IUTAM Symposium on Advances in Micro- and Nanofluidics: Proceedings of the IUTAM Symposium on Advances in Micro- and Nanofluidics, Dresden, Germany, September 6-8, 2007 (IUTAM Bookseries)

IUTAM Symposium on Advances in Micro- and Nanoﬂuidics IUTAM BOOKSERIES Volume 15 Series Editors G.M.L. Gladwell, Univ...

Author: Marco Ellero | Xiangyu Hu | Jochen Frohlich | Nikolaus Adams

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IUTAM Symposium on Advances in Micro- and Nanoﬂuidics

IUTAM BOOKSERIES Volume 15 Series Editors G.M.L. Gladwell, University of Waterloo, Waterloo, Ontario, Canada R. Moreau, INPG, Grenoble, France Editorial Board J. Engelbrecht, Institute of Cybernetics, Tallinn, Estonia L.B. Freund, Brown University, Providence, USA A. Kluwick, Technische Universit¨ at, Vienna, Austria H.K. Moﬀatt, University of Cambridge, Cambridge, UK N. Olhoﬀ, Aalborg University, Aalborg, Denmark K. Tsutomu, IIDS, Tokyo, Japan D. van Campen, Technical University Eindhoven, Eindhoven, The Netherlands Z. Zheng, Chinese Academy of Sciences, Beijing, China

Aims and Scope of the Series The IUTAM Bookseries publishes the proceedings of IUTAM symposia under the auspices of the IUTAM Board.

For other titles published in this series, go to www.springer.com/series/7695

Marco Ellero Xiangyu Hu Jochen Fr¨ohlich Nikolaus Adams

Editors

IUTAM Symposium on Advances in Micro- and Nanoﬂuidics Proceedings of the IUTAM Symposium on Advances in Micro- and Nanoﬂuidics, Dresden, Germany, September 6–8, 2007

Marco Ellero Technische Universit¨ at M¨ unchen Lehrstuhl f¨ ur Aerodynamik Boltzmannstrasse 15 85748 Garching Germany marco.ellero@ tum.de

Xiangyu Hu Technische Universit¨ at M¨ unchen Lehrstuhl f¨ ur Aerodynamik Boltzmannstrasse 15 85748 Garching Germany [email protected]

Jochen Fr¨ ohlich Technische Universit¨ at Dresden Professur f¨ ur Str¨ omungsmechanik George-B¨ ahr-Str. 3c 01062 Dresden Germany [email protected]

Nikolaus Adams Technische Universit¨ at M¨ unchen Lehrstuhl f¨ ur Aerodynamik Boltzmannstrasse 15 85748 Garching Germany [email protected]

ISSN 1875-3507 e-ISSN 1875-3493 ISBN 978-90-481-2625-5 e-ISBN 978-90-481-2626-2 DOI 10.1007/978-90-481-2626-2 Springer Dordrecht Heidelberg London New York Library of Congress Control Number: 2009926023 c Springer Science+Business Media B.V. 2009 No part of this work may be reproduced, stored in a retrieval system, or transmitted in any form or by any means, electronic, mechanical, photocopying, microﬁlming, recording or otherwise, without written permission from the Publisher, with the exception of any material supplied speciﬁcally for the purpose of being entered and executed on a computer system, for exclusive use by the purchaser of the work. Printed on acid-free paper Springer is part of Springer Science+Business Media (www.springer.com)

Preface

This volume contains papers presented at the IUTAM Symposium on Advances in Micro- and Nanoﬂuidics, held in Dresden, Germany, September 6– September 8, 2007. Micro- and nanoﬂuidic technologies have experienced a rapid development over the past few years. Since the physics of ﬂuids at the micro- and nanoscale can be quite different from that at the macroscale many problems have been studied and physical models and numerical-simulation approaches have been developed. The objective of the IUTAM Symposium was to promote discussions between researchers active in the ﬁeld and to provide a review on the state-of-the-art. Furthermore it should serve as a guideline to identify for the years to come important topics which require further research. Flows at the microscales are dominated by surface eﬀects and by interactions with immersed particles or structures. These eﬀects and interactions frequently are expressed on mesoscopic length and time scales and cannot be represented by a macroscale or continuum description in a straightforward manner. The description by molecular dynamics is unsuitable for reaching practically relevant time and length scales. Mesoscale modeling, on the other hand, requires further experimental and numerical investigations and theoretical modeling. The papers presented in this volume deal with multi-scale particle methods for numerical simulations, liquid-wall interactions and modeling approaches, modeling of immersed nano-scale structures, organized ﬂow behavior at microand nano-scales, and methods for control of micro- and nano-scale ﬂows.

Munich, October 26, 2008

Xiangyu Hu Marco Ellero Jochen Fr¨ ohlich Nikolaus A. Adams

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Contents

Part I Plenary Lectures Nonlinear Electrokinetic Flow: Theory, Experiment, and Potential Applications Sung Jae Kim, Jongyoon Han . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Fluid Particle Models for the Simulation of Microﬂuids Marco Ellero . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 Part II Micro-channel Flows Semi-analytical Solution of the Density Proﬁle for a Gas Close to a Solid Wall E.A.T. van den Akker, A.J.H. Frijns, S.V. Nedea, A.A. van Steenhoven 35 Comprehensive Analysis of Dewetting Proﬁles to Quantify Hydrodynamic Slip Oliver B¨ aumchen, Renate Fetzer, Andreas M¨ unch, Barbara Wagner, Karin Jacobs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 Variation of Transport Properties Along Nanochannels: A Study by Non-equilibrium Molecular Dynamics Filippos Sofos, Theodoros Karakasidis, Antonios Liakopoulos . . . . . . . . . . 67 Estimation of the Poiseuille Number in Gas Flows Through Rectangular Nano- and Micro-channels in the Whole Range of the Knudsen Number Stelios Varoutis, Dimitris Valougeorgis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79 Moving Contact Line with Balanced Stress Singularities X.Y. Hu, N.A. Adams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

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Contents

Part III Complex Fluids Clariﬁcation and Control of Micro Plasma Flow with Wall Interaction Takeshi Furukawa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97 Electrochemical Control of the Surface Energy of Conjugated Polymers for Guiding Samples in Microﬂuidic Systems Nathaniel D. Robinson . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113 Small Scale Cavitation Model Juan Miguel S´ anchez S´ anchez, Rafael Rodrigo Fern´ andez . . . . . . . . . . . . . 127 Experimental and Theoretical Approach for Analysis of Flow Induced by Micro Organisms Existing on Surface of Granular Activated Sludge B.E. Zima-Kulisiewicz, W. Kowalczyk, A. Delgado . . . . . . . . . . . . . . . . . . . 145

Part IV Numerical Modeling Coupling Atomistic and Continuum Descriptions Using Dynamic Control E.M. Kotsalis, J.H. Walther, P. Koumoutsakos . . . . . . . . . . . . . . . . . . . . . . 157 Lattice Boltzmann Simulation of Pulsed Jet in T-Shaped Micromixer Md Ashraf Ali, Lyazid Djenidi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 Simulation of High-Speed Flow in μ-Rockets for Space Propulsion Applications Jos´e A. Mor´i˜ nigo, Jos´e Hermida-Quesada . . . . . . . . . . . . . . . . . . . . . . . . . . . 175 Numerical Study on the Flow Physics of a T-Shaped Micro Mixer J. Hussong, R. Lindken, M. Pourquie, J. Westerweel . . . . . . . . . . . . . . . . . 191 Splitting for Highly Dissipative Smoothed Particle Dynamics S. Litvinov, X.Y. Hu, N.A. Adams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207

List of Contributors

N.A. Adams Lehrstuhl f¨ ur Aerodynamik, Technische Universit¨at M¨ unchen 85748 Garching, Germany E.A.T. van den Akker Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Md Ashraf Ali Discipline of Mechanical Engineering, The University of Newcastle, 2308 NSW, Australia Oliver B¨ aumchen Department of Experimental Physics, Saarland University, D-66123 Saarbruecken, Germany A. Delgado Lehrstuhl f¨ ur Str¨ omungsmechanik, Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg, Cauerstr. 4, D-91058 Erlangen, Germany Lyazid Djenidi Discipline of Mechanical Engineering, The University of Newcastle, 2308 NSW, Australia

Marco Ellero Lehrstuhl f¨ ur Aerodynamik, Technische Universit¨at M¨ unchen, Boltzmannstr. 15, 85748 Garching, Germany Departamento de F´ısica Fundamental, UNED, Apartado 60141, 28080 Madrid, Spain. Pep Espa˜ nol Departamento de F´ısica Fundamental, UNED, Apartado 60141, 28080 Madrid, Spain Rafael Rodrigo Fern´ andez E.T.S.Ingenieros Navales, Universidad Politecnica de Madrid. Avda. del Arco de la Victoria s/n, 28040 Madrid, Spain Renate Fetzer Department of Experimental Physics, Saarland University, D-66123 Saarbruecken, Germany Ian Wark Research Institute, University of South Australia, Mawson Lakes SA 5095, Australia

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List of Contributors

A.J.H. Frijns Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Takeshi Furukawa Aerodynamic Design Technology Section Supersonic Transport Team, Aviation Program Group JAXA (Japan Aerospace Exploration Agency) 6-13 Osawa, Mitaka, Tokyo, Japan Jongyoon Han Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139 Jos´ e Hermida-Quesada Dept. Aerodynamics & Propulsion, National Institute for Aerospace Technology Madrid, Spain X.Y. Hu Lehrstuhl f¨ ur Aerodynamik, Technische Universit¨at M¨ unchen 85748 Garching, Germany J. Hussong Laboratory for Aero and Hydrodynamics Delft University of Technology Leeghwaterstraat 21 2628 CA Delft, The Netherlands Karin Jacobs Department of Experimental Physics, Saarland University, D-66123 Saarbruecken, Germany

Theodoros Karakasidis Laboratory of Hydromechanics and Environmental Engineering Department of Civil Engineering, School of Engineering, University of Thessaly Pedion Areos 38834 Volos, Greece Sung Jae Kim Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139 E.M. Kotsalis Computational Science and Engineering Laboratory, ETH Zurich, CH-8092, Switzerland P. Koumoutsakos Computational Science and Engineering Laboratory, ETH Zurich, CH-8092, Switzerland W. Kowalczyk Lehrstuhl f¨ ur Mechanik und Robotik, Universit¨ at Duisburg-Essen, Lotharstr. 1, D-47057 Duisburg, Germany Antonios Liakopoulos Laboratory of Hydromechanics and Environmental Engineering Department of Civil Engineering, School of Engineering, University of Thessaly Pedion Areos 38834 Volos, Greece R. Lindken Laboratory for Aero and Hydrodynamics Delft University of Technology Leeghwaterstraat 21 2628 CA Delft, The Netherlands

List of Contributors

S. Litvinov Lehrstuhl f¨ ur Aerodynamik, Technische Universit¨at M¨ unchen 85748 Garching, Germany Jos´ e A. Mor´i˜ nigo Dept. of Space Programmes, National Institute for Aerospace Technology Madrid, Spain [email protected] Andreas M¨ unch School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK S.V. Nedea Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands M. Pourquie Laboratory for Aero and Hydrodynamics Delft University of Technology Leeghwaterstraat 21 2628 CA Delft, The Netherlands Nathaniel D. Robinson Link¨oping University, Dept. of Physics, Chemistry and Biology, 581 83 Link¨oping, Sweden Juan Miguel S´ anchez S´ anchez E.T.S.Ingenieros Navales, Universidad Politecnica de Madrid. Avda. del Arco de la Victoria s/n, 28040 Madrid, Spain Filippos Sofos Laboratory of Hydromechanics and Environmental Engineering Department of Civil Engineering, School of Engineering, University of Thessaly Pedion Areos 38834 Volos, Greece

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A.A. van Steenhoven Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands Dimitris Valougeorgis Department of Mechanical and Industrial Engineering, University of Thessaly, Pedion Areos, Volos, 38334, Greece Stelios Varoutis Department of Mechanical and Industrial Engineering, University of Thessaly, Pedion Areos, Volos, 38334, Greece Barbara Wagner Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstrasse 19, D-10117 Berlin, Germany J.H. Walther Computational Science and Engineering Laboratory, ETH Zurich, CH-8092, Switzerland Dept. of Mech. Engng., Technical University of Denmark, DK-2800 Lyngby, Denmark J. Westerweel Laboratory for Aero and Hydrodynamics Delft University of Technology Leeghwaterstraat 21 2628 CA Delft, The Netherlands B.E. Zima-Kulisiewicz Lehrstuhl f¨ ur Str¨ omungsmechanik, Friedrich-Alexander-Universit¨ at Erlangen-N¨ urnberg, Cauerstr. 4, D-91058 Erlangen, Germany

Part I

Plenary Lectures

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Nonlinear Electrokinetic Flow: Theory, Experiment, and Potential Applications Sung Jae Kim, Jongyoon Han Massachusetts Institute of Technology, 77 Massachusetts Avenue, Cambridge, MA 02139, USA. [email protected], [email protected]

Summary. In present paper the theory and the experimental results on nonlinear electrokinetic ﬂow near nanochannels and near perm-selective membranes are discussed. It was found that the characteristics of non-linear, non-equilibrium electrokinetic are diﬀerent from classical electrokinetics theory. Concentration polarization (CP) method with the ionic concentration gradients in the bulk phase are presented. Summary of the more recent experimental/theoretical advances in the study of nonlinear electrokinetic ﬂows as well as the future outlook and implications on both science and engineering is given.

1 Introduction Recently, electrokinetic ﬂows have been drawn critical attentions with the advances in biological/chemical microﬂuidic devices [1] as an eﬃcient method of manipulating ﬂuids [2]. While microﬂuidic systems can utilize electric ﬁelds, pressure gradient or electro-capillarity on discrete droplets [3] as their ﬂuiddriving mechanisms, the electric ﬁeld driven operations have been preferred in microﬂuidic system by several advantages such as the elimination of external mechanical parts (valveless ﬂuid control [4]) and low shear stress on biological species. Traditional electrokinetic phenomena often used in microﬂuidic systems are categorized by ﬂuid motions (electroosmosis) [5, 6] and particle motions (electrophoresis [7, 8] and dielectrophoresis [9, 10]). Among them, the applicability of electroosmotic ﬂow (EOF) has been extensively studied since the very early days of microﬂuidics engineering. The EOF is generated by the ﬁeld-driven movement of accumulated ions at the interface of solution and stationary charged wall, which drags the ﬂuid column in micro-scale channels. While its plug-type ﬂow proﬁle can minimize a deleterious dispersion along the microchannel and shear stress by the parabolic pressure-driven ﬂow, poor ﬂuid mixings [11] and low ﬂow rates [12] caused numerous challenges for its general application in microﬂuidic systems. The velocity of EOF, U, can be characterized by well-known Smoluchowski equation, |U| = −ζ|E|/μ, where

M. Ellero et al. (eds.), IUTAM Symposium on Advances in Micro- and Nanoﬂuidics, IUTAM Bookseries 15, c Springer Science + Business Media B.V. 2009

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ζ is the zeta potential of charged wall and and μ are the permittivity and viscosity of buﬀer, respectively. The zeta potential ς represents the amount of charges in the charge double layer that can be mobilized under the external ﬁeld. Under a deﬁned ion concentration and a ﬁxed surface charge density, ζ is constant across the system. Therefore, the velocity is linearly proportional to the applied electric ﬁeld, unless there is a non-uniformity of surface charges or complex geometry [13–15]. This picture, however, readily breaks down when there is either buﬀer concentration gradient, or surface charge density gradient, or both, in the microﬂuidic system. Then, the value of ζ, which determines the electrokinetic response of the system, can have spatial variation over the length of a microﬂuidic channel or a particle under a ﬁeld. For example, diﬀusiophoresis [16, 17] and diﬀusioosmosis [18, 19], both occur in solutions with ion concentration gradient, have been known among colloid researchers. These are the particle motion (diﬀusiophoresis) and ﬂuid motion (diﬀusioosmosis) caused solely by solute concentration gradient of the solution used. These secondary electrokinetic phenomena are both complex and rich scientiﬁcally, generating interesting engineering ideas on the chemical-driven self-transporter particles [20]. Interests in these secondary electrokinetic phenomena were signiﬁcantly renewed recently, with the development of novel microﬂuidic systems that can actively modify the zeta potential proﬁles, in order to get greater electrokinetic response. Induced-charge electroosmosis [21, 22] or AC electroosmosis [23] is a nonlinear electrokinetic motion around a polarizable surface under weak dc or ac electric ﬁeld. The electrical double layer (EDL) near polarized surface (typically metal electrodes) can charge up (act as a capacitor) with suddenly applied ﬁeld, which will lead to stronger electrokinetic response and faster EOF. In such a case, the eﬀective zeta potential becomes proportional to the applied electric ﬁeld (ζ ∼ |E|), which makes the (induced) EOF quadratic to the ﬁeld (∼|E|2 ). Therefore, one can generate much higher EOF than that possible with normal EOF. The induced EOF can be used as an enhancing mechanism for microﬂuidic mixing and pumping [24] and implantable selfpowered microﬂuidic devices [25]. Yet another form of nonlinear electrokinetics that has drawn signiﬁcant attention is the non-linear EOF near permselective membrane. Operation of r inevitably causes charge acion permselective membranes such as Naﬁon cumulation, in a similar manner as the electrode, and accumulated charges can induce signiﬁcant changes in local ion concentration (concentration polarization (CP)) and signiﬁcant enhancement in electrokinetic ﬂow nearby. Recently, it was ﬁrmly determined that a regular nanoﬂuidic channels and pores, with the critical dimension of ∼10 nm, can exhibit signiﬁcant permselectivity [26–31], due to the EDL overlap inside the nanochannel. As a result, the nanochannel junctions can generate the same CP and nonlinear electrokinetic ﬂow [32] just as in the system containing perm-selective membranes like r. Naﬁon

Nonlinear Electrokinetic Flow

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In this review, we will focus on the theory and the recent experimental results on nonlinear electrokinetic ﬂow, both near nanochannels and near permselective membranes. We will establish the characteristics of such non-linear, non-equilibrium electrokinetic ﬂow, which are quite distinct from aforementioned classical electrokinetics theory. We will begin by presenting the CP which creates the ionic concentration gradients in the bulk phase and its relation between ion depletion/enrichment eﬀects, followed by the discussion of limiting/over-limiting current behavior. Then, we will summarize the more recent experimental/theoretical advances in the study of nonlinear electrokinetic ﬂows, with the future outlook and implications on both science and engineering.

2 Concentration Polarization (CP) and Over-Limiting Current The thickness of EDL under uniform surface charge distributions and electrolyte concentrations of physiological buﬀers, is in the order of a few nanometers [33]. Empty ﬂuidic channels can exhibit perm-selectivity when the channel dimension becomes comparable with the EDL thickness. Such regular nanoﬂuidic channels can be created either on silicon or glass substrates by microfabrication techniques [34, 35]. Moreover, membrane junctions made out of r [36, 37] or nanometer-sized ﬂuid gaps [38–41] can be recently fabriNaﬁon cated in a PDMS microchannel, as well as nanoporous hydrogels [42], packed nanoscale beads [43–47], and nanocapillary arrays [48], all to serve as ion perm-selective junctions. In these nanoﬂuidic structures, EDL can be ‘overlapped’ under moderate electrolyte concentrations, where only counter-ion species of surface charges can preferentially pass through the membrane. As illustrated in Fig. 1(a), the perm-selective junction (usually with negatively charged surface) can be conceptually considered as a cation-selective structure, with zero permeability to co-ionic species (anions). The ﬂuxes of each ion, N± , are given by ± ± ± N± = N± diﬀ + Ndrift = −D± c − c μ± E.

Here D± denotes the diﬀusivity, c± is the ionic concentration, and μ± is the ion mobility for each ion. E is the applied electrical ﬁeld. Under dc bias, cations can enter and pass through the nanostructure on the anodic side, while anions would be hindered from the permselective membrane junction. Thus the ﬂux of cation can be transported without any disturbance through the entire system, while maintaining net zero anion ﬂux. In such a case, in order to satisfy electro-neutrality, the concentrations of both cations and anions have to decrease in anodic side to form an ion depletion zone, while the concentrations increase at the cathodic side to form an ion enrichment zone. This is the mechanism of CP which can often be found in the charge transfer

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Fig. 1. (a) Schematic diagram of ion concentration distribution front and back of perm-selective material which only let cations pass through. Ions in anodic side were depleted while they were enriched in cathodic side. N+ and N− are the ﬂuxn of cation and anion, respectively, and subscript diﬀ and drift represent the driving forces of the ion transport, diﬀusion and electrical ﬁeld, respectively. (b) Current sweep plot showing the limiting current and over-liminting current pattern through perm-selective nanochannels. From [32], Copyright (2007) by the American Physical Society.

process across ion exchange/electrodialysis membranes. Pu et al. ﬁrst demonstrated the ion enrichment/depletion phenomenon associated with eight glass nanochannels experimentally, as shown in Fig. 2(a) [49].

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Fig. 2. Experimental observation of concentration polarization (depletion) near nanochannels. (a) Schematic diagram of two microchannels connected by nanochannels fabricated in boroﬂoat glass wafers. (b) Eﬀect of buﬀer concentration (thickness of EDL) on ion enrichment/depletion behaviors; A: 200 μM, B: 500 μM, C: 1 mM and D: 3 mM of sodium tetraborate buﬀer concentrations. Any voltage was applied (i) and after a voltage of 1000 V was applied for (ii) 5 s, (iii) 10 s, (iv) 20 s. They showed that the degree of ion enrichment/depletion is directly related to the extent of EDL overlap. From [49], Copyright (2004) by the American Chemical Society.

While the classical theory for CP [33] qualitatively agrees with the experimental behavior shown in Fig. 2, there are several notable discrepancies. For example, in the classical CP theory, a ﬁxed concentration boundary condition is deﬁned, at a ﬁxed distance from the membrane. This distance (often called as diﬀusion length) is typically considered as the distance scale relevant to the bulk convective mixing. In other words, after the diﬀusion length from the membrane (or nanochannel), concentration gradients generated by CP

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will be completely eliminated due to convective mixing in the bulk solution. Within the diﬀusion length, however, convective mixing is regarded small and generally ignored, which leads to linear concentration gradient. However, in Fig. 2 for example, it is not clear at all the concentration proﬁle is linear, nor it seems that the length scale of the depletion zone (approximate diﬀusion length) is constant. Indeed, typical experimental behavior observed in CP near nanochannels shows quite dynamic behaviors, often the apparent depletion zone extending quite a distance (a few millimeters, often all the way back to the reservoirs) depending on the electric ﬁeld used and other conditions. This is, in part, due to the fact that in microﬂuidic channels, ﬂuid convective mixing is signiﬁcantly suppressed. Still, it is quite clear that a direct application of classical CP theory to the experimental situations such as Fig. 2 would be inadequate. Another point to be noted in the CP is the complexity caused by concentration changes in the system. The current (cation ﬂux) through the membrane, according to the classical CP theory, will saturate at the point when the ion concentration near the anodic side of the membrane becomes zero. This is often called as the limiting current behavior, [33, 50] as shown Fig. 1(b). This will make the current-voltage behavior of the membrane junction signiﬁcantly deviate from that of Ohm’s law. At suﬃciently low electric potential, the current-voltage behavior still follows Ohm’s law, since the CP-generated concentration gradient is not signiﬁcant. When the ionic concentration approaches nearly zero at the anodic side of nanostructure by increasing the electric potential applied, the system reaches a limiting current regime, when the anodic side of the membrane is almost completely depleted of ions (near zero ion concentration). Such a strong CP near limiting current regime induces significant changes in the local conductivity and electric ﬁeld distribution. Lower ionic strength within the depletion zone means that local ion conductivity will be decreased signiﬁcantly, which can in turn render the local electric ﬁeld signiﬁcantly higher than the value predicted by Ohm’s law. This has been conﬁrmed via one-dimensional approximate simulation and experimental data by Tallarek and his coworker, which clearly supported the idea that the local electric ﬁeld inside the ion depletion zone can be much greater than one in bulk phase [42]. Therefore, one cannot simply use the Ohm’s law to predict the ﬁeld and potential distribution within the system in this limit, and the zeta potential ζ, the ion conductivity σ, and the local electric ﬁeld E of the system will all be coupled to the concentration proﬁles of the system, which is also dynamically changing.

3 Nonlinear Electrokinetic Flow Near Perm-Selective Nanochannels According to the classical CP theory, at the limiting current of the system, the anionic side of the membrane (nanochannels) is almost depleted of ions,

Nonlinear Electrokinetic Flow

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therefore, any further increase in applied potential would not induce further increase in ion current through the system. However, in reality, signiﬁcant overlimiting current can be measured in most perm-selective membranes at higher electric potential, both in macroscopic permselective membranes [51, 52] and in regular nanochannels [32]. The source and mechanism of this over-limiting current has long been a subject of debate in membrane science for the last several decades. Many researchers, including Strathmann et al. theoretically suggested a model which combines the calculation of the depletion zone thickness and the inﬂuence of the electric ﬁeld and the results indicated that the over-limiting current behavior was associated with water dissociation at the vicinity of the membrane [53] and the current was carried by protons and hydroxyl ions [54]. But, further investigation revealed that the contribution of water dissociation was very low, and more than 97% of the current was carried by salt ions [55]. In the same work, it was shown that the membrane perm-selectivity remains constant in the over-limiting current region, which showed it is virtually all carried by the salt ions. Rubinstein and his-coworkers theoretically suggested that there is a strong convective mixing created by an ampliﬁed electrokinetic response of ﬂuid layer right next to the membrane [51, 56, 57]. The ampliﬁcation of electrokinetic responses can be induced because of signiﬁcantly lowered ion concentration inside the ion depletion zone, therefore higher local zeta potential. The hydrodynamic and electrokinetic models and the mass and charge coupled transport models are often adapted to elucidate the nonlinear electrokinetic ﬂow near permselective membranes [51, 56, 58–61]. The complete set of governing equations associated with convective electrodiﬀusion of ions in incompressible Newtonian electrolyte solutions and conﬁned in micro/nanochannel is shown below: Fluid motions: 0 = −∇p + ∇2 v + ρe E, ∇ · v = 0. Electric potential: E = −∇φ, ρe = ∇2 φ. Ionic transport: ∂c± /∂t = D± ∇ · (∇c± ± c± ∇φ) − Pe(v · ∇)c± . Here, p and is the external pressure and permittivity of solution and Pe is the Peclet number which represented the ratio of mass transported by convection to mass transported by diﬀusion: |v|L/D± where L is the characteristic length of the system. Though Pe is usually quite large in most engineering applications (convection-dominant system), it is often small (diﬀusion-dominant

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Sung Jae Kim, Jongyoon Han

system) or co-exist in the system of short length scale such as microﬂuidic applications and, thus, the system response can be largely diﬀerent depending on Pe. At diﬀusion-dominant regime (low Pe), the over-limiting current is suppressed because the diﬀusive ﬂux, which drive Ohm’s response, is much higher than the limiting diﬀusion ﬂux [62]. Thus, the over-limiting behavior can clearly show that the system is located in high Pe regime, i.e. convection dominant regime. Rubinstein derived the two-dimensional non-equilibrium electroosmotic slip boundary condition through a ﬂat ion exchange membrane along with the linear stability as following: ∂2c

1 ∂x∂y v|s = − V 2 ∂c , 8 ∂y where x and y is the axis parallel and perpendicular to the ion-exchange membrane, respectively. Detailed derivation can be found in literature [63]. This boundary condition clearly imposed the strong and vortex-like circulation ﬂow ﬁeld near the membrane as shown in Fig. 3 [59]. Due to the electric ﬁeld acting in the EDL, electroosmotic slip can induce electro-convections and it mainly caused vigorous mixings in diﬀusion layer and an electroosmotic (or electroconvective) instability. Such strong ﬂow patterns were not usually found in irrotational electrokinetic ﬂow within conﬁned micro/nanostructure with constant surface charge. The strong, non-linear EOF was initially observed in the experiments using ion-selective granules [64, 65]. Further investigation of electroosmotic slip and instability predicted that the electroosmotic velocity is proportional to a cube of the applied voltage at moderate strength, while the square relationship is still valid in large applied voltage [60]. Recently, Jin et al. conducted two-dimensional numerical simulations in micro-nanoﬂuidic interconnect structures [66]. They also can observe an induced EOF of the second kind and complex circulation ﬂow in the micro/nanochannel junction regions for both positive and negative bias potentials. Tallarek and his coworkers experimentally investigated nonequilibrium electrokinetic eﬀects in porous media by ﬂuorescent tracer molecules. The thickness of nonequilibrium EDL strongly depended on the applied electric ﬁeld strength and space charge region was located in the bulk phase, not close to the solid-liquid interface [44]. They observed highly chaotic ﬂow patterns around porous particles due to the complex interactions of the space charge region and the bulk solution. For further investigation, they closely looked up induced charge EOF around the ion-selective glass bead using confocal laser scanning microscopy as shown in Fig. 4(a) [67]. Electrohydrodynamic instability was observed as turbulent ﬂows near the particles. Numerical simulations were performed for the spatial variations of chemical and electrical potential gradients in porous media to present the instability as shown in Fig. 4(c) [68]. Later on, Kim et al. published the ﬁrst experimental, microscopic study on nonlinear electrokinetic ﬂow generated near nano-fabricated channels [32]. They visualized both the concentration proﬁles and electrokinetic ﬂow pat-

Nonlinear Electrokinetic Flow

11

Fig. 3. Nonlinear electrokinetic ﬂows near perm-selective membrane, calculated by solving two-dimensional Navier-Stokes, Poisson-Boltzmann and Nernst-Planck coupled equations using an iterative ﬁnite diﬀerence method. Time revolving snapshots when D− /D+ = 0.1 of (a) streamline and (b) velocity ﬁelds at (1) t = 0.0001, (2) t = 0.2 and (3) t = 0.6. The ion-selective membrane is located at x = 0. From [59], Copyright (2005) by the American Physical Society.

Sung Jae Kim, Jongyoon Han 12

Fig. 4. The actual concentration polarization pattern and induced space charge eﬀects in a dense packing of particles. (a) Snapshots of the concentration distribution. (b) Temporal variation of concentration in a particular voxel denoted by the cross. With a high electric ﬁeld, the ion enrichment zone was destroyed by locally chaotic electrokinetic ﬂow. From [67], Copyright (2005) by the American Chemical Society. (c) Distribution of the axial and radial ﬂow velocities near the biporous particle. From [68], Copyright (2005) by the American Chemical Society.

Nonlinear Electrokinetic Flow

13

Fig. 5. (a) The linear velocity and angular velocity of the vortex as a function of applied voltage. The speed was 10 times or even higher than the equilibrium EOFs. The data were ﬁtted by both the 2nd and 3rd polynomials to reveal the proportionality to E. (b) Fast vortices at steady state in single-side nanochannel device and (c) since the ions were depleted through both walls, the four independent vortices were formed in the four divided regions in dial-side nanochannel device. (d) Suppressed vortices in shallow dual-side nanochannel device. The size of the vortex was about 2 μm, which was similar value of the microchannel depth. (e) The time required for the depletion boundary to reach opposite microchannel wall as functions of applied voltage at anodic side and buﬀer concentrations. The values in the box were the equilibrium EDL thickness. From [32], Copyright (2007) by the American Physical Society.

14

Sung Jae Kim, Jongyoon Han

terns inside and outside the ion-depletion region by tracking the ﬂuorescent nanoparticles and dye molecules in situ. As shown Fig. 5(a), the speed of circulating ﬂow was estimated to be over 1000 μm/s, which is at least 10 times higher than that of equilibrium EOF expressed by the Smoluchowski formula under the same electrical potential distribution. At the steady state, the counterrotating vortices beside the nanochannel were clearly observed shown in Fig. 5(b) and Fig. 5(c) as predicted by Rubinstein and his coworker [59]. One can independently ‘suppress’ the convective part of the phenomena by decreasing the microchannel thickness. The sizes of the vortices in shallow dual-side nanochannel shown in Fig. 5(d) were approximately 2 μm, which corresponded to the depth of the microchannels used. With those suppressed vortices, liquid mixing inside the ion depletion zone was minimized and thus the concentration gradient, which is the driving force of the perm-selective ion transport, was maintained more stable way. This result was also in line with experiments where agarose gel coating on a permselective membrane was shown to suppress the over-limiting current by eliminating electroconvections [69]. It was also demonstrated that the CP (ion depletion) is a positive feedback process as shown in Fig. 5(e): a weak and localized CP can be induced at higher buﬀer concentrations and this in turn leads to locally decrease ion concentration (localized ion depletion) which would increased perm selectivity of nanochannel. This is signiﬁcant because the (qualitatively) same, strong CP was achieved even at the ionic strength conditions that are not considered to be ‘overlapping’ the EDLs of channel walls in the nanochannel. The main reason for this is because the local double layer thickness (within the depletion zone) is rather independent of equilibrium, bulk double layer thickness, once the CP is induced by the perm-selective nanochannels. In addition, Kim et al. experimentally demonstrated the ampliﬁed electrokinetic response using particle tracking method in two-dimensional micro/nanochannel hybrid structure [70]. They observed that, once the particles passed the depletion zone and entered the downstream low concentration zone, the particles can travel at 25 times faster than in higher concentration zones. Compared to this, the mass conservation is required for the ﬂuid motions, thus the electrokinetic ﬂow ampliﬁcation through entire system can be less than 25 times because the high ﬂow velocity in lower concentration zone was retarded by the low ﬂow velocity in higher concentration zone.

4 Outlook The renewed interests in this non-linear electrokinetic phenomenon largely come from the potential of developing novel micro/nanoﬂuidic systems based on these interesting and rich phenomena. For example, Wang et al. developed biomolecule concentrators [71] based on CP caused by nanochannels, while Bazant and coworkers, as well as Ramos and coworkers, developed eﬃcient

Nonlinear Electrokinetic Flow

15

microﬂuidic pumping systems based on induced electroosmosis [21, 27]. It is also expected that the strong non-linear EOF, enabled by the nanochannel junctions, could turn out to be useful for various applications, such as ﬂuid mixing [72] and pumping [73]. Liu [74] and Nguyen [75] have tried to make energy harvesting device based on streaming current generation and CP was applied in the particulated ﬁxed beds to capillary electrochromatography [76], a continuous ﬂow microﬂuidic demixing processes utilizing electrically ﬂoating metal electrodes [77]. In addition, the idea of using nanochannels as solid-state perm-selective membrane materials is gaining momentums [78]. It is obvious that sound understanding some of these nonlinear electrokinetic phenomena would be essential in realizing these engineering potentials of advanced nanoﬂuidic systems. This article summarized the recent theoretical and experimental advances in studying and analyzing these phenomena. Still the challenges remain. From the theoretical point of view, dynamic coupling of local concentration/zeta potential/conductivity/electric ﬁeld renders the problem at hand quite challenging mathematically, since one has to solve the multi-dimensional coupled equations of Navier-Stokes, Poisson-Boltzmann and Nernst-Planck equations [79]. In addition, experimental characterization of the system needs to be done in a multi-modal way (measuring both concentrations and ﬂow development, for example) in order to provide any meaningful physical picture on the phenomenon at hand. MEMS-fabricated nanoﬂuidic systems, with a full optical, ﬂuidic and electrical access to the critical micro-nano junction, will be especially useful in understanding this complex and rich problem.

Acknowledgement The authors are thankful for the support from NIH (CA119402, EB005743) and NSF CAREER award (CTS-0347348), which enabled this review article.

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Fluid Particle Models for the Simulation of Microﬂuids Marco Ellero1,2 1

2

Lehrstuhl f¨ ur Aerodynamik, Technische Universit¨ at M¨ unchen, Boltzmannstr. 15, 85748 Garching, Germany. [email protected] Departamento de F´ısica Fundamental, UNED, Apartado 60141, 28080 Madrid, Spain.

Summary. In the present work, some of the mesoscopic particle methods generally used for the simulation of microstructured ﬂuids are reviewed. In particular, the Dissipative Particle Dynamics (DPD) method, designed by Hoogerbrugge and Koelman in 1992, oﬀers a good compromise of performance and ﬂexibility. Some aspects of the method are discussed as well as the main conceptual shortcomings which limit its current applicability to some micro-ﬂow conditions. Reﬁned models of DPD are therefore presented, i.e. Smoothed Dissipative Particle Dynamics (SDPD) (Espa˜ nol and Revenga, Phys. Rev. E 67:026705, 2003). The method is a thermodynamically consistent version of DPD and, at the same time represents a direct discretization of the continuous Navier-Stokes equations on a Lagrangian framework. This feature is common to another macroscopic particle method, i.e. Smoothed Particle Hydrodynamics (SPH) (Gingold and Monaghan, Mon. Not. R. Astron. Soc. 181:375, 1977). SDPD can be therefore understood as a mesoscopic version of SPH with thermal ﬂuctuations consistently included and provides the unifying multiscale framework linking DPD to SPH. Finally, applications of the model to microﬂuids are discussed. In particular, results for polymer molecules and colloidal particles suspended in Newtonian solvent are presented.

1 Introduction The increasing technological interest in the design of micro-scale ﬂow devices characterized by components of size smaller than 1 millimeter (microﬂuidics) or 1 micron (nanoﬂuidics) is providing strong stimuli in the understanding of the hydrodynamic processes occurring at the micro-scales [1]. Typical target areas in microﬂuidics include control ﬂow devices for cooling of electronic systems, multiphase ﬂows in lab-on-a-chip, combustors, micro-mixing devices as well as fabrication processes involving individual organic and/or inorganic constituents [2]. Among the latter ones, particular focus has been recently given to the development of sensors for the detection of biological cells, manipulation of single DNA molecules or other micron-sized components. It is

M. Ellero et al. (eds.), IUTAM Symposium on Advances in Micro- and Nanoﬂuidics, IUTAM Bookseries 15, c Springer Science + Business Media B.V. 2009

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therefore evident how the ability to improve the numerical models at this spatio-temporal level and for these class of complex microstructured ﬂuids would be of great beneﬁt. It is worth to notice that the Navier-Stokes equations describing the dynamics of a Newtonian liquid at the macroscopic level still remain valid at the microﬂuidics scales, therefore providing a natural framework based on the continuum description. On the other hand, it is also clear that, whenever the physical dimensions of the considered objects (i.e. polymer macromolecules, colloidal particles etc.) are in the sub-micrometer range, the surrounding ﬂuid starts to feel the presence of its underlying molecular structure and hydrodynamic variables will be inﬂuenced by thermodynamics ﬂuctuations according to the Landau and Lifshitz theory [3]. Standard macroscopic approaches, based for example on ﬁnite volumes or ﬁnite elements methods, are not suitable for this type of simulations. They neglect thermal ﬂuctuations which, as mentioned above, are the most crucial ingredient of the mesoscopic dynamics. On the other hand, direct microscopic approaches, such as molecular dynamics, are able to resolve the smallest details of the molecular structures but they are computationally very expensive and are limited by the available computer resources. Nowadays, these approaches are restricted to computational domains of length of the order of nanometers which represent only the smallest scales (dimensions of one nanostructure) in the range covered by the considered system. Dissipative Particle Dynamics (DPD) as originally invented by Hoogerbrugge and Koelman, is perhaps the most popular particle model for the simulation of Newtonian ﬂuids at mesoscopic scales [1, 2]. In DPD, a Newtonian ﬂuid is represented by a collection of points with prescribed stochastic interactions that conserve momentum and produce hydrodynamic behaviour at a coarse-graining level. Moreover, DPD includes thermal ﬂuctuations in a thermodynamically consistent way [2] and it is thus applicable to mesoscopic scales where diﬀusive processes are important. Since its development, the method has been applied to a wide class of problems and is now emerging as a powerful numerical technique for simulations in the area of micro/nano science. Despite its recent success, DPD suﬀers from a number of conceptual shortcomings which can limit its applicability and physical understanding. In particular, they are related to the following issues: (i) the resulting equation of state turns out to be quadratic in density; (ii) no direct connection between the model parameters and the transport coeﬃcients of the simulated ﬂuid (kinetic theory or preliminary runs are necessary to measure the transport coeﬃcients); (iii) unclear deﬁnition of the particle size. The latter point represents a big drawback preventing an “a priori” control of the spatio-temporal scales simulated. Indeed, due to the lack of a speciﬁc physical size associated to the particles, DPD is unable to characterize in a unambiguous way the external lengths of the problem under study. This is crucial, for instance, in the case of suspended colloidal particles or in mi-

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croﬂuidics applications where the physical dimensions of the external objects determine whether thermal ﬂuctuations come into play. In this work, we discuss in detail a modiﬁed DPD formalism, i.e. Smoothed Dissipative Particle Dynamics (SDPD), recently proposed by Espa˜ nol [9]. The new method is able to solve the problems mentioned above and, in addition, it helps to bridge the gap with another particle approach operating at the macroscopic level: Smoothed Particle Hydrodynamics (SPH) which represents a Lagrangian mesh-less discretization of partial diﬀerential equations [8]. This article is composed as follows. Section 2 introduces the DPD governing equations; we review the reﬁned particle SDPD model in Sect. 3. Section 4 is devoted to the modelling of complex microstructured ﬂuids, in particular the SDPD modelling of a colloidal particle and a polymer molecules suspended in a Newtonian ﬂuid will be discussed. Finally, in Sect. 5 we validate the model by performing simulations in both, Brownian and non-Brownian environments.

2 Dissipative Particle Dynamics Before discussing the problems of the methodology, let us resume brieﬂy the original equations of motions for the DPD particles [4, 5]. If we assume to have N particles of mass m distributed over a physical domain V , each of them follows the Newton’s equations of motion r˙ i = vi and mv˙ i = Fi for i = 1, . . . , N . Here ri and vi represent respectively position and velocity of particle i. Fi represents the net force acting on particle i and it is evaluated as the sum of conservative, dissipative and stochastic interparticle components −1/2 D R + F , where Δt is the time step and as follows: Fi = j FC ij ij + Fij Δt C FC ij = F (rij )eij ,

(1)

D FD ij = −γω (rij )(vij · eij )eij

(2)

R FR ij = σω (rij )ξij eij

(3)

and being rij = |ri − rj | the relative distance, vij = vi − vj the relative velocity and eij = rij /rij the unit vector joining particles i and j. As usual, ξij represent symmetric Gaussian random variables with zero mean and unit variance. In (2), γ is a friction parameter which is connected to the kinematic viscosity of the coarse-grained ﬂuid. Validity of the ﬂuctuation-dissipation theorem requires σ and γ to be linked by the relation σ 2 = 2γkB T , being kB the Boltzmann factor and T the system temperature [5]. In addition, thermodynamic consistency requires also that ω D (rij ) = [ω R (rij )]2 . Finally, for the conservative part of the force a Mexican function is usually adopted F C (rij ) = aij max{1 − (rij /rc ), 0}, where aij is a particle interaction constant and rc is a cutoﬀ radius.

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One of the main problems of DPD is not only related to the absence of a model parameter which determines a physical measure of the particle size but, more importantly, to the lack of connection of this particle size with the magnitude of thermal ﬂuctuations in (3), i.e. the speciﬁcation of σ. Indeed, although the choice of γ remains uniquely speciﬁed through the ﬂuctuationdissipation relation, σ remains arbitrary and it is usually speciﬁed in terms of the algorithm’s performance [6]. This represents a serious drawback of the technique because the level of ﬂuctuations which aﬀects a ﬂuid particle on a given scale is dictated by numerical, rather than physical motivations. It would be therefore highly desirable to develop a formalism where the particle’s thermal energy is uniquely linked to physical parameters, i.e. particle size. In order to correct these drawbacks, a new formulation of DPD (denoted as Smoothed Dissipative Particle Dynamics: SDPD) has been recently introduced [9]. In the following section it will be discussed and its advantages over the classical DPD highlighted.

3 Smoothed Dissipative Particle Dynamics The new mesoscopic model makes use of an additional extra variable for every ﬂuid particle, c.f. a thermodynamic volume, which enters directly in the deﬁnition of the noise, characterizing the physical particle length and, at the same time, prescribing their thermal energy uniquely. Furthermore, SDPD deepens the connection with another particle method known as Smoothed Particle Hydrodynamics (SPH) in the sense that interparticle forces will be now considered as numerical discretizations of suitable sets of continuum equations with prescribed transport coeﬃcients. SPH is a macroscopic particle method designed in the late seventies to study astrophysical ﬂow problems [7, 8]. The basic idea of SPH is to use an interpolant function to evaluate spatial derivatives of the ﬁeld at a given particle location. Therefore hydrodynamic equations, written in the form of an arbitrary set of partial diﬀerential equations, can be solved to the prescribed order of accuracy. For example, the gradients of a function f (r) deﬁned over a domain V are computed in SPH as ∇f (r) j φj fj ∇W (|r − rj |, h) where W (r, h) is a bell-shaped interpolant (i.e. Lucy function) with ﬁnite support h and normalized to unity, while φ j represents the volume associated to the particle j and is deﬁned as φj = ( k W (|rj − rk |, h))−1 . Note that in SPH, there is a direct deﬁnition of the particle size given in terms of its volume, that is l = φ1/3 . In a previous work [9] it has been shown how, by using the so-called GENERIC framework [10], it is possible to cast the original SPH model, in a form which encodes automatically the First and Second Laws of Thermodynamics. In particular, it allows to introduce thermal ﬂuctuations in a systematic way, which by construction satisﬁes the ﬂuctuation-dissipation theorem

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and where the ﬂuctuations are governed by the Einstein equilibrium distribution. According to [9], the new equations of motion for the ﬂuid particles are given below. Let us consider ﬁrst the deterministic part; they read r˙ i = vi , Pi Pj 5η ωij −ζ mv˙ i = + 2 ωij rij − vij 2 di dj 3 di dj j j η ωij eij eij · vij −5 ζ + 3 di dj j

(4)

where Pi is a pressure variable which can be related to the mass density via an arbitrary equations of state. In this work P (ρi ) = (c2s /2ρ0 )ρ2 is considered where cs is the input speed of sound. In (4) η and ζ are respectively the ﬂuid shear and bulk viscosity, and κ the thermal conductivity, all input parameters, while the geometrical factor ωij is given by ωij = −W (rij )/rij . Concerning the random terms, they can be introduced in the equations by postulating a tensorial generalization of the stochastic Wiener process which enters the momentum and internal energy equation in the following way: ij · eij , Aij dW md˜ vi = j

1 Ti dS˜i = − 2

ij : eij vij + Aij dW

j

(5) Cij dVij

j

ij = (1/2)[dWij + dWT ] is the symmetric part of dWij . In this where dW ij expression we have introduced, for each pair i, j of particles, a matrix of independent increments of the Wiener process dWij . In (5) we have also introduced an independent increment of the Wiener process for each pair of particles, dVij . For the amplitudes Aij , Cij of the noises we select the very speciﬁc form

40η Ti Tj ωij kB Aij = 3 Ti + Tj di dj

1/2

,

ωij Cij = κkB Ti Tj di dj

1/2 .

(6)

By postulating the noise terms in (5) with the amplitudes (6), the amount of energy produced is exactly the same as that dissipated in the deterministic part through viscous forces in (4). Some remarks are here in order: • Being the conservative forces between ﬂuid particles dependent on a additional pressure variable, arbitrary equations of state can be adopted which are not restricted to the quadratic form usually assumed in standard DPD. • Thanks to the analogy with SPH, the set of equations (4) represents a second-order accurate discretization of the Navier-Stokes (NS) equations in

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a Lagrangian framework. Analogously to standard mesh-based discretization schemes, “second-order” here means that the discrete equations (4) are consistent with the exact NS up to terms of order O(h2 ), being h the cutoﬀ radius of the kernel interpolant. This solves the problems with the derivation of the transport coeﬃcients which represent now input parameters of the simulation and remain constant, independently of the resolution considered. • The deﬁnition of a particle volume φ provides a typical length scale for the particle which is l = φ1/3 . Notice that in a macroscopic particle method this does not represent any physical quantities. For instance, in SPH particles are simple Lagrangian moving nodes on which the NS equations are discretized. By increasing the resolution, that is reducing the particle volumes, convergent results must be recovered if the method is consistent. However, due to the fact that di enters directly in the deﬁnition of the stochastic noise, particle volumes acquire in SDPD a physical meaning that is: the dynamics of the solvent is not longer scale-invariant. Indeed, it can be shown that the averaged particle velocity ﬂuctuations scale as Δvi = (kB T /ρ)φ−1/2 . The size of thermal ﬂuctuation is given by the typical length size of the ﬂuid particle scaled as the inverse of its square root, in accordance with usual concepts of equilibrium statistical mechanics. Therefore, for large enough ﬂuid particles, the thermal ﬂuctuations in the momentum and energy equation can be neglected. On the other hand, if we consider a submicron structure we will need to resolve the surrounding solvent liquid with ﬂuid particles of one order or more smaller than the typical size of the object, which will produce non-vanishing stochastic terms giving rise to its ultimate microscopic diﬀusional dynamics. Note also that, the numerical sizes of the ﬂuid particle are completely speciﬁed, being determined by the external lengths of the problem.

4 Microstructured Fluids Modelling 4.1 Colloidal Particle In order to model a colloidal particle, we select a certain number of SDPD solvent particles which lie within a spherical region and denote them as ‘boundary particles’. In the computation of the hydrodynamic forces, when a solvent particle interact with a boundary particle an artiﬁcial velocity is assigned to the boundary particle in order to have zero perpendicular and tangential velocity on the nominal boundary surface. This guarantees impermeability and no-slip boundary conditions for the ﬂuid at liquid-solid interface [11, 12]. In order to take into account the translational motion of the colloidal particle, the following procedure is considered: at every time step, the total force exerted by the ﬂuid on the solid particle is evaluated as

Fluid Particle Models for the Simulation of Microﬂuids

25

Fig. 1. Light-grey particles represent solvent particles, dark-grey particles represent either the boundary particles forming the colloid (left) or the monomers describing the polymer molecules (right).

F coll =

Fk

(7)

j∈Ω

where Ω is the domain represented by the colloidal particle (ﬁlled with SDPD boundary particles) and Fk represents the total force acting on the boundary particle k dueto the interactions with all its neighbouring solvent particles, that is: Fk = l∈V Fkl where V is the volume occupied by the ﬂuid. Once the total force on the colloid F coll is evaluated, we simply update its center-of-mass velocity and position according to a predictor-corrector scheme: this deﬁnes the nominal surface at the next time step which is needed to evaluate the artiﬁcial boundary velocities. Accordingly, all the boundary particles are translated as a rigid body. Rotational motion of the colloidal particle can be modelled in the same way, by evaluating a total torque and deﬁning an additional angular velocity for the colloid. A sketch of the model is shown in Fig. 1 (left). 4.2 Polymer Molecule A polymer molecule can be modelled as a linear chain composed of N monomers interacting by ﬁnitely extendable nonlinear elastic springs (FENE potential) r2 HR02 ln 1 − 2 , (8) U FENE (r) = − 2 R0

26

Marco Ellero

where H is the spring constant, r = tr(rij rij ) is the monomer-monomer distance and R0 represents the maximum extensibility. Monomers belonging to the polymer chain interact, besides the FENE forces (only adjacent ones), also by the usual hydrodynamic SDPD forces with the neighbouring ﬂuid particles and monomers. A typical conﬁguration is sketched in Fig. 1 (right). The physical basis of the model has been discussed in detail in [13].

5 Simulations 5.1 Colloidal Particle in Suspension In order to test the colloidal particle model, we ﬁrst consider the 2D case of a colloidal disk of radius R suspended in a deterministic Newtonian solvent (no thermal ﬂuctuations) deﬁned on a square domain of size L with periodic boundary conditions applied to every directions. The ﬂuid and the colloid are initially at rest. At time t = 0, the colloidal particle is perturbed by assigning it a constant velocity U (0) = U0 in the x-direction. We monitor the velocity of the colloidal particle V (t) as a function of time in the reference frame of the center of mass for the total system (colloid + ﬂuid), that is: V (t) = (U (t) − UCM )/(U0 − UCM ) where UCM is the center of mass velocity of the total system. The box domain is L = 2, while the colloidal radius is R = 0.1 giving a suﬃciently small concentration ratio φ = (πR2 )/L2 ≈ 0.0078: this enable us to neglect hydrodynamic interactions between images of the colloidal particle produced by the replicas of the main box. Speed of sound is cs = 1, initial particle velocity U0 = 0.01 and solvent kinematic viscosity is ν = 0.02. The dimensionless numbers characterizing this ﬂow problem are the Reynolds number Re = RU0 /ν = 0.05 and the Mach number Ma = U0 /cs = 0.01. Figure 2 shows a snapshot of the ﬂow ﬁeld around the colloidal particle suddenly after ﬂow start-up in the laboratory frame. The double vortex structures around the moving particle are well reproduced by the method. It has been shown that, due to the presence of hydrodynamic ﬂuid-particle interaction, the velocity decay is not exponential but, at suﬃciently long times a typical algebraical decay ∝ t−1 should be observed. Figure 3 shows the decay of the velocity in the center-of-mass frame vs. time. It can be seen that for t > 2, the decay is algebraical with exponent α = −1 (see enlarged box). This shows that HI between particle and surrounding ﬂuid are properly taken into account in our model. As a further test, we show here the results of the diﬀusional motion of a colloidal particle suspended in a Brownian solvent. The solvent is modelled with SDPD particles whose dynamics is governed by the equations (4) with the random terms given in (5). No external perturbations are considered here.

Fluid Particle Models for the Simulation of Microﬂuids

27

Fig. 2. SDPD simulation (no thermal ﬂuctuations) of the ﬂow ﬁeld around a moving disk. The double vortex structures behind the disk are clearly visible.

Fig. 3. Decay of the normalized colloidal particle’s velocity. In the enlarged box, the typical algebraical decay at long times can be appreciated.

Temperature of the solvent is T0 = 1. According to the Einstein’s distribution function, a colloidal particle of mass M suspended in a Brownian solvent should be characterized by a Gaussian velocity probability distribution function (PDF) with variance given by V2 = D

kB T . M

(9)

We have performed a simulation and computed explicitely the equilibrium distribution function of the velocity of the colloidal particle. Figure 4 shows the histogram of the x-component of its velocity. There is a very good matching between the normalized histogram and the theory. It should be noticed that the result does not depend on the particle resolution used to simulate

28

Marco Ellero

Fig. 4. Velocity probability distribution function of the colloidal particle compared with the analytical solution.

the problem. Indeed, SDPD posses a correct scaling of the thermal ﬂuctuations [14]. 5.2 Polymer Molecule in Suspension In this section, the conformational properties of the polymer molecule are investigated in a 2D case. We apply the SDPD method to the study of a polymer molecule in an inﬁnite Brownian solvent medium under zero ﬂow condition. Under these conditions, the ﬂow is isotropic and theoretically predicted universal scaling laws for several polymer properties can be tested numerically. Conformational properties of a polymer chain, in particular deformation and orientation, can be analyzed by monitoring the evolution of several tensorial quantities as, for instance the gyration tensor: G ≡ (1/2N 2 ) i,j rij rij or the end-to-end tensor deﬁned as R ≡ (rN − r1 )(rN − r1 ) here, rij = rj − ri with ri being the position of the i-th monomer in the chain. The indices i, j run from 1 to N , the are the radius √ total number of beads. Related quantities √ of gyration RG = tr G and the end-to-end radius RE = tr R. The eﬀect of the number of monomers N on RG and RE is known to follow the analytical expressions RE = aE (N − 1)ν , RG = aG [(N 2 − 1)/N ]ν

(10)

Fluid Particle Models for the Simulation of Microﬂuids

29

Fig. 5. Scaling of the radius of gyration RG for several chain lengths corresponding to N = 20, 30, 40, 50, 60, 80, 100 beads. The dotted line represents the best ﬁt consistent with the theory (RG ∝ N ν ) and gives a static exponent ν = 0.76 ± 0.012.

where ν is called static factor exponent and, according to the Flory’s formula, it assumes the value ν ≈ 0.75 in two dimensions with aE and aG being suitable constants [15]. In order to extract the exponent ν, SDPD simulations have been carried out with ﬁve diﬀerent chain lengths characterized by N = 20, 40, 60, 80, 100 beads. In all cases the time-averaged values of the gyration radius RG has been evaluated from several independent steady-state polymer conﬁgurations. Figure 5 shows a log-log plot of the time-averaged RG versus N . Error bars are within point dimensions. The results can be ﬁtted (dotted line in the ﬁgure) by a power-law with exponent ν = 0.76 ± 0.012 which is in good agreement with theoretical results. It should be noticed that this way to evaluate ν is quite time consuming since simulations at large N are necessary in order to ﬁt accurately the data in Fig. 5. An alternative way to extract ν is, instead of using the scaling law (10), by employing the static structure factor deﬁned as: S(k) ≡

1 exp(−ik · rij ). N i,j

(11)

In the limit of small wave vector |k|RG 1, the structure factor can be approximated by S(k) ≈ N (1 − k2 RG /3), while for |k|RG 1 holds S(k) ≈ 2N/k2 RG . The intermediate regime |k|RG ∼ 1 contains information about the intramolecular spatial correlations. In absence of external perturbation and close to equilibrium, S(k) is isotropic and therefore depends only on the magnitude of the wave vector k = |k|. S(k) probes therefore diﬀerent length

30

Marco Ellero

˜ Fig. 6. Normalized equilibrium static structure factor S(k) = S(k)/S(0) versus RG k corresponding to several chain lengths. All the curves collapse on a master line for ˜ ∝ k −1/ν with ν = 0.75 (dotted 2 < RG k < 8 (scaling regime). In this region S(k) line).

scales even for a single polymer and in the intermediate regime is shown to behave like S(k) ∝ k −1/ν . (12) Figure 6 shows a log-log plot of S(k) vs. RG k. From this ﬁgure it is possible to see how curves evaluated from simulations with diﬀerent chain lengths (N ) collapse on a single curve for 2 < RG k < 8, the slope of the linear region being −1/ν. The dotted line in the ﬁgure represents the theory with ν = 0.75 and shows very good agreement with the SDPD results.

6 Conclusions In this work, a reﬁned particle model for the description of mesoscopic complex ﬂows has been discussed. The method is known as Smoothed Dissipative Particle Dynamics and represents a generalization of Smoothed Particle Hydrodynamics for micro-ﬂows. Several advantages of the method over standard mesoscopic DPD techniques have been highlighted which include: (1) ﬂexibility in the use of an arbitrary equation of state; (2) control of the transport coeﬃcients and (3) physical determination of particle size. Furthermore, thermal ﬂuctuations in SDPD depend on the particle volumes in such a way that they can be neglected for suﬃciently coarse ﬂuids while they are automatically present whenever the physical dimension of the problem under study become small (i.e. microﬂuidics conditions). This feature of the scheme makes SDPD

Fluid Particle Models for the Simulation of Microﬂuids

31

a good candidate for the simulations of multiscale/multi-resolution phenomena. In order to validate the numerical method, applications in the area of microﬂuids have been considered. In particular, modelling of colloidal particle and macromolecule suspended in a Newtonian liquid have been discussed. Finally, simulations of these systems in Brownian and non-Brownian environments showed good agreement with the theory and/or previous numerical results.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15.

Karniadakis GE, Beskok A (2002) Springer, New York Stone HA, Kim S (2001) AIChE J 47:1250 Landau LD, Lifschitz EM (1959) Pergamon, Elmsford Hoogerbrugge PJ, Koelman J (1992) Europhys Lett 19:155 Espa˜ nol P, Warren P (1995) Europhys Lett 30:191 Groot RD, Warren PB (1997) J Chem Phys 117:4423 Gingold RA, Monaghan JJ (1977) Mon Not R Astron Soc 181:375 Monaghan JJ (1992) Annu Rev Astron Astrophys 30:543 Espa˜ nol P, Revenga M (2003) Phys Rev E 67:026705 ¨ Grmela M, Ottinger HC (1997) Phys Rev E 56:6620 Morris JP, Fox PJ, Zhu Y (1997) J Comput Phys 136:214 Ellero M, Kr¨ oger M, Hess S (2006) Multiscale Model Simul 5:759 Litvinov S, Ellero M, Hu X, Adams NA (2008) Phys Rev E 77:066703 Vazquez-Quesada A, Ellero M, Espa˜ nol P (2009) J Chem Phys 130:034901 de Gennes PG (1979) Cornell University Press, Ithaca

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Part II

Micro-channel Flows

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Semi-analytical Solution of the Density Proﬁle for a Gas Close to a Solid Wall E.A.T. van den Akker, A.J.H. Frijns, S.V. Nedea, A.A. van Steenhoven Department of Mechanical Engineering, Eindhoven University of Technology, P.O. Box 513, 5600 MB Eindhoven, The Netherlands. [email protected]

Summary. To reduce the CPU time needed for Molecular Dynamics (MD) simulations or Direct Simulation Monte Carlo (DSMC), an eﬀort is made to reduce the equilibration period. A semi-analytical model for the equilibrium situation of particle simulations is developed, such that the initial positions and velocities in a particle model such as MD or DSMC can be chosen close to the equilibrium situation. A time-averaged intermolecular force and a time-averaged pressure are derived from the density distribution. It is shown that the intermolecular force is shortranged and a result of density variations. For uniform densities, the time-averaged pressure is shown to correspond to the empirically determined equation of state for hard sphere particles. For non-uniform densities, the approximation is no longer valid and a new linear approximation is derived. It is shown that this linear approximation is valid even for dense gases. With this information, a semi-analytical model for the equilibrium density is derived and solved numerically in detail for the problem of particles close to a hard wall. In this situation density oscillations occur. The numerical results also show these density oscillations, indistinguishable from MD. The CPU time needed to generate the density proﬁle of particles close to a wall was less than one second, whereas the CPU time needed to perform the Molecular Dynamics simulations to reach the same equilibrium with the same accuracy was several hours. Therefore this method can be used to reduce computation time of simulations.

1 Introduction The trend in manufacturing of electronic components is that these components become smaller. In this miniaturization process, heat management becomes a very important problem, because the power consumption in electronic components increases with a factor of 10 every 6 years [1]. New solutions for cooling are needed, as present methods limit the performance and lifetime of the new technology. In this new technology, the heat is generated in local heat sources, so local cooling is preferable. A single-phase ﬂow can achieve local cooling. In microchannels, because of the large area to volume-ratio, the heat is eﬃciently absorbed by the ﬂow. M. Ellero et al. (eds.), IUTAM Symposium on Advances in Micro- and Nanoﬂuidics, IUTAM Bookseries 15, c Springer Science + Business Media B.V. 2009

36

E.A.T. van den Akker, A.J.H. Frijns, S.V. Nedea, A.A. van Steenhoven

In a two-phase ﬂow, the cooling can be even larger when the phase transition is close to the local heat source. Results for these techniques are promising, both experimentally [2] and numerically [3]. Still the ﬂuid ﬂow and heat transfer characteristics in these microchannels are not fully understood. More knowledge about this can be used for better design of microsystems like micro-reactors or microchannels. When a gas or vapour is ﬂowing through a microchannel, the system can be too small for the Navier-Stokes approach to be valid [4], and the governing equations change to the Boltzmann equations [4, 5]. In the Boltzmann equations, not only the macroscopic quantities, but also molecular quantities are important. Modelling techniques that include the particle nature of the ﬂuids should be used. Two important particle simulation methods are Molecular Dynamics (MD) [6] and Direct Simulation Monte Carlo (DSMC) [5]. In MD molecules move and collide according to the forces exerted on the particles. In DSMC, one molecule or a cluster of molecules is concentrated into a large particle, and collisions between these large particles are stochastically determined. As a result, MD simulations are more accurate on extremely small scales but very time-consuming, while DSMC simulations are many times faster but less accurate [7]. The eﬀect in simulation time is seen best in a dense gas, where the MD method is very time consuming, and the DSMC method generates the same results in a fraction of the MD simulation time. Lately, these methods have been researched intensively [8]. Nevertheless, due to the fact that particles have to be simulated explicitly, the simulation times are too long to be able to cover realistic systems and to generate macroscopic properties. For example, in a typical MD simulation that lasted approximately 8 hours on one CPU, around 4000 particles are simulated for 10.000.000 time steps, in a simulation box of 20 × 40 × 40 molecular diameters. If the particles are thought to be Argon-atoms, this corresponds to a simulation of 2.8 × 10−19 g of ﬂuid argon for 66 ns in a simulation box of 1.26 × 10−8 μm3 . In low densities, a DSMC simulation can reduce simulation times by a factor 1000 [7]. In experimental setups the typical microchannel cross-section area is 200 × 50 μm2 [9], so a reasonable simulation box would need to have a volume of 106 μm3 . Currently, the experiments deal with length and time scales that are several orders of magnitude larger than those in the numerical calculations performed by DSMC and MD methods. Further experimental research will reduce the length and timescale, and the simulation methods will become faster in the future, but the gap is still large. Because particle-based methods involve simulations of many particles over many time steps, statistical mechanics can be used to analyze the problems. This has been used to analyze the surface tension and distribution function for a gas close to a solid wall [10]. A diﬀerent way of approach is using Density Functional Theory (DFT) [11]. DFT is based on making a thermodynamic quantity (usually the Helmholtz free energy) a functional of the average density proﬁle, and minimizing that quantity. Over the years, the density proﬁle for a liquid-solid interface has become a standard problem in DFT [12, 13].

Semi-analytical Solution of the Density Proﬁle for a Gas Close to a Solid

37

An analytical explanation for the occurring density oscillations has been given, analyzing up to 4 aligned particles [14]. In this paper, a method is developed to ﬁnd the equilibrium situation of the DSMC and MD methods semianalytically. This is done by analyzing the equilibrium properties of the MD and DSMC simulations, deriving equations for the equilibrium and solving these equations. The developed theory is tested for the standard problem of a gas close to a wall, such that it can be compared to other models that solve this problem. It is shown that for a steady-state problem the equilibrium density and all other equilibrium properties can be calculated in a fraction of the simulation time.

2 MD Derivation of Equation of State 2.1 Internal Force The equilibrium properties of a medium depend on the interactions of the particles with each other and the environment. Here, the assumption is made that the particles are described by their positions, velocities and forces. At every time step, particle i has position xi , velocity v i , and feels a force F i . In a system with N particles, at position x1 , x2 , . . . , xN , the force on particle i is given by + F ext (1) F i = F int i i , where F ext is the external force on particle i, by e.g. walls and gravity, and i is the internal force, originating from the interaction between particles. F int i In most cases the external force is conservative (wall forces, gravity, etc.), so it holds that (2) F ext i (xi ) = −∇Vext (xi ), where Vext is the external potential energy function. is the result of the interaction between particle i and all Because F int i other particles, this can be written as = F int i

N

F ij ,

(3)

j=1,j=i

where F ij is the force between particle i and j. Only pairwise interactions are considered here. Because F ij = −∇V (xi − xj ),

(4)

where V is the potential energy function, the internal force can be written as [15]

int F (x) = − neﬀ (u, x)∇V (x − u)du. (5) Ω

38

E.A.T. van den Akker, A.J.H. Frijns, S.V. Nedea, A.A. van Steenhoven

Because repulsive forces between particles prevent the particles from overlapping, the eﬀective number density neﬀ is not equal to the number density; in the direct vicinity of a particle’s center there is not likely to be another particle’s center. This is expressed by means of the radial distribution function [16] g(r, n), such that (6) neﬀ (u, x) = n(u)g u − x, n(u) . No exact equation for this radial distribution function is known, although several analytical approximations for this function are used [16]. It is known that this radial distribution function can be expanded in powers of the density n as V (r) (7) g(r, n) = e− kT 1 + ng1 (r) + n2 g2 (r) + · · · , where g1 and g2 can be expressed in cluster integrals [17]. 2.2 Pressure Similar to the force, also the pressure can be deﬁned in the continuous limit. First an expression for a pressure in a particle system has to be known; for this the Irving-Kirkwood [15] method can be used, which gives the expression 1 1 2 mi (v(r i , t)) + r ij (t)F ij (t) , (8) p(r, t) = 3V 2 ij i where v(r i , t) is the velocity of a particle at position r i at time t. In a static isothermal continuous problem, this becomes in the continuous limit [15]

n(r) neﬀ (x, r)F (x, r)(x − r)T dx, (9) p(r) = n(r)kT + 6 where neﬀ (x, r) again is the eﬀective number density at position x, given that there is a particle at position r, and F (x, r) is the force between a particle at position x and a particle at position r. With equations (7), (6) and (4), this becomes

V (x−r) n(r)kT ∇ e− kT p(r) = n(r)kT + 6 · · · (n + n2 g1 (x − r) + n3 g2 (x − r) + · · · )(x − r)T dx. (10) To compare this with other equations of state, a gas consisting of hard-sphere particles with radius rc with a constant number density n is considered. The potential energy function of a hard-sphere particle with radius tc is given by [17] VHS (x) = ∞ if x < rc ,

VHS (x) = 0

if x > rc .

(11)

Semi-analytical Solution of the Density Proﬁle for a Gas Close to a Solid

39

For a gas with such a potential energy function, equation (10) reduces to p = nkT + n2 kT

2πrc3 2πrc3 2πrc3 + n2 g1 (rc )nkT + n3 g2 (rc )nkT + · · · . (12) 3 3 3

It is known [17] for a hard-sphere gas that g1 (rc ) = 5π/12rc3 and g2 (rc ) = 1.258702rc6 . With these values the relation becomes p = 1 + 4η + 10η 2 + 18.3648η 3 + · · · , nkT

(13)

where the reduced density η is deﬁned by η = (πnrc3 )/6. Compared to the equation of state-approximation by Carnahan and Starling [18], 1 + η + η2 − η3 p = = 1 + 4η + 10η 2 + 19η 3 + · · · , nkT (1 − η)3

(14)

it is seen that (10) reduces to a correct result when the density is constant. 2.3 MD Calculation Method In the case of density ﬂuctuations close to a wall, density clearly is not constant. Finding the exact relation between pressure proﬁle and density proﬁle is harder then. In equilibrium however, similar to the macroscopic case, the balance of momentum in equilibrium states ∇p = n(F int − ∇Vext ).

(15)

This fact is used to calculate a data set of pressures and densities, using Molecular Dynamics. In our simulations, the truncated shifted Lennard-Jones particles (tsLJ-particles) [3] are used, which are deﬁned by their potential energy function VtsLJ , given by

VLJ (r) − VLJ (rc ), if r ≤ rc , (16) VtsLJ (r) = 0, if r > rc , where rc is the cutoﬀ radius, and VLJ (r) is given by VLJ (x) = 4ε

σ x

12

−

σ x

6 ,

(17)

where σ is a characteristic length (the distance between two particles at which the potential energy is zero) and ε is the well-depth energy. This potential energy function describes particles with an attractive force between them when they are far away, and a repulsive force when they are close. In our simulations with tsLJ-particles, the cutoﬀ radius is rc = 21/6 σ, with σ = 2RvdW , twice the Van der Waals-radius of the particles. In this way, the

40

E.A.T. van den Akker, A.J.H. Frijns, S.V. Nedea, A.A. van Steenhoven

only forces between particles are repulsive forces, such that the particles are similar to hard-sphere particles, as used in DSMC-simulations. With these tsLJ-particles, MD-simulations were performed for particles close to a wall, for two diﬀerent densities. In the simulations, the volume was ﬁxed at 40RvdW × 40RvdW × 40RvdW , while the number of particles was 3280 and 4800. Initially, the particles were placed in a f cc-lattice. In one dimension, the boundary conditions were reﬂecting walls, whereas the other two dimensions had periodic boundary conditions. The simulation setup can be interpreted as a gas between two inﬁnite planes. The simulation was ﬁrst run for 100000 time steps (where 1 time step corresponds to 0.005RvdW m/ε, approximately 0.011 ns for Argon-molecules), long enough to be sure that the situation was in equilibrium, after which the simulation was run 500000 more time steps. The simulations resulted in density oscillations. Using (15), the pressure at each point was calculated. This resulted in a data set of densities and corresponding pressures. 2.4 Equation of State Results In Fig. 1, the variation in pressure is compared to variation in number density for our MD simulation results (dots). Pressure is made dimensionless by −3 , when ε is a characteristic strength of the potential scaling with p0 = 8εRvdW energy (see (17)), approximately 1.66×10−21 J for Argon-molecules, and RvdW is the Van der Waals-radius of the particles (approximately 3.4 × 10−10 m for Argon-molecules). The number density is made dimensionless by scaling with −3 n0 = 8RvdW . In this example, the number density ﬂuctuates around n = 0.41n0 . Interpreted as a gas made out of Argon-molecules, this corresponds to n ≈ 1028 m3 . Because the mass of one Argon-molecule is 6.63 × 10−26 kg, this corresponds to a density = 674 kg/m3 . The density at atmospheric pressure and room-temperature is around 1.78 kg/m3 , so the simulated gas is dense. Therefore a large pressure is expected. Also shown is the relation according to the Carnahan-Starling equation (14). Clearly this equation overestimates the slope. The reason for this is that the Carnahan-Starling equation assumes a constant density, whereas the density here is ﬂuctuating (the density more than doubles in a range of an atomic diameter). The pressure is a result of the forces resulting from the neighboring particles. At a point where the density distribution is in a local maximum, the density of neighboring particles will be lower than the density in the maximum. Therefore, an equation of state that is based on a uniform density will overestimate the pressure. Similarly, at a point where the density distribution is in a local minimum, the density of neighboring particles will be larger than the density in the minimum. Therefore, an equation of state based on a uniform density will underestimate the pressure in a density minimum. Overall, the equation of state based on a uniform density will overestimate the slope, as is demonstrated in Fig. 1.

Semi-analytical Solution of the Density Proﬁle for a Gas Close to a Solid

41

Fig. 1. The relation between number density and pressure, when the dimensionless number density is around 0.4. Simulation results from Molecular Dynamics are shown in dots, the relation (14) is given by the solid line, the linear approximation (19) is shown by the dotted line. The linear approximation holds in this regime.

Unfortunately (10), although exact, is not usable in general for non-uniform densities, because higher order expansion terms g2 , g3 , . . . are not all known explicitly. Equation (10) is an expansion around n = 0, whereas the interest of this article is the ﬂuctuations around n = n∞ , the bulk density. Therefore another expansion is used around n = n∞ : p = n∞ kT + A(n − n∞ ) + · · · ,

(18)

where A is an unknown constant. Molecular Dynamics simulations with several densities suggest that this constant A can be approximated when n∞ < 0.5n0 by kT A= , (19) 2n∞ Veﬀ where n∞ is the mean number density of the system and Veﬀ is the eﬀective volume of a particle. From now on, this empirical equation of state will be used, with higher order terms neglected. The approximation for n∞ = 0.41 can be seen in Fig. 1, where it is shown as the dotted line. This approximation is working for dilute gases as well as for dense gases, although the correspondence gets worse for denser gases. As an example, consider the result when the dimensionless number density is ﬂuctuating around dimensionless bulk density n∞ = 0.6, shown in Fig. 2. This ﬁgure shows that for a higher density, the relation between pressure and density can still be considered linear, but equation (19) slightly underestimates the slope. Because (19) has a very simple form and is a good approximation for n < 0.5, the relation is not changed here.

42

E.A.T. van den Akker, A.J.H. Frijns, S.V. Nedea, A.A. van Steenhoven

Fig. 2. The relation between number density and pressure, when the dimensionless number density is around 0.6. Simulation results from Molecular Dynamics are shown in dots, the relation (14) is given by the solid line, the linear approximation (19) is shown by the dotted line. The linear approximation holds in this regime, although the slope is slightly underestimated.

3 Analytical Derivation of Equilibrium State 3.1 Method to Calculate the Equilibrium Density To ﬁnd an equation for the equilibrium density proﬁle, the balance of momentum (15) and the equation of state (19) can be combined into ∇n = n

2n∞ Veﬀ (F int (n) − ∇Vext ). kT

(20)

This is almost enough information to solve the number density proﬁle, only one boundary condition is needed. This can be that the total mass is known, or that the number density is known in one point. The ﬁxed total mass condition is the most physical one, because in most cases the total mass of the system is known. However, because the ﬂuctuations of a gas close to a wall are investigated here, the bulk density n∞ far away from the wall is described here. The combination of (20) and one of these boundary equations completely speciﬁes the problem. Equation (20) can be rewritten into

2n∞ Veﬀ x n(z)(F int (z) − ∇Vext (z))dz, (21) n(x) = g(n(x)) := n(y ∗ ) + kT ∗ y for some ﬁxed y ∗ . To solve this, a relaxed iterative scheme is used: ns+1 (x) = (1 − α)ns (x) + αg(ns (x)),

0 < α < 1,

(22)

Semi-analytical Solution of the Density Proﬁle for a Gas Close to a Solid

43

where α is the relaxation parameter [19]. For small α, it can be shown that (22) converges to the n(x) deﬁned by (21). After ﬁtting this parameter, a value of α = 0.15 showed a fast convergence for all calculations done so far. Iteration (22) has to start with an initial density proﬁle n0 . Because the approximation in equation (18) is best when the density oscillations are minimal, the iteration is started with the solution where the number density is everywhere equal to a reference number density n∞ , here equal to the bulk density. The iterative scheme (22) is valid for a 3-dimensional problem, and it can be used to numerically generate equilibrium conditions for DSMC or MD-simulations in complex conﬁgurations. 3.2 Simpliﬁcations at the Wall In the analysis of gases close to a hard wall, the calculation presented above can be simpliﬁed. First of all, because the main interest are dilute gases, all higher order terms can be neglected in (7), which leads to

− V (x−u) kT e − 1 ∇ndu. (23) F int (x) = −kT Ω

Secondly, the problem of interest here, a gas close to a hard wall, is one dimensional. If the problem is written using cylinder coordinates and the potential energy function is rotational symmetrical, (23) simpliﬁes to

∞ V(z) F (xez ) = 2πkT n(x + z)z e− kT − 1 dz. (24) −∞

In this way, only a one-dimensional convolution integral has to be calculated, which simpliﬁes the numerical calculations. The potential energy function V can be chosen freely; here the truncated shifted Lennard-Jones potential (16) is chosen to let the calculation correspond to the simulations. 3.3 Numerical Aspects Because it is known from previous simulations that close to a wall there are density oscillations that damp out after several particle diameters, the grid is restricted to 0 ≤ x ≤ 16RvdW (note that RvdW is the eﬀective radius of the particles). For the numerical computation, an equidistant grid is deﬁned, with grid size dx = 0.08RvdW . Grid points are then deﬁned as xi = idx, where 0 ≤ i ≤ 200. On every grid point i, the dimensionless density ni is 3 . Similarly, a dimensionless force Fi is deﬁned deﬁned by ni = n(xi )RvdW −1 as Fi = F (xi )RvdW ε . Because the wall only interacts with particles at position x = 0, at any other position there is no external potential energy due to the wall, so Vext = 0. Equation (21) reduces with this notation in the one-dimensional case to

44

E.A.T. van den Akker, A.J.H. Frijns, S.V. Nedea, A.A. van Steenhoven

2n∞ Veﬀ kT

g(ni ) := n(y ∗ ) +

x

n(z)F int (z)dz.

(25)

y∗

Because the density far away from the wall is taken equal to the bulk density, the boundary conditions y ∗ = x200 and n(y ∗ ) = n∞ are used, which gives 200 2n∞ Veﬀ k k g(ni ) := n∞ + n F dx. kT

(26)

k=i

In every iteration step j, ﬁrst the force is calculated. This is done by

∞ V (z) i Fj = 2πkT n(x + z)z e− kT − 1 dz ≈ 2πkT

−∞ ∞

∞ V (ldx) ni−l l e− kT − 1 dx2 = ni−l Ml ,

l=−∞

where

(27) (28)

l=−∞

V (ldx) Ml := 2πkT l e− kT − 1 dx2

(29)

is independent of the density proﬁle. Some special care has to be given to the summation in (28), because the summation refers to densities of points outside the grid; this can easily be solved by noting that for x < 0, there are no particles, so ni = 0 if i < 0, and for x > 16RvdW , the density is assumed to be constant, so ni = n∞ if i > 200. After this, (26) is used: gji = n∞ +

200 2n∞ Veﬀ l l nj Fj dx, kT

(30)

l=i

and ﬁnally, the density is updated according to (22). To check convergence, the error ei at time step i was estimated by ei =

200

nil − ni−1 . l

(31)

l=i

This error is rapidly decreasing, as can be seen in Fig. 3. This ﬁgure shows the convergence error for a simulation as described below for a bulk number den−3 , where the iteration was performed 300 times. After about sity of 0.60RvdW 200 iterations, the maximal precision was reached, therefore the iteration procedure can be stopped after 200 iterations. 3.4 Equilibrium Results In the test computation, the standard problem of particles close to a reﬂecting wall is considered. Simulations of this problem have been used to test the DSMC-method [5, 20] and density functional calculations [13].

Semi-analytical Solution of the Density Proﬁle for a Gas Close to a Solid

45

Fig. 3. The error as deﬁned in (31) for 300 iterations. After 200 iterations, the maximal precision was reached.

To ﬁnd the number density proﬁle for this problem, a Molecular Dynam3 has been ics simulation of 11835 particles in a volume of 18 × 40 × 40RvdW performed. At x = 0 (and x = 18RvdW ) the particles are reﬂected (simulating a hard wall at x = −1/2RvdW and x = (18 + 1/2)RvdW ). The initial temperature of the particles is set at T = 1.0ε/k and remains constant in the simulation. The results show that boundary eﬀects are negligible at a distance of more than 5RvdW from the wall, so the distance of 18RvdW between the walls was large enough for the eﬀects close to the wall to be independent. Therefore the simulation results were analyzed as if the simulation box was 3 , inﬁnitely large. Because there were 11835 particles in a volume of 28800RvdW −3 the overall number density was n∞ = 0.41RvdW . The particles used in the ﬁrst test are deﬁned by the truncated shifted Lennard-Jones (tsLJ) potential (16). This tsLJ-potential resembles a particle with only repulsive forces. The simulation was run for 500000 time steps. This was more than enough for the equilibrium situation to develop, so this equilibrium situation was determined with great accuracy. To compare this to the solution presented here, the iteration described in the Sect. 3.1 was performed, over the domain 0 < x < 16RvdW , with n(16RvdW ) = 0.41. The result is shown in Fig. 4. The dots are the Molecular Dynamics simulation results, the line shows the iterative solution found by the method described above; both methods give the same result. The Molecular Dynamics simulations lasted several hours, whereas the iterative solution was found in 0.7 s. In Fig. 5, the force and pressure proﬁles are shown for iterative solution found by the method described above as well as for the Molecular Dynamics simulations; both methods agree in results.

46

E.A.T. van den Akker, A.J.H. Frijns, S.V. Nedea, A.A. van Steenhoven

Fig. 4. The number density n proﬁle of a particle-based gas with a bulk number −3 close to a reﬂective wall. The Molecular Dynamics simuladensity n∞ = 0.41RvdW tion results are shown as dots, the result from the semi-analytical method described in this paper is shown with a line.

Fig. 5. The internal force f (top) and pressure proﬁle p (bottom) of a particle−3 close to a reﬂective wall. based gas with a bulk number density n∞ = 0.41RvdW The Molecular Dynamics simulation results are shown as dots, the result from the semi-analytical method described in this paper is shown with a line.

Semi-analytical Solution of the Density Proﬁle for a Gas Close to a Solid

47

Fig. 6. The number density n proﬁle of a particle-based gas with a bulk number −3 close to a reﬂective wall. The Molecular Dynamics simuladensity n∞ = 0.60RvdW tion results are shown as dots, the result from the semi-analytical method described in this paper is shown with a line. −3 A similar test was done for bulk density n∞ = 0.60RvdW , with the same conditions except more particles (17319). Here, the CPU-time was also 0.7 s. The positions of the oscillations are the same, but because the iteration scheme slightly underestimates the dependency of pressure on density (see Fig. 2), the amplitude of these oscillations are overestimated in the iterative calculation. In Fig. 7, the force and pressure proﬁles are shown for iterative solution found by the method described above as well as for the Molecular Dynamics simulations. Again, the positions of the oscillations are the same, but the oscillations are overestimated in the iterative solution. The density proﬁle of a gas close to a solid wall has also been calculated with Density Functional Theory for hard spheres [11], but also for Lennard-Jones ﬂuids [13]. The truncated shifted Lennard-Jones (16) particles with cut-oﬀ radius rc = 21/6 σ used here are similar to hard-sphere particles because they only show repulsive forces, therefore the density proﬁles in Figs. 4 and 6 can be compared to hard-sphere particle calculations in DFT. Compared to the hard-sphere calculations in [13], our density proﬁles show −3 , and DFT calculations give better results for densities below n∞ ≈ 0.6RvdW better results for higher densities. An important observation however is that in [13], the positions of the maxima are overestimated, whereas the results from Sect. 3.4 predicts the positions of the maxima correct. Over the years, diﬀerent DFT-models have been developed that give improved results, see for example [21, 22]; the results in Sect. 3.4 correspond to these improved DFT results.

4 Summary In Molecular Dynamics simulations or Direct Simulation Monte Carlo, an equilibration period in simulations is needed to transform the arbitrarily chosen initial conﬁguration into an equilibrium conﬁguration. Because no useful

48

E.A.T. van den Akker, A.J.H. Frijns, S.V. Nedea, A.A. van Steenhoven

Fig. 7. The internal force f (top) and pressure proﬁle p (bottom) of a particle−3 close to a reﬂective wall. based gas with a bulk number density n∞ = 0.60RvdW The Molecular Dynamics simulation results are shown as dots, the result from the semi-analytical method described in this paper is shown with a line.

information can be derived from the equilibration period, a method is derived here to shorten the equilibration period. At any time step in a simulation, the intermolecular force can be expressed in the positions of the particles and the potential energy function, therefore the time-averaged intermolecular force can be expressed in the density proﬁle and the potential energy function. Similarly, the time-averaged pressure can be expressed in the eﬀective particle density. It is shown that this is compatible with the empirically determined equation of state for hard-sphere particles in a uniform density. Unlike the force, the pressure is not a result of variations in number density, but a result of number density itself. The low-density expansion is not so useful for problems with denser gases, so a diﬀerent linear expansion around the bulk density is used. This linear expansion is shown to give a far better approximation in problems with large density oscillations than the original expansion. For dimensionless number densities up to n = 0.4, the linear approximation used here agrees with Molecular Dynamics simulation

Semi-analytical Solution of the Density Proﬁle for a Gas Close to a Solid

49

results very well. For larger number densities, up to n = 0.6, the Molecular Dynamics simulation still show a linear relation between density and pressure for small deviations in density, although with a slightly diﬀerent slope than predicted by the linear approximation used here. The force and pressure are coupled by the balance of momentum. Because the time-averaged force and the time-averaged pressure are both expressed in the density proﬁle, this results in an integral equation for the density proﬁle. Together with one boundary condition, the problem is completely determined. This problem is solved numerically by a relaxed iterative scheme. The iterative scheme is tested on a one-dimensional problem of a gas close to a hard wall. The iterative scheme predicts that so-called density oscillations close to the wall will occur for some molecular diameters, and that the height of the oscillations increases with the number density. These results are conﬁrmed by Molecular Dynamics simulation results. For number densities n = 0.4, the predicted density proﬁle is indistinguishable from the simulated density proﬁle. For larger densities, n = 0.6, the predicted density proﬁle overestimates the amplitude of the density oscillations, although the positions of the oscillations are predicted well. For simple problems, like the density distribution of a gas between two parallel plates, the method described here can eliminate the need for simulation completely. For more complex dynamic processes, for example evaporation of a ﬂuid moving through a micro-tube or the wetting of a realistic wall, the method presented here gives information about density and velocity distribution, which might be used to generate initial conﬁgurations for MD and DSMC simulations closer to equilibrium, saving valuable computation time.

Acknowledgment This project is ﬁnancially supported by MicroNed.

References 1. Schmidt R, Notohardjono B (2002) IBM J Res Develop 46:739–751 2. Colgan E, Furman B, Gaynes M, Graham W, LaBianca N, Magerlein J, Polastre R, Rothwell M (2005) 21st IEEE SEMI-THERM Symposium 3. Markvoort A, Hilbers P, Nedea S (2005) Phys Rev E 71:066702–1–9 4. Bird G (1994) Molecular gas dynamics and the direct simulation of gas ﬂows. Clarendon, Oxford 5. Frezzotti A (1997) Phys Fluids 9:1329–1335 6. Frenkel D, Smit B (1996) Understanding molecular simulation: from algorithms to applications. Academic Press, San Diego 7. Nedea S, Frijns A, van Steenhoven A, Markvoort A, Hilbers P (2005) Phys Rev E 72:016705–1–10

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8. Nedea S, Markvoort A, Frijns A, van Steenhoven A, Hilbers P (2006) ASME fourth international conference on nanochannels, microchannels and minichannels 9. Wu H, Cheng P (2003) Int J Heat Mass Transfer 46:2547–2556 10. Henderson J, van Swol F (1984) Mol Phys 51:991–1010 11. Evans R (1979) Adv Phys 28:143–200 12. Curtin W (1987) Phys Rev Lett 59:1228–1231 13. Tarazona P, Evans R (1984) Mol Phys 52:847–857 14. Nedea S, Frijns A, van Steenhoven A, Jansen A, Markvoort A, Hilbers P (2006) J Comput Phys 219:532–552 15. Irving J, Kirkwood J (1950) J Chem Phys 18:817–829 16. Goldman S (1979) J Phys Chem 83:3033–3037 17. Nijboer B, van Hove L (1953) Phys Rev 85:777–783 18. Carnahan N, Starling K (1969) J Chem Phys 51:635–636 19. Bertsekas D, Tsitsiklis J (1989) Parallel and distributed computation. Prentice Hall, New York 20. Frezzotti A (1999) Eur J Mech B Fluids 18:103–119 21. Zhou S, Jamnik A (2005) J Chem Phys 123:124708 22. Zhou S, Jamnik A (2006) Phys Chem Chem Phys 8:4009–4017

Comprehensive Analysis of Dewetting Proﬁles to Quantify Hydrodynamic Slip Oliver B¨ aumchen1 , Renate Fetzer1,2 , Andreas M¨ unch3 , Barbara Wagner4 , 1 Karin Jacobs 1

2

3

4

Department of Experimental Physics, Saarland University, 66123 Saarbruecken, Germany. [email protected] Present address: Ian Wark Research Institute, University of South Australia, Mawson Lakes, SA 5095, Australia. School of Mathematical Sciences, University of Nottingham, University Park, Nottingham, NG7 2RD, UK. Weierstrass Institute for Applied Analysis and Stochastics (WIAS), Mohrenstrasse 19, 10117 Berlin, Germany.

Summary. Hydrodynamic slip of Newtonian liquids is a new phenomenon, the origin of which is not yet clariﬁed. There are various direct and indirect techniques to measure slippage. Here we describe a method to characterize the inﬂuence of slippage on the shape of rims surrounding growing holes in thin polymer ﬁlms. Atomic force microscopy is used to study the shape of the rim; by analyzing its proﬁle and applying an appropriate lubrication model we are able to determine the slip length for polystyrene ﬁlms. In the experiments we study polymer ﬁlms below the entanglement length that dewet from hydrophobized (silanized) surfaces. We show that the slip length at the solid/liquid interface increases with increasing viscosity. The correlation between viscosity and slip length is dependent on the type of silanization. This indicates a link between the molecular mechanism of the interaction of polymer chains and silane molecules under ﬂow conditions that we will discuss in detail.

1 Introduction 1.1 Slippage at Solid/Liquid Interfaces The control of the ﬂow properties at the solid/liquid interface is important for applications ranging from microﬂuidics, lab-on-chip devices to polymer melt extrusion. Slippage would, for instance, greatly enhance the throughput in lab-on-chip devices and extruders. Usually slippage is characterized by the so-called slip length which is deﬁned as the distance between the solid/liquid interface and the point within the solid where the velocity proﬁle of the liquid extrapolates to zero. During the last years, several diﬀerent techniques have

M. Ellero et al. (eds.), IUTAM Symposium on Advances in Micro- and Nanoﬂuidics, IUTAM Bookseries 15, c Springer Science + Business Media B.V. 2009

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O. B¨ aumchen, R. Fetzer, A. M¨ unch, B. Wagner, K. Jacobs

been established to probe the slip length of diﬀerent systems. These methods can be classiﬁed into two groups, direct and indirect measurements of the ﬂow velocity at the solid/liquid interface. Direct measurements are based on particle imaging velocimetry techniques [1–3] that utilize tracer particles or ﬂuorescence recovery after photobleaching [4, 5]. Colloidal probe atomic force microscopy [6, 7] and surface forces apparatuses [8, 9] are more indirect techniques to probe slippage. Detailed information concerning slippage and techniques to measure liquid velocities in the vicinity of solid/liquid interfaces are reviewed in recent articles by Neto et al. [10], Lauga et al. [11] and Boquet and Barrat [12]. 1.2 Dewetting Dynamics of Polymer Films Dewetting takes place whenever a liquid layer can reduce energy by retracting from the contacting solid [13]. Dewetting starts by the birth of holes, which can be generated by three diﬀerent mechanisms, spinodal dewetting, homogeneous and heterogeneous nucleation [14]. The holes grow in size, cf. Fig. 1, until neighboring holes touch. The quasi-ﬁnal state is a network of droplets [15]. The equilibrium state would be a single droplet, yet this state is usually not awaited since it may take years for viscous liquids such as our polystyrene. Since we are only interested in the growth of holes and their rim morphology, the underlying mechanism of their generation is irrelevant. However, the most likely process in our system is nucleation. The type of nucleus is unclear and also of no relevance for our studies. It can be a dust particle or a heterogeneity in the ﬁlm or on the substrate. After holes are generated, they instantly start to grow until they touch neighboring holes and coalesce. We study the ﬂow dynamics of thin polymer ﬁlms on smooth hydrophobic substrates. The driving force for the dewetting process can be characterized by the spreading parameter S, which depends on the surface tension of the liquid γlv (30.8 mN/m for polystyrene) and the Young’s contact angle Θ of the liquid on top of the solid surface: (1) S = γlv (cos Θ − 1). The system also dissipates energy, namely by viscous friction within the liquid and sliding friction at the solid/liquid interface. A force balance between driving forces and dissipation determines the dewetting rate. Conservation of mass leads to a rim that surrounds each hole. We show that the shape of this rim is not only sensitive to the chain length of the polymer melt [16] but is also to the underlying substrate. The use of recently developed models [17] enables us to extract the slip length b as well as the capillary number Ca from the rim proﬁles. The latter is given by Ca =

η s˙ . γlv

(2)

Here, s˙ is the current dewetting velocity, which can be obtained by a series of optical images of the hole growth before the sample is quenched to

Analysis of Dewetting Proﬁles

53

room temperature and imaged by atomic force microscopy (AFM). From the capillary number, the viscosity η of the melt can be calculated and compared to independent viscosity measurements, cf. Sect. 3.4.

2 Experimental Section 2.1 Our System Atactic polystyrene (PS) obtained from PSS Mainz with a molecular weight of 13.7 kg/mol (Mw /Mn = 1.03) was used as a liquid in our experiments. As substrates, we used Si wafers (Siltronic AG, Burghausen, Germany) that were hydrophobized by two diﬀerent types of silanes following standard methods [18]. Thin PS ﬁlms were prepared by spin coating a toluene solution of PS onto mica, ﬂoating the ﬁlms onto Millipore water, and then picking them up with the silanized silicon wafers. The ﬂoating step is necessary since on the spin coater, a drop of toluene solution would just roll oﬀ the hydrophobized surface. All PS ﬁlms in this study have a thickness of 130(5) nm. We utilized two diﬀerent silane coatings on the Si wafer (2.1 nm native oxide layer): octadecyltrichlorosilane (OTS) and the shorter dodecyltrichlorosilane (DTS), respectively. The thicknesses of these self-assembled monolayers are dOTS = 2.3(2) nm and dDTS = 1.5(2) nm as determined by ellipsometry (EP3 by Nanoﬁlm, Goettingen, Germany). The contact angle hysteresis of water is very low (6◦ in case of OTS and 5◦ in case of DTS), while the advancing contact angles are 116(1)◦ (OTS) and 114(1)◦ (DTS). Surface characterization by AFM (Multimode by Veeco, Santa Barbara, CA, USA) revealed RMS roughnesses of 0.09(1) nm (OTS) and 0.13(2) nm (DTS) on a 1 μm2 area, and an (static) receding contact angle of polystyrene droplets of 67(3)◦ on both substrates. Surface energies cannot be determined directly from contact angle measurements of PS only, due to the fact that polystyrene has polar contributions. The advancing contact angles of apolar liquids like bicyclohexane vary slightly on both coatings. We ﬁnd 45(3)◦ on OTS and 38(4)◦ on DTS. The surface energy of the substrate is linked to the contact angle of apolar liquids via the Good-Girifalco equation [19]. Consequently, we ﬁnd a slightly larger surface energy for DTS (γDTS = 26.4 mN/m) than for OTS (γOTS = 23.9 mN/m) substrates. Identical contact angles of PS on both substrates therefore lead to diﬀerent energies at the OTS/PS and DTS/PS interface due to Young’s equation. 2.2 Hole Growth Dynamics To induce dewetting, the ﬁlms were heated to diﬀerent temperatures above the glass transition temperature of the polymer. After a short time circular holes appear and instantly start to grow (see Fig. 1).

54

O. B¨ aumchen, R. Fetzer, A. M¨ unch, B. Wagner, K. Jacobs

Fig. 1. Series of optical images of a growing hole in a PS ﬁlm on DTS at 120◦ C.

Fig. 2. Hole radius versus time on OTS and DTS at 110◦ C and 120◦ C.

Holes in the PS ﬁlm were imaged by optical microscopy captured by an attached CCD camera, and hole radii were measured. As shown in Fig. 2, dewetting progresses much faster on DTS than on OTS coated substrates. We explain these results as follows: At the same temperature, the liquids on both samples have exactly the same properties: the viscosity as well as the surface tension do not depend on the substrate underneath. Additionally, the contact angle of polystyrene on both surfaces is the same within the experimental error. Therefore, the spreading coeﬃcient S, that is the driving force of the dewetting process, is identical on both substrates. Hence, the diﬀerent dewetting velocities observed on OTS and DTS indicate diﬀerent energy dissipation pathways on these coatings. Viscous friction within the liquid is expected to be identical for both surfaces since we compare liquids of the same viscosity and with the same dynamic contact angle [20]. The latter can be probed by AFM scans in the vicinity of the three-phase contact line. In situ imaging reveals that the dynamic contact angle stays constant at 56(2)◦ during hole growth. Consequently, friction at the solid/liquid interface and therefore slippage must be diﬀerent on OTS and DTS. In the next section we show that not only the hole growth dynamics but also the shape of rims is aﬀected by the underlying substrate.

Analysis of Dewetting Proﬁles

55

Fig. 3. AFM image of a rim on OTS: scan size 10 × 10 μm2 . The dewetting was performed at 130◦ C. The white line represents a single scan line of the rim perpendicular to the three-phase contact line.

2.3 Rim Shapes For the characterization of the shape of the rims surrounding the holes, the samples were quenched to room temperature after the holes have reached a diameter of 12 μm. AFM scans were then taken in the glassy state of PS. An example is shown in Fig. 3. Figure 4 demonstrates that the type of substrate aﬀects the rim proﬁle: On OTS covered substrates, the rim of the dewetting PS ﬁlm exhibits an oscillatory shape, whereas on DTS covered surfaces, at the same temperature, a monotonically decaying function is observed. The insets to Fig. 4 shall clarify the term “oscillatory rim shape” on OTS. Furthermore, Fig. 4 shows that temperature inﬂuences the shape of the proﬁle: the higher the temperature the more pronounced are the oscillations on OTS and even on DTS, an oscillatory shape is recorded for T = 130◦ C. Comparing the impact of the substrate on the rim morphology to the hole growth experiments in the previous section, we observe a correlation between the dewetting velocity and the shape of corresponding rims: high dewetting velocities lead to monotonic rim proﬁles while lower dewetting speeds tend to result in oscillatory rim shapes. In the previous section we argued that diﬀerent sliding friction at the solid/liquid interface might be responsible for the diﬀerent dewetting speeds on OTS and DTS. In the next section, the theoretical expectation of the inﬂuence of slippage on the rim shapes will be explained. Moreover, the experimental results are compared to theoretical predictions.

3 Theoretical Models and Data Analysis 3.1 Lubrication Model The theoretical description is based on a lubrication model developed by M¨ unch et al. [21] for systems showing strong slip at the solid/liquid inter-

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O. B¨ aumchen, R. Fetzer, A. M¨ unch, B. Wagner, K. Jacobs

Fig. 4. Rim proﬁles on OTS and DTS at 110◦ C, 120◦ C and 130◦ C.

face. Starting point for this description are the Navier-Stokes equations in two dimensions for a viscous, incompressible Newtonian liquid: −∇[p + φ(h)] + η∇2 u = ρ(∂t u + u · ∇u),

∇ · u = 0,

(3)

with pressure p, disjoining pressure φ(h), viscosity η, density ρ and the velocity u = (u, w) of the liquid. By applying the Navier slip boundary condition u η (4) b= = , ∂z u z=0 κ where η is the viscosity of the melt and κ the friction coeﬃcient at the solid/liquid interface, and using a lubrication approximation (which assumes that the typical length scale on which variations occur is much larger in lateral direction than in vertical direction) we get the following equations of motion: 2b bh ∂x (2ηh∂x u) + ∂x (γ∂x2 h − φ (h)), η η ∂t h = −∂x (hu). u=

(5) (6)

Here, the slip length b is assumed to be much larger than the average ﬁlm thickness H (i.e., for b/H 1). In order to perform a linear stability analysis

Analysis of Dewetting Proﬁles

57

Fig. 5. Comparison of the theoretically expected transition from oscillatory proﬁles to monotonic ones for the strong-slip lubrication model (dashed line) and the thirdorder Taylor expanded Stokes model (solid line).

of the ﬂat ﬁlm of thickness H we introduce a small perturbation δh H traveling in the frame of the moving rim h(x, t) = H + δh exp(kξ)

(7)

ξ = x − s(t)

(8)

with where s(t) is the position of the three-phase contact line. This ansatz leads to the characteristic polynomial of third order in k: (Hk)3 + 4Ca(Hk)2 − Ca

H = 0. b

(9)

One of the three solutions for k is real and positive and therefore does not connect the solution to the undisturbed ﬁlm for large ξ. Hence, this solution is not taken into account any further. The remaining two solutions k1 and k2 are either a pair of complex conjugate numbers with negative real part or two real numbers < 0. These two diﬀerent sets of solutions correspond to two diﬀerent morphologies of the moving rim, i.e., either an oscillatory (k1 and k2 are complex conjugate) or a monotonically decaying (k1 and k2 are real) shape. These morphologies are separated by a distinct transition as indicated by the dashed line in Fig. 5. If both the capillary number Ca and the slip length b are given for a certain ﬁlm thickness H, it is possible to qualitatively predict the shape of the rim. This is in good agreement with our experimental observations: On DTS we observe fast dewetting and expect large slip lengths, and we indeed observe mainly monotonically decaying rims; on OTS we ﬁnd

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O. B¨ aumchen, R. Fetzer, A. M¨ unch, B. Wagner, K. Jacobs

slower dewetting and expect comparatively small slip lengths, and, again in agreement with the theoretical prediction, we observe oscillatory proﬁles. Furthermore, if two solutions k1 and k2 are known (or extracted from experimentally obtained rim proﬁles), the characteristic polynomial can by solved for the unknown slip length b and the capillary number Ca. By that we obtain the solution for b and for Ca from the strong-slip lubrication model: 1 k12 + k1 k2 + k22 , 4H k12 k22 H k12 + k1 k2 + k22 =− . 4 k1 + k 2

blub = Ca lub

(10) (11)

Besides this strong slip model, further lubrication models have been developed [17, 21]. The corresponding weak slip (i.e., for b/H < 1) model always leads to complex solutions for k1 and k2 and therefore predicts only oscillatory rim shapes. 3.2 Stokes Model As already mentioned, the former introduced lubrication model is just valid for large slip lengths compared to ﬁlm thickness. To handle smaller slip lengths it is necessary to use a more recently developed model [17] which is a thirdorder Taylor expansion of the characteristic equation gained by linear stability analysis of a ﬂat ﬁlm using the full Stokes equations: H H H 3 (Hk) + 4Ca 1 + (Hk)2 − Ca = 0. (12) 1+ 3b 2b b This equation again predicts a transition between oscillatory and monotonically decaying rims. In Fig. 5, the respective transition line (solid line) is compared to the one of the strong-slip lubrication model (dashed line). As expected, signiﬁcant deviations between the two models occur for moderate and weak slippage, while good agreement is given for strong slippage. As shown in [17], the transition line for the Taylor expansion is, even for small slip lengths, quite close to the one predicted by the full Stokes equations. Solving (12) for b and Ca leads to additional contributions compared to the strong-slip lubrication model, (10) and (11): 1 k12 + k1 k2 + k22 H − , 4H k12 k22 2 2 2 H k1 + k 1 k 2 + k 2 H 3 k12 k22 =− + . 4 k1 + k 2 6 k1 + k 2

bTaylor = Ca Taylor

(13) (14)

Note that the slip length obtained from the third-order Taylor expansion of the characteristic equation of the full Stokes model equals the slip length

Analysis of Dewetting Proﬁles

59

obtained from the strong-slip lubrication model minus a shift of half of the ﬁlm thickness. We like to point out that, in contrast to previous publications [22, 23], all data sets presented in the following sections were evaluated by applying the third-order Taylor expanded Stokes model, i.e., (13) and (14). This enables us to exclude deviations that occur from the strong-slip lubrication model, which is not valid for smaller slip lengths. 3.3 Method to Extract Slip Length and Capillary Number To ﬁt the proﬁle of a rim obtained by AFM we choose data points on the “wet” side of the rim ranging from about 110% of the ﬁlm thickness to the ﬁlm thickness itself (see Fig. 6). Oscillatory proﬁles are ﬁtted by a damped oscillation as described by (15) with δh0 , kr , ki and φ as ﬁt parameters: δhosci = δh0 exp(kr ξ) cos(ki ξ + φ)

(15)

kr is the real part and ki the imaginary part of k1 and k2 : k1,2 = kr ± iki .

(16)

In case of monotonically decaying rims we deal with a superposition of two exponential decays given by (17): δhmono = δh1 exp(k1 ξ) + δh2 exp(k2 ξ).

(17)

Here we obtain the amplitudes δh1,2 and the decay lengths k1,2 as ﬁt parameters. In some cases, the ﬁtting procedure may not be able to distinguish between two decaying exponentials, leading to two identical decay lengths. In

Fig. 6. AFM cross-section of a rim proﬁle of a hole (about 12 μm radius) in a PS(13.7k) ﬁlm on OTS dewetted at 120◦ C and corresponding ﬁt.

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O. B¨ aumchen, R. Fetzer, A. M¨ unch, B. Wagner, K. Jacobs

Fig. 7. Capillary number Ca versus the ratio of slip length b and ﬁlm thickness H on OTS and DTS obtained from the third-order Taylor expanded Stokes model.

that case, an independent calculation of b and Ca is not possible. However, if the capillary number Ca is known (for example if the viscosity is known and the capillary number was calculated via (2)), one decay length is suﬃcient to extract the slip length. From the parameters k1 and k2 gained by ﬁtting the respective function (15) or (17) to the rim proﬁles, we determined slip lengths via (13) and capillary numbers via (14) for the PS ﬁlms on our substrates at diﬀerent temperatures, i.e., for diﬀerent melt viscosities. As shown in Fig. 7, the capillary number clearly increases non-linearly with the slip length. In the transition region from oscillatory proﬁles to monotonic decaying rims, we have to deal with the fact that proﬁles with just one clear local minimum, but no local maximum in between the minimum and the undisturbed ﬁlm, can be ﬁtted by (17) as well as by (15). In the ﬁrst case, this corresponds to one of the amplitudes δh1,2 being negative. We emphasize that exclusively (15) leads to slip lengths that do not depend on the hole radius R, cf. the recent study [17]. This criterion enables us to justify the choice of (15) as the appropriate ﬁtting function in this region. 3.4 Experimental Tests The independent extraction of capillary number Ca and slip length b from the rim proﬁles allows us to check the consistency of the applied model. Comparing for instance the viscosities gained by rim shape analysis (via the calculated capillary numbers, the dewetting velocities s˙ from hole growth experiments and (2)) to independent viscosimetry measurements shows excellent agreement for both types of substrates, cf. Fig. 8. Furthermore we evaluated certain rim shapes of the same hole but at diﬀerent hole radii, cf. [23]. Although the rim grows in size during dewetting,

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61

Fig. 8. Viscosities obtained from rim proﬁle analysis and the third-order Taylor expanded Stokes model compared to viscosimetry data.

the extracted slip length stays constant within the range of our variation of rim size. The extracted capillary number, however, decreases for increasing hole radius. On the other hand, dewetting slows down with increasing rim size, cf. Fig. 2. The ratio of the measured dewetting velocities s˙ and the extracted capillary numbers gives viscosity data independent of the hole size, which again is in agreement with the expectation. To conclude, all these tests underline the applicability of the former introduced models to our system.

4 Results and Discussion Analysis of rim shapes for diﬀerent temperatures above the glass transition temperature gives slip lengths ranging from less than 100 nm up to about 5 microns. As shown in Fig. 9, the slip length on DTS is about one order of magnitude larger than on OTS. On both substrates, the slip length decreases for higher temperatures. These results are in good agreement with slip lengths obtained by hole growth analysis studies [24]. Plotting the slip length versus melt viscosity, determined by rim analysis, shows non-linear behavior (see Fig. 10). This is at variance with the Navier slip condition (4), where a linear dependency of slip length on viscosity is expected. To evaluate this discrepancy, experiments were performed with polymer melts of diﬀerent molecular weights below the entanglement length. The results are shown in Fig. 10. Variation of molecular weight of the melt gives data that fall on a master curve for each substrate. This means that friction has to be stronger for higher viscosities. This fact is possibly a valuable hint to the molecular mechanism of friction and slippage at the solid/liquid interface and will be further discussed in the following.

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Fig. 9. Slip length b on OTS and DTS versus the dewetting temperature. Rim proﬁle analysis was performed by applying the third-order Taylor expansion of the Stokes model on holes of 12 μm radius in a 130 nm PS(13.7k) ﬁlm.

Fig. 10. Slip length data from Fig. 9 versus the respective ﬁlm viscosity in logarithmic scale.

A scenario described in literature [25–27] for similar systems concerns melt chains penetrating between silane molecules and adhering to the underlaying high-energy silicon substrate. This may lead to polymer chains slipping over grafted polymer chains of the same polymer. For this situation it is reported that ﬂow is expected be strongly dependent on shear rate and grafting density. The density of adhered chains is assumed to be dependent on the perfectness of the monolayer of the silane molecules. From the surface energies we obtained for OTS and DTS, we might expect OTS to build up a denser layer compared to DTS. Higher shear rates obtained for larger dewetting rates i.e. temperatures may lead to a dynamic desorption of adhered chains and, thus,

Analysis of Dewetting Proﬁles

63

change friction at the interface. Molecular dynamics simulations of ﬂowing polymer chains interacting with grafted chains of the same polymer by Pastorino et al. [28] have shown that interpenetration and subsequent anchoring as well as the grafting density inﬂuences slippage. They observe pronounced slippage for lower density of anchored chains. Our further studies concerning the molecular mechanism of slippage focus on X-ray and neutron reﬂectivity experiments of (deuterated) polystyrene melts on Si substrates covered with self-assembled monolayers. These techniques are known to be very sensitive on changes at the solid/liquid interface. First X-ray reﬂectivity measurements on bare silanized wafers show that the silane molecules form very dense monolayers and stand upright on top of the underlying Si substrate [29]. Therefore, the previously discussed explanation of interdigitation does not seem to be very likely. Other explanations assuming structural changes of the substrate, such as bending or tilting of silane molecules, producing a temperature-dependent slip length also seem not to be a major issue in the temperature range of our experiments, though we can not absolutely exclude them. Further, roughness has been shown in miscellaneous studies to inﬂuence slippage [27, 30]. However, due to the fact that our surfaces are extremely smooth, cf. Sect. 2.1, the inﬂuence of roughness might be safely excluded. Other parameters such as the polarizability of the liquid have been shown to inﬂuence slippage dramatically. Cho et al. [31] observed lower slip length for higher polarizable liquid molecules. In addition, the shape of the liquid molecules can also be relevant for diﬀerences in slip lengths [5]. Using the same liquid, the latter two aspects can not be responsible for the diﬀerent slip length on OTS and DTS. The most probable scenario concerning the origin of the huge slip lengths is the formation of a so-called lubrication layer, i.e., a liquid layer of reduced viscosity close to the substrate, that may build up due to migration of low molecular-weight species to the solid/liquid interface or an alignment of liquid molecules at the interface. This slip plane could cause large apparent slip lengths. Systematic ellipsometry measurements on spots where a ﬁlm front has passed the substrate and potentially left a remaining liquid layer, may corroborate this point of view. If lubrication layers are important, then polydispersity of the liquid should play a crucial role on slippage. Dewetting experiments dealing with mixtures or double-layers of polymers with diﬀerent chain lengths are planned. At last, this argument cannot yet aﬀord explaining the diﬀerence in slippage on OTS and DTS of about one order of magnitude. Nevertheless, we can speculate that the formation of this lubrication layer may depend on the interfacial energies of PS and silane brushes, γPS/OTS and γPS/DTS , which are known to be slightly diﬀerent, cf. Sect. 2.1.

5 Conclusion We could show that slippage may strongly aﬀect proﬁles of moving rims surrounding growing holes in thin liquid ﬁlms. Furthermore, the connection of

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ﬁtting parameters concerning the shape of the rims and system parameters as slip length b and capillary number Ca via a theoretical model allows us to quantify slippage. We observe a slip length which is about one order of magnitude larger on DTS than on OTS and which decreases with increasing temperature. Moreover, we record experimentally a non-linear dependency of the viscosity on the slip length, which is at variance with the expectation due to the Navier-slip condition. Friction at the solid/liquid interface has to be enhanced for higher viscosity of the melt. From variation of the molecular weight we know that temperature can be assumed to be an indirect parameter in our system inﬂuencing slippage via the direct parameter viscosity. To conclude, we have presented a method to extract slip lengths from the analysis of rim proﬁles of dewetting polymer ﬁlms. This method is a powerful tool to characterize slippage of dewetting liquid ﬁlms. Further studies will focus on the variation of substrate and liquid properties. Viscoelastic properties of the liquid, i.e., high molecular-weight polymer chains above the entanglement length, may be topic of future research.

Acknowledgments This work was ﬁnancially supported by DFG grants JA 905/3 and MU 1626/5 within the priority program SPP 1164, the European Graduate School GRK 532 and the Graduate School 1276. We acknowledge the generous support of Si wafers from Siltronic AG, Burghausen, Germany.

References 1. Tretheway DC, Meinhart CD (2002) Phys Fluids 14:9–12 2. Tretheway DC, Meinhart CD (2004) Phys Fluids 16:1509–1515 3. Lumma D, Best A, Gansen A, Feuillebois F, R¨ adler JO, Vinogradova OI (2003) Phys Rev E 67:056313 4. Pit R, Hervet H, L´eger L (2000) Phys Rev Lett 85:980–983 5. Schmatko T, Hervet H, L´eger L (2005) Phys Rev Lett 94:244501 6. Craig VSJ, Neto C, Williams DRM (2001) Phys Rev Lett 87:054504 7. Vinogradova OI, Yakubov GE (2003) Langmuir 19:1227–1234 8. Cottin-Bizonne C, Jurine S, Baudry J, Crassous J, Restagno F, Charlaix E (2002) Eur Phys J E 9:47–53 9. Zhu Y, Granick S (2002) Langmuir 18:10058–10063 10. Neto C, Evans DR, Bonaccurso E, Butt H-J, Craig VSJ (2005) Rep Prog Phys 68:2859–2897 11. Lauga E, Brenner MP, Stone HA (2007) Microﬂuidics: The no-slip boundary condition. In: Tropea C, Yarin AL, Foss JF (eds) Springer handbook of experimental ﬂuid mechanics. Springer, Berlin, Heidelberg, New York 12. Bocquet L, Barrat J-L (2007) Soft Matter 3:685–693 13. Ruckenstein E, Jain RK (1974) J Chem Soc Faraday Trans II 70:132–147

Analysis of Dewetting Proﬁles 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31.

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Seemann R, Herminghaus S, Jacobs K (2001) Phys Rev Lett 86:5534–5537 Reiter G (1992) Phys Rev Lett 68:75–78 Seemann R, Herminghaus S, Jacobs K (2001) Phys Rev Lett 87:196101 Fetzer R, M¨ unch A, Wagner B, Rauscher M, Jacobs K (2007) Langmuir 23:10559–10566 Wasserman SR, Tao Y-T, Whitesides GM (1989) Langmuir 5:1074–1087 Good RJ, Girifalco LA (1960) J Phys Chem 64:561–565 Brochard-Wyart F, De Gennes P-G, Hervet H, Redon C (1994) Langmuir 10:1566–1572 M¨ unch A, Wagner BA, Witelski TP (2005) J Eng Math 53:359–383 Fetzer R, Jacobs K, M¨ unch A, Wagner B, Witelski TP (2005) Phys Rev Lett 95:127801 Fetzer R, Rauscher M, M¨ unch A, Wagner BA, Jacobs K (2006) Europhys Lett 75:638–644 Fetzer R, Jacobs K (2007) Langmuir 23:11617–11622 Migler KB, Hervet H, L´eger L (1993) Phys Rev Lett 70:287–290 Hervet H, L´eger L (2003) C R Phys 4:241–249 L´eger L (2003) J Phys: Condens Matter 15:S19–S29 Pastorino C, Binder K, Kreer T, M¨ uller M (2006) J Chem Phys 124:064902 Magerl A (2007) Personal communication Cottin-Bizonne C, Barrat J-L, Boquet L, Charlaix E (2003) Nat Mater 2:237– 240 Cho JJ, Law BM, Rieutord F (2004) Phys Rev Lett 92:166102

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Variation of Transport Properties Along Nanochannels: A Study by Non-equilibrium Molecular Dynamics Filippos Sofos, Theodoros Karakasidis, Antonios Liakopoulos Laboratory of Hydromechanics and Environmental Engineering, Department of Civil Engineering, School of Engineering, University of Thessaly, Pedion Areos, 38834, Volos, Greece. [email protected] Summary. In the present work we calculate the transport properties of liquid argon for Poiseuille ﬂow in a system conﬁned by krypton walls using non-equilibrium molecular dynamics (NEMD) simulations where atoms interact via a Lennard-Jones potential. We examine the eﬀect of channel width, system temperature and external force that drives the ﬂow on diﬀusion coeﬃcient, shear viscosity and thermal conductivity both as total average values for the whole channel, as well as local values across the channel. All transport properties are found to be signiﬁcantly affected by the presence of the solid walls since their values in regions adjacent to the walls are diﬀerent compared to those in layers near the channel centerline. In addition, for small channel widths where wall-ﬂuid interaction aﬀects most of the ﬂuid region, transport properties present diﬀerent behavior in comparison to bulklike behavior. Following the nanoscale methodology described in the paper we can extract transport properties that can be used as input in macroscopic or multiscale simulations.

1 Introduction Interest has grown lately on the ﬂow characteristics of liquids in micro- and nano-scale devices. Experimental studies at the atomic scale (of few atomic diameters) are diﬃcult to perform and, thus, atomistic simulation techniques provide the means for exploring regions that cannot be accessed by experiment, as well as to better understanding of experimental results. Non-Equilibrium Molecular Dynamics (NEMD) is a widely used simulation method for the examination of all the microscopic phenomena taking place inside nanochannels [1, 2]. NEMD is also an eﬀective simulation method for the calculation of transport properties of liquids. The study of transport properties, such as diﬀusion coeﬃcient, shear viscosity and thermal conductivity, presents particular interest for technological applications as well as basic research.

M. Ellero et al. (eds.), IUTAM Symposium on Advances in Micro- and Nanoﬂuidics, IUTAM Bookseries 15, c Springer Science + Business Media B.V. 2009

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Many references for the calculation of diﬀusion coeﬃcient of several monatomic bulk liquids in equilibrium such as argon, neon, krypton etc., can be found in [3–5]. As far as nanoﬂuidics is concerned, a useful review on diﬀusivity issues in slit cylindrical pores can be found in [6]. Shear viscosity and thermal conductivity are calculated in an equilibrium system with the Green-Kubo formalism [7] and good agreement between simulation and experimental data is found. Bitsanis et al. [8] found that the shear viscosity of a model liquid in Poiseuille ﬂow is indistinguishable from the homogenous ﬂuid viscosity down to channel widths of 6–7σ, while, below 4σ, viscosity vs. channel width becomes an oscillating function. Evans [9] found that thermal conductivity for argon in a conﬁned system is linearly dependent to the external ﬁeld applied to drive the ﬂow. Murad et al. [10] related thermal conductivity with the value of the elastic spring constant force used in simulations to keep wall atoms vibrating around their positions. In fact a higher constant results in a lower value for thermal conductivity. In another work [11] it was shown that there is a critical limit in nanochannels’ dimensions, below which the ﬂow properties of liquid argon seem to deviate signiﬁcantly from a continuum behavior. Wall-ﬂuid interactions, as well as the eﬀect of system temperature and the magnitude of the external driving force are of great signiﬁcance in the shape of density, velocity and temperature proﬁles in channel widths from 2.65σ to 18.58σ. Thus, it is expected that the calculation of transport properties in such small channels, especially at layers adjacent to the walls, will be of great interest. In this work we calculate the transport properties of liquid argon for Poiseuille ﬂow between two inﬁnite krypton walls using NEMD simulations. We examine the eﬀect of the channel width, h, from 2.65 to 18.58σ, system temperature from 100 to 150 K and magnitude of the external driving force on diﬀusion coeﬃcient, shear viscosity and thermal conductivity. All these transport properties are calculated both as local values in distinct layers along the channel as well as total average values over the whole computational domain. The paper is organized as follows. In Sect. 2 details on the molecular system modeled are presented. Computational details for the calculation of transport coeﬃcients are presented in Sect. 3. Results are shown and discussed in Sect. 4, and Sect. 5 contains concluding remarks.

2 Description of the Simulation System The NEMD technique is used to simulate planar Poiseuille ﬂow of liquid argon between two inﬁnite plates made of krypton. The simulation step for the system is 10−2 ps. In the beginning ﬂuid and wall atoms are located on fcc sites. Fluid atoms are given appropriate initial velocities in order to reach the desired temperature. Wall atoms are kept around their original fcc sites by the use of an elastic spring force F = −K(r(t) − req ), where r(t) is the position of an atom at time t, req is its initial lattice position and K is the spring constant.

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Fig. 1. The atomic system under examination.

Nos´e-Hoover thermostats at the thermal walls are invoked in order to keep the system’s temperature constant [12, 13]. The system reaches equilibrium state after a run of 105 timesteps (NVE). Then, NEMD simulations are performed with duration of 105 timesteps. The geometry of the simulation system is presented schematically in Fig. 1. Periodic boundary conditions are used along the x- and y-directions. Atomic interactions between similar ﬂuid argon atoms, as well as between wall krypton and ﬂuid argon atoms, are described by a Lennard-Jones 12-6 potential: 12 6 σ σ LJ . (1) − u (rij ) = 4 rij rij As we increase the channel width h from 2.65σ to 18.58σ (or, from 9.04 to 63.26 ˚ A), we keep ﬂuid density constant (ρ∗ = 0.642, in units of σ −3 ). A), and the number of wall atoms Dimensions Lx and Ly (Lx = Ly = 36.15 ˚ (N = 288) are kept constant. The magnitude of the external force varies from 0 to 3.615 pN and temperature ranges from 100 K to 150 K.

3 Computational Details The diﬀusion coeﬃcient can be obtained using the Einstein or the Green-Kubo relation [1], that provide the equivalent results. In our system we employed the Einstein relation, 1 MSD(t) (2) D = lim t→∞ 2dt where d is the dimensionality of the system (d = 1 in one direction, d = 2 in two directions and d = 3 in three directions) and MSD is the Mean Square Displacement N 1 2 (3) MSD(t) = [r j (t) − r j (0)] N j=1 where r j is the position of jth atom and N the number of ﬂuid atoms. In order to use the above relationships in the case of a ﬂow, one has to extract the drift contribution [6].

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Shear viscosity and thermal conductivity are calculated using the GreenKubo formalism as described in [7]. These relations can be used without modiﬁcation in NEMD as long as the system is in the linear regime close to equilibrium [14]. When the system is not in equilibrium, due to the existence of an external force, the magnitude of the external force should be small enough for linearity to hold [4, 15]. Our studies indicate that the system is in the linear regime for Fext = 1.8075 pN. Shear viscosity ηs for a pure ﬂuid is

∞ 1 ηs = dtJpxy (t) · Jpxy (0) (4) V KB T 0 where T is the system temperature, V the ﬂuid volume, KB the Boltzman’s constant and Jpxy the oﬀ-diagonal component of the microscopic stress tensor Jpxy

=

N

mi υix υiy

i=1

−

N N i=1 j>1

x rij

∂u(r ij ) y ∂rij

(5)

u(rij ) is the LJ potential of atom i interacting with atom j, r ij is the distance vector between atoms i and j, and υiα is the α-component (α = x, y or z) of the velocity of atom i. On the other hand, thermal conductivity λ can be calculated by the integration of the time-autocorrelation function of the microscopic heat ﬂow Jqx ,

∞ 1 dtJqx (t) · Jqx (0) (6) λ= V KB T 2 0 where the microscopic heat ﬂow Jqx is given by Jqx =

N N N 1 ∂u(r ij ) rij : mi (υi )2 υix − − I · u(r ) · υix ij x 2 i=1 ∂r ij i=1 j>1

(7)

where υi is the speed velocity magnitude of atom i and I is the unitary matrix.

4 Results and Discussion All transport properties are computed both in distinct layers as local values and as total average values over the whole channel. Each channel is partitioned in layers depending on the width of the channel. Two layers are considered in the channel of h = 2.65σ, three layers for h = 4.42σ, four layers for h = 7.9σ and ﬁve layers for h = 18.58σ. In Fig. 2 we present the layer partitioning for each case. In Fig. 3, we present the calculated diﬀusion coeﬃcient values in ﬁve layers (L1–L5) across the 18.58σ-wide channel at 120 K, under the eﬀect of external driving forces of diﬀerent magnitude. We observe that diﬀusion coeﬃcient in

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71

Fig. 2. Layer partitioning along each channel for the computation of transport properties.

Fig. 3. Diﬀusion coeﬃcient in layers along an 18.58σ-wide nanochannel under the eﬀect of two forces of diﬀerent magnitude at 120 K.

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Filippos Sofos, Theodoros Karakasidis, Antonios Liakopoulos

Fig. 4. Estimation of diﬀusion coeﬃcient in the x-, y- and z-directions at 120 K, for channel widths ranging from 2.65 to 18.58σ.

the two layers adjacent to the walls (L1, L5) is signiﬁcantly smaller compared to the respective values in the three inner layers (L2–L4). This means that diﬀusivity is aﬀected by the wall-ﬂuid interaction in regions near the walls, while in the layers close to the centerline diﬀusivity is close to the bulk value for argon. This is also observed in all the other channel widths examined (2.65, 4.42 and 7.9σ) and, thus, the corresponding results are not presented here. In addition, the magnitude of the external force has no impact on diﬀusion coeﬃcient values, at least in the range studied in the present work. It is of interest to summarize (Fig. 4) the calculated average Dx , Dy and Dz diﬀusion coeﬃcient for the whole channel region at 120 K. We observe that diffusion is isotropic along the x- and y- directions. However, its value along the z-direction (Dz ) is signiﬁcantly smaller for h = 2.65σ and its value increases approaching the values of Dx and Dy as the channel width increases. All diffusion coeﬃcient components converge to the bulk value at about h = 18.58σ and, furthermore, diﬀusion becomes isotropic. Similar behavior is observed for the other two temperatures studied here (100 and 150 K). This behavior indicates the existence of a characteristic channel width below which the diﬀusion behavior is highly anisotropic. In Fig. 5 we summarize the calculated average diﬀusion coeﬃcient D = Dx +Dy +Dz across the channel at each temperature studied. We can see that 3 diﬀusivity is increasing as the channel width increases. In the region between 2.65 to 4.42σ, the slope is steep and diﬀusivity values are practically doubled.

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73

Fig. 5. Total average diﬀusion coeﬃcient at 100, 120 and 150 K, for channel widths ranging from 2.65 to 18.58σ.

Between 4.42 and 7.9σ, the slope is increasing in a slighter way, and from 7.9 to 18.58σ is more moderate as we approach the bulk value at 18.58σ. In Fig. 6, results for shear viscosity in a 4.42σ-wide channel in three layers (L1–L3) under the eﬀect of external driving forces of diﬀerent magnitude at 120 K are summarized. In all layers, the obtained values are greater than the bulk value. However, shear viscosity is slightly smaller near the walls (layers L1, L3) compared to the innermost layer L2. This behavior is attributed to the diﬀerent kind of interaction that ﬂuid atoms encounter in the two cases. Fluid atoms in L2 are surrounded only by similar argon atoms, while ﬂuid atoms in L1 and L3 close to the walls interact both with Kr-wall and Ar-ﬂuid atoms. Krypton atoms have a stronger repulsive interaction than argon atoms and, as a result, atoms adjacent to the solid boundary are forced to move away. For channel widths above 6–7σ shear viscosity is nearly uniform across the channel. In Fig. 7 we present shear viscosity results for h = 7.9σ. Finally, we mention that the magnitude of the external force (in the range of our study) has no impact on shear viscosity values (Figs. 6–7), as we have also seen in the case of diﬀusion coeﬃcients. In Fig. 8, we summarize calculated average shear viscosity values for each channel width studied. In this ﬁgure, average values of shear viscosity over the whole extend of the channel are depicted. Shear viscosity decreases as channel width increases from 2.65 to 7.9σ. At about this channel width, shear viscosity approaches its bulk value for liquid argon [7] and maintains this value

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Fig. 6. Shear viscosity in three layers along a 4.42σ-wide nanochannel under the eﬀect of two forces of diﬀerent magnitude at 120 K.

Fig. 7. Shear viscosity in four layers along a 7.9σ-wide nanochannel under the eﬀect of two forces of diﬀerent magnitude at 120 K.

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75

Fig. 8. Total average shear viscosity at 100, 120 and 150 K, for channel widths ranging from 2.65 to 18.58σ.

up to h = 18.58σ. We attribute this behaviour at small channel widths to the signiﬁcant eﬀect of the walls on ﬂuid atoms. Thermal conductivity results in three distinct layers across a 4.42σ-wide channel under the eﬀect of external driving forces of diﬀerent magnitude at 120 K are presented in Fig. 9. It is evident that the inner channel layer (L2) conducts heat at a higher rate compared to the two layers adjacent to the walls (L1 and L3), since thermal conductivity in L2 has greater value compared to L1 and L3. Thermal conductivity is smaller in the layers adjacent to the walls for h = 7.9σ and 18.58σ, as well. The magnitude of the external force has no impact on thermal conductivity values. To conclude on average thermal conductivity behavior in relation with the channel width, h, we present the calculated thermal conductivity values for each channel studied in Fig. 10. Thermal conductivity increases slightly as h increases from 2.65 to 7.9σ at 100 K, while at 120 and 150 K it is an oscillating function of h. From this channel width and above, thermal conductivity has reached its bulk value for liquid argon [7] and is not aﬀected by the increase of h. It is clear that the existence of the solid boundary diminishes thermal transport, as it is expressed by thermal conductivity, and when solid/ﬂuid interaction is negligible, e.g., away from the walls in channels of width greater than 7.9σ, thermal conductivity is greater. This means that thermal conductivity has its bulk value in regions near the center of the channel, though near the walls its value is signiﬁcantly smaller.

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Fig. 9. Thermal conductivity in three layers along a 4.42σ-wide nanochannel under the eﬀect of two forces of diﬀerent magnitude at 120 K.

Fig. 10. Total average thermal conductivity at 100, 120 and 150 K, for channel widths ranging from 2.65 to 18.58σ.

Variation of Transport Properties Along Nanochannels

77

5 Conclusions We have presented NEMD simulations of planar Poiseuille nano-ﬂow of liquid argon between krypton walls. The eﬀect of channel width and system temperature on transport properties is studied for channel width, h, in the range 2.65–18.58σ, for temperatures in the range 100–150 K, and external forces in the range 0–3.615 pN. We calculated the diﬀusion coeﬃcient, shear viscosity and thermal conductivity both in layers as well as total average values for the whole channel. Diﬀusivity has an isotropic behavior along the directions parallel to the walls, since Dx and Dy are practically the same (x- is the direction along the ﬂow and y- the direction normal to ﬂow but parallel to the walls). On the contrary along the z-direction (normal to the walls) the diﬀusion coeﬃcient Dz Dz (or, similarly, D ) increases as Dz is smaller than Dx and Dy . The ratio of D x y the channel width increases and reaches unity, i.e. isotropic diﬀusion for h = 18.58σ. The calculation of the diﬀusion coeﬃcients in distinct layers parallel to walls reveals that diﬀusion is more important in layers close to the center of the ﬂow and decreases in the layers adjacent to the walls. This behavior is attributed to the fact that these layers “feel” the presence of walls in contrast to the central layers that are surrounded by ﬂuid atoms. As the channel width increases bulk-like behavior is approached. The diﬀusion coeﬃcient increases as temperature increases since the atomic mobility increases and is not aﬀected by the magnitude of the external driving force, at least in the range of the present study. Shear viscosity values are aﬀected by the channel width, h. The smaller the channel width the larger its value. However, above 6–7σ attains its bulk value. Along the channel layers considered, shear viscosity is nearly uniform and is not signiﬁcantly aﬀected by the solid/ﬂuid interaction for h 7σ. Shear viscosity in the cases we examined is larger in the layers near the middle of the channel for h 6σ. Furthermore, shear viscosity results are not aﬀected by the magnitude of the external driving force. Thermal conductivity, as an average total value, increases as channel width rises for h 7σ. For h 7σ it attains its bulk value for liquid argon. The study of thermal conductivity in layers across the channel reveals that its value increases as we approach the center of the channel. For every channel width, thermal conductivity diminishes as temperature increases, while it is not signiﬁcantly aﬀected by the magnitude of the external driving force. The results show that transport properties present a signiﬁcantly diﬀerent behavior than that of the bulk ﬂuid below a critical channel width which depends on the type of interaction of the ﬂuid with the walls and in our case seems to be located in the region 7–18σ. The fact that transport properties are signiﬁcantly diﬀerent than the values obtained for a bulk ﬂuid, as well as that their values are quite diﬀerent close to the walls due to the interaction with wall atoms, are of particular importance for the understanding and design of nanoﬂuidic devices.

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Acknowledgement This research project (PENED) is co-ﬁnanced by E.U.-European Social Fund (75%) and the Greek Ministry of Development-GSRT (25%).

References 1. Rapaport DC (1995) The art of molecular dynamics simulation. Cambridge University Press, Cambridge 2. Allen MP, Tildesley TJ (1987) Computer simulation of liquids. Clarendon, Oxford 3. Fernandez GA, Vrabec J, Hasse H (2004) Int J Thermophys 25:175–185 4. Pas MF, Zwolinski BJ (1990) Mol Phys 73:471–481 5. Meier K, Laesecke A, Kabelac S (2001) Int J Thermophys 22:161–173 6. Karniadakis GE, Beskok A, Aluru N (2002) Microﬂows: fundamentals and simulation. Springer, New York 7. Fernandez GA, Vrabec J, Hasse H (2004) Fluid Phase Equilib 221:157–163 8. Bitsanis I, Vanderlick TK, Tirell M, Davis HT (1988) J Chem Phys 89(5):1733– 1750 9. Evans DJ (1986) Phys Rev A 34:1449–1453 10. Murad S, Ravi P, Powles JG (1993) Phys Rev E 48:4110–4120 11. Sofos F, Karakasidis T, Liakopoulos A (2007) Phys Rev E 79:026305 12. Evans DJ, Holian BL (1985) J Chem Phys 83:4069–4074 13. Holian BL, Voter AF (1995) Phys Rev E 52(3):2338–2347 14. Todd BD, Evans DJ, Daivis PJ (1995) Phys Rev E 52(2):1627–1638 15. Hurst JL, Wen JT (2005) American control conference, Portland, OR, USA, pp. 2028–2033

Estimation of the Poiseuille Number in Gas Flows Through Rectangular Nano- and Micro-channels in the Whole Range of the Knudsen Number Stelios Varoutis, Dimitris Valougeorgis Department of Mechanical and Industrial Engineering, University of Thessaly, Pedion Areos, Volos, 38334, Greece. [email protected] Summary. Based on a recent work (Valougeorgis, Phys. Fluis 19(9):091701.1– 091701.4, 2007), the Poiseuille number for gas ﬂows through channels of orthogonal cross sections is estimated in the whole range of the Knudsen number from the free molecular through the transition and slip regimes all the way up to the hydrodynamic limit. The ﬂow problem is solved via kinetic theory, based on the BGK kinetic equation subject to Maxwell diﬀuse specular boundary conditions. Tabulated results of the Poiseuille number are presented for various aspect ratios and accommodation coeﬃcients. At the hydrodynamic limit the well known results are recovered.

1 Introduction Gas ﬂows through nano- and micro-channels, in many occasions, do not have local equilibrium. In these cases the hydrodynamic equations do not form anymore a closed set, while the traditional no slip boundary conditions break down even before the linear constitutive laws become invalid. Non-equilibrium gas ﬂows may be investigated at a kinetic (mesoscale) level, based on the Boltzmann equation or on simpliﬁed kinetic models [2, 3]. Then, the basic unknown is the particle distribution function and the macroscopic quantities of practical interest are obtained by taking moments of the distribution function. It is noted that approaches based on kinetic theory are capable of providing reliable results with modest computational eﬀort in the whole range of the Knudsen number. Over the years, kinetic type approaches have been used by several authors to study rareﬁed gas ﬂows through channels of various cross sections [4–6]. However, it is interesting to point out that although kinetic solutions are available for a relative large number of internal rareﬁed ﬂows, no estimates on the Poiseuille numbers of such ﬂows have been reported. In a recent work an expression for estimating the Poiseuille number for fully developed gas ﬂows,

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based on the corresponding kinetic solution, has been proposed [1]. Here, this expression and the associated kinetic solution are implemented to obtain the Poiseuille number for the non-equilibrium gas ﬂow through rectangular channels. The results are given in terms of the Knudsen number, the aspect ratio of the cross section and the accommodation coeﬃcient specifying the type of gas-surface interaction.

2 Flow Conﬁguration Consider the non-equilibrium ﬂow of a gas through a long orthogonal duct of length L, connecting two vessels maintained at pressures P1 and P2 , with P1 > P2 . The cross section of the duct is deﬁned by its height H and width W so that −W/2 ≤ x ≤ W/2 and −H/2 ≤ y ≤ H/2. The perimeter and the area of the cross section are deﬁned by Γ = 2(H + W ) and A = H × W respectively, while the hydraulic diameter of the duct, deﬁned by Dh =

4A 2HW , = Γ H +W

(1)

is taken as the characteristic macroscopic length of the problem. The ﬂow is considered as fully developed in the longitudinal direction z (Dh L) and end eﬀects in that direction are neglected. Therefore, the only nonzero component of the macroscopic velocity is the one in the z direction and it is denoted by u (x , y ). Other macroscopic distributions of practical interest are the shear stresses τxz (x , y ) and τyz (x , y ). The main ﬂow parameter is the Knudsen number. However, for purposes related to the more comprehensive presentation of the results, the so-called rarefaction parameter, deﬁned as √ π 1 Dh P0 (2) = δ= μ0 v0 2 Kn is used [5]. Here, in addition to the hydraulic diameter Dh , which is the characteristic macroscopic length, P0 = (P1 + P2 )/2 is a reference √ pressure, μ0 is the gas viscocity at reference temperature T0 and v0 = 2RT0 is the characteristic molecular velocity, with R = k/m denoting the gas constant (k is the Boltzmann constant and m the molecular mass). As it is seen the rarefaction parameter is deﬁned in terms of measurable quantities and it is proportional to the inverse Knudsen number. It is convenient to introduce the non-dimensional spatial variables x = x /Dh , y = y /Dh and z = z /Dh . In addition, we deﬁne the aspect ratio H/W of the duct and the dimensionless cross section A = A /Dh2 and perimeter Γ = Γ /Dh . The macroscopic distributions of the velocity u (x , y ) and shear

Estimation of the Poiseuille Number in Gas

81

stresses τxz (x , y ), τyz (x , y ) are non-dimensionalized as u = u /(v0 XP ), τ = /(2P XP ), where τxz /(2P XP ) and τ = τyz

XP =

Dh dP 1 dP = P0 dz P0 dz

(3)

is the dimensionless local pressure gradient causing the ﬂow. This problem has been solved using kinetic theory by several authors [4, 6, 7]. However, for completeness purposes and in order to be consistent with the introduced non-dimensional analysis we present in the next section the formulation of the problem.

3 Formulation The main unknown is the so-called reduced distribution function φ = φ(r, c), where r = (x, y) is the position vector and c = (ζ, θ) is the molecular velocity vector with 0 ≤ ζ < ∞ and 0 ≤ θ ≤ 2π denoting the magnitude and the polar angle respectively. The ﬂow may be simulated by the linearized reduced BGK kinetic equation given by [6] ∂φ 1 ∂φ + sin θ + δφ = δu − , (4) ζ cos θ ∂x ∂y 2 where u(x, y) =

1 π

2π 0

∞

φζe−ζ dζdθ. 2

(5)

0

is the macroscopic velocity. At the boundaries the gas-surface interaction is modelled according to Maxwell diﬀuse-specular scattering law as φ(+) = (1 − α)φ(−) ,

c · n > 0.

(6)

The superscripts (+) and (−) denote distributions leaving from and arriving to the boundaries respectively, while n is the unit vector normal to the boundaries and pointing towards the ﬂow. The coeﬃcient 0 ≤ α ≤ 1 is the momentum accommodation coeﬃcient and corresponds to the percentage of diﬀuse reﬂection of the gas at the wall. The linear integro-diﬀerential problem deﬁned by (4) and (5), with the boundary condition (6) is discretized in the phase space and then it is solved in an iterative manner. The description of the discretization procedure is omitted here, since it has been repeatedly described and applied in previous work [6, 8–10]. The kinetic solution depends on three parameters, namely the rarefaction parameter δ, the aspect ratio H/W and the accommodation coeﬃcient α. The Poiseuille number Po of the ﬂow, is deﬁned as [1, 11, 12] Po =

8τ w Dh , μ0 u

(7)

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Table 1. The Po number in terms of the rarefaction parameter δ for an orthogonal duct with aspect ratio H/W = 1 and accommodation coeﬃcient α = 1, 0.85, 0.7. Po α = 0.85 0.00377 0.0383 0.405 1.27 2.15 4.32 6.37 8.30 11.8 14.8 17.5 19.8 21.9 23.8 25.5 27.0 28.4 30.8 32.8 36.8 41.7 44.7 46.7 51.3 54.0 55.7 56.3 · · 56.9

δ α=1 0.00478 0.0483 0.504 1.56 2.62 5.21 7.63 9.88 13.9 17.3 20.3 22.8 25.1 27.0 28.8 30.4 31.8 34.2 36.2 39.9 44.5 47.1 48.8 52.6 54.7 56.0 56.4 · · 56.9

0.001 0.01 0.1 0.3 0.5 1 1.5 2 3 4 5 6 7 8 9 10 11 13 15 20 30 40 50 100 200 500 1000 · · ∞

α = 0.7 0.00287 0.0293 0.316 1.00 1.71 3.47 5.15 6.74 9.67 12.3 14.6 16.7 19.0 21.2 22.8 24.3 25.6 29.1 31.1 35.1 40.2 43.4 45.5 52.0 69.2 55.3 56.1 · · 56.9

where τ w is the mean wall shear stress and u is the mean velocity. The mean quantities τ w and u are computed by integrating accordingly the corresponding distributions over the perimeter and the area of the ﬂow cross section. Following the procedure described in [1, 13] it is deduced that Po =

2δ , u

(8)

where the dimensionless mean velocity is estimated by u=

u 4H/W = v0 X P (1 + H/W )2

W/(2Dh )

H/(2Dh )

u(x, y) dy dx. −W/(2Dh )

−H/(2Dh )

(9)

Estimation of the Poiseuille Number in Gas

83

Table 2. The Po number in terms of the rarefaction parameter δ for an orthogonal duct with aspect ratio H/W = 0.5 and accommodation coeﬃcient α = 1, 0.85, 0.7. δ 0.001 0.01 0.1 0.3 0.5 1 1.5 2 3 4 5 6 7 8 9 10 11 13 15 20 30 40 50 100 200 500 1000 · · ∞

α=1 0.00464 0.0469 0.492 1.53 2.59 5.17 7.62 9.91 14.0 17.6 20.7 23.5 25.9 28.0 29.9 31.7 33.2 35.9 38.1 42.4 47.6 50.6 52.6 57.1 59.5 61.1 61.6 · · 62.2

Po α = 0.85 0.00365 0.0371 0.395 1.25 2.12 4.28 6.35 8.30 11.9 15.0 17.8 20.3 22.5 24.5 26.3 28.0 29.5 32.1 34.4 38.8 44.4 47.9 50.2 55.6 58.7 60.8 61.5 · · 62.2

α = 0.7 0.00278 0.0284 0.308 0.98 1.68 3.43 5.12 6.73 9.70 12.4 14.8 17.0 19.0 20.8 22.5 24.0 25.4 28.0 30.2 34.7 40.7 44.5 47.2 53.7 57.6 60.3 61.2 · · 62.2

It is seen that once the kinetic solution is obtained the Poiseuille number of the ﬂow is easily estimated in the whole range of rarefaction. It is also noted that the dimensionless mean wall shear stress is [1, 13] τw =

τw 1 A = . = 2P XP 2Γ 8

(10)

This result, since it is obtained by applying basic principals, is always valid independently of the rarefaction parameter δ, the aspect ratio H/W and the accommodation coeﬃcient α and therefore it is used as a benchmark to test the accuracy of the kinetic calculations.

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Table 3. The Po number in terms of the rarefaction parameter δ for an orthogonal duct with aspect ratio H/W = 0.1 and accommodation coeﬃcient α = 1, 0.85, 0.7. δ 0.001 0.01 0.1 0.3 0.5 1 1.5 2 3 4 5 6 7 8 9 10 11 13 15 20 30 40 50 100 200 500 1000 · · ∞

α=1 0.00367 0.0376 0.416 1.37 2.39 5.02 7.64 10.2 14.9 19.2 23.1 26.5 29.7 32.5 35.1 37.4 39.6 43.3 46.5 52.8 60.7 65.5 68.7 76.0 80.1 82.8 83.7 · · 84.7

Po α = 0.85 0.00285 0.0294 0.333 1.11 1.95 4.12 6.30 8.42 12.4 16.1 19.5 22.5 25.3 27.9 30.3 32.5 34.5 38.1 41.3 47.5 55.9 61.2 64.9 73.6 78.8 82.2 83.4 · · 84.7

α = 0.7 0.00215 0.0224 0.260 0.88 1.55 3.29 5.05 6.77 10.0 13.1 15.9 18.5 21.0 23.2 25.4 27.3 29.2 32.5 35.5 41.7 50.3 56.1 60.2 70.4 76.9 81.4 83.0 · · 84.7

4 Results and Discussion Based on (8) and the associated dimensionless mean bulk velocity obtained by the kinetic solution (9), results for the Poiseuille number are presented in Tables 1, 2 and 3 for orthogonal channels with aspect ratios H/W = 1, 0.5 and 0.1, respectively. In each table results for three diﬀerent values of the accommodation coeﬃcient namely α = 1, 0.85 and 0.7 are included, while the rarefaction parameter δ varies from 10−3 up to 103 . The accuracy of the results has been conﬁrmed in several ways. Depending upon the values of δ, H/W and α the discretization has been progressively reﬁned to ensure grid independent results up to several signiﬁcant ﬁgures. For

Estimation of the Poiseuille Number in Gas

85

Fig. 1. The Po number in terms of the rarefaction parameter δ for orthogonal ducts with various aspect ratios H/W and accommodation coeﬃcient α = 1.

each set of parameters, the dimensionless mean wall shear stress is computed by the kinetic algorithm and in all cases, the analytical result, given by (10), is obtained. Finally, at large values of δ, there is very good agreement between the kinetic results and the ones based on the analytical slip and hydrodynamic solution, given in [12]. Therefore, the tabulated estimates of the Po number are considered as accurate to all three signiﬁcant ﬁgures shown within ±1 to the last one. From Tables 1, 2 and 3 the following remarks can be made. For 10−3 ≤ δ ≤ −1 10 (free molecular regime), the Po number is increased directly proportional to δ. Then, for 10−1 < δ < 10 (transition regime), the Po number keeps increasing as δ is increased but in a slower pace. Finally, for δ ≥ 10 (slip regime), as δ is increased, the Po number is increased very slowly and ﬁnally it approaches asymptotically the analytical results at the hydrodynamic limit (δ → ∞, last row in Tables 1, 2 and 3). These remarks apply to all values of the aspect ratios H/W and accommodation coeﬃcients α. Also, for the same δ, as H/W is decreased the Po number is increased, while as α is decreased the Po number is also decreased. Finally, the behavior of the Po number in terms of δ is clearly shown in Fig. 1. Here, in addition to the three speciﬁc aspect ratios examined before, results for aspect ratios H/W = 0.055, 0.2 and 0.3 are also included.

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5 Concluding Remarks The Poiseuille number for fully developed ﬂows in nano- and micro-channels with various orthogonal cross sections has been provided. The results are based on kinetic theory and therefore are valid in the whole range of the Knudsen number. The type of gas-surface interaction has been also considered. The tabulated results may be useful to comparisons with experimental work as well as to the design and optimization of micro-devices. Also, the implemented methodology may be applied in a straightforward manner to channels of other cross sections provided that the corresponding kinetic solution is available.

References 1. Valougeorgis D (2007) A study on the friction factor of a rareﬁed gas ﬂow in a circular tube. Phys Fluids 19(9):091701.1–091701.4 2. Ferziger JH, Kaper HG (1972) Mathematical theory of transport processes in gases. North-Holland, Amsterdam 3. Cercignani C (1988) The Boltzmann equation and its application. Springer, New York 4. Aoki K (1989) Numerical analysis of rareﬁed gas ﬂows by ﬁnite-diﬀerence method. In: Muntz EP, Weaver DP, Campbell DH (eds) Rareﬁed gas dynamics, vol 118. AIAA, Washington, p 297 5. Sharipov F, Seleznev V (1998) Data on internal rareﬁed gas ﬂows. J Phys Chem Ref Data 27(3):657–706 6. Sharipov F (1999) Rareﬁed gas ﬂow through a long rectangular channel. J Vac Sci Technol A 17(5):3062–3066 7. Valougeorgis D, Naris S (2003) Acceleration schemes of the discrete velocity method: Gaseous ﬂows in rectangular microchannels. SIAM J Sci Comput 25:534–552 8. Sharipov F (1999) Non-isothermal gas ﬂow through rectangular microchannels. J Micromechanics Microengineering 9(4):394–401 9. Naris S, Valougeorgis D, Sharipov F, Kalempa D (2004) Discrete velocity modelling of gaseous mixture ﬂows in MEMS. Superlattices Microstruct 35(3–6):629– 643 10. Naris S, Valougeorgis D, Sharipov F, Kalempa D (2005) Flow of gaseous mixtures through rectangular microchannels driven by pressure, temperature and concentration gradients. Phys Fluids 17(10):100607.1–100607.12 11. White FM (1974) Viscous ﬂuid ﬂows. McGraw-Hill, New York 12. Morini GL, Spiga M, Tartarini P (2004) The rarefaction eﬀect on the friction factor of gas ﬂow in microchannels. Superlattices Microstruct 35:587–599 13. Breyiannis G, Varoutis S, Valougeorgis D (2008) Rareﬁed gas ﬂow in concentric annular tube: Estimation of the Poiseuille number and the exact hydraulic diameter. Eur J Mech B, Fluids. doi:10.1016/g.euromechﬂu.2007.10.002

Moving Contact Line with Balanced Stress Singularities X.Y. Hu, N.A. Adams Lehrstuhl f¨ ur Aerodynamik, Technische Universit¨ at M¨ unchen, 85748 Garching, Germany. [email protected]

Summary. A diﬃculty in the classical hydrodynamic analysis of moving contactline problems, associated with the no-slip wall boundary condition resulting in an unbalanced divergence of the viscous stresses, is reexamined with a smoothed, ﬁnitewidth interface model. The analysis in the sharp-interface limit shows that the singularity of the viscous stress can be balanced by another singularity of the unbalanced surface stress. The dynamic contact angle is determined by surface tension, viscosity, contact-line velocity and a single non-dimensional parameter reﬂecting the lengthscale ratio between interface width and the thickness of the ﬁrst molecule layer at the wall surface. The widely used Navier boundary condition and Cox’s hypothesis are also derived following the same procedure by permitting ﬁnite-wall slip.

1 Introduction Immiscible two-phase ﬂows with moving contact lines occur in a variety of applications, such as coating and biological processes. The moving contact line problem, however, has for many years remained a partially open issue. One of the problems is the validity of the no-slip wall boundary condition, which arises with classical hydrodynamics, where for a no-slip wall an unbalanced divergence of the viscous stress occurs for the discontinuity of ﬂuid velocity, which leads to a violation of the contact-angle condition at a moving contact line [5, 8]. There have been many attempts to resolve the problem by introducing slip boundary condition, such as [6, 10, 14, 15, 20]. Studies by molecular dynamics show that even though considerable contact-line velocities can be obtained [11, 18], the maximum shear rate is still many orders less than that which can violate the no-slip wall boundary condition considerably [19]. If the contact-line velocity is explained as a result of slip boundary condition, it is diﬃcult to give a reasonable explanation why a comparably small shear rate can still result in a large slip. In order to avoid this contradiction, the slip is formulated to be proportional to the sum of viscous stress and/or uncompensated surface stress [14, 15]. Other hand, the moving contact line problem has be explained by no-slip boundary condition, such as the apparent slip M. Ellero et al. (eds.), IUTAM Symposium on Advances in Micro- and Nanoﬂuidics, IUTAM Bookseries 15, c Springer Science + Business Media B.V. 2009

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model (no-slip at the wall) [17] and the diﬀusive interface model [3] in which the viscous-force singularity is circumvented by introducing new dissipation assumptions, such as large interface-relation time or strong interface diﬀusion eﬀects. However, the validity of these assumptions is still in controversial [2, 16].

2 Smoothed, Finite-Width Interface Model for Moving Contact Line In this Letter, we reexamine the hydrodynamics of a ﬂuid/ﬂuid/solid system with a steady moving contact line. Instead of considering sharp interfaces directly, our analysis starts from smoothed, ﬁnite-width interfaces. Given the continuous interfacial free energy density with the form [9] f = 12 σ|∇C|2 + Ψ (C), where C is a color function, σ is a coeﬃcient and Ψ (C) is the bulk energy density, at the state which minimizes F = f dV the interface reaches its equilibrium proﬁle. In this case the surface stress Πij in a two dimensional Cartesian coordinate system is given by ∂C ∂C ∂C ∂C , i, j, k = 1, 2 (1) = σ δ − Πij ij ∂xk ∂xk ∂xi ∂xj where δij is the Kronecker delta. One important property of the surface stress is that one of the principle axes x1 is aligned with the gradient of the color function, the other principle axis x2 is aligned with the interface tangential direction, along which the only non-zero component of the surface stress is the positive normal stress (tension), Π2 2 = σ|∇C|2 . The relation between the surface tension γ and σ for an inﬁnite plane interface is given by

+∞ Π2 2 dx1 . (2) γ=σ −∞

We consider a steady moving contact line with dynamic contact angle α, and velocity Us , as shown in Fig. 1. Around the contact line there are three phases: ﬂuid 1, ﬂuid 2 and the static wall. We deﬁne the color function as

1 in phase l, kl C = k, l = 1, 2, w. (3) 0 else, Note that the color function is discontinuous across the interfaces. In order to obtain a ﬁnite, continuous surface stress, we introduce a two-dimensional smoothing-kernel function [12, 13] W (x, ξ) =

1 ξ2π

e−x

2

/ξ 2

(4)

Moving Contact Line with Balanced Stress Singularities

89

Fig. 1. Smoothed, ﬁnite width interface model for moving contact line problem.

in which ξ is smoothing length and ξ L, L is the characteristic length scale of the system. W (x, ξ) is radially symmetric and has the properties W (x − x , ξ)dx = 1 and limξ→0 W (x − x , ξ) = δ(x − x ). After convolution with the kernel function, the smoothed gradient of the color function pointing towards phase l at a point x in phase k is

kl ∇C (x ) = C kl (x)∇W (x − x , ξ)dx, l = k. (5) Assuming that the smoothed proﬁle deﬁned by ∇C kl (x ) is the interface proﬁle corresponding to an equilibrium form of the bulk energy density, the total surface stress at a point in phase k is can be calculated from (1), by Πij = kl Π ij (∇C ). It is easy to verify by (2) that, for a inﬁnite plane interface l=k between phase k and phase l, the surface tension is σkl . γkl = √ 2πξ

(6)

Note that σkl is a parameter which is proportional to the product of interface thickness and surface tension, i.e. σkl ∝ ξγkl . This relation suggests that when σkl and ξ go to 0 with the same speed, there is a distinguished limit which keeps the surface tension unchanged. Figure 1 indicates the regions of non-vanishing ∇C kl for diﬀerent phase pairings. Note that there are overlap regions near the contact line. In order to study the contact-line dynamics, as shown in Fig. 1, we deﬁne a small square control volume with side length 2ε, ε ξ, with one side on the wall surface so that the interface between the ﬂuid 1 and ﬂuid 2 goes through the control volume. Assuming incompressibility and straight interfaces with α not far from π2 , we obtain

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X.Y. Hu, N.A. Adams

l 2εΠ11

+

w Π21 dx1

+

l 2ετ11

+

w τ21 dx1

=

r 2εΠ11

+

+

f r Π21 dx1 + 2ετ11

f τ21 dx1 ,

(7)

by considering the force balance on the control volume in tangential (wall parallel) direction, where the superscripts l, w, r and f represent the left, wall, right and upper faces of the control volume. Π11 and Π21 are the tangential components of surface stress. τ11 and τ21 are the tangential components of viscous stress. As ε is small and ε ξ, the gradients of the color functions at a point on the face of control volumes can be approximated with the representative values on the contact line. If the contact line is deﬁned to be at the origin of a two-dimensional polar coordinate system, the gradient is given by

∇W (x, ξ)dx, l = k (8) ∇C kl (x → 0) = Ω(θ,θ )

where Ω(θ, θ ) represents the sector between polar angles θ and θ in twodimensional polar coordinates, and θ, θ = 0, α, π depending on the choice of phase pairs. It can be readily obtained that Ω(θ,θ ) ∇W (x, ξ)dx = (sin θ − sin θ , cos θ − cos θ) 2√1πξ . With (1), the tangential surface-stress components σ12 σ12 σ12 12 2 12 21 in (7) are Π11 = 4πξ 2 (1−cos α) , Π21 = 4πξ 2 sin α(cos α −1), Π11 = 4πξ 2 (1+ σ σ σ 21 1w 1w 2w 12 1w 2w = 4πξ cos α)2 , Π21 2 sin α(1 + cos α), Π11 = πξ 2 , Π21 = 0, Π11 = πξ 2 and 2w Π21 = 0, where σ1w , σ1w and σ12 are the coeﬃcients between ﬂuid 1 and wall, ﬂuid 2 and wall, and ﬂuid 1 and ﬂuid 2. Hence, using (6), (7) becomes γ1w +

w f r ∂u1 ∂u1 ∂u1 l ∂u1 1 π ξμ = γ2w + γ12 cos α (9) − +( )− 2 ∂x2 ∂x2 ∂x1 ∂x1 2

where γ1w , γ1w and γ12 are the surface tensions between ﬂuid 1 and wall, ﬂuid 2 and wall, and ﬂuid 1 and ﬂuid 2, respectively. Note that, the shear rates on the faces of the control volume are approximated with the values on face-centers, and that the viscosities of ﬂuid 1 and ﬂuid 2 are assumed to have the same value μ. When the ﬂuids are in static equilibrium the second term on the left-hand-side disappears, (9) becomes 1 γ1w = γ2w + γ12 cos α 2

(10)

where α is the static contact angle. Equation (10) implies that the static contact angle is diﬀerent from that obtained by Young’s relation, except α = π2 . This is not unexpected since we consider the force balance locally within the region with non-vanishing surface stress, the equilibrium contact angle is measured in a scale much smaller than ξ. This value is not necessarily equal to that of Young’s relation in which the contact angle is measured in a scale much larger than ξ. Note that for α = α , an unbalanced surface stress along the

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tangential direction arises in (7), and is balanced by the diﬀerences between the shear stresses. To study the details of the balance between surface forces and viscous forces, it is convenient to deﬁne a layer, as shown in Fig. 1, which has a small thickness of ε and a velocity Uε in the center of the control volume. As Uε → Us for ε → 0, one can study the force balance exactly at the contact line. For any location other than the contact line there is no unbalanced surface stress as in (7) and hence viscous forces are continuous. Therefore, it is straightforward to assume that the ﬂuid velocity on the left and right faces of the control volume match continuously with the wall velocity, and the shear rate on the upper face of the control volume match continuously with that of the bulk ﬂow. Here, three types of wall boundary conditions with diﬀerent wall-slips, i.e. no-slip, ﬁnite-slip and free-slip, are to be considered.

3 Discussion If a no-slip wall boundary condition is applied, the viscous stress is calculated from viscosity and shear rate. As ε is small, a linear approximation of the shear rates is suﬃcient, then (9) can be rewritten as f √ ∂u1 = σ12 (cos α − cos α ), Γ μUε − 2πξμ ∂x2

(11)

√ where Γ = 2π ξε 1 is a non-dimensional parameter. Note that the normal viscous stresses here cancel out because of opposite directions and same magnitudes. For the distinguished limit of a sharp interface Γ 1 as ξ, ε → 0, one has (12) Γ Ca = cos α − cos α 1 f ) ∼ U where the capillary number is deﬁned by Ca = μUs /γ12 , since ( ∂u ∂x2 L is 9π ξ ﬁnite and Uε → Us . Now that the same form as (12) with Γ = 2 ε can be derived if the viscosity of ﬂuid 2 is neglected. Here, the problem discussed in [8] and [5] can be solved: for inﬁnite viscous stress in (7) caused by the discontinuity of ﬂuid velocity in the limit ε → 0, (12) implies that there still is a contact-angle condition at the contact line for a non-vanishing contact-line velocity Us . The reason is that, for ξ → 0 the unbalanced surface stress in (7) becomes inﬁnite as well, and (12) gives the condition for which the two inﬁnite stresses are in equilibrium. Note that the problem of a singular viscous force remains only if ε → 0 and ξ does not vanish. This limit results in Us → 0, i.e. the contact line does not move. A straightforward interpretation of (12) with respect to the microscopic length scales indicates that ξ corresponds to the physical width of the interface and ε to the thickness of the ﬁrst molecules layer at the wall surface, and Γ is just the ratio between the two length scales. Since the thickness of

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the ﬁrst molecule layer is quite close to the molecule size and the interface width is at least several times the molecule size, our result Γ 1 is physically meaningful. Equation (12) is derived from a classical hydrodynamic analysis in which the only considered dissipation mechanism is the viscous force. It is quite surprising that (12) has the same form as a linearized formulation of the molecular-kinetic model [1] which was proposed to discard dissipation due to viscous ﬂow: Γ Ca = (cos α − cos α ), where Γ = η/μ, η is the coeﬃcient of wetting-line friction. Note that η has the units of the viscosity μ, and is always much larger than μ [2], which is in agreement with our result Γ 1. It is also interesting that (12) has the same form as the small-velocity-approximation relation of Shihkmuraev’s interface relaxation model [17]. However, Shihkmuraev obtained a zero contact-line velocity for negligible interface relaxation, as opposite to ﬁnite contact-line velocity obtained in our current analysis. Our results are also in agreement with the molecular dynamics results in [15], where the predicted ratio between the interface width and the thickness of the ﬁrst molecule layer is about 5 and a relation very similar to (12) is found from their simulations, and gives Γ approximately 3. If a ﬁnite-slip wall boundary condition is applied, since there is a velocity discontinuity between the ﬂuid and the wall, the viscous stress on the wall for ε → 0 is given as βUs , where β is the slip-coeﬃcient. In this case (9) becomes βUs − μ

∂u1 ∂x2

f =

√

2π

σ12 (cos α − cos α ). ξ

(13)

Equation (13) is in agreement with the slip-wall boundary condition of Qian, Wang & Sheng [14], which states that the wall slip is proportional to the sum of the viscous stress and the uncompensated Young stress. An important result diﬀerent from that of the sharp interface model is that for a given contactline velocity the dynamic contact angle is strongly aﬀected by the shear rate of the bulk ﬂow. Note that (13) is valid only for ﬁnite-thickness interfaces and implies a surface-force singularity if the interface thickness tends to zero. To eliminate the singularity the dynamic and static contact angles should be equal α = α , which explains the underlying reason of Cox’s hypothesis [4] for a macroscopic analysis stating that wall-slip is permitted and the contact angle is independent of contact-line velocity. A result of the surface-force balance with α = α from (13) is the widely used Navier boundary condition Us = 1 f λ( ∂u ∂x2 ) , where λ = μ/β is the slip length. Our analysis does not allow for a free-slip boundary condition [7] because free slip would result in the ﬁrst term on the left-hand-side of (13) to vanish, which may lead to an un-physical an decrease of the contact angle for an advancing contact line. For the sharp-interface limit with free slip there is no force balance, no matter whether the dynamic and static contact angles are the same or not.

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4 Conclusion To summarize, we have studied the force balance at a moving contact line with diﬀerent boundary conditions. It is found that, for the sharp-interface limit, both the ﬁnite-slip and the no-slip wall boundary conditions are possible. With the ﬁnite-slip wall assumption, the analysis explains that the previously used Cox’s hypothesis and the Navier boundary condition are essential for a force balance. However, without extra interface relaxation or diﬀusion assumption, the analysis also suggests that the no-slip wall boundary conditions is still valid along with a contact-angle condition, which agrees with several previous studies on dynamic contact angles. More importantly, since it is in agreement with the molecule slip condition [19], the no-slip wall boundary condition can serve for obtaining more reliable numerical predictions of moving contact line problems.

References 1. Blake TD (1993) Dynamic contact angles and wetting kinetics. In: Berg JC (ed) Wettability. Dekker, New York 2. Blake TD (2006) The physics of moving wetting lines. J Colloid Interface Sci 299:1–13 3. Chen HY, Jasnow D, Vi nals J (2000) Interface and contact line motion in a two phase ﬂuid under shear ﬂow. Phys Rev Lett 85:1686–1689 4. Cox RG (1986) The dynamics of spreading of liquids on a solid surface. Part 1. Vicous ﬂow. J Fluid Mech 168:169 5. Dussan VEB (1979) On the spreading of liquid on solid surface: Static and dynamic contact lines. Annu Rev Fluid Mech 11:371–400 6. Hocking LM (1977) A moving ﬂuid interface. Part 2. The removal of the force singularity by a slip ﬂow. J Fluid Mech 79:209–229 7. Huh C, Mason SG (1977) The steady movement of a liquid meniscus in a capillary tube. J Fluid Mech 81:401–419 8. Huh C, Scriven LE (1971) Hydrodynamic model of steady movement of a solid/liquid/ﬂuid contact line. J Colloid Interface Sci 35:85–101 9. Jacqmin D (1999) Calculation of two-phase Navier-Stokes ﬂows using phase-ﬁeld modeling. J Comput Phys 155:96–127 10. Jacqmin D (2000) Contact-line dynamics of a diﬀusive ﬂuid interface. J Fluid Mech 402:57–88 11. Koplik J, Banavar JR, Willemsen JF (1988) Molecular dynamics of poiseuille ﬂow and moving contact lines. Phys Rev Lett 60:1282 12. Monaghan JJ (1992) Smoothed particle hydrodynamics. Annu Rev Astron Astrophys 30:543 13. Natanson IP (2000) Theory of function of a real variable, vol II. Frederick Ungar, New York 14. Qian TZ, Wang XP, Sheng P (2003) Molecular scale contact line hydrodynamics of immiscible ﬂows. Phys Rev E 68:016306 15. Ren W, E W (2007) Boundary conditions for moving contact line problem. Phys Fluids 19:022101

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16. Robbins M (2005) Flow boundary conditions for ﬂuid mixtures at solid walls and moving contact lines. APS Meeting Abstracts, pages F2+, November 2005 17. Shihkmuraev YD (1997) Moving contact lines in liquid/liquid/solid systems. J Fluid Mech 334:211–249 18. Thompson PA, Robbins MO (1989) Simulations of contact-line motion: Slip and the dynamic contact angle. Phys Rev Lett 63:766 19. Thompson PA, Troian SM (1997) A general boundary condition for liquid ﬂow at solid surface. Nature 389:360 20. Zhou MY, Sheng P (1990) Dynamics of immiscible-ﬂuid displacement in a capillary tube. Phys Rev Lett 64:882–885

Part III

Complex Fluids

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Clariﬁcation and Control of Micro Plasma Flow with Wall Interaction Takeshi Furukawa Aerodynamic Design Technology Section, Supersonic Transport Team, Aviation Program Group, JAXA (Japan Aerospace Exploration Agency), 6-13 Osawa, Mitaka, Tokyo, Japan. [email protected] Summary. Micro plasma ﬂow generated in the channel of a hall-eﬀect accelerator is clariﬁed using unsteady numerical analysis and with experimental data by highlighting on propellant temperature, plasma parameters and channel-wall condition in order to suppress plasma instability and magnetic-pole overheating, and the following ﬁndings are obtained: “Average length of ionization-zone” introduced as a new physical parameter, which could be varied by neutral particles velocity, is correlated conversely with large-amplitude of plasma instability that should be solved to improve the operational stability and the system durability. Then we propose an enhanced hall-eﬀect accelerator with new design concepts giving simultaneous solution for reducing the instability and the accelerator core overheating. And the mechanisms of controlling amplitude are clariﬁed from the viewpoint of the spatiotemporal variations of plasma parameters and electromagnetic ﬁeld when neutral species temperature is changed.

Keywords: Micro plasma ﬂow, Unsteady numerical analysis, Hall-eﬀect accelerator, Micronization, Instability, Overheating.

1 Introduction 1.1 Electric Propulsion As space propulsion, electric propulsion converts sunlight energy or the like into electric energy, uses the electric energy to turn a propellant into plasma through various methods, accelerates the generated plasma in various forms, and generates propulsion. Electric propulsions can be largely divided into three types, namely an electrostatic acceleration type, an aerothermal acceleration type, and an electromagnetic acceleration type, in accordance with diﬀerences in the thrust generation mechanism. An ion engine, representing the electrostatic acceleration type, generates plasma through direct current discharge or the like, and obtains thrust by

M. Ellero et al. (eds.), IUTAM Symposium on Advances in Micro- and Nanoﬂuidics, IUTAM Bookseries 15, c Springer Science + Business Media B.V. 2009

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accelerating and injecting ions in the generated plasma using an electrostatic ﬁeld (of approximately 1000 V) applied to a porous grid. A considerably higher speciﬁc impulse (between 2000 and 7000 sec) than that of a chemical propulsion can be achieved with high eﬃciency (up to 80%), but due to the restrictions of the space charge limited current rule, the thrust density is comparatively small (thrust = several mN to 200 mN), and in the low speciﬁc impulse range, propulsion eﬃciency deteriorates dramatically. Several types of plasma generation methods, including an RF-type method, have been proposed. A thrust generation mechanism of an arc jet-type electric propulsion, which serves as an aero-thermal acceleration-type propulsion, subjects a propellant to ionization and Joule heating through an arc discharge formed between a rod-shaped cathode and a ring-shaped anode disposed coaxially with the rod-shaped cathode, and then expands and accelerates the heated plasma using a supersonic nozzle. High thrust density (thrust = 150 mN to 2 N) is obtained, but heat loss onto the wall is large, and therefore the propulsion eﬃciency is low (30 to 40%) in comparison with an electrostatic accelerationtype propulsion, and the speciﬁc impulse (between 500 and 2000 seconds) is not especially high. As regards commercial viability, the following important problems remain unsolved: cathode wear, which determines durability, reaches 5 μg/C during a steady state operation, and this wear must be reduced; heat loss must be improved. A MPD (Magneto-Plasma-Dynamic)-type electric propulsion, which is propulsion representing the electromagnetic acceleration type, has a similar basic structure to the arc jet-type electric propulsion. The propellant is heated and turned into plasma by arc discharge, whereupon a high discharge current in the order of kA is caused to ﬂow between electrodes to induce a magnetic ﬁeld in a circumferential direction. The generated plasma is accelerated in an axial direction by a Lorentz force, which is the interaction between the induced magnetic ﬁeld and the current, and as a result, thrust is obtained. A feature of the MPD-type electric propulsion is that it obtains the highest thrust (up to 10 N) of all electric propulsions, and is therefore promising as propulsion for interplanetary navigation of the future. The obtained speciﬁc impulse has a wide range of approximately 1000 to 6000 sec, but at present, the typical propulsion eﬃciency of approximately 10 to 50% remains low. 1.2 Hall-eﬀect Accelerator Finally, the hall-eﬀect accelerator according to the present research will be described. A hall-eﬀect accelerator has a ring-shaped, axisymmetrical acceleration channel that turns a neutral particle (propellant) introduced through an anode hole into plasma and accelerates a generated ion. When the acceleration channel length is designed to be shorter than the ion cyclotron radius and longer than the cyclotron radius of an electron (which is emitted from a cathode and caused to ﬂow in reverse through the acceleration channel in an

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anode direction), an electron is subjected to E × B drift in the circumferential direction by the interaction between an axial electric ﬁeld E and a radial external magnetic ﬁeld B is induced. By accelerating the ion using an electric ﬁeld generated through the electromagnetic interaction between the hall current and the externally applied magnetic ﬁeld B, the hall-eﬀect accelerator acts in an identical manner to the “electrostatic acceleration type”, and yet the hall-type electric propulsion also shares features with the “electromagnetic acceleration type” in that the accelerated ion is neutralized using an electron from the cathode and a high thrust density is obtained regardless of the space charge limited current rule by maintaining the quasi-neutrality of the acceleration-zone. Hall-eﬀect accelerator or hall-eﬀect accelerator which could product ion beam density over 103 times as high as that of electrostatic accelerator which is used regularly as beam heating device, is expected as ion beam source for nuclear fusion, because it is proven that the beam heating method could accelerate ion to high energy beam by electric ﬁeld and heat plasma to ultra high temperature of 100 million degrees or more [1]. Moreover it is expected as plasma rocket engine generating high-thrust eﬃciency over 50% in speciﬁc impulse 1000–3000 sec. Hall-eﬀect accelerator has the possibility of achieving thrust density identical to the electromagnetic-acceleration-type propulsion without restriction of space charge limited current rule. In particular, high-power hall-eﬀect accelerator of 50–100 kW class have been developed to achieve high-thrust, giving 1500 mN in SPT290 for example. It is expected to shorten the ﬂight time in Mars mission in the future and has already sustained an endurance test in NASA. Micro hall-eﬀect accelerator is a great deal of attention for micro/nano satellite station-keeping and orbit-correction, as sub-several kW class low-power thruster giving thrust of tens-hundreds mN. Hall-eﬀect accelerator is called hall accelerator or closed electron drift thruster et al. The name ‘hall’ originates in the hall current based on the electrons azimuthal drift and ‘closed electron drift’ the azimuthal drift of electrons. The two main modern variants of hall-eﬀect accelerator are classiﬁed the linearmodel and the sheath-model: The linear-model whose acceleration channel length is longer than the channel width and whose acceleration channel wall is made of ceramics such as Boron-Nitride and alumina. On the other hand, the sheath-model is whose acceleration channel length is shorter than the channel width and acceleration channel is wall electric conductor as copper [2–4]. 1.3 Phenomena in Micro Plasma Flow Plasma instabilities/oscillations of various frequencies generate in hall-eﬀect accelerator. In particular, the large-amplitude plasma instability in the tens of kHz at high-voltage mode of DC regime has been a serious problem that should be solved to improve the operational stability and the system durability since it is a main cause for operational instability and stoppage. It is an urgent item for development hall-eﬀect accelerator design to probe the

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physical mechanisms of this instability since a long-time stable operation is required in a space mission [2–5]. From the previous works [5], it was found that low-frequency oscillation was caused by the ionization instability, i.e., the periodic ionization. Necessity of physics-related research on plasma characteristics in hall-eﬀect accelerator has originated from such backgrounds. But coupling between electric and magnetic ﬁelds, and among electron, ion and neutral transports are rather complicated in hall-eﬀect accelerator where the plasma properties are far from clear understanding [2]. In this research, the dependencies of both plasma instability and the accelerator performances (thrust, speciﬁc impulse and thrust eﬃciency) on varied operational parameters (discharge voltage, neutral species temperature and magnetic ﬁeld proﬁle) are estimated using both unsteady numerical analysis and experimental data. Besides, we introduce a new physical parameter “average length of ionization-zone”, to suggest the operational condition and the design concept for making it possible to operate with the smaller amplitude. In addition, the spatiotemporal variations of plasma properties and electromagnetic ﬁeld in acceleration channel are investigated at the peculiar times in single-cycle of the plasma instability viz. a period for both the generation/acceleration of plasma.

2 Numerical Analysis 2.1 Fundamental Equations The ionization and acceleration mechanisms of hall-eﬀect accelerator are extremely complicated as shown in Fig. 1. A set of ﬂow-ﬁeld equations composed of the following (1), (2), (5) and (6) are solved using the 3rd-order upwind scheme for space-integration and the 4th-order Runge-Kutta-Gill method for time-integration:

Fig. 1. Schematic view of acceleration part of hall-eﬀect accelerator (left) and photo of operation (right).

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a) Conservation equations Conservation equations for ions and neutral species with the production/loss terms of plasma are given by (1) and (2): ∂Γi ∂ni + = ne (νP − νL ), ∂t ∂z ∂Γn ∂nn + = −ne (νP − νL ). ∂t ∂z

(1) (2)

Here Γ and n are ﬂux and number density. And subscripts e, i and n mean electron, ion and neutral species, respectively. t is time and z is position from anode (3) νp = nn σion 8kTe /πme (1 + eVion /kTe ) exp(−eVion /kTe ) and

νL = 2(router − rinner ) kTe /mi exp(−0.5)

(4)

are ion-production rate per unit volume and ion-loss rate per unit volume. rinner and router are inner radius and outer radius of acceleration channel, respectively. σion , k, T , m, e (not subscripts) and Vion are ionization crosssection, Boltzmann constant, temperature, particle mass, elementary charge andionization voltage. b) Momentum equations Ions motion is hardly ever aﬀected by magnetic ﬁeld and ions are accelerated toward the acceleration channel exit in a quasi-collisionless way by the large electric ﬁeld that is induced by the decrease of plasma conductivity due to radial magnetic ﬁeld [6]. Motion of ions can be described by the momentum equation (5) ∂Γi vi ni eE ∂Γi + = + ne (νP − νL )vn . (5) ∂t ∂z mi Here, v and E are velocity and electric ﬁeld, respectively. Also the ﬂow of electrons toward the anode across radial magnetic ﬁeld lines is given by the following momentum equation (6), taking into account of electric ﬁeld, divergences of density and electron temperature. Γe = −μe ne E − DeD

∂ne 1 ∂Te − DeT ne . ∂z Te ∂z

(6)

Here, μe is electron mobility. DeD and DeT = Te (∂DeD /∂Te ) are densitydiﬀusion coeﬃcient and thermal-diﬀusion coeﬃcient, respectively. c) Electron energy balance equation

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Energy balance equation for electrons is written as follows: 3 ∂(kTe ne ) 5 ∂(kTe Γe ) + 2 ∂t 2 ∂z = −ζeΓe E − (1 + β)ne νP εion −

1 − 2δ 2 + χ ne νL kTe . 1−δ 1−δ

(7)

Here, ζ is ratio of energy given to thermal electron to input power. β excitation parameter. εion ionization energy. δ secondary-electron emission coeﬃcient = 0.141(kTe /e)0.576 for BN and xenon. χ = (1/4) 8mi /πme × ln(1 − δ) exp(0.5). 2.2 Boundary Conditions Results of calculation are strongly dependent on boundary conditions. Referencing to the experiment [4], the ion density at anode is set to 5.0 × 1016 m−3 and the electron temperature at channel exit 3.5 eV; which is one tenth of the voltage still remaining at exit of channel 35 V. This assumption looks valid from the viewpoint that the electron temperature in hall-eﬀect accelerator is nearly one tenth of the voltage. Also, note that the equilibrium discharge current after a quasi-steady state is reached independent of initial conditions.

3 Results and Discussion 3.1 Plasma Instability The mechanism of plasma instability in the tens of kHz is as follows [5]: 1) By ionization, plasma density increases and neutral species density decreases in acceleration channel as shown in Fig. 2. 2) Under electric ﬁeld, the reduction of plasma particle is faster than the supply of neutral. 3) Neutral species is supplied from anode oriﬁce (all the while, ionization is hardly occurred since electron-neutral ionization collision is low). 4) Neutral species gets to a threshold for ionization collision and then ionization suddenly cause, and it goes back to 1). The plasma instability calculated for varied discharge voltages Vd are shown in Fig. 3. Calculation conditions are as following; neutral species temperature Tn = 1000 K, magnetic ﬁeld strength B = 0.062 T (value of peak strength in magnetic ﬁeld proﬁle which is almost constant in acceleration channel) and mass ﬂow rate of propellant m ˙ = 1.36 mg/s in Xenon. By introducing density-diﬀusion term into electron momentum and energy balance equations, the waveform of discharge current in calculation closely approaches almost the same U-shaped wave as one observed experimentally [5]. Accordingly density-diﬀusion besides electric ﬁeld plays an important role in the motion of electron in acceleration channel; thus, it is absolutely required in the study on the plasma instability. On the other hand, little diﬀer-

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Fig. 2. Cyclical phenomenon in acceleration channel.

Fig. 3. Time evolutions of plasma instabilities for varied discharge voltage in numerical analysis.

ence is seen in frequency fd and amplitude Ad (≡ Id,max (maximum discharge current)−Id,min (minimum discharge current)); amplitude between both ends is adopted because the waveform of instability is U-shaped one, even if the thermal-diﬀusion term is included. As shown in Fig. 4, experimentallyobserved frequency (fd = 15 ∼ 20 kHz) and amplitude (Ad = 1 ∼ 2 A) are almost agreement with calculations (fd = 35 ∼ 40 kHz and Ad = 2 ∼ 5 A) in Vd = 180 ∼ 220 V: Some diﬀerences are likely to arise from the ﬁxed value of parameters β, ζ, and boundary condition as voltage at channel exit. 3.2 Average Length of Ionization-Zone In discussing Ad in an unsteady numerical analysis, we introduce a new physical parameter, “average length of ionization-zone”. In our hall-eﬀect accelerator, ionization arises immediately behind anode. So the position where 5% of the initial neutral species supplied from anode has been consumed by ionization is assumed to be the ionization beginning position zion,beg. . And the position where 95% of the initial neutral species supplied from anode have been consumed by ionization is assumed to be the ionization completion

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Fig. 4. Dependencies of calculated and experimentally-observed amplitude and frequency on discharge voltage.

Fig. 5. Deﬁnition of average length of ionization-zone.

position zion,com. . Here zion,beg. and zion,com. turns out to be a function of time as shown in Fig. 5. So the time average value of zion,beg. and zion,com. is deﬁned as average ionization beginning position zion,beg.av . and ionization completion position zion,com.av ., respectively. Accordingly the distance between zion,beg.av . and zion,com.av . is deﬁned as average length of ionizationzone Lion,av . (≡ zion,com.av . − zion,beg.av . ). 3.3 Parametric Estimations of Instability and Performances a) Dependencies on neutral species temperature Firstly, the dependencies of plasma instability on neutral species temperature Tn are discussed. As shown in Fig. 6, fd and Ad in both calculated and experimental results increase with Tn . On the other hand, Lion,av . increases for the larger Tn . Now, interpreting this result in a practical way, Ad may be

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Fig. 6. Deﬁnition of average length of ionization-zone.

decreased if Tn is raised. Namely, the higher Tn increases the sound speed of neutral species vn and Lion,av . ; consequently, the rapid rise of plasma density np at time of ionization is suppressed and Ad decreases. Thus, the changing of Tn can be used as a method of controlling Ad . Next, Fig. 6 also shows the dependencies of performances (thrust F and eﬃciency η) on Tn . F , which is the product of ion velocity vi , ion density ni , cross-sectional area of acceleration channel exit and elementary charge, increases with Tn . This is because ion velocity vi or density ni increases with Vd in terms of the deﬁnition of F . On the other hand, η hardly changes for the larger Vd . This reason is because the increase in the product of Vd and average of discharge current Id,av . is roughly equivalent to the increase in F squared when Vd increases from the deﬁnition of η. b) Dependencies on magnetic ﬁeld conﬁguration The results mentioned in a) were calculated in the magnetic ﬁeld proﬁle of Case (c); which was based on the values measured at the radial midpoint of the acceleration channel when coil current for generating magnetic ﬁeld equaled 4 A. The magnetic ﬁeld proﬁle is expressly important parameter deciding all sorts of performances, as is known well. So in this section it is discussed that the eﬀects of the magnetic ﬁeld proﬁle on instability and performances. In particular, only the magnetic ﬁeld proﬁle is varied in each case compared Channel Exit B(z)dz = const., namely the total under the following condition, Anode of magnetic ﬁeld applied in the acceleration channel is constant. Five cases diﬀered in curvature as shown in Fig. 7 are compared: Case (a): the magnetic ﬁeld proﬁle of the present hall-eﬀect accelerator: Characteristic in magnetic ﬁeld proﬁle is small curvature and the largest magnetic ﬁeld intensity is at the center in the axial direction of the acceleration channel, or at the position 8 mm from anode.

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Fig. 7. Magnetic ﬁeld proﬁles with various curvatures.

Case (b): the magnetic ﬁeld proﬁle that the strength is uniform [ﬂat type]. Case (c): the magnetic ﬁeld proﬁle of SPT-type. Case (d) and (e): the peak position of the magnetic ﬁeld strength is 2 mm and 12 mm, respectively when the total of magnetic ﬁeld applied in the acceleration channel is the same as that of the present hall-eﬀect accelerator. Case (f): the magnetic ﬁeld proﬁle with the larger curvature than the present hall-eﬀect accelerator. As shown in Fig. 8, from the comparison among Case (a), Case (d) and Case (e), as the peak position of the magnetic ﬁeld strength is set to at the position closer to anode, thrust becomes larger and amplitude is suppressed but propulsion eﬃciency is lower. For the larger curvature the larger thrust and propulsion eﬃciency are but amplitude gets smaller, from the comparison between Case (a), Case (b) and Case (f). The smaller curvature the magnetic ﬁeld proﬁle is set to, amplitude gets smaller. The smaller curvature increases magnetic ﬁeld strength near anode. Hence the ionization beginning position zion,beg. where 5% of the initial neutral species is consumed by ionization closes to anode. Also since zion,com. hardly change, ionization-zone length increase; consequently, and then the rapid rise of plasma density at time of ionization is suppressed and Ad decreases. Accordingly, the curvature of magnetic ﬁeld proﬁle can be used as a method of controlling Ad . In the same way, F and Isp. increase and η decrease when magnetic ﬁeld proﬁle has the small curvature. This seems because vi or ni increases with Vd in terms of the deﬁnitions of F and Isp. . On the other hand, η hardly changes for the larger discharge voltage. This reason is because the increase in the product of Vd and Id,av . is roughly equivalent to the increase in F squared when Vd increases from the deﬁnition of η. Moreover the magnetic ﬁeld proﬁle of the SPT-type accelerator installed in commercial satellite (Case (c)) has the smaller thrust and propulsion eﬃciency and the larger amplitude. As a result, it is found that Case (d) has the best performance in thrust and amplitude.

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Fig. 8. Dependencies of maximum, minimum, average and amplitude of discharge current, and average length of ionization-zone, and thrust and thrust eﬃciency on magnetic ﬁeld proﬁle.

3.4 Spatiotemporal Variations of Plasma Parameters and Electromagnetic Field In this section, the distributions of plasma parameters and electromagnetic ﬁeld in the acceleration channel in single-cycle of plasma instability are examined. a) Time evolutions Figures 9 (a)–(d) show the spatial distributions of electron temperature Te , np , vi and electric ﬁeld E in channel at the peculiar times of plasma instability in Fig. 3. First, the position of the highest Te at each time, which is in nearby not 10 mm but 5 mm from anode at the time II, for example (which is the turning point where Te changes from the increasing tendency to the decreasing one viewing from channel exit, which is caused by consuming, for ionization, the energy that the electrons moving toward anode have acquired from electric ﬁeld) closes to anode in order of I, II, III and ?V. Also, the maximum value of Te becomes smaller in the same order. Next, the value of the largest np , whose position are nearly in agreement with the position of highest Te , has the following relationship; II < I ≈ III < IV. And that’s why ionization might start at I and ﬁnish at IV approximately. Moreover, the turning positions of vi , which are in nearby 8 mm from anode at II, are almost the same positions of maximum of Te . Finally, regarding to E, the turning positions are all but in agreement with the maximum position of Te . The order of the maximum value of E is equal to that of Te . b) Dependencies on neutral species temperature

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Fig. 9. Spatial distributions of electron temperature, plasma density, ion velocity and electric ﬁeld in acceleration channel at the peculiar times ?-?V of plasma instability in Fig. 3.

Fig. 10. Spatial distributions of plasma density and ion velocity for both maximum and minimum neutral species temperatures (600 K and 1400 K) at the distinctive times ?I and IV in Fig. 11.

Figures 10 show the spatial distributions of np and vi for both maximum and minimum Tn (600 K and 1400 K) at the distinctive times ?I and IV in Fig. 11. For the higher Tn , np increases at the time II and decreases at the time IV. On the other hand, vi decreases at the time II and almost remains

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Fig. 11. Time evolutions of discharge currents for both maximum and minimum neutral species temperatures (600 K and 1400 K).

Fig. 12. Dependencies of frequency, amplitude on β (left) and ζ (right).

unchanged at the time IV. Ad decreases 0.7 A; Id,max decreases 0.3 A and Id,min increases 0.4 A. Hence Ad is closely connected with the diﬀerence between np in the times II and IV. Consequently, to minimize this diﬀerence is desirable for decreasing Ad and increasing F , Isp. and η. 3.5 Inﬂuence of Fixed Parameters upon Calculations Qualitative disagreements between calculations and experimental data are discussed in this section. Calculated f (35–40 kHz) and Ad (2–5 A) are larger than experimentally-observed ones (15–20 kHz and 1–2 A), as shown in Fig. 4. Main reason for the disagreement seems to be based on the following ﬁxed parameter; (1) excitation parameter β or (2) distribution function of electrons ζ: a) Although β is set to 2.0 in this analysis, f decreases and Ad gets smaller if β is set larger than 2.0, as shown in Fig. 12. As a result, the calculated results approach the experiments. b) Also, if the value of ζ is less than 1.0, calculations are closer to experiments (Fig. 12).

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As other reasons, besides diﬀusion coeﬃcient set to 1/(16B), it can be considered that the potential still remaining at exit of acceleration channel Vexit is ﬁxed with regularity (35 V): It turns out experimentally that Vexit increases in connection with Vd . Therefore, the changing width of Vd (180 V to 220 V) in calculation may be taken larger than the one of actuality. 3.6 New Concept for the Higher Performance We propose a hall-eﬀect accelerator with new design concepts giving simultaneous solution for reducing the instability and the accelerator core overheating caused by plasma in acceleration channel. Cross-sectional view of this hall-eﬀect accelerator is shown in Fig. 13. Neutral species supplied from the external tube is introduced to anode oriﬁce through the circulation ducts made in the internal core of accelerator. Neutral species is heated through heat-exchange between the accelerator core overheated by plasma in acceleration channel and the neutral species passing in the circulation ducts. As a result, neutral species temperature is increased and the accelerator core is cooled. Also, this hall-eﬀect accelerator has belongs to the linear-model whose channel length is longer than channel width. The choking portion in anode is formed by extending the anode and reducing the size of the anode hole. The propellant can be choked, enabling an increase in acoustic velocity. The magnetic ﬂux line distribution of the ionization/acceleration channel is formed so as to optimize the ion acceleration

Fig. 13. Cross sectional view of hall-eﬀect accelerator of new concept.

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vector, whereupon the propellant conduit is disposed in the magnetic pole of the propulsion, or more speciﬁcally in the vicinity of the acceleration channel. Thus, the magnetic pole, which is overheated by the generated plasma, can be cooled, and at the same time the propellant can be heated. Furthermore, the heated propellant is choked immediately before being introduced into the ionization/acceleration channel by the throat region or throttling hole provided immediately before the ionization/acceleration channel, and as a result the acoustic velocity of the propellant (neutral particles) is increased. Moreover, operational instability, which is a problem of conventional hall-type electric propulsions, is caused when rapid ionization (an increase in plasma density) occurs in the ionization/acceleration channel such that the ionized ions are moved rapidly from the ionization-zone by the electric ﬁeld, but due to the increase in acoustic velocity in the inventions described above, rapid ionization of the neutral particles can be suppressed (the ionization-zone can be extended), and as a result rapid ionization is alleviated, thereby alleviating instability during ionization and providing operational stability. Furthermore, the concepts described above do not require new, complicated systems. A choking portion for choking the propellant is formed (manufactured) by a throat having a gap that reduces steadily instead of a region having a ﬁxed ﬂow passage gap. In this manner, stagnation of the ﬂow near the corners of the plenum chamber can be avoided, and the neutral particles of the propellant, which are preheated in the propellant ﬂow propellant conduit, can be led to the anode after being rectiﬁed. Hence, the hall-eﬀect accelerator also realizes overheating protection and operational stability. When boron nitride is used as the material for the acceleration channel wall surface rather than an alumina-type ceramic (3Al2 O3 2SiO2 or the like), the discharge current value required to obtain identical thrust can be reduced. Further, following longterm use, a stepped groove forms in the surface of the insulator, and when this groove increases in depth, the acceleration channel deforms, leading to a reduction in the ion extraction performance. In the present invention, however, wall surfaces having a material that is suited to each of the acceleration-zone and the ionization-zone are selected, enabling improvements in eﬃciency and durability (sputtering suppression). In the micro hall-eﬀect accelerator, the wall surface of the acceleration channel is formed by a plurality of acceleration channel walls. By selecting materials respectively suited to the respective wall surfaces corresponding to the internal acceleration-zone and ionizationzone, improvements in eﬃciency and durability (sputtering suppression) can be achieved. For example, the acceleration channel wall corresponding to the ionization-zone is formed from an alumina-type ceramic (3Al2 O3 2SiO2 ) material or the like, whereas the acceleration channel wall corresponding to the acceleration-zone is formed from a boron nitride material or the like. The magnetic force lines applied to the acceleration channel interior are formed to be perpendicular to the acceleration channel axial direction. Therefore, the acceleration vector for accelerating the generated ions becomes perpendicular to the applied lines of magnetic force distribution (= parallel to the accel-

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eration channel axial direction) such that theoretically, the ions are emitted to the exterior of the channel collisionlessly and generate much thrust. When the lines of magnetic force distort, the wall surface sputtering ratio of the generated ions increases, leading to reductions in propulsion eﬃciency and durability.

4 Conclusions In order to enhance the performance of hall-eﬀect accelerator for nuclear fusion, the reduction technology of plasma instability was estimated by numerical analysis and experimental data to give the following ﬁndings: a) Average length of ionization-zone introduced as a new physical parameter, which could be varied by neutral particles velocity, is correlated conversely with amplitude. b) To increase neutral species velocity-inlet in acceleration channel through preheating neutral species could bring about the lower amplitude of plasma instability and the higher performance. c) The mechanisms of controlling amplitude are clariﬁed from the spatial distributions of plasma parameters and electromagnetic ﬁeld at the peculiar times in single-cycle of periodic ionization when discharge voltage, neutral species temperature and spatial distribution of magnetic ﬁeld are changed.

Acknowledgments This research was partially supported by the Ministry of Education, Science, Sports and Culture, Grant-in-Aid for Young Scientists (A) 17686074.

References 1. Hirano K (1994) Plasma electromagnetic accelerator. Patent H6-196298 2. Choueiri EY (2001) Phys Plasmas 8:5025–5033 3. Jankovsky R, Tverdokhlebov S, Manzella D (1999) High power hall thrusters. 35th joint propulsion conference, AIAA-99-315745 4. Fife JM, Martinez-Sanchez M, Szabo J (1997) A numerical study of low-frequency discharge oscillations in hall thrusters. 33th joint propulsion conference, AIAA97-3052 5. Furukawa T, Miyasaka T, Sakurai Y, Fujiwara T (2000) Measurement and modeling of low frequency oscillation in a hall thruster. 22th international symposium on space technology and science, ISTS-00-b-22 6. Boeuf JP, Garritues L (1998) Low frequency oscillation in a stationary plasma thruster. J Appl Phys 84:3541–3554

Electrochemical Control of the Surface Energy of Conjugated Polymers for Guiding Samples in Microﬂuidic Systems Nathaniel D. Robinson Link¨ oping University, Dept. of Physics, Chemistry and Biology, 581 83 Link¨ oping, Sweden. [email protected] Summary. In the literature there are several methods for electronically guiding aqueous samples through capillary systems such as electrowetting or using various sorts of mechanical valves. Electrowetting typically requires rather large voltages to be applied to the sample itself, which can cause unwanted reactions in the sample. In the current paper the pre-programming the surface energy of a conjugated polymer is presented as a method to eliminate this risk. We found also that mechanical valves, such as pneumatic, magnetic, or based on electroactive polymers are all considerably more diﬃcult to manufacture than the devices presented in this work. Our experiments show that electrochemically modifying the redox state of several conjugated polymers can be used to control the surface energy of the polymer’s surface. The materials involved lend themselves to low-cost manufacturing, making disposable devices for applications such as in-home medical diagnostics plausible.

1 Introduction Control of the energy at the interface between a ﬂuid and a solid has hundreds of applications. Of these, microﬂuidics is one of the most interesting to study, particularly because of the potential impact the ﬁeld can have in areas like health care, drug discovery and public safety. It is well-established that interfacial energy plays a dominant role in the transport of, for example, drops of ﬂuid that are smaller than 1 mm in diameter. In the following pages, a technique for electronically controlling the interfacial energy between aqueous samples and the surface of a π-conjugated polymer will be described, including a demonstration of how the technique can be applied in a microﬂuidic system. There are currently several methods for electronically guiding aqueous samples through capillary systems such as electrowetting [1] or using various sorts of mechanical valves. Electrowetting is not that diﬀerent from the technique that will be presented in the coming pages as far as the need for electronic connections and the complexity of manufacturing are concerned. M. Ellero et al. (eds.), IUTAM Symposium on Advances in Micro- and Nanoﬂuidics, IUTAM Bookseries 15, c Springer Science + Business Media B.V. 2009

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However, electrowetting typically requires rather large voltages (e.g. 20 V) to be applied to the sample itself, which can cause unwanted reactions in the sample. Pre-programming the surface energy of a conjugated polymer, as presented here, eliminates this risk. Mechanical valves, whether they are pneumatic, magnetic, or based on electroactive polymers (similar to the conjugated polymers used here), are all considerably more diﬃcult to manufacture than the devices presented in this work. Before describing the operation of the device at the focus of this manuscript, a brief background in the materials used, conjugated polymers, is presented in the coming section. This is followed by a description of the electrochemical process that allows control over the redox state of the polymer, and a discussion of the chemistry behind the change in wettability observed. Finally, two diﬀerent devices capable of guiding an advancing air/liquid interface are presented in Sect. 5.

2 π-Conjugated Polymers Conjugated polymers, or conducting plastics as they are often called, are deﬁned by their chemical structure; they contain a string of alternating double and single carbon-carbon bonds along their backbone as shown in polyacetylene in Fig. 1. These bonds form a chain of neighboring π orbitals that interact, eﬀectively allowing electrons to travel the entire length of the polymer (the conjugation length). In the neutral state, conjugated polymers are semiconducting (or even non-conducting). However, the addition or removal of one or more electrons from the π-system creates a charge carrier that can move freely along the chain, increasing the conductivity many orders of magnitude. This mobility allows the charge to delocalize, thus reducing the energetic cost of creating the charge in the polymer, making the charged system more stable than it would be if the π bonds wre not connected (conjugated). The simplest conjugated polymers, for example polyacetylene, are highly conductive when doped, but rather unstable, requiring them to be studied under vacuum or in well-controlled dry and oxygen-free environments. However, more complicated materials such as polyaniline (PANI), poly(3,4ethylenedioxythiophene) (PEDOT) and derivatives of poly(hexylthiophene) are relatively stable in air, and even in contact with water. Although each of the latter materials contains a ring in the monomer unit, the alternating single/double bonds, and therefore the conjugated π-bonds, can still be found, as shown in Fig. 1.1 1

The N in PANI has lone pair electrons that conjugate with the adjacent aromatic rings.

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Fig. 1. Chemical structure of several conjugated polymers in their neutral (undoped) form.

3 Electrochemical Doping Charge injection, which is the process of adding or removing an electron from a conjugated polymer, is typically driven by an electric ﬁeld, e.g. from a nearby electrode. However, if the charge created in the polymer is then locally balanced by an ion (called a counter-ion) of the opposite charge, then the process can also be called electrochemical doping. In electrochemical terms, the polymer is oxidized or reduced as it is doped, depending on whether an electron was removed (oxidation) or added (reduction). The counter-ion balances the charge on the polymer, maintaining charge neutrality in the polymer ﬁlm and reducing the energetic cost of the charge injection, thus

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stabilizing the doped polymer segment. A long polymer chain can stabilize several charges along its length.2 3.1 Chemical Process A typical p-type doping half-reaction (oxidation) is shown in (1). Here, P 0 represents the undoped polymer (or polymer segment), X − represents an anion (such as Cl− ), P + represents the p-doped polymer (segment) and e− represents an electron. The conjugated polymers commonly available today are stable in air and water when p-doped, but not when n-doped. The remainder of this discussion will be limited to p-doped materials. P 0 + X − −→ P + + X − + e− .

(1)

As shown in the half-reaction (1), the polymer is transformed from a neutral molecule into an ion. This is accompanied by signiﬁcant changes in its chemical properties. 3.2 Chemical Property Changes When doped, a conjugated polymer often changes color, density, and conductivity as well as molecular conformation and chemical character. As an example, Table 1 describes some of the physical changes that occur in poly(3hexylthiophene) (P3HT) when it is p-doped. P3HT is a polythiophene with a six-carbon alkyl chain attached to the thiophene ring, as shown in Fig. 2. The focus of this work takes advantage of the last item on the list in Table 1, namely the chemical character of the polymer backbone. Undoped P3HT is a relatively hydrophobic hydrocarbon consisting of neutral thiophene rings and alkyl chains. However, when a positive charge is introduced through p-doping (removing an electron from the π-band) the material gladly forms a salt with negative ions such as Cl− if such an ion is available, e.g. from an adjacent electrolyte. Table 1. Observable properties of P3HT in the neutral and p-doped (oxidized) states. Property Color Conductivity Chemical nature 2

Neutral Deep pink/purple Semiconducting Non-polar

p-doped (Oxidized) Nearly transparent Conducting Ionic

It turns out that injected charges are often most stable as pairs, but this is beyond the scope of this manuscript.

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Fig. 2. The structure of (a) poly(3-butylthiophene) (P3BT), (b) poly(3-hexylthiophene) (P3HT), and (c) poly(3-octylthiophene) (P3OT).

Fig. 3. A schematic of a typical solid-state polymer ﬁlm electrochemical cell. Ions are transported between the electrodes via the polymer electrolyte, while electrons are driven through the electrical connection (in this case, metal wires connected to a power supply) between them.

3.3 Device Structure The electron in (1) must have a destination, and the anion must have a source; somewhere a reduction must take place to balance the oxidation occurring in the polymer. This typically happens at a counter electrode, the other half of an electrochemical cell. A typical polymer ﬁlm electrochemical cell is shown schematically in Fig. 3. If the polymer is oxidized by applying a positive voltage to the polymer relative to the counter electrode as shown, then electrons ﬂow through the power supply from the polymer to the counter electrode while anions ﬂow from the counter electrode to the working electrode through the electrolyte. A reductive process (e.g. AgCl → Ag0 + Cl− ) must occur at the counter electrode to maintain charge and species balance.

4 Wettability 4.1 Direct How a water droplet interacts with a P3HT surface depends dramatically on the electrochemical state of the polymer. On neutral P3HT, a water droplet forms a large contact angle as it attempts to avoid the non-polar, hydrophobic

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Fig. 4. Water on neutral (left) and p-doped (right) P3HT [2].

polymer surface. When P3HT is doped, however, the polar surface attracts the water droplet, causing it to spread and form a smaller contact angle. Figure 4 (from Robinson et al.) shows images of water droplets on each surface [2]. The mechanism behind the diﬀerence in wettability, i.e. that the interaction between the ionic polymer backbone and the polar water molecule causes the decrease in contact angle, can be veriﬁed by varying the chemical structure of the polymer. Since conjugated polymers are chemically synthesized, the hexyl chains in P3HT can be exchanged for other groups simply by starting with the appropriate monomer. Two examples that have been studied are P3BT (with a butyl group instead of the hexyl group) and P3OT (containing an octyl group instead of the hexyl group). These materials are illustrated in Fig. 2. As the aklyl side-chain length increases, one might speculate that 1) the contact angle of water droplets on the neutral polymer will increase and 2) the diﬀerence between the contact angle of water droplets on the doped and undoped polymer surfaces will decrease. Both of these eﬀects are caused by the increasing inﬂuence of the hydrophobic side-chains as they protrude further into the water droplet, eﬀectively “shielding” the water from the polymer backbone. In fact, one ﬁnds that both of these results are true. Figure 5, originally from Robinson et al. [2], shows the measured contact angle for water droplets on P3BT, P3HT and P3OT in the oxidized and reduced states in a device having the structure shown in Fig. 3. For reference, the contact angle of water on each polymer in the neutral state on a silicon surface (instead of the sandwich structure shown in Fig. 3) is also included. 4.2 Surfactant The poly(alkylthiophene) family exhibits a relatively straight-forward interaction with water. However, more complex situations can arise. One example is doped PANI with the surfactant dodecylbenzylsulfonate (DBS) as a counterion. DBS has a polar head (the sulfonate group) and a hydrophobic alkyl tail, as shown in Fig. 6. PANI is a rather complex conjugated polymer exhibiting multiple redox states and diﬀerent structure and properties depending on whether it is protonated. For simplicity, we will consider the transition between neutral and doped (pernigraniline) PANI, shown in (2). − PANI + 2DBS− −→ PANI+2 DBS− 2 + 2e .

(2)

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Fig. 5. Water contact angles measured on P3BT, P3HT and P3OT, in the pristine (neutral) and p-doped (oxidized) states. The angle of water on neutral material coated onto a bare silicon wafer instead of the polymer electrolyte is provided for reference. See Robinson et al. [2].

Fig. 6. Chemical structure of anionic DBS.

When PANI is in the neutral form, the DBS is not bound to the surface and is free to orient randomly, while its large size slows its departure from the polymer ﬁlm. When PANI is oxidized, the sulfonate group on the DBS orients towards the polymer backbone leaving the hydrophobic hydrocarbon chain exposed at the surface [3]. This hydrophobic contribution of the dodecylalkyl chain overcomes the hydrophilic contribution from the ionic polymer backbone, resulting in an increase in contact angle upon p-doping of the PANI. Thus, the special surfactant counter-ion in the PANI:DBS system eﬀectively reverses the direction of the change in contact angle from that seen for polyalkylthiophenes. DBS is often commonly used as a counter-ion to doped polypyrrole (PPY) as well. The conjugated polymers in PANI:DBS, PEDOT:PSS and PPY:DBS are all partially doped when they are synthesized. That is to say the addition of the counter-ion is part of the synthesis process. To switch them to the neutral state, they must be chemically or electrochemically reduced. Large counterions like PSS and DBS are often added to make the resulting polymer blend soluble or suspendable in a solvent.

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Table 2. Experimentally measured advancing water contact angles (in degrees) for various conjugated polymers. Material

Polymer State Neutral p-doped

Contact Angle 77 95

Measurement Method Water drop on surface

P3BT

Neutral p-doped

88 71

Water drop on surface

Robinson et al. [2]

P3HT

Neutral p-doped

102 89

Water drop on surface

Robinson et al. [2]

P3HT

Neutral p-doped

110 87

0.05 M LiClO4 in a Wilhelmy balance

Ingan¨ as et al. [4]

P3OT

Neutral p-doped

105 96

Water drop on surface

Robinson et al. [2]

PPY:DBS

Neutral p-doped

69 95

Hydrazine(aq) on surface

Causley et al. [5]

PEDOT:tosylate

Neutral p-doped

50 80

0.05 M LiClO4 in a Wilhelmy balance

Ingan¨ as et al. [4]

PEDOT:PSS

Neutral p-doped

50 65

0.05 M LiClO4 in a Wilhelmy balance

Ingan¨ as et al. [4]

PANI:DBS

Source Isaksson et al. [3]

4.3 Experimental Data Table 2 summarizes the measured contact angle of water on various conjugated polymer materials reported in the literature. The materials shown here can each be classiﬁed as having a direct or counter-ion-based change in surface energy depending on whether p-doping causes a decrease or an increase in water contact angle. The popular conjugated polymer system poly(3,4-ethylenedioxythiophene) blended with poly(styrene sulfonate) PEDOT:PSS is not particularly useful for control of interfacial energy, probably because the electrochemically inactive excess PSS used to make the material processable covers the exposed surface.

5 Application in Fluidics Electronic control of the wettability of a surface has obvious applications in ﬂuidic systems. At least two research groups have demonstrated this at the proof-of-concept level. First, Causley et al. showed that capillaries coated with PPY:DBS could be addressed to control the motion of a water/air interface in

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Fig. 7. Microﬂuidic device developed by Causley et al. [5].

the capillary [5]. Later, Robinson et al. demonstrated that the device structure shown in Fig. 9 could be used to direct an advancing water/air interface in an arbitrary network of channels [6]. The device from Causley et al. uses the surfactant eﬀect of the counter-ion DBS described in Sect. 4.2, while the device from Robinson et al. is based on the direct change in polarization of the polymer backbone (described in Sect. 4.1). The second contrast between the devices involves the electrolyte, the part of the electrochemical cell which allows ion transport between the working electrode and the counter electrode. Robinson et al. employed a gelled polymer electrolyte to switch the polymer before introducing water to the system, while Causley et al. used the water in the channel itself as the electrolyte. 5.1 Analyte as Electrolyte A schematic of the device employed by Causley et al. [5] is shown in Fig. 7. This device structure is much simpler than that used by Robinson et al. in that it requires no polymer electrolyte. Instead, the counter and working electrodes are ionically connected by the ﬂuid to be manipulated, a 0.1 M KCl solution in the published work. In this device, a capillary, internally coated with PPY:DBS on Pt, is connected to an electrolyte reservoir containing counter and reference electrodes. The electrolyte is enticed into the capillary by applying a reducing (negative) potential to the PPY:DBS relative to the reference electrode (current is driven through the counter electrode instead of the reference electrode by a potentiostat). As the PPY is reduced, it becomes increasingly hydrophyllic and the electrolyte progresses along the capillary. Using the analyte as the electrolyte limits the device to controlling the ﬂow of solutions containing small mobile anions able to interact with the conducting polymer ﬁlm. The concentration of such ions will be important in devices with long channels, as the electrochemical process associated with the advancing water/air interface consumes the ions in the electrolyte, potentially depleting the electrolyte locally until more ions can diﬀuse or electromigrate

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Fig. 8. Schematic of the device demonstrated by Robinson et al.

from the bulk. Fortunately, biological samples (e.g. cell media), which are a popular application for microﬂuidic systems, often contain relatively high concentrations of small salts. Causley et al. attribute the advancement of the water/air interface in their device entirely to the change in surface energy of the polymer ﬁlm. It should be noted that the ions being transported from the bulk of the electrolyte to the advancing water/air interface are carried in a shell of solvent molecules which is eventually delivered to the polymer/water interface. Thus, the transport associated with the electrochemical process which changes the polymer’s surface energy likely enhances the motion of the aqueous solution down the channel. 5.2 Secondary Electrolyte The ﬂuidic device demonstrated by Robinson et al. has a structure shown in Fig. 8. A Ag/Ag+ counter electrode and thin ﬁlm P3HT working electrode sandwich a polymer gel electrolyte. A 3-walled PDMS channel system (formed through a standard templating procedure) placed on top of the exposed P3HT surface completes the device. Where desired, the electrochemical cell can be activated by applying a potential between the P3HT (using the exposed edges of the ﬁlm) and Ag/Ag+ electrodes to oxidize the P3HT. The oxidation of the P3HT progresses in a front, moving away from the metal probe used for contact [7]. This allows the oxidation to be process to be driven until the desired area has reacted and then halted. Where oxidized, the entire depth of the P3HT thin ﬁlm is converted, meaning that the exposed surface in the (as yet) empty ﬂuidic channel becomes relatively hydrophilic. After the P3HT ﬁlm has been oxidized, the electrochemical cell will maintain its state for at least several minutes as long as the electronic circuit between the working and counter electrodes is open. Thus, the device is eﬀectively “programmed” to direct the aqueous sample down a speciﬁc path. At the laboratory technician’s convenience, water is introduced into the network and

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Fig. 9. Electrochemically-controlled ﬂuidic device demonstrated by Robinson et al. Dyed water ﬂows in the channels under which the P3HT has been oxidized electrochemically by applying a potential between the conjugated polymer ﬁlm and the Ag counter electrode underneath. Regions of oxidized (doped) and neutral P3HT are outlined with dotted and dashed lines respectively. The extent that the water sample travelled into each channel is circled. The preferred pathway is indicated by the arrow. [6]—Reproduced by permission of The Royal Society of Chemistry (RSC).

prefers to ﬂow into the channels with hydrophyllic (oxidized) P3HT rather than the hydrophobic (neutral) P3HT channels. A photo of a device in use is shown in Fig. 9. The ability to program this device before use is a signiﬁcant advantage over using the aqueous sample as an electrolyte, as is the case in Causley’s device and in electrowetting devices, since no electrical connections are required when the aqueous sample to be guided is present. However, the polymer electrolyte under the conjugated polymer ﬁlm is problematic. After long-term exposure (many minutes), water penetrates the P3HT layer and dissolves the electrolyte, causing the device to fall apart. This challenge can likely be overcome by using an irreversibly cross-linked polymer electrolyte instead of the hydrogel presented in the article [6].

6 Extension to Smaller Dimensions The devices described in this work all have widths on the 100 μm to 1 mm scale. However, there is no obvious reason that electrochemically controlling the surface energy of conjugated polymers cannot be scaled down to lengths less than 1 μm. Devices have been patterned at these length scales using, for example, soft lithography [8, 9] and the polymers appear to be homogenous down to the order of hundreds of nanometers. At lengths below 100 nm or so the phase-separation of some of the materials discussed here (e.g. PEDOT:PSS) will come into play.

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7 Integration with Sensing Devices The real advantage of microﬂuidic devices based on conjugated polymers is their use for guidance of samples based on in-situ evaluation of the sample. Driving the electrochemical cells described in this manuscript with the output of, for example, chemical sensors placed upstream results in a system capable of sorting samples for further processing downstream. Such sensors can be complex electrochemical-based systems employing crystalline materials such as silicon and metal electrodes, however, the simplicity and low-cost of producing devices based on conjugated polymers [10] oﬀers the potential for disposable plastic sample processing systems. For example, the electrochemical transistor-based [11] glucose sensor developed by Malliaras et al. [12, 13] could be employed upstream of a wettability switch in a microﬂuidic system [14] to sort samples based on the concentration of glucose in each. The possibilities in this area are just beginning to be explored.

8 Challenges Currently, there are two obvious disadvantages that may inhibit the acceptance of electrochemically-controlled conjugated polymer surface energy devices into a wide array of ﬂuidic systems. First, the polymer surfaces are not re-usable. Once wet, the aﬃnity water has for the otherwise hydrophobic form of each material increases. Thus, these devices are appropriate for gating the ﬂow of an advancing water/air interface in a ﬂuidic system, but not for breaking up the ﬂow or generating droplets or ﬂuid segments. Second, and related to the ﬁrst, the receding contact angle for a drop on these surfaces is much smaller than the advancing contact angle. This diﬀerence exceeds the diﬀerence between the contact angles of water on the oxidized and reduced states of the polymer. Thus, the spreading of a water droplet can be controlled to a limited degree (see for example Isaksson et al. [15]), but individual droplets cannot be guided through capillaries or over surfaces. Overcoming this challenge, e.g. by cleverly patterning the polymer surface with nano-scale features, would be a dramatic breakthrough.

9 Conclusion In summary, electrochemically modifying the redox state of several conjugated polymers can be used to control the surface energy of the polymer’s surface. This technique has been demonstrated for use in ﬂuidic channels by at least two research groups. The materials involved lend themselves to low-cost manufacturing, making disposable devices for applications such as in-home medical diagnostics plausible.

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Acknowledgements NDR is supported by a grant from Norrkpings Kommun via the Forskning och Framtid program and by a grant from the Swedish Research Council (VR).

References 1. Mugele F, Baret JC (2005) J Phys: Condens Matter 17(28):R705 2. Robinson L, Isaksson J, Robinson ND, Berggren M (2006) Surf Sci Lett 600(11):L148 3. Isaksson J, Tengstedt C, Fahlman M, Robinson N, Berggren M (2004) Adv Mater 16(4):316 4. Wang XJ, Ederth T, Inganas O (2006) Langmuir 22(22):9287 5. Causley J, Stitzel S, Brady S, Diamond D, Wallace G (2005) Synth Met 151(1):60 6. Robinson L, Hentzell A, Robinson ND, Isaksson J, Berggren M (2006) Lab Chip 6:1277 7. Johansson T, Persson NK, Ingan¨ as O (2004) J Electrochem Soc 151(4):E119 8. Pisignano D, Persano L, Cingolani R, Gigli G, Babudri F, Farinola GM, Naso F (2004) Appl Phys Lett 84(8):1365 9. Zhang F, Nyberg T, Ingan¨ as O (2002) Nano Lett 2(12):1373 10. Berggren M, Nilsson D, Robinson ND (2007) Nat Mater 6(1):3. ISSN 1476-1122. doi:10.1038/nmat1817 11. Nilsson D, Robinson ND, Berggren M, Forchheimer R (2005) Adv Mater 17(3):353 12. Macaya DJ, Nikolou M, Takamatsu S, Mabeck JT, Owens RM, Malliaras GG (2007) Sens Actuators, B, Chem 123(1):374 13. Zhu ZT, Mabeck JT, Zhu CC, Cady NC, Batt CA, Malliaras GG (2004) Chem Commun 13:1556 14. Mabeck JT, DeFranco JA, Bernards DA, Malliaras GG, Hocd´e S, Chase CJ (2005) Appl Phys Lett 87:013503 15. Isaksson J, Robinson ND, Berggren M (2006) Thin Solid Films 515:2003

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Small Scale Cavitation Model Juan Miguel S´ anchez S´ anchez, Rafael Rodrigo Fern´andez E.T.S. Ingenieros Navales, Universidad Politecnica de Madrid, Avda. del Arco de la Victoria s/n, 28040 Madrid, Spain. [email protected], [email protected]

Summary. In present paper the mixture incompressible liquid/non interacting spherical vapor-gas bubbles is considered. The mathematical model under consideration is compressible viscous ﬂuid continuum with internal structure described by the local number density of bubbles function. The evolution of the microstructure of continuum is deﬁned by an integral micro mechanics equation. Two diﬀerent techniques to approximate numerically the micro evolution equation were designed. The model has been tested in a bubbly ﬂuid with a population of micro bubbles uniformly dispersed. The histories of pressure and the initial number distribution have been imposed. Our study suggests that the small scale cavitation model should be used in the microscopic aspects of cloud cavitation in a single ﬂuid multi phase method for hydrodynamics cavitating ﬂows.

1 Introduction A mixture of an incompressible Newtonian liquid and a dilute suspension of non interacting vapour/gas bubbles is considered as a single Newtonian compressible viscous ﬂuid continuum with internal structure.1 The constituents of the micro structure are the micron-sized spherical bubbles dispersed in the liquid, that in the micro motion are allowed to expand and contract independently, while they are dragged along the ﬂow of a regular macro motion. We consider a micro space attached to each point of the mixture in which the extra microscopic degrees of freedom needed to describe the suspension vary. The radius, or the volume, of each bubble is the only internal variable of our continuum. It varies in a closed interval that deforms with time. 1

The compressibility of vapour/gas within the eﬀective continuum causes the mixture of liquid and gas to behave as compressible Newtonian ﬂuid.

M. Ellero et al. (eds.), IUTAM Symposium on Advances in Micro- and Nanoﬂuidics, IUTAM Bookseries 15, c Springer Science + Business Media B.V. 2009

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The ﬂow of the micro motion is not necessarily 1:1; and consequently, the micro motion is not regular in general. In order to describe the time evolution of the micro bubble suspension, we introduce the time-dependent local number density of bubbles function. The local vapour/gas fraction per unit volume of bubbly liquid that is deﬁned in terms of the number density distribution is a macroscopic reﬂection of the evolutionary process that occurs at the micro scale, and links both scales. The micro mechanics evolution law is a conservation law stated as an objective time rate obtained Lie-dragging along a regular micro ﬂow the number density function. This time rate should equal zero in the no nucleation hypothesis or the time rate of nucleated bubbles when nucleation is considered. To approximate the number density function we write two diﬀerent and equivalent expressions of the evolution law as integral equations which we integrate numerically. This numerical process requires the knowledge of the initial number density function, experimentally provided,2 and a precise deﬁnition of the micro ﬂow that is obtained with the numerical integration of the IVP deﬁned by the inertially controlled Rayleigh-Plesset equation, with a given initial radius and zero initial velocity as initial conditions [2], via the Dormand-Prince DOP853 code [4]. In the case of a singular micro ﬂow, the regularity is locally recovered by deﬁning at each time an atlas associated to the critical points of the singular ﬂow. The local regular micro ﬂows, restriction of the global singular ﬂow to the domains of the charts of this atlas, satisfy the same equations as the global ﬂow, and with the information they provide we construct at any time the number density function and the vapour/gas volume fraction. The time dependence on the evolution of the bubble population equation comes through the stresses in the present model and thereby it is coupled to the Navier Stokes and the continuity equations of the continuum. Our numerical approximation treats the micro mechanics evolution model as uncoupled at each time step to the conservation laws by changing the coeﬃcients of the evolution equation in due manner at the next time step. The result is an updated version of our continuum at time t, which is used to update the local void fraction, the density and the viscosity of the eﬀective continuum, which will feed back the conservation laws program. Although it is clear how our model should be used at the micro mechanic scale in hydrodynamic cavitating ﬂows, it has been tested so far in a bubbly ﬂuid continuum with a uniformly dispersed population of micro bubbles for two diﬀerent histories of pressures with both regular and singular micro ﬂows, in which the constants and the initial number density distribution function have been imposed. The modelling of these complex multi phase ﬂows poses some challenging problems that have only been addressed in the last ﬁfteen years. The ﬁrst 2

Some commercial companies oﬀer instruments to measure bubble size distribution and void fraction in gas/liquid mixtures and ﬂows.

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important step in this direction was the numerical modelling by Kubota A. et al. [5]. They used the mixture ﬂuid method, a single compressible ﬂuid with separate continuity equations for two-phase analysis, and the Rayleigh’s equation to introduce bubble dynamics into the problem, however their assumption of constant bubble number density is not desirable in regions of higher void fraction, in which the number density should decrease if bubbles are large. In 1995 Chen Y. et al. [3] also using the mixture ﬂuid method introduced a pseudo density, the density of the mixture incompressible ﬂuid/vapour bubbles, and they wrote “This density could be computed exactly if we were able to resolve the location and size of all bubbles in the ﬂow. To trace the evolution of every bubble is deﬁnitely out of reach at current stage since a single bubble may need a fully 3D calculation”. After two failed attempts Singhal A.K. et al. [8] designed in 2002 the ﬁrst successful commercial general purpose CFD code for cavitation. Deﬁning the vapour volume fraction in terms of a “constant” number density, they simplify the phase change rate expression “in the absence of a general model for estimation of the number density” also assuming that all the bubbles at time t have the same radius which they called “the typical bubble size”. Rhee S.H. et al. [6] presented in the NSH8 Conference in 2003 a cavitation model based on a single ﬂuid multi phase ﬂow method implemented in the RANS solver. It is based in the Singhal’s full cavitation model extending their capabilities by accounting for the eﬀects of the inter phase slip velocities, however they adopted Singhal’s mass transfer between phases without change. Our model is also under the framework of the mixture ﬂuid method but the introduction of the time-dependent number density function allows a more precise description of the evolution of the local void fraction that intervenes in the conservation equations for the single ﬂuid continuum both through the density and the viscosity, and in the transport equation for the mass transfer.

2 Description of the Internal Structure of the Continuum The system liquid-vapour/gas bubbles occupies an open and bounded domain B ⊂ R3 in which the micro bubble suspension will be dragged along the ﬂow of a regular (macro) motion ϕt : B → Bt . The micro bubbles in the suspension vary in size and are random in space distribution. At time t = 0 and for a ﬁxed particle X ∈ B the radii of the bubbles in the suspension vary in the closed interval I0 ⊂ R+ , the reference conﬁguration of the micro motion. The micro motion is a mapping t → ψt of R+ into the set of all conﬁgurations of I0 , a time-dependent family of deformed conﬁgurations {It = ψt (I0 )}. For t ﬁxed and R ∈ I0 , R → ψt (R) = r ∈ It deﬁnes the micro ﬂow. Holding R ﬁxed, t → ψt (R) = r ∈ It is the evolution curve or trajectory described by R along the ﬂow.

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In general, the micro ﬂow is not regular, the inverse correspondence R = ψt −1 (r) is not a mapping.3 This diﬃculty requires especial care particularly in the actions of the ﬂow on tensor ﬁelds. When the micro motion is regular we write ψt,s : Is → It deﬁned by ψt,s = ψt ◦ ψs −1 so that ψt,0 = ψt . We assume the existence of a time-dependent real number density distribution function ht deﬁned at time t on Bt × It , representing the time evolution of the average density of bubbles of radius r per unit volume, per unit length, in the neighbourhood of a ﬁxed material particle X ∈ B. The initial number density distribution function h0 deﬁned at time t = 0 on B × I0 is experimentally provided and deﬁnes the reference conﬁguration I0 of the micro motion. The number density function ht is the unique component of a four differential form in the 4D manifold Bt × It and if X is taken ﬁxed in B, r → ht (ϕt (X), r)dr is the expression of a one diﬀerential form αt ∈ T ∗ It with respect to the canonical atlas. We also assume the knowledge of the time-dependent real function Γt (x, r) representing at time t for a ﬁxed particle x = ϕt (X) the estimated time rate of nucleated bubbles per unit volume per unit length. Γt (x, r) is the unique component of a four diﬀerential form in Bt × It and if X is taken ﬁxed it deﬁnes a one diﬀerential form βt in It so that for all r ∈ It , βt (r) ∈ Tr∗ It and βt (r) = Γt (ϕt (X), r)dr. This function should be experimentally provided. The local volume/gas fraction per unit volume of bubbly liquid is given by

t 4π Rmax ϑ(x, t) = ht (x, r)r3 dr. (1) t 3 Rmin The density ρ(x, t) and the viscosity μ(x, t) of the bubbly ﬂuid are deﬁned by ρ(x, t) = ρL (1 − ϑ(x, t)) and μ(x, t) = (1 − ϑ(x, t))μL + μG ϑ(x, t) respectively, ρL is the liquid density, and, μL , μG are the liquid and gas dynamic viscosities.

3 The Micro Mechanics Evolution Equation 3.1 The Regular Micro Flow No Nucleation Hypothesis Assuming regular micro ﬂow and no nucleation of bubbles we can state the conservation law

d αt = 0 (2) dt ψt,s (P ) 3

At time t there are bubbles with radius r that had diﬀerent initial radii and ψt is not injective.

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where P is any nice open set of Is [7], meaning that the time rate of the number of bubbles in the population with radii in ψt,s (P ) at time t, per unit volume, per unit time, is zero. Using the generalised transport theorem [1] we have for any P

L w t αt = 0 (3) ψt,s (P )

where Lwt αt is the non-autonomous Lie derivative of αt with respect to wt , the material velocity of the motion. Condition (3) will be expressed in local diﬀerential form as (4) Lwt αt = 0. The second time-dependent Lie derivative theorem [1] relating the Liederivatives and the ﬂow allows to write d ∗ ∗ ψ αt = ψt,s Lwt αt = 0 (∀s). dt t,s

(5)

In particular when s = 0 integrating from 0 to t and using the Lie Transform method [1] we get our micro mechanics equation ψt∗ αt = α0 .

(6)

An equality in the cotangent bundle T ∗ I0 , expressing that the pull-back of the number density function to the reference conﬁguration equals the initial number density function. Integrating (6) in any nice set P of I0 we get

ψt∗ αt = α0 (7) P

that we can also write

P

αt = ψt (P)

P

α0 .

(8)

Both (7) and (8) are the integral equations that we use to approximate h. With Nucleation In this case, the conservation law is

d αt = βt dt ψt,s (P ) ψt,s (P )

(9)

expressing that the time rate of the number of bubbles of the population with radii in ψt,s (P ) at time t, per unit volume, per unit length, per unit time, equals the time rate of nucleated bubbles with radii in ψt,s (P ), per unit volume, per unit length, per unit time.

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Fig. 1. An example of the graph of ψt−1 in the singular case showing the atlas At of I0 − Ct associated to the critical points of ψt .

The application of the transport theorem and the localisation process give L wt αt = βt .

(10)

Using again the second time-dependent Lie derivative theorem and the Lie transform method we get the micro mechanics equation

t ∗ ∗ ψt,0 βt . (11) ψt,0 αt − α0 = 0

Finally integrating in any nice open set P of I0 we get

t

t

∗ ∗ ∗ ψ t αt = α0 + ψt βt dt = α0 + ψt βt dt, (12) 0 0 P P P P P t

t

αt = α0 + βt dt = α0 + βt dt . (13) ψt (P)

P

0

ψt (P)

P

ψt (P)

0

Although the model is ready to consider nucleation, at the present time no experimental information about the time rate of nucleation is available consequently, we will consider here only the no nucleation case. 3.2 The Non Regular Micro Flow When the micro ﬂow is singular, we deﬁne an atlas At of I0 − Ct where Ct ⊂ I0 is the ﬁnite set of the critical points of ψt for each t (see Fig. 1). The atlas At = (Iti , id|Iti ) is such that the restriction of the micro ﬂow to the domain of each chart ψt |Iti = ψti is a diﬀeomorphism from Iti onto ψti (Iti ) = Jti . In the domain Iti of each chart we will apply the same theorems as in the regular case with the same conclusions, consequently we have in this case a local micro mechanics equation in the domain of each chart of At

Small Scale Cavitation Model ∗

ψti (αti ) = α0 |Iti

∀i = 1, . . . , card Ct + 1

and the two corresponding local integral equations

∗ ψti (αti ) = α0 |Iti , Pi P

i αti = α0 |Iti .

133

(14)

(15) (16)

Pi

ψti (Pi )

From the local family of 1 forms {αti }, deﬁned in Jti (i = 1, . . . , card Ct +1) (Fig. 1), we deﬁne the global density αt by αt =

card Ct +1

αti .

(17)

i=1

4 Numerical Approximation In order to approximate the density function h at time t for a given history of pressures we need to compute the micro ﬂow ψt . This information is provided by the numerical integration of the IVP deﬁned by the generalised RayleighPlesset equation, with a given initial radius and zero initial velocity as initial conditions [2], for a range of bubbles size deﬁned by the initial number distribution. 4.1 Direct Method We consider in each chart of At the regular micro ﬂow ψti = ψt|I i . t The time-marching algorithm starts at time t = 0 with the initial size distribution α0 . At time t a grid in the manifold I0 − Ct is deﬁned. We split the domain Iti of every chart of the atlas At associated to the critical points of ψt into a i j = 1, . . . , Ni . high number of equally spaced subintervals denoted by It,j Taking the canonical atlas in Iti and Jti we have α0 (R) = h0 (R)dR and αti (r) = hit (r)dr where we have denoted hit the unique component of αti . With this, recalling that αti is a density, the integral micro mechanics equation (15) gives i

dψ hit (ψti (R)) t dR = h0 (R)dR. (18) i i dR It,j It,j Using the mean value theorem for integrals

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Fig. 2. Inverse method. Finding ψt−1 (It,i ).

hit

dψti i i dR (Rj )lj = i h0 (R)dR It,j h0 (R)dR Ii hit (rji ) = t,j i dψ | dRt (Rij )|lji

ψti (Rij )

=⇒

(19)

i i where lji is the length of It,j with Rij ∈ It,j and rji = ψti (Rij ). i i We assigned to every element of ψt (It,j ) the density value hit (rji ) as an average of the contribution of the domain Iti relative to the local diﬀeomorphism ψti . Varying j we deﬁne a piecewice approximation of αti . Considering now all the charts of At we get the family {αti }i=1,...,card Ct +1 that deﬁnes using (17) an approximation of the global number density distribution αt . We iterate the process at time t + Δt.

4.2 Inverse Method Again a time-marching algorithm that starts at time t0 = 0 with α0 . At time t we deﬁne a grid of It −Vt , where Vt is the set of the critical values of ψt , by splitting this manifold into N open subintervals It,i i = 1, . . . , N . The set ψt−1 (It,i ) is an open subset of I0 and a disjoint union of subintervals j {Ii } each of which is a subset of the domain of diﬀerent charts of the atlas At of I0 − Ct (see Fig. 2). Using (16) for Pi = Iij and recalling that ψtj (Iij ) = It,i (Fig. 2), we get

h (R)dR R∈Iij 0 j αt = α0 =⇒ hi,j (t) = (20) j Lt,i It,i Ii where hi,j (t) is an average value of h over It,i induced by Iij , the unique component of αtj over It,i , and Lt,i is the length of It,i . Adding the contributions of all the Iij j = 1, . . . , mi , we get the average value of h over It,i

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Fig. 3. Initial number density distribution function. It loosely resembles the shape of the cavitation nuclei number density distribution function measured by holography in the ocean oﬀ L.A.(Ca.).

hi (t) =

mi

mi hi,j (t) =

j=1

j=1 R∈Iij

h0 (R)dR

Lt,i

.

(21)

At time t + Δt the process is repeated using r = ψ(R, t + Δt).

5 Case Studies To test our model, we study the rheology of a dilute suspension of spherical vapour/gas bubbles uniformly dispersed in an incompressible viscous Newtonian ﬂuid, subject to two diﬀerent pressure histories. The data will be in both cases p∞ (0) = T∞ = pV (T∞ ) = S= ρ=

2200 Pa, 293 K, 2300 Pa, 0.0717 N/m, 1000 kg/m3 ,

ν = 8.662 · 10−7 m2 /s, p∗∞ = 2000 Pa. The initial size distribution function (Fig. 3) is imposed. See [2] Fig. 18. The radii of the micro bubbles in the initial population range from the 0 0 minimum value Rmin = 10−5 m to the maximum Rmax = 10−3 m so that −5 −3 I0 = [10 , 10 ].

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Fig. 4. t → r = ψt (R) for seven representative bubbles marked with diﬀerent colours in the illustration. The graph of the critical radii t → Rc (t) is also included.

5.1 Linearly Decreasing History of Pressures. Explosive growth The suspension of micro bubbles is convected into a region of low pressure within the macro ﬂow. The pressure decrease according to the linear law t → p∞ (t) = 2200 − 105 t

(22)

and p∞ (t) < pV (∀t > 0) for explosive growth to occur. The micro ﬂow (R → r = ψt (R)) is in this case regular. In Fig. 4, the evolution curves of the size of seven representative bubbles of the population in the time interval t ∈ [0, 0.009] are represented. The Blake’s critical radius curve [2] (t → Rc (t)) overlays these maps. This decreasing curve separates approximately the area of growth of the bubbles. Bubbles with radius r < Rc (t) are stable and don’t grow whereas those with radius r > Rc (t) are unstable, grow explosively and cavitate. The regular micro motion is represented in Fig. 5. The trajectories of 100 equally spaced values of R with t ∈ [0, 0.009] are included. We can see the reference conﬁguration I0 and the sequence of all the deformed conﬁgurations.

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Fig. 5. t → r = ψt (R) for 100 equally spaced values of R ∈ I0 . At each time t, the t t , Rmax ] is given by the intersection of the vertical deformed conﬁguration It = [Rmin line by t and the curves.

The existence of an instantaneous critical radius somewhat smaller than the Blake’s critical radius is evident. As time evolves the critical radius decreases so that new bubbles of the population start growing. t increases with time while their The maximum value of the radii Rmax t remains practically unchanged during the experiment. minimum value Rmin At time t we can identify three diﬀerent zones in the deformed conﬁguration It . •

Zone A corresponds to the radii below Rc (t). These bubbles should not grow or collapse, but due to the numerics, their radii oscillate almost imperceptibly about its initial value attempting to keep it constant. • Zone B where the bubbles start growing. As a consequence of the slow decrease of Rc (t), only a few new bubbles start growing. • Zone C with a high number of radius values, coloured red in Fig. 5. Practically the majority of the trajectories belong to this zone. These three zones will have their correspondent in the number density function curve at time t, in which we identify in a loose way • •

Area A close to Rc (t) that decreases in size with the critical radius causing a displacement of A to the left. Area B corresponding in the radius versus time maps with the zone where the bubbles start growing, in which the density is very low and decreasing.

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Fig. 6. Linearly decreasing history of pressures.

•

Area C in which the bulk of the number density curve concentrates. The increase of the maximum radius produces a displacement of C to the right.

In the left column of Fig. 6 we represent the graphs of the inverse micro ﬂow function ψt−1 at the three instants t = 0.0035, t = 0.0056 and t = 0.009 seconds and these are joined in pairs to the corresponding graphs of the number density function in the right column. Except for the small oscillations at time t = 0.0035 s in zone 1, close to the critical radius, which are later displaced to the right attenuating until they fade away, the density curves are smooth and show the features pointed out before. An increasing maximum radius, a decreasing critical radius and the three areas A, B, C marked only at t = 0.0056 s.

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Fig. 7. Linearly increasing history of pressures.

5.2 Linearly Increasing History of Pressures. Collapse and Growth In this case-study, the pressure increases linearly according to the law 8 p∞ (t) = 2200 + 105 t. 3

(23)

The bubbles in the population collapse catastrophically and this is followed by successive attenuated rebounds and collapses as a consequence of the viscosity. This is quite clear in Figs. 7 and 8 in which the evolution curves in the time interval t ∈ [0, 0.009] of the radii of six diﬀerent bubbles of the suspension and those relative to 150 equally spaced values of R ∈ I0 are respectively represented. Deﬁnition 1. An area of concentration will be an area of the graph of the inverse correspondence ψt−1 in which several bubbles of very diﬀerent initial radii share at time t the same or very similar radius. They correspond to the critical points in the micro ﬂow. In the neighbourt hood of these points the magnitude of the derivative | dψ dR (R)| is close to zero. The denominator in (19) is very small and the density is very large.

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Fig. 8. Linearly increasing history of pressures.

Deﬁnition 2. We call overlapping area an area of the graph of the inverse correspondence ψt−1 delimited by successive critical values of ψt in which the subset ψt−1 ({r}) of I0 has more than one element for each r. As in the ﬁrst case-study in the left column of Fig. 9 we represent the graphs of the inverse micro ﬂow function ψt−1 at the three instants t = 0.001287, t = 0.00279 and t = 0.00384 seconds, joined in pairs to the corresponding graphs of the number density function in the right column. At time t = 0.001287 s we have an example of area of concentration. Bubbles with initial radii in the interval [3.40 × 10−4 , 4.38 × 10−4 ] of length 0.98×10−4 have at this time their radii in the interval [2.77×10−4 , 2.99×10−4 ] of much smaller length 0.22 × 10−4 . A straight consequence of the almost vertical slope of ψt−1 . At time t = 0.00279 s we can see a big overlapping area deﬁned by the interval [2.71 × 10−4 , 3.36 × 10−4 ] where we add the densities of bubbles in diﬀerent subintervals of I0 . The same interval in the number density graph deﬁnes two areas of concentration in the neighbourhood of the vertical tangents and a connecting valley characteristic of an overlapping area. At time t = 0.00384 s we see in the graph, a huge overlapping area deﬁned by the interval [2.125 × 10−4 , 5.67 × 10−4 ] that contains several smaller overlapping areas with their peaks and valleys. The oscillating characteristic of these areas is a consequence of the successive collapses and rebounds of the evolution maps.

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Fig. 9. Linearly increasing history of pressures.

6 Conclusions The mixture incompressible liquid/non interacting spherical vapour-gas bubbles is considered as a single compressible viscous ﬂuid continuum with internal structure described by the local number density of bubbles function. The evolution of the microstructure of our continuum is deﬁned by an integral micro mechanics equation. We have designed two diﬀerent techniques to approximate numerically the micro evolution equation. The model has been tested in a bubbly ﬂuid with a population of micro bubbles uniformly dispersed. The histories of pressure and the initial number distribution have been imposed. The small scale cavitation model should be used in the microscopic aspects of cloud cavitation in a single ﬂuid multi phase method for hydrodynamics cavitating ﬂows.

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Nomenclature B an open and bounded subset of R3 ϕt the ﬂow of the regular macro motion Bt = ϕt (B) the deformed conﬁguration at time t ψt the ﬂow of the micro motion wt the material velocity of the micro motion I0 a closed interval of R+ , reference conﬁguration of ψt It = ψt (I0 ) the deformed conﬁguration at time t R ∈ I0 the radius of the bubbles in the reference conﬁguration r = ψt (R) ∈ It the radius of the bubbles in the current conﬁguration ht the number density distribution function αt the 1 form r → ht dr Γt the time rate of nucleated bubbles βt the 1 form r → Γt dr ϑ(x, t) the local volume/gas fraction Lwt the non-autonomous Lie derivative along the micro ﬂow Ct ⊂ I0 the ﬁnite set of the critical points of ψt for each t At the canonical atlas of I0 − Ct associated to the critical points of ψt (Iti , id|I i ) the charts of At t

Iti the domain of the chart id|I i the canonical injection of Iti t

ψti = ψt|I i the restriction of ψt to Iti t

Jti = ψti (Iti ) αti a local family of 1 forms deﬁned in Jti hit the unique component of αti i It,j j = 1, . . . , Ni an equally spaced grid of Iti i i lj the length of It,j i i Rj an element in It,j rji = ψti (Rij ) Vt the set of the critical values of ψt It,i i = 1, . . . , N an equispaced grid of It − Vt Iij ⊂ Iti such that ψti (Iij ) = It,i hi,j (t) average value of h over It,i induced by Iij Lt,i the length of It,i hi (t) average value of h over It,i p∞ (t) the remote pressure, the local pressure of the liquid surrounding the bubbles T∞ the remote constant temperature pV (T∞ ) the vapour pressure at T∞ Rc the Blakes critical radius

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References 1. Abraham R, Marsden JE, Ratiu TS (1982) Manifolds, tensor analysis, and applications. Addison-Wesley, Reading 2. Brennen CE, Earls C (1995) Cavitation and bubble dynamics. Oxford University Press, London 3. Chen Y, Heister SD (1995) Two-phase modeling of cavitated ﬂows. Comput Fluids 24(7):799–809 4. Hairer E, Nørsett SP, Wanner G (1993) Solving ordinary diﬀerential equations. Springer, New York 5. Kubota A, Kato H, Yamaguchi H (1992) Cavity ﬂow predictions based on the Euler equations. J Fluid Mech 240:59–96 6. Rhee SH, Kawamura T, Li H (2003) A study of propeller cavitation using a RANS CFD method. In: Proceedings of a the NSH8, Busan, Korea, p. 25 7. S´ anchez JM (2003) A micro mechanics model for cavitation. In: Proceedings of a the NSH8, Busan, Korea, pp. 274–280 8. Singhal AK, Athavale MM, Li H, Jiang Y (2002) Mathematical basis and validation of the full cavitation model. J Fluids Engr 124:617

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Experimental and Theoretical Approach for Analysis of Flow Induced by Micro Organisms Existing on Surface of Granular Activated Sludge B.E. Zima-Kulisiewicz1 , W. Kowalczyk2 , A. Delgado1 1

2

Lehrstuhl f¨ ur Str¨ omungsmechanik, Friedrich-Alexander-Universit¨ at, Erlangen-N¨ urnberg, Cauerstr. 4, 91058 Erlangen, Germany. [email protected] Lehrstuhl f¨ ur Mechanik und Robotik, Universit¨ at Duisburg-Essen, Lotharstr. 1, 47057 Duisburg, Germany.

Summary. Numerous processes in the chemical engineering are supported by the ﬂuid ﬂow at micro scale. As an example, aerobic granulation as promising technique in the biological puriﬁcation of wastewater can be pointed out. Granular Activated Sludge (GAS), in comparison to Conventional Activated Sludge (CAS) has denser, compacter structure and higher biomass retention. Due to those properties GAS has a great application potential in biological puriﬁcation of wastewater. However, granulation is a complex process and its mechanism has not been fully understood yet. Additional to biochemical, chemical and ﬂuid dynamical factors in macro scale also characteristic microorganismic ﬂow generated by cilia beats of Opercularia Asymmetrica which live on granules surface in micro scale should be taken into account during investigations of the granulation process. Thus, in the present work micro Particle Image Velocimetry investigations are carried out.

1 Introduction Sequencing Batch Reactor (SBR) process can be treated as optimal way for production of Granular Activated Sludge (GAS). Granules due to high density (ca. 1.05 g/ml), ellipsoidal form with length up to 5 millimetres, high settling ability can be successfully used in biological puriﬁcation of wastewater [4]. However, granulation process is not completely understood yet. Many factors inﬂuence the formation and structure of aerobic granular sludge in the SBR. Composition of the substrate and its concentration has important impact on granules formation [3, 4]. Also the Superﬁcial Gas Velocity (SGV) has a crucial eﬀect on granulation [1, 10]. With higher SGV granules obtain spherical, more compacted structures. Additionally, Extracellular Polymer Substances

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Fig. 1. Granules growth (source: [11]).

(EPS) play a decisive role in the formation, maintenance of the granules structure, their architecture and stability. EPS bridge the bacterial cells and hold granules together. Moreover, feast-famine regime and suﬃcient settling time are necessary for correct granules formation [7]. Fluid dynamical investigations carried out by [15] show that buoyancy forces, drag forces as well as collisional forces (particle-wall, particle-particle collision) play important role for granulation process. The induced, normal and tangential strains aﬀect granules formation and destruction. However, it should not be forgotten that biogranulation is a multiscale phenomenon. Thus, investigations in micro scale should be also taken into account. According to [11], the granules development takes place with the aid of ciliates in three diﬀerent phases (see Fig. 1). At the beginning ciliates settle on other organisms or particles (Fig. 1 A) and bulky growth of ciliates commences e.g. Epistylis sp. (Fig. 1 B). Stalks and zooids are colonized by bacteria. Cilia beats of the ciliates providing a continuous nutrient ﬂux toward bioﬁlm improve colonization process. In the second phase the granule grows and the core zone is developed. Here, a lot of ciliate cells are completely overgrown by bacteria and die. Consequently a dense core of bacteria and remains of ciliate stalks is formed (Fig. 1 C). Gradually a mature granule is developed. Finally, granules are composed of two zones: core zone (red part) and loose structured fringe zone (grey part), see Fig. 1 E, and serve as new substrate for swarming ciliates (Fig. 1 D). Above description emphasises decisive ciliates role on granulation process. However, ﬂuid ﬂow induced by Opercularia asymmetrica is poorly investigated in the literature and intensive research is necessary for making progress towards a better understanding of these natural phenomena being optimized in the course of evolution. In order to visualise ﬂuid ﬂow induced in biological

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systems powerful imaging methods for ﬂow analysis are required. It is of crucial importance that the measuring and ﬂow visualisation techniques employed guarantee biocompatibility, i.e. they do not aﬀect investigated biosystems. Unfortunately, this restricts possibilities for optimizing the image generation in comparison to other ﬂow ﬁelds in which no biological systems are present. In consequence, images of lower quality leading to erroneous artefacts are obtained. Thus, either novel detection techniques that are able to overcome these disadvantages or advanced evaluation methods enabling the sophisticated analysis and description of ﬂow ﬁelds are requisite. As shown by [8], predicting and correcting artefacts in ﬂow ﬁeld evaluation can be performed by the hybrid method using a priori knowledge of the ﬂow physics formulated in numerical expressions combined with Artiﬁcial Neuronal Networks (ANN) trained to detect erroneous visualisation results. In the present work detailed micro-ﬂow investigations with biotic seeding particles are proposed.

2 Materials and Methods Microorganisms are selected from granules which grow in laboratory scale SBR [3]. Flow induced by Opercularia asymmetrica during feeding movement is investigated with a help of μ-PIV. Micro-ﬂuid ﬂow is observed by using transmitted light microscope Axiotech 100 (Carl Zeiss) with 20- and 50-fold optical magniﬁcation and the phase contrast method (diﬀerential interference contrast—DIC). Here, GAS probe taken out from SBR with a certain amount of seeding particles is placed on the glass plate. Prepared sample is covered with a cover plate. Following, probe is analysed with a microscope. Investigations carried out by [5, 6, 8] show that biocompatibility of measurement technique in microorganismic ﬂow belongs to the most important issues. Eﬀective results can be obtained only with appropriate seeding biotracers. In the current work yeast cells (Saccharomyces cerevisiae, dimension approx. 3–10 μm) as well as milk being emulsion with scattering particles (fat and proteins, dimension 0.3–3 μm) enable ﬂow visualization. Moreover, in order to ensure illumination acceptable by microorganisms, built in microscope white light with moderated intensity is applied as light source. Images are recorded by a high speed CCD camera (Mikrotron GmbH) with two diﬀerent speeds 25 frames/seconds and 65 frames/seconds. Images have resolution of 860 × 1024 pixels (i.e. 818 μm × 975 μm for 20-fold optical magniﬁcation and 323 μm × 385 μm for 50-fold optical magniﬁcation). Figure 2 depicts the used μ-PIV system. The calculation of the ﬂuid velocity is carried out with help of software PIVview2C (PIVTEC GmbH), developed by [2]. In order to extract particle displacement the cross correlation mode is used. Built in PIVview2C software multiple-pass interrogation algorithm increases the data yield due to higher amount of matched particles and reduces the bias error [12]. In the current

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Fig. 2. Microscope with CCD camera.

work the interrogation window size is chosen as 32 × 32 pixels, the grid size is 20 × 20 pixels. Sub-pixel displacement of the correlation peak is obtained by 3-Point Gauss Fit. It selects the four closest neighbours of a correlation maximum and ﬁts a 3-point Gaussian curve each of the major axis [13]. Further, velocities data from PIVview2C are processed with Tecplot (Amtec Engineering).

3 Results The results are presented in dimensionless form. The dimensionless representation allows reduction of the number of parameters as well as a clear understanding of experimental results. Dimensionless velocity is calculated as a ratio of ﬂuid velocity and maximum velocity observed within the series of experiments. The maximum reference velocity amounts in this case 132 μm/s. Analysing ﬂow induced by ciliates, a characteristic micro ﬂow pattern with two vortices generated by cilia beats can be observed (see Fig. 3). Present investigations conﬁrm the ﬁrst studies with ﬂow patterns induced by Vorticella carried out by [9]. Former studies of Delgado and coworkers with Opercularia asymmetrica indicate yeast aqueous solution of 1:100 as optimal tracer concentration for μ-PIV analyses. Concerning milk solution, good quality results are obtained for several concentrations of 1:1, 1:2, 1:3 and 1:4 (milk to water). However, μ-PIV investigations with dilution higher than 1:4 are impossible, e.g. studies with milk to water concentration of 1:5 show that low tracer particles numbers prevents correct recognition of the ﬂow structure. As shown by [8] experiments with yeast cells as seeding particles enable obtaining ﬂow pattern for 10-fold optical magniﬁcation. However, more precise analysis is impeded in that case due to behaviour and size of tracer particles.

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Fig. 3. Characteristic ﬂow pattern at 50-fold optical magniﬁcation.

Fig. 4. Velocity ﬁeld observed by 20-fold optical magniﬁcation for colony of ciliates (a) seeding with yeast cells, (b) seeding with milk.

In the present work, detailed investigations with higher optical magniﬁcations (20- and 50-fold optical magniﬁcations) are carried out. Exemplary experiments with yeast cells (1:100) as well as with milk (1:3) for both cases are compared. Based on the experiments with 20-fold optical magniﬁcation with use of yeast cells (a) and milk (b) it can be observed that obtained results reveal some signiﬁcant diﬀerences in the ﬂuid ﬂow pattern in respect to the applied seeding (see Fig. 4). Although the order of magnitude of ﬂuid velocity remains at the same level, more detailed presentation of the ﬂuid ﬂow ﬁeld concerning visualization artefacts is obtained for the probe with milk solution. The maximal dimensionless velocities for the investigations with yeast cells are slightly lower than with milk and amount up to 0.19 for the former and 0.21 for the latter. However, smoothing eﬀect of milk on results of ﬂow visualisation is clearly seen. More detailed ﬂuid ﬂow visualization could be done with 50-fold optical magniﬁcation. In this case ﬂow induced by single ciliate is analysed with the same seeding substances as previously. Figure 5 proves that investigations with

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Fig. 5. Velocity ﬁeld observed by 50-fold optical magniﬁcation for one ciliate (a) seeding with yeast cells, (b) seeding with milk.

yeast cells at higher optical magniﬁcation are strongly limited. Calculated values of the velocity (maximal velocity amounts 0.07) are underestimated and loaded with high error. Moreover, it is impossible to obtain the ﬂow ﬁeld in a suﬃcient quality and quantify the velocity. Due to low density and large dimension of tracer particles many spurious vectors appear. However by using milk as seeding substance a detailed visualisation of the ﬂow close to the body of ciliates is enabled. The maximal noted dimensionless velocity amounts 0.34. Above studies show that detailed ﬂow analysis is enabled for 50-fold optical magniﬁcation with milk as seeding substance. Studies with diﬀerent seeding concentrations reveal crucial diﬀerences. As example ﬂow induced by one ciliate for diﬀerent milk to water proportions of 1:1, 1:2 and 1:4 is shown in Fig. 6. An increasing velocity magnitude with increasing dilution of seeding milk is observed, e.g. for the highest milk concentration (1:1) velocity has the lowest value of umax = 0.17 while for the lowest concentration (1:4) velocity reaches the highest value of umax = 1.00. Moreover, the number of ciliates inﬂuences signiﬁcantly the analysed ﬂow pattern. As example a comparison of velocity distribution for one ciliate, two ciliates as well as for colony with milk to water concentration of 1:1 is done (see Fig. 7). Figure 7 shows an increasing tendency of velocity with increasing ciliates number. The lowest value of maximal dimensionless velocity equal to umax = 0.20 is discovered for the single ciliate while the highest value of umax = 0.86 appears for the colony. Above comparison shows that cooperative colony work inﬂuences the ﬂow velocity displaying bio-synergetic eﬀect. The characteristic ﬂow pattern with two vortices can be seen only for the ﬁrst case with single organism. In the second and third case instead of typical ﬂow

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Fig. 6. Velocity distribution observed by 50-fold optical magniﬁcation with diﬀerent milk concentration (a) 1:1, (b) 1:2, (c) 1:4.

every ciliate produces one vortex. Additionally, synergetic vortex belonging partially to two diﬀerent ciliates is recognized.

4 Conclusions Granulation process is a multiscale phenomenon. As shown by [11, 15] ciliates play a crucial role for granules formation, they can be treated as backbone for GAS development. Moreover, cilia beats of the ciliates providing nutrient ﬂux toward bioﬁlm improve colonization process. Thus, presented μ-PIV

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Fig. 7. Velocity distribution observed by 50-fold optical magniﬁcation for (a) one, (b) two ciliates and (c) colony.

studies contribute to better understanding of granulation process. However, only appropriate experimental methods provide the biocompatible conditions. Detailed ﬂow visualization is enabled with milk as seeding substance. μ-PIV investigations indicate existence of characteristic two vortices generated by single protozoa. Furthermore, the number of ciliates as well as seeding particles concentration have signiﬁcant inﬂuence on the ﬂow pattern.

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References 1. De Kreuk MK, de Bruin LMM, van Loosdrecht MCM (2004). Aerobic granular sludge, from idea to pilot plant. IWA workshop aerobic granular sludge. ISBN 1-84339-509-6 2. Raﬀel M, Willert ChE, Kompenhans J (1998) Particle image velocimetry. A practical guide. Springer, Berlin 3. Buen JJ, Hendriks A, van Loosdrecht MCM, Morgenroth E, Wilderer PA, Heijnen JJ (1999) Aerobic granulation in a sequencing batch reactor. Water Res 33:2283–2290 4. Etterer T, Wilderer PA (2001) Generation and properties of aerobic granular sludge. Water Sci Technol 43:19–26 ¨ ¨ Petermeier H, Fried J, Delgado A (2007) Analysis of 5. Hartmann C, Ozmutlu O, the ﬂow ﬁeld induced by the sessile peritrichous ciliate Opercularia asymmetrica. J Biomech 40:137–148 6. Kowalczyk W, Zima BE, Delgado A (2007) A biocompatible seeding particle approach for -PIV measurements of a ﬂuid ﬂow provoked by microorganisms. Exp Fluids 43:147–150 7. McSwain BS, Irvine RL, Wilderer PA (2004) The eﬀect of intermittent feeding on aerobic granule structure. Water Sci Technol 49:19–25 8. Petermeier H, Kowalczyk W, Delgado A, Denz C, Holtmann F (2007) Detection of microorganismic ﬂows by linear and nonlinear optical methods and automatic correction of erroneous images artefacts and moving boundaries in image generating methods by a neuronumerical hybrid implementing the Taylor’s hypothesis as a priori knowledge. Exp Fluids 42:611–623 9. Sleigh MA, Barlow D (1976) Collection of food by Vorticella. Trans Am Microsc Soc 95:482–486 10. Tay JH, Liu QS, Liu Y (2001) The eﬀects of shear force on the formation, structure and metabolism of aerobic granules. Appl Microbiol Biotechnol 57:227– 233 11. Weber S, Ludwig W, Schleifer KH, Fried J (2007) Microbial composition and structure of aerobic granular sewage bioﬁlms. Appl Environ Microbiol 73:6233– 6240 12. Westerweel J, Dabiri D, Gharib M (1997) The eﬀect of a discrete window oﬀset on the accuracy of cross-correlation analysis of digital particle image velocimetry. Exp Fluids 23:20–28 13. Willert C, Gharib M (1991) Digital particle image velocimetry. Exp Fluids 10:181–193 14. Zima BE, Diez L, Kowalczyk W, Delgado A (2007) Sequencing Batch Reactor (SBR) as optimal method for production of Granular Activated Sludge (GAS)ﬂuid dynamic investigations. Water Sci Technol 55:151–158 15. Zima-Kulisiewicz BE, D´ıez L, Kowalczyk W, Hartmann C, Delgado A (2008) Bioﬂuid mechanical investigations in Sequencing Batch Reactor (SBR). Chem Eng Sci 63:599–608

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Part IV

Numerical Modeling

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Coupling Atomistic and Continuum Descriptions Using Dynamic Control E.M. Kotsalis1 , J.H. Walther1,2 , P. Koumoutsakos1 1

2

Computational Science and Engineering Laboratory, ETH Zurich, 8092, Zurich, Switzerland. Dept. of Mech. Engng., Technical University of Denmark, 2800 Lyngby, Denmark. [email protected]

Summary. We propose control algorithms to enhance the eﬃciency of a hybrid model coupling continuum and atomistic descriptions of dense liquids. Time and length scales are decoupled by using an iterative Schwarz domain decomposition algorithm. In this algorithm, the lack of periodic boundary conditions in the MD simulations leads to spurious density ﬂuctuations at the continuum-atomistic interface. We remedy this problem by using an external boundary force determined by a simple control algorithm that acts to cancel the density ﬂuctuations. The conceptual and algorithmic simplicity of the method makes it suitable for any type of coupling between atomistic, mesoscopic and continuum descriptions of dense liquids.

1 Introduction The modeling and simulation of systems such as biosensors embedded in aqueous environments [1–4], microﬂuidic channels with nanopatterned walls or bluﬀ bodies with superhydrophobic surfaces [5], requires a multiscale approach. Hybrid computations have been proposed in order to couple eﬀectively atomistic and continuum descriptions for dense ﬂuids. The atomistic eﬀects are modeled using Molecular Dynamics (MD) while macroscale phenomena are described by the discretized, incompressible Navier-Stokes equations. Hybrid techniques can be -distinguished on the way information is exchanged between the two descriptions. In ﬂux exchange schemes [6–9] the two descriptions communicate at an interface requiring a conservative exchange of ﬂuid properties, while Schwartz domain decompositions [10–12] require an overlap region where the atomistic and continuum descriptions coevolve. In both algorithms, a critical issue is the elimination of periodicity from the MD system that is often associated with the appearance of density disturbances close to the boundary. Repulsive wall potentials [6, 7] and buﬀer regions [8, 9, 11] have been proposed in order to circumvent this diﬃculty. Werder et al. [12] combined a hard wall with boundary potentials based on the radial distribution

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function of the system that is being simulated in order to impose the local system pressure. This scheme was found to signiﬁcantly reduce the density perturbations in the molecular system compared to existing algorithms and has been used in [13] to conduct multiscale simulations of a Lennard-Jones ﬂuid ﬂowing past and along the axis of a carbon nanotube, coupling MD to Lattice Boltzmann models. Here we employ a control algorithm [14] to adjust the boundary force in order to eliminate the oscillations in the density when exchanging information between atomistic and continuum domains.

2 Methodology The atomistic region is described by MD simulations subject to non-periodic boundary conditions (NPBC). The position ri = (xi , yi , zi ) and velocities vi = (ui , vi , wi ) of the i-th particle evolve according to Newton’s equation of motion: d ri = vi (t), dt d mi vi = Fi = − ∇U (rij ), dt j=i

where mi is the mass and Fi the force on particle i. The interaction potential U (rij ) models the physics of the system. Here we consider the monoatomic ﬂuid of argon. Thus: U (rij ) = U12−6 (rij ) + Um (rw ; ρ, T ), where U12–6 is the 12–6 Lennard-Jones (LJ) potential: U12−6 (rij ) = 4

σ rij

12

−

σ rij

6 ,

(1)

and rij denotes the distance between the i and j atom, and σ and are the length and energy scale of the LJ potential (for argon = 0.996 kJ mol−1 and σ = 0.340 nm). The term Um (rw ; ρ, T ) accounts for the interaction of the atomistic region with the surrounding medium. It depends on the distance to the outer boundary of the atomistic domain rw , the local density ρ, and the local temperature T of the ﬂuid. All interaction potentials are truncated for distances beyond a cutoﬀ radius (rc ) of 1.0 nm. We note that increasing the cutoﬀ from 1 nm to 2 nm does not aﬀect the quality of the results. The equations of motion are integrated using the leap-frog scheme with a time step of 10 fs. We perform the MD simulations at diﬀerent state points of the ﬂuid and report quantities in reduced units (T ∗ = kB T /, ρ∗ = ρσ 3 , and P ∗ = P σ 3 ). In hybrid algorithms, the elimination of periodic boundary conditions in the atomistic domain hinders the maintenance of a uniform

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Fig. 1. Integration domains for the eﬀective boundary force (2). The force contributions along z are integrated over the shaded area. The number of atoms in the inﬁnitesimal ring element is 2πρn g(r)xdxdz, where ρn is the average number density and g(r) the radial distribution function. ΩA and ΩC denote the atomistic and the continuum domains, respectively.

density across the domain and the proper calculation of the virial pressure. In order to correct for the “missing component” of the virial pressure, a boundary force is applied in the atomistic domain [6, 7, 11]. Hence, we impose NPBC in the x-direction with a wall force Fm to exert the correct mean virial pressure (PU ) on the MD system along with a specular wall to impose the ideal kinetic part (PK ) of the system pressure:

rc Fm (r)dr, P = PK + PU = kB T ρn + ρn 0

where kB is the Boltzmann constant. In [12] it was proposed to compute the wall force (Fm ) from the pair potential (1) and the pair correlation function (g(r)) of the working ﬂuid. This technique was shown to alleviate many of the drawbacks of existing methods and it constitutes the basis of the present algorithm. Thus the Lennard-Jones force of each particle weighted by g(r) is integrated over the part of the cutoﬀ sphere that lies outside of the atomistic domain, cf. Fig. 1,

rc √rc2 −z2 ∂U12−6 (r) z xdxdz. (2) g(r) Fm (rw ) = −2πρn ∂r r z=rw x=0 At the supercritical state point (T ∗ = 1.8, ρ∗ = 0.6) [15] this approach was found to reduce drastically the spurious density ﬂuctuations [12]. We examine the validity of this method in one additional state point in the liquid regime, namely (T ∗ = 1.1, ρ∗ = 0.81). The size of the computational domain is 5 nm × 5 nm × 5 nm. The periodicity is broken in the x-direction. The system is weakly coupled to a Berendsen thermostat [16] with a time constant of 0.1 ps. After equilibration we heat only the atoms located in the cells close to the x boundary. For this new state point the method is found

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Fig. 2. The reduced density values (ρ+ = ρ/ρbulk ) of argon conﬁned between hard walls and subject to the wall potential model of Werder et al. [12] up to a distance of a cutoﬀ from the boundary. By reducing the temperature, while increasing the density, the amplitude of the oscillations becomes higher. —+— (T ∗ = 1.8, ρ∗ = 0.60), — ×— (T ∗ = 1.1, ρ∗ = 0.81).

Fig. 3. Schematic of the control algorithm for reducing density ﬂuctuations.

to encounter diﬃculties when lowering the temperature, while increasing the density, at constant pressure, leading to density oscillations close to the boundary (Fig. 2). The amplitude of these oscillations amounts to 8% and is well below previously reported values in hybrid simulations [12] but they may still cause unnecessary disturbances to the atomistic system. In order to eliminate the oscillations we apply a control algorithm for the mean external boundary force applied to the MD system. The control approach is sketched in Fig. 3. Each iteration involves the following steps: we start by applying the external boundary force as proposed in (2). Then we measure the density in short time intervals ﬁltering away high frequency noise. The density ρm is measured with a spatial resolution δx of 0.0166 nm in time intervals of 30 ps and processed twice through a Gaussian ﬁlter resulting ρm as:

1 (x − y)2 ρm (x) exp − dy, ρm (x) = 2

1 (x − y)2 ρm (x) = ρm (x) exp − dy, 2

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Fig. 4. (a) The initial external boundary force computed taking the ﬂuid structure into account and the resulting one after applying the control algorithm at the state point (T ∗ = 1.1, ρ∗ = 0.81). (b) The corresponding uncontrolled (—) and controlled 3 mol reduced density values (- - -). The value used for KP is 0.083 nm and both the amu kJ force and density have been sampled over 10 ns.

where = 2δx. The cutoﬀ used for the discrete evaluation of the convolution is 3δx. We then evaluate the error as: e(rw ) = ρt − ρm (rw ),

(3)

where rw is the distance to the boundary, ρt the desired constant target density and ρm the measured ﬁltered value. We compute the gradient of this error as (rw ) = ∇e(rw ) = −∇ρm (rw ) and amplify this with a factor KP to obtain the changes ΔF in the boundary force as: ΔFi = KP i , for each ith bin. The boundary force is ﬁnally computed as:

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Finew = Fiold + ΔFi , After the root mean square (RMS) of the errors becomes less than a prescribed value, here 1%, we consider that the method has converged and we can start measuring the density for assessing the quality of the result. The controller online keeps acting on the system. We test this approach for the state point (T ∗ = 1.1, ρ∗ = 0.81), where the mean force algorithm of Werder et al. [12] 3 mol failed to fully eliminate the ﬂuctuations. We set KP = 0.0830 nm amu kJ and the results shown in Fig. 4 demonstrate that the method converges and eliminates the density oscillations. When compared with the initial force, we observe a decrease of the magnitude of the force close to the boundary and a shift for the location of the minimum. At larger distances from the wall (rw > 0.6 nm) the shape of the force is not signiﬁcantly altered. We ﬁnd a value of KP = 3 mol 0.0830 nm amu kJ to guarantee good stability properties and fast convergence.

3 Couette Flow Finally, we validate the control algorithm for the case of Couette ﬂow of liquid argon (T ∗ = 1.1, ρ∗ = 0.81) conﬁned between two graphite surfaces. A sketch of the ﬂow geometry is shown in Fig. 5. The size of the computational domain is 30.0 nm×4.3 nm×4.9 nm, small enough to allow a fully atomistic simulation which we will use as a reference. The number of computational boxes used for heating is 30 × 1 × 1. The same resolution is used to sample the velocities that serve as a boundary condition (BC) for the continuum solver. The ﬂow is imposed by moving the upper wall with a velocity v = 100 m s−1 . In the hybrid approach we apply the Schwarz alternating method with an overlap region of 4 cells (4 nm). Details about the exchange of boundary

Fig. 5. Sketch of the hybrid simulation for the Couette ﬂow. Shaded regions denote the domain of MD simulations and grid cells indicate simulations using the NavierStokes equations.

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Fig. 6. (a) Reduced density proﬁles in the Couette ﬂow of the reference MD simulations and of the hybrid controlled (—) and uncontrolled (− − −) cases. Density oscillations of the order of 10% in the uncontrolled case are eliminated by the controller. (b) Resulting velocity proﬁles in the Couette ﬂow of the reference MD simulations and the hybrid controlled (—) and uncontrolled (− − −) cases.

conditions between the MD and the continuum region, described by incompressible Navier-Stokes (NS) equations, can be found in [12]. In the present case the solution to the NS equations is a linear streamwise velocity proﬁle. In the hybrid approach the system contains a graphite surface and the argon atoms form layers in its vicinity. During equilibration we d Each MD subdomain in the hybrid case has the dimensions 10.0 nm × 4.3 nm × 4.9 nm (10 boxes in x and 1 box in y- and z-direction), large enough to resolve the extensive physical perturbations in the ﬂuid density at the ﬂuid-solid interface. In a cycle of the hybrid algorithm, we impose the boundary condition from the continuum to the MD, equilibrate the MD system for 20 ps to reach a new quasi-steady state and subsequently sample the velocities for 100 ps to extract the BC for the continuum. The ﬂow reaches a steady state after

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10 ns and we sample the results from 10 ns to 35 ns every 1 ps with a total of 25000 samples. In Fig. 6 we show the velocity and density proﬁles obtained from the hybrid simulations. We observe large physical perturbations in the density at the graphite surface and spurious oscillations at the continuum-MD interface in the uncontrolled case which are eliminated in the controlled one. The velocity proﬁles are similar in both cases.

4 Conclusions We have presented a control algorithm to eliminate density ﬂuctuations in the coupling of atomistic models with continuum descriptions of dense liquids using Schwarz domain decomposition. The algorithm is validated for ﬂuids at rest and it is shown to eliminate density oscillations with amplitude of the order of 8%. Finally we demonstrated the capability of the algorithm for multiscale simulations by successfully coupling of MD simulations to a continuum description for the Couette ﬂow problem.

References 1. Chen RJ, Bangsaruntip S, Drouvalakis KA, Kam NWS, Shim M, Li YM, Kim W, Utz PJ, Dai HJ (2003) Noncovalent functionalization of carbon nanotubes for highly speciﬁc electronic biosensors. Proc Natl Acad Sci USA 100(9):4984–4989 2. Li J, Ng HT, Cassell A, Fan W, Chen H, Ye Q, Koehne J, Han J, Meyyappan M (2003) Carbon nanotube nanoelectrode array for ultrasensitive DNA detection. Nano Lett 3(5):597–602 3. Lin Y, Taylor S, Li H, Shiral Fernando KA, Qu L, Wang W, Gu L, Zhou B, Sun Y (2004) Advances toward bioapplications of carbon nanotubes. J Mater Chem 14:527–541 4. Zheng M, Jagota A, Semke ED, Diner BA, McLean RS, Lustig SR, Richardson RE, Tassi NG (2003) DNA-assisted dispersion and separation of carbon nanotubes. Nat Matter 2:338–342 5. Watanabe K, Takayama T, Ogata S, Isozaki S (2003) Flow between two coaxial rotating cylinders with a highly water-repellent wall. AIChE J 49(8):1956–1963 6. O’Connell ST, Thompson PA (1995) Molecular dynamics-continuum hybrid computations: A tool for studying complex ﬂuid ﬂow. Phys Rev E 52(6):R5792– R5795 7. Flekkøy EG, Wagner G, Feder J (2000) Hybrid model for combined particle and continuum dynamics. Europhys Lett 52(3):271–276 8. Flekkoy E, Delgado-Buscalioni R, Coveney P (2005) Flux boundary conditions in particle simulations. Phys Rev E 72:026703 9. De Fabriitis G, Delgado-Buscalioni R, Coveney P (2006) Multiscale modeling of liquids with molecular speciﬁcity. Phys Rev Lett 97:134501 10. Hadjiconstantinou NG (1999) Hybrid atomistic-continuum formulations and the moving contact-line problem. J Comput Phys 154:245–265

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11. Nie XB, Chen SY, E WN, Robbins MO (2004) A continuum and molecular dynamics hybrid method for micro- and nano-ﬂuid ﬂow. J Fluid Mech 500: 55–64 12. Werder T, Walther JH, Koumoutsakos P (2005) Hybrid atomistic-continuum method for the simulation of dense ﬂuid ﬂows. J Comput Phys 205:373–390 13. Dupuis A, Kotsalis E, Koumoutsakos P (2007) Coupling lattice Boltzmann and molecular dynamics models for dense ﬂuids. Phys Rev E 75:046704 14. Kotsalis E, Walther J, Koumoutsakos P (2007) Control of density ﬂuctuations in atomistic-continuum simulations of dense liquids. Phys Rev E 76:016709 15. Johnson JK, Zollweg JA, Gubbins KE (1993) The Lennard-Jones equation of state revisited. Mol Phys 78(3):591–618 16. Berendsen HJC, Postma JPM, van Gunsteren WF, DiNola A, Haak JR (1984) Molecular dynamics with coupling to an external bath. J Chem Phys 81(8):3684– 3690

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Lattice Boltzmann Simulation of Pulsed Jet in T-Shaped Micromixer Md Ashraf Ali, Lyazid Djenidi Discipline of Mechanical Engineering, The University of Newcastle, Newcastle, 2308 NSW, Australia. [email protected], [email protected]

Summary. A numerical study using the lattice Boltzmann method (LBM) is carried out to investigate mixing enhancement in a T-shaped micromixer using pulsed jet at low Reynolds number. The objective of this study is to determine whether the use of pulsed jet can enhance mixing. The same non reacting iso-thermal and incompressible ﬂuid is used for the two ﬂuid streams as our study focuses on the mixing through mechanical process (folding and stretching). Although diﬀusivity ultimately dictates the mixing at low Reynolds number, diﬀusivity is not an issue in our study as stretching and folding is independent of diﬀusivity. The simulation is carried out for single jet, opposite jets and parallel jets. Three dimensional study is also done where a combination of active (pulsed jet) and passive mixers (ﬂat plate) are used to enhance mixing. For opposite and parallel jets conﬁgurations, simulations are done for inphase and out of phase jets. Study is also done varying the positions of the jets relative to each other. Pulsed jets seems to enhance mixing by creating vortical motion. These vortices increases the interfacial surface between the two streams, thus enhancing molecular diﬀusion.

1 Introduction A numerical study using the lattice Boltzmann method (LBM) is carried out to investigate mixing enhancement in a T-shaped micromixer using pulsed jet at low Reynolds number. The simulation is carried out for single jet, opposite jets and parallel jets. Eﬀects of synthetic jet geometry conﬁguration to create vortical motion is studied. Three dimensional study is also done where a combination of active (pulsed jet) and passive mixers (ﬂat plate) are used to enhance mixing. For opposite and parallel jets conﬁgurations, simulations are done for inphase and out of phase jets. Study is also done varying the positions of the jets relative to each other. The main objective of our research is to improve mixing eﬃciency in microchannel using either active or passive mixers or a combination of both. The Lattice Boltzmann Method is used to carry out the simulations. One of our complementary objectives is to develop an accurate, improved and modular code to simulate ﬂuid ﬂow. M. Ellero et al. (eds.), IUTAM Symposium on Advances in Micro- and Nanoﬂuidics, IUTAM Bookseries 15, c Springer Science + Business Media B.V. 2009

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2 Numerical Procedure 2.1 Lattice Boltzmann Method The numerical simulations are performed using the Lattice Boltzmann simulations method (LBM) [1, 2]. The LBM is based on kinetic theory. Rather than solving the governing ﬂuid equations (Navier-Stokes equations), the LBM solves the Boltzmann equation on a lattice. The basic idea of the LBM is to construct a simpliﬁed kinetic model that incorporates the essential physics of microscopic average properties, which obey the desired (macroscopic) Navier stokes equations [3]. With suﬃcient symmetry of the lattice, the LBM inherently solves these latter equations with second order accuracy. For present calculations, for 2D simulation each computational node consists of a two dimensional lattice composed of 8 moving particles and a rest particle. This model is called D2Q9 model (Fig. 1(a)). For 3D simulations D3Q19 model (Fig. 1(b)) is used which is composed of 18 moving particles and a rest particle. The Boltzmann equation is discretized on that lattice and results in the lattice Boltzmann equations, which governs the time and space variations of the single-particle distribution fi (x, t) at the lattice site x: 1 fi (x + ei Δt, t + Δt) − fi (x, t) = − [fi (x, t) − fieq (x, t)] τ

(i = 0, 1, . . . , n) (1)

where n = 8 for D2Q9 model and n = 18 for D3Q19 model, λ is the relaxation time, Δt is the time step, ei is the particle velocity in the i direction and fieq is the equilibrium single-particle distribution: 9 3 2 eq 2 fi = ρωi 1 + 3ei .u + (ei .u) − u (2) 2 2 where ρ(= i fi ) is the ﬂuid density, u(= i fi ei ) is the local ﬂuid velocity and ωi are the corresponding weights. For D2Q9 model ωi = 4/9 for i = 0, 1/9 for i = 1 to 4 and 1/36 for i = 5 to 8. For D3Q18 model ωi = 1/3 for i = 0,

Fig. 1. (a) D2Q9 Model, (b) D3Q19 Model.

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11/18 for i = 1 to 6 and 1/36 for i = 7 to 18. The relaxation time is related to kinematic viscosity of the ﬂuid via the relation: ν=

2τ − 1 . 6

(3)

The time step Δt and lattice spacing √ (Δx, Δy, Δz) are set to one. With this = 1 to 4 and 2 for i = 5 to 8 for D2Q9 model. |ei | = 1 setting |ei | = 1 for i √ for i = 1 to 6 and 2 for i = 7 to 18 for D3Q18 model. For both model |e0 | = 0. The left side of (1) is so-called streaming operation, which means that particles move to the nearest neighbors along their velocity direction. The right hand side is the collision term, here modeled by the BGK collision operator, which describes the redistribution of the particles at each node [2]. Thus (1) is solved according to these two rules: collision and streaming. The collision step is described by: finew = fi (x, t) −

Δt (fi (x, t) − fieq (x, t)) τ

(4)

where fieq are calculated using (2). The streaming step is described by fi (x + ei Δt, t + Δt) = finew (x, t)

(5)

The collision are entirely local, making the LBM eﬃciently parallelized. At time t, the particle distribution are updated based on (4) and then at time t + Δt, the particles propagate according to (5).

3 Results 3.1 Two Dimensional Pulsed Jet The Computational Details The computational domain for 2-D simulation of pulsed jet in T-shaped micromixer is shown in Fig. 2(a). The domain consisting of uniform Cartesian mesh has 180 × 90 grid points with Δx = Δy = 1. This 180 × 90 grid is chosen as it proved to be small enough in order to reduce the computation time and large enough so that the possible eﬀects of the inlet and outlet boundary conditions on the simulation are minimized. The Reynolds number Re(Uin ×D/ν) of the main channel is 100. The wall jet is generated by applying an oscillating velocity proﬁle at the lower part of the jet inlet. This proﬁle is speciﬁed as ujet = u0 cos2 (wt), where u0 is the jet velocity amplitude, ω is the jet frequency and t is the time. No slip boundary condition is used at the solid walls. A constant velocity is imposed at the inlets of the channel and a no ﬂux (zero gradient) boundary condition is used at channel outlet.

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Fig. 2. Computation domain (a) 2D simulation, (b) 3D simulation.

Single Pulsed Jet Simulations for diﬀerent velocity jet amplitudes and pulse values have been carried out to determine the eﬀect of these parameters on the mixing of the two streams of the ﬂuid. Simulations are done using u0 /v0 = 0, 1, 2, 5, 10 and ω = 0.001. Here v0 is velocity amplitude at the inlets. Figure 3 shows the velocity vectors and the streamlines at maximum jet exit velocity. For small jet velocity amplitudes (u0 /vo = 1) (Fig. 3a) the jet can not aﬀect the main channel ﬂow that much as it can not create any vortices in the main channel ﬂow. As the jet amplitude increases (u0 /vo = 2, 5, 10) (Figs. 3(b), (c) and (d)) rotational motion increases in size and intensity. For u0 /v0 = 5, 10, the rotational motion occupy half the height of the channel. Clearly mixing improve with the occurrence of the vortical motion. Indeed this vortices increases the interfacial surface between the two streams, thus enhancing molecular diﬀusion. The jet alters the velocity gradients of the mixing channel strongly by creating vortices. Opposite Pulsed Jets In these conﬁgurations two pulsed jets are placed on the opposite walls of the mixing channel. Two diﬀerent cases are studied. In one case both jets are pulsing at same time (in phase). In the second case the jets are pulsing at alternative times (out of phase). The position of one jet is varied relatively to other as pictured in the Fig. 4. Figure 4 shows velocity vectors and streamlines when the jets are in phase. In Fig. 4(a) the jets are inline that means they are facing each other. In Fig. 4(b) the jet on the upper wall is shifted downstream by a distance of 0.5D. From the Fig. 4 we can see that two jets seem to enhance further mixing since an extra vortical motion is created. Velocity gradients of the mixing channel is altered much more compared to using a single pulse jet. From ﬁgures we also can say that shifting one jet downstream to other can create more rotational motions. Figure 5 show velocity vectors

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Fig. 3. Plot of instantaneous velocity vectors and streamlines at maximum jet velocity exit. (a) u0 /vo = 1, (b) u0 /vo = 2, (c) u0 /vo = 5 and (d) u0 /vo = 10.

Fig. 4. Plot of velocity vectors and streamlines at maximum jet velocity exit for opposite and in-phase pulsed jets.

and streamlines of ﬂow in mixing channel when the inline jets are pulsing at alternating time form the opposite walls of the mixing channel. Figure 6 shows the same date when shifted jets are pulsing at alternating time. In Figs. 5(a) and 6(a) the top jet is at maximum jet exit velocity and in Figs. 5(b) and

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Fig. 5. Plot of velocity vectors and streamlines at maximum jet velocity exit for out of phase pulsed jets positioned at the same points of opposite wall of the mixing channel.

Fig. 6. Plot of velocity vectors and streamlines at maximum jet velocity exit for out of phase pulsed jets positioned at diﬀerent points of opposite wall of the mixing channel.

6(b) the bottom jet is at maximum jet exit velocity. From the ﬁgures we can see that pulsing the jets alternatively is not creating more rotational motion compared to single pulsed jet. So from our simulation we can say that using two jets in opposite wall of the mixing channel and pulsing them at the same time can enhance the mixing more compared to using a single jet or double jet at alternating times. Pulsed Jets in Parallel In parallel pulsed jets conﬁgurations two jets are placed on the same wall of the mixing channel. Two diﬀerent cases are studied. In one case both the jets are in phase that means they are pulsing at the same time. In the second case the jets are pulsing in alternative times. Figure 7(a) shows velocity vectors and streamlines of ﬂow in mixing channel when the jets are pulsing at the same time form the same wall of the mixing channel. Figure 7(b) shows same

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Fig. 7. Plot of velocity vectors and streamlines at maximum jet velocity exit for parallel pulsed jets. (a) Both jets are pulsing at the same time (In-phase). (b) Jets are pulsing at alternative time (Out of phase).

data when the jets are pulsing at alternating times. When the jets are inphase, they seems to enhance mixing further since an extra vortical motion is created. When the pulsing are done in alternating time parallel jets conﬁguration is not creating any extra rotational motion compared to the single jet conﬁguration.

3.2 Three Dimensional Pulsed Jet Mixing enhancement in a T-shaped micromixer with a rectangular cross sections (Fig. 2(b)) where a combination of active (pulsed jet) and passive mixers (ﬂat plate) are used is studied. The simulations are carried out in three dimensions where the width of the inlet and the mixing channels is 20 mesh points and the height is 10 mesh points, the inlet channels have 15 mesh points along the length and the main channel contains 110 mesh points. The Reynolds number Re(Uin D/ν) of the main channel is 100. The velocity of the pulsed jet is 5 times that of the main channel ﬂow. The inlet velocity proﬁle for jet is speciﬁed as ujet = u0 cos2 (wt). We want to induce a spanwise motion which combined with the main ﬂow will generate a swirl motion. Djenidi and Moghtaderi [4] showed that the eﬀective mixing can be achieved by the addition of swirl motion. To create a swirling motion a cap like structure made of a ﬂat plate is mounted a few distance above the jet exit in the channel. Same boundary conditions used in 2d simulations are used here. Figure 8(a) shows velocity vectors and streamlines in YZ plane with no cap like structure Fig. 8(b) shows the same with a cap like structure mounted on top of the jet exit. Looking at the streamlines in the ﬁgure we can say that the cap is helping to create a rotational motion perpendicular to the main channel ﬂow direction in the mixing channel. Without using the cap there is some rotational motion in the main channel but its only near the wall but if cap is used the rotational motion is throughout the width of the mixing channel.

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Fig. 8. Plot of velocity vectors and streamlines at maximum jet velocity exit (a) Without the cap like structure. (b) With the cap like structure.

4 Conclusions Study of pulsed jet in T-shaped micromixer to enhance mixing is done. Simulations are carried out for single jet, opposite jets and parallel jets. For single jet conﬁgurations it is found that jet velocity amplitude needs to be high enough to create rotational motion, which in turn improves mixing. If the jet velocity amplitude is low compared to main inlet ﬂow velocity, it can not alter the velocity gradient of the mixing channel thus cannot aﬀect the mixing. For opposite pulsed jets conﬁguration if both the jets are pulsing at the same time than they create more rotational motions but if the jets are pulsing at an alternative time than they are not creating more rotational motion compared to single jet conﬁguration. Putting the top jet further downstream of the bottom jet in the mixing channel instead of putting them at the same point but opposite wall creates more rotational motions. In case of parallel jets, jets pulsing at same time have more aﬀect on mixing compared to alternating pulsing. In 3D simulation using a cap like structure setup can enhance mixing more compared to the setup without the cap.

References 1. Succi S (2001) The lattice Boltzmann equation for ﬂuid dynamics and beyond. Oxford University Press, London 2. Chen S, Doolen GD (1998) Lattice Boltzmann method for uid ﬂows. Ann Rev Fluid Mech 30:329–364 3. Frisch U, Hasslacher B, Pomeau Y (1986) Lattice-gas automata for the NavierStokes equation. Phys Rev Lett 56(14):1505–1508 4. Djenidi L, Moghtaderi B (2006) Numerical investigation of laminar mixing in a coaxial microreactor. J Fluid Mech 568:223–242

Simulation of High-Speed Flow in μ-Rockets for Space Propulsion Applications Jos´e A. Mor´i˜ nigo1 , Jos´e Hermida-Quesada2 1

2

Dept. of Space Programmes, National Institute for Aerospace Technology, Madrid, Spain. [email protected] Dept. Aerodynamics & Propulsion, National Institute for Aerospace Technology, Madrid, Spain. [email protected]

Summary. The simulation of micron-sized rockets for space propulsion requires the accurate modelling of the highly coupled physical processes that take place. The purpose of this investigation is the assessment of the continuum-based Navier– Stokes approach with slip boundary conditions in “large” MEMS, where slip-ﬂow is expected to occur in the μ-nozzle. Non-equilibrium near the walls is addressed with the implementation of a 2nd-order slip-model for the velocity and temperature at the solid-gas interface, where thermal coupling is enforced. Initial validation with axisymmetric and three-dimensional DSMC data of μ-rockets is provided. Numerical results at operational ﬂow conditions show that transitional ﬂow is reached near the exit of the nozzle and that strong rarefaction occurs ahead the nozzle-lip and in the gas plume. The eﬀects of viscous dissipation and heat transfer upon the ﬂowﬁeld and performance are discussed in detail.

1 Introduction Micropropulsion constitutes an enabling technology for the new space missions relying on very small spacecrafts, that demand in-orbit maneuvers like stationkeeping or ﬁne attitude control. For these tasks, tiny impulse bits and thrust levels within the range of micro- to a few milli-Newtons are required. Among the variety of solutions, chemical μ-rockets arise as an attractive concept since it is simple, versatile and fulﬁlls the constrains of low mass and power, as well as ease of integration and fabrication. Hence, it meets the criterion of lowcost, so paramount to this class of applications (e.g., small satellites under 20 kg mass). Furthermore, its feasibility has been demonstrated with both cold and hot expanding gas through miniaturized nozzles. In particular, the development of arrayed solid-propellant μ-rockets is reported in [1, 3]. In this context, the prediction of their performance in steady and transient operation is aimed for obvious reasons and the accurate modelling is a challenge in these devices. Previous numerical studies [4, 5] have shown that μ-nozzles

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ﬂows may be resolved with moderate success considering only the rareﬁed ﬂow uncoupled from the wall thermal response. Nevertheless, heat transfer in μ-systems is a major issue that must be taken into account in micropropulsion studies [6]. Thus, the present work presents a methodology that incorporates the coupling of gasdynamics with the solid thermal response. The paper is organised as follows. Section 2 provides an overall description of the numerical method and the discrete 3D version of the slip-model and heat-ﬂux at the solid-gas interface. Section 3 addresses its validation with available Direct Simulation Monte Carlo (DSMC) data of two μ-nozzles with conical and ﬂatshaped divergent contours [7]. In the next section, the results from steady and unsteady coupled solid-gas simulations at operation conditions are discussed. Here, various cases are analysed: cold- and hot-gas ﬂow with a silicon- (Si) or glass-wafer. Finally some conclusions are given.

2 Numerical Method 2.1 Continuum-Based Simulation The μ-rockets considered may be classiﬁed in the “large” MEMS group, of characteristic dimension # ∼ 100 μm and where the supersonic expanding ﬂow experiences small to moderate rarefaction (quantiﬁed by the Knudsen number, Kn). One important aspect of such ﬂows is the disparity in Kn, as it is seen in Fig. 1, where chamber and nozzle-lip conditions are plotted. While ﬂow in the chamber behaves as continuum, the slip-ﬂow regime (10−2 < Kn < 0.15) is reached inside the μ-nozzle and even transitional ﬂow may occur at the lip vicinity. In slip-ﬂow (Table 1), rarefaction is conﬁned to a thin region near the walls, so local non-equilibrium makes the ﬂow to depart from the continuum-based Navier–Stokes (NS) description. To ﬁx this behaviour, the

Fig. 1. Flow regimes for systems at standard temperature. The μ-rocket location (chamber and nozzle-lip) is plotted at validation ﬂow conditions.

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Table 1. Geometrical parameters of the conical (2D-Axi) and ﬂat-shaped (3D-Flat) μ-nozzles. Geometry Divergent half-angle Convergent half-angle Divergent length Throat diameter Exit to throat area ratio Chamber to throat area ratio 1 Throat width of the 3D-Flat nozzle.

αdiv αconv Ldiv Dth Ae /Ath Ac /Ath

2D-Axi 15◦ 22◦ 5038 μm 300 μm 100 4.9

3D-Flat 15◦ 22◦ 5038 μm 300 μm1 10 2.2

Fig. 2. Layout of the Knudsen layer λ (a) and nomenclature of the computational cell (b). Uw and Tw correspond to the velocity and temperature of the wall.

NS equations should be supplemented with modiﬁed boundary conditions (BCs) at the walls, to account for the velocity slip (Uslip ) and temperature jump (Tslip ). These BCs are the so-called slip-model described in the next subsection, which has been implemented in the FLUENT code to perform compressible NS simulations coupled with solving the heat equation in the solid region. The set of governing equations is discretized using a 2nd-order upwind algorithm. Gas (N2 ) is assumed ideal. The existing small Reynolds number makes the assumption of laminar ﬂow plausible and the viscosity obeys the Sutherland’s law. Steady and time-accurate simulations have been carried out with time-marching and dual-time stepping integration, respectively. 2.2 Higher-Order Slip-Model The implementation corresponds to the 2nd-order (in Kn) slip-model of Karniadakis and Beskok [8], with the slip velocity Uslip (see Fig. 2a) given by #Kn ∂U k ∂T 2 − σV 3γ−1 Pr + (1) Uslip − Uw = σV 1 − B(Kn)Kn ∂n w 4 γ ρRg Tw ∂s w being the 2nd term on the right hand the thermal creep (Ucreep ), Uw the wall velocity, Pr = μCp /k the Prandtl number and Kn = λ/# the local Knudsen

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number with λ = μ π/2ρp the local mean free path and # the GradientLength Local (GLL) [9], deﬁned as T ρ Us , , (2) # = min |∇Us · s| |∇T · s| |∇ρ · s| where Us , T, ρ stand for tangential to wall velocity, temperature and density (tangent and normal vectors are indicated in Fig. 2). This local length provides an improved scale information in supersonic ﬂows, where large variations occur. The term B(Kn) is approximated by the ﬁrst coeﬃcient of its Taylor expansion B(Kn) = b + cKn + O(Kn 2 ), then # d2 Us /dn2 . (3) B≈b= 2 dUs /dn w

Formally this substitution leads to a 2nd-order in Kn condition. The temperature jump is proposed in analogy to Uslip from Smolukowski (1898) 2 − σT 2γ # Kn ∂T Tslip − Tw = (4) σT γ + 1 Pr 1 − B(Kn)Kn ∂n w

with accommodation coeﬃcients σV = 0.8 for momentum (corresponding to N2 in contact with silicon) and σT = 1.0 for energy. Besides, thermal coupling at the solid-gas interface implies to enforce the temperature jump BC and the heat-ﬂux continuity across it at each iteration. Thus, making use of the notation of Fig. 2(b), the heat-ﬂux balance at the solid (s) and gas (g) sides is written g Tslip − Tcell ∂T ∂T T s − Tw ≈ kg = ks ≈ ks cell . (5) kg ∂n g dg ∂n s ds Hence, Tw is computed in the simulation. It is noted that only heat conduction is included in the heat-ﬂux balance set at the solid-gas interface. Its extension to handle more complex situations (e.g., supply or removal of energy at the wall due to external heating or active cooling) is not considered in this study. The implementation of the slip-model and heat-ﬂux condition in FLUENT as test-bench has been done with the User Deﬁned Functions [10]. The discretized version of the slip-model reads FV Ucreep FV + , 1+ Uslip = Ucell d Ucell d (6) FT FT 1+ Tslip = Tw + Tcell d d where 2 − σV #Kn 2 − σT 2γ # Kn , FT = . (7) σV 1 − bKn σT γ + 1 Pr 1 − bKn In 3D, the slip velocity vector at the boundary V BC is build from the projection of the cell velocity along the local wall streamline: V BC = Uslip V s /V s , with V s = V cell − V cell · n. FV =

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3 Flow Conditions and Model Validation In a preceding work [11], the authors have accomplished the assessment of the slip-model by comparison with the DSMC data of Alexeenko et al. [7] for an axisymmetric μ-nozzle with isothermal walls and N2 as agent gas. This section extends the comparison to a DeLaval 3D(-Flat) nozzle and revisits the axisymmetric case (Fig. 3), both with thermal coupling. The 3D-Flat nozzle has been generated by a 300 μm extrusion of the meridian cut from the 2D-Axi nozzle. A sharp, squared throat-section and smooth walls are assumed. The computational domain comprehends twofold the nozzle length streamwise and up to the wafer edge crosswise in the outer region. Then, the gas expansion and turning ﬂow at the lip vicinity can be captured as part of the solution. More details of the mesh size and grid convergence can be found in [11]. Prescribed ﬂow conditions for validation purposes are shown in Ta-

Fig. 3. Half-view of the μ-rocket with the DeLaval axisymmetric conical (left-side) and 3D-Flat (right-side) nozzle. Fluid and wafer zones are visible. Table 2. Slip-model validation and operational ﬂow conditions. Agent gas: molecular nitrogen (N2 ) Speciﬁc heat ratio Freestream static pressure Chamber conditions for validation: Stagnation pressure Stagnation temperature Operation chamber conditions: Stagnation pressure Stagnation temperature

γ = Cp /Cv p∞

1.4 10 Pa

pt,o Tt,o

10000 Pa 300 K

pt,o Tt,o

50000 Pa 2000 K (HOT) 300 K (COLD)

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ble 2 and correspond to cold-ﬂow at 300 K and throat-based Reynolds number Re th = 461. The operation ﬂow conditions discussed in the following section are included in the table. 3.1 Conical Nozzle Steady-state solutions have been computed with the axisymmetric formulation of the NS equations in the case of the 2D-Axi nozzle. As initial test of the model validity, the Knudsen number along the nozzle wall and axis has been plotted (see Fig. 4) taking two diﬀerent length scales #: the GLL information (Kn GLL ) and the local nozzle radius (Kn R ). The Kn GLL plot conﬁrms the intensiﬁcation of the gradients near the nozzle exit, that leads the ﬂow into the transitional regime and it alerts on the foreseeable marginal validity of the slipmodel in this zone for such a low supply pressure (0.1 bar) at the μ-chamber. In addition, the comparison with the DSMC data plotted (see Figs. 5–6) shows a rather good agreement in the nozzle core, though a departure is visible in the plume. This progressive rarefaction of the expanding gas makes the prediction to deteriorate, as it is seen in Fig. 6, where the axial velocity proﬁle is compared at three cross-sections. At x/Ldiv = 1 the continuum simulation underpredicts the slip velocity, but the boundary layer thickness is fairly well resolved. Regarding the origin of the disagreement, it should be noticed that thermal coupling yields Tw ∼ 198 K, whereas the DSMC simulations have been performed with a diﬀuse reﬂection model (σV = 1.0) at the wall and Tw = 300 K. Hence, it seems reasonable to expect some deviation caused by the energy balance at the solid-gas interface.

Fig. 4. Knudsen number at the wall and axis of the 2D-Axi nozzle, corresponding to validation ﬂow conditions pt,o = 10000 Pa, Tt,o = 300 K (Kn GLL : Gradient-Length Local-based Knudsen; Kn R : Local Radius-based Knudsen).

Fig. 5. Normalized density (left) and temperature (right) at the 2D-Axi nozzle centreline, corresponding to validation ﬂow conditions. DSMC data from [7].

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Fig. 6. Normalized velocity proﬁle at the 2D-Axi nozzle centreline (left) and crosssections (right) for x/Ldiv = 0.5, 0.75 & 1, corresponding to validation ﬂow conditions (r: radial coordinate; Re : nozzle exit radius. DSMC data from [7]).

Fig. 7. Knudsen number Kn GLL contours at the 3D-Flat nozzle-lip (left) and crosssections (right) for x/Ldiv = 0.5, 0.75 & 1, corresponding to validation conditions.

3.2 3D-Flat Nozzle Similarly to the 2D-Axi nozzle, the Knudsen number is plotted in Fig. 7 at the xy-centreplane and four cross-sections. Rarefaction experiences a signiﬁcant increase in the end quarter of the divergent section, close to the lip walls. However, the much lower expansion ratio of the 3D-Flat nozzle makes the maximum Kn GLL attained inside the nozzle to be within the slip-ﬂow regime (for the same chamber supply conditions than 2D-Axi). The comparison of the velocity and temperature proﬁles along the nozzle centreline (Fig. 8) shows that over- and under-prediction, respectively, exists. Besides, the mismatch begins just after the throat, where gas rarefaction is rather small and the NS description accurate. Hence, the major origin of this behaviour is attributed to the energy exchange at the wall: the colder Tw obtained in the simulations with thermal coupling provokes a more eﬃcient expansion downstream the throat and the development of thinner boundary layers. Compared with the 2D-Axi case, the expansion inside the 3D-Flat μnozzle is much weaker and the gentle slope of the pressure curve in Fig. 9 (right) reveals an ineﬃcient divergent portion.

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Fig. 8. Normalized velocity (left) and temperature (right) at the centreline of the 3D-Flat nozzle, corresponding to validation ﬂow conditions. DSMC data from [7].

Fig. 9. Steady-state Mach contours (left) and normalized static pressure at the centreline (right) of the 2D-Axi and 3D-Flat nozzles, for validation ﬂow conditions.

4 Results and Discussion A series of steady-state (2D-Axi, 3D-Flat) and transient (2D-Axi) simulations have been run for the operation (cold- and hot-) conditions speciﬁed in Table 2 to investigate the ﬂowﬁeld and characterize the performance. The μ-rocket transients mimic the rapid opening of a valve (time-varying inlet BC) that injects the N2 into the chamber. 4.1 Viscous Heating Four ﬂuid-solid conﬁgurations have been explored for the 2D-Axi case, namely: cold-ﬂow with solid zone (wafer) made of silicon (Si) and thermally insulated (adiabaticity BC set at the external-wall); hot-ﬂow with adiabatic Si-wafer; hot-ﬂow with cooled Si-wafer (forced convection BC at the external-wall); and hot-ﬂow with adiabatic glass-wafer. The evolution of the Mach number along the axis, depicted in Fig. 10 for Si-wafer, reveals the existence of a weak shock wave running from the throat edge. This structure is better discerned in the case of cold-ﬂow with steady-state iso-Mach maps and Mach proﬁles, which exhibit a sudden drop at the location x/Ldiv ≈ 0.12 that corresponds to

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Fig. 10. Transient Mach number proﬁle at the axis (left) and steady-state Mach ﬁeld (right) of the 2D-Axi nozzle with Si-Wafer, for cold- and hot-ﬂow, at operation ﬂow conditions (time instants = 1, 10, 40 & 100 s).

the shock reﬂection on the axis. Furthermore, the iso-Mach contours permit to compare the growing of the boundary layers across the nozzle, notably thicker in hot-ﬂow of Re th = 233 (Re th = 2140 in cold-ﬂow). Here, the lower Mach number attained in the core is an obvious consequence of the stronger viscous eﬀects. Figures 11 and 12 illustrate a time sequence of the μ-rocket transient by means of temperature snapshots for the four cases analysed. Its examination reveals that wafer heating is maximum for the insulated (non-cooled) Si-wafer. On the contrary, cooling impedes the material temperature to rise over the melting point of silicon during long transients and it favours the gas expansion. The results for the glass-wafer show that the low thermal conductivity of the glass delays the heating of the divergent portion, then it acts improving the expansion process during short transients. The disparity between expansion and heat-transfer characteristic times explains the long time needed to reach the steady-state according to Fig. 10. The temperature maps deserve more attention. Hence, Fig. 13 gathers the temperature proﬁle at four crosssections, for various time instants. The main observation is the formation of a temperature peak in hot-ﬂow due to the action of viscous heating, which means that a portion of the heat generated by friction is transferred to the wall. The situation is diﬀerent in cold-ﬂow as a result of its higher Reynolds number, so the maximum temperature is reached at the wall at each time instant. To better understand this behaviour, it is convenient to recall the energy equation in boundary layer form ρCp

DT dp = Us − ∇q + ε Dt ds

(8)

where D is for total derivative, Us = V cell ·s and ε = μ(∂Us /∂n)2 is the viscous dissipation. Taking orders of magnitude for ε and heat conduction leads to

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Fig. 11. Snapshots of static temperature in the 2D-Axi nozzle with insulated upperwall for Si-Wafer with cold-ﬂow (left) and Glass-Wafer with hot-ﬂow (right). Time instants: 1, 10, 40 & 100 s. U2

μ R2c ε γ−1 ≈ Tthc = Pr Mc2 . ∇q k R γ th

(9)

Therefore, for Mc ∼ 2 is ε/∇q ∼ O(1), that indicates the dominant role of the viscous dissipation in supersonic ﬂows of even moderate Mach number. Hence, ε must be considered in the modelling. 4.2 Nozzle Performance Besides the viscous heating, other losses are accounted for in the thrust (E), massﬂow rate (G) or speciﬁc impulse Isp = E/Ggo delivered by a μ-rocket. In particular, the 1D approximation of the computed to ideal ratio of Isp 2Cp (Tt,o − Te ) 2q˙ Ae (pe − p∞ ) Isp Divergence = + + − (10) losses Isp,id Isp,id GIsp,id GIsp,id includes the losses terms due to heat exchange (q) ˙ at the solid-gas interface (i.e. q˙ < 0: gas → wafer), under-expansion at the nozzle exit (pe > p∞ ) and ﬂow divergence (radial velocity at exit). This expression states that the augmentation of the static temperature at exit (Te ) caused by the viscous dissipation,

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Fig. 12. Snapshots of static temperature in the 2D-Axi nozzle with Si-Wafer and hot-ﬂow for insulated upper-wall (left) and cooled upper-wall (right). Time instants: 1, 10, 40 & 100 s.

makes the Isp to drop. Table 3 summarizes the steady-state performance of the 2D-Axi nozzle. The comparison evidences the beneﬁcial eﬀect of cooling on the discharge coeﬃcient (G/Gcr ,id ), providing a massﬂow rate over the isentropic (Gcr ,id ). However, the eﬃciency and Isp decrease as a consequence of the energy released to the wafer. A solution to this drawback may be the use of a glass-wafer, which protects the material from the heat-load. Regarding the hot-ﬂow simulations, a remark should be pointed out. Melting of the substrate restricts the operation to short transients, say under one second lasting. Hence, the steady-state data of Table 3 should be considered as reference values. In fact, the major interest of glass-wafers aims at the early phase of the transient, where the thermal state of the material is colder and better performances are expected. The evolution of Tslip for the glass- and cooled-wafer (see Fig. 13) supports this aspect. Steady-state performance of the 3D-Flat nozzle has been computed for both cold- and hot-ﬂow and insulated Si-wafer. Figure 14 illustrates a sideby-side view of the Mach ﬁeld at the xy- and xz-centreplanes. From a qualitative standpoint, a deceleration (in Mach number) takes place along the divergent portion in both cases, but to a lesser extend with cold-ﬂow. A closer insight reveals that whereas the cold-ﬂow remains supersonic in the entire divergent length, the hot-ﬂow exhibits a shock-free deceleration that drives the

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Fig. 13. Normalized static temperature at cross-sections x/Ldiv = 0.25, 0.5, 0.75 & 1 of the 2D-Axi nozzle (r: radial coordinate; Rw : wall local radius; To : inlet static temperature).

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Table 3. 2D-Axi nozzle performance. Flow/Wafer G/Gcr ,id E(mN ) COLD/Si 0.968 5.50 HOT/Si-insulated 0.933 4.21 HOT/Si-cooled 1.069 4.22 HOT/Glass 1.007 4.06 1 Speciﬁc impulse Isp = E/Ggo , where go = 9.81 ms−2 .

Isp (s)1 71.6 146.5 128.1 130.9

Isp /Isp,id 0.919 0.738 0.645 0.659

Table 4. 3D-Flat nozzle performance. Flow/Wafer COLD/Si HOT/Si-insulated

G/Gcr ,id 0.943 0.815

E(mN ) 5.80 4.49

Isp (s) 61.0 140.3

Isp /Isp,id 0.793 0.707

Table 5. 3D-Flat to 2D-Axi nozzle comparison. Flow/Wafer COLD/Si HOT/Si-insulated

GFlat /GAxi 1.237 1.113

Flat Axi CE /CE 0.828 0.837

Flat Axi Isp /Isp 0.852 0.958

Fig. 14. Mach number contours at xy- and xz-centreplanes for the 3D-Flat nozzle with Si-wafer, corresponding to cold- and hot-ﬂow operation conditions.

gas from supersonic to subsonic. This phenomenon seems to be provoked by the signiﬁcant viscous dissipation in the core ﬂow with low Reynolds number. The spreading of the BLs is shown in Fig. 15 and the diﬀerences are evident. In fact, apart from the thicker BLs of hot-ﬂow (they occupy the entire ﬂow area at the nozzle exit), the side-walls give rise to a higher massﬂow blockage than the ﬂap-walls. The overall blockage can be inferred from the discharge coeﬃcient given in Table 4. Obviously, the growing rate of BLs is related to the gas expansion and the very thick BLs ﬂatten the pressure curve, as it is seen in Fig. 16. In Table 5, it is interesting to note that the massﬂow rate

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Fig. 15. Velocity contours at xy-centreplane and throat section for the 3D-Flat nozzle with Si-wafer, corresponding to cold- and hot-ﬂow operation conditions.

Fig. 16. Normalized static pressure along the centreline of the 2D-Axi (left-side) and 3D-Flat nozzle (right-side), for cold- and hot-ﬂow at operation conditions.

ratio GFlat /GAxi is under the factor 4/π (3D-Flat to 2D-Axi area ratio at the throat) since the 3D-Flat blockage is more severe than in axisymmetric as a consequence of the higher BLs growing rate. Additionally, lower thrust Flat Axi Flat Axi /CE and Isp /Isp , respectively) are attained in eﬃciency and Isp (CE the 3D-Flat nozzle.

Conclusions Steady and transient simulations of two μ-rockets for space propulsion have been carried out for realistic ﬂow conditions following a continuum-based approach with a 2nd-order slip-model and solid-gas coupling. An initial validation with DSMC data has been accomplished, showing appealing agreement. The approach conﬁrms its robustness in 3D simulations. Nevertheless, Knudsen number points out to the marginal validity of the continuum description

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near the nozzle exit. Simulations show that signiﬁcant viscous heating occurs in supersonic hot-ﬂows, that modiﬁes the heat balance at the wall and drives a shockless transition to subsonic inside the nozzle. This leads to a performance drop that overcomes the beneﬁt of the nozzle expansion.

Acknowledgements This research was supported as part of the micropropulsion activities in the Small Satellites Programme, funded by the Spanish Ministry of Defence. The authors thank Dr. F. Caballero-Requena of INTA for his insight and fruitful discussion in the course of the present work.

References 1. Lewis DH, Janson SW, Cohen RB, Antonsson EK (2000) Sens Actuators, A 80:143–154 2. Rossi C, Do Conto T, Est`eve D, Larangˆ ot B (2001) Smart Mater Struct 10:1156– 1162 3. Rossi C, Briand D, Dumonteuil M, Camps Th, Quyˆen-Pham Ph, Rooij NF (2006) Sens Actuators, A 126:241–252 4. Alexeenko AA, Levin DA, Gimelshein SF, Collins RJ, Markelov GN (2002) J Thermophys Heat Transf 16(1):10–16 5. Zhang KL, Chou SK, Ang SS (2006) J Propuls Power 22(1):56–63 6. Alexeenko AA, Levin DA, Fedosov DA, Gimelshein SF (2005) J Propuls Power 21(1):95–101 7. Alexeenko AA, Levin DA, Gimelshein SF, Collins RJ, Reed BD (2002) AIAA J 40(5):897–904 8. Karniadakis G, Beskok A (2002) Micro-ﬂows: fundamentals and simulation. Springer, Berlin, Heidelberg, New York 9. Wang WL, Boyd ID (2002) AIAA paper 2002–651. In: 40th aerospace sciences meeting and exhibit, AIAA, Reno 10. Fluent Inc (2006) Fluent 6.3 UDF manual, Lebanon 11. Mor´i˜ nigo JA, Hermida-Quesada J, Caballero-Requena F (2007) J Therm Sci 16(3):223–230

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Numerical Study on the Flow Physics of a T-Shaped Micro Mixer J. Hussong, R. Lindken, M. Pourquie, J. Westerweel Laboratory for Aero and Hydrodynamics, Delft University of Technology, Leeghwaterstraat 21, 2628 CA Delft, The Netherlands. Tel. +31-15-2785797. [email protected] Summary. A T-shaped micro mixer has two characteristic ﬂow regimes that dependent on the Reynolds number and the geometry in which the mixing of the ﬂuids entering in the two channels is determined by diﬀusion or by convective transport. In one regime the ﬂow in the T-shaped micro mixer is plane-symmetric with respect to one symmetry plane of the T mixer and mixing of ﬂuids from the two inlet channels is determined by diﬀusion. This regime is referred to as the diﬀusion regime in the remainder of this paper. In the other regime the ﬂow is symmetric with respect to the mixing channel’s centerline, and the mixing of the ﬂuids from the two inlet channels is primarily determined by convection. The aim of this work is to study numerically the ﬂow topology in the transition from the ﬂow regime of diﬀusive to convective mixing. Therefore a systematic study was performed to evaluate the inﬂuence of the discretization scheme, the spatial resolution and the choice of channel inlet lengths on the ﬂow topology. The systematic investigation showed that an improper choice of spatial resolution as well as an insuﬃcient channel inlet length can lead to a complete elimination of the Reynolds number dependent onset of convective mixing in the ﬂow. The T-shaped micro mixer ﬂow is represented by vortex core regions which are deﬁned by a λ2 -criterion. They show that the secondary vortical structures in the ﬂow consist of two Dean vortex pairs, both in the ﬂow regime of diﬀusive and convective mixing. The convective regime distinguishes itself from the diﬀusive one by an unequal swirling strength of the Dean vortices which leads to the characteristic co-rotating vortices in the mixing channel. We show that stationary ﬂow states exist in a narrow Reynolds number range that link the ﬂow regime of diﬀusive and convective mixing. From our investigation we conclude that the transition from a diﬀusion to a convection dominated T-mixer ﬂow is continuous.

1 Introduction The T-shaped micro mixer is widely used because it has a very simple geometry and it is a continuous mixer that has many advantages in comparison to a batch-operated mixer [9]. Two ﬂuids enter the micro mixer through the side channels. They meet in the junction region and enter the mixing channel under a 90◦ ﬂow direction change. The basic geometry of a T-shaped micro M. Ellero et al. (eds.), IUTAM Symposium on Advances in Micro- and Nanoﬂuidics, IUTAM Bookseries 15, c Springer Science + Business Media B.V. 2009

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mixer as we used for our study is shown in Fig. 1. Static micro mixers work within a laminar ﬂow regime due to their small dimensions and mixing can be diﬀusion or convection-dominated. Convective mixing starts in the T-shaped micro mixer at a geometry-dependent Reynolds number where the stationary ﬂow pattern changes in the mixing channel from parallel co-ﬂow to an engulfment of the entering ﬂow streams. The mixing eﬃciency of the T-shaped micro mixer has been studied intensively the last years [5, 7–11]. Those authors studied the micro mixer’s engulfment ﬂow regime. The physics of the reﬂectional-symmetric secondary ﬂow pattern has been described by Kockmann [6]. He gave an overview of ﬂow regimes for Reynolds numbers ranging from 10 < Re < 500 [13]. Despite a signiﬁcant number of simulations of the T-shaped micro mixer ﬂow only Kockmann [3] and Wong [4] investigated the transformation of the reﬂectional-symmetric ﬂow state to an engulfed ﬂow pattern. An evaluation of their numerical results is diﬃcult since the models they used are diﬀerent in a number of crucial details. Even though the same ﬂow was simulated signiﬁcant diﬀerences in the results appeared. The computations of Kockmann were performed with the commercial simulation tool CFD-ACE+ from CFDRC. Unfortunately no further speciﬁcations of the discretization scheme are given. Wong used the commercial software package FLUENT with a second order upwind diﬀerencing scheme. Furthermore both authors have chosen diﬀerent channel inlet lengths and mixing channel cross sections. The goal of the present work is therefore to investigate which model parameters will lead to numerical results that are independent of the discretization scheme, the grid spacing and the inlet channel length. The ﬁrst part of the presented results focuses on the inﬂuence of the numerical modeling on the ﬂow solution. In Sect. 3.1 simulation results performed with a central diﬀerencing scheme (CDS) and an upwind diﬀerencing scheme (UDS) will be compared. This is followed by a study on the spatial resolution in Sect. 3.2 and on the inﬂuence of the chosen channel inﬂow length on the simulation results in Sect. 3.3. The second part of the results deal with the ﬂow topology of the T-mixer ﬂow. A Reynolds number range from Re = 120 to 180 was investigated. For the given geometry the transition from diﬀusion-dominated mixing to convection-dominated mixing takes place within this Reynolds number range. After the presentation of the diﬀusive and convective ﬂow regimes in Sects. 4.1 and 4.2, the intermediate ﬂow regime at the boundary between diﬀusive and convective mixing will be analyzed more closely in the following two Sects. 4.3 and 4.3. Conclusions are drawn in Sect. 5.

2 Method 2.1 The Model For the present study we chose a T-mixer geometry that is equal to the geometry in which μPIV measurements have been done in a prior investigation [12].

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Fig. 1. Schematic drawing of the T-shaped micro mixer geometry. The reference length d equals d = 200 μm. The point of origin is located in the center of the junction region. A-A is the cutting plane shown in Fig. 6.

The geometry is shown in Fig. 1. An aspect ratio of the inlet width to the mixing channel width of two was chosen. The channel height is d = 200 μm. The geometry was created and meshed with the GAMBIT code. All models were meshed with hexagonal cells of the same equidistant spacing. The model was imported into FLUENT and boundary conditions were determined. A noslip boundary condition at the solid wall was applied. The analytical velocity proﬁle of a fully developed Poiseuille ﬂow in a rectangular duct [2], was set at the inlet surface of the entrance channels. The series solution includes the ﬁrst forty terms to ensure suﬃcient accuracy of the velocity proﬁle symmetry. At the outlet of the mixing channel the pressure was set to the ambient pressure. For each calculation a suﬃcient number of iterations has been performed to obtain a residual of less than 10−13 . 2.2 Numerical Schemes A direct numerical simulation analysis with the commercial code FLUENT 6.3 was performed to compute the ﬂow and pressure ﬁeld in a T-mixer. FLUENT uses a control-volume-based technique. Test simulations of the T-shaped micro mixer ﬂow have been performed with a steady and unsteady discretization solver. In both cases the ﬂow solution converged to the same stationary ﬂow solution. Therefore the steady-state continuity and momentum equations were solved in all following simulations. For a steady-state case the transport of a scalar quantity can take place by convection or diﬀusion, if no source terms are present in the ﬂow. FLUENT stores discrete scalar values at the centers of the grid cells. The discretized transport equation requires face values of the grid cells for the convection terms. Therefore an interpolation of the cell values from the cell center to its surfaces is necessary. This can be done by an upwind or central diﬀerencing scheme. When using an upwind scheme the cell face values are derived from quantities in the cell upstream relative to the direction of the normal velocity. A central diﬀerencing scheme approximates the value at the control volume face through a linear interpolation between the two nearest nodes. We used second order schemes. To prevent that the FLUENT code

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switches to a lower order discretization scheme the limiter was switched oﬀ for all computations. If the limiter has not been deactivated it will automatically change the discretization scheme when the solution does not converge monotonically. The diﬀusion terms in FLUENT are central-diﬀerenced and are always second-order accurate. The linear system of discretized and linearized transport equations for all cells is solved by FLUENT using a point implicit (Gauss-Seidel) linear equation solver in conjunction with an algebraic multigrid (AMG) method.

3 Results 3.1 The Diﬀerencing Scheme The same T-mixer ﬂow was simulated with an upwind diﬀerencing scheme (UDS) and a linear central diﬀerencing scheme (CDS). The diﬀusive error of both schemes is compared. In both simulations a Reynolds number of Re = 150 was chosen. At this Reynolds number the ﬂow is in the ﬂow regime of convective mixing, which can be seen by the velocity proﬁle of the w-velocity component along the y-axis. The location of the coordinate system and the above mentioned velocity proﬁle w(y) = w(x = 0, − d2 < y < d2 , z = 0) are illustrated on the left side of Fig. 2. On the right side of Fig. 2 the proﬁles w(y) resulting from both simulations can be seen. Even though Fig. 2 shows that the spatial resolution is poor, we can see that the deviation of velocity between both integration schemes is comparatively small. The CDS diﬀerencing scheme gives slightly smaller velocity contributions. Rotational symmetry of both proﬁles is maintained and the rotation point for both interpolation schemes stays exactly in the origin of the reference system. The deviations scale with the velocity magnitude and reach a maximum deviation of 7.35% at the maximum velocity w, see Fig. 2. Obviously the higher numerical diffusion of the upwind diﬀerencing scheme leads to an over-estimation of the velocity w(y). Nevertheless Fig. 2 shows that the ﬂow characteristics are independent of the interpolation scheme. In this work we therefore accepted a slightly higher numerical diﬀusion and decided to use an upwind diﬀerencing scheme. 3.2 The Spatial Resolution A suﬃcient spatial resolution is achieved if the numerical result is independent of the chosen grid. To estimate the discretization error the ﬂow has been computed with four diﬀerent grid densities. The discretization error was estimated on the ﬁnest grid. The reﬁnement was done in a substantial way which means that the number of nodes in all coordinate directions was doubled with each step. The left graph in Fig. 3 illustrates the velocity proﬁles w(y) at x = 0, z = 0 for diﬀerent mesh sizes. Qualitative diﬀerences can be recognized

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Fig. 2. Left: Schematic drawing of the T-shaped micro mixer’s junction region and the location of the point of origin; Right: Velocity distribution in z-direction along the y-axis w(y) = w(x = 0, − d2 < y < d2 , z = 0) for Re = 150 for an upwind and a central diﬀerencing scheme.

Fig. 3. Left: w(y) = w(x = 0, − d2 < y < d2 , z = 0) for diﬀerent spatial resolutions; Right: Double logarithmic plot of the discretization error n as a function of the grid spacing.

between the velocity proﬁles given by N = 21 and N = 41 nodes on the one hand and those of N = 6 and N = 11 on the other hand. With 6 and 11 nodes the change of the direction of the velocity close to the top and the bottom wall can not be resolved. The ﬂow is erroneous with ﬁve or less grid cells. A symmetry breaking of the ﬂow is completely suppressed for 6 nodes over the channel height as shown by the straight line in the left graph of Fig. 3. For N = 41 nodes which are equidistantly distributed over the channel height, the velocity proﬁle is smooth with a distance of 10 μm to the wall. Outside this region the resolution is poor and the proﬁle shows sharp bends. The discretization error n was computed (1) with the mean of the velocity w(y).

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N n 1 1 n = |w(yi )| − |w(yi )| . N i=1 n i=1

(1)

In (1) n gives the number of grid cells over the channel height d = 200 μm and N is the maximum number of cells, which is N = 40 for the present results. For the maximum number of N = 40 cells the cell edge length is Δx = 5 μm. The computed error is plotted in a double-logarithmic plot over the cell size Δx on the right side of Fig. 3. The error performed with cells of an edge length of Δx = 10 μm is almost 7 percent higher than with a doubled cell density. To check if the applied scheme converges to second order accuracy the slope given for the ﬁrst and second reﬁnement step can be compared with the slope of a second order function. Therefore a second order function y = Cx2 with the constant C = 10−4 was plotted into the double logarithmic plot shown on the right side Fig. 3. We can see that in the second reﬁnement step a slightly steeper slope than two was computed. This is due to the fact that the applied data set is based on very few grid cell values of the model. A coarse grid of 20 nodes per reference length has been chosen in order to save CPU time. This is suﬃcient for the investigations presented here since the main characteristics of the ﬂow can be resolved. 3.3 The Channel-Inlet Length In prior numerical studies T-shaped micro mixers had been simulated with inlet channel lengths from three up to four times the channel height [3–6, 10]. In this chapter we will investigate the inﬂuence of the inﬂow length on the simulation results. The purpose is to determine the minimum required inlet length that ensures a solution independent of the inlet length. The T-mixer ﬂow was computed for a constant Reynolds number of Re = 150 for twelve diﬀerent inlet lengths of zero up to three times the channel height. A fullydeveloped velocity proﬁle was prescribed at the channel inlet surface for each computation. The geometry variation is illustrated on the left side in Fig. 4. The simulation results are presented in the graph shown on the right side of Fig. 4. The absolute value of the z-velocity along the y-axis w(y) in the mixing channel is plotted against increasing inlet channel lengths. A zero velocity contribution of |w(y)| is equivalent with a ﬂow that is reﬂectional-symmetric to the y-x plane (Fig. 1). Hence the ﬂow is in the diﬀusive mixing regime. A nonzero contribution of |w(y)| indicates a ﬂow across the y-x plane, as it is the case for the ﬂow regime of convective mixing. The plot shows that the ﬂow swaps for the same Reynolds number for diﬀerent channel inlet length between the ﬂow regimes of diﬀusive and convective mixing. The change of the ﬂow topology takes place between an inlet length of 0.25 to 0.5 times the channel height. For shorter inﬂow lengths a symmetry breaking is suppressed. A solution that is independent of the inﬂow condition can be expected when the velocity contribution w(y) stays constant for increasing channel inlet lengths. The graph in Fig. 4 indicates that this is given for inﬂow lengths l which are

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Fig. 4. Left: Diﬀerent channel inlet lengths indicated in the geometry; Right: Average absolute velocity |w(y)| = |w(x = 0, − d2 < y < d2 , z = 0)| in the mixing channel is plotted over diﬀerent inlet lengths.

Fig. 5. Static pressure along the inlet channel centerline for diﬀerent channel inlet lengths. The positions z of the pressure points are given in the coordinate system scaled with the channel height d.

at least three times the channel height l ≥ 3d. The reason for the inﬂuence of the inﬂow conditioning on the ﬂow results can be better understood by plotting the static pressure along the inlet channel centerline pstat (0, 0, z) up to the mixing channel center y-x plane. The graph is shown in Fig. 5. The computed static pressure is plotted as a function of the channel inlet position scaled with the channel height. The data are plotted with reference to the coordinate system: z/d = −1 is the end of the inlet and the beginning of the mixing channel, and z/d = 0 is the center of the mixing channel. The entrance

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of the inlet channel with a length of three times the channel height is located at z/d = −4. In Fig. 5 the static pressure is plotted along the center line of the inﬂow channels for channel lengths of 0.25, 0.5, 1.0 and 3.0 times the channel height. The graphs for inﬂow lengths of three and one times the channel height fall reasonably on top of each other. The static pressure along the channel centerlines deviates for shorter inﬂow lengths. This conﬁrms the results shown in Fig. 4. The graphs in Fig. 5 show that the pressure drop is constant for all simulations in the inlet channel up to a downstream position −1.5z/d which is at a distance of 0.5d before the mixing channel entrance. A constant pressure drop shows that the pressure gradient is in balance with the wall shear stresses. The ﬂow is thus fully developed. Further downstream the static pressure increases with a varying slope up to the x-y middle plane (z/d = 0) that lies in the mixing channel. A non-constant pressure gradient is the result of a non-developed ﬂow with inherent disturbances. The graph shows that these disturbances reach from the mixing channel 0.5d upstream into the inlet channels. If the inlet channel is chosen so short that disturbances can reach upstream up to the inlet channel entrance, a prescribed fully developed proﬁle does not apply since it suppresses the upstream deviations. That is why the ﬂow can be forced back to an reﬂectional-symmetric ﬂow state, if a fully developed velocity proﬁle is prescribed too close to the mixing channel inlets.

4 The Flow Topology 4.1 The Reﬂectional-Symmetric Flow The ﬂow regime of diﬀusive mixing is characterized by its stationary, reﬂectional-symmetric ﬂow pattern as can be seen in Fig. 6 for Re = 120. The left side of Fig. 6 shows a cut through the mixing channel along line A − A (Fig. 1) in a plane at a location x = 200 μm downstream from origin. The left side of Fig. 6 shows the velocity components that lie in the y-z plane. The right side of Fig. 6 shows from the same viewing direction an iso-vorticity plot where the vortices were identiﬁed according to the λ2 method of Jeong and Hussain [1].

Fig. 6. yz-projection of the ﬂow for Re = 120. Left: In plane velocity vector ﬁeld; Right: λ2 iso surface plot indicating vortex core regions.

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Fig. 7. Left: xz-projection of the isosurfaces of λ2 vortex core regions for Re = 120. Right: xy -projection of the isosurfaces of λ2 vortex core regions for Re = 120.

At positions z = −d and z = d two large regions of vorticity are identiﬁed by the λ2 method. These regions are located behind the corners (position x = [d/2] and z = [d; −d]) which can be observed in Fig. 7. Two Dean vortex pairs are visible in the λ2 iso-surfaces further towards the middle of the mixing channel. They can also be clearly seen in the vector plot in Fig. 6 left. The left picture in Fig. 7 shows two completely separated λ2 iso-surface regions with an isosurface level of λ2 = −25 × 106 for the ﬂuid originating from the inlet channels. The right picture in Fig. 7 reveals that they form two U-shaped λ2 iso-surface regions on the left and right mixing channel half. The Dean vortices in the upper channel half (y ≥ 0) are linked to the vortices in the lower channel half (y ≤ 0) by corner vortices. The λ2 plots on the right side of Fig. 6 and Fig. 7 display them as vortex core regions of smaller diameter. The diameter of a vortex core region presented by the λ2 criterion is proportional to the absolute azimuthal velocities of the ﬂuid moving helically around the rotation axis of the vortex [1]. Since the λ2 iso-surfaces in Fig. 7 end all at the same x-position downstream in the mixing channel the azimuthal velocities of the ﬂuid, and thus the swirling strength of all four Dean vortices is the same. 4.2 The Rotational-Symmetric Flow The ﬂow regime of advective mixing is characterized by a stationary ﬂow pattern that is rotationally symmetric with respect to the channel center line (Fig. 8 for Re = 180). A velocity vector plot in a plane normal to the main ﬂow direction at x = 200 μm downstream is shown on the left side of Fig. 8. The right side of Fig. 8 shows a isosurface plot of λ2 = −45 × 106 from the same viewing direction. A comparison of the vortex core regions for Re = 120 (Fig. 6 right) and Re = 180 (Fig. 8 right) shows common features for both ﬂow regimes of the vortical structures. Dean vortices and corner vortices are present in the ﬂow at both Reynolds numbers. Also two U-shaped vortex core regions can be found in both ﬂow regimes as can be seen from the right pictures

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Fig. 8. yz-projection of the ﬂow for Re = 180. Left: In plane velocity vector ﬁeld; Right: λ2 iso surface plot indicating vortex core regions.

Fig. 9. Left: xz-projection of the isosurfaces of λ2 vortex core regions for Re = 180. Right: xy-projection of the isosurfaces of λ2 vortex core regions for Re = 180

of Figs. 8 and 9. However the reﬂection-symmetry of the isosurface structures found in the ﬂow regime at Re = 120 changed to a rotational-symmetry at Re = 180. This is due to the Reynolds number dependent change of the ﬂow ﬁeld. The velocity vector plot in Fig. 8 shows how the ﬂuid streams of both channel inlet arms engulf in the mixing channel. Fluid entering the mixing channel from the left side reaches far into the right channel half. At the same time it is displaced upwards (Fig. 8 left) giving space to the ﬂuid entering the mixing channel from the opposite side which likewise crosses the geometry middle axis (z = 0). If ﬂuid crosses the middle axis of the mixing channel (y = 0) it merges into one Dean vortex on the opposite mixing channel half. Two of four vortices are thus fed with liquid from both channel sides which increases the amount of ﬂuid entering these vortices. The more ﬂuid merges into a vortex, the higher the absolute velocity, which increases the swirling strength of a viscous vortex. The left picture in Fig. 8 shows two co-rotating vortices. In Fig. 9 vertical structures are displayed in λ2 iso-surfaces. The top and side view show that the two co-rotating vortices reach deeper downstream into the mixing channel due to a swirling strength that is higher than it is the case for the reﬂectional-symmetric ﬂow. It explains the characteristic ﬂow pattern of two co-rotating vortex pairs in the mixing channel of a T-shaped micro mixer in the advective ﬂow regime.

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Fig. 10. Absolute velocity component |w(y)| averaged along the y-axis as a function of the Reynolds number.

4.3 The Reynolds Number of Flow Transition This section focuses on the description of the characteristic ﬂow states of a T-shaped micro mixer geometry as it is shown in Fig. 1. A Reynolds number range of 120 < Re < 145 was investigated. The aim is to reveal Reynolds number dependent topological changes in the ﬂow close to the transition region. With the same geometry several independent computations have been performed for an increasing Reynolds number. Figure10 summarizes the result of the parameter study in which the averaged absolute velocity component |w(x ˆ = 0, y, z = 0)| along the y-axis (z = 0, x = 0) of each computation is plotted versus the according Reynolds number. Up to a Reynolds number of 138.6 the ﬂow is reﬂectional-symmetric. Within a small Reynolds number range of 138.6 < Re < 140 small velocity contributions normal to the middle plane (z = 0) appear. The ﬂow looses its reﬂection-symmetry and small non-zero velocity values in z-direction can be recognized. They lead to a distribution of the averaged velocity |w(y)| as plotted in the graph of Fig. 10. For Reynolds numbers Re ≥ 140 a sudden increase in z-velocity on the middle plain causes a rapid increase of |w(y)| in Fig. 10 and the engulfment process begins. The Transition Region Between Diﬀusive and Convective Mixing An analysis of the transition from the reﬂectional-symmetric to the rotationsymmetric ﬂow helps to understand better how the ﬂow regimes of diﬀusive and convective mixing are linked. Figures 11–13 show streamline plots of the stationary T-mixer ﬂow at Reynolds numbers of Re = 120, 140 and 145 from three diﬀerent viewing angles. On the left an isometric view is shown, on the

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Fig. 11. Streamline plot of the T-mixer ﬂow at a Reynolds number of Re = 120. Left: Isometric view; Right top: In the yz-plane; Right bottom: In the xz-plane.

Fig. 12. Streamline plot of the T-mixer ﬂow at a Reynolds number of Re = 140. Left: Isometric view; Right top: In the yz-plane; Right bottom: In the xz-plane.

Fig. 13. Streamline plot of the T-mixer ﬂow at a Reynolds number of Re = 145. Left: Isometric view; Right top: In the yz-plane; Right bottom: In the xz-plane.

top and bottom right plots in the yz- and xz-planes are shown. Since the symmetry breaking of the ﬂow is the central aspect in this section only streamlines starting from the two channel inlets in the plain that lies at y = 0 have been plotted. For a reﬂectional-symmetric ﬂow as shown in Fig. 11 the streamlines

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remain in the xz-plane (y = 0). Only streamlines that enter the corner vortex are all collapsing onto the corner vortex center line where they leave the xz-plane (Fig. 11). Axi-symmetry is conserved in the whole ﬂow. Figure 12 shows the T-mixer ﬂow for Re = 140. Streamlines entering the mixing channel from the left side are deﬂected upwards in the corner vortex region and close to the stagnation point, while those streamlines entering from the opposite inlet channel are deﬂected downwards. Thus the ﬂow has transformed from a reﬂectional-symmetric to a rotationally symmetric ﬂow. But the ﬂow is still far from the characteristic ﬂuid engulfment process that creates a convectionbased mixing process. A comparison of the streamline plots in the yz-plane of Figs. 12 and 13 show that with an increase of velocity the ﬂuid interface that lies in the xy-plane is rotated around the x-axis. At Re = 145 (Fig. 13) streamlines are not deﬂected near the interface, but they leave the middle xz-plane directly after entering the mixing channel. Streamlines from the left and right inlets pass by each other under a certain deﬂection angle before engulﬁng and forming the characteristic ﬂow topology of two dominant co-rotating vortex pairs further downstream. The early deﬂection of the ﬂuid allows a smoother direction change of the centerline streamlines.

5 Conclusions Within this work a systematic study indicates the model requirements for a CFD analysis in FLUENT of the T-shaped micro mixer. The focus was to determine the sensitivity of the vector ﬁeld with respect to symmetry breaking for the discretization scheme, the spatial resolution and the channel entrance length. A systematic reﬁnement of the grid revealed that the ﬂow is qualitatively erroneous with ﬁve grid cells or less. Across the channel height a symmetry breaking of the ﬂow is suppressed for such a coarse mesh. Previous authors claimed to receive a grid-independent solution for models with only 173,000 to 200,000 cells [3, 4]. We showed that more than n = 2.3 × 106 cells for a geometry as shown in Fig. 1 is necessary to reach a fully resolved velocity ﬁeld if an equidistant mesh is used. The critical Reynolds number of ﬂow transition is expected to be slightly lower for an upwind diﬀerencing scheme compared to a central diﬀerencing integration scheme due to numerical diﬀusion. A minimum channel inlet length is required even if the velocity proﬁle of a fully developed channel ﬂow is given at the inlet surfaces. This is due to the presence of upstream disturbances with respect to the T-mixer mid-plane. We showed that the minimum required length is one times the channel height for a quadratic inlet cross section. A complete inlet-independent solution can be only expected, if the inlet length is chosen to be longer than 3d. For arbitrary channel geometries where no analytical solution of the fully developed velocity proﬁle is available this length has to be added to the channel entrance length to receive a solution independent of the inlet condition. The ﬂow ﬁeld solution of the ﬂow regime of diﬀusive and convective mixing was presented

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in a three dimensional way by identifying connected vortex core regions with a Galilean invariant method [1]. Plots show that for the symmetric and asymmetric ﬂow state the corner vortices in the separation region are connected with the large swirl structures located downstream in the mixing channel. In both ﬂow regimes two Dean vortex pairs could be identiﬁed. The vortex core representation indicated that the typical stationary co-rotating vortex pair in the mixing channel region is due to a shift in swirling strength. Two vortices of same rotation sense gain higher swirling strength. They penetrate deeper into the mixing channel than the vortex pair of opposite rotation sense. The Reynolds number region in which the symmetry breaking of the ﬂow ﬁeld conﬁguration starts was found to be between a Reynolds number of 138 and 140. The parameter study showed that the reﬂectional-symmetric ﬂow is linked to the rotational-symmetric ﬂow ﬁeld through a narrow Reynolds number region of stationary ﬂow states. The existence of these intermediate stationary ﬂow solutions shows that the transition of the ﬂow from an axis-symmetric to a rotational-symmetric ﬂow state is continuous. The investigations conﬁrmed the result given by Kockmann [3] that the symmetry breaking in the ﬂow must be a ﬂuid dynamical phenomena since it is derived for diﬀerent interpolation schemes and spatial resolutions and thus independent of the numerical conditions.

References 1. Jeong J, Hussain F (1995) On the identiﬁcation of a vortex. J Fluid Mech 285:69– 94 2. Spurk J (1996) Stroemungslehre, 1st edn. Springer, Berlin/Heidelberg 3. Kockmann N, Engler M et al (2003) Liquid mixing in static micro mixers with various cross sections. Int conf micro and minichannels, ASME, pp. 911–918, Rochester, New York 4. Wong S, Ward M, Wharton C (2004) Micro T-mixer as a rapid mixing micromixer. Sens Actuators, B, Chem 100:359–379 5. Kockmann N, Engler M, Woias P (2004) Theoretische und Experimentelle Untersuchungen der Mischvorgaenge in T-foermigen Mikroreaktoren, Teil 3: Konvektisches Mischen und Chemische Reaktionen. Chemie Ingenieur Technik 76(12):1777–1783 6. Kockmann N, Engler M et al. (2005) Fluid dynamics and transfer processes in bended microchannels. Heat Transf Eng 26(3):71–78 7. Bothe D, Stemich C, Warnecke H (2005) Fluid mixing in a T-shaped micromixer. Chem Eng Sci 61:2950–2958 8. Kockmann N, Kiefer T et al. (2006) Convective mixing and chemical reactions in microchannels with high ﬂow rates. Sens Actuators, B, Chem 117:495–508 9. Hoﬀmann M, Schlueter M, Raebinger N (2006) Experimental investigation of liquid-liquid mixing in T-shaped micro-mixers using μ-LIF and μ-PIV. Chem Eng Sci 61:2968–2976 10. Bothe D, Stemich C, Warnecke H (2006) Mixing in a T-shaped microreactor: mechanisms and scales. Eur ﬂ eng summer meeting, ASME, FEDSM2006-98140, Miami, Florida

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11. Manhart M, Peukert J, Schwarzer H (2006) Numerical and experimental investigation of the turbulent flow and mixing in a static T-mixer. Eur ﬂ eng summer meeting, ASME, FEDSM2006-98246, Miami, Florida 12. Lindken R, Westerweel J, Wieneke B (2006) Stereoscopic micro particle image velocimetry. Exp Fluids 41:161–171 13. Kockmann N (2008) Transport phenomena in micro process engineering, 2nd edn. Springer, New York

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Splitting for Highly Dissipative Smoothed Particle Dynamics S. Litvinov, X.Y. Hu, N.A. Adams Lehrstuhl f¨ ur Aerodynamik, Technische Universit¨ at M¨ unchen, 85748 Garching, Germany. [email protected]

Summary. The method of smoothed dissipative particle dynamics (SDPD) is a novel coarse grained method for simulation of complex ﬂuids. It has some advantages over more traditional particles based methods (Espanol and Warren, Europhys. Lett. 30(4):191–196, 1995). But one of the problems common for particle based simulations of microﬂuid system takes place also for SDPD: it fails to realize Schmidt number of O(103 ) typical of liquids. In present paper we apply the implicit numerical scheme that allows signiﬁcantly increase time step in SDPD and perform simulation for larger Schmidt number. Simulations using this methods show close agreement with serial solutions for Couette and Poiseuille ﬂows. The results of benchmarks based on temperature control are presented. The dependence of self-diﬀusion coeﬃcient D on kinematic viscosity is examined and found to be in agreement with empirical observations (Li and Chang, J. Chem. Phys. 23(3):518–520, 1955).

1 Introduction A few years ago a new generalization of Smoothed Particle Hydrodynamics methods (SPH) [3] was introduced [4]. This method is called Smoothed Dissipative Particle Hydrodynamics (SDPD) by the authors and claimed to be “improved version” of Dissipative Particle Dynamics (DPD). SDPD has the following features: •

the method is based on a second-order discretization of the Navier-Stokes equations • transport coeﬃcients can be used as input parameters • hydrodynamics behavior is obtained at length scales of the same order of particle dimension and no coarse-grained assumption is needed But beside mentioned advantages we found that SDPD shares some problems known for DPD. In present paper we address one of such issues known as Schmidt number problem.

M. Ellero et al. (eds.), IUTAM Symposium on Advances in Micro- and Nanoﬂuidics, IUTAM Bookseries 15, c Springer Science + Business Media B.V. 2009

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Schmidt number is deﬁned as the ratio of momentum diﬀusivity (viscosity) and mass diﬀusivity. ν (1) Sc = D where ν is a kinematic viscosity and D is a diﬀusion coeﬃcient. And it was augured [5] that it is essential to guarantee acceptable liquid behavior to realize Sc of O(103 ) in the simulations. However in varies hydrodynamic simulation done with DPD Sc number was identiﬁed as a typical for gases O(1). We would like to mentioned that there is another opinion in the literature that the self-diﬀusion coeﬃcient of the particles cannot be equated with those of individual molecules. For example Jiang and coauthors [6] found that for low Schmidt number the hydrodynamic interactions are developed and reveals the correct polymer dynamics. However as it was shown in the [7] and in [8] that the agreement with experiment is better if Sc approaches liquid values. As far as in SDPD transport coeﬃcient is a parameter of the model the higher Schmidt number can be achieved by considering smaller Reynolds number. And the problem of Schmidt number is mostly a technical one: the properties of the standard integration schemes make it computational very costly to run simulation for such a high viscosity. The aim of this paper is to show that the implicit schemes known from the literature [9, 10] allow to increase signiﬁcantly time step in comparison with velocity-Verlet algorithms without any further assumption and modiﬁcation of the physical model. The structure of the report is the following one: in Sect. 2 we brieﬂy describe the SDPD method, in Sect. 3 we describe the time step limitation typical for velocity-Verlet and Predictor-Corrector schemes, in Sect. 4 the proposed integration scheme is described, Sects. 5 and 6 dedicated to application of the scheme to macroscopic and microscopic ﬂow respectively.

2 Model 2.1 Mesoscopic Modeling of the Liquids Let assume an isothermal Newtonian solvent described by the following NavierStokes equations written in a Lagrangian framework dρ = −ρ∇ · v, dt 1 η dv = − ∇p + ∇2 v dt ρ ρ

(2)

where ρ is material density, v is velocity, p is pressure and η is dynamic viscosity. The SPH discretization of the Navier-Stokes equations are

Splitting for Highly Dissipative Smoothed Particle Dynamics

ρi = mi

209

Wij ,

j

dvi 1 pi pj ∂Wij =− + eij dt mi j σi2 σj2 ∂rij +

(3)

η 1 1 vij ∂Wij + . mi j σi2 σj2 rij ∂rij

Here, mi is the mass of a particle, Wij is a kernel function, σi is the inverse of particle volume, eij and rij are the normalized vector and distance from particle i to particle j, respectively. To close the equation, the equation of state is given as γ ρ +b (4) p = p0 ρ0 where p0 , ρ0 , b and γ are parameters which may be chosen based on a scale analysis so that the density variation is less than a given value. Since a stiﬀ equation of state, usually γ = 7, is used, any penetration between particles are not allowed. In the SDPD formulation [4, 11, 12], (3) represents the deterministic part of the particle dynamics. Using GENERIC formalism [13, 14] the thermal ﬂuctuations can be taken into account by postulating the mass and momentum ﬂuctuations with dm ˜i = 0, Bij dW ij eij dP˜i =

(5)

j

where dW ij is the traceless symmetric part of independent increment of a Wiener process and Bij is deﬁned as 1/2 1 ∂W 1 1 Bij = −4kB T η + . σi2 σj2 rij ∂rij

(6)

Some important properties of this particular set of equations are: • • •

The total mass and momentum are exactly conserved; The linear momentum is also locally conserved due to the anti-symmetric property of the discretization; Because the formalism has been build with GENERIC framework, it is possible to show that the system conserves the total energy, and the total entropy is a monotonic increasing function of time [15].

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3 Time Step Limitations For stability of the velocity-Verlet numerical scheme the time step must satisfy the following conditions. CFL condition Δt ≤ 0.25

h c

(7)

where c is the speed of sound, h is the minimum distance between particles in the initial conﬁguration. And the viscous diﬀusion, Δt ≤ 0.125

h2 . ν

(8)

Let use the following notation: δtcNy is the CFL condition time limitation and δtvNy is the viscous diﬀusion limitation. Obviously that for higher resolution the and for higher viscosity of the liquids: δtvNy < δtcNy .

(9)

As was noted by Morris and coauthors [16] the condition 9 holds for typical microﬂuid simulations. It is therefor desire to integrate the equations implicitly so that the δtVNy is not limiting time step. This is usually can be dealt with an implicit numerical method. But straightforward approach has been tried by Pagonabarraga and coauthors [17] but found to be expensive to run. Another implicit method was attempted by Shardlow [9], based on replacing relative velocity at time step n with a semi-implicit velocity vin − vin+1 . But the poor behavior was observed. Monaghan [10] described an implicit integration scheme for handling interaction between dust and gas particles. The key feature of the method is to sweeping over all pair of interacting particles a number of time. In [9] the similar method was studied and applied to DPD. In the work of Nikunen and coauthors [18] the quality and performance of the Shardlow scheme was tested and found to be above the others traditional and novel schemes. Below we present method which is a combination of the Shardlow and Monaghan scheme applied to SDPD. We study the performance of the method for the conditions typical for microﬂuid simulations. And especially we would like to address the mentioned issuer of the Schmidt number [8].

4 Implicit Integration Scheme In the description of the integration scheme the following notation will be used: √ 1 D R (FC dt ), (10) dvi = i dt + Fi dt + Fi mi dri = vi dt

(11)

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{C,D,R} {C,D,R} where Fi = j Fij is a total conservation, dissipative and random force acting on particle i expressed as a sum of contribution of interaction with all particles. From mathematical point of view the system (10) and (11) is a system of stochastic diﬀerential equations. It is√important to note that in the ﬁrst equation the random force has the factor dt instead of dt. It is justiﬁed by a Wiener process used in GENERIC formalism. The velocity-Verlet scheme is characterized be explicit calculation of all forces on right hand side. It uses provisional values of the velocities for the force calculations, which are corrected at the end of each time step (PredictorCorrector approach). The method gives satisfactory results in many application, but the time step must be small for the method to be stable. In this paper an alternative time mashing scheme is described. To solve the system of (10) and (11) we used the following integration scheme based on the methods purposed by Monaghan [10] and Shardlow [9]. The key idea is to split the integration process in such a way that the conservative forces are calculated separately from dissipative and random terms. For conservative term the technique traditional for SPH can be used. But for ﬂuctuation-dissipation part we consider one pair of particles at a time. The structure of the pair equations is very simple and it is possible to choose implicit method that conserve invariance of the momentum. The scheme of the method is given below: • For all pairs of particles (repeated Ns times) (I) Explicit part of the sub-step vi ←− vi +

1 1 D Δt 1 1 R Δt F + F , 2 mi ij Ns 2 mi ij Ns

(12)

vj ←− vj −

1 1 D Δt 1 1 R Δt F − F . 2 mj ij Ns 2 mj ij Ns

(13)

D is a linear function of the velocity on (II) Implicit part of the sub-step. F ij the next time step and overall system of equations can easily be solved: vi = vi +

Δt 1 1 R Δt 1 1 FD + F , 2 mi ij Ns 2 mi ij Ns

(14)

vj = vj −

Δt 1 1 R Δt 1 1 FD − F . 2 mj ij Ns 2 mj ij Ns

(15)

• vi ←− vi + •

1 1 C F Δt. 2m i

ri ←− ri + vi Δt.

(16) (17)

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Calculate FC i (rij ).

• vi ←− vi +

1 1 C F Δt. 2m i

(18) (19)

5 Validation of the Scheme for Macroscopic Flow Simulations of the Couette and Poiseuille ﬂow is provided to validate the scheme described above. We measured the overall accuracy by L1 errors N L1 =

th − U SDPD | i=1 |U N th i=1 |U |

(20)

where U and U SDPD are the simulated and theoretical velocity respectively; N is the number of particles. 5.1 Poiseuille Flow The ﬁrst test is Poiseuille ﬂow is the laminar ﬂow in the space between two parallel plates y = 0 and y = L. The ﬂow initial is in rest and is driven by body force F parallel to x-axis. Following Morris [16] we chose the following parameters ν = 10−6 kg/m3 , F = 10−4 ms− 2, L = 10−3 m, ρ = 103 kg/m3 . Normalisation with maximum velocity Vmax =

F L2 8ν

(21)

gives a Reynolds number of 1.25 × 10−2 . The speed of sound was chosen in order to keep the Mach number less then 0.1: c = 10Vmax .

(22)

We performed simulations for the following number of particles spanning the channel: (23) Ny = 15; 30; 60; 120. We ran simulations using scheme described in the Sect. 4. And the time step which is larger then viscosity diﬀusion time in 5–200 times was used. In the Fig. 1 the comprising between velocity proﬁles obtained using serial solution and SDPD solution. For SDPD solution the proﬁle at tm = 0.63 s is shown which is close to steady state solution t = ∞. In Fig. 2 and Fig. 3 the conversion of the error of the numerical solution with increase of space and time resolution is shown.

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Fig. 1. Comparison of SDPD and theoretical solution for Poiseuille (top) and Couette ﬂow (bottom) (Ny = 30).

5.2 Couette Flow Couette ﬂow is the laminar ﬂow in the space between two parallel plates, one of which is moving. We located the moving plate at y = L and denote the constant velocity as V0 . Following Morris [16] we chose the parameters values typical for microﬂuid systems ν = 10−6 kg/m3 , V0 = 1.25 × 10−5 m/s, L = 10−3 m, ρ = 103 kg/m3 . The corresponding Reynolds number is Re =

V0 L = 1.25 × 10−2 . ν

(24)

The speed of sound was chosen in order to keep the Mach number less then 0.1: c = 1.25 × 10−4 m/s.

(25)

We performed simulations for the following number of particles spanning the channel: (26) Ny = 15; 30; 60; 120. For those simulations the characteristics velocity is the same as for simulations of the Poiseuille ﬂow. To make a conclusion about conversion of the method we chosen a moment of time tm = 0.16 s and compared the diﬀerences between SDPD solution and analytical solution using L1 -norm (20) for given time. In Fig. 2 and Fig. 3 the conversion of the error of the numerical solution with increase of space and time resolution is shown.

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Fig. 2. Comparison of SDPD and theoretical solution for Poiseuille (top) and Couette (bottom).

Fig. 3. Comparison of SDPD and theoretical solution for Poiseuille (top) and Couette (bottom).

5.3 Temperature Control The temperature control is an important test of the methods. We expect time averages of the kinetic temperature to converge to input temperature. We used the following parameters: ρ = 103 kg/m3 , T = 300 K, kinematic viscosity ν = 1 × 10−3 m2 /s. Domain box is 9.96 × 10−07 m and the number of particles used 15 × 15 × 15 = 3375. The conversion of the average kinetics temperature with increase of the number of sweeps Ns is shown in the Fig. 4.

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Fig. 4. The average kinetic temperature for SDPD vs. number of sweeps.

6 Schmidt Number It has been known [2] that the diﬀusion coeﬃcient of the solute and the viscosity of the solvent are related by 1 Dη = kT 6πa

(27)

where D is the diﬀusion coeﬃcient, η is the viscosity of the solvent, k is the Boltzmann constant, T is the absolute temperature, a is the radius of the particle of the solute. We assume no sliding friction. Using deﬁnition of Schmidt number (1), one can derive the following relationship: 6πaη 2 . (28) Sc = ρkT Here we are presenting a derivation of the self-diﬀusion coeﬃcient for SDPD particles based on Langevin equation. For a single particle long-time solutions represent the balance between viscous force and random Brownian force. With a relaxation time given by

1 2η ∞ 2 ∂W 1 = . (29) τ ρ 0 σ ∂rij rij For the 3D case the kernel typical takes the form rij 1 . W (rij , h) = 3 f h h Denoting s = rij /h one can obtain

(30)

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Fig. 5. Diﬀusion coeﬃcient plotted against reversed viscosity (top), Schmidt number plotted against reversed viscosity (bottom).

D=

8kB T kB T τ =− m ηρh

+∞

f (s) ds

(31)

0

for quintic spline function used as a kernel [16] equation (31) takes the form D=

16πkB T . ηρh

(32)

It is important to note that the similar conclusion may be obtained using dimensional analysis. The model is fully described by the following physical parameters: ρ, kB T , h and μ. Thus we have dimensionless parameters: h , (33) μ=μ ρkB T ρh . (34) D=D kB T And parameter μ fully characterize the system (we assume that in our simulations the ﬁnite box size eﬀect is negligible). We perform the series of simulation of liquid in order to compute Sc for wide range of μ. The eﬀect of the domain box and the time step was found to be negligible. We found that D ∝ μ1 is a good approximation to the data obtained in the simulations (Fig. 5). That leads to the conclusion D=C

kB T ηρh

(35)

Splitting for Highly Dissipative Smoothed Particle Dynamics

217

where C is the dimensionless constant depending on the kernel function. The equations (32) and (35) show the connections between diﬀusion coeﬃcient and the size of the discretization element used in SDPD simulations. We ﬁnd that presented implicit method shows a great potential in addressing the issue of relatively large Schmidt number. All simulations (see Fig. 5) required the same computational CPU time but the several order of magnitude larger values of Schmidt number of O(105 ) can be achieved in comparisons with traditional velocity-Verlet algorithm which makes impractical simulations of system with Schmidt number higher then O(1) [8].

References 1. Espanol P, Warren P (1995) Statistical mechanics of dissipative particle dynamics. Europhys Lett 30(4):191–196 2. Li J, Chang J (1955) Self-diﬀusion coeﬃcient and viscosity in liquids. J Chem Phys 23(3):518–520 3. Monaghan JJ (2005) Smoothed particle hydrodynamics. Rep Prog Phys 68:1703– 1759 4. Espanol P, Revenga M (2003) Smoothed dissipative particle dynamics. Phys Rev E 67:026705 5. Groot RD, Warren PB (1997) Dissipative particle dynamics: Bridging the gap between atomistic and mesoscopic simulation. J Chem Phys 107:4423–4435 6. Jiang WH, Huang JH, Wang YM, Laradji M (2007) Hydrodynamic interaction in polymer solutions simulated with dissipative particle dynamics. J Chem Phys 126:044901 7. Symeonidis V, Karniadakis GE, Caswell B (2005) Dissipative particle dynamics simulations of polymer chains: Scaling laws and shearing response compared to dna experiments. Phys Rev Lett 95:076001 8. Symeonidis V, Karniadakis GE (2006) A family of time-staggered schemes for integrating hybrid dpd models for polymers: Algorithms and applications. J Comput Phys 218:82–101 9. Shardlow T (2003) Splitting for dissipative particle dynamics. SIAM J Sci Comput 24(4):1267–1282 10. Monaghan J (1997) Implicit sph drag and dusty gas dynamics. J Comput Phys 138(2):801–820 11. Hu X, Adams N (2006) A multi-phase sph method for macroscopic and mesoscopic ﬂows. J Comput Phys 213:844–861 12. Hu XY, Adams NA (2006) Angular-momentum conservative smoothed particle dynamics for incompressible viscous ﬂows. Phys Fluids 18:101702 ¨ 13. Grmela M, Ottinger H (1997) Dynamics and thermodynamics of complex ﬂuids. I. development of a general formalism. Phys Rev E 56(6):6620–6632 ¨ 14. Ottinger H, Grmela M (1997) Dynamics and thermodynamics of complex ﬂuids. II. Illustrations of a general formalism. Phys Rev E 56(6):6633–6655 15. Serrano M, Espa˜ nol P (2001) Thermodynamically consistent mesoscopic ﬂuid particle model. Phys Rev E 64(4):46115 16. Morris JP, Fox PJ, Zhu Y (1997) Modeling low Reynolds number incompressible ﬂows using sph. J Comput Phys 136:214–226

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17. Pagonabarraga I, Hagen M, Frenkel D (1998) Self-consistent dissipative particle dynamics algorithm. Europhys Lett 42(4):377–382 18. Nikunen P, Karttunen M, Vattulainen I (2003) How would you integrate the equations of motion in dissipative particle dynamics simulations? Comput Phys Commun 153:407–423

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