Group Title: optical technique for measuring force between a colloidal particle and a flat surface
Title: An optical technique for measuring force between a colloidal particle and a flat surface
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Title: An optical technique for measuring force between a colloidal particle and a flat surface
Physical Description: Book
Language: English
Creator: Clapp, Aaron Robert, 1974-
Publisher: University of Florida
Place of Publication: Gainesville Fla
Gainesville, Fla
Publication Date: 2001
Copyright Date: 2001
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Subject: Chemical Engineering thesis, Ph. D   ( lcsh )
Dissertations, Academic -- Chemical Engineering -- UF   ( lcsh )
Genre: government publication (state, provincial, terriorial, dependent)   ( marcgt )
bibliography   ( marcgt )
theses   ( marcgt )
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Summary: ABSTRACT: The behavior of colloidal particles in solution is greatly affected by the interaction the particles have with their surroundings. This may take the form of particle--particle interactions or the interaction of a single particle with a nearby solid wall. These interactions are characterized by the forces that are generated as a function of the separation distance between two surfaces. Fundamental phenomena such as particle deposition, solution rheology, and even microbial adhesion primarily depend on the magnitude and range of these fundamental forces as the particles move through the fluid. Several experimental techniques can measure these small forces directly. However, there is no existing technique for measuring forces on particles having diameters on the order of 1 micron or less. This size range is especially important for studies of bacterial or viral adhesion mechanisms where the nominal diameter can be much smaller than 1 micron. This dissertation describes a novel technique for measuring the static and dynamic forces that arise between a single colloidal particle and a flat plate. A single-beam gradient optical trap is used as a sensitive force transducer and evanescent wave light scattering is used to determine the particle position within the trap. The static force is measured by observing the equilibrium position of the particle within the trap, while the dynamic force is measured from the relaxation time of the particle fluctuations near the equilibrium position. Each force contribution is measured as a function of the particle--surface separation distance by moving the particle toward the surface in nanometer-sized increments.
Summary: ABSTRACT (cont.): Absolute separation distances are determined by curve fitting the viscous force data to hydrodynamic theory in regions where the static force is negligible. Measurements of static force agree well with classical Derjaguin--Landau--Verwey--Overbeek theory over the entire range of separation distances. Measured dynamic force agrees well with hydrodynamic theory until there is appreciable overlap of the electrical double layers at close separations. This departure may be due to a coupling of hydrodynamic and electrical phenomena that greatly enhances the viscous drag.
Summary: KEYWORDS: colloids, surface forces, optical trap, evanescent wave light scattering, hydrodynamic drag
Thesis: Thesis (Ph. D.)--University of Florida, 2001.
Bibliography: Includes bibliographical references (p. 115-116).
System Details: System requirements: World Wide Web browser and PDF reader.
System Details: Mode of access: World Wide Web.
Statement of Responsibility: by Aaron Robert Clapp.
General Note: Title from first page of PDF file.
General Note: Document formatted into pages; contains xii, 117 p.; also contains graphics.
General Note: Vita.
 Record Information
Bibliographic ID: UF00100782
Volume ID: VID00001
Source Institution: University of Florida
Holding Location: University of Florida
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Resource Identifier: oclc - 50743427
alephbibnum - 002729330
notis - ANK7094

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AN OPTICAL TECHNIQUE FOR MEASURING FORCE
BETWEEN A COLLOIDAL PARTICLE AND A FLAT SURFACE


















By

AARON ROBERT CLAPP


A DISSERTATION PRESENTED TO THE GRADUATE SCHOOL
OF THE UNIVERSITY OF FLORIDA IN PARTIAL FULFILLMENT
OF THE REQUIREMENTS FOR THE DEGREE OF
DOCTOR OF PHILOSOPHY

UNIVERSITY OF FLORIDA


2001




























Copyright 2001

by

Aaron Robert Clapp


























This dissertation is dedicated to my parents, Robert and Deborah Clapp.















ACKNOWLEDGMENTS

I would first like to express sincere gratitude to my advisor, Professor Richard

Dickinson, for his support, encouragement, trust, and dedication. His intuition,

constructive criticism, and infectious optimism have motivated me throughout. I would

also like to thank the other members of my advisory committee, Professor Chang-Won

Park, Professor Spyros Svoronos, and Professor Ben Koopman for their time and

guidance in this work.

I also gratefully acknowledge financial support and colleagues at the Engineering

Research Center (ERC) for Particle Science and Technology. In particular, I would like

to thank Professor Brij Moudgil, Dr. Kevin Powers, and Dr. Anne Donnelly for their

tireless efforts.

I have worked with a number of undergraduate students in the chemical

engineering department and in collaboration with the ERC. Notably, the contributions of

Melissa Sullivan and Bennie Munnerlyn have added to this dissertation.

I would like to acknowledge the support of my laboratory co-workers: Dr. Alina

Ruta, Dr. Jennifer Myles, Dr. Brian Burgess, Jonah Klein, Jamaica Prince, Jong-Hoon

Lee, Jeff Sharp, Jessica Brown, and Huilian Ma. Additionally I wish to thank Professor

Anthony Ladd, Dr. Jorge Jimenez, Dr. Jason de Joannis, Dang Nhan, and Anand

Jaganathan for their helpful discussions and advice.









Lastly, I thank Dr. Pavel Zemanek and his colleagues at the Institute for Scientific

Instruments, Academy of Sciences of the Czech Republic, for supplying various

materials and offering expert opinion in the area of beam optics and optical trapping.
















TABLE OF CONTENTS

page

A C K N O W L E D G M E N T S ................................................................................................. iv

LIST O F FIG U RE S .......................... ...................... .. .. ........... ..... .. ....ix

A B STR A C T ................................................... ....... ............ ................. . xi

CHAPTERS

1 IN TR O D U C TIO N ....................... ........................... .. ........ ..............

1.1 Surface Forces and DLVO Theory ................................. ....................... 1
1.2 Surface Force M easurement Techniques............................ ................................. 2
1.3 Force Measurement Techniques for Micron-Sized Particles............................ 5
1.4 A New Technique for Colloidal Force Measurement....................................... 8
1.5 O outline of the D issertation ................................................... ........................ 9

2 T H E O R Y ........................................................................................................1 1

2.1 E electrical D ouble L ayer ................................................ ............................. 11
2.1.1 Poisson-B oltzm ann E quation............................................... ... ................. 12
2.1.2 Debye-Hiickel Approximation.................... .................................. 13
2.1.3 G ouy- C hapm an T heory ........................................................ .... .. .............. 14
2.1.4 Stern M odel ................................ .............................. ....... 15
2.1.5 C alculating Force and E energy ............................................... .... .. .............. 17
2.2 van der W aals Forces ........................................................................ 19
2.2.1 M olecular Dipole Interactions .............. ...... .......................... ........... 20
2.2.2 M acroscopic Interactions ........................................ .......................... 22
2.3 D L V O Theory ......................................... .......... ...... ........ .. 24
2.4 H ydrodynam ics of a D iffusing Sphere.............................................. ... ................. 27
2.5 Evanescent W ave Light Scattering .................................... ................................ 31
2.6 O optical Trapping ............ ........................... .. ........ .............................. 34
2.6.1 O ptical Trapping M odels ............................ ........................... .............. 35
2 .6 .2 T rap C alibration ........................................ .......................... ................ .. 37

3 DESCRIPTION OF THE TECHNIQUE ........................ .............................41

3 .1 A p p aratu s ................................................................... 4 1
3 .2 P ro ced u re ................................................................... 4 3









3.3 D ata A nalysis............................................ 44
3.3.1 Signal Processing ............ ...... .. .... ............ .... .... .. .......... .. 44
3.3.2 Brownian Motion in a Potential Energy Well......................................... 47
3.3.3 Calibrating the O ptical Trap ........................................ ....................... 51
3.3.4 M easuring Static F orces........................................................ .... .. .............. 53
3.3.5 M easuring D ynam ic Forces ...................................... ................ ... ........... 55
3.3.6 Determining Absolute Separation Distances ............................................... 56

4 INITIAL FORCE MEASUREMENTS AND EVIDENCE OF INTERFERENCE
E F F E C T S ........................................................................5 8

4.1 Initial Static Force Experim ents.................................... ........................... ........ 58
4 .1.1 Static F force R esu lts ............................................................... .................... 59
4.1.2 Discussion of Static Force Results .............. ........................................... 62
4.2 Effect of Trapping Beam Alignment ....................................................... 67
4.3 Reflection Effects in Optical Trapping .............. .......................................... 72
4.3.1 Standing W ave Trapping .......................................... .......... .......... ... 72
4.3.2 Experimental Evidence of a Standing Wave............................ ......... ..... 73
4.3.3 Modeling the Trapping Beam Profile Near a Reflective Surface ............... 80

5 STATIC AND DYNAMIC FORCE MEASUREMENTS USING A REVISED
METHODOLOGY .................. ................................... ........... .................8 81

5.1 Description of the Revised Methodology ........................................................... 81
5.2 Static Force M easurem ents .............. ........................................................... 82
5.2.1 Procedure D details .................. ............................ ........ ... .......... 82
5.2.2 Static Force R results ............................................... ...... .. .. .......... 83
5.2.3 D discussion of Static Force R esults............................................................... 87
5.3 Dynamic Force Measurements...................... ..... .......................... 89
5.3.1 D ynam ic Force R results ......................................... .............................. 89
5.3.2 Discussion of Dynamic Force Results ......................................................... 92

6 SIMULATING AND MODELING EXPERIMENTS .............. ............... 94

6.1 Brownian Dynamics Simulations of Trapping Experiments ............................... 94
6.1.1 Sim ulation Procedure .................................................................. .............. 95
6.1.2 Simulation Results ...................... ... ..... .................. 96
6.2 Modeling Statistical Data Generated from Force Measurements ...................... 98
6.2.1 M odeling Procedure ......................... ................................. ......................... 99
6 .2 .2 M odeling R results ........................................ ............................................ 100

7 CONCLUSIONS AND SUGGESTIONS FOR FUTURE WORK.............................104

7.1 Static F orce M easurem ents .................................................................................. 104
7.2 Dynamic Force M easurements........... ........................... ................ ...... 105
7.3 Suggestions for Future W ork................................... ..... ... .............. 105
7.3.1 Assembly of an Improved Optical Trap................................................. 106









7.3.2 Reduce Noise in Measurements ........................................... .............. 107
7.3.3 Investigate Possible Electroviscous Effects.............................. ................. 108
7.3.4 Measure Specific Interactions of Bacteria with Coated Surfaces ................ 109
7.3.5 Force Measurements with Sub-Microscopic Particles.............................. 109
7.3.6 Measure Complex Properties of the Interface.................... ....... ..... 110

APPENDIX

LABVIEW CODE FOR DATA ACQUISITION................................112

L IST O F R E FE R E N C E S ...................................................................... ..................... 115

BIOGRAPHICAL SKETCH ........................................................................117
















LIST OF FIGURES


Figure Page

2-1. Schematic of the Stem model for the electrical double layer................. ........... 16

2-2. DLVO interaction energy between a sphere and a flat plate. ......................................24

2-3. Drag force correction factor for a sphere moving normal to a solid surface.................30

3-1. Schematic of the experimental apparatus. ........................................ ............... 41

4-1. Position histograms of a particle trapped (a) far from and (b) near the plate surface. ..60

4-2. Experimental force-distance profiles for 1.5 pm silica near a glass slide. ( 0.1
mM NaC1, 0 0.18 mM NaC1, D 0.40 mM NaC1) .............. ............... ...61

4-3. Experimental and theoretical mean intensity versus trap position for 1.5 pm silica
sphere. (0.1 mM NaCl solution; + raw data, 0 background level
subtracted, solid line model prediction) ................. ................................ 65

4-4. Experimental mean intensity versus trap position for a 1.5 Jpm silica sphere ..............66

4-5. Schematic of LaserTweezers unit placed within the microscope...............................68

4-6. Mean intensity versus trap position for two locations of the trapping unit. ..................71

4-7. Position histograms for three slightly different locations (separated by 20 nm) of the
trap center in a standing wave arrangement (1.0 pm silica).............................74

4-8. Mean intensity as a function of trap position for a 1.5 pm silica particle near a
reflective glass slide (R=25% )............................ ................................... 75

4-9. Mean relative separation distance as a function of trap position for a 1.5 pm silica
particle near a reflective glass slide (R=25%). .......................................76

4-10. Variance of separation distance as a function of trap position for a 1.5 pm silica
particle near a reflective glass slide (R=25%). .......................................77









4-11. Axial trap stiffness as a function of trap position for a 1.5 Jrm silica particle near a
reflective glass slide (R=25% )............................ ................................. 78

5-1. Measured histograms of particle positions and estimated potential energy profiles
corresponding to most probable separation distances of 543.7 and 56.2 nm.......83

5-2. Static force measurements for a 1.0 pm diameter silica sphere near a glass plate in
0.23 mM (0) and 1.0 mM (A) NaC1. DLVO theory predictions are plotted
for both ionic strengths (0.23 mM solid line, 1.0 mM dashed line)..............85

5-3. Static force measurements for a 1.5 pm diameter silica sphere near a glass plate in
0.23 mM (0) and 1.0 mM (A) NaC1. DLVO theory predictions are plotted
for both ionic strengths (0.23 mM solid line, 1.0 mM dashed line)..............86

5-4. A plot of the intensity autocorrelation function (0) at two separation distances for a
1.5 pm diameter silica sphere. The separation distances correspond to those
shown in Figure 5-1. ............................................... .. ......... 89

5-5. Drag coefficient measurements for a 1.0 pm diameter silica sphere near a glass plate
in 0.23 mM (0) and 1.0 mM (A) NaC1. Hydrodynamic theory predictions
are plotted based on the fitted particle radius far from the surface (0.23 mM -
solid line, 1.0 mM dashed line) ......................... ......... ..... ... ............ 90

5-6. Drag coefficient measurements for a 1.5 pm diameter silica sphere near a glass plate
in 0.23 mM (0) and 1.0 mM (A) NaC1. Hydrodynamic theory predictions
are plotted based on the fitted particle radius far from the surface (0.23 mM -
solid line, 1.0 mM dashed line) ........................... .................. 91

5-7. Dimensionless departure of the experimental drag coefficient from hydrodynamic
theory ............................................................................92

6-1. Simulated experimental drag (0) versus Brenner's correction to Stokes' law.............96

6-2. Model prediction (0) and experimental data (0) of the mean intensity profile...........01

6-3. Model prediction (0) and experimental data (0) of the intensity variance profile......102

7-1. Schematic for a custom optical trapping system. .................................. .................106

A-1. LabVIEW wire diagram for the data acquisition program. .................... ................112

A-2. LabVIEW panel display for the data acquisition program. .................... ................113















Abstract of Dissertation Presented to the Graduate School
of the University of Florida in Partial Fulfillment of the
Requirements for the Degree of Doctor of Philosophy

AN OPTICAL TECHNIQUE FOR MEASURING FORCE BETWEEN A COLLOIDAL
PARTICLE AND A FLAT SURFACE

By

Aaron Robert Clapp

May 2001

Chairman: Richard Dickinson
Major Department: Chemical Engineering

The behavior of colloidal particles in solution is greatly affected by the interaction

the particles have with their surroundings. This may take the form of particle-particle

interactions or the interaction of a single particle with a nearby solid wall. These

interactions are characterized by the forces that are generated as a function of the

separation distance between two surfaces. Fundamental phenomena such as particle

deposition, solution rheology, and even microbial adhesion primarily depend on the

magnitude and range of these fundamental forces as the particles move through the fluid.

Several experimental techniques can measure these small forces directly.

However, there is no existing technique for measuring forces on particles having

diameters on the order of 1 pm or less. This size range is especially important for studies

of bacterial or viral adhesion mechanisms where the nominal diameter can be much

smaller than 1 pm.









This dissertation describes a novel technique for measuring the static and dynamic

forces that arise between a single colloidal particle and a flat plate. A single-beam

gradient optical trap is used as a sensitive force transducer and evanescent wave light

scattering is used to determine the particle position within the trap. The static force is

measured by observing the equilibrium position of the particle within the trap, while the

dynamic force is measured from the relaxation time of the particle fluctuations near the

equilibrium position. Each force contribution is measured as a function of the particle-

surface separation distance by moving the particle toward the surface in nanometer-sized

increments. Absolute separation distances are determined by curve fitting the viscous

force data to hydrodynamic theory in regions where the static force is negligible.

Measurements of static force agree well with classical Derjaguin-Landau-

Verwey-Overbeek theory over the entire range of separation distances. Measured

dynamic force agrees well with hydrodynamic theory until there is appreciable overlap of

the electrical double layers at close separations. This departure may be due to a coupling

of hydrodynamic and electrical phenomena that greatly enhances the viscous drag.














CHAPTER 1
INTRODUCTION


1.1 Surface Forces and DLVO Theory

Surface forces-the interactions that arise between exposed material surfaces in

solution-directly influence and control the behavior of colloidal particles (with a

diameter of 1 pm or less) suspended in a liquid. For example, stability of a particle

dispersion, theological behavior, and adhesion of particles from solution to other surfaces

are all mediated by surface forces. A better understanding of these and other phenomena

clearly necessitates a quantitative characterization of surface forces on the colloidal scale.

These interactions are classified as either conservative (in the case of static forces) or

non-conservative (in the case of drag forces) depending on the origin of the interaction.

In the last half-century, significant attention has been given to developing

accurate predictive theories for conservative colloidal forces. Typically, these theories

have been adaptations of the seminal work of Derjaguin and Landau (Russia) [1], and

Verwey and Overbeek (Holland) [2], known collectively as the DLVO theory. Originally

developed to predict the stability of colloidal suspensions, DLVO theory characterizes the

equilibrium interaction energy between two bodies as the additive contributions of

screened electrostatic and van der Waals interactions. Using various system-specific

parameters (e.g., surface charge, solution ionic strength, particle size), DLVO theory

predicts a potential energy profile for two interacting surfaces where a key result is the

effective maximum energy required for the surfaces to contact. This maximum energy is









often referred to as the energy barrier to attachment and is directly related to the

probability that the two surfaces will contact each other within a certain time interval.

DLVO theory has been used to describe the rheology of concentrated particle

suspensions. Whereas more primitive models consider particles to be simple hard

spheres, the inclusion of surface forces accounts for particle "softness" where the particle

has a larger effective radius due to the double layer ions collected near its charge surface.

Within the last thirty years, DLVO theory has also been used extensively to describe the

phenomenon of bacterial adhesion [3]. Though bacterial cells are far more complicated

than colloidal particles with their irregular shape, polymeric appendages, and

heterogeneous structure, DLVO theory can be a useful first approximation for estimating

the forces that exist between a cell and a nearby surface.

Though many predictive theories exist for colloidal forces, their success depends

on the material system considered and the availability of certain parameter values which

are often difficult to determine. In extreme cases, as may be the situation with

suspensions of bacteria, for example, the chosen theory may be inappropriate altogether

due to the invalidation of one or more critical assumptions. Often, it is desirable and

more convenient, even for relatively simple systems, to measure these forces

experimentally rather than rely solely on theory. In addition to the aforementioned

theoretical work, there has been a similar emphasis on developing accurate experimental

methods for measuring surface forces directly.


1.2 Surface Force Measurement Techniques

Three relatively recent techniques have emerged which account for most of the

experimental surface force measurements performed to date. The most commonly used









of these is the surface forces apparatus (SFA) of Israelachvili and Adams [4]. The SFA

has been used in numerous studies to measure the interaction forces between mica sheets

arranged in a crossed-cylinder geometry. The surfaces may also be coated, which greatly

expands the versatility of the technique. The interaction force is measured using a

sensitive spring in combination with an interferometry technique to determine absolute

separation distances down to the angstrom level. Since the radii of the cylinders are on

the order of centimeters, the measured forces are orders of magnitude greater than those

seen in colloidal systems. Therefore, other techniques are necessary for measuring

surface forces where one of the materials has colloidal dimensions.

One of these colloidal force measurement techniques, developed by Ducker and

co-workers [5], uses atomic force microscopy (AFM) [6] to measure the interaction force

of a colloidal probe particle, attached to the end of a solid cantilever, with a flat surface.

Since the imaging principle of atomic force microscopy is inherently based on the

existence of surface forces, quantitative measurements are readily made using the force

mode of AFM. Deflections of the cantilever are accurately measured using a laser that

reflects off the back surface of the cantilever and strikes a position sensitive

photodetector. Though it is widely described as a colloidal force measurement technique,

truly colloidal spheres (with a diameter of 1 pm or less) are rarely used because of the

challenge in mounting the particle to the cantilever. AFM also has a sensitivity limit of

about 10-10 N which is relatively large for colloidal systems [7]. The advantage of AFM,

however, is that the particle can be reversibly attached and removed from the surface due

to the stiffness of the cantilever.









The other common experimental technique for colloidal measurements is total

internal reflection microscopy (TIRM) developed by Prieve and co-workers [8]. Unique

to any of the methods previously described, TIRM directly measures the potential energy

of a particle near a wall rather than the force. It does this by applying Boltzmann's law,

which is relates the probability of finding a particle at a given location to the potential

energy at that location. In contrast to other techniques, TIRM is considered to be "non-

invasive" because the particle is free to diffuse within the suspending liquid. The

particle-surface separation distance is instantaneously found by measuring the scattered

light emitted from the particle in an evanescent wave; we therefore call this technique

evanescent wave light scattering. Later generations of TIRM included the use of a

radiation pressure (two-dimensional) optical trap to confine the lateral movements of the

particle [9]. This improved the measurements greatly by holding the scattering particle in

the field of view for the duration of the experiment. Since the technique does not impose

external control of the particle in the axis normal to the flat surface, the maximum

measurable energy is limited by the inherent thermal energy of the particle. This means

that TIRM can only effectively measure energy on the order of a few kT (where 1 kT =

4.1 x 10-21 J at 298 K). Although TIRM measures potential energy directly, the

corresponding force profile is found by evaluating the negative slope of the measured

energy profile. It should be noted that the radiation pressure optical trap is able to exert a

constant force on the particle in addition to constraining its lateral motion. By adjusting

the power of the beam, it is possible to shift the particle to regions of greater potential

energy. This is especially useful if the beam exerts a pressure on the particle toward the









flat surface since this is typically in opposition to the electrostatic repulsive force.

However, this method is usually not attempted in TIRM experiments.


1.3 Force Measurement Techniques for Micron-Sized Particles

Although studies of surface forces have been greatly enhanced by the techniques

previously described, there still remain many colloidal systems of various size and nature

that elude accurate measurement. In particular, force measurements using micron-sized

particles are either difficult or impossible using existing techniques. While it would be

desirable to develop a technique capable of measuring interactions for any colloidal

system, the interactions between a micron-sized particle and a flat surface are particularly

relevant to the study of bacterial adhesion. Many common bacteria species have nearly

spherical shapes with diameters that are near (and often below) 1 pm in size. For this

reason, bacteria are often considered to be simple colloidal particles even though their

internal and external physical structure is far more complicated. The success of DLVO

theory as a quantitative tool is usually quite poor for this reason. Most theories for

bacterial adhesion are based on the original DLVO framework, but are modified to

account for additional interactions; for example, steric forces are often considered as

well. Of course this can lead to theories that become too complicated or

phenomenological to be of much practical use, employing many more parameter values

than can be readily determined or justified. For these complicated systems, direct

measurements are even more important.

Until recently, TIRM has been the most suitable technique for measuring colloidal

interactions. (Note, however, that the current literature contains no studies of particles

with a diameter below 1 / m.) Since the measurements are non-invasive and rely only on









the ability of the particle to scatter light, particles of nearly any diameter may be used (in

practical terms, this range is about 0.2 to 20 pm). Since TIRM deduces the potential

energy of interaction directly, there is no preset limitation on the measurable force aside

from that imposed by the resolution of particle position. However, there is a strict

limitation on the maximum energy that is measurable, which is effectively set by

Boltzmann's law. This inherent energy limit hinders the usefulness and practicality of

TIRM measurements since the energy profile of a particle approaching a surface may far

exceed these measurement limits. In addition, as the diameter of a particle diminishes,

the diffusive movements increase according to the Stokes-Einstein relation. Thus, for

relatively small particles (less than about 5 pm in diameter), it is difficult to record

complete measurements because the particle will simply leave the active detection region

before the experiment is finished.

Addressing these issues, Brown and co-workers [9, 10] adapted the original

TIRM technique by adding a radiation pressure optical trap to localize the lateral position

of the particle and to push the particle axially toward the flat surface into regions of

higher energy. While this two-dimensional optical trap was effective at constraining

transverse fluctuations, the axially directed radiation pressure force was less useful

because it was a non-linear function of laser power. In order to scan the entire potential

energy profile, from far away to very near the surface, the laser power would have to be

discretely varied in order to advance the particle toward the surface. This is possible, but

impractical because the resulting potential energy data would need to be corrected for the

effect of the trap and the discrete sections pieced together to form the complete profile.









Ashkin [11] first reported development of the radiation pressure optical trap in the

early 1970s at Bell Laboratories. Since that time, various trapping schemes have been

implemented for a variety of research endeavors including the study of surface forces.

The utility of an optical trap for the study of colloidal forces is primarily two-fold: it is

essentially non-invasive to the sample and is able to impart small-scale forces to the

trapped particle ranging from 10-12 N to 10-15 N. Unfortunately, the two-dimensional

radiation pressure optical trap is unable to confine the particle along the beam axis and

provides a constant force in the direction of propagation. However, in 1986, Ashkin and

co-workers [12] introduced a single-beam gradient optical trap that was able to precisely

control a particle in three-dimensions. With its simple design, one can easily build a

gradient optical trap into a laboratory microscope with only minor modifications. This

led to the commercialization of gradient optical trapping units, known commonly as

"laser tweezers," which easily insert into many popular models of microscopes.

The most useful aspect of the gradient optical trap, in the context of force

measurement, is the ability to readily quantify the three-dimensional forces acting on the

particle, including those imparted in the axial direction. The intensity profile created by

a tightly focused beam imposes a three-dimensional harmonic potential energy profile on

the particle (the theory of which is yet to be described). That is, for any reasonably small

deviation of the particle center from the region of highest intensity, there is an

approximately linear restoring force that pushes the particle back to the equilibrium

location. The most useful analogy to consider is that of a mass attached to three

orthogonal springs, each with a characteristic spring constant and equilibrium

(unstressed) position. By observing the motion of the trapped particle over time, the









equilibrium distribution of positions taken by the particle allows these effective spring

constants (or alternatively, trap stiffness values) to be determined experimentally; this is

known as calibrating the trap.

If the trap stiffness and focus location are known, the optical trap can be used as a

force transducer in a fashion analogous to AFM force measurements with a known

cantilever stiffness. In either method, the force is measured by performing a simple force

balance, where the force of interaction between the particle and surface is equal and

opposite to the force applied by the optical trap or cantilever at the new equilibrium

position. The balance is somewhat complicated by the Brownian fluctuations of the

particle, but the net forces are zero at the peak, or mode, of the equilibrium distribution of

particle positions (i.e., at the minimum of total combined potential energy of the trap and

the surface). The concept of using an optical trap as a force transducer is relatively

simple, but it requires an accurate method of determining the particle position along the

axis of interest. It is possible to monitor small movements of a particle using microscope

image analysis with a resolution better than 10 nm, but this is far less effective for

movements made along the microscope objective axis (into and out of the viewing plane).

Image analysis is also insufficient for monitoring rapid Brownian movements.


1.4 A New Technique for Colloidal Force Measurement

To accurately measure the forces between a single micron-sized particle and a flat

surface, we have developed a new technique [13] that combines a single-beam gradient

optical trap as a force transducer and evanescent wave light scattering for precise

measurement of particle position. In contrast to TIRM, the technique measures force

directly (rather than potential energy) in a manner similar to AFM. As the trapping beam









focus is stepped toward the surface, the most probable particle position is measured at

each trap position to determine the force-distance profile. The trapping force can move

the particle to high-energy regions that are inaccessible through purely diffusive

movements. Moreover, the time required to perform a complete force profile

measurement is far less than with TIRM, even though the sampled spatial region can be

much larger with the new technique. This is because a diffusing particle held by an

optical trap samples the accessible energy landscape more quickly than a freely diffusing

particle.

We have validated the technique by measuring conservative (static) forces

between colloidal silica spheres (-1 pm in diameter) and a flat glass plate in aqueous

solutions of varying ionic strength. These forces usually include screened electrostatic

(double layer) and van der Waals interactions, or those typically described by DLVO

theory. In addition, we have extended the technique to simultaneously measure the non-

conservative forces that arise as a particle nears a flat surface. Non-conservative

(dynamic) forces depend on the motion of the particle as well as its position from the

surface. Usually, this is simply the hydrodynamic drag force as the particle moves

through the fluid. We have compared our experimental force results with DLVO theory

predictions (for static forces) and a modified version of Stokes' law (for dynamic forces)

in order to assess the accuracy of the data and validate the technique.


1.5 Outline of the Dissertation

Chapter 2 discusses necessary theoretical background required for the

understanding of essential concepts of electrical double layer forces, van der Waals

forces, colloidal hydrodynamics, evanescent wave light scattering, and optical trapping.









Chapter 3 describes our new technique with details of the experimental procedure and

data analysis. Chapter 4 gives experimental results of the initial static force

measurements and a discussion of persistent interference effects that significantly altered

the results. Chapter 5 describes a modified methodology for measuring static and

dynamic forces more accurately. The results of these experiments were shown to validate

the technique's accuracy. Chapter 6 describes methods for simulating and modeling the

experiments in order to make predictions about the experimental data. Finally, Chapter 7

summarizes some key observations from our experiments and offers some suggestions

for future work.














CHAPTER 2
THEORY


2.1 Electrical Double Layer

Electrostatic forces arise between materials immersed in solution when there is a

net charge on the surfaces. The interceding medium between the exposed surfaces

generally contains dissolved ions that serve to screen the purely coulombic interactions.

Before we can predict the forces that arise between two surfaces in solution, we must first

consider how the electrostatic potential varies with distance from the surface. This

requires an examination of the charges that exist on the surface and how ions in solution

are distributed.

Exposed surfaces in solution can acquire a net charge through a variety of

mechanisms. When a solid is brought into contact with a polar medium like water,

charge may be acquired through ionization of surface groups, isomorphic substitution in

the solid lattice structure, or adsorption of potential determining ions. The resulting

surface charge determines the distribution of ions in the polar medium attracting ions of

opposite charge (counterions) toward the surface, and repelling ions of similar charge

(co-ions) away from the surface. The redistribution of ions near a surface creates an

electrical double layer, a conceptual division of the charged region (including the surface

and nearby ions) into distinct bulk phases that carry equal and opposite charge.

Typically, though, the term "double layer" simply describes the diffuse ion atmosphere

near the surface. We will adopt this latter usage and use it throughout.









2.1.1 Poisson-Boltzmann Equation

The variation of potential from a surface is a well-known problem in classical

electrostatics. It is described by the Poisson equation:



P*
V 2y = (2.1)




where V2 is the Laplacian operator, yf is the electrostatic potential, p* is the charge

density, and e is the dielectric constant of the liquid. In order to solve equation (2.1) for

potential, there must be an expression for the charge density as a function of the potential.

Using the thermodynamic result known as Boltzmann 's relation to describe the

probability of finding an ion at a particular distance from a flat surface where the

potential is Vy, we arrive at




p*= z, e n, exp (2.2)




where z, is the valance number of ion type i, e is the proton charge, n, is the number of

ions far from the surface, k is Boltzmann's constant, and Tis the absolute temperature.

The combined form of equations (2.1) and (2.2) is known as the Poisson-Boltzmann (PB)

equation:




d2W e zew
d2L-- Z, n,. exp ze (2.3)
dx 2 kT









where x is the spatial dimension in the above one-dimensional form. (Note that future

sections will have z as the distance variable, but x is chosen here to avoid confusion with

the valence number, z,.) While the PB equation fully describes the variation of potential

from the surface, it is usually difficult to solve since the Boltzmann factor introduces an

exponential term rendering the equation non-linear; in fact, no generalized analytical

solution exists. There are, however, other solutions for the potential as a function of

distance from the surface for certain limiting cases discussed below.

2.1.2 Debye-Hickel Approximation

It is possible to solve the non-linear Poisson-Boltzmann (NLPB) equation

numerically, but this usually is not attempted since it is often cumbersome and would

have to be solved each time the conditions are varied. If we take a series expansion of

equation (2.2) and keep only the first term, we can solve a linearized form of the PB

equation known as the Debye-Hickel approximation:




dX2 fT Y
-^ -z2*n,^-^ =z. (2.4)



where Kis a constant parameter, the inverse of which is known as the Debye length or the

double layer thickness, K 1 [15]. Note that, at constant temperature, the Debye length is

purely a function of the electrolyte content of the solution. The solution to the linearized

PB equation is simply


V= /o exp(- Mx).


(2.5)









The Debye-Huckel approximation is accurate for surface potentials below kT/e

(-25 mV at 298 K). In practical terms, it remains reasonably accurate for potentials up to

about 60 mV. The solution given by equation (2.5) also assumes a fixed surface

potential, t0o, at one boundary, and y -> 0 as x -> oc at the other. In this model, the

relationship between surface charge density, a *, and surface potential, f0o, is assumed

linear, which is analogous to a parallel plate capacitor with a separation distance of K-1

2.1.3 Gouy-Chapman Theory

The usefulness of the Debye-Hickel approximation is hampered by many

simplifying assumptions, so we turn to a more generalized solution of the PB equation,

known as Gouy-Chapman theory, to describe the potential profile for any value of the

surface potential. The derivation of the final Gouy-Chapman result is somewhat lengthy

so it is not shown here, but the final expression is important to note. Returning to

equation (2.3), we allow the surface potential to take any value and integrate the equation

assuming the solution electrolyte is symmetric (z : z) and such that the two boundary

conditions of the equation are imposed at infinite distance from the surface where

Vy(x) and y'(x) both approach zero. The final result is



(ze __ ze=>
tanh e tanh exp(- ), (2.6)
4kT) 4kT



which can be simplified to


Y = Yoexp(- KX).


(2.7)









Note that the Gouy-Chapman expression is an analytical solution, but it requires a

symmetric electrolyte. Similar to the Debye-Huckel result in equation (2.5), the above

equation shows an exponential variation with distance, however it is now a

transcendental function of the potential, Y that is exponentially varying rather than the

potential itself. As we might expect, as the surface potential tends below 25 mV,

equation (2.7) approaches equation (2.5) demonstrating that the Debye-Hiickel

approximation is a subset of the more general Gouy-Chapman theory.

2.1.4 Stern Model

To this point, we have only considered ions in solution that are "indifferent" to

the surface. In other words, the models have assumed that the ions are not able to attach

to the surface. In addition, we have not accounted for the size of the ions, which becomes

important for larger potentials where ions will tend to adsorb and saturate binding sites on

the surface. In the model proposed by O. Stern [16], the double layer extending into

solution is divided by an imaginary boundary known as the Stern surface. The Stem

surface, also called the inner Helmholtzplane (IHP), runs through the adsorbed ions at

distance from the solid surface. Just beyond the IHP is the outer Helmholtzplane

(OHP) which defines the shear surface beyond which the ions are freely diffusing (as in

the Gouy-Chapman model). Figure 2-1 illustrates the Stern model of the electrical

double layer.











SURFACE STERN
CHARGE - PLANE

0 0
SOLID




DIFFUSE
LAYER


S SHEAR
S-J PLANE

I-

DISTANCE


Figure 2-1. Schematic of the Stern model for the electrical double layer.







Within the Stern layer, the potential is simply modeled as a parallel plate


capacitor with a linear potential drop from the solid surface out to distance 1. Beyond the


Stem surface, the potential follows a model for the diffuse double layer like the Gouy-


Chapman result in equation (2.7). In practice, the Stem model is almost never used


because several parameters remain unknown. However, the model does suggest that


adsorption is important to the formation of the double layer, and that the actual value of


the surface potential may be markedly different from values measured by experimental


methods. For this reason, it is customary practice for the surface potential to be


characterized by the potential at the shear plane rather than at the solid surface. The


potential at the shear plane is called the zeta potential and may be found using common


electrokinetic experimental techniques like electrophoresis and streaming potential


measurements.









2.1.5 Calculating Force and Energy

Now that we have developed expressions for the potential as a function of

distance from the surface, we can find equations for the interaction force and energy

between two charged surfaces in solution when their double layers overlap. The total

force on an infinitesimal volume element, dV, at some location between the surfaces is

the addition of the osmotic pressure (along the x-direction) and the Maxwellpressure

(electrical stress) contributions. At equilibrium, the total force equals zero and we can

integrate an expression to find the force per unit area between the surfaces as a function

of the distance between the flat surfaces, h. The result is




F,(h)& 64kTn tanh2 Z' exp(- id), (2.8)




which is an approximate expression that assumes slightly overlapping double layers and

identical surface potentials. This assumption, called the superposition approximation,

allows us to estimate the potential distribution between the surfaces using the Debye-

Hickel result for single surfaces rather than requiring us to resolve for the potential

between two surfaces explicitly. Since force and energy are related by F = -VO, we can

integrate equation (2.8) to find the interaction energy per unit area:




R(h>) 64kTn 1tanh C ZelO8 exp- ). (2.9)
4kT



Unfortunately, equations (2.8) and (2.9) are not particularly useful for colloidal

interactions where at least one surface has a small radius of curvature. In most cases, we









are interested in the interactions between two spheres in solution, or a sphere and a flat

plate, where the surfaces may have differing surface potentials. To account for this, the

usual case is to solve some form of the PB equation (typically a power series expansion

of the NLPB equation) in the space between the surfaces where the boundary conditions

are y = Vo, at x = 0 and yf = o, at x = h. The energy per unit area is calculated

directly and expressed in integral form by a charging process, described by Verwey and

Overbeek [2], where each surface potential remains fixed. To find the interaction energy

between two spheres, we use Derjaguin 's approximation [17], which integrates the

interaction energy produced by opposing infinitesimal rings, each of which are

considered to be flat surfaces:




2(h)_ 2 fa 4(h' h', (2.10)
a, +a2 h



where al and a2 are the radii of the spheres, and ^ is the interaction energy per unit area.

Note that we can consider a sphere interacting with a flat plate by allowing one radius to

become infinite. Derjaguin's approximation assumes the radii to be much larger than

both the double layer thickness, K 1, and the separation distance, h.

Hogg, Healy, and Fuerstenau [18], and later Wiese and Healy [19], derived

expressions for the interaction potential between two dissimilar spheres using the Debye-

Hickel approximation:



Eazay +1) i2 +et)
1 (h) = In +n 2 -r2 112 (--I lnI -e I) (2.11)
h 4(al +a2) ( +01 LI-2I e 1











where the "+" symbol becomes positive for constant surface potential or negative for

constant surface charge. Constant surface potential assumes that surface charge is

acquired through the adsorption of potential-determining ions, whereas constant surface

charge assumes that charge arises due to the ionization of surface groups. For typical

colloidal materials, a constant surface charge assumption is more suitable although many

more sophisticated models for double layer forces are based on a constant potential

assumption; the reason for this is not clear. The two cases given in equation (2.11) are

limiting cases of intermediate charge regulation models that may be more complicated.

For most practical situations, equation (2.11) is sufficient to estimate the

interaction energy between two surfaces. There are, of course, many other equations for

double layer interactions that are more recent and, unfortunately, far more complicated

than those presented here. Some interesting examples are given by Ohshima et al. [20],

Grant and Saville [21], and Sader et al. [22]. Usually, these equations are valid over a

broader range of conditions (e.g., surface potential, separation distance) and consider

other complicating factors.


2.2 van der Waals Forces

The tendency of the electrons in molecules to have an uneven spatial charge

distribution leads to the formation of dipoles. The ability of these molecular dipoles to

interact based on atomic polarizability and the generation of electric fields is a

phenomenon classified as van der Waals interactions. A defining characteristic of van

der Waals forces is that they are ubiquitous; electrostatic forces, for example, require a

net charge on each surface. In addition, van der Waals forces are nearly always attractive









whereas electrostatic forces may be either attractive or repulsive depending on the surface

chemistry.

2.2.1 Molecular Dipole Interactions

Molecular-level van der Waals forces determine interactions between

macroscopic bodies. One type of molecular interaction occurs between two permanent

dipoles. The dipoles are assumed to be isolated in a vacuum with sufficient thermal

energy to ensure free rotation. Using a Boltzmann average over all possible

configurations and giving more weight to favorable energies, the free energy of

interaction between two freely rotating dipoles is



2,u 2 P 2 CK
K(r)= 22 = (2.12)
3(4o, )2kTr6 r6



where u, is the dipole moment of molecule i, r is the intermolecular distance, and CK is

a characteristic constant. The above expression was first derived by W. H. Keesom [23]

and is known as the Keesom equation.

A second type of interaction occurs between an induced dipole and a permanent

dipole. P. Debye [24] derived the following result:



2 2
D(r) U12 + P2 CD (2.13)
(4 mo)26 r



where a is the atomic polarizability of molecule i, and CD is a characteristic constant.


This expression is known as the Debye equation.









The last type of interaction occurs between two induced dipoles. London first

solved the quantum mechanical model of two dissimilar hydrogen-like atoms to yield



3aro2hvlv2 CL
L(r) = 312hv12 (2.14)
2(47,)2(v + V2)r6 r6



where h is Planck's constant, v, is the oscillatory frequency of molecule i, and CL is a

characteristic constant. This expression is known as the London equation for dissimilar

molecules. This type of interaction is sometimes referred to as the dispersion interaction

because its role in a phenomenon related to the scattering of light (not the dispersion of a

colloidal solution).

Upon inspection of the preceding three equations, we notice that they all share the

same power law dependence, r 6, and can be combined to form the overall van der

Waals equation for intermolecular free energy:



(( CK+ CD+ CL) Cvd (2.15)
OvdW 6 6( )
r r



Regardless of the material, there will always be a contribution from the London

(dispersion) interaction since it does not depend on the existence of a permanent dipole

(i.e., a non-zero value of p). In general, the London interactions dominate the other two.

However, water molecules present a notable exception to this tendency where the

Keesom interactions account for most of the overall van der Waals attraction.









2.2.2 Macroscopic Interactions

With expressions for the molecular interactions between dipoles, we can integrate

all pairwise interactions occurring between the molecules in two macroscopic bodies.

This is the approach used by H. C. Hamaker [25] for solving the macroscopic interaction

energy between objects of various geometries. For the case of two interacting spheres,

the van der Waals interaction energy is




vdW (h)= Aaa2 (2.16)
6h(a1 + a2)



where A is the Hamaker constant, h is the minimum separation distance, a, and a2 are

the radii of the spheres. The Hamaker constant characterizes the magnitude of the

attraction and varies by material. As with the double layer energy equation for two

spheres, a sphere-plate geometry is considered by allowing one radius to become infinite.

The value of the Hamaker constant depends on the types of materials considered,

including the interceding medium between the two bodies. In general, the Hamaker

constant Ak represents the interaction between materials i and k across medium j. If we

have Hamaker constant values for all three materials interacting with themselves in

vacuum ( A,, A22, and A33), we can estimate the effective Hamaker constant for the

overall system:



A132 (A-22 A- 2)(A A'2), (2.17)


which is based on a chemical reaction analogy, the details of which are omitted.









Equation (2.16) is commonly used to describe the van der Waals attraction

between macroscopic bodies, but it is not particularly accurate for separation distances

above about 10 nm this is due to a retardation of the London interactions [7]. The

interaction between induced dipoles is significantly affected by the distance between the

materials because of the finite propagation speed of the electric field and the temporal

nature of the induced dipole. Even though the propagation speed is equal to the speed of

light, c, the induced dipoles themselves are rapidly fluctuating. This introduces a lag

effect that reduces the interaction of the temporary dipoles as the distance between them

increases. To account for this, a time-dependent quantum analysis is used for the London

interactions, which yields a power law of r 7 beyond a characteristic distance.

The most rigorous method of calculating the van der Waals attraction between

macroscopic bodies is known generally as Lifshitz theory [26]. Lifshitz theory uses bulk

dielectric properties to evaluate van der Waals energy rather than the sum of all

molecular pairwise interactions proposed by Hamaker. This approach has built into it

retardation effects and the effect of the interceding medium we mentioned earlier. As a

result, the calculated Hamaker constant is a misnomer: it varies with increasing

separation distance and specific dielectric properties.

While it is by far the most accurate means of calculating van der Waals

interactions, Lifshitz theory is usually very difficult to implement. The calculations are

tedious and rely on dielectric data that is often unavailable. This has led to the

development of approximate expressions that reasonably accurate for most practical

applications. Gregory [27] presents several of these approximate expressions with a

discussion of their applicability.










2.3 DLVO Theory

In the 1940s, two groups of scientists independently developed quantitative

theories for the kinetic stability of colloids. They included Russian physicists B. V.

Derjaguin and L. D. Landau [1], and Dutch chemists E. J. W. Verwey and J. Th. G.

Overbeek [2]. Hence, theories that describe the interaction energy that arises between

surfaces are collectively known as DLVO theory. DLVO theory assumes that there are

two primary contributions to the interaction energy, the electrostatic (double layer) and

van der Waals interactions, and that these contributions are additive:




ODLVO = OR + OvdW (2.18)




where OR and OvdW are appropriate expressions of energy for the geometry and materials


of interest. A typical plot of the DLVO interaction energy is shown in Figure 2-2.




25

20

15

LJ
10-
0
t 5
O



5
C 0


0 50 100 150
Separation Distance (nm)


Figure 2-2. DLVO interaction energy between a sphere and a flat plate.











The precise shape of the potential energy curve depends on the parameter values

chosen. The plot in Figure 2-2 considers a silica sphere interacting with a flat silica

surface in water where the following values are used: = 7 nm, f01, = /0,2 = -10 mV,

A131 = 8.3 x 10-21 J (silica-water-silica), and a = 0.5 pm.

The parameter values should represent actual experimental conditions, so it is

useful to understand how these values can vary. The most readily adjustable parameter is

the Debye length, which was noted previously to be a function of ionic strength:



3.04 x 10 10
i -[mi] z Mv2 (2.19)




where K1 is in meters, Mis the molarity of the solution in moles/dm3, and z is the

valence number of a symmetric electrolyte [14]. By simply changing the concentration

of salt in the bulk solution, we can greatly alter the interaction energy between suspended

materials. Consequently, this is a convenient way of controlling the behavior of a

colloidal suspension.

The surface potential can be adjusted by adding potential determining ions,

specifically adsorbing ions, or by changing the pH of the solution. If the charge on the

surface is due to ionization, the latter method is a useful means of controlling the double

layer interactions. A change in pH can easily render a stable suspension unstable (or vice

versa) by adjusting the surface potentials. An especially difficult parameter to adjust is

the Hamaker constant since it depends on the dielectric properties of the materials rather

than the solution conditions. While the addition of salt to the solution does little to affect









the Hamaker constant, it does affect the screening of van der Waals forces whereby the

salt ions tend to reduce the attractive interactions. The final parameter to mention is the

radius. Expressions for double layer and van der Waals interactions show that the overall

magnitude of the DLVO energy is proportional to the radius of curvature.

It is useful to point out several key features of the DLVO curve shown in Figure

2-2. At large separations, there is no interaction between the surfaces and the energy

tends to zero. As the negatively charged surfaces approach each other, however, there is

typically an appreciable energy barrier (maximum) to overcome in order for attachment

to occur. The steep descent of the energy very near the surface forms theprimary energy

minimum. Higher values for the surface potential and Debye length tend to increase the

repulsive energy and the net energy barrier; this acts to stabilize a colloidal system,

encouraging particles to remain dispersed in solution. A large value of the Hamaker

constant will lower the repulsive energy and net energy barrier, but its value is usually

fixed for a given system, unlike the other parameters mentioned.

For a specialized combination of parameter values, the potential energy curve can

show a secondary energy minimum. Unlike the primary minimum which is essentially

infinitely deep, the secondary minimum usually has a shallow depth on the order of the

thermal energy. This means that the surfaces may loosely aggregate around the location

of the minimum, but the lack of depth of the minimum will allow them to be easily

separated again. In the case of a sphere-plate interaction, a colloidal particle may diffuse

into the secondary minimum, spend some time there, and then naturally diffuse away.

The probability that a particle can escape an energy well with a particular depth in a finite

amount of time is determined by the height of the energy barrier over which the particle

must move to escape. For energy barriers on the order of a few kT, the probability is









reasonably high for escape. However, even if the barrier a modest value of 10 kT, the

escape probability for the particle is extremely small.


2.4 Hydrodynamics of a Diffusing Sphere

Colloidal particles are small enough such that the persistent collisions from the

surrounding fluid molecules will induce a net random movement known as Brownian

motion. The magnitude of motion can be modeled as a series of independent random

walks, where each step in the walk is described by classical mechanics. If a particle

acquires an initial velocity, v0, at the start of a random walk, the subsequent motion is

described by



d2x dx
m =-8 (2.20)
dt2 dt



where m is the mass of the particle, x is the position of the particle from the origin, and 5

is the drag coefficient. This equation shows a balance between the inertial force and the

viscous drag force. From equation (2.20) we see that the velocity of the particle decays

exponentially with time:



v(t)= vo exp(- t/T,) (2.21)



where T, is the characteristic relaxation time of the particle's momentum (rT, m/5 ).

Although the movement is purely random, the average movement away from the initial

position is zero, and the mean-square displacement of the particle in any one direction is

simply












x2) = 2Dt


(2.22)


where D is the diffusion coefficient of the particle. The diffusion coefficient and drag

coefficient are related by the Stokes-Einstein relation:


kT
D =
8


(2.23)


Many colloidal particles can be treated as perfect spheres, which allows for

relatively straightforward hydrodynamic modeling. Stokes' law gives the drag force on a

rigid sphere moving in a quiescent fluid at low Reynolds numbers (Re < 1):


Fdrag = v,


(2.24)


and the drag coefficient is defined as


3 = 6mr7a


(2.25)


where 77 is the fluid viscosity and a is the radius of the sphere. Stokes' law is only

rigorously valid for an uncharged particle moving within an unbounded medium. In

terms of experimental force measurements, however, the particle will be diffusing very

near a solid wall. This situation requires a correction to Stokes' law:









Fdag = 6 7pav (2.26)



where A is a dimensionless correction factor whose expression depends on the direction

of the movement and the type of boundary. For our purposes, we are concerned with the

sphere's motion with respect to a solid flat surface. There are two expressions to

consider in this geometry: corrections for the normal and parallel translations of the

sphere near the flat wall. Because force profiles are measured as a function of normal

separation distance, z, we restrict our attention to the normal correction factor:



Z 4 .sh n(n + ) 2sinh(2n + )a+(2n + 1)sinh 2a 1 (2
A(z)=-smhal y x 2- --- )--- -1 (2.27)
3 i (2n -1)(2n + 3) 4sinh 2(n + (2n +)2sinh2a



where




a= cosh1 z +a. (2.28)
Ia


The correction factor is highly sensitive to separation distance as shown by Figure 2-3.










30


25


20
0

L 15

10

0

5



0 02 04 06 08 1 1 2 1 4 1 6 1 8 2
Normalized Separation (z/a)


Figure 2-3. Drag force correction factor for a sphere moving normal to a solid surface.






At large separations, the correction factor is unity and Stokes' law holds (dotted line).

However, as the separation distance diminishes to zero (particle-surface contact), the

correction factor approaches infinity. Even for separations on the order of several


particle diameters (z/a ~ 20), the correction factor is significant and the effect of the


solid wall cannot be neglected [33].

The double layer also affects the drag force on a diffusing colloidal particle. The

ions contained within the shear plane move with the particle, effectively increasing its

radius. Additionally, the diffuse ions beyond the shear plane have a deformed

distribution with movement of the particle, thereby creating a dipole. Several authors

have derived correction factors to account for the effect of the double layer on the drag

force. Ohshima et al. gave a recent expression for this correction:











= 1+ b f(Ka) (2.29)




where b is a value related to the average ion size, D is the reduced zeta potential

(= zeg/kT), andfis a function of Ka. The functionfis a maximum at ca = 1 and tends

to zero as ca approaches either zero or infinity. Typical colloidal systems used during

force measurements have a large value of ca (>10) such that this correction can be safely

neglected. However, if the sphere approaches a flat surface such that the opposing

double layers begin to overlap, there may be a significant effect of this interaction on the

drag force. Presently, little is known about this effect from either a theoretical or

experimental perspective.


2.5 Evanescent Wave Light Scattering

Experimental techniques that measure force-distance profiles generally require a

means for measuring position accurately. In AFM, a laser beam is reflected off the back

of the cantilever to monitor minute fluctuations of the cantilever over time. Using a

sensitive quadrapole (position sensitive) photodiode, fluctuations on the order of 1 nm

can be measured. SFA uses an even more accurate position detector based on

interferometry with resolution on the order of 1 A The task of measuring the trajectory a

freely diffusing micron-sized sphere in solution is a bit more complicated, however-a

useful solution to this problem, as we will see, is one of the novel aspects of TIRM.

Total internal reflection of light at the interface between two transparent media of

differing refractive index leads to the formation of an evanescent wave in the optically

rarer medium [28]. (Note that experiments often use a flat glass plate in contact with









water, so we will consider this special case henceforth.) The intensity of the resulting

evanescent wave decays exponentially with distance from the interface. Solving the

generalized Mie scattering problem, Chew et al. [29] showed that a spherical particle

with a refractive index dissimilar to that of the suspending medium scatters light with an

intensity that diminishes exponentially with increasing separation:



I(z)= I0 exp(- 3z), (2.30)



where z is the particle-surface separation distance, and I0 is the scattered intensity at

zero separation (particle-surface contact). The characteristic decay constant is




3 = [n2 sin2o,- n 2, (2.31)
AO 1 a



where A0 is the wavelength of reflected light a in vacuum, n, and n2 are the refractive

indices of the flat surface and the surrounding medium, respectively, and 0, is the

incident angle. Notice that adjusting the incident angle of the light (typically a laser

beam) directly affects the value of 3 For physical significance, we commonly refer to

the inverse of the decay constant, 3 1, as the penetration depth.

Rearranging equation (2.30) to find separation distance yields


z = -P3 1In+I+ pl Inlo.


(2.32)









This result shows that we can directly relate the measured intensity to the separation

distance of the particle from the surface. In general, the value of I, is difficult to

determine experimentally since the particle must overcome the energy barrier and contact

the surface irreversibly. However, in the absence of an intensity measurement at contact,

equation (2.32) can be used to find relative separation distances from an arbitrary

reference position. (Later, we will describe methods for determining absolute separation

distances.)

The above method of calculating position from scattered intensity measurements

is called evanescent wave light scattering (EWLS). Depending on the measurement

equipment used (i.e., photodetector, data acquisition system), the technique is able to

resolve distances on the order of 1 nm. The first practical use of EWLS for position

measurement was seen with the development of TIRM. TIRM uses Boltzmann's law

where the potential energy profile between a sphere and a plate is directly determined by

measuring the equilibrium distribution of particle positions from the surface:



2(z)= -kTln( p(z))+ C (2.33)



where p(z) is theprobability density of locating the particle between z and z + dz from

the surface, and C is a constant required to normalize the distribution. EWLS is used to

experimentally determine the equilibrium probability density p(z) by measuring the

movement of a particle for a time period much larger than the relaxation time of particle

fluctuations. By converting time-series intensity data to distances using equation (2.32),

and generating a histogram of measured positions, we can find p(z). Given the

equilibrium distribution, the interaction energy is easily calculated using equation (2.33).









2.6 Optical Trapping

The ability of light to impart forces on macroscopic bodies is well-known.

Centuries ago, Johannes Kepler first postulated that the sun exerted a radiation pressure

on orbiting comets, leading to the formation of their distinctive dust tails. Classical

physics can help our understanding of this behavior. Since photons possess momentum

as they propagate, Newton's second law suggests that a change in the momentum of a

photon requires a force. The phenomena of refraction, absorption, and reflection of

electromagnetic radiation therefore all confer a force to the material with which they are

interacting.

In 1970, Ashkin [11] found that a laser beam of modest power could be used to

manipulate particles of colloidal dimensions, especially if the particles were neutrally

buoyant (or nearly so) in a suspending solution. His early work produced a radiation

pressure optical trap that could confine a particle laterally in the axis of the beam and

simultaneously accelerate the particle along the direction of beam propagation. This trap

is considered to be two-dimensional because it can only hold the particle in the radial

direction along the beam axis. If desired, a particle could be stabilized by using gravity

to oppose a constant upward radiation pressure, or by using two counter-propagating

traps to create a stable trapping location between them.

By 1986, Ashkin and co-workers [12] had developed a single-beam gradient trap

that could confine a particle in three-dimensions. In this version of the optical trap, a

high numerical aperture lens is used to focus the beam down to a diffraction-limited spot.

The gradient force exceeds the scattering force along the beam axis to confine the particle

to a position near the focal point. Since its introduction, the single-beam gradient trap









has been used extensively (more so that its radiation pressure counterpart) in a variety of

applications including force measurement.

2.6.1 Optical Trapping Models

The scattering of photons causes a radiation pressure force that tends to push a

particle in one direction. In a single-beam gradient trap, a significant gradient force is

generated to overcome tendency to destabilize and accelerate the particle along the beam

axis. The scattering and gradient force components are often defined as




Fscat Qcan (2.34)
c


Fgrad Qgradn2 (2.35)




where Q is an efficiency term for the momentum transfer, n2 is the refractive index of

the surrounding medium, P is the laser power, and c is the speed of light in a vacuum

[30].

The gradient force arises from the interaction of dipoles within a dielectric

medium with a strong electric field. A tightly focused beam of light produces strong

electric field gradients in three-dimensions near the focal point. The gradient force is

proportional to the intensity gradient acting in the direction of the gradient:




Fgrad(z,r)= 2n2a3 2- VI(z,r), (2.36)
c m2 + (2.36)2









where a is the sphere radius, m is the ratio of the sphere's refractive index to that of the

surrounding medium (= n /n2 ), and I is the intensity distribution in cylindrical

coordinates. The gradient force tends confine a particle near the region of maximum

intensity (in the usual case where m > 1) while the scattering force tends to disrupt stable

trapping. This competition between forces usually results in an equilibrium trapping

location a small distance away from the focal point along the beam axis.

Most theoretical treatments of optical traps consider two limiting size regimes.

The large particle limit, known as the ray optics (RO) or Mie regime, assumes the particle

to be much larger than the wavelength of light (a >> A). In this model, the focused

beam is composed of individual light rays that are focused to a point, ignoring diffraction

effects. The rays refract and reflect at the surface of the particle where the change in

momentum of the ray, as it interacts with the particle, determines the applied force. The

overall force on the particle is simply a vector sum of the forces applied by all the rays

present in the convergent beam. If we have values for the Fresnel reflection coefficient

(the fraction of the ray reflected at the interface), and the refractive indices of both media,

we can determine the trapping efficiency parameter Q by simply applying Snell's law at

the boundaries. We can determine the trapping force, which is independent of particle

size, as a function of particle position within the trap by solving the RO model for various

locations of the particle.

In many cases the RO model is unsatisfactory, especially if and the beam is

tightly focused. In this case we cannot ignore diffraction effects, and usually we must

also account for the complex electromagnetic (EM) fields that are created near the focal

point. This leads us to the other limiting case, known as the Rayleigh regime, which

assumes that the particle radius is much smaller than the wavelength of light (a << A).









Using an EM model, we attempt to approximate the electric and magnetic fields formed

by the focused laser beam and solve for the interaction force by integrating the Maxwell

stress tensor over the surface of the particle. In fact, regardless of the particle size, the

rigorous method of solving for the force is to use the generalizedLorentz-Mie scattering

theory (GLMT) based on an EM perspective. The Rayleigh limit is particularly useful

because the particle is considered to be a simple dipole, which simplifies the calculations

dramatically and yields accurate theoretical forces. In this limit, the force scales with a3,

or the volume of the particle. Difficulties appear in the intermediate size regime where

a A. Here, interference effects become important, and even higher-order

approximations of the fields do not give satisfactory results. Unfortunately, many

experiments are performed in this intermediate regime where neither RO nor EM theory

works particularly well.

Recently, Tlusty et al. [31] developed a model that extended the dipole

approximation to a particle of any size. In this model, the interactions between the

particle and focused beam are considered to occur in a localized region equal to the spot

size, 2o0, of the focused beam, where coo is the beam waist radius. Within this localized

region, the phase of the fields does not vary appreciably, and thus the contributions to the

interactions are far less than those due to variations in amplitude. Their result has been

used widely as a means of predicting trapping forces for any size of particle.

2.6.2 Trap Calibration

The properties of the laser spot at the focal point determine how the particle will

behave in the trap. For a tightly focused beam, the light converges to a theoretical point

(RO description), but the width of the beam at the focal point is actually finite due to

diffraction effects. Equation (2.36) tells us that the force on the particle is proportional to









the intensity gradient, so the three-dimensional contour of the intensity profile is what

ultimately determines the trapping force. Most lasers used in optical traps have a radial

(or transverse) Gaussian profile (TEMoo mode). This profile is usually maintained even

as the beam is focused to a spot on the order of microns. In many cases, the axial

intensity distribution is also modeled as having a Gaussian profile with a half-width some

multiple larger than the transverse direction, where eis the eccentricity. In most cases,

eis a value of about three or more [31]. Since the distribution is narrower in the

transverse direction (e > 1), the transverse trapping force is usually stronger than in the

axial direction.

The approximately Gaussian intensity distribution leads to a linear force-distance

relationship for the trapped particle for reasonably small displacements of the particle

from the trap center. This suggests a Hookean spring model for the trapping force in all

three dimensions, although we are typically only interested in the axial (z-direction) force

component:



F,, (z)= -y (z z0), (2.37)



where y7 is the axial trap stiffness, and z0 is the axial location of the trap center. The

Hookean analogy simplifies the description of the trap by using just two parameters (in

any one dimension) to quantify the trapping force. This relationship generally holds for

displacements that are on the order of the spot size. Because of the eccentricity in the

intensity distribution, the linear regime is often larger for the axial force. Beyond the

linear regime, the force-distance relationship reaches a maximum and quickly falls off to

zero. For force measurements, we are interested in the linear regime for simplicity of









calibration and in the data analysis. With values for y7 and z0 for the trap, we can

readily apply a simple force balance to the particle as it finds a new equilibrium location

in response to external forces such as double layer repulsion.

To calibrate the trap, we observe the Brownian fluctuations of the trapped particle

over time. The calibration procedure occurs far from any surface to ignore contributions

from surface forces. The trajectory of the particle is governed by the (one-dimensional)

Langevin equation:



d2Z dz
m z Z ZO)- 5d- + (t), (2.38)
dt dt



where m is the mass of the particle, Sis the drag coefficient, and 4 (t) is a random

Brownian force. In most cases, the mass of the particle is small, so we can neglect the

left-hand side (inertia term) of the Langevin equation. The exact motion of the particle is

random, and the time-series solution requires a somewhat complicated Brownian

dynamics simulation (which we have attempted and will show later). However, the

equilibrium statistics of the particle's motion are relatively easy to find since the random

force term is normally distributed around a zero mean. For a Hookean spring model of

the trapping force, the trap stiffness is simply



kT
= 2k, (2.39)
(7









where 0o2 is the variance of axial position. The trap center is the peak of the distribution

and is equal to the mean axial position:



o = (z). (2.40)



With a calibrated method of detecting axial position, like EWLS, we can in turn

calibrate the optical trap and use it as a sensitive force measurement device. The details

of how force is measured using a gradient optical trap and EWLS are described in the

following section.
















CHAPTER 3
DESCRIPTION OF THE TECHNIQUE


3.1 Apparatus

A schematic diagram of the experimental setup is shown in Figure 3-1.


-............
slide ""

particle-- ev. wave

trapping -
rays

cover slip
dichroic
mirror


prism


Figure 3-1. Schematic of the experimental apparatus.


He-Ne beam
(, = 632.8 nm)


motorized stage


beam splitter









The optical trap is a Cell Robotics LaserTweezers 100 which employs a 100 mW diode

laser in the near infrared (o = 830 nm). The trap housing is mounted in the

epifluorescence port of a Nikon TE300 inverted microscope. The trapping beam is

collimated and slightly overfills the rear opening of a Nikon 100x oil immersion

objective (plan fluor, 1.3 NA) and is focused to a diffraction-limited spot within the fluid

cell. The particle suspension is placed upon a pre-cleaned glass microscope slide (75 x

25 x 1 mm) and enclosed with a coverslip (25 x 25 x 0.13 mm). With the sample

mounted to a three-axis motorized stage (Ludl Electronic Products), the objective lens is

optically coupled to the glass coverslip with index matching oil (n = 1.515 at 230C).

The evanescent wave is formed by directing a Melles Griot 35 mW He-Ne laser

beam (Ao = 632.8 nm, linearly polarized) into the 450 hypotenuse face of a BK-7 glass (n

= 1.515 at 632.8 nm) dove prism, which is optically coupled to the glass slide with index

matching oil. The He-Ne beam angle is precisely positioned with a rotation stage that is

accurate to within 0.1. This allows the evanescent wave penetration depth, 1, to be

adjusted by varying the incident angle of the beam, as indicated by equation (2.20). A

fraction of the scattered intensity from a particle near the glass slide is collected by the

objective, and the image is visualized using a color CCD camera and a dedicated monitor.

The magnitude of the scattered intensity is measured by a side-on photomultiplier tube

(PMT; Oriel Instruments) mounted to the side port of the microscope. The PMT is fitted

with an adjustable iris (typically forming a 1 mm diameter aperture) and bandpass filter

(633 + 2 nm) which isolates the scattered light from the trapped particle. The PMT signal

is sent to a current pre-amplifier where it is electronically averaged by a 150 pis time

constant RC circuit (low-pass filter) and then digitally sampled by a National Instruments









12-bit data acquisition board. The time series intensity data are saved to a PC file for

later analysis.


3.2 Procedure

The following procedure was used for nearly all measurements performed over

the course of this work. Particle solutions were prepared using Barnstead Nanopure

ultrafiltered water (18.2 MQ-cm resistivity). Silica microspheres were purchased from

Geltech as dry powders and suspended in electrolyte solutions at low particle densities

(~105 particles/mL). A similar density of 9.14 + 1.24 pm diameter polystyrene

microspheres (Polysciences) served as gap spacers. All solutions were prepared the day

of an experiment to limit contamination. The solutions were each sonicated for one hour

to break up particle aggregates and then agitated briefly with a touch mixer just prior to

an experiment to resuspend settled particles.

A small amount (-10 pL) of the particle suspension was placed on the center of

the glass slide and overlaid with a pre-cleaned glass coverslip. Excess solution was

withdrawn from the slide using bibulous paper. The edges of the coverslip were then

sealed using a minimum amount of insoluble silicone vacuum grease to prevent the liquid

from evaporating during the course of the experiment. With the sample mounted, a

single diffusing silica sphere was isolated in the field of view of the microscope and the

optical trap was activated to confine the particle in the fluid gap. The He-Ne laser was

activated at least 30 min prior to any measurements to ensure stability of the beam

intensity. The particle was moved toward the glass slide surface until it began to scatter

light from the evanescent wave. The incident angle of the He-Ne beam was adjusted to a

value exceeding the critical angle (61.60) for total internal reflection. The penetration









depth, P 1, for these experiments ranged from 150 to 230 nm. A mirror mounted to a

precision rotation stage (0.1 increments) adjusted the angle of the beam. The stage angle

depended on the geometry of the system, but was calculated using equation (2.30).

Shorter depths offer better distance resolution whereas larger depths lead to a measured

signal and provide information about movements made farther from the slide surface.

The lateral position of the beam was also adjusted to maximize the scattered intensity.

With the He-Ne beam properly aligned, the particle was moved away from the

glass slide surface to a position where only background levels of intensity were

measurable (several microns from the slide surface). A program was written in

LabVIEW (National Instruments) to sample PMT measured intensities at varying

positions from the surface. At each step in the scan, the program acquired an adjustable

number of intensity samples (-104 to 105) at an adjustable rate (-2 to 50 kHz), recorded

the data to a file, and moved to the next position toward the slide surface in adjustable

increments (-10 to 30 nm). This process was repeated until the particle was as near to

the surface as possible with the optical trap. The measurement procedure was repeated

several times for the same particle-solution system in order to assess the quality and

reproducibility of the data. The glass slides were discarded following an experiment, and

each new particle-solution sample was mounted on a new glass slide.


3.3 Data Analysis

3.3.1 Signal Processing

The actual voltage signal obtained by the data acquisition system is not strictly

proportional to the scattering intensity from the particle, I(t), used in the above analysis.

This is because the measured signal is persistently affected by a background intensity









level and uncorrelated (white) noise. An RC circuit associated with the pre-amplifier also

filters the signal by removing high frequency fluctuations.

For typical experimental conditions, noise contributes less than 5% of the total

signal magnitude, but accurate correction for this small noise is important for calibration

of the optical trap and subsequent force measurements. Fortunately, most contributions

to the total noise can be isolated. The first noise contribution is the fluctuating

background intensity, IB (t), due to scattered light from surface irregularities and the

residual background reading from the PMT. The second source is known as "shot noise,"

Is (t), where the discrete number of photons incident on the photodetector (described

mathematically as a Poisson process) contributes to a distribution of measured intensities

around a mean value. Assuming intensity contributions to be additive, the total measured

intensity at some instant in time is



I (t)= I(t)+ I,(t)+ I (t). (3.1)



The mean background intensity, PB = (B), is measured when the particle is far enough

from the surface as to not scatter the evanescent wave. This allows the correction of the

subsequent positions using the background-corrected intensity, Ic (t):


I,(t) IT(t)- I(t)+ I (t)


(3.2)









where IN(t)- I (t)+ I (t)- p~ is the total fluctuating noise contribution. Because

(Is) = 0, then (Ic)= (I) = p.

Assuming that fluctuations in I(t) and I,(t) are uncorrelated, the autocorrelation

function of Ic(t) is given by



Gc,(r)_- (Ic(t + )Ic (t))- 2 = G, ()+ GCTr) (3.3)



where G (r)= (I(t + z)I(t)) 2 and GNr) (I, (t + z)I (t)). The relaxation time of

the fluctuations in I (t) is much smaller than that of Ic(t), which allows a substantial

amount of noise to be removed by the RC filter without corrupting the desired lower

frequency fluctuations of I(t). In most cases, the time constant, z of the RC filter was

set to 150 ps. The autocorrelation function of the filtered signal is given by




Gf(r) G '(r) e Vdr' + ,2 ITV
2- 2N (3.4)



where a-2 is the variance of the filtered noise [32]. Extracting force data from the

measurements using equation (3.4) requires a model for the fluctuating particle position.

Our approach is to model these fluctuations as Brownian motion in a potential energy

well.









3.3.2 Brownian Motion in a Potential Energy Well

The goal of the measurements is to simultaneously extract force-distance profiles

for the conservative (static) forces, which depend only on separation distance, and non-

conservative (dynamic) forces, which depend on the position and velocity of the particle.

The conservative forces are characterized by the gradient of the potential energy, O(z):



F(z)= -0'(z). (3.5)



The simplest example of a non-conservative force is the low Reynolds number viscous

force, which is directly proportional to the particle velocity:



dz
Fd, (z) -8(z)- (3.6)
dt



where 8(z) is the drag coefficient in the direction normal to the flat surface. For large

separations, 8(z) approaches the Stokes' law value, 8(z)= 677a, where 7 is the fluid

viscosity and a is the particle radius. However, for separations on the order of several

particle radii and less, there is sufficient viscous coupling with the solid wall such that the

drag coefficient becomes a strong function of separation:


5(z)= 6m7aA(z)


(3.7)









where A(z) is a correction factor that depends on separation distance. For particle

motion in a Newtonian fluid with no-slip boundary conditions, A(z) has an analytical

solution that becomes infinite as the separation vanishes, and tends to unity (as in Stokes'

law) for infinite separation. This result was first derived by Brenner [33] in 1961.

In these experiments, the particle's range of motion is limited by a potential

energy well created by the optical trap alone or together with the surface. The probability

density of a Brownian particle in a potential energy well can be described by the one-

dimensional Fokker-Planck equation,



ap(z, t Iz',O) a (z) t ',)+ kT Op(z, t z',0) (3.8)
8t 8z L (z) (z) 8z



where p(z, t I z',O)dz is the probability of finding the particle between z and z + dz at

time t given the particle was at position z' at time zero, <,,, (z) is the total potential

energy, k is Boltzmann's constant, and Tis the absolute temperature. In the case of a

Newtonian fluid, the calibration and measurement procedures assume the validity of

equation (3.8) for the trapped particle. The primary assumptions underlying this equation

are the validity of equation (3.6) (viscous force proportional to velocity) and a fast

relaxation of velocity fluctuations relative to the time scale of interest. Based on a root

mean square velocity, V, = 2kT/m the Reynolds number is estimated to be less than

0.004, validating the Stokes flow assumption. The velocity relaxation time, equal to

m/5, is estimated to be less than 107 s for typical measurements. Also, the persistence

length associated with the velocity fluctuations, estimated by Vrms, m/, is less than 0.5








nm, which indicates that any persistent motion of the particle due to a finite relaxation

time of the velocity can be neglected.

Solving equation (3.8) for the stationary probability density yields


e -, (zykT
p,(z)= (T (3.9)
Je (z')/kTdz'




which is a statement of Boltzmann's law. A key property of equation (3.9) used in the

force measurements is that, because p'(z)= -p,(z),'o (z)/kT, 0,(z) and p,(z) share

extrema with respect to z. In other words, the maximum in the measured position

histogram and the minimum in the total potential profile occur at the same location. The

autocorrelation function of the particle position, given by



G()- (z(t+)-(z)Xz(t)-(z)))= dz' dz p(z')p(z,-r z',Oz'-(z)2, (3.10)



can also be obtained from equation (3.9) and the solution to equation (3.8).

In the special case where o, (z) is approximately parabolic of the form

ot (z) _= 't" (z)(z z p)2, centered around position z then



p (z) (z)2 exp (3.11)
\ 2nkT ) 2kT









from which the mean position, (z) = z,, and the variance, az2 = kT/O", (z), can be

obtained. Furthermore, when 5(z) is nearly constant and approximately equal to 8(zp)

over the local range of fluctuations, then equation (3.8) can be solved and introduced into

equation (3.10) to yield



G(rT)-- z2 -V'" (3.12)



where Tr = &o- /kT = 5/o", (z) is the characteristic relaxation time of the fluctuating

position. We show in the Appendix that equation (3.12) remains a good approximation

even when the potential well is slightly skewed and 8 1(z) (or the particle diffusivity)

varies linearly with position.

Because we are directly measuring the intensity rather than the particle position,

we need to relate the statistical properties of the two variables using equation (2.19).

From equations (2.19) and (3.11), the mean intensity, p and variance, o-2, are given by




p = JI(z)p,(z)dz =IOe -oe1'212 (3.13)



and




o = If2(z)p,(z)dz =p(e ), (3.14)









respectively. Moreover, it can be shown that the autocorrelation function of intensity is

given by



GI(r)- (I(t + )(t))- 2 = 2 [exp( 02G(r)) 1] (3.15)



where G,(z) is given by equation (3.12).

Equation (3.15) is the key result to be used in the calibration of the trap and in the

measurement of dynamic forces, which requires estimation of a 2 and Tz from the

measured autocorrelation function of intensity. However, the actual signal is represented

by equation (3.4) due to the low-pass filter. Upon introducing the series expansion of the

exponent in equation (3.15) into the integral in equation (3.4), we find



G,(r)= G I(T)+ E2 _2 2G G(T)( )]-2Gr+ 2 t -[ 2 pe 2 2 +2 ] + O(E3)(3.16)



where e -- f/cr is the ratio of the filter time constant to the relaxation time of particle

fluctuations. Because T~ is typically at least one order of magnitude larger than fy, the

higher-order terms in equation (3.16) are safely dropped.

3.3.3 Calibrating the Optical Trap

Since the stiffness of the optical trap may be sensitive to experimental conditions,

an independent calibration is performed for each experimental run. For separation

distances where the surface forces are negligible, the trapping force is the only









conservative force acting on the particle. The restoring force due to the trap is

approximately linear for axial displacements on the order of a particle diameter:



F,, (z)= -y (z zo) (3.17)



where yz is the axial trap stiffness and z0 is the trap center (neither is known apriori).

Since the location of the particle is specified by the separation distance, z, the convention

is adopted to define the trap center as the equilibrium separation distance of the particle

in the absence of surface forces. This means that the actual location of the trapfocal

point and the so-called "trap center" are offset by a distance approximately equal to the

particle radius, a. A linear model for the trapping force implies a simple harmonic

potential energy profile; integrating equation (3.17) with respect to separation distance

gives




0,,(ra)= -(Z- z)2 (3.18)
2



which is the potential energy relative to the local minimum value at z0.

If no other forces are present, then Pr,, (z) = yo, (z), and the stationary density is

simply the Gaussian distribution given in equation (3.11). The trap stiffness,

y = tap (z), and the trap center, z0, are determined by the first two statistical moments

(i.e., the mean, (z), and variance, a2 ) of the particle's position within the trap. This

calibration is performed over a range of trap positions far from the surface where only the









trapping force is significant and the position variance, 0-2, is a constant and equal to

kT/7y. The position variance, a is determined at each calibration position by fitting

equation (3.16) to the measured autocorrelation function by non-linear least-squares

regression, using o2, rT, and a as fitted parameters. This provides a precise, noise-

corrected estimate of the trap stiffness, 7y, and ultimately the drag coefficient, 8(z).

The calibration is performed by at known trap positions {zo,J } initially defined

relative to an arbitrary zero point. The calibration procedure also determines the intensity

at zero separation, I0 = I(z = 0), on this scale. This can be estimated by rearranging

equation (3.13) to relate I0 to z0 and the measured mean intensity, p then taking the

appropriate average from the measurements:



1 ( k2T
lnIo = 3z, + Inp 2 (3.19)




where p, is the measured mean intensity at trap position z0, .

3.3.4 Measuring Static Forces

Once the optical trap is calibrated, it is used as a sensitive force transducer. Of

particular interest are the DLVO-type forces which include the screened electrostatic (i.e.,

double layer) and van der Waals interactions of the particle with the flat surface. These

are considered to be static forces because they are conservative over the entire range of

separation distances and are insensitive to the particle's Brownian motion. In the

calibration region, the static forces are negligible due to the relatively large separation









distance; here, only the force due to the trap is significant. However, as the particle

approaches the surface, the surface forces begin to perturb the equilibrium position of the

particle within the trap. Assuming that the potential energy contributions from the trap,

t,,p (z), and particle-surface interactions, p(z), are additive, the total potential energy of

the particle is



o, (z)= O(z)+ a,,p (z). (3.20)



The local minima of ,,,o(z) represent separation distances where there is a stable balance

between the trapping force and surface forces such that



t',(zP)= 0=-F(zp)+ 7(zP zo). (3.21)



Therefore, once z, is identified, the force can be calculated directly from



F(zp)= -Ftp (Zp)= z(zp zo) (3.22)



for any trap position. The details of the procedure for estimating zp from the raw

intensity measurements are presented elsewhere [13]. Briefly, z, is obtained from the

minimum of a fourth-order polynomial fit to the negative logarithm of the measured

distribution of positions, i.e., lnp(z(I))c Oo, (z). By moving the location of the trap

center, z0, toward the surface in small increments, the interaction force as a function of









separation distance is readily determined from an equilibrium force balance at each new

location. A smoothly varying force-distance profile can be integrated numerically to

yield an estimate of the potential energy,




O(z)= fF(zpz', (3.23)




over the same range of separation distances. This offers a comparison with potential

energy data produced by TIRM measurements.

3.3.5 Measuring Dynamic Forces

In order to determine the drag force as a function of separation distance, we again

examine the autocorrelation function at each new trap position. A key assumption is that

equation (3.12) and, consequently, equation (3.16) remain good approximations even

when particle is within a potential energy well created by the trap and the surface

together. This requires that the particle fluctuations are reasonably symmetric about the

potential energy minimum and the drag coefficient does not deviate largely over the

range of fluctuations from its value at zp. A Taylor series expansion of the potential

energy centered at the most probable position, z yields




o, (Z)- "() (z Zp )2 (3.24)
2









where we have noted that 0'(z )= 0 at equilibrium. Equation (3.24) assumes that the

total potential profile, to, (z), is approximately harmonic (parabolic in shape) near z,

such that the effective stiffness, combining trap and surface force effects, is



y ),ef (z,)= 0"(z)= y F'(z,). (3.25)



Equation (3.25) shows that repulsive surface forces that decrease with separation distance

(i.e., F(z )> 0 and F'(z )< 0) enhance the effective stiffness and narrow the total

potential energy well. Conversely, weakly attractive forces (F(z,)< 0, F'(z,)> 0) tend

to decrease the effective stiffness and broaden the potential energy well. We also assume

that 8 1(z) (or the particle diffusion coefficient) depends weakly and linearly on z over

the range of fluctuations to justify the use of equations (3.12) and (3.16) to estimate the

position variance, a 2(zp)= kT/7yf and the relaxation time, rc(zp)= 8/Y,ff as

functions of z,. As with the calibration procedure, these parameters are estimated at

each trap location by first measuring Gf (z), and using equation (3.16) to fit to the data

via weighted non-linear least-squares regression, with oa-, T and ao as fitted

parameters.

3.3.6 Determining Absolute Separation Distances

As mentioned above, the measured separation distances using this technique are

initially expressed relative to an arbitrary reference point since we lack a direct estimate

of the intensity at zero separation distance. To determine absolute separation distances,

we compare the drag force data with theory such that the position offset can be inferred









from a fitting procedure. Whereas DLVO theory depends on several unknown

parameters (the surface potential is often the most difficult to estimate for a particular

system), the hydrodynamic drag far from the surface theoretically depends only on the

particle radius, fluid viscosity, and absolute separation distance. This allows for a rather

simple method of determining absolute separation distances using the measured drag

force profile. Similar procedures of using the drag coefficient (or, equivalently, the

apparent diffusion coefficient) as a means of determining absolute distances has been

applied previously in TIRM measurements [9, 10].

All dynamic force data reported here were ascribed absolute distances determined

from a fit of equation (3.7) to the data. Because our measured drag coefficient deviated

significantly from equation (3.7) in the region of double layer overlap, we performed the

fit using only data from larger distances where there the measured static force was near

zero and the overlap of the double layers is negligible. Nevertheless, the large number of

data points remaining at these larger separation distances provides a good estimate of the

absolute separation distance.














CHAPTER 4
INITIAL FORCE MEASUREMENTS AND EVIDENCE OF INTERFERENCE
EFFECTS


4.1 Initial Static Force Experiments

This section summarizes the first set of static force measurements made using the

new technique. This study was intended to demonstrate the technique's accuracy for

measuring static interactions between a micron-sized particle and a flat glass plate.

Experiments were performed using 1.5 pm silica spheres suspended in solutions of

varying ionic strength. Since the characteristic distance of electrostatic interactions, the

Debye length, varies with the ionic content of the solution, this is a convenient way to test

the accuracy of the technique. As will be shown, however, the measurements produced

unexpected results that did not agree well with theoretical predictions. At first it was

thought that these measurements were accurate because they follow DLVO theory,

although the Hamaker constant required for such agreement was unusually high.

Regardless, the measured double layer repulsion agreed well with theory.

Following this study, further investigations eventually led to the hypothesis that

reflection effects at the glass-water interface might be influencing the experimental

results. This was verified by adjusting the entry point of the trapping beam to the

objective lens which would reduce or enhance this effect depending on the beam

placement. Additional studies using slides with reflective dielectric coatings further

showed that this was the source of the anomalous data. The details that lead to this

conclusion are presented in this chapter as well.









4.1.1 Static Force Results

In order to validate the technique experimentally, we chose to study a well-

characterized system that could be described by classical DLVO theory. The simplest

system we can consider is a spherical particle interacting with a flat plate immersed in an

aqueous solution of known ionic strength. We chose silica microspheres because of their

spherical shape, monodisperse size distribution, availability in the micron-size range, and

tabulated material properties. Standard microscope glass slides were used as the test

surfaces and were found to be reasonably smooth for experiments (surface roughness of

about 2 nm RMS as measured by AFM). Solutions were prepared with NaCl added to

ultra-purified water to a specified concentration. The ionic strength of the solutions was

measured using a conductivity bridge. Using equation (2.19), we were able to specify

several values for the Debye length, K 1. The measured exponential force-distance

relationship created by the overlapping double layers offers the most robust test of the

technique's accuracy since the Debye length is only a function of the electrolyte

concentration. Attractive forces due to van der Waals interactions are usually not

measurable for most common systems of low ionic strength where the double layer forces

dominate.

Measurements were made using 1.5 pm silica microspheres interacting with a

soda-lime glass slide in solutions of varying NaCl concentration. We prepared solutions

of 0.10, 0.18, and 0.40 mM NaCl in doubly distilled water (corresponding to three equal-

spaced Debye lengths), each suspended with a low density of silica particles and

polystyrene spacers (-105 particles/dm3). We made several runs of the same system to











assess repeatability of the data, and saved all data to PC files for later analysis using an

algorithm written in MATLAB. The details of the analysis are given in detail previously.

The applied trap potential is assumed to be harmonic, and we can demonstrate this

by looking at the position histograms generated by a trapped particle. Figure 4-1 is a

histogram plot for two different locations of the trap center.



1400
0
1200
(a)
1000

800

I (b) o
o 600 I
0 0
400 -
0 00
0
200 0


0 100 200 300 400 500
Relative Separation Distance (nm)

Figure 4-1. Position histograms of a particle trapped (a) far from and (b) near the plate
surface.






Histogram (a) shows a trapped particle sufficiently far from the plate surface such that the

interaction forces are negligible. Fitting this histogram with a Gaussian distribution

function (solid line) shows a good agreement with the harmonic trap model for

displacements up to about 100 nm. From this data, we cannot tell if the model fails for

larger displacements within the trap since the particle does not naturally sample these

regions. Histogram (b) shows the same particle trapped much closer to the plate surface

where interaction forces become significant. Rather than having a Gaussian shape, the









distribution is now skewed to the right due to the presence of strong double layer

repulsion.

The histograms in Figure 4-1 represent the collected data for a given location of

the trap center. To calculate a complete force-distance profile, many such histograms are

generated as the trap center is moved toward the plate surface. For each of these

locations, a force balance is made at the peak of these distributions knowing the location

of the trap center. The experimental force-distance profiles are shown in Figure 4-2.



4

3-

z
o2-

0 1




-1
0 100 200 300
Separation Distance (nm)

Figure 4-2. Experimental force-distance profiles for 1.5 pm silica near a glass slide.
(O 0.1 mM NaC1, 0 0.18 mM NaC1, D 0.40 mM NaC1)




The symbols in Figure 4-2 represent measured data points, and the lines are a DLVO

theory fit generated from a Hogg-Healy-Fuerstenau (HHF) expression [18]. In all three

measurements, there was good agreement between the expected Debye length and the

observed decay constant of the data. However, there is an obvious region for the two

higher electrolyte concentrations where the forces are attractive (negative), signifying the









existence of an unusually deep secondary energy minimum. Note that the reasonable fit

of the data using the HHF expression is only achieved by choosing a relatively high value

for the Hamaker constant (for a silica-water-glass system). In this form of the DLVO

theory, the van der Waals forces are calculated using equation (2.16). This expression for

the van der Waals energy does not assume retardation effects, however, which will be

significant above separations of about 10 nm [7]. If we use a more rigorous Lifshitz

expression for the van der Waals interactions, the predicted attractive force will be much

lower for our system than the measurements in Figure 4-2 would indicate, over the same

range of separation distances. In effect, a rigorous version of DLVO theory suggests that

our system should generate purely repulsive interactions over the range of separation

distances and concentrations measured.

4.1.2 Discussion of Static Force Results

It is clear that the measured attractive forces are unusually large for these systems,

so we need to consider some reasonable explanations for the discrepancy. One possible

explanation is to assume that our measurements of large attractive forces are legitimate

and that the theory somehow does not apply in this case. This explanation is tempting,

but is probably presumptuous since a large number of studies-many using SFA-have

shown the Lifshitz theory to be very accurate [4, 34]. Another possible explanation

would be the presence of depletion interactions, which can induce significant attractive

forces, but this is unlikely since it requires the addition of small particles or long-chain

molecules to the solution in sufficient concentration.

If we assume that DLVO theory is accurate, our attention turns to the assumptions

about optical trap. It is possible that the trap does not act as a linear spring over a large

range of axial displacements from the trap center, as it is assumed. We might infer from









our measurements of strong attractive forces that the trap is somewhat weak in the

reverse direction (i.e., opposite to the direction of beam propagation) as opposed to the

forward direction. This may allow the particle to be disrupted from the trap more easily

if there are moderate attractive forces. One argument against this view is that Lifshitz

theory predicts van der Waals attractive forces to be significant only within 100 nm, and

even then these forces are very small for a micron-sized particle (recall that van der

Waals forces scale with particle radius). Although our force-distance profiles in Figure

4-2 reflect relative separation distances, clearly there appear to be strong attractive

interactions occurring at separations much beyond 100 nm. For these unusually large

attractions to occur, even with weak reverse trapping, there would still need to be a

significant attractive force that extends significantly beyond 100 nm. This hypothesis is

not supported by Lifshitz theory.

Thus far, our attempts to explain the experimental force data have been

inadequate. It appears as though we are accurately measuring double layer repulsion

(note the 0.1 mM data in Figure 4-2), but the relatively strong attractive forces seemingly

defy explanation using reasonable arguments. There are cases in the literature where

attractive forces appear to be higher than usual, and we might consider our measurements

to support these findings. Of note are results from recent experiments by Velegol et al.

[35, 36] using differential electrophoresis where the electric field required to separate

two particles of differing surface charge, loosely bound in mutual secondary minima, was

found to be much higher than that predicted by DLVO theory. A possible explanation for

this effect, however, might be that the applied separation force is not well known in

electrophoresis. The resolution of force in such a study would be far less than that found









using an optical trap. This experimental result alone is not enough to bolster our

findings.

The lack of an obvious explanation for the poor agreement between theory and

data led us to look at the data more carefully. Based on DLVO theory and an accurate

description of the trapping energy, we can simulate an experiment and determine how the

data should appear for a certain particle-surface system. Using the overall potential

energy for the particle as the sum of DLVO-type interactions and trap contributions, we

can use Boltzmann's equation to predict how the histograms should appear as we move

the trap center toward the plate surface. These histograms can be difficult to compare

with a large set of experimental histogram data, so it is often useful to compare the

statistics of these distributions. The raw data sampled by the data acquisition program is

measured in terms of a voltage level that is proportional to the scattered intensity of the

particle, so it is convenient to analyze the mean and variance intensity profiles as the

particle approaches the surface. This is a convenient way of comparing the measured

data with predicted quantities. It is a simple matter to calculate the expected intensity

mean and variance from the theoretical distributions by first using the EWLS relation in

equation (2.32) to convert distances to intensities, and then integrating the curves to

generate the first two statistical moments. A comparison of the mean intensity profiles is

shown in Figure 4-3.















S 0.35
-10.30 0
0.25
S 0.20
U 0.15
0.10
0.05 -

0.00
-1000 -500 0 500 1000 1500 2000
Relative Trap Position (nm)

Figure 4-3. Experimental and theoretical mean intensity versus trap position for 1.5 /m
silica sphere. (0.1 mM NaC1 solution; + raw data, O background level subtracted,
solid line model prediction)






With the background level subtracted, the mean intensity profile appears to follow

the model prediction (solid line) very well. The plot clearly shows an exponential rise of

the average intensity as the trap center is moved toward the surface (leftward on this

plot). The exponential rise is then followed by a curve inflection and finally a leveling of

the intensity. We can interpret this data as an indication of repulsive forces for smaller

separations because the curve deviates far below an exponential dependence. In the

absence of surface forces, the intensity data should vary exponentially until contact. We

can divide the curve into two distinct regions: a calibration region where the surface

forces are negligible (exponential rise of mean intensity), and a measurement region

where surface forces are significant to oppose the trapping force and displace the particle

from the trap center (portion of the curve to the left of the inflection point).











The data shown in Figure 4-3 corresponds to the force curve shown Figure 4-2 for


a 1.5 pm silica sphere in 0.1 mM NaC1. In both plots, the data is consistent with DLVO


theory predictions, showing a purely repulsive interaction over the range of

measurements. Unfortunately, this agreement is not easily reproducible. In most cases,

there is significant deviation from DLVO theory predictions. Without having to

rigorously analyze the data to produce force-distance profiles (the ultimate end-result of

measurement), the raw data of mean intensity versus trap position can suggest if the

measured data follows DLVO theory predictions or not. In many cases, the measured

profile of the mean intensity has certain unexpected features. A typical example is shown

in Figure 4-4.



04

035
S 03
03 00
0
t 025 0

S 02
C o
t"- 0
015 0



005

-800 -600 -400 -200 0 200 400 600 800 1000 1200

Relative Trap Position (nm)

Figure 4-4. Experimental mean intensity versus trap position for a 1.5 pm silica sphere.






In comparison to the plot in Figure 4-3, the data shown in Figure 4-4 shows two unique


features. First, there is an unusual "bump" in the data occurring in the exponential rise

portion of the data (from about 200 to 400 nm). This feature cannot be explained using









DLVO theory. Second, the maximum slope of the curve is so extreme that the curve

appears to be nearly discontinuous. Since the data depict discrete 20 nm movements of

the trap center toward the plate surface, this discontinuity would appear to be due to an

unusually strong attractive force. Most of the static force measurements using colloidal

particles ranging from 0.5 to 5.0 pm in diameter have shown these effects to varying

degrees.

In order to analyze this data, we had to account for these unexpected features.

The simplest explanation seemed to be that the small "bump" was due to some artifact

that did not affect the force measurement, and that the discontinuity in the mean intensity

was due to a strong attractive force. As such, we could disregard the data to the right of

and including the "bump," and perform a calibration using a smoothly varying portion of

the exponentially varying region. Notice that the force data shown thus far have been

over a limited range of separation distances, up to about 300 nm. This is primarily due to

the presence the "bump" where we have omitted anomalous data. Isolating the cause of

these unexpected features has been extraordinarily difficult. These effects appear to be

independent of the sample preparation and experimental conditions. Fortunately, though,

a reasonable explanation was discovered for this unexpected result. This is the subject of

the sections to follow.


4.2 Effect of Trapping Beam Alignment

An additional adjustable parameter not considered in the previous experiments is

the position of the trapping laser beneath the objective lens. The LaserTweezers 100 unit

is designed to simply slide into place within the epifluorescence port of the microscope

without end-user adjustment. It was determined that the position of the trap along the









guide rail was important only after the trap was inserted and removed from the

microscope (before and after experiments) with greater regularity. Only then was the

connection drawn between the quality of the measured data the location of the trapping

unit. This observation led to the conclusion that there may be significant reflection of the

trapping beam at the glass-water interface. It is not intuitively obvious that this should

be the case since the interface should reflect a minute fraction of the total beam intensity

(<1%). Nonetheless, experiments were run to deduce the dependence of the trapping unit

location within the microscope, which affects the entry point of the beam to the back

aperture of the objective lens. The results of these studies are discussed below.

A schematic of the optical trapping unit (Cell Robotics LaserTweezers 100)

placed within the microscope is shown in Figure 4-5.




/















Figure 4-5. Schematic of LaserTweezers unit placed within the microscope.









The trapping laser is contained within a small plug-in unit that is specifically designed to

fit within the epifluorescence port of an inverted microscope. The unit collimates an

infrared beam and reflects the light to the rear opening of a high numerical aperture

objective. The collimated beam intentionally overfills the rear opening (by about 20%)

such that the marginal rays are occluded; this significantly improves the axial trapping

strength. The entire unit slides into position on a dovetail-shaped rail that restricts

movements along one axis (as indicated by the arrows in Figure 4-5). A small setscrew is

used to fix the position of the unit along this axis. Ideally, the beam axis should be

concentric with the axis of the objective. This corresponds to the maximum power of the

beam exiting the objective and the maximum trapping force.

The most precise way of aligning the trap is to measure the light intensity emitted

from the objective. Since the laser operates in the near-infrared (IR) spectrum, the beam

can be imaged using an IR-sensitive card or a CCD camera. The properly aligned

position will yield a maximum intensity spot. A more practical approach is to align the

unit using a trapped particle as a guide. As the trapping unit slides along the guide rail,

the particle will also show small micron-scale movements along one axis. The details of

this method are somewhat difficult to describe and require hands-on experience.

Essentially, the method involves correlating the particle's movements to the ideal

location of the trapping unit within the microscope. Although difficult to describe here,

this method is far more convenient than measuring the beam output each time.

The so-called "artifacts" in the initial force measurements appeared be attributable

to a yet unknown physical phenomenon since the particle was visually observed to make

unexpected movements as it neared the flat plate surface, corroborating the trends seen in









the anomalous data. Recall that this particularly strange observation was shown in the

region where the intensity jumped discontinuously as a function of trap position (see

Figure 4-4). Here it was noted that the particle became somewhat unstable within the

trap and moved almost erratically towards and away from the surface. This was

qualitatively observed under the microscope as large variations in the intensity.

Experimentally, the sampled data clearly showed a larger than expected peak in the

intensity variance, oI than would be predicted by DLVO theory. This observation leads

to two possible explanations: either the attractive surface forces sufficiently broaden the

overall potential profile, or the trap itself is weakened significantly in this region. If we

are reasonably confident that DLVO theory is accurate for this system, we should

naturally suspect that the trap is responsible the unexpected results. At this point, it is not

as important to describe a precise mechanism for this result (to be addressed in a later

section) as it is to show the effect of beam alignment on the data.

To test the effect of the beam alignment, intensity mean and variance plots were

generated for different locations of the trapping unit beneath the objective.

Representative mean intensity plots are shown in Figure 4-6.






71






0
..- .









Relative Axial Trap Position (nm)

Figure 4-6. Mean intensity versus trap position for two locations of the trapping unit.





The data show an obvious dependence on beam alignment. Curiously, "anomalous" data

results when the beam is precisely centered beneath the objective. Conversely, when the

beam is moderately off-axis, (i.e., intentionally misaligned) the data are consistent with

DLVO theory predictions. This result is both exciting and frustrating: we have

implicated the alignment of the trap as the cause of our inaccurate measurements, but the

solution to this problem makes little sense. Why should an aligned beam, and hence a

well-formed trap, lead to inaccurate measurements? This is the subject of the next

section.

While it is not yet described why an off-axis beam gives improved results, it is a

useful pragmatic approach that significantly improves experimental static force

measurements. Unfortunately the measurements are extremely sensitive to the beam

location beneath the objective, so even minor movements of the trapping unit can

influence the data greatly. This represents a significant obstacle to achieving repeatable

measurements using this system.









4.3 Reflection Effects in Optical Trapping

That an aligned beam would give the most unusual results while a misaligned

beam would yield agreement with theory seems counterintuitive. The most rational

explanation would be that the description of the optical trap is inadequate in some

fundamental way. One explanation previously conjectured for this discrepancy is that the

presence of the flat plate in proximity to the beam focus somehow perturbs the

electromagnetic field distribution. Since the beam propagates normal to the plate, it is

possible that effects due to beam reflection could be responsible for a more complicated

field distribution as the particle nears the surface.

4.3.1 Standing Wave Trapping

Interestingly, there is a related technique called standing wave trapping (SWT)

developed by Zemanek and co-workers [39, 40]. This technique intentionally generates

interference between the incoming trapping beam and its reflection at the plate surface,

thereby forming a standing wave that confines micron-sized particles near the plate

surface. Rather than generating a single focal point to which a particle is attracted, the

standing wave is comprised of several periodic intensity maxima that can collect several

particles at regular intervals of one-half the wavelength of the light, AL/2. Sufficient

reflection of the beam is achieved by coating a glass plate with several layers of

alternating refractive index materials. In some cases, the reflected intensity can reach

values near 99%. Although our technique does not encourage reflections, it is possible

that SWT characteristics are inherent into our technique. This may fully explain the

unexpected results.









There are two primary differences between the standing wave trap and our

technique. First, SWT uses a beam that is usually not focused as tightly as that found in a

gradient trap. In fact, the beam is more characteristic of a two-dimensional or radiation

pressure trap as described previously. Second, the glass surface in SWT is coated with

reflective layers to encourage the formation of a standing wave. While the techniques are

not precisely the same, we might infer some general similarities between the two since

the arrangements are nearly the same (focused trapping beam normally incident at a

glass-water interface). For this reason, we would expect to see some reflection of the

gradient trapping beam, especially as the focal point nears the plate surface, but the

reflected amount is typically expected to be about 0.4% for an unmodified glass-water

interface. In that case, we would not expect reflection to be an important consideration

for our technique, but there is compelling experimental evidence that suggests otherwise.

4.3.2 Experimental Evidence of a Standing Wave

Examining the data for which there are obvious artifacts ("bumps" or

discontinuities seen in plots of the mean intensity versus trap position-as shown in

Figure 4-6), we see that these features appear at periodic intervals of about 400 nm. This

corresponds well with the expected interval of A/2 for a standing wave trap. Further

evidence of this effect is provided by the individual position histograms where these

intensity discontinuities occur. The appearance of two distinct potential energy minima

is shown in Figure 4-7 for three slightly different locations of the trap center (offset by

-20 nm).














II II II II I


o



-.




Relative Separation Distance (nm)


Figure 4-7. Position histograms for three slightly different locations (separated by 20
nm) of the trap center in a standing wave arrangement (1.0 pm silica).





For critical locations of the trap center, it is possible to create two local energy minima

(corresponding to intensity maxima) of equal depth such that the particle will spend an

equal amount of time in each. This result is possible if we consider an unusually deep

secondary energy minimum, but this is expected from DLVO theory for the particle

systems considered here. In addition, the effect occurs at regular intervals, lessening as

the separation distance increases, which DLVO theory could never predict. The

experimental evidence seems to support the generation of a standing wave as the cause of

the unusual data.

To definitively prove the importance of reflections at the interface, the trajectory

of a trapped particle was observed as the trap focus was stepped toward slide surfaces

having reflective dielectric coatings. If experiments using uncoated glass slides show

evidence of a standing wave, then a more reflective slide should show an exaggerated









effect. Several experiments were run to deduce the effect of a reflective surface.

Zemanek and co-workers supplied us with reflective glass slides using multiple layers of

dielectric materials (SiO2 and TiO2). We were provided two sets of coated slides having

1% and 25% reflectivity values (R). Experiments were performed using the standard

measurement procedures detailed previously. The trap center was moved in increments

of 40 nm toward the slide surface. A plot of the mean intensity data using these reflective

slides is shown in Figure 4-8.


0 1000 2000 3000 4000 5000 6000 7000
Trap Position (nm)


Figure 4-8. Mean intensity as a function of trap position for a 1.5 pm silica particle near
a reflective glass slide (R=25%).





The data show obvious discrete jumps of the particle from one stable trapping position to

the next as the trap focus is moved toward the slide. Notice also that as the focal point







76


moves beyond the interface, the particle actually moves away from the surface to a

location near its initial position. This is because there is sufficient reflection of the beam

to create a focus that moves away form the surface. The basic features of this data

demonstrate the clear existence of a standing wave since the particle essentially can only

find discrete positions with respect to the surface. Notice that this plot is an exaggeration

of the trends seen in Figure 4-6. The spacing of these discrete movements is found by

converting the intensity data in Figure 4-8 to position data, shown in Figure 4-9.


1800

1600 -

1400

1200 -

1000

C 800 -

2 600

400 -

200


0 1000 2000 3000 4000 5000 6000 7000
Trap Position (nm)


Figure 4-9. Mean relative separation distance as a function of trap position for a 1.5 pm
silica particle near a reflective glass slide (R=25%).






The three stable locations closest to the surface appear to be spaced nearly equally (-300


nm); this interval is consistent with the distance of A/2 seen in SWT (recall that the







77


wavelength of the trapping light in water is roughly 624 nm, since the frequency is

unchanged as it propagates through a dielectric material). In this arrangement, rather

than follow a single beam focus that moves in 40 nm increments, the particle is finding

the antinode with the highest intensity. Since the locations of the antinodes remain fixed

regardless of the location of the theoretical focal point, the particle is restricted to make

discrete movements from one antinode to the next as the intensity distribution shifts.

The variance of separation distance also shows interesting behavior as the particle

makes its discrete movements. This data is shown in Figure 4-10.


0 1000 2000 3000 4000
Trap Position (nm)


6000 7000


Figure 4-10. Variance of separation distance as a function of trap position for a 1.5 pm
silica particle near a reflective glass slide (R=25%).


0 0
o

0 0
O
d% o

o o

0%
00


O Oi

O i^^'^MlWBi







78


The variance of particle fluctuations decreases markedly as the particle approaches the

surface. This indicates that the effective energy wells trapping the particles become

increasingly sharp at shorter separations. This is expected since the interference of

trapping light would be maximized when the focal point is very near the interface.

Because the trap stiffness is inversely proportional to the position variance, the data are

re-expressed as stiffness values in Figure 4-11.




x 10

? o
0


6 O
z
0

4-
ar o








0 1000 2000 3000 4000 5000 6000 7000

Trap Position (nm)


Figure 4-11. Axial trap stiffness as a function of trap position for a 1.5 pm silica particle
near a reflective glass slide (R=25%).






The trap stiffness gives a measure of the steepness of the potential well that holds the

particle at any given antinode. This plot reiterates that the steepest well is located near

the surface. The variability in the data is due to the sensitivity of the calculation upon the









measured variance since the stiffness grows quickly for small values of the variance. The

magnitude of the trap stiffness seen in these measurements far exceeds the typical values

observed for a usual force measurement experiment. Typical force measurements with

micron-sized silica using the LaserTweezers 100 optical trap have consistently shown

stiffness values ranging from 1 10 6 to 3 x 10 6 N/m (1 to 3 fN/nm). Figure 4-11 shows

a maximum stiffness value that is at least an order of magnitude larger than what is

typical for our technique. This maximum value may be elevated due to the presence of

the particle-surface interactions, but even locations far from the plate show large stiffness

values. These higher stiffness values are a novel aspect of SWT especially considering

that a strong axial force can be created without requiring a high numerical aperture lens

(i.e., a tight beam focus) or a large increase in laser power.

Similar results were obtained for studies using coated slides where R=l%, where

the standing wave effects were less pronounced. The effects were greater than those seen

for an uncoated glass slide, however. It is surprising that such a low reflectivity (R<1%)

slide could give rise to these effects, but clearly the technique is highly sensitive to any

such reflections. In fact, we ran additional experiments using an "anti-reflective" slide

(R<0.15%) and saw no major improvements over measurements made with uncoated

slides. The reason for this is likely that the reflectivity of the trapping beam is a strong

function of incident angle. The predicted reflectivity values for these coatings assume at

most a 30 deviation from normal incidence. The microscope objective lens (1.3 NA)

used in these experiments, however, has a maximum convergent angle that is far in

excess of this limit, so it may be that the actual reflectivity is much higher than the

predicted value.









4.3.3 Modeling the Trapping Beam Profile Near a Reflective Surface

It is possible to predict standing wave behavior theoretically by using beam optics

to describe the intensity profile near the glass-water interface. This is the subject of a

publication by Zemanek et al. [39] where the force-distance profile of a Rayleigh sphere

(a << A ) is predicted using a paraxial (PA) approximation of the light rays. In the case

of a tightly focused beam, however, the PA assumption becomes quite poor. For highly

convergent rays, higher-order corrections are required to accurately describe the EM

fields of a Gaussian beam near the focal point. To our knowledge, no previous work

exists that provides a theoretical description of a standing wave using a tightly focused

beam (i.e., single-beam gradient trap).














CHAPTER 5
STATIC AND DYNAMIC FORCE MEASUREMENTS USING A REVISED
METHODOLOGY


5.1 Description of the Revised Methodology

Chapter 4 discussed the problems associated with reflection of the trapping beam

as its focus approaches the flat surface. Significant reflection of the beam can generate a

standing wave that compromises the accuracy of the technique. Rather than trapping the

particle into a single beam focal point, the actual electromagnetic field distribution may

be far more complex and lead to unexpected movements of the particle as the trap focus

is moved toward the flat surface. Though this is a problem when the trapping beam is

centered below the objective, the standing wave effects can be largely eliminated if the

beam is positioned to enter the objective slightly off-axis. We can adjust beam entry

point by sliding the trapping unit along the guide rail. A beam that is concentric with the

objective aperture gives the greatest standing wave effects and is considered undesirable

in force measurements. However, if the trapping unit is moved a small distance (a few

millimeters) in either direction from this location, the beam axis is incident at the slide

surface at a slight angle. This reduces the amount of retro-reflected light that can

interfere with the incoming beam.

The ideal position of the beam to reduce standing wave effects appears to depend

upon several factors related to the sample. The particle type and the precise placement of

the glass slide upon the microscope stage may both influence this ideal position. For the

results shown in this chapter, we adopted a trial-and-error approach to minimizing the









reflections. This is done by trapping a particle, positioning the trapping unit at some off-

axis position, and then taking rapid samples of the intensity as a function of the trap focus

position. If there is a significant standing wave effect, it is manifests as discontinuities in

the statistics of the measured intensity. The trapping unit can then be repositioned and

the diagnostic sampling repeated until an optimum location is found. In general, this

optimum location will vary only slightly depending on the sample such that the trapping

unit can be left in place to achieve satisfactory results. With this additional parameter

optimized, the experiments proceed exactly as described in Chapter 3. As will be shown,

data generated using this revised methodology are far superior to earlier measurement

results where the trapping unit position was unaltered.


5.2 Static Force Measurements

5.2.1 Procedure Details

The measurements reported in this section precisely follow the experimental

procedure described previously in Chapter 3. The specifics of this procedure are noted

here. Two solutions of NaCl (0.23 and 1.0 mM) were prepared using Barnstead

Nanopure ultrafiltered water. Silica microspheres of 1.0 and 1.5 pm nominal diameter (+

0.1 pm) were purchased from Geltech as dry powers and suspended in the NaCl solutions

at low particle densities (-105 particles/mL) for a total of four different samples. A

similar density of 9.14 pm diameter polystyrene spacer particles (Polysciences) served as

gap spacers. Solutions were prepared the day of the experiment to eliminate

contamination of the samples. The samples were sonicated to break up aggregates and

mixed just prior to an experiment to resuspend the particles in solution. The He-Ne laser

beam was a set to an incident angle of 63.7. This gives an evanescent wave penetration











depth, 1, of 184.4 nm. The data acquisition program was set to acquire 65536


intensity samples at a sampling rate of 20 kHz for each trap position. The trap was


moved by 20 nm increments until the particle was found to achieve its minimum


accessible separation distance. The data was continuously saved to a file and analyzed at


a later time using the MATLAB analysis program.


5.2.2 Static Force Results


Figure 5-1 shows histograms of particle positions for a 1.5 pm silica sphere at


equilibrium separation distances of 56.2 and 543.7 nm, which correspond to regions


where the surface forces are appreciable and negligible, respectively.


6000





w 4000

0
0


2000


30





I--
20

a,
W

m
10
0-
a_


0 l . . IL. . . . .. / I.. J , 10
-400 -200 0 200 400 600 800

Separation Distance (nm)


Figure 5-1. Measured histograms of particle positions and estimated potential energy
profiles corresponding to most probable separation distances of 543.7 and 56.2 nm.









The dashed lines are the predicted potential energy profiles in the absence of surface

forces, equivalent to 0,ra (z). Note that it is possible to have a trap potential minimum

theoretically located within the solid plate. The static force is calculated directly from the

distance between the hypothetical trap minimum and the actual minimum. At z, = 543.7

nm, the histogram is centered over the trap potential profile indicating a negligible static

force. However, at z, = 56.2 nm, there is a large deviation between the trap potential

minimum and the most probable separation distance. Assuming that the trap potential is

harmonic, the static force is simply proportional to the observed distance deviation as in

equation (2.37).

Figures 5-2 and 5-3 show experimental force-distance profiles obtained for 1.0

and 1.5 pm silica spheres, respectively, interacting with a flat glass plate in 0.23 and 1.0

mM NaCl solutions.















3




Z

LL

0






-1
0 100 200 300 400
Separation Distance (nm)


Figure 5-2. Static force measurements for a 1.0 pm diameter silica sphere near a glass
plate in 0.23 mM (0) and 1.0 mM (A) NaC1. DLVO theory predictions are plotted for
both ionic strengths (0.23 mM solid line, 1.0 mM dashed line).

















S3

0 2
UL



oao ao ,a

-1
0 100 200 300 400
Separation Distance (nm)


Figure 5-3. Static force measurements for a 1.5 pm diameter silica sphere near a glass
plate in 0.23 mM (0) and 1.0 mM (A) NaC1. DLVO theory predictions are plotted for
both ionic strengths (0.23 mM solid line, 1.0 mM dashed line).






The absolute separation distances were obtained by fitting the viscous drag coefficient

data to equation (3.7), as discussed previously and shown below. From conductivity

measurements, we were able to confirm the ionic strengths and accurately estimate the

Debye lengths, 1, which represent the characteristic exponential decay of the double

layer forces as a function of separation distance (K 1 = 20.0 nm for 0.23 mM NaC1, and

KC = 9.6 nm for 1.0 mM NaC1). In each case we found very good agreement with

DLVO theory (shown as solid and dashed lines in the static force plots), accounting for

the combined effects of double layer and van der Waals interactions. The double layer

force model is based upon the Debye-Hickel approximation, which assumes low surface

potentials (absolute value of -25 mV or less) and slightly overlapping double layers









(principle of superposition). Derjaguin's approximation is used to account for the

curvature of the silica particle. The van der Waals model is based on Hamaker theory

and neglects retardation and screening effects. The van der Waals force should be small

enough in this range of separation distances such that a more accurate model is not

required.

5.2.3 Discussion of Static Force Results

For the systems tested in this study, the dominant static force is generated by an

overlap of the diffuse double layers where the repulsive force is sufficient to keep the

particle from finding regions very near the surface where van der Waals attractive forces

become appreciable. The double layer repulsive force, then, provides a convenient test of

the accuracy of our technique since the agreement with theory does not strictly depend on

the absolute separation distance. The DLVO theory profile was generated using literature

parameter values and an experimentally derived value for the particle radius. By

assuming shear plane potentials of-15 mV, the DLVO theory profile agreed well with

our static force measurements. This value of the potential provided good agreement for

initial experiments and was therefore used throughout as a fixed parameter. Although the

actual shear plane potentials were not be verified independently for the silica particles

and the glass slide directly, a shear plane potential of -15 mV is reasonable for Si02 at

neutral pH conditions.

The static force results demonstrate the ability of the optical trap to apply a linear

force to the particle for relatively large displacements from the trap center. Since the

intensity profile of the focused trapping beam decays more gradually in the axial

direction, the linear force regime can extend well beyond a particle radius. For the

experiments reported here, we observed a linear response up to about one particle









diameter for a 1.5 Jm diameter sphere. The linear force approximation was assumed to

be valid for regions where the static force profile agreed well with DLVO theory.

Measured forces clearly beyond the linear regime were omitted.

These revised measurements, which optimize the off-axis position of the trapping

beam, are greatly improved over the initial results shown in Chapter 4. The standing

wave effects are almost completely eliminated using this new methodology. However,

some slight effects remain no matter how carefully the off-axis position is chosen. Upon

closer examination, the static force data in Figure 4-2 and 4-3 show non-zero force

measurements in regions where the double layer and van der Waals forces should be

negligible. Consistently, there are small positive forces (-0.2 pN) that appear for

separations of about 300 nm. Also, there are small negative forces that appear just

beyond 100 nm. This leads to a slight wavy appearance of the overall force-distance

profile, although the errors are minimal. While this is likely due to slight interference

effects, it does not seem to disrupt the force data where significant forces (>0.5 pN) are

measurable. Overall, the measurements agree very well with DLVO theory predictions

which validates the accuracy of the technique for these particle systems. Because the

behavior of trapped particle within this complex electromagnetic field depends on the

particle size and material, the results may be better or worse. At present, it is difficult to

predict which particle systems will behave well within this trap although it appears that

higher refractive index particles (e.g., polystyrene) suffer greater effects due to the

standing wave. In fact, it was this early observation that led us to study silica particles

rather than equivalently sized polystyrene microspheres.




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