Manipulation of Microparticles Using a Piezoelectric Actuator

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Manipulation of Microparticles using a Piezoelectric Actuator

A Thesis

Submitted to the Faculty

Drexel University

Johann deSa

in partial fulfillment of the

requirements for the degree

Doctor of Philosophy

November 2009

III

Dedications

This thesis is dedicated to my parents, Ninette and Michael, my siblings, Colin, Anil, David and Anisha, and to my girlfriend Elena, for their unwavering love and support.

Acknowledgements

I would like to express my sincere appreciation to my advisor Dr Ryszard Lec, for his continuous guidance and support during the course of my PhD. I would also like to thank all my committee members, Dr Todd Doehring, Dr Alan Lau, Dr Peter Lewin, Dr Marek

Swoboda, Dr Guoliang Yang and Dr Qiliang Zhang, for their time, advice, reviews and discussions on my thesis topic.

I would like to extend a special thanks to Dr Qiliang Zhang for always making the time and effort to observe, analyze, and discuss several key aspects of my thesis work and to Dr

Guoliang Yang for his guidance in using the AFM and analyzing its results.

There are several colleagues from the Biosensors Lab., who, over the course of my PhD have become more like family to me, namely, Nishit Mehta, Robert Hart, Himanshu Mehta, Matias

Hochman, Mark Mattuchi, Dr Sun Kwoun, Agastya Anishetty, Jimit Doshi, Atman Shah,

Trevor King, and many more. Special thanks to Ertan Ergezen for his input and discussions on my thesis and several other projects that we worked on together. Besides always being available for technical discussions these colleagues/friends provided the necessary support framework needed to make my PhD a success.

TABLE OF CONTENTS

LIST OF TABLES...... X

LIST OF FIGURES ...... XI

LIST OF SYMBOLS ...... XVIII

ABSTRACT ...... XXIII

1. INTRODUCTION ...... 1

1.1. Background...... 1

1.2. Problem statement...... 2

1.3. Existing manipulation techniques ...... 3

1.3.1. Optical Tweezers ...... 4

1.3.2. Magnetic Tweezers ...... 5

1.3.3. AC Electrokinetic Manipulation...... 6

1.3.4. Atomic Force Microscopy (AFM)...... 7

1.3.5. Acoustic Particle Manipulation...... 8

1.4. Proposed Acoustic Interfacial Force Manipulation Technique...... 10

1.5. Acoustic Wave Transducers (AWT)...... 12

1.6. Acoustic Waves and their interaction with particles...... 13

1.7. Piezoelectric transducers...... 14

1.7.1. Piezoelectric Effect ...... 14

1.7.2. Piezoelectric transducer materials and their modes of vibration ...... 16

1.7.3. Material properties of some piezoelectric transducers...... 18

1.8. Finite element analysis of piezoelectric transducers...... 20

1.9. Conceptual model of a Piezoelectric Interfacial Particle Manipulator (PIPM)...... 21

1.10. Significance...... 23

1.11. Objectives of the thesis ...... 25

1.12. Organization of the thesis ...... 26 VI

2. STUDY OF PARTICLE-PIPM INTERFACIAL FORCES ...... 28

2.1. Interfacial adhesion forces ...... 29

2.1.1. Gravitational force ...... 30

2.1.2. Electrostatic force ...... 32

2.1.3. Van der Waal’s (VDW) force...... 32

2.1.4. Capillary force ...... 35

2.1.5. Comparison of adhesion forces...... 37

2.1.6. Friction force...... 39

2.2. Interfacial manipulation forces ...... 42

2.2.1. Inertial manipulation force...... 43

2.2.1.1. Critical dynamic displacement and frequency of vibration...... 46

2.2.2. Impulsive manipulation force ...... 48

2.2.2.1. Determination of the final velocities of the particle and the surface: 49

2.3. Mechanism of manipulation...... 54

2.3.1. Projectile motion of the particle...... 61

2.4. Theoretical limits of particle manipulation...... 64

2.5. Summary...... 66

3. FINITE ELEMENT METHOD (FEM) ANALYSIS OF THE PIPM ...... 68

3.1. Theory of finite element analysis of a piezoelectric disc...... 68

3.2. FEA results...... 77

3.2.1. Radial mode of vibration ...... 79

3.2.1.1. Study of maximum normal and tangential displacements as a function of damping factor...... 86

3.2.1.2. Study of maximum normal and tangential displacements as a function of excitation voltage...... 88

3.3. Study of particle motion in 1 and 2-D space...... 92

3.3.1. Particle moving in 1-D space...... 92 VII

3.3.2. Particle moving in 2-D space...... 98

3.4. Summary:...... 105

4. EXPERIMENTAL METHODS...... 106

4.1. AFM force and roughness measurements...... 106

4.1.1. Atomic Force Microscopy (AFM) setup for interfacial adhesion force ...... measurement...... 106

4.2. PIPM displacement calibration using AFM...... 110

4.3. Particle manipulation setup...... 111

4.4. Ring PIPM Sensing Structure set up...... 115

5. EXPERIMENTAL RESULTS...... 117

5.1. AFM surface topography and adhesion force results...... 117

5.1.1. Surface scan of particle and PIPM surfaces...... 118

5.1.1.1. Particle roughness measurement ...... 118

5.1.1.2. PIPM surface roughness measurement:...... 119

5.1.2. Measurement of adhesion force binding stainless steel particles to the PIPM ... surface...... 122

5.2. PIPM displacement calibration using AFM...... 127

5.2.1. FEM with calibrated loss factor...... 130

5.3. Experimental results of particle manipulation ...... 133

5.3.1. Manipulation force as a function of location and PIPM frequency ...... 134

5.3.1.1. Particle manipulation on a 30 mm PIPM...... 138

5.3.1.2. Particle manipulation on a 20 mm PIPM...... 138

5.3.1.3. Particle manipulation on a 10 mm PIPM...... 140

5.3.2. Particle translation speed as a function of PIPM excitation voltage...... 144

5.3.3. Particle manipulation at a 56kHz natural mode of vibration ...... 147

5.3.4. PIPM multi-modal operation...... 151

5.3.5. Application of particle manipulation in Biosensing ...... 153 VIII

5.3.5.1. FEA of ring PIPM and ring PIPM sensing structure ...... 154

5.3.5.2. Improvement in detection of 10 m polystyrene particles ...... 159

5.3.5.3. Improvement in detection of 5 m silica particles ...... 161

5.3.5.4. Improvement in detection of Escherichia coli...... 162

5.4. Advantages and disadvantages of the PIPM technique...... 164

5.5. Summary...... 165

6. CONCLUSIONS AND FUTURE WORK ...... 167

6.1. Conclusions...... 167

6.2. Contribution...... 168

6.3. Future work...... 170

6.3.1. Theoretical and experimental study of the manipulation of deformable objects ...... 170

6.3.2. Manipulation applied to a biochip ...... 170

6.3.3. Acousto-magnetic manipulation ...... 170

6.3.4. Building of pre-programmed materials...... 170

6.3.5. Particle manipulation on demand, inverse problem...... 171

6.3.6. Closed loop control to improve accuracy of particle manipulation ...... 171

7. REFERENCES ...... 172

Appendix 1. Analytical expressions for free vibrations of a piezoceramic disc...... 182

Appendix 2. Interfacial forces...... 186

Appendix 3. AFM spring constant calibration...... 192

Appendix 4. Electrical analysis of the PIPM ...... 194

Appendix 5. Particle motion due to momentum transfer ...... 205

Appendix 6. Angle of applied force...... 213

Appendix 7. Program for the processing of AFM data files...... 216

Appendix 8. Program to calculate the particle motion due to momentum transfer ...... 222

Appendix 9. Program to process AFM data for PIPM displacement calibration...... 229 IX

Appendix 10. FEM program code for vibration of piezoelectric plates ...... 230

Appendix 11. FEM program code for vibration of Ring PIPM ...... 237

Appendix 12. Piezoelectric ceramic material properties...... 245

VITA 247

LIST OF TABLES

Table 1.1 List of material properties for three piezoelectric materials, Pz-27, BaTiO3, LiNBO3 and ZnO...... 19

Table 2.1 Listing of the interfacial forces acting on the particle ...... 29

Table 2.2 The effect of different manipulation frequencies and vibration amplitudes on the minimum particle size that can be detached from the PIPM...... 45

Table 2.3 Adhesion forces and corresponding critical displacement that the PIPM will need to generate in order to detach the particles from the surface...... 47

Table 3.1 Equations for the determination of normalized ux and uz at any spatial location x along the x-axis from center of the discs to the edge ...... 82

Table 3.2 Particle and surface velocities before and after elastic collision leading to a projected height h and a time of flight TOF...... 95

Table 3.3 Comparison table of the velocities before and after collision with a surface as the amplitude of vibration is increased ...... 96

Table 3.4 Initial and final velocities for 4 successive inelastic collisions with a vibrating surface ...... 97

Table 5.1 Peak to peak roughness, separation distance and image RMS roughness for the scanned surfaces...... 122

Table 5.2 Summary of the adhesion force between stainless steel particles and the rough silver surface of the PIPM...... 127

Table 5.3 Normal and tangential displacements, for an excitation voltage of 1V, determined using 4.25% material damping...... 131

Table 5.4 PIPM specifications ...... 136

Table 5.5 Excitation voltages measured to initiate the motion of 20-30 m stainless steel particles at the edge and center of 10, 20 and 30 mm discs. Normal and tangential components of manipulation forces corresponding to the excitation voltages, calculated for a 25 m particle, are also shown ...... 142

Table 5.6 Peak voltages needed to initiate motion of 20 – 30 m on a 30 mm piezoelectric PIPM excited at 56.79 kHz ...... 148 XI

LIST OF FIGURES

Figure 1.1 Optical trap by Bukusoglu et al. [13] ...... 4

Figure 1.2 Magnetic tweezers by Gosse et al.[32]...... 5

Figure 1.3 EP and DEP particle manipulation by Voldman et al [33]...... 6

Figure 1.4 Schematic of particle manipulation using AFM by Baur et al [36]...... 7

Figure 1.5 Schematic of acoustic standing wave manipulation by Kozuka et al [24] ...... 8

Figure 1.6 Schematic of the particle-manipulator set-up. The spring element represents the particle-surface binding force, while the dotted lines represent the vibration of the manipulator leading to the generation of resultant manipulation forces Fr...... 10

Figure 1.7Direct piezoelectric effect with compressive or tensile stress ...... 15

Figure 1.8 Converse piezoelectric effects with alternating electric fields ...... 15

Figure 1.9 Three important modes of mechanical motion in the piezoelectric effect...... 17

Figure 1.10 Conceptual model of the PIPM for the manipulation of single and multiple particles. fig 1.10(a) schematic of the forces experienced by the particle, fig 1.10 (b) & (c) manipulation force distribution experienced by the particle at mode M1 and M2, fig. 1.10(d)-(g) single particle manipulation. Fig. 1.12(h)-(k) multiple particle manipulation...... 22

Figure 2.1 Schematic of the particle-PIPM interfacial interaction. The force balance diagram shown on the right. (Image not to scale) ...... 28

Figure 2.2 Schematic of the effect of roughness on van der Waals force. Geometry proposed by Rabinovich et al [76]...... 33

Figure 2.3 Theoretical estimation of the particle-plate adhesion force. Calculated for a 25 m stainless steel particle and a rough surface using equation 2.5, where,  is varied from 1-5000 nm, ymax is varied from 1-2500nm, with d = 0.3nm, A = 4x10-19J. 34

Figure 2.4 A schematic of the liquid meniscus formed between a partilce and a plate. rc and l are the two principal radii of curvature for the water meniscus, 1 and 2 are the contact angles for water on the sphere and the plate, respectively...... 35

Figure 2.5 Comparison of particle-surface adhesion forcses as a function of particle radius. 38 XII

Figure 2.6 Schematic representing the angle of applied force. (a)Force balance diagram in which F is the applied force attempting to move the particle when applied at an angle  with respect to the x-axis. (b) Plot of applied force as a function of the angle at which it is applied...... 41

Figure 2.7 Contour map of a radial mode of vibration showing the manipulation force, Fmanipulation exerted on a particle placed on its surface ...... 42

Figure 2.8 Comparison of the forces exerted by the PIPM when excited at different frequencies and amplitudes, with adhesion forces...... 45

Figure 2.9 Free body diagram of two colliding rigid bodies, namely the particle and the PIPM, in x-z plane. (Not to scale) ...... 50

Figure 2.10 Schematic showing the mechanism of particle manipulation using the PIPM. (a)Contour plots of the resultant and x, y, and z components of the magnitude of PIPM displacement. (b ) & (c) Normal and tangential components of surface displacement plotted across the diameter (along the x-direction) of the PIPM showing the gradient of manipulation force experienced by the particle...... 57

Figure 2.11 Schematic of multimodal particle manipulation. (a) normalized resultant magnitude of force, experienced by a particle, plotted along the diameter of PIPM in the x-direction for natural Modes A and B, (b) & (c) represent the corresponding contour plots of modes A and B...... 59

Figure 2.12 Vector representation of a particle position two dimensional space ...... 61

Figure 2.13 Particle projectile motion in 2-D space ...... 62

Figure 2.14 Percentage change in the final velocity of the manipulator after collision with stainless steel particles of increasing radii...... 65

Figure 2.15 The adhesion force binding particles to the surface of the PIPM determines the lower limit of particle manipulation. The minimum partricle size that can be manipulated at a specific frequency of vibration is plotted as a function of adhesion force...... 66

Figure 3.1 Schematic of a disc PIPM of thickness h that is excited by applying an electric potential  across its top and bottom surfaces...... 69

Figure 3.2 Radial mode of vibration of a disc shaped PIPM sectioned along the x-axis. (a) radially expanded and (b) radially contracted...... 76

Figure 3.3 Radial mode resonance frequency of piezoelectric ceramic, Pz 27, as a function of disc diameter ...... 78

Figure 3.4 Contour plots of the magnitude of displacement and its x, y and z components... 80

Figure 3.5 Normalized tangential displacement, ux, plotted along the x-axis...... 81 XIII

Figure 3.6 Normalized vertical displacement, uz, plotted along the x-axis...... 81

Figure 3.7 The angle which the magnitude of resultant displacement makes with the horizontal axis plotted as a function of spatial location for the radial mode of a 30mm piezoceramic disc...... 83

Figure 3.8 Distribution of the resultant harmonic displacement at 4 locations on the PIPM surface, namely, A, B, C, & D, (a) A schematic showing the locations of the 4 ellipse’s plotted in in fig. 3.8 (b)...... 84

Figure 3.9 Maximum normal displacements plotted as a function of of stiffness proportional damping factor, , represented as % damping for 4 PIPM’s excited with a peak voltage of 1V...... 87

Figure 3.10 Maximum tangential displacements plotted as a function of of stiffness proportional damping factor, , represented as % damping for 4 PIPM’s excited with a peak voltage of 1V ...... 87

Figure 3.11 Maximum normal displacements plotted as a function of peak excitation voltage, extracted for a damping factor of 5%...... 88

Figure 3.12 Plot of uz_max vs peak voltage for all four disc PIPM’s, indicating that they all have the same displacement sensitivity at the center of the disc...... 89

Figure 3.13 Plot of maximum tangential displacement, ux_max, vs peak excitation voltage, Inverse of the slope gives the displacement sensitivity at the edge of each disc. ... 90

Figure 3.14 Slope of ux_max and uz_max vs peak voltage for 5% damping...... 91

Figure 3.15 Particle projected vertically (1-D) from a surface exhibiting thickness vibrations. Collisions at t1 and t2 are fully elastic...... 93

Figure 3.16 Path of a particle projected vertically after successive elastic collisions with the vibrating surface...... 94

Figure 3.17 Effect of amplitude of vibration on particle projection. Particle projected due to elastic collision with a surface vibrating with a peak amplitude of 1, 5 and 10nm showing that the particle is projected to an increasing height as the surface amplitude is increased...... 95

Figure 3.18 1-D particle projection after inelastic collisions with the vibrating surface. Coefficient of restitution, e = 0.9 ...... 97

Figure 3.19 Comparison of elastic and inelastic 1-D collisions with a vibrating surface. Combined plots of figs 20 and 22 showing the effect of a coefficient of 0.9 on the vertical height the particle is projected after collision...... 98

Figure 3.20 Particle motion in 2-D space with elastic collisions. No energy loss at the interface is considered, therefore coefficient of restitution, e = 1, and dynamic frictional constant,  = 0...... 100 XIV

Figure 3.21 Particle motion in 2-D space with inelastic collisions. No energy loss in the x- direction is considered , therefore dynamic frictional constant,  = 0, however a loss in the z-direction is introduced, coefficient of restitution, e = 0.9...... 101

Figure 3.22 The time for a 25 m particle to travel from the edge (~15 mm) to the center of the PIPM (~2mm) is plotted for different coefficients of restitution, with the frictional loss set to zero...... 102

Figure 3.23 Particle motion in 2-D space with frictional losses. No energy loss in the z- direction is considered here, therefore coefficient of restitution, e = 1, however a

loss in the x-direction is introduced, dynamic frictional constant,  = 0.1...... 103

Figure 3.24 The time that it takes for a 25 m particle to travel from the edge (~15 mm) to the center of the PIPM (~2mm) is plotted for different coefficients of friction, for elastic collisions...... 104

Figure 4.1 AFM experiemental setup and schematic (a) Image of a BioScope Atomic Force Microscope used for experimental measurements and (b) schematic of the AFM setup for interfacial force measurement...... 107

Figure 4.2 A typical AFM Force-Distance curve ...... 109

Figure 4.3 Design drawings of the bottom plate of the PIPM holder. (a) Top view of the bottom plate, (b) crossectional view of the bottom plate and (c) the fitting rings enabling the bottom plate to be used for different disc sizes...... 112

Figure 4.4 Design drawings of the top plate of the PIPM holder. (a) Top view of the top plate, (b) Crossectional view of the top plate...... 113

Figure 4.5 Block diagram of the PIPM...... 114

Figure 4.6 Schematic of the Ring PIPM sensing structure (a) Structure construction, (b) Finished structure ...... 115

Figure 4.7 Ring PIPM sensing structure experimental setup...... 116

Figure 5.1 3-D surface plot of a stainless steel micro particles using AFM operated in contact mode...... 118

Figure 5.2 Particle surface roughness determination. (a) 2-D image (b) Roughness along image sections ...... 119

Figure 5.3 50m x 50m AFM scan showing the microstructure of the silver electroded piezo-manipulator. The grain size of the piezoelectric ceramic ranges from 1-5 m ...... 120

Figure 5.4 Three dimensional plot of the PIPM surface...... 120

Figure 5.5 PIPM surface roughness scan (a) 2-D image of the surface (b) Roughness along image sections ...... 121 XV

Figure 5.6 Optical images of 11, 16, 21, 24, 26 and 32m stainless steel particles attached to AFM cantilevers, figs 5.13(a) – (d)...... 123

Figure 5.7 Theoretical van der Waals force between a 25 m stainless steel particle and a rough grounded silver PIPM electrode, with ymax, and  given in table 5.1...... 125

Figure 5.8 Adhesion force measured for 6 particle radii, 11, 16, 21, 24, 26 and 32m at 4 different locations...... 126

Figure 5.9 Calibration curve showing peak deflection of the AFM laser as a function of known the piezo motor displacement...... 128

Figure 5.10 Calibration of PIPM displacement using AFM. The displacement of a 16x1mm piezoelectric manipulator disc obtained using the AFM operated in contact mode. The deflection data measured is converted to displacement using the AFM laser deflection sensitivity of 35mV/nm...... 129

Figure 5.11 Experimental and theoretical comparison of the PIPM displacement for a 16 mm disc simulated with 4.25% damping...... 130

Figure 5.12 The magnitude of maximum tangential displacements, ux_max, of four discs as a function of excitation voltages, obtained for a Pz 27 material with 4.25% damping...... 132

Figure 5.13 Flowchart of PIPM based particle manipulation ...... 133

Figure 5.14 Normalized magnitude of resultant force experienced by a 25m stainless steel particle when placed on 30, 20 and 10 mm PIPM’s...... 135

Figure 5.15 Peak excitation voltage needed to initiate the motion of 20-30 m particles at a series of locations from the edge (15mm) to the center (0 mm) of a 30 mm disc excited at its radial resonance frequency, 65 kHz...... 138

Figure 5.16 Magnitude of maximum normal, Fmanipulation_z, and tangential, Fmanipulation_x, components of manipulation forces exerted on a 25 m stainless steel particle at excitation voltages shown in figure 5.15. The shaded region represents the adhesion force that the manipulator has to overcome...... 138

Figure 5.17 Peak excitation voltage needed to initiate the motion of 20-30 m particles at a series of locations from the edge (10 mm) to the center (0 mm) of a 20 mm disc excited at its radial resonance frequency, ~98 kHz...... 139

Figure 5.18 Magnitude of maximum normal, Fmanipulation_z, and tangential, Fmanipulation_x, components of manipulation forces exerted on a 25 m stainless steel particle at excitation voltages shown in figure 5.16. The shaded region represents the adhesion force that the manipulator has to overcome...... 139

Figure 5.19 Peak excitation voltage needed to initiate the motion of 20-30 m particles at a series of locations from the edge to the center (0 mm) of a 10 mm disc excited at its radial resonance frequency, ~199 kHz...... 140 XVI

Figure 5.20 Magnitude of maximum normal, Fmanipulation_z, and tangential, Fmanipulation_x, components of manipulation forces exerted on a 25 m stainless steel particle at excitation voltages shown in figure 5.18. The shaded region represents the adhesion force that the manipulator has to overcome...... 140

Figure 5.21 Peak excitation voltage needed to initiate motion of 20 – 30 m stainless steel particles at the center of 10, 16, 20 & 30 mm discs having radial resonance frequencies ~ 200, 125, 100 & 65 kHz respectively...... 143

Figure 5.22 Inertial manipulation force, Fmanipulation_z, plotted as a function of radial resonance frequency for 10, 16, 20 and 30 mm PIPM’s. The two solid horizontal lines with the square and triangular markers represent theoretical adhesion force of 100 and 400 nN respectively...... 144

Figure 5.23 Optical images of a 25 m particle being translated from the edge of a PIPM towards its center when excited at its radial mode of vibration...... 145

Figure 5.24 Average particle translation speed for 20 – 30 m particles plotted as a function of excitation voltage. The particles travel a distance of 15 mm, from the edge to the center of a 30 mm disc, excited with different peak voltages, at its radial mode of vibration...... 146

Figure 5.25 Contour displacement maps of a 30 x 1mm disc excited at 56 kHz...... 147

Figure 5.26 Normalized resultant displacement with its normal and tangential components plotted along the x-axis for a piezoelectrically excited mode at 56 kHz...... 148

Figure 5.27 Normalized peak excitation voltage and normalized resultant manipulation force plotted as a function of location along the x-axis...... 149

Figure 5.28 Experimental and FEA simulated results showing particles orienting along the nodal lines of the piezoelectrically excited natural modes of vibration at the indicated frequencies...... 150

Figure 5.29 Multi-modal particle manipulation of a 8 m particle by switching between 53 and 56 kHz modes. The contour maps representing the magnitude of displacement are shown on the left of the image. The normalized magnitude of displacement for both modes are plotted along the diameter and shown. (a)-(e) represent experimental results of an 8 m particle being translated between the high displacement regions of the two modes...... 152

Figure 5.30 Schematic of a ring PIPM...... 154

Figure 5.31 FEA of the radial mode of vibration of the ring actuator, excited at 220 kHz, with an excitation voltage of 1Vp ...... 155

Figure 5.32 Schematic of the ring PIPM sensing structure...... 156

Figure 5.33 Contour map of the magnitude of displacement of the SAHS excited at the radial resonance frequency of the ring actuator (~220kHz, 1Vp) ...... 157 XVII

Figure 5.34 Top view of the sensor actuator structure, showing the magnitudes of the x, y and z components of displacement ...... 157

Figure 5.35 The magnitudes of the resultant, normal and tangential components of displacement plotted along the x-axis of the ring PIPM sensing structure excited at 220 kHz...... 158

Figure 5.36 Improvement of a TSM biosensor using the ring PIPM sensing structure for the detection of 10 m polystyrene particles (a) Frequency response of the TSM sensor to loading of 10 m (diameter) polystyrene particles, without concentration (PIPM OFF) and with concentration (PIPM ON). (b) – (e) represent the progression of the concentration of particles...... 160

Figure 5.37 Improvement of a TSM biosensor using ring PIPM sensing structure for the detection of 5 m silica particles (a) Frequency response of the TSM sensor to loading of 5 m (diameter) Silica particles, without concentration (PIPM OFF) and with concentration (PIPM ON). (b) – (e) represent the progression of the concentration of particles...... 161

Figure 5.38 Improvement of a TSM biosensor using ring PIPM sensing structure for the detection E. Coli. Frequency shift as a function of time, of a TSM sensor in response to the addition of E-Coli solution. The control represents the response without concentration of E. Coli, whereas the dotted line represents the ring PIPM sensing structure, which is turned on at the 15 minute mark...... 163

Figure 5.39 Frequency response of the ring PIPM sensing structure due to uniformly distributed and concentrated E. Coli spores...... 164

XVIII

LIST OF SYMBOLS

Fmanipulation – Inertial manipulation force experienced by the particle

Fmomentum – Impulsive manipulation force experienced by the particle

Fmanipulation_z – Vertical/Normal component of manipulation force

Fmanipulation_x,y – Horizontal/Tangential components of manipulation force

Fadhesion – Force binding a particle to a surface

Ffriction – Frictional force between a particle and a surface

Fvdw – van der Waals force

Fes – Electrostatic force

Fgrav – Gravitational force

Fcap – Capillary force

Fcomp – Compressional wave force mp – mass of a particle

mm - mass of the manipulator

d – Separation distance between a particle and a surface

R – Radius of the particle

p – Material density of the particle

g – Acceleration due to gravity

f – Fluid density

VCP – Contact potential in volts

VBC – Potential difference due to bulk charge difference

0 – Permittivity of free space

A – Hamaker constant XIX

ymax – Peak surface roughness

 – Roughness peak to peak separation distance

 - Surface tension of the liquid

rc – Principal radius of curvature of the liquid with the curved surface

l – Principal radius of curvature of the liquid with the planar surface

P – Difference in pressure between the liquid and the surrounding vapor phase

1 – Contact angle of the liquid with the sphere

2 – Contact angle of the liquid with the plate

Rg – Gas constant

T – Temperature in Kelvin

Vm – Molar volume of water

P – Vapor pressure of a vapor in equilibrium with a curved surface

P0 – Saturation vapor pressure over the planar liquid surface

RH – Relative Humidity

 s - Static coefficient of friction

 k - Kinetic coefficient of friction

u – Instantaneous displacement

v – Instantaneous velocity

a – Instantaneous acceleration

Subscript i – x, y and z components

 - Angular frequency

 - Phase the phase angle between normal and tangential components of u, v, and a az_crit – Critical acceleration of the manipulator surface

uz_crit – Critical displacement of the manipulator surface

P – Momentum XX

When 3 subscripts are used, the first represents particle (p) or manipulator (m), the second represents initial (i) or final (f) and the third (i) represents the x, y or z component. For example, vpfi – final velocity of the particle, (p-particle, f-final velocity after impact, i- x,y,z components) e – Coefficient of restitution e, d, g, h – Piezoelectric constants

 - Dielectric constant

V – Applied voltage  rp - Particle position vector h – Maximum height that the particle is projected

TOF – Time of flight

R1 – Horizontal distance traveled by the projected particle

hm – Thickness of the PIPM rm – Radius of the disc PIPM

 - Electric potential

[ ] – Matrix

{ } - Vector

S – Mechanical Strain

T – Mechanical stress

E – Electric field

D – Charge density cE – Elastic constant at constant electric field

S – Dielectric constant at constant strain

{f} – Mechanical displacement vector

[N] – Displacement shape function XXI

{} – Displacement vector (PIPM)

[M] – Mass matrix

[K] - Stiffness matrix

F – Mechanical force

Q – Electrical charge

 - Time independent amplitude vector (Eigen vector)

R – Mass proportional damping

R – Stiffness proportional damping

I – Critical damping fR – resonance frequency of the PIPM

Y – Young’s Modulus

 - Poisson’s ratio

kc – Spring constant of an AFM cantilever fc – Resonance frequency of the AFM cantilever

fpc – Resonance frequency of the AFM cantilever with particle attached to it a – Outer radius of the Ring PIPM b – Inner radius of the Ring PIPM c – Radius of the TSM sensor attached to the Ring PIPM ts – Thickness of the TSM sensor

D - Flexural rigidity of PIPM

kp – Planar electromechanical coupling coefficient

Rc – Reflection coefficient

Tc – Transmission coefficient

Z – Impedance

Pm is the maximum value of the pressure XXII c – Speed of sound in the medium (air or water)

 - Angular frequency of the acoustic wave

smax – Maximum displacement of the air element adjacent to the PIPM

I – Intensity of the periodic acoustic wave

PD – penetration depth of the shear acoustic wave

η - Shear viscosity of air

Am – Area of the electroded region of the PIPM

vm – Velocity of the vibrating PIPM surface

C0 – Static capacitance

βm – Wave number of the acoustic wave inside the PIPM

NT - Transformer turn ratio

R1 – Series resistance

L1 – Series Inductance

C1 – Series capacitance

R0 – Impedance of the signal generator

PS – Power supplied to the PIPM

PL – Power consumed by the PIPM

ηP – Efficiency of power transfer XXIII

ABSTRACT

The manipulation of micro and sub-micrometer, inorganic and organic particles at a solid- sample interface is important in many modern biomedical technologies, including biosensors and biochips, tissue engineering, drug delivery and MEMS. Currently available manipulation techniques such as Atomic Force Microscopy (AFM) and the techniques based on optical, magnetic, electrokinetic, and acoustic phenomena exhibit limitations related to stringent

requirements on the properties of the particles, samples, as well on operational, experimental

and environmental conditions. To overcome some of these limitations, a simple and versatile,

technique called the Piezoelectric Interfacial Particle Manipulator (PIPM) is proposed. The

PIPM principle of operation is based on an efficient generation of interfacial manipulation

forces, at distances ranging from 0.5-50nm from the manipulator surface, which are capable

of producing a broad range of particle motion in 3-D space. These forces, piezoelectrically

generated by multi-modal surface vibration, can be applied to any type of the substrate on

which manipulation is required. A theoretical study of the interfacial forces between the

particle and the PIPM surface, and a finite element modeling of the PIPM-generated force

pattern provide both formal and phenomenological models that enable understanding of the

mechanism of particle manipulation as well deliver the criteria for a PIPM design. The PIPM

was successfully tested with inorganic (5-50m) and organic particles (0.5-5m) that

included stainless steel and silica microparticles as well bacteria spores. The PIM results

were positively confirmed using standard laboratory instrumentation like atomic force

microscope (AFM). The integration of the PIPM with a biosensor (TSM thickness shear

mode sensor) demonstrated an improvement of up to 40% in analyte detection. The obtained

results demonstrate that the PIPM provides the means for reproducible, high throughput

manipulation of microparticles without stringent environmental and particle property XXIV requirements. Specifically, it provides a large array of manipulation modalities such as controllable sample delivery, concentration control, mixing and sorting on a single substrate

that should find applications in development of broad range of industrial and biomedical

devices, including various sensing structures based on piezoelectric, optical or magnetic

principles of operation, biochips and bio-actuators.

Keywords: Manipulation, AFM, FEA, TSM, piezoelectric 1

1. INTRODUCTION

1.1. Background

The ability to organize, pattern and rearrange micrometer and sub-micrometer sized objects, both inorganic and organic, is critical to numerous applications such as Lab-on-a-Chip (LOC)

[1-2], tissue engineering [3-5], biosensors [6-7], and drug discovery [1, 8-9].

A LOC includes several key components, namely, sample delivery, sample processing, biosensing layers, biochips, and electronic display and control, each of which plays an important role in the seamless functioning of the LOC. A complex network of microfluidic channels deliver the sample to the sample processing component which is required to provide features such as, sample concentration, multiple sample mixing and sample splitting. The

sample processing unit acts to, improve the efficiency of the LOC as features such as mixing

can improve binding kinetics [10], lower the detection limit by concentrating the sample at

regions of high sensitivity [11], and enable reduction in sample volume required by splitting

the sample once it is mixed. The sample is then is then delivered to the biosensing layers on

the biochip which provides information on the presence or absence of specific biological

objects.

This thesis focuses on sample handling and the various functionalities required of it. Similar

sample processing features, in addition to sample patterning, are necessary in the fields of

tissue engineering and drug discovery thus greatly increasing the scope this research.

Sample handling can be broadly termed as sample manipulation. We therefore define “sample

manipulation”, in the context of this thesis, as, the controlled, non-destructive transport,

concentration, mixing and splitting of samples. We go further to define the “sample” as

inorganic/organic microparticles that are present in a fluid media, namely air or water,

which are to be manipulated. 2

Although the demand for techniques to provide sample manipulation is great, only a handful of techniques are currently available. Based on optical, electrical, magnetic and acoustics

technologies, researchers have developed a techniques to achieve sample manipulation,

namely, optical tweezers [12-14], electro/dielectrophoresis [15-16], magnetic tweezers [17-

18], hydrodynamic flow [19-20], and acoustic techniques [2, 21-25], the pros and cons of

which will be briefly discussed in the following sections.

1.2. Problem statement

At the micrometer and sub-micrometer scales a large segment of research is conducted at a solid-fluid interface, the solid being the manipulator surface and the fluid being air or liquid.

For example, the adhesion of surfaces and the prevention of adhesion of dust particles is of great importance for the quality control in the fabrication of semiconductor devices and micro-electro-mechanical-systems (MEMS); [26], biosensors, biochips, and lab-on-a-chip; are based on detecting phenomena occurring at a sensors surface [6-7], the discovery, delivery, separation and mixing of drug particles are conducted at drug-surface interfaces

[27], and the adhesion of cells to surfaces and their interaction with the surrounding scaffolds, is one of the most important aspects of tissue engineering [4] etc.

The development of the current manipulation technique is driven by the growing need to handle, control and process samples at this solid-liquid interface.

In order to develop this manipulation technique, it is therefore necessary to develop a clear understanding of particle-surface interfacial adhesion forces. This will enable the determination of the forces needed to be developed by the manipulator to facilitate sample manipulation. It is also important to determine the effect of manipulation forces on the integrity of the sample. 3

This project is focused on the study of micrometer sized inorganic and organic particles

(radius 0.5 – 50 m) on a metallic manipulator surface. The adhesion forces binding these

particles to the manipulator surface could range from pico Newton’s to hundreds of nano

Newton’s depending on the particle and manipulator properties such as size, shape, density,

elasticity etc., surface conditions such as roughness, the medium in which the interaction

takes place, such as vacuum, air or liquid and environmental conditions such as temperature

and humidity.

In order to facilitate sample manipulation, it is necessary to generate manipulation forces on

the same order of magnitude as those binding the sample to the manipulator. Furthermore, it

is necessary to develop multiple complex distributions of the surface force thus allowing for

the translation of the sample from one location to the next.

Here the development of a sample manipulation technique capable of generating interfacial

forces for the non-destructive, controlled manipulation of micrometer and sub-micrometer

sized particles is discussed. Its applications range from the mixing and splitting of groups of

particles, for biochip applications, to the concentrating of particles to regions of high

sensitivity, for biosensor applications.

Prior to discussing the details of the manipulation technique used in this thesis it is necessary

to first review some of the existing manipulation techniques, each of which is capable of

generating the necessary particle manipulation forces.

1.3. Existing manipulation techniques

The demand for manipulation techniques is ever increasing as the focus of medicine and healthcare is moving towards personalized medicine. In order to achieve this life changing but challenging goal of personalized medicine it is necessary to develop techniques that are capable of handling and processing small sample volumes containing micrometer and sub- 4 micrometer sized biological objects. A few of these techniques have been developed and are now briefly discussed.

1.3.1. Optical Tweezers

Optical Manipulation of particles was first demonstrated by A. Ashkin in the 1970’s [12], wherein optical tweezers were used to levitate and guide and trap particles. This technique has been used to manipulate particles ranging from 10nm to several micrometers [13]. It can be used to determine mechanical properties of biopolymers, cell membranes, protein fibers

etc and also to characterize

forces generated by

molecular motors (myosin,

kinesin, etc) [14].

Figure 1.1 Optical trap by Bukusoglu et al. [13]

As shown in fig. 1.1, this technique uses a highly focused beam of light to create a 3-D gradient of intensity about the focal point where the particles are trapped. In this way a force is exerted on the particle, and this force can be measured from the displacement of the particle from the center of the trap. This technique is capable of applying forces on the pN range [13, 28].

Advantages: High resolution manipulation (few nm’s), particle mechanical properties can be determined from the force exerted to manipulate the particle.

Disadvantages: The downside of this technique is, expensive and complex experimental setup, the distance particles can be manipulated is limited by the focusing requirements [29]. 5

Also, for certain optical tweezer configurations, photodamage of biological objects has been observed [30].

1.3.2. Magnetic Tweezers

Magnetic forces have been used to manipulate particles with permanent magnetic moment.

By attaching biological molecules on these particles, magnetic tweezers allow the measurement of forces on individual macromolecules.

Figure 1.2 Magnetic tweezers by Gosse et al.[31]

As shown in figure 1.2 the pole pieces produce a horizontal magnetic filed at the center of the sample. The magnetic moment of the particle (2-5m) aligns it with the magnetic field lines while the vertical magnetic field gradient exerts a force on the particle raising it up. Particles are reproducibly trapped in this way [31]. By attaching biological objects like DNA to the particles, Chiou et al, 2004, [18] was able to rotate and stretch single DNA molecules, furthermore mechanical properties of single DNA strands could be determined [17].

Advantages: Magnetic forces have been used for separation techniques in biotechnology.

These tweezers can be used to isolate rare biologically active compounds from difficult-to- handle samples [16]. These isolation techniques are able to perform the adsorption process or affinity interaction and subsequent adsorbent separation even in the presence of particulate 6 diamagnetic contaminants. The tweezers can exert a force of about 100 pN on micrometer sized particle [16].

Disadvantages: Requirement of incorporating electromagnets on the device, thus making them relatively large in size. The magnetic field is difficult to control at the point of the particles. The sample being manipulated has to be mixed with magnetic particles, which could have toxic effects on certain biological samples furthermore the sample is lost after the

experiment.

1.3.3. AC Electrokinetic Manipulation

This technique involves manipulation of particles using an ac electric field [15].

Figure 1.3 EP and DEP particle manipulation by Voldman et al [32]

Figure 1.3 represents the basic principle of EP and DEP techniques. Interaction of the

particle’s charge with the electric field results in electrophoretic (EP) manipulation fig 1.3a

(left), whereas interaction of a particles dipole with the spatial gradient of an electric field

results in dielectrophoretic (DEP) manipulation of particles, fig 1.3b [32]. This technique can

lead to several phenomena, namely, electrorotatoion, dielectrophoresis, and traveling-wave

dielectrophoresis. 7

Advantages: This technique of manipulation has high throughput and its resolution can be improved using optically induced DEP [29]. Relative to other manipulation techniques, it has a simple set-up and depending on the electrokinetic effect used, particles ranging from 10’s

nm’s to several micrometers can be manipulated [32].

Disadvantages: Electrode-electrolyte interaction could lead to gas generation (H2 or O2) or

corrosion in EP, which makes is unsuitable for biological particles which require a controlled

pH environment. The need to apply a strong electric field for particles (< 1 m) could lead to

localized Joule heating and convection of the ionic solution [33]. Another limitation is that

particle translation is limited by the electrode pattern.

It should be noted that for manipulation of particles less than 1micro-meter in diameter, bulk

fluid motion can be induced due to the motion of ions in solution which in turn carries small

particles with it.

1.3.4. Atomic Force Microscopy (AFM)

The Atomic Force Microscope is a well developed tool for the manipulating of micro and nano particles and sensing of molecular forces [34].

Figure 1.4 Schematic of particle manipulation using AFM by Baur et al [35]

As shown in figure 1.4, this technique performs particle manipulation by physically pushing the particles with the AFM cantilever tips. It has been used to build complex patterns at the nanometer scale[35]. Furthermore, the non- contact AFM (NC-AFM) can be used for non- destructive imaging and interfacial force measurement [36-37]. By having a real time feed- 8 back system, the AFM tip can be controlled precisely in the attractive or repulsive operating distance over the sample, thus being able to predictably manipulate particles [35].

Advantages: The same tip can be used in situ for manipulation, force measurement and imaging in a real-time fashion. Thus knowledge of the position of the particle for manipulation is known almost simultaneously. It is a high resolution manipulation technique.

Disadvantages: The drawbacks of the system are, manipulation of only one particle at a time is possible thus having low efficiency (low throughput). Physically pushing on biological samples could lead to the damage and loss of sample. Lastly it has an expensive and complex setup.

1.3.5. Acoustic Particle Manipulation

The trapping and manipulation of particles using acoustic radiation force was first introduced by Ashkin in 1970.

Figure 1.5 Schematic of acoustic standing wave manipulation by Kozuka et al [24]

As seen in figure 1.5, acoustic standing wave manipulation works on the principle of trapping particles in the nodal planes of a standing wave formed between an acoustic actuator and an appropriately placed reflector. Several researches have been able to apply this technique to the trapping of micro particles at the nodal planes of the acoustic field [24, 38]. It has been used for the separation of particles and cells [22, 39-40], as well as the deposition of cells 9

[23]. Studies have also been conducted using phenomena such as acoustic streaming and radiation pressure to concentrate micrometer sized objects [21, 25] and surface acoustic waves to pattern micron sized organic & inorganic objects [2].

Advantages: This technique of trapping and manipulation of particles using acoustic fields is advantageous over some of the above mentioned manipulation techniques, because it is possible to achieve non-contact manipulation, improved detection sensitivity, reduced detection time and efficient trapping of biological particles [41]. Acoustic techniques are also capable of non-destructive manipulation [2]. Surface acoustic wave techniques have the advantage of ease of integration on more complex systems for Lab on Chip applications since their fabrication is closely related to those used for MEMS devices and in the silicon industry.

Disadvantages: Currently available acoustic wave manipulation techniques trap and

manipulate particles in the vicinity of the nodal points of a standing wave, thus a very limited

range of manipulation [2, 24, 38]. Techniques based on surface acoustic waves are limited in

terms of particle translation by the interdigital electrode design, effectively rendering them a

trapping technique [2]. It is necessary to have a reflector whose distance from the actuator is critical for formation of the standing wave[24, 38]. Lastly, except for surface acoustic wave technique, most other acoustic wave techniques operate in the bulk of solution and not at the solid fluid interface, therefore it would be difficult to use them for application in tissue engineering or Lab on Chip applications.

The manipulation technique discussed in this thesis is also based on acoustic technology. It is outlined next.

1.4. Proposed Acoustic Interfacial Force Manipulation Technique

The technique discussed in this thesis is based using acoustic waves within a structure to generate interfacial manipulation forces. Its simplicity in design and operation can make it a key player in the manipulation of micrometer and sub-micrometer sized particles.

A schematic of the particle-manipulator set-up is shown in figure 1.6.

The encircled numbers in figure 1.6 represent individual components of the setup each of

which is listed in the adjoining table.

This manipulation technique can be operated in vacuum, air or fluidic media, shown as

encircled 1 in the figure. The properties of each medium will impact the particle-manipulator

interaction in terms of the forces involved in manipulation.

The encircled number 2 represents the particles that can be manipulated. The particle sizes

that can be manipulated range from 0.5-50m in radius. Although the foregoing studies are

mainly conducted using elastic rigid spheres that can be either inorganic or organic, it will be

extended to demonstrate the manipulation of deformable objects in future studies. The surface 11 topography of the particle, such as surface roughness, can greatly impact the adhesion force binding the particle to the manipulator due to the change in effective contact area between the particle and the manipulator.

Encircled 3 represents the adhesion of the particle to the manipulator. There always exists a finite separation distance between a particle placed on the manipulator and the surface of the manipulator ranging from 0.1 – 10’s nm. This separation distance plays a vital role in the magnitude of the adhesion force binding the particle to the manipulator. The adhesion force is composed of several intermolecular electromagnetic forces and gravitational forces, the total magnitude of which depends on the particle and manipulator material and geometric properties as well as the environmental conditions. The adhesion force can therefore vary from a few pico Newton’s to 100’s of nano Newton’s. These forces will be discussed in more detail in chapter 2.

Lastly, encircled 4, represents the manipulator. The surface conditions can be either smooth or rough which, as with the particle roughness, will impact the adhesion force binding the particle to the manipulator and therefore the force the manipulator will need to generate to facilitate manipulation. The excitation of the manipulator, leads to it periodically expanding and contracting, as shown in fig 1.6. This motion of the manipulator leads to exertion of a resultant force on to particles placed on its surface facilitating the translation of particles from one location to the next. Due to the ability of the manipulator to generate a variety of complex surface manipulation force distributions, several modalities of manipulation can be achieved such as, levitation, linear motion, concentration, splitting and mixing. The forces exerted on particles occur at the particle manipulator interface, and can be designed to enable

the motion of particles in various directions at relatively high speeds. These forces act to reduce and/or overcome adhesion forces thus allowing for the particle to be manipulated to regions of interest. 12

Advantages: Simple experimental setup, inexpensive components, interfacial manipulation forces ranging from 0.1 – 1000 nN can be generated. In contrast to standing wave acoustic manipulation techniques no reflector is necessary. Particles can be translated over distances

100’s of times their diameters. Non-destructive manipulation, thus the sample can be recovered. Can provide features such as interfacial force monitoring, particle concentration and splitting. It can be easily applied for the improvement of the performance of biosensors

and Lab on Chips.

Disadvantages: Since it operates at the interface, surface topography and contamination as

well as environmental conditions can influence manipulation results. [20, 50, 51]

1.5. Acoustic Wave Transducers (AWT)

A transducer is a device that converts one form of energy to another. An acoustic wave transducer (AWT) is one that converts the input energy, electrical/mechanical/electromagnetic, to acoustic energy. Thus, AWT’s refer to a group of transducers, namely sensors and actuators that utilize acoustic waves as the transduction element. When an acoustic wave propagates within a transducer, or from one medium to the next, the mechanical properties (i.e. velocity and amplitude) of the wave are determined by

the characteristics of the medium in its propagation path, such as viscosity, density, and

elasticity. Therefore AWT’s that function as sensors detect any change in the mechanical

property of the wave due to changes in the characteristics of the medium and those AWT’s

transducers that function as actuators utilize the mechanical properties of the waves within

the transducer to exert forces on the adjacent medium and any objects that might be present in

it. 13

1.6. Acoustic Waves and their interaction with particles

Thus as mentioned above, acoustic waves either propagate or penetrate the surrounding media. Any object placed in an acoustic field experiences a steady force called acoustic radiation pressure [42]. King, 1934 documented the effect of the acoustic radiation pressure on rigid spheres suspended in fluid [43]. The effect of the fluid viscosity on the acoustic radiation pressure on rigid spheres was later studied and documented by Doinikov [42].

Extensive theoretical work has been done on the force exerted by both plane [44-45] and

spherical acoustic waves on spheres suspended in fluid [46]. These waves exert an acoustic

pressure on the surrounding media and on any particles in its path.

A detailed study of the non-linear effects of acoustic fields such as acoustic streaming (a

steady current in a fluid driven by the absorption of high amplitude acoustic oscillations) on

the interaction of particles with acoustic fields was performed by Lighthill [47]. A recent

study was conducted on the interfacial interactions of solid particles with shear acoustic wave

sensors by Zhang in 2006 [6].

The above mentioned pioneering studies led to the development of several applications of acoustic manipulation in chemical, biological and industrial areas [48]. Biological particles

such as animal cells were trapped in standing acoustic waves [49] and the separation of live

and dead cells was performed by Gaida in 1996 [50]. Several high intensity acoustic fields

were used for purposes of levitation [51-52], where objects experienced a focused acoustic

pressure, leading to predictable control of these objects. Anderson 2002 utilized acoustic

radiation pressure to concentrate 20um droplets of water, in air, at frequencies as low as 60

kHz [21]. However, in order to utilize acoustic radiation pressure, high displacement

amplitude of vibration has to be generated, thus requiring the use of high power transducers. 14

Virtually all the AWT’s are made of piezoelectric materials. By utilizing the piezoelectric effect, an acoustic wave is generated when an alternating electrical signal is applied to an acoustic wave transducer.

1.7. Piezoelectric transducers

Piezoelectric materials are widely used to excite various modes of vibration thus leading to

the generation of acoustic waves. We now discuss the piezoelectric effect.

1.7.1. Piezoelectric Effect

Piezoelectricity was first discovered in the 1880’s by the brothers Pierre and Jacques Curie in

1880. Piezoelectricity is the ability of certain materials to generate a voltage when mechanically stressed. It is a reversible effect leading to the material being deformed when a voltage is applied to it. This arises from the fact that there is a non-uniform charge

distribution within the crystal, therefore, when it is mechanically deformed, the positive and

negative charge centers get displaced by different amounts, resulting in an electric

polarization within the material. The positive and negative charges inside the atomic lattice of

a non-centrosymmetric material form electric dipoles. When the material is stress free, the

electric dipoles are balanced and the overall dipole moment is zero. If a stress is applied to

the material, the material exhibits deformation and the overall dipole moment becomes non-

zero. The stronger the electro-mechanical coupling of the material, the higher is the

piezoelectric effect. Furthermore, piezoelectric materials can be operated to produce a

deformation that is linearly proportional to the excitation voltage[53].

An important feature of the piezoelectric effect is that it is reversible. This means that the

excitation and response can be interchanged between mechanical stress/strain and electric 15 field/dipoles. This feature results in two types of piezoelectric effects: direct piezoelectric effect and converse piezoelectric effect. Direct piezoelectric effect refers to the appearance of an electric field inside the piezoelectric material that is under mechanical stress. Changing the direction of the stress can alter the polarities of the electrical field/charges, as shown in

Figure 1.7. When a compressive stress is applied, electrical charges appear on the two opposite sides of the piezoelectric material. If a tensile stress is applied, the polarities of the charges alter, as well as the direction of the generated electric field.

Figure 1.7Direct piezoelectric effect with compressive or tensile stress

Converse piezoelectric effect is also called reciprocal or inverse piezoelectric effect, i.e. when an electric field is applied across two surfaces of a piezoelectric material it strains. By alternating the direction of the electric field, the strain will also change [54]. Figure 1.8 shows the strain of a piezoelectric material under alternating electric fields

Figure 1.8 Converse piezoelectric effects with alternating electric fields

In both direct and converse piezoelectric effects, the relations between the electric field/charges and the stress/strain are determined by the mechanical, electrical and 16 piezoelectric properties of the material. The behaviors of a piezoelectric material are described by the constitutive relations describing the piezoelectric effect.

1.7.2. Piezoelectric transducer materials and their modes of vibration

There are several materials that exhibit piezoelectric effect, namely quartz, Rochelle salt, lead titanate zirconate (PZT), barium titanate, lithium niobate, lithium tantalite, polyvinylidene fluoride etc. These materials can be cut or poled in different directions with respect to their crystallographic co-ordinate system, causing them to vibrate in different modes when an AC voltage is applied to electrodes placed on the surface of the transducer. This vibration leads to the generating acoustic waves that travel in different directions. These acoustic waves either propagate (compressional waves) or penetrate (shear waves) the surrounding medium while simultaneously applying a force on this medium. Compressional deformation is associated with structural relaxation process in the medium, while shear deformation is coupled to the viscoelastic properties of the medium, making them sensitive to interfacial molecular processes taking place adjacent to the surface of the transducer.

There are three main modes of mechanical vibration due to piezoelectric effects: longitudinal, transverse and shear motion. In a longitudinal motion, the direction of the deformation is parallel to the direction of the applied electric field. In a transverse motion, the direction of the deformation is perpendicular to the direction of the electric field. While in a shear motion, the deformation is in the shear direction and the maximum deformation happens at the two surfaces of a piezo-material. The three motions are shown in Figure 1.9 a), b) and c), respectively. 17

Figure 1.9 Three important modes of mechanical motion in the piezoelectric effect

These are the fundamental motions in a piezoelectric effect. The actual motion of a piezoelectric material usually consists of a combination of the three modes and is determined by the mechanical, dielectric and piezoelectric properties of the material and the applied electric field.

The acoustic waves generated can be broadly classified as bulk and surface generated waves.

Bulk acoustic waves are excited by metalized bulk piezoelectric structures such as discs or rods, where as surface waves are excited by an interdigital array of electrodes deposited on the surface of the piezoelectric structure. Two of the most common configurations of piezoelectric structures are, thin electroded discs, for the excitation of bulk waves in the

Thickness Shear Mode (TSM), and interdigital transducers (IDT), for the excitation of

Surface Rayleigh Waves (SRW), Surface Transverse Waves (STW), Shear Horizontal

Acoustic Plate Modes (SH-APM), and Flexural Plate Waves (FPW). STW’s form a large family of various waves which include Surface Horizontal SAW (SH –SAW), Surface

Skimming Bulk Waves (SSBW) and Love wave modes [55-57].

Most of these modes of vibration have the potential to manipulate particles in a medium, by carefully analyzing the properties of these modes, the current manipulation technique is proposed using bulk modes for the generation of acoustic waves for manipulation.

1.7.3. Material properties of some piezoelectric transducers

Many single crystals, ceramic material and polymers exhibit piezoelectricity. Some of these materials exist in nature while others are fabricated to exhibit predetermined properties. Out of 32 classes of crystalline materials, 20 exhibit piezoelectric effects. Due to the large difference in their electrical, mechanical and piezoelectric properties it is important to understand the advantages and disadvantages of each material. Here only a few materials are discussed.

Quartz is one of the first piezoelectric materials to be discovered and is found in abundance in nature. It was first implemented in sonars in the 1910s and has since been incorporated in almost every facet of modern life. Other important piezoelectric crystals include lithium niobate (LiNbO3) and lithium tantalate (LiTaO3) that are often used in SAW devices [58]

Single crystals, especially quartz crystal, have excellent long term stability and are normally resistive to chemicals and environmental changes therefore making them suitable for sensing applications.

Some naturally occurring ceramic materials, such as Rochelle salt or potassium sodium tartrate tetrahydrate (KNaC4H4O6·4H20) were also found to exhibit piezoelectric properties.

Now day’s most piezoelectric ceramics are synthesized, some important piezoceramics are, barium titanate (BaTiO3), lead titanate (PbTiO3) and lead zirconate titanate (PZT). These ceramic materials are fabricated to have higher electromechanical coupling coefficients and sensitivities than single crystals like quartz but also have higher mechanical losses than crystalline piezoelectrics. They also tend to be brittle and susceptible to temperature fluctuations. Due to their high sensitivities piezoceramics are widely used as precision actuators and motors.

Certain polymers like polyvinylidene fluoride (PVDF) also exhibit piezoelectric effect and are widely used as piezoelectric materials. They have high coupling coefficients (almost 19 twice as high as quartz) and higher piezoelectric stress constants (a parameter to evaluate the response of the material under stress) than ceramics therefore making them better sensors than ceramics [59].

The choice of piezoelectric material is ultimately determined by the specific application. For example, crystalline piezoelectrics are frequently used for narrowband applications due to their high quality factors whereas piezoceramics are more commonly used for broadband applications [60]. The principle parameters that are useful in choosing a piezoelectric material for a specific application are listed in table

Table 1.1 List of material properties for three piezoelectric materials, Pz-27, BaTiO3, LiNBO3 and ZnO

As can be seen each material has a unique set of properties making them suitable for different applications. In this project it was desirable to have a material with high coupling coefficient and voltage coefficient, therefore Pz27 was selected.

The simplicity of this manipulation technique stems from the utilization of natural vibrational modes that can be piezoelectrically excited. The study of vibrational modes of membranes, shells, plates, bars, and beams has been carried out for centuries, by Poisson and Cauchy in the 1820’s, Kirchoff, Rayleigh, Lamb, Timoshenko, Mindlin etc., relevant reference 20 information can be found in [61-62]. Several analytical solutions for the vibration of structures are available for isotropic materials, however, solving these equations of motion analytically for anisotropic materials is complex and thus numerical methods are used to approximate solutions to the equations of motion. The current study involves the use of piezoelectric materials to excite the necessary modes of vibration that will enable the

manipulation of particles. Piezoceramic plates, can be approximated to be transversely isotropic when studying their transverse modes of vibration [63], however here too the solutions to the equations of motion are relatively complex. Therefore Finite Element Method

(FEM) software, ABAQUS is employed to perform a modal analysis of the manipulator.

1.8. Finite element analysis of piezoelectric transducers

Finite Element Method (FEM) is a widely used tool for solving complex structural analysis problems. It was first introduced by Hrennikoff in 1941 and by Courant 1942 [64-65]. From

1950-1962 M.J.Turner worked on developing FEM for everyday use. Most of this pioneering work was done in the aviation industry and later was incorporated into civil engineering. In

FEM a complex physical system is broken down into simpler sub-domains, called (finite) elements, each having a finite number of degrees of freedom (DOF). The mechanical response of each element is characterized in terms of these DOF’s, which are the unknown functions at the node points of each element. Each element response is defined by algebraic equations constructed from mathematical or experimental arguments. The response of the original system is then approximated, by the discretized model, by reassembling the individual elements.

FEM has been used to solve problems in structure and solid mechanics, heat conduction, acoustic fluids, flow, electrostatics, magnetostatics and piezoelectric structures.

Using Maxwell’s and the piezoelectric constitutive equations for linear piezoelasticity, the vibrational characteristics of piezoelectric materials can be analyzed; however, a complete 21 analytical solution can only be obtained for simple geometries [63]. Therefore, we turn to a numerical approach using FEM. One of the first numerical approaches used was the variational approach, [66] to determine the short circuit resonance frequencies, mechanical displacements and electric potentials of piezoelectric transducers of comparable diameter and thickness dimensions. This was followed by a FEM approach by Holland 1967 for the analysis of the resonance properties of rectangular piezoelectric plates [67]. Kunkel et al and

Guo et al used this approach to perform a modal analysis of circular and annular piezoelectric ceramics structures made of Lead Zirconate Titanate (PZT) for various diameter-to-thickness ratios (d/t) [68-69]. Several other studies were performed to analyze the radial and longitudinal modes of vibration [70], and piezoelectric elements coupled to other structures

[71]. Huang C. et al published a series of papers on the analysis of the mode shapes and corresponding resonance frequencies of piezoelectric ceramics, with a comparison to optical data of the vibrational mode shapes.

In this thesis, a FEM modal analysis of a circular piezoelectric ceramic disc (Pz 27), which is used to generate a manipulation force field for the controlled translation of particles on its surface, is proposed. This will provide a map of the force distribution over the actuator surface with which the direction of motion and the final destination of the particles can be predicted. The approximate values of the forces generated at the nodal points will also be extracted and compared with the interfacial force measurement done using the AFM.

1.9. Conceptual model of a Piezoelectric Interfacial Particle Manipulator (PIPM)

The previous sections discussed the need to generate interfacial manipulation forces to enable sample/particle processing which involves certain manipulation functions such as translation, concentration, mixing and splitting of the sample.

In order to generate these manipulation forces we have adopted an acoustic approach wherein an piezoelectric transducer is forced to vibrate at specific modes of vibration. Thus using 22 these modes of vibration, a particle place on the surface of the transducer will experience a complex distribution of interfacial manipulation forces leading to the translate of the particles. We call this technique a Piezoelectric Interfacial Particle Manipulator (PIPM)

Figure 1.11 represents the concept behind the proposed PIPM.

Figure 1.10 Conceptual model of the PIPM for the manipulation of single and multiple particles. fig 1.10(a) schematic of the forces experienced by the particle, fig 1.10 (b) & (c)

manipulation force distribution experienced by the particle at mode M1 and M2, fig. 1.10(d)- (g) single particle manipulation. Fig. 1.12(h)-(k) multiple particle manipulation

As seen in the schematic shown in fig. 1.10(a), a particle placed on the PIPM experiences adhesion forces opposing its motion, namely Fadhesion and Ffriction and manipulation forces that act to enable particle manipulation, namely Fmanipulation_z and Fmanipulation_x. When the PIPM is

excited at specific modes of vibration, the particles experience a resultant force, Fmanipulation,

acting in opposition to the adhesion and friction forces leading to the translation of the 23 particles from regions of high force to those of low force. This can lead to single, fig. 1.10

(d)-(g), or multiple, fig. (h)-(k), particle manipulation.

Consider a single particle placed at the edge of the PIPM, fig 1.10 (d). When the PIPM is excited at a vibration mode M1, the particle experiences a manipulation force distribution shown in fig. 1.10(b). This leads to the particle being translated to the center of the PIPM, fig.

1.10(e). Now if the PIPM is excited at another vibration mode, M2, fig. 1.10(c), assuming the force distribution is symmetric (same along the y-direction), the particle will now be forced along the y-direction, towards the edge of the PIPM, fig. 1.10(f). It can once again be brought to the center by switching back to mode M1, fig. 1.10(g). Analogous to the concept behind single particle manipulation is that behind multiple particle manipulation, fig. 1.10 (h)-(k).

Thus exciting the PIPM at mode M1, the particles will be concentrated at the center of the

disc, fig. 1.10(i), if it is then excited at mode M2, the particles will be reoriented in to a ring

formation near the circumference of the PIPM, fig 1.10(j). To demonstrate the patterning

capability, if the particles are randomly distributed as in fig 1.10(h), and then activated at

mode M3, particles will distributed as shown in fig. 1.10(k).

Therefore, as can be seen from fig. 1.10, if the PIPM is capable to overcome/reduce adhesion forces it will lead to the translation of particle(s) down the gradient of surface manipulation force thus enabling particle manipulation modalities such as translation, concentration and mixing, patterning and splitting. This technique is extremely versatile allowing for flexibility in controlling the direction, speed and orientation of particles.

1.10. Significance

The manipulation of micrometer and sub-micrometer sized, inorganic/organic particles is of growing demand in fields of personalized medicine, biosensors, tissue engineering, interfacial phenomena, MEMS and pharmaceuticals. Furthermore particle/sample manipulation at the 24 solid-fluid interface can greatly impact most of the above mentioned fields that more often than not operate at the interface. It is however, challenging to operate at the interface as required the manipulation forces can be relatively high (on the order of 10’s – 100’s nN) as compared to those needed to manipulate samples above the interface (pN range).

This is a significant challenge for most of the currently available techniques which operate on samples above the interface, namely several optical and acoustic techniques. Furthermore, existing techniques that do operate at the interface often have stringent requirements of particle type, magnet particles for magnetic tweezers, or certain properties of the medium, conductivity requirements of solutions for EP/DEP techniques. Also, several techniques act more like particle traps rather than manipulators, EP/DEP, acoustic standing wave and SAW techniques, which can be advantageous for particle/sample patterning applications in tissue engineering and drug discovery, however if particle/sample translation is required, these techniques would fall short. Lastly for drug discovery and tissue engineering applications, sample recovery, integrity and minimum contamination are important requirements.

Techniques like magnetic tweezers, wherein magnetic particles are mixed with the sample,

EP/DEP wherein electrode corrosion and strong electric fields can lead to sample degradation and loss of sample.

The current technique is focused at operating at the solid-fluid interface, i.e. particle- separation distance of 0.1-10’s nm. It is capable of generating interfacial forces ranging from

1-100’s nN for the manipulation of particles ranging in radius from 0.5-50m. Since it

operates by exerting mechanical forces on the sample at the interface due to low amplitude

vibrations (1-100 nm) of the PIPM, it can be operated in vacuum, air or liquid. There are no

stringent particle property requirements, although the particle material and surface properties

will play a significant role in determining the magnitude of manipulation forces required to

manipulate them. This requirement of different manipulation forces for different particle sizes

and/or properties can be used to selectively manipulate particles based on their properties on 25 the same substrate, essentially enabling sorting of particles, for detection, filtering, and interfacial force measurement applications. In addition to particle translation, this technique also enables particle/sample concentration, mixing and splitting. Particle/sample concentration can greatly impact biosensors, biochip and Lab on Chip applications wherein

small volume low concentration samples need to be processed and detected. The ability to

concentrate samples at regions of high detection sensitivity will improve the detection

kinetics and detection limit of the sensing elements. Furthermore by simply switching the

mode of vibration, the sample can be split in to groups or mixed with another sample thus

allowing for automated sample processing on a single substrate for LOC applications. Lastly,

this system has a simple inexpensive set up that can be easily miniaturized and implemented

to improve specific application.

From the above discussion it can be deducted that this technique can impact a variety of

fields and can prove to be an important technique for sample processing in the future.

There are several applications that arise from this technique, some of which are:

– The study of interfacial adhesion forces

– A hybrid sensor-PIPM can be constructed to improve the performance of the sensor in

terms of its detection kinetics and detection limit.

– The trapping of biological samples (cells (2-25m), bacteria (0.5-2m) or functionalized

microparticles (1-10m)) for tissue engineering and drug discovery applications.

– Incorporating some of the manipulation features, such as concentration, mixing and splitting

in a Lab on Chip device.

1.11. Objectives of the thesis

This thesis is focused on the theoretical and experimental study of the manipulation of micro- particles by the generation of interfacial manipulation forces by a piezoelectric actuator. The 26 results of this work should demonstrate the potential of this manipulation technique for a variety of applications.

The objectives of this thesis are:

– To identify and evaluate the interacting forces that exists between micro-particles (0.5 – 50

m) and the PIPM surface

– To model the PIPM and the mechanism by which particles are manipulated

– To design and fabricate the PIPM

– To experimentally test the PIPM and study the influence of its operating conditions on

particle manipulation.

– To apply the PIPM for improving the performance of a thickness shear mode (TSM)

acoustic biosensor.

By achieving these goals, a novel manipulation technique capable of reproducibly translating

particles, will be developed.

1.12. Organization of the thesis

The main components of the thesis are shown in figure 1.11

Figure 1.11 Block diagram showing the main components of the thesis and the corresponding chapter numbers 27

The current chapter deals with the background information. Chapter 2 involves a theoretical study of the interfacial forces, whereas in chapter 3 a FEM analysis of the PIPM and an insight into the mechanism of particle manipulation is studied. This is followed by chapter 4 where in experimental methods are described. Chapter 5 encompasses the experimental study carried out to address the objectives of this thesis along with an application of this technique. 28

2. STUDY OF PARTICLE-PIPM INTERFACIAL FORCES

The particle manipulation technique discussed in this thesis is dependent on the ability to generate interfacial manipulation forces that can reduce/overcome those forces that bind particles to surfaces. Thus, a study of the forces involved will define the capabilities of the

PIPM, such as, the size and type of particles that can be manipulated and the translation distance.

Figure 2.1 represents a schematic of the dominant forces binding the particles to the PIPM surface and the opposing forces generated by the PIPM.

Figure 2.1 Schematic of the particle-PIPM interfacial interaction. The force balance diagram shown on the right. (Image not to scale)

The manipulation experiments discussed in this document were conducted using metallic micro-particles on a metallic surface, in air, therefore the dominant forces acting between the particle and the manipulator under these conditions are discussed here. Extension of this technique applied to aqueous environment with biological samples is briefly discussed at the end of the document, preliminary studies can be found in Desa 2008 [11].

The dominant forces can be split into 2 main categories, namely, interfacial adhesion forces,

Fadhesion, that act to oppose particle motion, and interfacial PIPM forces generated by the 29 piezoelectric manipulator in order to detach and translate the particles. These can be seen in table 1.

Table 2.1 Listing of the interfacial forces acting on the particle

The listed forces in table 2.1 are now briefly described:

There are several interfacial forces acting between a micro-particle and a surface. Interaction

of the particle with the surface is the result of all these interfacial forces. Several factors

affect the magnitude of these forces such as, particle size, surface roughness, humidity or

temperature, etc.

In most applications, the focus is on the overall effect of all the forces. The goal of this

section is to analyze the interaction between a particle and a surface, thus developing the

necessary understanding of the forces that need to be generated by the PIPM. This is followed

by a study of the PIPM forces that enable the translation of particles from one location to the

next.

First, it is necessary to identify the various forces existing between a micro-particle and a

surface.

2.1. Interfacial adhesion forces

Interactions between bodies are broadly classified into 4 fundamental interactions: strong, weak, electromagnetic and gravitational interactions [72]. Interactions between neutrons, 30 protons, electrons and other elementary particles are termed as strong and weak interactions and act at distances < 10-5 nm. On the other hand the electromagnetic and gravitational interactions act between atoms and molecules as well as between elementary particles and are effective from the subatomic to infinite distances. The electromagnetic force is the force that an electromagnetic field exerts on a charged particle. It is the source of all intermolecular interactions, determining the physical, chemical, mechanical and electrical properties of matter, acting from 0.2-100nm. The gravitational force is and attractive force acting over extremely large distances, for example, it accounts for tidal forces and many cosmological phenomena.

In the study of the interfacial interaction of a microparticle with a surface, the separation between the particle and the surface is usually in the range of 0.1-1000 nm. The strong and weak interactions are extremely weak at these distances, whereas the gravitational and electromagnetic forces dominate particle-surface interaction. Therefore, only electromagnetic force and gravitational force are considered.

The interfacial adhesion forces include normal adhesion forces that bind the particle to the surface, namely, gravitational, van der Waals, capillary and electrostatic forces as well as those that act tangentially in opposition to the relative horizontal motion of the interacting bodies, namely the friction force. The sum of the normal adhesion forces act to oppose the vertical motion of the particle, in this thesis we call this sum, Fadhesion, whereas the tangential adhesion force is referred to as Ffriction.

2.1.1. Gravitational force

Gravitational force exists between any two object having non-zero masses. For two objects having masses, m1 and m2, the force of gravity, a universal force, is directly proportional to the product of the masses and inversely proportional to the square of the distance separating their centers, r, 31

m1m2 Fgrav  G 2 r 2.1(a)

Where, G is the gravitational constant having a value of 6.673 x 10-11Nm2/kg2. For a mass on the surface of the earth, ~6.38 x 106 m from its center, with the mass of the earth ~ 5.98 x 1024

kg, equation 2.1 (a) reduces to,

4 F  m g  R 3  g 2.1(b) grav p 3 p

Where, mp is the mass of the object, g is the acceleration due to gravity normally taken as, 9.8

2 m/s , R is the radius of the particle and p, is the material density of the particle.

An object immersed in a fluid, like air or water, experiences an opposing force from the fluid

that it displaces, tending to reduce its effective mass. Thus an object in a fluidic media

experiences a net force given by,

4 F  R 3    g 2.1(c) net 3 p f

Where, f, is the fluid density. Equation 2.1(c) reduces to eq. 2.1(b) if p >>f, for example, a

3 3 stainless steel (f = 8000 kg/m ) particle in air (~1.18 kg/m ). However for a polystyrene

particle (1050 kg/m3) in water (1000 kg/m3), the net force decreases by 95% from its value in

air or vacuum. 32

2.1.2. Electrostatic force

The electrostatic force of attraction can be categorized by two interactions; long range

Coulombic attraction due to bulk excess of charge in the particle or the surface, and, short range interactions due to a difference in work functions of each surface resulting in a contact

potential difference [73]. These two forces can be represented by the following expressions;

2 R FES  0VCP d  R d 2.2

2 2  R    0VBC   d  R  R  d  2.3

Where, VCP is the contact potential in volts, VBC is the potential difference due to bulk charge

difference, 0 is the permittivity of free space. As can be seen from equation 2.2 and 2.3, FES

2  1/d at small separation distances (d<R)

[74].

2.1.3. Van der Waal’s (VDW) force

The VDW force like the gravitational force, acts between all atoms and molecules. This electromagnetic force, has three sources: atom-atom interactions, dipole-dipole interactions, dipole-induced dipole interactions.

The VDW force between a particle and a surface is inversely proportional to the square of the particle-surface separation distance [75] as given in eq. 2.4.

AR Fvdw  2 6d 2.4

Where, A, is the Hamaker constant whose value ranges from 0.4 – 5 x 10-19J, R, is the radius of the particle and d, is the separation distance between the particle and the manipulator surface. Although Eq. 2.4 applies to atomically smooth interacting surfaces; several studies have been conducted to study the effect of surface and particle roughness on the van der

Waals force of attraction [64-67]. These studies show that surface roughness, even at the nano-scale, can greatly diminish the magnitude of the VDW force.

Rabinovich et al derived an expression taking into account the effect of surface roughness, equation 2.5 [76]:

Figure 2.2 Schematic of the effect of roughness on van der Waals force. Geometry proposed by Rabinovich et al [76]

    AR 1 1 F  F *b     VDW _ rough VDW _ smooth 2  32Ry 2  6d max  ymax  1 2 1        d   2.5

In eq. 2.5, the effect of roughness is studied by introducing a roughness factor, b, which depends upon the radius of the particle, R, peak surface roughness, ymax, the peak-peak

separation distance, , and the particle-surface separation distance, d, as can be seen in the

schematic, fig 2.2. The first term in the square brackets represents the contact interaction

between an adhering spherical particle and a rough surface, while the second term represents

the non-contact interaction of the particle with the surface. The contribution of the second 34 term is found to be small for surface roughness greater than 2nm and in those cases can be ignored.

Equation 2.5 is plotted as a function of  ranging from 1-5000nm, and ymax ranging from 1-

2500nm.

As seen in figure 2.3, the surface roughness can reduce the magnitude of the van der Waal’s

force, by upto 3 orders of magnitude as compared to that of smooth interacting surfaces.

Although the Rabinovich model takes into account surface roughness, it was observed that

experimental values could be upto 2 orders of magnitude larger than those calculated

theoretically.

When the particle-surface interaction is studied in high vacuum, the dominant adhesion forces

are van der Waals and electrostatic forces, however, in ambient air environmental conditions

(Temperature 300K and Relative Humidity (RH) 15-90%) the detachment of a particle from a

surface is often dominated by the forces due to a liquid bridge formation between interacting

surfaces, namely capillary forces [77-78]. 35

2.1.4. Capillary force

As the humidity is increased, a nanoscale liquid vapor film forms on surfaces. The thickness and formation of this layer is dependent on the properties of the surface (surface energy, surface roughness) and the relative humidity (RH). As a particle approaches a surface, due to the VDW force of attraction, the liquid between the particle and the surface is squeezed out until the particle-surface reaches an equilibrium state as shown in figure 2.4, leading to the formation of a liquid meniscus between the particle and the surface [79]. If the particle is in contact with the surface and the RH increases, then too a liquid meniscus is formed due to water condensing into the gap at the contact region between the particle and the surface.

Figure 2.4 A schematic of the liquid meniscus formed between a partilce and a plate. rc and l

are the two principal radii of curvature for the water meniscus, 1 and 2 are the contact angles for water on the sphere and the plate, respectively.

In general, the total capillary force is given by the sum of the capillary pressure force caused by a reduced pressure inside the meniscus and a surface tension component. For a rotational symmetric geometry (particle on plate) the capillary force is given by [77],

F  2l  l 2 P cap 2.6

Where,  is the surface tension of the liquid, rc and l are the principal radius of curvature of

the liquid with the curved and planar surfaces respectively and P is the difference in

pressure between the liquid and the surrounding vapor phase given by the Young-Laplace

equation:

1 1  2.7 P       l rc 

The curvature of the meniscus is characterized by the radii, rc and l. The azimuthal radius l is positive as it is concave with respective to the liquid, whereas the meridional radius rc, is

counted negative. More detailed studies of the capillary force between particles and surfaces

can be found in, [80-81]. As discussed in [77] the capillary force calculations can be

simplified to give:

 d    2.8 Fcap  2Rcos1  cos2    rc 

Where, 1 and 2 are the contact angles of the liquid with the sphere and plate respectively.

The assumptions made by Butt 2009 to derive equation 2.8 apply to most general situations and are therefore incorporated in the current study. Furthermore, when the particle-surface is in equilibrium and at constant vapor pressure, the radius of curvature, rc, is constant for a

particular relative humidity and can be calculated using the Kelvin equation.

V r   m R T ln(P / P ) g 0 2.9

Where, Vm is known as the Kelvin length, and is calculated to be 0.52nm at 25C for Rg T

water.  is the surface tension of water, Rg is the gas constant, T is the temperature in Kelvin, 37

Vm is the molar volume of water, P is the vapor pressure of a vapor in equilibrium with a curved surface, and P0 is the saturation vapor pressure over the planar liquid surface. Using equations 2.8 and 2.9 the capillary force due to a liquid meniscus between a particle and a planar surface can be calculated as a function of relative humidity and particle radius.

2.1.5. Comparison of adhesion forces

The above mentioned forces act bind the particle to the surface and constitute the adhesion force experienced by the particle normal to the surface. Thus the combination of the interfacial adhesion forces, given by equations 2.1, 2.2, 2.4 or 2.5 and 2.8, can be represented by,

F  F  F  F  F adhesion grav vdw es cap 2.10

In order to observe the effect of each of these normal adhesion forces, a theoretical study is performed. The adhesion forces between stainless steel particles and the silver PIPM surface, assuming ideally smooth surfaces, are plotted in fig 2.5 using the following conditions: A = 4

-19 x 10 J [82], d = 0.3 nm, RH = 50%, Vcp = 0.1V (calculated based on the work functions of stainless steel and silver), 1 = 89 [83]and 2 = 80 [84].

Figure 2.5 Comparison of particle-surface adhesion forcses as a function of particle radius.

Fig. 2.5 represents a theoretical plot of the forces binding the particle to the PIPM surface. It is observed that the van der Waals force is dominant for the conditions set for the theoretical calculations. However, it should be noted that the VDW force decays rapidly with the increase in separation distance as shown in eq. 2.4 & 2.5, whereas the capillary force could act over much larger d at high RH due to the formation of the liquid bridge. It should be noted that the theoretical force curves in fig 2.5 are plotted for smooth surfaces; however,

surfaces used for experimental measurements always exhibit some level of roughness which

can greatly reduce the values of the measured interfacial forces [76, 85-88]. Therefore a

surface roughness study will have to be performed to better understand the range of particle-

surface adhesion forces.

In this thesis the quantification of Fadhesion is performed via the use of an Atomic Force

Microscope (AFM). The AFM is used to measure both, the force with which the particle is attracted to the surface and, the force needed to detach the particle from the surface.

In order to facilitate vertical motion of the particle, for example particle bouncing, the PIPM will need to overcome the adhesion forces that bind the particle to the surface, given by equation 2.10. On the other hand horizontal translation of the particle does not necessitate the 39 overcoming of adhesion force but rather the reduction of adhesion force and/or the overcoming of tangential adhesion forces, namely frictional forces discussed next.

2.1.6. Friction force

Friction force opposes the relative motion of two bodies in contact with each other. As given in the ASTM standard, G-40-93, Friction force -- “the resisting force tangential to the

interface between two bodies, when under action of external force, one body tends to move relative to the other”. It is directly proportional to the normal component of force binding the bodies together and is related to the normal component via the frictional constant, also defined in the ASTM standard G-40-93, coefficient of friction – “the ratio of the force resisting tangential motion between two bodies to the normal force pressing the bodies together”. Friction force is an intermolecular force between surfaces of materials and is determined empirically. Unlike gravitational or electrostatic forces, frictional force is not conserved as some amount of energy is lost during the breaking of intermolecular bonds.

The force of friction can be categorized as static friction, Ffriction_s, the force that is just

sufficient to resist the onset of relative motion, and kinetic friction, Ffriction_k, which is the

force that slows bodies in relative motion. For micro-particles the normal force binding the

particle to a surface is no longer dominated by the weight of the particle alone but is given by

equation 2.10. Thus the static and dynamic friction are given by [89]

F   F friction _ s s adhesion 2.11 F   F friction _ k k adhesion 2.12

Where s and  k represent the static and kinetic coefficients of friction which are dependent

on the materials of the interacting surfaces and are determined empirically. Static dry friction 40 between metals can range from 0.3 – 0.7. Surface conditions such as contamination, humidity and roughness can impact the coefficients of friction and is therefore unique for two surfaces in contact.

As deducted from eq.’s 2.11 & 2.12, the friction force is a fraction of the adhesion force since the frictional constant is less than one. In order to translate a particle from one location to the next, it would therefore require less force to overcome frictional force and slide the particle from one location to the next rather than vertically lifting it off the surface followed by horizontal translation.

However it is found that the minimum force needed to initiate particle motion is not equal to the force applied exactly opposite to the friction force, but is obtained when applied at a particular angle to the horizontal axis as shown in appendix 6.Consider a particle interacting with a planar surface shown in figure 2.6 (a)

The equation relating the applied force, F, and the angle, , at which it is applied is given by,

 F F  s adhesion cos   sin s 2.13

Assuming a friction constant, s = 0.5, and adhesion force, Fadhesion = 100nN, the force needed to move the particle is plotted as a function of angle with the horizontal x-axis, equation 2.13

As seen in figure 2.6, at  = 0 degrees, the force needed is equivalent to overcoming the friction force, Ffriction_s = 0.5 x Fadhesion = 0.5 x 100 = 50 nN. As the angle of force is increased,

the force needed to move the particle decreases to a minimum of ~44nN at ~ 25 degrees after

which it once again increases and reaches a maximum value of 100 nN at  = 90 degrees when it is acting vertically in opposition to Fadhesion.

The basis for the magnitude of the applied force being less than that of the friction force is

that applying a force at an angle reduces the adhesion force which in turn reduces the friction

force thus reducing the force required to move the particle. Therefore applying a force at

certain angles to the horizontal axis can reduces the force needed to manipulate particles.

Also, knowing the frictional constant, the minimum force necessary to manipulate a particle

is given by,

s Fadhesion Fmin  (1  2 ) s 2.14

Further details on how we come about this equation can be found in appendix A6.

In this project, a PIPM is used to generate the necessary interfacial manipulation forces to move the particles. These forces can be resolved into their normal and tangential components which act synergistically to over come adhesion and frictional forces. 42

2.2. Interfacial manipulation forces

The manipulation of micro- and sub-micrometer sized particles requires the generation of interfacial forces to reduce and/or overcome adhesion and frictional forces binding particles to surfaces, followed by a means to translate these particles from one location to the next.

In this project particle manipulation is achieved by using an harmonically vibrating actuator to generate the necessary manipulation forces. As an example consider a disc vibrating at its radial resonance mode thus having a surface displacement map shown in figure 2.7.

Figure 2.7 Contour map of a radial mode of vibration showing the manipulation force,

Fmanipulation exerted on a particle placed on its surface

At each location on the vibrating surface, the instantaneous displacement, velocity and acceleration can be defined as:

ui  ui0 (x, y)cos(t ) u v  i  u (x, y)sin(t ) i t i0 2.15(a), (b), & (c) v a  i  u (x, y) 2 cos(t ) i t i0

Where, the subscript i represents the x, y, and z axes components, ui0 (x, y) represents the

maximum instantaneous displacement at each location on the surface which varies with the

spatial location, x, y,  is the angular frequency, and  is the phase angle between normal and 43 tangential components. In this technique, the manipulation force is characterized by its surface displacement, velocity or acceleration given by equations 2.15 (a), (b), & (c) respectively.

A particle placed on a disc vibrating with the steady state mode shape shown in figure 2.7 experiences a resultant manipulation surface force, Fmanipulation, normal to the surface at that

location.

The force exerted on particles by the vibrating PIPM is categorized as inertial and impulsive

forces.

Inertial manipulation forces are generated due to the acceleration of the vibrating surface

relative to the particle placed on its surface whereas the impulsive manipulation forces are

generated by the transfer of momentum when the particle collides with the vibrating PIPM.

These manipulation forces are now discussed.

2.2.1. Inertial manipulation force

The inertial manipulation force is defined as the force generated by the PIPM, due to its acceleration, acting in opposition to the interfacial adhesion forces. This force can be caused due to horizontal or vertical (normal) surface accelerations. Using eq. 2.15, it can be mathematically represented as,

F  m a  m u  2 cos(t  ) manipulation _ i p i p i0 2.16

Where mp is the mass of the particle, a, is the surface acceleration and the subscript i

represents the x, y and z components of force, acceleration and displacement. This force can

be resolved into its normal/vertical and tangential/horizontal components as shown in figure

2.7. 44

The tangential component inertial force, Fmanipulation_x,y, acts in opposition to frictional force

and can cause the PIPM surface to slide with respect to the particle, thus leading to slip

conditions.

On the other hand the normal component of inertial force, Fmanipulation_z acts in the in the

vertical direction and can lead to the surface overcoming adhesion forces, thus causing the

detachment of the particle.

In this project we focus on the vertical/normal inertial manipulation force as the required

condition for particle manipulation since manipulation is carried on a harmonically vibrating

plate and the horizontal inertial force will not allow for the translation of the particle from

one location to the next. A more detailed explanation is given in the next section on the

mechanism of manipulation.

Therefore we define necessary condition for particle manipulation as,

F  F manipulation _ z adhesion 2.17

Satisfying condition given in eq. 2.17 will lead to the detachment of the particle from the

PIPM surface. This condition will be utilized to get an approximation of the adhesion force

binding the particle to the surface by recording acceleration/velocity/displacement of the

surface at the point of detachment.

The adhesion force binding micro and nano particles to a surface is dominated by

intermolecular forces as explained in the previous section. As shown in figure 2.5, for a

stainless steel particle having radius, R < 60 m, adhesion force is dominated by van der

Waals force, eq. 2.4, which is proportional to the radius, R. However, the inertial force is proportional to the mass and therefore R3, thus, as the particle size is decreased, the inertial

force decreases more rapidly than Fadhesion . Therefore, as seen in eq’s 2.17, either the

amplitude or frequency of vibration, or both, will need to be increased to satisfy equation

2.17. During the theoretical adhesion forces study in the previous section it was shown that 45 the van der Waals force is the dominant interfacial adhesion force. Therefore assuming

Fadhesion = Fvdw, the adhesion force for a range of particle sizes, 0.1 - 100m is plotted in

figure 2.7. Also shown is the PIPM inertial force, Fmanipulation_z, plotted for 4 combinations of

displacement amplitude and frequency.

Figure 2.8 Comparison of the forces exerted by the PIPM when excited at different frequencies and amplitudes, with adhesion forces.

Table 2.2 The effect of different manipulation frequencies and vibration amplitudes on the minimum particle size that can be detached from the PIPM

The vertical lines 1-4 are drawn passing through the intersection of each of the manipulation

force curves with the Fadhesion curve. They indicate the region where the manipulation force becomes larger than the adhesion force. For each vertical line, particle sizes to the right of the line can be detached by operating the PIPM at the selected frequency and amplitude combinations. The minimum particle sizes that can be detached for each case are shown in table 2.2. It should be noted that the adhesion force was calculated based on smooth interacting surfaces causing it to be up to 2 orders of magnitude higher than that observed experimentally, wherein the surface roughness greatly reduces its magnitude. Owing to the 46 fact that both particle and surface always exhibit some level of surface roughness, the adhesion force would, decrease if the contact area is reduced, and vice versa. As can be seen from fig 2.8 and table 2.2, as the amplitude or frequency of vibration is increased, the inertial force generated by the PIPM increases according to equation 2.16. We can therefore define 2 critical manipulation parameters which need to be satisfied to facilitate particle manipulation.

2.2.1.1. Critical dynamic displacement and frequency of vibration

For each particle-surface interaction, there exists an adhesion force binding the particle to the surface. In order to facilitate particle manipulation we can now define the critical PIPM acceleration, az_crit, and therefore the critical displacement and frequency, uz_crit and fcrit

defining the minimum requirements of the manipulator. From equation 2.17 one can infer that

in order to detach a particle from the PIPM, the critical manipulator acceleration is given by,

F a (x, y)  adhesion z _ crit m p 2.18

This in turn can be given in terms of the displacement of the PIPM surface and the frequency of vibration,

F  2u (x, y)  adhesion z _ crit m p 2.19

Therefore it can be seen that there are two PIPM parameters that can be modified to enable particle manipulation, the displacement and the frequency. At any one manipulation mode the displacement of the surface can be increased by increasing the excitation voltage.

When the PIPM is excited at a particular frequency the critical displacement at any location on the manipulator surface is given by,

F u (x, y)  adhesion z _ crit  2m p 2.20

Therefore, at each excitation frequency, a range of particle sizes can be potentially manipulated. Furthermore, in addition to increasing the displacement, it can be seen that the manipulation force is proportional to the square of the excitation frequency. Thus by operating at higher frequencies, stronger adhesion forces can be overcome and/or smaller particles can be manipulated. It should be noted that modifying surface conditions, like roughness, hydrophobicity, etc., can also be used to aid in decreasing adhesion forces.

The current technique is based on generating interfacial forces to overcome adhesion forces.

Table 2.3 lists a few examples, from literature, of the forces binding particles to surfaces and the critical displacement the PIPM will need to generated to manipulate them when operated

at 65x10-3,2 x10-1, 1 and 100 MHz.

Table 2.3 Adhesion forces and corresponding critical displacement that the PIPM will need to generate in order to detach the particles from the surface.

The references shown in table 2.3 correspond to [36, 78, 85, 90-94]

The details of the listed forces in table 2.3, can be obtained from the references listed. It

should be noted that the indicated adhesion forces are dependent upon the experimental

conditions, such as surface and environmental properties, and measurement techniques such 48 as, AFM, vibrational method and optical tweezers. What is common to each technique is that each force is measured at the particle-surface interface. Therefore, in order for the current manipulation technique to facilitate particle manipulation, it would need to overcome the listed forces. In order to do so, the PIPM is operated at specific frequencies; therefore the critical displacement that would have to be generated is indicated in table 2.3. As can be seen, the PIPM operated at 65 or 200 kHz would be incapable of manipulating small biological objects such as proteins. However, it is possible to attempt overcoming the binding force by increasing the frequency of operation of the PIPM to 100MHz at which point the critical displacement needed to detach proteins reduces to ~125nm.

Now that the inertial manipulation component has been addressed, the minimum displacement and/or frequency of particle detachment can be determined. The next step is to translate the particle from one location to the next, which is achieved by impulsive manipulation forces experienced by the particle.

2.2.2. Impulsive manipulation force

Impulsive manipulation force is defined as the force experienced by the particle due to the momentum transferred to it at the point of impact with the vibrating PIPM. If the condition for particle manipulation is satisfied, eq. 2.17, the PIPM distances itself from the particle. The

particle then undergoes free fall and collides with the vibrating manipulator. At the point of

collision, the particle experiences a transfer of momentum from the vibrating surface, whose mass, mm, is much larger than that of the particle, mp, i.e. mm >> mp .

The particle thus experiences an impulsive manipulation force given by,

Pi m p (v pfi ) Fmomentum _ i   t t 2.21

Where, the subscript i represents x, y and z components, P is the momentum of the particle

after collision and v pfi , represents the final velocity, tangential and/or normal component, of

the particle after collision with the PIPM surface.

The collision of the particle with the surface leads to the particle being projected off the

surface with the direction of projection being dependent on the initial normal and tangential

velocities of the particle and the surface at the point of collision. If the final particle velocity

is directed along only the z-axis, this will lead to the bouncing of the particle. However a final

velocity having both normal and tangential components will lead to the simultaneous vertical

and horizontal projection of the particle down the gradient of surface force from regions of

high to low force.

2.2.2.1. Determination of the final velocities of the particle and the surface:

The collision of bodies is a complex event that involves a very short duration of interaction during which large magnitude of impulsive forces can be generated. Furthermore, during this collision event, phenomena like vibrational waves propagating through the bodies, local deformations in the regions of contact and frictional and plastic dissipation of mechanical

energy can take place [95].

Figure 2.9 Free body diagram of two colliding rigid bodies, namely the particle and the PIPM, in x-z plane. (Not to scale)

Figure 2.9 represents a free body diagram of the two bodies, with the x-z coordinates chosen such that the lines through the particle and PIPM centers are along the z-axis (normal axis).

The velocity symbols have three subscripts, the first subscript indicating particle ‘p’ or

manipulator ‘m’, the second subscript indicating initial ‘i’ or final ‘f’ velocities, and the third

indicating normal ‘z’ or tangential ‘x’ direction. For example, vpix is the initial tangential

velocity of the particle.

Elastic Collisions:

We first consider one dimension elastic collision that occurs along the z-axis,

Considering only normal impact of the two bodies, the following terms are defined: mp mass of the particle

mm mass of the surface vpiz and vmiz – Initial (original) velocities of particle, ‘p’ and manipulator surface ‘m’

vpfz and vmfz – Final velocities after collision

For a fully elastic collision, the energies and momentum of the colliding bodies are conserved, 51 i.e.

m v2 m v2 m v2 m v2 p piz  m miz  m mfz  p pfz 2 2 2 2 2.22

m v  m v  m v  m v p piz m miz p pfz m mfz 2.23

Solving the simultaneous equations 2.22 and 2.23 algebraically (Appendix A5) we can find

the final velocities after the collision.

mpvpiz  mmvmiz  mp (vpiz  vmiz ) vmfz  2.24 mp  mm

mpvpiz  mmvmiz  mm (vmiz  vpiz ) vpfz  2.25 mp  mm

Please note that equations 2.24 and 2.25 represent the final velocities due to momentum

transferred after a head on collisions.

At the micro- and nano-scale, the process of detachment of particles from a surface involves

the loss of energy, thus leading to a decrease in the final particle velocity after impact. This

loss of kinetic energy leads to the collision being inelastic.

Collisions with normal losses:

We will now reformulate the problem to represent 2 dimensional collision where the coefficient of restitution, e, is used to define the energy loss during normal impacts (along z- axis) and a coefficient of friction term is introduced to account for tangential energy losses.

An impulse-momentum formulation is adopted as it provides simple algebraic equations from which the initial and final velocities can be determined.

Here we discuss rigid body collisions between a particle of mass mp and the PIPM of mass

mm. We assume no spin is introduced at the point of collision, which, if did occur, would need to be represented as a change in the angular velocity of the particle at the point of collision. 52

Once again, as with the elastic case, momentum is conserved in the normal direction. To account for the loss of energy in the energy in the normal direction, the coefficient of restitution is defined as,

v  v e  pfz mfz v  v miz piz 2.26

The value of e lies between 0 and 1, i.e. 0  e  1, where, e = 1, represents a fully elastic

collision and for e = 0, the collision is purely inelastic or plastic collision.

The final normal component of velocities for an inelastic collision has been extracted in

Appendix 5, and is found to be,

m (1 e)(v  v ) v  v  m miz piz pfz piz m  m p m 2.27

m (1 e)(v  v ) v  v  p miz piz mfz miz m  m p m 2.28

Since mm >> mp, the final velocity of the surface can be assumed to remain unchanged. Thus equations 2.27 and 2.28 represent the final particle and surface velocities after an inelastic collision.

In order to translate the particle from one location to the next, the surface would have to apply a force to the particle at some angle to the horizontal (x) axis, i.e. a resultant impulsive manipulation force. Thus the particle would now have both normal and tangential components of velocity after the collision.

As shown in figure 2.9, collision of the two bodies can lead to the generation of an impulse force in both the normal and tangential directions.

Collision with tangential losses:

The loss in the tangential energy is governed by the tangential impulse, Px, developed during

the collision. An equivalent coefficient of friction, , is therefore defined as,

P   x 2.29 Pz

Once again, the momentum is conserved in the tangential direction. The final tangential velocities of the PIPM and particle are given by,

m (1 e)(v  v ) v  v  p miz piz mfx miz m  m p m 2.30

m (1 e)(v  v ) v  v  m miz piz pfx pix m  m p m 2.31

Since mm >> mp, the normal and tangential velocities of the PIPM remain unchanged after the collision, however the particle’s normal and tangential final velocities are given by equations

2.28 and 2.31 respectively [96]. For the current study, in order to demonstrate the mechanism of particle motion along the surface of the PIPM it is sufficient to limit the study to conditions of no spin, but it is not entirely complete. A study involving particle spin will be undertaken in the future. This phenomenon has however been addressed by several authors, [95-96] etc.

It should also be noted here that the impact takes place at a single point, however in reality it takes place over a small impact area which could vary based on the stiffness of the materials.

The Johnson Kendall R model is widely used to study the deformation of objects at the point of impact and the energy lost during the deformation [97].

It should also be noted that particle manipulation conducted in a fluidic media, leads to the generation of acoustic wave forces in addition to the surface vibration forces discussed above. 54

In fluid, the non-uniform force distribution over the PIPM surface also leads to the generation of shear and compressional acoustic waves into the adjacent media. However, experiments discussed in this document were conducted in air wherein the RH was 15-60% at the particle- manipulator interface. In these environmental conditions, the acoustic wave forces experienced by particles on the PIPM surface were calculated and found to be negligible in

comparison to the inertial force and force due to momentum transferred to the particles.

These forces were therefore not considered here.

Now knowing the governing equations and effect of the inertial and impulsive manipulation forces the mechanism of particle manipulation can be discussed.

2.3. Mechanism of manipulation

As discussed in the above sections, the manipulation of particles can be achieved by the generation of surface forces such that the adhesion forces binding particles to the surface are overcome, followed by the controlled guidance of the particles to predetermined locations, which is facilitated by well defined surface force distribution.

Here we present a simple means to manipulate particles, using inherent natural modes of vibration of structures to generate the required manipulation forces.

Early studies by the German experimentalist, Chladni, 1787, demonstrated the visualization of nodal lines of vibration modes, an extensive theoretical and experimental work has since been performed on the vibration of structures like, beams, plates, shells etc. [98-99]

In this project, we utilize the natural modes of vibration to generate interfacial particle manipulation forces. Every structure has inherent natural modes of vibration; however the excitation of these modes requires the harmonic excitation of the structure at its natural frequency. This we achieve by utilizing piezoelectric materials that can be excited over a broad range of frequencies. 55

The principle of acoustic manipulation is based using the mechanical stresses developed at the PIPM’s surface when it is excited at its natural frequencies, to exert inertial and impulsive

manipulation forces on particles placed on its surface.

The mechanism of particle manipulation behind this current technique can be explained as a

two step process:

Step 1: The particle is detached from the manipulator surface  Inertial manipulation force

(Fmanipulation_z)

Step 2: The particle is projected down the gradient of surface manipulation force 

Impulsive manipulation force (Fmomentum_i)

Step 1 is achieved by satisfying the criteria given by eq. 2.17, i.e., the surface is be capable of

overcoming the adhesion force thus distancing itself from the particle (detachment). This sets

the stage for step 2, wherein the falling particle collides with the vibrating PIPM surface

resulting in it experiencing the impulsive manipulation force. It then gets projected with a

final velocity that is a function of the velocities of the PIPM and the particle just before the

collision.

We now list the main conditions for particle manipulation. Assume that initially both the

PIPM and particle are at rest. When the PIPM is activated:

Condition 1: If Fmanipulation _ z  Fadhesion  particle and surface move together with same velocity

Condition 2: If Fmanipulation _ z  0 and Fmanipulation _ x, y  Ffriction  slip condition with no net

particle translation for a harmonically vibrating PIPM. However if the PIPM is tilted at an

angle to the horizontal, or is excited with an asymmetric signal, the particle can be translated

in one direction.

Condition 3: If Fmanipulation _ x,y  0 and Fmanipulation _ z  Fadhesion  particle detachment and

vertical projection (bouncing) 56

Condition 4: If Fmanipulation _ x,y  0 and Fmanipulation _ z  Fadhesion  particle detachment and

simultaneous vertical & horizontal projection

In order to achiever particle manipulation, the current technique utilizes condition 4, where in the particle is simultaneously detached and projected down the gradient of surface force, thus enabling particle manipulation.

As per the above conditions, during particle manipulation, the particle can exist in three main states:

State 1: No motion relative to the PIPM: If Fmanipulation _ z  Fadhesion , there no relative motion

with respective to the PIPM and the particle moves with the PIPM.

State 2: Free fall and projectile motion: When condition 3 or 4 is satisfied, the particle

undergoes free fall following its detachment from the PIPM. Also, after collision with the

vibrating surface it is projected, either vertically or at an angle with respect to the surface.

The status of the particle during free fall or after it is projected is governed by kinematic

projectile motion equations, discussed next.

Status 3: Impact with the vibrating surface: After free fall, the particle collides with the

vibrating surface leading to a transfer of momentum between the particle and the surface as

discussed in section 2.2.2.1. During the collision, both the magnitude and direction of particle

velocity are modified leading to its motion due to impulsive manipulation forces.

Impact and centrifugal techniques, based on inertial force, have been used to detach particles

from surfaces, however they are unable to translate the particles [78, 100]. Translation via

sliding over a surface has been demonstrated using shear inertial forces; however particle

motion is limited by the direction of shear displacement [101]. In this thesis we provide an

alternative technique based on natural modes of vibration which can detach as well as

manipulate particles thus providing a high degree of flexibility. 57

As an example we demonstrate particle manipulation using the radial mode of vibration of a thin piezoceramic disc shown in fig. 2.10

As can be seen from the contour plot in fig. 2.10(a), the surface exhibits a non-uniform distribution of displacement over its surface, which can be resolved into tangential and normal components, namely, ux, uy and uz respectively. By analyzing the displacements along

the x-axis, it is observed that the y-component is small compared to the x and z components

of displacement, i.e., uy << ux & uz. We therefore focus on the uz and ux components along the

x-axis. A particle placed at any location on the vibrating PIPM, would experience a

combination of inertial and impulsive forces due to the surface motion, given by equations 58

2.16 and 2.21. These manipulation forces can be resolved into their normal and tangential components that act synergistically to manipulate the particles.

As shown in fig. 2.10(b), the normal component of manipulation force, Fmanipulation_z, acts to

detach the particle from the PIPM surface, whereas the tangential component, Fmanipulation_x,

acts to translate the particle down the gradient of manipulation force, fig. 2.10(c). Therefore,

for the radial mode of vibration, assuming condition 4 mentioned above is satisfied, a particle placed at the edge of the PIPM would be simultaneously detached and translated to the center of the PIPM, effectively trapping the particles at this location. It should be noted that the force generated by the PIPM is a function of excitation voltage, therefore the rate of particle manipulation can be increased/decreased by adjusting the PIPM excitation voltage.

It should also be noted that it is necessary for the particle diameter to be much smaller than the wavelength of the elastic wave in the solid at the frequency of manipulation. This enables the particle to experience the gradient of surface force when projected off the PIPM surface.

Here this procedure was applied to the radial mode of vibration, however it can be applied to any mechanical mode of vibration used for particle manipulation.

The particle trapped at the center of the PIPM (radial mode), can then be translated to a new location by simply switching the frequency of excitation to one that corresponds to another mode, this is briefly discussed next.

Multimode particle manipulation

As explained above, particles can be manipulated using a single mode of vibration; however the particles would then be trapped at nodes of that vibrational mode. In order to then manipulate the particles to new locations, the frequency of vibration can be switched to correspond to another natural mode of vibration exhibiting a different vibrational displacement distribution and therefore driving particles to new predetermined locations, 59 namely the closest nodes of that particular mode. This is demonstrated using two modes of a

30 mm piezoceramic disc PIPM shown in fig. 2.11 below,

Figure 2.11 Schematic of multimodal particle manipulation. (a) normalized resultant magnitude of force, experienced by a particle, plotted along the diameter of PIPM in the x- direction for natural Modes A and B, (b) & (c) represent the corresponding contour plots of modes A and B.

As can be seen in figure 2.11, the two modes, A and B, observed at 56 and 65 kHz respectively have different distributions of surface forces. Mode A has the first nodal circle at a distance of ~ 5mm around the center, as can be seen from fig. 2.11 (b), whereas Mode B has a node at the center of the PIPM, fig 2.11 (c). Therefore if a particle is placed at the edge of the manipulator and it is activated at 65 kHz which corresponds to Mode B, the particle will be driven to the center of the PIPM. The frequency can now be switched to correspond to

Mode A, i.e. 56 kHz, which will lead to the particle being driven from the center to a location on the nodal circle around the circle. In this way, using multiple modes, single or multiple particles can be manipulated reproducibly to predetermined locations. 60

Further flexibility in its ability to generate the necessary forces is obtained by either increasing the excitation amplitude of a single mode of vibration or by switching to a mode that exists at a higher frequency of vibration.

The minimum size of particles that can be manipulated at a particular mode is dependent on the ability of the PIPM to generate sufficient force to overcome interfacial adhesion forces.

The adhesion forces are dependent on particle and surface geometric and material properties as well as environmental conditions, whereas the PIPM surface forces experienced by the

particles are dependent on the frequency and amplitude of excitation.

We can therefore write a generalized relation defining the current manipulation technique:

Fmanipulation = f(umanipulator, fR, Fadhesion, mp) 2.32

Where, umanipulator is the resultant displacement of the PIPM, given by,

umanipulator = f(c, e, d, g, h, k,, V, fR, mode) 2.33

where, c is the material elastic constant, e, d, g, h are the piezoelectric constants,  is the

dielectric constant, V is the applied voltage, fR is the frequency of vibration and mode

represents the mode of vibration of the PIPM.

Thus as is seen from expression 2.32, the manipulation force is a function of the particle-

PIPM adhesion force, mass of the particle, frequency of vibration and the PIPM

displacement,. This manipulation technique is based on generating interfacial manipulation

forces due to the vibration of the PIPM. The displacement of the harmonically vibrating

PIPM is therefore defined in expression 2.33, as a function of the piezoelectric properties,

applied voltage, frequency of vibration and the mode(s) of vibration that is used for particle

manipulation.

2.3.1. Projectile motion of the particle

To better understand the motion of a particle we first consider conventional Galilean projectile motion. The trajectories of the particle, while it is under free fall, and after it is projected from the vibrating surface, are both governed by equations of projectile motion

which are described next.

We now discuss a few basic concepts of projectile motion of a particle in two dimensional

space:

Figure 2.12 Vector representation of a particle position two dimensional space

Consider fig 2.12, representing the path of a particle moving in 2-D space. The position of a  particle in the xz-plane is given by position vector, rp . The particle displacement is defined

as the change in its position,    rp  rpi  rpf 2.34

For a particle moving in xz-plane, the position and velocity vectors can be written as:

r  x iˆ  z ˆj p p p 2.35   ˆ  ˆ v p  v pi  v p j 2.36

Where iˆ and ˆj are unit vectors and vp is the instantaneous velocity of the particle. 62

Considering the condition of constant acceleration due to gravity, the kinematical equations are written as,

1 r  r  v t  at 2 2.37 pf pi pi 2

   vpf  vpi  at 2.38

We can now describe the motion of the particle in 2-D space

Assumptions:

Particle experiences a constant acceleration (acceleration due to gravity) and drag force due to air friction is assumed to be negligible  Consider a particle projected with an initial velocity, vpi , at an angle  with respect to the x-

axis.

Figure 2.13 Particle projectile motion in 2-D space

The velocity at each point in the particles trajectory can be resolved into its components, i.e.

  x-component: v pix  v pi cos and the z-component: v piz  vpi sin . Here we consider a

particle under free fall, wherein the horizontal and vertical motions can and are addressed

independent of each other.

The particle experiences constant velocity in the horizontal, x, direction and constant

acceleration in the vertical, z, direction 63

Under the current assumptions, the acceleration is zero in the x-direction, thus the horizontal

velocity, v pix , is constant throughout the particles trajectory, as is seen in fig 2.13.

The position of the particle is therefore given by eq 2.37, which is the vector sum of three

positions, the initial position, the position resulting from initial velocity and the position

resulting from acceleration, g , whose sign is set as negative in the downward direction

(towards the earth).

Thus using equations 2.37 and 2.38 the particle’s position and velocity in 2-D space is fully

defined. Furthermore, for symmetric trajectories, the maximum height, h, that the particle

reaches, the time for it to return to the same starting height, called the Time of Flight (TOF),

and the horizontal distance reached, R1, can be easily determined. Using the fact that the

particle’s velocity is zero at the maximum height of its trajectory, the TOF, h, and R1 are

calculated as,

2 2 v sin v h  pi i  piz 2.39 2g 2g

2v sin 2v TOF  pi i  piz 2.40 g g

Since the horizontal acceleration is 0, R1 is given by

2v sin v cos 2v v R  TOF.v cos  pi i pi  pix piz 2.41 1 pi i g g

Naturally, it can be seen from eq.’s 2.41-43, the initial normal component of velocity with

which the particle is projected determines ‘h’ and thus the ‘TOF', whereas the horizontal

distance, R1, that it travels is dependent on both the initial normal and tangential velocities of projection.

Assuming that the x-axis, in fig. 2.13, represents a rigid surface, it can be seen that the particle would collide with the surface at point D. This could be a fixed surface or a vibrating 64 surface, as is the current case. The velocity of the particle after the collision is determined by impulse momentum equations described in section 2.2.2.1

2.4. Theoretical limits of particle manipulation

The upper and lower limits of particle manipulation using the PIPM are dependent on several factors, namely, the mass of the PIPM, the mass of the particle, the frequency and amplitude of excitation and environmental conditions.

Upper limit of PIPM based particle manipulation:

The upper limit of this manipulation technique is momentum transfer limited. This technique is based on the assumption that the particle-PIPM collision does not significantly impact the vibration of the PIPM, i.e. the velocity of the vibrating PIPM before and after collision does not differ by more than 0.1%. However, when the mass of the particle increases, by either an increase in size or property or both, a point is reached where the above assumption is no longer valid. At this point, the momentum transferred to the PIPM would cause a change in the frequency of vibration of the PIPM which in turn would lead to the prevention of particle motion.

Consider a 30 x 1mm disc PIPM (Pz27) excited at, 65 KHz with a maximum displacement of

1nm, used to manipulate stainless steel particles (density 8000Kg/m3) in air/vacuum. The

change in velocity of the PIPM, vpmf, after collision with the particle is plotted as a function

of particle radius,

Figure 2.14 Percentage change in the final velocity of the manipulator after collision with stainless steel particles of increasing radii.

As can be seen in fig. 2.14 the upper limit of this technique corresponds to a stainless steel particle having radius of approximately 600m. Above this particle size the collision of the

particle with the PIPM would potentially lead to a change in the frequency of vibration and

thus preventing particle motion.

Lower limit of PIPM based particle manipulation:

The lower limit of this technique of particle manipulation is defined by the ability of the manipulator to overcome adhesion force binding the particle to the manipulator surface. As

has been discussed, the condition for particle manipulation is Fmanipulation_z>Fadhesion, which can

then be used to determine the minimum particle size that can be manipulated given by,

1 3Fadhesion Rmin  3 . Therefore, for the same particle type, the minimum particle size, Rmin,  4u z

that can be manipulated is directly related to the cube root of Fadhesion. Furthermore, it can be

seen that particles of smaller radii can be manipulated by increasing the frequency and/or

amplitude of vibration. Consider a PIPM used to manipulate stainless steel particles in air. 66

Calculations are performed for 3 different frequencies of vibration, 65, 200 and 500 KHz, with a vibration amplitude of 1 nm (p-p). Rmin, and are plotted as a function of adhesion force,

Fadhesion,

As can be seen in fig. 2.15 , for each frequency of vibration, the minimum particle size

decreases with a decrease in the adhesion force. It can also be observed that the minimum

particle size can also be reduced by operating at a higher vibration frequency. i.e. for an

adhesion force of 0.1nN, the minimum particle size that can be manipulated is approximately

1 m at 65KHz and can be reduced to 300 nm by operating the PIPM at 500KHz.

2.5. Summary

In order to manipulate particles in the micrometer and sub-micrometer scale, it is necessary to develop an understanding of the dominant adhesion forces at this scale. Here, a study of the interfacial adhesion forces namely, van der Waals, electrostatic, capillary, gravitational and 67 frictional forces binding micro-particles to surfaces was conducted. This study determined the minimum requirements of the forces that the PIPM will need to generate to manipulate the particles. The interfacial manipulation forces generated by the PIPM when it is excited at its natural modes of vibration were then studied. This manipulation force was subcategorized as inertial and impulsive manipulation forces that act synergistically to manipulate particles.

The mechanism by which these forces enable the manipulation of particles, namely via Single and Multiple mode manipulation, as well as the various conditions for manipulation were then discussed. A study of the theoretical limits of particle manipulation was then conducted, demonstrating that the upper limit is momentum limited whereas the lower limit is dependent on the adhesion forces. In this way this chapter provided some of the basic knowledge necessary to better understand this manipulation technique. 68

3. FINITE ELEMENT METHOD (FEM) ANALYSIS OF THE PIPM

3.1. Theory of finite element analysis of a piezoelectric disc

The previous chapter discussed the various forces binding particles to a surface, in this project these particles are manipulated on the surface of a piezoelectric manipulator (PIPM).

In order to overcome the normal adhesion forces and the tangential frictional forces, the

PIPM needs to generate a resultant force at its surface to overcome the binding intermolecular forces. Furthermore, to facilitate the motion of particles from one location to the next, it needs to generate a non uniform distribution of these resultant surface forces. In this project, this non-uniform surface force map is generated by a piezoelectric ceramic actuator excited at a series of piezoelectrically excited natural frequencies. Each natural frequency exhibits a

unique surface force map having the potential to manipulate particles, the capability of which

is determined by the amplitude and frequency of vibration and the environment in which it is

operated.

The vibration characteristics of piezoelectric structures can be determined from the three-

dimensional equations of linear elasticity, Maxwell equations, and the piezoelectric

constitutive equations and the appropriate boundary conditions [68, 102-103]. For

transversely isotropic plates having simple geometries, the governing equations of motion of

freely vibrating plates have been widely studied and solved to extract the resonance

frequencies and corresponding shape functions [98-99, 104]. However, for anisotropic

materials this analysis becomes increasingly complex, owing to which numerical techniques

such as Finite Element Method (FEM) have been employed to study three dimensional

vibrational characteristics of piezoelectric materials [68].

Consider a disc shaped PIPM of thickness, hm, and radius, rm, shown in figure 3.1, 69

Figure 3.1 Schematic of a disc PIPM of thickness h that is excited by applying an electric potential  across its top and bottom surfaces

The constitutive equations of linear piezoelectricity in matrix form is given by,

T  c E S  e E 3.1(a) & (b) D  e S   S E

Where { } denotes a vector, [ ] denotes a matrix and superscript T denotes the transpose of a vector or matrix. {S} and {T} are the mechanical strain and stress vectors; {E} and {D} are the electric field and charge density vectors respectively; [cE], [S] and [e] are the elastic

constant matrix, dielectric constant matrix at constant strain and piezoelectric constant matrix,

respectively.

The stress and strain terms in equation 3.1 are given by,

 2 f   T 3.2 t 2

S  B f f 3.3

Where {f} = {u v w}T is the mechanical displacement vector along the x, y and z axes, and,

   x 0 0 0   z   y  B   0   y 0   z 0   x  The electrical terms in equation 3.1 are f    0 0   z   y   x 0 

related by Gauss’s law, with the assumption that the piezoelectric material is an insulating

material and that no flow of charge occurs inside the transducer. Thus, 70

 D  0 3.4

The electric field is related to the electric potential  by,

E  B  3.5

 Where B is given by, B   x  y  z

A linear piezoelectric material is completely modeled by eqs 3.1-3.5 by applying appropriate

boundary conditions.

T  Let the generalized stress vector be,  G    , and the generalized strain vector be, D

 S   G    . Therefore the generalized stress-strain relationship can be written as  E

c E e  C . Where [C] is the generalized elasticity matrix given by,  G   G C   T S  e   

and the strain-displacement relationship is given by,  G  BG f G 

  Where {fG} is the generalized displacement vector, such that f G  f  u v w  , 

B f 0  is the electric potential. The generalized B matrix is given by, BG     0 B 

In setting up a finite element analysis, the generalized displacement at any point within the

structure is expressed in approximate terms using displacement shape function [N] and vector

{} containing a finite number of known nodal displacements. Therefore, f G  N . 71

For each node, i, of a piezoelectric material there are 4 degrees of freedom, with the fourth

  being the electric potential, therefore,  i  f i i   ui vi wi i 

Equation of motion:

The governing dynamic equation of piezoelectric materials without considering damping is given by,

M ff 0 f K ff K f  f  F            3.6  0 0 K f K   Q

Where, {f}={u v w}T is the mechanical displacement vector, {} is the electrical potential

vector, [Mff] is the mass matrix, [Kff], [Kf] and [K] are mechanical, piezoelectric and dielectric stiffness matrices, respectively, and {F} and {Q} are the mechanical force and electrical charge vectors respectively.

Modal analysis:

Boundary conditions:

A piezoelectric disc is excited via the application of a voltage across its top and bottom electrodes. Therefore electrical boundary conditions can be set as follows:

The bottom electrode is set as ground, therefore, B = 0. The vector of the electrical potential,

{B}, corresponding to the finite element nodes on the bottom electrode is,

B  0 3.7

While the top surface has a potential of , the nodal electrical potential vector is given by,

P  I P  3.8

Where {IP} is a unit vector in which the components correspond to the positions of the finite element nodes on the top surface. The total electrical charge on the bottom grounded surface,

QB, is equal and opposite to that on the top surface, QT-, so QB  QT

 and QT  I P QP

Where, {QP} is the nodal charge on the top electrode surface. Here the mechanical boundary conditions are assumed to be stress free conditions, i.e. . T  0 h ,T  T  0 2 2 2 3 j z m 1 j 2 j x  y r 2 m

Equation 3.6 now becomes,

  M ff 0 0 f  K ff K f K f  f   F     i P          0 0 0    K  K K    0 3.9  i  fi ii iP  i              T   0 0 0P  K f K K P  QP      P iP P    

where the subscript i denotes the components corresponding to the electrical potential

degrees of freedom of the non-electroded nodes, and subscript p denotes the components

corresponding to the electrical potential degrees of freedom of the ungrounded electroded

nodes. Similarly, {i} is the electrical potential vector corresponding to the non-electroded

nodes, and {P} is the electrical potential vector corresponding to the nodes on the

ungrounded electrode surface.

Natural frequency extraction:

In order to determine all the natural modes of vibration of the piezoelectric disc, F is set to 0, thus the eigenproblem for free vibrations is given by,

M 0 f  K K  f  ff   ff fi           0 3.10  0 0  K  K    i   fi ii  i 

Equation 3.10 is a homogeneous equation, writing eq. 3.10 in terms of generalize mass [M] and stiffness [K],

 2 M  K   0 3.11 t 2

Assuming a solution of the form,  (t)  e jt

Where  is a time independent amplitude vector of order N consisting of both mechanical

 f    jt and electrical terms such that,     f e and  is the angular frequency of vibration. i   i 

Thus the generalized eigenproblem is then given by,

K   2 M   0 3.12

2 The solution of equation 3.12 gives N pairs of eigenvalues and eigenvectors, (1 ,{}1),

2 2 (2 ,{}2),….. (N ,{}N), where N is the total number of degrees of freedom.

Here a Lanczos method has been applied to convert the general eigenproblem, eq. 3.12 to a

standard form and is solved using the Lanczos solver available in ABAQUS. The

eigenvectors obtained have the following properties:

T T  M   I &  K  2  3.13(a) & (b)

Where, [I] is a diagonal unit matrix of the order n, (n is the number of mechanical degrees of freedom), and  2  is the diagonal matrix of order N which is the total number of degrees of

freedom of the system.

Thus all the possible natural frequencies within the selected frequency range can be extracted.

These modes include radial modes (R), edge modes (E), thickness shear modes (TS),

thickness extensional modes (TE) and high frequency radial modes (A). Each of these mode

types exhibit a unique surface deformation map, thus leading to the generation of a complex

surface force map each which can be harnessed for the manipulation of particles.

This document is focused on the fundamental radial mode of vibration of the piezoelectric

ceramic disc, however the analysis of the surface forces and their application to particle

manipulation is intended to be applicable to other, E, TS, TE and A modes.

Steady State Analysis

Next, the linear response of a structure subjected to continuous harmonic excitation is conducted by performing a steady state linear dynamic analysis. ABAQUS offers the option of using the modes extracted in the previous, natural frequency step, to calculate the steady state solution as a function of the frequency of applied excitation, or to use the “direct” steady state linear dynamic analysis where in the equations of steady harmonic motion of the system are solved directly without using the modes extracted in a natural frequency step.

With F = 0, no damping, but with a voltage excitation, equation 3.9 can be written as,

K K  K   M ff 0 f  ff fi  f   f p p             3.14 0 0 K K   K   p  i   fi ii  i   i p 

Which can be written in the generalized form, without damping

2 M  K   [R] 3.15 t 2

The eigenvectors obtained in the natural frequency extraction step are used to express the generalized nodal displacement vector,  (t) , in terms of the generalized modal

displacement/participation vector, z(t),

 (t)   z(t) 3.16

Using equation 3.16 in eq 3.15 and pre multiplying 3.15 by T , we get the forced response,

T T T  2  M  z   K z  [R 3.17

Using the properties of the eigenvectors, equation 3.13, in equation 3.17, we get,

T 2  2 z   [R] 3.18

Using equation 3.18, the modal displacement and therefore the generalized nodal displacements can be determined.

The radial mode of vibration is extracted using a direct steady state linear dynamic analysis, where in the disc is subjected to a continuous harmonic excitation. In this step Rayleigh damping is introduced to obtain quantitatively accurate surface displacements. Rayleigh damping is defined by a damping matrix formed as a linear combination of the mass and the stiffness matrices [Abaqus Manual].

It is given by,

G   R M   R K 3.19

Where R is the mass proportional damping and R is the stiffness proportional damping.

For a given mode i the fraction of critical damping, i , can be expressed in terms of the

damping factors R and R as:

 R  Ri  i   3.20 2i 2

where i is the natural frequency at this mode. This equation implies that, generally speaking, the mass proportional Rayleigh damping, R, damps the lower frequencies and the stiffness

proportional Rayleigh damping, R, damps the higher frequencies.

In this study the stiffness proportional damping R is employed to introduce damping

proportional to the strain rate and is therefore proportional to the elastic material stiffness.

Radial mode of vibration:

An example of a radial resonance mode of vibration of a piezoceramic disc, diameter 30mm, thickness 1mm, poled in the z-direction, shown in figure 3.2.

(a)

(b)

Not to scale Figure 3.2 Radial mode of vibration of a disc shaped PIPM sectioned along the x-axis. (a) radially expanded and (b) radially contracted.

Fig. 3.2 represents the radial mode of vibration of a piezoceramic disc, sectioned along the x- axis to view the expansion and contraction of the disc along the diameter. As seen in fig.

3.2(a), the disc is fully expanded in the x-direction with a thickness contraction at the center in the z-direction. The opposite can be observed in fig. 3.2(b) wherein the disc is contracting radially, therefore leading to a thickness expansion in the center. The dotted lines represent the equilibrium position without harmonic excitation.

Finite element analysis of these vibrational modes provides an in depth understanding of the electrical and mechanical nature of each mode. Furthermore, with the selection of appropriate boundary conditions and mesh size, FEM can provide an excellent approximation to experimentally relevant solutions [63, 68].

3.2. FEA results

This project involves the use of a piezoelectric manipulator, excited at various natural and piezoelectric modes of vibration, to generate a complex surface force map which is harnessed for the manipulation of micro-particles. To better understand this manipulation technique and predict the motion of the particles, a finite element analysis is conducted of the vibrating structure. The criteria for manipulating particles was discussed in chapter 2, section 2.3. Here

a few FEM results are analyzed and the necessary understanding is developed to enable

predictable particle manipulation along with some insight to the pros and cons of this

technique.

In order to perform this study, simple disc shaped piezoelectric structures, material Pz 27, are

selected, and their resonance volumetric radial modes are studied. These modes are radially

symmetric allowing for identical particle motion along any diameter of the disc PIPM.

Four discs, having thickness 1mm and diameters ranging from 10 – 30 mm are studied.

The resonance frequencies of the radial modes can be determined from equation 3.21 :

c f  3.21 R 2D

Where fR is the resonance frequency, D is the disc diameter, c is the longitudinal speed of

Y E (1 E ) sound in the ceramic given by, c  11 12 , Y E , is the Young’s modulus, E E 11 (1 12 )(1 2 12 )

E 12 , is the Poisson’s ratio, and  is the density of the piezo-ceramic [105]. Thus using

equation 3.21, the radial mode resonance frequency vs disc diameter is shown in fig. 3.3

4.5E+05 4.0E+05 -9.9927E-01 3.5E+05 y = 1.9854E+06x 2 3.0E+05 R = 9.9903E-01 2.5E+05 (Hz)

R 2.0E+05 f 1.5E+05 1.0E+05 5.0E+04 0.0E+00 0 102030405060 Disc diameter (mm)

Figure 3.3 Radial mode resonance frequency of piezoelectric ceramic, Pz 27, as a function of disc diameter

From equation 3.21 and fig 3.3, it can be seen that the radial resonance frequency is inversely

proportional to the diameter of the disc. Therefore, to study the effect of increased frequency

on the force exerted on particles, the diameter of the disc is reduced from 30mm to 10 mm.

We now discuss the FEA of the radial mode of vibration for the four discs. This is followed a

section on the motion of the particle using the force generated by the discs studied here. 79

3.2.1. Radial mode of vibration

Using FEM analysis of the four discs, a large number of natural modes of vibration for each disc can be extracted. Each of these modes, when excited, has the potential to manipulate and pattern particles, wherein the size and type of particles being manipulated by each mode is contingent on the ability of the PIPM to overcome adhesion forces.

To extract the approximate solution of the wave equation describing vibration of the piezoelectric disc, FEM software, ABAQUS, was used. This is performed in a two-step process:

Step1: Natural Frequency Extraction using Lanczos eigensolver

Step 2: Steady State Direct Dynamic Analysis.

Step 1 performs an eigen value extraction, which is used to extract the natural frequencies of vibration and their corresponding mode shapes of the piezoceramic disc. Step 2 is used to calculate the steady-state dynamic linearized response of the PIPM to harmonic excitation at frequencies extracted in step 1. In this step an electrical load is applied to the surface of the piezoelectric thin disc owing to which piezoelectrically excited natural modes, within the specified frequency range, are extracted. Both electrical and mechanical parameters of the piezoelectrically excited modes can be extracted and analyzed.

To demonstrate particle manipulation radial modes of the four discs are extracted.

Care has been taken to ensure the mesh size chosen incorporates at least 5-7 nodes per wavelength of the mode beings simulated. The displacement contour maps of a radial mode of vibration exhibit are shown in figure 3.4

Figure 3.4 Contour plots of the magnitude of displacement and its x, y and z components.

Particles placed at any location on the PIPM surface experience a resultant force due to all three components synergistically. As can be seen from the y-component of velocity, uy, is

minimum is along the x-axis. Therefore, by studying the displacement along the x-axis, the

analysis can be reduced to two dimensional space, wherein only the x and z components are

analyzed.

Although the resonance frequency of the discs can be calculated, quantitatively accurate

amplitudes of vibration for the discs at their resonance frequencies can be obtained after

introducing a suitable damping factor in the simulations. Therefore the displacement of each

disc was studied as a function of excitation voltage and material damping.

The displacement is resolved into its x, y and z components and studied along the x-axis. The

x and z displacement, ux and uz, of the discs are normalized to the maximum displacement and shown in figure 3.5 and 3.6 respectively.

Figure 3.5 Normalized tangential displacement, ux, plotted along the x-axis

Figure 3.6 Normalized vertical displacement, uz, plotted along the x-axis

As seen in figures 3.5, each diameter disc exhibits the same tangential displacement distribution at their radial resonance modes of vibration. The vertical component of displacement was normalized to the maximum tangential displacement of each disc and therefore appears as in figure 3.6. For later reference, the normalized displacements are plotted from center to edge and curve fitted. A cubic curve fit gives a correlation coefficient

of ~1 for each case, the equations of the lines are shown in table 3.1 82

Table 3.1 Equations for the determination of normalized ux and uz at any spatial location x along the x-axis from center of the discs to the edge

Therefore using the equations in table 3.1, the normalized x and z displacements at any

location along the x-axis can be obtained.

As can be seen in figures 3.5 and 3.6, the tangential displacement is maximum at the edge,

ux_max, and minimum at the center of the disc whereas the vertical displacement has a maximum value at the center, uz_max, of the disc and minimum at the edge. However

quantitatively accurate displacement values is dependent on an accurate choice of damping

factor that needs to be introduced while performing the simulations.

The ratio of the magnitudes of the normal to tangential components at each location gives the

angle of resultant displacement and therefore the angle at which the resultant force is applied

to the particle.

Figure 3.7 The angle which the magnitude of resultant displacement makes with the horizontal axis plotted as a function of spatial location for the radial mode of a 30mm piezoceramic disc.

Figure 3.7 shows that the angle increases from ~0.1 degrees at the edge (15 mm) to ~90 degrees at the center (0 mm) of the disc PIPM, and is shown for a 30 mm disc, the same can

be shown for discs of other diameters. At an angle of 45 degrees, the normal and tangential

components have the same magnitude at ~1.5mm from the center of the PIPM. Thus the

angle of force that would be experienced by the particle is almost parallel to the PIPM surface

at the edge (opposing frictional force) and increases towards the center where the particle

experiences a normal force (opposing adhesion force). It should be noted that each location

on the PIPM surface is harmonically vibrating at the frequency of excitation. Due to the

simultaneous radial expansion and thickness contraction or vice versa, the resultant

displacement at each point on the PIPM surface is not purely tangential or normal but traces

an ellipse while going through 360 degrees of each cycle of vibration. This is shown in

below 84

Figure 3.8 Distribution of the resultant harmonic displacement at 4 locations on the PIPM

surface, namely, A, B, C, & D, (a) A schematic showing the locations of the 4 ellipse’s

plotted in in fig. 3.8 (b).

As seen in figure 3.8, each location on the PIPM is vibrating harmonically; therefore the resultant displacement at each location traces an ellipse whose major axis makes an increasing angle with the x-axis you move from center to the edge of the PIPM (i.e. from location AD, fig 3.8(a)).

The ellipse traced at the four locations A, B, C, & D are re-plotted in figure 3.8(b). Each plot

is divided into 4 quadrants, I, II, III, & IV. With respect to the normal component of

displacement, quadrants I and II represent the thickness expansion of the PIPM whereas

quadrants III and IV represent thickness contraction, Similarly, with respect to the tangential

component of displacement, quadrants I and IV are representative of the radial expansion, 85 whereas quadrants II and III represent the radial contraction of the PIPM. Thus by observing the any point on the ellipse both the direction and magnitude of resultant displacement at any location on the PIPM surface can be extracted. Figure 3.8 represents the surface resultant displacement distribution of a typical radial mode of vibration. The distribution of the ellipse’s in the 4 quadrants is unique for each mode of vibration.

At location A, shown in fig. 3.8(b), the resultant displacement mainly lies in quadrants II and

IV. In quadrant II, the resultant displacement is. At this location, in quad. II, it can be seen that the magnitude of the tangential component can be an order of magnitude larger that the normal component leading to the resultant displacement being directed towards the center of the PIPM. Resultant displacement lying in quadrant I is directed out of the PIPM and will only impact particle manipulation if the normal component is larger than the critical displacement, as discussed in section 2.2.1.1, and if the particle collides with the PIPM during this phase of its vibration cycle. Similarly during quadrants III and IV, the resultant displacement is directed into the PIPM, owing to which a particle colliding with the manipulator in this phase would adhere to it or be projected depending on the ratio impulsive to adhesion forces. On the other hand a particle initially in contact with the PIPM could be detached from the surface in all 4 quadrants if the criterion for critical displacement is satisfied.

It can be seen that the probability of the resultant displacement pointing towards the center is dependent on the width of the minor axis of the ellipse (i.e. increases if the width is reduced) and this width decreases as we move from location A D.

Going through locations B-D, the only change is in the ratio of the normal to tangent

components leading to the change in the angle of projection of the particle. It is seen that at

location D, the resultant displacement is directed along the z-axis thus representing only

thickness extension and contraction, and therefore particles would be trapped in this location

as they do not experience any significant tangential force driving them to another location. 86

Thus by observing the normal and tangential components of displacements simultaneously the direction of particle manipulation can be predicted. This same procedure can be extended to other modes of vibration which will exhibit different ratios of the two components of displacement.

In order to quantify the manipulation force exerted on a particle, it is necessary to determine

the magnitude of displacement, velocity and acceleration at each location on its surface.

Values for these parameters obtained from FEM analysis will only be accurate if the

appropriate material damping is introduced in the simulations. Since experimentally relevant

stiffness proportional damping is not known, we now perform a brief study of the particle

displacement, ux_max and uz_max, as a function of damping factor:

3.2.1.1. Study of maximum normal and tangential displacements as a function of

damping factor.

The radial mode of vibration was simulated for a peak voltage of 1V and damping ranging from 0.1 – 5% of critical damping and plotted as a function of ux_max and uz_max, as shown in

figures 3.9 and 3.10.

Figure 3.9 Maximum normal displacements plotted as a function of of stiffness proportional damping factor, , represented as % damping for 4 PIPM’s excited with a peak voltage of 1V

Figure 3.10 Maximum tangential displacements plotted as a function of of stiffness proportional damping factor, , represented as % damping for 4 PIPM’s excited with a peak voltage of 1V

As can be seen in fig. 3.9 the uz_max for all four discs can be fit by a single power curve.

Equations representing individual fits for each disc are shown in the figure, indicating that the

variation in the power curve fit is small and can be approximated by a single power curve-fit.

However, for each of the 4 discs, the maximum tangential displacement, ux_max, needs to be

fitted with a unique power curve, equations of which are shown in figure 3.10.

Since the effect of damping on the maximum normal displacement, uz_max, is independent of

disc size, measuring uz_max, which is at the center of the disc for a radial mode, will be used to

determine the damping for the discs. 88

3.2.1.2. Study of maximum normal and tangential displacements as a function of

excitation voltage

Now a brief study of the maximum normal and tangential displacements as a function of excitation voltage is conducted to extract the displacement sensitivities.

Figure 3.11 Maximum normal displacements plotted as a function of peak excitation voltage, extracted for a damping factor of 5%.

Figure 3.11 represents the maximum normal component of displacement, uz_max, plotted as a

function of excitation voltages ranging from 0.1 - 4Vpeak. The displacements shown are

extracted from simulations wherein 5% of critical damping was used. As is seen, for each

excitation voltage, uz_max is almost constant with a variation of <1%, indicating that the

normal displacement at the center of the disc is independent of disc size and therefore

frequency.

This can be seen by observing the displacement sensitivity at the center of the disc:

Figure 3.12 Plot of uz_max vs peak voltage for all four disc PIPM’s, indicating that they all have the same displacement sensitivity at the center of the disc.

As seen in figure 3.12, the increase of the displacement at the center of the disc, namely, uz_max, with an increase in peak excitation voltage show very little variation from disc to disc,

i.e. each disc exhibits approximately the same z-displacement sensitivity at the center. The

equation in fig. 3.12 represents the equation of a line drawn through the average of the

displacements of each disc for each excitation voltage. The inverse of the slope represents the

z-displacement sensitivity, wherein the deviation between displacements of the discs for a

particular voltage did not exceed 0.75% of the average. Please note, since the actual damping

factor is not yet extracted, here we perform this study with a randomly selected damping of

5%. The same procedure was conducted for a number of damping factors with the

displacement sensitivity varying with damping factor but always constant as a function of

disc diameter.

Since the force is proportional to the product of the displacement and the square of the

frequency, as we go from a larger diameter disc, 30 mm, to a smaller diameter disc, 10 mm,

i.e from a lower to a higher resonance frequency, for the same excitation voltage, the

difference in the surface force generated is directly proportional to the difference in the 90 square of the frequencies. Therefore, the surface force generated at the center of each disc for the radial mode can be normalized to the frequency.

The maximum tangential displacement, ux_max,, on the other hand, varies from disc to disc for a particular excitation voltage. This can be seen in figure 3.13

Figure 3.13 Plot of maximum tangential displacement, ux_max, vs peak excitation voltage, Inverse of the slope gives the displacement sensitivity at the edge of each disc.

Although each disc exhibits unique ux_max sensitivity, the sensitivity varies linearly with disc

diameter and can be used to estimate the maximum displacement for a particular voltage and

diameter once the damping is determined. 91

Figure 3.14 Slope of ux_max and uz_max vs peak voltage for 5% damping

As can be seen in fig. 3.14, the maximum tangential displacement sensitivity varies linearly with the disc diameter, whereas the maximum normal displacement sensitivity is

approximately constant for discs ranging from 10 – 30mm. Therefore this property of the

normal component of displacement at the center of the disc will be used to predict the

displacement of discs of different diameters; furthermore, by substituting an experimentally

extracted displacement for a particular excitation voltage into the plot in fig 3.9, the

approximate damping factor can be determined. Using this damping constant the maximum

tangential displacement for each of the discs can be determined following which

experimentally relevant PIPM displacements velocities and accelerations can be obtained.

Due to the dynamic nature of the PIPM exhibiting a complex distribution of surface

accelerations, a force is exerted on a particle placed on the PIPM. The means by which these

surface forces can be used for 1-D and 2-D motion of micro-particles is now addressed. 92

3.3. Study of particle motion in 1 and 2-D space

The Projectile motion of the particle and the equations governing the particles trajectory have

been discussed in chapter 2, section 2.3.1. Also, the collision of the particle with the vibrating

PIPM surface leads to the transfer of momentum described in section 2.3.1.1. The collision of

the particle with the surface could be described as an elastic collision, described next or an

inelastic collision, discussed in the following section. Here, the particle projected in 1-D

space is described, followed by a discussion of its motion in 2-D space.

3.3.1. Particle moving in 1-D space

One-Dimension particle motion with elastic collisions

Assumptions: mm >> mp, surface has uniform thickness vibration, surfaces are atomically

smooth, single point contact, rigid interacting bodies, maximum peak displacement of the

surface is 1nm, Fmanipulation_z>Fadhesion, momentum is conserved, energy is conserved. i.e. no

losses due to intermolecular binding or deformation.

For the following plots the particle and surface properties are as follows:

Particle material: Stainless steel

Particle radius: 25m

Particle mass: 5.24x10-7g

Planar surface material: Silver coated Pz 27

Planar surface mass: 5.4g

Vibration frequency: 65 kHz

The velocity of the surface is described by equation 2.17(b), v  u .sin(.t  ) , i i0

where i = z for one dimensional particle motion, and the transfer of momentum in the z-

direction is given by eq 2.25. 93

In this case, the surface exhibits uniform normal vibration (in z-direction). A particle thus placed on the vibrating surface of would experience a free-fall, region A in fig.3.15 wherein the surface exhibits a velocity rate change high enough to distance itself from the particle, i.e., Fmanipulation_z>Fadhesion. The particle then undergoes free-falls for a short period of time

following which it collides with the vibrating surface at time t1.

Figure 3.15 Particle projected vertically (1-D) from a surface exhibiting thickness vibrations.

Collisions at t1 and t2 are fully elastic.

The particle free-fall as well as the projectile motion are described by equations 2.35 and

2.38, while the maximum height, h, the time of flight and the TOF, are determined using equations 2.39 and 2.40. The collision of the particle with the manipulator surface leads to a transfer of momentum to the particle whose final velocity is calculated using eq. 2.25.

The first collision takes place at time t1 after the particle under went a free-fall for 0.123s.

The maximum height, h, is calculated from the point of collision to the point where the particle velocity reduces to 0, for the current case it determined to be 42nm. The TOF is the time from the first collision at t1 to the second at time t2, and is found to be 0.184 ms.

Of course h and TOF are dependent on the final velocity of the particle after collision, which in turn is dependent on the initial velocities of both the particle and the surface. Owing to the 94 surface is vibration of 65 kHz, the initial surface velocity at each particle-surface collision, plays a vital role in determining the final velocity of the particle. The particle’s trajectory for a series of collisions is demonstrated in figure 3.16.

Figure 3.16 Path of a particle projected vertically after successive elastic collisions with the vibrating surface.

    The final velocities, vpf 1 , v pf 2 , v pf 3 , and , v pf 4 , depend on the initial velocities of the

particle and the surface at each collision point and are calculated using eq. 2.25. As seen in

fig. 3.16, the particle is projected and reaches varying heights with each projection; this is due

to the particle impacting the surface at different phases of the surface vibration cycle, thus

leading to different initial surface velocities as shown in table 3.2.

Table 3.2 Particle and surface velocities before and after elastic collision leading to a projected height h and a time of flight TOF

As seen in fig. 3.16 and table 3.2, the TOF varies from 0.186-0.35 ms while h varies from 42-

154nm for the first 4 collisions. It is therefore important to accurately determine the initial surface and particle velocities at the point of collision.

Since the velocity of the particle is a function of the velocity of the surface vibration, increasing the peak displacement would increase the momentum transferred to the particle, this is demonstrated in fig. 3.17 and table 3.3.

Figure 3.17 shows that increasing the amplitude of surface vibration from 1nm to 5 nm and then to 10 nm, leads to an increase in momentum transferred and there fore projecting the particle to an increasing height. Initial and final velocities as well as h and TOF are shown in table 3.3.

Table 3.3 Comparison table of the velocities before and after collision with a surface as the amplitude of vibration is increased

Thus, as seen in table 3.3, increasing the peak amplitude of vibration of the surface from 1nm

to 5nm, leads to a 2.5 fold increase in the final velocity of the particle at the first collision and

thus leading to a 7.5 fold increase in h. This shows that an increase in the peak displacement leads to an increase in force exerted on the particle thus projecting it higher. For 2-D motion of particles, it would lead to the particles traveling a particular distance in a shorter period of time.

One-Dimension particle motion with inelastic collisions

The current project involves the manipulation of rigid micro-particles on a rigid surface. As particle size is reduced below 100m, macro-scale forces like gravitational force play a less

dominant role than intermolecular forces such as van der Waals, capillary and electrostatic

forces. Therefore at this scale the collision of particles with a surface leads to the loss of

energy in the detachment of the particle from the surface [83, 84, 89]. These interactions need

to be modeled as inelastic collisions governed by eq. 2.28. 97

Figure 3.18 represents successive inelastic collisions of a 25m (radius) particle with a

surface vibrating at 65 kHz with peak amplitude of 1nm. The system was modeled using a

coefficient of restitution of 0.9 to demonstrate the effect of a loss of energy during the

collision.

Figure 3.18 1-D particle projection after inelastic collisions with the vibrating surface. Coefficient of restitution, e = 0.9

The values of the initial surface and particle velocities, as well as the final velocities of the particle are shown in table 3.4.

Table 3.4 Initial and final velocities for 4 successive inelastic collisions with a vibrating surface

A comparison of tables 3.3 and 3.4 can be visualized by observing figure 3.19.

As is seen in figure 3.19, e = 0.9, represents a decrease in the final velocities at the point of collision as compared to the case for elastic collisions (blue curve). Owing to this loss, the velocities with which the particle is projected and hence the maximum height and TOF are both reduced, the values of which can be seen in table 3.4.

We now discuss the effect of two dimensional particle motion.

3.3.2. Particle moving in 2-D space

The previous section described the motion of a micro-particle in 1-D space, with only vertical

motion, i.e. vertical bouncing. A vibrating surface whose frequency and amplitude of vibration satisfying the condition given in equation 2.17, has the potential to distance itself

from the particle due to inertial force, eq. 2.16, the particle then collides with the vibrating

surface and experiences an impulsive force, eq. 2.17, leading to the projection of the particle.

This applies to 1 and 2-dimensional particle motion.

A vibrating surface having uniform thickness vibration, such that, Fmanipulation  Fmanipulation _ z ,

leads to a vertical projection of particles, discussed in the previous section. However, in order

to translate the particle from one location to the next, a horizontal velocity has to be imparted 99 to the particle. In this project this feature is achieved by the surface exhibiting 3-dimensional deformation. Furthermore, it exhibits a predictable non-uniform deformation and therefore, a particle placed at any location on the PIPM surface experiences a resultant force driving it

from regions of high to low force. Here we demonstrate the translation of a particle using a

piezoceramic manipulator excited at its radial mode of vibration. At this mode, the tangential

and normal components of displacement were plotted along the x-axis as shown in figs. 3.5 &

3.6. Since the y-component of displacement is an order of magnitude smaller than the x and

z-components, the 3-dimensional motion of the disc is approximated to a 2-D vibration where only the x and z components are considered. As discussed in the previous section, the direction of particle projection is dependent on the direction of resultant manipulation force exerted on it. The magnitude and direction of this force depends on the particle and PIPM velocities at the point of impact and will be determined using momentum transfer equations.

The motion of the particle is influenced by several parameters such as, the geometries of the interacting bodies, their surface condition, roughness, and material properties, as well as

environmental conditions. The assumptions made to enable a reasonable study of the particle

motion are given in the previous section of 1-D motion.

The following discussion is based on the fundamental radial mode of vibration of a 30 x 1

mm piezoceramic disc that exhibits maximum surface displacement of 47 nm when harmonically excited at 65 kHz with a peak voltage of 1V. As discussed previously, the x and z surface displacements in fig 2.10(b) are normalized to the maximum displacement, thus re- scaling these curves by multiplying by the frequency and 47 nm gives the surface

displacement at every point along the x-axis. The normal and tangential surface

displacements are curve fitted and the equations of the fitted lines are used to determine the

magnitude of surface displacement at each location, eq’s in table 3.1. This is simply

converted to surface velocity by taking the first derivative of the displacement with respect to 100 time. The phase difference between the normal and tangential components, at each location, is also taken into consideration when calculating the transfer of momentum to the particle.

Two-Dimension particle motion with elastic and frictionless collisions, e =1, and  = 0

The first case involves lossless collision of the particle with the surface. The horizontal velocity is set to be equal to the horizontal velocity of the surface at the point of first collision. Since this case is also frictionless, the horizontal velocity is constant throughout the

particles path. Therefore, using equation 2.25 with e =1, and eq. 2. 31 with  = 0, the transfer

of momentum for elastic collisions can be calculated, while using equations 2.37 and 2.39,

the normal and tangential motions of the particles can be plotted. Figure 3.20 represents the

first 5 particle trajectories in 2-D space.

Figure 3.20 Particle motion in 2-D space with elastic collisions. No energy loss at the interface is considered, therefore coefficient of restitution, e = 1, and dynamic frictional constant,  = 0.

As mentioned in the 1-D case, the height the particle is projected and the time of flight depends only on the normal component of velocity. The encircled numbers represent the collisions of the particle with the vibrating surface. Collision 1 represents the first collision of the particle after experiencing free fall (assuming the inertial force is larger than the force of 101 adhesion binding the particle to the surface), while collision 2, 3, 4, 5, and 6 represent successive collisions with the PIPM as the particle is driven from the edge to the center of the disc.

The current case assumes no energy loss at the interface, however, in all real experimental conditions; there exists both, normal and tangential energy loss at point of impact [96] . The following cases discuss normal component loss via the introduction of the coefficient of restitution and tangential component losses via the introduction of friction.

Two-Dimension particle motion with inelastic and frictionless collisions, e =0.9, and  =

The case of normal losses, during the collision of a particle with the surface, is demonstrated by the introduction of a coefficient of restitution as was demonstrated for the 1-D motion of a particle. We now do the same for 2-D motion by calculating particle motion using e = 0.9

Figure 3.21 Particle motion in 2-D space with inelastic collisions. No energy loss in the x- direction is considered , therefore dynamic frictional constant,  = 0, however a loss in the z- direction is introduced, coefficient of restitution, e = 0.9.

102

Figure 3.21 shows an increase in the number of collisions to travel the same distance. It is difficult to compare the heights that the particle reaches as the energy loss at the point of collision leads to the particle impacting the vibrating surface at a different phase in its vibration cycle and thus leading to it being projected to a different height. Therefore, instead of comparing individual collisions, the time to travel the same distance for the same PIPM parameters (frequency and amplitude of excitation) is studied as a function of the coefficient of restitution, while setting the coefficient of friction, , to 0.

Figure 3.22 The time for a 25 m particle to travel from the edge (~15 mm) to the center of the PIPM (~2mm) is plotted for different coefficients of restitution, with the frictional loss set to zero.

Figure 3.22 (a) represents the time it takes a 25 m stainless steel particle to be manipulated

from edge of the PIPM (~14.5 mm) to a region near its center (~ 2 mm) as the coefficient of

restitution, e, is varied from e = 1 (elastic collision) to e = 0.5. Each point in each curve

corresponds to the time between successive collisions with the manipulator. The final

translation time is plotted as a function of coefficient of restitution in figure 3.22 (b). As can

be seen, the time for the particle to travel the same distance increases from ~0.3s to 1.2s as

the inelasticity of the collision is increased by decreasing e from 1 to 0.5. Furthermore as

expected, the number of collisions increased from 196 collisions for e = 1 to 3306 for e = 0.5.

103

Two-Dimension particle motion with elastic collisions involving friction, e =1, and  = 1 x 10-3

In the previous case, the particle had a constant horizontal velocity since the coefficient of friction was set to 0; however, due to frictional loss at the point of collision, the horizontal velocity after each collision is modified. This can be demonstrated by introducing a dynamic friction constant and calculating the final horizontal velocity using eq. 2.41 wherein  = 1 x

10-3 . The particle trajectory is now plotted in fig 3.23.

Figure 3.23 Particle motion in 2-D space with frictional losses. No energy loss in the z- direction is considered here, therefore coefficient of restitution, e = 1, however a loss in the x-

direction is introduced, dynamic frictional constant,  = 0.1.

As is seen in fig 3.23, introducing a frictional loss impacts horizontal distance the particle travels. It should be noted that the horizontal velocity of the particle after impact is now modified and therefore the particle-PIPM collision will take place at the different phase of the manipulator vibration cycle, which then leads to the particle being projected to a different height and horizontal distance. Furthermore, since the PIPM surface is harmonically vibrating, at certain frictional constant values it is possible for the particle to be projected to

the right or to the left as is shown by the red trajectory after collision 4, as seen in figure 3.23. 104

However, due to the gradient of surface force directed towards the center of the PIPM, a majority of the collisions project the particle to the center leading to its final destination being the node of the selected mode, which is the center for the radial mode.

Figure 3.24 The time that it takes for a 25 m particle to travel from the edge (~15 mm) to the center of the PIPM (~2mm) is plotted for different coefficients of friction, for elastic collisions.

Figure 3.24 (a) represents the time it takes a 25 m stainless steel particle to be manipulated

from edge of the PIPM (~14.5 mm) to a region near its center (~ 2 mm) as the coefficient of

friction, , is varied from  = 1 x 10-4 to  = 1 x 10-3 with the coefficient of restitution kept

constant at 1. Each point in each curve corresponds to the time between successive collisions

with the PIPM. The final translation time is plotted as a function of coefficient of friction in

figure 3.24 (b). As the frictional loss is increased, the particle translation time increases from

~1 s for  = 1 x 10-4 to ~2 s for  = 1 x 10-3.

In this way, the particle motion due to the vibration of the PIPM can be studied to provide a

better understanding of the mechanism of manipulation of this technique. The resultant

component of velocity of the vibrating manipulator leads to the projection of particles down

the gradient of surface velocity. This particle motion could encompass as many at 5000

collisions for a 30 mm disc, the number of collisions and the time to travel the specified

distance is dependent on several factors, such as, particle and PIPM properties, surface 105 conditions, manipulator surface velocity, the coefficient of restitution and frictional constant.

It should be noted that in the current model, the colliding bodies are assumed to be rigid with no deformation at the point of impact, however the model can be extended to include deformation of the particle (when used to predict the motion of deformable particles) which can be calculated by JKR or Hertz theories.

3.4. Summary:

Vibrational analysis of the piezoceramic manipulator was discussed via the use FEM software package ABAQUS. The harmonically varying resultant displacement for the radial mode was discussed at different locations on the PIPM surface explaining the basis of the mechanism behind this technique. The study can be extended to any other mode of vibration.

The displacement of the PIPM was studied as a function of excitation voltage and damping factor, for the radial mode of vibration of four discs ranging in diameter from 10 – 30 mm, having a thickness 1mm. This study set up the basis for the experimental determination of the displacement of the PIPM, and the manipulation forces that a particle on its surface would experience.

Following the numerical study, the 1-D motion and 2-D translation of particles was demonstrated, incorporating losses at the point of impact via the introduction of coefficient of restitution and friction. The displacement and velocity distribution obtained from the FEM of the discs was used to as inputs to determine the motion of the particle. In this way a theoretical study of the mechanism of particle manipulation was conducted. 106

4. EXPERIMENTAL METHODS

The previous chapters dealt with the necessary background and theoretical basis for the current acoustic manipulation technique. The next chapter deals with the experimental results of particle manipulation. Prior to the experimental results, the experimental set-up, materials and methods used are described here in this chapter.

4.1. AFM force and roughness measurements

The manipulation of micrometer and sub-micrometer sized particles necessitates an understanding of interfacial adhesion forces and surface topography. In this project, these studies are conducted using an Atomic Force Microscope. Furthermore, the AFM is also used to calibrate the displacement of the PIPM, thus enabling the determination of the force experienced by the particle due to the current manipulation technique.

4.1.1. Atomic Force Microscopy (AFM) setup for interfacial adhesion force

measurement.

One of the most widely used techniques for the quantification of interfacial forces is the

Atomic Force Microscope [34]. It is also widely used to measure the roughness and material properties of surfaces thus providing vital information for their characterization.

Here the AFM is used to measure the force binding stainless steel particles to the PIPM surface, however, as mentioned in section 2.1, the surface roughness can greatly modify the magnitude of this adhesion force. Therefore, prior to measuring the adhesion force, the surface roughness of both the PIPM surface and that of the particles are measured using the

AFM.

The roughness of the manipulator was performed by operating the AFM in contact mode with

Silicon Nitride cantilevers, Vecco NanoProbe, DNP, having dimensions L x W = 196 x 107

41m. An area of 50m2 consisting of 512 lines with 512 samples/line was scanned at 3 Hz,

and the peak-peak & RMS roughness, as well as the distance between peaks along several

sections of the surface were measured.

To get an estimate of the particle roughness, a layer of particles were glued onto a glass

substrate, following which surface scans were performed of several particles and their

average roughness measured.

In order to quantify the interfacial forces to be overcome by the piezoceramic manipulator, a

study of the force needed to pull the particle off (detach) the surface was conducted by using

AFM. A schematic of the experimental set-up is shown in Figure 4.1.

The experimental setup consists of an integrated BioScope AFM and a NanoScope IIIa controller, Veeco Instruments Inc., with an inverted optical microscope as shown in fig

4.1(a). The setup was built on a vibration isolation table to stabilize the AFM system.

To measure the adhesion force between stainless steel micro particles and the PIPM surface, micro-particles had to be attached to the tips of the AFM cantilevers:

Proper attachment of the stainless steel spheres to the AFM cantilevers is extremely important to the repeatability and reproducibility of the AFM force measurement results. The Stainless 108 steel particles, Duke Scientific Corp., consisted of particles ranging from 15m to 65m in

diameter. The calibration and attachment of the particles to the AFM cantilevers were

performed under an optical microscope as follows: AFM cantilever, Silicon Nitride, Vecco

NanoProbe, DNP-S, L x W = 196 x 41m, was first calibrated using the auto-tune option on

the AFM in order to determine its resonance frequency fc. Next, the cantilever tip was

carefully dipped in UV-glue, DYMAX, with care being taken to avoid wicking of the glue

onto the reflective side of the cantilever. The cantilever with glue was then brought in to

contact with pre-selected stainless steel particle. The glue was then cured by exposing the

cantilever pressing against the particle to ultraviolet light. The cantilever was once again

calibrated to obtain its new resonance frequency, fpc. The above procedure was repeated for 6-

particles having radii 11, 16.5, 21.5, 24, 26, and 32.5 m.

Prior to performing AFM force measurements, the silver piezoceramic surface was cleaned

with acetone, washed with distilled water and dried in an oven at 75C for 40 minutes. AFM

force experiments were performed under ambient environmental conditions, 20-25C, 15-

25% RH. A schematic of the force measurement is shown in fig 4.1(b). The Force-Distance

(F-D) curves were recorded using 30,000 sample points for each F-D curve, with a vertical

scan rate of 0.2 Hz. Force measurements were performed at a minimum of 4 locations with

10 measurements being made at each location for each particle size. The data files in ASCII

format were then exported and the processing of the retract data gave the overall particle-

surface adhesion force results.

Short description of the Force-Distance curve:

The AFM is used to measure atomic scale interactions by monitoring the deflection of a laser beam off the back of a cantilever that is interacting with the sample of interest.

Attractive/repulsive forces between the sample and the cantilever tip lead to the bending of 109 the AFM cantilever which is detected as deflection of the laser beam. As the sample and the

AFM cantilever tip interact, the force F on the cantilever, at the position of the tip, is related to the distance z that the cantilever’s free end bends up or down from its equilibrium

position. It can be calculated using Hooke’s law;

F  kc  z 4.1

Where, kc is the cantilever spring constant. Therefore, knowing the spring constant of the

cantilever and the AFM tip – surface separation distance, the force attracting the particle to

the surface, mainly Fvdw, and the force needed to detach the particle from the surface, Fadhesion,

can be determined. A more detailed description of the AFM force-distance curve can be

found in a previous paper [Zhang 2005]. A typical AFM force distance (F-D) curve is shown

in figure 4.2.

Figure 4.2 A typical AFM Force-Distance curve

The AFM Force-Distance (F-D) curve consists of 2 sets of curves, the approach curve, where in the AFM tip approaches the sample, and the retract curve, wherein the AFM tip is retracted from the sample surface. These curves can be subcategorized into 6 regimes as shown in Fig.

4.2. Regime 1 and 6 represent the AFM tip far from the surface, where the tip is not in 110 contact with the surface. Regime 2 represents the attractive VDW force wherein the tip is pulled toward the sample surface due to van der Waals forces of attraction causing it to bend and finally jumping to contact the surface, regime 3 and 4 represent the AFM tip and surface in contact and therefore moving together, and regime 5 represents the AFM tip being pulled off the surface in the process of which it overcomes all adhesive forces and breaks away from the surface. In this paper we are concerned with the force needed to detach the particle from

the surface and is given by regime 5. The force corresponding to regime 5 is a cumulative effect of all interfacial forces, namely, van der Waals, capillary, electrostatic and gravitational forces. For the purpose of particle manipulation we would like to quantify the force needed to detach a particle from the PIPM surface, which in turn determines the minimum force that the

PIPM will need to exert on the particle to facilitate its motion. This information is obtained by measuring the force corresponding to regime 5 of the F-D curve.

4.2. PIPM displacement calibration using AFM

As mentioned in section 3.3, in order to obtain quantitatively accurate results from the FEM simulations, a damping factor needs to be included in the simulation of the PIPM’s. The determination of the damping factor is performed based on the displacement of the manipulator which needs to be measured. Here we use the AFM to measure the displacement at the center of the piezoceramic manipulator excited at its resonance radial mode of vibration.

As mentioned above, the AFM is operated in the contact mode, with the AFM cantilever held in contact with the manipulator surface.

The first step is to quantify the deflection signal from the AFM cantilever.

A 16 x 1mm (diameter x thickness) disc manipulator having its radial mode of vibration at

125 kHz is placed in the sample holder of the AFM. With the manipulator OFF, the AFM cantilever is lowered to come in contact with the manipulator surface and operated in regime 111

3, in figure 4.2, corresponding to the cantilever being pressed against the manipulator surface.

Care is taken to ensure that the cantilever is pressing against the surface for piezo motor displacement ~150 nm. Therefore if the ramp size is reduced to zero, we are assured that the cantilever is pressing against the manipulator surface. Next, setting the scan rate to 28 Hz, the ramp size is varied from 0 – 50 nm. The vertical deflection signal corresponding to each ramp size is recorded via a Data Acquisition Card (DAC) capable of recording 150000 points at a sampling frequency of 300 kHz. A FFT of the data is then performed in MATLAB and a curve giving the deflection sensitivity can be obtained.

Next, the ramp size is reduced to zero and the manipulator is turned ON. The excitation voltage is increased from 0.1 – 3.3 Vp with the deflection data being recorded for each

excitation signal. Using the measured deflection sensitivity, the deflection due to vibration of

the manipulator surface can be converted into displacement at that location. In this way the

displacement sensitivity of the PIPM can be determined.

4.3. Particle manipulation setup

A few manipulation techniques were mentioned earlier; most of them have complex and expensive setups. In contrast, the current technique has a relatively simple experimental setup utilizing an inexpensive Lead Zirconate Titanate (PZT) actuator to manipulate micrometer and sub-micrometer sized particles.

The piezoceramic manipulator was placed in a brass holder described below:

A brass PIPM holder was designed to accommodate piezoceramic disc PIPM’s ranging in diameters from 10 – 30 mm having thickness ranging from 0.5 – 2 mm. The drawings are shown below: 112

The bottom plate is designed to accommodate different PIPM disc sizes, namely 10, 16, 20 and 30 mm discs by using fitting rings having different inner diameters that can be interchanged when working with different disc sizes. Provisions are made for SMA connectors for electrical connections. The bottom plate is designed to fit on the x-y stage of an Olympus microscope for observation, and recording of particle manipulation videos and images. The top plate is shown below: 113

Figure 4.4 Design drawings of the top plate of the PIPM holder. (a) Top view of the top plate, (b) Crossectional view of the top plate

Four top plates were designed to fit the four PIPM disc sizes mentioned earlier with the diameters of the circular opening having the dimensions shown. Between the top and bottom plates of the PIPM a soft foam cushion is placed and the holder is assembled. The PIPM is loosely held between the top and bottom plates of the holder and the foam cushions and care is taken to minimize stresses on the disc which would lead to modification of mechanical boundary conditions and therefore manipulation modes of vibration.

The experimental setup is relatively simple as show in, fig 4.5.

114

Figure 4.5 Block diagram of the PIPM

The current study involves the manipulation of stainless steel particles (15-65m in diameter,

Duke Scientific) in ambient air with temperature at 20-25 C and relative humidity (RH) at

15-65%. Piezoceramic discs (Ferroperm Inc., PZ-27, 30, 20, 16 & 10 mm diameter x 1 mm thick, screen printed silver electrodes) were used to generate the interfacial forces to manipulate particles. The manipulators were excited at various modes of vibration ranging from 40 kHz – 2MHz. In order to simplify the study, the radial resonance mode was selected for demonstration. The radial resonance frequency for the 20, 30, 16 and 10 mm discs were,

65, 98, 125, & 200 kHz respectively. The were excited at amplitudes ranging from 0.1 – 6 Vp generated by a signal generator, HP 8648B The impedance and scattering parameters of the disc were monitored using an Impedance/Network/Spectrum Analyzer, HP 4395A. The discs were mounted on a non-conductive base plate allowing for free radial mode vibration. This setup was mounted on a 3-D moving stage of an optical microscope adapted with a CCD camera for the capture of particle manipulation images and videos. Each disc was cleaned with acetone and dried with Nitrogen gas prior to placing particles on its surface and conducting manipulation experiments on it.

115

4.4. Ring PIPM Sensing Structure set up

The feasibility testing of particle manipulation in ambient air led to development of a sensor actuator hybrid structure based on the PIPM aimed at improving the performance of a

Thickness Shear Mode (TSM) quartz biosensor.

The structure is composed of a 6.35mm (Outer Diameter) x 2.4mm (Inner Diameter) x 1mm

(Thickness) ring actuator (Ferroperm, Pz 27) affixed to a 100MHz AT-Cut quartz sensor using a UV cured epoxy as shown in Figure 4.6a. The finished sensor-actuator structure is seen in figure 4.6 (b).

(a) (b) Figure 4.6 Schematic of the Ring PIPM sensing structure (a) Structure construction, (b) Finished structure

The experimental set-up is shown in figure 4.6, it is essentially the same as that for the PIPM disc discussed in the previous section except for the sensor response monitoring block. The forward transmission parameter (S21) of the 100MHz TSM sensor is monitored on a Network

Analyzer, HP4395A and controlled via a Labview program on the computer. The ring actuator is excited at its radial mode (220 kHz at 1.5Vp) of vibration using a Signal Generator,

HP8648B. A CCD camera is used to capture images and record videos of particle motion on

the sensing surface. 116

Figure 4.7 Ring PIPM sensing structure experimental setup

The Ring PIPM sensing structure was cleaned with acetone, rinsed with DI water and dried using Nitrogen gas. All experiments were conducted using DI as the buffer solution and therefore all detection results were compared to a DI baseline.

With the ring actuator OFF, 5l of DI water is first placed on the Ring PIPM sensing structure

and the response of the TSM sensor is monitored using a LabView program to acquire the

data from a Network Analyzer. Once the response of the sensor stabilizes (~ 15 minutes), the

program is paused and the solution containing the sample is introduced.

For the results discussed in this document, a solution containing 6 x 103 polystyrene

partilces/5l of DI water (10m in diameter, Polysciences) was placed on the sensing

electrode of the sensing element. The particles were allowed to sediment and the response to

the particles randomly (and more or less uniformly) distributed on the sensing electrode was

recorded. Next the ring actuator was turned on for 5 seconds leading to the concentration of

the particles on the sensor surface. The motion of the particles was recorded using the CCD

camera setup while simultaneously recording the response of the sensor to particles

concentrated on its surface. Each experiment was repeated 3 times.

The above procedure was repeated using silica particles, 2.7 x 104 particles/ 5l ,(5 m in

diameter, Polysciences), and Escherichia Coli bacterial spores, 2 x 104 spores/ 5l, ( 2m in

length and 0.8m in diameter) in DI water. 117

5. EXPERIMENTAL RESULTS

The manipulation of micro-particles requires a detailed study of the various contributing factors that facilitate as well as hamper the functioning of this technique. Previous chapters discussed the theoretical basis for the manipulation of particles using a piezoelectric actuator.

Right from the forces dominating the adhesion of particles to a surface, to the manipulating

forces generated by the piezoelectric PIPM and finally to the motion of the particles has

approached from the theoretical perspective. An experimental study demonstrating

capabilities of this manipulation technique is now performed.

5.1. AFM surface topography and adhesion force results

The manipulation of particles involves the predictable translation of particles from one location to the next. It is therefore necessary to overcome or weaken the adhesion forces that bind the particles to the surface preventing their motion. The quantification of these adhesion forces will enable estimation of the order of the force that the PIPM will need to generate to initiate particle motion.

This quantification of interfacial adhesion force is performed using an AFM. Since we are mainly interested in the total adhesion force, only the pull-off forces corresponding to the detachment of the particle from the surface will be addressed. This adhesion force, Fadhesion, is

a combination of van der Waal’s, gravitational, capillary and electrostatic forces discussed in

chapter 2. However, the magnitudes of these forces are highly dependent on the surface

conditions such as surface roughness and contamination, as well as environmental conditions

and material and geometrical properties of the interacting surfaces. The first step in

determining the pull off forces is to determine the surface condition of the particle and the

PIPM. 118

5.1.1. Surface scan of particle and PIPM surfaces

In order to quantify the particle-PIPM adhesion force, the AFM is used to perform a surface scan of the stainless steel particle and silver electrode of the manipulator.

5.1.1.1. Particle roughness measurement

A 5m x 5m scan of a stainless steel particle is shown in figure 5.1.

Figure 5.1 3-D surface plot of a stainless steel micro particles using AFM operated in contact mode.

The surface of the particle, exhibits a high surface roughness which will greatly modify the

interfacial adhesion force due to the complex contact region between the particle and the

surface.

The peak to peak surface roughness can be measured by sectioning the scanned surface image

as shown in fig 5.2. From the surface scan, the rms roughness can also be extracted, however

this value can be misleading since similar rms values can lead to different peak to peak

roughness [106]. 119

Figure 5.2 Particle surface roughness determination. (a) 2-D image (b) Roughness along image sections

As can be seen in figure 5.2, the surface exhibits a varying peak to peak roughness. Several particles were imaged and an average of over 30 sections of the images gave an average peak-peak roughness of 133  38 nm with a peak-peak separation distance of 1.434  0.54

m. An average image RMS value of 75  16 nm was also measured.

As mentioned in chapter 2, the dominant adhesion force for relative humidity, RH < 65% is

the van der Waal’s force which is inversely proportional to the square of the separation

distance between the particle and the surface and directly proportional to the radius of the

particle. The peak to peak surface roughness acts to increase the effective particle surface

separation distance, thus decreasing the van der Waal’s force.

5.1.1.2. PIPM surface roughness measurement:

Next , a surface scan of the PIPM was performed. 120

5m

Figure 5.3 50m x 50m AFM scan showing the microstructure of the silver electroded piezo-manipulator. The grain size of the piezoelectric ceramic ranges from 1-5 m

Figure 5.4 Three dimensional plot of the PIPM surface.

Figure 5.3 is a deflection plot from which the grain size of the piezoceramic, Pz27 can be observed. The grain size ranges from 1-5 m, and determines the final surface roughness of the PIPM. The manipulator is excited by applying a potential to silver electrodes deposited on the top and bottom surface of the transducer. The thickness of this silver electrode ranges from 100-150nm and conforms to the rough grainy boundary of the ceramic. A large surface 121 scan of 50 m x 50m showing the height data of fig 5.3 is seen in fig. 5.4, indicating that the

PIPM surface exhibits large roughness that varies with spatial location.

Figure 5.5 PIPM surface roughness scan (a) 2-D image of the surface (b) Roughness along image sections

The peak to peak roughness height and separation distance was recorded along three sections of fig 5.5(a) and shown in 5.5(b). Up to 30 more sections were then taken and the peak-peak height, separation between peaks and image RMS were recorded to give, 1026  615 nm,

2.88  1.2 m and 316  50 nm respectively.

Owing to the large variation in the p-p roughness and p-p separation distance, the adhesion force between the particle and the surface can exhibit large variation from location to location.

The roughness of the particle and the PIPM surface is summarized in table 5.1

122

Table 5.1 Peak to peak roughness, separation distance and image RMS roughness for the scanned surfaces

Using the values in table 5.1, the theoretical estimation of the adhesion force between stainless steel particles and silver PIPM surface can be made by substituting the listed values

in the equation used to calculate the van der Waals force of rough surfaces, eq. 2.5

5.1.2. Measurement of adhesion force binding stainless steel particles to the

PIPM surface.

In order to measure the adhesion force using the AFM, the AFM cantilevers were modified by attaching stainless steel micro-particles to their tips. The attachment procedure was

discussed in chapter 4, section 4.1.1. 123

Figure 5.6 Optical images of 11, 16, 21, 24, 26 and 32m stainless steel particles attached to AFM cantilevers, figs 5.13(a) – (d)

Figure 5.6 consists of 6 optical images of stainless steel particles each of whose average radii range from 11 – 32 m, allowing for a 5% error in size measurements, attached to the tips of

AFM cantilevers.

The addition of the particles to the cantilevers constitutes an additional mass on it, leading to a possible change in the spring constant of the cantilevers. By recording the resonance frequencies before and after the particles are attached to the AFM tip and using optically measured particle size to determine its mass, the modified spring constant for each cantilever was determined. [Appendix 3]

Using the calculated effective spring constants for each tip, the deflection of the cantilever due to adhesion force, measured using the AFM, can now be converted to force values using

Hooke’s law.

The piezoceramic manipulator, having a silver electrode deposited on its top and bottom surfaces, is used for the manipulation of micro-particles. Here stainless steel particles are utilized for the purpose of testing this manipulation technique. The determination of the 124 adhesion force binding the stainless steel particles to the PIPM surface will enable the quantification of the minimum force needed to manipulate the particles. This is performed using an atomic force microscope in contact force mode as explained in chapter 4. Below are the results of the force required to pull off particles from the silver electrode of the piezoceramic manipulator. We first use the surface roughness conditions to obtain theoretical adhesion force results.

The roughness of the surface acts to reduce the adhesion force and has been studied by several authours [27, 72, 76, 82, 85-87]. Since the experiments were conducted in RH of 15-

20 % it has been shown that the van der Waals force is the main contributing factor to the

adhesion force [72, 78, 86, 106-107]. Thus, taking the surface roughness into consideration

while calculating the van der Waals force, a theoretical prediction of the force can be made.

As shown in section 5.1.1, the surface exhibits a roughness that varies with location; it is

therefore difficult to predict the roughness of the surface at the location where the pull out

force was measured. However, using the range of surface roughness parameters, ymax and ,

the range of theoretical van der Waals forces, eq. 2.5, for a particular surface roughness can

be estimated. 125

Figure 5.7 Theoretical van der Waals force between a 25 m stainless steel particle and a rough grounded silver PIPM electrode, with ymax, and  given in table 5.1

The figure 5.7 shows the variation of the van der Waals force with surface roughness parameters extracted by performing a surface scan of the silver electrode of the manipulator, values of which can be obtained from table 5.1. The particle roughness was not considered as its maximum peak-peak roughness is an order of magnitude less than that of the silver electroded PIPM surface. As can be seen in figure 5.7 the theoretical van der Waals force ranges from 15-633 nN for experimental surface conditions. Therefore, based on the location where the pull out force is measured, the value of the force can show large variation.

Furthermore, particles of different sizes can have similar adhesion forces due to the variation in the surface roughness.

At any one particular location, the adhesion force measured is reproducible, however its value varies from location to location. This can be seen from the experimental results of adhesion force shown below, 126

Figure 5.8 Adhesion force measured for 6 particle radii, 11, 16, 21, 24, 26 and 32m at 4 different locations.

As can be seen, the measured adhesion force at any one location does not vary by more than

10-15% from the average of 10 measurements, however, the measured force can vary as much as a 95% from one location to the next. (Please note that the force measurement for each particle size at the 4 locations shown in fig. 5.8, was not measured at the same location each time.)

A linear decrease in adhesion force as a function of decreasing particle radii is expected on smooth interacting surfaces, however, owing to the surface roughness in the current experimental conditions, this linear particle-surface adhesion force relation is not observed.

The adhesion force was previously tested on a mica surface, known to be atomically smooth, and the tested range of particle sizes did exhibit a linear decrease with decrease in particle radius [36].

The adhesion force between stainless steel particles and the silver electrode of the PIPM are summarized in table 5.2 127

Table 5.2 Summary of the adhesion force between stainless steel particles and the rough silver surface of the PIPM

As is seen in table 5.2, the force of adhesion of particles, in the size range of 11 -32 m, to

the rough silver surface varies greatly from location to location. However the force measured

does lie in the expected theoretical range of 15 - 633 nN, as shown in fig. 5.7, for the

roughness exhibited by the PIPM surface.

In the discussions that follow, we therefore refer to a range of particle sizes, rather than any

one particular size. Since the adhesion force can show large variations, we also propose a

range of 15-200nN as the minimum force range that the PIPM will need to generate to initiate

particle motion.

5.2. PIPM displacement calibration using AFM

The manipulation of particles is performed using surface forces generated by a piezoelectric actuator. One of the modes used for manipulation, namely the radial mode, was studied in detail in chapter 3, by discussing the normalized displacements, velocities and accelerations.

However, the quantification of these quantities requires the knowledge of the amplitude of vibration of the surface. This is performed using the AFM in contact mode, a method which was discussed in section 4.2. The first step is to quantify the deflection of the AFM with 128 known surface displacement. This is performed using the piezo-motor of the AFM, operated at voltages where its displacement is linearly proportional to the excitation signal. Since the displacement for a particular excitation voltage is known, the peak deflections of the laser corresponding to known periodic displacements at 28 Hz are first recoded. The deflection observed on the array detector is collected via a LabView controlled data acquisition system.

The deflection data is then processed using a simple code written in Matlab to take the

Fourier transform (FFT) of the data. The amplitude of the peak at 28 Hz was recorded as the peak deflection voltage and plotted as a function of piezo displacement.

Figure 5.9 Calibration curve showing peak deflection of the AFM laser as a function of known the piezo motor displacement.

A simple linear regression of the data gives a correlation coefficient of 0.99 and a deflection sensitivity of ~35mV/nm. The AFM deflection was calibrated by exciting the piezo-motor at

28Hz, with the signal to noise ratio (s/n) dropping from ~56 at 50nm displacement to ~2 for a

2 nm displacement. It is however possible to observe displacements < 2 nm by operating in liquid or at a higher frequency of excitation. Assuming that the curve in figure 5.9 can be extrapolated to represent the deflections corresponding displacements less than 2 nm, the 129 equation of the linear regression fit to the data is used to calibrate the displacement of the

PIPM.

Figure 5.10 represents the displacement measured at the center of a 16mm piezoelectric manipulator excited at 124 kHz. A linear regression of the data gave a correlation coefficient

of 0.97. The deflection data was recorded 3 times for each excitation voltage with the data having maximum standard error of 1.3%. The secondary y-axis represents the deflection of

the AFM laser caused due the displacement of the PIPM surface when excited at voltages

shown on the x-axis. It was observed that the noise decreased from peak value of 27 mV,

observed at 28Hz, to ~1mV at 124 kHz, therefore data points corresponding to deflections as

low as 1.8mV have been shown in figure 5.10.

The PIPM displacement sensitivity of ~0.21V/nm is obtained which is now used to determine

the material damping of the piezoelectric PIPM as discussed in chapter 3 on theoretical

results. 130

5.2.1. FEM with calibrated loss factor

A study of the normal and tangential components of displacement as a function of damping factor and excitation voltage was performed in chapter 3, section 3.2.1. Using the experimentally determined displacements, fig. 5.10, an estimate of the average percentage damping, was found to be 4.34  0.2% of critical damping. In section 3.2.11, the normal and tangential components of displacement of 10, 16, 20 and 30mm diameter discs was studied as a function of percentage of critical damping for excitation voltage of 1V, shown in figures 3.9

& 3.10. Therefore determining the experimental displacement corresponding to an excitation voltage of 1V from figure 5.10, and inserting it in to the equation of the power curve of figure

3.9, an estimate of the material damping is made. Due to the linear relationship between the theoretical displacement and the excitation voltage, the same percentage of critical damping would be obtained if extracted at different voltages. The extracted value of critical damping is verified by re-simulating the 16mm piezoceramic manipulator with 4.25% damping and comparing the displacement sensitivity to that obtained in the previous section.

Figure 5.11 Experimental and theoretical comparison of the PIPM displacement for a 16 mm disc simulated with 4.25% damping.

131

A steady state analysis of the 16 mm piezoelectric disc is performed using FEM software,

ABAQUS. The normal component of displacement, uz_max, at the center of the disc is plotted

as a function of excitation voltage and compared to that observed experimentally, as shown in figure 5.11. The experimental displacement shows good agreement with the theoretical values differing by < 1%.

As discussed in section 3.2.1.2, for radial modes of vibration, the normal component of displacement, uz_max, at the center of the discs, is independent of the disc diameters. Therefore

the displacement sensitivity of 0.21 V/nm, which is the inverse of the slope in figure 5.20, is

representative of the displacement, uz_max, for all four discs under consideration here, namely,

10, 6, 20, and 30mm discs.

Thus using the extracted damping of 4.25%, the maximum normal and tangential

displacements of the discs can be obtained by substituting the damping value in to the

equations in figs 3.9 & 3.10. An example for an excitation voltage of 1V is shown in the table below,

Table 5.3 Normal and tangential displacements, for an excitation voltage of 1V, determined using 4.25% material damping

Table 5.3 shows the maximum normal and tangential displacements of four discs, excited at

their radial modes of vibration as discussed in section 3.2. The values in the ‘Expected’

columns are determined using the equations in figures 3.9 & 3.10 by substituting 4.25% for 132 the damping value, whereas the values in the ‘Simulated’ columns are obtained by performing a dynamic steady state analysis of the discs using a FEA software ABAQUS, wherein a stiffness proportional damping of 4.25% is introduced in the material properties.

The table shows that the displacements obtained via calculation and simulation show good correlation with an error of less than 2%.

The equations shown in table 3.1 represent the normal and tangential components of displacements as a function of spatial location, from the edge to the center of the disc, which

were normalized to the maximum tangential displacement of each disc (because the material

damping was not known at that point). However now that the material damping has been

determined, the maximum tangential displacements as a function of excitation voltage for the

four discs can be obtained.

Figure 5.12 The magnitude of maximum tangential displacements, ux_max, of four discs as a function of excitation voltages, obtained for a Pz 27 material with 4.25% damping.

133

Thus using the equations of the linear regression fit of the data, shown in figure 5.12, ux_max,

for a particular excitation voltage can be determined which is then used to rescale the

normalized equations shown in table 3.1 Thus, quantitatively accurate displacements as a

function of spatial locations can be reconstructed. The velocities and accelerations can also be

determined using the scaled displacements and corresponding disc frequencies in equations

2.15 (b) & (c), while taking the phase relation between displacement, velocity and

acceleration into account.

5.3. Experimental results of particle manipulation

The entire process involved in PIPM based particle manipulation can be outlined in the form of a flowchart:

Figure 5.13 Flowchart of PIPM based particle manipulation

As is seen, the current manipulation technique has three main components, i.e. mechanical model of the PIPM, AFM calibration of the PIPM displacement and the experimental study of particle manipulation.

Prior to the experimental study, the PIPM is modeled using a FEM analysis wherein the normalized displacement distribution of the PIPM is extracted at each mode of vibration.

These results were discussed in chapter 3, table 3.1. 134

Next the PIPM is calibrated in order to determine the displacement sensitivity of each mode.

In the current study this is performed using an AFM and the results for the radial mode of vibration were discussed in sections 5.2 & 5.3.

Finally, the PIPM is tested experimentally wherein the following information is recorded, the excitation voltage needed to initiate particle manipulation (V), the frequency of PIPM vibration (f), the particle properties (size and material), and the location of the particle on the

PIPM surface (loc (x,y)).

Here we experimentally demonstrate the use of inertial manipulation force exerted by the

radial mode of vibration on to particles placed on its surface.

First, the voltages needed to initiate particle motion at different spatial locations on the PIPM

surface are recorded and analyzed. This study is performed on three discs, 10, 20, & 30 mm in diameter and 1mm thickness. As the disc diameter is reduced, the radial resonance frequency increases as shown in figure 3.5. Next, the speed for a particle to travel from the edge to the center of the disc is studied.

5.3.1. Manipulation force as a function of location and PIPM frequency

In section 5.1 it was shown that the adhesion forces in the range of 15 –200 nN bind particles ranging in radius of 16 – 32 m to the silver PIPM surface, thus the PIPM will need to

generate interfacial forces in this range to initiate particle motion.

In this section, the voltages needed to initiate particle motion at different spatial locations,

along the x-axis, on the PIPM surface are recorded and related to the normal and tangential

manipulation surface forces at those locations. Owing to the variation in surface roughness

from location to location, the adhesion force exhibits large spatial variation, therefore the

voltage needed to initiate particle motion of a range of particle sizes, 20-30 m, is addressed

rather than any one particular size. 135

When the PIPM is excited at any natural mode of vibration, the radial mode in this case, it exhibits a non-uniform distribution of displacement and therefore inertial force (since

2 Fmanipulation = mpuR , where mp is the mass of the particle, uR is the resultant displacement of

the PIPM and  is the radial frequency of vibration), given by eq. 2.16. The normalized surface resultant force experienced by a 25 m particle as a function of spatial location is

shown below,

Figure 5.14 Normalized magnitude of resultant force experienced by a 25m stainless steel particle when placed on 30, 20 and 10 mm PIPM’s

As seen in fig. 5.14, the force generated by the PIPM is maximum at its edge and decreases to a minimum near the center. Since the displacement of the PIPM surface is linearly proportional to the excitation voltage, as seen in figs., 5.11 and 5.12, and the manipulation force is directly proportional to the displacement of the PIPM, it is expected that the excitation voltage needed to initiate particle motion will to be minimum at the edge and increase towards the center.

Therefore it is expected that the excitation voltage to initiate particle motion would vary inversely with the surface force distribution. This is now tested experimentally using PIPM’s having the following specifications 136

Table 5.4 PIPM specifications

Table 5.4 represents the basic specifications of the PIPM used for the current experiments.

The recorded voltage needed to initiate particle motion (V) is then fed to the calibration

module in order to determine the corresponding PIPM displacement. Using this displacement

and the FEM model, experimentally relevant displacement distribution over the PIPM surface

can be determined. This displacement distribution is then converted to a force distribution

using the frequency of vibration and the mass of the particle. Lastly, knowing the location

(x,y) of the particle, the force needed to initiate particle motion and the direction of

manipulation can be determined.

The disc is excited at its radial mode of vibration and the peak excitation voltage at which the

particles just begin to move is recorded. The voltage recorded at each location is then converted into normal and tangential components of surface force as follows:

1. Record the particle location and peak voltage to initiate motion

2. Use the voltage at each location to calculate ux_max by substituting the voltage in

equation for the corresponding disc in figure 5.12 137

3. Scale the normalized displacements in table 3.1 by multiplying by ux_max and

substituting the recorded particle location. Thus determine ux(x) and uz(x).

4. Determine the normal and tangential components of the inertial force, Fmanipulation_z

and Fmanipulation_x, using equation 2.16, where mp is the mass of a stainless steel particle

(average radius of 25 m and density of 8000Kg/m3) and the acceleration is

determined from the displacement multiplied by the square of the radial frequency

5. Plot and compare the voltage vs x-location and force vs x-location from the edge to

the center of each of the discs.

For each disc we have 2 plots, the first represents the excitation voltage vs spatial location and the second represents the inertial force exerted on the particle due to the applied voltage, calculated as per the above listed steps. Furthermore, the rate of particle translation can also be modeled knowing the particle properties and the displacement distribution over the PIPM surface. The current study utilizes an open loop system wherein knowledge of the initial particle position is used to predict the final location of the particle, however in future studies a closed loop approach will also be adopted to enable improved control of this particle manipulation technique.

138

5.3.1.1. Particle manipulation on a 30 mm PIPM

5.3.1.2. Particle manipulation on a 20 mm PIPM

139

5.3.1.3. Particle manipulation on a 10 mm PIPM

As expected, the excitation voltage required to initiate the motion of 20 -30 m stainless steel

particles decreases from edge, to a region close to the center of each of the 3 PIPM discs and

can be seen in figures 5.15, 5.17, & 5.19. However, it is observed that the voltage needed to

initiate particle motion is reduced near the center of the discs. This region of maximum

voltage, after which it decreases, is associated with the location where the normal and

tangential components of surface force are equal in magnitude. From this location towards

the center, the normal component begins to dominate the force exerted on the particle and

reaches a maximum value at the center where the tangential component reduces to zero. Thus

the voltage needed to initiate particle motion at the center of each PIPM is associated with the

force needed to overcome adhesion force.

This excitation voltage is then converted to normal, Fmanipulation_z and tangential, Fmanipulation_x,

components of inertial force experienced by a 25 m stainless steel particle and shown in

figures 5.16, 5.18, & 5.20. Due to the harmonic vibration of the PIPM’s, these forces vary

periodically at the frequency of excitation. Therefore, the exact forces that cause the particle

motion cannot be determined, but an estimate of the maximum magnitude of the

manipulation force at the measured excitation voltage can be given. The normal component

of force acts to reduce and/or overcome the adhesion force and is proportional to the critical

displacement, eq. 2.20, needed for particle manipulation. The tangential component acts to

overcome frictional force between the particle and the PIPM attempting to drive the particle

towards the center. It should also be noted that the angle of the resultant PIPM force

experienced by the particle can be determined by taking the inverse tangent of the ratio of the

normal to tangential components of force. The forces experienced by the particle at the center

and edge of the PIPM are summarized below,

142

As can be seen from the force plots, figures 5.16, 5.18 & 5.20, and table 5.4, the voltage at which the particle begins to move, corresponds to a normal component of force that is greater than that of the adhesion force binding particles to the surface which ranges from 15 – 200 nN, shown as the shaded region in the plots. Thus for all three discs, the condition for particle manipulation given by equation 2.17, is satisfied for all locations along the radius of the disc.

From table 5.5 it can be seen that the tangential force at the edge of the discs, can be up to one order of magnitude larger than the normal forces, and act to rapidly translate the particles from the edge to the center of the discs. The speed of particle translation is discussed in the next section.

The radial mode resonance frequency is inversely proportion to the diameter of the disc, therefore decreasing the PIPM diameter leads to the generation of increased manipulation force which is proportional to the square of the radial resonance frequency. Therefore, to initiate particle motion a lower voltage is expected to be required as the frequency of vibration is increased. To test this hypothesis the voltage needed to initiate particle motion at the center of the 4 PIPM discs, having diameters, 10, 16, 20 and 30 mm, corresponding to

200, 125, 100 and 65 kHz respectively is studied. 143

As seen in figure 5.21, the peak excitation voltage required to initiate particle motion decreases with increasing radial resonance frequency (corresponding to PIPM’s of smaller diameters). Therefore, this study proves that operating at higher frequencies can be used to generate higher manipulation forces which in turn can be used to manipulate smaller particles.

Each peak voltage measured corresponds to the PIPM displacement which can be then used to estimate the normal component of manipulation force knowing the size and properties of the particle.

For a 25 m stainless steel particle, the inertial manipulation force, Fmanipulation_z, that it

experiences is plotted as a function of radial resonance frequency of the 4 discs,

144

As shown in figure 5.22, an average manipulation force needed to move a 25 m particle is ~

240 nN measured using the 4 PIPM’s. For ideally smooth interacting surfaces we would

expect to have no variation in the measured force, however, as mentioned in section 5.1, the

manipulators and the particles exhibit a high degree of surface roughness. Also variation in

roughness between manipulators leads to a difference in adhesion forces and therefore a

difference in measured inertial manipulation force as is seen in fig 5.22. Although an exact

value of adhesion force cannot be provided, an estimate of the range of adhesion forces can

be made. For a 25 m stainless steel particle the adhesion force is measured to lie between

150 and 300 nN which is a good approximation of the forces measured using the AFM.

5.3.2. Particle translation speed as a function of PIPM excitation voltage

The above section demonstrated the ability of the PIPM to generate the necessary interfacial forces to detach a particle from its surface. The combination of normal and tangential forces act synergistically to translate particles from one location to another, discussed in the section 145 on mechanism of manipulation, 2.3. The motion of a particle on an ideal, smooth vibrating surface has been discussed in section 3.3.

In this section we demonstrate that the PIPM has the ability to translate particles over distances a few hundred times its radius in a short period of time. Since the displacement (and therefore the force experienced by the particle) of the PIPM increases linearly with the excitation voltage, it is expected that the average speed of particle translation over the same distance would also increase linearly with excitation voltage. This is demonstrated using a 30 mm diameter, 1mm thickness, Pz 27 disc, excited at it radial mode of vibration (65 kHz).

Stainless steel particles, having radius ranging from 20 – 30 m, are placed at the edge of the disc, following which the PIPM is excited at a series of excitation voltages and the time for the particles to be translated to the center of the disc is recorded.

Figure 5.23 Optical images of a 25 m particle being translated from the edge of a PIPM towards its center when excited at its radial mode of vibration.

Figure 5.23, represents four still images of a 25 m particle being forced towards the center

of a 30 mm disc. The particles are capable of traveling at high speeds from edge to center and

tracking their motion over distances of 15 mm is extremely challenging. Therefore, the

average speed to travel from edge to the center of the disc is presented. As discussed in the 146 previous section, the voltage needed to initiate particle motion increases towards the center.

Therefore the excitation voltages chosen for these particle translation experiments were selected such that the normal component of surface force was larger than that of adhesion force at all locations on the surface, thus ensuring particle detachment and translation to the center of PIPM.

Experimental results in figure 5.24 were performed in air at RH of 40-50%. It is seen that the

average particle speed, to travel a distance of 15mm, increases linearly with excitation

voltage. Furthermore, the PIPM is capable of translating a particle, having a radius of ~25

m, at an average speed of 5mm/s, when excited at 1.6Vp. It is therefore able to translate the

particle a distance 600 times its radius in just under 3 seconds. Similarly it can be shown that

higher frequency PIPM’s would exhibit a linear relationship of average particle speed to 147 excitation voltage, and that the translation speed would be increased since the momentum transferred to the particles increases linearly with frequency.

5.3.3. Particle manipulation at a 56kHz natural mode of vibration

The above sections discuss the use of the resonance radial mode for the unidirectional translation of particles. This analysis can be extended to study other piezoelectrically excited natural modes for the manipulation of particles. Here we demonstrate the use of a transverse mode of a 30 x 1 mm disc excited at 56 kHz.

Figure 5.25 Contour displacement maps of a 30 x 1mm disc excited at 56 kHz

As done for the case of the radial mode, the displacement along the radius, from edge to the center, along the x-axis is studied. In doing so, the y-component of displacement, which is an order of magnitude smaller than the x or z components along the x-axis, will not be considered when studying the distribution over the surface in this direction. The variation of the x and z components of displacement, as a function of location along the x-axis, is shown in figure 5.26.

148

Figure 5.26 Normalized resultant displacement with its normal and tangential components plotted along the x-axis for a piezoelectrically excited mode at 56 kHz.

The components of displacement along the x-axis are shown in figure 5.26; this mode has a more complex distribution of displacements as compared to the radial mode discussed previously. It exhibits three nodal regions wherein particles would be trapped at, causing them to orient in the pattern of concentric rings. The resultant displacement, shown as a dashed line is also plotted in figure 5.26, a particle placed at any location on the disc would

2 experience a resultant inertial force given by, Fmanipulation = mpur , where mp is the mass of

the particle and ur is the resultant displacement and  is the angular frequency of vibration.

Stainless steel particles (20 – 30 m) were placed on the PIPM at various locations and once

again the voltage needed to initiate their motion was recorded and listed in table 5.6.

Table 5.6 Peak voltages needed to initiate motion of 20 – 30 m on a 30 mm piezoelectric PIPM excited at 56.79 kHz

149

The excitation voltages are normalized to the maximum voltage and plotted along the x-axis in fig 5.27. The normalized excitation voltage and resultant manipulation force are now compared:

Figure 5.27 Normalized peak excitation voltage and normalized resultant manipulation force plotted as a function of location along the x-axis

As seen in figure 5.27, an inverse relation is observed, i.e. the voltage needed to initiate particle motion is minimum at locations where the resultant manipulation force exerted on the particles is maximum. The dashed line represents the force, distribution, while the solid line represents the experimentally measured excitation voltages.

This shows that the analysis applied to the radial mode of vibration can be extended to other natural modes of vibration that are used for particle manipulation. Furthermore, the ability to 150 excite 2 or more modes on a single substrate enables controllable particle motion by operating the PIPM at different modes sequentially. The sequence of mode excitation is to be designed based on the initial and final locations of the particle(s).

These forces generated by the PIPM are also capable of manipulating a large number of particles simultaneously. This can be observed by uniformly distributing particles on the surface of the disc and exciting the disc at various piezoelectrically excited natural modes of vibration.

Figure 5.28 Experimental and FEA simulated results showing particles orienting along the nodal lines of the piezoelectrically excited natural modes of vibration at the indicated frequencies.

Exciting the PIPM at the frequencies shown in figure 5.28, the particles quickly orient along the nodal lines of each mode. Figure 5.28 demonstrates the ability of the piezoelectric PIPM to excite the displayed natural modes of vibration and the ability of the forces generated by 151 each mode to reorient particles. The experimental results correlate with the mode shapes extracted using FEM analysis of the PIPM.

We have shown the ability of individual modes to generate sufficient force to initiate particle motion as well as translate the particles to the nodal regions of the selected mode. We now demonstrate the use of two or more of these modes to manipulate particles by moving them to and from predetermined locations.

5.3.4. PIPM multi-modal operation

In the above sections, the unidirectional translation of particles from regions of high displacement (anti-nodes) to regions of low displacement (nodes) was discussed. We also demonstrated several modes capable of manipulating particles; however another attractive feature of this technique is the ability to switch between modes of vibration by simply changing the frequency of excitation. In doing so, particles that come to rest or that are trapped at the nodes of one mode will experience a force driving them to the node of the second mode. Switching between modes allows for the reproducible translation of particles to

a particular location and back to its original location by the careful excitation of a sequence of

modes.

Here we demonstrate the planar motion of a 8 m particle, by switching between modes at 54

and 56 kHz.

152

A particle placed randomly on the PIPM will be driven to regions of low displacement when excited at a particular modal frequency. The images of the particle shown in fig 5.29 represents a few frames of a video taken as the particle traveled between the nodes of the two modes show. Consider an 8m particle located at a location, A0, on the PIPM that is OFF.

When turned ON at 53 kHz, the particle is driven to location A2 which is the node of the 53

kHz mode, shown as image sequence (c). The distance traveled is ~5 mm. The frequency is

then switched to 56 kHz, leading to the particle now being forced back to location A0 where

the nodal circle of the 56 kHz mode is located, sequence (e). The particle can be reproducibly

driven back and forth between the two modes. 153

It should be noted that the particle could potentially be forced to low displacement regions other than that at the center. This can be avoided by using the resonance radial mode, 65 kHz and the 56 kHz modes to drive the particle back and forth, between the center and the nodal circle of the 56 kHz mode. For translation over longer distances the sequence mode excitation frequencies can be selected such that the force experienced by the particle keeps driving it towards its final destination.

Now that this technique was successfully tested for particle manipulation, it was applied for the improvement of the performance of a biosensor.

5.3.5. Application of particle manipulation in Biosensing

Thickness Shear Mode (TSM) sensors are widely employed as biosensors. One of the most important operational parameters of biosensors is their sensitivity. Due to the fact that TSM sensors have maximum sensitivity at the center of the sensing electrode, there has been increased research efforts focused on the development of techniques for controlling the distribution of the measurand over the sensor surface. Here the improvement of TSM sensor performance is achieved via the construction of a simple, inexpensive, Sensor-Actuator

Hybrid Structure using a ring PIPM and a TSM biosensor. The ring PIPM sensor structure

consists of a piezoelectric ceramic, ring-shaped, actuator affixed to a TSM AT-cut quartz sensor. The ring actuator operating at a given frequency generates a specific force-pattern over the TSM sensor surface. A Finite Element Analysis (FEA) is used to simulate various force patterns, identify the appropriate ones and determine the corresponding driving frequencies of the ring actuator. The simulation results show that the ring PIPM sensor structure is capable of concentrating micron and sub-micron sized particles to high sensitivity locations at and around the center of the sensor. A structure incorporating a ring-actuator

(6.35 x 2.4 x 1 mm) with a TSM sensor, operated at 100 MHz (diameter 5mm x thickness 16

m), has been experimentally tested with micrometer sized inorganic particles, namely, 154 polystyrene and silica, and biological bacterial spores, Escherichia Coli. The response of the sensor, to particle loading, has been improved by means of manipulation and clustering of particles. Furthermore, particle distribution over the sensors surface was recorded and was consistent with the FEA simulation results.

5.3.5.1. FEA of ring PIPM and ring PIPM sensing structure

A schematic of the ring PIPM having thickness, h, outer radius, a, and inner radius, b, excited with an electric potential =Vejt is shown below [54],

Figure 5.30 Schematic of a ring PIPM

The equations of motion and boundary conditions describing a ring actuator, are given by

equations 5.1 and 5.2

Tij  f j  .u 5 5.1 j

T  0 h ,T  T  0 2 2 2 2 2 2 3 j z m 1 j 2 j x  y a or,x  y b 2 jt 5.2   Ve h ,  0 hm 2 2 2 2 z m b2 x2  y2 a2 z b x  y a 2 2

Where, T, is the stress tensor, f is the body force, u is the displacement  is the densit, and 

is the electric potential. 155

Using a FEA ABAQUS, the solution to the above equation of motion for a piezoelectric ring actuator, outer diameter 6.35 mm, inner diameter 2.4 mm and thickness 1mm is approximated and the steady state radial mode of vibration is shown in figure 5.30.

Figure 5.31 FEA of the radial mode of vibration of the ring actuator, excited at 220 kHz, with an excitation voltage of 1Vp

Figure 5.31 represents the first radial resonance mode of the ring actuator. Steady state analysis simulation of the actuator with an excitation voltage of 1V(peak) enabled the extraction of the radial resonance mode at ~220 kHz with a peak displacement of 10 nm.

The next step is to model the ring PIPM sensing structure. Here a FEA of a 100MHz AT-Cut

Quartz, attached to the top of the piezo-ceramic ring actuator (6.35 x 2.4 x 1 mm) is performed. The construction of the ring PIPM sensing structure was described in chapter 4, section 4.4. In order to simulate the structure, a homogeneous glass plate having the thickness of the 100 MHz TSM sensor (~16 m), with an outer diameter of 5 mm is attached to the

surface of the ring actuator by performing a ‘tie-function’ wherein the nodes of the two

intersecting surface are unconditionally bound together. The mesh elements used for the ring

actuator were 3-D quadratic hexahedral piezoelectric elements, C3D20RE, whereas the

sensor acting as a membrane was meshed with 3-D quadratic hexahedral elements, C3D20R

stress elements. An electric potential boundary condition ( 1V) was applied to the top and 156 bottom surface of the ring actuator and a steady state response analysis was conducted. Due to the attachment of the TSM sensor to the ring PIPM a new set of boundary conditions are now defined. The schematic of the structure is identical to figure 5.30 with the addition of the sensor of radius c and thickness ts. It is shown below,

Figure 5.32 Schematic of the ring PIPM sensing structure

The equation of motion is again given by equation 5.1, however the boundary conditions are

now modified to be,

T  0 h h h 3 j z m t x2  y2 c2 ,z m b2 x2  y2 a2 ,z m c2 x2  y2 a2 2 s 2 2 T  T  0 h h 5.3 1 j 2 j m z m t x2  y2 c2 ,x2  y2 a2or,x2  y2 b2 2 2 s jt   Ve h ,  0 hm 2 2 2 2 z m b2 x2  y2 a2 z b x  y a 2 2

The ring PIPM sensing structure was then simulated with the above boundary conditions.

Figure 5.33 shows the SAHS vibrating at the radial mode of vibration of the ring actuator.

157

Figure 5.33 Contour map of the magnitude of displacement of the SAHS excited at the radial resonance frequency of the ring actuator (~220kHz, 1Vp)

Figure 5.33 represents the distribution of the magnitude of displacement of the structure. It can be seen that the maximum magnitude of displacement is ~15nm at the center, as compared to ~10 nm for the ring actuator alone, fig 5.31. This is because the sensing plate attached to the ring PIPM behaves like a membrane capable of exhibiting large displacement due to the radial expansion and contraction of its edges. Due to the conservation of mass, the forced response of the ring PIPM sensing structure can be seen in figure 5.33.

The displacement can be resolved into its x, y and z components, namely, ux, uy, and uz, as

shown below.

Figure 5.34 Top view of the sensor actuator structure, showing the magnitudes of the x, y and z components of displacement

158

The displacement uz is directly proportional to the normal component of force, whereas, ux

and uy are proportional to the tangential components of force exerted by the actuator on to

objects placed on its surface. In order to better understand the force generated by the ring

PIPM sensing structure and thus the force experienced by a particle placed on its surface, the

magnitudes of displacement, normalized to the maximum displacement, is plotted along the

diameter and shown in figure 5.35.

Figure 5.35 The magnitudes of the resultant, normal and tangential components of displacement plotted along the x-axis of the ring PIPM sensing structure excited at 220 kHz.

It can be seen that there exists a non-uniform displacement distribution over the sensing surface. As with the radial mode descried in the previous sections, the vibrating surface exhibits high tangential displacement near the edge (uz  0.35ux) of the structure, and high

normal component of displacement near the center (uz  95ux).

A particle placed on the surface of this structure SAHS will experience a combination of normal-force, tending to reduce/break adhesion forces by forcing the particle vertically off the surface, and tangential forces, tending to move the particles horizontally over the surface.

The particles would therefore be clustered around the center of the TSM sensor. Particles uniformly distributed on an in-active structure, would be expected to be driven to regions of 159 low magnitude of displacement once activated at the radial mode of the ring actuator, thus forming a ring, having a radius of ~0.7 mm, around the center of the ring PIPM sensing structure.

5.3.5.2. Improvement in detection of 10 m polystyrene particles

The designed structure was tested with both, non-biological and biological particles. The first tests were performed with 10m – diameter polystyrene particles in DI water. The

concentration of particles was 6 x 103 partilces/5l of DI water. After dispensing 5l of the

solution on the sensing surface of the sturcture and allowing the particles to sediment, the

resonance frequency of the TSM was recorded and compared to a DI water baseline. The ring

PIPM was then turned on and the progression of the concentration of particles was monitored

optically as well as acoustically (monitoring TSM frequency response). 160

It was observed that frequency shifted from a DI water baseline by 2000 Hz, (f1 = 2000 Hz)

and can be seen in the bar graph in fig. 5.36(a), represented as the bar named “PIPM OFF”.

A CCD camera was used to take an image of the particles sedimented on the actuator surface

and is shown in fig. 5.36(b), thus confirming that the frequency shift, (f1), corresponds to a

uniform distribution of particles on the sensor surface. Next, the ring actuator was excited at

220 kHz with amplitude of 1 Vp, for a 5 second duration. The particles now experienced a force driving them from regions of high force to regions of low force, thus clustering in the form of a ring around the center of the sensing electrode, as seen in fig. 5.36(e). This 161 concentrating of particles led to a further 500 Hz increase in the frequency shift,

(f2 = 500Hz), corresponding to a 25% improvement in the observed response of the TSM

sensor to 10 m polystyrene particle loading.

5.3.5.3. Improvement in detection of 5 m silica particles

The ring PIPM sensing structure was then tested with 5m-diameter Silica particles,

following the same procedure as was done for the polystyrene particles, and the results can be

seen in fig 5.37.

162

The concentration of Silica particles used was 2.7 x 104 particles/ 5l. The same procedure as

mentioned above, for particle dispensing and sedimentation, was followed. After

sedimentation, with the PIPM OFF, a frequency shift of 600Hz, (f1=600Hz), was observed

with respect to a DI water baseline as can be seen from the bar graph, fig. 5.37 (a). The

corresponding optical image of uniformly distributed particles on the sensor surface can be

seen in fig. 5.37(b). Activation of the ring actuator led to the concentration of particles, as

seen from the optical images show in figure 5.37 (c) – (e). The concentration of particles

produced an additional 275 Hz shift (f2=275Hz) in resonance frequency which corresponds

to a 45% improvement in the TSM-sensor response of the SAHS to the silica micro particles.

Both polystyrene and silica particles have low densities, namely 1050 & 1500 Kg/m3,

respectively, and are dispersed in DI water, density ~1000 Kg/m3. Therefore, the observed

relative frequency change for a 100 MHz sensor was small, ~ 2% for polystyrene and 0.6%

for silica particles. However the signal was differentiable from the noise with an s/n of 13 for

polystyrene and 4 for silica. Thus any improvement in the response of TSM to the loading of

these particles plays an important role in the detection of these particles. The concentration of

the particles using this simple SAHS structure was able to improve the response of the sensor

to these relatively low density particles.

5.3.5.4. Improvement in detection of Escherichia coli

The above results of particle concentration and improvement of the response of the TSM led to the testing of the ring PIPM sensing structure with Escherichia Coli (E-Coli) bacterial spores.

163

Figure 5.38 represents an experimental result for the testing of the SAHS with E. Coli. The sensor was loaded with a 2 x 104particles/5l of DI water. For these experiments, the control represents a TSM sensor without a ring actuator attached to it. The sensor responds to the presence of E. Coli spores almost instantaneously and produces a resonance frequency shift of approximately 13 kHz, (f1=13kHz) in both, the Control and the ring PIPM sensing

structure. At time = 15 minutes, as shown, the SAHS is turned on, frequency = 220 kHz &

1Vp, for a period of 5 seconds. The ring PIPM sensing structure responds to the concentration

of particles with an additional frequency shift of 2.4 kHz (f2=2.4 kHz).

164

Figure 5.39 Frequency response of the ring PIPM sensing structure due to uniformly distributed and concentrated E. Coli spores.

As can be seen in fig. 5.39, there is a 22% improvement in the frequency response of the

TSM sensor due to the concentration of the E. Coli spores by the ring PIPM sensing structure.

5.4. Advantages and disadvantages of the PIPM technique

Based on the conducted studies, both theoretical and experimental, the main advantages and disadvantages of a PIPM are now listed:

Advantages:

 Can generate interfacial forces for particle manipulation

 Can be used to measure interfacial adhesion forces

 Can achieve single and multiple particle manipulation

 Predictable and reproducible manipulation is achieved via multimodal manipulation

 Can be extended to a multitude of structures as it uses inherent natural modes of

vibration of the structure to manipulate particles

 Can be used to design high sensitivity structures by placing sensing elements at nodes

of manipulation modes

 Can achieve high throughput manipulation 165

 Can be used for sorting of particles by size and/or property by proper calibration of

the manipulator and using selected modes at selected frequencies to manipulate

different particles.

 Can operate in liquid, air or vacuum.

Disadvantages:

 Limited by the available modes for each PIPM structure.

 Minimum particle size that can be manipulated is determined by ability to overcome

adhesion force

 Surface properties (such as roughness, contaminants etc.) can greatly impact

reproducibility of this technique

 The particle size should be much smaller than wavelength of sound in PIPM

5.5. Summary

The adhesion force binding stainless steel micro-particles to the PIPM surface was measured using an AFM and found to range from 15-200 nN. The PIPM material damping was determined to be 4.2% of critical damping, and was used to calibrate the displacement distribution over the PIPM surface. Using this damping factor, the displacement of the PIPM extracted via a FEA showed good correlation (less than 1% error) with the displacement

measured using the AFM over the range of excitation voltages used for manipulation

experiments. Measuring the voltage needed to initiate particle manipulation and converting it

to manipulation force exerted on stainless steel particles, 20 - 30m, it was found that the

adhesion force binding particles to the PIPM could be determined. The adhesion force

measured using this technique gave values that ranged from 150 -300 nN. It was also shown

that the force exerted on particles increases with increase in frequency of vibration, thus 166 indicating that operating at higher frequencies enables the manipulation of smaller particles.

The repeatable manipulation of particles by switching from one mode of vibration to the next was also demonstrated. Furthermore, it was shown that particles can be translated distances, several hundred times their radius at relatively high speeds of 5mm/s.

This manipulation technique was then incorporated with a biosensor to demonstrate its applicability for the improvement of the performance of the biosensor. The hybrid ring PIPM sensing structure, was simulated using a FEA and the modes of vibration, corresponding to particle concentration, were extracted. The hybrid structure was then constructed and tested with 10 m polystyrene, 5 m silica and E. coli bacterial spores, wherein the PIPM was used to concentrate particles at regions of high sensor sensitivity (near the center of the sensing electrode). The response of the sensor to the three types of particles showed an improvement due to the concentration of the particles as was demonstrated in the last section of this chapter. As was predicted theoretically, using FEA, the particles were forced to concentrate in a ring around the center of the biosensor. 167

6. CONCLUSIONS AND FUTURE WORK

6.1. Conclusions

The study of interfacial properties and the tools to operate at the interface and manipulate these properties is of growing importance. Here we demonstrated the ability of a piezoelectric

PIPM for the generation of interfacial forces that are used to estimate adhesion forces as well as manipulate micrometer and sub-micrometer sized particles.

The objectives outlined in chapter 1 have been achieved. The interfacial adhesion forces between micro-particles and the PIPM surface, used in the experimental study, have been quantified using an AFM. They are found to range in value from 15 – 200 nN for stainless steel particles ranging in radius from 11 – 32 m. These adhesion force values fall within the

force range that was theoretically estimated for the experimental surface conditions.

The PIPM displacement was calibrated using the AFM in contact mode. This calibration

enabled the determination of the PIPM material damping which in turn enabled the

quantification of the forces experienced by a particle when placed on the PIPM surface. The

FEM results enabled resolving the force experienced by a particle at different locations in to

its normal and tangential components, thus exposing the mechanism by which particles are

manipulated from one location to the next.

It was shown, theoretically and experimentally, that the PIPM is capable of generating the

required interfacial forces needed to manipulate particles ranging in radius of 20 – 30 m. It

was also demonstrated that operating at higher frequencies led to an increase in interfacial

manipulation forces, a property that can be utilized to manipulate particles of smaller size or

experiencing stronger adhesion forces. High particle translation speeds of approximately

5mm/s were observed, indicating that this technique is capable of high throughput 168 manipulation. Switching between modes of vibration was shown to provide a simple means to reproducibly manipulate particles between the nodes of the modes.

Lastly the manipulation technique was tested to improve the performance of a biosensor.

Experimental results showed good agreement with numerical expectations of particle

manipulation, which led to the improvement of the biosensor’s response to the concentration

of particles.

6.2. Contribution

The manipulation of micro and sub-micrometer particles is a challenging field sporting only a few available techniques. Here a novel technique based on basic principles of mechanical vibration of structures is demonstrated for the generation of interfacial manipulation forces to facilitate the translation of micrometer sized particles

A theoretical study outlining the forces that need to be generated to enable particle manipulation at a solid interface was conducted. Based on this study it was found manipulation forces of the order of 10’s -100’s of nN are required to be generated at the

interface. However, most current manipulation techniques generate forces on the order of

pN’s and are operated in the bulk of solution at a certain distance above the interface.

Therefore a relatively simple means to generate interfacial manipulation forces was proposed.

The current technique is based on the generation of low amplitude (1-100nm), high frequency

(30 kHz-10’sMHz) vibrations using acoustic actuators. This surface vibration led to the

generation of inertial manipulation forces of the order of 10’s -100’s nN (dependent on the

sample being manipulated) enabling the PIPM to detach particles from its surface. At these

operational parameters the minimum frequency and/or amplitude of displacement required to

enable particle manipulation was extracted. Furthermore it was shown that the detachment of

particles can be used to estimate the interfacial adhesion forces (providing an additional

feature to particle manipulation). 169

It was then demonstrated that generation of dynamic non-uniform surface displacement distribution can be used to exert forces at an angle to the PIPM. This led to the exertion of

impulsive manipulation forces on particles leading to their projection down the gradient of

the surface displacement (and therefore force). Furthermore, this non uniform displacement

distribution is generated by exciting inherent natural modes of vibration of the PIPM structure

(disc in the case of this thesis). Natural modes of vibration of structures have been widely

studied; however their use for the manipulation of micro- and sub-micrometer sized particles

has not been addressed.

In this thesis an experimental study of particle manipulation demonstrated the capability of

this technique to reproducibly manipulate micrometer sized particles while simultaneously

showing good correlation with theoretical expectations.

Beyond particle manipulation, the application of this technique for the improvement of a

thickness shear mode biosensor was demonstrated. The use of a ring shaped PIPM enabled

the generation of manipulation forces without significantly affecting the response of the

sensor. Improvement in detection of both biological and non-biological samples was shown.

The versatility of this technique was thus demonstrated, illustrating that the use of forced

natural modes of vibration are excellent candidates for particle manipulation.

Due to its simplicity in design, and flexibility in construction, this manipulation technique can

be applied to a wide variety of fields, such as, biosensors and biochips by providing features

of sample handling and improved sensing performance. It can impact the field of tissue

engineering by enabling non-destructive cell adhesion studies as well as the study of cellular

responses to interfacial forces. It can also be applied for the trapping and/or detachment of

drug particles for screening purposes in pharmaceuticals. Thus it can be seen that the studied

manipulation technique has the potential to have far reaching effects and only a small

percentage of its potential has been tapped. 170

6.3. Future work

6.3.1. Theoretical and experimental study of the manipulation of deformable

objects

The study in this thesis focused on rigid body manipulation, the scope of this technique will be greatly increased by performing a theoretical and experimental study of the manipulation of deformable objects.

6.3.2. Manipulation applied to a biochip

As discussed in the results wherein the PIPM was applied to a biosensor, here the application will be extended to a quartz biochip to provide features such as mixing and separation of bioparticles, as well as improving the performance of the sensing elements.

6.3.3. Acousto-magnetic manipulation

The current manipulation experiments with metallic particles will be operated in conjunction

with an external magnetic field. By controlling the strength of the magnetic field, the particle

surface separation distance and therefore adhesion force can be controlled. Thus for situations

in which the adhesion force is too large for the current manipulation technique to overcome,

the magnetic field force can be used to reduce the adhesion force, thus now allowing for the

manipulation of the particles. It will also enable improved quantification of the interfacial

forces.

6.3.4. Building of pre-programmed materials

There has been an increased interest in flexible materials, sensors, and displays which has boosted the development of composite materials [Seigel 2007]. Here we propose to use the nodal patterns of each mode to build materials from the bottom-up possessing predetermined properties in a particular direction. Thus by building structures, layer by layer, with the PIPM 171 being used to align particles in a specified direction for each layer, thin film having unique material properties can be developed.

6.3.5. Particle manipulation on demand, inverse problem

This study will be extended to provide a manipulation tool tailored to the user’s manipulation requirements. Here an attempt will be made to solve the inverse problem, wherein the user will be required to input the particle type, size and manipulation distance into the developed software. The user will then receive the optimal specifications of the manipulation tool to address his/her manipulation needs

6.3.6. Closed loop control to improve accuracy of particle manipulation

The particle manipulator in its current form is operated as an open loop system wherein knowledge of the initial location of the particle and the selected mode is used to predict the final particle destination (for that mode). However improved control of the particle motion can be achieved by introducing a feedback system wherein the particle position is tracked and continuously fed back to a central controller. This set-up will enable high resolution control of the particle motion while simultaneously allowing for improved efficiency in manipulation of particles via complex travel paths. 172

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182

Appendix 1. Analytical expressions for free vibrations of a piezoceramic disc

Figure A1.1 Schematic of a PIPM

Consider a thin disc having a radius, rm , and thickness, hm as shown in figure A1.1. The origin of the cylindrical coordinate system, (r,, z) coincides with the center of the disk half way through its thickness.

The transverse motion of a freely vibrating isotropic plate have been widely studied by several authors and the governing equations are well known [98-99, 104]

This analysis can be extended to piezoceramic plates which are considered to be transversely isotropic. Huang et al. modified the equations for isotropic thin plates and applied the same analysis to a piezoelectric ceramic disk [63]. The piezoceramic disk is polled in the z- direction and the following assumptions are made: the electrical potential, , varies

2 quadratically with the thickness, =0 +z1 + z 2, where 0 ,1,and 2 are constants, and the electrical displacement Dz is constant with respect to the thickness. When the piezoelectric plate is excited by an alternating voltage, Veit, the governing equation of motion is.

 2 D 4  h  0 A1.1 t 2

where  = (r,, z) is the displacement of the plate in the z-direction,  is the material density, D , is the equivalent flexural rigidity which is given by:

3 2 h 2  (1 p )k p D   E 2 2 A1.2 12 2s11(1 p )(1 k p ) 183

E 2 s11 2d31 Where,  p   E and k p  T E are the planar Poisson’s ratio and planar s12  33s11 (1 p )

E E electromechanical coupling coefficients, s11 and s12 are compliance constants measured at

T constant electric field, d31 , is the piezoelectric constant and, 33 , is the dielectric constant

measured at constant stress. 4 is the biharmonic operator, in cylindrical coordinates it is

given by:

  2 1  1  2    2 1  1  2   4   2  2            2 2 2   2 2 2   r r r r    r r r r  

Now assuming that the vibration of the middle of the plate is non-axisymmetric, an arbitrary

initial displacement  can be approximated as:

 (r,,t)  (r, )eit A1.3

Substituting equation A1.3 into equation A1.1, the governing equation becomes:

4 2 D   hm   0 A1.4

The solution of which is given by,

(n) (n) (r, )  [C1 J n ( 2r)  C2 I n ( 2r)]cosn, A1.5 n  0,1,2,3,....,

(n) (n) th Where C1 and C2 are constants of integration, Jn and In are n order Bessel functions of

h  2 the first and modified first kind respectively and  4  m 2 D

At r = rm, the circumferential bending moment is given by,

 2  1  1  2  M  D     A1.6 r  2 p  2 2   r  r r r  

And the Kelvin-Kirchhoff edge reaction is given by, 184

2   2 1 p   1   Vr  D       A1.7 r r r  r  

For traction free boundary conditions, Mr = 0 and Vr = 0. Substituting equating A1.5 in

equations A1.6 & A1.7 and using traction free boundary conditions, the characteristic

equation of transverse vibration is obtained:

(1 )  1 k 2  p 2  p 2 [n J n (m )  J n (m )]    2 m J n (m ) (1 p ) (1 p ) 2(1 k p ) (1 )  1 k 2  p 2  p 2 [n I n (m )  I n (m )]    2 m I n (m ) (1 p ) (1 p ) 2(1 k p )   A1.8 (1 )  1 k 2  p 2  p 3 n [J n (m )  m J n (m )]    2 m J n (m ) (1 p ) (1 p ) 2(1 k p )    (1 )  1 k 2  p 2  p 3 n [I n (m )  m I n (m )]    2 m I n () (1 p ) (1 p ) 2(1 k p )

Where m =2rm.

Huang et al showed that by using an effective Poisson’s ratio:

2 2  (1 p )(2  k p )   2 A1.9 2  (1 p )k p

The frequency equation (equation A1.8) for a Piezoceramic disk takes the same form as that

for an isotropic disk:

(1 )[n2 J ( )   J ( )]   2 J ( ) (1 )n2[J ( )   J ( )]   3 J ( ) n m m n m m n m  n m m n m m n m 2 2 2 3 (1 )[n I n (m )  m I n (m )]  m I n (m ) (1 )n [I n (m )  m I n (m )]  m I n (m ) A1.10

The transverse resonance frequencies of the piezoceramic disk are found to be:

2 2 m hm 2  (1 p )k p f R  2 E 2 2 A1.11 2rm 24s11 (1 p )(1 k p )

As is seen in equation A1.11, the transverse resonance frequencies are dependent on the geometric and material properties of the plate. 185

The analytical expressions for transverse vibration, given by equations A1.10 & A1.11 , can be obtained, however, it is not convenient to solve the transcendental equation for a large number of mode. Finite Element Method offers an effective and efficient means to perform a natural frequency, steady state and/or transient analysis of the piezoceramic disc. The vibration of the disc in fluid medium can lead to the generation of acoustic waves

(longitudinal/dilatational and shear) that propagate and/or penetrate the medium adjacent to the PIPM.

186

Appendix 2. Interfacial forces

A 2.1 Acoustic forces

The vibration of the PIPM at the excited frequencies can lead to the generation of acoustic waves transmitted into the medium adjacent to the PIPM surface. A particle placed on the

PIPM surface would experience an acoustic wave force due to the synergistic action of shear and compressional wave which are now treated individually.

The transmission of acoustic energy from the PIPM to the adjacent medium is dependent on the difference in acoustic impedances of the two media and is given by,

Tc 1Rc A2.1

Where T is called the transmission coefficient and R is called the reflection coefficient given

by,

2  Zceramic  Zmedium  Rc    A2.2 Zceramic  Z medium 

where, Zceramic, Zmedium, are the impedances of the piezoelectric PIPM and medium

respectively. The acoustic impedances for air and water and piezoelectric ceramic are given

in table A1

Table A1.1 Acoustic impedances and transmission coefficients for an acoustic wave

generated by the piezoelectric ceramic and traveling into air and water [Kinsler]

187

As can be seen, only a fraction of the input energy is transmitted from the PIPM to the air- molecules above the actuator, whereas almost 18% of the energy is transmitted if the medium is water.

A 2.1.1 Compressional wave force: Fcomp:

The compressional force is generated due to the compressional deformation of the PIPM.

This wave travels in the z-direction direction exerting a normal force on a particle placed on the surface of the PIPM as shown in figure A2.1

Figure A2.1 The Compressional wave motion exerts a force in the vertical direction.

The particles sizes under consideration in this project are much smaller (2 orders of

magnitude smaller) than the wavelength of the acoustic wave force that they experience. With

respect to the particle, the compressional wave can be considered as a plane wave. The force

per unit area exerted by the compressional wave on the particles along the direction of

propagation is given [105] by the expression:

 P m   c  s max A2.3

Where, Pm is the maximum value of the pressure or force F, per unit area A, ρ is the mass

density of the medium, c is the speed of sound in the medium (air or water),  is the angular

frequency of the acoustic wave and smax is the maximum displacement of the air element adjacent to the PIPM.

The intensity of the compressional wave is as shown in equation A2.4 188

1 I   c  s 2 A2.4 2 max

where, I is the intensity of the periodic acoustic wave.

Then the pressure, or force per unit area, at any time, t, and at any point along z direction

(which is the direction of propagation of the compressional wave), is given as:

Pi (z,t)  Pm cos(k  z  t) A2.5

The compressional force acting on the surface of the particle between z and z + dz is:

dFcomp (z,t)  Pm cos(k  z  t)(2  R sin )  R cos  d A2.6

Finally, the compressional force acting on the whole particle is found by summing the effect

on each layer performed by,

 2 Fcomp (t)  Pm cos(k  z  t)2  R sin cos d 0

 2  (Pm  2  R ) cos(k  z  t)sin cos d 0

2 2 R 2 R  z z  2z R  (Pm  2  R ) cos(k  z  t)    dz R R R 2R  R 2  z 2 z

R  (Pm 2 ) cos(k  z  t) z dz, R

Solving the above equation we get the periodic compressional wave force,

4 1 F (t)   P sint [R  cos(kR)  sin(kR)] A2.7 comp k m k

Since this is a periodic force, it could have positive or negative value. Using equations A2.3,

A2.4 and A2.7, the compressional wave force exerted on a particle can be determined.

Assuming continuity at the PIPM-air interface, the displacement of the air element is

assumed to be that of the adjacent PIPM surface.

Consider a 25 m radius particle placed on the PIPM surface, vibrating at 65 kHz with a peak displacement of 1nm, it is found that the periodic compressional wave force in air is:

-11 Fcomp(t)  1.32 x 10 sin (t) 189

As has been shown, the compressional wave force is about 2 orders of magnitude smaller than that of the adhesion force binding the particle to the surface and therefore would not be capable of detaching it from the surface. In a denser medium like water, it was calculated to be,

-9 Fcomp(t)  10 x 10 sin (t)

The periodic compressional wave force in water is of about same order of magnitude at that of the adhesion force and therefore could aid in detaching the particle from the PIPM surface.

It should be noted that the net effect of a periodic force is zero, however acting in conjunction with the inertial force it could aid in particle detachment.

A2.1.2 Shear Force

The horizontal vibration of the manipulator could lead to the generation of a shear wave that penetrates the adjacent medium. This is an evanescent wave that decays within a short distance above the PIPM as shown in Fig. A2.2. This acoustic wave force could potentially act to reduce the frictional force between the particle and the surface.

Figure A2.2 (a) Shear acoustic wave penetrating the adjacent medium (air in this case)

applying a force parallel to the PIPM surface to the adjacent medium. (b) Depth of

penetration of the shear wave is inversely proportional to the square root of the frequency of

the input signal.

The distance that the wave penetrates the adjacent medium (air in our case) is known as the

Penetration Depth (PD). It is given by the expression (A2.8). 190

2 PD  A2.8  where, η is the shear viscosity of air,  is the angular frequency and  is the density of air.

The penetration depth is inversely proportional to the square root of the frequency and thus,

operating at higher frequencies reduces the depth of penetration. The penetration depth in air

for frequencies ranging from 0.04 – 2 MHz is plotted in figure A2.3

Figure A2.3 The penetration depth (PD) of a shear acoustic wave in air and water as a

function of frequency. At 65 kHz, the PD in air and water is 8.6 and 2 m respectively.

As seen from Fig. A2.3 of penetration depth vs. frequency, the shear wave penetrates a

distance of 8.6 and 2 m in air and water respectively when operated at 65 kHz. The shear

force exerted on a particle placed on the PIPM surface is calculated as follows:

The shear force per unit area or stress is given by:

v (z,t) Shear Stress is T (z,t)    x A2.9 xz z where, , is the dynamic viscosity of the medium, and vx(z,t) is the tangential/shear velocity

of the medium given by expression A2.10. 191

z vx (z,t)  vx0 e cos( t  k  z) A2.10

Differentiating with respect to z gives the shear stress shown in expression (A2.11)

z Txz (z,t)  vx0   e (k sin( t  k  z)  cos( t  k  z)) A2.11

where, k is the wave number and  is the angular frequency.

Assuming continuity of tangential displacement and shear stress at the PIPM-air interface, the velocity of the air element adjacent to the PIPM, v0x, is equal to that of the PIPM surface.

Assuming a uniform shear vibration of the surface, the shear wave force on a particle can be

calculated by sectioning the particle into rings and summing up the force experienced by

individual rings

The magnitude of the shear force on a 25 m particle in air was calculated to be 5 orders of

magnitude smaller than the frictional force, whereas in water it was found to be 3 orders

lower.

Therefore it has been shown that the acoustic wave forces do not play a significant role in

particle motion at the particle-PIPM interface.

However, if the PIPM is operated to exhibit large displacement, the non-linear acoustic wave

effects close to the PIPM surface could impact the motion of the particles. 192

Appendix 3. AFM spring constant calibration

In chapter 5, the adhesion force binding stainless steel particles was presented. Since the addition of the particles to the cantilevers constitutes an additional mass on it, this could lead to a possible change in the spring constant of the cantilevers. We therefore calibrated the spring constant of the cantilevers that were used to measure the adhesion force. The calibration process is now briefly outlined.

By recording the resonance frequencies before and after the particles are attached to the

AFM tip and using optically measured particle size to determine its mass, the modified spring constant can be determined using the Cleveland method,

2 (2 ) M p keff  A3.1  1 1   2  2   f pc f c 

Where, Mp is the mass of the particles, fc and fpc are the resonance frequencies of the

cantilever without and with the attached particle respectively. Equation A3.1 is only valid if a

-2 linear relationship between added mass (Mp) and (2fpc) is observed [108]. This is

demonstrated below,

193

1200 R2 = 0.9805 1000

800

600

Mp (ng) 400

200

0 01234567

-2 -9 2 (2 fcp) (x10 ) (s )

-2 Figure. A3.1 A plot of added mass vs (2fpc) . Linear regression of the data, gives a

correlation coefficient of 0.98.

Thus as seen in fig A3.1, a linear relationship is observed, thus enabling us to determine the effective spring constant using equation A3.1, the values of which are shown in table A3.1.

Table A3.1 The data consisting of the resonance frequencies of the cantilever before, fc, and

after particles are added, fcp, to the cantilever tip, which are then used to calculate the

effective spring constants, keff, for each modified cantilever.

Using the calculated effective spring constants for each tip, the deflection of the cantilever due to adhesion force, measured using the AFM, was converted to force values using

Hooke’s law. 194

Appendix 4. Electrical analysis of the PIPM

Piezoelectric transducers can be represented by a three-port network as shown in Figure A4.1.

The characteristics of the three-port network are represented by the parameters at the three ports, the force and velocity at the acoustic port 1 and port 2, and the current and voltage at the electrical port 3.

Figure A4.1 Acoustic PIPM modeled as a three port network

The forces and particle velocities at the two acoustic ports are defined as F1, F2 and v1 , v2,

respectively. Based on the relations among the stress, velocity and displacement of the

acoustic wave, the forces and velocities at the two surfaces of the piezoceramic manipulator,

z = -hm/2 and z = hm/2, can be defined as 195

F  A T h A4.1 1 m m z m 2

F  A T h A4.2 2 m m z m 2

v  v h 1 m z m A4.3 2

v  v h 2 m z m A4.4 2

Where, hm, is the thickness of the piezoceramic disc, Am is the area of the top/bottom of the

disc that has silver deposited for electrical connections.

The mechanical impedance at ports 1 and 2 are given by,

A T h m m z m F1 2 Z1    A4.5 v v h 1 m z m 2

A T h m m z m F1 2 Z1     A4.6 v v h 1 m z m 2

According to Gauss’s law, the current at the electrical port, I3 , is linearly proportional to the

electric displacement field inside the piezoelectric transducer, D, given by,

I 3  jAm D A4.7

The electrical impedance at port 3 is given by,

V3 Z 3  A4.8 I 3

The relations between these parameters at three ports determine the characteristics of the piezoelectric manipulator. By considering the piezoelectric constitutive equations and the boundary conditions, the relations can be found as [Kino ] 196

 hT  Z m cot m hm Z m csc  mhm  F1   V1     h    T  F2    j Z m csc m hm Z m cot  mhm V2  A4.9       V   hT hT 1 V   3    3     C0  where, ω and βm are the angular frequency and wave number of the acoustic wave inside the

PIPM, respectively; Zm is the acoustic impedance of the quartz plate; C0 is the static

capacitance of the of the piezoelectric transducer; hT is the transmitting constant. These

parameters are defined as,

   m   A4.10 v D m cm  m

D Z m  Am Z 0  Am cm  m A4.11

S  m Am C0  A4.12 hm

em hT  S A4.13  m

In the above equations, ρm and em are the mass density and the piezoelectric stress constant of

S the manipulator, respectively.  m is the permittivity of the piezoceramic material at constant

D strain, S, cm is the elastic constant of the PIPM when the electric displacement field, D, is

E zero or constant. Another form of the elastic constant is cm , represents the elastic constant

with constant electric field, E. These two elastic constants are linked by the piezoelectric

2 coupling constant, Km , by the following equations, 197

 e 2  c D  c E 1 m   c E 1 K 2 A4.14 m m  E S  m m  cm  m 

2 2 em K m  E S A4.15 cm  m

Algebraically solving the matrix, the electrical impedance at port 3, defined in Eq. 2.20, can be related to the mechanical impedance at the two acoustic ports, Z1 and Z2, by

 2 2  1 K m 2Z q cos mhm 1 jZ q Z1  Z 2 sin mthm  Z3  1  2  A4.16 jC0   mhm Z q  Z1  Z 2 sin m hm  jZ q Z1  Z 2 cos  mhm 

K 2 2 m Where K m  2 A4.17 1 K m

Based on the three-port network model of a piezoelectric plate, a Transmission Line Model

(TLM) can be derived from Eq. A4.9

First, the 3 × 3 matrix on the right side of Eq. A4.9 can be considered as the impedance

matrix relating the velocity or current, v1, v2 and I3, to the force or voltage, F1, F2 and V3, at

the three ports. The four parameters on the top left corner of the impedance matrix are

defined as

Z11  Z 22   jZ m cot( m hm ) A4.18

Z12  Z 21   jZ m csc( m hm ) A4.19

Therefore F1, F2, and V3 can now be written as 198

h F  Z  Z v  Z v  v  T I A4.20 1 11 12 1 12 1 2 j 3

h F  Z  Z v  Z v  v  T I A4.21 2 22 12 2 12 1 2 j 3

hT 1 V3  v1  v2  I 3 A4.22 j jC0

The first term on the right hand side of Eq. A4.20 represents the voltage generated by a current, v1, through the impedance, Z11 – Z22. Similarly, the second term represents the

voltage generated by a current, v1 + v2, through the impedance, Z12. The third term represents

the voltage proportional to the current, I3, seen at port 3. By analyzing all three equations Eq.

A4.20, 21 & 22, a TLM named Mason series equivalent circuit is constructed as in Figure

A4.2 [109].

Figure A4.2 Mason series equivalent circuit

This model makes use of a transmission line representing the propagation of an acoustic wave within a piezoelectric transducer and a transformer representing the electro-mechanical coupling between the acoustic ports 1, 2 and the electrical port 3. The turn ratio of the transformer, NT, is given by 199

S em  m Am em Am NT  hT C0  S  A4.23  m hm hm

When port 3 is open circuit or I3 = 0, the third terms in both Eqs. A4.20 and A4.21 equal to zero and the Mason equivalent circuit can be represented by a T–network equivalent of a coaxial transmission line with impedance of Zm.

A Lumped Element Model (LEM) of can be created by analyzing the Mason series equivalent circuit model in Figure A4.2 and the electrical impedance at port 3 given in Eq. A4.16. Based on the LEM of an unloaded acoustic transducer, models for different types of loadings have been reported [110].

LEM for an Unloaded PIPM (the microparticles used for experimental purposes do no represent a load because the particle mass<1000*PIPM mass)

Here, both top and bottom PIPM surfaces are stress free, Tm1 = Tm2 = 0. The mechanical

forces at port 1 and 2 become zero, F1 = F2 = 0, based on Eq’s A4.1& A4.2. Similarly, the

acoustic impedance at port 1 and port 2 are also found to be zero, Z1= Z2 = 0, from Eq’s A4.5

& A4.6. Therefore, the Mason equivalent circuit in Figure A4.2 can be simplified for an

unloaded resonator, as shown in Figure A4.3

200

Figure A4.3 Mason series equivalent circuit for an unloaded transducer

The electrical impedance at port 3 for an unloaded resonator is now given by,

   m hm   tan  1 2 Z C Z mot  2   0 A4.24 Z 3  1 K m  jC0   m hm  Z C  Z mot   0  2  where, ZCo is the impedance of the static capacitance, C0, given by Eq. A4.12 and Zmot is the

motional impedance. The expressions for the two impedances are given by,

1 Z  A4.25 C0 jC0

  h   m m  1  1 2  A4.26 Z mot  2 1 jC0  K m   m hm    tan     2  

Therefore, the electric impedance of an unloaded piezoelectric transducer can be represented by two impedances in parallel, ZCo and Zmot. In the vicinity of the resonance mode of vibration

the transducer can be operated as a resonator. Thus the impedance Zmot can be approximated

by the total impedance of a series LCR resonator shown in Eq.A4.26. 201

1 Z mot  jL1   R1 A4.27 jC1

Where,

8K2 C  m C A4.28 1 N 2 0

1 L1  2 A4.29 N C1

m R1  A4.30 cmC1 where, μm is viscosity of the piezoelectric manipulator; N is the order of harmonic frequency,

th N = 1, 3, 5… , ωN is the N harmonic frequency.

D cm vm m N  N  N A4.31 tm tm

Therefore, a lumped element model for an unloaded piezoelectric resonator, called

Butterworth-Van Dyke (BVD) model, is constructed and shown in Figure A4.4.

Figure A4.4 A lumped element model of an unloaded piezoelectric resonator

t resonance, the transducer exhibits zero motional reactive impedance, the inductive and

capacitive terms in the motional arm cancel out giving, Z mot  R1 0 202

Therefore figure A4.4 reduces to,

Figure A4.5 LEM of an unloaded transducer excited at resonance, V is the source voltage, R0

is the source resistance.

/ / As shown in figure A4.5, the parallel R1 and C0 is equivalent to C 0, and R 1 in series with the

source resistance R0,

R  1 R1  2 2 2 A4.32 1 0 R1 C0

1  2 R 2C 2  0 1 0 C0  2 2 A4.33 0 R1 C0

Therefore the impedance ZL is,

1 Z L  R1  A4.34 jC0

We can now determine the efficiency of the power transferred to the piezoelectric transducer

when excited at its resonance frequency.

Consider the circuit shown in figure A4.5, where R0 is the output impedance of the signal generator, the power supplied to the PIPM is given by, 203

2 Vp p PS  A4.35 8(R0  Z L )

Whereas the power consumed by the PIPM at resonance when supplied with a voltage Vp-p, is

given by,

2 Vp p PL  2 Z L A4.36 8(R0  Z L )

The efficiency is defined as the ratio of power dissipated by the transducer to power developed by the source, given by,

PL  P  A4.37 PS

The transducers used for particle manipulation were operated at their radial resonance frequencies wherein the following electrical parameters and efficiency was measured.

Table A4.1 Measured electrical circuit parameters of piezoceramic discs excited at their

radial modes of vibration

Since the transducers are operated at their resonance frequency the transducer offers almost

zero motional reactive impedance and most of the power is dissipated in the resistance R1. It

can be seen from table that each transducer has a different efficiency and therefore while 204 studying particle manipulation as a function of excitation voltage care was taken to record the voltage drop across the transducer and not the excitation voltage.

Efficiency off resonance: due to high reactive impedance (mostly capacitive) below resonance and above anti-resonance, the excitation of modes in these frequencies leads to lower efficiency. 205

Appendix 5. Particle motion due to momentum transfer

Momentum transfer equations for pure elastic collisions

For one dimension collisions, considering only normal impact of the two bodies, we have not

introduced the direction along which the collision takes place. In the next section however,

these normal collisions will be referred to as occurring along the z-axis.

mp mass of the particle

mm mass of the surface vpiz and vmiz – Initial (original) velocities of particle, ‘p’ and surface ‘s’

vpzf and vmfz – Final velocities after collision

For a fully elastic collision, the energies and momentum of the colliding bodies are conserved, i.e.

m v2 m v2 m v2 m v2 p piz  m miz  m mfz  p pfz A5.1 2 2 2 2

mpv piz  mmvmiz  mpv pfz  mmvmfz A5.2

Solving the simultaneous equations A5.1 & 2 algebraically we can find the final velocities

after the collision. Redefining eq A5.1 and A5.2 in terms of final particle velocity, vpf.,

2 2 2 2 mpv piz  mmvmiz  mmvmfz v pfz  mp

A5.3

2  m v  m v  v2   p piz m miz  A5.4 pfz    mp 

206

Equating A5.3 and A5.4

2  m v  m v  m v2  m v2  m v2  p piz m miz   p piz m miz m mfz A5.5    mp  mp

Since the masses of the particle and the surface, and the initial velocities are known, it can be

seen in equation A5.5 that the only unknown is the final velocity of the surface. Expanding

2 the LHS of equation A5.5 multiplying both sides with mp we get

2 2 2 mm vmiz  mm vmfz  2m p mm v piz vmiz  2mm vmiz vmfz  2m p mm v piz vmfz A5.6

2 2  mm m p vmiz  mm m p vmfz  0

 2  Finally rearranging equation A5.6 in the quadratic form, avsf  bvsf _ c  0 , we have,

2 2 2 mm  mm m p vmfz   2mm vmiz  2m p mm v piz vmfz A5.7 2 2  mm vmi  2m p mm v piz vmiz  mm m p vmiz  0

Let,

2 a  ms  ms m p 

2 b   2mmvmi  2m p mmv pi

2 2 c  mmvmiz  2m p mmv piz vmiz  mm m pvmiz

The roots of equation A5.7 are found using the quadratic formula:

 b  b2  4ac v  A5.8 sf 2a

Finding individual terms of equation A5.8, 207

2 2  b   2mmvmiz  2m p mmv piz  2mmvmiz  2m p mmv piz

2 2 2 2 2 2 3 b  2mmvmiz  2m p mmv piz  4mmvmiz  m p mmv piz  2mmm pvmiz v piz A5.9

4 2 2 2 2 3  4mmvmiz  4m p mmv piz  8mmm pvmiz vmiz

2 2 2 4ac  4mm  mmmp mmvsiz  2mpmmvpizvmiz  mmmpvmiz 

4 2 3 2 3 2 2 3 2 2 2 2  4mmvmiz  mmmpvmiz  2mmmpvpivmiz  2mmmpvpizvmiz  ms mpvmiz  mmmpvmiz A5.10

4 2 3 2 2 2 2 2  4mmvmiz  8mmmpvpivmiz  8mmmpvpizvmiz  4mmmpvmiz

2 4 2 2 2 2 3 4 2 3 b  4ac  4mmvmiz  4mpmmvpiz  8mmmpvmizvpiz  4mmvmiz  8mmmpvpivmiz

 8m2 m2v v  4m2 m2v2 m p pi miz m p miz A5.11

2 2 2 2 2 2 2 2  4mpmmvpiz  4mmmpvmiz  8mmmpvpivmiz

2 2 2  4mpmm vpiz  vmiz

We can now get the final velocities of the particle and the surface using A5.9-11.

 b  b2  4ac v  mfz 2a

2m2 v  2m m v  4m2 m2 v  v 2 v  m miz p m piz p m piz miz mfz 2m2  m m m m p m2 v  m m v  m2 m2 v  v 2  m miz p m piz p m piz miz mm mm  m p

Therefore the final velocity of the surface is,

m pv piz  mmvmiz  m p (v piz  vmiz ) vmfz  A5.12 m p  mm

The final particle velocity is given by the mirror image, 208

mpvpiz  mmvmiz  mm (vmiz  vpiz ) vpfz  A5.13 mp  mm

Thus equations A5.9 & 10 represent the final velocities of two objects undergoing head-on collision. For experimental conditions, mm is the mass of the PIPM which is much larger than

that of the particle, i.e., mm >> mp, therefore, the above equations reduce to

vmfz  vmiz A5.14

v pfz  2vmiz  v piz A5.15

Please note that equations for the final velocities, derived above, was performed for pure

elastic collisions wherein both momentum and kinetic energy is conserved during the

collision. However, at the micro- and nano-scale, the process of detachment of particles from

a surface involves the loss of energy, thus leading to a decrease in the final particle velocity

after impact. This loss of kinetic energy leads to the collision being inelastic.

We will now reformulate the problem to represent 2 dimensional collision where the

coefficient of restitution, e, is used to define the energy loss during normal impacts (along z-

axis) and a coefficient of friction term is introduced to account for tangential energy losses.

Momentum transfer equations with coefficient of restitution and friction for two body collisions

energy can take place [95].

An impulse-momentum formulation is adopted as it provides simple algebraic equations from

which the initial and final velocities can be determined. 209

Here we discuss rigid body collisions between a particle of mass mp and the PIPM of mass

mm. Here we assume no spin is introduced at the point of collision, which, if did occur, would need to be represented as a change in the angular velocity of the particle at the point of collision.

Figure A5.1 Free body diagram of two colliding rigid bodies, namely the particle and the

PIPM, in x-z plane. (Not to scale)

Figure A5.1 represents a free body diagram of the two bodies, with the x-z coordinates

chosen such that the lines through the particle and PIPM centers are along the z-axis (normal

axis).

The velocity symbols have three subscripts, the first subscript indicating particle ‘p’ or

manipulator ‘m’, the second subscript indicating initial ‘i’ or final ‘f’ velocities, and the third

indicating normal ‘z’ or tangential ‘x’ direction. For example, vpix is the initial tangential

velocity of the particle.

The conservation of momentum for the two bodies along the z-axis is given by, 210

m pv pfz  mmvmfz  m pv piz  msvsiz A5.16

The conservation of momentum for the two bodies along the x-axis is given by,

m pv pfx  mmvmfx  m pv pix  ms vsix A5.17

The loss of energy at the point of collision is commonly represented by the coefficient of

restitution,

v  v e  pfz mfz A5.18 vmiz  v piz

Which can be rewritten as,

v pfz  vmfz  e(v piz  vmiz ) A5.19

The value of e lies between 0 and 1, i.e. 0  e  1, where, e = 1, represents a fully elastic collision and for e = 0, the collision is purely inelastic or plastic collision.

The loss in the tangential energy is governed by the tangential impulse, Px, developed during

the collision. Brach 1984 [96]defined an equivalent coefficient of friction, , as,

P   x A5.20 Pz

Where the tangential and normal impulses are given by,

Pz  m p (v pfz  v piz )  mm (vmfz  vmiz ) A5.21

Px  m p (v pfx  v pix )  mm (vmfx  vmix ) A5.22

The linear combination of equations A5.20-22 gives,

m p v pfz  m p v pfx  mm vmfz  mm v pfx A5.23

 m p v piz  m p v pix  mm vmiz  mm v pix

Relative tangential motion always exists at the beginning of contact as long as

vmix  v pix  0 If the relative tangential motion ceases during contact then, 211

v mfx  v pfx  0 A5.24

Thus we now nave four unknowns, i.e., the final velocities, and four linear algebraic equations, A5.16, A5.17, A5.19 and A5.23 or A5.24.

These four equations, A5.16, A5.17, A5.19 and A5.23, are put in a matrix form for more convenient viewing,

 mp 0 mm 0 vpfz   mp 0 mm 0 vpiz         0 m 0 m v   0 m 0 m   p m  pfx  p m vpix        A5.25     1 0 1 0 vmfz  e 0 e 0 vmiz        m  m  m  m  v m  m  m  m   p p m m  mfx   p p m m vmix 

The solution can be written in terms of the four unknowns, namely the final velocities,

mm (1 e)(vmiz  v piz ) v pfz  v piz  A5.26 m p  mm

mm (1 e)(vmiz  v piz ) v pfx  v pix  A5.27 m p  mm

m p (1 e)(vmiz  v piz ) vmfz  vmiz  A5.28 m p  mm

m p (1 e)(vmiz  v piz ) vmfx  vmiz  A5.29 m p  mm

If the relative motion between the two bodies ceases during the collision, the governing equations are, A5.16, A5.17, A5.19 and A5.24, and thus the final tangential velocities take a different form.

mp 0 mm 0 vpfz  mp 0 mm 0 v piz         v   v  0 mp 0 mm  pfx  0 mp 0 mm  pix         A5.30 1 0 1 0 vmfz   e 0  e 0 vmiz          v    0 1 0 1 mfx   0 0 0 0 vmix  212

The solutions for equation A5.30 for the normal final velocities are identical to equations

A5.1 26 and A5.28. The final tangential velocity component solutions are,

mm (vmix  v pix ) v pfx  v pix  A5.31 m p  mm

m p (vmix  v pix ) vmfx  vmix  A5.32 m p  mm

In order to determine whether to use equations A5.27 & A5.29 or equations A5.31 & A5

32, a maximum friction coefficient m is defined as,

vmix  v pix 1  m   A5.33 vmiz  vmiz (1 e)

Thus if  <m sliding exists at separation and equations A5.26-29 give the final velocities,

whereas if   m sliding ceases during collision and the final velocities are given by

equations A5.26, A5.27, A5.31 & A5.32.

It should be noted that in the equations above, the spin of the particle has not been included.

Although spin could play a role during the collision of the particle with the PIPM, it

introduces another dimension of complexity to the problem. For the current study, in order to

demonstrate the mechanism of particle motion along the surface of the PIPM it is sufficient,

but not entirely complete, to limit the study to conditions of no spin. This has however been

addressed by several authors, Brach 1984, Wang 1992 etc.

213

Appendix 6. Angle of applied force

Figure A6.1 Force balance diagram in which F is the applied force attempting to move the particle when applied at an angle  with respect to the x-axis.

As seen in figure A6.1, “F” is an external force applied to the particle at an angle “”. Here

we show how to determine the angle at which the minimum magnitude of the force “F” is

required to initiate particle motion. [111]

The coefficient of static friction is given by the ratio of the static friction force to the normal

force,

Ffriction _ s s   tan  A6.1 FN

Where  is called the angle of friction, and is defined as the angle that the maximum contact

force, Ffriction_s (called the static friction force), makes with the direction of normal force.

Consider the diagram above, instead of the downward force being only that due to

gravitational force, we have now used “Fadhesion”, as we are working with microparticles and

the force binding the particle to the surface is a combination of van der Waal’s, capillary,

electrostatic and gravitational forces. FN is the normal reaction force exerted by the surface on the particle.

In equilibrium, 214

 Fx  F cos  s FN  0 A6.2

 F cos  s FN A6.3

F cos  FN  A6.4 s

 Fz  F sin  Fadhesion  FN  0 A6.5

 FN  Fadhesion  F sin A6.6

 F F  s adhesion A6.7 cos  s sin

Figure A6.2 Force as a function of the angle at which it is applied (wrt x-axis)

Assuming a friction constant, s = 0.5, and adhesion force, Fadhesion = 100nN, the force needed to move the particle is plotted as a function of angle with the horizontal x-axis, equation A6.7.

It is seen that at  = 0 degrees, the force needed is equivalent to overcoming the friction force

= 0.5 x Fadhesion = 0.5 x 100 = 50 nN. As the angle of force is increased, the force needed to move the particle decreases to a minimum of ~44nN at ~ 25 degrees after which it once again increases and reaches a maximum value of 100 nN at  = 90 degrees when it is acting vertically in opposition to Fadhesion.

This minimum value of applied force can be found theoretically as follows: 215

Equation A6.7 can be further simplified using eq. A6.1

F tan F  adhesion A6.8 cos  tan sin

Therefore,

F sin F  adhesion A6.9 cos( )

The angle at which F is minimum is obtained when the denominator of eq. A6.9 is maximum,

i.e.  = , this give,

Fmin  Fadhesion sin  A6.10

1 1 1 Since tan =s, and cos    sec 2 2 1 tan  1  s

Therefore we get,

 sin  1 cos 2   s A6.11 2 1  s

Substituting eq. A6.11 in eq. A6.10 we get,

 F F  s adhesion A6.12 min 2 (1 s )

Equation A6.12 represents the minimum force needed to move the particle.

216

Appendix 7. Program for the processing of AFM data files

% Data processing for multiple data files. % Output the processed data into a .txt file. clear all; fprintf('\n\n------\n') fprintf(' Program Starts here!\n\n')

% Search for the Number_Of_Rows_of_Index % num_row_index = 736; % number of rows of index % number_of_data_rows = 512; % number of rows of data % number_of_data_columns = 5; % number of data columns spring_constant = 0.12; % spring constant of AFM cantilever, unit: N/m or nN/nm data_file_path = 'F:\091006\091006_1_txt\'; % directory in which data files are stored. data_file_date = '091006_1.'; % data files are named as YYMMDDxxx, where YY is year, MM is month, DD is day, xxx is sequence no. data_file_start = 75; number_of_data_files = 250; % total number of data files generated in the same day. output_file_path ='F:\091006\091006_txt\test\'; % directory in which output file will be stored. output_file_name = [data_file_date '.txt']; output_file = [output_file_path output_file_name]; fid_output = fopen(output_file, 'w+'); fprintf(fid_output, 'data decreasing decreasing increasing increasing decreasing increasing decreasing decreasing decreasing '); fprintf(fid_output, 'increasing increasing increasing deflection deflection deflection deflection\r\n'); fprintf(fid_output, 'file start stop start stop correlation correlation slope intercept sensitivity slope intercept sensitivity '); fprintf(fid_output, 'sensitivity voltage distance force\r\n'); fprintf(fid_output, 'name point point point point R^2 R^2 (V/nm) (V) (nm/V) (V/nm) (V) (nm/V) (nm/div) (V) (nm) (nN)\r\n'); for i = data_file_start : data_file_start + number_of_data_files - 1

% create data file name if i < 10

data_file_name = [data_file_date '00' num2str(i)]; elseif i < 100 data_file_name = [data_file_date '0' num2str(i)]; elseif i < 1000 217

data_file_name = [data_file_date num2str(i)]; end

data_file = [data_file_path data_file_name '.txt']; fid_data = fopen(data_file, 'r');

if fid_data == -1

fprintf(fid_output, '%s %s %s \r\n', 'Data File ', data_file_name, ' Can Not Be Opened!'); fprintf('%s %s %s \n', 'Data File ', data_file_name, ' Can Not Be Opened!');

else

counter_of_rows = 0; label_extend = 'Time_s'; %'Extend'; %label_retract = 'Retract';

% Find the row number where data starts and the number of rows for index while 1

temp = fgetl(fid_data);

if strcmp(temp(1:6), label_extend) == 0

counter_of_rows = counter_of_rows + 1;

else

counter_of_rows = counter_of_rows + 1; % fprintf('Search for Extend Data is finished.\n'); break;

end end

number_of_index_rows = counter_of_rows ; start_row_of_data = counter_of_rows + 1;

% Find the row number where data ends and the number of rows for data while 1

temp = fgetl(fid_data);

if ischar(temp)

counter_of_rows = counter_of_rows + 1;

else

break; 218

end end

end_row_of_data = counter_of_rows; number_of_data_rows = end_row_of_data - start_row_of_data + 1; total_number_of_rows = counter_of_rows;

% Retrieve data from original data file

frewind(fid_data); temp_extend = number_of_index_rows;

while temp_extend-1

buffer = fgetl(fid_data); temp_extend = temp_extend - 1;

end

labels = fgetl(fid_data); %legend_extend = fgetl(fid_data); data = fscanf(fid_data,'%f'); % Load the numerical values into one long vector number_of_data = length(data); % total number of points of retract data number_of_data_columns = round(number_of_data / number_of_data_rows);

if number_of_data ~= number_of_data_rows * number_of_data_columns

fprintf('\n Error in extend data %s: number of data points is not equal to number of rows X number of columns', data_file_name); fprintf('\n %d != %d x %d\n', number_of_extend_data, number_of_extend_data_rows, number_of_extend_data_columns); error_data = 1;

else

data = reshape(data, number_of_data_columns, number_of_data_rows)';

end

% Define the definition of each data column for j = 1 : number_of_data_rows

z_modified(j) = data(number_of_data_rows, 3) - data(j, 3);%z or ramp extend 219

z_modified(j + number_of_data_rows) = data(number_of_data_rows-j+1,4); %z or ramp retract deflection_voltage(j) = data(j, 5); %deflection v extend deflection_voltage(j + number_of_data_rows) = data(j, 6);

end

% figure (1) % plot(time, z_modified); % xlabel('Time (s)') % ylabel('Modified Z-distance (nm)') % Title('Modified Z-distance vs. Time')

% figure (2) % plot(z_modified, deflection_voltage); % xlabel('Modified Z-distance (nm)') % ylabel('Deflection Voltage (V)') % Title('Deflection Voltage vs. Modified Z-distance')

decrease_start_flag = 0; decrease_stop_flag = 1; increase_start_flag = 1; increase_stop_flag = 1;

for k = number_of_data_rows + 1 : 2 * number_of_data_rows – 1

if decrease_start_flag == 0 & deflection_voltage(k) > deflection_voltage(k + 1)

decrease_start_point = k + 2; % 'i + 1' is ok. 'i + 2' for better linearity. decrease_start_flag = 1; decrease_stop_flag = 0;

end

if decrease_stop_flag == 0 & deflection_voltage(k) <= deflection_voltage(k + 1)

for m = 1 : 10

if deflection_voltage(k + m) <= deflection_voltage(k + m + 1)

decrease_stop_flag = 1;

else

220

decrease_stop_flag = 0;

break; end end if decrease_stop_flag == 1

decrease_stop_point = decrease_start_point + 300; % decrease_stop_point = k - 1; % 'i + 1' is ok. 'i + 2' for better linearity. increase_start_flag = 0;

end

if increase_start_flag == 0 & deflection_voltage(k) <

deflection_voltage(k + 1) increase_start_point = k; increase_start_flag = 1; increase_stop_flag = 0;

end

if increase_stop_flag == 0

if deflection_voltage(k) >= deflection_voltage(k + 1) | (k - increase_start_point) > 4 increase_stop_point = k;

increase_stop_flag = 1; break;

end

if decrease_stop_flag == 0

fprintf('Can not find decrease_stop_point\n');

else

fprintf(fid_output, '%s %d %d %d %d ', data_file_name, decrease_start_point, decrease_stop_point, increase_start_point, increase_stop_point); end

221

% Find deflection sensitivity in unit of (V/nm) x_decrease = (z_modified(decrease_start_point : decrease_stop_point))'; y_decrease = (deflection_voltage(decrease_start_point : decrease_stop_point))'; size_decrease = size(x_decrease);

% covariance_decrease = cov(x_decrease, y_decrease) correlation_coefficient_decrease = corrcoef(x_decrease, y_decrease); correlation_decrease = correlation_coefficient_decrease(1,2)^2; fprintf(fid_output, '%s %f ', data_file_name, correlation_decrease);

linear_fitting_decrease = polyfit(x_decrease, y_decrease, 1); slope_decrease = linear_fitting_decrease(1); intercept_decrease = linear_fitting_decrease(2); deflection_sensitivity_decrease = abs(1/slope_decrease); fprintf(fid_output, '%f %f %f ', slope_decrease, intercept_decrease, deflection_sensitivity_decrease);

y_min = slope_decrease * z_modified( increase_start_point ) + intercept_decrease ; y_max = deflection_voltage( increase_stop_point) ; effective_deflection_voltage = y_max - y_min ; effective_deflection_distance = effective_deflection_voltage * deflection_sensitivity_decrease; effective_deflection_force = effective_deflection_distance * spring_constant; fprintf(fid_output, '%5.1f %5.1f %5.1f\r\n', effective_deflection_voltage, effective_deflection_distance, effective_deflection_force); fprintf('%s %s %s \n', 'Data File ', data_file_name, ' Is Processed Successfully!'); fclose(fid_data);

end end fclose(fid_output); fprintf('\n Program ends here!') fprintf('\n------\n\n')

222

Appendix 8. Program to calculate the particle motion due to momentum transfer

clear format long; d1=30e-3; %diameter of disc in m t1=1e-3; %thickness of disc in m density1=7700; %density of disc in Kg/m3 d2=50e-6; %diameter of particle in m density2=8000; %density of particle in m f=65000; %frequency of surface vibration in Hz u10=1e-9; %peak maximum displacement of the surface v2i0=0; %initial velocity of the particle before falling g=9.8; %acceleration due to gravity m/s2 m2=(4/3)*pi*(d2/2)^3*density2; %mass of particle in Kg m1=pi*(d1/2)^2*t1*density1; %mass of surface in Kg w=2*pi*f; %angular frequency T=1/f; %Period for u_max = [38.5e-9 61e-9] ;

for e = [1 0.9 0.7 0.5]

dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\x_c.txt',e,'delimiter','\t','precision', '%.9f','newline', 'pc','-append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\tnew.txt',e,'delimiter', '\t','precision', '%.9f','newline', 'pc','-append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\upz_pzmaxx.txt',e,'delimiter', '\t','precision', '%.9f','newline', 'pc','- append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\upx_pxmaxx.txt',e,'delimiter', '\t','precision', '%.9f','newline', 'pc','- append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\vpz_pii.txt',e,'delimiter', '\t','precision', '%.9f','newline', 'pc','- append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\vpx_pii.txt',e,'delimiter', '\t','precision', '%.9f','newline', 'pc','- append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\TOFF.txt',e,'delimiter', '\t','precision', '%.9f','newline', 'pc','- append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\umax.txt',e,'delimiter', '\t','precision', '%.9f','newline', 'pc','-append');

for f_constant = [0 1e-4 1e-3 1e-2 1e-1]

disp('u_max: '); disp(u_max); 223 tic; x_i=14.5e-3;%initial particle position (problem with 10e-3) no_cycle=1000; %# of cycles n_pts=2000; %#points per cycle t=0:T/n_pts:no_cycle*T; x=0:.01e-6:15e-3; ind = find(x==x_i); %index corresponding to particle location aa=find(t==T/2); usx_0i = u_max*(1.5736E+05*x_i.^3 - 1.4549E+03*x_i.^2 + 1.2292E+02*x_i + 1.5357E-05); % tangential displacement at particle location usz_0i = u_max*(1.6729E+04*x_i.^3 - 6.1887E+02*x_i.^2 + 4.6878E-01*x_i + 9.9633E-02); % normal displacement at particle location phd_z = 1.9949E+11*x_i.^5 - 5.2058E+09*x_i.^4 + 5.2326E+07*x_i.^3 - 1.5242E+05*x_i.^2 + 2.7605E+02*x_i - 6.1067E-03; %phase change of uz along x- axis in degrees ph_z = phd_z*T/360; usx= -(usx_0i*cos(w*t)); usz=usz_0i*cos(w*t+ph_z); vsx=(usx_0i*w*sin(w*t)); vsz=-usz_0i*w*sin(w*t+ph_z); upx = 0; %particle only falls vertically if acc of surf > acc of adhesion vpx = 0; upz = usz_0i - (1/2)*g*t.^2; vpz = -g*t; %since the particle is large and its contact area >>u_max, I use only z component to calculate collision point. However if the particle size is of the order of u_max the x and z collision points will differ. a=(n_pts*0.02):length(t)/no_cycle; % performing this step to exclude t=0 to find collision time t_c=interp1(upz(a)-usz(a),t(a),0,'nearest'); index=find(t==t_c); vsz_i = vsz(index); vpz_i=vpz(index); vsx_i=vsx(index); vpx_i=vsx_i; vpz_pi = (vpz_i*(m2-e*m1)+vsz_i*(m1+e*m1))/(m1+m2); f_constant_m = (vsx_i-vpx_i)/((vsz_i-vpz_i)*(1+e));

if f_constant >= f_constant_m

vpx_pi= (vpx_i*(m1+m2)+m1*(vsx_i-vpx_i))/(m1+m2);

else

vpx_pi = (vpx_i*(m1+m2)+f_constant*(1+e)*(vsx_i-vpx_i))/(m1+m2);

end tpz_zmax = vpz_pi/g; % time for particle to reach max z height tpx_xmax = 2*tpz_zmax; %time for ptcle to travel horizontally where velo = -vpz_pi TOF = 2*tpz_zmax; 224 upz_pzmax = (vpz_pi^2)/(2*g); %max z height (at this pt z velo=0) upx_pzmax = vpx_pi*tpx_xmax/2; % distance particle travelled horizontally to point where z is max (where v=0) upx_pxmax = vpx_pi*tpx_xmax;%horizontal (x) distance where velo = -vpz_pi

for i=1:4000

x_c(1)= x_i; TOFF(1)=TOF; t_cz(1) = t_c; upz_pzmaxx(1) = upz_pzmax; upx_pzmaxx(1) = upx_pzmax; % displ at max z height upx_pxmaxx(1) = upx_pxmax; vpz_pii(1) = vpz_pi; vpx_pii(1) = vpx_pi; vpz_ii(1) = vpz_i; vsz_ii(1) = vsz_i; vsx_ii(1) = vsx_i; tpz_zmaxx(1) = tpz_zmax; t_shift(1)=t_cz(1)+0.5*TOFF(1); tnew(1)= t(index); p_angle(i)=atand(vpz_pii(i)/vpx_pii(i)); %angle of projection (-ve towards center, +ve outwards) upz_c = upz_pzmaxx(i) - (1/2)*g*t.^2; %Find the distance the particle will travel in the horizontal direction. Find the maximum amplitude of x and z at that point with the assumption that the change in the max ampl about that location is very small for +/- 1 nm of horizontal position.

if p_angle > 0 %projected away from center

x_c(i+1) = x_c(i) + abs(upx_pxmaxx(i)); %approximate location of impact (accecptable since the difference bet approx and exact displacements is small)

else

%if -90 <= p_angle <= 0 % projected towards center x_c(i+1) = x_c(i) - abs(upx_pxmaxx(i));

end

if x_c(i+1)>2e-3&&x_c(i+1)<15e-3 usx_0c(i,:) = u_max*(1.5736E+05*x_c(i+1).^3 - 1.4549E+03*x_c(i+1).^2 + 1.2292E+02*x_c(i+1) + 1.5357E-05); usz_0c(i,:) = u_max*(1.6729E+04*x_c(i+1).^3 - 6.1887E+02*x_c(i+1).^2 + 4.6878E-01*x_c(i+1) + 9.9633E-02); phd_zc(i,:) = 1.9949E+11*x_c(i+1).^5 - 5.2058E+09*x_c(i+1).^4 + 5.2326E+07*x_c(i+1).^3 - 1.5242E+05*x_c(i+1).^2 + 2.7605E+02*x_c(i+1) - 6.1067E-03; %phase change of uz along x-axis in degrees ph_zc(i,:) = phd_z*T/360; 225

usx_c = -(usx_0c(i)*cos(w*(t+t_shift(i)))); usz_c = usz_0c(i)*cos(w*(t+t_shift(i))+ph_zc(i));

if upz_pzmaxx(i)<=usz_0c %loop to ensure correct collision time incase starting amplitude of particle is less than or equal to max surface ampl

t_cz(i+1) = interp1(upz_c(a)-usz_c(a),t(a),0,'nearest'); else

diff=abs(upz_c-usz_c); asc=sort(diff,'ascend');

for aa=1:10

inds=find(diff==asc(aa));

end

c_ind=min(inds); t_cz(i+1)=t(c_ind); end vsx_ii(i+1) = (usx_0c(i)*w*sin(w*(t_cz(i+1)+t_shift(i)))); vsz_ii(i+1) = -usz_0c(i)*w*sin(w*(t_cz(i+1)+t_shift(i))+ph_zc(i)); vpz_ii(i+1) = -g*t_cz(i+1); tnew(i+1)=tnew(i)+tpz_zmaxx(i)+t_cz(i+1); TOFF(i)=t_cz(i+1)+tpz_zmaxx(i); %proper TOF tpx_xmaxx(i)= TOFF(i); upx_pxmaxx_correct(i)=vpx_pii(i)*tpx_xmaxx(i); x_c_proper(i+1)=x_c(i)+upx_pxmaxx_correct(i); vpz_pii_a(i) = (vpz_ii(i+1)*(m2-e*m1)+vsz_ii(i+1)*(m1+e*m1))/(m1+m2); if vpz_pii_a(i) < 0 % if vpz_pii_a is more negative (particle has higher velo downwards) than vsurface --> particle moves with surface/sticks % if vpz_pii_a is less negative (ptcle has lower velo downwards than surface velo) then it can impact again if vpz_pii_a(i))>abs(vsz_ii(i+1))

usz_cc1(i,:) = usz_0c(i)*cos(w*(t_cz(i+1)+t_shift(i))+ph_zc(i)); %displ of surface at point of collision upz_cc = usz_cc1(i,:) + vpz_pii_a(i)*t - (1/2)*g*t.^2; f_constant_m = (vsx_ii(i)-vpx_pii(i))/((vsz_ii(i+1)-vpz_pii_a(i))*(1+e));

if f_constant >= f_constant_m

vpx_pii_a(i)= (vpx_pii(i)*(m1+m2)+m1*(vsx_ii(i)-vpx_pii(i)))/(m1+m2);

else

vpx_pii_a(i) = (vpx_pii(i)*(m1+m2)+f_constant*(1+e)*(vsx_ii(i)- vpx_pii(i)))/(m1+m2); 226

end usz_cc = usz_0c(i)*cos(w*(t+(t_shift(i)+t_cz(i+1)))+ph_zc(i)); x=5:length(t)/(no_cycle); diff1=abs(upz_cc-usz_cc); asc1=sort(diff1,'ascend'); for aa1=1:10

inds1=find(diff1==asc1(aa1)); end c_ind1=min(inds1); t_cz_neg(i)=t(c_ind1); vsx_ii1(i+1) = (usx_0c(i)*w*sin(w*(t_cz_neg(i)+(t_shift(i)+t_cz(i+1))))); vsz_ii1(i+1) = -usz_0c(i) * w * sin(w*(t_cz_neg(i)+(t_shift(i)+t_cz(i+1)))+ph_zc(i)); vpz_ii1(i+1) = vpz_pii_a(i)-g*t_cz_neg(i); vpz_pii(i+1) = (vpz_ii1(i+1)*(m2-e*m1)+vsz_ii1(i+1)*(m1+e*m1))/(m1+m2); f_constant_m = (vsx_ii1(i)-vpx_pii_a(i))/((vsz_ii1(i+1)-vpz_pii(i+1))*(1+e));

if f_constant >= f_constant_m

vpx_pii(i+1)= (vpx_pii_a(i)*(m1+m2)+m1*(vsx_ii1(i+1)- vpx_pii_a(i)))/(m1+m2);

else

vpx_pii(i+1) = (vpx_pii_a(i)*(m1+m2)+f_constant*(1+e)*(vsx_ii1(i+1)- vpx_pii_a(i)))/(m1+m2);

end else

vpz_pii(i+1) = (vpz_ii(i+1)*(m2- e*m1)+vsz_ii(i+1)*(m1+e*m1))/(m1+m2); f_constant_m = (vsx_ii(i+1)-vpx_pii(i))/((vsz_ii(i+1)-vpz_pii(i+1))*(1+e));

if f_constant >= f_constant_m

vpx_pii(i+1)= (vpx_pii(i)*(m1+m2)+m1*(vsx_ii(i+1)- vpx_pii(i)))/(m1+m2);

else

vpx_pii(i+1) = (vpx_pii(i)*(m1+m2)+f_constant*(1+e)*(vsx_ii(i+1)- vpx_pii(i)))/(m1+m2);

end 227

end tpz_zmaxx(i+1) = vpz_pii(i+1)/g; % time for particle to reach max z height tpx_xmaxx(i+1) = 2*tpz_zmaxx(i+1); TOFF(i+1) = 2*tpz_zmaxx(i+1); t_shift(i+1)=t_shift(i)+t_cz(i+1)+0.5*TOFF(i+1); upz_pzmaxx(i+1) = (vpz_pii(i+1)^2)/(2*g); %max z height (at this pt z velo=0) upx_pzmaxx(i+1) = vpx_pii(i+1)*tpx_xmaxx(i+1)/2; % distance particle travelled horizontally to point where z is max (where v=0) upx_pxmaxx(i+1) = vpx_pii(i+1)*tpx_xmaxx(i+1);%horizontal (x) distance where velo = -vpz_pi disp('e :'); disp(e); disp('friction :'); disp(f_constant); disp('bounce #: '); disp(i); disp('Particle location: '); disp(x_c(i)); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\x_c.txt',x_c(i),'delimiter', '\t','precision', '%.9f','newline', 'pc','-append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\tnew.txt',tnew(i),'delimiter', '\t','precision', '%.9f','newline', 'pc','-append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\upz_pzmaxx.txt',upz_pzmaxx(i),'delimiter', '\t','precision', '%.9f','newline', 'pc','-append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\upx_pxmaxx.txt',upx_pxmaxx(i),'delimiter', '\t','precision', '%.9f','newline', 'pc','-append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\vpz_pii.txt',vpz_pii(i),'delimiter', '\t','precision', '%.9f','newline', 'pc','-append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\vpx_pii.txt',vpx_pii(i),'delimiter', '\t','precision', '%.9f','newline', 'pc','-append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\TOFF.txt',TOFF(i),'delimiter', '\t','precision', '%.9f','newline', 'pc','-append'); Elapsed_Time=toc; disp('Time Elapsed: '); disp(Elapsed_Time); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\Runtime.txt',Elapsed_Time,'delimiter', '\t','precision', '%.9f','newline', 'pc','-append'); else 228

break

end

dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\x_c.txt',' ','delimiter', '\t','precision', '%.9f','newline', 'pc','- append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\tnew.txt',' ','delimiter', '\t','precision', '%.9f','newline', 'pc','- append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\upz_pzmaxx.txt',' ','delimiter', '\t','precision', '%.9f','newline', 'pc','-append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\upx_pxmaxx.txt',' ','delimiter', '\t','precision', '%.9f','newline', 'pc','-append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\vpz_pii.txt',' ','delimiter', '\t','precision', '%.9f','newline', 'pc','- append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\vpx_pii.txt',' ','delimiter', '\t','precision', '%.9f','newline', 'pc','- append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\TOFF.txt',' ','delimiter', '\t','precision', '%.9f','newline', 'pc','- append'); dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\umax.txt',' ','delimiter', '\t','precision', '%.9f','newline', 'pc','- append');

end dlmwrite('C:\Documents and Settings\Johann Desa\My Documents\momentum prog\results\100209\friction.txt',f_constant,'delimiter', '\t','precision', '%.9f','newline', 'pc','-append');

end end

229

Appendix 9. Program to process AFM data for PIPM displacement calibration

clear; for c=0:1:41

fmin=20; fmax=30; range=[2 c 149000 c]; A=dlmread('C:\Documents and Settings\Johann Desa\Desktop\manip data trial\calib\calib.txt','\t',range); % Read data Fs=300000; % Sampling frequency T = 1/Fs; % Sample time L=length(A); % Length of signal b=blackmanharris(L); t = (0:L-1)*T; % Time vector y=A.*b; % plot(t,y); % xlabel('time (seconds)'); % plot(Fs*t(1:50),y(1:50)) % title('Signal Corrupted with Zero-Mean Random Noise') % NFFT = 2^nextpow2(L); % Next power of 2 from length of y % Y=fft(y); NFFT=pow2(nextpow2(L)); Y = fft(y,NFFT)/L; f = Fs/2*linspace(0,1,NFFT/2); % y = abs(fft(u,nfft)) % Plot single-sided amplitude spectrum. ampl=2*abs(Y(1:NFFT/2)); aa=round(fmin*length(f)/max(f)):round(fmax*length(f)/max(f)); [maximum index]=max(ampl(aa)); ind_max=find(ampl==maximum); f_max=f(ind_max); disp(f_max); disp(maximum); % plot(f,ampl); % axis([10 100 0 2000]); % title('Single-Sided Amplitude Spectrum of y(t)'); % xlabel('Frequency (Hz)'); % ylabel('|Y(f)|'); dlmwrite('C:\Documents and Settings\Johann Desa\Desktop\manip data trial\calib\calib_defl.txt',maximum,'precision', '%.6f','delimiter', ' ','-append'); dlmwrite('C:\Documents and Settings\Johann Desa\Desktop\manip data trial\calib\calib_freq.txt',f_max,'precision', '%.6f','delimiter', ' ','-append'); end

230

Appendix 10. FEM program code for vibration of piezoelectric plates

*Heading ** Job name: 30mm_0p1_0p5_1_3V_ Model name: 30mmDisc *Preprint, echo=NO, model=NO, history=NO, contact=NO ** ** PARTS ** *Part, name=DISC30MM *Node Generate 4131 nodes *Element, type=C3D20RE

*Nset, nset=_PickedSet8, internal, generate

*Elset, elset=_PickedSet8, internal, generate *Nset, nset=_PickedSet10, internal, generate

*Elset, elset=_PickedSet10, internal, generate *Nset, nset=_PickedSet11, internal, generate *Elset, elset=_PickedSet11, internal, generate *Orientation, name=Ori-2 1., 0., 0., 0., 1., 0. 3, 0. ** Section: ceramic DISC30mm *Solid Section, elset=_PickedSet8, orientation=Ori-2, material=PZ27 0.001, *End Part **

** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=DISC30MM-1, part=DISC30MM *End Instance ** *Nset, nset=_PickedSet149, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet149, internal, instance=DISC30MM-1, generate

*Nset, nset=_PickedSet150, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet150, internal, instance=DISC30MM-1, generate

*Nset, nset=_PickedSet152, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet152, internal, instance=DISC30MM-1

*Nset, nset=_PickedSet153, internal, instance=DISC30MM-1

231

*Elset, elset=_PickedSet153, internal, instance=DISC30MM-1, generate *Nset, nset=_PickedSet154, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet154, internal, instance=DISC30MM-1 *Nset, nset=_PickedSet157, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet157, internal, instance=DISC30MM-1

*Nset, nset=_PickedSet159, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet159, internal, instance=DISC30MM-1, generate

*Nset, nset=_PickedSet160, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet160, internal, instance=DISC30MM-1, generate *Nset, nset=_PickedSet161, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet161, internal, instance=DISC30MM-1, generate *Nset, nset=_PickedSet163, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet163, internal, instance=DISC30MM-1, generate *Nset, nset=_PickedSet164, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet164, internal, instance=DISC30MM-1, generate *Nset, nset=_PickedSet165, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet165, internal, instance=DISC30MM-1, generate *Nset, nset=_PickedSet166, internal, instance=DISC30MM-1

*Elset, elset=_PickedSet166, internal, instance=DISC30MM-1, generate *Nset, nset=_PickedSet167, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet167, internal, instance=DISC30MM-1, generate

*Nset, nset=_PickedSet168, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet168, internal, instance=DISC30MM-1, generate *Nset, nset=_PickedSet171, internal, instance=DISC30MM-1

*Elset, elset=_PickedSet171, internal, instance=DISC30MM-1, generate *Nset, nset=_PickedSet172, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet172, internal, instance=DISC30MM-1, generate *Nset, nset=_PickedSet177, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet177, internal, instance=DISC30MM-1, generate *Nset, nset=_PickedSet179, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet179, internal, instance=DISC30MM-1, generate *Nset, nset=_PickedSet180, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet180, internal, instance=DISC30MM-1, generate *Nset, nset=_PickedSet181, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet181, internal, instance=DISC30MM-1, generate *Nset, nset=_PickedSet182, internal, instance=DISC30MM-1

*Elset, elset=_PickedSet182, internal, instance=DISC30MM-1, generate

*Nset, nset=_PickedSet183, internal, instance=DISC30MM-1 *Elset, elset=_PickedSet183, internal, instance=DISC30MM-1, generate *End Assembly ** ** MATERIALS ** *Material, name=PZ27 232

*Damping, beta=2.0497e-07 *Density 7700., *Dielectric, type=ORTHO 1.59e-08, 1.59e-08, 1.59e-08 *Elastic, type=ORTHOTROPIC 1.47e+11, 1.05e+11, 1.47e+11, 9.37e+10, 9.37e+10, 1.13e+11, 2.12e+10, 2.3e+10 2.3e+10, *Piezoelectric 0., 0., 0., 0., 11.64, 0., 0., 0. 0., 0., 0., 11.64, -3.09, -3.09, 16., 0. 0., 0. *Material, name=Teflon *Density 2200., *Elastic 5e+08, 0.46 ** ** INTERACTION PROPERTIES ** ** ** PHYSICAL CONSTANTS ** *Physical Constants, absolute zero=0., stefan boltzmann=0. ** ** BOUNDARY CONDITIONS ** ** Name: BC-3 Type: Electric potential *Boundary _PickedSet159, 9, 9 ** ------** ** STEP: Step-1 ** *Step, name=Step-1, perturbation *Frequency, eigensolver=Lanczos, acoustic coupling=on, normalization=displacement, number interval=1, bias=1. , 120000., 130000., , , ** ** OUTPUT REQUESTS ** *Restart, write, frequency=1 ** ** FIELD OUTPUT: F-Output-1 ** *Output, field *Node Output EPOT, RF, RM, RT, TF, U, UR, UT *Element Output, directions=YES E, EE, EFLX, ENER, EPG, S *End Step 233

** ------** ** STEP: Step-2 ** *Step, name=Step-2_53k, perturbation *Steady State Dynamics, direct, frequency scale=LINEAR, friction damping=NO 66800., 66900., 20, 1. ** ** BOUNDARY CONDITIONS ** ** Name: BC-3 Type: Electric potential *Boundary, op=NEW ** Name: BC-neg_1V Type: Electric potential *Boundary, op=NEW, load case=1 _PickedSet172, 9, 9, -1. *Boundary, op=NEW, load case=2 _PickedSet172, 9, 9 ** Name: BC-pos_1V Type: Electric potential *Boundary, op=NEW, load case=1 _PickedSet171, 9, 9, 1. *Boundary, op=NEW, load case=2 _PickedSet171, 9, 9 ** ** OUTPUT REQUESTS ** ** ** FIELD OUTPUT: F-Output-2 ** *Output, field *Node Output A, AR, AT, EPOT, U, UR, UT, V VR, VT *Element Output, directions=YES E, ECD, EFLX, EPG, MISESMAX, S *Contact Output ECD, ** ** HISTORY OUTPUT: H-Output-2 ** *Output, history *Energy Output ALLAE, ALLCD, ALLEE, ALLFD, ALLIE, ALLJD, ALLKE, ALLPD, ALLSD, ALLSE, ALLVD, ALLWK, ETOTAL *End Step ** ------** ** STEP: Step-3 ** *Step, name=Step-3_66k, perturbation *Steady State Dynamics, direct, frequency scale=LINEAR, friction damping=NO 66800., 66900., 20, 1. 234

** ** BOUNDARY CONDITIONS ** ** Name: BC-neg_3V Type: Electric potential *Boundary, load case=1 _PickedSet177, 9, 9, -3. *Boundary, load case=2 _PickedSet177, 9, 9 ** Name: BC-pos_3V Type: Electric potential *Boundary, load case=1 _PickedSet180, 9, 9, 3. *Boundary, load case=2 _PickedSet180, 9, 9 ** ** OUTPUT REQUESTS ** ** ** FIELD OUTPUT: F-Output-2 ** *Output, field *Node Output A, AR, AT, EPOT, U, UR, UT, V VR, VT *Element Output, directions=YES E, ECD, EFLX, EPG, MISESMAX, S *Contact Output ECD, ** ** HISTORY OUTPUT: H-Output-2 ** *Output, history *Energy Output ALLAE, ALLCD, ALLEE, ALLFD, ALLIE, ALLJD, ALLKE, ALLPD, ALLSD, ALLSE, ALLVD, ALLWK, ETOTAL *End Step ** ------** ** STEP: Step-4 ** *Step, name=Step-4_66k, perturbation *Steady State Dynamics, direct, frequency scale=LINEAR, friction damping=NO 66800., 66900., 20, 1. ** ** BOUNDARY CONDITIONS ** ** Name: BC-neg_0p5V Type: Electric potential *Boundary, load case=1 _PickedSet179, 9, 9, -0.5 *Boundary, load case=2 _PickedSet179, 9, 9 ** Name: BC-pos_0p5V Type: Electric potential 235

*Boundary, load case=1 _PickedSet181, 9, 9, 0.5 *Boundary, load case=2 _PickedSet181, 9, 9 ** ** OUTPUT REQUESTS ** ** ** FIELD OUTPUT: F-Output-2 ** *Output, field *Node Output A, AR, AT, EPOT, U, UR, UT, V VR, VT *Element Output, directions=YES E, ECD, EFLX, EPG, MISESMAX, S *Contact Output ECD, ** ** HISTORY OUTPUT: H-Output-2 ** *Output, history *Energy Output ALLAE, ALLCD, ALLEE, ALLFD, ALLIE, ALLJD, ALLKE, ALLPD, ALLSD, ALLSE, ALLVD, ALLWK, ETOTAL *End Step ** ------** ** STEP: Step-5 ** *Step, name=Step-5_66k, perturbation *Steady State Dynamics, direct, friction damping=NO 66800., 66900., 20, 1. ** ** BOUNDARY CONDITIONS ** ** Name: BC-neg_0p1V Type: Electric potential *Boundary, load case=1 _PickedSet183, 9, 9, -0.1 *Boundary, load case=2 _PickedSet183, 9, 9 ** Name: BC-pos_0p1V Type: Electric potential *Boundary, load case=1 _PickedSet182, 9, 9, 0.1 *Boundary, load case=2 _PickedSet182, 9, 9 ** ** OUTPUT REQUESTS ** ** ** FIELD OUTPUT: F-Output-2 236

** *Output, field *Node Output A, AR, AT, EPOT, U, UR, UT, V VR, VT *Element Output, directions=YES E, ECD, EFLX, EPG, MISESMAX, S *Contact Output ECD, ** ** HISTORY OUTPUT: H-Output-2 ** *Output, history *Energy Output ALLAE, ALLCD, ALLEE, ALLFD, ALLIE, ALLJD, ALLKE, ALLPD, ALLSD, ALLSE, ALLVD, ALLWK, ETOTAL *End Step

237

Appendix 11. FEM program code for vibration of Ring PIPM

Link will be uploaded on the library website. *Heading ** Job name: SAHS_0p1_1_3_5V_4p2pc Model name: SAHS *Preprint, echo=NO, model=NO, history=NO, contact=NO **

** PARTS **

*Part, name= RING ACTUATOR *Node Generate 2520 nodes *Element, type=C3D20RE Assign elements *Nset, nset=_PickedSet23, internal, generate 1, 2520, 1 *Elset, elset=_PickedSet23, internal, generate 1, 448, 1 *Nset, nset=_PickedSet24, internal, generate 1, 2520, 1 *Elset, elset=_PickedSet24, internal, generate 1, 448, 1 *Orientation, name=Ori-3 1., 0., 0., 0., 1., 0. 3, 0. ** Section: pz27 *Solid Section, elset=_PickedSet23, orientation=Ori-3, material=PZ27 0.0001, *End Part **

*Part, name=Biosensor *Node Generate 11716 nodes *Element, type=C3D20R Assign elements *Nset, nset=_PickedSet2, internal, generate 1, 11716, 1 *Elset, elset=_PickedSet2, internal, generate 1, 2060, 1 ** Section: Section-9 *Solid Section, elset=_PickedSet2, material=Biosensor 1., *End Part 238

** ** ASSEMBLY ** *Assembly, name=Assembly ** *Instance, name=Ring-1, part= RING ACTUATOR *End Instance ** *Instance, name=Biosenosr -1, part=Biosensor 3.91404032708959e-08, 0., 0.001 *End Instance ** *Nset, nset=_PickedSet150, internal, instance= RING ACTUATOR -1

*Elset, elset=_PickedSet150, internal, instance= RING ACTUATOR -1, generate

*Nset, nset=_PickedSet152, internal, instance= RING ACTUATOR -1 *Elset, elset=_PickedSet152, internal, instance= RING ACTUATOR -1

*Nset, nset=_PickedSet161, internal, instance= RING ACTUATOR -1 *Elset, elset=_PickedSet161, internal, instance= RING ACTUATOR -1, generate *Nset, nset=_PickedSet164, internal, instance= RING ACTUATOR -1 *Elset, elset=_PickedSet164, internal, instance=RING ACTUATOR-1, generate *Nset, nset=_PickedSet166, internal, instance=RING ACTUATOR-1 *Elset, elset=_PickedSet166, internal, instance=RING ACTUATOR-1, generate *Nset, nset=_PickedSet168, internal, instance=RING ACTUATOR-1 *Elset, elset=_PickedSet168, internal, instance=RING ACTUATOR-1, generate *Nset, nset=_PickedSet172, internal, instance=RING ACTUATOR-1 *Elset, elset=_PickedSet172, internal, instance=RING ACTUATOR-1, generate *Nset, nset=_PickedSet190, internal, instance=RING ACTUATOR-1 *Elset, elset=_PickedSet190, internal, instance=RING ACTUATOR-1, generate *Nset, nset=_PickedSet191, internal, instance=RING ACTUATOR-1 *Elset, elset=_PickedSet191, internal, instance=RING ACTUATOR-1, generate

*Nset, nset=_PickedSet194, internal, instance=RING ACTUATOR-1 *Elset, elset=_PickedSet194, internal, instance=RING ACTUATOR-1, generate

*Nset, nset=_PickedSet195, internal, instance=RING ACTUATOR-1 *Elset, elset=_PickedSet195, internal, instance=RING ACTUATOR-1, generate *Nset, nset=_PickedSet196, internal, instance=RING ACTUATOR-1

*Elset, elset=_PickedSet196, internal, instance=RING ACTUATOR-1, generate *Nset, nset=_PickedSet197, internal, instance=RING ACTUATOR-1 *Elset, elset=_PickedSet197, internal, instance=RING ACTUATOR-1, generate 225, 336, 1 *Nset, nset=_PickedSet198, internal, instance=RING ACTUATOR-1 *Elset, elset=_PickedSet198, internal, instance=RING ACTUATOR-1, generate

*Nset, nset=_PickedSet199, internal, instance=RING ACTUATOR-1 239

*Elset, elset=_PickedSet199, internal, instance=RING ACTUATOR-1, generate *Elset, elset=__PickedSurf187_S2, internal, instance=Biosensor-1, generate *Surface, type=ELEMENT, name=_PickedSurf187, internal __PickedSurf187_S2, S2 *Elset, elset=__PickedSurf188_S2_1, internal, instance=RING ACTUATOR-1, generate *Surface, type=ELEMENT, name=_PickedSurf188, internal __PickedSurf188_S2_1, S2 *Elset, elset=__PickedSurf189_S2, internal, instance=Biosenosr-1, generate

*Surface, type=ELEMENT, name=_PickedSurf189, internal __PickedSurf189_S2, S2 *Elset, elset=__PickedSurf192_S2, internal, instance=RING ACTUATOR-1, generate *Surface, type=ELEMENT, name=_PickedSurf192, internal __PickedSurf192_S2, S2 *Elset, elset=__T0_ Biosenosr -1_M_S2, internal, instance= Biosenosr -1, generate *Surface, type=ELEMENT, name=_T0_ Biosenosr -1_M, internal __T0_ Biosenosr -1_M_S2, S2 *Elset, elset=__T0_Biosensor-1_S_S1, internal, instance= Biosenosr -1, generate *Surface, type=ELEMENT, name=_T0_ Biosenosr -1_S, internal __T0_ Biosenosr -1_S_S1, S1 *Tie, name=_T0_Biosensor-1, position tolerance=0.00707107 _T0_ Biosenosr -1_S, _T0_ Biosenosr -1_M ** Constraint: Constraint-1 *Tie, name=Constraint-1, adjust=yes, type=SURFACE TO SURFACE _PickedSurf189, _PickedSurf188 *End Assembly

** ** MATERIALS ** *Material, name=Biosensor *Density 2300., *Elastic 6.3e+10, 0.2 *Material, name=PZ27 *Damping, beta=6.1492e-08 *Density 7700., *Dielectric, type=ORTHO 1.59e-08, 1.59e-08, 1.59e-08 *Elastic, type=ORTHOTROPIC 1.47e+11, 1.05e+11, 1.47e+11, 9.37e+10, 9.37e+10, 1.13e+11, 2.12e+10, 2.3e+10 2.3e+10, *Piezoelectric 0., 0., 0., 0., 11.64, 0., 0., 0. 0., 0., 0., 11.64, -3.09, -3.09, 16., 0. 0., 0. *Material, name=Teflon *Density 240

2200., *Elastic 5e+08, 0.46 ** ** INTERACTION PROPERTIES ** *Surface Interaction, name=IntProp-1 1., *Surface Behavior, pressure-overclosure=HARD *Surface Interaction, name=IntProp-2 1., *Surface Behavior, no separation, pressure-overclosure=HARD ** ** PHYSICAL CONSTANTS ** *Physical Constants, absolute zero=0., stefan boltzmann=0. ** ** BOUNDARY CONDITIONS ** ** Name: BC-3 Type: Electric potential *Boundary _PickedSet190, 9, 9 ** ** INTERACTIONS ** ** Interaction: Int-1 *Contact Pair, interaction=IntProp-1, type=SURFACE TO SURFACE, adjust=0.0 _PickedSurf187, _PickedSurf192 ** ------** ** STEP: Step-1 ** *Step, name=Step-1, perturbation *Frequency, eigensolver=Lanczos, acoustic coupling=on, normalization=displacement, number interval=1, bias=1. , 150000., 230000., , , ** ** OUTPUT REQUESTS ** *Restart, write, frequency=1 ** ** FIELD OUTPUT: F-Output-1 ** *Output, field *Node Output EPOT, RF, RM, RT, TF, U, UR, UT *Element Output, directions=YES E, EE, EFLX, ENER, EPG, S *End Step ** ------** 241

** STEP: Step-2_0p1V ** *Step, name=Step-2_0p1V, perturbation *Steady State Dynamics, direct, frequency scale=LINEAR, friction damping=NO 222440., 222540., 10, 1. ** ** BOUNDARY CONDITIONS ** ** Name: BC-3 Type: Electric potential *Boundary, op=NEW ** Name: BC-neg_0p1V Type: Electric potential *Boundary, op=NEW, load case=1 _PickedSet172, 9, 9, 0.1 *Boundary, op=NEW, load case=2 _PickedSet172, 9, 9 ** Name: BC-pos_0p1V Type: Electric potential *Boundary, op=NEW, load case=1 _PickedSet191, 9, 9, -0.1 *Boundary, op=NEW, load case=2 _PickedSet191, 9, 9 ** ** OUTPUT REQUESTS ** ** ** FIELD OUTPUT: F-Output-2 ** *Output, field *Node Output A, AR, AT, EPOT, U, UR, UT, V VR, VT *Element Output, directions=YES E, ECD, EFLX, EPG, MISESMAX, S *Contact Output ECD, ** ** HISTORY OUTPUT: H-Output-2 ** *Output, history *Energy Output ALLAE, ALLCD, ALLEE, ALLFD, ALLIE, ALLJD, ALLKE, ALLPD, ALLSD, ALLSE, ALLVD, ALLWK, ETOTAL *End Step ** ------** ** STEP: Step-3_1V ** *Step, name=Step-3_1V, perturbation *Steady State Dynamics, direct, frequency scale=LINEAR, friction damping=NO 222440., 222540., 10, 1. ** ** BOUNDARY CONDITIONS 242

** ** Name: BC-3 Type: Electric potential *Boundary, op=NEW ** Name: BC-neg1V Type: Electric potential *Boundary, op=NEW, load case=1 _PickedSet195, 9, 9, -1. *Boundary, op=NEW, load case=2 _PickedSet195, 9, 9 ** Name: BC-pos1V Type: Electric potential *Boundary, op=NEW, load case=1 _PickedSet194, 9, 9, 1. *Boundary, op=NEW, load case=2 _PickedSet194, 9, 9 ** ** OUTPUT REQUESTS ** ** ** FIELD OUTPUT: F-Output-2 ** *Output, field *Node Output A, AR, AT, EPOT, U, UR, UT, V VR, VT *Element Output, directions=YES E, ECD, EFLX, EPG, MISESMAX, S *Contact Output ECD, ** ** HISTORY OUTPUT: H-Output-2 ** *Output, history *Energy Output ALLAE, ALLCD, ALLEE, ALLFD, ALLIE, ALLJD, ALLKE, ALLPD, ALLSD, ALLSE, ALLVD, ALLWK, ETOTAL *End Step ** ------** ** STEP: Step-4_3V ** *Step, name=Step-4_3V, perturbation *Steady State Dynamics, direct, frequency scale=LINEAR, friction damping=NO 222440., 222540., 10, 1. ** ** BOUNDARY CONDITIONS ** ** Name: BC-3 Type: Electric potential *Boundary, op=NEW ** Name: BC-neg3V Type: Electric potential *Boundary, op=NEW, load case=1 _PickedSet196, 9, 9, -3. *Boundary, op=NEW, load case=2 243

_PickedSet196, 9, 9 ** Name: BC-pos3V Type: Electric potential *Boundary, op=NEW, load case=1 _PickedSet197, 9, 9, 3. *Boundary, op=NEW, load case=2 _PickedSet197, 9, 9 ** ** OUTPUT REQUESTS ** ** ** FIELD OUTPUT: F-Output-2 ** *Output, field *Node Output A, AR, AT, EPOT, U, UR, UT, V VR, VT *Element Output, directions=YES E, ECD, EFLX, EPG, MISESMAX, S *Contact Output ECD, ** ** HISTORY OUTPUT: H-Output-2 ** *Output, history *Energy Output ALLAE, ALLCD, ALLEE, ALLFD, ALLIE, ALLJD, ALLKE, ALLPD, ALLSD, ALLSE, ALLVD, ALLWK, ETOTAL *End Step ** ------** ** STEP: Step-5_66k ** *Step, name=Step-5_66k, perturbation *Steady State Dynamics, direct, friction damping=NO 222440., 222540., 10, 1. ** ** BOUNDARY CONDITIONS ** ** Name: BC-3 Type: Electric potential *Boundary, op=NEW ** Name: BC-neg5V Type: Electric potential *Boundary, op=NEW, load case=1 _PickedSet199, 9, 9, -5. *Boundary, op=NEW, load case=2 _PickedSet199, 9, 9 ** Name: BC-pos5V Type: Electric potential *Boundary, op=NEW, load case=1 _PickedSet198, 9, 9, 5. *Boundary, op=NEW, load case=2 _PickedSet198, 9, 9 ** 244

** OUTPUT REQUESTS ** ** ** FIELD OUTPUT: F-Output-2 ** *Output, field *Node Output A, AR, AT, EPOT, U, UR, UT, V VR, VT *Element Output, directions=YES E, ECD, EFLX, EPG, MISESMAX, S *Contact Output ECD, ** ** HISTORY OUTPUT: H-Output-2 ** *Output, history *Energy Output ALLAE, ALLCD, ALLEE, ALLFD, ALLIE, ALLJD, ALLKE, ALLPD, ALLSD, ALLSE, ALLVD, ALLWK, ETOTAL *End Step

245

Appendix 12. Piezoelectric ceramic material properties

Piezoelectric constitutive relations.

E S  ij  cijkl Skl  ekij Ek , Di  eikl skl   ik Ek

E E cijkl  c pq ,ekij  eiq , ij  Tp

E S Tp  c pq Sq  ekp Ek , Di  eiq Sq   ik Ek

Piezoelectric ceramic material matrices

E E E c11 c12 c13 0 0 0    cE cE cE 0 0 0   12 11 13   E E E  c13 c13 c33 0 0 0 cE    ij  0 0 0 cE 0 0   44   0 0 0 0 cE 0   44  cE  cE  0 0 0 0 0 11 12   2 

 0 0 0 0 e15 0   eip   0 0 0 e15 0 0     e31 e31 e31 0 0 0

S 11 0 0     S   0  S 0  ij  11   S   0 0 11  E E cij  c ji

Material properties for Pz 27 piezoelectric ceramic: 246

E 10 2 c11 (10 N / m ) 14.7 E c12 10.5 E c13 9.37 E c33 11.3 E c44 2.3 E E E c66  (c11  c12 ) 2 2.12

e31 (N /Vm)  3.09

e33 16

e15 11.64 S 11 1130 S  33 914 (kg / m3 ) 7700 247

VITA

Johann deSa is a Ph.D. candidate studying in the School of Biomedical Engineering, Sciences and Health Systems at Drexel University. He received his Bachelor’s of Science degree in

Physics from St. Xavier’s College, Mumbai, and a Bachelors of Science and Technology in

Electronic Instrumentation from Watumull Institute of Technology, Mumbai, India. Since arriving in the U.S. he has been working in the Biosensors Research Laboratory, mainly focusing on piezoelectric transducers for sensing and actuating applications. Currently he is working on the development of a microparticle manipulation technique using piezoelectric actuators and applying this technique for biosensor and biochip applications. 248