Effects of Electric Fields on Forces between Dielectric Particles in Air. Ching-Wen Chiu. Master of Science In Chemical Engineering

Transcription

1 Effects of Electric Fields on Forces between Dielectric Particles in Air Ching-Wen Chiu Thesis submitted to the faculty of the Virginia Polytechnic Institute and State University in artial fulfillment of the requirements for the degree of Master of Science In Chemical Engineering William A. Ducker James R. Heflin Chang Lu Stehen M. Martin May 1st, 213 Blacksburg, VA Keywords: Electric field-induced force, dielectrohoresis, Maxwell-Wagner interfacial relaxation, AFM

2 Effects of Electric Fields on Forces between Dielectric Particles in Air Ching-Wen Chiu Abstract We develoed a quantitative measurement technique using atomic force microscoy (AFM) to study the effects of both DC and AC external electric fields on the forces between two dielectric microsheres. In this work we measured the DC and AC electric field-induced forces and adhesion force between two barium titanate (BaTiO 3 ) glass microsheres in a low humidity environment by this technique. The objective here is to find out the correlation between these measured forces and alied field strength, frequency, and the searation distance between the two sheres was studied. Since the sheres would oscillate under an AC field, the AC fieldinduced force was divided into dynamic comonent (i.e., time-varying term) and static comonent (i.e., time-averaged term) to investigate. The oscillatory resonse occurs at a frequency that is twice the drive frequency since the field-induced force is theoretically roortional to the square of the alied field. This behavior can be observed in the fast Fourier transformation (FFT) sectra of the time series of the deflection signal. The magnitude of the vibration resonse increases when the frequency of the drive force is near resonant frequency of the article-cantilever robe. The amlitude of this vibration increases with roximity of the two articles, and ultimately causes the articles to reeatedly hit each other as in taing mode AFM. The effect of the Maxwell-Wagner interfacial relaxation on the DC electric field-induced force was discovered by monitoring the variation of the field-induced force with time. The static comonent of the AC electric field-induced force does not vary with the alied frequency in the range from 1 to 1 khz, suggesting that the crossover frequency may equal to or less than 1 khz and the ermittivities of the BaTiO 3 glass microsheres and medium dominate the field-

3 induced force. The AC field-induced force is roortional to the square of the alied electric field strength. This relationshi ersists even when the searation between the sheres is much smaller than the diameter of the microsheres. The large magnitude of the force at small searations suggests that the local field is distorted by the resence of a second article, and the continued deendence on the square of the field but the measured force is much larger than the theoretical results, suggesting that the local electric field around the closely saced sheres is distorted and enhanced but the effects of the local field distortion may have not much to with the alied electric field. Comared with the calculated results from different models, our results demonstrate that the field-induced force is much more long-range than exected in theory. In addition, the DC field-induced adhesion force is larger than the AC field-induced one due to the interfacial charge accumulation, agreeing with the discovery of the Maxwell-Wagner interfacial relaxation effect on the DC field-induced force. No obvious correlation between the fieldinduced adhesion and the alied frequency is found. However, both the DC and AC fieldinduced adhesion forces dislay the linearity with the square of the alied electric field strength as well. iii

4 Acknowledgements I would like to thank Dr. William Ducker for his guidance and generous funding from ACS PRF. In addition, I would like to thank Diane, Tina, and Nora for the hel with the school and administrative matters. Thanks my labmates and Riley for the hel and assistance with the exerimental matters. Last I would like to thank my friends and arents for their suort during these years. iv

5 Table of Contents Abstract... ii Acknowledgements... iv List of symbols... vii List of Figures... xi List of Tables... xvii Chater 1 Introduction Literature Review and Thesis Motivation Objectives and Outline of This Thesis... 5 Chater 2 Electric Polarization and Relaxation of Solid Dielectrics The Mechanisms of Electric Polarization... 7 Chater 3 Electromechanics of Particles General Overview of Dielectrohoresis (DEP) Analysis of DEP by Effective Moment Method General Overview of Calculation of DEP force by Effective Moment Method Frequency-Deendent DEP Models for Layered Sherical Particles Maxwell-Wagner Model of Interarticle Force in AC-DC Electric Fields Equivalent Circuit Model of Interarticle Force in AC-DC Electric Fields Chater 4 Atomic Force Microscoy (AFM) and the Technique of Force Measurements Basic Concets and Modes of AFM Technique of Force Measurement Chater 5 Exerimental Section AFM Setu Samle Prearation Alignment of Two Particles Force Curve Collection and Analysis Chater 6 Results and Discussions Cantilever Contribution to the Total Measured Electric Field-Induced Force AC Electric Field-Induced Oscillation Behavior of a Colloidal Probe Electric Field-Induced Force between Two Dielectric Microsheres v

6 6.4 Electric Field-Induced Adhesion Force Chater 7 Conclusions References vi

7 List of symbols A, B coefficients of diole solution for otential in Eq. (3.5) and Eq. (3.6) A ' area of the electrode late and the cantilever in Eq. (6.2) A c effective caacitance area C caacitance of caacitor C contact caacitance at the interface between articles c C self-article caacitance D electric flux density vector or electric dislacement vector (unit: Cm -2 ) D distance between the two subjects in Eq. (5.7) E electric field vector E external alied electric field E root mean square value of electric field rms E normal electric field comonents in fluid E rm r F DEP normal electric field comonents in article dielectrohreitc force vector F DC DC electric force net force vector on an infinitesimal diole in a non-uniform electric field F diole F el ext electrostatic force F external force on article F in F ij F ot F ω F2 ω AC electric field-induced force field-induced force vector between two articles i and j in Maxwell-Wagner model force of the otical tra acting the same force with oosite direction to balance the external force force at electric ulsation ω force at electric ulsation 2ω G ga between the two electrodes K Calusius-Mossotti function as defined by Eq. (3.12) K( ω ) comlex Calusius-Mossotti function as defined by Eq. (T3.2) ( n) K Calusius-Mossotti function of n th order linear multiole defined by Eq. (T3.6) constants account for effective caacitance area, distributed caacitance, curvature K 1, K 2 of articles, and the average contact searation distance R radius of sherical article R 1, R 2 radii of the two subjects resectively R article contact resistance (i.e., resistance of the air between the two articles R T c single article resistance Temerature vii

8 U V V V 1 el 1, V 2 energy stored in a arallel lated caacitor alied voltage equilibrium voltage to levitate article for DEP levitation voltage between the cantilever and the bottom electrode V maximum voltage between the cantilever and the bottom electrode 2, voltage between the cantilever and the to electrode V maximum voltage between the cantilever and the to electrode V c AC (or DC as ω ) voltage dro across the contact caacitance voltage of contact otential difference (CPD) between the ti and the samle V CPD surface V DC-voltage DC V AC AC-voltage W interaction free-energy a ratio of radii of sherical shell d vector distance between charges in finite diole d article diameter d searation or distance between the colloidal robe and the surface e i unit vector in the i direction f Frequency drive/alied frequency f f r resonant frequency g gravitational acceleration (9.81 m/s 2 ) g ' ga between the cantilever and the bottom electrode i, j notations of two identical sheres in Maxwell-Wagner model j square root of minus one; j = 1 k stiffness/sring constant of a AFM cantilever k otical tra stiffness m ot b buoyant mass of article in fluid electric diole moment vector ( n) moment of n th order liner multiole eff effective electric diole moment q electric charge r osition vector ' r radius of effective caacitance area A c r c r ij x article contact radius searation between the centers of the articles i and j in Maxwell-Wagner model dislacement of article from its equilibrium osition viii

9 x x c calibrated and converted deflection signal searation distance between the closest aexes of the articles. z dislacement of the cantilever Ω sloe of the constant comliance region α fluidized bed voidage coefficients deending on the geometry and sacing of the electrodes in the DEP α, α 1 levitation δ, β, γ constants in Eq. (3.36) and Eq. (3.37) β eff effective relative olarizability as defined by Eq. (3.21) and Eq. (3.22) δ 1, δ δ 2 1 and δ 2 are defined as tanδ1 = 1/ ωτ b and tanδ 2 = 1/ ωτ ε Permittivity ε comlex ermittivity ε ' real art f comlex ermittivity ε " negative imaginary art of comlex ermittivity ε ermittivity of free sace ( F/m) ε ε ε ε 1 2 m ε θ θ ij ρ ρ b ρ σ σ σ σ σ σ τ 1 2 m s b ermittivity of ohmic dielectric fluid where the articles susend ermittivity of ohmic dielectric sherical article ermittivity of lossless dielectric fluid where the articles susend ermittivity of lossless dielectric sherical article corresonding bulk owder ermittivity olar angle in sherical coordinates angle between the line-of-center of the articles i and j and the electric field in Maxwell-Wagner model volume charge density aarent bulk owder resistivity aarent (single) article resistivity electrical conductivity electrical conductivity of ohmic medium where the articles susend electrical conductivity of ohmic sherical article electrical conductivity of lossless dielectric fluid where the articles susend electrical conductivity of lossless dielectric sherical article surface electrical conductivity of article article-article relaxation time for transient decay of charge between two articles τ Maxwell-Wagner relaxation time τ MW self-article charge relaxation time for transient decay of charge within a article c τ article contact relaxation time φ scalar electric otential (unit: volts) ix

10 φ diole diolar electric otential φ φ m electric otential within fluid electric otential within article φ ti work function of ti φ smale work function of samle ϕ volume fraction of noninteracting sherical articles in a medium ω ω AC radian frequency of electric field frequency of an AC electric field to excited the ti in a Kelvin Probe Force Microscoy (KPFM) x

11 List of Figures Figure 1-1 Schematic of a digital active feedback controlled DEP levitation system (rerinted from ref. 8)... 2 Figure 1-2 Schematic of an aaratus for measuring electric field-induced force between two articles with a comuter-controlled elevator to adjust the searation between the articles (Rerinted from Ref. 9)... 3 Figure 1-3 The rincial of measuring an external force on a article by an otical tra. (A) There is no external force acting on the article so the article is at its equilibrium osition and the net force from the otical tra is zero. (B) The article deviates from its equilibrium osition due to an external force ( F ext ) and the force of the otical tra ( F ot ) acting the same force with oosite direction to balance the external force. Fext = Fot = k x, where k ot is the otical tra stiffness. (Rerinted from Ref. 16)... 4 Figure 1-4 (A) Scheme of Mizes setu to study the electrostatic adhesion of microarticles (rerinted from Ref. 18); (B) The modified setu based on Mizes setu to study DC electric field-induced force of a microarticle between the arallel condenser electrodes (rerinted from Ref. 2) Figure 2-1 Real and imaginary arts of comlex ermittivity and dielectric disersion of a heterogeneous lossy dielectric versus the alied frequency (Modified from Ref. 28)... 8 Figure 3-1 Schematic of the olarization of an uncharged article under an electric field when (a) the olarizability of the article is greater than the susending medium or (b) the olariability of the article is less than the susending medium. (Rerinted from Ref. 27) Figure 3-2 Numerically calculated electric field lines of four different situations: (a) the article is more olarizable than the medium and the field is uniform; (b) the article is less olarizable than the medium and the field is uniform; (c) the article is more olarizable than the medium and the field is non-uniform; (d) the article is less olarizable than the medium and the field is non-uniform. (Rerinted from Ref. 27) Figure 3-3 Two dimensional electric field distributions around two identical articles align with an axis connecting the centers of the articles arallel to the direction of an alied uniform electric field: (a) articles are more olarizable than the medium so ositive DEP occurs; (b) xi

12 articles are less olarizable than the medium so negative DEP occurs. (Rerinted from Ref. 27) Figure 3-4 Two dimensional electric field distributions around two identical articles align with an axis connecting the centers of the articles erendicular to the direction of an alied uniform electric field: (a) articles are more olarizable than the medium so ositive DEP occurs; (b) articles are less olarizable than the medium so negative DEP occurs. (Rerinted from Ref. 27) Figure 3-5 The two charges in a diole exeriences different forces in magnitudes of a nonuniform electric field. (Rerinted from Ref. 7) Figure 3-6 A dielectric article with radius R and ermittivity susending in a fluid with ermittivity under a uniform electric field in z direction. (Rerinted from Ref. 7) Figure 3-7 Calculated DEP sectra of homogeneous dielectric sheres with ohmic loss but no dielectric loss when (a) / 2.5 ε ε =, ε / ε = 1., m σ m = S/m, 8 σ = 1 S/m and R = 5 µm, and (b) m / 1. ε ε =, ε / ε = 1., 8 σ 1 m S / m =, σ 7 = 1 S/m and 5 R = µm. (Modified from Ref. 7) Figure 3-8 (a) Schematic of the relacement of multilayered sherical article with an equivalent homogeneous shere of the same radius but with the effective ermittivity. (Rerinted from Ref. 7) (b) The DEP sectrum of the tow-layered sherical article. The solid line reresents Re[ K ] and the dot line reresents Im[ K ]. (Rerinted from Ref. 27) Figure 3-9 (a) Electric field-induced interaction between two identical sheres i and j in an electric field. r ij is the searation between the centers of the articles and θ ij is the angle between the line-of-center and the electric field. (b) Calculated square of the effective relative olarizability 2 βeff versus dimensionless frequency in terms of the Maxwell-Wagner relaxation time: (I) β = 1β ; (II) β =.1β. (Modified from Ref. 5) d c d c Figure 3-1 An equivalent circuit for formulating the interarticle force of a one-diemensional article chain in AC-DC fields. (Rerinted from Ref. 31) Figure 4-1 (a) The behavior of the cantilever in an AM mode (Modified from ref. 43) (b) Scheme of an AFM setu in the AM mode. (Rerinted from ref. 41) xii

13 Figure 4-2 (a) The behavior of the cantilever in an FM mode. (Adated from Ref. 43) (b) Scheme of an AFM setu in the FM mode. (Rerinted from Ref. 41) Figure 4-3 Scheme of the oeration rincile of a KPFM. (Rerinted from ref. 44) Figure 4-4 On the left, a tyical raw data diagram of the deflection of the cantilever (i.e., the deflection of the laser beam for the catnielver) vs. the iezo osition (i.e., the dislacement of the iezo) from a force measurement by AFM is shown. On the right, the scheme of the behaviors of a cantilever with resect to a flat samle in a force measurement is illustrated. (Modified from ref. 34)... 4 Figure 4-5 The rocedure for converting a cantilever deflection vs. iezo dislacement curve to a force vs. distance curve. (Rerinted from Ref. 46) Figure 5-1 Schematic of aaratus. The shere attached to the cantilever is referred to as the to shere whereas the shere attached to the scanner is referred to as the bottom shere Figure 5-2 Effect of air-flow on deflection of AFM cantilever in glove bag. Plot shows deflection of cantilever as a function of time with no air flow (to) and with air flow (bottom). 44 Figure 5-3 (a) Raw data showing deflection iezo osition with the field on or off. (b) Net Force as a function of searation after the zero field data has been subtracted Figure 6-1 The three different systems considered here (a) Two sheres system: The fieldinduced force between two BaTiO 3 glass microsheres was measured. Note that the measured force in this system contains the contribution from the cantilever. (b) Cantilever-only system. This is used to determine the effect of the field on the cantilever. (c) Cantilever-bottom shere system. This is used to determine the interaction between the cantilever and bottom shere. The dash circles reresent the locations of the sheres in the two sheres system, which means the searations between the cantilever and the bottom electrode in these three systems are the same Figure 6-2 (a) Comarison of the force under an electric field (at 6 V and 5 khz) to the force without alying electric filed from the cantilever-only system. The searation here is the distance between the cantilever and the bottom electrode. (b) The comarisons of three searation-deendent forces from three different systems at 6 V and 5 khz. Note the searation here is the distance between two sheres in the two sheres system. For the cantilever-only system, the searation should be searation + 6 μm ; for the cantilever-bottom system, it should be searation + 3 μm xiii

14 Figure 6-3 Time series data of the deflection signal of the two sheres system when the alied voltage is 6 V and the alied frequency is: (a) 1 khz; (c) 26 khz; (e) 27 khz and the corresonding FFT sectra (b), (d) and (f) resectively. The resonant frequency of the colloidal robe is 53 khz Figure 6-4 Time series data of the deflection signal of the two sheres system when the alied voltage is 6 V and the alied frequency is: (a) 4 khz; (c) 5 khz; (e) 55 khz; (g) 82 khz and the corresonding FFT sectra (b), (d), (f) and (h) resectively. The resonant frequency of the colloidal robe is 53. khz Figure 6-5 The comarison of the magnitude of different eaks that resonse at (i.e., static comonent), f, 2 f and f r at different alied frequencies ( f ) Figure 6-6 The scheme of the configuration of the cantilever and electrodes, where g is the ga between the cantilever and bottom electrode, G is the ga between the two electrodes, g ' is the ga between the cantilever and the bottom electrode, z is the dislacement of the cantilever, V is the alied voltage, V 1 is the voltage between the cantilever and bottom electrode, and V 2 is the voltage between the cantilever and to electrode. Note that V=V 1 +V 2. The bottom electrode was totally fixed. Although the to electrode was fixed at one end, it still can be taken as totally fixed due to its large mass inertia. The cantilever was only fixed at one end Figure 6-7 Time series data of the deflection signal of the cantilever-only system when the alied voltage is 6 V and the alied frequency is: (a) 75 khz; (c) 8 khz; (e) 82 khz and the corresonding FFT sectra (b), (d) and (f) resectively. The resonant frequency of the cantilever is 164 khz Figure 6-8 The comarison of the magnitude of different eaks that resonse at (i.e., static comonent), f and 2 f at different alied frequencies ( f ) Figure 6-9 The correlation between the alied electric field strength and the amlitude of the eak at (a) the alied frequency f and (b) twice the alied frequency 2 f in the two sheres system Figure 6-1 The variation of the AC electric field-induced force with the searation between two BaTiO 3 glass microsheres at the alied voltage: (a) 3 V and (b) 6 V when the alied frequency is around half the resonant frequency of the colloidal robe (i.e., 53 khz) xiv

15 Figure 6-11 The comarison of the magnitude of the colloidal robe vibration resonses at twice the alied frequency when the alied voltages are 3, 6 and 1 V Figure 6-12 The variation of the deflection signal at the searation between the two BaTiO 3 glass microsheres ranged from (a).9 to 1. µm; (b).4 to.5 µm as the alied voltage is 6 V and the alied frequency is 5 khz and at the searation ranged from.9 to 1. µm when the alied voltage is 1 V and the alied frequency is (c) 5 khz; (d) 1 khz Figure 6-13 The variation of the deflection signal with the searation between the two BaTiO 3 glass microsheres at different alied voltages and frequencies: (a) 6 V, 5 khz; (b) the enlargement of the bouncing oint of (a); (c) 1 V, 5 khz; (d) 1 V, 1 khz Figure 6-14 (a) The variation of the AC electric field-induced forces with the searation between two BaTiO 3 glass microsheres at different frequencies when the alied voltage is 6 V. (b) The variation of the AC electric field-induced forces with frequencies at different alied voltages when the searation between the two dielectric microsheres is 2.5 μm Figure 6-15 The variations electric field-induced force between two dielectric microsheres with time when the alied voltage is (a) 6 V and (b) 12 V and the distance between the two sheres is around 6 μm. The electric fields were alied after AFM started to record the data for about 5 seconds. (c) The variation of normalized DC electric field-induced force with time at difference alied voltages Figure 6-16 Relationshi of the electric field-induced forces between two BaTiO 3 glass microsheres with the square of the alied electric field at different searation as the alied frequency is 1 khz Figure 6-17 (a) Comarison of the measured electric field-induced force with calculated results from different models and the fit to a simle ower law, b F / E = a x, where a and b are constants and b= (b) The enlargement of (a) at small searation Figure 6-18 The linearity between ln F and ln x c Figure 6-19 The variation of the AC electric field-induced adhesion with different alied frequencies when the alied voltages are 1, 6 and 3, resectively. The dash lines reresent the average values of each set of data Figure 6-2 (a) Adhesion force with field on and off. The solid sots means field off and the hollow sot means field on. (b) Correlation between the number of time of two sheres contact and the adhesion force c xv

16 Figure 6-21 Relationshi between the electric field-induced adhesion force and the square of the alied electric field strength under a DC and an AC electric field, where the frequency of the AC field is 1 khz. The dash lines are the linear fitting results xvi

17 List of Tables Table 3-1 Different behaviors and forces for a article in different conditions Table 3-2 Mathematical exressions for the moment, electric otential, and z-directed force due to the linear multioles, or any axisymmetric charge distribution. (Rerinted from Ref. 7) Table 3-3 Summary of some mathematical exressions based on the effective moment method for some secific cases Table 6-1 Brief summary of the bouncing oint at different alied voltages and frequencies xvii

18 Chater 1 Introduction 1.1 Literature Review and Thesis Motivation The forces between microarticles induced by an external electric field lay an imortant role in many areas, such as owder contamination in the semiconductor industry, xerograhy, 1 electric field stabilized fluidized bed, 2 electrostatic searation, 3 electrorheological fluids 4,5 and micro- and nano-electromechanical systems (MEMS and NEMS). 6 Although there are numerous studies on surface and interfacial henomena of articles, not many are related to the effects of external electric fields, and even fewer studies on the closely saced microarticles under an electric field where the oint-diole aroximation may fail. One of the reasons for the lack of study in this area is the limits of the measurement techniques. So far, the measurement techniques for studying the electric-field induced interactions and/or forces are mainly dielectrohoretic (DEP) levitation, otical tweezers, atomic force microscoy (AFM) and some home-built aaratus. Dielectrohoretic (DEP) levitation is rincially used to study the field-induced effective diole moment and Calusius-Mossotti function of single microarticles and microarticle chains, which both are roortional to the electric field-induced force (i.e, dielectrohoretic force, which will be introduced in detail in Chater 3). 7,8 The rincile of the DEP levitation is to adjust the alied voltage and frequency so that the electric field induced force would be equal to the buoyant force with oosite direction, levitating the article(s). 7 The reduced equation of motion for the levitated article(s) in an equilibrium condition can be exressed as where 4 Re[ ( )] 3 2 = mg b + πrε1 Kω αα 1V (1.1) m b is the buoyant mass of article(s), g is gravitational acceleration, R is it the radius of the levitated articles, ε 1 is dielectric constant of the medium where the articles susend, K( ω ) is the Calusius-Mossotti function and it is a comlex number if the alied field is an AC field, Re[ K( ω )] means the real art of the Calusius-Mossotti function, α and α 1 are the coefficients deending on the geometry and sacing of the electrodes, and V is the alied voltage. 7 Figure 1-1 deicts a tyical DEP levitation system. However, the density of individual microarticles used for calculating the buoyant mass is hard to determine due to their small size. 7 Furthermore, 1

19 the DEP levitator cannot control the searation between the articles; it is thus not able to study the correlation of the article chains and the searation between/among the articles. Figure 1-1 Schematic of a digital active feedback controlled DEP levitation system [Tombs, T. N.; Jones, T. B., DIGITAL DIELECTROPHORETIC LEVITATION. Review of Scientific Instruments 1991, 62 (4), ] 8 (Used under fair use, 213) Wang et al. have develoed a home-built aaratus to control the searation between the articles and which is able to directly measure the electric field-induced forces between two articles in both liquid and gaseous media as shown in Figure ,1,11 Nevertheless, the size of the article they used is around 6 mm and the smallest of the searation between the articles is.1 mm, which cannot truly show the field-induced interaction between closely saced microarticles. Rankin et al. have modified a rheometer for the alication of an electric field so as to study the electric field olarization mechanisms of electrorheological behavior. 12 The arameter they measured is the yield stress of a solution of BaTiO 3 with diameters in the range 53-9 µm and silicone oil under an electric field. However, the modified rheometer only can be used to study the collective behavior of microarticles in fluids, not the behavior of individual articles. Besides, the searation between the articles cannot be controlled by this technique. 2

20 Therefore, it is very difficult to extract the fundamental force-searation data from their exeriment. Figure 1-2 Schematic of an aaratus for measuring electric field-induced force between two articles with a comuter-controlled elevator to adjust the searation between the articles [Wang, Z. Y.; Peng, Z.; Lu, K. Q.; Wen, W. J., Exerimental investigation for field-induced interaction force of two sheres. Alied Physics Letters 23, 82 (11), ] 9 (Used under fair use, 213) Several research grous 13,14,15 have adated otical tweezers to truly investigate the interaction between individual microarticles under an electric field. Otical tweezers basically use a laser beam to maniulate articles ranged in diameter from ~2 nm to several micrometers. 16,17 Because of the dual nature of light, the mechanism of maniulating articles by a laser beam can be exlained from the article and wave viewoints resectively. From the asect of the article nature of light, when light irradiates articles, light would be scattered and absorbed by the articles, deending on the refractive indexes of the articles and the medium. The change in direction and magnitude of hoton momentum causes an equal and oosite change in the article momentum, alying a force to the article. 16 The dislacement of the articles from their own equilibrium osition is roortional to the force of the otical tra so the tra can be taken as a linear sring and used to measure external forces on the articles, as shown in Figure 1-3. The sring constant of the otical tra (i.e., otical tra stiffness) deends on the otical gradient (i.e., how tightly the laser is focused), the laser ower, the size and otical roerties of the traed articles and the otical roerties of the medium. 16,17 However, the magnitude of the force that otical tweezers can measure is ranged from femtonewtons to iconewtons. 17 Thus, it is not suitable to use otical tweezers to study strong electric fieldinduced force. Also, local heating due to the high ower laser beam may damage the articles 3

21 and affect flow and the viscosity if the medium is liquid, thus causing artificial effects in the measuring results. In addition, the commercial otical tweezers usually do not have the osition detection so extra modification and setu are necessary. Figure 1-3 The rincial of measuring an external force on a article by an otical tra. (A) There is no external force acting on the article so the article is at its equilibrium osition and the net force from the otical tra is zero. (B) The article deviates from its equilibrium osition due to an external force ( F ext ) and the force of the otical tra ( F ot ) acting the same force with oosite direction to balance the external force. Fext = Fot = kot x, where k ot is the otical tra stiffness. [Jonas, A.; Zemanek, P., Light at work: The use of otical forces for article maniulation, sorting, and analysis. Electrohoresis 28, 29 (24), ] 16 (Used under fair use, 213) Another otential technique able to truly study the olarization and interaction of microarticles under an electric field is atomic force microscoy (AFM). The rincile of the surface force measurement by AFM is that a microarticle is glued to a cantilever, creating a colloidal robe. The forces on the microarticle are determined by measuring the deflection of the sring. A laser beam is reflected from the sring and a built-in hotodiode reads the deflection of the sring, which is roortional to the force. The measured force can range from iconewtons to micronewtons, deending on the stiffness of the chosen cantilever, and the force measurement can be emloyed either in a gaseous medium or in a liquid medium. In contrast to the aforementioned techniques, the searation between articles can be changed in an AFM easily by using its built-in iezo and feedback loo system. Mizes et al. at Xerox have modified an AFM to investigate the effects of external direct-current (DC) electric field on the adhesion of different individual microarticles to an electrode (Figure 1-4 (A)) so as to understand the mechanism of toners behaviors in the xerograhic rocess. 18,19 Later on, Kwek et al. modified Mizes setu to study DC electric field-induced force of a charged and a uncharged article 4

22 resectively by varying the osition of the microarticles between the arallel condenser electrodes as shown in Figure 1-4 (B). 2 All the measurements by Mizes et al. and Kwek et al. were in ambient environment, which are strongly influenced by water adsorbed on the article, and were measured in the resence of a DC field, not an alternating-current (AC) field, which make them strongly influenced by surface conductivity. 21,22,23 (A) (B) Figure 1-4 (A) Scheme of Mizes setu to study the electrostatic adhesion of microarticles [Mizes, H. A., ADHESION OF SMALL PARTICLES IN ELECTRIC-FIELDS. Journal of Adhesion Science and Technology 1994, 8 (8), ] 18 (Used under fair use, 213); (B) The modified setu based on Mizes setu to study DC electric field-induced force of a microarticle between the arallel condenser electrodes. [Kwek, J. W.; Vakarelski, I. U.; Ng, W. K.; Heng, J. Y. Y.; Tan, R. B. H., Novel arallel late condenser for single article electrostatic force measurements in atomic force microscoe. Colloids and Surfaces a-physicochemical and Engineering Asects 211, 385 (1-3), ] 2 (Used under fair use, 213) Based on the above introduction, we believe that AFM enables to measure the fieldinduced force between closely saced microarticles varying with the searation between the articles, thus heling to clarify the real henomena and mechanisms that the studies by other measurement techniques have simlified and/or overlooked. 1.2 Objectives and Outline of This Thesis The objective of this work is to study olarization forces between articles, articularly at small searations (less than the article diameter). To maximize the olarization, we used two BaTiO 3 glass microarticles. To minimize the effects of surface conductivity, we used a low 5

23 humidity environment. We used an AFM to measure the forces, and built a air of arallel electrodes in an AFM similar to Mizes aaratus. In contrast to Mizes work, we study both AC and DC field-induced forces. The resonse to a DC field is dominated by surface conductivity whereas the resonse to an AC field is dominated by olarization. We have three fundament variables: (1) the frequency of the alied field, which exlores the frequency deendence of olarization; (2) the alied field strength, which hels to determine the mechanism of the force (as described later, a olarization force should be roortional to the square of the field) and (3) the searation between the articles, which also hels us to understand the erturbation of the alied field by a second article. The use of high frequency fields induced high frequency forces, which required us to synchronize two different data acquisition methods with different samle rates. Through these exeriments, we hoe that this study will imrove the understanding of the olarization and interaction between microarticles under an electric field, facilitating the develoment of alications involving field induced forces. This thesis will show that the electrical field-induced forces are much, much larger than the traditionally studied van der Waals and double-layer surface forces and thus offer a greater ability to control microarticles. In the following chaters of this thesis, we will first introduce basic definitions of solid dielectric materials and some basic dielectric theory in Chater 2, including the mechanisms of electric olarization and the frequency resonse of dielectrics under an AC electric field, which are required for the analysis of electric field-induced interactions and forces between dielectrics. Then the theories of olarization and interaction of articles under an electric field will be exlained in Chater 3. Chater 4 will describe the rinciles of AFM and the technique of force measurement by AFM in detail. Chater 5 will describe the home-built aaratus for this work, samle rearation, some exerimental methods and methods for data analysis in detail. The exerimental and theoretical results accomanying with discussion will be shown in Chater 6. 6

24 Chater 2 Electric Polarization and Relaxation of Solid Dielectrics Ideal dielectrics do not ermit the assage of any articles, including electrons (i.e., the conductivity is zero) so they are also referred to ideal insulators and taken as caacitors electrically. 24,25 Nevertheless, real dielectrics always contain some free electrons but far fewer than conductors so that real dielectrics have finite conductivity. The differences between real dielectrics and semiconductor are that semiconductors have higher conductivity and the dominate charge carriers in semiconductor are attributed to thermal excitation (i.e., electron-hole generation) but in dielectrics are due to other sources, such as charge injection and otical excitation. 24 When the dielectrics have finite conductivity, the dielectrics can be said lossy and are electrically equivalent to an ideal lossless caacitor in arallel with a resistor. 26 The ermittivity, ε, of a lossy dielectric is a comlex term exressed as where ε = ε' jε" j σ ω (2.1) reresents the ositive square root of -1, σ is conductivity of the lossy dielectric and ω is the frequency of external source (e.g., electric field). 7 The real art of the ermittivity, ε ', in Eq. (2.1) is related to the storage of energy in the electric field and the imaginary art to the energy dissiation. 26 Both ε ' and ε " are a function of frequency. 27 Note that the two imaginary terms in Eq. (2.1) reresent two different mechanisms of energy dissiation. 27 The third term on the right hand side of Eq. (2.1) reresents that the energy is lost by Joule heating because the lossy dielectric has a finite conductivity 27 and this mechanism is referred to as ohmic loss 7. The origin of the second term on the right hand side of Eq. (2.1) is that the dioles or ions dissiate energy during rotation or other motion. 27 Such energy dissiation is called dielectric loss. It is negligible in may circumstances 7, including as we show, in the exeriments in this thesis. In the following sections, we will introduce the olarization mechanisms of dielectrics subjected to different frequencies of an AC electric field. 2.1 The Mechanisms of Electric Polarization When the alied electric field is much lower than the inner atomic or molecular fields and the low conductivity of the dielectric can be neglected, there are three different olarization mechanisms would revail, deending on the corresonding behavior of electrons or dioles to an AC electric field. 24 Figure 2-1 shows how the real ε ' and the imaginary ε '' arts of the 7

25 comlex ermittivity vary with different frequency due to loss of a series of resonse mechanisms. 28 ~1 3 ~1 9 ~1 15 Figure 2-1 Real and imaginary arts of comlex ermittivity and dielectric disersion of a heterogeneous lossy dielectric versus the alied frequency [Hao, T.; Kawai, A.; Ikazaki, F., Mechanism of the electrorheological effect: Evidence from the conductive, dielectric, and surface characteristics of waterfree electrorheological fluids. Langmuir 1998, 14 (5), ] 28 (used under fair use, 213) The resonse mechanisms are: 1. Electronic olarization: At UV frequencies (~1 15 to ~1 16 Hz), the electron clouds of atoms or molecules are dislaced with resect to the inner ositive nucleus. This olarization exists in any kind of dielectrics. 2. Atomic or ionic olarization (vibrational olarization): The ions or atoms of a olyatomic molecule are moved relative to each other due to the external electric field, causing a change in diole moment. 3. Orientational olarization (or Debye relaxation): This olarization only haens in olar dielectric. The electric field would drive the ermanent dioles to align toward the direction of the electric field. 8

26 There is a fourth tye olarization that occurs when the low conductivity of the dielectric cannot be neglected (or the alied field is high enough): the migration of charge carriers. When the migration of charge carriers is imeded at the interface between two unlike dielectrics and/or grain boundaries of the dielectric, then an accumulation of charge occurs at the interface. 24 This is so-called interfacial or sace charge olarization 24 and is also referred to as Maxwell- Wagner interfacial olarization since the theory was develoed by James Clerk Maxwell and later on refined by K.W. Wagner 29. This olarization exists in amorhous or olycrystalline solids or materials consisting of traed charges. 24 The movement of charge to the surface takes time, so this mechanism only oerates at low frequencies or in a DC electric field. Jones has investigated this time-deendent henomenon by deriving the time-deendent effective diole moment of a homogeneous ohmic dielectric shere with ermittivity ε 2, conductivity σ 2, and radius R in a ohmic dielectric fluid with ermittivity ε 1 and conductivity σ 1 (Eq. (2.2)). 7 3 σ σ ε ε ( ) eff t = πε1r E[1 ex( t/ τ MW )] + 4πε1R Eex( t/ τ MW ) σ2 + 2σ1 ε2 + 2ε (2.2) 1 ε2 + 2ε1 where τ MW = is Maxwell-Wagner charge relaxation time associated with the charge σ + 2σ 2 1 accumulation at the surface of the shere. When t << τ (i.e., high frequency), the first term on the right hand side of Eq. (2.2) would be negligible and the effective moment is dominate by the ermittivities, meaning no charge accumulation at the surface of the shere. However, when t >> τ MW (i.e., low frequency), the first term on the right hand side of Eq. (2.2) dominates if charge accumulation can occur at the surface. Since the field-induced force is roortional to the effective diole moment, we can exect that the field-induced force would have a similar frequency resonse. The relationshi between the field-induced force and the effective moment would be introduced in detail in Chater 3. MW 9

27 Chater 3 Electromechanics of Particles Every material has its own secial electrical roerties, thus having different resonses, such as olarization and field-induced forces, to an external electric field. In Chater 2, different electric olarizations and the corresonding mechanisms have been introduced. In this chater, we focus on the forces that the articles with diameter range from 1 to 1 µm exerience through the action of an external electric field and/or an induced-field by other nearby olarized articles. The forces can be categorized based on the tyes of the alied electric field and the articles surroundings (e.g., high conductivity electrolyte or air) as electrohoresis (EP), dielectrohoresis (DEP), electrorotation (ROT), electroosmosis (EO), electrohydrodynamics (EHD), and so forth. 27 Herein we mainly cover DEP since in our study it is the main mechanism and the others do not occur or can be ignored. First the fundamentals and some equations of the DEP are introduced. Then two analytical models of electric field-induced forces between two identical articles develoed from different oints of view are reviewed. We will use these models to interret our exerimental observations in Chater General Overview of Dielectrohoresis (DEP) DEP refers to the translation motion of an uncharged dielectric and/or conductive article caused by olarization effects in a non-uniform electric field. 7,29 EP is the movement of a charged article resulted from the Coulomb force between the article and an electric field. 27 Different from EP, DEP is basically attributed to the interaction of an electric field with the electric field-induced diole moments. 27 In addition, DEP can occur not only in a DC electric field but also in an AC electric field as long as the electric field is non-uniform, which can be from the directly alied a non-uniform external electric field and/or indirectly generated by other nearby olarized articles 7,27 as is the case in the work described in this thesis. Although a uniform electric field can also induce diole moments in an uncharged article, the vector sum of these dioles is zero, causing no motion of the article. On the other hand, EP can only occur in a DC electric field since the direction of the EP force deends on the olarity of the electric field and an AC field would cause the charged article to oscillate around an axis but not move from that average oint over time; that is, the EP force is averaged to zero in an AC field. 29 Therefore, 1

28 we can conclude different behaviors and forces of a article under an electric field as follows (Table 3-1). Table 3-1 Different behaviors and forces for a article in different conditions DC field AC field Uncharged Particle Both DEP and EP occur in a non-uniform filed. EP occurs due to the accumulated charges at the interface of the article. In a uniform field, only EP occurs. Only DEP occurs if the field is non-uniform. Otherwise, neither DEP nor EP will occurs. The article will oscillate. Charged article Both DEP and EP occur in a non-uniform field and only EP occurs in a uniform filed. Only DEP occurs if the field is non-uniform. Otherwise, neither DEP nor EP will occurs. The article will oscillate. One of interesting features of DEP is that the olarized article is not always attracted to the region of the stronger electric field. It is oosite under certain conditions instead. When the article is more olarizable than its environment (medium), more induced charge searation takes lace in the article than in the medium, causing higher charge density inside the interface than outside and the effective diole across the article is in the same direction as the electric field, as shown in Figure 3-1 (a). If the alied field is non-uniform, then the article is attracted and to the high-field region, which is termed ositive DEP. This is the case in the exeriments described in this thesis, where we have a high dielectric article in air, and the article is near another article. On the contrary, if the article is less olarizable than the medium, the effective diole across the article in the oosite direction to the electric field (Figure 3-1 (b)). If the alied field is non-uniform, the article would be reelled from the high-field region. In other words, the article is attracted to the low-field region in a non-uniform electric field when the olarizability of the article is less than the medium. The henomenon is termed negative DEP. 11

29 Figure 3-1 Schematic of the olarization of an uncharged article under an electric field when (a) the olarizability of the article is greater than the susending medium or (b) the olariability of the article is less than the susending medium. [Morgan, H.; Green, N. G., AC Electrokinetics: colloids and nanoarticles. Research Studies Press: Philadelhia, PA, 23.] 27 (Used under fair use, 213) Figure 3-2 Numerically calculated electric field lines of four different situations: (a) the article is more olarizable than the medium and the field is uniform; (b) the article is less olarizable than the medium and the field is uniform; (c) the article is more olarizable than the medium and the field is non-uniform; (d) the article is less olarizable than the medium and the field is non-uniform. [Morgan, H.; Green, N. G., AC Electrokinetics: colloids and nanoarticles. Research Studies Press: Philadelhia, PA, 23.] 27 (Used under fair use, 213) 12

30 Figure 3-2 shows the electric field lines of the above situations. Note that the electric field line on both sides of the article is symmetric when the electric field is uniform no matter whose olarizability is greater (Figure 3-2 (a) and (b)), meaning the net DEP force exerted on the article is zero in a uniform field. Further, as the olarizability of the article greater than the medium, the electric field lines inside the article is near zero and the lines outside it would bend to and/or be erendicular to the surface of the article, just like a metal shere in an electric field (Figure 3-2 (a) and (c)). By contrast, the article would act as an insulator as its olarizability less than the medium. The outside electric field lines would bend around the article and the density of the inside lines is similar to those outside. If the article and the medium have the same olarizability, there is no effective diole across the article and no DEP occurs. In addition to the electrical roerties of the article and the medium and the nonuniformity and strength of the electric field, the frequency of the alied field can control the direction and the magnitude of the DEP force as well. At low frequencies, the accumulation of the free charge at the interface between the article and the medium (i.e., a Maxwell-Wagner interfacial olarization as mentioned in Chater 2) dominates the DEP force. The amount of the accumulated charge on either side of the interface deends on the imedance of these materials and the frequency of the alied field. In other words, the conductivities I, 3 of the article and the medium dominate the DEP force. At high frequencies, the ermittivities of the article and the medium dominate the DEP force since the charge within the article is not able to resond to the high frequency field. At the intermediate frequency range, the transition from ositive DEP to negative DEP or from negative DEP to ositive DEP takes lace. This is referred to as dielectric disersion. 7,27,29 The detail of frequency-deendent DEP would be reviewed in Section So far we have discussed DEP on a single article. But article-article interactions are often imortant in alications, and are the subject of this thesis. Here two cases of articlearticle interactions are stated and their calculated electric field distributions are shown in Figure 3-3 and Figure 3-4, resectively. The first case is that the axis connects the centers of two identical articles (i.e., line-of-centers) arallel to the direction of an alied uniform electric I The more accurate terminology should be admittance, not conductivity. The admittance in an AC electric field is analogous to the electrical conductance in a DC field. 3 The admittance is comlex but usually the real art, which is the electrical conductance, is considered in most discussions. Further, the electrical conductance is roortional to the electrical conductivity so conductivity is used in most literatures and models. 13

31 field, as shown in Figure 3-3. The darker art in the figures means lower field strength and the lighter art means higher field strength. In Figure 3-3 (a), the article are more olarizable than the medium so they exerience ositive DEP. Note that the field strength between the articles is higher than those of the other sides of the articles and the articles would move to high-field region so they would move to each other. On the other hand, the articles are less olarizable than the medium in Figure 3-3 (b) so they exerience negative DEP. Interestingly the article move to each other as well since the field strength between the articles is lower than those of the other sides of the articles. As a result, two identical articles would attract to each other when their line-of-centers arallels to the direction of the alied electric field no matter whether Figure 3-3 Two dimensional electric field distributions around two identical articles align with an axis connecting the centers of the articles arallel to the direction of an alied uniform electric field: (a) articles are more olarizable than the medium so ositive DEP occurs; (b) articles are less olarizable than the medium so negative DEP occurs. [Morgan, H.; Green, N. G., AC Electrokinetics: colloids and nanoarticles. Research Studies Press: Philadelhia, PA, 23.] 27 (Used under fair use, 213) the olarizability of the articles is greater or smaller than the medium. The second case is similar to the first one but the direction of the electric field is erendicular to the line-of-centers of the articles (Figure 3-4). Contrary to the first case, two identical articles would reel each other and move to the region with their favorable field strength (e.g., articles exeriencing ositive DEP move to high-filed regions) to reach a stable state (i.e., like the arrangement in the first case) no matter whether their olarizability is greater or smaller than the medium. In conclusion, for identical articles under an electric field, the direction of force is determined by the angle between the electric field and the line-of-centers of the articles, not the olarizability. 14

32 Figure 3-4 Two dimensional electric field distributions around two identical articles align with an axis connecting the centers of the articles erendicular to the direction of an alied uniform electric field: (a) articles are more olarizable than the medium so ositive DEP occurs; (b) articles are less olarizable than the medium so negative DEP occurs. [Morgan, H.; Green, N. G., AC Electrokinetics: colloids and nanoarticles. Research Studies Press: Philadelhia, PA, 23.] 27 (Used under fair use, 213) We have briefly introduced some features of the DEP from a henomenological ersective. The following will introduce some related equations of DEP in the electromagnetic and mathematical oint of view. 3.2 Analysis of DEP by Effective Moment Method General Overview of Calculation of DEP force by Effective Moment Method To derive the equation of the DEP force, first we need to consider a force on an infinitesimal diole in a non-uniform electric field since the origin of the DEP is the interaction of a non-uniform electric field with the induced field-diole. As shown in Figure 3-5, a diole consists of two charges of equal magnitudes and oosite signs searated by a vector distance d in an electric field E. Assume that the charges are stationary or with negligible velocities and accelerations so the movement of the charges does not change the electric field much and thus is negligible, which is referred to as a quasi-electrostatic system. 7 15

33 Figure 3-5 The two charges in a diole exeriences different forces in magnitudes of a non-uniform electric field. [Jones, T. B., Electromechanics of Particles. Cambridge University Press: New York City, NY, 1995.] 7 (Used under fair use, 213) Thus, the net force on the diole is F = qe( r + d) qe() r (3.1) where r is the osition of q. If d is smaller than the characteristic dimension of the electric field nonuniformity (i.e., the electric field is nearly uniform across the diole), the electric field can be exanded around r by a vector Taylor series, given by Er ( + d) = Er ( ) + d Er ( ) +... (3.2) Substituting Eq. (3.2) into Eq. (3.1) gives F= qer () + qd Er () qer () +... qd Er () (3.3) where the higher order terms (i.e., d 2, d 3, and so on) is neglected. For a finite diole, Fdiole = E (3.4) Eq. (3.4), referred to as dielectrohoretic aroximation, can be alied to any hysical dioles, such as a olar molecule or a olarized article with finite size. 7 Note that this aroximation demonstrates that the force on a diole is zero in a uniform electric field, corresonding to the revious statement. However, higher order terms in Eq. (3.3) should be taken into consideration when the characteristic dimension of the electric field nonuniformity is comarable to the dimension of the object we concern, such as the case of closely saced articles where the dimension of the nonuniformity of the local electric field induced by the articles is comarable to the article size. Herein the effective moment method is introduced to aroximate the in Eq. (3.4) more 16

34 accurately for the calculation of the DEP force of a article. The idea of this method is to hyothesize that the DEP force of a article can be exressed by the effective moments derived from solving the induced electrostatic field due to the article. 7 The reasons for this choice are that its ease of imlementation and validity in many cases where is hard to emloy the derivations based on the Maxwell stress tensor. 7 Here the derivation of the effective moment in the simlest case is reviewed first since this can derive the effective moments of other more general cases simly by adjusting some arameters to the corresonding situations. Subsequently, the general exression of effective moment and the limitation of this method are stated. Let s consider a homogeneous lossless dielectric article of radius R and ermittivity ε is susending in a lossless dielectric fluid of ermittivity ε m under a uniform electric field E in z direction, as illustrated in Figure 3-6. Assume that the article does not significantly disturb the source of the alied electric field. eff Figure 3-6 A dielectric article with radius R and ermittivity ε susending in a fluid with ermittivity ε m under a uniform electric field E in z direction. [Jones, T. B., Electromechanics of Particles. Cambridge University Press: New York City, NY, 1995.] 7 (Used under fair use, 213) We know Gauss s law is ρ E =, where ρ is the volume charge density and ε is ε ermittivity of free sace ( F/m). The scalar electric otential φ (Volts) can be 17

35 related to the vector electric field by E = φ. Thus, we can get φ ρ ε 2 =, which is referred to as Poisson s equation. As the charge density is zero, we obtain 2 φ =, which is termed Lalace s equation. Due to the conservation of an electrostatic field 26 and no existence of free charge (i.e., both the article and the medium are lossless) in this case, the curl and the divergence of the electric field are zero. As a result, the electric otential satisfies Lalace s equation everywhere for this case. Assume the diole moment is the dominate contribution comared with other multioles, which can be ignored. Hence the general solutions for Lalace s equation are assumed 7 (, ) cos Acosθ φ, m rθ Er θ r R r = + > 2 (3.5) φ ( r, θ ) = Br cos θ, r < R (3.6) where φ m and φ are the otential within the fluid and the shere, resectively and A and B are unknown constants. The first term on the right hand side of Eq. (3.5) reresent the alied electric field and the second term reresents the induced diole due to the article. To secify the solutions, we need to find out the boundary conditions at the article surface (i.e., r = R). The first boundary condition is the continuity of the normal electric field comonents across the article-fluid boundary, which can be rewritten as φm( r = R, θ) = φ( r = R, θ) (3.7) The second boundary condition is the continuity of the normal comonents of the electric flux density (or electric dislacement) D (Cm -2 ) across the boundary. D is defined as D= ε E. Thus the second boundary condition can be written as εmerm( r = R, θ) = ε Er ( r = R, θ) (3.8) φm where E rm = r and E = φ r are the normal electric field comonents in the fluid and the r article, resectively. Substituting Eq. (3.5) and (3.6) into Eq. (3.7) and (3.8), we obtain 18

36 ε εm A= ε + 2ε m 3 RE 3ε m and B= E ε + 2ε (3.9) m Based on the concet of the effective moment method, the effective diole moment of the olarized article in this case can be taken as an equivalent, free-charge, oint diole offering the same diolar electric otential φdiole II, which is φ diole cosθ eff = 2 4πε mr (3.1) Table 3-2 Mathematical exressions for the moment, electric otential, and z-directed force due to the linear multioles, or any axisymmetric charge distribution. [Jones, T. B., Electromechanics of Particles. Cambridge University Press: New York City, NY, 1995.] 7 (Used under fair use, 213) Comaring the second term on the right hand side of Eq. (3.5) with Eq. (3.1), we can get II The exressions of the electric otential by linear multioles are summarized in Table 3-2 and the detailed derivations by Legendre olynomials can be found in Ref. 7 and 26. Here we directly use the derived results. 19

37 ε ε = 4πε A= 4πε RE = 4πε KRE + (3.11) 3 m 3 eff m m m ε 2εm ε εm K( ε, ε ) = ε + 2ε (3.12) K is known as the Clausius-Mossotti function, reresenting the strength of the effective olarization of a sherical article. 7 Note that K can oint out the relative direction of the alied electric field and the effective diole moment. As K > (i.e., ε > εm), the diole and alied electric field oint in the same direction; however, as K < (i.e., ε < εm), the diole oints toward the oosite direction to the electric field, corresonding to the statement in section Finally, substituting Eq. (3.11) into Eq. (3.4), dielectric aroximation gives m ε ε F DEP = 2 πε R ( ) E m 3 m 2 ε + 2εm (3.13) However, Eq. (3.13) only can be alied to the case we secified. Table 3-3 summarizes the effective moments, Calusius-Mossotti function, electric otentials, and forces for more general cases. The detailed derivations of these exressions can be found in Ref. 7. Note that these mathematical formulas look very similar to Eq. (3.11) and (3.13) and they all can be derived by the equations in Table 3-2. As for lossy materials, the ermittivities should be in the comlex form, which is reresented by underlining ε, so does Calusius-Mossotti function. In addition, the lossy roerties of materials are significant under an AC electric field; thus the DEP behaviors of lossy materials under an AC field are discussed, so do that in Table 3-2. Thus the concet of hasor, a reresentation of a comlex number in terms of a comlex exonential by Euler formula, is introduced to rewrite time-varying arameters, such as electric field and otential for the connivance of calculation. For examle, E = Ecosωtez = Re[ Eex( jωt)] ez and E = Eex( jωte ) z is the vector hasor exression of a sinusoidal AC electric field. Only the real art, which is denoted Re[ ], is meaningful. That is why the real art of the force is considered in Table 3-3. Furthermore, the force under an AC electric field consists of a constant (average) term and a time-varying term. Since the motion due to the time-varying term is usually 2

38 damed heavily by the viscosity of the susension medium for the article size range from 1 to 1 μm 7, Table 3-3 only ays attention on the time-average term, which is designated < >. However, the medium we used in our study is air so the time-varying term should be studied as well. We will discuss the time-varying term in detail in Chater 6. Note that the effective moment method is not rigorous for lossy dielectrics since otential energy arguments are not legitimate. 7 Even so, the effective moment method is still very useful because in most circumstance, dielectric loss is negligible and there is no much difficulty to adjust the method when dielectric loss cannot be neglected Frequency-Deendent DEP Let s now consider the case of a homogenous article with ohmic loss but no dielectric loss susended in a dielectric medium under an electric field. Eq. (T3.4) and (T3.1) oint out 21

39 Table 3-3 Summary of some mathematical exressions based on the effective moment method for some secific cases. 7 Lossless dielectric homogeneous shere Only consider diole moment eff = 4πε (3.11) 3 mkr E ε εm K( ε, ε ) = ε + 2ε φ cosθ m (3.12) eff diole = (3.1) 2 4πε mr ε ε F DEP = 2 πε R ( ) E m 3 m 2 ε + 2εm (3.11) Consider both diole and other mutioles 4πε K R E ( n 1)! z ε ( ) ε n m K = nε + ( n+ 1) ε ( n) 2n+ 1 n 1 ( n) m z = n 1 n Ez ( n+ 1)! q φn = = n n 2 z 4πε ζ + m m (T3.5) (T3.6) (T3.7) ζ : the distance between the electric field source and the center of the article ( n) n n E z Fn = qn ( dn ) E = e n z n! z ( n) 2n+ 1 n 1 2πε mk R E z = n 1 n!( n 1)! z z 2 e z (T3.8) Ohmic lossy dielectric homogeneous shere eff = 3 4πε m KR E (T3.1) ε ε m σ σm jωτ + 1 K( ω) = = ε + 2ε m σ + 2σ m jωτ MW + 1 ε + 2εm ε εm τ MW =, τ = (T3.3) σ + 2σm σ σm 1 * Ft ( ) = Re[ E] = 2πε mr Re[ K( ω)] E eff 2 (T3.4) (T3.2) 3 2 rms F ( n) z K ε ε = nε + ( n+ 1) ε ( n) m m (T3.9) 2n+ 1 n 1 2πε mr ( n) E z, rms = Re[ K ] n 1 n!( n 1)! z z 2 (T3.1) 22

40 that Re[ K] is the factor to determine the frequency deendence of the time-averaged DEP force. To study the frequency-deendent DEP, we can merely focus on Re[ K ]. From Eq. (T3.2), we can get ε ε 3( εσ εσ ) Re[ K] = + ε + ε τ σ + σ + ωτ (3.14) m m m m MW ( 2 m) (1 MW ) where τ MW deicts the decay of a diolar distribution of free charge at the surface of the article. 7 Alying two frequency-limitations to the K( ω ) gives σ σm lim Re[ K] = σ + 2σ (3.15) ωτ MW ωτ m ε εm lim Re[ K] = MW ε + 2ε (3.16) Eq. (3.15) and (3.16) confirm two different DEP behaviors at low and high frequencies, resectively, that is, the conductivities of the shere and medium govern the low-frequency DEP m σ < σ m ε < ε m σ < σ ε < ε m m Figure 3-7 Calculated DEP sectra of homogeneous dielectric sheres with ohmic loss but no dielectric 8 8 loss when (a) εm / ε = 2.5, ε / ε = 1., σ m = 4 1 S/m, σ = 1 S/m and R = 5 µm, and (b) 8 εm / ε = 1., ε / ε = 1., σ 1 7 m = S / m, σ = 1 S/m and R = 5 µm. [Jones, T. B., Electromechanics of Particles. Cambridge University Press: New York City, NY, 1995.] 7 (Used under fair use, 213) 23

41 behavior and the ermittivities of the shere and medium governs the high-frequency DEP behavior. Figure 3-7 shows the variation of Re[ K ] versus the alied electric field frequencies in different conditions. Note that at a certain frequency, Re[ K ] becomes zero and there is no F DEP. This frequency is so-called crossover frequency since the force has oosite sign on either side of this frequency. 29 For the work in this thesis, the medium is always air, so the dielectric constant and conductivity of the articles is always greater than the medium Models for Layered Sherical Particles So far, we have only concerned homogeneous sheres. As for heterogeneous sheres, we briefly introduce the article with shell structure here. The rincile of the model for layered sherical articles is to relace a layered sherical article with an equivalent homogeneous article of the same radius but with the effective ermittivity, as shown in Figure 3-8 (a), so that the mathematical exressions for a homogenous shere above are still valid for the layered shere as long as the ermittivity of the homogeneous shere is relaced by the effective ermittivity of the layered shere. 7 So the general exression of the effective ermittivity is ε 3 ε N+ 1 ε N a + 2 ε + 2ε N = ε N a ε N ε N ' N+ 1 N ε N 3 ε N+ 1 (3.17) where a = RN / R N + 1 and N+1 indicates the most inner layer so N reresents the layer shells the most inner layer. Eq. (3.17) can be alied reeatedly for multilayer structure. As shown in Figure 3-8 (a), we can use ε 3 and ε 4 to calculate the effective ermittivity ε ' 3 first and then use ' ' ε 3 and ε 2 to calculate ε 2. In addition, Eq. (3.17) is for a shere with ohmic loss but no dielectric loss and Maxwell-Wagner relaxation haens for each interface. For a lossless shere, we can just relace the comlex form of ermittivities by the real form of ermittivies. In fact, the derivation of Eq. (3.17) is similar to that in section so it is valid only for a uniform or nearly uniform electric field since only the diole is concerned. For a layered shere with ohmic loss but no dielectric loss in a nonuniform field, higher mutilolar terms should be concerned, which gives 24

42 2n+ 1 ( n) ' a ( n+ 1) K 2 n 2n+ 1 ( n) ( ε ) = a nk (3.18) where ( n) K is defined as Eq.(T3.9). Similarly, for a lossless shere, we can just relace ( n) K defined as Eq. (T3.6). ( n) K by (a) (b) Re[K] and Im[K] Figure 3-8 (a) Schematic of the relacement of multilayered sherical article with an equivalent homogeneous shere of the same radius but with the effective ermittivity. [Jones, T. B., Electromechanics of Particles. Cambridge University Press: New York City, NY, 1995.] 7 (Used under fair use, 213) (b) The DEP sectrum of the tow-layered sherical article. The solid line reresents Re[ K ] and the dot line reresents Im[ K ]. [Morgan, H.; Green, N. G., AC Electrokinetics: colloids and nanoarticles. Research Studies Press: Philadelhia, PA, 23.] 27 (Used under fair use, 213) It is noteworthy that Eq. (3.17) is the same as Maxwell s classic mixture formula for the effective ermittivity of a mixture consisting of noninteracting sherical articles in a medium with very low volume fraction (i.e., ϕ << 1) 7,27, which is usually written as 25

43 ε mix ε ε m 1+ 2ϕ ε 2 ε + m = ε m ε ε m 1 ϕ ε 2 ε + m (3.19) 3.3Maxwell-Wagner Model of Interarticle Force in AC-DC Electric Fields Maxwell-Wagner Model is the simlest model to simulate the electric-field interarticle force resulting from both the erimittivites and condcutivities of identical dielectric sheres with ohmic loss and an ohmic lossy dielectric medium by combining the effective moment method with the oint-diole aroximation. 5 The derivation of this model starts at solving Lalace s equation of an isolated dielectric shere with ohmic loss in a uniform AC electric field and assuming the induced diole moment is the dominate contribution of the article olarization, like the derivation in section Combining the concet of the oint-diole aroximation, the time-averaged force on a shere i at the origin due to another identical shere j at ( r, θ ) as, shown in Figure 3-9 (a), can be exressed ij ij 6 12πε mr Fij rij θij = β 4 eff ω Erms θij er + θij eθ (, ) ( ) [(3cos 1) (sin 2 ) ] r (3.2) where E = E / 2, and the effective relative olarizability β eff is rms where β 2 eff β + β ( ωτ ) = c d mw ( ωτ mw) (3.21) β d ε εm =, ε + 2ε m σ σm βc = σ + 2σ (3.22) m Interestingly, if we aly two frequency limits to Eq. (3.21), we can find similar results to Eq. (3.15) and (3.16). That is, ωτ lim β ( ω) = β MW ωτ MW 2 2 eff d lim β ( ω) = β 2 2 eff c (3.23) (3.24) 26

44 which also demonstrate that the conductivities of articles and medium govern the field-induced force in a DC and/or low-frequency AC electric fields and the ermittivities of articles and medium govern the force in a high-frequency AC fields. Likewise, to see the frequency deendence of the electric field-induced interarticle force, we merely focus on the variation of β 2 ( ω ) with frequency, which is lotted in Figure 3-9 (b). eff (a) (b) (I) E r ij (II) Figure 3-9 (a) Electric field-induced interaction between two identical sheres i and j in an electric field. r ij is the searation between the centers of the articles and θ ij is the angle between the line-ofcenter and the electric field. (b) Calculated square of the effective relative olarizability β 2 eff versus dimensionless frequency in terms of the Maxwell-Wagner relaxation time: (I) β = 1β ; (II) β =.1β. [Parthasarathy, M.; Klingenberg, D. J., Electrorheology: Mechanisms and models. Materials d c Science & Engineering R-Reorts 1996, 17 (2), ] 5 (Used under fair use, 213) d c 3.4 Equivalent Circuit Model of Interarticle Force in AC-DC Electric Fields Different from the electromagnetic oint of view above, Colver has emloyed the concet of equivalent circuits to develo an interarticle force model of for electric field stabilized gas-fluidized beds 31, as illustrated in Figure 3-1. The resistance-caacitance comonents in this model result in two time constants: self-article charge relaxation time τ and article-article relaxation time τ b for transient decay of charge within a article and between two articles, resectively. To formulate the model, Colver came u some assumtions as follows. 27

45 (1) The concerned sherical owders are in a cubic acking. (2) Particles are semi-insulating with a small but finite surface electrical conductivity (i.e., ohmic loss) and finite contact resistance at oint of contact. (3) Self-article charge decay is no influenced by neighboring articles and can be evaluated using single-article relaxation theory. The single article charge relaxation time is d τ = (3.25) ( ε 2 ' + ε) 4σ s where d is the article diameter and σ s is the surface conductivity of the article, which can be given as d d σ s = ln( ) πρ r c (3.26) where ρ is the aarent (single) article resistivity and r c is the article contact radius. (4) Particle-article charge decay can be evaluated using continuum bed relaxation theory. The bulk owder charge relaxation time is given as where τ ' b = ε ρb (3.27) ρ b is the aarent bulk owder resistivity and ε is the corresonding bulk owder ermittivity for bed voidage α, given as Eq. (3.28) and (3.29) resectively RH = ( ) ( ) (3.28) ρ b E T d e where T is temerature (K) and RH is relative humidity (%) and this equation has been tested u to 4% RH. ε ε ε ε α ε ε = ε 3+ 1 α ε (3.29) 28

46 (5) The equivalent circuit is comarable small to the wavelength of the alied signals so all electrical signals travel through the circuit quickly, affecting every oint in the circuit simultaneously (i.e., lumed arameter circuit theory). 3 Thus distributed caacitance and resistance over the surface of a article can be assumed having single values. (6) Circuit comonents are linear, which means any steady-state outut of the circuit (e.g., voltage and current) is also a sinusoidal signal with frequency f when the inut signal is a sinusoidal function with frequency f but they may not be in hase. Figure 3-1 An equivalent circuit for formulating the interarticle force of a one-diemensional article chain in AC-DC fields. [Colver, G. M., An interarticle force model for ac-dc electric fields in owders. Powder Technology 2, 112 (1-2), ] 31 (Used under fair use, 213) Clover also assumed the caacitors in the circuit are simle arallel late caacitors. Therefore the contact caacitance C c at the interface between the articles and the self-article caacitance C are given as C c ε A επr '2 c = = K1, xc xc C επr '2 = K2 (3.3) d ' where x c is the searation distance between the articles, Ac is the effective caacitance area, r is the radius of A c and the constants K 1 and K 2 account for effective caacitance area, distributed caacitance, curvature of articles, and the average contact searation distance. 29

47 For the alication of the model to both AC and DC fields, a sinusoidal alied field Vt ( ) = Vex( jωt) is assumed. Based on Kirchhoff s law, we can obtain the differential equation deicting the AC (or DC as ω ) voltage dro across the contact caacitance V c dv V V c c c [1 ] j ω jωτ t e dt + τ = τ + AC or DC ( ω ) b c (3.31) where the time constants are defined as 1 RR c τ b = [ C + Cc] = [ C + Cc] 1/ R 1/ R + c R + Rc R R c Rc = τ c = τ + τc R + R R + R R + R c c c (3.32) where Rc is the article contact resistance, R is the single article resistance τ = R[ C + C] = τ + RC, τ = RC and τ c = RC c c is the article contact relaxation time. c c c Solving Eq. (3.32) gives 2 τ 1 + ( τω ) b 2 b c 1 ( b ) t Vc = Aex + Vex j( ωt+ δ1 δ2) τ τ + τω (3.33) where δ 1 and δ 2 are defined as tanδ1 = 1/ ωτ b and tanδ 2 = 1/ ωτ, resectively. On the right hand side of Eq. (3.33), the first term is the transient resonse and the second term is the steadyeriodic resonse. The interarticle force can be taken as the force acting on the lates of the contact caacitor. Since the transient term would diminish over time, we can ignore it and also the hase angle (i.e., ωt + δ1 δ2 = ), obtaining the eak force er unit area F E 1 + ( τω) V = = A, AC/DC (3.34) 2 2 max ε c ε τb 2 c ( τω b ) τ c xc 2 2 ε 1 + ( τω) τb d = 2 E ( τω b ) τ c xc '2 where Ac = π r and the searation distance between the articles is assumed smaller than the diameter of the article so the alied voltage can be rewritten in terms of the electric field 2 3

48 strength in the far-field. Further, the time-averaged interarticle force er unit area can be exressed F 2 2 avg ε 1 + ( τω) τb d Fmax = 2 E rms = c ( τω b ) τ c xc 2Ac A, AC (3.35) where E = E / 2 is the rms value for the alied sinusoidal field. rms To cover both AC and DC fields, Colver further combined DC field interarticle force equation Eq. (3.36) with Eq. (3.34), giving more general AC-DC interarticle force equation Eq. (3.37). F d E E 2 2 dc = δπε β β bk DC (3.36) ( τω) 2 2 β Fmax = δπε d E 2 bk E 1 + ( τω b ) γ β (3.37) where δ, β, and γ are constants and sheres. E bk is the electric breakdown strength III between the Although Colver s model is based on the concet of the equivalent circuit, some features of his model corresond or similar to some conclusions from the effective moment method and Maxwell-Wagner model, which are summarized as follows. (1) The mathematical exressions of the electric field-induced force from these three models all show the force is roortional to the square of the rms value of the alied electric field. (2) Colver s model also shows that the force would either increase of decrease with frequency. Colver mentioned that it is the relative magnitude of τ and τ b determines the trend of the force with the frequency. Note that Eq. (3.25) Eq. (3.27) can be used to calculate these two time constants based on the assumtions (3) and (4). In addition, the ermittivity and conductivity of the articles would always greater than the that III Electric (or dielectric) breakdown strength is the maximum electric field strength that an insulating material can withstand without any current leakage. 25 Once the electric field strength larger than this value, the insulating material would not be insulating anymore, but conductive. 31

49 medium in Colver s model since he derived his model for air medium. Thus we can infer that the trend of the force deends on the relative magnitude of the ermittivity and surface conductivity of the articles and the resistivity and the ermittivity of bulk owder in Colver s model. (3) Besides τ and τ b, Colver also ointed out that C / forces at large values of b τω and R / c C governs the magnitude of the c R governs the force at small values of b τω. Note that C / Cc = ( xc / d)( ε / ε) as K1 = K2 = K for simlification. Then in other words, we can interret that the force is governed by the erimitivities of the articles and air at high frequency and by the conductivities of the articles and the air at low frequency when the searation distance between the two articles are fixed. We will use these models to interret our exerimental results in Chater 6. 32

50 Chater 4 Atomic Force Microscoy (AFM) and the Technique of Force Measurements AFM is well-known for its ability to image surface toograhy at the micro- and/or nanoscale. However, what AFM can really do is far more than this by a well-designed exeriment, the choice of oeration modes, instrumental modification, and/or methods of data acquisition. 32 Besides imaging, AFM has used in the measurements of different surface forces (e.g., van der Waal and adhesion forces) 33,34 and nanoscale dielectric constant, 35 the quantitative study of elastic and viscoelastic roerties and tribology, including friction, wear and lubrication, 36,37 icogram mass sensors, 38 nanoscale fabrication, 39 and so on, shown the versatility of this instrument. In this charter, we will focus on the technique of force measurement, which is used in the work. But in the beginning, we will briefly introduce basic concets and modes of AFM before talking about the measurement of forces. 4.1 Basic Concets and Modes of AFM There are mainly five arts in AFM: 1) cantilevers and the attached tis, 2) laser beam 3) hotodiode (i.e., hoto dectactor), 4) iezoelectric ositioner (usually shorten to iezo), and 5) a feedback loo, as shown in Figure 4-1. Before the measurement, the laser beam is focused at the end of the cantilever and deflects to the hotodiode. When the ti on the cantilever feels or senses some forces between itself and the samle, the cantilever would start to bend, thus changing the osition of the deflected laser beam on the hotodiode. At this moment, the signal detected by the hotodiode would be fed into a DC feedback amlifier in the feedback loo, comaring with the set oint determined by the oerator. The resulting difference (i.e., error signal) is sent to a roortional integral differential (PID) controller to make the iezo adjust the searation between the cantilever and the samle, keeing the deflection of the laser beam equal to the set oint. The adjustment of the iezo generates the image of the samle surface. 4 Note that the stiffness of the cantilever would affect how small the force can be reflected to the hotodiode. The softer the cantilever is, the smaller the force can be detected, which can be exlained by Hooke s law and the stiffness of the cantilever is also called the sring constant of the cantilever. In addition, the aex radius and the geometry of the ti would affect how small the indentation and the convexity at the surface of the samle can be robed by the ti. Thus, the combination of the cantilever and ti would determine the resolution of an AFM image since the 33

51 sensitivity and the quality of the hotodioe, iezo, and the feedback loo for commercial AFMs are quite similar nowadays. The oeration of AFM can be divided into several modes based on whether the cantilever oscillates or not and the tyes of the set oint. Here, a static mode contact mode and two dynamic modes amlitude modulation (AM) and frequency modulation (FM) modes are introduced. Besides those three basic modes, we will also introduce a Kelvin robe force microscoy (KPFM), which is related to this work. Contact mode The features of the contact mode are that the cantilever does not oscillate, the ti would directly touch the samle while imaging and can be used in both ambient conditions and liquids due to its simlicity. The set oint of this mode is the deflection of laser beam on the hotodiode. Although this mode is mostly used, it would cause the damage either of the samle or of the ti, deending on which one is harder, thus distorting the features of generated image and not suitable for imaging soft materials, such as biomaterials and olymers. 4 Amlitude-Modulation (AM) Mode The rincile of the AM mode is shown in Figure 4-1. First the cantilever is mechanically excited at or near its free resonant frequency (i.e., the resonant frequency without the effect of ti-samle interaction). When the ti aroaches the samle and starts to sense the force, the deflection of the laser beam from the cantilever would change and is detected by the hotodiode. The detected signal is sent to a lock-in amlifier to get the information of the amlitude and hase of the oscillating cantilever for comaring with the set oint, a chosen amlitude of the oscillation, usually is 7-8% of the free amlitude (the amlitude at the free resonant frequency). As state before, the calculated error signal from the DC feedback amlifier is fed into the PID controller to control the iezo, keeing the detected signal from the hotodiode (or the detected amlitude) is the same as the set oint. 41 The adjustment of the iezo here is the toograhy of the samle surface as well. In addition, the difference between the detected hase and the hase from the driving force for exciting the cantilever can give the ma of the material roerties of the samle, such as soft or hard. 42 The AFM mode is usually used in ambient 34

52 conditions and liquids, but not in the ultra high vacuum (UHV), which would slow down the seed of imaging (a) (b) Figure 4-1 (a) The behavior of the cantilever in an AM mode [Melitz, W.; Shen, J.; Kummel, A. C.; Lee, S., Kelvin robe force microscoy and its alication. Surface Science Reorts 211, 66 (1), 1-27.] 43 (Used under fair use, 213) (b) Scheme of an AFM setu in the AM mode. [Schirmeisen, A.; Anczykowski, B.; Hölscher, H.; Fuchs, H., Dynamic Modes of Atomic Force Microscoy. In Nanotribology and nanomechanics. Volume 1, Measurement techniques and nanomechanics, Bhushan, B., Ed. Sringer: Berlin ; Heidelberg ; New York 211; ] 41 (Used under fair use, 213) Conventionally, the AM-mode is also called the taing mode since the cantilever is oscillating at or near its free resonant frequency, causing the ti touch the samle shortly but 35

53 frequently. 4 However, the AM-mode can be used in a non-contact mode as well, 42 which is confusing; thus, it is better to name the modes based on the tye of the set oint and the oeration rincile of the feedback loo. Frequency-Modulation (FM) Mode Similar to the AM-mode, the cantilever is excited at its free resonant frequency in the FM mode as well. Nevertheless the set oint of the FM mode is the free resonant frequency of the (a) (b) Figure 4-2 (a) The behavior of the cantilever in an FM mode. [Melitz, W.; Shen, J.; Kummel, A. C.; Lee, S., Kelvin robe force microscoy and its alication. Surface Science Reorts 211, 66 (1), 1-27.] 43 (Used under fair use, 213) (b) Scheme of an AFM setu in the FM mode. [Schirmeisen, A.; Anczykowski, B.; Hölscher, H.; Fuchs, H., Dynamic Modes of Atomic Force Microscoy. In Nanotribology and nanomechanics. Volume 1, Measurement techniques and nanomechanics, Bhushan, B., Ed. Sringer: Berlin ; Heidelberg ; New York 211; ] 41 (Used under fair use, 213) 36

54 cantilever. The DC feedback amlifier would calculate the difference between the detected resonant frequency and the free resonant frequency of the cantilever as the control signal to adjust the searation between the ti and samle. In addition, the FM mode has an additional feedback loo, as shown in Figure 4-2, to kee the amlitude of the oscillating cantilever constant so that the searation between the ti and the samle would not be affected by the amlitude. 41 That is why the FM-mode is also called the non-contact mode or the self-excitation mode. 41,42 The FM-mode is designed for imaging in UHV to achieve high-resolution image and to comensate the disability of the AM-mode in UHV Kelvin Probe Force Microscoy (KPFM) KPFM was derived from both the dynamic modes of AFM and the macroscoic Kelvin Probe technique, which is used to measure surface otential, to ma the work function or surface otential of the samle with high-resolution at and the micro- and nanoscale. 43,44 Different from the forementioned modes, an DC-voltage ( V DC ) is alied to comensate the contact otential difference (CPD) between the ti and the samle surface and an AC-voltage ( V sin( ω t) ) is alied to excited the ti in a KPFM. As a result, the force sensed here is mainly the electrostatic force. The electrostatic force can be derived as follows by regarding the ti-samle system as a arallel late caacitor due to the very short distance between them. 43,44 AC AC The energy stored in a arallel lated caacitor is force is given as Uel = CV, thus the electrostatic 1 C 2 r 2 Fel = Uel = V CV Since the voltages here are distance-indeendent and the direction of them is erendicular to the samle surface, which is z-direction, the Eq. (5.1) can be simlified as V r (4.1) 1 C 2 1 C Fel = V = VDC ± VCPD + VAC ACt 2 z 2 z [ sin( ω )] 2 (4.2) where V CPD is defined as 37

55 where φti V CPD φ φti φ = = e e smale (4.3) and φsmale are work function of the ti and samle and e is the elementary charge. Note that the sign of VCPD in Eq. (4.2) is determined by how V DC is alied to the samle or the ti. Based on Eq. (4.3), we can calculate the work function of the samle by using the ti with a known work function in KPFM. Further, F el can be divided into three forces by rearranging Eq. (4.2). C FDC = ( VDC ± VCPD ) + VAC z 2 4 C Fω = ( VDC ± VCPD ) VAC sin( ωact) z (4.4) (4.5) C VAC F2 ω = cos(2 ωact) (4.6) z 4 The oeration rincile of KPFM is based these three forces, F DC, F ω and F2 ω. We would also use similar concets to exlain our results in chater 8. Since KPFM is based on the dynamic modes of AFM, it can be oerated in either AM or FM mode as well. The rinciles and feedback loos designs of AM- and FM-KPFM are show in Figure 4-3. In AM-KPFM, the amlitude of the oscillating cantilever at ω AC is taken as control signal. Once the feedback loos get the signal of amlitude, it would adjust V DC to minimize the difference between V DC and V CPD, thus nullifying F ω and the amlitude. The adjustment of V DC is can be used to ma the CPD of the samle. 43 In FM-KPFM, the frequency shift of the oscillation cantilever from ω AC is the control signal instead. Similarly, the feedback loos would adjust VDC to reduce the frequency shift and the adjustment of V DC can be converted to the image of CPD since they ossess the same magnitude but oosite sign. 38

56 Figure 4-3 Scheme of the oeration rincile of a KPFM. [Sadewasser, S., Exerimental Technique and Working Modes. In Kelvin robe force microscoy: measuring and comensating electrostatic forces, Sadewasser, S.; Glatzel, T., Eds. Sringer-Verlag Berlin Heidelberg: Heidelberg ; New York, 212; 7-24.] 44 (Used under fair use, 213) 4.2 Technique of Force Measurement Both the contact mode and dynamic modes can be used in surface force measurement. 33,41 Here we only introduce the fore measurement technique in contact mode since we use this mode in this work. The rincile of surface force measurement by AFM, as shown in Figure 4-4, is different from that of imaging. For imaging, the cantilever usually with a shar ti would scan the samle surface, which is controlled by a feedback loo. However, for force measurement, a microshere would be attach to the end of the cantilever, used as a ti/robe, and the cantilever or the samle, deending on where the iezo is located, would aroach and retract the samle surface as illustrated in the right-hand side of Figure 4-4. The cantilever with a microshere is so-called the colloidal robe, which is develoed to quantitatively measure the surface forces. 33 The reasons for using a microshere rather than a shar ti are as follows. First, the force between a colloidal robe and samle is larger than that between a shar ti, which usually only has a 5-1 nm radius, and samle based on the Derjaguin aroximation so the measurement can be more accurate. The Derjaguin aroximation is exressed as 39

57 FD RR π R1+ R2 1 2 ( ) 2 WD ( ) where D is the distance between the two subjects (e.g., the colloid and the samle), R 1 and R2 the radii of the two subjects, resectively, and W the interaction free-energy. 45 Second, more quantitative analysis, such as calculating the surface energy by the Derijaguin aroximation can be roceeded since the geometry of the colloidal robe is known and it is easy to determine the radius of a microshere. Also, the surface roughness of a microshere can be controlled and examined easier than that of a small, shar ti during manufacturing. Furthermore, we can choose to use the microsheres made from different materials and/or coated with different chemicals/materials, such as hydrohobic functional grous and gold to study surface forces in different systems. 33 (4.7) V Figure 4-4 On the left, a tyical raw data diagram of the deflection of the cantilever (i.e., the deflection of the laser beam for the catnielver) vs. the iezo osition (i.e., the dislacement of the iezo) from a force measurement by AFM is shown. On the right, the scheme of the behaviors of a cantilever with resect to a flat samle in a force measurement is illustrated. [Ralston, J.; Larson, I.; Rutland, M. W.; Feiler, A. A.; Kleijn, M., Atomic force microscoy and direct surface force measurements - (IUPAC technical reort). Pure and Alied Chemistry 25, 77 (12), ] 34 (Used under fair use, 213) A tyical raw data curve from the force measurement by AFM is shown in the left-hand side of Figure 4-4. There are five regions in this curve corresonding to different behaviors of the cantilever which are illustrated on the right of Figure 4-4. In region I and V, the distance between the robe and the samle is so large that the colloidal robe does not sense any force and no bending of the cantilever. In region II, however, the distance is quite short; thus the colloidal robe starts to sense some forces, either attractive (in Figure 4-4) or reulsive (in Figure 4-5), and the cantilever deflection is not flat anymore. In region III, the robe contacts the 4

58 samle and moves comliantly with the iezo. This linear region is called the constant comliance region and reflects the otical sensitivity of the hotodiode with resect to the cantilever deflection, which is usually used to calibrate the cantilever deflection. 34,46 In region IV, the cantilever still contacts the samle but the iezo moves in an oosite direction. Once the restoring force of the cantilever is large enough to ull off the cantilever itself from the samle surface, the cantilever would jum back to the original osition (i.e., region V) since the distance between the colloidal robe and the samle is large and no surface force is sensed by the colloidal robe. Figure 4-5 The rocedure for converting a cantilever deflection vs. iezo dislacement curve to a force vs. distance curve. [Senden, T. J., Force microscoy and surface interactions. Current Oinion in Colloid & Interface Science 21, 6 (2), ] 46 (Used under fair use, 213) 41

59 The standard rocedure for converting the raw data to a force curve is illustrated in Figure 4-5. Usually the cantilever deflection at large distance ( z in Figure 4-5) is assumed no measured surface force and is used to define the zero of cantilever deflection signal. As mentioned before, the sloe of the constant comliance region ( Ω ) is used to calibrate and convert the cantilever deflection signal into units of distance. Then the calibrated and converted deflection signal ( x ) would be multilied by the calibrated sring constant of the cantilever with the microshere ( k ) to get the surface forces between the colloidal robe and the samle. The searation or distance ( d ) between the colloidal robe and the surface is the relative movement between the cantilever ( x ) and the iezo ( D ) and the contact osition ( c ) is used to define the zero searation. 46 However, in this work, we modify this rocedure since the electric fieldinduced force is a very long-range force and we need to use the data collected under no electric field to define the zero of the cantilever deflection signal. We will state our modify rocedure in Chater 6 for details. 42

60 Chater 5 Exerimental Section 5.1 AFM Setu This work was mainly erformed with a Cyher TM AFM (Asylum Research, Santa Barbara, CA). To measure the F DEP between the two articles, we modified the AFM as shown in Figure 5-1. Flexible Glove Chamber Laser Photodiode Dry Air Piezo ITO coated slide (transarent) Cantilever holder Cantilever Dielectric sheres E Rubber ro Voltage Amlifier Humidity meter x, y, z-scanner Functional Generator Figure 5-1 Schematic of aaratus. The shere attached to the cantilever is referred to as the to shere whereas the shere attached to the scanner is referred to as the bottom shere The electric field was created using two electrodes searated with a ga of 7. mm by a rubber ro. Indium tin oxide (ITO) coated slides which are transarent but highly conductive were chosen as the electrodes to allow the laser beam ass through the to electrode as shown in Figure 5-1. The alied voltage was generated by a function generator (Mode MHz Synthesized Arbitrary Waveform Generator, Wavetek-Datron) and amlified 1 by a voltage amlifier (Model 21HF High Frequency High-Voltage Power Amlifier, Trek Inc.). The voltage was varied from 6 to 22 V rms (root-mean-square). All voltages in this thesis are rms values. The frequency was in the range from DC to 1 MHz. To create a low humidity environment, the whole AFM was enclosed in a flexible glove chamber (GLOVE BAG Inflatable Glove Chamber, Glas-Col LLC) sulied with dry air 43

61 continuously circulated. The RH can go down to less than 3% within around 1 to 2 hours but the measurement was executed after about 8 hours of continuous flow to ensure the surface of the microsheres and their surroundings are dry enough. During the measurement, dry air still circulated to maintain the low humidity environment. The air flow did not significantly affect AFM oeration which has been examined by measuring the variation of the deflection of laser beam with time (Figure 5-2). The RH and temerature in the chamber was monitored by a humidity meter (THWD-3 Relative Humidity and Temerature Meter, Amrobe). All measurements of field-induced interactions between two sheres were conducted at 2 ± 1 % RH and 25 ± 3 C. 1.5 no air flow 1. Variation of Deflection (nm) air flow Time (s) Figure 5-2 Effect of air-flow on deflection of AFM cantilever in glove bag. Plot shows deflection of cantilever as a function of time with no air flow (to) and with air flow (bottom). 5.2 Samle Prearation A barium titanate (BaTiO 3 ) glass microshsere (MO-SCI Secialty Products, L.L.C.) was attached to the end of a silicon rectangular shaed cantilever (tiless, NSC12/no backside metal coating, MikroMasch) with heat-softening eoxy resin (Eikote 14) by micromaniulators under an otical microscoe. The attachment was confirmed by an otical microscoe. The normal sring constants and resonant frequencies of the colloidal robes and cantilevers alone were calibrated by the Hutter method 47. The surface roughness of the microsheres on the 44

62 cantilever was examined by inverse imaging with a TGT1 grating (NT-MDT). The microsheres had a roughness less than 5 m rms over the areas of 1 µm x 1 µm and 5 nm x 5 nm. For studying field-induced interactions between two sheres, the BaTiO 3 glass sheres were attached to a thin glass slide with eoxy resin by using micromaniulators. The sizes of the microsheres used in this study verified by using the camera module of the AFM were aroximately 3 μm in diameter. 5.3 Alignment of Two Particles To investigate field-induced intershereical interaction, the two microsheres were first aligned manually using a video microscoe that is art of the AFM, and then aligned more accurately by imaging the bottom shere with the to shere. After imaging the shere on the glass slide on the bottom electrode by the colloidal robe, the commercial AFM software can identify and control the relative osition of the to and bottom sheres. Thus, we can align the two microsheres and measure the force as a function of distance between the aexes of the sheres via the software Force Curve Collection and Analysis Since the electric field-induced force is a long range force, we modified the normal rocedure. In the normal method, the zero of force is determined from the force (deflection) at large searation, where there is no surface force. When there is a very long range and slowly varying force, it is difficult to know at what searation zero force occurs. In addition, there is a comlicating effect, the virtual deflection 48, where the signal from the deflection varied slowly with distance, even when there is no change in force. Fortunately, the field-induced force can be conveniently turned on and off with a switch, so we have established the zero force with the field turned off. Thus, for each condition (e.g., at 6 V and 5 khz), two successive force curves, one with and one without an alied voltage, were collected for around one second of each other. The measurement without an alied voltage was used to determine zero deflection as in the normal method as shown in Figure 5-3 (a). The two measurements (with and without the alied field) were each analyzed in the normal way, i.e. to roduce a force searation lot, thereby ensuring that the two curves have the same zero of searation, and were not shifted due to slight changes in temerature. The deflection signal in the 45

63 (a) Deflection (nm) Jum-to-Contact Piezo Position (µm) no E 6 V, 5 khz Deflection (nm) Piezo Position (µm) (b) 1 6 V, 5 khz Force (nn) Searation (µm) Figure 5-3 (a) Raw data showing deflection iezo osition with the field on or off. (b) Net Force as a function of searation after the zero field data has been subtracted. absence of the field was then subtracted from the deflection signal in the field. An additional effect of this subtraction is that all other contributions to the measured force are removed. We have exlicitly assumed that these other contributions are additive. These other contributions include measurement artifacts due to variation in the laser signal with searation (virtual 46

64 deflection and otical interference between reflections of light from the cantilever and samle) and real forces, such as the van der Waals force. Thus, we assume that the van der Waals force is not affected by the alied field. In ractice, this is a small assumtion because the van der Waals force is very weak and short range (Figure 5-3 (a)) comared to the field-induced force, and is usually only significant at searations where we are unable to measure data because of a mechanical instability. 47

65 Chater 6 Results and Discussions As described in section 3.2.1, an AC electric field-induced force consists of a constant (time-averaged) term and a time-varying term. By rewriting the Maxwell-Wagner model, we can get 6 12πε 2 2 mr 2 F = β 4 eff ( ω) E[(3cos θij 1) er + (sin 2 θij ) eθ ] r 6 12πε mr = β ( ) 4 eff ω E (cos ωt)[(3cos θij 1) er + (sin 2 θij ) eθ ] r 6 12 πε mr 2 2 (1+ cos 2 ωt) 2 = β ( ) 4 eff ω E [(3cos θij 1) er + (sin 2 θij ) eθ ] r πε mr = β ( ) 4 eff ω E /2+ E r (cos 2 ωt) / 2 [(3cos θij 1) er + (sin 2 θij ) eθ ] 6 12πε mr = β ( ) (cos 2 ) [(3cos 1) (sin 2 ) ] 4 eff ω Erms + Erms ωt θij er + θij eθ r When the two articles align with the direction of the alied electric field (i.e., θ ij = ), the AC electric field-induced force between two identical articles can be exressed as πε mr πε mr 2 2 F = β ( ) ( ) (cos2 ) 4 eff ω Erms + β 4 eff ω Erms ωt (6.1) r r 49, IV The first term on the right hand side of Eq. (6.1) is referred to as static comonent of F DEP (i.e., the time-averaged term) which is related to the translation behavior of articles under an electric field. The second term is called dynamic comonent IV of F DEP (i.e., the time-varying term) reresenting the vibration behavior of articles under an AC electric field. Note that the amlitude of the dynamic comonent is the same as the static comonent. Most studies ay attention on the static comonent of F DEP and scarcely discuss the dynamic comonent due to the heavy daming effect for small articles in liquid media or no obvious oscillation for large 7 articles with large mass inertia in gas media. In this study, we have develoed a secial IV The static comonent is also called DC comonent or DC offset, which is the average value of a signal. 49 And the dynamic comonent is called AC comonent. 49 To avoid confusion DC and AC will only be used to describe the alied field, while static and dynamic are used to describe the resultant force. 48

66 exerimental setu and measurement techniques by AFM to observe the translation and vibration behaviors of a microarticle in air simultaneously. In addition to study the time-averaged electric field-induced force between two closely saced sherical dielectric microarticles, we have also found that the vibration behavior of small articles would affect their translation behaviors while they are close enough in air. Therefore, we also describe and discuss the interesting dynamic comonent. We believe both static and dynamic comonents of the electric field-induced force of two closely saced microarticles are imortant for better understanding the interfacial and surface henomena under an electric field. Our analyses and discussion are divided into several sections as follows. The contribution from the cantilever to the total force measured by AFM will be introduced and discussed first in section 6.1. Then, we will discuss the static and dynamic behavior of the colloidal robe to determine the behavior of small articles in air under an AC electric field in section 6.2. We will then discuss how the electric field affects the time-averaged electric field-induced force in section 6.3 and field-induced adhesion in section Cantilever Contribution to the Total Measured Electric Field-Induced Force The main objective of this thesis is to understand how an external electric field affects the forces between two sherical articles. However, our measurement necessitates the use of an AFM cantilever to determine the force, and this cantilever can also be olarized and can affect the olarization of the sheres, so it was necessary to first characterize the behavior of the cantilever. The behavior of the cantilever is interesting in its own right because an alied filed can be used to drive the oscillation of a cantilever. The three exerimental systems are shown in Figure 6-1: (a) cantilever with shere attached (colloidal robe) interacting with shere; (b) cantilever only and (c) cantilever interacting with shere. When comaring results from these three systems, our objective is to finally understand the two shere exeriment, so we always describe a range of searations that are relevant to that system. Thus the cantilever is always the same distance from the lower late when we comare different exeriments. For examle, if the range of searations of interest is 6 µm in a shere-shere (diameter 3 µm) exeriment, then the cantilever is 6 66 µm from the 49

67 late during the exeriment. Thus the relevant searation for the cantilever in the cantilever-only exeriment is also 6 66 µm from the late. Figure 6-2 (a) shows the forces between the cantilever and the bottom electrode under an electric field and no electric field, resectively. The results demonstrate that even when the searation is quite large (from 6 to ~66 µm), the cantilever still can be olarized and sense a strong attractive force, imlying the field-induced force is a strong long-range force. In addition, force for system (c) is greater than (b) so the cantilever interacts significantly with the bottom shere. (a) (b) (c) POSITION SENSITIVE DETECTOR LASER POSITION SENSITIVE DETECTOR LASER POSITION SENSITIVE DETECTOR LASER CANTILEVER CANTILEVER CANTILEVER PIEZOELECTRIC TRANSLATION STAGE PIEZOELECTRIC TRANSLATION STAGE PIEZOELECTRIC TRANSLATION STAGE Figure 6-1 The three different systems considered here (a) Two sheres system: The field-induced force between two BaTiO 3 glass microsheres was measured. Note that the measured force in this system contains the contribution from the cantilever. (b) Cantilever-only system. This is used to determine the effect of the field on the cantilever. (c) Cantilever-bottom shere system. This is used to determine the interaction between the cantilever and bottom shere. The dash circles reresent the locations of the sheres in the two sheres system, which means the searations between the cantilever and the bottom electrode in these three systems are the same. In more detail, the field-forces of the cantilever-only system and the cantilever-bottom shere system (Figure 6-2 (b)) have no significant searation deendence. This shows that there is little variation in the field at this large searation and/or that the force has a short range. This is useful because the shae of the shere-shere force curve is not affected by the cantilever. When we bring the cantilever much closer to the bottom late (searation 5 μm), the force decreases exonentially with distance. Figure 6-2 (b) dislays the comarison of the forces of three different systems at 6 V and 5 khz. As exected, the force of the two sheres system is the largest since it contains the 5

68 (a) (b) -5 Force (nn) V at 5 khz Two sheres Cantilever-only Cantielver-bottom shere Searation (µm) Figure 6-2 (a) Comarison of the force under an electric field (at 6 V and 5 khz) to the force without alying electric filed from the cantilever-only system. The searation here is the distance between the cantilever and the bottom electrode. (b) The comarisons of three searation-deendent forces from three different systems at 6 V and 5 khz. Note the searation here is the distance between two sheres in the two sheres system. For the cantilever-only system, the searation should be searation + 6 μm ; for the cantilever-bottom system, it should be searation + 3 μm. contributions from the cantilever and sheres. As the searation is large (e.g., 5 μm in the two sheres system), the field-induced force of the cantilever-bottom system ( nn at 5 μm) is larger than half that of the two sheres system (-2.78 nn at 5 μm), indicating that the contribution from the cantilever to the measured field-induced force is significant at large searation. However, the field-induced force of the cantilever-bottom system is equal to and/or less than half the forces of the two sheres while the searation is equal to and/or smaller than 3 µm. Similar results were obtained from a series of different alied voltages and frequencies. 51

69 Thus, we would not consider the forces at the searation larger than 3 µm in the following discussion. 6.2 AC Electric Field-Induced Oscillation Behavior of a Colloidal Probe To study the vibration behaviors, we collected time series of the deflection signal at samle rate 5 MHz while the colloidal robe was aroached to and retracted from the bottom BaTiO 3 glass microshere with different alied voltages and frequencies. The small data blocks (4 samles) at the searation from around 2 μm to 1.85 μm while the robe was aroached to the bottom shere were chosen to convert to amlitude-frequency sectra by fast Fourier transformation (FFT) with the alication of a Hanning window (IGOR Pro 5.2, WaveMetrics, Portland, OR) so that the effect of cantilever can be neglected in the data analyses. (a) (b) (c) (d) (e) (f) Figure 6-3 Time series data of the deflection signal of the two sheres system when the alied voltage is 6 V and the alied frequency is: (a) 1 khz; (c) 26 khz; (e) 27 khz and the corresonding FFT sectra (b), (d) and (f) resectively. The resonant frequency of the colloidal robe is 53 khz. 52

70 (a) (b) (c) (d) (e) (f) (g) (h) Figure 6-4 Time series data of the deflection signal of the two sheres system when the alied voltage is 6 V and the alied frequency is: (a) 4 khz; (c) 5 khz; (e) 55 khz; (g) 82 khz and the corresonding FFT sectra (b), (d), (f) and (h) resectively. The resonant frequency of the colloidal robe is 53. khz. From equation 6.1, we exect that the time series data will show a eak at two times the drive frequency ( 2 f ). The measured time series of the deflection data is shown in Figure 6-3 Figure 6-4, along with Fourier transforms. The measured data can be more comlex, because (i) the cantilever has its own resonse function and (ii) there could be other forces. The resonse function of the colloid robe can be determined from the thermal noise sectrum, which shows that the cantilever resonance ( f r ) is at 53 khz. Thus, we always exect a constant resonse 53

71 function below 53 khz, a stronger resonse at 53 khz and a weaker resonse at higher frequencies. The thermal noise is resent during all exeriments so we always exect to see a eak at 53 khz regardless of the drive frequency. This can be seen in Figure 6-3 and Figure 6-4. We also see the resonse at 2 f in all data, confirming the alicability of Eq. (6.1). However, there are also other harmonics ( f, 3 f, 4 f, etc.). The amlitude of different eaks at different alied frequency is summarized in Figure Static Comonent f 2f f r =53 khz Amlitude (V) Alied Frequency, f (khz) Figure 6-5 The comarison of the magnitude of different eaks that resonse at (i.e., static comonent), f, 2 f and f r at different alied frequencies ( f ). Both the Maxwell-Wagner Model (Eq ) and Colver s equivalent circuit model (Eq ) oint out that the electric field-induced force is roortional to the square of the electric field. By rearranging the equations (e.g., Eq. (6.1)), we can conclude that articles under an AC electric field would vibrate at 2 f in theory. In addition, the cantilever would vibrate under an AC electric field at 2 f as well. The reason for the cantilever vibration is the cooeration of the mechanical sring force from the cantilever itself and the AC electric field-induced force between the cantilever and the electrodes. Take the electrodes and the cantilever as two arallellate caacitors in series as illustrated in Figure 6-6 and assume that they have the same area and the fringing fields can be neglected. 5,51 Since the cantilever is olarized by the electrodes, 54

72 Fixed electrode V V 2 One end is fixed the other end is moveable Behave like a sring z G-g V 1 g Fixed electrode Figure 6-6 The scheme of the configuration of the cantilever and electrodes, where g is the ga between the cantilever and bottom electrode, G is the ga between the two electrodes, g ' is the ga between the cantilever and the bottom electrode, z is the dislacement of the cantilever, V is the alied voltage, V 1 is the voltage between the cantilever and bottom electrode, and V 2 is the voltage between the cantilever and to electrode. Note that V=V 1 +V 2. The bottom electrode was totally fixed. Although the to electrode was fixed at one end, it still can be taken as totally fixed due to its large mass inertia. The cantilever was only fixed at one end. electric fields would be built between the cantilever and the to and bottom electrodes resectively. Assuming that the voltage between the cantilever and the bottom electrode is V1 = V1, cos( ωt) and that between the cantilever and the to electrode is V 2 = V 2, cos( ω t) ; thus V = V cos( ω t) = V + V. As a result, the net force of the configuration illustrated in Figure can be exressed as 5,51 1 εa' 2 1 εa' 2 Fnet = V V kz 2( g' z) 2( G g' + z) 1 εa' εa' (6.2) 2 2 = V 2 1, cos ( ωt) + V 2 2, cos ( ωt) + kz 2( g' z) 2( G g' + z) where A ' reresents the area of the electrode late and the cantilever resectively and the ositive sign is for the forces that increase the ga between the cantilever and the bottom electrode. On the right hand side of Eq. (6.2), the first term is the electrostatic force between the cantilever and bottom electrode, the second term is the electrostatic force between the cantilever and to electrode and the third term is the sring force of the cantilever. In Eq. (6.2), the magnitude of the electrostatic forces change with the alied frequency and time at a fixed magnitude of the alied voltage, thus changing the corresonding sring force according to the Newton s third law. Therefore, the frequency of the net force waveform would be equal to 2f theoretically. 55

73 (a) (b) (c) (d) (e) (f) Figure 6-7 Time series data of the deflection signal of the cantilever-only system when the alied voltage is 6 V and the alied frequency is: (a) 75 khz; (c) 8 khz; (e) 82 khz and the corresonding FFT sectra (b), (d) and (f) resectively. The resonant frequency of the cantilever is 164 khz. Our results (Figure 6-3, Figure 6-4 and Figure 6-5) demonstrate that the AC electric fieldinduced force indeed resonse at 2 f. Nevertheless, we found that the force would also resonse at f r and other nf ( n is a non-zero ositive integer) as well, which cannot be exlained by the above theories. We also collected time series of the deflection signal of the cantilever-only system as shown in Figure 6-7. Similar henomena can be observed in the cantilever-only system as well even if the f r of the cantilever without mounted shere is 164 khz, which is quite different from that of the colloidal robe. The detected eak at f r is quite small and mostly is buried in the noise floor as shown in Figure 6-7 (b) and (f). Besides, no eak revails above the noise floor at 3 f and 4 f in the cantilever-only system, which may be buried in the noise floor as well. The magnitude of different eaks at different resonse frequencies and different alied frequencies is summarized in Figure 6-8. Likewise, the magnitude of the static 56

74 comonent is always the largest excet at around half f r of the cantilever (i.e., 82 khz). As for the dynamic comonent, the eak that resonses at 2 f is the largest, esecially at half f r. The magnitude of the eak at f has no big difference at different f, for f does not cover f r of the cantilever. 1 4 Static comonent f 2f 1 3 Amlitude (V) Alied Frequency, f (khz) Figure 6-8 The comarison of the magnitude of different eaks that resonse at (i.e., static comonent), f and 2 f at different alied frequencies ( f ). The ossible reason for the aearance of the eak at 4 f in the two shere system may be the local electric field around the sheres due to the olarized charge on the two sheres. Since the to shere mainly oscillates at 2 f, the bottom shere would sense a field with frequency 2 f due to the olarized charges on the to shere and thus has olarized charges alternating at 2 f. Likewise, the to shere would sense a field with frequency 2 f due to the olarized charge on the bottom shere and thus would resonse at twice the 2 f ; that is, 4f. Note that the eak at 4 f is smaller than the eak at 2 f in Figure 6-3 (b), (d) and (f) since the olarized charges induced AC local field is much smaller than the externally alied field. This also exlains why no eak revails above the noise floor at 6 f in the two sheres system. In the 57

75 cantilever-only system, only a cantilever is under an electric field so no local field induced from the olarized charge on other objects and hence no eak revails at 4 f. (a) Voltage (V) Peak at f 1 khz 1 khz Amlitude (V) Electric Field (V/mm) (b) Peak at 2f 1 khz 1 khz 7 Amlitude (V) E 2 (V 2 /mm 2 ) Figure 6-9 The correlation between the alied electric field strength and the amlitude of the eak at (a) the alied frequency f and (b) twice the alied frequency 2 f in the two sheres system. 58

76 As for the eak at f in the FFT sectra of both the two sheres system and the cantilever-only system, Kaler et al. have reorted similar results for a 45-μm-diameter rotolasted tobacco lant cell susended in sorbitol solution under a low-frequency electric field. 52 They suggest that the eak at f is attributed to the Coulomb interaction between the (a) 4 2 Force (nn) V 25 khz 26 khz 27 khz Searation (µm) (b) Force (nn) V 25 khz 26 khz 27 khz Searation (µm) Figure 6-1 The variation of the AC electric field-induced force with the searation between two BaTiO 3 glass microsheres at the alied voltage: (a) 3 V and (b) 6 V when the alied frequency is around half the resonant frequency of the colloidal robe (i.e., 53 khz). 59

77 surface charges on the cells in the aqueous medium and the external electric field (i.e., electrohoresis). Based on their results, they found that the magnitude of the oscillation is roortional to the alied voltage at frequency above ~1 Hz when the alied frequency is ranged from ~1 to ~1 Hz. Nevertheless, based on our results, we do not find any correlation between the alied field strength and the amlitude of the eak at f (Figure 6-8 (a)) whereas the amlitude of the eak at 2 f is roortional to the square of the alied field strength, agreeing with the Maxwell-Wagner and Colver s models (Figure 6-8 (b)). We are not sure whether the eak at f is attributed to the surface charge or not. It may also result from the nonlinearity of our systems. Further study is necessary to understand these henomena. 1 4 Peak at 2f 1 V 6 V 3 V Amlitude (V) Alied Freuqency, f (khz) Figure 6-11 The comarison of the magnitude of the colloidal robe vibration resonses at twice the alied frequency when the alied voltages are 3, 6 and 1 V. Note that the eak at 2 f abrutly increases and become the largest eak (larger than the static comonent) when f is equal to half f r in the two shere system and cantilever-only system. This is because the colloidal robe/cantilever vibrates at 2 f and when f is equal to half f r of the colloidal robe/cantilever, 2 f is equal to f r and resonance of the colloidal robe/cantilever haens. Figure 6-1 shows the force curves of the static comonent of the two 6

78 shere system when the alied frequency is around half the resonance frequency. As shown in Figure 6-1, the colloidal robe is not able to measure the force due to its resonance. Therefore, while the colloidal robe resonates (i.e. the alied frequency is at 25, 26, or 27 khz), the measured forces are not legitimate and we would not concern these into our discussion. We also study the effect of the alied electric voltage on the amlitude of the deflection signal at different alied frequency. Figure 6-11 shows the comarison of the eak at 2 f at different alied voltages and alied frequencies. The magnitude of the amlitude increases with the alied voltage. As the alied voltage increases, the resonance still occurs when the alied frequency is half f r of the colloidal robe. This indicates that no electrostatic sring softening/hardening effect haens, which would decrease/increase the resonant frequency of the resonator. 53 (a) (b) (c) (d) Figure 6-12 The variation of the deflection signal at the searation between the two BaTiO 3 glass microsheres ranged from (a).9 to 1. µm; (b).4 to.5 µm as the alied voltage is 6 V and the alied frequency is 5 khz and at the searation ranged from.9 to 1. µm when the alied voltage is 1 V and the alied frequency is (c) 5 khz; (d) 1 khz. In addition to the time series of the deflection signal, we also analyze the variation of the deflection signal with the searation between the two BaTiO 3 glass microsheres. Figure 6-12 (a) and (b) show the variation of the deflection signal at two different regions of the searation between the two BaTiO 3 glass microsheres when the alied voltage is 6 V and the alied frequency is 5 khz. As the range of the searation is between.9 and 1. µm, the amlitude of 61

79 the deflection is around 4 nm. However, the amlitude becomes around 6 nm as the range of the searation is between.4 and.5 µm. This may be because the local electric field between the two sheres becomes stronger while the searation between the sheres decreases. When the two sheres get closer with each other, the degree of the olarization increases since the olarized sheres can olarize each other as well, causing the electric field around the closely saced sheres in fact is stronger than the alied one. The amlitude is roortional to the AC electric field-induced force so the amlitude would increase with the increase of the local electric field strength. Also, the amlitude would incrase with the alied voltage (Figure 6-12 (a) and (c)) and change with the alied frequency (Figure 6-12 (c) and (d)), following the trend of the curves in Figure 6-11 due to the nature of vibration. Since the colloidal robe under an AC electric field would vibrate while it is aroaching to the bottom shere, it would bounce back when the searation between the two sheres is comarable to the amlitude of the colloidal robe vibration as shown in Figure 6-13 (a) and (b). The increase of the amlitude of the deflection signal with the decrease of the searation (a) (b) (c) (d) Figure 6-13 The variation of the deflection signal with the searation between the two BaTiO 3 glass microsheres at different alied voltages and frequencies: (a) 6 V, 5 khz; (b) the enlargement of the bouncing oint of (a); (c) 1 V, 5 khz; (d) 1 V, 1 khz. between the two microsheres would aggravate this henomenon, giving rise to the electric field deendency of the bounce (Figure 6-13 (a) and (c)). Furthermore, the bounce is alied 62

80 frequency-deendent as well due to the frequency-deendent amlitude as shown above. In Figure 6-12, we oint out where the colloidal robe would bounce back, which is referred to as bouncing oint and summarize them at some conditions in Table 6-1. At a fixed alied frequency, the bouncing oint increases with the alied voltage. At a fixed alied voltage, the bouncing oint would increase with the alied frequency when it is below or equal to the resonant frequency of the colloidal robe but decrease with the alied frequency when it is above the resonant frequency, like the trend of curves in Figure Note that there is no aarent bounce as the alied frequency is 1 khz. For examle, the bouncing oint is 295 nm as the alied voltage is 1 V and the alied frequency is 1 khz. Thus, we would discuss the data at 1 khz from 1 V to 3 V for the static comonents in the following sections since this set is affected by the vibration of the colloidal robe much less. Also the smallest searation we would concern is 3 nm. Table 6-1 Brief summary of the bouncing oint at different alied voltages and frequencies. Alied Voltage and Frequency bouncing oint (nm) 12, 1 khz V, 5 khz V, 1 khz V, 5 khz V, 1 khz V, 5 khz None 3 V, 1 khz None 6.3 Electric Field-Induced Force between Two Dielectric Microsheres We measured the variation of the AC electric field-induced force ( F in ) with different searations between the two BaTiO 3 glass microsheres and different alied voltages/alied electric field strengths ( E ) at different frequencies. Figure 6-14 shows the frequency deendence of the field-induced force and the negative sign of the force means the measured force is an attractive force. In Figure 6-14 (a), the field-induced force does not change with frequency at different searations. In Figure 6-14 (b), there is no obvious variation of the fieldinduced force from 1 khz to 1 khz at difference voltages. Thus, our results demonstrate that the AC field-induced force between two dielectric microsheres at 1-1 khz is indeendent of 63

81 (a) Force (nn) V 1 khz 2 khz 3 khz 4 khz 5 khz 6 khz 7 khz 8 khz 9 khz 1 khz Searation (µm) (b) µm 1 V 6 V 3 V Force (nn) Freqeuency (khz) Figure 6-14 (a) The variation of the AC electric field-induced forces with the searation between two BaTiO 3 glass microsheres at different frequencies when the alied voltage is 6 V. (b) The variation of the AC electric field-induced forces with frequencies at different alied voltages when the searation between the two dielectric microsheres is 2.5 μm. the frequency. Based on the Maxwell-Wagner model, 5,12 the frequency-deendent term in the Maxwell-Wagner model can be exressed by combining Eq. (3.21) and Eq. (3.22) 64

82 β 2 eff ( σ σ ) + ω ( ε ε ) = ( σ 2 σ ) ω ( ε 2 ε ) m m m + + m 25, 54,V Since the dielectric constant of BaTiO 3 glass microsheres (6.3) and the range of the frequency we used are quite high, we can ignore the terms of conductivity in Eq. (6.4), resulting in indeendent of the frequency. In addition, the frequency-deendent terms in Colver s model 31 (Eq. (3.35)) can be neglected as well due to the large dielectric constant of BaTiO 3 and alied frequencies. Hence, our results about frequency deendence of the AC field-induced force agree not only with Maxwell-Wagner model but also with Colver s model. Although some research results 31,1 demonstrate that the AC field-induced force is frequency-deendent, the range of the frequencies they used is so small that the conductivity would dominate the field-induced force, not the dielectric constant. Some charges would easily accumulate at the material surfaces under a low-frequency or DC field; that is the Maxwell-Wagner interfacial Relaxation, 7,27 causing the time-varying field-induced force even if the humidity is well-controlled. To verify that our results do not have the Maxwell-Wagner interfacial relaxation, we measured the variation of the field-induced force with time when the colloidal robe was ositioned around 6 μm above the bottom microshere and hovered for over 1 seconds. 18 We also measured the force without any electric field as a reference. Figure 6-15 shows our results of examining on Maxwell-Wagner Relaxation effect. In Figure 6-15 (a) and (b), the force varies with time as a DC field is alied but the force maintains constant as 1 khz electric field is alied. This examination oints out that the ermittivities of the articles and the medium dominate the field-induced force, not their conductivities under the alied AC field in our exeriment, imlying the crossover frequency is lower than the 1 khz for our system. We also study the effect of the electric field strength on the rate of the charge accumulation by comaring the variation of the normalized force with time (Figure 6-15 (c)). Surrisingly, there is no big difference when the alied voltage is doubled. The above examination and analysis of the Maxwell Wagner Relaxation effect suggests that AFM has otential to quantitatively study this V The BaTiO 3 glass microsheres we used are not made of BaTiO 3. The comosition (by weight) of the microsheres is SiO 2 (1~2%), B 2 O 3 (1~2%), ZrO 2 (1~1% ), BaO (4~7% ), and TiO 2 (4~7%). 54(a) Thus the relative ermittivity of the article we used may be less than 8 54(b) but still much higher than that of air and of cantilever. And the electrical conductivity of oxides are usually quite low (< ~1-3 S/m) 54(c). Further, BaTiO3 is ferroelectric so the secial henomena of the ferroelectric materials under an electric field are not taken into account here since the microsheres are not made of ferroelectric materials. 2 β eff 65

83 effect at micro- and/or nanoscale, which may hel the study of high-dielectric electronic materials. 55 (a) (b) (c) Figure 6-15 The variations electric field-induced force between two dielectric microsheres with time when the alied voltage is (a) 6 V and (b) 12 V and the distance between the two sheres is around 6 μm. The electric fields were alied after AFM started to record the data for about 5 seconds. (c) The variation of normalized DC electric field-induced force with time at difference alied voltages. Beside the frequency deendence, we have also studied the relationshi between the field-induced force and the alied electric field strength at different searations as shown in Figure Our analyses manifest that the field-induced force would increase with the alied 66