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 ChangWon
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 coworkers: Dr. Alina
Ruta, Dr. Jennifer Myles, Dr. Brian Burgess, Jonah Klein, Jamaica Prince, JongHoon
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 MicronSized 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 PoissonB oltzm ann E quation............................................... ... ................. 12
2.1.2 DebyeHiickel 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 SubMicroscopic 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
21. Schematic of the Stem model for the electrical double layer................. ........... 16
22. DLVO interaction energy between a sphere and a flat plate. ......................................24
23. Drag force correction factor for a sphere moving normal to a solid surface.................30
31. Schematic of the experimental apparatus. ........................................ ............... 41
41. Position histograms of a particle trapped (a) far from and (b) near the plate surface. ..60
42. Experimental forcedistance 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
43. 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
44. Experimental mean intensity versus trap position for a 1.5 Jpm silica sphere ..............66
45. Schematic of LaserTweezers unit placed within the microscope...............................68
46. Mean intensity versus trap position for two locations of the trapping unit. ..................71
47. 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
48. Mean intensity as a function of trap position for a 1.5 pm silica particle near a
reflective glass slide (R=25% )............................ ................................... 75
49. 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
410. 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
411. Axial trap stiffness as a function of trap position for a 1.5 Jrm silica particle near a
reflective glass slide (R=25% )............................ ................................. 78
51. Measured histograms of particle positions and estimated potential energy profiles
corresponding to most probable separation distances of 543.7 and 56.2 nm.......83
52. 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
53. 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
54. 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 51. ............................................... .. ......... 89
55. 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
56. 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
57. Dimensionless departure of the experimental drag coefficient from hydrodynamic
theory ............................................................................92
61. Simulated experimental drag (0) versus Brenner's correction to Stokes' law.............96
62. Model prediction (0) and experimental data (0) of the mean intensity profile...........01
63. Model prediction (0) and experimental data (0) of the intensity variance profile......102
71. Schematic for a custom optical trapping system. .................................. .................106
A1. LabVIEW wire diagram for the data acquisition program. .................... ................112
A2. 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 particleparticle
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 singlebeam
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 nanometersized
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 DerjaguinLandau
VerweyOverbeek 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 forcesthe interactions that arise between exposed material surfaces in
solutiondirectly 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
nonconservative (in the case of drag forces) depending on the origin of the interaction.
In the last halfcentury, 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 systemspecific
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 crossedcylinder 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
coworkers [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 1010 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 coworkers [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
particlesurface 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 (twodimensional) 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 1021 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 MicronSized 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 micronsized
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 micronsized 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 noninvasive 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 StokesEinstein 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 coworkers [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 twodimensional optical trap was effective at constraining
transverse fluctuations, the axially directed radiation pressure force was less useful
because it was a nonlinear 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 twofold: it is
essentially noninvasive to the sample and is able to impart smallscale forces to the
trapped particle ranging from 1012 N to 1015 N. Unfortunately, the twodimensional
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
coworkers [12] introduced a singlebeam gradient optical trap that was able to precisely
control a particle in threedimensions. 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 threedimensional forces acting on the
particle, including those imparted in the axial direction. The intensity profile created by
a tightly focused beam imposes a threedimensional 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 micronsized particle and a flat
surface, we have developed a new technique [13] that combines a singlebeam 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 forcedistance profile. The trapping force can move
the particle to highenergy 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. Nonconservative
(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
(coions) 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 PoissonBoltzmann Equation
The variation of potential from a surface is a wellknown 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 PoissonBoltzmann (PB)
equation:
d2W e zew
d2L Z, n,. exp ze (2.3)
dx 2 kT
where x is the spatial dimension in the above onedimensional 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 nonlinear; 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 DebyeHickel Approximation
It is possible to solve the nonlinear PoissonBoltzmann (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 DebyeHickel 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 DebyeHuckel 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 K1
2.1.3 GouyChapman Theory
The usefulness of the DebyeHickel approximation is hampered by many
simplifying assumptions, so we turn to a more generalized solution of the PB equation,
known as GouyChapman theory, to describe the potential profile for any value of the
surface potential. The derivation of the final GouyChapman 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 GouyChapman expression is an analytical solution, but it requires a
symmetric electrolyte. Similar to the DebyeHuckel 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 DebyeHiickel
approximation is a subset of the more general GouyChapman 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 GouyChapman model). Figure 21 illustrates the Stern model of the electrical
double layer.
SURFACE STERN
CHARGE  PLANE
0 0
SOLID
DIFFUSE
LAYER
S SHEAR
SJ PLANE
I
DISTANCE
Figure 21. 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 xdirection) 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 LI2I 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 potentialdetermining 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
Molecularlevel 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 hydrogenlike 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 nonzero 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 sphereplate 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 (A22 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 timedependent 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 22.
25
20
15
LJ
10
0
t 5
O
5
C 0
0 50 100 150
Separation Distance (nm)
Figure 22. 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 22 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 1021 J (silicawatersilica), 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
22. 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 sphereplate 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 meansquare 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 StokesEinstein 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 23.
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 23. 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 (particlesurface 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 forcedistance 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 micronsized sphere in solution is a bit more complicated, howevera
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 particlesurface separation distance, and I0 is the scattered intensity at
zero separation (particlesurface 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 timeseries 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 wellknown.
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 twodimensional 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 counterpropagating
traps to create a stable trapping location between them.
By 1986, Ashkin and coworkers [12] had developed a singlebeam gradient trap
that could confine a particle in threedimensions. In this version of the optical trap, a
high numerical aperture lens is used to focus the beam down to a diffractionlimited 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 singlebeam 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 singlebeam 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 threedimensions 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 generalizedLorentzMie 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 higherorder
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 threedimensional 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 halfwidth 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 forcedistance
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 (zdirection) 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 forcedistance 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 (onedimensional)
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
lefthand side (inertia term) of the Langevin equation. The exact motion of the particle is
random, and the timeseries 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 31.
............
slide ""
particle ev. wave
trapping 
rays
cover slip
dichroic
mirror
prism
Figure 31. Schematic of the experimental apparatus.
HeNe 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 diffractionlimited spot within the fluid
cell. The particle suspension is placed upon a precleaned 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 threeaxis 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 HeNe laser
beam (Ao = 632.8 nm, linearly polarized) into the 450 hypotenuse face of a BK7 glass (n
= 1.515 at 632.8 nm) dove prism, which is optically coupled to the glass slide with index
matching oil. The HeNe 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 sideon 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 preamplifier where it is electronically averaged by a 150 pis time
constant RC circuit (lowpass filter) and then digitally sampled by a National Instruments
12bit 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 MQcm 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 precleaned 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 HeNe 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 HeNe 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 HeNe 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 particlesolution system in order to assess the quality and
reproducibility of the data. The glass slides were discarded following an experiment, and
each new particlesolution 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 preamplifier 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 backgroundcorrected 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 a2 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 forcedistance 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 nonconservative 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 noslip 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 FokkerPlanck 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, o2, 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 lowpass 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
higherorder 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 socalled "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, 02, 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 nonlinear leastsquares
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 DLVOtype 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 particlesurface 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 fourthorder 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 forcedistance 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 nonlinear leastsquares 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 micronsized 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 glasswater 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 micronsize 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
ultrapurified 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 forcedistance
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
sodalime 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 41 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 41. 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 41 represent the collected data for a given location of
the trap center. To calculate a complete forcedistance 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 forcedistance profiles are shown in Figure 42.
4
3
z
o2
0 1
1
0 100 200 300
Separation Distance (nm)
Figure 42. Experimental forcedistance 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 42 represent measured data points, and the lines are a DLVO
theory fit generated from a HoggHealyFuerstenau (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 silicawaterglass 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 42 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 studiesmany using SFAhave
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 longchain
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 micronsized particle (recall that van der
Waals forces scale with particle radius). Although our forcedistance profiles in Figure
42 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 42), 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 particlesurface system. Using the overall potential
energy for the particle as the sum of DLVOtype 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 43.
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 43. 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 43 corresponds to the force curve shown Figure 42 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 forcedistance profiles (the ultimate endresult 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 44.
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 44. Experimental mean intensity versus trap position for a 1.5 pm silica sphere.
In comparison to the plot in Figure 43, the data shown in Figure 44 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 enduser 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 glasswater 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 45.
/
Figure 45. Schematic of LaserTweezers unit placed within the microscope.
The trapping laser is contained within a small plugin 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 dovetailshaped rail that restricts
movements along one axis (as indicated by the arrows in Figure 45). 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 nearinfrared (IR) spectrum, the beam
can be imaged using an IRsensitive 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 micronscale movements along one axis. The details of
this method are somewhat difficult to describe and require handson 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 socalled "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 44). 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 46.
71
0
.. .
Relative Axial Trap Position (nm)
Figure 46. 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 offaxis, (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
wellformed trap, lead to inaccurate measurements? This is the subject of the next
section.
While it is not yet described why an offaxis 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 coworkers [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 micronsized 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 onehalf 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 twodimensional 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
glasswater 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 glasswater
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 positionas shown in
Figure 46), 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 47 for three slightly different locations of the trap center (offset by
20 nm).
II II II II I
o
.
Relative Separation Distance (nm)
Figure 47. 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 coworkers 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 48.
0 1000 2000 3000 4000 5000 6000 7000
Trap Position (nm)
Figure 48. 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 46. The spacing of these discrete movements is found by
converting the intensity data in Figure 48 to position data, shown in Figure 49.
1800
1600 
1400
1200 
1000
C 800 
2 600
400 
200
0 1000 2000 3000 4000 5000 6000 7000
Trap Position (nm)
Figure 49. 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 410.
0 1000 2000 3000 4000
Trap Position (nm)
6000 7000
Figure 410. 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
reexpressed as stiffness values in Figure 411.
x 10
? o
0
6 O
z
0
4
ar o
0 1000 2000 3000 4000 5000 6000 7000
Trap Position (nm)
Figure 411. 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
micronsized 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 411 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 particlesurface 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 "antireflective" 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 glasswater interface. This is the subject of a
publication by Zemanek et al. [39] where the forcedistance 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, higherorder 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., singlebeam 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 offaxis. 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 retroreflected 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 trialanderror 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 HeNe 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 51 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 51. 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 52 and 53 show experimental forcedistance 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 52. 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 53. 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 DebyeHickel 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 of15 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 offaxis 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 offaxis position is chosen. Upon
closer examination, the static force data in Figure 42 and 43 show nonzero 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 forcedistance
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.
