Soft Materials Interactions, Structure, and Microscopic ...wxjs.chinayyhg.com/upload/Files/Soft_Materials/2003-Volume-1/2/167... · content=t713597297 Interactions, Structure,

PLEASE SCROLL DOWN FOR ARTICLE

This article was downloaded by:On: 15 January 2011Access details: Access Details: Free AccessPublisher Taylor & FrancisInforma Ltd Registered in England and Wales Registered Number: 1072954 Registered office: Mortimer House, 37-41 Mortimer Street, London W1T 3JH, UK

Soft MaterialsPublication details, including instructions for authors and subscription information:http://www.informaworld.com/smpp/title~content=t713597297

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

Full terms and conditions of use: http://www.informaworld.com/terms-and-conditions-of-access.pdf

This article may be used for research, teaching and private study purposes. Any substantial orsystematic reproduction, re-distribution, re-selling, loan or sub-licensing, systematic supply ordistribution in any form to anyone is expressly forbidden.

The publisher does not give any warranty express or implied or make any representation that the contentswill be complete or accurate or up to date. The accuracy of any instructions, formulae and drug dosesshould be independently verified with primary sources. The publisher shall not be liable for any loss,actions, claims, proceedings, demand or costs or damages whatsoever or howsoever caused arising directlyor indirectly in connection with or arising out of the use of this material.

http://www.informaworld.com/smpp/title~content=t713597297

http://dx.doi.org/10.1081/SMTS-120022462

http://www.informaworld.com/terms-and-conditions-of-access.pdf

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

Downloaded At: 17:20 15 January 2011

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

168 Furst


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


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]).

170 Furst


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).



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]

172 Furst


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



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.

174 Furst


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.



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.

176 Furst


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.



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.

178 Furst


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



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.

180 Furst


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.

REFERENCES

1. Ashkin, A. Acceleration and trapping of particles by radiation pressure. Phys. Rev.

Lett. 1970, 24, 156–159.2. Ashkin, A. Optical levitation by radiation pressure. Appl. Phys. Lett. 1970, 19,

283–285.3. Ashkin, A. Applications of laser radiation pressure. Science 1980, 210, 1081–1088.4. Ashkin, A.; Dziedzic, J.M.; Bjorkholm, J.E.; Chu, S. Observation of a single-beam

gradient force optical trap for dielectric particles. Opt. Lett. 1986, 11, 288–290.5. Ashkin, A.; Dziedzic, J.M.; Yamane, T. Optical trapping and manipulation of single

cells using infrared laser beams. Nature 1987, 330, 769–771.



6. Finer, J.T.; Simmons, R.M.; Spudich, J.A. Single myosin molecule mechanics:

piconewton forces and nanometre steps. Nature 1994, 368, 113–119.7. Schnitzer, M.J.; Block, S.M. Kinesin hydrolyses one atp per 8-nm step. Nature

1997, 388, 386–390.8. Visscher, K.; Schnitzer, M.J.; Block, S.M. Single kinesin molecules studied with a

molecular force clamp. Nature 1999, 400, 184–189.9. Yin, H.; Wang, M.D.; Svoboda, K.; Landick, R.; Block, S.M.; Gelles, J.

Transcription against an applied force. Science 1995, 270, 1653–1657.10. Dai, J.; Sheetz, M.P. Membrane tether formation from blebbing cells. Biophys. J.

1999, 77, 3363–3370.11. Svoboda, K.; Schmidt, C.F.; Schnapp, B.J.; Block, S.M. Direct observation of

kinesin stepping by optical trapping interferometry. Nature 1993, 365, 721–727.12. Perkins, T.T.; Smith, D.E.; Chu, S. Direct observation of tube-like motion of a

single polymer chain. Science 1994, 264, 819–822.13. Perkins, T.T.; Quake, S.R.; Smith, D.E.; Chu, S. Relaxation of a single DNA

molecule observed by optical microscopy. Science 1994, 264, 822–826.14. Perkins, T.T.; Smith, D.E.; Larson, R.G.; Chu, S. Stretching of a single tethered

polymer in a uniform flow. Science 1995, 268, 83–87.15. Grier, D.G. Optical tweezers in colloid and interface science. Curr. Opin. Colloid

Interface Sci. 1997, 2, 264–270.16. Ashkin, A. Optical trapping and manipulation of neutral particles using lasers. Proc.

Natl. Acad. Sci. U.S.A. 1997, 94, 4853–4860.17. van de Hulst, H.C. Light Scattering by Small Particles; Wiley: New York, 1957.

18. Ashkin, A. Forces of a single-beam gradient laser trap on a dielectric sphere in the

ray optics regime. Biophys. J. 1992, 61, 569–582.19. Ren, K.F.; Grehan, G.; Gouesbet, G. Prediction of reverse radiation pressure by

generalized Lorenz-Mie theory. Appl. Opt. 1996, 35, 2702–2710.20. He, H.; Heckenberg, N.R.; Rubinsztein-Dunlop, H. Optical particle trapping with

higher-order doughnut beams produced using high efficiency computer generated

holograms. J. Mod. Opt. 1995, 42, 217–223.21. Gahagan, K.T.; Swartzlander, G.A., Jr. Optical vortex trapping of particles. Opt.

Lett. 1996, 21, 827–829.22. Rubinsztein-Dunlop, H.; Nieminen, T.A.; Friese, M.E.J.; Heckenberg, N.R. Optical

trapping of absorbing particles. Adv. Quantum Chem. 1998, 30, 469–491.23. Simpson, N.B.; Allen, L.; Padgett, M.J. Optical tweezers and optical spanners with

Laguerre-Gaussian modes. J. Mod. Opt. 1996, 43, 2485–2491.24. Wohland, T.; Rosin, A.; Stelzer, E.H.K. Theoretical determination of the influence

of the polarization on forces exerted by optical tweezers. Optik 1996, 102, 181–190.

25. Friese, M.E.J.; Enger, J.; Rubinsztein-Dunlop, H.; Heckenberg, N.R. Optical

angular-momentum transfer to trapped absorbing particles. Phys. Rev. A 1996, 54,1593–1596.

26. Svoboda, K.; Block, S.M. Biological applications of optical forces. Annu. Rev.

Biophys. Biomol. Struct. 1994, 23, 247–285.27. Simmons, R.M.; Finer, J.T.; Chu, S.; Spudich, J.A. Quantitative measurements

of force and displacement using an optical trap. Biophys. J. 1996, 70, 1813–1822.

182 Furst


28. Furst, E.M. Optical trapping and scattering studies of field-induced micromecha-

nics, interactions and dynamics in a colloidal suspension. Ph.D. thesis; Stanford

University: Stanford, CA, 2000.

29. Mio, C.; Gong, T.; Terray, A.; Marr, D.W.M. Design of a scanning optical trap for

multiparticle manipulation. Rev. Sci. Instrum. 2000, 71, 2196–2200.30. Walz, J.Y.; Prieve, D.P. Prediction and measurement of the optical trapping forces

on a microscopic dielectric sphere. Langmuir 1992, 8, 3073–3082.31. Crocker, J.C.; Grier, D.G. Methods of digital video microscopy for colloidal

studies. J. Colloid Interface Sci. 1996, 179, 298–310.32. Allersma, M.A.; Gittes, F.; deCastro, M.J.; Stewart, R.J.; Schmidt, C.F. Two-

dimensional tracking of ncd motility by back focal plane interferometry. Biophys. J.

1998, 74, 1074–1085.33. Gittes, F.; Schnurr, B.; Olmsted, P.D.; MacKintosh, F.C.; Schmidt, C.F. Mic-

roscopic viscoelasticity: shear moduli of soft materials determined from thermal

fluctuations. Phys. Rev. Lett. 1997, 79, 3286–3289.34. Furst, E.M.; Gast, A.P. Micromechanics of dipolar chains using optical tweezers.

Phys. Rev. Lett. 1999, 82, 4130–4133.35. Furst, E.M.; Gast, A.P. Micromechanics of magnetorheological suspensions. Phys.

Rev. E 2000, 61, 6732–6739.36. Bonnecaze, R.T.; Brady, J.F. Yield stresses in electrorheological fluids. J. Rheol.

1992, 36, 73–115.37. Weitz, D.A.; Oliveria, M. Fractal structures formed by kinetic aggregation of

aqueous gold colloids. Phys. Rev. Lett. 1984, 52, 1433–1436.38. Buscall, R.; Mills, P.D.A.; Goodwin, J.W.; Lawson, D.W. Scaling behavior of the

rheology of aggregate networks formed from colloidal particles. J. Chem. Soc.,

Faraday Trans. 1988, 84, 4249–4260.39. de Hoog, E.H.A.; Kegel, W.K.; van Blaaderen, A.; Lekkerkerker, H.N.W. Direct

observation of crystallization and aggregation in a phase-separating colloid-

polymer suspension. Phys. Rev. E 2001, 64, 021407.40. Verhaegh, N.A.M.; Asnaghi, D.; Lekkerkerker, H.N.W. Transient gels in colloid-

polymer mixtures studied with fluorescence confocal scanning laser microscopy.

Physica, A 1999, 264, 64–74.41. Iler, R.K. Relation of particle size of colloidal silica to amount of a cationic

polymer required for flocculation and surface coverage. J. Colloid Interface Sci.

1971, 37, 364.42. Fleer, G.J.; Lyklema, J. Polymer adsorption and its effect on stability of hy-

drophobic colloids. 2. Flocculation process as studied with silver iodide poly-

vinyl alcohol system. J. Colloids Interface Sci. 1974, 46, 1–12.43. Santore, M.M.; Russel, W.B.; Prud’homme, R.K. A two-component model for the

phase behavior of dispersions containing associative polymer. Macromolecules

1989, 22, 1317–1325.44. Mio, C.; Marr, D.W.M. Tailored surfaces using optically manipulated colloidal

particles. Langmuir 1999, 15, 8565–8568.45. Kantor, Y.; Webman, I. Elastic properties of random percolating systems. Phys.

Rev. Lett. 1984, 52, 1891–1894.46. Krall, A.H.; Weitz, D.A. Internal dynamics and elasticity of fractal colloidal gels.

Phys. Rev. Lett. 1998, 80 (4), 778–781.



47. Potanin, A.A.; De Rooij, R.; Van den Ende, D.; Mellema, J. Microrheological

modeling of weakly aggregated dispersions. J. Chem. Phys. 1995, 102, 5845–5853.48. Potanin, A.A.; Russel, W.B. Fractal model of consolidation of weakly aggregated

colloidal dispersions. Phys. Rev. E 1996, 53, 3702–3709.49. Verma, R.; Crocker, J.C.; Lubensky, T.C.; Yodh, A.G. Attractions between hard

colloidal spheres in semiflexible polymer solutions. Macromolecules 2000, 33,177–186.

50. Lin, K.; Crocker, J.C.; Zeri, A.C.; Yodh, A.G. Colloidal interactions in suspensions

of rods. Phys. Rev. Lett. 2001, 87, 088301.51. Crocker, J.C.; Grier, D.G. Microscopic measurement of the pair interaction

potential of charge-stabilized colloids. Phys. Rev. Lett. 1994, 73, 352–355.52. Sugimoto, T.; Takahashi, T.; Itoh, H.; Sato, S.; Muramatsu, A. Direct measurement

of interparticle forces by the optical trapping technique. Langmuir 1997, 13, 5528–5530.

53. Furst, E.M.; Gast, A.P. Dynamics and lateral interactions of dipolar chain. Phys.

Rev. E 2000, 62, 6916–6927.54. Crocker, J.C.; Grier, D.G. When like charges attract: the effects of geometrical

confinement on long-range interactions. Phys. Rev. Lett. 1996, 77, 1897–1900.55. Squires, T.M.; Brenner, M.P. Like-charge attraction and hydrodynamic interaction.

Phys. Rev. Lett. 2000, 85, 4976–4979.56. Amblard, F.; Maggs, A.C.; Yurke, B.; Pargellis, A.N.; Leibler, S. Subdiffusion and

anomalous local viscosity in actin networks. Phys. Rev. Lett. 1996, 77, 4470–4473.57. Amblard, F.; Yurke, B.; Pargellis, A.; Leibler, S. A magnetic manipulator for

studying local rheology and micromechanical properties of biological systems. Rev.

Sci. Instrum. 1996, 67, 818.58. Valentine, M.T.; Ou-Yang, H.D. Forces on a colloidal particle in a polymer

solution: a study using optical tweezers. J. Phys. Condens. Matter 1996, 8, 9477–9482.

59. Mason, T.G.; Gang, H.; Weitz, D.A. Diffusing-wave-spectroscopy measurements of

viscoelasticity of complex fluids. J. Opt. Soc. Am. 1997, 14, 139–149.60. Mason, T.G.; Weitz, D.A. Optical measurements of frequency-dependent linear

viscoelastic moduli of complex fluids. Phys. Rev. Lett. 1995, 74, 1250–1253.61. Palmer, A.; Mason, T.G.; Xu, J.; Kuo, S.C.; Wirtz, D. Diffusing wave spectroscopy

microrheology of actin filament networks. Biophys. J. 1999, 76, 1063–1071.62. Gisler, T.; Weitz, D.A. Scaling of the microrheology of semidilute f-actin solutions.

Phys. Rev. Lett. 1999, 82, 1606–1609.63. Yamada, S.; Wirtz, D.; Kuo, S.C. Mechanics of living cells measured by laser

tracking microrheology. Biophys. J. 2000, 78, 1736–1747.64. Fabry, B.; Maksym, G.N.; Butler, J.P.; Glogauer, M.; Navajas, D.; Fredberg, J.J.

Scaling the microrheology of living cells. Phys. Rev. Lett. 2001, 87, 148102.65. Valentine, M.T.; Kaplan, P.D.; Thota, D.; Crocker, J.C.; Gisler, T.; Prud’homme,

R.K.; Beck, M.; Weitz, D.A. Investigating the microenvironments of inhomo-

geneous soft materials with multiple particle tracking. Phys. Rev. E 2001, 64,061506.

66. McGrath, J.L.; Hartwig, J.H.; Kuo, S.C. The mechanics of f-actin microenviron-

ments depend on the chemistry of probing surfaces. Biophys. J. 2000, 79, 3258–3266.

184 Furst


67. Lu, Q.; Solomon, M.J. Probe size effects on the microrheology of associating

polymer solutions. Phys. Rev. E 2002, 66, 061504.68. Velegol, D.; Lanni, F. Cell traction forces on soft biomaterials. I. Microrheology of

type I collagen gels. Biophys. J. 2001, 81, 1786–1792.69. Valberg, P.A.; Albertini, D.F. Cytoplasmic motions, rheology, structure probed by a

novel magnetic particle method. J. Cell Biol. 1985, 101, 130–140.70. Dichtl, M.A.; Sackmann, E. Microrheometry of semiflexible actin networks through

enforced single-filament reptation: frictional coupling and heterogeneities in

entangled networks. Proc. Natl. Acad. Sci. U.S.A. 2002, 99, 6533–6538.71. Schneider, J.P.; Pochan, D.J.; Ozbas, B.; Rajagopal, K.; Pakstis, L.; Kretsinger, J.

Responsive hydrogels from the intramolecular folding and self-assembly of a

designed peptide. J. Am. Chem. Soc. 2002, 124, 15030–15037.

Received March 21, 2003

Accepted March 28, 2003



Documents

Soft Materials Interactions, Structure, and Microscopic ...wxjs.chinayyhg.com/upload/Files/Soft_Materials/2003-Volume-1/2/167... · content=t713597297 Interactions, Structure,