Title Page

국문초록

Contents

Nomenclature 17

I. Introduction 21

1. Scope of research 21

a) Physics at micro scale 22

b) Role of electrokinetics in microsystems 23

2. Introduction to electrokinetics 25

a) Historic Background 25

b) Structure of the electric double layer 29

c) Electroosmosis and electrophoresis 32

d) Electrohydrodynamic flow in dielectric liquids 34

II. Implementation of immersed boundary method 39

1. Scope of numerical simulation 39

2. Fractional-step Navier-Stokes solver 40

3. Introduction to immersed boundary methods 41

a) Discrete forcing IBM 43

b) Validation of discrete forcing IBM solver 43

c) Continuous forcing IBM 49

d) Validation of continuous forcing IBM solver 50

III. AC/DC electroosmosis around a pair of cylindrical electrodes 52

1. Background 52

2. Objectives and outline of the chapter 53

3. Physical and mathematical model 54

4. Numerical schemes 59

5. IB method for Robin type no-flux boundary condition 60

6. Simulation results 63

a) Numerical validation with electroosmosis around rectangular electrodes 63

b) DC electroosmosis around circular electrodes 67

c) AC electroosmosis around circular electrodes 71

d) Effect of AC frequency 76

IV. Electrophoretic interaction between a cylindrical colloidal particle and a planar wall 78

1. Introduction 78

2. Objectives and outline of the chapter 80

3. Physical and mathematical model 81

4. Numerical method and validation 87

a) Chimera-grid-based PNP solver 87

b) Validation 89

5. Electrostatic effects 98

a) Constant surface potential 98

b) Constant surface charge 104

6. Electrophoretic motion 110

a) Formulation 110

b) Effect of EDL thickness 113

c) Effect of particle surface zeta potential 115

V. Electrohydrodynamic flow of dielectric liquid in an annulus 118

1. Introduction 118

2. Objectives and outline of the chapter 120

3. Mathematical model and numerical method 120

4. Simulation results 125

a) Onset of electro-convection 125

b) Effect of qs on Tc(이미지참조) 126

c) Effect of ri on Tc(이미지참조) 128

d) Electro-convection with inner injection 129

e) Electro-convection with outer injection 137

VI. Experimental and numerical study on electrohydrodynamic flow around a sandwiched wire electrode 139

1. Motivation 139

2. Objectives and outline of the chapter 141

3. Mathematical model 141

4. Experimental setup 145

5. Experimental and simulation results 147

VII. Summary of the thesis 152

1. DC/AC electroosmosis around a pair of cylindrical electrodes 152

2. Electrophoretic interaction between a cylindrical colloidal particle and a planar wall 153

3. Electrohydrodynamic flow of dielectric liquid in an annulus 155

4. Experimental and numerical study on electrohydrodynamic flow around a sandwiched wire electrode 156

References 157

Abstract 169

Table 1. Comparison of wake length and separation angle for flow over a circular cylinder at different Reynolds numbers. 46

Table 2. Comparison of drag coefficient (CD) for different Reynolds numbers with those in the literature.(이미지참조) 51

Table 3. Variation of wave number k with T for the inner injection for different ri ; "―" represents the hydrostatic 1D conduction state.(이미지참조) 133

Table 4. Variation of wave number k with various T and ri for the outer injection; "―" represents the hydrostatic 1D conduction state. 135

Table 5. Properties of the working fluid (Dodecan + 5%wt SPAN 80) used for numerical simulation; (V), properties taken from internet; (O), properties taken from references [121], [124]. 148

Table 6. Derived properties for the working fluid (Dodecan + 5%wt SPAN 80). 148

Fig. 1. Helmholtz's rigid electric double layer. 27

Fig. 2. A schematic diagram of the electric double layer (EDL) and distribution of electric potential in it for a flat solid wall in contact with an aqueous solution (when the wall is negatively charged). 31

Fig. 3. Schematic diagram of DC electroosmotic flow in a plane channel. 33

Fig. 4. AC electroosmotic flow over a pair of integrated flat plate electrodes. 33

Fig. 5. Mechanism of free ion formation through reverse micelle. 36

Fig. 6. Schematic diagram of physical domain for the simulation of flow over circular cylinder. 44

Fig. 7. Streamline patterns of the steady flow over a circular cylinder at 45

Fig. 8. Schematic diagram showing the wake length Lw and the separation angle θ .(이미지참조) 46

Fig. 9. Instantaneous streamlines pattern of steady periodic flow observed at Re＝100. 46

Fig. 10. Flow characteristics of the backward-facing step flow at Re＝100 48

Fig. 11. Comparison of flow velocities along an axial line starting at (4, 0.5, 2) for backward-facing-step flow at Re＝100, obtained from the present numerical code and the commercial code CFX. 49

Fig. 12. Schematic diagram of the computational domain showing the position of the boundaries surrounding a circular cylinder. 50

Fig. 13. Schematic diagram of the computational domain showing (a) 3D structure of the microchannel with axial electrodes and the cross-sections with (b) square and (c) circular electrodes. 56

Fig. 14. Schematic diagram of the interpolation scheme for the concentration source for the case of (a) rectangular and (b) circular electrodes. 62

Fig. 15. Comparison of numerical results of decoupled and coupled solvers; distributions, along the horizontal centerline, of (a) nondimensional cation concentration and (b) nondimensional potential. 65

Fig. 16. Comparison of numerical results of the decoupled solver with the Poisson- Boltzmann model; variation of induced potential with the distance from the surface of the Ieft electrode in the negative x direction at steady state. 66

Fig. 17. Comparison of the time evolved horizontal velocity at a point (4.0, 3.1) under AC field for the rectangular electrodes，obtained from the fractional-step based primitive-variable solver and the vorticity-stream-function based solver. 66

Fig. 18. Distribution of the equi-potential (nondimensional) lines around the electrodes under DC field. 69

Fig. 19. The equi-potential and equi-concentration lines around the circular electrodes depicting EDL around them under DC field 69

Fig. 20. Time evolution of flow velocities and ion concentrations at a point (4.4, 2.5) for circular electrodes under DC field. 70

Fig. 21. Variation of potential and concentration at a point (4.1, 2.5) near the left electrode during an AC period for circular electrodes. 72

Fig. 22. The distribution of variables along the horizontal center line during different instances of an AC period 73

Fig. 23. Streamline plots showing the variation of instantaneous flow field during an AC period. (streamline values are different for each plot). 74

Fig. 24. Time evolution of the horizontal velocity profile along the upper-half vertical centerline of the left electrode; the profile along the lower-half is symmetric with this. 74

Fig. 25. Time-averaged flow field 75

Fig. 26. Effect of AC frequency on the electroosmotic velocity (u* is measured at (6.5, 3.25) and v* at (5, 1.5)).(이미지참조) 77

Fig. 27. Schematic diagram of the physical domain showing a colloidal particle near a planar wall. 81

Fig. 28. Schematic of Chimera grids near the particle and three different regions. 88

Fig. 29. Enlarged view of Chimera grids near the grid interface highlighting interpolation points. 88

Fig. 30. (a) Electric field distribution in the domain for the constant surface 'zeta' potential case (κ＝2, Cartesian 100x100, polar 15x90), (b) Enlarged view of the right bottom region near the particle. 90

Fig. 31. Schematic diagram of the ionic flux interpolation (FI) at the inner grid interface. 92

Fig. 32. Variation of EDL interaction force with the gap between the particle and the planar wall. The plot also shows comparison of EDL interaction forces given by the FI-PNP, CI-PNP and PB solvers. 92

Fig. 33. Grid independence test: Distribution of the potential in the gap between the particle and the planar wall for κ＝2 and for three different grid sizes (△y＝0.02, 0.025 and 0.033). 94

Fig. 34. Comparison of the electric potential distribution along a radial direction obtained from the PNP solver with that obtained from the 1D PB model in cylindrical coordinates for a colloidal particle in an unbounded domain. 94

Fig. 35. Geometrical construction for the calculation of the EDL interaction force between a cylindrical particle and a planar wall. 95

Fig. 36. Comparison of the EDL interaction force between the particle and the planar wall obtained from the FV-based PB solver and the corresponding estimate obtained using the Derjaguin approximation. 96

Fig. 37. Comparison of the EDL interaction force between the particle and the planar wall obtained from the FV-based PNP solver and the corresponding estimate obtained using the FV-based PB solver. 97

Fig. 38. Distribution of (a) internal electric potential, (b) cation concentration and (c) anion concentration in the EDL for the constant surface potential case obtained from the PB solver at κ＝2 and H＝0.5. 99

Fig. 39. Distribution of (a) internal electric potential, (b) cation concentration and (c) anion concentration in the EDL for the constant surface potential case obtained from the PNP solver at κ＝2 and H＝0.5. 100

Fig. 40. Distribution of the radial component of the internal-electric-field distribution around the particle obtained from the PNP model in comparison with that from the PB model for the constant surface potential case at κ＝2 and H＝0.5. 102

Fig. 41. Comparison of the PNP model and the PB model in the variation of the horizontal electrostatic-force component with the gap width H under a constant surface potential. 102

Fig. 42. Comparison of the PNP model and the PB model in the variation of the vertical electrostatic-force component with the gap width H under a constant surface potential. 103

Fig. 43. Distribution of the (a) internal electric potential, (b) cation concentration and (c) anion concentration in the EDL for the constant surface charge case obtained from PNP model at κ＝2 and H＝0.5. 106

Fig. 44. Internal potential on the particle surface and ionic concentration at the first radial grid near the particle as functions of θ. 107

Fig. 45. Variation of the horizontal component of the electrostatic force with the gap width H for different inverse EDL thickness κ under a constant surface charge. 107

Fig. 46. Variation of the EDL interaction force with the gap between the particle and the planar wall for different EDL thickness under a constant surface charge. 109

Fig. 47. Relative velocity vector plot showing electroosmotic flow field around the particle and the plane under a constant surface potential. 112

Fig. 48. Variation of the horizontal component of electrophoretic velocity of the particle with H for different κ. 114

Fig. 49. Variation of the vertical component of electrophoretic velocity of the particle with H for different κ. 114

Fig. 50. Variation of the rotational speed of the particle with H for different κ. 115

Fig. 51. Variation of the horizontal component of electrophoretic velocity of the particle with H for different ςp.(이미지참조) 116

Fig. 52. Variation of the vertical component of electrophoretic velocity of the particle with H for different ςp.(이미지참조) 116

Fig. 53. Comparison of numerical results of the one-dimensional hydrostatic (base) solutions ΨB(r) and qB(r) obtained from 1D and 2D numerical methods for the inner injection at ri＝0.1 and qs＝10.(이미지참조) 126

Fig. 54. Effect of the injection strength on Tc at ri＝0.1 for both inner and outer injection.(이미지참조) 127

Fig. 55. Effect of the inner cylinder radius on Tc at qs＝10. The region of even wave number is indicated by solid curves and that of the odd wave number by dashed curves.(이미지참조) 128

Fig. 56. Map of flow patterns for the inner injection at qs＝10 ; curve, 1D linear stability analysis; symbols, 2D numerical solution. Symbol classification: ■, 1D hydrostatic regime; ◆, critical; 口, stationary electro-convection; ◇, oscillatory electro-...(이미지참조) 130

Fig. 57. Typical stationary electro-convection for the inner injection at ri＝0.1, qs＝10 and T＝ 100 (이미지참조) 130

Fig. 58. Oscillation behavior of Ek for various T for the inner injection at ri＝0.1 and qs＝10.(이미지참조) 131

Fig. 59. Instantaneous charge-density contours (a) at t＝20 and (b) at t＝25 representing typical oscillatory electro-convection for the inner injection at, ri＝0.1, qs＝10, and T ＝ 500.(이미지참조) 131

Fig. 60. (a) Stationary contours of charge density and (b) streamlines for the inner injection at ri＝0.5, qs＝10 and T＝200.(이미지참조) 134

Fig. 61. Instantaneous contours of charge density for the inner injection at ri＝0.5, qs＝10 and T＝1000(이미지참조) 134

Fig. 62. Map of flow patterns for the outer injection at qs＝10 ; curve, 1D linear stability analysis; symbols, 2D numerical solution. Symbol classification:■, 1D hydrostatic regime; ◆, critical; 口, stationary electro-convection; △, chaotic electro-...(이미지참조) 135

Fig. 63. Typical stationary electro-convection for the outer injection at ri＝ 0.1, qs＝10 and T＝410(이미지참조) 136

Fig. 64. Instantaneous contours of (a) charge density and (b) streamlines for the outer injection at ri＝0.1, q＝10 and T＝ 500.(이미지참조) 136

Fig. 65. Mean wavelength of stationary electro-convection at various ri for the inner and outer injections. Dash-dot line corresponds to λm ＝ λfc.(이미지참조) 137

Fig. 66. Physical domain with the boundary conditions. 142

Fig. 67. Experimental setup 146

Fig. 68. Comparison of experimental and numerical streamlines at the applied potential difference between the electrodes set at 1 kV . 149

Fig. 69. Effect of ionic radius on the EHD flow velocity. 149

Fig. 70. Effect of electric field on the EHD flow velocity. 150

 MEMS나 μTAS와 같은 미세유체 시스템의 성능은 펌프, 혼합, 반응물과 바이오 샘플의 검출 및 분리 등과 같은 기능을 구현하는 메커니즘에 의해 결정된다. 동전기학은 이러한 기능들을 수행할 수 있게 하는 도구 충 하나이다. 본 학위논문은 여러 가지 동전기적 현상에 대한 수치해석적 및 실험적 연구를 다룰 것인데, 특히 동전기현상의 물리학적 특성과 미세유체 시스템에서의 응용 가능성에 초점을 맞출 것이다.

1) 한 쌍의 실린더 전극 주위의 교류/직류 전기삼투: 여기서는 한 쌍의 실린더 형전극 사이에 직류(DC) 혹은 교류(AC) 전기장이 가해질 때 실린더 주위에서 나타나는 이온 수송과 그로 인해 야기되는 전기삼투 유동을 다루었다. 완전히 극성화되는 전극 주위의 이온 수송을 지배하는 Poisson-Nernst-Plank (PNP) 방정식은 Stern 층을 무시한 채 수치해석에 의해 해를 구한다. 전극 표면에서 이온 농도의 플럭스가 0이 되는 경계조건을 만족시키기 위한 새로운 가상경계법(IB)을 개발하였으며, 유도 전기삼투 유동에 대해서는 Stokes 방정식을 수치해석에 의해 풀어서 해를 구하고 그 특징을 조사한다. AC 하에서는 전기삼투 유동이 외부 교류 진동 수의2배에 해당하는 진동 수로 진동한다. 또한 0이 아닌 정상 속도장이 얻어진다. 속도장의 크기는 교류 주파수에 크게 의존한다.

2) 실린더 형 콜로이드 입자와 평면 벽 사이의 전기 영동: 여기서는 전하를 띈 평면 벽 근처에 실린더 형 콜로이드 입자가 놓여 있고 외부로부터 전기장이 벽면에 나란히 인가되는 경우 입자에 가해지는 정전기력과 유체역학적 힘을 계산하였고, 전기영동에 따른 입자의 운동은 정전기력과 유체역학적 힘의 균형으로부터 결정하였다. 정전기력은 입자 주위의 전기장 분포에 의해 결정되며 본 연구에서는 PNP방정식을 Chimera 격자 상에서 유한 체적법에 의한 수치해석적 방법으로 해를 구한다. 전기삼투에 의해 야기되는 유체역학적 힘은 Stokes 방정식의 수치해로부터 구해지는 속도장과 압력 분포에 의해 계산된다. 전하를 띈 평면 벽은 입자에 가해지는 정전기력의 수평방향 성분을 감소시키는 효과를 보이며, 전기삼투를 통해 그것이 입자의 운동에 미치는 효과는 매우 크다.

3) 환상공간 내에서 절연 유체의 전기동역학적 유동: 여기서는 두 개의 동심원전극 사이에 존재하는 절연 유체가 전극에서 발생되는 편극성 전하 분출에 의해 전기一대류 현상이 나타나며 나아가 복잡한 유동 현상을 보이는 유동 문제를 다루었다. 전하 밀도의 수송을 지배하는 NP 방정식, 전기포텐셜을 결정하는 Poisson 방정식, 그리고 유체유동을 지배하는 Navier-Stokes 방정식을 유한체적법으로 풀어서 해를 구한다. 해석결과로부터 안정한 유체정역학적 상태와 전기-대류의 상태를 구분 짓는 파라미터 공간을 구한다. 전기-대류의 영역은 다시 정지, 진동 및 카오스의 세가지 소 영역으로 나누어진다.

4) 평판 전극사이의 와이어 전극 주위의 전기동역학적 유동에 대한 실험적 및 수치해석적 연구:동일한 극성을 가진 두 개의 평판 전극 사이에 다른 극성을 가지는 와이어 전극이 존재하는 경우에 발생하는 Onsager효과로 인해, 전극 주위에 형성되는 절연유체의 전기동역학적 유동에 관하여 연구하였다. 상용 소프트웨어인 COMSOL MULTIPHYSICS를 사용하여 Onsager 효과에 기인하는 전하 생성의 수학적 모델을 수치적으로 해석한다. 계면활성제가 혼합된 도데칸 용액으로 실험을 수행하고 그 결과를 수치해와 비교한다. 두 결과는 모두 속도의 크기가 외부로부터 인가한 전위차이의 세 제곱에 비례함을 보였다.