Microparticle Dynamics in Electrostatic and Flow Fields 9811608423, 9789811608421

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Springer Theses Recognizing Outstanding Ph.D. Research

Sheng Chen

Microparticle Dynamics in Electrostatic and Flow Fields

Springer Theses Recognizing Outstanding Ph.D. Research

Aims and Scope The series “Springer Theses” brings together a selection of the very best Ph.D. theses from around the world and across the physical sciences. Nominated and endorsed by two recognized specialists, each published volume has been selected for its scientific excellence and the high impact of its contents for the pertinent field of research. For greater accessibility to non-specialists, the published versions include an extended introduction, as well as a foreword by the student’s supervisor explaining the special relevance of the work for the field. As a whole, the series will provide a valuable resource both for newcomers to the research fields described, and for other scientists seeking detailed background information on special questions. Finally, it provides an accredited documentation of the valuable contributions made by today’s younger generation of scientists.

Theses may be nominated for publication in this series by heads of department at internationally leading universities or institutes and should fulfill all of the following criteria • They must be written in good English. • The topic should fall within the confines of Chemistry, Physics, Earth Sciences, Engineering and related interdisciplinary fields such as Materials, Nanoscience, Chemical Engineering, Complex Systems and Biophysics. • The work reported in the thesis must represent a significant scientific advance. • If the thesis includes previously published material, permission to reproduce this must be gained from the respective copyright holder (a maximum 30% of the thesis should be a verbatim reproduction from the author’s previous publications). • They must have been examined and passed during the 12 months prior to nomination. • Each thesis should include a foreword by the supervisor outlining the significance of its content. • The theses should have a clearly defined structure including an introduction accessible to new PhD students and scientists not expert in the relevant field. Indexed by zbMATH.

Sheng Chen

Microparticle Dynamics in Electrostatic and Flow Fields Doctoral Thesis accepted by Tsinghua University, Beijing, China

Author Dr. Sheng Chen Department of Energy and Power Engineering Tsinghua University Beijing, China

Supervisor Prof. Shuiqing Li Department of Energy and Power Engineering Tsinghua University Beijing, China

School of Energy and Power Engineering Huazhong University of Science and Technology Wuhan, China

ISSN 2190-5053 ISSN 2190-5061 (electronic) Springer Theses ISBN 978-981-16-0842-1 ISBN 978-981-16-0843-8 (eBook) https://doi.org/10.1007/978-981-16-0843-8 Jointly published with Tsinghua University Press The print edition is not for sale in China (Mainland). Customers from China (Mainland) please order the print book from: Tsinghua University Press. © Tsinghua University Press 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publishers, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publishers nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publishers remain neutral with regard to jurisdictional claims in published maps and institutional affiliations. This Springer imprint is published by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Dedicated to my family.

Supervisor’s Foreword

Adhesive particle flow arises in many applications in industry, nature, and life sciences and has driven great research interests in areas of aerosol filtration, dust mitigation, nanoparticle deposition, ceramics manufacturing, fouling of MEMS devices, sediment transport, and production of fuel cells. An in-depth understanding of the relationship between microscopic interparticle interactions and the collective behavior of a large number of particles would be helpful to understand and further design large-scale devices. However, linking the microscopic properties of discrete particles to the macroscopic behaviors of particle flow systems is never a simple task. The difficulty lies in the complicated interacting modes between particles, namely the electrostatic interaction, the hydrodynamic interaction, and the contact interactions, across several orders of magnitude in time and length scales. Within the past few decades, the discrete element method (DEM), in which the motion, collision, and adhesion of individual particles are resolved in time and space, has been developed to model particle collective dynamics from singleparticle level. DEM coupled with computational fluid dynamics (i.e., CFD-DEM) has shown powerful capabilities in investigating particle-laden flows. Moreover, there has recently been rapid progress on understanding the physics related to the intermolecular and surface forces, which enable us to develop more rational adhesive contact models. Scalable and efficient computational frameworks have also been proposed for handling long-range many-body interactions and for collision resolution. It is recognized that merging the expertise across various disciplines of fluid and solid mechanics, condensed matter physics, materials science, and applied mathematics will significantly improve our understanding of particle dynamics in electrostatic and flow fields. The objective of this thesis is to propose new approaches for modeling contacting interactions and electrostatic interactions between microparticles in the framework of discrete element methods and to present an insightful view on the agglomeration, migration, and deposition of microparticles in electrostatic and flow fields. The first chapter discusses various applications of adhesive particle flows. Chapter 2 starts with a simple case of binary collisions of adhesive particles to show how the discrete element method gives the information on the force, the displacement, and the energy vii

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Supervisor’s Foreword

conversion. A novel fast DEM based on the reduced particle Young’s modulus is then proposed to accelerate the computation. In Chap. 3, the fast DEM is coupled with direct numerical simulation to investigate the agglomeration of particles in homogeneous isotropic turbulence. The structure and the size distribution of agglomerates are obtained. The agglomeration and collision-induced breakage rates are formulated based on the classic theory for particle collisions in turbulence. In Chap. 4, the evolution of spherical clouds of charged particles that migrate in a uniform external electrostatic field is then investigated by Oseen dynamics and a continuum approach, and the scaling laws for evolution of cloud radius and particle number density are derived. Finally, in Chaps. 5 and 6, an elaborate investigation of the deposition of charged particles on a flat plane and fibers is presented. The findings, together with previous results for neutral particles, form a more complete picture of filtration and deposition of microparticles. I believe that the results in this book will substantially impact the field relevant to adhesive particle flows. Beyond that, the findings here may also have broader implications for granular fluidization, liquid–solid suspensions, and colloidal gels, where complicated particle–particle interactions exist. Beijing, China January 2021

Prof. Shuiqing Li

Parts of this thesis have been published in the following journal articles • S. Chen, S. Li. Collision-induced breakage of agglomerates in homogenous isotropic turbulence laden with adhesive particles, Journal of Fluid Mechanics, 2020, 902, A28. • S. Chen, S. Li, J. S. Marshall. Exponential scaling in early-stage agglomeration of adhesive particles in turbulence, Physical Review Fluids, 2019, 4(2): 024304. • S. Chen, W. Liu, S. Li. A fast adhesive discrete element method for random packings of fine particles, Chemical Engineering Science, 2019, 193(16), 336– 345. • S. Chen, T. Bertrand, W. Jin, M. D. Shattuck, C. S. O’Hern, Stress anisotropy in shear-jammed packings of frictionless disks, Physical Review E, 2018, 98(4): 042906. • S. Chen, W. Liu, S. Li, Scaling laws for migrating cloud of low-Reynolds-number particles with Coulomb repulsion, Journal of Fluid Mechanics, 2018, 835: 880– 897. • S. Chen, W. Liu, S. Li, Effect of long-range electrostatic repulsion on pore clogging during microfiltration. Physical Review E, 2016, 94(6): 063108. • S. Chen, S. Li, W. Liu, H. A. Makse, Effect of long-range repulsive Coulomb interactions on packing structure of adhesive particles. Soft Matter, 2016, 12:1836– 1846. • S. Chen, S. Li, M. Yang, Sticking/rebound criterion for collisions of small adhesive particles: Effects of impact parameter particle size. Powder Technology, 2015, 274:431–440.

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Acknowledgements

First and foremost, I would like to thank my advisor Prof. Shuiqing Li for acting as an inspirational and motivational figure throughout my time as a Ph.D. student. Professor Li impacted me by his deep knowledge, intuition, and passion for research, his confidence and calmness under all circumstances, and his trust and support for his students. Professor Li can always see through the essence of scientific or engineering problems. I am grateful to him for the time he devoted in my research despite the heavy teaching and administrative workloads. I wish I could incorporate some of his merits and good qualities. I would like to express special thanks to my co-advisor Prof. Corey O’Hern for his support and guidance during my one-year visit to Yale University. I would like to thank Prof. Qiang Yao for his support and valuable advices. His clear logic and encyclopedic knowledge on research and his enthusiasm for education have deeply affected me. I also thank Prof. Qiang Song and Prof. Jiankun Zhuo for the continuous support. I also thank my colleagues at PACE research group, Yiyang Zhang, Runru Zhu, Gengda Li, Guilong Xiong, Mengmeng Yang, Huang Zhang, Hongsheng Chen, Xing Jin, Yichen Zong, Ye Yuan, Wenwei Liu, Qian Huang, Yihua Ren, Xuhui Zhang, Ran Tao, Qi Gao, Xuan Ruan, Zeyun Wu, and others. The happy days in PACE group will always be a wonderful memory in my life. I thank my family for always supporting, understanding, and encouraging me. Lastly, and most importantly, I thank my wife Jingying Xu for being the permanent source of support, love, and inspiration. I acknowledge the National Scientic Fund in China (NSFC) (Grant No. 51725601 and 51390491), the National Key Research and Development Program of China (Grant No. 2016YFB0600602), and the China Scholarship Council (CSC) for supporting my research.

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Contents

1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Adhesive Particle Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Example Systems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Collision and Agglomeration of Particles in Turbulence . . . . . . . . . 1.4 Migration of Microparticles in an Electrostatic Field . . . . . . . . . . . . 1.5 Deposition of Microparticles and Clogging Phenomenon . . . . . . . . 1.6 Discrete Element Methods for Adhesive Particles . . . . . . . . . . . . . . 1.7 A Road Map to Chaps. 2–6 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 2 4 6 9 12 13 14

2 A Fast Discrete Element Method for Adhesive Particles . . . . . . . . . . . . 2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Discrete Element Method for Adhesive Particles . . . . . . . . . . . . . . . 2.3 Critical Sticking Velocity for Two Colliding Particles . . . . . . . . . . . 2.3.1 Temporal Evolution of the Collision Process . . . . . . . . . . . . 2.3.2 Prediction of the Critical Sticking Velocity . . . . . . . . . . . . . 2.3.3 Effect of Particle Size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 A Fast Adhesive DEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4.1 Accelerating Adhesive DEM Using Reduced Stiffness . . . . 2.4.2 Modified Models for Rolling and Sliding Resistances . . . . 2.5 Determination of Parameters in Adhesive DEM . . . . . . . . . . . . . . . . 2.5.1 An Inversion Procedure to Set Parameters in Adhesive DEM . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5.2 Comparison Between Experimental and DEM Results . . . . 2.6 Test on Packing Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.1 Packing Fraction and Coordination Number . . . . . . . . . . . . 2.6.2 Local Structure of Packings . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6.3 Interparticle Overlaps and Normal Forces . . . . . . . . . . . . . . 2.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

17 17 18 20 22 25 29 31 31 34 36 36 39 40 43 45 46 48 49

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Contents

3 Agglomeration of Microparticles in Homogenous Isotropic Turbulence . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.1 Fluid Phase Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.2 Equation of Motion for Solid Particles . . . . . . . . . . . . . . . . . 3.2.3 Multiple-time Step Framework . . . . . . . . . . . . . . . . . . . . . . . . 3.2.4 Simulation Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.5 Identification of Collision, Rebound and Breakage Events . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2.6 Smoluchowski’s Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Collision Rate, Agglomerate Size and Structure . . . . . . . . . . . . . . . . 3.4 Effect of Stokes Number . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.5 Exponential Scaling of Early-Stage Agglomerate Size . . . . . . . . . . 3.6 Agglomeration Kernel and Population Balance Modelling . . . . . . . 3.7 Effect of Adhesion on Agglomeration . . . . . . . . . . . . . . . . . . . . . . . . 3.8 Effect of Adhesion on Breakage of Agglomerates . . . . . . . . . . . . . . 3.9 Formulation of the Breakage Rate . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.10 Agglomerate Size Dependence of the Breakage Rate . . . . . . . . . . . 3.11 Role of Flow Structure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.12 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

57 59 60 62 62 64 65 68 68 76 76 78 79

4 Migration of Cloud of Low-Reynolds-Number Particles with Coulombic and Hydrodynamic Interactions . . . . . . . . . . . . . . . . . . 4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 Formulation of Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3 Effect of Coulomb Repulsion on Cloud Shape . . . . . . . . . . . . . . . . . 4.3.1 Cloud Shape . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.2 Effect of Fluid Inertia . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.3.3 Stability of the Cloud . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.4 Evolution of Particle Cloud Under Strong Repulsion . . . . . . . . . . . . 4.4.1 Scaling Analysis and Continuum Description . . . . . . . . . . . 4.4.2 Prediction of Cloud Size and Migrating Velocity . . . . . . . . 4.4.3 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

81 81 81 84 84 87 88 91 91 93 97 98 99

5 Deposition of Microparticles with Coulomb Repulsion . . . . . . . . . . . . . 5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Models and Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.1 Simulation Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Forces on Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Average-Field Calculation for Coulomb Interactions in 2D Periodic System . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.3 Effects of Coulomb Interaction on Packing Structure . . . . . . . . . . .

51 51 52 52 53 54 55

101 101 102 102 103 103 106

Contents

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5.4 Scaling Analysis of the Interparticle Force . . . . . . . . . . . . . . . . . . . . 5.5 Governing Parameters for the Packing Structure . . . . . . . . . . . . . . . 5.6 Phase Diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109 112 115 117 118

6 Deposition of Charged Micro-Particles on Fibers: Clogging Problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Models and Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.1 Simulation Conditions: Two Fiber System . . . . . . . . . . . . . . 6.2.2 Gas Phase Simulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.2.3 Solid-Phase: Discrete-Element Method (DEM) . . . . . . . . . . 6.2.4 Governing Parameters . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 Clogging/Non-clogging Transition . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Measurement of Particle Capture Efficiency . . . . . . . . . . . . . . . . . . . 6.4.1 Repulsion Effect: The Critical State . . . . . . . . . . . . . . . . . . . 6.4.2 Structure Effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.5 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

119 119 120 120 121 122 123 124 127 128 130 133 134

7 Conclusions and Perspective . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.1 Conclusions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

135 135 137 138

Nomenclature

English Characters (Lowercase) a bˆ dt F dt P dt C e e0 f i k kN kT m g(r) n n˙ c p q r rp rf t u u us v vn

Radius of contact region between two colliding particles (m) Impact parameter Fluid time step (s) Particle convective time step (s) Collision time step (s) Coefficient of restitution Elementary charge (1.6 × 10−19 C) Friction factor for viscous drag Particle ID (i) Wave number; (ii) permeability (m2 ) Elastic coefficient in normal direction (N/m) Elastic coefficient in tangential direction (N/m) Mass (kg) Radial distribution function Particle number density (m−3 ) Collision rate of particles per unit volume (s−1 m−3 ) (i) Pressure (Pa); (ii) dipole strength (Cm) (i) Turbulent kinetic energy (m2 /s2 ); (ii) particle charge (C) Position vector (m) Particle radius (m) Fiber radius (m) Time (s) Flow velocity, (m/s) Turbulent velocity fluctuations (m/s) Slip velocity (m/s) Particle velocity (m/s) Normal component of the colliding velocity (m/s)

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Nomenclature

English Characters (Uppercase) A Ad El Df E E ER F FC Fl I M NC NS NR NB R0 Rij Re Rep Reλ Rg S St Stk Te V CN V C0 Wi W ij Z

Agglomerate size Adhesion parameter Elasticity Parameter Fractal dimension of particle agglomerate Particle Young’s modulus (Pa) Electric field (V/m) Reduced particle Young’s modulus (Pa) Force on particles (N) Critical pull-off force (N) Lubrication force (N) Moment of inertia of particles (kgm2 ) Torque on particles (Nm) Number of collision events Number of sticking events Number of rebound events Number of breakage events Radius of particle cloud (m) (i) Reduced particle radius (m); (ii) radius of collisional sphere (m) Reynolds number Particle-scale Reynolds number Taylor-scale Reynolds number Gyration radius of particle agglomerate (m) Size ratio between a particle and a cloud Stokes number Kolmogorov-scale Stokes number Large eddy turnover time (s) Normal component of critical sticking velocity (m/s) Critical sticking velocity for head-on collisions (m/s) Volume of Voronoi cell Kernel for particle–particle hydrodynamic interaction Coordination number

Greek Symbols α β γ  a δ

Damping coefficient for particle collisions (i) Particle size ratio; (ii) parameter for the agglomerate size distribution Particle surface energy density (J/m2 ) Particle collision kernel (m3 /s) Particle agglomeration kernel (m3 /s) Overlap between contact particles (m)

Nomenclature

δc δh ε η

κ λ μ ν  ρf ρp σ τk τP φ ϕ ϕI ϕR ϕD χ ωr  ψ 

Critical overlap between contact particles (m) Regularized delta function Dissipation rate of turbulent kinetic energy (m2 /s3 ) (i) Kolmogorov length scale (m); (ii) particle capture efficiency Sticking probability for colliding particle (i) Charge parameter; (ii) parameter of agglomerate size distribution Aspect ratio of particle cloud (i) Fluid dynamic viscosity (Pas); (ii) friction coefficient Fluid kinetic viscosity (m2 /s) Particle capture efficiency Fluid density (kg/m3 ) Particle density (kg/m3 ) Poisson’s ratio Kolmogorov time scale (s) Particle response time (s) Particle packing fraction Particle collision efficiency Particle collision efficiency due to particle inertia Particle collision efficiency due to intersection Particle collision efficiency due to diffusion Reduced ratio of particle stiffness Relative radial velocity (m/s) Particle angular velocity (rad/s) Fine-grained probability of breakage Fraction of breakage events for particles in turbulence

Superscript N R S T Coul

Normal direction Rolling motion Sliding motion Twisting motion Coulomb interaction

Subscript crit p 0

Normal direction Particle Initial state

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Chapter 1

Introduction

1.1 Adhesive Particle Flow An adhesive particle flow is a system in which a moving fluid and a large number of discrete adhesive particles coexist. Particle adhesion originates from a variety of different mechanisms, which can be categorized as: dispersive adhesion due to van der Waals (VDW) attraction of molecules in the contact region between the particles; capillary adhesion due to the liquid bridge between wet particles; electrostatic adhesion due to the attraction of charged or polarized particles [1]; chemical adhesion due to swapping or sharing of atoms of the two particles, etc. In this thesis, we focus on the transport of micron particles, for which the dispersive adhesion due to van der Waals attraction plays a dominant role. Micron particles are encountered in an extraordinarily broad area of industrial, biological, and environmental processes, encompassing fields of study such as aerosol dynamics, sediment transport, colloidal dispersions, fluidized beds, and cohesive granular flows. In these systems, the van der Waals adhesion acting on micron particles is substantially stronger than the gravitational force and the elastic force and leads to the sticking of particles upon collisions and the formation of agglomerates. Micron particles are usually charged through the triboelectrification, the field charging, and the diffusion charging [2–5]. Compared with the van der Waals force, the electrostatic forces can exert their influence across a much longer distance. These long-range forces can cause profound changes in the behavior of a particulate system, such as the particle clustering [6], blockage, self-assembly [7] as well as levitation [8, 9], and offer the ability to manipulate the particles at microscales. An in-depth understanding of the relationship between the adhesive/electrostatic forces and the collective behavior of a large number of particles would be helpful to design and optimize industrial processes. This work focuses on the transport of microparticles in the presence of the electrostatic and flow fields.

© Tsinghua University Press 2023 S. Chen, Microparticle Dynamics in Electrostatic and Flow Fields, Springer Theses, https://doi.org/10.1007/978-981-16-0843-8_1

1

2

1 Introduction

1.2 Example Systems Figure 1.1 presents several example systems involving the transport of adhesive particles in the presence of the electrostatic and flow fields. In an aerosol agglomerator (Fig. 1.1a), the turbulent flow field and the electrostatic field are used to coagulate submicron particles to larger ones. In Fig. 1.1b, the electrophoretic deposition (EPD) process is sketched, which has been widely used to deposit various metal oxide particles to form functional films for different applications. The transport of adhesive particles also exists in space. The dust particles on Mars are electrically charged primarily through triboelectric effects that result from the frequent planetary dust storms. These charged dust particles are expected to cause instrument failure and severe health problems for astronauts. Figure 1.1c shows that dust particle sticks on the solar panels of NASA’s Mars exploration rover Opportunity, reducing the amount of electrical power the rover can generate from sunlight. In Fig. 1.1d, a picture of the fibrous filtration process is presented, which has been widely used in devices for aerosol control and water treatment. The deposited microparticles pack closely together, reducing the effective pore size of the filters leading to a high separation efficiency.

Fig. 1.1 Example systems involving the transport of particles in the presence of the electrostatic and flow fields: a bipolar parallel-plate agglomerator and agglomerates of fly ash particles (Reprinted from Jaworek et al. [10], with permission from Elsevier); b sketch of the electrophoretic deposition (EPD) process and the metal oxide film of solid oxide fuel cells fabricated by EPD (Reprinted from Hu et al. [11], with permission from Elsevier); c dust deposited on NASA’s Mars Exploration Rover Opportunity; d a fibrous membrane before and after particle filtration (Reprinted from Gopal et al. [12], with permission from Elsevier)

1.2 Example Systems

3

Another typical system involving the transport of microparticles in electrostatic and flow fields is the electrostatic precipitator (ESP). ESP is one of the most efficient devices used for the removal of fly ash particles from the flue gases produced by coal-fired boilers in power plants [13]. In an ESP, particles are charged by the corona surrounding the high-voltage electrodes and subject to a transverse electrostatic force, which drives the particles toward the collecting plates. The overall particle collection efficiency is usually higher than 99% for ESP. However, various measurements have shown that the collection efficiency drops for particles with a size ranging from 0.2 to 2.5µm [14, 15]. To improve the performance of an ESP or other gas cleaning devices, it is of great importance to understand the characteristics of the particle motions. The modes of particle transport can be roughly summed up into the following aspects 1. Agglomeration of microparticles in turbulence. Strong turbulence with a large Reynolds number (Re > 104 ) can always be observed in the main flow of a particle collection device. The clustering effect of the turbulent flow on particles can significantly improve the collision rate of particles. Based on this effect, turbulent agglomerators are developed to coagulate submicron particles to larger ones, which can then be removed more easily by a conventional ESP or a bag filter. The agglomeration of particles in turbulence also plays a role in the initiation of raindrops [16] and in flocculation during water treatment [17]. The key point to understand the agglomeration is to build the relationship between the agglomeration rate and (a) the particle-scale interactions (i.e., adhesion, repulsion, friction, etc.) and (b) the properties of the turbulent flow. 2. Migration of a cloud of charged particles. In an ESP, the charged fly ash particles are forced to move towards the collection plate by an external electrostatic field. The design of an ESP usually relies on the Deutsch equation, in which the particle collection efficiency is related to the velocity of particles moving toward the collection plate (termed as the migrating velocity). However, due to the interparticle interactions, predicting the migrating velocity of a particle cloud is quite challenging. Another example system related to the migration of charged particles is the electrophoretic deposition (EPD) process. EPD technology is derived from the transport of charged suspended particles under the influence of an external field and has been widely applied to the fabrication of wear-resistant coatings as well as functional nanostructured films for electronic, biomedical, and electrochemical applications [11, 18, 19] (see Fig. 1.1b). Agreements have been reached that the deposition rate would significantly affect the quality of the deposits. Therefore, it is of vital importance to know the correlation between the migrating velocity and the interparticle electrostatic and hydrodynamic interactions. 3. Deposition of charged microparticles on surfaces. The migrating charged fly ash particles would finally deposit on a flat surface in an ESP or on cylindrical fibers in a bag filter (Fig. 1.1d). It is well accepted that the structure of the deposits on a filter significantly affects the pressure drop and the particle collection efficiency [12, 20]. Moreover, the deposition of charged particles can be related to the classic packing problem, which has been studied to understand the microstructure and bulk properties of liquids, glasses, and crystals, as well as granular matter

4

1 Introduction

[21–23]. Understanding the complex, collective behavior of particles during the packing process and bridging the gap between the macroscopic packing structure and the microscopic interparticle forces are helpful for the design, control, and optimization of the deposition and filtration systems. In the following sections, we briefly review the research progress of the agglomeration, migration, and deposition of microparticles in the presence of electrostatic and flow fields.

1.3 Collision and Agglomeration of Particles in Turbulence Clustering of particles suspended in turbulence has been extensively studied in experiments [24, 25], in simulations [26–28], and by theoretical approaches [29–31]. To predict the evolution of cluster or agglomerate size, Smoluchowski’s equation, built on statistical collision kernels, is one of the few theoretical tools that can be applied to large-scale systems [32–34]. The Smoluchowski’s equation is written as 

∂n(k) ∂t

 coag

∞  1  = (i, j)n(i)n( j) − n(k) (i, k)n(i), 2 i+ j=k i=1

(1.1)

where (i, j) is the collision kernel, which measures the averaged rate for agglomerates of size i colliding with agglomerates of size j. It is defined as (i, j) ≡

n˙ c,i j , n(i)n( j)

(1.2)

with n˙ c,i j being the collision rate per unit volume and n(i) being the average number concentration of size group i. The first term on the right-hand side of Eq. 1.1 is the source term that accounts for the rate at which agglomerates of size k are created. The second term is a sink that describes agglomerate disappearance due to its coalescence with other agglomerates. It is generally accepted that the turbulent flow first brings two initially separate particles at a sufficiently close distance (collision process), and microphysical mechanisms (collisional dissipation, hydrodynamic interactions, surface effects, etc.) then determine whether the two approaching particles can form an agglomerate. Most research has focused on the collision process. The collision kernel (i, j) has been formulated based on the collision sphere, as illustrated in Fig. 1.2. For two spherical particles of radius r p,i and r p, j , the collision kernel is the rate at which the separation vector between the centers of two particles cross a sphere of radius Ri j = r p,i + r p, j . The collision rate is proportional to the area of the collision sphere, 4π Ri2j , and to the radial component of the relative velocity of the two particles on the collision sphere, |wr |. Moreover, inertial particles have a tendency to cluster together, the effect of the nonuniform distribution on the collision kernel can be quantified by the radial

1.3 Collision and Agglomeration of Particles in Turbulence

5

Fig. 1.2 Geometrical description for the sphere formulation of the collision kernel

distribution function (also known as the two-point correlation function), g(Ri j ), for particles with the separation Ri j . Taking all these factors together, the collision kernel can be modelled as   (1.3) (i, j) = 2π Ri2j |wr | g Ri j . For zero-inertia particles, the particles follow flow streamlines and the quantities in Eq. 1.3 can be statistically determined from those of turbulence flows [35]. Further assuming a Gaussian distribution of the flow velocity gradient, a formula of the collision rate for zero-inertia particles was derived by Saffman and Turner [35]:  3 0 = 2r p



8π ε 15 v

1/2 ,

(1.4)

where ε is the dissipation rate of the turbulence kinetic energy per unit mass and ν is the kinematic viscosity of the flow. With the help of direct numerical simulations (DNS), the kernel functions are further extended to reflect the influence of particle inertia, identifying the effect of preferential concentration [24, 31, 36, 37] leading to an inhomogeneous particle distribution and sling or caustic effect [29, 34, 38], which causes the inertial particles to collide with large velocity differences. The Kolmogorov scale Stokes number St is defined as St =

τp , τK

(1.5)

which is the ratio between the response time of the particles τ p and the Kolmogorov time τk and provides a way to quantify the effect of inertia.

6

1 Introduction

Most DNS studies focus on the collision statistics of particles and do not pay much attention to the particle-scale interactions or the post-collision behaviors. When particles collide, it is normally assumed that particles can pass through each other without any modification to their trajectories (i.e., ghost collision approximation) or the colliding particles are simply removed from the simulation domain. To simulate the coagulation or agglomeration processes, it is generally assumed that the particles are spherical and the colliding particles merge immediately to form new larger spherical particles. These assumptions are not applicable to the agglomeration of solid non-coalescing microparticles. Solid microparticles have two significant differences from Brownian nanoparticles or coalescing droplets: (1) The interparticle adhesion due to van der Waals attraction is short-ranged and relatively soft [39]. It leads to the sticking and rebound behavior of colliding particles (i.e., nonunity coagulation efficiency). (2) The formed agglomerates are usually non-spherical, whose structure will evolve due to restructuring and breakage. Recent studies have suggested that even the simplest interparticle interactions, including the elastic repulsion [40, 41] and the electrostatic interactions [25, 42], give rise to nontrivial collision phenomena that cannot be predicted from the ghost collision approximation. Constructing a kernel function that can reflect the influence of complicated interparticle interactions is a crucial problem that has not been settled.

1.4 Migration of Microparticles in an Electrostatic Field The migration of microparticles in an electrostatic field is ubiquitous in engineering processes, including aerosol removal in an electrostatic precipitator [39], fabrication of functional films using electrophoretic deposition [43], and dust devils in Martian atmosphere [44, 45], where both long-range hydrodynamic and electrostatic interactions among particles result in complex collective dynamics. In most of these processes, the relevant Reynolds number can be very small, thus the hydrodynamic interaction can be simulated by solving the Stokes/Oseen equations [46], which read 1 0 = − ∇ p + v∇ 2 u ρ  ∞  1 U · ∇ u = − ∇ p + v∇ 2 u ρ

(Stokes),

(1.6a)

(Oseen).

(1.6b)

In Stokes equation, the convection term in the Navier-Stokes equation ((u · ∇)u) has been neglected, which is a reasonable assumption for flows with a sufficiently small Reynolds number (Re ≈ 0). Instead of setting the fluid velocity u = 0, the Oseen equation assumes a nearly uniform flow (U ∞ ) in the far field and is applicable for flows with small but finite Reynolds number. Solving the Stokes/Oseen equations, one could obtain the disturbance flow field of a moving particle. The disturbance of every individual particle on the flow field then can be linearly summed to obtain the flow field at the position of each particle [46]. As illustrated in Fig. 1.3, the results given by the Oseen dynamics can well reproduce the experimental observations [47].

1.4 Migration of Microparticles in an Electrostatic Field

7

Fig. 1.3 Evolution of a settling particle cloud: a simulation results from Oseen equation; b Experimental results (©Cambridge University Press, reproduced with permission from Pignatel et al. [47])

Earlier studies of sedimenting clouds have been carried out in the absence of electrostatic interactions. A common feature therein is that, while settling down, most particles tend to stay as a cohesive blob, resulting in a mean settling velocity that can be several times larger than the Stokes velocity of an isolated particle. The settling velocity and the shape of a particle cloud during gravity settling are first studied in conditions where the fluid inertia is negligible [48, 49]. An initially spherical cloud is found to evolve into a torus shape and eventually break up into two secondary clouds, each of which will further break again. For the conditions with a small but finite Reynolds number, the inflow at the rear of the cloud plays a key role in the evolution of a cloud into a torus shape, and the deformation is accelerated as the fluid inertia increases [47]. Subramanian and Koch [50] were one of the first to consider the inertial clouds and organized their behavior into different regimes (see Fig. 1.4). They also presented a theoretical prediction of the long-time dynamics of a sedimenting cloud. The cloud is characterized by a number density field n and a corresponding induced velocity field u r (only the radial component appears due to symmetry). According to their prediction, both planar axisymmetric clouds and spherical clouds undergo a selfsimilar expansion. Recently, the effects of the fluid inertia, the shape of the particle cloud, and the size polydispersity have been investigated by Oseen simulations [52, 53]. The settling velocity of the particle cloud, which is defined as the average settling velocity of all particles in the cloud, at the initial stage is proposed as follows [53]:   Uc = N S f R ∗ + 1, U0

(1.7)

where U0 is the settling velocity of an isolated particle, S is the size ratio between a particle and the cloud and is a factor accounting for the cloud shape effect. For a spherical cloud, = 1.2, whereas for a cylindrical cloud, = 1.5 ln[(1 + h)/ h],

8

1 Introduction

Fig. 1.4 Regimes of evolution for a migrating cloud characterized by particle Reynolds number Re p and cloud-to-particle size ratio R0 /r p . The points are the specific cases studied in this thesis (Chap. 4) (©Cambridge University Press, reproduced with permission from Chen et al. [51]) Fig. 1.5 Cloud settling velocity as a function of the aspect ratio of the cloud. The points are simulation results and the lines are predictions from Eq. 1.7 (Reprinted from Yang et al. [53], with permission from Elsevier)

with h being the column aspect ratio. The function f (R ∗ ) is a correction for the fluid inertia with R ∗ being the inertia length scale. As displayed in Fig. 1.5, the results predicted from Eq. 1.7 matches the simulation data very well. In the absence of electrostatic interactions, a cloud containing a sufficiently large number of particles is quite unstable. The instability and breakup phenomena are found to be sensitive to the polydispersity (size or density) of the particles: a greater degree of polydispersity tends to cause the cloud to persist as a cohesive entity for a shorter period of time [54, 55]. High instability poses huge challenges in predicting the long-time dynamics of a migrating cloud.

1.5 Deposition of Microparticles and Clogging Phenomenon

9

For charged microparticles, the electrostatic force is one of the most effective ways to drive particles’ motion. Intuitively, the Coulomb repulsion between charged particles may result in a faster expansion of the migrating cloud and stabilize the cloud shape [56]. However, compared with the gravitational settling of neutral particles, the migration behavior of charged particles with electrostatic interaction is less investigated.

1.5 Deposition of Microparticles and Clogging Phenomenon In an ESP, the migrating fly ash particles will deposit on the collection plate. The back corona and the re-entrainment of particles occurring in the deposits are key issues to improve the efficiency of an ESP. Both phenomena are closely related to the structure and the mechanical stability of the deposits. Moreover, in fibrous filters, the particles are continuously captured by the staggered arrays of fiber. A dust cake is then formed and blocks the gap between fibers. Such a clogging phenomenon can significantly increase the filtration efficiency as well as the pressure drop of a filter. Therefore, it is necessary to know how the deposits and clogs are formed and what are the key factors that determine the structures. The deposition of particles on a flat surface can be closely related to the classic packing problems [21–23]. Most published studies have focused on high-density jammed packings, which, in the case of uniform frictionless spheres, are found at the random close packing (RCP) density φ RC P ≈ 0.64 [22, 57, 58]. This maximally dense sphere packing has been interpreted thermodynamically as the infinite pressure limit of liquid and metastable glass phases [22, 59, 60]. In the presence of friction, packings can reach the random loose packing (RLP) limit with volume fraction φ R L P ≈ 0.55 [57, 61]. RLP is usually regarded as the lower limit for mechanically stable packings [62–64]. An analytical representation of the equation of state for packings is developed based on Edwards’ ensemble approach, where the relationship between the packing fraction φ and the average coordination number Z  is proposed [57]. The VDW adhesive force between microparticles usually hinders the particles from compaction during the packing process. Therefore, the packings formed by microparticles usually have a looser structure than those formed by nonadhesive particles. Previous studies have found that the packing fraction of adhesive microparticles varies in the range of φ = 0.165 − 0.622 using a discrete element method (DEM) [65]. Packings with φ = 0.20 − 0.55 were also obtained for 45 µm particles both in experiments and simulations [66]. The latest work of Liu et al. [67] combined the effects of particle kinetic energy and interparticle adhesion, identifying a universal regime of adhesive loose packings (ALP) for microparticles. Together with previous results from [57, 68–70], a phase diagram in the Z  − φ plane was presented for packings of frictionless, frictional, adhesive/nonadhesive spheres, as well as non-spherical particles (see Fig. 1.6).

10

1 Introduction

Fig. 1.6 Equation of state for particle packings (Reproduced from Liu et al. [67] with permission from The Royal Society of Chemistry)

Fig. 1.7 Illustration of clogging mechanisms

The clogging phenomena can be regarded as packings of particles on the wall of channels. As shown in Fig. 1.7a–c, clogging mechanisms can be categorized into three groups [71]. (1) The simplest one is the steric effect, where particles block the pores that are smaller than their diameter [72]. (2) Clogging also happens if the size of the pore is only a few times larger than the particles and the particle flow is jammed due to a sufficiently high local volume fraction of particles [73, 74]. (3) Even at a low volume fraction, particles with adhesion may continuously stick onto the wall or onto deposited particles, forming particle bridges that span across the channel and eventually block the channel. Since the gap between fibers is normally much larger than the particle diameter, clogging of a filter by microparticles belongs to the third category. The particles in the flow field first collide with the fiber due to the effects of particle inertia, intersection, and Brownian diffusion. The contact forces then determine whether the colliding particle will stick to the fiber. For an isolated fiber at the initial stage of the filtration, the particle collision efficiency (ϕ) can be calculated by ϕ = 1 − (1 − ϕ R ) (1 − ϕ D ) (1 − ϕ I ) ,

(1.8)

1.5 Deposition of Microparticles and Clogging Phenomenon

11

where ϕ R , ϕ D and ϕ I are the collision efficiencies due to the intersection, the Brownian diffusion, and the inertial impact mechanisms, respectively. The particle deposition efficiency is the production of the particle collision efficiency (ϕ) and the sticking probability ( ) [75], which is defined as the ratio between the number of sticking particles to the number of collisions. A relationship between and the surface energy of particles has been proposed based on the DEM simulation results [75]. Most of the existing studies in aerosol filtration focus on either the capture behavior of a single fiber or the filtration performance of a macroscale filter. Few of them have dealt with the problem of the nonclogging-clogging transition at the single-pore level. With the help of microfluidic technology, there have been several observations of clogging at the single-pore level in colloidal systems in recent years. It was found that the variation of the strength of particle-particle adhesive or repulsive interactions significantly changes the clogging probability as well as the structures of the clogs [76–78]. However, for most of the experimental observations, only macroscopic properties can be measured. The inability to obtain detail information about the various forces exerted on particles and the structure of deposited aggregates has greatly limited the understanding of these microscale clogging processes. Particle dynamics simulation in the framework of the Lagrangian method, where interparticle forces are considered explicitly at the particle level, offers a helpful tool to understand microscale clogging. Kim and Zydney [79] analyzed numerically a two-particle behavior when the particles flow through a pore in the presence of both electrostatic repulsion and Brownian motion. The calculations demonstrated that the presence of a second particle can significantly alter the particle trajectory through long-range interactions. More recently, based on a direct calculation of the coupled equations of the motion of particles and fluid, Agbangla et al. [80] investigated the dynamics of clogging at a single pore with both adhesive and repulsive forces. It was demonstrated that the particle aggregates formed at the pore entrance change from dendritic structures (for low repulsion) to dense aggregates (for high repulsion). In most of the simulation studies, it is often assumed that particles are “frozen” onto the wall or onto other deposited particles when contact occurs. This is suitable only if the adhesion is significantly larger than any other forces. However, the flow stress is normally comparable with the adhesion during the filtration of aerosols and rearrangement and resuspension of deposited particles are often observed [81, 82]. Recently, Yang [83] and Tao et al. [20] performed computational fluid dynamics with a discrete element simulation to investigate the clogging transition in a twofiber system. According to the evolution of the pressure drop, the formation of the clogs was divided into three stages: the clean filter stage, the transition stage, and the cake filtration stage (see Fig. 1.8). In the clean filter stage, the incoming particles are mostly captured by the fibers or the deposited particles on the upstream side. Both the capture efficiency and the pressure drop are low in this stage. In the transition stage, the particle chains then start to lodge down into the gap between fibers to form bridges, which significantly enhance the particle capture efficiency. A clogged state, in which the incoming particles can no longer penetrate through the filter, is then built and the system enters the cake filtration stage. The pressure drop then increases linearly with the number of deposited particles. It has been shown that the CFD-DEM

12

1 Introduction

Fig. 1.8 a Pressure drop as a function of the dimensionless time (C0 T ). Snapshots for b the clean filter stage, c the transition stage and d the cake filtration stage (Reprinted from Tao et al. [20], with permission from Elsevier)

coupling method can comprehensively account for the two-way interaction between the particles and the fluids and the dynamic behavior of deposited particles, providing information on the interparticle forces and microscopic structures, which cannot be observed by experiments. The computational error due to the “frozen” assumption can also be prevented. However, there have been few studies on the CFD-DEM simulation on the clogging transition for charged particles, where the short-range contact forces and the long-range electrostatic forces coexist.

1.6 Discrete Element Methods for Adhesive Particles A common feature for the agglomeration, migration, and deposition of microparticles is that the interactions between particles, including the contact interaction, the electrostatic interaction, and the hydrodynamic interaction, play essential roles in determining the particles’ behavior. The discrete element method, which solves for motions of individual particles, probably is the most suitable tool to account for these interactions. DEM was first proposed by Cundall and Strack [84] and has been widely used for simulating the particle motions in debris flows, fluidized bed reactors, pneumatic conveying, and particle separators. For two contacting microparticles, the Johnson-Kendall-Roberts (JKR) model, which couples the elastic repulsion and the VDW adhesion, was proposed to calculate the normal force. The friction force in the tangential direction was modelled by [85, 86]. A detailed description of the force models is given in Chap. 2.

1.7 A Road Map to Chaps. 2–6

13

Fig. 1.9 Modes of particle interaction: a normal impact; b sliding; c twisting; and d rolling (Reprinted from Marshall [87], with permission from Elsevier)

Recently, a JKR-based soft-sphere discrete-element method was developed by Li and Marshall [81] and Marshall [87] for adhesive particle flows, which simultaneously solves the normal forces, the tangential friction, the rolling, and twisting torques between two contacting particles (see Fig. 1.9). Model parameters are determined from the atomic force microscope measurements [88] or the particle-wall collision experiments [89, 90]. This novel DEM has also included a fast multipole method (FMM) to accelerate the calculation of electrostatic interactions [91] and has been applied successfully to the studies of particle capture by a fiber with or without electrostatic interactions [75, 81, 92]. The predictions for the dendrite length and the capture efficiency are found to agree well with the experimental data [93]. Although, the interactions between particles have been well formulated by the adhesive DEM, a large-scale CFD-DEM coupling simulation is usually computationally expensive. Since the collision time scale (∼ 10−8 s) is several orders of magnitude smaller than the typical flow time scale (∼ 10−4 s). It is necessary to reduce the calculation cost while ensuring an accurate computation. The simplest way for the speedup of DEM is introducing a reduced particle stiffness. It allows for larger time steps and therefore fewer total iterations in a simulation. Although this approach works well for dry nonadhesive particles [94, 95], it has been shown that for microparticles with adhesion, a reduction of stiffness can substantially change the simulation results [95, 96]. Besides, a simple and applicable principle to set the parameters in adhesive DEM is also lacking.

1.7 A Road Map to Chaps. 2–6 In this thesis, we propose new approaches for modeling the contacting interactions and electrostatic interactions between particles in the framework of the discrete element method and systematically investigate the agglomeration, migration, and deposition of microparticles in the presence of electrostatic and flow fields. The overall objective is to understand the relationship between the macroscale collective behaviors and the particle-scale interactions. The contents are organized as follows:

14

1 Introduction

1. Chapter 2 introduces the discrete element method for adhesive particles. A fast DEM is proposed based on the reduced particle Young’s modulus in the framework of Johnson-Kendall-Roberts (JKR)-based contact theory. A novel inversion method is then presented to help users quickly determine the input parameters in the fast DEM. This chapter also investigates the dynamics of three-dimensional collisions between two microparticles. The energy dissipation pathways are analyzed and the effect of particle size and impact angles on the critical sticking velocity is discussed. 2. Chapter 3 investigates the agglomeration of microparticles in homogeneous isotropic turbulence using a DNS-DEM coupling method. The dynamics of every individual particle are tracked both while they are traveling alone through the fluid and while they are colliding with other particles. The agglomeration rate and the collision-induced breakage rate are then formulated based on the Smoluchowski equations. The role of the flow structure on the collision-induced breakage is also examined. 3. Chapter 4 studies the evolution of spherical clouds of charged particles that migrate under the action of a uniform external electrostatic field. Hydrodynamic interactions are modelled by Oseen equations and the Coulomb repulsion is calculated through a pairwise summation. The evolutions of the cloud shape and the migrating velocity are presented with a coupling effect of both hydrodynamic and electrostatic interactions. 4. Chapter 5 discusses the packing of charged micron-sized particles. An averagefield method is proposed to calculate the Coulomb interaction in two-dimensional periodic systems. The formation process and the final structures of ballistic packings are then studied to show the effect of interparticle Coulomb force. Analysis from both dynamic and statistical mechanical aspects is then conducted to clarify the connections and the differences between the effects of the short-range van der Waals force and the long-range Coulomb force. 5. In Chap. 6, we perform simulations on the clogging process of charged microparticles at a single-pore level. The effect of the long-range Coulomb repulsion on the clogging transition is characterized by the bulk permeability, the number of penetrating particles, and the particle capture efficiency. A clogging phase diagram is constructed to quantify the non-clogging/clogging transition and the morphological changes of the deposits are shown through varying the strength of the long-range Coulomb repulsion and the short-range adhesion.

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1 Introduction

50. 51. 52. 53. 54. 55. 56. 57. 58. 59. 60. 61. 62. 63.

G. Subramanian, D.L. Koch, J. Fluid Mech. 603, 63 (2008) S. Chen, W. Liu, S.Q. Li, J. Fluid Mech. 835, 880 (2018) H. Chraibi, Y. Amarouchene, Phys. Rev. E 88(4), 042204 (2013) M. Yang, S.Q. Li, J.S. Marshall, J. Aerosol Sci. 90, 154 (2015) M. Faletra, J.S. Marshall, M. Yang, S.Q. Li, J. Fluid Mech. 769, 79 (2015) T.X. Ho, N. Phan-Thien, B.C. Khoo, Phys. Fluids 28(6), 063305 (2016) W. Yan, J.F. Brady, Phys. Rev. E 96(6), 060601 (2017) C. Song, P. Wang, H.A. Makse, Nature 453(7195), 629 (2008) C.S. O’Hern, L.E. Silbert, A.J. Liu, S.R. Nagel, Phys. Rev. E 68, 011306 (2003) T. Aste, A. Coniglio, Europhys. Lett. 67(2), 165 (2004) R.D. Kamien, A.J. Liu, Phys. Rev. Lett. 99(15), 155501 (2007) L.E. Silbert, Soft Matter 6(13), 2918 (2010) G.R. Farrell, K.M. Martini, N. Menon, Soft Matter 6(13), 2925 (2010) M. Jerkins, M. Schröter, H.L. Swinney, T.J. Senden, M. Saadatfar, T. Aste, Phys. Rev. Lett. 101(1), 018301 (2008) G.Y. Onoda, E.G. Liniger, Phys. Rev. Lett. 64(22), 2727 (1990) R.Y. Yang, R.P. Zou, A.B. Yu, Phys. Rev. E 62(3), 3900 (2000) E.J. Parteli, J. Schmidt, C. Blümel, K.E. Wirth, W. Peukert, T. Pöschel, Sci. Rep. 4, 6227 (2014) W. Liu, S. Li, A. Baule, H.A. Makse, Soft Matter 11(32), 6492 (2015) A. Baule, R. Mari, L. Bo, L. Portal, H.A. Makse, Nat. Commun. 4, 2194 (2013) Y. Jin, H.A. Makse, Phys. Stat. Mech. Its Appl. 389(23), 5362 (2010) A. Baule, R. Mari, L. Bo, L. Portal, H.A. Makse, Soft Matter 10, 4423 (2014) E. Dressaire, A. Sauret, Soft Matter 13(1), 37 (2017) A. Sauret, E.C. Barney, A. Perro, E. Villermaux, H.A. Stone, E. Dressaire, Appl. Phys. Lett. 105(074101) (2014) A. Kunte, P. Doshi, A.V. Orpe, Phys. Rev. E 90(020201(R)) (2014) K. To, P.Y. Lai, H.K. Pak, Phys. Rev. Lett. 86(1) (2001) M. Yang, S. Li, Q. Yao, Powder Technol. 248, 44 (2013) J.L. Perry, S.G. Kandlikar, Microfluidics and Nanofluidics 5, 357 (2008) H.M. Wyss, D.L. Blair, J.F. Morris, H.A. Stone, D.A. Weitz, Phys. Rev. E 74(061402) (2006) P. Bacchin, A. Marty, P. Duru, M. Meireles, P. Aimar, Adv. Colloid Interface Sci. 164, 2 (2011) M.M. Kim, A.L. Zydney, Chem. Eng. Sci. 60, 4073 (2005) G.C. Agbangla, P. Bacchin, E. Climent, Soft Matter 10(33), 6303 (2014) S. Li, J.S. Marshall, J. Aerosol Sci. 38, 1031 (2007) C. Henry, J.P. Minier, Prog. Energy Combust. Sci. 45, 1 (2014) M. Yang, Discrete element simulation of fine particles in fabric filtration by incorporating multiscale fluid-particle interactions. Ph.D. thesis, Tsinghua University, Beijing (2015) P.A. Cundall, O.D. Strack, Geotechnique 29(1), 47 (1979) C. Thornton, Z. Ning, Powder Technol. 99(2), 154 (1998) C. Dominik, A.G.G.M. Tielens, Astrophys. J. 480(2), 647 (1997) J.S. Marshall, J. Comput. Phys. 228(5), 1541 (2009) B. Sümer, M. Sitti, J. Adhes. Sci. Technol. 22(5–6), 481 (2008) B. Dahneke, J. Colloid Interface Sci. 51(1), 58 (1975) G. Liu, S. Li, Q. Yao, Powder Technol. 207(1–3), 215 (2011) G. Liu, J. Marshall, S. Li, Q. Yao, Int. J. Numer. Methods Eng. 84, 1589 (2010) M. Yang, S. Li, G. Liu, Q. Yao, in American Institute of Physics Conference Series, vol. 1542, pp. 943–946 (2013) B. Huang, Q. Yao, S.Q. Li, H.L. Zhao, Q. Song, C.F. You, Powder Technol. 163(3), 125 (2006) R. Moreno-Atanasio, B. Xu, M. Ghadiri, Chem. Eng. Sci. 62(1–2), 184 (2007) Y. Gu, A. Ozel, S. Sundaresan, Powder Technol. 296, 17 (2016) T. Kobayashi, T. Tanaka, N. Shimada, T. Kawaguchi, Powder Technol. 248, 143 (2013)

64. 65. 66. 67. 68. 69. 70. 71. 72. 73. 74. 75. 76. 77. 78. 79. 80. 81. 82. 83. 84. 85. 86. 87. 88. 89. 90. 91. 92. 93. 94. 95. 96.

Chapter 2

A Fast Discrete Element Method for Adhesive Particles

2.1 Introduction Collision and coagulation of micron particles, existing in a variety of areas of engineering, biology, astrophysics, and environmental science, play a central role in determining the structure and the growth rate of particle agglomerates or deposits [1–3]. In these applications, it is of central importance to predict whether two colliding particles will stick together to form an agglomerate or not [4, 5]. Previous studies mostly focused on normal collisions between particles or between a particle and a surface. However, the collisions between two spheres are generally oblique with non-zero impact angles, which may give rise to highly coupled inter-particle motions in the normal and tangential directions. Despite the preliminary efforts made to model the tangential friction caused by the oblique collision [6], an explicit expression of the sticking criterion, which is of great importance for incorporation with CFD codes to describe the particle deposition and coagulation processes, is still lacking. As we introduced in Sect. 1.6, the discrete element method (DEM) has been regarded as a powerful tool to resolve particle-particle collisions. However, the required computational time is much higher than other methods. It imposes challenges in implementing DEM for large numbers of particles. Introducing a reduced particle stiffness in DEM allows for bigger time steps and therefore fewer total iterations in a simulation. Although this approach works well for dry nonadhesive particles, it has been shown that for fine particles with adhesion, system behaviors are drastically sensitive to particle stiffness [7, 8]. To counterbalance the deviation that arises from the reduced stiffness, a modification of the adhesive forces is needed. In this chapter, we introduce the discrete element method (DEM) for adhesive particles, which is the primary methodology adopted in this thesis. This DEM is then employed to investigate oblique collisions of micron-sized particles. The central issue addressed is whether particles of a certain size will stick or not upon collisions with a given impact parameter and how the kinetic energy of colliding particles is dissipated. © Tsinghua University Press 2023 S. Chen, Microparticle Dynamics in Electrostatic and Flow Fields, Springer Theses, https://doi.org/10.1007/978-981-16-0843-8_2

17

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2 A Fast Discrete Element Method for Adhesive Particles

Moreover, we propose a fast DEM based on scaling laws to reduce particle Young’s modulus, surface energy and to modify rolling and sliding resistances simultaneously in the framework of Johnson-Kendall-Roberts (JKR)-based contact theory. It allows for bigger time steps and therefore fewer total iterations in a simulation. A novel inversion method is then presented, by which the parameters in DEM can be set according to a prescribed particle-wall collision result.

2.2 Discrete Element Method for Adhesive Particles The discrete element method is a framework that solves Newton’s second law of each particle. The translation and rotation motions of particles can be integrated according to dvi di = F i , Ii = Mi , (2.1) mi dt dt where m i and Ii are the mass and moment of inertia of particle i and F i and M i are forces and torques induced by both the fluid flow, the electrostatic field, and the interparticle contact. The particles are regarded as soft bodies and the forces and torques between contact particles are resolved. The current work adopted DEM models based on Johnson-Kendall-Roberts (JKR) contact theory [9–11], which assumes that the length scale of elastic deformation is large compared to the length scale of the adhesive force (with the particles’ Tabor parameter larger than unity) [12]. When the contact between two particles i and j are formed at time t0 , the normal force F N , the sliding friction F S , the twisting torque M T , and the rolling torque M R acting on particle i from particle j are expressed as   3/2 Fi j N = FiNj E + FiNj D = −4FC aˆ i3j − aˆ i j − η N v i j · ni j , ⎤ ⎡  FiSj = −min ⎣k T v i j (τ ) · ξ S dτ + ηT v i j · ξ S , FiSj,crit ⎦ , ⎡

(2.2a) (2.2b)

t0 t

⎤ 2 a η T iTj (τ ) · ni j dτ + iTj · ni j , MiTj,crit ⎦ , 2 t0 ⎤ t 3/2 MiRj = −min ⎣4FC aˆ i j v iLj (τ ) · t R dτ + η R v iLj · t R , MiRj,crit ⎦ . kT a2 MiTj = −min ⎣ 2 ⎡

t

(2.2c)

(2.2d)

t0

The normal force FiNj contains an elastic term FiNj E derived from the JKR contact theory and a damping term FiNj D , which is proportional to the rate of deformation. The elastic term F N E combines the effects of van der Waals attraction and the elastic deformation and its scale is set by the critical pull-off force, FC = 3π Ri j γ , where

2.2 Discrete Element Method for Adhesive Particles

19

−1 −1 Ri j = (r −1 is the reduced particle radius and γ is the surface energy density p,i + r p, j ) of the particle. The surface energy density γ is defined as half the work required to separate two surfaces that are adhesively bound per unit area. √ The normal dissipation coefficient η N in Eq. 2.2a is given as η N = α m ∗ k N , where the coefficient α is a function of a prescribed value of the coefficient of restitu−1 is the effective mass of the two coltion e0 (see Marshall [11]), m ∗ = (m i−1 + m −1 j ) liding particles, and the normal elastic stiffness k N is expressed as k N = (4/3)E i j ai j . The tangential stiffness k T is expressed as k T = 8G i j ai j and the effective elastic modulus E i j and shear modulus G i j are functions of particle’s Young’s modulus E i and Poisson ratio σi ,

1 − σ j2 1 − σi2 1 = + , Ei j Ei Ej

2 − σj 1 2 − σi = + Gi j Gi Gj

(2.3)

where G i = E i /(2(1 + σi )) is the particle’s shear modulus. The radius of contact area ai j is related to the value at the zero-load equilibrium state ai j,0 through ai j = aˆ i j ai j,0 , where ai j,0 is given as ai j,0 = (9π γ Ri2j /E i j )1/3 and aˆ i j is calculated inversely from the particle overlap, δ, through [9, 11, 13]

4 δ 1 1 2 3 2 = 6 2(aˆ i j ) − (aˆ i j ) , δC 3

(2.4)

where δC is the critical overlap and is given by δC =

ai2j,0 1

2(6) 3 Ri j

,

(2.5)

The contact between the particles is built up when the overlap δ > 0 and is broken when δ < −δC . For the tangential dissipation coefficient ηT in Eqs. 2.2b and 2.2c, we simply set ηT = η N [14]. The rolling viscous damping coefficient η R in Eq. 2.2d is a function of the coefficient of restitution e0 , the normal elastic force FiNj E , and the effective mass of the two colliding particles m ∗ . For details, see [11]. The sliding friction F S , twisting torque M T , and rolling torque M R (Eqs. 2.2b– 2.2d) are all calculated based on spring-dashpot-slider models, where v i j · ξ S , iTj , and v iLj are the relative sliding, twisting, and rolling velocities. When these resistances reach their critical limits, namely FiSj,crit , MiTj,crit , and MiRj,crit , irreversible relative sliding, twisting, and rolling motions will take place between a particle and its neighboring particle. The critical limits are expressed as [11]:

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2 A Fast Discrete Element Method for Adhesive Particles

      3/2 FiSj,crit = μ S FC 4 aˆ i3j − aˆ i j + 2 , , 16 3/2 = 4FC aˆ i j θcrit Ri j .

MiTj,crit = MiRj,crit

3πai j FiSj,crit

(2.6a) (2.6b) (2.6c)

Here μ S (= 0.3) is the friction coefficient and θcrit (= 0.01) is the critical rolling angle. We set these values according to the experimental measurements [15]. The soft-sphere DEM for adhesive particles has been successfully applied to simulations of various systems, including the particle-wall collisions [16] and the deposition of particles on a fiber [17] or on a plane [18], and the agglomeration of particles in a pressure-driven duct flow [19], with a series of experimental and theoretical validations.

2.3 Critical Sticking Velocity for Two Colliding Particles Understanding the collision dynamics for two adhesive particles is of fundamental importance to further understand the complicated agglomeration and deposition behaviors. Previous studies mostly focused on head-on collisions between two particles or between a particle and a surface. However, the collisions between two spheres are usually oblique with non-zero impact angles, which gives rise to highly coupled particle-particle interaction modes. In this section, we focus on oblique collisions between micron particles by employing the three-dimensional adhesive DEM. A generalized expression for the sticking/rebound criterion of collisions covering a wide range of impact angles will be given. Collisions between a particle and a wall, which can be regarded as a sphere with infinite radius and mass, are also considered. The collision geometry is displayed in Fig. 2.1. Firstly, a reference frame that is at rest relative to the target particle (particle 2 in Fig. 2.1a) is selected so that the relative collision velocity is equal to the velocity of the colliding particle (particle 1). The angle θ between the velocity of particle 1 and the line connecting the centers of the two particles determines the geometry of a collision. The dimensionless impact parameter bˆ then can be calculated from θ through bˆ =

b = sin θ, r1 + r2

(2.7)

where b is the vertical distance from the center of particle 2 to the path of particle 1, r1 and r2 are the radii of particle 1 and particle 2, respectively. The impact parameter bˆ ranges between 0 and 1, where bˆ = 0 indicates a head-on collision and bˆ → 1 corresponds to the state that particles barely touch each other. For the collision between a particle and a wall (Fig. 2.1b), the angle θ is defined in the same way and the impact parameter is also given by bˆ = sin θ .

2.3 Critical Sticking Velocity for Two Colliding Particles

21

Fig. 2.1 Illustration of the collision geometry for collisions between a two particles and b a particle and a wall. Reprinted from Chen et al. [20], with permission from Elsevier

The two colliding particles touch each other at t = 0 with the initial conditions

1 r2 − r1 ˆ R |V |bt =− v i j (0) = V , 2 r2 + r1 δ N (0) = δT (0) = ξ(0) = 0, v iLj (0)

(2.8a) (2.8b)

where v i j is the relative velocity between the two particles, v iLj is the relative rolling velocity, δ N is the deformation (overlap) in the normal direction, δT and ξ are the relative sliding and rolling displacements. Given the equation of motion in Eq. 2.1, the forces and torques in Eq. 2.2, the initial conditions in Eq. 2.8 and the bond breakage criterion in Eq. 2.5, the detailed information of the collision can be obtained. The key parameters used in the DEM simulation are listed in Table 2.1. Collisions between particles are always accompanied by various energy-loss mechanisms. How fast the initial kinetic energy is dissipated has a tremendous influence on the magnitude of the critical sticking velocity and the coefficient of restitution for bouncing collisions. In DEM simulation, the energy conversion and dissipation can be calculated by integrating the forces along the corresponding displacement. The initial kinetic energy dissipated or converted by the normal force F N , the sliding friction F S , and the rolling resistance M R can be expressed as

Table 2.1 Parameters in the DEM simulation of two particle collisions Parameters Values Particle radius (r p ) Particle density (ρ p ) Surface energy (γ ) Young’s modulus (E) Coefficient of restitution (e0 )

≥ 1 µm 2500 kg/m3 15 × 10−3 J/m2 2 × 108 Pa 0.6

Reprinted from Chen et al. [20], with permission from Elsevier

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2 A Fast Discrete Element Method for Adhesive Particles

t

E N = E ne + E nd =



 F N E + F N D δ˙ N (τ )dτ,

(2.9a)

t0

t

E S =



 F S v i j · ξ S + ri F S (ni j × ξ S ) ·  dτ,

(2.9b)

t0

t

Er =

M R t R · (τ )dτ.

(2.9c)

t0

The energy converted by the normal force E N contains two parts: E ne , which is the energy stored within the contact region in the form of the elastic potential energy and the surface energy, and E nd , which is the energy dissipated by the normal damping force.

2.3.1 Temporal Evolution of the Collision Process In order to illustrate how the adhesive DEM can generate microscopic information that can help to improve the understanding of collision problems, simple cases of oblique collisions between a 1 µm particle and a wall (Fig. 2.1b) are investigated. We consider two typical cases with the same impact angle θ = 45◦ . The impact velocities are set as VN = VT = 0.8 m/s for the sticking case and VN = VT = 1.0 m/s for the rebound case. The evolutions of displacements, including the normal overlap δ N , the sliding displacement δT , and the rolling displacement ξ , for the case with the lower impact velocity are shown in Fig. 2.2a. One can see that the kinetic energy of the particle is dissipated during the oscillation till the final captured state is achieved. The time and displacements √ in the figure have been normalized by the typical collision time scale TN = 2π/ FC /mδC − α Ea0 /3m for adhesive particles and the particle radius r p , respectively. Due to the dissipation of the kinetic energy, the particle-wall overlap (δ N ) cannot exceed the critical value δC (indicated by the horizontal dash line in Fig. 2.2a), at which point the neck connecting the contact particles will suddenly break. The particle then oscillates back and forth towards the equilibrium point δ N = δ0 . As for the sliding motion, the displacement increases rapidly and then reaches an equilibrium value at approximately T ∗ = 0.62. The rolling motion lasts for a much longer time compared with the sliding and the final state is achieved at about T ∗ = 3.5. To further illustrate the coupling between sliding and rolling motions, we plot the evolution of the tangential velocity VT and the rotating velocity, which is defined as the product of particle’s rotating rate  and the particle radius r P in Fig. 2.2b. The whole process can be divided into three stages. At the first stage (labeled with A), VT

2.3 Critical Sticking Velocity for Two Colliding Particles

23

Fig. 2.2 Temporal evolution of a the normal overlap δ N , the sliding displacement δT , and the rolling displacement ξ (scaled by the particle radius r p ), b the tangential velocity VT and the rotating velocity  · r p . c Energy conversion during a typical sticking collision with initial velocities VN = VT = 0.8 m/s. The horizontal dashed line in (a) indicates the critical overlap δC . Reprinted from Chen et al. [20], with permission from Elsevier

is obviously larger than  · r P , which indicates that there is a relative sliding between the particle and the wall. In this stage, the sliding resistance plays a dominant role in dissipating the kinetic energy of the particle. The sliding force acts at the contact point in a direction opposite to VT , leading to the decrease of VT and the increase of  · r P . The collision process then enters a transition stage (labeled with B in Fig. 2.2b), during which VT is very close but not identical to  · r P . At this stage, both sliding and rolling resistances simultaneously dissipate the particle’s kinetic energy. During the final stage labeled with C, the two lines coincide with each other, indicating a pure rolling motion of the particle. The rolling resistance becomes the main mechanism to dissipate the kinetic energy until the final state is achieved. The conversion of the energy during the collision, normalized by the initial kinetic energy of the particle, is shown in Fig. 2.2c. The initial kinetic energy is steadily dissipated through the normal damping force, the sliding resistance and the rolling resistance, which is shown as the continuous increase of E nd , E S , and Er in Fig. 2.2c. The conservative part of the transferred kinetic energy E ne , which is the sum of the released surface energy and the stored elastic potential energy, fluctuates as the colliding particle oscillates back and forth towards the final state. In the current

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2 A Fast Discrete Element Method for Adhesive Particles

Fig. 2.3 Temporal evolution of a the normal overlap δ N , the sliding displacement δT , and the rolling displacement ξ (scaled by the particle radius r p ), b the tangential velocity VT and the rotating velocity  · r p . c Energy conversion during a typical rebound collision with initial velocities VN = VT = 1.0m/s. The vertical dashed lines indicate the moment when the contacting bond between the particle and the wall breaks. Reprinted from Chen et al. [20], with permission from Elsevier

collision case, the normal damping force dissipates most of the energy (∼ 60% of the initial kinetic energy) and the sliding and rolling resistances each account for the ∼ 20% of the dissipation. For the impact with a higher velocity, shown in Fig. 2.3, the kinetic energy of the particle is larger than that could be dissipated by the contact interactions. The particle-wall overlap (δ N ) crosses over the critical value −δC , which results in the breakage of the bond between the particle and the wall. In this rebounding case, the evolutions of VT and  · r P are limited in the first stage (Fig. 2.3b). Furthermore, one can observe a sudden conversion of the energy from E ne to E nd at the separation point in Fig. 2.3c. The necking between contact surfaces disappears at this moment and the energy stored in the contact region is instantaneously released leading to the so-called first-contact loss.

2.3 Critical Sticking Velocity for Two Colliding Particles

25

Fig. 2.4 The critical sticking velocity VC and its normal component VC N as functions of the impact angle θ. The inset shows the relative deviation of VC N from the value of the head-on (i.e. θ = 0) case VC0 as a function of 1 − bˆ in double logarithmic coordinates. VC∗ N is defined as

VC∗ N = (VC N ,0 − VC N )/VC N ,0 and bˆ is the dimensionless impact velocity. Reprinted from Chen et al. [20], with permission from Elsevier

2.3.2 Prediction of the Critical Sticking Velocity Depending on the magnitude of the impact velocity, two colliding particles will either stick together or rebound from each other. The impact velocity that separates the two conditions is termed as the critical sticking velocity. An accurate prediction of the critical velocity can help design the aerosol agglomerators and filters. In the current subsection, we discuss the relationship between the critical sticking velocity VC and the impact angle θ . As shown in Fig. 2.4, for particles with radius r P = 1 µm, the critical sticking velocity VC increases with the impact angle θ . The red line in Fig. 2.4 presents how the normal component of the impact velocity VC N varies with θ . It can be seen that VC N almost keeps as a constant at small value of θ and quickly drops when θ > 65◦ . In previous work, the relative sliding and rolling motions between the colliding particles are often neglected and VC N is normally regarded as a constant when the impact angle varies (see Ref. [1]). Here, we show that this assumption is valid only when the impact angle is not too large. The results in Fig. 2.4 suggest that the role of sliding and rolling motions should be considered when the VC N drops to 90% of the its value at θ = 0. The value of θ at this moment is denoted as θ90 . For the case shown in Fig. 2.4, θ90 is around 73◦ . To predict the value of VC N at different θ , we start from the critical sticking velocity for head-on collisions (denoted as VC N ,0 ) and then figure out the relationship between VC N /VC N ,0 and θ . A theoretical model for VC N ,0 was proposed by Thornton and Ning [21], in which the force-displacement relationship is described by JKR model, shown as the red line in Fig. 2.5. The force is normalized by the critical force FC = 3π γ Ri j and the normal overlap is scaled by the critical value δC . The loading

26

2 A Fast Discrete Element Method for Adhesive Particles

Fig. 2.5 Evolution of the normalized contact force F/FC versus the normalized overlap δ N /δC with viscoelastic effect taken into account (black line). Prediction from the JKR model is also plotted (red line). Reprinted from Chen et al. [20], with permission from Elsevier

phase covers only δ N ≤ 0 (curve AB), while the unloading process (curve BAC) continues till the critical separation point δ N = −δC . The difference between the loading and unloading processes results in the first-contact energy loss, which can be expressed as 5 4 1/3 c w R FJ K R dδ N = 7.09 , (2.10)

E ad = E2 A

where w = 2γ is the work of adhesion. Assuming that the first-contact energy loss is identical to the initial kinetic energy of the particles in the critical sticking case, Thornton and Ning [21] proposed the following expression for the sticking/rebound criterion



1/6 14.18 1/2 w5 R 4 . (2.11) VC1 = m∗ E2 Plugging the simulation parameters into Eq. 2.11, one can obtain the critical sticking velocity VC = 0.34 m/s, which is considerably smaller than the DEM result due to the neglect of the energy dissipation from the viscoelastic effect. Such effect together with the first-contact energy loss accounts for the dissipation of the particle’s kinetic energy. Taking the viscoelastic effect into account, the loading/unloading path for the two colliding particles is described by the black line in Fig. 2.5, which covers A1 BA2 C. For a given value of the overlap δ N , the force in the loading process is larger than that in the unloading process since the damping force F N D points in opposite directions during the two different processes. Here, we use F to denote the difference between F in the loading and unloading phases at the same δ N value. The energy dissipation, which includes both the first contact loss and the viscoelastic effect, thus can be expressed as

2.3 Critical Sticking Velocity for Two Colliding Particles

27

Fig. 2.6 The difference between the normal force in the loading and unloading processes F as a function of the overlap δ N (black solid line). The black dashed line indicates the force of JKR model. The region enclosed by the red dashed line is an approximation of the energy dissipated by the viscoelastic effect (shaded area). Reprinted from Chen et al. [20], with permission from Elsevier

δmax

E k =

Fdδ N .

(2.12)

−δC

To better illustrate the energy dissipation, we plot F as a function of δ N in Fig. 2.6. The force difference F and the overlap δ N have been normalized by the critical force FC and the critical overlap δC , respectively. The energy loss E k is the sum of the area enclosed by the solid black line and the x-axis. The adhesion part

E ad , which is indicated by the area under the JKR line, was used by Thornton and Ning [21] to calculate the critical sticking velocity. As shown in Fig. 2.6, E ad is much smaller than the energy dissipated by the viscoelastic effect E diss (area of the shaded region), indicating that the viscoelastic effect cannot be neglected. The energy conservation yields 1 ∗ 2 1 2 m VC N ,0 = E ad + E diss = m ∗ VC1 + E diss , 2 2

(2.13)

  −1 −1 is the reduced mass of the two colliding particles. VC1 where m ∗ = m −1 1 + m2 is the critical sticking velocity calculated from Eq. 2.11, which only considers the first-contact loss. E diss is given by 0

E diss = −δC

δmax

F (δ N ) dδ N . [ F (δ N ) − FJ K R (δ N )] dδ N +

(2.14)

0

The accurate value of the integration in Eq. 2.14 cannot be obtained directly. An approximation, therefore, is made as follows. First, we adopt F (δ N = 0) as the first-order approximation of F (δ N ) when δ N > 0 and assume a linear relationship between F and δ N when δ N < 0. The shaded area in Fig. 2.6 is approximated as the area enclosed by the red dash lines (triangle for δ N < 0 and rectangle for

28

2 A Fast Discrete Element Method for Adhesive Particles

δ N > 0). Furthermore, we use the Hertzian relation to calculate δmax , which is δmax =  2/5 √ 15m ∗ VC2 N ,0 /(16 R E) . Equation 2.14 thus is approximated as

E diss ≈

δC ( F (0− ) − FJ K R (0)) + F (0+ ) δmax .    2  

(2.15)

E diss (δ N >0)

E diss (δ N θ90 , the normal component of the critical sticking velocity is considerably smaller than VC N ,0 . Based on the DEM results in Fig. 2.4, we then fit the relationship between the ratio VC N /VC N ,0 and the impact parameter bˆ using

2.3 Critical Sticking Velocity for Two Colliding Particles

29

Fig. 2.7 Plot of Vin − VC2 4/5 and H Vin as functions of the impact velocity Vin . The critical sticking velocity predicted by Thornton and Ning [21] is also marked in the figure. Reprinted from Chen et al. [20], with permission from Elsevier

ˆ ˆ ≡ VC N (b) = f (b) VC N ,0



ˆ −1.6 0 ≤ bˆ ≤ bˆ90 1 − 0.0011(1 − b) ˆ −0.4 bˆ90 < bˆ ≤ 1 1 − 0.0454(1 − b)

(2.20)

which is shown as the red dashed line and blue solid line in the inset of Fig. 2.4. In this case, bˆ90 = 0.955, which is indicated by the vertical dashed line in the figure.

2.3.3 Effect of Particle Size The size of the particle is one of the most important parameters that affect the behavior of particle systems. The real systems are mostly involved with polydisperse particles. Here, we discuss how particle size affects collision dynamics by investigating the binary collision between particles with different sizes. The radius of the colliding particle is recorded as r1 and that of the target particle is set as r2 . The size ratio β, defined as β = r2 /r1 , is assumed to be larger than unity without loss of generality. We first fixed the size ratio as β = 1, implying r1 = r2 = r p , to investigate the size effect on the collision between two identical particles and then vary β to look at the effect of the size ratio. The effect of r p on the normal component of the critical sticking velocity VC N is displayed in Fig. 2.8a. Results are calculated under three representative impact parameters. It is shown that VC N decreases with the particle radius r p . A dimensionless adhesion parameter Ad, which is defined as the ratio of the particle surface energy 2γ π R 2 and the particle kinetic energy m ∗ V 2 /2, is then proposed to quantify the size effect [10]. For a collision between particles with radius r1 = r p and r2 = βr p , the expression for Ad can be written as Ad =

1 + β3 3γ 2γ π R 2   = 1 ∗ 2 ρV 2 r p 1 + β 2 β m V 2

(2.21)

30

2 A Fast Discrete Element Method for Adhesive Particles

Fig. 2.8 Plots of the normal critical velocity VC N for a collisions with identical particle radius r1 = r2 = r p as a function of r p , and for b collisions with a fixed radius r1 = 1μm of the smaller particle as a function of the size ratio β. The variation of the adhesion parameter Ad is also plotted. Reprinted from Chen et al. [20], with permission from Elsevier

The value of Ad is also plotted in Fig. 2.8a as a function of the radius r p . The Ad curve follows a similar trend as the VC N curves under the given collision parameters. Thus, the normal critical velocities for collisions between two identical particles nearly scale as r −1 p . We then look at how the size ratio β affects the collision results. The radius of particle 1 is fixed as r1 = 1 µm, and β varies from 1 to 50. The results are shown in Fig. 2.8b. VC N first decreases and then increases as β increases. The curves of VC N for different bˆ in Fig. 2.8b show a good consistency with the line of Ad in low β region but deviate from Ad at large β. The reason is that during an oblique collision between two particles with a large size ratio, the relative rolling and sliding motions are triggered to dissipate the energy, which makes the collision process less predictable. With the effects of the particle size and the impact parameter taken into account, an analytical expression for the normal component of the critical sticking velocity can be written as 3   ˆ ˆ r p0  1 + β  VC N ,0 , r p , β VC N ,0 = f (b) VC N = f (b)φ r p 1 + β2 β

(2.22)

where r p is the radius of the smaller particle, VC N ,0 is the critical velocity for headon collisions between two particles with identical radius r p0 and can be calculated ˆ accounts for the effect of impact parameter and is expressed in from Eq. 2.18. f (b) ˆ r p , β) describes the influence of particle size and size ratio on VC N . Eq. 2.20. φ(b, To summarize, the critical sticking velocity can be predicted in five steps: 1. Predict the critical velocity VC1 from Eq. 2.11, which accounts for the first-contact energy loss; 2. Calculate the critical unloading velocity VC2 at δ N = 0 using

2.4 A Fast Adhesive DEM

31

VC2 =

 2 α (k N /m ∗ )1/2 VC1 δC + VC1 .

(2.23)

 1/3 Here, k N = 4Ea (δ N = 0) /3, a (δ N = 0) = (4/9)1/3 a0 = 4π γ R 2 /E . The damping coefficient α is determined by a predefined coefficient of restitution e through α = 1.2728 − 4.2783e + 11.087e2 − 22.348e3 + 27.467e4 − 18.022e5 + 4.8218e6 [11]; 3. Determine the value of H through H = 3.43αm ∗−1/10 R 2/15 E −1/15 γ 1/6 and then solve Eq. 2.18 for VC N ,0 , which is the critical velocity for head-on collisions between two particles with identical radius r p0 . 4. Calculate the normal component of the critical sticking velocity VC N for the ˆ C N ,0 with f (b) ˆ oblique collision with impact parameter bˆ from VC N = f (b)V given by Eq. 2.20. 5. Determine the normal component of the critical sticking velocity VC N for particles with radius r p and βr p according to Eq. 2.22. The critical sticking velocity here has taken the properties of adhesive particles (e.g., elasticity and surface energy) and the collision conditions (colliding velocity and angle) into account. For large-scale particle flows, such as the flow in an electrostatic precipitator or in a fluidized bed, a collision-resolved DEM simulation is computationally expensive since the time step for the collision process is usually several orders of magnitude smaller than the time step for computational fluid dynamics (CFD). The sticking/rebound criterion here can be readily implemented in CFD codes and provide a feasible way to determine whether two colliding particles will stick together or rebound from each other. However, for simulation of systems that contain particle agglomerates or deposits, one needs to know the evolution of their structures. In this case, DEM simulation is inevitable. In the following subsection, we propose an accelerated algorithm for the adhesive discrete element method (DEM) by introducing reduced particle properties. It allows one to use bigger time steps and therefore fewer total iterations in a simulation and reproduce essentially the same results as those calculated with real particle properties.

2.4 A Fast Adhesive DEM 2.4.1 Accelerating Adhesive DEM Using Reduced Stiffness The typical collision time tC , which is defined as the time associated with the elastic response during the collision between two particles, can be generally estimated as tC = r p (ρ 2p /E 2 U )1/5 [10]. Here, r p is the particle radius, ρ p is the particle density, E is the particle Young’s modulus and U is the particle collision velocity. To resolve the collision, one should use a time step dtC = f C tC with f C much less than unity. Acceleration of the simulation can be achieved by choosing a reduced Young’s modulus E R that satisfies the condition E R  E O (hereafter, we use the subscript O

32

2 A Fast Discrete Element Method for Adhesive Particles

Fig. 2.9 A graphical representation of accelerating DEM with reduced stiffness. The top dark blue bar indicates the entire simulated process, which has a time span Ttot . The green bars stand for collision events calculated using the original stiffness of the particles, which have a typical timescale tC,O , and the light blue bars are collision events calculated using the reduced stiffness of the particles, which have a timescale tC,R . Each collision event is resolved by the time step dtC,O or dtC,R , indicated by the discretized grids. Reprinted from Chen et al. [22], with permission from Elsevier

to indicate the original particle properties and R to indicate reduced properties). It allows one to use a larger time step to resolve the collision event. The speedup of DEM is of prime importance when the simulated system contains numerous collision events. A graphical representation of this idea is displayed in Fig. 2.9. The time span Ttot is usually set by macroscopic parameters, such as the total mass loading of deposited particles in filtration/deposition systems or the total amount of gas in fluidization system, and thus is independent of the particle stiffness. When the original Young’s modulus E O is used in the simulation, the collision events (indicated by green bars) take place over the typical collision time tC,O . In contrast, if a reduced Young’s modulus E R is assigned to the particles, the collision events (indicated by light blue bars) will have a much larger timescale tC,R . As a result, a larger time step dtC,R ( dtC,O ) can be used to resolve the collision events and the total number of iterations decreases. Since the collision time is usually several orders of magnitude smaller than the typical particle transport time. It is reasonable to assume that the extension of the collision time does not apparently affect the subsequent collision events. Such an approach for speedup of DEM has been tested and found widespread uses in the simulation of nonadhesive particles [7, 23, 24]. However, for fine particles with van der Waals adhesion or wet particles with cohesion, a reduction of stiffness in DEM models can substantially change the simulation results [7]. Intuitively, with a smaller stiffness, the particles in contact tend to have a larger deformation along the direction of compression and an enlarged area of the contact region, which leads

2.4 A Fast Adhesive DEM

33

to an overestimation of the adhesive effect [8, 25]. To counterbalance the deviation that arises from the reduced stiffness, a modification of the adhesive force is needed. We consider a simple case of the particle-wall normal collision, where the particle radius is r and the impact velocity is dx/dt = −v0 . The state of the particle can be described using the equation of the overlap δ(t). According to Newton’s second law, 2 the temporal evolution of δ(t) is given by m ddt2δ = F, where the force is taken from the JKR model (Eq. 2.2a). The evolution of δ(t) then follows  4FC  3 η N dδ d2 δ 3 2 (δ) = 0. + a ˆ + (δ) − a ˆ dt 2 m dt m

(2.24)

Normalizing the overlap by its critical value δC (Eq. 2.5) and the time t by T0 = δC /v0 , Eq. 2.24 can be written in the dimensionless form ˆ d2 δˆ 1/2 dδ ˆ = 0, + A a ˆ + Bg(δ) dtˆ2 dtˆ

(2.25)

ˆ = aˆ 3 − aˆ 3/2 can be derived from the relationship between where the function g(δ) aˆ and δˆ in Eq. 2.4. The dimensionless factors, A and B, are functions of particle properties and the impact velocity v0 , and can be rewritten as A = 2.515α B=

E ρv02

− 13

γ ρv02 r

56

,

(2.26a)

3.633 2 A . α2

(2.26b)

ˆ ˆ tˆ(0) = 1. The contact The initial conditions for Eq. 2.25 are δ(0) = 0 and dδ/d between the particles is built up when the overlap δ > 0 and is broken when δ < −δC . One can easily find that the results are determined by the damping coefficient and a dimensionless parameter A∗ , which is given by ∗

A = H (E, γ , ρ, v0 ) =



E ρv02

− 13

γ ρv02 r

56

≡ (El)− 3 (Ad) 6 . 1

5

(2.27)

We have removed all other coefficients in the governing equation and in the initial conditions through scaling. The first parameter in Eq. 2.27, El = E/(ρv02 ), is called elasticity parameter, which can be regarded as the ratio of the elastic force to the particle inertia [10]. The second parameter is the adhesion parameter, Ad = γ /(ρv02 r ), defined as the ratio of the adhesive energy and the particle kinetic energy [10]. When a reduced particle Young’s modulus E R is used, the surface energy should be modified accordingly such that the parameter A∗ is unchanged. Thus, the reduced surface energy can be calculated as:

34

2 A Fast Discrete Element Method for Adhesive Particles

Fig. 2.10 a The coefficient of restitution e and b the collision time ts as functions of the impact velocity v0 . Reprinted from Chen et al. [22], with permission from Elsevier

γR =

ER EO

25

2

γO ≡ χ 5 γO ,

(2.28)

where χ = E R /E O is the reduction ratio. The particle-wall collision is simulated under both real parameters (E O = 1 × 109 Pa and γ0 = 15 mJ/m2 ) and two sets of reduced parameters. The coefficients of restitution e for different impact velocities are shown in Fig. 2.10a. The result demonstrates that the scaling yields e − v curves identical to those calculated with the original parameters. We also display the physical time of collisions ts , which is defined as the time interval between the moments of contact formation and separation, as a function of the impact velocity in Fig. 2.10b. For a given velocity, the collision takes place over a much longer time when the reduced Young’s modulus is used.

2.4.2 Modified Models for Rolling and Sliding Resistances A proper description of the adhesive rolling and sliding resistances is of significance to predict the formation of agglomerates and the structure of particle deposits. For adhesive microparticles, rolling is generally the preferred deformation mode, which gives rises to the rearrangement of packing structures [1, 26, 27]. To accurately simulate the rolling motion, the adhesive rolling model needs to be modified in the framework of JKR-based DEM with reduced stiffness. The same idea can be readily applied to modify the sliding resistance.

2.4 A Fast Adhesive DEM

35

R /r as a function of particle size r at different stiffnessFig. 2.11 The critical rolling force Mcrit p p reduced ratio χ = E R /E O . The black circles are experimental results from [15]. The solid lines are calculations of Eq. 2.6c using the surface energy γ R = χ 2/5 γ O and the critical rolling angle θcrit = ξcrit /r p = 0.0085, with γ O and ξcrit the same as those measured in experiments [15]. The inset shows the set-up of the measurements. Reprinted from Chen et al. [22], with permission from Elsevier

Assume a simple case where a particle is in a mechanical equilibrium with a wall and an external force Fext , which is parallel to the wall, is then applied on the center R /r p , the particle rolls of the particle. If Fext is smaller than the critical value Mcrit R /r p , the over a small distance and reaches a mechanically stable state. If Fext > Mcrit particle will roll irreversibly. According to the experimental measurements of [15] using polystyrene microparticles, the critical rolling angle θcrit = ξcrit /r p is nearly constant, θcrit = 0.0085. As displayed in Fig. 2.11, using the same parameters as in [15], Eq. 2.6c gives a good prediction of the particle size dependence of the critical R /r p is underestimated when a reduced particle stiffness rolling force. However, Mcrit E R = χ E O and the corresponding reduced surface energy γ R = χ 2/5 γ O are used. The reason is that the critical rolling resistance in Eq. 2.6c is proportional to the surface energy but is independent of the particle stiffness. An easy and intuitive way to retain the original value of the critical rolling resistance is to use the real surface energy γ O to calculate the rolling resistance. Substituting FC in Eq. 2.6c with FC = 3π γ O Ri j , we have: 3/2

R Mcrit = 12π γ O Ri j aˆ i j θcrit Ri j .   

(2.29)

rolling stiffness kr

In some particular cases, where the friction coefficient is small enough (usually S R smaller than 0.05) to yield Fcrit < Mcrit /r p , irreversible sliding will be triggered before rolling [26, 27]. In such conditions, one should calculate the critical sliding S in Eq. 2.6a using the original value of particle properties, i.e., forces Fcrit

36

2 A Fast Discrete Element Method for Adhesive Particles

      3/2 S Fcrit = μ · (3π γ O Ri j ) · 4 aˆ i3j − aˆ i j + 2 ,

(2.30)

where the critical pull-off force, FC , in Eq. 2.6a is again calculated using the original value of the particle surface energy FC = 3π γ O Ri j .

2.5 Determination of Parameters in Adhesive DEM 2.5.1 An Inversion Procedure to Set Parameters in Adhesive DEM A principle for setting parameters in the framework of fast adhesive DEM with reduced stiffness is proposed in this section. Based on Eq. 2.26, the parameters to be determined include the damping coefficient α, a reduced particle Young’s modulus E R and a reduced surface energy γ . Other parameters in Eq. 2.26 can be easily determined from the direct measurement (particle density ρ and radius r ) or are regarded as an input parameter for simulations (such as the initial velocity v0 ). In Eq. 2.26, α and 1/A∗ are the only parameters that affect the result (i.e., the coefficient of restitution e). The contour plot in Fig. 2.12 shows the value of restitution coefficient e as a function of the damping coefficient α and 1/A∗ . We use 1/A∗ instead of A∗ because 1/A∗ scales as 1/A∗ ∼ v0 and the initial collision velocity v0 is usually a well-controlled parameter in experiments. Several interesting features can be observed: (1) there is a sticking region (e = 0) when both α and A∗ are large values; (2) with a large value of 1/A∗ , e has a weak dependence on 1/A∗ and is mainly determined by the dissipation coefficient α. For instance, at 1/A∗ > 40, the contour lines with e = 0.6 and e = 0.8 are nearly parallel to the abscissa axis. For any given e, the contour lines approximately follow an exponential form. Based on this observation, we assume an exponential relationship between α and 1/A∗ with the fitting parameters ε, ω and α∞ determined by e.  ω α = α∞ − εexp − ∗ . A

(2.31)

We fit the contour lines for given values of e in Fig. 2.12 using Eq. 2.31 with e varying from ∼ 0 to 0.9. The fitting parameters ε, ω, and α∞ are all inversely calculated from e through three-order polynomial fittings (as shown in Fig. 2.13). ε(e) = −0.2302e3 + 0.9806e2 − 2.026e + 1.294,

(2.32a)

ω(e) = −0.1504e + 0.110e + 0.05783e + 0.04534,

(2.32b)

3

2

α∞ (e) = −0.3325e + 1.279e − 2.094e + 1.157. 3

2

(2.32c)

Based on Eqs. 2.31 and 2.32, an inversion procedure to determine the value of α and A∗ in DEM from the experimental data is proposed as:

2.5 Determination of Parameters in Adhesive DEM

37

Fig. 2.12 Coefficient of restitution e as a function of the damping coefficient α and the inverse of the parameter A∗ . The value of e is indicated by the color scale with red contour lines. The dashed red line separates the sticking region (e = 0) and the rebound region (e > 0). Reprinted from Chen et al. [22], with permission from Elsevier

1. Use Eq. 2.32c to determine α(= α∞ ) at high-velocity state ( A∗−1 → ∞) according to the value of e. 2. Pick another typical point on e − v curve, (vt , et ), and calculate corresponding values of εt , ωt and α∞,t through Eq. 2.32. 3. Using the values of α obtained from step (1) and the parameters εt , ωt and α∞,t from step (2), calculate A∗t (α; εt , ωt , α∞,t ) inversely from Eq. 2.31: A∗t = −ωt ln−1 ( α∞,tεt−α ). 4. Choose reduced Young’s modulus E R and surface energy γ R , which are usually −1/3 5/6 much smaller than their original values E O and γ O , and make sure E R γ R = A∗t ρ 1/2 vt r 5/6 (see Eq. 2.27). Step (1) is extended from the e − α relationship in non-adhesive collision cases, where e is almost a constant that is determined by the damping coefficient α. Therefore, for nonadhesive particles, one can calculate the damping coefficient α inversely from e. In Fig. 2.13c, we plot such a correlation: α = 1.2728 − 4.2783e + 11.087e2 − 22.348e3 + 27.467e4 − 18.022e5 + 4.8218e6 , which is proposed by [11]. When an adhesive particle collides with a wall, e is zero if v0 is smaller than the critical sticking velocity vc . As v0 increases, e will first increase ∂e ∂e ∂(1/A and then enter a plateau, corresponding to the region ∂α ∗ ) in Fig. 2.12. In the high-velocity state, the amount of energy dissipated due to the viscoelasticity is much larger than that of first-contact loss (i.e., the necking effect). As a result, the function e(α, A∗ ) is reduced to a single-parameter function e∞ (α) and we relate α to e∞ through Eq. 2.32c (Step (1)). As shown in Fig. 2.13c, there is only a slight difference between the α − e curve calculated from adhesive DEM and that from Hertz model. Recall that A∗ = H(E, γ , ρ, v0 ) is a function of particle properties and the velocity. One may expect to determine A∗ according to the real physical properties of the particle and further predict the restitution coefficient e. However, these parameters are usually not readily available. For example, the surface energy γ is strongly affected by the surface roughness and the ambient humidity and is usually hard to determine. In addition, to accelerate the computation, a reduced Young’s modulus instead of

38

2 A Fast Discrete Element Method for Adhesive Particles

Fig. 2.13 Fitting parameters ε, ω, and α∞ in Eq. 2.31 as functions of the restitution coefficient e (data points). Dashed lines are the three-order polynomial fittings of Eq. 2.32. Solid line in (c) is the relationship between α and e in [11], which is derived for non-adhesive particles based on Hertz model. Reprinted from Chen et al. [22], with permission from Elsevier

its true value is often needed. From step (2) to (4), we suggest one to alternatively select another typical point on a prescribed e − v curve (vt , et ) that is outside the high-velocity region and use Eq. 2.31 to obtain the corresponding A∗t (et , α) and to −1/3 5/6 −1/3 5/6 further get the value of E R γ R through E R γ R = A∗t ρ 1/2 vt r 5/6 , which can reproduce the prescribed e − v curve. In Fig. 2.14 we present an example of the inversion procedure based on the experimental data in literature [28]: (a) Use Eq. 2.32c and the coefficient of restitution in the high-velocity region, e = 0.96, to obtain α∞ = 0.0321; (b) Pick a typical point (vt , et ) on e − v curve—here we use the point (2.454, 0.848), indicated by the triangle in Fig. 2.14—and then calculate the fitting parameters εt , ωt , and α∞,t at et = 0.848, then (c) solve Eq. 2.31 to obtain A∗t = 0.11. (d) Determine the value −1/3 5/6 of E −1/3 γ 5/6 through: E R γ R = A∗t ρ 1/2 vt r 5/6 = 5.92 × 10−5 N1/2 m−1/6 . The value obtained in this way is quite close to the value calculated using physical prop−1/3 5/6 erties of polystyrene particles (PSL): E O γ O = (3.8 GPa)−1/3 (0.05 J m−2 )5/6 = −5 1/2 −1/6 [15]. At last, pick a reduced Young’s modulus E R and calcu5.28 × 10 N m

2.5 Determination of Parameters in Adhesive DEM

39

Fig. 2.14 Comparison of the coefficient of restitution e calculated by JKR-based DEM with the experimental measurements from [28]. Parameters used in DEM are determined through the proposed inversion procedure. The red triangle stands for the typical point at v0 = 2.454 and e = 0.848. Reprinted from Chen et al. [22], with permission from Elsevier

late the modified surface energy γ R . As shown in Fig. 2.14, the e − v curve calculated −1/3 5/6 from E R γ R well reproduces the experimental measurements [28]. A large number of research studies have reported experimental results of e − v curves [29–33], and the proposed inversion procedure can assist the selection of contact parameters before large-scale DEM simulations.

2.5.2 Comparison Between Experimental and DEM Results We compare the results of particle-wall collisions from DEM simulations with those measured in experiments. It should be noted that the plastic deformation and the breakage of particles that result from high-velocity impacts are not covered in the current work. We thus select experimental data in the low or the moderate velocity regimes for comparison. The parameters in DEM simulations are determined inversely from selected data points (i.e., e∞ and (vt , et )) in experiments and the −1

5

combined parameter E R 3 γ R6 is then calculated. The radius and density of particles are the same as those in experiments. Tables 2.2 and 2.3 summarize the experimental conditions and the DEM parameters, respectively. We plot the coefficient of restitution e as a function of the impact velocity v0 in Fig. 2.15. The results from DEM calculation well reproduce the experimental data for different particle materials and sizes, indicating that the DEM model in the current work is able to account for the complicated interactions between colliding particles. Besides, comparing the e − v0 curves for different experimental conditions, we summarize the following rules for collisions between a micron particle and a wall: 1. Due to the adhesive contact forces, the particle will stick to the wall when the impact velocity is smaller than the critical sticking velocity; 2. The restitution coefficient quickly increases with impact velocity v0 once v0 is larger than the critical value; 3. As v0 further increases, the restitution coefficient enters a plateau and does not obviously vary with v0 .

40

2 A Fast Discrete Element Method for Adhesive Particles

Table 2.2 Parameters in experiments of particle-wall collisions Source Particle materials Wall materials

Dahneke [28] Kim and Dunn [32] Li et al. [31] Dong et al. [34] Dong et al. [34] Wall et al. [29] Wall et al. [29] Wall et al. [29]

PSL Ag-coated glass Stainless steel Ash from Zhundong Coal Ash from Fushun Coal Ammonium fluorescein particle Ammonium fluorescein particle Ammonium fluorescein particle

Particle radius r p (µm)

Particle density ρ (kg/m3 )

Quartz (polished) Silica Quartz (polished) Stainless steel Stainless steel Silica

1.27 40 55 7.0 7.0 2.58

1026 1350 7920 2953 2680 1350

Silica

3.44

1350

Silica

4.90

1350

Table 2.3 Parameters in DEM simulation determined inversely from experimental data  1  1 −1 5 Experimental data e∞ (vt , et ) E R 3 γ R6 N 2 m− 6 Dahneke [28] Kim and Dunn [32] Li et al. [31] Dong et al. [34] Dong et al. [34] Wall et al. [29] Wall et al. [29] Wall et al. [29]

0.96 0.86 0.7 0.37 0.43 0.76 0.79 0.83

(2.45, 0.85) (0.09, 0.26) (0.39, 0.45) (1.10, 0.07) (1.08, 0.11) (5.18, 0.38) (4.03, 0.44) (2.17, 0.37)

5.92 × 10−5 1.45 × 10−4 9.28 × 10−4 1.70 × 10−4 1.76 × 10−4 5.52 × 10−4 5.57 × 10−4 5.07 × 10−4

2.6 Test on Packing Problem To check if the fast adhesive DEM can reproduce the results from simulations with the original particle properties in cases associated with aggregates. We run a large number of simulations on the packing problem. As shown in Fig. 2.16, we consider the ballistic falling of N (= 2000) particles. Particles have radius r p and initial velocity U 0 (= (U0 , 0, 0)) and are randomly added into the computational domain from an inlet plane at height L x (= 160r p ). Periodic boundary conditions are set along y and z directions with box length L y = L z = 28r p . The physical parameters used in our simulations are set according to the properties of polystyrene (PS) particle in [15], which has the density ρ = 1000 kg/m3 , Young’s modulus E O = 3.8 × 109 Pa, surface energy γ O = 0.05 J/m2 , friction coefficient μ f = 0.3, and the critical rolling angle θcrit = 0.0085. A vacuum condition is assumed to filter out the fluid effect. Gravity effect can be neglected since the Froude number, Fr = U0 /(gL x )1/2 of

2.6 Test on Packing Problem

41

Fig. 2.15 The coefficient of restitution e as a function of the impact velocity v0 . Scatters are experimental results and curves are DEM simulations. Red triangles indicate the typical points (vt , et )

our system satisfies Fr 1. This ballistic packing system has been widely used in both experimental [35, 36] and numerical [18, 26, 27, 37–39] studies and has been proved to be useful to bridge the gap between the particle-level interactions and the macroscopic structure of aggregates [40]. To understand how to simulate the packing process in the framework of fast adhesive DEM with reduced stiffness, we set four series of computational experiments (listed in Table 2.4): in the cases of series S, we use the original value of the elastic modulus E O and surface energy γ O and the results can be regarded as a benchmark case; in series A, reduced elastic modulus E R is used without modification of the surface energy; in series B, we use the same elastic modulus as those in A and modify the surface energy according to γ R = χ 2/5 γ O ; series C is essentially the same as series B except that the rolling stiffness is calculated based on the original sur3/2 face energy, i.e., kr = 12π γ O Ri j aˆ i j . For each case, at least ten configurations are obtained to provide a meaningful average and standard deviation. According to the analysis in Sect. 2.4, the packing structure is determined by three parameters: the damping coefficient α, which is fixed here, the dimensionless adhesive parameter Ad, and the elasticity parameter El. To separately tune the value of Ad at given El, we fixed the velocity U0 and varied the particle size r p in our simulation.

42

2 A Fast Discrete Element Method for Adhesive Particles

Fig. 2.16 Schematic of simulation setup. Reprinted from Chen et al. [22], with permission from Elsevier

Table 2.4 Parameters used in DEM simulations of microparticle packings Parameters E (Pa) γ (J/m2 ) kr El

Ad

3/2 12π γ O Ri j aˆ i j 3/2 12π γ O Ri j aˆ i j

1.69 × 106

0.1 − 33

4.44 × 105 2.22 × 105 4.44 × 104

Ad S

3/2

4.44 × 105 2.22 × 105 4.44 × 104

Ad S · χ 2/5

3/2

4.44 × 105 2.22 × 105 4.44 × 104

Ad S · χ 2/5

S

3.8 × 109

0.05

A-1 A-2 A-3

1.0 × 109 5.0 × 108 1.0 × 108

0.05

B-1 B-2 B-3

1.0 × 109 5.0 × 108 1.0 × 108

0.0293 0.0222 0.0117

12π γ R Ri j aˆ i j

C-1 C-2 C-3

1.0 × 109 5.0 × 108 1.0 × 108

0.0293 0.0222 0.0117

12π γ O Ri j aˆ i j

The parameters used in case S are the same as those in the experiments [15] and are regarded as original particle properties. In series A, three reduced particle moduli are used without modification of the surface energy. In series B, the surface energy is modified according to γ R = χ 2/5 γ O . In case C, we modified the surface energy in the same fashion as in case B and use the original surface 3/2 energy γ O to calculate the rolling stiffness, i.e. kr = 12π γ O Ri j aˆ i j . Reprinted from Chen et al. [22], with permission from Elsevier

2.6 Test on Packing Problem

43

2.6.1 Packing Fraction and Coordination Number Figure 2.17a–c show the variation of the packing fraction φ and the mean coordination number Z as functions of adhesion parameter Ad for series A, B and C, respectively. To avoid the wall effect, both φ and Z are calculated from the middle part of the packing (0.15h ≤ x p ≤ 0.85h, with h being the packing height). The blue circles in the three panels are data for cases S. From Fig. 2.17a, one can draw the conclusion that the reduction of the particle stiffness obviously decreases the packing fraction. This effect is more prominent in the range of moderate Ad. With a low adhesion number Ad(< 0.2) and a high Ad(> 10), the packing fraction converges to the random close packing limit (RCP) and the adhesive loose packing limit (ALP), respectively, [18, 27] and the difference in φ due to stiffness is substantially prevented. In contrast to φ, the coordination number Z only has a weak dependence on particle stiffness. This interesting phenomenon can be understood through an analysis of the mechanical equilibrium of the packing. For a given contact network of a packing, a mechanical stable condition is achieved with force- and torque-balance on all particles under the constraint F < Fcrit (F is FiSj , MiRj or MiTj ). According to Eqs. 2.6a and 2.6c, the critical value Fcrit is independent of the particle stiffness. A packing of harder particles can have each particle balanced at a similar coordination number as the packing of softer particles. However, before the final mechanical equilibrium is achieved, the kinetic energy of particles needs to be dissipated. A softer particle has a better capability of energy dissipation and is more likely to stick onto packed particles upon collisions, limiting its movement along the x direction. In contrast, a particle with higher stiffness needs more collisions to be captured, which leads to a compaction of the packing. When the surface energy is modified according to Eq. 2.28, both the packing fraction and the coordination number increase for particles with reduced stiffness and the case with higher reduced ratio χ tends to have a denser structure. Note that, in Fig. 2.17b and c, we choose to use A∗ (= El −1/3 Ad 5/6 ) instead of Ad as the abscissa, because the modification of surface energy will shift the data points in φ − Ad plane. The difference in φ due to the reduction of the stiffness is, to some extent, balanced by the modification of the surface energy. However, there still remains considerable discrepancy. This discrepancy again can be attributed to the mechanical equilibrium: a reduction of surface energy causes the decrease of the critical value of rolling resistances MiRj,crit , which practically puts stricter constraints on the forceand torque-balance of particles. A packing with smaller MiRj,crit generally needs more contacts to achieve mechanical equilibrium. These results, combined with the e − v curves in Fig. 2.10, indicate that an exactly same particle-particle normal collision behavior does not ensure the same results of packing structure. In the last case, we modify the surface energy when calculating the normal forces but maintain the original value of the rolling stiffness kr . As shown in Fig. 2.17c, the packings simulated with reduced stiffness well reproduce the structure, both φ and Z , of the original packings. This result confirms our statement that the critical value of the rolling resistance strongly affects the mechanical equilibrium of a packing. In

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2 A Fast Discrete Element Method for Adhesive Particles

Fig. 2.17 a Packing fraction φ and coordination number Z as functions of adhesion parameter Ad for packings with El = 1.69 × 106 (case S, circles), 4.44 × 105 (case A-1, squares), 2.22 × 105 (case A-2, diamonds) and 4.44 × 104 (case A-3, triangles). b φ and Z as functions of parameter 1 5 A∗ (= El− 3 Ad 6 ) for packings in series B, modified surface energy γ R = χ 2/5 γ O are used and the rolling stiffness is calculated as 3/2 kr = 12π γ R Ri j aˆ i j . c φ and Z as functions of parameter A∗ for packings in series C, modified surface energy are used and the rolling stiffness is calculated as 3/2 kr = 12π γ O Ri j aˆ i j . Reprinted from Chen et al. [22], with permission from Elsevier

the framework of adhesive DEM with reduced stiffness, similarities in both particleparticle collision behavior and mechanical constraints are necessary to simulate a packing process. The friction coefficient μ is kept unchanged during the entire simulation since the value (μ = 0.3) we use is large enough to ensure that the rolling is the dominant mode of deformation of the packing. If the particles have a small friction coefficient, which is usually smaller than 0.05, the sliding motion between contact particles will become non-negligible [26, 27], and one should modify the S according to Eq. 2.30. critical value of Fcrit

2.6 Test on Packing Problem

45

2.6.2 Local Structure of Packings To further validate the fast adhesive DEM, we do statistics of the local structure of each particle inside a packing. We calculate the local packing fraction of each particle, which is expressed as φlocal,i =

Vp , Vvor,i

(2.33)

where V p is the volume of a particle and Vvor,i is the volume of its Voronoi cell. Figure 2.18 shows the probability density distributions (PDFs) of φlocal,i and coordination number Z of each particle for the case S, C-2, and C-3 at A∗ = 0.035. We choose this value because it locates in the transition region between RCP and ALP and packings in this region are more sensitive to the particle stiffness. There is a very good agreement between the PDFs obtained from original packings and from packings with reduced stiffness.

Fig. 2.18 Distribution of the local volume fraction φlocal and the coordination number Z of each particle. For each reduced ratio χ, the PDF is averaged over 10 packings. Reprinted from Chen et al. [22], with permission from Elsevier

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2 A Fast Discrete Element Method for Adhesive Particles

2.6.3 Interparticle Overlaps and Normal Forces One of the most important properties that needs to be checked is the interparticle overlap, which usually puts a restriction on reducing the particle stiffness. Interparticle overlap significantly affects the heat or charge transfer between heated particles or charged particles [41–43]. However, there is no universal criterion for choosing a limit of interparticle overlap. For example, it has been pointed out that the flow pattern on a bumpy inclined chute was not sensitive to the particle stiffness when the interparticle overlap is smaller than 1% of the particle diameter [44]. In a measurement of the angle of repose, an overlap smaller than 0.34% was suggested to retain the results [24]. Based on a broad review of different simulation tasks, [45] argued that, when the particle overlap is kept smaller than 1% of the particle diameter, there would be no major change in the simulation results. In Fig. 2.19, we show the distributions of interparticle overlaps for Case S, C-1, C-2, and C-3 (corresponding to χ = 1, 0.26, 0.132, and 0.026) at A∗ = 0.035. Two extra reduced ratios, χ = 0.053 and 0.034, are also added. It is easy to understand that, as particle stiffness decreases, the distributions move to larger δ N . The interparticle overlaps are almost symmetrically distributed around the equilibrium value δ0 (indicated by the dashed vertical lines), which results from the balance between van der Waals attraction and the elastic repulsion. δ0 is calculated as: 1

δ0 = 3.094γ 3 Ri3j E − 3 . 2

2

(2.34)

The symmetry in the distributions of interparticle overlap and normal force is a key feature of a static packing of strong adhesive particles [26]. In Fig. 2.19, the values of δ O increases from δ0 /r p = 0.08% at χ = 1 to δ0 /r p = 0.34% at χ = 0.026, which is still within the range suggested in [45]. From Eq. 2.34, one can easily evaluate the effect of reduction of stiffness on interparticle overlaps

Fig. 2.19 Distribution of the scaled interparticle overlaps δ N /r p . Curves from left to right, correspond to packings with χ = 1, 0.26, 0.13, 0.053, 0.034, and 0.026, respectively. For each reduced ratio χ, the PDF is averaged over 10 packings. The dashed lines indicate overlaps in the equilibrium state (δ0 /r p ). Reprinted from Chen et al. [22], with permission from Elsevier

2.6 Test on Packing Problem

47

Fig. 2.20 Distribution of the scaled normal force F N /|F N | for packings with χ = 1 (circles), 0.26 (squares), 0.13 (diamonds), 0.053 (triangles), 0.034 (axes), and 0.026 (pluses). |F N | is the mean value of the magnitude of the normal force. Dashed lines are guides for the eye. Reprinted from Chen et al. [22], with permission from Elsevier

δ N ,R = χ −2/5 δ N ,O .

(2.35)

This scaling allows users to determine a feasible amount of stiffness reduction once the constraint is put on the interparticle overlap. It is also of great interest to know the force distribution in packings, especially, in loose packings with adhesive particles. Here, we measure the normal force of each contact in the same packings as those in Fig. 2.19. As displayed in Fig. 2.20, the forces could be both attractive (negative F N ) and repulsive (positive F N ). After normalizing F N in each case with the corresponding mean value of its magnitude, |F N |, distributions with different χ nicely collapse onto a single curve. The normalized distributions are almost symmetrical around F N /|F N | = 0, which is in good agreement with previous results on the packing of strong adhesive particles [26]. The results again verify that the fast adhesive DEM with reduced particle stiffness can retain both the structural and mechanical properties of the contact network in a packing. At last, we report the timing results for the simulation of packings in Fig. 2.19. The computational time is measured on a computing node with 20-core Intel (R) Xeon (R) E5-2660 V3 running at 2.60 GHz and 128 GB memory. The results in Fig. 2.21 indicate that reducing the particle stiffness by 1 or 2 orders of magnitude can shorten 5 times the computation time, however, further reduction in χ does not guarantee an obvious speedup. Combining the timing results and the scaling of interparticle overlap, we suggest that a reduction of stiffness by 1 − 2 orders of magnitude can remarkably accelerate the simulation and retain both micro- and macroscopic properties of a static packing of adhesive particles.

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2 A Fast Discrete Element Method for Adhesive Particles

Fig. 2.21 Timing results (in seconds) for N = 2000 packings with the reduced ratio χ(= E R /E O ) of particle stiffness. Each data point is averaged over 10 runs. Reprinted from Chen et al. [22], with permission from Elsevier

2.7 Summary This chapter investigates the dynamics of three-dimensional collisions between two microparticles using adhesive discrete element method. The energy dissipation pathways are analyzed and the effect of particle size and impact angles on the critical sticking  velocity is discussed. An explicit formula in the form of ˆ r p , β VC N ,0 is put forward as a sticking/rebound criterion for colliVC N = f (b)φ sions of micron-sized particles covering different impact angles, particle sizes and size ratios. Based on the dimensionless equation describing the collision between a particle and a wall, we have been able to propose a scaling relationship to reduce the particle’s stiffness (i.e., particle’s Young’s modulus) and the surface energy simultaneously. It allows one to use larger time steps to resolve the collision and ensure that the results stay the same. A novel inversion method, which helps users to set the damping coefficient, the particle stiffness, and the surface energy to reproduce a prescribed e − v curve is also presented. This inversion method is different from previous calibration approaches, in which an iterative procedure is normally used and the parameters are tuned to match the bulk response of the material to measured results [46]. Compared with these calibration approaches, our approach uses practical formulas for a direct calculation avoiding complicated iteration process. Indeed, one can also determine the parameters based on a direct measurement of them at particle or contact level. However, experimental measurements are usually limited by particle sizes, and parameters like damping coefficient cannot be directly measured. We suggest that the proposed inversion method should be used in combination with direct measuring approach. Parameters such as particle size and density are usually measured directly from experiments.

References

49

With a simple but indispensable modification of the rolling and sliding resistances, the accelerated JKR-based DEM can be feasibly applied to simulations of static packings of adhesive particles. Structural proprieties, including the overall packing fraction, the averaged coordination number, and the distributions of local packing fraction and coordination number of each particle, are in good agreement with the packings simulated using the original parameters.

References 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30. 31. 32. 33. 34. 35. 36. 37. 38. 39.

C. Dominik, A.G.G.M. Tielens, Astrophys. J. 480(2), 647 (1997) A.G. Konstandopoulos, Powder Technol. 109(1–3), 262 (2000) G.I. Bell, Science 200(4342), 618 (1978) F. Brauer, C. Dullemond, T. Henning, Astron. Astrophys. 480(3), 859 (2008) D. Paszun, C. Dominik, Astron. Astrophys. 507(2), 1023 (2009) C. Dominik, A. Tielens, Philos. Mag. A 72(3), 783 (1995) Y. Gu, A. Ozel, S. Sundaresan, Powder Technol. 296, 17 (2016) T. Kobayashi, T. Tanaka, N. Shimada, T. Kawaguchi, Powder Technol. 248, 143 (2013) K. Johnson, K. Kendall, A. Roberts, Proc. R. Soc. Lond. A Math. Phys. Sci. 324(1558), 301 (1971) S. Li, J.S. Marshall, J. Aerosol Sci. 38, 1031 (2007) J.S. Marshall, J. Comput. Phys. 228(5), 1541 (2009) J.S. Marshall, S. Li, Adhesive Particle Flow: A Discrete-element Approach (Cambridge University Press, Cambridge, 2014) A. Chokshi, A. Tielens, D. Hollenbach, Astrophys. J. 407, 806 (1993) Y. Tsuji, T. Tanaka, T. Ishida, Powder Technol. 71(3), 239 (1992) B. Sümer, M. Sitti, J. Adhes. Sci. Technol. 22(5–6), 481 (2008) G. Liu, S. Li, Q. Yao, Powder Technol. 207(1–3), 215 (2011) M. Yang, S. Li, Q. Yao, Powder Technol. 248, 44 (2013) W. Liu, S. Li, A. Baule, H.A. Makse, Soft Matter 11(32), 6492 (2015) W. Liu, C.Y. Wu, AIChE J. e16974 (2020) S. Chen, S. Li, M. Yang, Powder Technol. 274, 431 (2015) C. Thornton, Z. Ning, Powder Technol. 99(2), 154 (1998) S. Chen, W. Liu, S. Li, Chem. Eng. Sci. 193, 336 (2019) R. Moreno-Atanasio, B. Xu, M. Ghadiri, Chem. Eng. Sci. 62(1–2), 184 (2007) S. Lommen, D. Schott, G. Lodewijks, Particuology 12, 107 (2014) P. Liu, C.Q. LaMarche, K.M. Kellogg, C.M. Hrenya, Chem. Eng. Sci. 145, 266 (2016) W. Liu, S. Li, S. Chen, Powder Technol. 302, 414 (2016) W. Liu, Y. Jin, S. Chen, H.A. Makse, S. Li, Soft Matter 13(2), 421 (2017) B. Dahneke, J. Colloid Interface Sci. 51(1), 58 (1975) S. Wall, W. John, H.C. Wang, S.L. Goren, Aerosol Sci. Technol. 12(4), 926 (1990) P.F. Dunn, R.M. Brach, M.J. Caylor, Aerosol Sci. Technol. 23(1), 80 (1995) X. Li, P. Dunn, R. Brach, J. Aerosol Sci. 30(4), 439 (1999) O. Kim, P. Dunn, J. Aerosol Sci. 39(4), 373 (2008) C. Sorace, M. Louge, M. Crozier, V. Law, Mech. Res. Commun. 36(3), 364 (2009) M. Dong, X. Li, Y. Mei, S. Li, J. Aerosol Sci. 117, 85 (2018) J. Blum, R. Schräpler, Phys. Rev. Lett. 93(11), 115503 (2004) E.J. Parteli, J. Schmidt, C. Blümel, K.E. Wirth, W. Peukert, T. Pöschel, Sci. Rep. 4, 6227 (2014) R.Y. Yang, R.P. Zou, A.B. Yu, Phys. Rev. E 62(3), 3900 (2000) S. Yang, K. Dong, R. Zou, A. Yu, J. Guo, Granular Matter 15(4), 467 (2013) W. Liu, S. Chen, S. Li, AIChE J. 63(10), 4296 (2017)

50 40. 41. 42. 43. 44. 45. 46.

2 A Fast Discrete Element Method for Adhesive Particles A. Baule, F. Morone, H.J. Herrmann, H.A. Makse, Rev. Mod. Phys. 90(1), 015006 (2018) G. Batchelor, R. O’Brien, Proc. R. Soc. Lond. Ser. A Math. Phys. Sci. 313–333 (1977) P. Moysey, M. Thompson, Powder Technol. 153(2), 95 (2005) X. Jin, J.S. Marshall, J. Electrostat. 87, 217 (2017) D.M. Hanes, O.R. Walton, Powder Technol. 109(1–3), 133 (2000) M. Paulick, M. Morgeneyer, A. Kwade, Powder Technol. 283, 66 (2015) C. Coetzee, Powder Technol. 310, 104 (2017)

Chapter 3

Agglomeration of Microparticles in Homogenous Isotropic Turbulence

3.1 Introduction For particles in turbulence, the collision kernel is usually expressed as the production of the mean relative radial velocity and the radial distribution functions (RDFs) of particle pairs at the distance of contact. For zero-inertial particles, these two quantities can be statistically determined from those of turbulence flows [1]. In contrast, inertial particles preferentially sample certain regions of the flow due to the centrifugation effect, giving rise to higher values of both relative radial velocity and spatial concentration [2–6]. As the inertia of particles further increases, particles from different regions of the flow come together. A larger relative velocity, consequently a larger collision rate, is then observed. Such an effect is termed as “caustics” [7, 8] or “sling effect” [9]. Given the models of geometric collision rate, Smoluchowski’s theory can be used to describe the growth of clusters assuming that colliding particles merge immediately to form new larger spherical particles. As we mentioned in Sect. 1.3, the assumption of unity coagulation efficiency is normally valid for droplets. However, it is not applicable to the agglomeration of solid non-coalescing adhesive particles. Such systems are quite ubiquitous, ranging from electrostatic agglomerators [10], flocculation during water treatment [11], assemblage of preplanetary grains [12] to the growth of dendrites during aerosol filtration [13]. Constructing a kernel function that can reflect the influence of complicated inter-particle interactions is a crucial problem that has not been settled. Moreover, recent work has revealed that a stronger clustering effect gives rise to a higher collision velocity, which increases the breakage rate of agglomerates [14]. The collision-induced breakage is usually important for gas-solid systems containing small but heavy particles (with high Stokes numbers). The competition between agglomeration and deagglomeration provides an explanation for the saturation of agglomeration levels in these gas-solid systems. However, the formulation of the collision-induced breakage rate is still far from perfect.

© Tsinghua University Press 2023 S. Chen, Microparticle Dynamics in Electrostatic and Flow Fields, Springer Theses, https://doi.org/10.1007/978-981-16-0843-8_3

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3 Agglomeration of Microparticles in Homogenous Isotropic Turbulence

In this chapter, we investigate the agglomeration and the collision-induced deagglomeration of solid adhesive particles in homogeneous isotropic turbulence (HIT). Direct numerical simulation (DNS) and the fast adhesive DEM are employed to fully resolve the turbulent flow and the translational and rotational motions of all particles. We first introduce how to identify various events, including the sticking event, the rebound event, the collision-induced breakage, and the shear-induced breakage of agglomerates, in simulations and then try to quantify the possibility of the occurrence of these events. An agglomeration kernel is then constructed containing the information of agglomerate structures and the sticking probability. The collision-induced breakage rate is also formulated based on the Smoluchowski equation and a breakage fraction. Moreover, the role of the flow structure on the collision-induced breakage is also discussed.

3.2 Methods 3.2.1 Fluid Phase Calculation To investigate the agglomeration of microparticles, we consider non-Brownian solid particles suspended in an incompressible isotropic turbulent flow, which is calculated by direct numerical simulation (DNS) on a cubic, triply periodic domain with 1283 grid points. A pseudospectral method with second-order Adams-Bashforth time stepping is applied to solve the continuity and momentum equations ∇ · u = 0, 1 ∂u + (u · ∇)u = − ∇ p + ν∇ 2 u + f F + f P . ∂t ρf

(3.1a) (3.1b)

Here, u is the fluid velocity, p is the pressure, ρ f is the fluid density, and ν is the fluid kinematic viscosity. The small wavenumber forcing term f F is used to maintain the turbulence with approximately constant kinetic energy. f P is the particle body force, which is calculated at each Cartesian grid node i using f p (x i ) =   N − n=1 F nF δh x i − X p,n . Here, x i is the location of grid node i, F nF is the fluid force on particle n located at X p,n and δh x i − X p,n is a regularized delta function, which is given by  n bi   if x i ∈ N B (3.2) δh x i − X p,n = Ng Nb 0 if x i ∈ / NB Here, N B is the set consisting of Nb = 27 adjoining grid cells (three in each direction) and each cell has N g = 8 grid nodes. The cell containing the particle locates in the center of the 27 cells. For a given grid node xi , the number of cells in the set N B that contains this specific node is recorded as n bi . For the nodes defining the center cell in

3.2 Methods

53

N B (i.e., the cell containing the particle), each of the nodes is shared by eight different nodes receive the same cells in N B , therefore, n bi = 8. It indicates that the eight load. The summation of δh over all grid nodes is unity, i.e., x i δh x i − X p,n = 1, indicating that the choice of the delta function is conservative in force. All the parameters in the simulation have been non-dimensionalized by typical length, velocity and mass scales that are relevant to the agglomeration of microparticles. Specifically, the typical length scale is L 0 = 100r p = 0.001 m, where r p = 10 µm is the particle radius. The box size for the simulation is set as 2π L 0 . The velocity scale is set as U0 = 10 m/s which is the typical value for the gas flow in a turbulent-mixing agglomerator [10]. The typical mass is M0 = ρ f L 30 = 10−9 kg, where ρ f = 1 kg/m3 is the fluid density. The typical time scale is given by T0 = L 0 /U0 = 10−4 s. Other dimensional input parameters are the fluid viscosity μ = 1.0 × 10−5 Pa · s, the particle density ρ p = 10 − 320 kg/m3 , and the particle surface energy γ = 0.01 − 5 J/m2 . Hereinafter, all the variables appear in their dimensionless form and, for simplicity, the same notations as the dimensional variables are used. One could obtain the physical values of the dimensionless variables by multiplying the dimensionless values with the typical scales.

3.2.2 Equation of Motion for Solid Particles The fast adhesive DEM (described in Chap. 2) is employed to track the dynamics of every individual particle. We integrate the linear and angular momentum equations of particles m i v˙ i = F iF + F iC , ˙ i = M iF + M iC . Ii 

(3.3a) (3.3b)

where m i and Ii are mass and moment of inertia of particle i and v i and i are the translational velocity and the rotation rate of the particle. The forces and torques are induced by both the fluid flow (F iF and M iF ) and the interparticle contact (F iC and M iC ). In this work, the dominant fluid force/torque  is theStokes drag given by F drag = −3π μd p (v − u) f and M drag = −π μd 3p  − 21 ω , where u and ω are velocity and vorticity of the fluid, μ is the fluid viscosity, and d p is the particle diameter. Each particle in the flow is surrounded by other particles and the presence of surrounding particles will influence the drag force for any given particle. The friction factor f , given by Di Felice [15], is used to correct for the crowding of particles. For particle Reynolds number in the range 0.01 to 104 , f can be written as   2 1 . f = (1 − φ)1−ζ , ζ = 3.7 − 0.65 exp − 1.5 − ln Re p 2

(3.4)

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3 Agglomeration of Microparticles in Homogenous Isotropic Turbulence

Here, φ is the local particle volume fraction which is determined by the concentration blob method [16, 17] and Re p is the particle Reynolds number, which is defined as Re p = d p |v − u|/ν. In addition to the Stokes drag, we also include the Saffman and Magnus lift forces in F iF [18, 19]. Two approaching particles interact with each other through the fluid squeeze film between them. Such near contact interaction significantly reduces the approach velocity and further influences the collision and agglomeration process. In this work, a viscous damping force derived from the classical lubrication theory is included, given by 3π μr 2p dh . (3.5) Fl = − 2h dt Here, Fl is initiated at a surface separation distance h = h max = 0.01r p and a minimum value of h, h min = 2 × 10−4 r p , is set at the instant of particle contact according to experiments [20, 21]. The maximum value h max = 0.01r p is selected such that the particles are close enough that the lubrication theory is valid. The value of h max is assigned according to previous work on particle-wall collision [20, 22], in which simulation results yield a good fit to the experimental data of the restitution coefficient. The minimum separation distance h min is set to avoid singularity. It is normally accepted that the fluid density and viscosity can increase significantly at a small value of h, making the fluid within the contact region behave in a more solid-like manner and limiting the value of h. Surface roughness also imposes a lower limit on the value of h [23]. The contact mechanics are then activated when h < h min . Setting a small gap between contacting particles has been widely adopted in contact theories (see Ref. [24] and references therein). The hydrodynamic force is then neglected when the two particles are in contact with each other since the contacting forces are normally much larger than the hydrodynamic force. When two particles i and j are in contact at t0 , the normal force F N , the sliding friction F S , the twisting torque M T , and the rolling torque M R acting on particle i from particle j are determined according to Eq. 2.2. For details of the contact forces models, we refer to Chap. 2.

3.2.3 Multiple-time Step Framework The DNS-DEM computational framework is designed with multiple-time steps [16, 25, 26]. The flow field is updated using a fluid time step dt F = 0.005. To correctly identify the inter-particle collisions, a smaller particle convective time step dt P = 2.5 × 10−4 is adopted to update the force, velocity, and position of particles that do not collide with other particles. Such a small dt p ensures that the distance each particle travels during a time step is only a small fraction of the particle or the grid size. In addition, we build a local list at each fluid time step to record the neighboring particles that each particle may collide as it is advected over a fluid time step. Once a particle is found to collide with other particles during a particle

3.2 Methods

55

Fig. 3.1 Multiple-time step framework for the DNS-DEM simulation in this chapter

time step, we then recover its information (i.e., its force, velocity, and position) to the start of this particle time step and instead advect it using a collision time step dtC = 6.25 × 10−6 . The value of dtC is small enough to resolve the rapid variation of the contact forces, velocity, and position of the particles (Fig. 3.1).

3.2.4 Simulation Conditions Monodisperse particles are randomly seeded into the domain after the turbulence reaching the statistically stationary state. The statistical properties of the turbulent flow are fixed. Dimensionless flow parameters include the Taylor Reynolds number Reλ = 93.0, the fluctuating velocity u  = 0.28, the dissipation rate  = 0.0105, the kinematic viscosity ν = 0.001, the Kolmogorov length η = 0.0175, the Kolmogorov time τk = 0.31, and the large-eddy turnover time Te = 7.4. These parameters together with typical scales and particle properties are listed in Table 3.1 in both dimensional and dimensionless forms. Gravity is neglected here. One of the most important parameters governing the clustering of particles is the Kolmogorov-scale Stokes number, St = τp /τk , where τ p = m/(6πr p μ) is the particle response time and τk = (ν/)1/2 is the Kolmogorov time. In the classical theory of turbulent collision of nonadhesive particles, St significantly influences the value of the collision kernel. On the one hand, the turbulent flow brings separate

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3 Agglomeration of Microparticles in Homogenous Isotropic Turbulence

Table 3.1 Physical and dimensionless values of the parameters in the simulation Parameters Physical value Dimensionless value Typical scales Length, L 0 Velocity, U0 Time, T0 Mass, M0 Fluid properties Dynamic viscosity, μ Kinematic viscosity, ν Taylor Reynolds number, Reλ Fluctuating velocity, u  Dissipation rate,  Kolmogorov length, η Kolmogorov time, τk Large-eddy turnover time, Te Integral length , l0 Particle properties Particle number, N Particle radius, r P Average particle volume fraction, φ¯ Particle density, ρ P Surface energy, γ

0.001 m 10 m/s 10−4 s 10−9 kg

1 1 1 1

10−5 Pa · s 10−5 m2 /s – 2.8 m/s 1.05 × 104 m2 /s3 1.75 × 10−5 m 3.1 × 10−5 s 7.4 × 10−4 s 1.03 × 10−3 m

– 0.001 93.0 0.28 0.0105 0.0175 0.31 7.4 1.03

4 × 104 5.0–12.5 µm 8.4 × 10−5 –1.3 × 10−3

– 0.005–0.0125 –

10–320 kg/m3 0.01–5 J/m2

10–32 0.1–50

© Cambridge University Press, reproduced with permission from Chen and Li [35]

particles together to form agglomerates in the presence of adhesion. A sufficiently high collisional impact velocity between particles, on the other hand, gives rise to the breakage of agglomerates (collision-induced breakage, see Fig. 3.2a). The adhesion parameter Ad = γ /(ρ p u 2 r p ), defined as the ratio of interparticle adhesion to particle’s kinetic energy, is normally used to quantify the adhesion effect [16, 25]. The surface energy density γ is determined according to experimental measurements [27, 28] or calculated from the Hamaker coefficients of the materials [16]. For two colliding particles, a modified adhesion number Adn = γ /(ρ p vn2 r p ), which is defined based on the normal impact velocity vn , is often used to predict the post-collision behavior. The determination of Adn requires the information of the normal impact velocity vn , which is usually obtained from the post-processing of the simulation. One can also adopt analytical expressions to model vn (see Refs. [29, 30]) so that the value of Adn can be estimated before the simulation. The parameter Ad (Adn ) has been successfully used to estimate the critical sticking velocity of two colliding particles (see Chap. 2), agglomeration efficiency of particles in turbulence, the aerosol capture efficiency during fiber filtrations [31, 32] and the packing structure of adhesive

3.2 Methods

57

particles [33, 34]. In this work, we systematically vary Ad (Adn ) to show the effect of adhesion on the agglomeration and the collision-induced breakage.

3.2.5 Identification of Collision, Rebound and Breakage Events Figure 3.2a presents a typical collision-induced breakage event from the DNS-DEM simulation, where a doublet containing particles 1 (P1) and 2 (P2) collides with a third particle (P3) and then breaks into two singlets. The evolutions of the interparticle overlap (scaled by the particle radius r p ) between P1 and P2 and that between P2 and P3 are shown in Fig. 3.2b. The vertical dashed lines, from left to right, mark the moment at which the contact between P2 and P3 is formed, the bond between P2 and P3 and that between P1 and P2 break. The contact duration τ of each bond thus can be calculated. For instance, τ23 in Fig. 3.2b indicates the contact duration between P2 and P3. To accurately interpret the breakage mechanism and formulate the agglomeration and breakage rate of agglomerates in turbulence, it is of crucial importance to identify various events in the simulation, including the sticking of particles upon collision, the rebound, the collision-induced breakage and shear-induced breakage of agglomerates. We determine all these events according to the following criteria (a) If the contact duration τ between two colliding particles is smaller than a critical value τC , we regard it as a rebound event. In this case, there is no agglomerate formed by these two colliding particles. A rebound event normally happens when the collisional velocity is large [36, 37]. (b) If the bond between two colliding particles does not break within τC , we regard it as a sticking collision. An agglomerate is then formed (or grows in size) upon the collision. (c) When a breakage of a certain bond, whose contact duration is larger than τC , leads to the fragmentation of an agglomerate, we regard it as a breakage event. For each breakage case, two different breakage mechanisms are further identified: If the broken agglomerate is collided by other particles right before its breakage, we consider the breakage event as a collision-induced breakage, otherwise, the breakage event is regarded as a shear-induced breakage. To determine the value of τC , we plot the probability density function of the contact duration τ for the interparticle bonds in two typical cases in double logarithmic coordinates (see Fig. 3.3). There is an obvious scale separation between the contact duration in rebound events and breakage events. In the current work, the critical value τC = 0.005 (indicated by the vertical dashed line) was chosen to separate the rebound events (τ < τC ) and the breakage events (τ > τC ). The following quantities thus can be recorded in each simulation run: the number of collisions NC ; the number of sticking events N S ; rebound events N R ; and breakage events N B .

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Fig. 3.2 a Trajectories of an agglomerate (doublet) and a particle from DNS-DEM simulation. Here, 1, 2 and 3 are initial positions of the particles; 1 , 2 and 3 are corresponding particles at the collision moment; 1 , 2 and 3 are corresponding particles at the end of trajectories. 82,000 collision time steps are used to resolve the process in panel (a), and the position of the particles at each 2000 time steps is presented by a grey sphere. b Evolution of the interparticle overlap, where the contacting bond between particle 2 and 3 are formed at δ23 = 0 (indicated by the vertical dashed line on the left-hand side) and the bonds between particle 2 and 3 and particle 1 and 2 break at δ23 = −δC and δ12 = −δC , indicated by the vertical dashed lines in the middle and on the right-hand side, respectively. © Cambridge University Press, reproduced with permission from Chen and Li [35]

Fig. 3.3 Scaled probability distribution of the contact duration τ for the interparticle bonds in two typical cases with St = 5.8 and a Ad = 0.64, and b Ad = 6.4. The vertical dashed line indicates the critical value τC = 0.005, which separates the rebound events (τ < τC ) and the breakage events (τ > τC ) © Cambridge University Press, reproduced with permission from Chen and Li [35]

3.2 Methods

59

3.2.6 Smoluchowski’s Theory Before showing the DNS-DEM results, we introduce the Smoluchowski coagulation equation and discuss how to apply the theory to the agglomeration of non-coalescing adhesive particles. In Smoluchowski’s theory, the growth of agglomerates can be described using the population balance equation (PBE) [38] n(A) ˙ =

∞  1 (i, j)n(i)n( j)−n(A) (i, A)n(i), 2i+ j=A i=1

(3.6)

where (i, j) is the averaged rate (kernel) for agglomerates of size i colliding with agglomerates of size j. The collision kernel should reflect all the factors affecting collisions. It is defined as (i, j) ≡ n˙ c,i j /(n(i)n( j)) with n˙ c,i j being the collision rate per unit volume and n(i) being the average number concentration of size group i. The first term on the right-hand side of Eq. 3.6 is the source term that accounts for the rate at which agglomerates of size A are created. The second term is a sink that describes agglomerate disappearance due to its coalescence with other agglomerates. PBE can be readily used to predict the growth of droplets in clouds with an underlying assumption—colliding particles coalesce instantaneously to form larger particles [39]. Therefore, the growth rate of agglomerates is equivalent to the collision rate. The collision between adhesive non-coalescing microparticles, however, does not ensure the growth of an agglomerate. Both sticking and rebound could happen as a result of the competition between the particles’ kinetic energy and the surface energy. Thus, it is natural to introduce a sticking probability, , defined as the ratio of the number of sticking collisions to the total number of collisions. We then have an agglomeration kernel, which reads a (i, j) = (i, j), ∀i, j.

(3.7)

The sticking probability has a minimum value 0 for non-adhesive particle systems and a maximum value 1 for the hit-and-stick case. We can then simulate the agglomeration with different adhesion levels, by simply replacing (i, j) in Eq. 3.6 by a (i, j). We will show below that such simple modification can well reproduce the DNS-DEM results in a statistical manner. The structure of agglomerates is another crucial factor affecting the agglomeration rate. For non-coalescing adhesive particles, the formed agglomerates usually have fractal structures, which distinguishes our system from those of droplets [40, 41]. In systems involving Brownian nanoparticles, theoretical collision kernels can be extended to fractal agglomerates when substituting the particle radius with the radius of effective collision spheres (ECSs) for agglomerates [42, 43]. We will show below that the idea of the effective radius can also be applied to non-Brownian adhesive particles.

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3 Agglomeration of Microparticles in Homogenous Isotropic Turbulence

3.3 Collision Rate, Agglomerate Size and Structure We first measure the temporal evolution of the collision kernel in a system with St k = 5.8 and Ad varying from 0.013 to 128. To show the effect of adhesion, here we simply regard the system as a monodisperse system and count the collisions between every primary particle. The effective collision kernel is then calculated as  = 2n˙ c /n 20 , where n 0 is the number density of primary particles. The temporal evolution of the collision kernel (t), normalized by the collision kernel for zero-inertia particles 0 = (8π /15ν)1/2 (2r p )3 [1], is shown in Fig. 3.4a. When the adhesion is extremely weak (Ad = 0.013 and 1.3), the collision kernel rapidly reaches a statistically steady state with (t)/ 0 = 11.1. This value is quite close to the previous DNS results for nonadhesive particles with the same inertia [4]. As Ad increases, the collision kernel is significantly reduced and the system is pushed away from equilibrium. Since the adhesion number only affects the interaction between contacting particles, we attribute these phenomena to adhesion-enhanced agglomeration. When Ad is larger than 64, further increase of Ad does not change the curve of collision kernel. Because in this strong-adhesion limit, the sticking probability for colliding particles is essentially unity and every collision event will lead to agglomeration. In Fig. 3.4b, the agglomeration at t = 15 is clearly displayed in the form of the fraction of particles P(A) contained in an agglomerate of size A. The agglomerate size A is defined as the number of primary particles contained in that agglomerate. For small Ad, most particles remain as singlets (A = 1) and only a small number of particles (∼ 4%) are contained in agglomerates with the size A ≥ 2. In contrast, cases with large Ad yield a considerable number of agglomerates with size A up to 20.

Fig. 3.4 a Temporal evolution of the collision kernel (t)/ 0 for cases with St k = 5.8 and Ad = 0.013 (circles), 1.3 (left-pointing triangles), 13 (diamonds), 64 (upward triangles), and 128 (squares). b Fraction of particles, P(A), contained in agglomerates of size A at t = 15 for Ad = 1.3 and 64. Reprinted figure with permission from Chen et al. [44] Copyright by the American Physical Society

3.3 Collision Rate, Agglomerate Size and Structure

61

Fig. 3.5 a Gyration radius of agglomerates Rg (A)/r p as a function of agglomerate size A at t = 15 for the cases with St k = 5.8 and Ad = 13 (diamonds), 64 (triangles), and 128 (squares). The solid line shows Eq. 3.8 with χ = 1.64 and D f = 1.64. b An agglomerate produced in the simulation with St = 5.8 and Ad = 64 with its equivalent sphere with radius of gyration (shaded region). Reprinted figure with permission from Chen et al. [44] Copyright by the American Physical Society

To model the agglomeration process in the framework of Smoluchowski’s equation, a measure of the agglomerate structure in the form of the equivalent sphere is necessary. One such quantity is the radius of gyration, defined for an agglomerate with 3 or more primary particles (A ≥ 3) by Rg (A) = (1A |X i − X¯ i |2 /A)1/2 , where X i denotes the position of i th particle within the agglomerate and X¯ i is the center of mass of the agglomerate. For√agglomerates with two primary particles, we use the explicit expression Rg (2) = 1.6r p suggested by Waldner et al. [45]. In Fig. 3.5b, we show an agglomerate generated from the DNS-DEM simulation and its equivalent sphere with the radius of gyration. We calculate Rg for all the agglomerates produced in the simulations in Fig. 3.4a at t = 15 and plot the ratio Rg /r p as a function of agglomerate size A in Fig. 3.5a (large size agglomerates with A > 12 only contain 0.2% particles thus are neglected here). The results fall onto a power-law curve D1 Rg (A) A f = , for A > 2, (3.8) rp χ with the factor χ = 1.64 and the fractal dimension D f = 1.64. The D f value measured here is consistent with experimental measurements of Waldner et al. [45], who measured the radius of gyration for early-stage agglomerates formed in a stirred tank using small angle static light scattering [45]. The value of fractal dimension fitted from experimental results is D f = 1.7 ± 0.1, which is consistent with results of our simulations. Selomulya et al. [46] adopted the same experimental technique to measure the shear-induced agglomeration of latex particles and reported values of D f between 1.7 and 2.1. Their results are close to but slightly larger than the values of D f measured in our DNS-DEM results. The possible reason for the deviation is that Selomulya et al. [46] assumed the factor χ to be 1.01 in their measurements. Such a small value of χ may give a D f value that is larger than its actual value.

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3 Agglomeration of Microparticles in Homogenous Isotropic Turbulence

3.4 Effect of Stokes Number The temporal evolution of the collision kernel (t)/ 0 , the fraction of particles, P(A), contained in agglomerates of size A, and the gyration radius of agglomerates Rg (A)/r p for cases with different Stokes number Stk and adhesion parameter Ad are plotted in Fig. 3.6. For particles with small inertia (Stk = 0.72), the increase of adhesion parameter only has a limited effect on the temporal evolution of the collision kernel (Fig. 3.6a). Moreover, there is no obvious statistical steady state for the system with St k = 0.72. The reason is that the lubrication force between particles near contact significantly reduces the collision rate for particles with small inertia [20] and the collision rate is too small to form a considerable number of agglomerates even if the adhesion is strong. The system thus behaves like a monodisperse system. This is further displayed in the form of the fraction of particles P(A) contained in an agglomerate of size A (Fig. 3.6b). In both strong and weak adhesion cases, most particles remain as singlets. For particles with higher Stokes number (St k = 12 or 23), similar results are observed as those for St k = 5.8 in Fig. 3.4. In both cases, a statistically steady state can be identified in the temporal evolution of the collision kernel (t)/ 0 at the small Ad limit (Fig. 3.6d and g). When Ad > 64, further increase of Ad does not change the (t)/ 0 − t curves. The results once again confirm the existence of the strong adhesion limit. In this limit, one can simply adopt the hit-and-stick assumption—two particles will stick together once there is a contact between them—to simulate the agglomeration without performing DEM calculations. In Fig. 3.6e and h, we observe similar results as those for St k = 5.8 in Fig. 3.4b. For all three values of St k , the radius of gyration for agglomerates of different sizes can be well described using the power-law function in Eq. 3.8 (see Fig. 3.6c, f and i). For a given St k , the factor χ and fractal dimension D f are insensitive to the value of adhesion parameter Ad. It suggests that the interparticle adhesion strongly affects the growth rate of early-stage agglomerates but has no obvious impact on their structures. Interestingly, as we mentioned in the previous subsection, the agglomerates formed in different experimental conditions also have similar values of D f , which further implies that the influences of flow conditions and interparticle adhesion on the structure of agglomerates may be significant only if the size of agglomerates is sufficiently large [47].

3.5 Exponential Scaling of Early-Stage Agglomerate Size Figure 3.7 shows the distributions of number density of agglomerates as a function of size A at early-stage (t ≤ 20). These distributions, when scaled by the initial number density of primary particles n 0 , follow an exponential equation (solid lines in Fig. 3.7)

A n(A) , = β exp − n0 κ

(3.9)

3.5 Exponential Scaling of Early-Stage Agglomerate Size

63

Fig. 3.6 Left (panels a, d and g): Temporal evolution of the collision kernel (t)/ 0 . Middle (panels b, e and h): Fraction of particles, P(A), contained in agglomerates of size A at t = 15 for Ad = 1.3 and 64. Right (panels c, f and i): Gyration radius of agglomerates Rg (A)/r p as a function of agglomerate size A at t = 15. The solid lines in c, f, and i are fitting results from Eq. 3.8 with c χ = 1.80 and D f = 1.54, f χ = 1.70, D f = 1.60, and i χ = 1.49, D f = 1.71. Different rows stand for results for different St k , top: St k = 0.72, middle: Stk = 12, and bottom: Stk = 23. Reprinted figure with permission from Chen et al. [44] Copyright by the American Physical Society

with the coefficients β and κ depending on time. Based on the conservation of the total number of primary particles, 1∞ A ∗ n(A) = n 0 , the prefactor β can be expressed as β(κ) = 2 cosh (κ −1 ) − 2. Therefore, the size distribution of early-stage agglomerates is determined by a single scale parameter κ, which gives a typical value of the size of agglomerates. A larger value of κ means that there are more particles contained in agglomerates with a larger size and the growth of early-stage agglomerates can be characterized by the increase of κ. In the inset of Fig. 3.7, the number density distributions for cases with St k = 5.8, 12, and 23 and Ad = 1.3, 13, and 64 are plotted in a rescaled form, n(A)/(n 0 β) ∼ A/κ. Except for the deviation in the tail caused by agglomerates with n(A)/n 0 < 0.3%, the results center around the curve y = exp(−x), suggesting that the exponential scaling for the early-stage agglomeration is valid for inertial particles across a wide range of adhesion force magnitudes.

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3 Agglomeration of Microparticles in Homogenous Isotropic Turbulence

Fig. 3.7 Scaled number density n(A)/n 0 of agglomerates of size A for the case with St k = 5.8 and Ad = 64 at t = 5 (circles), 10 (squares), 15 (exes), and 20 (triangles). The solid lines are fits to Eq. 3.9. Inset: scaled number density n(A)/(n 0 β) as a function of A/κ for Stk = 5.8 (circles), 12 (triangles), and 23 (squares). For each St k , results are shown for Ad = 1.3 (black), 13 (blue), and 64 (red), at t = 5, 10, 15. The solid line is the exponential scaling y = exp(−x). Reprinted figure with permission from Chen et al. [44] Copyright by the American Physical Society

A comparison between the exponential distribution and the well-known selfpreserving size distribution for Brownian nanoparticles [48–50] would be of interest. If the collision kernels are homogeneous functions of the volume of colliding particles and the degree of homogeneity is smaller than unity, the particle size distribution will reach a self-preserving shape (normally bell-shaped). In that case, tracking the evolution of the mean agglomerate size is sufficient to describe the growth of agglomerates. Although, both the exponential distribution in Eq. 3.9 and the self-preserving size distribution are single-parameter distributions, there is a fundamental difference between them. The exponential distribution describes the transition behavior at the early stage of the agglomeration when most particles remain as singlets and is no longer valid when there is a considerable number of large agglomerates. In contrast, the self-preserving size distribution is an asymptotic limit which is invariant with time.

3.6 Agglomeration Kernel and Population Balance Modelling Now we introduce how to construct the agglomeration kernel that can be applied to Smoluchowski’s theory based on DNS-DEM results. We first look at the strong adhesion case by assuming that particles will stick together upon collisions (i.e.,  = 1) and then show how adhesion influences the sticking probability. For spherical particles, (i, j) can be modelled by [5]

3.7 Effect of Adhesion on Agglomeration

(i, j) = 2π Ri2j |wr |g(Ri j ),

65

(3.10)

where Ri j = r p,i + r p, j is the collision radius, |wr | is the average radial relative velocity and g(Ri j ) is the radial distribution function at contact. Explicit expressions of these quantities are summarized in [5]. Since turbulence parameters are fixed here, Ri j , wr , and g(Ri j ) are determined by particle size and St k . For collisions between agglomerates, we simply use the radius of gyration in Eq. 3.8 with known values of χ and D f instead of the particle radius r p to calculate all the quantities in Eq. 3.10 [42, 51, 52]. For instance, the collision radius for an agglomerate with i primary particles and that with j primary particles is calculated as Ri j = Rg (i) +√Rg ( j). The gyration radius Rg (i) is given by Eq. 3.8 when i > 2 and Rg (i) = r p and 1.6r p for i = 1 and i = 2, respectively. Given the initial conditions, n(1) = n 0 and n(i) = 0 for i > 1, PBEs in Eq. 3.6 are numerically integrated using a sufficiently small time step with the agglomerate size truncated at i C = 50 (i.e., assuming n(i) = 0 for i > 50). As a result, we can get the evolution of the number density n(i) for each size group. PBE calculations are much faster than DNS-DEM since PBEs only solve for the number density n(i) at each time step rather than resolve the motion of every particle. We plot the scaled number density n(A)/n 0 calculated from PBE in Fig. 3.8. It is shown that results from PBE well reproduce the results of DNS-DEM in Fig. 3.7 when t ≤ 15. We then fit the scaled distribution n(A)/n 0 using the Eq. 3.9 at each t and get the evolution of the scale parameter κ, which is in good agreement with the DNS results when t ≤ 15 (see the inset of Fig. 3.8b). It indicates that the kernel (i, j) constructed in the form of gyration radius readily reflects the effect of the fractal structure of agglomerates on the agglomeration. At t = 20, the distribution of n(A)/n 0 from PBE still follows the exponential form, however, a non-negligible deviation between PBE results and those from DNS-DEM is observed. Such deviation may be attributed to two reasons. First, (i, j) does not contain the information of breakage or rearrangement, which is expected to be significant for large-size agglomerates [41]. Moreover, statistics may also get worse when the total number of agglomerates 1∞ n(A) reduces.

3.7 Effect of Adhesion on Agglomeration When designing large-scale devices, one does not need to know the information of every single particle, instead, knowing the size distribution is enough. In those cases, solving the population balance equations would be more feasible. Therefore, it is of significance to check if the complicated effect of particle-particle contacting interactions on the growth kinetics of agglomerates can be captured by the sticking probability  (given in Eq. 3.7). We solve PBE using agglomeration kernel a (i, j) with  increasing from 0 to 1 (see Eq. 3.7). The evolution of the scale parameter κ is shown as solid lines in Fig. 3.9. It is evident that a smaller sticking probability  leads to a lower growth rate of

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Fig. 3.8 Scaled number density n(A)/n 0 v.s. agglomerate size A calculated from the population balance equations. In the inset, we show the temporal evolution of the scale parameter κ from DNS-DEM simulations (circles) and from PBE (solid line). Reprinted figure with permission from Chen et al. [44] Copyright by the American Physical Society

agglomerates. We also plot corresponding results from DNS-DEM simulations with different values of adhesion parameter Ad as data points in Fig. 3.9. For Ad = 0.013, κ(t) is close to the PBE results with sticking probability  = 0, indicating that almost no agglomerates are formed given such a weak adhesive force. As Ad increases beyond ∼ 64, the κ(t) curves converge to the PBE result with sticking probability  = 1. This strong adhesion case corresponds to the conventional PBE simulations, where the hit-and-stick assumption is made. Our results here suggest that PBE can also simulate the agglomeration process for particles with relatively weak adhesion once the sticking probability  is adopted. Describing turbulence-induced agglomeration using PBE requires knowledge of the sticking probability  a priori. Therefore, it is of significance to relate  to the particle properties. We consider a head-on collision between two primary particles with vcn being the relative collision velocity. According to the analysis in Sect. 2.4.1, the result of a collision is determined by a dimensionless elasticity parameter 2 r p ). We measure the value of El ≡ ρvE 2 and the adhesion parameter Ad ≡ γ /(ρ p vcn 0 vcn for every collision event in each simulation run and use the mean value vcn  as the typical velocity scale. An effective adhesion parameter then is given as Adn =

γ . ρ p vcn 2 r p

(3.11)

In Fig. 3.10, we plot the simulation results in the  − Adn plane and all the data points collapse onto two curves:  = 0.017Adn , for 1 < Adn < 30, and  = 1 for Adn > 50.

(3.12)

3.7 Effect of Adhesion on Agglomeration

67

Fig. 3.9 Temporal evolution of the parameter κ for DNS-DEM simulation with St k = 5.8 and Ad = 0.013 (circles), 1.3 (left-pointing triangles), 13 (diamonds), 64 (upward triangles), 128 (squares), and 256 (axes). The solid lines spanning from light to dark color are results from PBE with the sticking probability  = 0, 0.2, 0.4, 0.8, and 1. Reprinted figure with permission from Chen et al. [44] Copyright by the American Physical Society

Fig. 3.10 Sticking probability  as a function of the adhesion parameter Adn , which is defined based on the averaged normal collision velocity vcn , for St k = 2.9 (circles), 5.8 (triangles), 12 (diamonds), and 23 (squares). The solid line is  = 0.017Adn and the horizontal dashed line is  = 1. Reprinted figure with permission from Chen et al. [44] Copyright by the American Physical Society

The results in Fig. 3.10 indicate that the mean relative collision velocity is an appropriate choice to scale the effect of adhesion and the sticking probability  can be well estimated once Adn is known. Here, the data points for cases with Adn < 1 are neglected, since the sticking probability is less than 10−2 , which is too small to ensure good statistics.

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3.8 Effect of Adhesion on Breakage of Agglomerates We have neglected the breakage of the agglomerates in the previous sections. In the sections below, we discuss how the breakage affects the growth of agglomerates. Specifically, we focus on the collision-induced breakage and formulate the collisioninduced breakage rate based on the Smoluchowski equation and a breakage fraction function. In Fig. 3.11a–c, we show the temporal evolution of the number of overall collisions NC , the number of sticking collisions N S , rebound events N R , and breakage events N B for St = 5.8 and three different values of effective adhesion parameter Adn , 2 which is defined based  on the square root of the average value of vn over all collision events, i.e. v¯n = vn2 . When the adhesion is extremely weak (Adn = 0.73), NC increases linearly with time. It indicates that the collision kernel  almost keeps as a constant, which is consistent with previous DNS results for non-adhesive particles [4]. N R is close to NC and both N S and N B are nearly zero. Agglomerates therefore can barely be formed given such a weak adhesion. For the case with a relatively stronger adhesion (Adn = 7.3), agglomeration between colliding particles can be clearly observed. However, the agglomeration at this Adn value is still quite limited, since the sticking probability is small (∼ 0.4). When Adn further increases to 70, adhesion plays a dominant role. As illustrated in Fig. 3.11c, N S ≈ NC , implying that almost all collisions lead to the agglomeration of colliding particles. Consequently, a considerable number of breakage events can only be observed at a moderate value of Adn . We normalize the number of sticking collisions N S , rebound collisions N R , and breakage events N B with the total number of collisions NC and plot them against Adn in Fig. 3.12. Three different regimes can be identified: a rebound regime with Nˆ R > 95%; a sticking regime with Nˆ S > 95%; and a transient regime between the two regimes. The critical Adn values dividing the three regimes are approximately 1.5 and 35. Simulation results for different St collapse, implying that the possibility of occurrence of sticking, rebound, and breakage event can be well quantified by the dimensionless adhesion number Adn .

3.9 Formulation of the Breakage Rate In the current subsection, we focus on the formulation of the rate of collision-induced breakage of agglomerates. In turbulent flow laden with particles, the growth or collision-induced breakage of agglomerates results from two successive processes. First, the turbulent flow brings two initially separate agglomerates (or particles) close enough to initiate collisions. Second, the two colliding agglomerates will either merge into a large one, rebound from each other, or break up into fragments.

3.9 Formulation of the Breakage Rate

69

Fig. 3.11 Temporal evolution of the number of collisions NC , the number of sticking collisions N S and rebound collisions N R , and the number of breakage events N B for St = 5.8 and a Adn = 0.73, b Adn = 7.3, and c Adn = 70. © Cambridge University Press, reproduced with permission from Chen and Li [35]

Fig. 3.12 Normalized number of sticking collisions Nˆ S (blue), rebound collisions Nˆ R (red), and breakage events Nˆ B (purple) over the entire simulation as functions of Adn . Results for three different Stokes numbers are shown: St = 2.9 (circles), St = 5.8 (triangles), and St = 12 (diamonds). © Cambridge University Press, reproduced with permission from Chen and Li [35]

For the first step (i.e., collision), we introduce the classic statistical model of the collision rate in particle-laden turbulence. The collision rate for agglomerates of size i, n˙ C (i), can be expressed as n˙ C (i) =

∞  j=1

(i, j)n( j)n(i),

(3.13)

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3 Agglomeration of Microparticles in Homogenous Isotropic Turbulence

where (i, j) is the collision kernel between agglomerates of size i and agglomerates of size j and n(i) is the average number concentration of size group i (see Eq. 3.10). The breakage rate due to the collisions with other particles or agglomerates can be expressed as the product of the collision rate n˙ C (i) and the fraction of collision events resulting in breakage  [53] as follows: ∞

f br (i) =

  n˙ C (i) = (i, j)n( j). n(i) j=1

(3.14)

The fraction of breakage events  is defined as the ratio of the breakage number to the overall collision number and should include the influence of both turbulent transport and particle scale interactions. Here, we introduce a statistical framework to calculate  in terms of well-known impact velocity distributions PC (vn ). For collision events with impact velocity vn , the fine-grained probability of breakage is recorded as ψ(vn ). Thus, the probability density function (p.d.f.) of the impact velocity for breakage event is given by PC (vn )ψ (vn ) , PB (vn ) =  ∞ 0 PC (v)ψ (v) dv

(3.15)

where the denominator is the normalization coefficient. ψ (v) can be regarded as a transfer function, which relates the p.d.f. of the breakage to the p.d.f. of impact velocity. For particles with a given adhesion value, ψ(vn ) is expected to be zero as vn tends to zero (sticking regime) and rises to unity as vn increases, given that all colliding agglomerates will break when the impact velocity is sufficiently large. Knowing the value of ψ (vn ), one can directly obtain the fraction of breakage  through ∞ PC (vn )ψ (vn ) dvn .

=

(3.16)

0

Substituting Eq. 3.16 into Eq. 3.14 further gives the breakage rate. To validate the statistical framework above and to give a specification of the transfer function ψ(vn ), we obtain the statistics of doublet breakage from DNSDEM simulation and compare them with the theoretical descriptions in Eq. 3.14. The breakage of doublets has been widely adopted as the prototype of agglomerates that break into two fragments. For doublets, the breakage rate in Eq. 3.14 reduces to ∞

f br (2) =

  n˙ C (2) = (2, j)n( j). n(2) j=1

(3.17)

At the early-stage of agglomeration most particles remain as singlets [14] and the equation above can be further simplified as

3.9 Formulation of the Breakage Rate

f br (2) ≈ (1, 2)n(1) = n(1)S12 (1, 1).

71

(3.18)

On the right-hand side of the equation, we relate the singlet-doublet collision kernel (1, 2) to the singlet-singlet kernel through (1, 2) = S12 (1, 1), where the constant S12 is the correction for collisional cross-sectional areas for singlet-doublet collisions. Although the expression in Eq. 3.18 only gives low-order statistics for the breakage of doublets, it provides valuable insights: the breakage rate scales linearly to the number concentration and the effect of turbulent transport are included in both (1, 1) and the breakage fraction ; contacting interactions affect the breakage rate by changing  through the transfer function ψ(vn ) in Eq. 3.16. In order to obtain the transfer function ψ(vn ), we track all the collision events in the simulation and record whether the collision leads to the breakage of the agglomerate according to the criterion in Sect. 3.2.5. The p.d.f. of the impact velocity PC (vn ) for singlet-doublet collision events are then measured at different St and Ad values (as shown in Fig. 3.13a–c). For the cases with weak adhesion (Ad = 0.64), most particles remain as singlets and the number of singlet-doublet collision events observed within a large-eddy turnover time is quite limited. We thus run three simulations with different initial random positions of particles to obtain more collision events. It ensures a good statistic on the collision velocity for the singlet-doublet collision events and the breakage events. For a given value of St, varying Ad does not obviously affect PC (vn ). In contrast, a strong dependence on St can be observed. For collisions that result in the breakage of a doublet, we also plot the corresponding p.d.f. of the impact velocity, PB (vn ), in Fig. 3.13d–f. One can easily find a strong correlation between PB (vn ) and Ad. Particles with stronger adhesion tend to stick together upon collisions. The breakage events, therefore, are more likely to happen with a higher impact velocity. We then calculate the transfer function ψ(vn ) inversely from PC (vn ) and PB (vn ) according to Eq. 3.15. As shown in Fig. 3.14a, despite the inconsistency in PC (vn ), ψ(vn ) for different St collapses nicely. In contrast, the adhesion strongly affects ψ(vn ). Although, there is considerable scatter in the data at large vn due to the limited sample size of the energetic collision events, the transfer function ψ(vn ) at a given Ad value is roughly linear to the collision velocity vn . The results in Fig. 3.14a suggest that the transfer function may only depend on the short-range contacting interactions, whereas the effects of turbulent transport and hydrodynamic interactions are included in the probability density functions of the impact velocity PC (vn ). To validate this argument, we run simulations with different particle radii (ranging from 0.0075 to 0.0125) and with/without the hydrodynamic damping force (Eq. 3.5) at a fixed St value. As seen in Fig. 3.14a, the measured transfer function ψ(vn ) does not show obvious dependence on the particle size and the hydrodynamic interaction, confirming that the transformation function ψ(vn ) is determined by the short-range contacting interactions. According to the results in Fig. 3.14a, we propose a linear relationship between ψ and vn as follows:

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3 Agglomeration of Microparticles in Homogenous Isotropic Turbulence

Fig. 3.13 Probability density functions of the collision velocity (normal component) vn for singletdoublet collision events (a–c) and collision-induced breakage events (d–f). Statistics are made over approximately a large-eddy turnover time t ∈ [15, 25]. Different columns are results for different Stokes numbers: St = 2.9 (left), St = 5.8 (middle), and St = 12 (right). For each Stokes number, we show results from different Ad values: Ad = 0.64 (squares), Ad = 1.3 (circles), Ad = 6.4 (upward triangles), and Ad = 12 (downward triangles). © Cambridge University Press, reproduced with permission from Chen and Li [35]

ψ(vn ) =

⎧ ⎨ 0, ⎩

1 (v vC2 −vC1 n

1,

for vn < vC1 , − vC1 ), for vC1 ≤ vn ≤ vC2 , for vn > vC2 .

(3.19)

Two typical values of collision velocity vC1 and vC2 are indicated by Eq. 3.19. Breakage does not happen when the collision velocity between two agglomerates, vn , is smaller than vC1 . On the other hand, if vn > vC2 , the colliding doublets always break. We then fit the measured values of the transfer function ψ(vn ) (linear part) using Eq. 3.19 for all the cases presented in Fig. 3.14a and plot the fitting parameters vC1 and the slope (vC2 − vC1 )−1 as a function of Ad. It is seen that the fitted values of the slope for different cases center around a logarithmic curve (Fig. 3.14b), which reads

Ad −1 . (3.20) (vC2 − vC1 ) = −2.1 ln 13 Several interesting features are indicated by Eq. 3.20. First, the slope (vC2 − vC1 )−1 diverges in the small adhesion limit (Ad → 0), indicating that there is a critical collision velocity separating the breakage and non-breakage collisions. This is in accordance with the theoretical model proposed by Liu and Hrenya [14], in which a Heaviside function H (v − vb,crit ) is proposed to transform the p.d.f. of normal impact velocity PC (vn ) into the p.d.f. of impact velocity for breakage events PB (vn ). We show here that such a transfer function is reasonable only when the adhesive interaction is extremely weak. As Ad increases, the slope of ψ(vn ) considerably

3.9 Formulation of the Breakage Rate

73

Fig. 3.14 a Transfer function ψ (vn ) versus collision velocity vn for different Stokes numbers: St = 2.9 (squares), 5.8 (circles), and 12 (triangles), and different Ad values: Ad = 1.3 (light blue), 6.4 (yellow), 12 (dark blue). Results with different particle radii (ranging from 0.0075 to 0.0125) and with/without the hydrodynamic damping force (Eq. 3.5) at St = 2.9 are also included. Scatters are results calculated from PDFs in Fig. 3.13, and dashed lines are linear-fittings from Eq. 3.19. b and c Fitting parameters (vC2 − vC1 )−1 and vC1 as functions of Ad. Legends are the same as in (a). © Cambridge University Press, reproduced with permission from Chen and Li [35]

decreases and there is no sharp transition between the breakage and the non-breakage collision velocities. Although the data points for the minimum breakage velocity vC1 are relatively dispersed when plotted as a function of Ad, a quadratic curve, vC1 = aAd2 with a = 7.4 × 10−4 , can roughly describe the variation of vC1 (see Fig. 3.14c). At large adhesion limit vC1 diverges, implying that all collisions give rise to the growth of agglomerates when the adhesion is sufficiently strong. To further validate the model of the transfer function, we present an example of the model prediction for cases with St = 2.9 in Fig. 3.15a. First, the p.d.f. of the normal collision velocity PC (vn ) is measured from the simulation with a small Ad value (Ad = 1.3). The breakage fraction  is then calculated by substituting Eq. 3.19 and the measured PC (vn ) into Eq. 3.16. One can also adopt models of PC (vn ) obtained from simulations with non-interacting particles to estimate the breakage fraction  [54–56]. Such approximation does not bring large errors since PC (vn ) is almost independent of the adhesive interactions (see Fig. 3.13). The results generated from the model together with predictions for St = 1.4, 5.8, and 11.5 are plotted as a dashed line in Fig. 3.15b. We see that the model predictions are in accordance with DNS-DEM simulations. The deviation between the model and the simulations in Fig. 3.15b may result from the linear assumption of the transfer function ψ(vn ) (Eq. 3.19), in which a sharp transition is assumed between the linear part ((vn − vC1 )/(vC2 − vC1 )) and unity. The simulation data in Fig. 3.14a, in contrast, shows a much slower approach to unity, indicating that the model in Eq. 3.19 overestimates ψ(vn ) when vn → vC2 . Despite this deviation, the model well captures the variation of the breakage fraction  with adhesion parameter Ad. Moreover, the Stokes number dependence of  can be observed in Fig. 3.15b. Since the breakage fraction  here is calculated from a universal transfer function, the St number dependence of  originates from

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3 Agglomeration of Microparticles in Homogenous Isotropic Turbulence

Fig. 3.15 a Probability density functions PC (vn ) of the normal collision velocity for St = 2.9 and Ad = 1.3 (left-hand axis) and the transfer function ψ(vn ) modeled by Eq. 3.19 (right-hand axis). Color code spans from blue to yellow with increasing Ad (from 0.1 to 10). b Fraction of collision-induced breakage of doublets  at different Ad values. Points are DNS-DEM results and dashed lines are results calculated from PC (vn ) and the modeled ψ(vn ) (Eq. 3.19). © Cambridge University Press, reproduced with permission from Chen and Li [35]

the difference in PC (vn ): the hydrodynamic damping force significantly reduces the relative approaching velocity of colliding particles with small St. The collision-induced breakage rate of the doublets f br (2) is calculated from Eq. 3.18 and compared with DNS-DEM results in Fig. 3.16a. Quantitative agreement is observed, indicating that the analytical model well captures the effects of the particle inertia and the adhesive interaction on the breakage. Since the adhesion parameter Ad does not include the effect of particle inertia, there is considerable distinction in results for different St at the same Ad. We stress again that particle inertia affects the breakage rate through its influence on the statistics of the collision velocity. One simple way to include both effects of the particle inertia and the adhe(see Eq. 3.11), which scales the sion is to use the modified adhesion parameter Adn  adhesion using St-dependent average velocity v¯n = vn2 . The normalized breakage rate, when plotted as a function of Adn , collapses nicely onto the exponential curve (see Fig. 3.16b): f br τk = 86 exp(−0.12Adn ). (3.21) 3 r p n(1)  The result indicates that v¯n = vn2 is an appropriate choice to scale the effect of adhesion and the collision-induced breakage rate can be well estimated once Adn is known. It is of great importance to know how the breakage rate scales with particle size r p and particle number density n. We measure the doublet breakage at different particle sizes (r p = 0.005 − 0.015) and particle numbers (N = 19,600 − 40,000) for typical St and Ad values (shown in Fig. 3.17). The results are plotted in a scaled form: fˆbr = f br (r p )/ f br (r p,0 ), rˆ p = r p /r p,0 in Fig. 3.17a and fˆbr = f br (n(1))/ f br (n m (1)), n(1) ˆ = n(1)/n m (1) in Fig. 3.17b. Here, f br (r p,0 ) is the doublet breakage rate for the case with r p,0 = 0.01 and f br (n m (1)) is the breakage rate for the case with the maximum value of singlet number density n m (1). As displayed in Fig. 3.17, DNS-

3.9 Formulation of the Breakage Rate

75

Fig. 3.16 a Normalized breakage rate f br τk /(r 3p n(1)) for doublets as a function of Ad. The data points are DNS-DEM results and the dashed lines are predictions from Eq. 3.18, in which the breakage fraction  is calculated from the p.d.f. of the normal collision velocity through ∞  = 0 PC (vn )ψ(vn )dv (see Eq. 3.16) and the transfer function ψ(vn ) is modeled by Eq. 3.19. b Normalized breakage rate as a function of Adn . The dashed line is exponential fittings using Eq. 3.21. © Cambridge University Press, reproduced with permission from Chen and Li [35]

Fig. 3.17 Scaled breakage rate fˆbr as a function of a scaled particle size rˆ p , and b scaled number density of singlet n(1) ˆ at St = 2.9, 5.8, and 12 and Ad = 1.3 and 3.8. Dashed lines in a and b indicate power law functions with exponents 2 and 1, respectively. © Cambridge University Press, reproduced with permission from Chen and Li [35]

DEM results follow the power laws fˆbr ∝ rˆ 2p and fˆbr ∝ nˆ 1 (1) when particle size and singlet number density are varied. The n(1) dependence is easy to understand from Eq. 3.18. The rˆ 2p scaling originates from the r p dependence of the collision kernel (1, 1) in Eq. 3.18. For inertial particles (St  1), the approaching velocity of colliding particles is decorrelated from the local fluid gradient, thus is not affected by the particle size. The rˆ 2p scaling enters (1, 1) through the effective collision area.

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3 Agglomeration of Microparticles in Homogenous Isotropic Turbulence

3.10 Agglomerate Size Dependence of the Breakage Rate The breakage rate of agglomerates with size A is calculated from DNS-DEM simulations according to Nbr (A) (3.22) f br (A) = N (A)t where N (A) is the number of agglomerates of size A averaged over the time range t ∈ [30, 40], Nbr (A) is the number of breakage events of agglomerates with size A and t = 10. As shown in Fig. 3.18a, a stronger adhesion promotes the formation of larger agglomerates. In contrast, the number of breakages decreases with Ad (see Fig. 3.18b). To provide meaningful statistics, we only calculate f br (A) when Nbr (A) is larger than 20. The results are normalized by the mean shear rate G and plotted as a function of size A in Fig. 3.18c. It is seen that the breakage rate depends linearly on the agglomerate size with the slope being a function of Adn . Fitting the data at different Adn and St according to f br (A) = ζ (St, Adn ) · A + χ G

(3.23)

gives us the values of the slope ζ (St, Adn ). As shown in Fig. 3.18d, when plotted as a function of Adn , ζ for different St centers around a universal curve, which is analogous to the f br dependence in Fig. 3.16b. The universal curve has a power-law . These results once again confirm that the modified adhesion form: ζ = 0.012Ad−0.81 n parameter Adn is an appropriate choice to reflect the combined effect of the particle inertia and adhesive interactions on the breakage.

3.11 Role of Flow Structure In this subsection, we quantify the correlation between the structures of turbulent flow and the breakage of agglomerates with different St and Ad values. We identify the flow structures based on the second invariant ofthe velocity   gradient T strain rate tenor S = A + A /2 and the tensor Q = R2 − S 2 /2, where the  rotation rate tensor R = A − AT /2 are symmetric and antisymmetric parts of the velocity gradient tensor A ≡ τk ∇u (normalized by the Kolmogorov time τk ), respectively. Figure 3.19a presents the contour plots of Q, showing the vortex tubes   with Q > 3.3 Q 2  and the straining sheets with Q < −2.5 Q 2 , and the corresponding two-dimensional slice at y = 0. One can clearly see the red vortex tubes surrounded by blue straining sheets (vortex-strain worm-rolls), which implies that intense structures typically occur near each other [57]. We calculate the average Q, sampled by the singlet-doublet collisions, at different St and Ad values in Fig. 3.19b. The results for non-interacting particles based on the ghost collision approximation are also included [57] (only data at St > 0.5 are shown

3.11 Role of Flow Structure

77

Fig. 3.18 a Number of agglomerates of size A averaged over the time range t ∈ [30, 40]. b Number of breakages of agglomerates of size A measured within t ∈ [30, 40] for St = 5.8. c Breakage rate f br (A), normalized by the shear rate G, as a function of agglomerate size A. The dashed lines are linear fittings from Eq. 3.23. d The fitting values of the slope ζ for the linear relationship between f br /G and A at different Adn and St values. The dashed line indicates the power function ζ = 0.012Ad−0.81 . © Cambridge University Press, reproduced with permission from Chen and n Li [35]

here). One can notice that as St increases from 0.5 to 20, Q increases from a negative value to zero, implying that finite-inertia particles (St ∼ 1) tend to collide in the straining zone whereas particles with large inertia collide more uniformly. According to Picardo et al. [57], decreasing St also leads to the approach to zero of Q and the largest absolute value of Q occurs at St ≈ 0.3. Such flow structure dependence can be understood from two aspects. First, particles with finite inertia (St ≈ 1) tend to accumulate in straining regions outside vortices due to the centrifugal effect (known as preferential concentration). Moreover, particle inertia also increases the relative approaching velocity between particles. Such an effect also prevails in straining zones [57]. Here, we show that varying particle-particle contact interactions (Ad) does not obviously affect the structure dependence of collisions. The average Q, sampled by the singlet-doublet collision-induced breakage events, shows a strong dependence on Ad (Fig. 3.19c). Doublets with larger Ad values are more difficult to break thus needing higher impact velocities. For particles with moderate inertia (St ≈ 1), violent collisions are more likely caused by particles ejected

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3 Agglomeration of Microparticles in Homogenous Isotropic Turbulence

Fig. 3.19 a Contour plot of the second invariant of  the velocity gradient tensor Q and the two Q 2  are colored in red and straining sheets dimensional slice at y = 0. Vortex tubes with Q > 3.3  2 with Q < −2.5 Q  are colored in blue. b Average Q, sampled by singlet-doublet collision events, and c average Q sampled by doublet breakage events as functions of St. d Average Q for breakage events as a function of Ad at different St values: St = 1.4 (squares), St = 2.9 (circles), St = 5.8 (upward triangles), and St = 12 (downward triangles). The straight dashed lines are linear fittings. © Cambridge University Press, reproduced with permission from Chen and Li [35]

rapidly from strong vortices and happen in straining sheets (with smaller negative Q) that envelope the vortices. As St increases, the relative velocity between colliding particles becomes less sensitive to the underlying flow, both collision events and breakage events distribute more uniformly in the flow. As shown in Fig. 3.19d, the relationship between Q and Ad at given St can be well described by linear functions.

3.12 Conclusions In summary, by means of DNS and multiple time scale DEM, we are able to resolve all the collision, rebound, and breakage events for adhesive particles in turbulence. We find that the collision rate is significantly reduced due to the adhesion-induced agglomeration. As the value of adhesion parameter Ad increases, the system reaches a strong-adhesion limit, in which the sticking probability for colliding particles is unity and further increase of Ad does not affect the dynamics of agglomeration. We

References

79

also find that the size distribution of early-stage agglomerates follows an exponential equation n(A)/n 0 = β(κ) exp(−A/κ) regardless of the adhesion force magnitude. The transient dynamics of agglomeration at the early stage thus can be characterized using a single scale parameter κ. The evolution of κ then serves as an indicator for the quantitative comparison between DNS-DEM and PBE simulations. We show that, by introducing an agglomeration kernel constructed in terms of the gyration radius of agglomerates and a sticking probability , PBE can well reproduce the results of DNS-DEM. A relationship between the sticking probability and particle properties is then proposed based on the scaling analysis of the equation for head-on collisions. We have also shown that the collision-induced breakage rate of agglomerates can be modelled based on the statistics of the collision rate and a breakage fraction function . The fraction function  is further expressed as a function of the wellknown distributions of the impact velocity and a universal transfer function ψ(vn ), which is shown to rely on particle-particle contacting interactions and is independent of particle inertia St, particle size, and hydrodynamic interactions. Based on a large number of simulations, we propose an exponential function of adhesion parameter Adn for the breakage rate of doublets and show that the breakage rate increases linearly as the agglomerate size increases. The framework in the current chapter enables one to estimate the agglomeration and breakage rates for agglomerates in homogenous isotropic turbulence.

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Chapter 4

Migration of Cloud of Low-Reynolds-Number Particles with Coulombic and Hydrodynamic Interactions

4.1 Introduction The migration of charged particles in an electrostatic field is ubiquitous in engineering processes, including dust removal in an electrostatic precipitator [1], films fabrication using electrophoretic deposition [2], and self-assembly of colloidal particles [3]. During these processes, both the hydrodynamic and electrostatic interactions between multiple particles result in a wealth of complex collective behaviors of the particle clouds. As we introduced in Sect. 1.4, most studies pay attention to the settling of clouds under gravity, while the migration of charged particles in an electrostatic field is rarely involved. The long-range repulsion between charged particles has been found to have significant effects on particle packing structure [4], clustering of particles in turbulence [5], shear-induced ordering in suspensions [6], and clogging/non-clogging transition in colloids [7]. However, the migration behavior of particle clouds with a coupling effect of both hydrodynamic and electrostatic interactions is still less investigated. In this chapter, we incorporate electrostatic interactions in a rigorous manner and perform Oseenlet simulations of migrating clouds in a wide range of particle Reynolds number Rep and cloud-to-particle size ratio R0 /r p to show how the Coulomb repulsion affects the evolution of a migrating cloud.

4.2 Formulation of Problem As shown in Fig. 4.1, We consider a spherical cloud with initial radius R0 containing N unipolarly charged non-Brownian particles (with charge number q0 ). The particles are randomly seeded in the cloud with an initial volume fraction φ = 2 × 10−6 and are assumed to be neutrally buoyant to avoid the gravity effect. In such a dilute condition, particle-particle collisions can be prevented, and higher-order multipoles, e.g. dipoles or quadrupoles, which decay sufficiently fast with distance, can be ignored. The cloud is immersed in an unbounded stationary fluid and migrates under the action of a © Tsinghua University Press 2023 S. Chen, Microparticle Dynamics in Electrostatic and Flow Fields, Springer Theses, https://doi.org/10.1007/978-981-16-0843-8_4

81

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4 Migration of Cloud of Low-Reynolds-Number Particles …

Fig. 4.1 Schematics of the simulation setup and the particle-particle electrostatic and hydrodynamic interactions

uniform electric field E 0 (in the vertical x direction). The hydrodynamic interactions among particles are modelled by Oseen equations, and we simultaneously consider the long-rang electrostatic forces: a uniform external electrostatic force (E 0 q0 ) and a pairwise Coulomb repulsion (q02 /4π ε f ri2j ), where ε f = 8.85 × 1012 F/m is the   permittivity of the fluid and ri j =  r i − r j  is the distance from the centroid of a source particle to that of a target particle. The phase diagram, adapted from Ref. [8], in Fig. 4.2 shows different regimes of evolution for a settling cloud of particles. In Fig. 4.2, all the physical quantities that influence the properties of a migrating cloud, including the particle radius r p and density ρ p , the fluid viscosity μ f and density ρ f , as well as the cloud radius R0 , can be grouped into two dimensionless parameters: one is particle Reynolds number Re p , which is related to the driving force through Re p = E 0 q0 ρ f /(6π μ2f ); the other is the cloud-to-particle size ratio R0 /r p . Since we have fixed the volume fraction φ, 3  parameters like particle number N = φ R0 /r p and dimensionless inertial length   l ∗ = r p /R0 / Re p can be directlyderived from Rep and R0 /r p . It is noted that the Stokes number St = (2/9) ρ p /ρ f Re p , which is kept in the range of 10−4 − 10−2 , does not play a key role here. We pick out nine points located in four different regimes in the regime map (Rep = 2.54 × 10−4 , 1.27 × 10−3 , 2.03 × 10−2 and R0 /r p = 400, 800, 1600) and systematically perform particle dynamic simulations by changing the strength of long-range Coulomb repulsion. The equation of motion for the particle can be written as ui = u f,i + us,i =

 j=i

⎛

W i j us, j

⎞ 2  q0 r i j ⎠ 1 ⎝ E 0 q0 + . + 6π μ f r p 4π ε f ri3j j=i

(4.1)

Here, u f,i is the fluid velocity at the center of particle i and us,i is the particle slip velocity. Since the Stokes number is sufficiently small, the particle inertia is calculated from the external force acting on the negligible, and us can be directly   particle through us = F ext / 6π μ f r p . Vector r i j = r i − r j is that from source point j to target point i. The interaction kernel W i j = W ri , r j is a function of the positions of particles and can be obtained from the Oseen solution for the flow around a particle:

4.2 Formulation of Problem

83

Fig. 4.2 Regimes of evolution for a migrating cloud characterized by particle Reynolds number Re p and cloud-to-particle size ratio R0 /r p . The points are the specific cases studied in this Chapter (©Cambridge University Press, reproduced with permission from Chen et al. [9])

u r,oseen =

u θ,oseen

 r cos θ θ)r Res r (1+cos θ) − p 2r − 3(1−cos exp −

 4r p  2r p Res r (1+cos θ) 3 1 − exp − + 2 Re 2r p s

u s r 2p r2

us r p sin θ =− r



  Res r (1 + cos θ ) 3 + exp − 4r 2 4 2r p r 2p

(4.2)

(4.3)

with the polar axis (θ = 0) coincident with the direction of particle motion. Re S = 2r p u s ρ f /μ f is the instantaneous particle Reynolds number based on its slip velocity u s . Normalizing the velocity by the migrating velocity of an isolated particle, which is U0 = E 0 q0 /(6π μ f r p ), and the length by the initial radius of the cloud R0 , Eq. 4.1 becomes  ˆ i j uˆ s, j + ex + κq i . (4.4) uˆ p,i = W i=i

    Here, i =(1/N ) j=i rˆ i j /ˆri3j is a function of the position vectors rˆ 1 , rˆ 2 , . . . , rˆ N of the particles. A dimensionless charge parameter is indicated as κq =

q0 N , 4π ε f E 0 R02

(4.5)

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4 Migration of Cloud of Low-Reynolds-Number Particles …

which is interpreted as the ratio of the velocity caused by particle-particle Coulomb repulsion to the velocity driven by the external field. The 3N coupled equations of motion of the particles Eq. 4.4 are solved by the Adams-Bashforth method with sufficiently small time steps. The information of all the particles is simultaneously recorded and thus the temporal evolution of the migrating cloud can be easily captured. Before presenting the results from particle dynamic simulations, we introduce a continuum description of cloud expansion in terms of the number density field n(ˆr , tˆ) ˆ r , tˆ). A continuity equation can be built based on the and induced velocity field u(ˆ conservation of particle number [8]: ∂ ˆ r , tˆ)n(ˆr , tˆ)) = 0 n(ˆr , tˆ) + ∇ · (u(ˆ ∂t

(4.6)

A similarity solution of Eq. 4.6 for a spherical cloud settling under gravity was given in the condition of a sufficiently low number density (N /R0  U0 /v f , v f = μ f /ρ f ). Under such a constraint, most particles in the cloud are outside the wake of any other particle, and the evolution  of the cloud is governed by repulsive source-field interactions u f,r = 3v f r p / 2r 2 [10]. Under such conditions, the expansion rate uˆ r ( Rˆ cl , tˆ) and the relative increase of the migrating velocity Uˆ = (U − U0 ) /U0 (Here, U is the defined as the average velocity of all particles) of the cloud can be estimated as    (4.7) uˆ r Rˆ cl , tˆ = 3r p N v f / 2R02 U0  1,       Uˆ = (6/5)N r p /R0 3/ 3 + Re p R0 /r p ≈ 18r p N v f / 5R02 U0  1.

(4.8)

It indicates that the hydrodynamic interaction between particles is extremely weak and the particles behave like isolated ones. Moreover, as mentioned in Sect. 1.4, the settling cloud is quite unstable and tends to break when the particle number density is high. It restricts the scope of application of the continuum description. We will show below that the Coulomb repulsive interaction makes the migrating cloud more stable so that the continuum description can be readily applied to predict the long-time evolution of the migrating velocity and the cloud shape.

4.3 Effect of Coulomb Repulsion on Cloud Shape 4.3.1 Cloud Shape We first consider two migrating clouds in the Stokes regime (Re p = 2.54 × 10−4 )   with the same R0 /r p and initial cloud configuration rˆ 1 , rˆ 2 , . . . , rˆ N . In the first case, we set the interparticle Coulomb repulsion to zero, whereas a strong repulsion is employed in the second case. The typical evolution of the two clouds is displayed

4.3 Effect of Coulomb Repulsion on Cloud Shape

85

Fig. 4.3 Typical evolution of clouds in the Stokes regime with Re p = 2.54 × 10−4 , R0 /r p = 800, and with a κq = 0 and b κq = 1.0 (©Cambridge University Press, reproduced with permission from Chen et al. [9])

in Fig. 4.3. The cloud with κq = 0 is seen to flatten and to expand in the horizontal direction. A circulation of the particles inside the cloud is also observed, which can be regarded as a typical feature for Stokes clouds or drops [11]. At the same time, a leakage of particles occurs at the rear of the cloud (Fig. 4.3a). The flat cloud further expands and breaks up into two secondary clouds, each of which further breaks again. All these features qualitatively resemble the numerical and experimental results of Stokes cloud settling under gravity [12, 13], indicating that, with extremely weak Coulomb repulsion (κq → 0), the effect of the external electrostatic field (E 0 ) is similar to that of gravity. In the presence of strong Coulomb repulsion (Fig. 4.3b), the cloud undergoes an expansion without breakup during a long-term observation (tˆ ∼ 100). After a compression in the vertical direction in the initial stage, the cloud then undergoes a self-similar expansion with size increased and geometrical shape unchanged. Figure 4.4 clearly shows the distribution of particles in the two clouds. In the case without Coulomb repulsion, two distinct secondary clouds with high local volume fraction are displayed. Both secondary clouds have rugged surfaces, which may lead to further destabilization. In contrast, with a strong repulsion, the cloud has a relatively smooth boundary and a uniform particle distribution.

86

4 Migration of Cloud of Low-Reynolds-Number Particles …

Fig. 4.4 Scatter plots of particle dispersion in migrating clouds with Re p = 2.54 × 10−4 , R0 /r p = 800, and with a κq = 0 and b κq = 1.0. The particle position relative to the center of mass of the cloud with their local volume fraction is plotted at tˆ = 60. Top and bottom panels correspond to top views and side views of the clouds, respectively (©Cambridge University Press, reproduced with permission from Chen et al. [9])

In Fig. 4.5, the evolution of horizontal-to-vertical aspect ratio λ = Rh /Rv for clouds with different repulsion is shown. The horizontal radius Rh is defined as the average of the maximum distance from the center of mass over four quadrants in the horizontal plane. The vertical radius Rv is defined as the distance from the front leading particle to the center of mass of the cloud. It can be seen that the aspect ratio increases in all cases due to the expansion in horizontal direction. Obvious fluctuations appear in the case with κq = 0, which is a result of the coupling between the toroidal circulation of the cloud and the expansion in the horizontal direction [12]. When the cloud breaks up, particles in the region with a higher particle concentration has a larger migrating velocity, leading to a stretch of the cloud in the vertical direction. The aspect ratio λ drops remarkably at tˆ ≈ 30. In cases with strong Coulomb repulsions, the fluctuation in λ is significantly inhibited, and the growth rate of λ decreases as κq increases. When a sufficiently large repulsion is imposed, λ stays close to unity, indicating that the cloud can keep its spherical shape.

4.3 Effect of Coulomb Repulsion on Cloud Shape

87

Fig. 4.5 Evolution of the aspect ratio of clouds in the Stokes regime with Re p = 2.54 × 10−4 and R0 /r p = 800. We show results with four different strengths of Coulomb repulsion: κq = 0 (circles), κq = 0.11 (upward-pointing triangles), κq = 1.0 (squares), and κq = 11 (right-pointing triangles) (©Cambridge University Press, reproduced with permission from Chen et al. [9]) Fig. 4.6 The shapes of particle clouds under two different particle Reynolds numbers (Re p = 2.54 × 10−4 and Re p = 2.03 × 10−2 ) and two charge parameters (κq = 0 and κq = 0.12)

4.3.2 Effect of Fluid Inertia We then discuss how the fluid inertia affects the evolution of the cloud by varying the particle’s Reynolds number Re p . The shapes of cloud with Re p = 2.54 × 10−4 and Re p = 2.03 × 10−2 are shown in Fig. 4.6 under two different κ p values. For particles without Coulomb interaction, there is an obvious breakage of the cloud at a small Re p number. In contrast, for particles with a higher Re p value, the cloud does not break into secondary clouds and evolves into irregular shapes instead. When the Coulomb repulsion is strong, the particles are uniformly distributed in the clouds enclosed by sharp boundaries for both low- and high-Reynolds-numbers. The shape of low-Reynolds-number cloud is close to sphere. In contrast, the cloud with a high Reynolds number is slightly concave, which is caused by the inflow at the rear of the cloud.

88

4 Migration of Cloud of Low-Reynolds-Number Particles …

In Fig. 4.7, we plot the velocities of particles in clouds with the same configuration but different Re p and κq . When the Coulomb repulsion is extremely weak (κq = 0.01 in Fig. 4.7a–c), we found the following characteristics that are qualitatively similar to clouds settling under gravity: (1) a conspicuous circulation in a toroidal vortex is observed in the Stokes drop-like cloud; and (2) as the inertia increases, the inflow at the rear of the cloud becomes stronger and drastically diminishes the particle leakage [13, 14]. With a sufficiently strong repulsion (κq = 5.0), the velocity of particles becomes radial. With such a high κq , the direct Coulomb interaction term (κq i ) becomes dominant and the evolution of clouds is no longer affected by fluid inertia (see Fig. 4.7g–i). We check all the simulation runs with different R0 /r p and Re p and find that, when a sufficiently large κq is given, the clouds always remain in a stable shape. It is worth noting that the velocity vectors in Fig. 4.7f have a pattern that is closer to an isotropic form than those in Fig. 4.7d. It indicates that the cloud with higher Rep transits from the hydrodynamically controlled regime to the isotropic expansion regime at a relatively lower κq . This effect of fluid inertia can be understood by considering the hydrodynamic interaction between two identical particles. Figure 4.8a shows a simple case where particle i has a unit slip velocity and generates an induced velocity at the location of particle j. According to the Oseen solution in Eqs. 4.2 and 4.3, the induced velocity is a function of the ratio between the distance and the particle radius r/r p , the angle θ between the direction of slip velocity and the vector from particle i to particle j, and the particle Reynolds number Re  p . Here,  r/r p is fixed as uˆ r,Oseen  in Fig. 4.8b) and 10 and the magnitude of velocities in the radial direction (   for four distinct tangential velocity (uˆ θ,Oseen  in Fig. 4.8c) at different   θ areplotted Re p (0.001, 0.01, 0.1, and 0.5). It is evident that uˆ r,Oseen  (uˆ θ,Oseen ) reaches its maximum at θ = π (θ = π/2). The induced velocities in both radial and tangential directions decrease with increasing Re p (except at θ = π ). Given the same κq , clouds with higher Re p will have relatively weaker hydrodynamic interaction (i.e. smaller  ˆ ˆ s, j in Eq. 4.4). This result confirms the findings in Fig. 4.7 that clouds j=i W i j u with higher Re p transit into the isotropic expansion regime at a relatively lower κq .

4.3.3 Stability of the Cloud To further understand the stability of the cloud and the breakage mechanism, we show the angular distribution of the particles on y − z plane in Fig. 4.9. For particles without Coulomb interaction, the initial nonuniformity in the angular distribution will be enhanced. The reason is that the hydrodynamic interaction is stronger for particles in the high concentration region, which leads to a larger migrating velocity of these particles and a stretch of the cloud toward the corresponding direction. For the case shown in Fig. 4.9a, the initial particle concentration is slightly higher at 0◦ and 180◦ . As a result, at tˆ = 60, there is a remarkable accumulation of particles around 0◦ and 150◦ . For particles with strong Coulomb interaction, the initial nonuniformity of the

4.3 Effect of Coulomb Repulsion on Cloud Shape

89

Fig. 4.7 Velocity vectors of particles in clouds with the same configuration but different Reynolds number (Re p ) and repulsion (κq ). Each column has the same Re p (from left to right: Re p = 2.54 × 10−4 , 1.27 × 10−3 and 2.03 × 10−2 ); and each row has the same κq , (from top to bottom: κq = 0.01, 1.0 and 5.0). The cloud-to-particle size ratio is fixed as R0 /r p = 800. For clarity, only 300 particles near the plane of zˆ = 0 are drawn for each case (©Cambridge University Press, reproduced with permission from Chen et al. [9])

Fig. 4.8 a Alignment of two particles and the magnitude of velocities in the radial direction (b) and tangential direction (c) at the location of particle j induced by the slip velocity of particle i. In b and c θ stands for the angle between the direction of slip velocity and the vector from particle i to particle j. Results for four different particle Reynolds numbers are presented: Re p = 0.001 (blue solid lines), Re p = 0.01 (red dashed lines), Re p = 0.1 (yellow dotted lines) and Re p = 0.5 (green dash-dotted lines) (©Cambridge University Press, reproduced with permission from Chen et al. [9])

90

4 Migration of Cloud of Low-Reynolds-Number Particles …

Fig. 4.9 The angular distribution of particles in y − z plane for cases with a κq = 0 and b κq = 0.12

angular distribution is suppressed. The angular distribution of the particles inside the cloud is quite uniform at tˆ = 30 and tˆ = 60 (see Fig. 4.9b). These results indicate that, the particle-scale Coulomb interaction will reduce the nonuniformity of particle concentration and prevent the breakage of the cloud. We then evaluate an average of the Lyapunov exponent to characterize the chaotic nature of the particle motion. Two simulations with slightly different initial configurations (termed as A and B) were performed for a cloud. Configuration B was prepared from A by moving each particle in a random direction with a magnitude of σ = 10−8 . We then calculate the distance between these two configurations as   N 1   A 2  2  2  xi − xiB + yiA − yiB + z iA − z iB . l= N i=1

(4.9)

The evolutions of l for different κq values are displayed in Fig. 4.10. A stronger Coulomb interaction leads to a lower rate of l. Moreover, l nearly follows  increasing  an exponential trend l(tˆ) ∼ exp λ y tˆ and the divergence rate λ y is related to the Lyapunov exponent. The system with a larger value of λ y tends to be more chaotic. We obtain the value λ y = 0.31 for particles without Coulomb interaction. This value is close to λ y value (0.36) for particles settling under gravity [12]. When κq increases from 0 to 0.12, λ y decreases from 0.31 to 0.047, confirming that Coulomb repulsion makes the migrating cloud more stable.

4.4 Evolution of Particle Cloud Under Strong Repulsion

91

Fig. 4.10 The evolution of the distance l between two configurations A and B for different κq . The dashed lines are exponential fittings   l(tˆ) ∼ exp λ y tˆ and the values of λ y are also displayed in the figure

4.4 Evolution of Particle Cloud Under Strong Repulsion 4.4.1 Scaling Analysis and Continuum Description In this subsection, we will focus on the effect of Coulomb repulsion and try to answer the questions: how to characterize the transition from an unstable migrating cloud to a stable cloud without breakup, and what rules the stable cloud would follow during its evolution. The influence of Coulomb repulsion is evaluated by a scaling analysis of the particles’ equation of motion (Eq. 4.4). The simplest term in Eq. 4.4 is the mobility caused by the external field (ex ), which has no effect on the evolution of the cloud shape. Here, we compare the contribution from the remaining  two terms: the mobility due to Coulomb repulsion from other particles ((κq /N ) j=i rˆ i j /(ˆri3j )) and  ˆ i j uˆ s, j ). the fluid velocity induced by the slip velocities of other particles ( j=i W In the second term, uˆ s, j can be calculated from the external force and the repulsive forces on particle j. It thus can be regarded as an indirect effect of the imposed external field and the interparticle electrostatic repulsions. The relative magnitudes of these two terms can be estimated as:         ˆ i j uˆ s, j  ˆ i j ex + κq k= j rˆ ik3   j=i W  j=i W N rˆik  =    = i,ext + i,Coul . i =    κq  κq rˆ i j  rˆ i j  N    3 3 j=i rˆi j j=i rˆi j N (4.10) Here, i,ext is the ratio of the velocity caused by the indirect effect of the imposed external field to the velocity caused by the direct Coulomb repulsion, and i,Coul is the ratio between the velocities from the indirect and direct effects of Coulomb repulsion. Note that the only anisotropic term (ex ) is included in i,ext , and thus the clouds will undergo an isotropic expansion if i,ext  1. If i,Coul  1 as well, the isotropic expansion will be determined by the direct repulsive interaction.

92

4 Migration of Cloud of Low-Reynolds-Number Particles …

According to Eq. 4.10, i,ext and i,Coul can be written as i,ext Here, i =

1 N



    ˆ e W   i j x j=i 1 = and i, Coul = |i | κq

rˆ i j j=i rˆi3j

   ˆ i j  j   j=i W |i |

.

(4.11)

is a function of particles’ position. For simplicity, we ignore

ˆ i j is also determined by the particles’ position. the effect of fluid inertia here, so that W According to Eqs. 4.2, 4.3 and Fig. 4.8, the maximum induced radial velocity uˆ r,Oseen rˆ at the position of particle i due to particle j (at θ = π ) scales as O( rˆipj ); and the  rˆ ˆ maximum of uˆ θ,Oseen (at θ = π/2) also scales as O( rˆipj ). Thus j=i W i j can be  rˆ p rˆ i j estimated as j=i rˆ 2 . Here, we assume the target particle i is located close to the ij boundary of the cloud and use an integration to approximate the summation. Then ˆ i j can be estimated as:

i and j=i W          

 1   1   ˆ ˆ r ˆ r − r i j i

∼ ˆ rˆ  ∼ O Rˆ cl−2 | i | =  d V   3    rˆ i − rˆ 3  N j=i rˆi j   Vˆcl 

(4.12)

Vˆcl

      

       ˆ ˆ ˆ N r ˆ r r ˆ r − r p i j p i    

ˆ rˆ  ∼ O N rˆ p Rˆ cl−1 ˆ ij ≈  ∼ d V W     j=i j=i rˆ 2   Vˆcl    rˆ i − rˆ 2 ij   Vˆcl (4.13) Here, Rˆ cl = R(t)/R0 and R(t) is the cloud radius at time t. Then Eq. 4.11 takes the form 1 N rˆ p Rˆ cl ∝ R(t) and i,coul ≈ N rˆ p Rˆ cl ∝ R(t). (4.14) i,ext ≈ κq   

It is straightforward that the condition i,ext  1 can always be met through increasing κq . However, i,Coul is independent of κq , so that the relative magnitude of the direct and indirect contributions of the Coulomb repulsion is determined only by the cloud configuration. Both i,ext and i,Coul are proportional to cloud radius R(t), indicating that a repulsion-dominant cloud will evolve into the hydrodynamically ∼ 1 or i,Coul∼ 1) when its radius reaches a controlled regime (with either

 i,ext −1  −1 . , κq N rˆ p critical value Rˆ cl,crit ∼ min N rˆ p Substituting Rˆ cl with its initial value 1 into the direction-repulsion-dominant condition ( i,ext  1 and i,Coul  1), a new regime of isotropic expansion is identified as  2 R0 φ ≡ κq,t and κq,t  1. (4.15) κq rp

4.4 Evolution of Particle Cloud Under Strong Repulsion

93

A migrating cloud in this regime will undergo an isotropic expansion before its radius reaches a critical value Rˆ cl.crit . Equation 4.4 (in a reference frame moving with velocity U0 ) is simply reduced to uˆ p,i ≈ κq i , which can be transformed into a continuous form:  κq rˆ − rˆ     ˆ r , tˆ) = (4.16) u(ˆ   n rˆ , tˆ dV rˆ  rˆ − rˆ 3 N Vcl

It gives the induced velocity term in Eq. 4.6. Then, for a spherical cloud with initial density field independent of angular coordinates, Eqs. 4.6 and 4.16 take the form  ∂n 1 ∂  uˆ r (ˆr , tˆ)ˆr 2 n = 0, + 2 ˆ r ˆ ∂ r ˆ ∂t κq uˆ r (ˆr , tˆ) = N

Rcl π 0

0

    rˆ − rˆ cos β n rˆ , tˆ

2

 3/2 sin βdβ rˆ drˆ , 2

2

rˆ + rˆ − 2ˆr rˆ cos β

(4.17)

(4.18)

where β is the azimuthal angle between rˆ and rˆ .

4.4.2 Prediction of Cloud Size and Migrating Velocity To predict the evolution of size and migrating velocity for clouds with strong Coulomb repulsion, we introduce the standard similarity solution of Eq. 4.17. A similarity formulation is employed as n = f (tˆ)g( ˆ η) ¯ with the similarity variable η¯ = rˆ /tˆγ . Transforming Eq. 4.17, we obtain ⎤ ⎡ η¯ 4π κ d f (tˆ) ∂ f (tˆ) ∂ gˆ q ⎣g( = − f 2 (tˆ) 2 g( ˆ η¯ )η¯ 2 dη¯ ⎦ . gˆ − γ η¯ ˆ η) ¯ η¯ N ∂ η¯ dtˆ tˆ ∂ η¯

(4.19)

0

There should be no explicit time dependence in Eq. 4.19 to ensure the similarity solution to work, implying that f (tˆ) ∼ 1/tˆ. The integral constraint of a constant total number of particles in a cloud gives γ = 1/3. Then another non-dimensional similarity variable, defined as η = η/(4π ¯ κq /N )1/3 = rˆ /(4π κq tˆ/N )1/3 is used to form a dimensionless number density, given by g(η) = 4π gκ ˆ q /N , and Eq. 4.19 takes the form ⎤ ⎡ η  d d (gη3 ) = 3 ⎣g g(η )η 2 dη ⎦ . (4.20) dη dη 0

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4 Migration of Cloud of Low-Reynolds-Number Particles …

Integrating Eq. 4.20 and taking the limit η → ∞ to determine an integration constant, we can obtain an integral equation for g(η) η 3

gη 2 dη = η3 (η ≤ ηm ),

(4.21)

0

where ηm is the radial dimension of the cloud in similarity coordinates. The solution of Eq. 4.21, g = 1, is straightforward. Thus, the evolution of number density field and the cloud radius are obtained, which can be expressed as n(ˆr , tˆ) =

N 4π κq tˆ

 1/3 Rˆ cl (tˆ) = 3κq tˆ

(4.22) (4.23)

Applying Eqs. 4.22 and 4.23 in Eq. 4.18, the induced velocity field reduces to  uˆ r (ˆr , tˆ) =

κq 9tˆ2

 13

rˆ .  Rcl

(4.24)

These solutions build the relationship between the cloud evolution and the charge parameter κq . In Fig. 4.11, we plot the induced velocity (in radial direction) in clouds with weak (κq = 0.01κq,t ) and strong (κq = 5.2κq,t ) repulsions and the corresponding particle density profiles. The colored scatters and lines are results from particle dynamic simulations, and the black solid lines are the scaling laws in Eqs. 4.24 and 1/3 4.22. An evident result is that the normalized induced velocities uˆ r tˆ2/3 /κˆ q for clouds with strong repulsion all collapse onto a single universal curve (Fig. 4.11b) formulated by Eq. 4.24. The corresponding number density profiles, plotted in a rescaled form n · 3κq tˆ, at different time also collapse (Fig. 4.11d). The number density almost remains constant across the entire cloud, and its value can be simply derived from the initial condition of number density n 0 through n(tˆ) = n 0 / Rˆ cl3 (tˆ). These results indicate that a cloud with strong repulsion undergoes a self-similar expansion, which is well predicted by our scalings. When the repulsion interaction is weak (κq  κq,t ), the scaling may break down. In such circumstances, the particles’ motion is dominated by hydrodynamic interaction. Particles are circulating in the clouds which results in a significant scatter of the data points in Fig. 4.11a. The cloud will break into secondary clouds with a local high particle concentration (as shown in Fig. 4.4a), corresponding to peaks in density profile in Fig. 4.11c. Figure 4.12a show how the cloud radius Rˆ cl evolves as tˆ increases for different κq /κq,t . The analytical solution Eq. 4.23 can well predict the evolution of the radius for cloud with κq /κq,t = 5.2, whereas it underestimates the radius for cloud with

4.4 Evolution of Particle Cloud Under Strong Repulsion

95

1/3 Fig. 4.11 Normalized velocity uˆ r tˆ2/3 /κq as a function of rˆ / Rˆ cl for cloud with a κq = 0.01κq,t and b κq = 5.2κq,t ; c and d are corresponding rescaled density field n · 3κq tˆ as a function of rˆ / Rˆ cl . Color code spans from blue to yellow with increasing time (from tˆ = 20 to tˆ = 90 ) and black solid lines are theoretical predictions given by Eq. 4.24 and Eq. 4.22 (©Cambridge University Press, reproduced with permission from Chen et al. [9])

κq /κq,t = 0.01, where the separation between sub-clouds becomes the main cause of the increase of the cloud radius. Besides the cloud radius, the cloud migrating velocity Uˆ cl is also of great significance for design of industry unit. For neutral clouds, one can only predict the velocity at a very early stage before the breakup occurs. With a strong repulsion, the formation of relatively stable configurations enables us to predict the long-term evolution of Uˆ cl . Based on the analytical expression for cloud settling velocity under gravity in earlier study [11] and the prediction of cloud radius Rˆ cl in Eq. 4.23, we construct a new formula of the migrating velocity for the cloud in repulsion-dominant regime:   rˆ p 6 + 1, Uˆ cl (tˆ) = N f R ∗ 5 (3κq tˆ)1/3

(4.25)

Here, R ∗ = Re p Rcl /r p = 1/l ∗ is the instantaneous system inertia scale and f (R ∗ ) = 3/(3 + R ∗ ) is the correction factor for the fluid inertia. Figure 4.12b shows that the simulation results nicely collapse on the theoretical line Eq. 4.25 in the condition of strong repulsion. To generalize these results, we run a large number of simulations for clouds with different particle Reynolds number Re p and cloud-to-particle size ratio R0 /r p with κq /κq,t ranging from 10−3 to 101 . We rewrite Eq. 4.23 as Rˆ cl /(tˆκq,t )1/3 = (3κq /κq,t )1/3 and Eq. 4.25 as 5(Uˆ cl − 1)(tˆκq,t )1/3 /(6N f (R ∗ )ˆr p ) = (3κq /κq,t )−1/3 and plot all the simulation results at tˆ = 10 as a function of κq /κq,t in Fig. 4.13. It can be seen that for κq /κq,t  1, the scatters in the data are large. Whereas, for κq /κq,t > 1 the scaled cloud radius and migrating velocity all fall into a narrow range around the theoretical curves. Figure 4.13 also displays snapshots of clouds at different Re p . We observe that for κq /κq,t < 1, the clouds are compressed into

96

4 Migration of Cloud of Low-Reynolds-Number Particles …

Fig. 4.12 Evolution of a cloud radius Rˆ cl and b migrating velocity Uˆ cl with κq = 0.01κq,t , κq = 0.1κq,t and κq = 5.2κq,t . The particle Reynolds number is Re p = 2.54 × 10−4 and cloudto-particle size ratio is R0 /r p = 800. Red lines are theoretical predictions of Eqs. 4.23 and 4.25 (©Cambridge University Press, reproduced with permission from Chen et al. [9])

Fig. 4.13 Scaled cloud radius Rˆ cl /(tˆκq,t )1/3 and migrating velocity Uˆ cl∗ = 5(Uˆ cl − 1)(tˆκq,t )1/3 /(6N f (R ∗ )ˆr p ) as functions of κq /κq,t at tˆ = 10. The scatters are results of Oseenlet simulations and the black lines are corresponding theoretical predictions. Legends are the same as in Fig. 4.2 (©Cambridge University Press, reproduced with permission from Chen et al. [9])

different shapes, while the clouds with κq /κq,t > 1 stay spherical. Therefore, combining our results for Rˆ cl (tˆ) and Uˆ cl (tˆ) highlights a Coulomb repulsion controlled regime characterized by the dimensionless ratio κq /κq,t .

4.4 Evolution of Particle Cloud Under Strong Repulsion

97

4.4.3 Discussion For dilute clouds considered here, the isotropic expansion condition can also be understood through the force balance between the hydrodynamic drag  q02 r i j 1 6π μ f r p u r,Oseen and pairwise Coulomb repulsion 6πμ f r p j=i 4πε f ri3j . Substituting u r,Oseen with the maximum inward flow velocity at the rear boundary of  2 E 0 q 6Nr p R0 the cloud 6πμ [11], we can reproduce Eq. 4.15 as κ ∼ φ. A balance q,t rp f r p 5R0 ∗ 1/3 ˆ length scale can be calculated as lq ∼ q0 /(4π ε f E 0 r p N ), and the isotropic condition is satisfied if the typical distance between neighboring particles l is smaller than lˆq∗ . It is straightforward to see from Eq. 4.23 that lˆ increases with time as lˆ ∼ tˆ1/3 . Therefore, an initially isotropic expanding cloud will evolve into the hydrodynamiccontrolled regime after a sufficiently long time. Recall that the condition of a sufficiently low number density (N /R0  U0 /ν f ) indicates another length scale lˆ∗f ∼ N 2/3 r p /(Re p R0 ). If the transition happens at lˆ lˆ∗f , the wake interactions remain negligible and further evolution will still be controlled by O(1/r 2 ) type source interaction u f,r = 3ν f r p /(2r 2 ) [10]. Then the evolution of the cloud radius Rˆ cl is given as [8]   3N Q 1/3 ˆ Rcl (tˆ) = , (4.26) 4πU0 R02 with Q = 6π ν f r p . We have simulated some cases to observe this transition and find that an expanding cloud will finally resemble the behavior of a neutral cloud settling under gravity. In Fig. 4.14, we present a typical case of a long-time evolution of the cloud radius Rˆ cl to show the transition. The evolution only has a change in intercept when presented in log-log coordinates.

Fig. 4.14 Long-time evolution of cloud radius Rˆ cl . The solid line is the theoretical prediction for electrostatic-interaction-dominant expansion in Eq. 4.23 and the dashed line is the theoretical prediction, Eq. 4.26, of Subramanian and Koch [8] for a neutral cloud with O(1/r 2 ) type source interactions (©Cambridge University Press, reproduced with permission from Chen et al. [9])

98

4 Migration of Cloud of Low-Reynolds-Number Particles …

It should be noted that, owing to the low particle volume fraction in our simulation, it is reasonable to consider a particle as a point when calculating electrostatic and hydrodynamic interactions. However, for a sufficiently dense cloud, the finite size of the particles substantially affects the interactions between particles: (1) higherorder multipoles in electrostatic interaction, e.g. dipoles or quadrupoles, cannot be neglected; and (2) higher-order moments of the traction taken over the particle surface should be considered. A reasonable estimation of the volume fraction over which the point force condition is valid can be obtained by comparing the relative magnitude of: (1) the interaction between point charged particles (FCoul ) versus that between induced dipoles (Fdi pole ); and (2) the flow field induced by point force (U P F ) versus that induced by the degenerate quadrupole (U D Q ) (finite size term). These conditions can be written as r 2p Fext q02 3 p2 Fext and 4π ε f l 2 2π ε f l 4 4π μ f l 12π μ f l 3 !" # !" # !" # !" # FCoul

Fdipole

UP F

(4.27)

UD Q

Here, p = 4π ε f K r 3p E 0 is the dipole moment of a particle induced by the exterε −ε nal field [15], and the Clausius-Mossotti factor K = ε pp+2εf f is a function of particle permittivity ε p and the fluid permittivity ε f . The typical distance between two neighboring particles l scales as l = φ −1/3 r p . Therefore, the point force simplification is suitable when the volume fraction satisfies: $ φ

q0 10π ε f K r 2p E 0

%3 and

φ

1 . 3

(4.28)

4.5 Summary In this chapter, we impose electrostatic interactions into Oseenlet simulations to study the behavior of a migrating cloud containing charged particles. We found that, for particles with a weak Coulomb interaction, the migrating cloud expands and breaks up into secondary clouds, which resembles the numerical and experimental results of Stokes cloud settling under gravity. We have shown that the Lyapunov exponent decreases as the dimensionless charge parameter κq increases, indicating that the long-range Coulomb repulsion makes the cloud more stable. The simulated results with a strong repulsion can be described with a continuum convection equation, which predicts the evolution of the density field, the radius, and the migrating velocity of the cloud. A dimensionless charge parameter κq is proposed to quantify the effect of the repulsion, and the ratio κq /κq,t successfully captures the transition from the hydrodynamically controlled regime κq /κq,t < 1 to the repulsion-controlled regime κq /κq,t > 1.

References

99

References 1. M. Yang, S. Li, G. Liu, Q. Yao, in American Institute of Physics Conference Series, vol. 1542 (2013), pp. 943–946 2. J. Cordelair, P. Greil, J. Mater. Sci. 39(3), 1017 (2004) 3. A. Kumar, B. Khusid, Z. Qiu, A. Acrivos, Phys. Rev. Lett. 95(25), 258301 (2005) 4. A. Schella, S. Herminghaus, M. Schröter, Soft Matter 13(2), 394 (2017) 5. J. Lu, H. Nordsiek, E.W. Saw, R.A. Shaw, Phys. Rev. Lett. 104(18), 184505 (2010) 6. E. Nazockdast, J.F. Morris, Soft Matter 8(15), 4223 (2012) 7. G.C. Agbangla, P. Bacchin, E. Climent, Soft Matter 10(33), 6303 (2014) 8. G. Subramanian, D.L. Koch, J. Fluid Mech. 603, 63 (2008) 9. S. Chen, W. Liu, S.Q. Li, J. Fluid Mech. 835, 880 (2018) 10. G.K. Batchelor, An Introduction to Fluid Dynamics (Cambridge University Press, 2000) 11. M. Yang, S.Q. Li, J.S. Marshall, J. Aerosol Sci. 90, 154 (2015) 12. B. Metzger, M. Nicolas, E. Guazzelli, J. Fluid Mech. 580, 283 (2007) 13. F. Pignatel, M. Nicolas, E. Guazzelli, J. Fluid Mech. 671, 34 (2011) 14. T. Bosse, L. Kleiser, C. Härtel, E. Meiburg, Phys. Fluids 17(3), 037101 (2005) 15. T.B. Jones, Electromechanics of Particles (Cambridge University Press, 2005)

Chapter 5

Deposition of Microparticles with Coulomb Repulsion

5.1 Introduction The primary concern of the current chapter is the deposition of micro-sized charged particles, which is ubiquitous in areas of material, astrophysics, and environmental science [1–4]. For example, electrophoretic deposition (EPD) is one of the phenomena related to the packing of charged particles, in which the charged suspended particles are transported by an external field and has been widely applied to the fabrication of wear resistant coatings as well as functional nanostructured films [4, 5]. It is reported that both the interparticle electrostatic interaction and the applied field significantly affect the quality of the deposits [6, 7]. Electrostatic precipitators (ESP), one of the most commonly used dust-removal methods in industrial processes, is another related application. In an ESP, dust particles are charged and deposited on a collecting plate to form a dust cake [8]. It has been reported that the dust collection efficiency is closely related to the cake structure [9]. In addition, structure changes caused by the long-range electrostatic interactions also exist in the areas of colloidal suspensions [10, 11], protoplanetary disks [12, 13], and contamination of optical lens[2, 14] or integrated circuits [15]. Investigations of packing of charged particles are of importance to better understand these processes. Nonetheless, few publications deal with that by now. In this chapter, the fast adhesive DEM (as described in Chap. 2) is applied to ballistic packings of micron-sized charged particles, with the aim of elucidating the effect of interparticle Coulomb interaction on the packing structure. We try to extract simple but effective rules that can predict packing properties, through extensive simulations and in-depth scaling analysis. In particular, the fluid effect is filtered out by assuming a vacuum condition to develop an “ideal” system.

© Tsinghua University Press 2023 S. Chen, Microparticle Dynamics in Electrostatic and Flow Fields, Springer Theses, https://doi.org/10.1007/978-981-16-0843-8_5

101

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5 Deposition of Microparticles with Coulomb Repulsion

Table 5.1 Simulation parameters for deposition of charged particles Properties Value Unit Particle properties Particle radius, r p Density, ρ p Poisson’s ratio, σ Elastic modulus, E Restitution coefficient, e Friction coefficient, μ f Surface energy, γ Charge on particles, q Typical parameters Length, L Hight, H Initial velocity, U0 Particle number, Ntot

1.0, 2.0, 5.0 2500 0.33 2 × 108 0.7 0.3 5.0–15.0 0–500

µm kg/m3 – Pa – – mJ/m2 e0 (1.6 × 10−19 C)

28 160 0.5–2.0 2000

rp rp m/s –

Reproduced from Chen et al. [21] with permission of The Royal Society of Chemistry

5.2 Models and Methods 5.2.1 Simulation Conditions We consider a random free falling of 2000 spheres in x direction with an initial injection velocity U0 from a specific height H . The square plane for particle deposition in the bottom has a width of L = 28r P (r P is the particle radius). The value is large enough such that the packing structure does not depend on L [16]. Periodic boundary conditions are applied along the horizontal, y and z, directions to avoid lateral wall effect. The ballistic deposition condition can be directly compared with the aforementioned EPD and ESP processes, where U0 can be regarded as the migration velocity of particles derived from the balance between electrostatic forces and the fluid drag (see Chap. 4). Similar conditions have been widely adopted in both experimental [17, 18] and numerical [8, 16, 19] studies. The physical parameters of our simulations are listed in Table 5.1. Note that the surface energy γ is varied over the range 5 − 15 mJ/m2 , which is the typical range for silica microspheres, to reflect the effect of van der Waals adhesion [1]. The number of elementary charge e0 , which equals 1.6 × 10−19 C, on a particle is determined according to the typical surface charge density due to the diffusion and field charging for micro-particles [1]. Due to the low conductivity among dielectric particles, the charge on a particle is assumed to be unchanged during the packing process. Higher-order multipoles, e.g., dipoles or quadrupoles, decay sufficiently fast with the distance and are ignored in this work. Such interactions, which may have effects on particle dynamics in the presence of electrodes or strong external fields [11, 20], will be left for future investigation.

5.2 Models and Methods

103

5.2.2 Forces on Particles To investigate how the particle scale interactions affect the packing structure, we track the translational and rotational motions of each particle using the fast adhesive DEM. In this chapter, we consider two kinds of forces, i.e., the long-range Coulomb force and short-range contact forces, acting on each particle (as summarized in Fig. 5.1). The models for the short-range contact forces have be introduced in Chap. 2. Besides the contact forces, the presence of charged particles induces an electric field, which decays slowly with distance. The charged particles in turn bare the force exerted by the induced field, which can be rewritten as FiCoul = qi E = qi

 q j ri j , 4π ε0 ri3j j=i

(5.1)

where qi is the charge on particle i, ri j = xi − x j is the vector from the centroid of source particle j to the target particle i and ε0 is the permittivity of the vacuum. The long-range feature of the electrostatic forces poses challenges in calculating the pair interactions among thousands of particles. For a system with N particles, the cost of direct calculation of the pair-wise Coulomb interactions scales as O(N 2 ) leading to an unacceptably low simulation efficiency for systems with large N . This difficulty can be overcome by employing a fast multipole method (FMM) which provides an approximation for electrostatic forces on a target particle exerted by a group of particles located sufficiently far away. Particles are separated into boxes and the electric field generated by box l can be expressed in terms of the multipole expansion as E(r) =

+∞  +∞  +∞  (−1)m+n+k m=0 n=0 k=0

m!n!k!

Il,mnk

∂ m+n+k K(r − rl ), ∂ x m ∂ y n ∂z k

(5.2)

where Il,mnk is the box moment and the interaction kernel KCoul (r) = r/4π ε0 (r )3 depends only on the location of box centroid rl and the target point r. The computing cost thus is reduced to O(N log N ) , with the precision controlled by an analytic error bound [22]. For details see Refs. [1, 23].

5.2.3 Average-Field Calculation for Coulomb Interactions in 2D Periodic System The periodic images in the virtual domains, which are far away from the physical domain, are approximated by uniformly distributed charges with enough high precision. Compared with the direct calculation of each periodic images, this average-field method is less time-consuming and has a good applicability for simulation of packing

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5 Deposition of Microparticles with Coulomb Repulsion

Fig. 5.1 Schematic representation of the long-range Coulomb interaction and short-range contact interactions. Reproduced from Chen et al. [21] with permission of The Royal Society of Chemistry

systems. Consider a packing system of N charged particles with a periodic boundary condition mentioned previously. A periodic array of replicated systems is created as shown in Fig. 5.2. Because of the long-range nature of the Coulomb interaction, the force exerted on a particle includes contributions from the other particles inside the physical domain and all their replica images over the periodic array. The total field at location of the particle i is Ei =

N 1   ri j + Ln · qj, 4π ε0 n j=1 |ri j + Ln|3

(5.3)

where q j is the charge on particle j and L is the cell dimension. The sum is over all integer vectors n = (n y , n z ) with the term i = j omitted when |n| = 0. The whole periodic system is divided into two regions according to the distance from the central computational domain. Equation 5.3 then can be written as the summation of contributions from these two parts Ei = Ei,in + Ei,out =

1 4π ε0

 |n y |,|n z

1 + 4π ε0

N  ri j + Ln · qj |ri j + Ln|3 |≤1 j=1

 |n y |or|n z

N  ri j + Ln · qj. 3 |r i j + Ln| |>1 j=1

(5.4)

The first term on the right-hand side of Eq. 5.4 accounts for the contribution of particles inside the central computational domain and its neighboring virtual domains enclosed by the red square in Fig. 5.2. As stated in previous subsection, we use fast multiple method to calculate the electric field induced by this part of particles. On the other hand, the second term of Eq. 5.4 is not truncated and therefore extends over the entire system. For simplicity, these periodic images in the virtual domains outside

5.2 Models and Methods

105

Fig. 5.2 Illustration of the periodic boundary condition. The electric field induced by particles inside the physical domain and its neighboring virtual domains (the region inside the red square) is calculated by a fast multipole method (FMM). The field induced by the particles outside the red square is approximated by an average field. Reproduced from Chen et al. [21] with permission of The Royal Society of Chemistry

the red square are approximated by uniformly distributed charges and the summation is thus turned into an integral Ei,out =

1 4π ε0

1 ≈ 4π ε0

 |n y |or|n z



out

N  ri j + Ln · qj |ri j + Ln|3 |>1 j=1

r σ dv. |r|3

(5.5)

The integral is taken over the region occupied by periodic images of packed particles outside the red charge density is written as σ = N  Nsquare and the average 2 q /V , where q and V = L · H j bed j bed p are the sum of charges and the j=1 j=1 volume of packed bed inside each domain respectively. This average-field method accelerates the computation of the field induced by charged particles over periodic arrays since the only quantity needed to be updated during the computation process is the sum of particle charges and the integral can be determined without knowing the exact location of particles. The physical basis for this method is that when charged particles (the source) are sufficiently far from the target point, the charges on them can be redistributed over nearby regions without causing obvious changes in the field at the target point. For packing systems here, particles are uniformly distributed inside the domains and the charges on them thus can be well approximated by a homogeneous charge density σ . To show the accuracy of the average-field method, the Coulomb interaction between two charged particles at (0, 0, 0) and (X i , 0, 0) was calculated. The result obtained by a two-dimensional Ewald method (EW2D) [24] is also presented. The result in Fig. 5.3 shows that the average-field method obtain a sufficiently high accu-

106

5 Deposition of Microparticles with Coulomb Repulsion

Fig. 5.3 Coulomb force between two charged particles at (0, 0, 0) and (X i , 0, 0) as a function of X i with periodic boundary condition in the y − z directions. L is the cell dimension and the forces are normalized by the its value at X i = L. Solid line: direction summation of Eq. 5.3 with sufficiently large n; diamonds: results of the average-field method; crosses: 2-dimensional Ewald summation (EW2D). Reproduced from Chen et al. [21] with permission of The Royal Society of Chemistry

racy in this case. However, an analytical error bound is still needed to further evaluate this average-field method, which is left for future work.

5.3 Effects of Coulomb Interaction on Packing Structure The current section presents the results of the DEM simulations. We analyze the structure of packings obtained with different charge on particles in terms of the most commonly used concepts, such as the volume fraction φ, the coordination number Z and the radial distribution function g(r ). As shown in Fig. 5.4, with other parameters fixed, a higher charge of particles will lead to a looser packing structure. The expansion effect is further quantified in Fig. 5.5, where the volume fraction φ is plotted as a function of particle charge for three typical cases. The volume fraction for packings of neutral particles (termed as φ0 ) ranges from 0.270 to 0.363 here, which lies in the range of adhesive loose packings of non-charged particles reported in literature [16, 19]. As particle’s charge q increases, φ starts to decrease lightly and then rapidly drops as q further increases. We redraw the data in the form of φ0 /φ − 1, which can be regarded as the relative expansion of packed beds, in double logarithmic coordinates. It is found that, a line with a slope of two nicely describes the variation tendency of φ0 /φ − 1, implying that the expansion effect has a relation to the inverse square law of Coulomb interaction. In these cases, the relative decrease of φ can reach a maximum of 40%. Regarding the charge on particles as a controllable parameter, this expansion effect suggests the ability of manipulating the packing structures through tuning the interparticle interactions. It should be noted that the maximum value of q is limited by both the charging mechanisms [1, 25] and the onset of particle levitations where incident

5.3 Effects of Coulomb Interaction on Packing Structure

107

Fig. 5.4 Typical packing structures for r p = 2.0 µm, U0 = 1.0 m/s with a q = 0, b q = 260e0 , and c q = 400e0 . Different colors stand for different coordination number Z . Reproduced from Chen et al. [21] with permission of The Royal Society of Chemistry

Fig. 5.5 Volume fraction φ (left y-axis) and the relative variation of φ, which is calculated as φ0 /φ − 1, (right y-axis) as functions of particle charge for a r p = 1.0 µm, U0 = 1.0 m/s, b r p = 1.0 µm, U0 = 2.0 m/s, and c r p = 2.0 µm, U0 = 1.0 m/s. Reproduced from Chen et al. [21] with permission of The Royal Society of Chemistry

particles are repelled away from deposited particles before reaching the packed bed. Packing of particles with charges beyond this limit is not within the scope of this study.

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5 Deposition of Microparticles with Coulomb Repulsion

Fig. 5.6 Radial distribution function for the packings of neutral and charged particles, corresponding to the packing conditions of (a) and (c) in Fig. 5.4. The curves are average values taken from 5 samples for each case and the filled area around curves stands for the standard deviations. Reproduced from Chen et al. [21] with permission of The Royal Society of Chemistry

In addition to φ, we also consider the radial distribution functions (RDF), g(r ), to obtain the information on the microstructure of packings. The RDF is defined as the probability of finding a particle at a given distance r from a reference one and indicates structural changes when shoulders or peaks appear. The RDFs here are measured using a histogram of discretized pair separations: g(rn ) = N (rn )/4πrn2 ρ0 r , where rn = (n − 1/2)r and N (rn ) is the number of particle pairs (i, j) for which (n − 1)r ≤ ri j < nr [26]. We simulated 5 samples for each case so that the number of particle pairs counted into g(r ) is sufficiently large to reduce statistical fluctuations. Plots of the RDFs as well as the standard deviations for packings of both charged and uncharged particles are shown in Fig. 5.6. The sharp peak observed at r = 2r p corresponds to the first contact shell of particles that are in touch with the reference particle. The increase of g(rn ) from the r/2r p = 1.0 to the second peak at r/2r p = 2.0 directly describes a remarkably higher probability to find a particle just inside the second contact shell at r/2r p = 2.0, accounting for the chainlike structure shown in the inset of Fig. 5.6. The general trends of these two RDFs, including the narrow first peak and absence of any peaks after r/2r p = 2.0, are parallel to the existing results for micro-sized neutral particles [19]. Results here indicate that Coulomb interaction does not bring any detectable crystallization based on the observation of the absence of a peak at √ √ r/2r p = 2 or 5, which are typical of crystal packings [27]. The sharp decrease to g(r ) ≈ 1 after r/2r p = 2.0 means that all correlation is lost beyond a few particle diameters, confirming the absence of any long-range positional order in both charged or uncharged packing systems. Despite the fact that an obvious decrease of the volume fraction is caused by the Coulomb interaction, the RDFs remain unchanged, indicating a structural similarity between the charged and uncharged packing systems.

5.4 Scaling Analysis of the Interparticle Force

109

We also do statistics on the contact condition of particles inside the packings. Particles inside the packing are in contact with each other to maintain the stability of the structure. The coordination number Z , defined as the number of contacts of a particle in the packing, is an observable measurement of the packing structure. A quantitative description of the contact condition is given in the form of coordination number distribution f (Z ), which is shown in Fig. 5.7. It can be seen that, with other parameters fixed, the mean value of the distributions moves to the left with the increasing of the particle charge. The populations of Z = 4 − 6 on the right of the peaks shrink with higher q while the Z = 1 − 2 populations grow, indicating that chain-like agglomerates are more likely to be formed. This chain-like network acts as a skeleton to support the highly porous structure. Interestingly, decreasing the particle radius (compare Fig. 5.7a and c) or decreasing the injection velocity (compare Fig. 5.7a and b) has a similar effect on the contact condition as the interparticle Coulomb interaction, i.e., makes the distribution move toward left. This kind of similarity leads us to find a general principle that can bring together the role of interparticle interactions at different ranges (e.g., the short-range contact interactions and the long-range Coulomb interaction) and quantify the combined effect of the particle size, the velocity, the surface energy, and the charge, which will be discussed in Sect. 5.5.

5.4 Scaling Analysis of the Interparticle Force For the system considered here, the forces exerted on particles can be classified into two categories: the short-range contact forces and the long-range Coulomb repulsion. The former includes normal adhesive force Fne , damping force Fnd and resistances in tangential direction. Note that the interparticle van der Waals force decays quickly with the distance and has been integrated into the JKR model together with the normal elastic force. Thus, it is classified into the category of short-range forces. Intuitively, the repulsive Coulomb force between deposited particles may bring in an “expansion” effect to the packed bed. To test this, we charged the particles from 0 to 400e0 after the packings have reached an equilibrium state. During a sufficiently long-term observation (T /(L/U0 ) ≈ 104 ), no detectable change of the packing structure was found (see Table 5.2). It indicates that the loose structures are stable and have no characteristic of aging. Once the packing has been formed, all the particles inside the packing are bonded into the contact network and the strength of the repulsive Coulomb force is much weaker than the dominant van der Waals adhesion. This can also be inferred from a simple scaling analysis of the van der Waals adhesion force and the Coulomb force. Here, the typical van der Waals force can be represented by the critical pull-off force FC = 3π γ R while the Coulomb force is F Coul = q 2 /4π ε0 r 2p . With parameters listed in Table 5.1, we evaluate the ratio between these two forces and have FC /F Coul  1, which confirms the statement above.

110

5 Deposition of Microparticles with Coulomb Repulsion

Fig. 5.7 The distribution of the coordination number Z for different particle charges. a stands for packing with r p = 2.0 µm, U0 = 1.0 m/s, b stands for r p = 2.0 µm, U0 = 0.5 m/s, and c stands for r p = 1.0 µm, U0 = 1.0 m/s . Reproduced from Chen et al. [21] with permission of The Royal Society of Chemistry

Table 5.2 Packing structures when charged after formation Particle charges(e0 ) Volume fraction 0 260 400

0.3249 0.3248 0.3247

Coordination number 3.2056 3.2056 3.2056

Reproduced from Chen et al. [21] with permission of The Royal Society of Chemistry

Besides the macroscopic structure parameters presented in Table 5.2, we also present the forces carried by the contact networks with a visualization of the repulsive and attractive normal forces in Fig. 5.8. For clarity, a slice with a width of 3r p is taken from each packing. The force patterns are typical for packing of particles in the presence of adhesion [28]. When the repulsive Coulomb interaction is introduced into the equilibrated samples, only a small fraction of normal forces changes from repulsive ones to attractive ones and the main part of the force network remains essentially unchanged. To characterize the force patterns in a quantitative way, the probability density functions (p.d.fs) of the normal force for these two samples are plotted in Fig. 5.9. The forces have been normalized by the typical pull-off force

5.4 Scaling Analysis of the Interparticle Force

111

Fig. 5.8 Force-carrying structures in a neutral, and b charged packing samples. For clarity, the samples were slices with a width of 3r p taken from the packings. Red and green lines stand for compressive and tensile interactions respectively and the width of the lines is related to the magnitude of these forces. Reproduced from Chen et al. [21] with permission of The Royal Society of Chemistry

Fig. 5.9 Probability distribution functions of dimensionless normal force values for packings of neutral and charged particles. Reproduced from Chen et al. [21] with permission of The Royal Society of Chemistry

FC . The distributions are narrow (within the range of |F n /FC < 0.1|) and almost symmetric, which is a result of the force balance between the repulsive elastic force and the attractive adhesion force. After introducing the Coulomb force, only a small fraction of the force shifts from repulsive (positive) to attractive (negative) to adaptively balance the applied repulsive Coulomb force. Such an imperceptible change of the force distribution does not cause significant break-up of contact pairs or rearrangement of packing structures. Based on the results of packing structures and the scaling analyses of the forces, we infer that the long-range Coulomb interaction exerts its influence on particles before they are bonded into the force network. Figure 5.10 shows the evolution of the kinetic energy E k of an incident particle as a function of the distance between its centroid and the top of the packed bed. The kinetic energy and the distance have been normalized by the initial kinetic energy E k0 and the particle radius r p , respectively. The Coulomb force exerted by the deposited particles decelerates the incident one,

112

5 Deposition of Microparticles with Coulomb Repulsion

Fig. 5.10 The variation of the kinetic energy E k of an incident particle as a function of the distance between the particle and a deposited one. The vertical line at r/r p = 2 divides the contact state and the non-contact state. The JKR line stands for the energy transferred from the kinetic form to the sum of the released surface energy and the stored elastic energy. Reproduced from Chen et al. [21] with permission of The Royal Society of Chemistry

leading to the energy conversion from the kinetic form to the Coulomb potential. A higher charge results in a lower impact velocity at the moment of contact. The relationship between particle impact velocity and the packing structure has been discussed for neutral particles with the conclusion that a lower particle velocity results in a looser packing structure [16]. These results provide compelling evidence that, the long-range Coulomb interaction influences the packing structure indirectly through its influence on the kinetic energy of particles before they are bonded into contact networks.

5.5 Governing Parameters for the Packing Structure It is of great importance to build the relationship between the packing structure and the particle scale interactions. According to the results in Chap. 2, the outcome of a collision between two adhesive particles is determined by the dimensionless adhesive parameter, which is expressed as Ad =

γ . ρ p U02 r p

(5.6)

The surface energy γ measures the strength of the adhesion between two particles and the denominator is the typical value of particle’s kinetic energy. The parameter Ad combines the influences of the size, the velocity, the density and the surface energy of the particle and has been successfully used to quantify the volume fraction

5.5 Governing Parameters for the Packing Structure

113

and the coordination number of packings of neutral micro-particles [16]. A universal regime of adhesive loose packing for Ad > 1 is identified [16]. Taking the long-range Coulomb interaction into account, we need to modify the kinetic term to include the energy conversion from the kinetic form to the Coulomb potential. An easy and intuitive way is to take an average over all the particles’ kinetic energy at the moment of impact to get an effective value  Ntot  1 2 1  1 2 U = U . 2 ef f Ntot i=1 2 i

(5.7)

Here, Ui is the impact velocity of the ith particle, which can be approximately derived from the energy conversion 1 1 m p Ui2 ≈ m p U02 − H E i q, 2 2

(5.8)

where H is the height of the computational domain and can be regarded as a typical deposition distance. E i is the strength of the field induced by the deposited particles and can be approximated as E i = i · q/(2L 2 ε0 ) in the spirit of the average-field method. Therefore, the effective impact velocity for the present deposition conditions can be expressed in the form Ue2f f = U02 (1 − E ∗ ).

(5.9)

Here, the dimensionless charge parameter E ∗ is given as E ∗ = K e f f q 2 /(ε0 U02 r 4p ρ p )

(5.10)

It measures the relative importance of the Coulomb interaction compared with the particle kinetic energy. The coefficient K e f f is related to the total particle number 3 Ntot (H/r p )(L/r p )−2 ≈ 51 for and the domain dimension, and is fixed at K e f f = 8π all the cases in the present chapter. A modified adhesion parameter Ad∗ thus can be written as γ γ = . (5.11) Ad∗ = ρ p Ue2f f r p ρ p U02 (1 − E ∗ )r p We simulate a series of packings with different values of r p , U0 , γ and q. The variation of the volume fraction φ and the average coordination number Z are plotted as a function of Ad∗ in Fig. 5.11. It can be found that both φ and Z decrease monotonically as Ad∗ increases and all the data points center around the same line. The results imply that, for packing systems with charged micro-particles, the structure can be readily predicted once the value of Ad∗ is known. The decrease of φ and Z with Ad∗ can be understood as follows. When particles are being packed, the VDW adhesion force tends to attract particles and make them stick together while the particle inertia will urge them to move and impact with

114

5 Deposition of Microparticles with Coulomb Repulsion

Fig. 5.11 a Packing volume fraction φ, and b average coordination number Z as functions of the modified adhesion parameter Ad∗ . Reproduced from Chen et al. [21] with permission of The Royal Society of Chemistry

other particles. If adhesion is strong (corresponding to a large Ad∗ ), particles will be caught at the first moment of impact and hardly move or roll (termed as hit-and-stick phenomenon) so that a loose packing structure is easier to be formed [29]. With the increase of particle size or velocity, its kinetic energy becomes higher, leading to violent collisions. Consequently, the particle bed will rearrange upon collisions to form a denser packing. The long-range Coulomb interaction exerts its influence on a particle as soon as it enters the computational domain and continuously decelerates the particle. This effect causes an increase of Ad∗ and thus results in relatively loose packings. In this sense, the adhesion parameter Ad∗ successfully combines the effects of particle’s properties and bridges the gap between the macroscopic packing structure and the microscopic interparticle forces. It should be noted that all the cases in the present study are located in the regime of Ad∗ > 1, where the adhesion dominates the packing dynamics. When Ad∗ < 1, other interactions (including sliding and rolling frictions) may have a strong influence on the packing structure [30, 31]. Under this circumstance, the scaling may break down since Ad∗ does not include the effect of frictions. As Ad∗ further increases, both φ and Z deviate from the straight line and start to enter a prolonged plateau. This deviation is mainly due to the existence of an asymptotic adhesive loose packing (ALP) limit at Z = 2 and φ = 1/23 ,

5.6 Phase Diagram

115

Fig. 5.12 Illustration of Voronoi unit for particle i

which is an important conjecture in the latest packing studies [16]. This point is naturally related to the observation that φ = 1/2d is the lower bound of saturated sphere packings in d dimensions [32]. Moreover, Z = 2 is the minimal possible value of the coordination number for particles bonded into a network spanning the packing.

5.6 Phase Diagram Apart from the dynamic analysis above, we have also compared our simulation results with an analytical representation of the equation of state φ(Z ) for adhesive packing, which is developed based on the Edwards’ ensemble approach [16, 33–35]. The approach starts with the Voronoi tessellation of the total volume of the packing: N Wi , where Wi is the Voronoi volume of a reference particle i. As shown V = i=1 in Fig. 5.12, the Voronoi cell of a given particle i is a region consisting of all points of the space closer to the surface of particle i than to the surface of any other particles. The Voronoi volume Wi can be calculated through l i ( S) ˆ

Wi =

1 r dr ds = 3 2



ˆ 3 ds, li ( S)

(5.12)

0

ˆ is the distance from the centroid of particle i to the boundary of its where li ( S) ˆ The average value of Voronoi volume over different Voronoi cell along the direction S. particles is W = Wi . The packing fraction can be simply written as φ = V0 /W with V0 being the volume of a particle. W can be expressed in terms of the cumulative distribution function

116

5 Deposition of Microparticles with Coulomb Repulsion

P(c, Z ), which is the integration of the pair distribution function p(c, Z ) for finding the boundary of the Voronoi cell at a distance c from the sphere center. Then the average Voronoi volume W is ∞ W (Z ) = V0 + 4π

c2 P(c, Z )dc,

(5.13)

rp

For P(c, Z ) one can derive a Boltzmann-like form using a factorization assumption of the multi-particle correlation function into pair correlations to find ⎧ ⎪ ⎨



P(c, Z ) = exp −ρ ⎪ ⎩

(c)

⎫ ⎪ ⎬

drg2 (r, Z ) , ⎪ ⎭

(5.14)

where ρ = N /V = 1/W is the particle number density and g2 (r, Z ) is the pair correlation function of two spheres separated by r. The volume (c) is an excluded volume for the N − 1 spheres outside of the reference sphere, since otherwise they would contribute a Voronoi boundary smaller than c. The exponential form of Eq. 5.14 is the key assumption of the mean-field approach. To capture the substantial correlation between each particle and its neighbours in the packing, the p.d.f. needs to be modelled. We adopt the model for g2 (r, Z ) proposed by Liu et al. [16], which considers four distinct contributions to the value of g2 : (1) a delta peak due to contacting particles; (2) a power-law peak over a range  due to the near contacting particles; (3) a step function due to bulk particles; and (4) a gap of width b separating the bulk and (near) contacting particles. This gap captures the effect on correlations due to adhesion and is assumed to depend on Z . Overall, g2 (r, Z ) is expressed as   −v   Z  δ r − 2r p − σ r − 2r p  2r p +  − r ρλ ,    +  r − 2r p + b(Z )

cg2 (r, Z ) =

(5.15)

Plugging Eq. 5.15 into Eqs. 5.14 and 5.13 and evaluating their values with numerical integrations, we obtain the value of packing faction φ as a function of the average coordination number Z (shown as the dashed line in Fig. 5.13). In the adhesive loose packing regime, our simulation results are in substantially good agreement with the theory. These findings extend those of Liu et al. [16], confirming that the addition of the interparticle Coulomb interaction does not break the microstructural feature of adhesive packings.

5.7 Summary

117

Fig. 5.13 Packing states on the phase diagram for charged micro-sized particles. Points are simulation data and the dashed line is the theoretical prediction. Reproduced from Chen et al. [21] with permission of The Royal Society of Chemistry

5.7 Summary In this chapter, we have presented a computational study of the packing of charged micron-sized particles using JKR-based adhesive DEM. The effect of long-range Coulomb force on packing structure is quantified in terms of volume fraction, coordination number, and radial distribution function. Further analysis from both dynamic and statistical mechanical levels is conducted to clarify the connections and the differences between the effects of the short-range van der Waals force and the long-range Coulomb force. We found that the presence of the long-range Coulomb interaction results in a looser packing structure through its influence on particle inertia. The relative decrease of the volume fraction, denoted as φ0 /φ − 1, approximately follows a square law with the increase of particle charge up to a maximum of 40%. However, this effect is suppressed by the short-range adhesion once particles are bonded into the contact network. Furthermore, a modified Ad∗ was derived to clarify the combined effects of particle kinetic energy, adhesion, and Coulomb interaction. As Ad∗ increases, both the volume fraction φ and the average coordination number Z decrease monotonously. It suggests that Ad∗ can be used to predict and further design the macroscopic structure of packings for charged adhesive particles. We have also shown that the packing state of charged micron-sized particles can be well described by the latest derived adhesive loose packing (ALP) regime in the phase diagram and increasing the charge of particle makes packing states move toward the ALP point. The results confirm the universality of the analytical presentation for packings of adhesive or charged particles based on the Edwards’ ensemble approach.

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5 Deposition of Microparticles with Coulomb Repulsion

References 1. J.S. Marshall, S. Li, Adhesive Particle Flow: A Discrete-element Approach (Cambridge University Press, 2014) 2. K.M. Kinch, J. Sohl-Dickstein, J.F. Bell, J.R. Johnson, W. Goetz, G.A. Landis, J. Geophys. Res. Planets 112(E6) (2007) 3. M. Yang, S. Li, Q. Yao, Powder Technol. 248, 44 (2013) 4. A. Chavez-Valdez, M.S.P. Shaffer, A.R. Boccaccini, J. Phys. Chem. B 117(6), 1502 (2013) 5. J.H. Dickerson, A.R. Boccaccini, Electrophoretic Deposition Nanomater. (Springer, New York, 2012) 6. R.N. Basu, C.A. Randall, M.J. Mayo, J. Am. Ceram. Soc. 84(1), 33 (2001) 7. J. Cordelair, P. Greil, J. Mater. Sci. 39(3), 1017 (2004) 8. S. Yang, K. Dong, R. Zou, A. Yu, J. Guo, Granular Matter 15(4), 467 (2013) 9. S.H. Kim, K.W. Lee, J. Electrostat. 48(1), 3 (1999) 10. A. Kumar, B. Khusid, Z. Qiu, A. Acrivos, Phys. Rev. Lett. 95(25), 258301 (2005) 11. J.S. Park, D. Saintillan, Phys. Rev. E 83, 041409 (2011) 12. L.S. Matthews, V. Land, T.W. Hyde, Astrophys. J. 744(1), 8 (2012) 13. L.S. Matthews, T.W. Hyde, New J. Phys. 11(6), 063030 (2009) 14. J. Colwell, S. Batiste, M. Horányi, S. Robertson, S. Sture, Rev. Geophys. 45(2) (2007) 15. H. Lee, S.J. Yook, K.S. Lee, IEEE Trans. Semicond. Manuf. 27(2), 287 (2014) 16. W. Liu, S. Li, A. Baule, H.A. Makse, Soft Matter 11(32), 6492 (2015) 17. J. Blum, R. Schräpler, Phys. Rev. Lett. 93(11), 115503 (2004) 18. E.J. Parteli, J. Schmidt, C. Blümel, K.E. Wirth, W. Peukert, T. Pöschel, Sci. Rep. 4, 6227 (2014) 19. R.Y. Yang, R.P. Zou, A.B. Yu, Phys. Rev. E 62(3), 3900 (2000) 20. T.B. Jones, Electromechanics of Particles (Cambridge University Press, 2005) 21. S. Chen, S.Q. Li, W. Liu, H.A. Makse, Soft Matter 12(6), 1836 (2016) 22. J.K. Salmon, M.S. Warren, J. Comput. Phys. 111(1), 136 (1994) 23. G. Liu, J. Marshall, S. Li, Q. Yao, Int. J. Numeric. Meth. Eng. 84, 1589 (2010) 24. E. Spohr, J. Chem. Phys. 107(16), 6342 (1997) 25. L. Matthews, B. Shotorban, T. Hyde, Astrophys. J. 776(2), 103 (2013) 26. D.C. Rapaport, The Art of Molecular Dynamics Simulation (Cambridge University Press, 2004) 27. A. Donev, S. Torquato, F.H. Stillinger, Phys. Rev. E 71(1), 011105 (2005) 28. F.A. Gilabert, J.N. Roux, A. Castellanos, Phys. Rev. E 75(1), 011303 (2007) 29. C. Dominik, Astrophys. J. 480(2), 647 (1997) 30. G.R. Farrell, K.M. Martini, N. Menon, Soft Matter 6(13), 2925 (2010) 31. M. Jerkins, M. Schröter, H.L. Swinney, T.J. Senden, M. Saadatfar, T. Aste, Phys. Rev. Lett. 101(1), 018301 (2008) 32. S. Torquato, F.H. Stillinger, Exp. Math. 15(3), 307 (2006) 33. C. Song, P. Wang, H.A. Makse, Nature 453(7195), 629 (2008) 34. A. Baule, R. Mari, L. Bo, L. Portal, H.A. Makse, Nat. Commun. 4, 2194 (2013) 35. A. Baule, R. Mari, L. Bo, L. Portal, H.A. Makse, Soft Matter 10, 4423 (2014)

Chapter 6

Deposition of Charged Micro-Particles on Fibers: Clogging Problem

6.1 Introduction Clogging of pores caused by micro-particles universally exists in engineering processes, including the transport of biological cells [1], the aerosol filtration [2], assay applications of colloidal particles [3] and microreactors [4], where both the small size of the microchannel and their sensitivity to the particle adhesion significantly raise the tendency of clogging [5, 6]. During the microfiltration process, particles with sufficiently strong adhesion will stick onto the wall or onto other particles when they come into contact. Particle bridges are then formed, spanning across the channel entrance and eventually blocking the channel, even at a solid volume fraction that is far less than the jamming limit [7]. The previous chapter discusses the packing of charged particles on a flat surface, the clogging behavior studied here can be regarded as the packing of charged particles on cylinders. There are several key differences between the packing on a flat surface and the clogging during microfiltration. First, the flow field plays a significant role on the clogging process for microfiltration. As we mentioned in Chap. 1, the probability for a particle colliding on a fiber is a function of particle’s Stokes number. The fluid flow also affects the colliding velocity and the consequent sticking probability between the particle and the fiber. Moreover, the clogging transition does not always happen. Whether a clogging transition happens or not depends on the relative strength between the adhesive contact force and the flow stresses. To reasonably account for the role of fluid flow, we adopt computational fluid dynamics (CFD) coupling with the fast DEM (described in Chap. 2) to investigate the clogging transition during the filtration for charged microparticles. A brief description of the method and the two-way coupling between the particle and the gas phase are given in the following section. Then, we try to establish the relationship between microscopic adhesive/repulsive forces and the macroscopic clogging quantities, i.e. the flow permeability, the number of penetrating particles, the particle capture effi-

© Tsinghua University Press 2023 S. Chen, Microparticle Dynamics in Electrostatic and Flow Fields, Springer Theses, https://doi.org/10.1007/978-981-16-0843-8_6

119

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6 Deposition of Charged Micro-Particles …

ciency, and the structure of clogs. Finally, the morphological change of the deposits is shown through varying the strength of the long-range Coulomb repulsion and the short-range adhesion.

6.2 Models and Method 6.2.1 Simulation Conditions: Two Fiber System A sketch of the simulated system is shown in Fig. 6.1. The pore is formed by two parallel cylindrical fibers with radius r f = 10rp . This system can be regarded as a basic composition unit of widely used fibrous filtration systems. The dimensions of the simulation domain are X = 300rp , Y = 30rp , and Z = 40rp . Periodic boundary conditions are set for both fluid flow and particles along the Y and Z directions. A constant fluid velocity, U0 , in a laminar regime (Re  1) is imposed on the inlet and a non-slip boundary condition is applied on fibers’ surfaces. The constant rate condition can be directly compared with the applications of flue gas treatment for vehicles or power plants and has been widely used in both experimental [2, 8] and numerical studies [7, 9]. Particles are randomly seeded at the inlet with a constant volume fraction φ (< 1%) and are assumed to be neutrally buoyant to avoid gravity effect. The ratio of the pore gap (the minimum distance between the two surfaces) to particle radius is fixed as 10. Given such a large gap-to-particle size ratio and a low volume fraction, the clogging of the pore results only from the successive deposition of individual particle on the fibers [10, 11]. Other physical parameters of our simulations are listed in Table 6.1. Note that the surface energy γ is varied within the range 10−15 mJ/m2 to reflect the effect of van der Waals adhesion. The surface charge density of the particle is determined according to the typical value due to the diffusion and the field charging mechanisms for micro-particles [12].

Fig. 6.1 Sketch of the simulated system (Reprinted figure with permission from Chen et al. [13] Copyright by the American Physical Society)

6.2 Models and Method

121

Table 6.1 Parameters for computer simulations Parameters Value Particle Particle radius (rp ) Particle density (ρp ) Poisson’s ratio (σp ) Elastic modulus (E p ) Restitution coefficient (e) Friction coefficient (τf ) Surface energy (γ ) Charge density (q/4πrp2 ) Gas phase Gas density (ρf ) Dynamic viscosity (μf ) Permittivity (εf ) Inlet velocity (U0 )

Unit

1.0, 2.5, 3.0, 5.0 2500 0.33 2 × 108 0.7 0.3 1.0–30.0 0–10.0

µm kg/m3 – Pa – – mJ/m2 µC/m2

1.25 1.79 × 10−5 8.85 × 10−12 0.1

kg/m3 Pa · s F/m m/s

Reprinted table with permission from Chen et al. [13] Copyright by the American Physical Society

6.2.2 Gas Phase Simulation In our simulations, the mass and momentum conservation equations for the gas phase are solved in the framework of open-source MFIX code [14]. These equations are similar to those in single-phase CFD but with additional coupling terms due to the drag from the solid-phase. The governing equations of gas phase, which is assumed to be incompressible with a constant dynamic viscosity μ f and fluid density ρ f , are written as ∂(φ f ρ f ) + ∇ · (φ f ρ f v f ) = 0, ∂t (6.1) D ¯ ¯ ¯ (φ f ρ f v f ) = ∇ · (−P f I + τ¯ f ) − I fp . Dt Here, v f and P f are the volume-averaged gas-phase velocity and the gas-phase pressure, φ f = 1 − φ is the gas-phase volume fraction. The gas-phase shear stress tensor, τ¯¯ f is given by (6.2) τ¯¯ f = μ f [∇v f + (∇v f )T ]. The momentum transfer term I fp between the gas and the solid phase on the grid node located at x k is calculated as I kfp =

1  drag F i,(i∈k) K drag (x p,i , x k ), Vk i

(6.3)

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6 Deposition of Charged Micro-Particles … drag

where F i,(i∈k) is the drag force on the ith particle located at x p,i inside the kth computational cell, whose geometric volume is Vk . K drag (x p,i , x k ) is a generic kernel with compact support that determines the influence of the particle’s force on the grid node. The equations of gas-phase (Eq. 6.1) are solved using a staggered grid finite volume scheme. The computational details of the MFIX code can be found in the user manual [15, 16] as well as published papers with a series of verification tests [14]. This kind of two-way coupling method has been widely used to simulate the pore clogging in flowing colloids [7], particle motion in vortex [14], and gas-solid fluidized bed [17].

6.2.3 Solid-Phase: Discrete-Element Method (DEM) The fast DEM is adopted to solve the linear and angular momentum equations of particles, where the forces and torques exerted on a particle can be written as drag

m i v˙ i = F itot = F i

+ F iCoul + F icon ,

˙ i = M itot = M drag + M icon . Ii  i

(6.4)

In these equations, m i and Ii are the particle mass and moment of inertia, v and  are the translational velocity and the rotation rate of a particle. The total force/torque includes three terms: (i) the hydrodynamic drag (F drag /M drag ), (ii) the long-range Coulomb repulsive force (F Coul ) and (iii) forces/torques resulting from the contact with other particles or the fibers (F con /M con ). The dominant force/torque from the flow field at small Re is the viscous drag, which is given by the modified Stokes drag law [18]: F drag = −3π μ f dp (v − v f ) f, (6.5) 1 M drag = −π μ f dp3 ( − ω f ). 2 where ω f is the fluid vorticity. The friction factor, f = f (Re p , φ f ), accounts for the effect of particle crowding and fluid inertia. We adopt the correlation proposed by Benyahia et al. [19], which blends the Hill-Koch-Ladd (HKL) drag correlation with known limiting forms of the gas-solids drag function [20, 21] and is applicable to a wide range of Re (from 0.01 to 1000) and gas-phase volume fraction. The models for the adhesive contact forces/torques are given in Chap. 2. The Coulomb force on particle i from other charged particles is given by F iCoul = qi E = qi

 q j ri j , 4π ε f ri3j j=i

(6.6)

where qi is the charge on particle i, r i j = x p,i − x p, j is the vector from the centroid of a source particle j to the target particle i and ε f is the permittivity of the fluid.

6.2 Models and Method

123

6.2.4 Governing Parameters A variety of dimensionless parameters that influence clogging process have been proposed and discussed, including the flow Reynolds number Re, the particle Stokes number St, the particle volume fraction φ [7] and the ratio between pore size to particle size [10, 22]. Here, we introduce two additional parameters governing this process. The equation of motion, Eq. 6.4, for ith particle in a flow can be expanded as ⎛ ⎞ 2  ri j ⎠ q dv ⎝ = − 3π μ f dp (v − v f ) + m dt 4π ε f dp2 j=i ri j rˆi2j (6.7)    3/2 + 3π dp γ ni j (aˆ i3j − aˆ i j ) . i,cont

For the sake of simplicity, the friction factor in the drag force, f , is omitted here. Particles are assumed to be charged with the same value q. Only the conservative force in the normal direction, FiNj E , is selected as a representative of contact forces. Nondimensionalizing the velocity by U0 and time by r f /U0 , Eq. 6.7 becomes d˜v 1 = [(˜v f − v˜ ) + κq i ] + Ad i . St dt˜

(6.8)

Here, i and  i are functions of particles’ position vectors (r 1 , r 2 , · · · , r N p ). Two dimensionless parameters that influence clogging process are indicated by Eq. 6.8. The first is the charge parameter, κq , defined as κq =

q2 . 3 f ε f dp U0

12π 2 μ

(6.9)

The parameter κq is interpreted as the ratio of the magnitude of particle-particle Coulomb interaction to that of viscous drag in the flow field. It measures the ability of a particle to move relative to the local streamline due to the Coulomb force. The other parameter is the adhesion parameter, written as Ad =

γ , ρp U02 rp

(6.10)

which is defined as the ratio of interparticle adhesion to particle kinetic energy [9]. According to the results in Chap. 2, the adhesion parameter estimates the sticking probability of a particle when it collides with a surface.

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6 Deposition of Charged Micro-Particles …

6.3 Clogging/Non-clogging Transition We start by studying the influence of the strength of the long-range repulsive Coulomb interaction through varying the charge parameter κq . During our simulation, the information of all particles (position, velocity, and forces) and the flow field (velocity and pressure) were simultaneously recorded. Thus, the temporal evolution of the clogging process can be easily captured. Side views of the computational domains at different values of κq are shown in Fig. 6.2a. Particles are projected onto the x − y plane and the snapshots are plotted at t˜ = 120. It can be found that, a transition from a clogging state to a non-clogging one is obtained by increasing the charge parameter κq . Figure 6.2b, taken from the case of κq = 0 in Fig. 6.2a at different time, clearly shows the particle rearrangement during the clogging process. The red particle chains, initially aligning along the direction of the main flow, are gradually bent into short clusters. It indicates a compression of the particle cake caused by the flow stress. The dynamic behavior of particle agglomerates suggests that the assumption of “frozen” particles—deposited particles are fixed—does not apply to the filtration system. The pore clogging can be further quantified by the temporal evolution of the bulk permeability derived from the Darcy law (shown in Fig. 6.2c). Here, the permeability is defined as k(t˜) = μ f U0 X/P with P being the pressure difference between the inlet and the outlet. It has been scaled by its initial value k0 free of deposited particles. At the initial stage of t˜ < 75, the permeability for particles with different κq decrease at almost the same rate. The reason is that, during this stage, the repulsive interaction among particles is weaker than the flow stress due to a low concentration of particles in the bulk flow and the limited number of captured particles. Once the fiber surfaces are covered by a certain number of particles, a sufficiently strong local electrostatic field is established, which forces the particles to deviate from the stream

Fig. 6.2 a Side views of the particles and flow field projected onto the x − y plane at t˜ = 120 with Re = 0.087 and κq = 0, 0.079, and 0.14. Particles located at z < 0 are colored with light grey and particles at z > 0 are colored with dark grey. b Particle rearrangement during the clogging process. c Scaled permeability v.s. time for different values of κq , where k0 is the initial permeability without any deposited particles and the time has been normalized by r f /U0 (Reprinted figure with permission from Chen et al. [13] Copyright by the American Physical Society)

6.3 Clogging/Non-clogging Transition

125

Fig. 6.3 a Number of particles, N ∗ , that flow through the constriction prior to the clogging as a function of Coulomb repulsion κq at St = 0.39. b Normalized N˜ ∗ as a function of κq /κq,crit for different St. The solid curves are fittings using N˜ ∗ = 1 − (κq /κq,crit )λ and error bars have been removed for the sake of clarity (Reprinted figure with permission from Chen et al. [13] Copyright by the American Physical Society)

lines. The subsequent permeability reduction significantly depends on the strength of the repulsive interaction. For the case with κq = 0.14, the scaled permeability does not drop toward zero any longer and fluctuates around 0.4 instead. It indicates that no clogging occurs with a sufficiently strong Coulomb repulsion. We then measure the number of particles, N ∗ , that flow through the pore prior to clogging. Because of the variations from one simulation run to the next for the same physical parameters, three runs with different realizations of particle injection positions were performed to provide a meaningful average and standard deviation. The change of N ∗ with Coulomb repulsion κq in a typical case is shown in Fig. 6.3a, where a two-regime behavior can be observed. In the regime with a low κq , N ∗ is nearly a constant. In the second regime, N ∗ increases rapidly with κq . In this regime, the particle’s trajectory is obviously modified when it approaches the deposited particles due to the strong repulsive Coulomb barrier. When κq further increases to a certain level, no clogging occurs in all three simulation runs. This value can be regarded as the critical value κq,crit for the clogging/non-clogging transition. No clogging can be observed with κq larger than κq,crit . To further characterize the transition, the number of penetrating particles is normalized as N˜ ∗ = N0∗ /N ∗ (κq ), where N0∗ is the value for the case without Coulomb repulsion (κq = 0), and is plotted as a function of the scaled charge parameter κq /κq,crit in Fig. 6.3b. It is seen that the normalized N˜ ∗ for particles with different Stokes numbers have a similar trend and can be fitted with a power-low functions, (6.11) N˜ ∗ = 1 − (κq /κq,crit )λ , where the fitting parameters λ and κq,crit are related to Stokes number. The curves remain flat in the regime of small κq /κq,crit and then rapidly drops toward 0 when κq /κq,crit approaches unity, indicating a fast clogging/non-clogging transition.

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6 Deposition of Charged Micro-Particles …

Fig. 6.4 Clogging phase diagram in the κq ∼ 1/P0∗ plane obtained for particles with long-range repulsion. The dashed line is the power-law function κq,crit ∝ (1/P0∗ )−1.44 (Reprinted figure with permission from Chen et al. [13] Copyright by the American Physical Society)

It is of great importance to predict the critical charge parameter κq,crit for the clogging/non-clogging transition at various conditions. Here, we compared our simulation results with the qualitative representation of the clogging phase diagram [22]. In the clogging phase diagram, all the variables affecting the clogging process are grouped into three generic parameters: a length scale, such as the size ratio between the pore and the particle diameter; a compatible load parameter related to the pressure or other driving force; an incompatible load parameter related to the background noise that prevents the clogging formation. Then the transition from the clogging state to the non-clogging one can be sketched in the three-dimensional phase diagram. In our simulation, the characteristic length scale is fixed by setting a constant ratio between the pore size and the particle size. The three-dimensional phase diagram thus is reduced to a two-dimensional form and the clogging simply arises due to the competition between the compatible load and the incompatible load. It is generally accepted that, for micro-filtration systems, the pressure or the driving force is responsible for the development of clogging [23]. We consider the initial pressure drop of the pore completely free of particles (P0∗ ) as the compatible load. On the other hand, the pair-wise repulsive Coulomb force, which prevents particles from being captured, is supposed to be the incompatible load [11]. Then, a projection of the clogging phase diagram on κq ∼ 1/P0∗ plane is obtained (Fig. 6.4). Given a fixed value of 1/P0∗ , the increase of κq leads to a transition from clogged systems to nonclogging ones. The dependence of the critical value κq,crit for the transition on 1/P0∗ can be well described by a power law function, κq,crit ∝ (1/P0∗ )β , with the exponent β = −1.44. It indicates that charging the particle is an effective way to reduce the possibility of clogging. The results also extend the latest proposed clogging phase diagram to the specific situation of micro-filtration of charged particles.

6.4 Measurement of Particle Capture Efficiency

127

6.4 Measurement of Particle Capture Efficiency The capture efficiency is one of the most important properties for the filtration systems. Here, we introduce a method to calculate the instantaneous particle capture efficiency based on the simulation data. In this method, the positions of deposited particles and the flow field are extracted from a certain case at a given time and are fixed. Testing particles are then uniformly released at the inlet plane and their trajectories are followed to see whether they will be captured by the fibers or the deposited particles. There is no interaction between testing particles. The capture efficiency η is given as the ratio of the number of captured testing particles to that of total released ones. We set 20 releasing points in the z direction and 15 points in the y direction on the inlet plane. Consequently, the particle capture efficiency η can reach a resolution as high as 0.5%. Two typical trajectories for neutral and charged particles released at the same point are shown in Fig. 6.5a. The neutral particle directly hits the particle cake and is captured, while the charged particle penetrates the cake and escapes. As shown in Fig. 6.5b, the remarkable growth of the area of the white part when κq increases from 0 to 0.079 indicates an obvious decrease of the capture efficiency. Intuitively, the results in Sect. 6.3 are likely caused by two mechanisms. One is that the repulsive Coulomb interaction among charged particles changes the particles’ trajectory and inhibits the capture. Thus, particles with a higher κq are more likely to penetrate the deposited particle cake. Such effect is termed as “repulsion effect”. On the other hand, as we show in Chap. 5, the particle-particle interactions have a remarkable influence on the structure of deposits, which consequently affects the clogging process. It is regarded as “structure effect”. To separate the effect of longrange repulsion from the structural effect, two series of measurements of the particle capture efficiency are performed, which are summarized in Table 6.2. In Series 1, the effect of long-range repulsion is studied by measuring the capture efficiencies η for different κq based on an identical structure. Since the number of deposited particles (Ndep ) also influences the repulsion, we extract several cake structures with different Ndep from a base clogging case. For instance, we fix a cake structure with Ndep = 200 and then measure the capture efficiency using the aforementioned method with κq = 0, 0.035, 0.079, 0.107, and 0.14. Next, we repeat the measurements by fixing another cake structure with Ndep = 400 and this process is reproduced. In this way, the variation of η is independent of the structure and is only due to the effect of repulsion. In Series 2, four different structures are extracted from the cases with different Ad and κq . We then look at how the adhesion and long-range repulsion affect the structure. Next the four structures are fixed and the particle capture efficiency η is measured using neutral testing particles. The measurements in Series 2 can reflect the structure effect on the particle capture.

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6 Deposition of Charged Micro-Particles …

Fig. 6.5 a Two typical trajectories for natural and charged particles released at the same point. b Releasing points at the inlet plane with κq = 0 and κq = 0.079. White and blue parts stand for releasing points of escaped particles and captured particles, respectively (Reprinted figure with permission from Chen et al. [13] Copyright by the American Physical Society) Table 6.2 Parameters for computer simulations Cases Structures Series 1 (Fig. 6.6a)

Ndep = 0 − 800

Series 2 (Fig. 6.9a)

Ad = 32, κq = 0 Ad = 32, κq = 0.079 Ad = 480, κq = 0 Ad = 480, κq = 0.079

Repulsions κq κq κq κq κq κq

=0 = 0.035 = 0.079 = 0.107 = 0.14 =0

Reprinted table with permission from Chen et al. [13] Copyright by the American Physical Society

6.4.1 Repulsion Effect: The Critical State The effect of the long-range repulsion on the particle capture, from the measurements in Series 1, is shown in Fig. 6.6a. For particles without repulsive interaction, η increases monotonically as Ndep increases. This is due to the decrease of the passable channel area with the growth of the deposited layers [10]. In the presence of the repulsive interaction, η is obviously reduced and a decreasing-increasing trend appears when κq reaches 0.079. When κq gets to the critical value κq,crit (= 0.107 in this case), the capture efficiency η decreases toward zero. Here, we define the moment, when the particle capture efficiency η achieve its minimum value, as the critical state. If the capture efficiency η is close to zero at the critical state, the follow-

6.4 Measurement of Particle Capture Efficiency

129

Fig. 6.6 a Particle capture efficiency as a function of the number of deposited particles Ndep . The structures for different cases are all the same. b and c, The scaled capture efficiency η(κq )/η(0) as a function of the charge parameter, κq , for different Ndep . Here, η(0) corresponds to the capture efficiency without Coulomb repulsive interaction (Reprinted figure with permission from Chen et al. [13] Copyright by the American Physical Society)

ing particles cannot be captured by the filtration system leading to a non-clogging situation. Interestingly, with either a very small or a very large Ndep the effect of Coulomb repulsion is inhibited. To better illustrate this two-regime behavior, we replot the data in the form of the scaled capture efficiency η(Ndep , κq )/η(Ndep , 0) in Fig. 6.6b and c, where η(Ndep , 0) corresponds to the capture efficiency without repulsive interaction. For a given Ndep , this scaled efficiency is a function of the charge parameter, κq , and can be simply recorded as η(κq )/η(0). At the early stage, the particles’ trajectories are mainly determined by the flow stress and the repulsive interaction among particles has only a minor effect on the particle capture efficiency. Thus flat curves appear in Fig. 6.6b when Ndep is small. As Ndep increases, the fiber surfaces are gradually covered by the particles and a sufficiently strong local electrostatic field is established to force the particle move relatively to the fluid, leading to an increase of the curves’ slope. When Ndep reaches a certain critical value (approximately 230 here), further increase of Ndep will inhibit the effect of repulsion, which corresponds to the flattening of the curves in Fig. 6.6c. The reason is that the steric effect on particle capture gets stronger as the passable channel area decreases. When Ndep reaches 700, almost all the channels are blocked and a clogging state is nearly established. These results indicate that the long-range Coulomb interaction among particles has a dramatic effect on the particle capture only when the number of the deposited particles is moderate and a sufficiently strong repulsive force can push the particle through the pores inside the cake. Summarizing the results above, a qualitative description of the clogging process is proposed in Fig. 6.7. At the early stage of the filtration, the flow stress dominates

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Fig. 6.7 Mechanism of pore clogging with particles in the presence of long-range repulsive force (Reprinted figure with permission from Chen et al. [13] Copyright by the American Physical Society)

the particle motions and the repulsion barely affects the particle trajectory (Fig. 6.7a). As particles accumulate on the surface of the fiber, the local electrostatic field gets stronger and the capture efficiency decreases to its minimum (Fig. 6.7b). Then, two distinct scenarios can be identified according to the magnitude of long-range interaction. With a sufficiently strong repulsion, the capture efficiency almost drops to zero (corresponding to the case with κq = 0.14 in Fig. 6.6a) and the coming particles keep escaping through the gaps, resulting a non-clogging situation (Fig. 6.7c). On the other hand, if the repulsion is not strong enough, the capture efficiency at the critical state is significantly larger than zero. Particles will continue to deposit on the fiber surface and the process enters a steric effect dominated regime. The capture efficiency increases until the final clogging state is established (Fig. 6.7d and e). Therefore, the minimum value of the particle capture efficiency can be regarded as a clogging/nonclogging criterion during the filtration of particles with long-range repulsion. It is worth noting that, with a sufficiently small κq (e.g. κq = 0 and κq = 0.035), the capture efficiency increases monotonically with Ndep . In these cases, the initial state (Ndep = 0) can be regarded as the critical state.

6.4.2 Structure Effect The structures of the cake are closely related to particle-particle interactions and may significantly affect the clogging process. Here, we first characterize the clogging structure in terms of distributions of the local volume fraction of the deposited particles to find the correlation between structure and microscale forces and then demonstrate the effect of structure on the clogging process.

6.4 Measurement of Particle Capture Efficiency

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Fig. 6.8 a Distributions of the local volume fraction of clogging structures with Ad = 32, 480 and κq = 0, 0.035, and 0.079. In each frame, the black circles stand for data obtained at Ndep = 500 and the blue triangles and the green diamonds stand for data obtained at Ndep = 1000 and 1500, respectively. The dashed lines indicate the average volume fractions. b Voronoi tessellations for a dense particle cluster and a loose one (Reprinted figure with permission from Chen et al. [13] Copyright by the American Physical Society)

The local volume fraction φL for each particle inside the clogs is measured. φL is defined as the ratio between the particle’s volume Vp to its Voronoi volume Wi , as depicted in red in Fig. 6.8b. The distributions of φL , which have been grouped into sections with an increment of 0.1, are given in Fig. 6.8a. For each case, the distributions are measured at three typical snapshots taken at Ndep = 500, 1000, and 1500 to illustrate the structural evolution. It is easy to see the compression over time: the average volume fraction grows approximately 30% as the number of deposited particle increases. This phenomenon is mainly due to the compression exerted by the flow stress. By comparing the cases in the horizontal direction in Fig. 6.8a, we find that the long-range repulsion has a negligible effect on the clogging structure. It is quite different from the ballistic packings of charged particles under vacuum conditions, where the long-range repulsion leads to a significantly looser structure through its influence on particle inertia (see Chap. 5). The reason is that the change of particles’ kinetic energy, caused by the long-range repulsion, is smaller in the presence of the fluid than that in vacuum packing cases. Assume a simple case, where a charged particle with initial velocity U0 moves across a distance L in the presence of a typical electrostatic force Fe . In the vacuum environment, the energy balance yields 1 1 m i U02 − Fe L = m i U 2 . 2 2

(6.12)

The variation of particle velocity scales as Uvacuum ∼ m2Fi Ue L0 . In the presence of fluid, this variation can be directly derived from the force balance between viscous drag and the electrostatic force Fe , yielding Ufluid ∼ 3πμFef dp . With parameters listed in

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Table 6.1, we evaluate the ratio between the velocity variations with/without fluid and Ufluid ∼ 0.2, which confirms the statement above. On the other hand, for the have U vacuum vacuum packing process, the kinetic energy of the particles is immediately dissipated through the inelastic deformation, the relative sliding and rolling and a mechanically stable structure is easily formed. In contrast, for the constant rate filtration system, the pressure drop increases as the cake thickness grows and the cake undergoes a continuous compression, which lasts for the entire clogging process. A remarkable difference between the structures with Ad = 32 clogging and those with Ad = 480 can be found by comparing the top panels to the bottom ones in Fig. 6.8a. A stronger adhesive interaction results in a looser clogging structure. From the models for contacting forces in DEM (see Sect. 2.2), one can finds that the normal force, the critical sliding friction, and the critical rolling resistance have positive correlations with the surface energy. Therefore, particles with small adhesion (Ad = 32) are more likely to roll or slide irreversibly relative to its neighbors, which leads to a denser clogging structure [24]. For a detail discussion of the mechanical stability of a particle cake, the reader may refer to [25]. We then discuss the effect of cake structure on the particle capture efficiency η using the method introduced at the beginning of the current section. Four typical cake structures, which are listed as Series 2 in Table 6.2, as well as the corresponding flow fields were extracted at different Ndep from simulations. By removing the Coulomb repulsion on testing particles, the differences between the four η − Ndep curves reasonably reflect the influence of the cake structure, as shown in Fig. 6.9a. We found that the structures formed with different Ad have a strong influence on the evolution of η, whereas the curves with different κq almost coincide with each other, which is consistent with previous results that the cake structure is not obviously affected by the long-range repulsion. In order to quantitatively measure the difference, we have fitted the data using

Nc + η0 , (6.13) η = χ exp − Ndep where χ is a pre-factor and η0 is 0.123 for all four cases, referring to the identical initial capture efficiency when the fiber surfaces are free of attached particles. This expression also indicates a typical number of deposited particles, Nc , that needed to form a cake with a certain capture efficiency. The fitting results for the four cases are list in Table 6.3. It can be found that, Nc for Structure 1 and 2 are much smaller than the values for Structure 3 and 4, indicating a faster growth of particle deposits at small Ad values. This is because a denser clogging structure formed with small adhesion (Ad = 32) has a lower probability of particle penetration. We also found that, when Ad further decreases below 12, the deposited particles will be continuously resuspended into the fluid and no stable clogging can be formed. Based on the results above, a schematic representation of the clogging mechanism in the presence of both the long-range repulsion (κq ) and the short-range adhesion (Ad) is summarized in Fig. 6.9b. The presence of strong adhesion leads to the formation of dendrites and loose structures. For particles with a moderate adhesion, the deposit is compressed by hydrodynamic forces and its permeability is relatively

6.5 Summary

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Fig. 6.9 a Particle capture efficiency η as a function of the number of deposited particles Ndep . The structures are taken from the cases with Ad = 32, κq = 0 (blue solid circles), Ad = 32 κq = 0.079 (pink diamonds), Ad = 480 κq = 0 (black hollow circles), and Ad = 480, κq = 0.079 (green triangles). The curves are fittings by Eq. 6.13. b Schematic representation of the clogging mechanism in the presence of both long-range repulsion (κq ) and short-range adhesion (Ad) (Reprinted figure with permission from Chen et al. [13] Copyright by the American Physical Society) Table 6.3 Fitting parameters in Eq. 6.13 for capture efficiency for the four cases Structures Fitting expressions Ad = 32, κq = 0 Ad = 32, κq = 0.079 Ad = 480, κq = 0 Ad = 480, κq = 0.079

η = 1.17exp(−212/Ndep ) + 0.123 η = 1.24exp(−218/Ndep ) + 0.123 η = 1.50exp(−418/Ndep ) + 0.123 η = 1.40exp(−395/Ndep ) + 0.123

Reprinted table with permission from Chen et al. [13] Copyright by the American Physical Society

low. When the adhesion is sufficiently low, no stable deposits can be formed. Compared with the adhesive forces, the electrostatic repulsion exerts its influence across a much longer distance and mainly affects the particles’ capture through modifying their trajectories. However, it has little effect on the structure of deposits. With a sufficiently strong repulsion, the deposition of particles is inhibited once a strong local electrostatic field is established leading to a non-clogging situation.

6.5 Summary In this chapter, we adopt computational fluid dynamics (CFD) coupling with the fast DEM to study the clogging behavior with charged micro-particles. The effect of long-range Coulomb repulsion on clogging is characterized in terms of the bulk permeability, the number of penetrating particles, the particle capture efficiency as well as the distributions of local volume fraction. It is observed that the clog formation is delayed or totally prevented by the repulsion and the normalized number of penetrating particles can be well described by power-low functions. A clogging phase diagram, in the form of the driving pressure

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and a proposed charge parameter κq , is constructed to quantify the clogging/nonclogging transition. Furthermore, the instantaneous particle capture efficiency is calculated to resolve the temporal evolution of the clogging process. The results indicate that the long-range Coulomb repulsion among particles has a significant effect on the particle capture only when the number of the deposited particles is moderate. A critical state, where the capture efficiency is decreased to its minimum, is identified as a clogging/non-clogging criterion for repulsive particles. The distributions of local volume fraction show that the structure of clogs is mainly determined by the shortrange adhesion (Ad) whereas the long-range Coulomb repulsion has little effect on the structure of clogs. With a relative strong adhesion, a loose clog will be formed and is easier for particles to penetrate. Finally, a schematic representation of the clogging, considering effects of both the long-range repulsion (κq ) and the short-range adhesion (Ad), is proposed to show the relationship between the clogging results and the inter-particle interactions.

References 1. H. Harrison, X. Lu, S. Patel, C. Thomas, A. Todd, M. Johnson, Y. Raval, T.R. Tzeng, Y. Song, J. Wang et al., Analyst 140(8), 2869 (2015) 2. D. Thomas, P. Penicot, P. Contal, D. Leclerc, J. Vendel, Chem. Eng. Sci. 56(11), 3549 (2001) 3. Z.B. Sendekie, P. Bacchin, Langmuir 32(6), 1478 (2016) 4. P. Le-Clech, V. Chen, T.A. Fane, J. Membrane Sci. 284(1), 17 (2006) 5. C.Y. Tang, T.H. Chong, A.G. Fane, Adv. Colloid Interface Sci. 164, 126 (2011) 6. X. Zhu, M. Elimelech, Environ. Sci. Technol. 31, 3654 (1997) 7. G.C. Agbangla, P. Bacchin, E. Climent, Soft Matter 10(33), 6303 (2014) 8. S. Bourrous, L. Bouilloux, F.X. Ouf, P. Lemaitre, P. Nerisson, D. Thomas, J. Appert-Collin, Powder Technol. 289, 109 (2016) 9. S. Li, J.S. Marshall, J. Aerosol Sci. 38, 1031 (2007) 10. H.M. Wyss, D.L. Blair, J.F. Morris, H.A. Stone, D.A. Weitz, Phys. Rev. E 74(061402) (2006) 11. B. Dersoir, M.R.D.S. Vincent, M. Abkarian, H. Tabuteau, Microfluid. Nanofluid. 19, 953 (2015) 12. J.S. Marshall, S. Li, Adhesive Particle Flow: A Discrete-element Approach (Cambridge University Press, 2014) 13. S. Chen, W. Liu, S.Q. Li, Phys. Rev. E 94(6), 063108 (2016) 14. R. Garg, J. Galvin, T. Li, S. Pannala, Powder Technol. 220, 122 (2012) 15. M. Syamlal, et al., National Energy Technology Laboratory, Department of Energy, Technical Note No. DOE/MC31346-5824 (1998) 16. R. Garg, J. Galvin, T. Li, S. Pannala (2010) 17. Z. Peng, E. Doroodchi, C. Luo, B. Moghtaderi, AIChE J. 60(6), 2000 (2014) 18. E. Guazzelli, J.F. Morris, A Physical Introduction to Suspension Dynamics (Cambridge University Press, 2012) 19. S. Benyahia, M. Syamlal, T.J. O’Brien, Powder Technol. 162, 166 (2006) 20. R.J. Hill, D.L. Koch, A.J. Ladd, J. Fluid Mech. 448, 213 (2001) 21. R.J. Hill, D.L. Koch, A.J. Ladd, J. Fluid Mech. 448, 243 (2001) 22. I. Zuriguel, D.R. Parisi, R.C. Hidalgo, C. Lozano, A. Janda, P.A. Gago, J.P. Peralta, L.M. Ferrer, L.A. Pugnaloni, E. Clèment, Sci. Rep. 4, 7324 (2014) 23. M.E. Cates, J.P. Wittmer, J.P. Bouchaud, P. Claudin, Phys. Rev. Lett. 81(9), 1841 (1998) 24. W. Liu, S. Li, A. Baule, H.A. Makse, Soft Matter 11(32), 6492 (2015) 25. W. Liu, S. Li, S. Chen, Powder Technol. 302, 414 (2016)

Chapter 7

Conclusions and Perspective

7.1 Conclusions In this thesis, the agglomeration, migration and deposition of microparticles in the presence of electrostatic and flow fields are investigated by numerical modeling and theoretical analysis. A fast discrete element method is proposed for modeling the contact interactions. An average-field method is proposed to calculate the Coulomb interaction in two-dimensional periodic systems. The dynamics of three-dimensional collisions between two microparticles is investigated to reveal the energy dissipation pathways during the collision. The agglomeration of microparticles in homogeneous isotropic turbulence is then investigated using a DNS-DEM coupling method. The agglomeration rate and the collision-induced breakage rate are formulated. We then investigate the evolution of a cloud of charged particles that migrates under an external electrostatic field. Scaling laws for the evolution of the cloud shape and the migrating velocity are derived. Finally, we present an elaborate investigation of the deposition of charged particles on a flat plane and on fibers. The temporal evolution of the deposit structure, particle capture efficiency, and the pressure drop are displayed with varying values of Coulomb repulsion and adhesion magnitudes. The main findings are as follows: 1. The energy dissipation pathways during oblique collisions between two microparticles are presented and the effect of particle size and impact angles on the critical sticking velocity is analyzed. An explicit formula in the form of VC N =  ˆ f (b)φ r p , β VC N ,0 is put forward as a sticking/rebound criterion for collisions of micron-sized particles covering different impact angles, particle sizes and size ratios. Based on the dimensionless equation describing the collision between a particle and a wall, we propose a scaling relationship to reduce the particle’s stiffness and surface energy simultaneously. It allows one to use larger time steps to resolve the collision and ensure that the results stay the same. A novel inversion method, which helps users to set the damping coefficient, particle stiffness, and surface energy to reproduce a prescribed e − v curve is also presented. With a simple but indispensable modification of the rolling and sliding resistances, the © Tsinghua University Press 2023 S. Chen, Microparticle Dynamics in Electrostatic and Flow Fields, Springer Theses, https://doi.org/10.1007/978-981-16-0843-8_7

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accelerated JKR-based DEM can be feasibly applied to the simulations of packings of adhesive particles. Structural proprieties, including the overall packing fraction, the averaged coordination number, the distributions of the local packing fraction and the coordination number of each particle, are in good agreement with the packings simulated using the original particle parameters. 2. Coupling the fast DEM with the direct numerical simulation, we are able to resolve the collision, rebound, and breakage events for adhesive particles in turbulence. We find that the size distribution of the early-stage agglomerates follows an exponential equation n(A)/n 0 = β(κ) exp(−A/κ), regardless of the adhesion force magnitude. We show that, by introducing an agglomeration kernel constructed in terms of the gyration radius of the agglomerates and the sticking probability, PBE can well reproduce the results of DNS-DEM. A relationship between the sticking probability and particle properties is then proposed based on the scaling analysis of the equation for head-on collisions. We have also shown that the collision-induced breakage rate of agglomerates can be modelled based on the statistics of the collision rate and a breakage fraction function . The fraction function  is further expressed as a function of the well-known distributions of the impact velocity and a universal transfer function ψ(vn ), which is shown to rely on particle-particle contact interactions and is independent of the particle inertia, particle size, and hydrodynamic interactions. Based on a large number of simulations, we propose an exponential function of adhesion parameter Adn for the breakage rate of doublets and show that the breakage rate increases linearly as the agglomerate size increases. 3. The behavior of a migrating cloud containing charged particles is studied by the Oseen dynamics. It is found that, for particles with a weak Coulomb interaction, the migrating cloud expands and breaks up into secondary clouds, which resembles the behaviors of Stokes cloud settling under gravity. We have shown that the Lyapunov exponent decreases as the dimensionless charge parameter κq increases, indicating that the long-range Coulomb repulsion makes the cloud more stable. The simulated results with strong repulsion can be described with a continuum convection equation, which predicts the evolution of the density field, the radius, and the migrating velocity of the cloud. A dimensionless charge parameter κq is proposed to quantify the effect of the repulsion, and the ratio κq /κq,t successfully captures the transition from the hydrodynamically controlled regime (with κq /κq,t < 1) to the repulsion-controlled regime (with κq /κq,t > 1). 4. We then present a computational study of the packing of charged micron-sized particles. An average-field method is proposed to calculate the Coulomb interaction in two-dimensional periodic systems. The formation process and the final structures of ballistic packings are then studied to show the effect of interparticle Coulomb force. We found that the presence of long-range Coulomb interaction results in a looser packing structure through its influence on the particle kinetic energy. The relative decrease of the volume fraction, denoted as φ0 /φ − 1, approximately follows a square law with the increase of particle’s charge. However, this effect is suppressed by short-range adhesion once particles are bonded into the contact network. Furthermore, a modified adhesion number, Ad∗ , was derived to

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clarify the combined effects of particle’s kinetic energy, adhesion, and Coulomb interaction. As Ad∗ increases, both the volume fraction φ and the average coordination number Z  decrease monotonously. We have also shown that the packing state of charged micron-sized particles can be well described by the latest derived adhesive loose packing (ALP) regime in the phase diagram. Increasing the charge of particles makes the packing states move toward the ALP point. The results indicate the universality of the analytical presentation for packings of adhesive or charged particles based on the Edwards’ ensemble approach. 5. At last, we adopt computational fluid dynamics (CFD) and fast DEM to study the clogging behavior of the charged micro-particles. It is observed that the clog formation is delayed or totally prevented by the repulsion. The normalized number of penetrating particles can be well described by power-low functions. A clogging phase diagram, in the form of the driving pressure and the proposed charge parameter κq , is constructed to quantify the clogging/non-clogging transition. Furthermore, the instantaneous particle capture efficiency is calculated to resolve the temporal evolution of the clogging process. The results indicate that the long-range Coulomb repulsion among particles has a significant effect on the particle capture only when the number of the deposited particles is moderate. A critical state, where the capture efficiency is decreased to its minimum, is identified as a clogging/non-clogging criterion for repulsive particles. The distributions of the local volume fraction show that the structure of clogs is mainly determined by the short-range adhesion whereas the long-range Coulomb repulsion has little effect on the structure of clogs. With a relatively strong adhesion, a loose clog will be formed, which is easier for particles to penetrate. A schematic representation of the clogging process, considering the effects of both long-range repulsion (κq ) and short-range adhesion (Ad), is proposed to show the relationship between the clogging results and the inter-particle interactions.

7.2 Future Work The transport of microparticles is a complicated problem, often involving forces across different length and time scales. This thesis provides just a few typical problems in which DEM is able to provide insight into the particle transport. These problems, however, are still far from being fully understood. There are several interesting directions for future study. 1. The fast DEM approach developed in Chap. 2 has neglected the effect of external forces (Fext ), which actually exits in a variety of particulate systems. When the reduced stiffness is used in the DEM models, Fext should be simultaneously modified. A general principle to reduce the external force is not clear so far. Moreover, the so-called coarse-grain DEM models have been developed for nonadhesive gas-solid systems to reduce the computational cost [1]. In a coarse-grain DEM, coarse-grained particles are introduced to represent a group of original particles

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such that the total number of the simulated particles in the system is reduced. However, there have been few attempts to systematically study the coarse-grain DEM for adhesive particle flows. 2. In Chap. 3, we fix the value of Taylor-microscale Reynolds number Reλ of the turbulent flow at Reλ ≈ 93. It has been reported that the relative velocity and the collision rate for inertial particles increase strongly with the increase of Reλ [2–4]. However, a stronger clustering effect may suppress the agglomeration [5]. For high-Reynolds-number flows, other effects, including the correlated collision events [6] and multifractal statistics of velocity differences [7], may dominate the agglomeration. Therefore, a complete picture of agglomeration should include the role of both the turbulent transport (e.g., vortices and intermittency) and the microphysical mechanisms (particle-level interactions), which is an interesting direction for future research. 3. In Chaps. 4–6, the migration and deposition problems are investigated with the Coulomb repulsion and uniform external fields. In many real-world systems, higher-order multipoles or nonuniform external fields often lead to the phenomena of particle chaining and ordering [8, 9]. Expanding our model to include the effects of higher-order multipoles or nonuniform external fields, seems worth pursuing. Moreover, the size polydispersity and the shape of particles are also crucial factors that need to be considered as well, thus much more effort is necessary in relevant studies.

References 1. 2. 3. 4. 5. 6. 7. 8. 9.

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