Boundary Conditions and Multi-Scale Modeling for Micro-and Nano-flows by Lin Guo A dissertation submitted to The Johns Hopkins University in conformity with the requirements for the degree of Doctor of Philosophy. Baltimore, Maryland March, 2016 c Lin Guo 2016 All rights reserved Abstract The development of micro- and nanofluidic devices requires detailed knowledge of interfacial phenomena. This thesis addresses two important effects at wall-fluid interfaces, boundary slip and electroosmosis, through numerical simulations. The first study uses molecular dynamics (MD) simulations to probe the influence of surface curvature on the slip boundary condition for a simple fluid. The slip length is measured for flows in planar and cylindrical geometries. As wall curvature increases, the slip length decreases dramatically for close-packed surfaces and increases slightly for sparse ones. The magnitude of the variation depends on the crystallographic orientation and the flow direction. The different patterns of behavior are related to the curvature-induced variation in the ratio of the spacing between fluid atoms to the spacing between minima in the potential from the solid surface. The results are consistent with a microscopic theory for the viscous friction between fluid and wall that expresses the slip length in terms of the lateral response of the fluid to the wall potential and the characteristic decay time of this response. The second study performs MD simulations to explore the effective slip boundary conditions over surfaces with one-dimensional sinusoidal roughness for two differ- ent flow orientations: transverse and longitudinal to the corrugations, and different atomic geometries of the wall: smoothly bent and stepped. The results for the sparse ii ABSTRACT bent surfaces quantitatively agree with the continuum predictions with a constant local boundary condition. The effective slip length decreases with increasing corruga- tion amplitude, and the reduction is larger for the transverse direction. Atomic effects become significant for the close-packed bent and for the stepped surfaces, which may even enhance the effective slip along the longitudinal direction. In the third study, an efficient multi-scale method is developed to simulate elec- troosmotic flows. MD is used in the near wall region where the atomistic details are important, while continuum incompressible fluctuating hydrodynamics is applied in the bulk region. The two descriptions are coupled in an overlap region. Because of the low ion density and the long-range of electrostatic interactions, discrete ions are retained in the bulk region and simulated by a stochastic Euler-Lagrangian method (SELM). The MD and SELM descriptions seamlessly exchange ions in the overlap region. This hybrid approach is validated against full MD simulations for different geometries and types of flows. 1 Thesis Advisor: Professor Mark O. Robbins Thesis Advisor: Professor Shiyi Chen Thesis Reader: Professor Andrea Prosperetti 1This material is based on work supported by the National Science Foundation under Grants No. DMR-1411144 and No. DMR-1006805. iii Acknowledgments First, and foremost, I would like to express my sincere and deepest gratitude to my advisors, Professors Mark O. Robbins and Shiyi Chen. Their guidance, patience, and encouragement have supported me to grow throughout this dissertation process, as well as the entire graduate training. I would also like to thank all the faculty and staff members, both in the Mechanical Engineering and the Physics and Astronomy departments, for their kind help during my Ph.D. study. In particular, I would like to thank Professor Andrea Prosperetti for serving as my dissertation reader. I want to thank Professors Charles Meneveau, Joseph Katz, Omar M. Knio, Cila Herman and Lester Su, for spending time teaching me. Thank you to our research group members, Jin Liu, K. Michael Salerno, Ting Ge, Lars Pastewka, Tristan A. Sharp, Thomas O’Connor, Joel Clemmer, Vikram Jadhao, Joseph Monti and Marco Aurelio Galvani Cunha. I appreciate all their help and the motivating and fun environment in which to learn and grow. Finally, I would like to express my heartfelt gratitude to my family, particularly my wife, for their love and support along the way. iv Contents Abstract ii Acknowledgments iv List of Tables ix List of Figures x 1 Introduction 1 1.1 SlipBoundaryCondition. 2 1.2 EffectiveSlipBoundaryCondition. 4 1.3 ElectroosmoticFlow ........................... 6 1.4 Multi-ScaleSimulation .......................... 7 2 Slip Boundary Conditions over Curved Surfaces 9 2.1 Introduction ............................... 9 2.2 Detailsofmolecularsimulations . 12 2.2.1 Interaction Potentials and Equations of Motion . .. 12 2.2.2 PlanarGeometry ........................ 15 2.2.3 FluidStructurenearWall . 16 2.2.4 CylindricalGeometry . 16 v CONTENTS 2.2.5 CalculatingtheSlipLength . 23 2.2.5.1 PlanarCouetteflow . 23 2.2.5.2 CylindricalCouetteFlow . 27 2.2.5.3 AxialFlowinCylindricalGeometry . 30 2.3 Results .................................. 31 2.3.1 SlipLength ........................... 31 2.3.2 FluidStructure ......................... 33 2.3.3 RelatingSlipLengthtoStructure . 42 2.4 SummaryandConclusions . 50 2.5 AppendixA ............................... 55 2.6 AppendixB ............................... 61 3 Effective Slip Boundary Conditions for Sinusoidally Corrugated Sur- faces 64 3.1 Introduction ............................... 64 3.2 SimulationMethodsandAnalyticalModels . 67 3.2.1 MolecularDynamicsSimulations . 67 3.2.2 FluidStructurenearWall . 74 3.2.3 Determining Intrinsic and Effective Slip Lengths . ... 76 3.2.4 ContinuumSimulations . 81 3.2.5 AnalyticalModels ........................ 83 3.3 ResultsandDiscussion ......................... 84 vi CONTENTS 3.3.1 BentSurfaces .......................... 84 3.3.2 SteppedSurfaces ........................ 94 3.4 SummaryandConclusions . 101 4 Multi-Scale Simulation Method for Electroosmotic Flows 105 4.1 Introduction ............................... 105 4.2 SimulationMethods ........................... 108 4.2.1 OverviewoftheHybridScheme . 108 4.2.2 ParticleModel1: MolecularDynamics . 112 4.2.3 Particle Model 2: Particles of Stochastic Eulerian Lagrangian Simulations ............................ 116 4.2.4 Continuum Fluid Model: Incompressible Navier-Stokes Fluc- tuatingHydrodynamics . 126 4.2.5 Coupling Continuum Fluid Dynamics with Molecular Dynam- ics ................................. 131 4.2.5.1 Boundary Conditions for Continuum Fluid Dynamics fromMolecularDynamics . 132 4.2.5.2 Boundary Conditions for Molecular Dynamics from ContinuumFluidDynamics . 133 4.2.6 Exchanging Charges between Molecular Dynamics and Stochas- ticEuler-LagrangianSimulations . 137 4.2.7 TimeCouplingScheme . 139 vii CONTENTS 4.3 ResultsandDiscussion . 141 4.3.1 Incompressible Navier-Stokes Fluctuating Hydrodynamic Solver inBulkSystems ......................... 141 4.3.2 MatchingBulkDiffusioninSELMandMD . 146 4.3.3 DynamicalChannelFlow . 152 4.3.4 Diffusion of Charges Between Between MD and SELM Regions 155 4.3.5 ElectroosmoticFlowsinFlatChannels . 157 4.3.6 Electroosmotic Flow in Channels with Sinusoidal Corrugations 165 4.4 SummaryandConclusions . 177 Bibliography 179 Vita 215 viii List of Tables 2.1 The four sets of wall-fluid interaction parameters studied in this chapter 13 2.2 Resultsforplanarsurfaces . 61 2.3 Results for cylindrical surfaces with axis along the nearest-neighbor (110)direction............................... 62 2.4 Results for cylindrical surfaces with axis along the next-nearest-neighbor (100)direction............................... 63 3.1 Parameters of the five groups of solid bottom walls presented in this chapter................................... 75 ix List of Figures 2.1 Geometries for planar and cylindrical simulations . ....... 17 2.2 Close up side views of the wall geometry for planar and cylindrical simulations ................................ 18 2.3 Fluid density as a function of distance from wall surface . ...... 21 2.4 Velocity profiles for planer and cylindrical flows . ...... 25 2.5 Slip length as a function of surface curvature . ..... 32 2.6 Wallpotential............................... 35 2.7 In-plane order as characterized by normalized structurefactor . 37 2.8 Variation of S˜1(G~ main)/N1 withsurfacecurvature . 39 2.9 Variation of S˜(Gmain~ )/N1 and Ls with relative effective spacing . 41 2.10 Ls as a function of S˜1(G~ main)/N1 and tph as a function of relative effectivespacing.............................. 46 2.11 Cosine Fourier transform of the local area density and fluid profiles as afunctionofheight............................ 48 2.12 Variation of slip length with relative wall spacing . ........ 58 2.13 Ratio of drag coefficients as a function of the relative wallspacing . 60 3.1 GeometryofMDsimulation . 69 3.2 Snapshots of fluid atoms near a flat surface and three corrugated sur- faceofdifferenttypes........................... 73 3.3 Fluid density as a function of distance from wall . .... 75 3.4 Velocityprofilesasafunctionofheight . ... 78 3.5 Effective slip length as a function of normalized amplitude ...... 85 3.6 Variation of intrinsic slip length with surface curvature ........ 89 3.7 In-plane order as characterized by normalized structure factor on flat surfaces .................................. 93 3.8 Effective slip length as a function of normalized amplitude ...... 95 3.9 Different scaling of effective slip length between stepped and bent surfaces 96 3.10 Effective longitudinal slip length for stepped surfaces with different wavelengths