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Interactions, Structure, and Microscopic Response: Complex FluidRheology Using Laser TweezersEric M. Fursta
a Department of Chemical Engineering, University of Delaware, Newark, DE, USA
Online publication date: 06 November 2003
To cite this Article Furst, Eric M.(2003) 'Interactions, Structure, and Microscopic Response: Complex Fluid Rheology UsingLaser Tweezers', Soft Materials, 1: 2, 167 — 185To link to this Article: DOI: 10.1081/SMTS-120022462URL: http://dx.doi.org/10.1081/SMTS-120022462
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SOFT MATERIALS
Vol. 1, No. 2, pp. 167–185, 2003
TECHNICAL TUTORIAL
Interactions, Structure, and Microscopic Response:Complex Fluid Rheology Using Laser Tweezers
Eric M. Furst*
Department of Chemical Engineering, University of Delaware,
Newark, Delaware, USA
ABSTRACT
Optical trapping techniques are emerging as significant research tools in complex
fluids, offering the ability to probe nano- and microscopic interactions, structures, and
responses that govern the rheology of complex fluids. In combination with real-space
imaging, microstructural response of these fluids can be directly and quantitatively
correlated to imposed microscopic stresses and strains. Thus, laser tweezers are
enabling us to bridge multiple length scales in colloid and polymer rheology and
should be highly useful for investigating the mechanisms of linear and nonlinear
rheology. In this article, we briefly review the theory and practice of using optical
traps in complex fluids. We discuss the characteristics of the gradient force trap,
practical concerns in trapping experiments, and applications, including measurements
of micromechanics and microrheology in colloid and polymer gels.
Over 30 years ago, Arthur Ashkin demonstrated that radiation pressure could be
used to manipulate individual colloidal particles and cells, levitating them against
gravity, or trapping them between counterpropagating beams.[1 – 3] Shortly thereafter,
the first single-beam optical gradient force trap, or ‘‘laser tweezer,’’ was developed,
enabling precise, three-dimensional control of particles.[4,5] Optical trapping has since
*Correspondence: Eric M. Furst, Department of Chemical Engineering, University of Delaware,
Newark, DE 19716, USA; E-mail: [email protected].
167
DOI: 10.1081/SMTS-120022462 1539-445X (Print); 1539-4468 (Online)
Copyright D 2003 by Marcel Dekker, Inc. www.dekker.com
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become a highly effective research tool, particularly in biophysics, by providing a
means by which to manipulate cells, organelles, and particles at submicron precision,
and to quantify forces from tenths to hundreds of piconewtons. Optical tweezers have
played a central role in advancing the knowledge of motor protein mechanochem-
istry,[6 – 8] the forces of transcription,[9] and cytoskeletal-membrane interactions.[10] For
instance, laser tweezers have been used to measure the stall forces for single molecular
motor proteins, such as myosin and kinesin, as they walk along cytoskeletal micro-
filaments and tubules, in addition to measuring their nanometer step sizes, and the
processive movements that drive intracellular transport.[7,11]
Mirroring the development in biophysics, optical trapping techniques are emerging
as significant research tools in complex fluids, offering the ability to probe nano- and
microscale interactions, structures, and responses that govern the rheology of soft
materials. The direct in situ manipulation and force measurements accomplished with
optical tweezers are especially powerful in combination with concurrent real-space
imaging, such as video, fluorescence, and confocal microscopies. For instance, the
pioneering work of Chu and coworkers introduced single-polymer visualization and
manipulation to understand polymer dynamics in dilute and entangled solutions of
DNA.[12 – 14] Previous to these works, the physics that underlie many aspects of bulk
rheology were only accessible to simulation techniques or scattering experiments.
By providing a means of nanoscale force sensing, microscale directed assembly,
mechanical measurements, and microrheology, laser tweezers will enable us to bridge
multiple length scales to directly establish structure–response relationships in colloids
and polymers.
In this article, we briefly review the theory and practice of optical trapping to
measure microstructure and response in complex fluids. First, we discuss the
underlying theory and characteristics of the gradient force trap. Next, we discuss
practical experimental concerns, including common methods for creating and
controlling traps. Finally, we discuss several applications, including the use of
optical tweezers to measure microstructural response and interactions in colloidal and
polymeric materials.
THEORY OF THE GRADIENT OPTICAL FORCE TRAP
To generate an optical trap, a single laser beam is focused to a diffraction-
limited spot. Two regimes are convenient for describing the physical principles of
gradient optical trapping. In the Rayleigh regime, dielectric particles of a diameter
much less than the optical wavelength, d� l, minimize the energy density stored in
the electric field when they are at the center of the focus.[15] Thus, the particle
experiences a Lorentzian force from time-averaged electric field intensity pulling it
into the light gradient:
Fgrad ¼ 1
2ar E2
� �ð1Þ
where a is the polarizability of the neutral particle.[16] The particle also experiences
a scattering force Fscatter proportional to the rate of scattering momentum and
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absorption.[17] To successfully hold a particle in the propagation direction, the
gradient force must exceed the scattering force, Fgrad�Fscatter, otherwise, the particle
is pushed along the light propagation axis and out of the trap. The scattering force
increases with the particle refractive index contrast relative to the surrounding
medium as well as absorption.
In the ray-optic regime, d�l, a gradient trap can be depicted as individual rays
refracting through the particle, as shown in Figure 1. The change in momentum of a
photon refracted through the particle imparts a reactive force. For example, in Figure 1,
the imparted momentum pushes the particle toward the focal point. By summing the
momentum change from all refracted rays, a force profile of the trap can be found.[18]
The equilibrium particle position is offset from the beam focus in the direction of
propagation due to scattering and absorption. When the size of the particle is on the
order of the wavelength, resonant modes between scattering volumes complicate the
quantitative description.[19]
Optical tweezers are generally limited to trapping particles with a minimum
absorption at the laser wavelength and a relatively low refractive index contrast
with the suspending medium. Methods that utilize transverse laser modes other than
the Gaussian TEM00 are useful for trapping absorbing and highly-reflective
particles. In particular, the Laguerre–Gaussian LG03 mode, known as the ‘‘donut’’
or ‘‘optical vortex’’ mode, has been used to trap such particles.[20 – 22] The LG03
mode exhibits a phase singularity along the propagation axis which, by destructive
interference, causes the beam intensity to vanish. Laguerre–Gaussian modes can
also be used to impart controlled torques onto trapped particles.[23 – 25] The mag-
nitude of the torque depends on the topological charge l of the beam. Each photon
contributes lh angular momentum. Also, photons in circularly polarized beams carry
an additional ± h.
Figure 1. In the ray-optic regime, the gradient force is due to momentum transfer from the
refracted beam to the particle, creating a resultant force that, in the case shown, pulls the particle
down into the focus. Radiation pressure from scattering and absorption offsets the gradient force in
the direction of the light propagation.
Complex Fluid Rheology Using Laser Tweezers 169
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As mentioned above, the gradient trap can be used to manipulate and position
individual particles. More importantly, for applications in complex fluid rheology, the
displacement of the particle from the trap center acts as a sensitive in situ measure of
force acting on the particle. This enables us to measure nano- and microscopic
responses in complex fluids.
Trap Compliance and Maximum Trapping Force
A particle displaced from its equilibrium position in an optical trap experiences a
restoring force that pulls it back into the center of focus. The restoring force increases
with displacement until the maximum trapping force, or escape force FT,max (typically
from 1–100 pN) and corresponding displacement dmax are reached. At small lateral
displacements, the force profile of an optical trap is Hookean; however, the restoring
force of the trap becomes nonlinear as the displacement reaches lengths comparable to
the particle size. A calculated force profile due to Ashkin[18] is shown in Figure 2A for
the ray-optic regime. Although it is difficult to predict the force profile and escape
force from the first principles,[26] calculations capture the essential features observed
experimentally, most notably the range and nonlinearity at large displacements.[27,28]
The escape force can be estimated from the momentum transfer to the particle
FT,max � Q(nP/c), where P is the laser power, c/n is the speed of light in a material of
refractive index n, and Q is a dimensionless trapping force that can reach as high as
0.30 (Q = 2 for the radiation pressure exerted on a perfectly reflective plane).[18] While
the trapping strength is controlled primarily by the incident laser intensity, particle
geometry and refractive index contrast of the suspending medium influence it as
well.[19,24]
In the Rayleigh regime, the escape force scales with particle radius a as FT,max � a3,
while in the ray-optic regime, it is independent of a.[18] Escape forces and
displacements compiled by Simmons et al.[27] using a near-infrared laser (Nd:YAG,
l = 1064 nm) and polystyrene particles with 2a � l are consistent with Ashkin’s
calculations and are reproduced in Figure 2B. They show that FT,max for trapped
Figure 2. A. Optical trap force profile in the ray-optic limit (adapted from Ref. [18]). B.
Measured trap escape forces for polystyrene particles using a Nd:YAG laser (adapted from
Ref. [27]).
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polystyrene particles increases with laser intensity for each particle size. More
importantly, FT,max for 2a < l increases sharply, then begins to plateau for 2a > l.
DESIGN OF AN OPTICAL TRAPPING APPARATUS
Beam Steering
We constructed an optical trap apparatus, shown in Figure 3. We generate two
independent optical traps using polarizing beam splitters. The intensity to the traps is
controlled by the laser power, while the relative intensity of the traps can be
controlled using a half-waveplate. Trap positions in the microscope image plane are
controlled by changing the angle of the beam entering the back aperture of the
objective. In our dual-trap setup, we use a pair of perpendicular acousto-optic
deflectors (AODs, AA Opto-electronic) for the first trap and a motorized gimbal
mirror for the second. A telescope consisting of the microscope tube lens and a lens
at the side port images the AOD or gimbal onto the back aperture of the objective.
To maximize the angular range, we use a telescope magnification close to 1�.
Trapping efficiency is maximized by slightly overfilling the back aperture.[15] For this
reason, we selected an AOD pair with large TeO2 crystals (7 mm), choosing to
Figure 3. The optical trapping apparatus. A single laser beam (l = 1064 nm) is expanded and split
by polarization to form two independent traps. P-polarized beam is steered using a pair of AODs.
After recombining the beams, the laser enters the back aperture of a high NA objective. Forward-
scattered light can be collected and imaged onto a quadrant photodiode to measure residence times
and particle displacement in the trap at fast sampling rates (�10 kHz).
Complex Fluid Rheology Using Laser Tweezers 171
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expand and collimate the beam before the AOD. However, improved angular range
reduces the bandwidth of the AOD to about 100 kHz. The 48 mrad angular range of
the AOD enables us to generate an optical trap (or multiple time-shared traps)
throughout a 100�100 mm2 region using our 63� objective, and also enables force
clamping (constant stress optical tweezers) by rapidly changing the laser position to
maintain a constant separation between the trap and particle centers. The gimbal
mirror normally controls the position of a stationary trap, which is used for force
measurements. Although steering is generally slow with the gimbal, it is stable over
long times. Other beam-steering methods can be used, including translating lenses,
galvanometers and piezo-controlled mirrors.[26,29]
Laser
We employ a 4 Watt CW Nd:YAG laser (l= 1064 nm), chosen to minimize
damage to biological samples, including live cells and DNA.[5] Visible lasers are
often used for optical trapping (Ar+ at 488 and 514 nm, frequency doubled
Nd:YAG at 532 nm, HeNe at 632 nm), and can be significantly less expensive.
For instance, using a visible laser lowers the power requirement and minimizes
optical aberrations, because microscopes and objectives are optimized for these
wavelengths. Reducing the wavelength also shifts the Rayleigh, intermediate, and
ray-optic regimes by a factor of up to two, in theory. In our experience, however,
the trapping strength is greatly reduced by power limitations imposed by increased
absorption at these wavelengths. For instance, we find that polystyrene particles
degrade rapidly at laser powers 15 mW at l= 488 nm,[28] limiting maximum trap
strengths to approximately FT,max 10 pN. This is sufficient for particle positioning
applications but limits force measurements relevant to complex fluid rheology to
fairly small magnitudes. In addition, the traps would no longer be useful for
capturing cells or other biologicals, because significant damage results in cell
death.[5]
Microscope Objective
To maintain a sufficient axial gradient force to overcome scattering and absorption,
it is necessary to use a high numerical aperture objective (NA > 1). Oil immersion
objectives with numerical apertures as high as 1.4 are available from most
manufacturers and are reasonably inexpensive. However, in aqueous samples, the
mismatch between the refractive index of the immersion medium and the sample
induces aberrations, particularly spherical, that limit trapping to 10–20 mm from the
coverslip for a 100� NA 1.4 oil objective (Zeiss Plan-Apo). Improved trapping
strength and distance from the coverslip can be achieved by using a water immersion
objective, such as a 63� C-Apo. These can be highly advantageous when trying
to minimize the influence of a nearby interface. If full three-dimensional trapping
is unnecessary, two-dimensional traps can be created with inexpensive, low NA
objectives. These have been used successfully in total internal reflection microscopy
(TIRM) measurements of colloidal forces, to prevent lateral drift of particles and
levitate them against gravity with radiation pressure.[30]
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Position Detection
A final consideration in laser trapping is position detection of the trapped particle.
While it is not a significant concern for pure micromanipulation using optical traps, in
situ measurement of forces when probing nano- and microscale mechanical response
and interactions is accomplished by measuring the displacement of a particle from its
equilibrium position in the laser trap. Two techniques are predominantly used: particle
tracking and back focal plane interferometry. The first method simply requires the
acquisition of bright field or fluorescence images from the microscope. Particle centers
in the image can then be identified to within a subpixel resolution through a series of
image-processing operations.[31] Typically, the accuracy of image tracking is �30 nm,
however, time scales are limited by video frequencies (30 Hz). To overcome the spatial
resolution and bandwidth of video tracking, back focal plane (bfp) interferometry is a
convenient alternative.[32] As shown in Figure 3, laser light from the optical trap is
collected using a high numerical aperture condenser. A second lens images the back
focal plane of the condenser onto a quadrant photodiode, resulting in an interference
pattern that is used to determine the particle position relative to the trap.[33] In addition
to increasing the bandwidth to tens of kHz and the spatial resolution to �1 nm, bfp
interferometry allows active control of the trap to generate constant-stress or force-
clamp optical tweezers.[8,11]
APPLICATIONS TO COMPLEX FLUID RHEOLOGY
Interactions, Microstructure, and Rheology in Colloids
Micromechanics in Magnetorheological Suspensions
Laser tweezers enable us to measure the interactions and microstructural responses
that give rise to bulk rheology in colloidal and polymeric materials. The multiple length
scales accessible to laser tweezer experiments enable one to directly associate nanoscale
particle interactions to microstructure and macroscopic properties. A clear example is
the relationship between dipolar interactions, formation of particle chains, and
micromechanics that underlie the bulk rheology of magnetorheological suspen-
sions.[34,35] MR suspensions are colloidal-size paramagnetic particles dispersed in a
nonmagnetic fluid. When the dipolar interaction between particles induced by an
external magnetic field H exceeds thermal energy, MR particles aggregate into chains of
dipoles aligned in the field direction. The energy required to deform and rupture the new
microscopic structure results in the onset of a large, ‘‘tunable’’ yield stress. The ability
of MR suspensions to slowly store elastic energy in their microstructures and viscously
dissipate it on much faster time scales is common to materials that exhibit yield stress
behavior, including electrorheological (ER) suspensions, particulate gels, and foams.[36]
Using laser tweezers, we directly measured the micromechanical properties of
individual dipolar chains, as shown in Figure 4. The rupture tensions of chains scaled
as H2 (Figure 5) are in agreement with the shear stress for dilute suspensions at field
strengths below magnetic saturation. In addition, we found that the rupture tensions
could be calculated from a self-consistent point-dipole model of the interaction
Complex Fluid Rheology Using Laser Tweezers 173
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combined with a repulsive electrostatic double layer. In these studies, a significant
increase in microstructural strength was identified due to induction effects and
multiparticle interactions along dipolar chains. However, this results purely in an
enhancement of rupture tension and does not change the scaling of chain tensile
Figure 4. A chain of 1 mm superparamagnetic emulsion droplets is deformed perpendicular to a
magnetic field using two 3 mm ‘‘tether’’ particles. The rupturing angle and tension are dominated
by the dipolar interaction between particles. Such microstructural mechanics dominate the rheology
of MR suspensions in the presence of a magnetic field and lead to a large, tunable yield stress. The
scale bar is 10 mm.
Figure 5. Rupture strength versus dimensionless interaction potential l = �Umax/kBT for dipolar
chains, where Umax is the maximum attractive interaction between dipoles aligned in the field. The
measured rupture tensions are in excellent agreement with calculations based on an attraction
between point-dipoles and electrostatic double-layer repulsion (there are no adjustable parameters.)
In addition, the linear dependence of the rupturing tension on l gives rise to the same scaling of
the suspension yield stress, clearly linking the interactions, microstructural mechanics, and
bulk rheology.
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strength with H. Thus, it should not affect the field-dependence of the yield stress. In
the case of MR suspensions, laser tweezer experiments clearly identified the
relationship between nanoscale interactions and microstructural mechanics of
polarizable particles to the bulk rheology of MR suspensions.
Assembly and Bending of Gel Backbones
We are currently investigating the relationship between interactions, microstruc-
ture, and bulk rheology in other colloidal systems, including particulate gels. Like MR
fluids, the rheological behavior of particulate gels ultimately depends on the nature and
magnitude of nanoscale interparticle interactions. Attractions induced by van der Waals
forces,[37] depletion interactions due to nonadsorbing polymers,[38 – 40] and adhesion
caused by adsorbing or grafted polymers[41 – 43] cause particles to aggregate into highly
branched, tortuous structures.[37] Particle interactions, in turn, affect the microstructure,
the ability to rearrange, and the tensile strength and bending elasticity of the backbone.
At the point at which the microstructure forms a space-spanning network, the bulk
rheology exhibits a transition to elastic and yield behavior. We are interested in directly
measuring properties that are assumed in models of gel rheology, including the
mechanical properties of the gel backbone.
To start, we use time-shared optical traps to directly assemble mimics of gel
backbones, as shown in Figure 6. The time-shared traps are generated by rapidly
changing the beam angle using our AODs. The scan rate n for the laser must be
n ¼2kBT erf�1ðgÞ
� �2
3pZa3b2ð2Þ
where b is the number of radii a particle is allowed to diffuse, and g is the fraction of
particles that must remain within ba of the origin during one scan.[29] Glycerol is
Figure 6. Time-shared optical traps are used to assemble colloidal chains of 3.14 mm PMMA
particles in an aqueous 0.1 M MgCl2 solution. Upon contact, the particles aggregate into primary
minima to form a stable, permanent chain. Note that the assembly is performed far from the
interface to prevent particles from adhering to the coverslip. The scale bar is 10 mm.
Complex Fluid Rheology Using Laser Tweezers 175
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sometimes used to decrease the diffusivity of particles,[44] but the high bandwidth of
the AOD ensures that we can position and control particles far from the interface, even
in aqueous solutions. The trap displacement is then reduced until particles come into
contact. At sufficient ionic strengths, van der Waals interactions force the particles into
primary minima, creating a rigid chain that mimics the backbone. Note that the chain is
formed and held at �100 mm separation from the interface to prevent the particles from
adhering to the coverslip.
The mechanical properties can then be measured by optical tweezers to establish the
bending stiffness of the microstructure that gives rise to macroscopic elasticity in
gels.[45,46] An example of one such bending experiment is shown in Figure 7, where 3.14
mm poly(methyl methacrylate) (PMMA) particles have been assembled into a chain in
0.l M MgCl2. The chain is held with three time-shared optical traps—two stationary
traps at either end and a translating trap in the center. The relative strength of the traps is
adjusted so that the center trap has a lower maximum trapping force than the stationary
traps. The chain bends as the center trap is moved to the right. A low velocity (�0.1 mm/s)
of the moving trap ensures that mechanical equilibrium is achieved, and the applied
tension can be found by displacement from the stationary traps. The observed deforma-
tion is in good agreement with that of a bending rod under an applied load F,
yðxÞ ¼ 4F
pa4E
Lx2
2� x3
6
� �ð3Þ
for a rod of radius a, length L, and Young’s modulus E. Shown in Figure 7 is the position of
particle centers at the point of maximum deformation compared to the equation for a
bending rod. After the trap releases, the chain relaxes to the unbent configuration.
We are currently investigating how the mechanics of the gel backbone is
influenced by the nanoscale particle interactions. For instance, compare the stiff
behavior of chains aggregated into primary minima above to those formed at lower
ionic strength, where secondary minima dominate the interaction, as shown in Figure 8.
Figure 7. The mechanical properties of a chain assembled from 3.14 mm PMMA particles in an
aqueous solution of 0.l M MgCl2. The chain is held with three time-shared optical traps—two
stationary traps at either end and a translating trap in the center. The chain bends as the center trap
is moved to the right, in good agreement with the deformation of a bending rod.
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Particles interact via centrally acting forces but are free to rotate. When a compressive
stress is applied, the particles in singly bonded chains easily rotate around one another
until a multiply bonded structure is formed. Such particle rearrangements underlie the
low-shear viscosity of weakly aggregated gels[47,48] and contrast strongly aggregated
chains, which exhibit an Euler instability upon compression.
Direct Measurements of Particle Interactions
As demonstrated above, laser tweezers are effective tools for directly measuring
particle interactions associated with the mechanical properties of colloidal micro-
structures. Other methods developed to quantify particle pair potentials include
scanning line tweezers,[49,50] blinking tweezers,[51] and dual traps.[52] We also used
laser tweezers to measure interactions on mesoscopic scales, such as the fluctuation-
mediated attraction between dipolar chains that drives long-time microstructural
coarsening in MR suspensions.[53]
An example of a direct measurement of the interaction potential between colloids
is shown in Figure 9. Two particles are held by scanning the trap rapidly along a single
axis. The velocity of the trap is modulated along the line to create a weak parabolic
well, biasing the particles toward the center of the line. The distribution of particle
separations d is given by the Boltzmann distribution,
PðdÞ ¼ expf�½kðd=2Þ2 þ UðdÞ�=kBTgR10
dr expf�½klðr=2Þ2 þ UðrÞ�=kBTgð4Þ
and can be used to calculate the pair interaction U(d), using the variance of the center-
of-mass position of the particles hR2i= (kBT )/(2kl) to find the line tweezer compliance
kl. The weak repulsive barrier we measured in Figure 9 is consistent with an observed
aggregation time scale of several minutes for a particle pair.
Line tweezer measurements are particularly useful for measuring weak interactions
in complex fluids, such as those induced by polymer depletion. For instance, Yodh and
Figure 8. Compression is applied to a chain of weakly aggregated particles. The centrally acting
forces allow particles to roll past one another. The resulting multiply bonded structure resists
further deformation. Similar rearrangements underly the zero-shear viscosity of weakly aggregated
particulate gels. The scale bar is 10 mm.
Complex Fluid Rheology Using Laser Tweezers 177
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coworkers measured interactions between colloidal particles in dilute and entangled
DNA solutions and suspensions of fd virus, using a method similar to those outlined
above.[49,50] Similar to rapid scanning, repeated positioning and releasing of colloidal
particles using laser traps allows one to measure the pair interaction potential with high
resolution. Crocker and Grier used these ‘‘blinking’’ tweezers to generate trajectories
that allowed them to numerically solve the master equation for the spheres’ Markovian
dynamics.[51,54] Note, however, that special care must be taken when making inter-
particle interactions in order to avoid nonequilibrium effects. For instance, it was
found that the apparent long-range attractive interaction between two like-charged
colloids near interfaces was due to hydrodynamic interactions.[54,55]
Laser Tweezer Microrheology of Polymer Networks and Gels
Laser tweezers can also be used to measure microscopic and bulk structure and
response in polymer materials. For instance, probe particle microrheology has recently
emerged as a method of measuring the rheology of complex fluids. By driving the
motion of embedded particles with magnetic[56,57] or optical forces,[58] or measuring
displacement due to thermal motion,[33,59 – 61] small sample volumes can be measured
with minimal perturbation due to the extraordinarily small stresses and strains. Probe
Figure 9. Interparticle interactions measured for two PMMA particles in a parabolic line trap.
The top part of the figure shows the distribution of interparticle spacings r between particles,
measured for three particle pairs over a total of 9998 images. The interaction potential U(r) is
calculated after the trap compliance is found from the distribution of centers of mass R (inset). We
find a repulsive barrier of approximately 7 kBT between the particles, which is consistent with their
aggregation on time scales of several minutes. The scale bar is 10 mm.
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microrheology has been of particular interest for measuring bulk and local response in
polymeric systems, particularly in biological materials, such as the reconstituted
cytoskeleton,[33,61,62] and even the rheology of individual cells.[63,64] Typically, one
measures the thermal motion of embedded probe particles with time, known as tracer
particle microrheology.[59,60] Particle tracking is performed using videomicroscopy,[65]
laser tracking,[33] or light scattering,[59] depending on the length and time scales to be
probed. Tracer particle microrheology relies on a generalization of the well-known
Stokes–Einstien relationship for the displacement of a particle of radius a in response
to a force f(o):
_xðoÞ ¼ f ðoÞ6paioZ
ð5Þ
substituting the viscosity – ioZ with the complex shear modulus G*.[33,59,60] The
generalized Stokes–Einstein relationship (GSER) yields material responses over a wide
range of time scales; however, under some conditions, the GSER breaks down,[62,66]
probably due to nanoscale fluid structure surrounding the probe, such as depletion or
through enthalpically driven interactions.[67]
An alternative to tracking thermal motion of the probe particle is to drive particle
motion with an optical trap and measure the particle response.[58,68] Active
manipulation has been used to measure cell rheology,[69] the reconstituted cytoskele-
ton,[70] and membrane–cytoskeleton interactions;[10] however, most studies used
paramagnetic particles in a field gradient to probe local response, also known as
magnetic tweezers. A disadvantage of magnetic tweezers is the variation in
magnetization of the paramagnetic particles, which must be known to accurately
estimate the applied stress.[28] Laser tweezers are advantageous, because the properties
that govern the optical trapping force (refractive index contrast to the medium, particle
size, and shape) vary to a much smaller extent, enabling accurate calibration. The
equation of motion for the bead, neglecting inertial terms, is
6pZa _x þ ½2ð4mþ 2kÞ þ kT �x ¼ kT A cos 2pot ð6Þ
where Z is the solvent viscosity, kT is the trap compliance, m is the shear modulus, k is
the bulk modulus, and A and o are the amplitude and angular frequency of the forcing
function, respectively.[58] By measuring the in-phase and out-of-phase bead displace-
ment, it is possible to measure the frequency-dependent viscous dissipation and elastic
storage of the network. The particle response is x(o) = D(o)cos[o t + d(o)], where the
amplitude is
DðoÞ ¼ kT AffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiðkT þ 2ð4mþ 2kÞÞ2 þ ð6paZaoÞ2
q ð7Þ
and the phase angle is
dðoÞ ¼ tan�1 6paZokT þ 2ð4mþ 2kÞ ð8Þ
An example of the response is illustrated in Figure 10A for a 3 mm polystyrene (PS)
particle suspended in water. The bead is oscillated using a function generator to drive
Complex Fluid Rheology Using Laser Tweezers 179
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our AOD. The function generator signal was acquired in synchronization with a video
frame grabber, used to capture images of the bead and perform particle tracking in real
time. The measured phase angle between the laser trap and particle is small,
d� 4�10�3 rad/s. Contrast this data to the response measured for a 10 mm PS particle
in 0.1 wt% b-hairpin (D-Pro L-Pro),[71] shown in Figure 10B. The b-hairpin is a short
(40 amino acid) oligopeptide that folds under basic pH conditions. The folded peptides
then rapidly assemble into supramolecular structures, driven by amphiphilicity of the
Figure 10. (A) Response of a 3 mm PS particle (symbols) to an oscillating optical trap in water
(solid line). The optical trap leads the particle by a phase angle d, given by Eq. 8. (B)
Microrheology of the b-hairpin D-pro L-pro using a 10 mm probe particle. A strong elastic response
is evident by the small displacements the particle translates before pulling out of the trap and the
subsequent recovery. The particle jumps in on the return pass of the optical tweezer.
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amino acids on opposing faces of the folded molecule. The pH sensitivity and rich
microstructure make them potential materials for tissue scaffolds. Probe microrheology
demonstrates the local elasticity of this dilute gel. The particle is displaced by the trap
but is pulled out as the local elastic resistance increases past the maximum trapping
force. A fast elastic recovery is observed before the particle is pulled back into the trap
on its return (‘‘jump in’’). From Stokes law for particle response in a purely elastic
medium, we estimate that the local elastic modulus is 5 Pa.
CONCLUSIONS
This brief survey illustrates the current and future potential of optical trapping in
studies of complex fluid rheology. Laser tweezers will continue to expand our ability
to directly study structure–property relationships in colloids and polymers, bridging
nanoscale interactions to microstructure response and bulk rheology. Several
examples discussed in the case of colloids highlighted our ability to directly measure
interparticle interactions, assemble micro- and mesostructures, and measure their
mechanical properties. These provided insights into the mechanisms of yield and
elasticity in suspensions with strong attractive interactions, such as MR fluids and
particulate gels. Tweezer experiments will continue to provide new methods for
testing models of suspension rheology, especially the mechanisms that underlie
nonlinear response, such as strain hardening. Similarly, microrheological measure-
ments not only enable access to extraordinarily small quantities of materials, as
demonstrated by measurements of the rheological characteristics of individual cells,
but also the length scales probed will be significant for studying cell–material
interactions, particularly for understanding the role of substrate compliance and
remodeling in tissue engineering scaffolds.
ACKNOWLEDGMENTS
This work was funded by the generous support of the NSF (CTS-0209936 and
CAREER CTS-0238689), the University of Delaware Research Foundation, and NIH
COBRE.
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Received March 21, 2003
Accepted March 28, 2003
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