Characterization of microparticles with driven optical tweezers.

Abstract

We discuss how actively-driven optical tweezers may be used to characterize Brownian microparticles. Two experiments are described in detail. We follow the thermal fluctuations of a charged particle in an oscillatory electric field and demonstrate that charges as low as a few elementary charges can be measured accurately and reproducibly. Secondly, we measure the orientational dynamics of a trapped rotating droplet and use circular polarimetry within optical tweezers to determine in situ birefringence.

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Characterization of microparticles with driven
optical tweezersw
Tiffany A. Wood, G. Seth Roberts, Sarayoot Eaimkhong and
Paul Bartlett*
Received 16th March 2007, Accepted 3rd April 2007
First published as an Advance Article on the web 16th July 2007
DOI: 10.1039/b703994h
We discuss how actively-driven optical tweezers may be used to characterize
Brownian microparticles. Two experiments are described in detail. We
follow the thermal fluctuations of a charged particle in an oscillatory
electric field and demonstrate that charges as low as a few elementary
charges can be measured accurately and reproducibly. Secondly, we
measure the orientational dynamics of a trapped rotating droplet and use
circular polarimetry within optical tweezers to determine in situ
birefringence.
1.Introduction
One of the distinctive features of many micro-particulate systems is their diversity—
no two particles have precisely the same size, charge or optical properties. Recent
advances in soft matter that allow the interactions between individual pairs of
microparticles to be studied have spawned an increased interest in characterizing the
optical and electrical properties of single microparticles. Single particle detection and
imaging offers the promise of new insights into many fundamental physical processes
and a better understanding of the functionality of soft matter systems. This
information is frequently obscured by ensemble measurements that provide only
average material properties.
In this paper, we detail a general approach that utilizes the change in the random
walk of a submicron particle following the application of a periodic perturbation to
probe the optical and electrical properties of a single microparticle. The way in which
the random Brownian motion of a small sphere immersed in a fluid is modified by an
external force lies at the core of many diverse problems in physics, chemistry and
biology. Examples include the operation of molecular motors,1the phenomenon of
stochastic resonance2and the driven transport of colloids through a potential
landscape.3We employ a focused laser beam to confine a microparticle in an optical
trap and then track the Brownian fluctuations of the trapped particles with
nanometer resolution.4In the harmonic potential well, generated by the laser beam,
the microparticle oscillates around its equilibrium position as a consequence of the
random thermal motion of the solvent molecules. By measuring the subsequent
changes in these fluctuations produced by external periodic forces, the optical and
electrical properties of single microparticles are derived. We analyze the effects of
School of Chemistry, University of Bristol, Bristol, UK BS8 1TS. E-mail:
p.bartlett@bristol.ac.uk
{ The HTML version of this article has been enhanced with colour images.
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two classes of external perturbations: an oscillatory linear force, produced by
applying an alternating electric field to a charged particle, and a constant optical
torque, generated by illuminating a spherical birefringent microparticle with circu-
larly-polarized light. The absolute value of the charge and the birefringence of a
single microparticle are found from the autocorrelation of the particle’s coordinates.
We present a detailed analysis of the Brownian fluctuations of the trapped particle in
each of these two situations and show how our measurements provide new
quantitative tools for characterization of the optical and electrical properties of
micrometric systems.
The paper isorganised as follows:Section 2 describes the basis of the optical tweezers
technique, in Section 3 we place a trapped particle in an oscillatory electric field and use
the response to determine the charge on the particle, and in Section 4 we apply a
constant angular torque to a trapped particle and record its orientational dynamics.
2. Optical tweezers
Optical tweezers are formed by tightly focusing a laser beam with a high numerical
aperture objective lens.5The operation of the trap is based on the fact that a
dielectric material subject to an external electric field E is polarized and generates an
induced dipole moment per unit volume, P, given by P = wE, where w is the electric
susceptibility.6For a uniform dielectric sphere of radius a, w is isotropic and the
polarization P is parallel to the electric intensity E(r, t) and equal toz,
?
where nmis the index of refraction of the medium and m is the ratio of the index of
refraction of the particle to the medium (np/nm). There is no net force on the dipole in
the field E unless the field is non-uniform such that one end of the dipole is subjected to
a greater force than the other. In the non-uniform electric field generated at the focus of a
laser beam there is an instantaneous gradient force per unit volume on the dielectric of,
Pðr;tÞ ¼ 3n2
me0
m2? 1
m2þ 2
?
Eðr;tÞ;
ð1Þ
Fgðr;tÞ ¼ ½Pðr;tÞ ? r?Eðr;tÞ ¼ 3n2
me0
m2? 1
m2þ 2
??1
2rE2ðr;tÞ:
ð2Þ
The time-averaged gradient force that the particle experiences
fg¼
Z
dr ½Pðr;tÞ ? r?Eðr;tÞ
hi;
ð3Þ
is proportional to the intensity gradient and is directed towards the intensity maximum
for m 4 1. As a result, a highly-focused light beam will tend to confine a high refractive
index microparticle within its focal volume. The strength of the isotropic optical forces
is controlled by the spatial profile of the beam and the size and refractive index of the
particle.7
Angular trapping occurs in optical tweezers with spherical particles made from
materials such as nematic Liquid Crystals (LCs) that are birefringent. In this case,
the susceptibility is no longer isotropic and the expression for the polarization P
must be generalized to
P = wiEiıˆ + wjEj^ ^ ; ; + wkEkkˆ(4)
where ıˆ,^ ^ ; ; and kˆare unit vectors along the principal axes of the liquid crystal and wi, wj
and wkare the corresponding susceptibilities. A uniaxial birefringent material, such
{ Here, by way of illustration, we have assumed that the trapped sphere is much smaller than
the wavelength of the trapping laser, i.e. a { l. Particle shape effects are neglected. More
complete theories are required when the dimension of the trapped particle is comparable to the
wavelength of the trapping laser.
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as a nematic LC, can be described by two susceptibilities: an ordinary susceptibility,
wo, for electric fields normal to the optic axis, and an extraordinary susceptibility, we,
for fields parallel to the optic axis. Choosing the unit vector ıˆ parallel to the optic axis
gives, wi= weand wj= wk= wo. Since in general a birefringent nematic LC is more
polarizable along the director (the extraordinary axis) than the ordinary axes, the
electric polarization vector P is not parallel to the external electric field E but is tilted
towards the extraordinary axis. The misalignment of P and E means that in a
uniform electric field, a spherical birefringent particle experiences a torque
Z
that tends to align the ordinary axes of the nematic LC drop with the electric field
direction. The polarization state of the trapping beam controls the time-dependence
of the resulting torque imparted to the trapped particle. In linearly-polarized light,
the particle is angularly trapped and aligned along a particular orientation, and a
particular particle is always aligned along the same direction whenever trapped. By
contrast, in circularly-polarized light, where the plane of polarization rotates
continually, a birefringent particle experiences a constant torque and, if free, rotates
with a fixed frequency and angular speed.
t ¼
drPðr;tÞ ? Eðr;tÞð5Þ
2.1 Optical configuration
Fig. 1 shows a schematic of our optical tweezers system. A linearly-polarized TEM00
mode of a Ytterbium fibre-coupled laser of wavelength l = 1064 nm
(IPG Photonics, Germany) was focussed to a diffraction-limited beam spot with a
diameter of approximately of 0.9 mm by an oil-immersion microscope objective
(Plan-Neofluar, ?100, N.A. 1.3, Zeiss) mounted in an inverted microscope (Axiovert
S100, Zeiss). The intensity of the laser beam was varied using a combination of a l/2
waveplate and a polarizing beam-splitter cube placed in the beam path. For the
optical torque experiments an additional l/4 plate was placed immediately before the
objective to convert the linearly polarized light of the laser into the circularly-
polarized beam required to rotate the birefringent particles. The transmitted laser
beam and light scattered by the trapped microparticle was collected by a high
numerical aperture oil-immersion objective and projected onto a polarizing beam
splitting cube, before being sent to two orthogonal quadrant photodetectors (model
QD50-4X, Centronics, UK).
Fig. 1
were added for measurements under applied optical torque.
Schematic representation of the optical tweezer apparatus. Components shown dashed
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The signals from the two orthogonal detectors were analyzed in different
ways depending on the optical experiment. In the linear force experiments
detailed in Section 3, the in-plane position of the trapped particle was tracked with
the resolution of a few nanometers by back-plane optical interferometry.8
The quadrant photodetectors were aligned so that they recorded the intensity profile
of the pattern generated by the interference of scattered and transmitted light in a
plane conjugate to the focal plane. Appropriate combinations of the four
quadrant signals give a voltage proportional to the instantaneous in-plane x and y
displacement of the fluctuating particle, relative to the axis of the laser
beam. Positions measured in voltages were converted into displacements in
nanometers by recording the time-dependent mean-square voltage hDV2(t)i
of five particles from the same batch of particles, immediately before each set of
measurements. Assuming the voltage recorded is proportional to the displacement,
hDV2(t)i was fitted to the theoretical expression for the mean-squared displacement
hDx2(t)i of a Brownian sphere in a harmonic potential, to yield the detector
calibration and the optical trap stiffness k. In the optical torque experiments detailed
in Sec. 4, the voltages from the four sections of each detector were summed together.
Each of the two orthogonal detectors were separately calibrated so that they
provided the power of each polarization component, in units of the power at the
trap focus. The transmitted powers recorded by the x and y polarization detectors
(Pxand Py) were combined (see Section 4.1) to analyze the polarization change of
the laser beam after it had passed through the nematic LC drop. Data was acquired
by a Labview programme (National Instruments, Austin, Texas, USA) and
digitized using a high-speed data acquisition card (National Instruments model
PCI-MI0-16E-1).
3.Measurement of the charge on a single microparticle
Many key properties of colloidal suspensions are directly or indirectly controlled
by the electrical charge carried by their constituent particles. Consequently,
methods to measure the electrophoretic mobilities of colloidal particles have received
a lot of attention during the last couple of decades. Most techniques available
today, however, rely on ensemble averages and are based on monitoring the
linear motion of particles in an electric field. Laser Doppler Velocimetry (LDV) is
widely used9to investigate the light scattered by particles suspended in an
electric field and to infer their charge, but is limited to large samples of monodisperse
materials. Phase Analysis Light Scattering (PALS) is commonly used to
measure very low particle charges in either non-polar suspensions10or near the
isoelectricpoint but suffersfromsimilar
These methods, although fast and accurate, provide little or no information on
the details of the charge distribution. In many cases, however, knowledge of the
charge distribution is of vital importance. Colloidal suspensions have, for example,
proved to be valuable model systems in condensed matter physics. There is strong
evidence that the width of the charge distribution (the charge polydispersity)
significantly influences the glass transition and at high levels may suppress the
freezing transition in these systems.11It is therefore important that techniques are
available that provide an accurate characterization of the charge distribution of
microparticles.
Here, we report the development of an ultra-sensitive technique for the measure-
ment of the charge carried by a single colloidal particle in a non-polar suspension.
Our approach is to use the well-known phenomenon of the resonance of a harmonic
oscillator driven by a oscillatory linear force. A linearly-polarized laser beam is used
to trap a single microparticle. The laser and trapped particle constitute a harmonic
oscillator with a stiffness that is proportional to the laser power. The modulation of
the Brownian motion generated by an alternating electric field is measured with
nanometer sensitivity using an interferometric position detector and the charge is
practicallimitationsto LDV.
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obtained directly from the power spectrum of the thermal fluctuations. Charges as
low as a few elementary charges can be measured with an uncertainty of about
0.25 e.12This is significantly better than existing techniques and opens up new
possibilities for the study of non-polar suspensions.
3.1Experimental details
We use a model colloidal system of sterically-stabilized poly(methyl methacrylate)
spheres [PMMA] of radius a = 610 ? 30 nm.13Electron microscopy revealed
the particles were highly uniform in size with a radius polydispersity (root mean
square variation/mean radius) of 0.046. Single PMMA spheres were suspended in
dry dodecane and trapped in three dimensions using a linearly-polarized laser
beam in a purpose-built microelectrophoresis cell. The cell contained two flat
platinum foil electrodes separated by 128 mm ? 2 mm in a cylindrical glass
sample chamber. Voltages of up to 10 V were applied to the electrodes. The electric
field was estimated from the expression E = lV/d, where V is the applied
voltage, d is the plate separation and, l is a factor correcting for the finite size of
the electrodes. Finite element simulation of the electric field gave l = 0.93.
The confined microparticle was held about 70 mm above the base of the
cell and 218data points were collected at 10 kHz. The Brownian motion of the
particle was followed by back-plane optical interferometry using data collected
from just one quadrant photodetector. The duration of each measurement was
E26 s.
3.2Brownian motion of an oscillating microparticle
The total force on a charged microparticle in an optical trap has four elements:
a harmonic force, ?kx(t), arising from the optical trap, where k is the force constant
of the potential well; a viscous drag force ?xt_ x(t), where from Stokes law xt=
6pZa for a sphere of radius a; an oscillatory driving force fD(t) = A sin Ot defined by
the frequency O and amplitude A of the applied electric field; and Brownian forces
fB(t) characterizing the random fluctuating forces due to collisions with molecules of
the solvent. The Langevin force equation for the Brownian motion of a sphere of
mass m is,14
mx ¨ = ?kx ? xt_ x + A sin Ot + fB(t)(6)
where fB(t) is a white-noise Gaussian random process with time-averaged
correlations hfB(t) fB(t0)i = 2xtkBTd(t ? t0). To simplify the problem, we neglect
all inertial terms that are negligible at frequencies o { 106rad s?1. Defining the
amplitude of the oscillatory linear force as A = ZeE, where Z is the charge
on the microparticle and E is the amplitude of the electric field, then eqn (6)
reduces to
xt_ x + kx ? ZeE sin Ot = fB(t). (7)
Since the Brownian and periodic forces are uncorrelated, the general solution of
eqn (7) is the superposition,
x(t) = xB(t) + xD(t),(8)
where xBis the solution in the presence of random Brownian forces only (i.e. A = 0)
and xDis the solution where only driven forces act (i.e. T = 0). The factorization
evident in eqn (8) is universal, so, for instance, the spectral density SD(o) of the
particle’s positional fluctuations is simply a combination of the spectral densities for
purely Brownian and driven motion,
SDðoÞ ¼kBT
pxt
1
o2þ O2þ
kBTg2
2k
½dðo ? OÞ þ dðo þ OÞ?:
ð9Þ
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