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 Fröhlich 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 José A. Morí˜ nigo, José 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 Universität 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 Universität 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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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 Universität 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


S. Litvinov Lehrstuhl f¨ ur Aerodynamik, Technische Universität M¨ unchen 85748 Garching, Germany Jos´ e A. Morí˜ 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 Linköping University, Dept. of Physics, Chemistry and Biology, 581 83 Linköping, 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


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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Sung Jae Kim, Jongyoon Han

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

5

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,


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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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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-


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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.


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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


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


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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Beebe DJ, Mensing GA, Walker GM (2002) Annu Rev Biomed Eng 4:261–286 Stone HA, Strook AD, Ajdari A (2004) Annu Rev Fluid Mech 36:381–411 Pollack MG, Fair RB, Shenderov AD (2000) Appl Phys Lett 77:1725–1726 Seiler K, Fan ZH, Fluri K, Harrison DJ (1994) Anal Chem 66:3485–3491 Burgreen D, Nakache FR (1964) J Phys Chem 68:1084–1091 Rice CL, White R (1965) J Phys Chem 69:4017–4023 Dukhin SS, Derjaguin BV (1974) Electrokinetic phenomena. Willey, New York Russel WB, Saville DA, Schowalter WR (1989) Colloidal dispersions. Cambridge University Press, Cambridge 9. Pohl HA (1951) J Appl Phys 22:869–871 10. Pohl HA (1978) Dielectrophoresis the behavior of neutral matter in nonuniform electric fields. Cambridge University Press, Cambridge

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11. Nguyen NT, Wu Z (2005) J Micromechanics Microengineering 15:R1–R16 12. Manz A, Effenhauser CD, Burggraf N, Harrison DJ, Seiler K, Fluri K (1994) J Micromechanics Microengineering 4:257–265 13. Anderson JL, Idol WK (1985) Chem Eng Commun 38:93–106 14. Ajdari A (1995) Phys Rev Lett 75:755–758 15. Stroock AD, Weck M, Chiu DT, Huck WT, Kenis PJ, Ismagilov RF, Whitesides GM (2000) Phys Rev Lett 84:3314–3317 16. Lechnick WJ, Shaeiwitz JA (1985) J Colloid Interface Sci 104:456–470 17. Pawar Y, Solomentsev YE, Anderson JL (1993) J Colloid Interface Sci 155:488– 498 18. Keh HJ, Wei YK (2002) J Colloid Interface Sci 252:354–364 19. Wu JH, Keh HJ (2003) Colloid Surface A 212:27–42 20. Ajdari A, Bocquet L (2006) Phys Rev Lett 96:186102.1-4 21. Bazant MZ, Squires TM (2004) Phys Rev Lett 92:066101.1-4 22. Levitan JA, Devasenathipathy S, Studer V, Ben Y, Thorsen T, Squires TM, Bazant MZ (2005) Colloid Surface A 267:122–132 23. Ramos A, Morgan H, Green NG, Castellanos A (1998) J Phys D App Phys 31:2338–2358 24. Squires TM, Bazant MZ (2004) J Fluid Mech 509:217–252 25. Squires TM, Bazant MZ (2006) J Fluid Mech 560:65–101 26. Stein D, Kruithof M, Dekker C (2004) Phys Rev Lett 93:035901.1-4 27. Karnik R, Castelino K, Fan R, Yang P, Majumdar A (2005) Nano Lett 5:1638– 1642 28. Schoch RB, Lintel HV, Renaud P (2005) Phys Fluids 17:100604.1-5 29. Plecis A, Schoch RB, Renaud P (2005) Nano Lett 5:1147–1155 30. Daiguji H, Oka Y, Shirono K (2005) Nano Lett 5:2274–2280 31. Cheng LJ, Guo LJ (2007) Nano Lett 7:3165–3171 32. Kim SJ, Wang YC, Lee JH, Jang H, Han J (2007) Phys Rev Lett 99:044501.1-4 33. Probstein RF (2004) Physicochemical hydrodynamics: an introduction, 2nd edn. Wiley-Interscience, New York 34. Mao P, Han J (2005) Lab Chip 5:837–844 35. van Delft KM, Eijkel JCT, Mijatovic D, Druzhinina TS, Rathgen H, Tas NR, van den Berg A, Mugele F (2007) Nano Lett 7:345–350 36. Kim SJ, Han J (2008) Anal Chem 80:3507–3511 37. Lee JH, Song Y-A, Han J (2008) Lab Chip 8:596–601 38. Kim SM, Burns MA, Hasselbrink EF (2006) Anal Chem 78:4779–4785 39. Lee JH, Chung S, Kim SJ, Han J (2007) Anal Chem 79:6868–6873 40. Huh D, Mills KL, Zhu X, Burns MA, Thouless MD, Takayama S (2007) Nat Mater 6:424–428 41. Chung S, Lee JH, Moon M-W, Han J, Kamm RD (2008) Adv Mater 42. Dhopeshwarkar R, Crooks RM, Hlushkou D, Tallarek U (2008) Anal Chem 80:1039–1048 43. Khandurina J, Jacobson SC, Waters LC, Foote RS, Ramsey JM (1999) Anal Chem 71:1815–1819 44. Leinweber FC, Tallarek U (2004) Langmuir 20:11637–11648 45. Zeng CC, Wang SN, Lee L (2005) Mater Lett 59:3095–3098 46. Nischang I, Reichl U, Seidel-Morgenstern A, Tallarek U (2007) Langmuir 23:9271–9281 47. Nischang I, Holtzel A, Seidel-Morgenstern A, Tallarek U (2008) Electrophoresis 29:1140–1151

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Zhang Y, Timperman AT (2003) Analyst 128:537–542 Pu Q, Yun J, Temkin H, Liu S (2004) Nano Lett 4:1099–1103 Holtzel A, Tallarek U (2007) J Sep Sci 30:1398–1419 Rubinstein I, Shtilman L (1979) J Chem Soc Faraday Trans II 75:231–246 Rubinstein I, Zaltzman B, Pretz J, Linder C (2002) Russ J Electrochem 38:956– 967 Strathmann H, Krol JJ, Rapp HJ, Eigenberger G (1997) J Membr Sci 125:123– 142 Levich VG (1962) Physicochemical hydrodynamics. Prentice-Hall, New York Krol JJ, Wessling M, Strathmann H (1999) J Membr Sci 162:145–154 Rubinstein I, Zaltzman B (2000) Phys Rev E 62:2238–2251 Rubinstein I, Zaltzman B (2005) Phys Rev E 72:011505.1-19 Pennathur S, Santiago JG (2005) Anal Chem 77:6772–6781 Pundik T, Rubinstein I, Zaltzman B (2005) Phys Rev E 72:061502.1-8. http://linkapsorg/abstract/PRE/v72/e061502 Zaltzman B, Rubinsetain I (2007) J Fluid Mech 579:173–226 Rubinstein I, Zaltzman B (2007) Adv Colloid Interface 134–135:190–200 Ben Y, Demekhin EA, Chang HC (2004) J Colloid Interface Sci 276:483–497 Rubinstein I, Zaltzman B (2001) Math Mod Meth Appl S 11:263–300 Mishchuk NA, Takistov PV (1995) Colloids Surf A 95:119–131 Ben Y, Chang HC (2002) J Fluid Mech 462:229–238 Jin X, Joseph S, Gatimu EN, Bohn PW, Aluro NR (2007) Langmuir 23:13209– 13222 Leinweber FC, Tallarek U (2005) J Phys Chem B 109:21481–21485 Leinweber FC, Pfafferodt M, Seidel-Morgenstern A, Tallarek U (2005) Anal Chem 77:5839–5850 Maletzki F, Rossler HW, Staude E (1992) J Membr Sci 71:105–116 Kim SJ, Wang YC, Lee JH, Jang H, Han J (2006) Proceedings of microTAS 2006 Conference, vol 1, pp 522–524 Wang YC, Stevens AL, Han J (2005) Anal Chem 77:4293–4299 Kim D, Raj A, Zhu L, Masel R, Shannon MA (2008) Lab Chip 8:625–628 Miao J, Xu Z, Zhang X, Wang N, Yang Z, Sheng P (2007) Adv Mater 19:4234– 4237 Liu S, Pu Q, Gao L, Korzeniewski C, Matzke C (2005) Nano Lett 5:1389–1393 Nguyen NT, Chan SW (2006) J Micromechanics Microengineering 16:R1–R12 Tallarek U, Leinweber FC, Nischang I (2005) Electrophoresis 26:391–404 Leinweber FC, Eijkel JCT, Bomer JG, Berg AVD (2006) Anal Chem 78:1425– 1434 Siwy Z, Fulinski A (2002) Phys Rev Lett 89:198103.1-4 Hu G, Li D (2007) Chem Eng Sci 62:3443–2454


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


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Marco Ellero

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-

Fluid Particle Models for the Simulation of Microfluids

21

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


23

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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Marco Ellero

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


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

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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.


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)


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


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


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 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


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)


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


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.


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


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)


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


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].


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


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.


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


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


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


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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


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.


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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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


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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

58


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


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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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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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,


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, Léger L (2000) Phys Rev Lett 85:980–983 5. Schmatko T, Hervet H, Léger 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, Léger L (1993) Phys Rev Lett 70:287–290 Hervet H, Léger L (2003) C R Phys 4:241–249 Léger 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


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.


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Filippos Sofos, Theodoros Karakasidis, Antonios Liakopoulos

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.

Variation of Transport Properties Along Nanochannels

69

Fig. 1. The atomic system under examination.

Nosé-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


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.

72


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.


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

74


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.


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.

76


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.


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,


80

Stelios Varoutis, Dimitris Valougeorgis

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)

82


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)


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


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

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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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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


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


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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.

Electrochemical Control of Conjugated Polymer Surface Energy

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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)


119

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.

120


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


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


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



PEDOT:PSS

Neutral p-doped

50 65



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


121

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

122


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


123

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.

124


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.


125

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, Hocdé S, Chase CJ (2005) Appl Phys Lett 87:013503 15. Isaksson J, Robinson ND, Berggren M (2006) Thin Solid Films 515:2003


Small Scale Cavitation Model Juan Miguel S´ anchez S´ anchez, Rafael Rodrigo Fernández 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.


128

Juan Miguel S´ anchez S´ anchez, Rafael Rodrigo Fern´ andez

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.

Small Scale Cavitation Model

129

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.

130


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.


131

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.

132


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

134


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


135

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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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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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


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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B.E. Zima-Kulisiewicz, W. Kowalczyk, A. Delgado

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


Part IV

Numerical Modeling


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


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.

Lattice Boltzmann Simulation of Pulsed Jet in T-Shaped Micromixer

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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 José A. Morí˜ nigo1 , José 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 =


179

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, Estève D, Larangˆ ot B (2001) Smart Mater Struct 10:1156– 1162 3. Rossi C, Briand D, Dumonteuil M, Camps Th, Quyên-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. Morí˜ nigo JA, Hermida-Quesada J, Caballero-Requena F (2007) J Therm Sci 16(3):223–230


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].

Numerical Study on the Flow Physics of a T-Shaped Micro Mixer

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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

200


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.


201

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

202


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.



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


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.


208

S. Litvinov, X.Y. Hu, N.A. Adams

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].

210


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)


211

{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)

212

•


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.


213

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.

214


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.


215

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)

216


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)


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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