, some remarks concerning the experimental setup built in INFICON will be given.

4.2.1 Description of the rise-of-pressure technique

The present technique provides an indirect measurement of the mass flow rate. The pressure and temperature variations are measured and the mass flow rate is determined from the perfect gas law.

The experimental setup is represented schematically in Figure 4.1. Note that the components illustrated in the drawing are not necessary drawn to scale. The experimental setup consists of four separate functional groups: gas delivery system, volumes system, data acquisition system and microsystem.

The gas delivery system noted by (10) on Figure 4.1 consists of four bottles of helium, nitrogen, argon and carbon-dioxide gases at pressure of 200 bar with gas purity of 99.999 %, connected to Swagelok delivery system. The switch between gases can be handled by a simple operation of opening and closing valves. The gas delivery system is connected to the test section via a Swagelok flexible tube (11).

The system of volumes A and B contains several discrete components such as, tubes, valves (V 1 and V 2), reservoirs (5), connection elements (6, 9) and pressure transducers. The valves V 1 and V 2 (type: Bellows Swagelok valves, ref: SS-4BKT 1/4" [116]) serve to separate the systems volume A and B from the gas delivery system (10) and the vacuum pump (3). The tubes (4) are Swagelok 316L stainless steel 1/4" diameter. The connection between the different components, valves, tubes (4) and reservoirs (5) is made by Swagelok 1/4" (Nut, ferrule) Cross (6), Tee (7) and Elbows (8) connections. The pressure transducers (2) are connected to the systems volume by ISO-KF/Swagelok 16/ 1/4" adapter, where the tightness is ensured by Elastomer FPM O-ring and clamping ring. The system of volumes A and B can be modified depending on the mass flow to be measured. The minimum value is obtained by removing completely the reservoirs volume (5) and replacing them by a blank flange. We dispose of four reservoirs that can be added to the system of volumes A and B: two with a nominal volume of 55 cm3 and two other with nominal volume of 208 cm3. These reservoir’s volumes were home made, and were created using ISO-KF 16 or 25 mm 1 Tee connection sealed in two sides by welded blank flanges and in the third side Swagelok 1/4" tube is connected .

1The 16 mm Tee connection was used to create the 55 cm3 reservoir while the 25 mm was used to create the 208 cm3 reservoir. 70 Chapter 4. Description of the experimental approach

Pressure transducers A B C D Inficon CDG ([53]) 25-1000 T 25-100 T 25-10 T 45-1T Full Scale (FS) (P a) 133322. 13332.2 1333.22 133.322 Pression max (P a) 133322. 13332.2 1333.22 133.322 Pression min (P a) 13332. 1333.2 133.32 13.332 Precision 0.20 % of reading Temperature effect on zero 0.0050 % FS / ◦K 0.015 % FS / ◦K Temperature effect on span 0.01 % of reading / ◦K Resolution 0.0015 % FS 0.0025 % FS

Table 4.4: Technical data for the pressure transducers (CDG, Capacitance Diaphragm Gauge) [53].

The pressure transducers (2) used here are INFICON R Capacitance Diaphragm Gauge (CDG) connected to the system of volumes A and B by ISO-KF 16 mm connection and are chosen according to their pressure range. Table 4.4 gives the technical characteristics of the pressure transducers (A, B, C and D) used in the experimental setup. The acquisition of the signal delivered from the pressure transducers is assured by National Instrument data acquisition card (USB NI-cDAQ-9174), which is interfaced with GX60 DELLTMcomputer via LabViewTMsoftware. A program is created under LabView permitting the collection of the data from the pressure transducers and from the temperature sensor PT100 (4 wires), which is placed near the microsystem and measures the room temperature fluctuations.

One of the most important parts of the experimental setup is the microsystem. The fixation of the microsystem presents the most delicate step of the experimental setup prepa- ration. We used two simple and particulate support systems (1) to handle the microsystem:

• For the rectangular microchannels we realized a fixation system by modifying ISO- KF/Swagelok adapter. A metallic cylinder having a slot that corresponds to the wafer dimensions is welded to the adapter. The wafer is then slipped into the slot (Figure 4.2a) and the tightness is provided by the use of glue that polymerizes under U.V. light. Another ISO-KF/Swagelok adapter is also modified and sealed with the first adapter by Elastomer O-ring and clamping ring.

• For the microtube, an orifice corresponding to the microtube diameter is drilled in blank flange and the microtube is slipped into the hole, then it is sealed using Araldite 2011 glue (Figure 4.2b). 4.2. Mass flow measurement techniques 71

2 CO

Ar 11

2 N

3

He V 8

7

10 V1 A

5

4

1

2

B

9

6

5 Figure 4.1: Schematic representative of the experimental setup.

V2

4 V

3

72 Chapter 4. Description of the experimental approach

(a) (b)

Figure 4.2: Pictures of the fixation system for the rectangular microchannels (a) and micro- tubes (b).

The experimental setup is connected to vacuum pump (3), type: ADIXON DRYTEL 1025, which is a dry pumping system, includes secondary pump (molecular drag pump) and primary pump (oil-free forepump). The vacuum pump can reach a minimum pressure of order of 10−5 T orr with pumping speed of 4 l/s for helium gas.

4.2.2 Procedure

To measure the mass flow rate of the working gases (helium, nitrogen argon and carbon- dioxide) through the microsystems, we monitored the pressure and the temperature variations in time occurred in the inlet and outlet tanks (B). The mass flow rate is deduced from the equation of state for perfect gas. The technique used is called the rise-of-pressure technique or the constant volume technique. This technique involves the use of two constant volume tanks connected by microsystem. The flow through the microsystem is generated by setting a pressure difference between the inlet and outlet tanks. The volume of the tanks has to be much greater than the microchannel volume to guarantee that the flow parameters are independent of time, but remain detectable. The mass variations occurring in the tanks during the experiments do not call into question the stationary assumption, because the pressure variations are maintained less than 1% during the experiments.

The experimental procedure followed to measure the mass flow of gas is:

• First, the experimental loop (Figure 4.1) is pumped down using the vacuum pump (3) by opening all the valves V 1, V 2, V 3, V 4 during a time period (from one to two hours). This first step is done when the working gas is changed or after the opening of the experimental loop to air in order to change one of the components, like the pressure 4.2. Mass flow measurement techniques 73

transducer. We have to note that the experimental loop is also vacuumed every day for around 12 hours (during the night).

• Second, when the system is pumped down the leak and outgassing rates are checked. If their rates are negligible the experimental test is started.

• Third, the inlet pressure pin is set by closing the valve V 4 in order to isolate the test section from the vacuum pump and then the test section is pressurized with the working gas from the gas delivery system (10) to the desired inlet pressure value (pin), which is done by opening the leak valve V 1. when the inlet tank (A) pressure pin is set, the valves V 1 and V 3 are closed to isolate the tank A.

• Fourth, The outlet pressure is set by means of the vacuum pump. The valve V 4 is opened resulting in the decrease of the pressure in the outlet tank volume B to the desired value (pout).

• Finally, in order to start a test, the valves V 2 and V 4 are closed to isolate the tank B and the test section from the vacuum pump. The acquisitions of the temperature and pressure in both tanks A and B are started with the help of the LabView interface.

The duration of the data acquisition depends on the flow rate, a few seconds for high mass flow rate (10−7kg/s) and about 250s for low mass flow rate (10−13kg/s).

The temperature of the gas inside the system is assumed to be equal to the room temperature. To check this assumption we have integrate three temperature sensors PT100 in the experimental setup built in INFICON. The first and the second sensors are placed in the inlet and outlet tanks, respectively, while the third sensor was placed in the room to measure the variation of the room temperature. Figure 4.3 shows the variation in time of the outlet tank temperature and the room temperature. One can see that the difference between both measurements is negligible, less than the uncertainty on each measurement (0.1 K), therefore the assumption made above is valid. The experiments were carried out at room temperature around 297 K. In order to maintain the thermal stabilization during the experiments all the source of heat were neutralized. The variation of the temperature during the experimental time length is less than 0.1K.

The volumes of the inlet and outlet tanks are measured and the implemented technique is described in paragraph 4.4.

4.2.3 Leak and outgassing controls The leak testing is an engineering challenge. At very low pressure the leak outcome from the flow of air into the test section. The leak is caused by an open flow path, such as pinholes, 74 Chapter 4. Description of the experimental approach

2 4 . 2 T r o o m T 2 4 . 1 o u t

) 2 4 . 0 C

° ( T

2 3 . 9

2 3 . 8 0 5 0 1 0 0 1 5 0 2 0 0 2 5 0 3 0 0 t ( s )

Figure 4.3: Comparison between the temperature in the outlet tank and the room tempera- ture. broken seals or material porosity.

The leak can be detected by pumping down the test section until certain pressure value smaller than the atmospheric pressure, then closing all the valves of the system and monitoring the pressure rise. If pressure continues to rise at steady rate over time, it means that there is a leak. If pressure rise begins to tail of, the cause here should be an internal gas load often called "outgassing". This pumping down technique is not very efficient to detect the leak, however another technique that needs a leak detector can be used.

Often the leak detector uses helium as working gas, but other gases can be also used as nitrogen or air. In the experimental setup described in this thesis we have used a leak detector to check the presence of leak in our experimental loop. The leak detector is connected to the experimental loop at the level of the valve V 4 (see Figure 4.1). The pressure in the test section is decreased approximately to 0.01 P a using a primary palette and secondary turbo-molecular vacuum pumps. Helium is pressurized around the test section. If helium penetrates inside the test section, the mass spectrometer will detect it immediately and will give an estimation of the leak. This technique has allowed us to conclude that we have no measurable leak in our experimental loop. 4.2. Mass flow measurement techniques 75

Another problem that can alter the measurement of the mass flow rate is the presence of the water vapor in the test section’s internal surfaces. Since the water is present in the air as humidity, some sorption of the water will occur whenever the system is opened to air. The presence of the water vapor in the system will result in increasing of the pressure due to the slow desorption of water molecules. To prevent our experiment from this problem we have used several techniques.

The first technique consists in flushing the test section with dry gas, often nitrogen, then pumping down at low pressure. The combination of these two steps, flushing and pumping-down several times will remove some part of the sorbed water vapor layer.

The second technique consists in heating the test section at temperature around 120-130 ◦C (over the boiling temperature of the water) and pumping down the test section for several hours. Heating will transfer some thermal energy to the sorbed water molecules and helps them to desorb from the test section’s surfaces.

The last technique, which we have used, consists in flowing dry gas through the test section continuously at low pressure. The idea is to bring some energy to the water molecules by impact with the gas molecules, which helps the water vapor desorption. To make that the vacuum valve V 4 is fully opened and the leak valve V 3 is opened at level to maintain the test section at pressure about 1 to 10 T orr for 5 to 10 min. Argon is used instead of nitrogen because its molecule has the heaviest molecular weight, and it is likely to have more kinetic energy upon impact.

The combination of two of these techniques like flushing and heating, or flowing and heating, techniques leads to the maximum effect.

The last problem, we deal with it in our experiments, is the outgassing problem. This problem was the most difficult to handle. The exposition of the test section’s surfaces to the gases contained in air, when it is opened, causes the contamination of the surfaces. The molecules of gas adsorbed by the surface tend to escape from the surface at relatively low pressure, which increases the pressure in the test section. The rate of the pressure rise due to the outgassing can be comparable to the mass flow rate through the microchannel at low pressure (of the order of 10−13 kg/s). One of the solution to decrease the rate of outgassing is to pump down the system for a long time (several days) using the vacuum pump that in order to reach a pressure of 10−5 P a.

Heating the system to relatively high temperature presents an efficient outgassing tech- nique. However this technique can leads to some other problems as deforming the components of the experimental loop due to the expansion of the metal, which will cause the leak problem.

Figure 4.4 shows the pressure variation in the test section kept at pressure of around 0.15 P a. The acquisition of the pressure was started 20 min after the pressure setting and 76 Chapter 4. Description of the experimental approach

0 . 1 8 4

0 . 1 8 2

0 . 1 8 0

0 . 1 7 8 ) a 0 . 1 7 6 P ( P 0 . 1 7 4

0 . 1 7 2

0 . 1 7 0

0 1 0 0 2 0 0 3 0 0 4 0 0 5 0 0 6 0 0 t ( s )

Figure 4.4: Graph of the pressure rise during the outgassing check. The points represent the measured pressure values and the line is a representative fitting curve using the fourth order polynomial form. closing all valves of the test section. The data are fitted with a polynomial curve to see better the pressure variation. The pressure variation shown in this Figure 4.4 results from the combination of all the problems cited above (leak, water vapor sorption and outgassing). This pressure variation leads to a flow inferior to 2.9×10−15 kg/s, which can be considered negligible comparing to the majority of the mass flow rate values measured in this thesis (superior 10−13 kg/s). However in some cases (when the mass flow rate is of the order of 10−13 kg/s) this mass flow rate must be taken into account in the calculation of the uncertainty.

4.3 Mass flow rate calculation

The mass flow rate is defined as the mass (m) of gas passing through a system during a given time interval (t). The mass flow rate through the microsystems can be calculated from the equation of state for ideal gas by assuming steady conditions

pV = mRT, (4.4) where V represents the volume of a tank. p and T are, respectively, the pressure and tem- perature of gas. If we assume now small variation in time of the fluid proprieties (p, V , m 4.3. Mass flow rate calculation 77 and T ) in a tank, equation (4.4) can be written as follows

dm d  pV  p dV V dp pV dT = = + − . (4.5) dt dt RT RT dt RT dt RT 2 dt

If two of the three variables (p, V and T ) are kept constant while the third one varies during the experiment, the mass flow rate can be calculated. Equation (4.5) represents the basis of two techniques of the mass flow rate measurements. The first one is the constant pressure technique, see Eq. (4.2), where the pressure and temperature are kept constant and the variation of the volume is monitored. The second one is the constant volume technique where the volume and temperature are kept constant and the pressure varies. This last technique is considered in this thesis.

Using this constant volume technique or also called rise-of-pressure technique, the mass flow rate can be deduced from the Eq. (4.5) as

dm V dp dT/T = (1 − ε), ε = . (4.6) dt RT dt dp/p The relative variation of the temperature dT/T is calculated from the standard deviation v u n u 1 X s = t (T − T )2, (4.7) n − 1 i i=1 where T is the average temperature during the experimental time. Ti is the instantaneous temperature measurements and n is the number of measurement during the experimental time. This standard deviation s is smaller than 2 × 10−4 for all the experiment series. The relative pressure variation dp/p is of order of 10−2 because the pressure variation is maintained less than 1% of the mean pressure in the tank during the experiments (see Section 4.2.2). Therefore, ε is clearly less than 2×10−2. Thus, the mass flow rate M˙ exp can be written under the following form V dp M˙ exp = . (4.8) RT dt The measurement of M˙ exp is affected by a specific relative error of the order of 10−2 due to the neglecting of term ε in Eq. 4.6. This error will be taken into account when calculating the uncertainty of the mass flow rate.

In expression for the mass flow rate (4.8) the parameters (V and R) are known and remain constant during the experiments. The relative variation of the temperature T during the experiments is very small (less than 0.1K) so the mean value is considered in Eq. (4.8). To calculate the remained term (dp/dt) in equation (4.8) we use the registered data of the pressure variations pi at different instants ti. This variation is in order of 2%. The measured data of the pressure can be fitted with first order polynomial form using the least square method as follows p(ti) = ati + b, (4.9) 78 Chapter 4. Description of the experimental approach

V4 Ref Ref Tank Tank

V3 VR2 VR1

µ-channel Pout Pin Tank B Tank A V2 V1

CDG CDG Gas source

Figure 4.5: Simplified sketch of the experimental data used for the volumes measurement.

where the slope a of the function p(ti) is equal to the ratio dp/dt and it is used in equation (4.8) to calculate the experimental mass flow rate M˙ exp.

4.4 Volume measurement

An accurate measurement of the inlet and outlet volumes is necessary to reduce the uncer- tainty on the mass flow measurements (Eq. (4.8)). These volumes include the contribution of the tubes, pressure transducers and valves (see Figure 4.1). In our thesis we have used a simple and accurate technique to measure the inlet and outlet volumes. Figure 4.5 provides a simplified sketch of the experimental setup drawn in Figure 4.1, which explains how the volumes A and B are measured. A reference tank with known volume is connected to the inlet and outlet tanks A and B, respectively. The volumes of the reference tanks, including the tubes and the valves, are measured by weighting the tanks before and after they were filled by water. The temperature of the water is measured and its density is calculated. The volumes of the reference tanks are deduced from the weight of the water.

One of the corked microchannels by the gold layer takeoff (see Section 2.6.1) is used to avoid any flow between the tanks A and B. The reference tanks are filled with nitrogen at given pressure monitored by the CDG’s. Then the valves VR1 and VR2 are closed and another value of the pressure is set in the tanks A and B. After that the valves V 2 and V 3 are closed. Opening the valves VR1 and VR2 result in equilibrium of pressure between the reference thanks and the tanks A or B (see Figure 4.6).

Using the Boyle’s law we can write the following expression:

piVi + prefi Vrefi = peqi (Vi + Vrefi ). (4.10) 4.4. Volume measurement 79

pi pref i

peq i ) r r o

T (

p

t ( s )

Figure 4.6: Schematic representation of data from volume measurement.

The subscription i refers to the tanks A and/or B, and pref and Vref are respectively the pressure and the volume of the reference tanks. peq is the pressure of equilibrium recorded after opening the valves VR1 and VR2. From expression (4.10) the inlet and outlet tanks volumes can be easily deduced as

peqi − prefi Vi = Vrefi , i = A or B. (4.11) pi − peqi Some results of the volume’s measurements are given in Table 4.5. These values concern the volumes of the inlet and out tanks used in the experimental setup built in INFICON, where the volumes of the inlet and outlet tanks were maintained the same for all experi- ments. During the experiments carried out in IUSTI Laboratory the volumes were changed in function to the mass flow rate to be measured; larger volume for the highest mass flow rate.

From five runs of volume measurements for each volume (in and out), given in Table 4.5, the corresponding mean value is calculated. For the inlet and outlet volumes these values 3 3 are equal to Vin = 75.303 cm , Vout = 75.150 cm , respectively. The maximum deviation of each measurement from the mean value was less than 0.16%, which is less than the volume measurement uncertainty, which is detailed in the next paragraph.

4.4.1 Uncertainty on the volume measurement The essential part of uncertainty on the volume measurements comes from the reference volume and the pressure measurements. The uncertainty on the volume measurements, esti- 80 Chapter 4. Description of the experimental approach

◦ 3 3 Run N : Vin [cm ] Vout [cm ] 1 75.287 75.195 2 75.249 75.151 3 75.361 75.197 4 75.334 75.129 5 75.284 75.078 Vav 75.303 75.150

Table 4.5: Results of the inlet and outlet volumes measurements for the experimental setup built in INFICON.

mated from expression (4.11) using the root square sum (RSS) method, reads

!2 !2 !2 ∆V 2 ∆P ∆P 2 ∆P ∆V i = 2 eqi + i + refi + refi . (4.12) Vi Peqi Pi Prefi Vrefi The uncertainty on the measurement of the reference volume is estimated less than 1 % and the uncertainty on the pressure measurement is given by the manufacture of the pressure gauge and it is equal to 0.2 % (see Table 4.4). So the total uncertainty on the volume measurement is estimated to be less than 1.1 %.

4.5 Total uncertainty on mass flow

The uncertainty on the measurement of the experimental mass flow rate is evaluated using the RSS method as !2 ∆M˙ exp ∆V 2 ∆T 2 ∆a2 ∆ε2 = + + + , (4.13) M˙ exp V T a ε where

• ∆V/V is the uncertainty on the volume measurement estimated less than ±1.1% (Eq. 4.12).

• ∆T/T is the uncertainty on the temperature measurement estimated less than ±0.01%.

• ∆a/a is the uncertainty on the coefficient a = dp/dt estimated from standard deviation following the method used in [37] and its value is less than ±0.5%.

1/2  n  P 2  n (pi − pf (ti))   i=1  ∆a =   . (4.14)  n  n 2!  P 2 P  (n − 2) n ti − ti i=1 i=1 4.5. Total uncertainty on mass flow 81

• ∆ε/ε is the uncertainty coming from the non-isothermal effects (term ε in Eq. (4.6)) estimated less than ±2%.

In equation (4.14) n is the number of measurements, pi is the measured pressure and pf (ti) is the fitted value at the instant ti.

Therefore, the total uncertainty on the mass flow rate measurements calculated from equation (4.13) is less than ±2.4%. If we use the sum of uncertainty method the total uncertainty on the mass flow rate will be less than ±3.6%.

Chapter 5 Results in Microtubes

Flows of helium, nitrogen, argon and carbon-dioxide gases through two microtubes T 1 and T 2 having different internal surface materials are studied experimentally in the mean Knudsen number range [10−4, 3.3]. The microtube T 1 is made from stainless steel and is covered in the internal surfaces with a layer of Sulfinert R (see Section 2.6.2 for more details). The microtube T 2 is made from stainless steel without any coating of the internal surface. We have to recall that the present experiments were conducted in the laboratory of INFICON Company in Liechtenstein.

The experimental conditions for both microtubes T 1 and T 2 are summarized in Table 5.1. The pressure ratio P = pin/pout between upstream and downstream tanks was changed from the value of 1.5 in the hydrodynamic regime to the value of P' 4 in the transitional flow regime. The lower value of the pressure ratio was applied in the hydrodynamic regime to reduce the flow rate in order to capture better the pressure change in the tanks, and the higher value was applied in the transitional regime to make reasonable the experimental time length. It was shown in Sharipov & Seleznev [104] and Ewart et al.[38] that the non-dimensional mass flow rate does not depend significantly on the pressure ratio, therefore, this change in the pressure ratio does not have a large influence on the extracted velocity slip and accommodation coefficients.

Due to the large Knudsen number range investigated [10−4, 3.3] the analysis of the results was split in two ranges: for Knm < 0.1 the continuum approach was used, for Knm > 0.1 the kinetic approach based on the BGK equation was implemented.

5.1 Continuum and slip regimes

It is well known that for Knudsen number up to 0.1 the continuum approach based on the Navier-Stokes equation can be used with first order velocity slip boundary condition at walls.

Considering the flow through microtube, the system of the Navier-Stokes equations can be reduced to the Stokes equation as

1 ∂2u 1 dp z = , (5.1) r ∂r2 µ dz where r is the radial coordinate. 84 Chapter 5. Results in Microtubes

The first order velocity slip boundary condition in the microtube reads [45]

∂u z uztu = ±A1λ . (5.2) s ∂r s

The mass flow rate through the microtube of diameter D and length L is obtained with the Stokes equation (5.1) and the boundary condition (5.2) reads [39]

˙ tu ˙ tu M = MP (1 + 8A1Knm) , (5.3)

˙ tu where MP is the classical Poiseuille mass flow rates through a microtube defined as

πD4∆pp M˙ tu = m . (5.4) P 128µRTL

The analytical first order, according to the Knudsen number, expression for the dimen- T ˙ tu sionless mass flow rate Stu is obtained by dividing the analytical mass flow rate M (5.3) by ˙ tu the Poiseuille mass flow rate MP (5.4) as

T T T T ST = B0 + B1 Knm, with B0 = 1,B1 = 8A1. (5.5)

In order to compare the analytical expression for the mass flow rate (5.5) to the experi- mental results the measured values of the mass flow rate M˙ exp are normalized by the Poiseuille ˙ tu mass flow rate MP (5.4) according to expression exp ˙ exp ˙ tu S = M /MP . (5.6)

exp Then, the values of S (5.6) are fitted using the first order in Knm polynomial form following the method proposed by Maurer et al. [76]

exp exp exp Sf = B0 + B1 Knm, (5.7)

exp exp where the coefficients B0 and B1 are the fitting coefficients.

Comparing the analytical and experimental expressions of the dimensionless mass flow rates (5.5) and (5.7), respectively, the analytical and experimental coefficients may be related as following: exp exp B0 = 1,B1 = 8A1. (5.8) exp The fitting of the experimental data is carried out without setting the coefficient B0 equal to the value of 1, even if its "theoretical value" is equal to one, see equation (5.5).

exp The values of the coefficient B0 , obtained from the experimental fitting, were used to check the data of the tube diameter given by the provider. In the hydrodynamic flow regime, when the Knudsen number is very small, the mass flow rate must be equal to the Poiseuille ˙ tu mass flow rate MP , see Eq. (5.4). If we suppose that all measured quantities, except the tube diameter, in this Poiseuille expression do not have significant uncertainties, then the 5.1. Continuum and slip regimes 85

exp deviation of the coefficient B0 from the value of 1 may be attributed to an error in the diameter measurements. This technique was used here to adjust the indicative diameter given by the provider and equal to 250 µm. The values of the microtube diameter, adjusted by using the Poiseuille mass flow rate, are given in Table 2.2.

After this first step of the diameter adjustment, the measured mass flow rate was fitted with the first order polynomial form (5.7) in the mean Knudsen number range [10−4, 0.1] (see Figures 5.1 and 5.2). One can see that the analysis of the results was limited to the value of mean Knudsen number of 0.08 for helium, nitrogen and argon and to the value of 0.09 for carbon-dioxide. Therefore, we set the maximum value of 0.2 for the Knudsen number in the outlet tank in order to stay in the assumption of slip regime.

In order to evaluate the pertinence of the polynomial approximation we calculate several statistical parameters indicating the quality of the fitting: the determination coefficient r2, the square residual sum sr and the standard error.

• The determination coefficient r2 as defined in software’s like OriginLab or Matlab.

q 1 P 2 • The square residual sum sr is defined as following: sr = n−k ei , where n is the number of the measured points, e = Sexp − Sexp is the local difference between the i i fi measured values Sexp, normalized according to (5.6), and the fitted values Sexp (5.7), k i fi is the number of the unknown coefficients of the fitting model, k = 2 for the first order and k = 3 for second order fit (to be used in the Chapter6).

• The standard error or also called the statistical uncertainty, defined as q 1 P 2 exp exp Es = n−k ei /Sm , where Sm is the average value of the measured data: exp 1 Pn exp Sm = n i=1 Si . This statistical uncertainty will be compared to the experimental uncertainty defined by expression (4.13).

The obtained fitting coefficients and the statistical characteristics of the fitting procedure 2 r , Es and sr are given in Table 5.2, for the microtubes T 1 and T 2, respectively.

It is clear from the results presented in Table 5.2 that the procedure of the adjustment exp of the tubes diameter works well. The values of the coefficients B0 are very close to their theoretical values equal to one. From the same Table 5.2 one can see that the values of the determination coefficients r2 are all higher than 0.998, and those of the residual variance sr are smaller than 0.018 for both microtubes T 1 and T 2. Finally, comparing the experimental uncertainty (2.4%, see Eq. (4.13)) and the statistical uncertainty of the polynomial fitting Es, one can see that the values of Es are lower than the experi- mental uncertainty for both microtubes T 1 and T 2. Therefore, analyzing the statistical characteristics of the polynomial approximation we can conclude that the first order ac- 86 Chapter 5. Results in Microtubes

2 . 4 H e N 2 . 2 2 A r 2 . 0 C O 2 1 . 8

S 1 . 6

1 . 4

1 . 2

1 . 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 K n m

exp Figure 5.1: Dimensionless mass flow rate Stu as function of the mean Knudsen number for microtube T 1.

2 . 6 H e 2 . 4 N 2 2 . 2 A r C O 2 2 . 0

1 . 8 S 1 . 6

1 . 4

1 . 2

1 . 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 K n m

exp Figure 5.2: Dimensionless mass flow rate Stu as function of the mean Knudsen number for microtube T 2. 5.1. Continuum and slip regimes 87

Microchannel T 1 T 2 Quantity Min Max Min Max Mass flow rate M˙ exp (10−12kg/s) 1.1 229 000 0.3 403 000 Inlet pressure pin (Pa) 6.7 123 600 14.5 129 000 Outlet pressure pout (Pa) 1.5 78 700 3.3 89 700 −4 −4 Average Knudsen number Knm 1.8 10 3.2 2.1 10 3.3

Table 5.1: Experimental conditions for microtubes T 1 and T 2.

cording to the Knudsen number polynomial expressions represents well the experimental data.

5.1.1 Extraction of the slip and accommodation coefficients exp To extract the value of the velocity slip coefficient we use the value of the coefficient B1 and we compare it to its theoretical value equal to 8A1 (see Eq. (5.8)) where A1 = σp/kλ (see Eq. (3.12)). Then the coefficient σp can be written as

k σ = λ Bexp. (5.9) p 8 1

Here σp is the first order velocity slip coefficient and kλ is the coefficient depending on the intermolecular interaction model. The model retained here is the VHS model. The values of the coefficient kλ are given in Table 3.1, calculated according to Eq. (3.4).

The values of the velocity slip coefficients σp are given in Table 5.3 for both microtubes T 1 and T 2. The uncertainty on the coefficient σp in Table 5.3 derives from the uncertainty on exp the fitting coefficient B1 , see relation (5.7), and does not take into account the systematic uncertainty on the mass flow rate measurements, see Eq. (4.13), and the uncertainties related to the Poiseuille expression (5.4).

The values of the experimental velocity slip coefficients (see Table 5.3) are different from the theoretical value (σp = 1.016 [25]), showing that the accommodation is not complete in our experimental conditions.

When the first order velocity slip coefficient σp is known, the tangential momentum accommodation coefficient (TMAC) may be calculated. In the present work we use expres- sion (3.10), which relates the values of the first order velocity slip coefficient and TMAC. By resolving this expression (3.10) for a given value of σp the TMAC (α) value can be obtained.

Analyzing the TMAC values obtained for both microtube T 1 and T 2 in the range of mean Knudsen number [0.0, 0.1], we notice that the highest value for helium and carbon- dioxide, and the lowest value for nitrogen and argon are obtained. The measurements with monatomic gases (He, Ar) seem to correspond to the tendency, observed in previous 88 Chapter 5. Results in Microtubes

Microtube T 1 exp exp 2 Gas B0 B1 sr r Es(%) He 0.987± 0.002 11.806± 0.077 0.010 0.998 0.8 N2 0.989± 0.001 13.791± 0.063 0.008 0.999 0.6 Ar 0.998± 0.002 14.735± 0.090 0.011 0.998 0.9 CO2 1.002± 0.002 14.971± 0.068 0.009 0.999 0.7 Microtube T 2 exp exp 2 Gas B0 B1 sr r Es(%) He 0.990± 0.002 12.463± 0.050 0.007 0.999 0.6 N2 0.987± 0.002 13.804± 0.095 0.012 0.998 1.0 Ar 0.998± 0.003 15.137± 0.110 0.015 0.998 1.2 CO2 0.993± 0.002 16.107± 0.074 0.010 0.999 0.8

exp exp Table 5.2: Experimental coefficients B0 and B1 obtained from the first order polynomial approximation for the microtubes T 1 and T 2.

works [39, 37, 38, 44], showing that TMAC depends on the molecular mass, namely the accommodation coefficient of the lighter gas is greater than that of the heaviest ones. We note that in this case the weight and the spherical molecule size are directly related. Thus, we suggested that a large spherical size could increase the molecule specular reflec- tion by minimizing the wall asperity or structure effects (roughness and atomic arrangement).

In addition, the present results show that for polyatomic molecules this specular mass effect at the wall is moderate (N2 case) or inverted (CO2 case). The non-spherical structure of the polyatomic molecules, associated to the corresponding internal degrees of freedom, seems to favor a random direction of reflection and consequently a diffusive reflection. Furthermore, comparing the results in Table 5.3 this diffusive effect is larger when the molecular structure is complex.

Moreover, we have to recall that the length of the microtubes T 1 and T 2 was not the same when experiments were carried out for the carbon-dioxide gas. Their length was reduced by a factor ∼ 20 and therefore the experimental conditions for this gas were different. But we think that this change has not a significant influence on the interacting law.

Thus globally, the values of the TMAC obtained in the T 2 microtube with stainless steel internal surface lie in the narrow range [0.872, 0.902] centered on a less diffusive value when comparing with the T 1 (Sulfinert) internal surface. The T 2 (Stainless steel) wall tends to reduce the influence of the nature of interacting gas. It seems that at the contact of the stainless surface the gas is submitted to a physical or chemical influence so that its interacting behaviour becomes similar whatever its initial nature. 5.1. Continuum and slip regimes 89

Microtube T 1 Microtube T 2

Gas molecular mass (g/mol) σp α σp α He 4.00 1.160± 0.008 0.930± 0.003 1.225± 0.005 0.901± 0.002 N2 28.02 1.260± 0.006 0.886± 0.002 1.261± 0.009 0.886± 0.004 Ar 39.95 1.260± 0.008 0.886± 0.003 1.294± 0.009 0.872± 0.004 CO2 44.01 1.136± 0.005 0.941± 0.002 1.222± 0.006 0.902± 0.002

Table 5.3: Slip and tangential momentum accommodation coefficients obtained for microtubes T 1 and T 2.

5.1.2 Comparison between the HS and the VHS models As we mentioned above, the results for the microtubes T 1 and T 2 are obtained using the VHS model. Here we will apply the HS model to the results of microtube T 1 in order to highlight the influence of the intermolecular model on the extraction of the slip and tangential momentum accommodation coefficients.

Applying the HS model means changing the value of the viscosity µ and the coefficient kλ in expressions (5.4) and (5.9), respectively. The definition given by the HS model for the viscosity µ and the coefficient kλ are presented in Section 3.1.

exp exp Table 5.4 gives the results of the coefficients B0 and B1 , and of the slip and tangential momentum accommodation coefficients. Comparing the results given in this Table with those presented in Tables 5.2 and 5.3 for microtube T 1 using the VHS model, several remarks may be done:

exp • The coefficient B0 values found with both models are almost equal, only small differ- ence is observed for carbon-dioxide. This difference is due to the value of viscosity µ involved in expression for Poiseuille flow rate (5.4), which is different for both models.

exp • The obtained values of B1 coefficient using the HS model are different from those obtained by the VHS model. In Table 5.2, using the VHS model, it seems like the exp coefficient B1 values depend on the molecular mass of gas, such the lower value is obtained for the lighter gas (He). This dependence comes from the fact that the values

exp exp 2 Gas B0 B1 sr r Es(%) σp α He 0.987± 0.002 10.404± 0.073 0.009 0.998 0.8 1.153± 0.008 0.933± 0.004 N2 0.988± 0.001 11.383± 0.062 0.008 0.999 0.7 1.261± 0.007 0.886± 0.003 Ar 0.998± 0.002 11.442± 0.098 0.011 0.997 0.9 1.268± 0.011 0.883± 0.004 CO2 1.005± 0.002 10.049± 0.068 0.007 0.999 0.6 1.113± 0.008 0.951± 0.004

Table 5.4: Experimental results obtained from the first order polynomial approximation for the microtube T 1 using the HS model. 90 Chapter 5. Results in Microtubes

of the coefficient kλ and viscosity index ω are different for each gas. However, this exp dependence is less observed using the HS model, particularly for CO2, where its B1 value is the lowest one even if its molecular mass is the highest.

• The obtained values of σp coefficient, in case of VHS model, have not similar hierarchy exp with respect to the molecular mass as the coefficient B1 (see Tables 5.2 and 5.3). However, the same hierarchy is obtained in case of HS model (see Table 5.4).

• The values of σp coefficient and the corresponding TMAC values obtained with the VHS and the HS models are similar (see Tables 5.3 and 5.4), with maximum difference of order of 1% obtained for CO2.

It is clear from Tables 5.3 and 5.4 that the use of both VHS and HS models has not a significant influence on the extracted values of the velocity slip and tangential momentum accommodation coefficients.

5.1.3 Comparison with other works

The comparison of the TMAC obtained with those measured in the past is complicated due to several factors. In addition to their obvious dependence on particular gas and surface material: the coefficients are generally sensitive to the wall and the surrounding gas temperature, and the results should be analyzed for the same flow regime. The contam- ination and the roughness of surface play an important role in determining the TMAC values.

Table 5.5 shows a comparison of TMAC values calculated by other authors [39, 88, 92] with our results. The following differences between our experimental conditions and those of Refs. [39, 88, 92] are:

• The ratio L/D is more than 4 times larger for our microtubes.

• The diameters of the microtubes are different: Ref. [39] D = 25 µm, Ref. [88] D = 50 µm and Ref. [92] D = 100 µm.

• The materials of the microtubes surfaces are different: glass in Ref.[92] and silica in Refs. [39, 88].

The treatment of the experimental data was also different in Refs. [39, 88, 92]:

• The accommodation coefficient was calculated in Ref. [92] from the velocity slip coeffi- cient without taking into account the influence of the Knudsen layer.

• The Knudsen number range considered in Refs. [88, 39] was [0.0, 0.3] and the second order approximation was used. 5.1. Continuum and slip regimes 91

Material Helium Nitrogen Argon Porodnov and al [92] Glass 0.895± 0.004 0.925± 0.014 0.927± 0.028 Ewart and al [39] Silica 0.986± 0.009 0.981±0.041 0.942±0.017 Perrier and al [88] Silica 1.00± 0.019 0.961±0.005 0.954± 0.010 present results (T 1) Sulfinert 0.930± 0.003 0.886± 0.002 0.886± 0.003 present results (T 2) Stainless Steel 0.901± 0.002 0.886± 0.004 0.872± 0.004

Table 5.5: The tangential momentum accommodation coefficient (TMAC) obtained from the present experiments and in Refs. [39, 88, 92].

• The Hard Sphere model was used in Ref. [39] for the definition of the molecular mean free path.

The different TMAC values presented in Table 5.5 show high dependence of TMAC on the surface material. TMAC values obtained in Refs. [39] and [88], for the same surface material (Si), are very close for each gas if considering their uncertainties. The comparison of TMAC values obtained in our experiments for Sulfinert surface, which is silica based coating, with the results of Refs. [39] and [88] for silica surface shows a high difference. It seems that the chemical treatment made on the Sulfinert surface, to be inert to the sulfur and organic components, changes significantly the interaction of the different gases on this surface. We have to note that the surface roughness of the microtubes used in present experiments and in Refs. [39], [88] and [92] are not known. It is well known that the surface roughness has an important influence on TMAC value, thus the difference of TMAC value may be attributed to the difference of the surfaces roughness.

The tendency of TMAC decreasing with increasing of the gas molecular mass, observed in our experiments, is also reported in Refs. [39, 88]. The opposite behaviour is observed in Ref. [92]. In addition, in our experiments similar TMAC values for nitrogen and argon gases are found, which was also obtained in the works [39, 88, 92] if we consider the uncertainty.

In conclusion, the results obtained in our experiments for the Sulfinert and stainless steel surface are in good qualitative agreements with other works in literature.

We saw above that the analysis of the experimental data, done using the continuum approach, were limited only to the first order approximation of the boundary condition and we did not used the second order approximation, which allows to extend the Knudsen range to 0.3. Unfortunately, the experimental data obtained in this range were neither sufficiently numerous nor homogeneously distributed to allow a pertinent statistical treatment. Thus, we could not extract the theoretical coefficient (velocity slip and accommodation coefficients) with a sufficient quantitative reliability. The required accuracy would be all the more impor- tant here as our investigation shows qualitatively a very significant second order in this range. 92 Chapter 5. Results in Microtubes

5.2 Transitional and near free molecular regimes

The transitional and free molecular regimes are the most complicated regimes from experi- mental point of view. In these regimes, in addition to the problems of leakage and outgassing (see Section 4.2.3), the detection of flow parameters, especially, the pressure variation in tanks becomes difficult and requires a high sensitive pressure transducers. As example, the pressure variation in the outlet tank, for nominal pressure of 14.2 P a, is of the order of 0.28P a during the experimental time t = 250s. In our experiments we use a very high sensitive pressure transducer having a resolution of 1.5 × 10−5 of full scale and response time of 30 ms (see Table 4.4).

Contrarily to the slip flow regime an analytical expression for the mass flow rate does not exist in the literature for the transitional regime. Only the numerical results for the mass flow rate obtained by using different kinetic models may be found.

In the transitional and free molecular regimes (Knm > 0.1 corresponding to δm < 8.8) the mass flow rate calculated, using the linearized BGK kinetic equation, by Lo & Loyalka, 1982 [66] for diffuse reflexion at wall (α = 1) and by Porodnov & Tuchvetov, 1979 [94] for diffuse-specular reflexion (α 6= 1), is compared to the experimental mass flow rate.

In order to obtain the value of the mass flow rate from the linearized BGK equations, the following assumptions were made:

• The tube diameter D is small compared to its length L, D << L;

D dp • The dimensionless pressure gradient along the tube equal to p dz , is small compared to one.

The first hypothesis allows neglecting the ends effect, while the second one allows the linearization of the BGK equation.

By solving numerically the linearized BGK equations, the values of the dimensionless mass flow rate Qtu are obtained by Lo & Loyalka [66] for α = 1 and by Porodnov & Tuchvetov [94] for α 6= 1 in form of Tables for the pair (Qtu(δ), δ). The dimensionless mass flow rate Qtu was normalized following this expression: √ 2RT Qtu = − M˙ tu (5.10) π(D/2)3dp/dz

The values of the dimensionless mass flow rate was obtained in Refs. [66] and [94] under the assumption of small pressure gradients along the tube, therefore the rarefaction parameter δ (3.85) is supposed to be constant along the tube. However, it is not the case of our experimen- tal conditions. The pressure along the microchannel may vary with a ratio pin/pout up to 5, 5.2. Transitional and near free molecular regimes 93 so the rarefaction parameter is not constant in our microtubes. Then, a direct use of the val- ues of Qtu (5.10) taken for the mean value of the rarefaction parameter may be not very exact.

Another method can be used, which consists in the integrating the value of Qtu over the interval of δ following expression:

δ Zout tu 1 tu G (δin, δout) = Q (δ)dδ, (5.11) δout − δin δin

tu tu where Q (δ) is the value of Q for a given value of the rarefaction parameter δ and δin, δout are the values of the rarefaction parameter corresponding to the pressure of the inlet and outlet tanks, respectively. This method was proposed by Sharipov & Seleznev [103] and it was used by Ewart [36] and Pitakarnnop [89]. The values of the dimensionless mass flow rate given by expressions (5.10) and (5.11) are different, however this difference is not very significant. Ewart [36] and Pitakarnnop [89] showed that the use of both expressions gives close results with difference less than the uncertainty on the experimental mass flow rate. For this reason we will use directly the numerical values of Qtu (5.10) taken for the mean value of the rarefaction parameter calculated by Lo & Loyalka, 1982 [66] and Porodnov & Tuchvetov, 1979 [94] to compare them to the experimental values.

To obtain the corresponding dimensionless experimental mass flow rate, the following expression is used: √ exp L 2RT ˙ exp G = 3 M . (5.12) π(D/2) (pin − pout) Expression (5.12) can be obtained by simple integration of expression (5.10) according to relation (5.11). The dimensionless mass flow rates (experimental and numerical) are presented in Figures 5.3 and 5.4 as function of the mean rarefaction parameter δm for the microtubes T 1 and T 2, respectively. The mean rarefaction parameter δm (3.85), calculated from the mean value of the pressure between the inlet and outlet tanks, is obtained using the HS interaction model. In these Figures the minimal values of the mean rarefaction parameters reached for the working gases (He, N2 and Ar) are different. This difference is essentially due to the difference of the gases molecular mass, so that for the same value of the pressure, the rarefaction parameter for helium is 3 times smaller than for argon. This is not true in the case of carbon-dioxide, which has the minimum rarefaction parameter as helium, even if its molecular mass is 11 time higher. This is because the experimental conditions for carbon-dioxide were different from the other gases. The length of the microtubes T 1 and T 2 was reduced to around 10 cm in the case of experiments yields with carbon-dioxide, while this length was around 2 m for the other gases (see Table 2.2).

The Analysis of the curves on Figures 5.3 and 5.4 is divided in two parts: first, for δm > 10 and second, for δm < 10. When the rarefaction parameter δm is superior to 10, the gas flow is in the slip regime. In this part of the curves, the difference between the numerical results for the different values of TMAC (α) is very small, which makes difficult 94 Chapter 5. Results in Microtubes

6 H e N 2 5 A r C O 2 1 =؁ 4 9 . 0 =؁

8 . 0 =؁ G 3

2

1 0 . 1 1 1 0 ؄ m

Figure 5.3: Dimensionless mass flow rate G as function of the Knudsen number for microtube T 1.

6 H e N 2 5 A r C O 2 1 =؁ 4 9 . 0 =؁

8 . 0 =؁ G 3

2

1 0 . 1 1 1 0 ؄ m

Figure 5.4: Dimensionless mass flow rate G as function of the Knudsen number for microtube T 2. 5.2. Transitional and near free molecular regimes 95

the comparison just from a simple visualization. For δm > 10 the use of the continuum approach (see Section 5.1) is more accurate to determine the values of TMAC.

When the rarefaction increases, so δm decrease less than 10, the slip regime gives place to the transitional regime. In this regime (δm < 10) the difference between the numerical results for different values of TMAC begins to be visible, and then when the rarefaction parameter decreases this difference becomes more evident. The experimental points of nitrogen and argon obtained for the microtube T 1 (see Figure 5.3) start to move away the numerical curve α = 1, as the rarefaction parameter decreases, to reach the numerical curve α = 0.9. From Figure 5.3 one can see that the experimental points have exactly the same look, which confirms the result obtained in the slip regime (i.e. the same TMAC value can be attributed for nitrogen and argon). For helium and carbon-dioxide, in the range of δm [2, 10], the experimental points reach the numerical curve for α = 1, then, when δm is less than 2, a small deviation of carbon-dioxide experimental points from the numerical curve α = 1 is observed. However, for helium the deviation is more significant. Again, this result confirms what was found using the continuum approach (i.e. a higher value of TMAC is obtained for carbon-dioxide).

The experimental points of all gases for the microtube T 2 (see Figure 5.4), obtained for δm > 4, reach the numerical curve for α = 0.9. When δm is less than 4 a significant deviation is observed for nitrogen and argon. However, for helium and carbon-dioxide the experimental points continue to follow the same curve α = 0.9, which is in agreement with the results found in the continuum analysis. The results shown here confirm the same TMAC value for the couple helium/carbon-dioxide and couple nitrogen/argon.

Globally, the analysis made in the transitional regime using the BGK kinetic model con- firm the qualitative and quantitative results of TMAC found with the continuum approach, therefore the same TMAC value can be applied in all Knudsen flow regimes.

Chapter 6 Results in Rectangular Microchannels

In this section the results of the measurements series of the mass flow rate through rectan- gular cross-section microchannels are analyzed. The microchannels considered in this thesis have different materials of the internal surfaces, different surface roughness and different aspect ratios w/h. The technical characteristics of the rectangular microchannels are given in Table 2.1, page 17. The investigated Knudsen number range is large and covers all the flow regimes: from hydrodynamic to free molecular regimes.

Due to the large number of microchannels involved in this study (12) and the large Knudsen number range investigated the analysis and the presentation of the results are structured in three parts, following the microchannel groups: A, E and S, see Table 2.1. As it was previously explained in Section 2.6.1 the difference between these three groups consists in the difference of their surface materials and roughness. The microchannels A and E internal surfaces are covered with a layer of gold having a roughness of around 1 nm and 12 nm , respectively. The microchannels S internal surfaces are covered with a layer of Silica (SiO2) having a roughness of around 1.12 nm (see Table 2.1).

The experiments involving the microchannels A, E and S were conducted under the same experimental conditions for temperature (around 296 K) and pressure. It was shown in Sharipov & Seleznev, 1994 [104] and Ewart et al. 2007 [38] that there is not a significant dependence of the dimensionless mass flow rate on the pressure ratio, that is why, in our experiments, the pressure ratio pin/pout between the inlet and the outlet tanks was changed from the minimum value of 1.5, for experiments conducted in the hydrodynamic regime, to the maximum value of 5, for experiments conducted in the free molecular regime. Only for experiments conducted with the microchannel S2 (see Table 2.1) this ratio was maintained around a constant value equal to 4. The tables of the experimental conditions (given bellow) for each microchannel show the minimum and maximum values of the pressure in the inlet and outlet tanks, thus the maximum and minimum values of the pressure ratios can be calculated. In general rule, the maximum value of 5 for the pressure ratio was set for the microchannels with indexes 1 and 2 (e.g. A1, S2 ...etc.), while the maximum value of 3 was set for the microchannels with indexes 3 and 4 (e.g. F 3, E4 ... etc.).

The results obtained for each group (A, E and S) are examined with two different ap- proaches depending on the Knudsen number Knm (1.1), calculated from the mean value of the pressure between the inlet and outlet tanks. First, the stokes equation, subjected to the 98 Chapter 6. Results in Rectangular Microchannels

first order velocity slip boundary condition in the mean Knudsen number range [0, 0.1] and to the second order velocity slip boundary condition in the mean Knudsen number range [0, 0.3], is applied to simulate the gas flow through the microchannels in the hydrodynamic and slip regimes. Second, the kinetic approach, based on the linearized BGK model, is used to simulate numerically the gas flow through rectangular cross-section microchannels in the transitional and free molecular regimes (Knm > 0.1) with assumption of specular-diffuse reflection of the gas molecules from the wall.

6.1 Continuum and slip regimes

The continuum approach used here for modeling the flow through the rectangular microchan- nels is an original approach developed during this thesis and presented in details in Section 3.2. This approach leads to a complete analytical expression for the mass flow rate of second order in Knudsen number (see Eq. (3.77)). The previous approaches allowing the direct extraction of the flow characteristics (such as the velocity slip and tangential momentum accommodation coefficients) are of first order according to the Knudsen number.

In order to extract the velocity slip and tangential momentum accommodation coefficients from the measurement of the mass flow rate we proceed in the following way.

The values of the measured mass flow rate (4.8) are normalized with the expression of ˙ ∞ Poiseuille mass flow rate between two parallel plates Mp (3.21) and the coefficient Vn (3.79), which takes into account the influence of the lateral walls, according to the following expres- sion: exp ˙ exp ˙ ∞ S = M /(Mp · Vn). (6.1) Then, by the use of the least square method detailed in Maurer el al, 2003 [76] the values of the dimensionless mass flow rate Sexp (6.1) are fitted with a first order polynomial form in the mean Knudsen number range [0.0, 0.1] using expression:

exp exp exp Sf = B0 + B1 Knm, (6.2) and with a second order polynomial form in the mean Knudsen number range [0.0, 0.3] using expression: exp exp exp exp 2 Sf = B0 + B1 Knm + B2 Knm. (6.3) exp The coefficients of fitting Bi (i = 0, 1 or 2) are obtained by applying the non-linear square Marquard-Levenberg algorithm to the dimensionless experimental values of the mass flow exp exp rate S (6.1). The uncertainties on the coefficients Bi are calculated using the standard procedure. The pertinence of the first and second order polynomial approximations (6.2) 2 and (6.3) is analyzed using the same parameters of fitting (r , sr and Es) used to analyze the results obtained in the microtubes (see Section 5.1, page 85).

To extract the value of the velocity slip coefficient from the measured values of the mass flow rate, the experimental expressions for the dimensionless mass flow rate (6.2) and 6.1. Continuum and slip regimes 99

(6.3) are compared to the following analytical expressions of the mass flow rate M˙ ch (3.82) ˙ ∞ normalized with the Poiseuille mass flow MP (3.21) and the coefficient Vn (3.79)

T T T T 2 S = B0 + B1 Knm + B2 Knm, (6.4) with 4 T T Tn T A2π P + 1 B0 = 1,B1 = 6A1 ,B2 = ln P. (6.5) Sn 16Sn P − 1

Comparing expressions (6.3) and (6.4) the coefficients A1 and A2 (3.12) may be expressed under this form exp exp σp B1 Sn σ2p 16B2 Sn P − 1 A1 = = ,A2 = 2 = 2 . (6.6) kλ 6Tn kλ π ln P P + 1

Therefore the "experimental" slip coefficients of first and second order σp and σ2p, respectively, may be found from relations

exp exp 2 B1 kλSn 16B2 kλSn P − 1 σp = , σ2p = 2 . (6.7) 6Tn π ln P P + 1 It is clear from previous expressions that the lateral wall influence is present in both exp exp coefficients B1 and B2 through the coefficients Sn and Tn. The dependence of velocity slip coefficients (σp and σ2p) on the molecular interaction model comes from the coefficient kλ (see Eq. (6.7)). The value coefficient kλ is calculated using the VHS molecular interaction model (see Section 3.1).

T It is to be noted that the value of the analytical coefficient of zero order B0 involved exp in expression (6.4) is equal to 1, therefore the corresponding experimental value B0 (6.3) should be also equal to 1. When the measured values of the dimensionless mass flow rate exp exp Sch were fitted with expressions (6.2) and (6.3) the values of the coefficient B0 were not set to 1. Therefore we will see that its values are close to one in the most cases but in some ∞ cases its values deviate significantly from of 1. By this way the accuracy on the term Mp Vn can be estimated.

The value of the tangential momentum accommodation coefficients TMAC is calculated from the value of the velocity slip coefficient σp following expression (3.10) given in Section 3.2.1.

6.1.1 Validation of the continuum approach with practical example.

From the comparison of both dimensionless mass flow rate expressions (6.3) and (6.4), the "experimental" velocity slip coefficient, and then the "experimental" accommodation coefficient may be easily deduced.

T The velocity slip coefficient σp is proportional to coefficient B1 , see Eq. (6.5). There- 100 Chapter 6. Results in Rectangular Microchannels

w/h 1 2 5 10 20 50 100 ∞ Tn/Sn 1.333 1.092 1.028 1.013 1.006 1.003 1.001 1.000

T Table 6.1: The influence of the lateral walls on B1 . fore, when the slip coefficient is extracted from the mass flow rate measured through a microchannel of rectangular cross-section, it is important to use mass flow rate expression T (3.82) instead of (3.20), due to the changes introduced in the corresponding coefficient B1 . T If we compare Eqs. (3.20) and (3.82), the coefficients B1 differ by a factor of Tn/Sn. The values of this factor according to w/h are given in Table 6.1. It is possible to see that, for a channel with an aspect ratio w/h = 2, the two parallel plates approximation underestimates this coefficient by ∼ 9%.

In order to illustrate the application of the proposed approach, two experimental results of Graur et al.[44] will be compared with our analytical results.

Gas σp α ∗ N2 1.104 ± 0.010 0.956 ± 0.005 Ar∗ 1.205 ± 0.064 0.910 ± 0.028 ∗∗ N2 1.104 ± 0.010 0.956 ± 0.005 Ar∗∗ 1.244 ± 0.064 0.893 ± 0.027

Table 6.2: Slip and tangential momentum accommodation coefficients extracted from the experimental data [44]. The first two lines ∗ correspond to results [44] using the parallel plate expression (3.20). The second two lines ∗∗ represent the implementation of expression (3.82), which takes the influence of the lateral walls into account.

The mass flow rate through a silicon microchannel of dimensions h = 9.38µm, w = 492µm and L = 9.39mm was measured by Graur et al.[44]. The slip and accommodation coefficients were extracted using expression (3.20) for the mass flow rate between two parallel plates. When expression (3.82), which consider the influence of the lateral walls, is applied to the same experimental data, the values of the slip and accommodation coefficients are changed very slightly, by less than 3%, see Table 6.2. This slight variation, 0.039 for Ar remains smaller than the confidence interval, of 0.064. This is not surprising because the ratio w/h for this channel is 52.45 and the analysis of the analytical expressions (3.82), (3.20) confirms that, when w/h > 50, the implementation of expression (3.20), instead of (3.82), gives an error of the order of 1% (see Table 3.6). 6.1. Continuum and slip regimes 101

6.1.2 Microchannels A

Microchannel A1 A2 Quantity Min Max Min Max Mass flow rate M˙ exp (10−12kg/s) 0.4 16 000 0.1 34 400 Inlet pressure pin (Pa) 142.3 136 760 14.4 129 190 Outlet pressure pout (Pa) 28.4 88 220 3.3 93 300 −3 −3 Average Knudsen number Knm 1.7 10 6.8 1.8 10 70.4

Table 6.3: Experimental conditions for the microchannels A1 and A2.

Microchannel A3 A4 Quantity Min Max Min Max Mass flow rate M˙ exp (10−12kg/s) 1.9 135 000 3.4 285 000 Inlet pressure pin (Pa) 61.6 125 100 64.9 120 250 Outlet pressure pout (Pa) 22.1 79 200 25.7 79 500 −3 −3 Average Knudsen number Knm 1.7 10 14.7 2.1 10 14.6

Table 6.4: Experimental conditions for the microchannels A3 and A4.

The experimental conditions for the microchannels of the group A are summarized in Tables 6.3 and 6.4. This group of microchannels A1, 2, 3, 4 have the following aspect ratios w/h = 1.9, 3.9, 18.1, 39.0, respectively. As it was mentioned before, the internal surfaces of these microchannels are covered with a layer of gold having a roughness of around 1 nm.

The fitting curves of the dimensionless mass flow rate Sexp (6.1) as function of the mean Knudsen number using first and second order polynomial expressions (6.2) and (6.3), respectively, for the microchannels of the group A and for the gases helium, nitrogen and argon are shown in Figures 6.2, 6.3, 6.4 and 6.5, and their corresponding fitting coefficients exp Bi (i = 0, 1, 2) are given in Table 6.5. The first line in Table 6.5, for each gas, represents the fitting coefficients obtained with the first order polynomial approximation (6.2) in the mean Knudsen number range [0.0, 0.1] and the second line represents the fitting coefficients obtained with the second order polynomial approximation (6.3) in the mean Knudsen number range [0.0, 0.3].

exp From Table 6.5 one can see that the values of the coefficient B0 obtained for the microchannels of the group A deviate slightly from the theoretical value of 1. This deviation can be explained by an error in the parameters involved in the Poiseuille mass flow rate (3.21) or by the shape of the microchannels cross-section, which is not truly rectangular due to the problem of alignment of the two silicon wafer used to create the microchannels (see Section 2.5). The continuum approach, use here, was developed for a perfect rectangular channel, hence the use of this approach for a non perfect rectangular channels can induce 102 Chapter 6. Results in Rectangular Microchannels

exp exp exp 2 Channel Gas B0 B1 B2 sr r Es(%) 1.023± 0.008 10.625± 0.165 - 0.021 0.996 1.4 He 1.019± 0.007 10.666± 0.142 1.635± 0.536 0.020 0.999 0.9 1.009± 0.005 12.104± 0.111 - 0.016 0.998 1.1 A1 N 2 1.008± 0.006 12.181± 0.171 -1.555± 0.850 0.020 0.999 1.1 1.012± 0.009 12.621± 0.190 - 0.028 0.995 1.9 Ar 1.027± 0.014 11.731± 0.390 3.115± 1.863 0.044 0.997 2.3 1.001± 0.003 8.785 ± 0.056 - 0.008 0.999 0.6 He 1.000± 0.008 8.681 ± 0.183 2.324± 0.729 0.024 0.999 1.3 0.996± 0.004 9.168 ± 0.098 - 0.015 0.997 1.1 A2 N 2 0.999± 0.008 8.701 ± 0.195 7.575± 0.812 0.027 0.999 1.5 1.017± 0.005 9.580 ± 0.112 - 0.016 0.997 1.1 Ar 1.014± 0.006 9.448 ± 0.157 6.699± 0.645 0.021 0.999 1.1 1.011± 0.002 8.699± 0.046 - 0.006 0.999 0.5 He 1.013± 0.003 8.426± 0.062 3.981± 0.237 0.009 0.999 0.5 1.003± 0.002 9.413± 0.042 - 0.007 0.999 0.5 A3 N 2 1.006± 0.002 9.047± 0.038 4.944± 0.144 0.006 0.999 0.3 1.013± 0.003 10.078± 0.068 - 0.011 0.999 0.8 Ar 1.017± 0.002 9.593± 0.060 5.767± 0.256 0.008 0.999 0.5 0.988± 0.003 7.993± 0.073 - 0.008 0.998 0.6 He 0.984± 0.005 7.974± 0.106 4.072± 0.442 0.014 0.999 0.8 0.989± 0.003 9.110± 0.080 - 0.010 0.998 0.8 A4 N 2 0.993± 0.005 8.660± 0.121 6.555± 0.464 0.018 0.999 1.0 1.007± 0.003 9.326± 0.074 - 0.012 0.998 0.9 Ar 1.008± 0.007 8.784± 0.177 9.661± 0.670 0.025 0.999 1.4

Table 6.5: Fitting coefficients of the dimensionless mass flow rate Sexp (6.1) obtained with the first (first line) and second order (second line) approximations (6.2) and (6.3) in the mean Knudsen number ranges [0, 0.1] and [0, 0.3], respectively, for microchannels of the group A. 6.1. Continuum and slip regimes 103 some deviation from theory. For the microchannel A1 the ratio w/h is the smaller one, so the influence of the lateral walls in this case is the more important. Therefore, the deviation exp of the coefficients B0 from 1 is more important for this microchannel, where the obtained values deviate by more than 2% in case of helium and argon.

exp Another reason of the coefficients B0 deviation from 1 could be an error on one of the parameters involved in the normalization term (see Eq. 6.1) corresponding to the Poiseuille mass flow rate: the microchannels dimensions, the pressure or temperature measurements, ∞ which can influence the normalizing term MP Vn.

The pertinence of the first and second order approximations (6.2) and (6.3) in the range of the mean Knudsen number [0.0, 0.1] and [0.0, 0.3], respectively, can be analyzed from the 2 parameters of fitting r , sr and Es presented in Table 6.5.

The analysis of these coefficients shows that the determination coefficients r2 are closer to 1 for the second order approximation. For the microchannel A3 there is no difference in the values of r2, the same value of 0.999 is obtained with both approximations. However, the higher value of r2 for second order fitting is due to the higher number of measured points of the mass flow rate in the range of Knudsen number [0.0, 0.3] than in the range [0.0, 0.1]. Therefore a conclusion of the pertinence of the approximation order using only the determination coefficient is not accurate. Comparing the values of the residual sum sr one can see that in most cases their values are lower for the first order approximation than for the second order approximation, except for the microchannels A3, where their values are similar for both approximations.

Finally, comparing the statistical error Es made on the first and second order approxi- mations to the experimental uncertainty value of 2.4% (4.13) one can see that all the values of Es presented in Table 6.5 are lower than the experimental uncertainty.

Globally, based only on the statistical parameters of fitting we can say that both of first and second order polynomial approximations are pertinent in their respective Knudsen number ranges [0.0, 0.1] and [0.0, 0.3]. Only for the case of the microchannel A1 the second order approximation seems to be not pertinent due to the large errors and the negative value exp for the B2 coefficients.

6.1.2.1 First order coefficients analysis

Using expression (6.7) the values of the velocity slip coefficient σp for the microchannels of group A are calculated and presented in Table 6.6. The uncertainty on these coefficients exp (see Table 6.6) derives from the uncertainty on the coefficients B1 and does not take into account the experimental uncertainty on the mass flow rate (4.13).

The values of σp obtained in the mean Knudsen number ranges [0.0, 0.1] and [0.0, 0.3] 104 Chapter 6. Results in Rectangular Microchannels

Channel Gas molecular mass (g/mol) σp α σ2p 1.263± 0.020 0.885± 0.008 - He 4.00 1.268± 0.017 0.883± 0.007 0.053± 0.018 1.338± 0.012 0.855± 0.005 - A1 N 28.02 2 1.346± 0.019 0.851± 0.007 -0.043± 0.023 1.305± 0.020 0.867± 0.008 - Ar 39.95 1.213± 0.040 0.906± 0.018 0.075± 0.045 1.109± 0.007 0.953± 0.003 - He 4.00 1.096± 0.023 0.960± 0.011 0.095± 0.030 1.076± 0.011 0.969± 0.006 - A2 N 28.02 2 1.021± 0.023 0.997± 0.012 0.265± 0.028 1.052± 0.012 0.981± 0.006 - Ar 39.95 1.038± 0.017 0.989± 0.009 0.206± 0.020 1.132± 0.006 0.943± 0.003 - He 4.00 1.096± 0.008 0.960± 0.004 0.192± 0.011 1.138± 0.005 0.940± 0.002 - A3 N 28.02 2 1.094± 0.005 0.961± 0.002 0.203± 0.006 1.141± 0.008 0.938± 0.004 - Ar 39.95 1.086± 0.007 0.965± 0.003 0.208± 0.009 1.044± 0.009 0.986± 0.005 - He 4.00 1.041± 0.014 0.987± 0.007 0.198± 0.022 1.106± 0.010 0.955± 0.005 - A4 N 28.02 2 1.051± 0.015 0.982± 0.007 0.272± 0.019 1.060± 0.008 0.978± 0.004 - Ar 39.95 0.998± 0.020 1.009± 0.011 0.351± 0.024

Table 6.6: Slip and tangential momentum accommodation coefficients found experimentally from the first (first line) and second order (second line) approximations of the dimensionless mass flow rate Sexp (6.2) and (6.3), respectively, for microchannels of the group A. 6.1. Continuum and slip regimes 105 with first and second order approximations, respectively, are different from the theoretical T values obtained by Albertoni et al. 1963 [3] and Cercignani, 1964 [23](σp = 1.016) and by T Kogan, 1969 [60](σp = 1.012) using the BGK kinetic model with complete accommodation of the molecules at wall. Therefore, the accommodation of the molecules on the gold surface of the microchannels A is not complete. Only one exception is the value of σp obtained for argon in the microchannel A4 where its value is close to the theoretical values if considering its uncertainty.

The values of the tangential momentum accommodation coefficients α obtained from the values of the velocity slip coefficients σp using expression (3.10) are presented in Table 6.6.

TMAC values obtained with the first order approximation (6.2) in the Knudsen number range [0.0, 0.1] for the microchannels A1 are in the narrow range [0.85-0.89] with higher value for helium and lower value for nitrogen. The behaviour of TMAC decreasing with the molecular mass of gas reported in Ewart et al. 2007 [38] is observed here but, only for monatomic gases (He and Ar).

The opposite behaviour is observed for the microchannel A2 in the same Knudsen number range [0.0, 0.1], where a lower TMAC value is obtained for the lighter gas (He). The decrease of TMAC with the increase of the molecular mass of gases is not obtained in this case. However, the TMAC values obtained for A2 are also in narrow range [0.95, 0.98]. It seems that the smooth surfaces of the microchannels A1 and A2 make insignificant the influence of the gas type (molecular mass) on the momentum exchange of the gas molecules with the walls.

Comparing now the TMAC values obtained for the microchannels A1 and A2 we notice that higher values are obtained for the microchannels A2. Normally, the TMAC values should be the same for all microchannels A as they have the same material and roughness of the internal surfaces. This difference of the TMAC values is due, essentially, to the influence of the lateral walls on the gas flow. For the microchannel A1 the lateral walls represent around 35% of the interacting surface with the fluid while it is less than 14% for the microchannel A2. The technique of deposition (PVD technique, see Section 2.3) used to coat the microchannels of group A with the layer of gold does not guaranty, neither the same thickness of the gold layer on the bottom and on the lateral walls, nor the same roughness. In addition, the problem of alignment of silicon wafers to create the microchannels makes the lateral walls not truly vertical. Consequently, the TMAC is further affected by the lateral walls in the case of the microchannel A1 than in the case of the microchannel A2, which is the reason of the different TMAC values obtained.

The results of TMAC for the microchannel A3 confirm the finding in the microchannels A1 and A2. The lateral walls influence in this case (A3) is smaller (less than 5%). TMAC values obtained in the microchannel A3 for all gases are similar, in the range of their uncertainties, centered on the value of 0.940 ± 0.003. As the lateral walls influence decreases TMAC does not depend on the molecular mass of gas. In addition, the TMAC values obtained for the microchannel A3 are close to those obtained for the microchannel A2. 106 Chapter 6. Results in Rectangular Microchannels

The results found for the last microchannel A4, which is the largest one (the lateral walls surface represents less than 2.5% of the total interacting surface), do not have the same behaviour as that found for the microchannel A3. Although the influence of the lateral walls is not significant in this case, the TMAC values obtained for the different gases are not equal. However, these values remain in a narrow range [0.95, 0.98]. We have to remind that the surface roughness of the microchannel A4 is not the same as that of the other microchannels of the group A (see Table 2.1, page 17). In addition, this microchannel was not fabricated from the same silicon wafer and it has not been manufactured in the same conditions as other microchannels of the group A, which can explain the difference in the TMAC value obtained comparing to the microchannel A3.

Now we analyze the TMAC values obtained in the range of mean Knudsen numbers [0.0, 0.3] using the second order approximation (6.3). TMAC values obtained are similar or superior to those obtained in the mean Knudsen number range [0.0, 0.1] with the first order approximation (see Table 6.6). For the microchannel A1 the second order approximation does not appear to be very pertinent. The value of second order velocity coefficient σ2p obtained for nitrogen is negative and the uncertainty on this coefficient for argon is of the order of the coefficient itself. We suppose that this is due to lack of data in the range of mean Knudsen number [0.1, 0.3], which makes the statistical treatment not very accurate (see Figure 6.2, bottom graph).

For the other microchannels, the TMAC values obtained with the second order approx- imation confirm the results found with the first order approximation: no dependence of TMAC on the molecular mass of gas. The difference in the TMAC values obtained with the first and second order approximations (higher values for the second order approximation) is not very significant, almost included in the experimental uncertainty.

In Figure 6.1 the TMAC values obtained for microchannels of the group A are plotted for all gases. The upper and bottom graphs present the TMAC values obtained with the first and second order approximations, respectively. From these two graphs (Figure 6.1) it is easy to see that TMAC values obtained for the microchannel A1 are different from those obtained for other microchannels, which highlights the influence of the lateral walls. Furthermore, the TMAC values obtained for the microchannels A2, A3 and A4 are in narrow range [0.94, 1.00], which confirms that the TMAC does not depend on the molecular mass for the smooth surface of gold.

6.1.2.2 Second order coefficients analysis

exp The second order coefficients B2 and σ2p are presented in Table 6.5. As we discussed exp previously, the second order coefficients (B2 ) obtained for the microchannel A1 (see Table 6.5) are not pertinent, in particular, for nitrogen and argon gases, therefore, the analysis of the second order coefficients for the microchannel A1 is not carried out. 6.1. Continuum and slip regimes 107

1 . 0 4 A 1 A 3 1 . 0 2 A 2 A 4 1 . 0 0 0 . 9 8

) 0 . 9 6 α ( 0 . 9 4

C

A 0 . 9 2 M T 0 . 9 0 0 . 8 8 0 . 8 6 0 . 8 4 H e l i u m N i t r o g e n A r g o n

1 . 0 4 A 1 A 3 1 . 0 2 A 2 A 4 1 . 0 0 0 . 9 8

) 0 . 9 6 α ( 0 . 9 4

C

A 0 . 9 2 M T 0 . 9 0 0 . 8 8 0 . 8 6 0 . 8 4 H e l i u m N i t r o g e n A r g o n

Figure 6.1: Tangential momentum accommodation coefficients (TMAC) plotted as function of gas obtained with the first and second order approximations in microchannels of the group A. 108 Chapter 6. Results in Rectangular Microchannels

The obtained values of the second order velocity slip coefficients σ2p are different from the theoretical values proposed by Deissler [34](σ2p = 0.884) and Cercignani [23](σ2p = 0.766), see Table 3.2, page 28. However, these values are close to those obtained theoretically by Hadjiconstantinou [48](σ2p = 0.243) and Cecignani & Lorenzani [28](σ2p = 0.184) with the assumption of full accommodation of the molecules at walls (see Table 3.2).

The physical significance of the second order velocity slip coefficient and its dependence on the molecular mass of gas and aspect ratios of the channel cross-section was weakly investigated in literature [24], [110], [39], [44], [88]. The authors of Refs. [39], [44], [88] showed that the coefficient σ2p increases with the molecular mass of the gas. Ewart et al. 2007 [39] and Perrier et al. 2011 [88] conducted their study microtubes, while Graur et al. 2009 [44] studied the second order effect in plane microchannel.

Comparing our results of the coefficient σ2p with the results of these works [39], [88], [44] we observe some similarities. The behaviour of coefficient σ2p increasing with the molecular mass is also observed in our experimental results, particularly for the microchannel A4 (see Table 6.6). However, for the other microchannels A2 and A3 this behaviour is less obvious. The values of the coefficient σ2p obtained for the microchannel A3 are similar for all gases, centered around the value of 0.2 ± 0.01. For the microchannel A2 similar values are obtained for nitrogen and argon. Taking into account the fact that the surface roughness for the microchannel A4 is slightly higher than in other two microchannels A2 and A3, we can conclude that in the case of the smooth gold surface the second order coefficient does not depend on the molecular mass of gas.

The influence of the microchannel aspect ratio on the gas flow through microchannels is assumed to be taken into account by the factors Sn, Tn and Vn in expression for the mass flow rate (see Eq. 3.82). That means that if the bi-dimensional modeling assumptions are correct, the microchannels cross-section aspect ratio should not have a significant effect on the velocity slip coefficients σp and σ2p. However in our bi-dimensional modeling we supposed that all channel’s walls are homogeneous and the gas-surface interaction is the same for upper-bottom and lateral walls. Therefore when the influence of the cross-section aspect ratio appears in the behaviour of the first and second order velocity slip coefficient it may be explained by the difference of the gas-surface interaction between the upper-bottom and lateral walls of the channels or by the deformation of the cross-section shape.

In order to analyze the influence of the microchannels cross-section aspect ratio (the influence of the nature of the lateral walls) we compare only two microchannels A2 and A3 from the group A. The results for microchannels A1 and A4 are excluded exp from this analysis, because the second order coefficient B2 is not pertinent in the case of microchannel A1 and the surface roughness of microchannel A4 is slightly different. We exclude also from this analysis the results for helium in microchannel A2 because the statistical uncertainty for the coefficient σ2p is on the order of 30% of its value (see Table 6.5). 6.1. Continuum and slip regimes 109

Comparing the results for argon and nitrogen in microchannels A2 and A3, it is clear when changing the channel that there is no difference in the second order coefficients for argon and that only a slight difference (almost within the statistical uncertainty) appears for nitrogen in microchannel A2. This difference may be effectively attributed to the influence of the different nature of the lateral walls in microchannel A2, which has smaller aspect ratio (w/h = 3.9) in comparison to A3 (w/h = 18.1).

A last remark on the second order velocity slip coefficient can be given. According to the theoretical analysis presented in Ref. [28] the second order coefficient depends also on the TMAC. When the TMAC decreases the second order velocity slip coefficient decreases too (see Table 3.2). The same behaviour can be observed in microchannel A4, where the roughness is slightly greater than in other microchannels of the group A. 110 Chapter 6. Results in Rectangular Microchannels

2 . 2 H e N 2 2 . 0 A r

1 . 8

S 1 . 6

1 . 4

1 . 2

1 . 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 K n m

4 . 5 H e N 2 4 . 0 A r

3 . 5

3 . 0 S 2 . 5

2 . 0

1 . 5

1 . 0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 K n m

Figure 6.2: Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expressions for the microchannels A1. 6.1. Continuum and slip regimes 111

2 . 0 H e N 2 1 . 8 A r

1 . 6 S 1 . 4

1 . 2

1 . 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 K n m

4 . 5 H e N 4 . 0 2 A r 3 . 5

3 . 0

S 2 . 5

2 . 0

1 . 5

1 . 0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 K n m

Figure 6.3: Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expressions for the microchannels A2. 112 Chapter 6. Results in Rectangular Microchannels

2 . 0 H e N 2 1 . 8 A r

1 . 6 S 1 . 4

1 . 2

1 . 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 K n m

4 . 5 H e N 4 . 0 2 A r 3 . 5

3 . 0

S 2 . 5

2 . 0

1 . 5

1 . 0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 K n m

Figure 6.4: Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expressions for the microchannels A3. 6.1. Continuum and slip regimes 113

2 . 0 H e N 2 1 . 8 A r

1 . 6 S 1 . 4

1 . 2

1 . 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 K n m

4 . 5 H e N 4 . 0 2 A r 3 . 5

3 . 0

S 2 . 5

2 . 0

1 . 5

1 . 0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 K n m

Figure 6.5: Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expressions for the microchannels A4. 114 Chapter 6. Results in Rectangular Microchannels

6.1.3 Microchannels E

Microchannel E1 E2 Quantity Min Max Min Max Mass flow rate M˙ exp (10−12kg/s) 0.9 21 200 0.8 66 100 Inlet pressure pin (Pa) 342.1 131 850 139.4 129 720 Outlet pressure pout (Pa) 89.7 94 900 23.1 89 600 −3 −3 Average Knudsen number Knm 1.1 10 4.3 1.4 10 5.6

Table 6.7: Experimental conditions for the microchannels E1 and E2.

Microchannel E3 E4 Quantity Min Max Min Max Mass flow rate M˙ exp (10−12kg/s) 2.1 253 000 2.3 574 000 Inlet pressure pin (Pa) 63.7 105 850 27.2 114 650 Outlet pressure pout (Pa) 22.4 74 000 8.8 80 000 −4 −3 Average Knudsen number Knm 6.6 10 11.36 1.6 10 27.7

Table 6.8: Experimental conditions for the microchannels E3 and E4.

To accomplish one of the goals set in this thesis, which is the study of the surface roughness influence on the momentum transfer between a gas and a surface, we conducted series of mass flow rate measurements through rectangular cross-section microchannels having similar aspect ratios and the same material (Au) of the internal surfaces as microchannels of the group A. But these microchannels have surface roughness 12 times greater than that of the microchannels A (see Table 2.1). These microchannels are referenced by the letter E. There are four microchannels E1, 2, 3, 4 having the following aspect ratios w/h = 1.7, 2.9, 14.5, 29, respectively, which are comparable to those of microchannels of the group A.

The experimental conditions for the microchannels E are presented in Tables 6.7, 6.8. exp The values of the fitting coefficients Bi (i =1, 2, 3) obtained with the first and second order polynomial approximations are given in Table 6.9 with the statistical parameters of 2 fitting (sr, r and Es). The curves of the first and second order polynomial approximations for all the microchannels E and all gases (He, N2 and Ar) are shown in Figures 6.8, 6.9, 6.10 and 6.11.

The visual analysis of Figures 6.8, 6.9, 6.10 and 6.11 and the use of the same statistical 2 analysis of the fitting parameters sr, r and Es (see Table 6.9), made for the microchannels A, confirm the relevance of the first and second order approximations in their respective mean Knudsen number intervals [0.0 , 0.1] and [0.0, 0.3].

exp From Table 6.9 one can see that the values of the fitting coefficients (B0 ), obtained with 6.1. Continuum and slip regimes 115

exp exp exp 2 Channel Gas B0 B1 B2 sr r Es(%) 1.042± 0.001 8.133± 0.032 - 0.005 0.999 0.4 He 1.042± 0.003 7.978± 0.061 3.081± 0.244 0.009 0.999 0.5 1.037± 0.001 9.333± 0.033 - 0.005 0.999 0.4 E1 N 2 1.042± 0.003 8.926± 0.083 4.359± 0.329 0.013 0.999 0.7 1.048± 0.001 10.146± 0.035 - 0.005 0.999 0.4 Ar 1.054± 0.002 9.562± 0.053 6.682± 0.233 0.007 0.999 0.4 1.030± 0.001 7.227± 0.025 - 0.004 0.999 0.3 He 1.032± 0.001 6.972± 0.024 3.177± 0.098 0.003 0.999 0.2 1.027± 0.001 8.103± 0.023 - 0.004 0.999 0.3 E2 N 2 1.026± 0.003 7.976± 0.066 2.880± 0.269 0.010 0.999 0.6 1.039± 0.001 8.704± 0.035 - 0.006 0.999 0.4 Ar 1.039± 0.003 8.599± 0.073 2.536± 0.302 0.011 0.999 0.7 1.036± 0.001 6.728± 0.022 - 0.003 0.999 0.3 He 1.039± 0.001 6.424± 0.022 3.810± 0.090 0.003 0.999 0.2 1.033± 0.001 7.572± 0.025 - 0.004 0.999 0.3 E3 N 2 1.034± 0.001 7.354± 0.032 3.508± 0.130 0.005 0.999 0.3 1.045± 0.001 8.220± 0.034 - 0.005 0.999 0.4 Ar 1.046± 0.001 7.972± 0.033 3.773± 0.134 0.005 0.999 0.3 1.054± 0.002 6.839± 0.044 - 0.007 0.999 0.5 He 1.056± 0.003 6.612± 0.074 3.223± 0.290 0.011 0.999 0.7 1.055± 0.003 7.614± 0.078 - 0.011 0.997 0.9 E4 N 2 1.055± 0.005 7.344± 0.135 4.930± 0.546 0.020 0.999 1.2 1.062± 0.003 8.414± 0.071 - 0.010 0.998 0.8 Ar 1.066± 0.006 7.980± 0.160 5.238± 0.641 0.024 0.999 1.4

Table 6.9: Fitting coefficients of the dimensionless mass flow rate Sexp (6.1) obtained with the first (first line) and second order (second line) approximations (6.2) and (6.3) in the mean Knudsen number ranges [0.0, 0.1] and [0.0, 0.3], respectively, for microchannels of the group E. 116 Chapter 6. Results in Rectangular Microchannels the first and second order approximations (see Table 6.9), are different from the theoretical T value B0 = 1 (see Eq. 6.5). This surprising discrepancy (greater than experimental uncertainty) may be caused by an error in the measurement of microchannels dimensions (h and/or w) or by an error on the mass flow rate M˙ exp (4.8) measurements. The causes of this discrepancy are specified in the next paragraphs.

6.1.3.1 First order coefficients analysis

The values of the velocity slip coefficients σp and TMAC (α) calculated for the microchannels E are given in Table 6.10. The experimental values of σp are different from the theoretical values (1.016 [23] and 1.012 [60]). The values obtained are lower than the theoretical values calculated for diffuse scattering. Consequently, the TMAC (α) values are higher than 1 (value for the diffuse reflection), which is outside the interval determined by the Maxwell model used here (σp ≥ 1.016 and 0 < α ≤ 1). Only two exceptions are the values obtained for argon and nitrogen gases in the microchannel E1 with first order approximation (see Table 6.10).

The reasons for these surprising results are discussed below. Several assumptions are made in order to find the causes of such results.

exp 1. Due to the higher value of the coefficients B0 compared to 1 (see Table 6.9), the first hypothesis that comes to mind is an error in the measurements of the microchannels exp dimensions. By bringing back the values of the coefficients B0 to 1, either by increasing the value of the microchannels height h or width w, either both at the same time, the values of coefficient σp decrease and these of the coefficient α increase further. Thus, this hypothesis does not seem to solve the problem. In addition, the same method of dimensions’ measurements is used for the other microchannels of groups A and S, therefore we eliminate the hypothesis of an error in the microchannels dimensions.

2. With the second hypothesis, we tried to go further in our analysis of the problem. The steady assumption established to measure the mass flow rate using the constant volume technique (see Section 4.3) was suspected. In the experiments of the mass flow rate mea- surements conducted for small Knudsen numbers (i.e. in continuum and near slip regimes) the pressure ratio P = pin/pout was less than 2. The use of such pressure ratio induces variation in the value of the pressure difference ∆p, during the time of the experiment, of the order of 6% (±3% of the mean value). By using the continuum theory we assume that the term ∆p in expression for the Poiseuille mass flow rate (3.21) is constant, then such vari- ation (±3%) may leads to an error that can (maybe) explain the TMAC values greater than 1.

To check the validity of the mass flow rate measurements another method called ’unsteady method’ [92] was performed. This method was proposed by Knudsen and used by Porodnov et al. 1974 [92]. Using this method, the mass flow rate M˙ exp can be obtained (see Appendix B for more details) and the reduced mass flow rate Sexp is calculated from expression (6.1). 6.1. Continuum and slip regimes 117

Channel Gas molecular mass (g/mol) σp α σ2p 0.947± 0.004 1.038± 0.002 - He 4.00 0.928± 0.007 1.048± 0.004 0.094± 0.007 1.010± 0.004 1.003± 0.002 - E1 N 28.02 2 0.966± 0.009 1.027± 0.005 0.115± 0.009 1.027± 0.004 0.994± 0.002 - Ar 39.95 0.968± 0.005 1.026± 0.003 0.153± 0.005 0.899± 0.003 1.065± 0.002 - He 4.00 0.867± 0.003 1.084± 0.002 0.123± 0.004 0.937± 0.003 1.043± 0.002 - E2 N 28.02 2 0.922± 0.008 1.051± 0.004 0.095± 0.009 0.942± 0.004 1.040± 0.002 - Ar 39.95 0.931± 0.008 1.047± 0.005 0.074± 0.009 0.874± 0.003 1.080± 0.002 - He 4.00 0.834± 0.003 1.104± 0.002 0.181± 0.004 0.914± 0.003 1.056± 0.002 - E3 N 28.02 2 0.888± 0.004 1.072± 0.002 0.142± 0.005 0.929± 0.004 1.048± 0.002 - Ar 39.95 0.901± 0.004 1.064± 0.002 0.133± 0.005 0.892± 0.006 1.069± 0.003 - He 4.00 0.863± 0.010 1.087± 0.006 0.155± 0.014 0.923± 0.009 1.051± 0.005 - E4 N 28.02 2 0.891± 0.016 1.070± 0.010 0.203± 0.022 0.955± 0.008 1.033± 0.004 - Ar 39.95 0.906± 0.018 1.061± 0.011 0.189± 0.023

Table 6.10: Slip and tangential momentum accommodation coefficients found experimentally from the first (first line) and second order (second line) approximations of the dimensionless mass flow rate Sexp (6.2) and (6.3), respectively, for microchannels of the group E. 118 Chapter 6. Results in Rectangular Microchannels

The results are then fitted with the second order polynomial approximation (6.3). The method is applied to the flow of helium and argon through the microchannel E3. Figure 6.6 exp shows the dimensionless mass flow rate S as a function of the mean Knudsen number Knm for both methods (steady and unsteady). According to Figure 6.6 one can see that both methods give very similar values of Sexp (within the experimental uncertainties). The TMAC values obtained with both methods are calculated and they are also similar (e.g. the value obtained for argon with the steady method is 1.064 (see Table 6.10) and with the unsteady method is 1.060). These results allows us to validate the experimental measurements of the mass flow rate, but not to explain the TMAC values found for the microchannels of group E.

3. The third hypothesis is an error in the measurements of the experimental mass flow rate M˙ exp (4.8). There are two possible sources for such error: first, an error in the measurement of the parameters influencing the mass flow rate (essentially tanks volume), second, a leak or an outgassing problems caused by mishandling.

The possibility of an error in the measurements of the tanks volume was checked. The water weighting technique was used to measure the tanks volume, which is a very simple and accurate technique. The measurements were repeated several times each and the results of these measurements were different within the experimental uncertainty of the volume (less than 1.1%, see Section 4.4).

The second possible source of the problems is the outgassing or leak. As it had already been explained (see Section 4.2.3), the outgassing and the leak are checked before each experiment, then these two factors cannot be the sources of problem. Therefore, this third hypothesis can also be eliminated.

4. The fourth and the last hypothesis that we made, and which we think it can explain the problem, is the possible deformation of the cross-section of the microchannels E caused by a partial takeoff of the gold layer from the surfaces inside microchannels.

Multiple reasons push us to think that the gold layer takeoff is the cause of TMAC and exp B0 values greater than 1:

• First, after eliminating all other hypotheses, it is the only one that seems to be reason- able and tangible.

• Second, during our tests of microchannels we have encountered the same problem of gold layer takeoff with other microchannels, where the takeoff blocked completely the microchannels.

• Third, with the group of microchannels E, another group has been provided at the same time and had almost the same roughness and the same material (Au) of the internal surfaces as that of the microchannels E. The microchannels of this additional group had also a partial takeoff of the gold layer but visible in the microscope (see Figure 2.7b) and were excluded for this reason. 6.1. Continuum and slip regimes 119

3 . 5

U n s t e a d y 3 . 0 s t e a d y

2 . 5 p x e

S 2 . 0

1 . 5

1 . 0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 K n m

3 . 5

U n s t e a d y 3 . 0 s t e a d y

2 . 5 p x

e

S 2 . 0

1 . 5

1 . 0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 K n m

Figure 6.6: Dimensionless mass flow rate Sexp (6.1) as function of the mean Knudsen number obtained using the steady and unsteady methods in the microchannels E3. The upper graph is for helium and the bottom graph is for argon. 120 Chapter 6. Results in Rectangular Microchannels

For the microchannels E we think that the takeoff happened inside the microchannels, that is why the verification which we conducted with the microscope did not reveal anything. Further verification involves the use of very expensive cutting technique of the microchannels and could not be realized in the frame of our project.

5. Nevertheless, another verification, theoretical this time, was used and concerns the assimilation of the cross-section of the microchannels E to a circular section and to calculate exp the equivalent hydraulic diameter that makes the coefficient B0 equal to 1. Using this technique and basing on the same method of calculation used for microtubes T 1 and T 2 (see Section 5.1) the velocity slip coefficient (σp) and TMAC (α) are calculated. When this approach was used the values of σp and α coefficients become reasonable (i.e. in the interval determined by the Maxwell model). The resulting TMAC values are in the range [0.9, 1.0] for the microchannels E1 and E2 and are of the order of 0.7 and 0.6 for the microchannels E3 and E4, respectively. The low TMAC values obtained for the microchannels E3 and E4 are due to the large aspect ratio of these microchannels, hence, their assimilation to a circular section is not pertinent at all.

Finally, it was demonstrated (even if without certainty) that the probable factor influ- encing the results of the microchannels E is the gold layer takeoff inside the microchannels, which causes the change of their cross-section shape. The change of the microchannels cross-section means that the theoretical model used for rectangular channels is not adapted, exp which results in wrong values of the coefficients B0 , σp and α.

A last additional remark can be made. The microchannels of both groups (A and E) differ only by the roughness of their internal surfaces. The other characteristics, like the surface material and the gas species involved in this analysis, are the same. In the frame of this thesis we attempted to examine this influence of the surface roughness only through the accommodation coefficient, which leads in our particular case of the group E to the value of TMAC larger than one. Probably, an alternative model characterizing the gas-surface interaction might be developed in order to take into account the roughness as additional parameter of interaction.

In what follows, in order to get the maximum information from these microchannels (E) a qualitative analysis of the results is conducted as the pertinent quantitative analysis is impossible. This analysis is focused on the dependence of σp and α coefficients on the molecular mass of the gas.

It was seen for the microchannels of the group A that the dependence of TMAC on the gas molecular mass is not very significant, almost null for several cases. Comparing qualitatively the results obtained in the microchannels E with those of microchannels A, one can conclude that the roughness plays a very important role in the dependence of TMAC on the molecular mass of gas. When the surface roughness increases the gas molecules stand out by their molecular mass, such as the highest TMAC value is obtained for the lightest gas. The slight difference between the TMAC values obtained for nitrogen and argon 6.1. Continuum and slip regimes 121 can be due to their close molecular mass. It seems that the roughness of the microchan- nels E is not enough high to make a difference between the TMAC values obtained for nitrogen and argon, especially as nitrogen molecule is polyatomic so it may behave differently.

The influence of lateral walls on the TMAC values is also observed for the microchannels E. From Figure 6.7 one can see that the TMAC values obtained for the smallest microchannel E1 (w/h = 1.7) are different from the other microchannels. Practically the same values are obtained for the largest microchannels E3 (w/h = 14.5) and E4 (w/h = 29). This is not surprising because the influence of the lateral walls in the microchannels E3 and E4 is not significant.

A further investigation on TMAC in rougher microchannels may confirm the results found in the microchannels A and E.

6.1.3.2 Second order coefficients analysis

The values of the second order velocity slip coefficients σ2p, obtained for microchannels of the group E, are given in Table 6.10. These values are of the same order as those obtained for microchannels of the group A (see Table 6.6). It seems that the order of magnitude of the coefficients σ2p obtained for microchannels of the group E are not affected by the probable deformation of the microchannels cross-section. Despite this, only the comparison between the values obtained in the same microchannel for different gases is effectuated in order to investigate the dependence of σ2p on the gas molecular mass.

From Table 6.10 one can notice some dependence of the coefficients σ2p on the gas molecular mass in the microchannel E1, such as a higher value is obtained for argon and similar values (almost within their uncertainties) are obtained for helium and nitrogen. Whereas, for the microchannels E2 and E3, helium has the highest value of σ2p and the two other gases (nitrogen and argon) have similar values (within their uncertainties). Looking now to the results obtained for the last microchannel E4 (the largest one) one can see that the value of σ2p obtained for all gases are very close (within their uncertainties). Therefore, it is difficult to conclude on the dependence of the second order velocity slip coefficients on the molecular mass of gas from these results, but it appears that this dependence attenuates when the microchannel is large, which may mean that a unique value of σ2p can be obtained when the influence of the lateral walls is low. This unique value of σ2p may signify that the roughness of the upper and bottom surfaces of the microchannels E is not enough high to make the σ2p coefficient depending on the molecular mass as it is the case for the coefficient of first order (σp and α). An investigation in microchannels rougher than the microchannels E may clarify this question.

However, the changes in the tendency of the σ2p variation following the molecular mass could mean a significant dependence of σ2p on the aspect ratio. Therefore, an influ- ence of the section deformation or inhomogeneous roughness cannot be excluded at this point. 122 Chapter 6. Results in Rectangular Microchannels

1 . 1 8 E 1 E 3 1 . 1 6 E 2 E 4 1 . 1 4 1 . 1 2

) 1 . 1 0 α ( 1 . 0 8

C

A 1 . 0 6 M T 1 . 0 4 1 . 0 2 1 . 0 0 0 . 9 8 H e l i u m N i t r o g e n A r g o n

1 . 1 8 E 1 E 3 1 . 1 6 E 2 E 4 1 . 1 4 1 . 1 2

) 1 . 1 0 α ( 1 . 0 8

C

A 1 . 0 6 M T 1 . 0 4 1 . 0 2 1 . 0 0 0 . 9 8 H e l i u m N i t r o g e n A r g o n

Figure 6.7: Tangential momentum accommodation coefficients (TMAC) plotted as function of gas obtained with the first and second order approximations in microchannels of the group E. 6.1. Continuum and slip regimes 123

H e 2 . 0 N 2 A r 1 . 8

1 . 6 p x e S 1 . 4

1 . 2

1 . 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 K n m

4 . 5 H e N 4 . 0 2 A r 3 . 5

3 . 0 p x e 2 . 5 S

2 . 0

1 . 5

1 . 0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 K n m

Figure 6.8: Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expressions for the microchannels E1. 124 Chapter 6. Results in Rectangular Microchannels

H e 1 . 8 N 2 A r 1 . 6 p x e 1 . 4 S

1 . 2

1 . 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 K n m

H e 3 . 5 N 2 A r 3 . 0

2 . 5 p x e S 2 . 0

1 . 5

1 . 0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 K n m

Figure 6.9: Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expressions for the microchannels E2. 6.1. Continuum and slip regimes 125

1 . 8 H e N 2 A r 1 . 6 p

x 1 . 4 e S

1 . 2

1 . 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 K n m

H e 3 . 5 N 2 A r 3 . 0

2 . 5 p x e S 2 . 0

1 . 5

1 . 0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 K n m

Figure 6.10: Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expressions for the microchannels E3. 126 Chapter 6. Results in Rectangular Microchannels

1 . 8 H e N 2 A r 1 . 6 p

x 1 . 4 e S

1 . 2

1 . 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 K n m

H e 3 . 5 N 2 A r 3 . 0

2 . 5 p x e S 2 . 0

1 . 5

1 . 0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 K n m

Figure 6.11: Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expressions for the microchannels E4. 6.1. Continuum and slip regimes 127

6.1.4 Microchannels S

Microchannel S1 S2 Quantity Min Max Min Max Mass flow rate M˙ exp (10−12kg/s) 0.4 10 220 1.2 174 600 Inlet pressure pin (Pa) 234.8 130 220 105.6 125 000 Outlet pressure pout (Pa) 57.5 89 850 26.5 31 460 −3 −3 Average Knudsen number Knm 2.0 10 4.8 1.6 10 6.0

Table 6.11: Experimental conditions for the microchannels S1 and S2.

Microchannel S3 S4 Quantity Min Max Min Max Mass flow rate M˙ exp (10−12kg/s) 4.3 372 400 3.9 952 700 Inlet pressure pin (Pa) 66.3 131 000 25.9 124 380 Outlet pressure pout (Pa) 22.9 108 650 9.1 93 300 −4 −4 Average Knudsen number Knm 1.1 10 9.3 1.2 10 24.0

Table 6.12: Experimental conditions for the microchannels S3 and S4.

The microchannels of group S are covered with a layer of silica (Si02) on the internal surfaces. Their surface roughness is similar to that of microchannels of the group A. Therefore, a comparison of the TMAC results between these two groups can clarify the influence of the surface materials on the momentum exchange between gas and surface.

The experimental conditions for microchannels of the group S are given in Tables 6.11 and 6.12. The aspect ratios for the microchannels S1, 2, 3, 4 are, respectively, w/h = 2.1, 2.4, 11.9, 24. We have to note that the microchannel S1 was fabricated following the procedure explained in Chapter 2 (i.e. by assembling two wafers etched at the same width w and half of depth h/2), which caused some undesirable shape of the lateral walls when the wafers were aligned. However, the microchannels S2, 3, 4 were created by etching only one wafer to the depth h and covering it with another non-etched wafer, in order to avoid the problem of the lateral walls shape.

exp The coefficients of first and second order polynomial approximations Bi (i =0, 1, 2) are presented in Table 6.13. From this Table 6.13 one can see that the value of the coefficients exp B0 obtained for microchannels of the group S are all close to 1. If we go back to the exp problem of B0 values higher than one obtained in microchannels of the group E (see Table exp 6.9), the fact that the coefficients B0 obtained for microchannels of the group S are close to 1 gives us a new confirmation that the technique used to measure the microchannels exp dimension is valid. Therefore, this confirms that the values of B0 obtained higher than one in the group E are not related to the errors in the microchannels dimensions measurement. 128 Chapter 6. Results in Rectangular Microchannels

The first and second order approximations curves for all the microchannels S and all gases are given in Figures 6.13, 6.14, 6.15 and 6.16. The statistical analysis of the fittings 2 parameters (r , sr and Es) and the visual analysis of Figures 6.13, 6.14, 6.15 and 6.16 confirm the relevance of both first and second order approximations in their corresponding intervals of Knudsen number.

6.1.4.1 First order analysis

The first order velocity slip coefficients σp calculated for microchannels of the group S are given in Table 6.14. These coefficients differ from the theoretical values calculated for the assumption of diffuse scattering, which means that the accommodation of gas molecules on the surface is not complete.

The corresponding TMAC values obtained from expression (3.10) are given in Table 6.14. These values obtained for the microchannel S1 with first and second orders approximations are all similar within their uncertainties. The difference of TMAC between gases is not so significant, especially for N2 and Ar, however the value obtained for helium is slightly higher. This may be caused by the roughness and the shape of the lateral walls.

Comparing both microchannels S1 and S2 one can notice that the TMAC values obtained for microchannel S1 are higher than the microchannel S2, despite they have almost the same aspect ratio. This proves that the problem of the lateral walls shape (caused by the misalignment of the wafers), encountered in the microchannel S1, has an important effect on the mass flow rate, and thus on the TMAC. The same problem was encountered with the groups A and E where the TMAC obtained for the microchannels with the small- est aspect ratio w/h (A1 and E1) were different from the other microchannels of their groups.

Analyzing now the TMAC values obtained for the microchannel S2 separately. One can remark that the TMAC value of helium is the highest one, while the values of nitrogen and argon are very close for both approximations (first and second orders), which is the same tendency observed in the microchannel S1. It appears that the roughness of lateral walls has engendered this difference.

For the largest microchannels S3 and S4 the lateral walls influence is negligible, thus their results can be considered as the most pertinent when compared to those of the microchannels S1 and S2. By analyzing the TMAC values obtained for the microchannels S3 and S4 one can see that their values are very close to each other for all gases. All these values are centered on the value of 0.906 ± 0.016. The gas molecular mass in these cases appears to have no significant effect on the TMAC values.

Figure 6.12 shows the values of TMAC obtained with the first and second orders polynomial approximations as function of the gas type. This Figure 6.12 shows clearly the 6.1. Continuum and slip regimes 129

exp exp exp 2 Channel Gas B0 B1 B2 sr r Es(%) 0.989± 0.003 9.380± 0.072 - 0.010 0.999 0.7 He 0.988± 0.004 9.222± 0.085 3.497± 0.315 0.013 0.999 0.7 0.993± 0.003 10.203± 0.062 - 0.009 0.999 0.7 S1 N 2 0.984± 0.009 10.421± 0.220 3.158± 0.866 0.032 0.999 1.8 1.000± 0.005 10.983± 0.112 - 0.017 0.997 1.2 Ar 0.991± 0.011 11.098± 0.286 4.876± 1.138 0.042 0.998 2.2 0.997± 0.007 9.582± 0.135 - 0.017 0.996 1.1 He 0.997± 0.007 9.325± 0.147 4.825± 0.561 0.018 0.999 0.9 0.992± 0.004 10.941± 0.082 - 0.013 0.998 0.9 S2 N 2 0.989± 0.004 10.868± 0.104 4.101± 0.410 0.016 0.999 0.9 1.007± 0.003 11.493± 0.082 - 0.013 0.998 0.9 Ar 1.008± 0.005 11.313± 0.134 3.289± 0.537 0.021 0.999 1.1 0.998± 0.002 9.468± 0.033 - 0.006 0.999 0.4 He 1.002± 0.002 9.086± 0.042 4.918± 0.173 0.006 0.999 0.4 0.994± 0.003 10.085± 0.083 - 0.012 0.998 0.9 S3 N 2 0.993± 0.003 9.914± 0.082 5.369± 0.357 0.012 0.999 0.7 1.006± 0.003 11.071± 0.080 - 0.011 0.998 0.8 Ar 1.009± 0.003 10.722± 0.074 4.560± 0.315 0.011 0.999 0.6 0.995± 0.002 9.515± 0.039 - 0.006 0.999 0.5 He 0.996± 0.003 9.266± 0.080 4.015± 0.345 0.011 0.999 0.6 0.991± 0.002 10.096± 0.042 - 0.006 0.999 0.5 S4 N 2 0.993± 0.002 9.822± 0.049 4.284± 0.225 0.007 0.999 0.4 1.002± 0.003 10.654± 0.071 - 0.011 0.999 0.8 Ar 1.003± 0.003 10.386± 0.076 4.699± 0.330 0.011 0.999 0.7

Table 6.13: Fitting coefficients of the dimensionless mass flow rate Sexp (6.1) obtained with the first (first line) and second order (second line) approximations (6.2) and (6.3) in the mean Knudsen number ranges [0.0, 0.1] and [0.0, 0.3], respectively, for microchannels of the group S. 130 Chapter 6. Results in Rectangular Microchannels remarks made above on TMAC. First, the TMAC values obtained for the microchannel S1 are different (higher) than those obtained for the other microchannels, which confirms their lateral walls influence. Second, the TMAC values obtained for the microchannels S3 and S4 are very close within their uncertainties.

The results found for the microchannels S confirm the results obtained for the microchan- nels A, which are: • The smooth surfaces of microchannels A and S make the influence of the gas type (i.e. molecular mass) on the TMAC almost nonexistent.

• The lateral walls may have a very important effect on the momentum exchange if they are not controlled.

If comparing the TMAC values obtained for microchannels of the group S, especially those obtained for the microchannels S3 and S4, with those obtained for the microchannels A2 and A3 one can remark that lower values are obtained for the microchannels of the group S. Both groups of microchannels A and S have almost the same surface roughness, therefore it can be concluded from this comparison that the gold (Au) material is more diffusive than the silica (SiO2) material.

Another comparison can be made with the microtube T 1 (see Section 5.1, Table 5.3), which is coated with Sulfinert (bases silica material). The TMAC values obtained for both surfaces (silica and Sulfinert) are similar for nitrogen and argon, while the value obtained for helium is higher for the Sulfinert surface. This difference may be due to the surface roughness of the microtube T 1 (Sulfinert), which is not known, but we think that it is rougher than the silica surface of microchannels of the group S.

6.1.4.2 Second order analysis exp The values of the second order coefficients B2 and σ2p are given in Table 6.13. The uncertainty on these coefficients obtained for the microchannel S1 with nitrogen and argon are the largest, of the order of 25% of their values. For the other microchannels these uncertainties decrease.

By analyzing the behaviour of the second order velocity slip coefficients (σ2p) one can notice that the influence of the gas molecular mass is not significant: the values obtained for each microchannel are very close, almost within their uncertainties. It appears again that the smooth surfaces of the microchannels S make the dependence of the gas molecular mass on the second order velocity slip coefficients insignificant.

Let us compare now the values of σ2p obtained in the different microchannels. The values obtained for both microchannels S1 and S2 are comparable within their uncertainties. More- over, the values obtained for the microchannels S3 and S4 are also comparable. However, the 6.1. Continuum and slip regimes 131

Channel Gas molecular mass (g/mol) σp α σ2p 1.130± 0.009 0.944± 0.004 - He 4.00 1.111± 0.010 0.953± 0.005 0.120± 0.011 1.142± 0.007 0.938± 0.003 - S1 N 28.02 2 1.167± 0.025 0.927± 0.011 0.093± 0.025 1.151± 0.012 0.934± 0.005 - Ar 39.95 1.163± 0.030 0.928± 0.014 0.125± 0.029 1.168± 0.016 0.926± 0.007 - He 4.00 1.137± 0.018 0.940± 0.008 0.157± 0.018 1.240± 0.009 0.894± 0.004 - S2 N 28.02 2 1.232± 0.012 0.898± 0.005 0.115± 0.012 1.219± 0.009 0.903± 0.004 - Ar 39.95 1.200± 0.014 0.912± 0.006 0.081± 0.013 1.227± 0.004 0.900± 0.002 - He 4.00 1.178± 0.005 0.922± 0.002 0.232± 0.008 1.215± 0.010 0.905± 0.004 - S3 N 28.02 2 1.195± 0.010 0.914± 0.004 0.218± 0.014 1.248± 0.009 0.891± 0.004 - Ar 39.95 1.209± 0.008 0.908± 0.004 0.161± 0.011 1.240± 0.005 0.894± 0.002 - He 4.00 1.208± 0.010 0.908± 0.005 0.196± 0.017 1.223± 0.005 0.902± 0.002 - S4 N 28.02 2 1.190± 0.006 0.916± 0.003 0.180± 0.009 1.208± 0.008 0.908± 0.004 - Ar 39.95 1.178± 0.009 0.922± 0.004 0.172± 0.012

Table 6.14: Slip and tangential momentum accommodation coefficients found experimentally from the first (first line) and second order (second line) approximations of the dimensionless mass flow rate Sexp (6.2) and (6.3), respectively, for microchannels of the group S. 132 Chapter 6. Results in Rectangular Microchannels

σ2p values obtained for the microchannels pair (S1, S2) are lower than the values obtained for the pair (S3, S4). The difference in the value of σ2p obtained for the pair (S1, S2) and the pair (S3, S4) can be related to the aspect ratio (w/h), which is more than 5 times higher for the pairs (S3, S4). If the microchannels lateral walls roughness is different from the upper and bottom walls roughness, the gas flow through these microchannels and therefore the ex- tracted coefficients may be affected significantly, especially, for the microchannels with small aspect ratio (w/h). Nevertheless, a direct influence of the geometrical aspect ratio (w/h) on the values of σ2p cannot be completely excluded. 6.1. Continuum and slip regimes 133

1 . 0 4 S 1 S 3 1 . 0 2 S 2 S 4 1 . 0 0 0 . 9 8

) 0 . 9 6 α ( 0 . 9 4

C

A 0 . 9 2 M T 0 . 9 0 0 . 8 8 0 . 8 6 0 . 8 4 H e l i u m N i t r o g e n A r g o n

1 . 0 4 S 1 S 3 1 . 0 2 S 2 S 4 1 . 0 0 0 . 9 8

) 0 . 9 6 α ( 0 . 9 4

C

A 0 . 9 2 M T 0 . 9 0 0 . 8 8 0 . 8 6 0 . 8 4 H e l i u m N i t r o g e n A r g o n

Figure 6.12: Tangential momentum accommodation coefficients (TMAC) plotted as function of gas obtained with the first and second order approximations in microchannels of the group S. 134 Chapter 6. Results in Rectangular Microchannels

H e 2 . 0 N 2 A r 1 . 8

1 . 6 S 1 . 4

1 . 2

1 . 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 K n m

4 . 5 H e N 2 4 . 0 A r 3 . 5

3 . 0 S 2 . 5

2 . 0

1 . 5

1 . 0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 K n m

Figure 6.13: Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expressions for the microchannels S1. 6.1. Continuum and slip regimes 135

H e 2 . 0 N 2 A r 1 . 8

1 . 6 S 1 . 4

1 . 2

1 . 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 K n m

4 . 5 H e N 2 4 . 0 A r 3 . 5

3 . 0 S 2 . 5

2 . 0

1 . 5

1 . 0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 K n m

Figure 6.14: Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expressions for the microchannels S2. 136 Chapter 6. Results in Rectangular Microchannels

H e 2 . 0 N 2 A r 1 . 8

1 . 6 S 1 . 4

1 . 2

1 . 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 K n m

4 . 5 H e N 2 4 . 0 A r 3 . 5

3 . 0 S 2 . 5

2 . 0

1 . 5

1 . 0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 K n m

Figure 6.15: Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expressions for the microchannels S3. 6.1. Continuum and slip regimes 137

H e 2 . 0 N 2 A r 1 . 8

1 . 6 S 1 . 4

1 . 2

1 . 0 0 . 0 0 0 . 0 2 0 . 0 4 0 . 0 6 0 . 0 8 0 . 1 0 K n m

4 . 5 H e N 2 4 . 0 A r 3 . 5

3 . 0 S 2 . 5

2 . 0

1 . 5

1 . 0 0 . 0 0 0 . 0 5 0 . 1 0 0 . 1 5 0 . 2 0 0 . 2 5 0 . 3 0 K n m

Figure 6.16: Dimensionless mass flow rate Sexp (6.1) plotted as function of the mean Knud- sen number, fitted with first (6.2) and second (6.3) order polynomial expressions for the microchannels S4. 138 Chapter 6. Results in Rectangular Microchannels

6.2 Transitional and free molecular regimes

In the transitional and free molecular regimes, the measured values of the mass flow rate M˙ exp (4.8) are normalized according to the following expression: √ exp L 2RT ˙ exp G = 2 M . (6.8) h w(pin − pout)

exp The results of G are plotted as function of the mean rarefaction parameter δm (3.86), which is calculated from the mean pressure of the inlet and outlet tanks. The theoretical model to which we compare our experimental results is the BGK kinetic model. In this thesis we developed the numerical approach which allows to solve numerically the linearized BGK kinetic equation subjected to the Maxwellian diffuse-specular scattering (see Section 3.3.4). The linearization of the BGK operator was carried out under the assumption of small gradient of the pressure (3.83) along the channel, which is always true in our microchannels as h/L << 1. The numerical calculations of the dimensionless mass flow rate Qch are fulfilled for several values of the aspect ratios (h/w = 1, 0.5, 0.2, 0.1, 0.05, 0.02, 0.01) and for several values of TMAC (α = 0.8, 0.85, 0.90, 0.95, 1.0). The results are presented in Tables 3.9, 3.10, 3.11, 3.12, 3.13, 3.14 in non-dimensional form according to expression (3.84).

These results are obtained under the assumption of constant rarefaction parameter δ (3.85) along the channel. However, the measurements of the mass flow rate were carried out using a large pressure ratio pin/pout, therefore the change of the rarefaction parameter along the channel must be taken into account. This may be realized by integrating expression (3.84) along the channel as following:

δ Zout ch 1 ch G (δin, δout) = Q (δ)dδ. (6.9) δout − δin δin

However, as it was shown in Ewart [36] and Pitakarnnop [89], the results of this integration ch are very close to the expression Q (δm):

ch ch G (δin, δout) ' Q (δm), with δm = (δin + δout)/2. (6.10)

Therefore, in the following, for the comparison of the experimental and numerical values we will use the results provided in Tables 3.9- 3.14 for the mean value of the rarefaction parameter for each measurement.

It is mentioned above that the value of the theoretical dimensionless mass flow rate Gch (6.10) (see Section 3.3.4) are calculated for several values of the aspect ratio w/h. For each value of the microchannels aspect ratio, used in this thesis, the linear interpolation technique is used. It is explained in Section 3.3.4 that the linear interpolation gives a good estimation with uncertainty less than ±0.5%. 6.2. Transitional and free molecular regimes 139

The results obtained for the microchannels of groups A, E and S are presented separately. The comparison between experiments and numerical results is performed visually through the plot of the experimental points of the dimensionless mass flow rate Gexp (6.8) and the curves of the numerical dimensionless mass flow rate Gch as function of the mean value of the rarefaction parameters δm on the same graph. The TMAC value corresponding to each gas is deduced from the visual comparison between the experimental points and the numerical curves.

To avoid encumbering graphs the experimental error bars of the dimensionless mass flow rate Gexp are not be presented. To have an idea of the error bars on the experimental points Gexp, an example is given in Figure (6.17). The error bars drawn in Figure (6.17) are deduced from the uncertainty on the measured mass flow rate M˙ exp (4.8).

3 . 0 H e 1 =؁ 5 9 . 0 =؁ ؁ 2 . 5 = 0 . 9 0 5 8 . 0 =؁ 0 8 . 0 =؁ G 2 . 0

1 . 5

0 . 0 1 0 . 1 1 1 0 ؄ m

Figure 6.17: Graph illustrating the error bars on the points of the experimental dimensionless mass flow rate Gexp of helium through the microchannels A3. 140 Chapter 6. Results in Rectangular Microchannels

6.2.1 Microchannels A

The plots of the experimental and numerical dimensionless mass flow rates Gexp and Gch, for the microchannels A, are given in Figures 6.18 and 6.19, respectively. For these microchannels the experimental setup was pushed to its limit in order to investigate the flow in the free molecular regime, particularly for the microchannel A2, where the highest value of the mean Knudsen number is attended for helium (Knm = 70 which corresponds to δm = 0.013, see Table 6.3).

From the visual comparison of the experimental points and the numerical curves for the microchannel A1 (see upper graph in Figure 6.18) two regions can be differentiated according to the rarefaction parameter.

First, δm superior to 2 (i.e. slip and near transitional regimes), in this region the difference between the experimental points for the different gases is very small, almost not visible. These points are between both numerical curves corresponding to α = 0.95 and α = 0.85. It appears that the TMAC does not depends on the molecular mass of the gas, which was also obtained with the continuum approach in the slip regime.

Second, δm lower than 2, the experimental points of helium seems to follow on the numerical curve α = 0.85 and these of nitrogen and argon are higher, peel off from the numerical curve. The results obtained in this region are not very pertinent. This is due to the low values of the mass flow rate measured and to the long experiments time length. The influence of the lateral walls, especially their roughness and shape, may also have an important effect on the measured mass flow rate, and thus on the TMAC. It should be also recalled that the microchannel A1 was the first tested one, so we may have some errors related to the mishandling and temperature stability. We will see that, for the other microchannels, the results are better in this region.

The experimental results obtained for the second microchannel A2 (see bottom graph in Figure 6.18) reproduce very well the shape of the dimensionless numerical mass flow rate. For this microchannel, two regions can be also differentiated according to the rarefaction parameter. For δm superior to 2, one can see that the experimental points of all gases coincide with the numerical curve α = 1. The difference between gases is difficult to be observed, thus the use of the continuum approach in this region is better. For δm inferior to 2, the points of helium are between both numerical curves α = 0.95 and α = 1, which confirms the results on TMAC obtained in the slip regime (see Table 6.6). The same comment can be done for nitrogen and argon even if the minimum value of the rarefaction parameter reached for these two gases is around 0.2.

The best trend of the experimental results is obtained for the third microchannel A3 (see upper graph in Figure 6.19). We saw in the slip regime (see Section 6.1.2) that the TMAC values obtained for the microchannel A3 are similar for all gases. The results obtained for this microchannel in the transitional regime, presented in Figure 6.19, show the same behaviour. 6.2. Transitional and free molecular regimes 141

The experimental points obtained for the different gases coincide and they are in between the numerical curves for α = 1 and α = 0.95. The value of 0.96 for TMAC found in the slip regime using the second order approximation seems to be confirmed in the transitional regime. From these two observations and regarding the results obtained for the microchannel A2 (see previous paragraph) we can say that:

• First, the same results obtained in the slip regime (non-dependence of TMAC on the molecular mass of gas) are confirmed in the transitional regime.

• Second, a single value of TMAC may be used for all Knudsen flow regimes.

These last results are also confirmed in the microchannels A4 (see bottom graph on Figure 6.19). Considering the interval of rarefaction parameter [4, 10] (i.e. near transitional regime) the experimental points for all gases seem to coincide with the numerical curve α = 1. In the slip regime the TMAC values obtained with the second order approximation are all close to 1, which confirms the results found in the transitional regime. For the rest of the rarefaction parameter’s interval (δm < 4) one can notice a discrepancy between the experimental points obtained for the different gas. First, for helium, the experimental points are below the numerical curve α = 1, however within the error bars. The same results was obtained in the slip regime with the second order approximation, where the TMAC value obtained for helium was higher than 1 (αHe = 1.009 ± 0.011, see Table 6.6). For nitrogen and argon, the experimental points appear to coincide together, except for some points of nitrogen in the interval of δm [0.2, 0.4], where these points are lower. Again, the same results for nitrogen and argon are obtained in the transitional and slip regimes.

Finally, from the remarks and observations made above for microchannels of the group A we can say that the results of TMAC found in the transitional and free molecular regimes are in good agreement with those obtained in the slip regime. Investigations with other microchannels may confirm this conclusion. 142 Chapter 6. Results in Rectangular Microchannels

2 . 0 H e 1 =؁ N 5 9 . 0 =؁ 2 A r 0 9 . 0 =؁ 5 8 . 0 =؁ 0 8 . 0 =؁

1 . 5 G

1 . 0 0 . 0 1 0 . 1 1 1 0 ؄ m

2 . 5 H e 1 =؁ N 5 9 . 0 =؁ 2 A r 0 9 . 0 =؁ 5 8 . 0 =؁ ؁ 2 . 0 = 0 . 8 0 G

1 . 5

0 . 0 1 0 . 1 1 1 0 ؄ m

Figure 6.18: Experimental and numerical dimensionless mass flow rates as function of the mean rarefaction parameter δm, for the microchannels A1 (upper) and A2 (bottom). 6.2. Transitional and free molecular regimes 143

3 . 0 H e 1 =؁ N 5 9 . 0 =؁ 2 A r 0 9 . 0 =؁ 5 8 . 0 =؁ 5 . 2 0 8 . 0 =؁ G 2 . 0

1 . 5

0 . 0 1 0 . 1 1 1 0 ؄ m

3 . 0 H e 1 =؁ N 5 9 . 0 =؁ 2 A r 0 9 . 0 =؁ 5 8 . 0 =؁ 5 . 2 0 8 . 0 =؁ G 2 . 0

1 . 5

0 . 0 1 0 . 1 1 1 0 ؄ m

Figure 6.19: Experimental and numerical dimensionless mass flow rates as function of the mean rarefaction parameter δm, for the microchannels A3 (upper) and A4 (bottom). 144 Chapter 6. Results in Rectangular Microchannels

6.2.2 Microchannels E

For the microchannels of this group (E) let us first recall the obtained TMAC results in the slip regime using the continuum approach. The TMAC values obtained were superior to 1 for all microchannels. It has been explained previously (see Section 6.1.3) that these values are outside the interval [0, 1] suggested by the Maxwellian model used in this thesis.

A similar pattern emerges when examining the experimental results of the mass flow rate found in the transitional and near free molecular regimes using the kinetic BGK model. exp ch The curves of G (6.8) and G as function of the rarefaction parameter (δm) for the microchannels E1 and E2 are shown in Figure 6.20 (upper and bottom graphs are for the microchannels E1 and E2, respectively). From these two graphs (Figure 6.20) it may be observed that the experimental points of the dimensionless mass flow rate Gexp (6.8) are lower than the numerical curves, for both microchannels (E1 and E2). The TMAC values resulting from the comparison of the experimental points and the numerical curves would be greater than 1.

The same tendency is found for the microchannels E3 and E4 (see Figure 6.21), where experimental points are even farther away numerical curves than those of the microchannels E1 and E2. These results, for the group E, confirm the existence of the problem related to the TMAC value higher than one, encountered in the slip regime, also in the transitional and free molecular regimes, even using different modeling approach. One can say that the same results is obtained for all regimes, it is true, but the same strange results!!

Let us analyze now the graphs of the experimental and numerical dimensionless mass flow rate, drawn for all microchannels E (see Figures 6.20 and 6.21) from qualitative point of view. One can see the coincidence of experimental points for the different gas in all microchannels of the group E. A slight difference between the points of helium and the points of nitrogen and argon can be observed in the range of δm [1, 10]. The points of helium seem to be slightly lower than those of nitrogen and argon, which means that TMAC values of helium are higher than these of nitrogen and argon. A similar observation was made in the slip regime (see Table 6.10). However, when the rarefaction parameter is inferior to 1 (i.e. near free molecular regime) the difference between the experimental points of all gases attenuates. It appears that, when approaching the free molecular regime, the mo- mentum exchange of the gas molecule with walls becomes independent of gas molecular mass.

The detailed study of the possible problems, which can lead to the TMAC greater than one, allowed to conclude that the problem of gold layer takeoff inside the microchannels can be the cause. In the slip regime, we tried to assimilate the rectangular cross section of the microchannels to an equivalent circular cross section using the definition of the hydraulic exp diameter, which was adjusted from the condition B0 = 1 (see Section 6.1.3 for more details). Using the same technique, but this time applied to the transitional and near the free molecular regimes, the dimensionless experimental mass flow rate Gexp is calculated from expression (5.12) and is compared with the numerical dimensionless mass flow rate Gtu 6.2. Transitional and free molecular regimes 145 obtained, using the BGK kinetic equation, by Lo & Loyalka, 1982 [66] for diffuse reflexion at wall and by Porodnov & Tuchvetov, 1979 [94] for diffuse-specular reflexion (see Section 5.2 for more details).

exp tu The curves of G (5.12) and G as function of the rarefaction parameter (δm) obtained by the use of this technique are given in Figure 6.22, for the microchannels E1 and E2. From this Figure 6.22 it is noticeable that the experimental points are now in between the numerical curves α = 1 and α = 0.9. The TMAC values resulting from the visual comparison, for both microchannels (E1 and E2), are higher for the microchannel E1 (TMAC in the interval [0.94, 1.00]) than for the microchannel E2 (TMAC in the interval [0.90, 0.94]). We should note that TMAC values obtained with this technique can note be considered as the right values corresponding to the microchannels E1 and E2 surfaces, because we know that the theoretical approach used (for a circular cross-section) is not conform to shape of our microchannels.

For the two other microchannels E3 and E4 the experimental points obtained using the technique for circular cross-section (Figure not given here) are close to the numerical curve for α = 0.6. Comparable results are obtained in the slip regime, where the TMAC value were of the order of 0.7 and 0.6 for the microchannels E3 and E4, respectively (see Section 6.1.3).

Finally, to conclude for microchannels of the group E, we can say that the qualitative results obtained with those microchannels confirm these obtained with microchannels of the group A: the same value of TMAC may be applied for all Knudsen flow regimes. A new result has risen from these microchannels, which is the tendency of TMAC to not depend on gas molecular mass, when approaching the free molecular regime. This last result should be checked by investigating TMAC in rougher microchannels. 146 Chapter 6. Results in Rectangular Microchannels

2 . 0 H e 1 =؁ N 5 9 . 0 =؁ 2 A r 0 9 . 0 =؁ 5 8 . 0 =؁ ؁ 1 . 5 = 0 . 8 0 G

1 . 0

0 . 0 1 0 . 1 1 1 0 ؄ m

2 . 5 H e 1 =؁ N 5 9 . 0 =؁ 2 A r 0 9 . 0 =؁ 5 8 . 0 =؁ 0 . 2 0 8 . 0 =؁

G 1 . 5

1 . 0

0 . 0 1 0 . 1 1 1 0 ؄ m

Figure 6.20: Experimental and numerical dimensionless mass flow rates as function of the mean rarefaction parameter δm, for the microchannels E1 (upper) and E2 (bottom). 6.2. Transitional and free molecular regimes 147

3 H e 1 =؁ N 5 9 . 0 =؁ 2 A r 0 9 . 0 =؁ 5 8 . 0 =؁ 0 8 . 0 =؁

2 G

1 0 . 0 1 0 . 1 1 1 0 ؄ m

3 H e 1 =؁ N 5 9 . 0 =؁ 2 A r 0 9 . 0 =؁ 5 8 . 0 =؁ 0 8 . 0 =؁

2 G

1 0 . 0 1 0 . 1 1 1 0 ؄ m

Figure 6.21: Experimental and numerical dimensionless mass flow rates as function of the mean rarefaction parameter δm, for the microchannels E3 (upper) and E4 (bottom). 148 Chapter 6. Results in Rectangular Microchannels

3 . 5 H e 0 0 . 1 =؁ N 4 9 . 0 =؁ 2 A r 0 9 . 0 =؁ 0 . 3 0 8 . 0 =؁

2 . 5 G

2 . 0

1 . 5

0 . 0 1 0 . 1 1 1 0 ؄ m

3 . 5 H e 0 0 . 1 =؁ N 4 9 . 0 =؁ 2 A r 0 9 . 0 =؁ 0 . 3 0 8 . 0 =؁

2 . 5 G

2 . 0

1 . 5

0 . 0 1 0 . 1 1 1 0 ؄ m

Figure 6.22: Experimental and numerical dimensionless mass flow rates as function of the mean rarefaction parameter δm, for the microchannels E1 (upper) and E2 (bottom), obtained by assimilating the rectangular microchannels cross-section to a circular one. 6.2. Transitional and free molecular regimes 149

6.2.3 Microchannels S The experimental results obtained for the group of microchannels S confirm the results of TMAC obtained for the microchannels A and E: a unique TMAC value can be used in all Knudsen flow regimes.

Figure 6.23 (see upper graph) shows the results obtained in the microchannel S1. A difference between the points of helium and those of nitrogen and argon is observed; particularly when the rarefaction parameter is lower than 5. The experimental points of helium in this region (δm < 5) are the lowest, thus they have the highest TMAC value. While the points of nitrogen and argon are close each other, therefore they have similar TMAC values. This conclusion is the same as that obtained in the slip regime (see Table 6.14) with second order approximation. The TMAC value obtained for helium in the slip regime is αHe = 0.953 ± 0.005. The same value can be deduced, in the transitional regime (see Figure 6.23, upper graph), from the visual comparison of the numerical curve α = 0.95 and the helium experimental points. In the same graph (Figure 6.23) one can remark a discrepancy of three points of nitrogen and two points of argon, around the value of δm equal to 1, due to unknown reason.

For the second microchannel S2 the results are presented in Figure 6.23 (see bottom graph). Almost the same behaviour remarked for the microchannel S1 is observed for the microchannel S2. Over the value of rarefaction parameter (δm = 5) the experimental points of all gases seem to follow the numerical curve α = 0.95. When δm is inferior to 5, a discrepancy of the nitrogen and argon points can be observed from the points of helium, which are the lowest one, so they have the highest TMAC value. In the slip regime the TMAC value obtained for helium is αHe = 0.940 ± 0.008, while in the transitional regime the points of helium seem to tend to the numerical curve α = 0.90. However, if considering the experimental and the statistical uncertainties on TMAC values we can say that the conclusion of the same TMAC value, can be applied for all Knudsen flow regimes, is also valid for this microchannel (S2).

This conclusion is confirmed with the results obtained for the microchannels S3 and S4. For these two microchannels (see Figure 6.24) the behaviour of the experimental points is quite similar. The experimental points of the different gases appear to be close to the numerical curve α = 0.95. The TMAC values resulting from the visual comparison are in between the values 0.90 and 0.95 for the microchannel S3 and are equal to 0.95 for the microchannel S4. In the slip regime, using the second order approximation, the TMAC values obtained (see Table 6.14) are slightly lower the values obtained in the transitional regime. However, they are within the experimental uncertainties.

Finally, the results obtained in the microchannels S are in good agreements with those obtained for the other microchannels (A and E). The following conclusions can be made for the results obtained in the microchannels of the groups A, E and S: • The same TMAC value can be applied for all Knudsen flow regimes, from continuum 150 Chapter 6. Results in Rectangular Microchannels

to near free molecular regimes.

• The non-dependence of TMAC on the molecular mass of gas is observed for the mi- crochannels having a very smooth surfaces (A and S), whatever their surface materials. 6.2. Transitional and free molecular regimes 151

2 . 0 H e 1 =؁ N 5 9 . 0 =؁ 2 A r 0 9 . 0 =؁ 5 8 . 0 =؁ 0 8 . 0 =؁ 1 . 5 G

1 . 0 0 . 0 1 0 . 1 1 1 0 ؄ m

H e 1 =؁ N 5 9 . 0 =؁ 2 2 . 0 A r 0 9 . 0 =؁ 5 8 . 0 =؁ 0 8 . 0 =؁ G 1 . 5

1 . 0 0 . 0 1 0 . 1 1 1 0 ؄ m

Figure 6.23: Experimental and numerical dimensionless mass flow rates as function of the mean rarefaction parameter δm for the microchannels S1 (upper) and S2 (bottom). 152 Chapter 6. Results in Rectangular Microchannels

3 . 0 H e 1 =؁ N 5 9 . 0 =؁ 2 A r 0 9 . 0 =؁ 5 8 . 0 =؁ 5 . 2 0 8 . 0 =؁ G 2 . 0

1 . 5

0 . 0 1 0 . 1 1 1 0 ؄ m

3 . 0 H e 1 =؁ N 5 9 . 0 =؁ 2 A r 0 9 . 0 =؁ 5 8 . 0 =؁ 5 . 2 0 8 . 0 =؁ G 2 . 0

1 . 5

0 . 0 1 0 . 1 1 1 0 ؄ m

Figure 6.24: Experimental and numerical dimensionless mass flow rates as function of the mean rarefaction parameter δm for the microchannels S3 (upper) and S4 (bottom). Chapter 7 General Conclusion and Perspectives

This thesis was conducted in the framework of the GASMEMS project, financed by the European Commission (Marie Curie Actions) under the Seventh Framework Program (FP7). During three years of this thesis experimental, theoretical and numerical studies of isothermal gas flow through microchannels with rectangular and circular cross-sections have been carried out with the aim to highlight the aspect of the momentum exchange between solid and gas in the presence of rarefaction.

The experimental study has been performed through two experimental setups based on the rise-of-pressure technique to measure the stationary mass flow rate through mi- crochannels. The first setup was implemented in the IUSTI Laboratory to measure the mass flow rate through rectangular cross-section microchannels having various aspect ratios, surface materials (Au and SiO2) and roughness. The second setup was realized during a secondment of two months in INFICON Company to measure the mass flow rate through long industrial microtubes with different internal surface materials (Stainless Steel and Sulfinert), which are used for chromatography issues. Both setups allowed to investigate the mass flow rate in all Knudsen flow regimes: from continuum to free molecular flow regimes.

The problems related to the leak, outgassing and vapor adsorption were handled in order to achieve very low mass flow rate values of the order of 10−13 kg/s. The simple and accurate technique performed to measure the tanks volume has enabled decreasing the uncertainty of the mass flow rate compared to other studies (e.g. Ewart et al. [38]).

The theoretical study has focused on the modeling of the gas flow through a long rectangular cross-section microchannel in the continuum and slip flow regimes. An original approach based on the spectral properties of the Laplace operator was proposed. An explicit analytical second order, according to the Knudsen number, expression for mass flow rate in the slip flow regime was obtained. This expression takes into account the influence of the two-dimensional aspect of the rectangular cross-section.

The comparison of this expression with the fitted data of the measured mass flow rate has enabled the easy extraction of the gas flow characteristics such as the velocity slip and tangential momentum accommodation coefficients, without using numerical calculation as it is the case of the approach developed in Ref. [101]. 154 Chapter 7. General Conclusion and Perspectives

The numerical study was conducted due to the lack of the numerical results on the mass flow rate through rectangular microchannel using the BGK kinetic model for values of the accommodation coefficient α different from one in the transitional and free molecular regimes. The numerical simulations were carried out by solving the linearized BGK kinetic equation using the discrete velocity method (DVM) with the assumption of diffuse-specular reflexion of the molecules on the surfaces.

These numerical simulations were carried out for various cross-section aspect ratios of the microchannels and for different tangential momentum accommodation coefficients. The visual comparison between the network of the numerical curves of the mass flow rate obtained for different values of α and the measured mass flow rate has allowed us to deduce the values of TMAC in the transitional and free molecular regimes.

The obtained results of TMAC showed that the internal surface materials (Gold, Silica, Stainless Steel and Sulfinert) of the microchannels discussed in this thesis are not characterized by fully diffuse reflexion. Furthermore, the TMAC values deduced from the comparison between the experimental data of the mass flow rate and the network of the numerical curves in the transitional and free molecular regimes are similar to those obtained in the slip regime using the continuum approach. Thus, a unique value of TMAC can be used for all flow regimes.

In the slip regime, the result obtained for the velocity slip coefficient σp and the TMAC with the first and second order approximations are very close within the experimental uncertainty. Both approximations are pertinent in their respective Knudsen number intervals [0, 0.1] and [0, 0.3], except for some cases (microchannels A1, S1) and for the microtubes T 1 and T 2, where the second order approximation is not found pertinent due to the lack of the experimental data in the Knudsen number [0.1, 0.3]. In the transitional and free molecular regimes the shape of the numerical curves of the mass flow rate obtained for different values of TMAC using the linearized BGK kinetic equation in rectangular microchannels and in microtubes have the same shape as the experimental data.

The following conclusions can be drawn from this study:

Influence of the surface roughness: The results showed that the surface roughness seems to have an important role concerning the dependence of the TMAC and of the second order velocity slip coefficient σ2p on the molecular mass of the gas. When the microchannels surface roughness is very small (of the order of 1 nm in our case) the TMAC and the first and the second order velocity slip coefficients do not depend on the molecular mass of the gases analyzed in this thesis. When the surface roughness increases, the TMAC becomes different for the monatomic gases considered here, in such a manner: more accommodation occurs for the lighter gas. However, for the polyatomic gases this tendency is not confirmed, so that the TMAC values obtained for nitrogen and argon are close each other, and those obtained for carbon-dioxide and helium are also close. Moreover, it seems that rougher 155

surface is needed to more differentiate the coefficient σ2p with respect to the gas molecular mass.

Influence of the surface material: The analysis of the TMAC values obtained for microchannels of the groups A (Au) and S (SiO2), which have the same surface roughness, concludes that the gold (Au) surface is more diffusive than the silica (SiO2) surface. Moreover, the obtained TMAC results for microtubes T 1 (Sulfinert) and T 2 (Stainless steel) are similar for N2 and Ar, and different for He and CO2. It seems that the coating of the microtube T 1 with Sulfinert has differently changed the TMAC value depending on the gas.

From the comparison of the results obtained for silica (group S) and Sulfinert (microtube T 1), which is a silica based material, it was observed that the TMAC values are similar for the heaviest gases (N2 and Ar). Unfortunately, the roughness of the microtube T 1 is not known, thus its influence on the TMAC cannot be examined.

Influence of the microchannels lateral walls: The investigation of the lateral walls effect on the TMAC and σ2p in the rectangular microchannels is difficult to carry out, due to the fact that the lateral walls of the microchannels are not truly vertical and their surface roughness are not homogeneous with that of the upper and bottom surfaces. The TMAC values obtained for the microchannels with the smallest aspect ratio (w/h) are very different from the results obtained for other microchannels which have very close TMAC values.

The influence of the lateral walls on the second order velocity slip coefficient σ2p may be analyzed as follows. The results showed that the values of σ2p obtained for both microchannels of each group with the smallest aspect ratios (w/h) are close. For both microchannels of the same group with the highest aspect ratios the values of σ2p are also similar. But when passing from microchannel pair (A1, A2) to the microchannel pair (A3, A4) we observe a significant difference on σ2p. This variation is probably due to the influence of the roughness and the imperfect shape of the lateral walls. More investigation on microchannels having different aspect ratios with the same material and roughness on the lateral walls as the upper and bottom walls may clarify better the reason of this difference.

Other results: Some surprising results were obtained with microchannels of the group E where the values of TMAC were generally higher than one. The analysis of the problem showed that the probable reason may be the gold layer take-off inside the microchannels, but it is still only an assumption, as it was not possible to check it with a direct measurement technique. In our analysis of the problem we did not examined the pertinence of the Maxwellian model. The values of TMAC superior to one may be due to the phenomenon called "Backscattering". This phenomenon is not recognized by the Maxwellian model, but other models, such as the Cercignani-Lampis (CL) model, take in account of this 156 Chapter 7. General Conclusion and Perspectives phenomenon. The use of this model (CL) is more complicated because it involves two accommodation coefficients in order to better explain the momentum and energy exchanges of the molecules with the walls. Consequently, two types of measurements would be needed to obtain both accommodation coefficients. Nevertheless, the Maxwellian model is considered as working well when the values of the accommodation coefficient is close to one.

Some suggestions are made here as options for further investigations:

• Examine the influence of the degree of freedom of the gas on the TMAC and the first and the second order velocity slip coefficients by conducting experiments using polyatomic gases.

• Investigate the TMAC for the gas mixture using two gases having a known value of TMAC. The effect of the gas concentration on TMAC may be studied.

• A more systematic study of various roughness of the surface. Appendix A Publications

International journals

• M. Hadj-Nacer, I. Graur and P. Perrier Mass flow measurement through rectangular microchannel from hydrodynamic to near free molecular regimes La Houille Blanche Journal, n: 4, 2011, p. 49-54;

• J. G. Méolans, M. Hadj-Nacer, M. Rojas, P. Perrier, and I. Graur. Effects of two transversal finite dimensions in long microchannel: analytical approach in slip regime. Phys. Fluid Journal (accepted).

International conferences - Talks with proceedings

• M. Hadj-Nacer, I. Graur and P. Perrier. Mass flow measurement through rectangu- lar microchannels from hydrodynamic to near free molecular regimes. In Proceed- ings on CDROM of 2nd European Conference on Microfluidics (MicroFlu10), Toulouse (France), December 8-10, (2010).

• M. Rojas Cardenas, M. Hadj-Nacer, I. Graur, P. Perrier, and G. J. Méolans. Measure- ments of the thermal creep phenomenon. In Proceedings on CDROM of 2nd European Conference on Microfluidics (MicroFlu10), Toulouse (France), December 8-10, (2010).

• M. Rojas Cardenas, M. Hadj-Nacer, I. Graur, P. Perrier. Thermal creep effects in rarefied micro-flows. In proceeding of 2nd GASMEMS Workshop, Les Embiez (France), July 07-10, (2010).

• M. Hadj-Nacer, I. Graur and P. Perrier. Accommodation coefficients measurement in rectangular microchannels. In proceeding of 3rd GASMEMS Workshop, Bertinoro (Italy), June 9-11, (2011).

• M. Hadj-Nacer, I. Graur and P. Perrier. Mass flow rate measurement through rect- angular microchannels for large Knudsen number range. In proceeding of the ASME 2011 International Conference on Nanochannels, Microchannels, and Minichannels (IC- NMM), Edmonton (Canada), June 19-22, (2011).

• M. Hadj-Nacer, I. Graur and P. Perrier. Accommodation coefficient investigation in rectangular microchannels for large Knudsen number range. In proceeding of the 15th International Meeting on Thermal Sciences, Tlemcen (Algeria), September 24-26, (2011). 158 Appendix A. Publications

• M. Hadj-Nacer, P. Perrier, J. G. Méolans, I. Graur and M. Wüest. Experimental study of the gas flows through channels with circular cross sections. In proceeding of first International GASMEMS conference, Skiathos (Greece), June 5-8, (2012).

International conferences - Talks

• M. Hadj-Nacer, I. Graur and P. Perrier. Mass flow rate measurement through rectan- gular micro-channels in a large Knudsen number range. 2nd GASMEMS Workshop, Les Embiez (France), July 07-10, (2010).

• M. Hadj-Nacer, I. Graur and P. Perrier. Mass flow rate measurements through mi- crochannels with Gold and Silica surfaces in all flow regimes. 64th IUVSTA Workshop on Practical Applications and Methods of Gas Dynamics for Vacuum Science and Tech- nology, Leinsweiler (Germany), May 15-19, (2011).

International Conferences - Posters

• M. Hadj-Nacer, I. Graur and P. Perrier. Mass flow rate measurement through rectan- gular micro-channels in a large Knudsen number range. 2nd GASMEMS Workshop, Les Embiez, (France), July 07-10, (2010).

• M. Hadj-Nacer, I. Graur and P. Perrier. Accommodation coefficient investigation in all flow regimes: from hydrodynamic to near free molecular regimes. Proc. of the 4th International Conference on Heat Transfer and Fluid Flow in Microscale (HTFFM-IV), Fukuoka (Japan), September 4-9 (2011).

• M. Hadj-Nacer, P. Perrier, J. G. Méolans, I. Graur and M. Wüest. Tangential momen- tum accommodation coefficients in industrial microtubes. 28th Rarefied Gas Dynamic symposium, Seragoza (Spain), July 8-13, (2012).

• M. Hadj-Nacer, P. Perrier, J. G. Méolans, I. Graur and M. Wüest. Tangential mo- mentum accommodation coefficients in coated microtubes. 59th AVS International Symposium and Exhibition, Florida (US), October 28-November 2 (2012). Appendix B Unsteady technique

The unsteady technique used to measure the mass flow rate is similar to the rise of pressure technique described in Section 4.3. The difference between these two techniques is that in the rise-of-pressure technique the absolute variation in time of pressure in one tank (inlet or outlet) is measured, while in the unsteady method the variation in time of pressure difference (∆p(t)) between the inlet and outlet tanks is measured. Another difference is that, in the rise-of-pressure technique, small variation of the pressure in the tank is considered (of order of 1%), however, in the unsteady technique the experimental time must be longer to capture better the exponential behavior of variation of pressure difference in time.

From conservation equation for the number of gas molecule in the tanks [115], whose volumes are Vin, and Vout, it is easy to find the pressure difference between the tanks as a function of time [92]: ∆p(t) = ∆p0 exp(−t/τp), (B.1) where ∆p0 is the value of the pressure difference at the instant t = 0 and τp is the relaxation time. Using the least square method the relaxation time τp can be calculated from the measured value of the pressure difference.

From the law of perfect gases (4.4) and following the same procedure used in the rise-of- pressure technique (see Section 4.3) the mass flow rate through microsystem connecting two tanks can be written as d∆p(t) V M˙ unst = 0 , (B.2) dt RT where V0 = VinVout/(Vin + Vout) is the reduced system volume. The term d∆p(t)/dt is calcu- lated by deriving expression (B.1) regarding to time. It is obvious that the uncertainty on the mass flow rate M˙ unst (B.2) using this technique is higher than that of the rise-of-pressure technique, due to the uncertainty on the term V0 and temperature. We should note that the restriction on the temperature is the same as for the rise of pressure technique (see Section 4.3). However, the advantage of the unsteady technique is the possibility of measuring the mass flow rate at any time of the experimental time length.

Using the root square sum, the uncertainty on the mass flow rate M˙ unst (B.2) is less than 3.7% compared to 2.4% for the rise-of-pressure technique (see Section 4.5).

Bibliography

[1] Tison S. A. and Lu S. System and method for measuring flow. US Patent Apllication Publication, US2006/0123920 A1, 2006. (Cited on page 68.)

[2] A. Agrawal and S. V. Prabhu. Survey on measurement of tangential momentum ac- commodation coefficient. Journal of Vac. Sci. Technol., A26(4):634–645, 2008. (Cited on pages xxix,4, 61 and 66.)

[3] S. Albertoni, C. Cercignani, and L. Gotusso. Numerical evaluation of the slip coefficient. Phys. Fluids, 6(7):993–996, 1963. (Cited on pages 26, 29 and 105.)

[4] M. D. Allen and O. G. Raabet. Slip correction measurements of spherical solid aerosol particles in an improved Millikan apparatus. Journal of Aerosol Science and Technology, 4(3):269–286, 1985. (Cited on page 61.)

[5] T. A. Ameel, R. O. Warrington, R. S. Wegeng, and M. K Drost. Miniaturization technologies applied to energy systems. Energy Convers, 38:969–982, 1997. (Cited on page9.)

[6] E. B. Arkilic. Measurement of mass flow and tangential momentum accommodation coefficient in silicon micro-machined channels. PhD thesis, Massachusetts Institute of Technology, 1997. (Cited on pages 25 and 68.)

[7] E. B. Arkilic, K. S. Breuer, and M. A. Schmidt. Mass flow and tangential momentum accomodation in silicon micromachined channels. Fluid mechanics, 437:29–43, 2001. (Cited on pages xxix,4, 64, 65 and 68.)

[8] E. B. Arkilic, M. A. Schmidt, and K. S. Breuer. Gaseous slip flow in long microchannels. Journal Microelectomech Systeme, 6(2):167–178, 1997. (Cited on pages xxix,4, 30, 40 and 65.)

[9] C. Aubert and S. Colin. High-order boundary conditions for gaseous flows in rectangular microducts. Microscale Thermophys. Eng., 5:41–54, 2001. (Cited on page 24.)

[10] J. A. Bentz, R. V. Tompson, and S. K. Loyalka. The spinning rotor gauge measure- ments of viscosity, velocity slip coefficients, and tangential momentum, accommodation coefficients for N2 and CH4. Vacuum, 48(10):817–824, 1997. (Cited on pages xxix,4 and 63.)

[11] J. A. Bentz, R. V. Tompson, and S. K. Loyalka. Measurements of viscosity, velocity slip coefficients, gas and tangential momentum accommodation coefficients using a modified spinning rotor gauge. J. Vac. Sci. Technol. A, 19:317–324, 2001. (Cited on page 63.)

[12] R. F. Berg. Capillary flow meter for calibrating spinning rotor gauges. J. Vac. Sci. Technol., A 26(5), 2008. (Cited on pages xxix and4.) 162 Bibliography

[13] A. Beskok. Molecular-Based Microfluidic Simulation Models, chapter 8. CRC Press, Boca Raton, FL, 2001. (Cited on pages xiii and3.)

[14] A. Beskok and G. Karniadakis. A model for flows in channels, pipes, and ducts at micro and nano scales. Microscale Thermophysical Engineering, 3:43–77, 1999. (Cited on pages 28 and 29.)

[15] P. L. Bhatnagar, E. P. Gross, and M. Krook. A model for collision processes in gases. I. Small amplitude processes in charged and neutral one-component systems. Phys. Rev, 94(3):511–525, 1954. (Cited on page 48.)

[16] B. Bhushan. Springer Handbook of Nanotechnology. Springer- Verlag, 2007. (Cited on pages7,9 and 10.)

[17] G. A. Bird. Difinition of mean free path for real gases. Phys. Fluids, 26, 11 1983. (Cited on page1.)

[18] G. A. Bird. Molecular Gas Dynamics and the Direct Simulation of Gas Flows. Oxford Science Publications, 1994. (Cited on page 23.)

[19] G. Breyiannis, S. Varoutis, and D. Valougeorgis. Rarefied gas flow in concentric annular tube: estimation of Poiseuille number and the exact hydraulic diameter. Eur J Mech B/Fluids, pages 609–622, 2008. (Cited on page 46.)

[20] J. E. Broadwell. Shock structure in a simple discrete velocity gas. Phys. Fluids, 7(8), 1964. (Cited on page 51.)

[21] B. Y. Cao, M. Chen, and Z. Y. Guo. Temperature dependence of the tangential mo- mentum accommodation coefficient for gases. Appl. Phys. Lett., 86, 2005. (Cited on pages 66 and 67.)

[22] B. Y. Cao, J. Sun, M. Chen, and Z. Y. Guo. Molecular momentum transport at fluid- solid interfaces in MEMS/NEMS: A Review. Int. J. Mol. Sci., 10:4638–4706, 2009. (Cited on pages 61 and 62.)

[23] C. Cercignani. Higher order slip according to the linearized Boltzmann equation. In- stitute of Engineering Research Report AS-64-19, University of California, Berkeley, 1, 1964. (Cited on pages xiii, 24, 25, 26, 27, 28, 29, 34, 35, 55, 105, 108 and 116.)

[24] C. Cercignani. Mathematical methods in kinetic theory. Premuim Press, New York, London, 1990. (Cited on pages xxviii, 34 and 108.)

[25] C. Cercignani and A. Daneri. Flow of a rarefied gas between two parallele plates. Phys. Fliuds, 6:993–996, 1963. (Cited on pages 27, 46, 55 and 87.)

[26] C. Cercignani and C. D. Pagani. Variational approach to rarefied flows in cylindrical and spherical geometry. In New York Academic, editor, Rarefied gas dynamics, volume 1, pages 555–573, 1967. (Cited on page 46.) Bibliography 163

[27] C. Cercignani and F. Sernagiotto. Cylindrical Couette flow of a rarefied gas. Phys. Fluids, 10(6):1200–1204, 1966. (Cited on page 46.)

[28] S Cercignani, C. Lorenzani. Variational derivation of second-order slip coefficients on the basis of the boltzmann equation for hard-sphere molecules. Phys. Fluids, 22, 2010. (Cited on pages xiii, 27, 28, 35, 55, 108 and 109.)

[29] S. Chapman and T. G. Cowling. The mathematical theory of non-uniform gases. Uni- versity Press, Cambridge, 1970. (Cited on pages 23, 26, 29 and 45.)

[30] S. Colin, P. Lalonde, and R. Caen. Validation of a second-order slip flow model in a rectangular microchannel. Heat Transf Eng, 25(3):23–30, 2004. (Cited on pages xiii, xxix,4,7,8, 64, 65, 67 and 68.)

[31] F. G. Collins and E. C. Knox. Parameters of Nocilla gas/surface interaction model from measured accommodation coefficients. AIAA Journal, 32(4), 1994. (Cited on page 61.)

[32] G. Cosma, J. K. Fremerey, B. Lindenau, G. Messer, and P. Rohl. Calibrating of spinning rotor gas friction gauge against a fundamental vacuum pressure standard. J. Vac. Sci. Techno, 17:642–644, 1980. (Cited on page 63.)

[33] E. J. Davis. A history and state-of-the-art of accommodation coefficients. Atmospheric research, 82:561–578, 2006. (Cited on page4.)

[34] R. G. Deissler. An analysis of second order slip flow and temperature jump boundary conditions for rarefied gas. International Journal Heat and Mass Transfer, 45:681–694, 1964. (Cited on pages 24, 26, 27, 28, 29, 65 and 108.)

[35] W. A. Ebert and E. M. Sparrow. Slip flow in rectangular and annular ducts. Trans. ASME D J. Basis Engrg., 87:1018–1024, 1965. (Cited on page 24.)

[36] T. Ewart. Etude des écoulements gazeux isothermes en microconduits: du régime hy- drodynamique au proche régime molćulaire libre. PhD thesis, Provence University, 2007. [in French]. (Cited on pages 67, 93 and 138.)

[37] T. Ewart, P. Perrier, I. A. Graur, and J. G. Méolans. Mass flow rate measurements in gas micro flows. Experiments in Fluids, 41(3):487–498, 2006. (Cited on pages 67, 68, 80 and 88.)

[38] T. Ewart, P. Perrier, I. A. Graur, and J. G. Méolans. Mass flow rate measurements in microchannel, from hydrodynamic to near free molecular regimes. Fluid mechanics, 584:337–356, 2007. (Cited on pages xxx, 30, 64, 65, 68, 83, 88, 97, 105 and 153.)

[39] T. Ewart, P. Perrier, I. A. Graur, and J. G. Méolans. Tangential momentum accomoda- tion in microtube. Microfluid and Nanofluid, 3(6):689–695, 2007. (Cited on pages xviii, 64, 84, 88, 90, 91 and 108.)

[40] J. H. Ferziger. Flow of a rarefied gas through a cylindrical tube. Phys. Fluids, 10(7):1448–1453, 1967. (Cited on page 46.) 164 Bibliography

[41] G. W. Finger. Estimation of tangential momentum accommodation coefficient using molecular dynamics simulation. PhD thesis, University of Central Florida, Department of Mechanical, Materials and Aerospace Engineering, 2002. (Cited on page4.)

[42] D.H. Gabis, S.K. Loyalka, and T.S. Storvick. Measurements of the tangential mo- mentum accommodation coefficient in the transition flow regime with a spinning rotor gauge. J. Vac. Sci. Techno. A, 14:2592–2598, 1996. (Cited on page 63.)

[43] R. Gatignol. Kinetic theory for a discrete velocity gas and application to the shock structure. Phys. Fluids, 18, 1975. (Cited on page 51.)

[44] I. A. Graur, , P. Perrier, W. Ghozlani, and J. G. Méolans. Measurements of tangential momentum accommodation coefficient for various gases in plane microchannel. Physic. Fluids, 21, 2009. (Cited on pages xviii, xxxi, 64, 88, 100 and 108.)

[45] I. A. Graur, J. G. Méolans, and D. E. Zeitoun. Analytical and numerical description for isothermal gas flows in microchannels. Microfluid and Nanofluid, 2:64–77, 2006. (Cited on pages 30, 40 and 84.)

[46] T. Gronych, R. Ulman, L. Peksa, and P. Repa. Measurements of the relative momentum accommodation coefficient for different gases with a viscosity vacuum gauge. Vacuum, 73:275–279, 2004. (Cited on pages xxix and4.)

[47] M. Hadj-Nacer, I. A. Graur, and P. Perrier. Mass flow measurement in rectangular mi- crochannel from hydrodynamic to near molecular regime. Journal La Houille Blanche, 4:49–54, 2011. (Cited on page 68.)

[48] N. G. Hadjiconstantinou. Comment on Cercignani’s second-order slip coefficient. Phys. Fluids, 15(8):257–274, 2003. (Cited on pages xiii, 26, 27, 28, 29, 35, 55 and 108.)

[49] J. Harley, Y. Huang, H. Bau, and J. Zemel. Gas flows in microchannels. Fluid Mechanic, 284:257–274, 1995. (Cited on pages7 and 67.)

[50] K. A. Hickey and S. K. Loyalka. Plan poiseille flow: Ridgid sphere gas. J. Vac. Sci. Technol. A, 8:957, 1990. (Cited on page 46.)

[51] Y. T. Hsia and G. A. Domoto. An experimental investigation of molecular rarefac- tion effects in gas lubricated bearings at ultra low clearances. Journal of Lubrication, 105(1):120–130, 1983. (Cited on page 29.)

[52] S. S. Hsieh, H. H. Tsai, C. Y. Lin, and Huang C. F. Gas flow in a long microchan- nel. International Journal of Heat and Mass Transfer, 47:3877–3887, 2004. (Cited on pages xiii,7,8, 64 and 65.)

[53] INFICON. Catalog, 2003. (Cited on pages xvii and 70.)

[54] J. Jang and S.T. Wereley. Gaseous slip flow analysis of a micromachined flow sensor for ultra small flow applications. J. Micromech. Microeng., 17:229–237, 2007. (Cited on page 64.) Bibliography 165

[55] K. Jousten. Is the effective accommodation coefficient of the spinning rotor gauge temperature dependent? J. Vac. Sci. Technol. A, 21:318–324, 2003. (Cited on pages4, 63 and 66.)

[56] K. Jousten, G. Messer, and D. Wandrey. A precision gas flowmeter for vacuum metrol- ogy. Vacuum, 44(2):135–141, 1993. (Cited on page 67.)

[57] G. Karniadakis and A. Beskok. Microflows: Fundamentals and Simulation. Springer, 2002. (Cited on page 26.)

[58] G. R. Karr. Study of effects of the gas-surface interaction on spinning convex bodies with application to satellite experiments. PhD thesis, University of Illinois, Urbana, Illinois, 1969. (Cited on page 62.)

[59] E.D. Knechtel and W.C. Pitts. Experimental momentum accommodation on metal surfaces of ions near and above earth-satellite speeds. 6th International Symposium on Rarefied Gas Dynamics, pages 1257–1266, 1969. (Cited on page 62.)

[60] M. N. Kogan. Rarefied gas dynamics. Plenum Press New York, 1969. (Cited on pages 105 and 116.)

[61] I. Kuscer. Phenomenology of gas-surface accommodation. In 9em International Sympo- sium on Rarefied Gas Dynamics, volume E1, pages 1–21, 1974. (Cited on page xxviii.)

[62] P. Lalonde. Étude expérimentale d’écoulements gazeux dans les microsystèmes à fluides. PhD thesis, INSA, Toulouse, France, 2001. [in French]. (Cited on page 67.)

[63] H. Lang and S. K. Loyalka. Some analytical results for thermal transpiration and the mechanocaloric effect in cylindrical tube. Phys. Fluids, 27, 1984. (Cited on page 46.)

[64] W. L. Li and C. H. Weng. Modified average Reynolds equation for ultra-thin film gas lubrication considering roughness odentations at arbitrary Knudsen numbers. Wear, 209:292–300, 1997. (Cited on page 28.)

[65] S.M. Liu, P.K. Sharma, and E.L. Knuth. Satellite drag coefficients calculated from mea- sured distributions of reflected helium atoms. AIAA Journal, 17:1314–1319, Liu:1979. (Cited on page 62.)

[66] S. S. Lo and S. K. Loyalka. An efficient computation of near continuum rarefied gas- flows. J. Appl. Math. and Phys., 33(419-424), 1982. (Cited on pages 46, 92, 93 and 145.)

[67] S. S. Lo, S. K. Loyalka, and T. S. Stvorik. Kinetic theory of thermal transpiration and mechanocaloric effect. V. Flow of polyatomic gases in a cylindrical tube with arbitrary accommodation at the surface. J. Chem. Phys., 81:2439, 1984. (Cited on page 46.)

[68] D. A. Lockerby, J. M. Reese, D. R. Emerson, and R. W. Barber. Velocity boundary at solid walls in rarefied gas calculations. Physical Review E, 70(1), 2004. (Cited on page 29.) 166 Bibliography

[69] R.G. Lord. Tangential momentum accommodation coefficients of rare gases on polycrys- talline metal surfaces. In 10th symposium on Raretied Gas Dynamics,, pages 531–538, 1976. (Cited on pages 61, 63 and 66.)

[70] S. K. Loyalka and K. A. Hickey. Plan Poiseille flow: near continuum results for ridgid sphere gas. Physica A, 160:395–408, 1989. (Cited on page 26.)

[71] S. K. Loyalka, N. Petrellis, and S. T. Stvorick. Some numerical results for the BGK model: thermal creep and viscous slip problems with arbitrary accommodation of the surface. Phys. Fluids, 18:1094, 1975. (Cited on pages 26, 27, 29 and 46.)

[72] S. K. Loyalka, T. S. Stvorik, and H. S. Park. Poiseulle flow and thermal creep flow in long, rectangular channels in the molecular and transition flow regimes. J. Vac. Sci. Technol., 13(6):1188, 1976. (Cited on pages xvii, 24, 25, 46, 47 and 54.)

[73] W.J. Maegley and A.S. Berman. Transition from free - molecule to continuum flow in an annulus. Physic. Fluids, 15(5):780–785, 1972. (Cited on page 66.)

[74] N. Maluf and K. Williams. Introduction to Microelectromechanical Systems Engineering. Artech House, 2004. (Cited on page7.)

[75] F Martya, L Rousseaua, B Saadanya, B Merciera, O Franc¸aisa, Y Mitab, and T Bourouina. Advanced etching of silicon based on deep reactive ion etching for sili- con high aspect ratio microstructures and three-dimensional micro- and nanostructures. Microelectronics, 36:673–677, 2005. (Cited on page 10.)

[76] J. Maurer, P. Tabeling, P. Joseph, and H. Willaime. Second-order slip laws in mi- crochannels for helium and nitrogen. Phys. Fluids, 15:2613–2621, 2003. (Cited on pages xxix,4, 64, 65, 67, 84 and 98.)

[77] J. C. Maxwell. On stress in rarefied gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond., 170:231–256, 1878. (Cited on page xxix.)

[78] J. C. Maxwell. On stress in rarefied gases arising from inequalities of temperature. Phil. Trans. R. Soc. Lond., 170:231–256, 1879. (Cited on page 25.)

[79] J. C. Maxwell. The scientific papers of James clerk Maxwell. Dover Pulications, Inc., New York, 1965. (Cited on page 29.)

[80] R. A. Millikan. Coefficients of slip in gases and the law of reflection of molecules from the surfaces of solids and liquids. Physical Review, 21(3), 1923. (Cited on pages xxix and4.)

[81] Y. Mitsuya. Modified Reynolds equation for ultra-thin film gas lubrication using 1.5- order slip flow model and considering surface accommodation coefficient. Journal of Tribology, 115(2):289–294, 1993. (Cited on page 29.)

[82] G. L. Morini and M. Spiga. Slip flow in rectangular microtubes. Microscale Thermophys. Engrg., 2:273–282, 1998. (Cited on page 24.) Bibliography 167

[83] E. Y.-K. Ng and N. Liu. A multicoefficient slip-corrected Reynolds equation for micro- thin film gas lubrication. International Journal of Rotating Machinery, 2:105–111, 2005. (Cited on page 28.)

[84] P. Norberg, L. G. Petersson, and I. Lundström. Characterization of gas transport through micromachined submicron channels in silicon. Vacuum, 45(1):139–144, 1994. (Cited on page7.)

[85] T. Ohwada, Y. Sone, and K. Aoki. Numerical analysis of the Poiseuille and thermal transpiration flows between two parallel plates on the basis of the Boltzmann equation for hard sphere molecules. Phys. Fluids A, 1(12):2042–2049, 1989. (Cited on pages 26 and 46.)

[86] A. I. Omelik. Measurement of momentum transport to surfaces of different form in a hyper-sonic free molecular stream. Fluid Dynamics Journal, 12:633–636, 1977. (Cited on page 62.)

[87] L. Peksa, T. Gronych, M. Vicar, M. Jerab, P. Repa, J. Tesar, D. Prazak, and Z. Krajicek. Method of measuring the change in volume of a diaphragm bellows used in a volume displacer of a constant-pressure gas flowmeter (with a practical guide). Measurement, 44:1143–1152, 2011. (Cited on page 67.)

[88] P. Perrier, I. A. Graur, T. Ewart, and J. G. Méolans. Mass flow rate measurements in microtubes: From hygrodynamic to near free molecular regime. Physic. Fluids, 23(042004), 2011. (Cited on pages xviii, 64, 65, 90, 91 and 108.)

[89] J. Pitakarnnop. Analyse expérimentale et simulation numérique d’écoulements raréfiés de gaz simples et de mélanges gazeux dans les microcanaux. PhD thesis, Université de Toulouse, Institut National des Sciences Appliquées, 2009. [in French]. (Cited on pages 28, 93 and 138.)

[90] J. Pitakarnnop, S. Varoutis, D. Valougeorgis, S. Geoffroy, L. Baldas, and S. Colin. A novel experimental setup for gas microflows. Microfluid and Nanofluid, 8:57–72, 2010. (Cited on pages 64, 65 and 68.)

[91] K. Pong, C. Ho, J. Liu, and Y. Tai. Non-linear pressure distribution in uniform mi- crochannels. In ASME FED, editor, Application of microfabrication to fluid mechanics, volume 197, pages 51–56, 1994. (Cited on page7.)

[92] B. T. Porodnov, P. E. Suetin, S. F. Borisov, and V. D. Akinshin. Experimental inves- tigation of rarefied gas flow in different channels. J. Fluid Mech., 64(3):417–437, 1974. (Cited on pages xviii,4, 61, 64, 65, 90, 91, 116 and 159.)

[93] B. T. Porodnov and Tuchvetov F. T. Thermal transpiration in a circular capillary with a small temperature difference. Journal of Fluid Mechanics, 88(609-622), 1978. (Cited on page 46.) 168 Bibliography

[94] B. T. Porodnov and F. T. Tuchvetov. Theoretical investigation in non-isothermal flow of rarefied gas in a cylindrical capillary. J. Eng. Phys., 36:61, 1979. (Cited on pages 46, 92, 93 and 145.)

[95] C.T. Rettner. Thermal and tangential momentum accommodation coefficients for N2 colliding with surfaces of relevance to disk-drive air bearings derived from molecular beam scattering. IEEE T. Magn, 34:2387–2395, 1998. (Cited on page 62.)

[96] S. A. Schaaf and P. L. Chambre. Flow of Rarefied Gases. Princeton University Press, USA, 1961. (Cited on pages xxvi and2.)

[97] R. Schamberg. The fundamental differential equations and the boundary conditions for high speed slip-flow, and their application to several specific problems. PhD thesis, California Institute of Technology, 1947. (Cited on page 29.)

[98] M. Seidl and E. Steinheil. Measurement of momentum accommodation coefficients on surfaces characterized by Auger Spectroscopy, Sims and Leed. In International Symposium on Rarefied Gas Dynamics, pages E 9.1–E 9.12, 1974. (Cited on pages 62 and 66.)

[99] S. D. Senturia. Microsystem Design. Kluwer Academic, 2002. (Cited on pages7 and8.)

[100] E. M. Shakhov. Generalization of the Krook kinetic relaxation equation. Fluid Dyn., 3(5):95–96, 1968. (Cited on page 46.)

[101] F. Sharipov. Rarefied gas flow through a long rectangular channel. J. Vac. Sci. Technol. A, 17(5):3062–3066, 1999. (Cited on pages xvii, xxiii, xxxi, 24, 30, 31, 32, 34, 44, 46, 47, 54 and 153.)

[102] F. Sharipov. Application of the Cercignani-Lampis scattering kernel to calculations of rarefied gas flows. III. Poiseuille flow and thermal creep through a long tube. European Journal of Mechanics B/Fluids, 22:145–154, 2003. (Cited on page 46.)

[103] F. Sharipov and V. Seleznev. Data on internal rarefied gas flows. J. Phys. Chem. Ref. Data, 27(3):657–706, 1998. (Cited on pages 21, 27, 46, 47 and 93.)

[104] F. M. Sharipov and V. D. Seleznev. Rarefied gas flow through a long tube at any pressure ratio. J. Vac. Sci. Technol. A, 12(5):2933–2935, 1994. (Cited on pages 83 and 97.)

[105] F. M. Sharipov and E. A. Subbotin. On optimization of the discrete velocity method used in rarefied gas dynamics. Z angew Math Phys, 44, 1993. (Cited on pages 52 and 53.)

[106] S. Shen, G. Chen, R. M. Crone, and M. Anaya-Dufresne. A kinetic-theory based first order slip boundary condition for gas flow. Phys. Fluids, 19, 2007. (Cited on page 29.)

[107] F. D. Shields. An acoustical method for determining the thermal and momentum accommodation of gases on solids. J. Chem. Phys., 62, 1975. (Cited on page 66.) Bibliography 169

[108] F. D. Shields. More on the acoustical method of measuring energy and tangential momentum accommodation coefficients. J. Chem. Phys., 72, 1980. (Cited on page 66.)

[109] Y. Sone. Kinetic Theory and Fluid Mechanics. Birkhäuser, Boston, 2002. (Cited on page 24.)

[110] Y. Sone. Molecular Gas Dynamics. Birkhäuser, Boston, 2007. (Cited on page 108.)

[111] Y. Sone and Y. Onishi. Kinetic theory of evaporation and condensation hydrodynamic equations and slip boundary conditions. Journal of Physic Society of Japan, 44:1981, 1978. (Cited on pages 24 and 34.)

[112] Y. Sone and K. Yamamoto. Flow of rarefied gas through a circular pipe. Phys. Fluids, 11:1672–1678, 1968. (Cited on pages 25 and 34.)

[113] L. J. Stacy. A determination by the constant deflection method of the value of the coefficient of slip for rough and for smooth surfaces in air. Phys. Rev, 21, 1923. (Cited on page 66.)

[114] E. Steinheil, M. Scherber, and M. Seidl. Investigations on the interaction of gases and well-defined solid surface with respect to possibilities for reduction of aerodynamic friction and aerothermal heating. In Proc. Symp. Rarefied Gas Dynamics. N.Y.:AIAA, pages 589–602, 1977. (Cited on page 66.)

[115] P.E. Suetin, B.T. Porodnov, V.G. Chernjak, and S.F. Borisov. Poiseuille flow at ar- bitrary Knudsen numbers and tangential momentum accommodation. J. Fluid Mech., 60:581–592, 1973. (Cited on page 159.)

[116] Swagelok. Catalog, 2010. (Cited on page 69.)

[117] P. Tekasakul, J.A. Bentz, R.V. Tompson, and S.K. Loyalka. The spinning rotor gauge: Measurements of viscosity, velocity slip coefficients, and tangential momentum accom- modation coefficients. J. Vac. Sci. Technol. A, 14:2946–2952, 1996. (Cited on page 63.)

[118] J. M. Thevenoud, B. Mercier, T. Bourouina, F. Marty, M. Puech, and N. Launay. DRIE technology: from micro to nanoapplications. Technical report, Alcatel Micro Machining System, 2012. (Cited on page 10.)

[119] L.B. Thomas and R.G. Lord. Comparative measurements of tangential momentum and thermal accommodation on polished and roughened. In 8th Symposium on RarefiedGas Dynamics, 1974. (Cited on pages 61, 62, 63 and 66.)

[120] A. Ukhov, S. Borisov, and B. Porodnov. Surface chemical composition effect on internal gas flow and molecular heat exchange in a gas-solids system. In In 27th Symposium on Rarefied Gas Dynamics, 2010. (Cited on page 66.)

[121] D. Valougeorgis and J. R. Thomas. Exact numerical results for Poiseuille and thermal creep flow in a cylindrical tube. Phys. Fluids, 29, 1986. (Cited on page 46.) 170 Bibliography

[122] T. Veijola, H. Kuisma, and J. Lahdenpera. The influence of gas-surface interaction on gas-film damping in a silicon accelerometer. J. Sensor. Actuat. A-Phys., 66:83–92, 1998. (Cited on page 63.)

[123] C. Y. Wang. Slip flow in a triangular duct an exact solution. Z. Angew. Math. Mech., 83:629–631, 2003. (Cited on page 24.)

[124] L Wangkui, Z Dixin, L Qiang, L Shiliang, L Detian, and M Yang. A gas flow standard apparatus. Vacuum, 47(6-8):519–522, 1996. (Cited on page 67.)

[125] P. Welander. The temperature jump in a rarefied gas. Arkiv foer Fysik, 7(44):507–553, 1954. (Cited on page 29.)

[126] D. R. Willis. Comparison of kinetic theory analyses of lineralized Couette flow. Phys. Fluids, 5(2):127–135, 1962. (Cited on page 29.)

[127] L. Wu. A slip model for rarefied gas flows at arbitrary Knudsen number. Appl. Phys. Lett., 93, 2008. (Cited on page 29.)

[128] L Wu and D. B. Bogy. A generalized compressible Reynolds lubrication equation with bounded contact pressure. Phys. Fluids, 13, 2001. (Cited on page 29.)

[129] H. Xue and Q. Fan. A new analytic solution of the Navier-Stokes equations for mi- crochannel flows. Microscale Thermophysical Engineering, 4(2):125–143, 2000. (Cited on page 29.)

[130] H. Yamaguchi, T. Hanawa, O. Yamamoto, Y. Matsuda, Y. Egami, and T. Niimi. Exper- imental measurement on tangential momentum accommodation coefficient in a single microtube. Microfluid and Nanofluid, 11:57–64, 2011. (Cited on pages 64 and 65.)

[131] H. Yamaguchi, Y. Matsuda, and T. Niimi. Tangential momentum accommodation coefficient measurements for various materials and gas species. Journal of Physics: Conference Series, 362, 2012. (Cited on pages 64 and 65.)

[132] H. Yoshida, M. Shiro, K. Arai, M. Hirata, and H. Akimichi. Effect of surface material and roughness on conductance of channel between parallel disks at molecular flow. J. Vac. Sci. Technol, A 28(4), 2012. (Cited on page7.)

[133] S. Yu and T. Ameel. Slip flow heat transfer in rectangular microchannels. International Journal of Heat and mass transfer, 44:4225–4234, 2001. (Cited on page 24.)

[134] Y. Zohar, S. Y. K. Lee, W. Y. Lee, L. Jiang, and P. Tong. Subsonic gas flow in a straight and uniform microchannel. Fluid Mechanics, 472:125–151, 2002. (Cited on pages xxix, 4 and 67.)