# Light-scattering microscope.

**ABSTRACT** We demonstrate a new design for a light-scattering microscope that is convenient to use and that allows simultaneous imaging and light scattering. The design is motivated by the growing use of thermal fluctuations to probe the viscoelastic properties of complex inhomogeneous environments. We demonstrate measurements of an optically nonergodic sample, one of the most challenging light-scattering applications.

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**ABSTRACT:**We present a theoretical framework for field-based dynamic light scattering microscopy based on a spectral-domain optical coherence phase microscopy (SD-OCPM) platform. SD-OCPM is an interferometric microscope capable of quantitative measurement of amplitude and phase of scattered light with high phase stability. Field-based dynamic light scattering (F-DLS) analysis allows for direct evaluation of complex-valued field autocorrelation function and measurement of localized diffusive and directional dynamic properties of biological and material samples with high spatial resolution. In order to gain insight into the information provided by F-DLS microscopy, theoretical and numerical analyses are performed to evaluate the effect of numerical aperture of the imaging optics. We demonstrate that sharp focusing of fields affects the measured diffusive and transport velocity, which leads to smaller values for the dynamic properties in the sample. An approach for accurately determining the dynamic properties of the samples is discussed.Applied Optics 11/2013; 52(31):7618-28. · 1.69 Impact Factor - SourceAvailable from: pac2001.aps.anl.gov
##### Article: THE DYNAMICS OF MAGNETORHEOLOGICAL ELASTOMERS STUDIED BY SYNCHROTRON RADIATION SPECKLE ANALYSIS

W. F.schlotter, C.cionca, S. S.paruchuri, J. B.cunningham, E.dufresne, S. B.dierker, D.arms, R.clarke, J. M.ginder, M. E.nichols[Show abstract] [Hide abstract]

**ABSTRACT:**We introduce a new technique for probing the microscopic relaxation of magneto-viscoelastic materials consisting of magnetic particles embedded in a natural rubber matrix. Transversely coherent x-rays from a high brilliance synchrotron source are scattered by the magnetic particles, forming a speckle pattern at low scattering angles. The time dependence of this pattern is recorded with a CCD area detector while the sample is cyclically perturbed by a reversal of the magnetic field direction. The corresponding time-resolved scattering pattern probes both the dynamics of the particles and the relaxation of the matrix in which they are embedded. X-ray photon correlation spectroscopy (XPCS) reveals characteristic time scales for this relaxation by applying the intensity auto-correlation function to the time dependent speckle pattern. For low angle scattering, the wave vector dependence of the relaxation rate exhibits power law length scaling.International Journal of Modern Physics B 01/2012; 16(17n18). · 0.46 Impact Factor - [Show abstract] [Hide abstract]

**ABSTRACT:**Soft matter is studied with a large portfolio of methods. Light scattering and video microscopy are the most employed at optical wavelengths. Light scattering provides ensemble-averaged information on soft matter in the reciprocal space. The wave-vectors probed correspond to length scales ranging from a few nanometers to fractions of millimetre. Microscopy probes the sample directly in the real space, by offering a unique access to the local properties. However, optical resolution issues limit the access to length scales smaller than approximately 200 nm. We describe recent work that bridges the gap between scattering and microscopy. Several apparently unrelated techniques are found to share a simple basic idea: the correlation properties of the sample can be characterized in the reciprocal space via spatial Fourier analysis of images collected in the real space. We describe the main features of such digital Fourier microscopy (DFM), by providing examples of several possible experimental implementations of it, some of which not yet realized in practice. We also provide an overview of experimental results obtained with DFM for the study of the dynamics of soft materials. Finally, we outline possible future developments of DFM that would ease its adoption as a standard laboratory method.Journal of optics 08/2014; 16(8):083001. · 2.01 Impact Factor

Page 1

Light-scattering microscope

Peter D. Kaplan, Veronique Trappe, and David A. Weitz

We demonstrate a new design for a light-scattering microscope that is convenient to use and that allows

simultaneous imaging and light scattering.The design is motivated by the growing use of thermal

fluctuations to probe the viscoelastic properties of complex inhomogeneous environments.

strate measurements of an optically nonergodic sample, one of the most challenging light-scattering

applications.© 1999 Optical Society of America

OCIS codes:

120.5820, 290.5850, 300.6480.

We demon-

1.

Dynamic light scattering ?DLS? is a well-developed

and general technique for studying the dynamics of

small objects.Traditional applications have focused

onthedynamicsoftheparticlesthemselves;however,

recent developments suggest an alternate interpre-

tation of the data in which the dynamics of the par-

ticlesareusedasaprobeoftheviscoelasticproperties

of the surrounding medium.1

correct interpretation of DLS data from systems that

possess a combination of fluctuating and nonfluctu-

ating contributions; in this case, the light-scattering

signal is nonergodic in that the time and the ensem-

ble averages are different.

lenging task of correctly interpreting such nonergodic

scattering from highly constrained systems has been

addressed.2–4

These developments significantly ex-

tend the potential applications of DLS, particularly

as a probe of local viscoelasticity.

vate the extension of its use to probe small regions in

heterogeneous systems.For example, DLS could be

used to probe the properties of different regions

within biological cells.

other techniques that interpret thermal fluctuations

of probe particles as measures of local viscoelasticity;

these techniques include laser tweezers, polarization

interferometry, and imaging the particles directly

onto a position-sensitive photodiode.5–8

Introduction

Essential to this is the

Fortunately, the chal-

They also moti-

This would complement

To conve-

niently collect DLS data from subcellular volumes

requires an instrument that provides both a well-

defined scattering geometry and independent infor-

mation about the scattering volume, preferably from

an image, to correctly locate the region of interest.

An optical microscope is a good candidate platform

for such a system.In fact, there have been several

implementations of light scattering using optical mi-

croscopes.Some have used a strongly focused,

hence strongly diverging, beam9–11for illuminations,

thus complicating analysis of the scattering geometry

and results.Others have operated without regard

to the image,12thus diminishing the value of using

the microscope as a platform.

present a simple but robust implementation of DLS

on an optical microscope.

imaging with light scattering from well-defined re-

gions and at well-defined scattering angles.

strument can study particular small volumes,

perhaps subcellular volumes13,14; it also facilitates

DLS from highly absorbing samples by simplifying

the collection of data from samples so thin that ab-

sorption is not a problem.

here takes full advantage of both the formalism and

the applications of DLS and of the microscope as a

highly engineered, well-understood, and easy to op-

erate optical platform.

Previous implementations of DLS on a microscope

have had two basic designs.

laser beam of several millimeters in diameter is di-

rected by mirrors onto a microscope stage, and the

scattered light is collected in a plane conjugate to the

microscope’s back focal plane.

both imaging and scattering but suffers from the

problems of the confined geometry typical of micro-

scopes, complicating the alignment of the scattering

beam.In the second design,13,15,16the collimated

illuminating beam passes through either the con-

In this paper we

It combines high-quality

This in-

The instrument described

In the first design,12a

This scheme allows

The authors are with the Department of Physics and Astronomy,

University of Pennsylvania, 209 South 33rd Street, Philadelphia,

Pennsylvania 19104.P. D. Kaplan’s e-mail address is kaplan@

ziggy.lrsm.upenn.edu.

Received 7 January 1999; revised manuscript received 19 March

1999.

0003-6935?99?194151-07$15.00?0

© 1999 Optical Society of America

1 July 1999 ? Vol. 38, No. 19 ? APPLIED OPTICS4151

Page 2

denser or objective, forming a submicrometer spot in

the image plane.An aperture or fiber either in the

sample itself or in a conjugate image plane collects

scattered light. Although simplifying the alignment

of the scattering beam, because light is collected in

the image plane, the analysis of scattering departs

from that of traditional DLS17because the detected

lightcontainscontributionsfrommanydifferentscat-

tering vectors.Furthermore, there is a fundamen-

tal trade-off between the statistical advantages of

large sample volumes and the specificity of small

scattering volumes. This trade-off comes from two

factors—first, the diffraction limit requires that

smaller volumes have a larger range of input wave

vectors, smearing out the scattering geometry.

ond, for small numbers of scatterers, number fluctu-

ations are potentially problematic.

In our design we overcome the primary limitations

of both of these classes of design.

tage of modern optical components to mount the laser

source for the scattering directly on the microscope,

yielding a more robust and easily aligned arrange-

ment that exploits the high-quality optics already

incorporated in the microscope.

scattered light from the back focal plane of the mi-

croscope, ensuring that the scattering vectors are

well defined, thereby allowing a more straightfor-

ward interpretation of the DLS data.

directly using the microscope optics, we are able to

image and scatter simultaneously from the sample,

fully exploiting the advantages of using a microscope

for DLS.

Because one of the goals of this application of DLS

is to measure the local viscoelastic properties of ma-

terials, and because highly elastic materials will typ-

ically result in optically nonergodic scattering, we

also incorporate methods for obtaining a true ensem-

ble average of the scattering signal.

tant for an accurate interpretation of the data.

However, this is complicated for scattering from

small volumes of highly inhomogeneous samples.

Thus careful consideration of the correct averaging is

important.

Sec-

We take advan-

We also collect the

In addition, by

This is impor-

2.

We use an inverted microscope for our light scatter-

ing.A schematic of the optical path is shown in Fig.

1.The laser source for DLS follows a light path

beginning above the condenser.

mated laser beam at the sample, we focus the source

laser on the condenser aperture or iris; a beam fo-

cused at this point will be collimated in the sample

with a reasonable diameter and little divergence.

By contrast, if we had simply directed a collimated

beam down the condenser axis, the condenser would

focus it in the sample plane, resulting in a highly

divergent source comprising a large range of incident

angles. This is not optimal for DLS.

collimated beam with diameter d in the condenser

aperture is focused to a waist in the sample with

diameter ds? ?4????F?d ? ?, where ? is the photon

wavelength and F is the condenser focal length.

Optical Design

To obtain a colli-

In detail, a

By

way of contrast, in our design the beam is focused

onto the condenser aperture by placing a lens in the

field iris ?Fig. 1?. The condenser can be thought of as

a relay optic, producing an image of the collimated

beam waist at the field iris in the sample plane.

size of the reimaged spot is smaller by M, the easily

measured condenser magnification.

d?M ? 5 ? 100 ?m.To control the illumination

angle, we generally keep the illuminating beam co-

axial with the microscope’s optical axis.

minating beam is moved off of, but parallel to, the

optic axis, the beam waist in the sample will rotate

about the center of the image plane producing oblique

illumination13and allowing collection of a larger

range of scattering angles.

The collection optics begin with a microscope ob-

jective with a large numerical aperture ?NA?.

back focal plane ?BFP? of the objective, all parallel

rays are brought to a point, so by collecting light from

a point in the BFP, we collect light scattered through

the same scattering wave vector q.

illumination along the optical axis, the scattering an-

gle ? depends simply on the distance between the

collection fiber and the center of the BFP, ?x ? sin ?,

allowing us to calculate the scattering wave vector

q ? 4? sin???2??n, where n is the index of refraction

with the sample.Unfortunately, the BFP is incon-

The

That is, ds?

As the illu-

In the

Moreover, for

Fig. 1.

?514.5 nm? is delivered by an optical fiber, collimated, and then

focused on the condenser aperture of a Leica Model DM IRB?E

inverted microscope.The sample is first imaged, and then the

condenser aligned for Ko ¨hler illumination.

tion the condenser aperture is conjugate to the objective’s back

focal plane ?BFP?.In this configuration, the laser beam is colli-

mated in the sample plane ?SP?.

100?, 1.4 numerical aperture objective is imaged onto a transla-

tion stage that moves a single-mode fiber, varying the scattering

vector that is collected.The collection fiber is split to two photo-

multipliers ?Hammamatsu, Model H5783?.

output is filtered by Brookhaven amplifier discriminators and then

fed to a Model BI9000 correlator.

Design of the DLS microscope.An Ar?-ion laser beam

In Ko ¨hler illumina-

The back focal plane of our

The photomultiplier

4152 APPLIED OPTICS ? Vol. 38, No. 19 ? 1 July 1999

Page 3

veniently located slightly inside the exit pupil of most

objectives. We therefore construct a projection sys-

tem to make the BFP available in a more accessible

plane where a single coherent mode is collected by a

single-mode fiber. The fiber is mounted on a trans-

lation stage, allowing adjustment of ?x, hence the

scattering angle. Millimeter displacements of this

fiber produce changes in the scattering angle of the

order of 10°.

In detail, we mount the illumination on an inverted

microscope ?Leica, Model DM IRB?E? by building a

platform between the condenser and the field iris.

On this platform we mount our source laser.

some experiments, we use a laser diode, operating at

? ? 680 nm, mounted directly on the platform and

imaged with collimating optics and a dichroic beam-

splitting mirror onto the condenser aperture.

natively as depicted in Fig. 1, we use a single-mode

optical fiber to direct light from the 514.5-nm line of

an Ar?laser onto collimating optics mounted on the

platform; a dichroic beam-splitting mirror then di-

rects the beam to the condenser aperture.

cusing lens in the field iris makes a beam waist in the

condenser that is in turn relayed into a spot in the

sample of diameter ds? d?M.

nification M is typically approximately 25 for a large

NA oil-coupled condenser.

lens, a 1-mm-diameter beam of light with ? ? 500 nm

would be focused to a submicrometer spot in the focal

plane with a spread of input angles of nearly a full

radian. With the lens, a 1-mm-diameter beam

forms a spot of 40 ?m in diameter with a divergence

angle of less than 10 mrad.

late the beam size by adding enlarging or reducing

telescopes before the field diaphragm lens.

over, these rigidly mounted illumination optics are

not much more difficult to assemble than standard

goniometer-based DLS illumination systems; how-

ever, the microscope tends to be more robust and

stable.

The collection optics used in our apparatus begin

with a plan-apochromatic 1.4 NA, 100? lens.

objective collects light from input angles as large as

72° off normal. If this light has been refracted from

an aqueous sample, exit angles as large as 90° in the

sample will be collected.

mounted in the camera port of a trinocular head, an

image of the BFP is projected onto a film plane on a

standard camera lens mount.

is formed with a NA of approximately 0.03.

collected by an angularly underfilled single-mode fi-

ber of NA ? 0.1, mounted on a translation stage that

is in turn fitted to a camera lens mount.

adjustment of the translation stage allows determi-

nation of the location of the bright, unscattered beam

at q ? 0. By scattering off of a graticle with bars

positioned every 10 ?m, we calibrate the relation

between ?x and q, thereby determining the scattering

vector.To perform DLS experiments, we place a

sample on the microscope, visually focus, divert light

to the scattering port, and position our fiber at a

desired q. The detection fiber is split into two arms

For

Alter-

The fo-

The condenser mag-

Note that, without this

We can easily manipu-

More-

This

Using a phase telescope

The image of the BFP

Light is

Manual

that each illuminate a photomultiplier tube ?PMT?

?Hammamatsu, Model HC120?.

nected through an amplifier discriminator to an elec-

tronic correlater ?Brookhaven, Model BI9000?.

temporal correlation of a single PMT includes an ar-

tifact that is due to the production of extraneous af-

terpulses by electrons misnavigating the dynode

chain in a small fraction ??1%? of photon counts.

cross-correlating two signals, with uncorrelated af-

terpulsing events, the desired DLS signal is obtained,

essentially free from this artifact.

study of slow dynamics, the microscope must be me-

chanically rigid and vibration isolated.

dition to working on a vibration-isolated floating

table, we use rubber pads placed between the table

and the microscope that significantly reduce acoustic

coupling to the microscope’s natural frequency, which

is near 100 Hz.

The PMT’s are con-

The

By

To allow the

Thus, in ad-

3.

To demonstrate the utility of this arrangement for

DLS, we first investigate the scattering from a stan-

dard sample of polystyrene spheres.

weuseamixtureofwaterandglycerol,chosentoslow

the diffusion so that the measured correlation func-

tion decays on time scales similar to those of the

agarose sample discussed below.

0.5-?m-diameterpolystyrene

Model F-8812?, and the illumination source is the

514.5-nm line of an Ar?-ion laser coupled ?from an-

other lab? through a 45-m-long single-mode fiber.

We show some typical correlation functions, mea-

sured at different scattering vectors in Fig. 2, where

we plot the normalized intensity autocorrelation

function:

Experiment

As a solvent,

The spheres are

?MolecularProbes,

g2??? ? ?I?0?I??????I?0??2.

One advantage of the microscope as a scattering

platform is the ease with which thin samples can be

studied.This is particularly advantageous for

strongly scattering or absorbing systems.

we can see the results of scattering from a sample of

carbon black ?Model VulcanXCT2R from Cabot? at 4

wt. % in mineral oil ?viscocity of 55 cP at 25 °C?.

this concentration, the sample is completely black,

making DLS in a traditional setup impossible.

However, when inserted in a small capillary used for

microscopy, the absorption is reduced to a tolerable

level, and good quality data can be obtained.

carbon black consists of 30-nm particles assembled

into widely polydisperse half-micrometer aggregates

that in turn can form more-complex structures.

diffusion observed here is sufficiently slow that it

must be due to the motion of the large structures

rather than the 30-nm particles or the submicrome-

ter aggregates.Although DLS from this sample is

feasible with a traditional apparatus, it is particu-

larly straightforward with the microscope.

data show our ability to perform DLS on strongly

absorbing materials, a particularly challenging task

technically.

In Fig. 3

At

The

The

These

1 July 1999 ? Vol. 38, No. 19 ? APPLIED OPTICS4153

Page 4

We also measure the scattering from a sample

that is not optically ergodic to determine the feasi-

bility of using the microscope to exploit DLS to

measure the viscoelasticity of samples.

0.5-?m-diameter fluorescent polystyrene spheres

embedded in a 0.48 wt. % agarose gel at a volume

fraction of 0.2% and in the presence of 0.025% by

mass of the surfactant Triton X-100, added for sta-

bilization.Light scattering was performed at an-

gles between 20.6° and 55.1°.

constrained by the gel so that their maximum dis-

placement ?r2?? 3 ??? is less than 1?q2so the cor-

relation function does not decay completely.

light-scattering literature, samples such as this are

commonly, albeit not strictly correctly, referred to

as nonergodic. The correlation functions do not

fully decay and depend strongly on the particular

scattering volume being probed.

trated in Fig. 4 where we show several correlation

functions taken from different scattering volumes

at a scattering angle of 28°.

correlation functions appear to be an unrepeatable

measure of the sample; we show below that these

We use

The particles are

In the

This is illus-

By themselves, these

data can be corrected to account for the static com-

ponent of the scattered intensity.

4.

There are several important potential problems in

interpreting DLS data obtained with a microscope

that must be investigated.

can arise is the contribution of the non-Gaussian

number fluctuation effect, which occurs if the number

of particles in the scattering volume is small.

most likely to be a significant problem for the opti-

cally ergodic samples, as the particles in these sam-

ples are typically completely unconstrained and can

diffuse throughout the sample.

nonergodic sample such as the one measured in Figs.

4–6, the particles are typically sufficiently con-

Data Interpretation

The first problem that

This is

By contrast, for a

Fig. 2.

long times and large displacements by studying 0.5-?m spheres in

a glycerol and water mixture. Note the excellent agreement be-

tween measurements at different angles and the agreement with

the power law expected for purely diffusive motion.

figure displays the field correlation function ?solid curve? taken

from an average of 30, 11-s correlation functions and an exponen-

tial fit ?dashed curve? taken at a scattering angle of 44.7°.

error bars ?plotted at every fifth point? are statistical based on the

variation between individual correlation functions.

Here we can see the ability of the instrument to work at

The lower

The

Fig. 3.

black particles in oil.

scattering difficulties associated with absorption can be minimized

and useful dynamic data can be obtained.

?solid curve? is an average of 20, 30-s correlation functions obtained

from the same spot of one sample at a scattering angle of 44.7°.

The dashed curve is an exponential fit that describes the diffusion

of the particles fairly well.The error bars, plotted at every fifth

data point, are statistical based on the variation between individ-

ual correlation functions.

Scattering data taken from a dense suspension of carbon-

By working with thin samples, light-

Correlation function

Fig. 4.

prepared with 0.2 vol. % 0.46-?m-diameter polystyrene spheres

?Seradyn? and stabilized by adding 0.025 wt. % Triton X-100.

failure of repeated measurements at a 28° scattering angle to

show, even approximately, the same correlation function demon-

strates the strongly nonergodic nature of the sample.

A sample of agarose 0.48 wt. % ?FMC, SeaKem? was

The

4154 APPLIED OPTICS ? Vol. 38, No. 19 ? 1 July 1999

Page 5

strained spatially that they do not diffuse into or out

of the beam, thus there are no number fluctuations.

The problem of number fluctuations will be exac-

erbated when a small beam-spot size is used, as the

total scattering volume will then be limited.

will, however, most often be the case for highly

inhomogeneous samples.

homogeneous samples, the spot size can be larger,

and number fluctuations will not be severe.

for the DLS data from the polystyrene spheres in the

water and glycerol mixture, the beam size was ap-

proximately 50 ?m in diameter, and the sample

thickness was approximately 100 ?m.

tal illuminated volume was approximately 2 ? 106

?m3, which contains approximately 10,000 particles;

this should be more than sufficient to prevent num-

ber fluctuations from being a problem.

To demonstrate the lack of non-Gaussian contribu-

tions, we convert the data from the correlation func-

tions into the mean-square displacements of the

beads.We assume that all the particles are com-

pletely statistically independent and that each parti-

cle has sufficient mean-square displacement that

??r2?? 3 ??? ? ? 1?q2. Then the normalized field

autocorrelation function should be

This

By contrast, for more-

Thus

Thus the to-

g1??? ? exp??q2??r2?????6?, (1)

where the intensity and the field correlation func-

tions are connected by the Siegert relation g2? 1 ?

?g1

Here the factor ? determines the coherence of

the detection scheme; for the single-mode optical fi-

bers used here, ? ? 1. We use Eq. ?1? to invert the

data and calculate the mean-square displacement of

the particles.The results are plotted in Fig. 2,

where we show ??r2???? for all the different angles.

The data superimpose extremely well, confirming our

success in obtaining the correlation functions at dif-

ferent angles with the microscope.

data are perfectly linear on this logarithmic plot over

the full range accessible to our measurements.

confirms the lack of any contributions of number fluc-

tuations that would have distorted the inferred

??r2????.

The second potential problem for performing DLS

measurements with a microscope can arise when the

scattering is optically nonergodic, as is the case for

the agarose data discussed above.

tion functions must be corrected to properly account

for the contribution of the static scattering intensity,

which varies strongly with the location of the detector

and the position of the sample.

have been proposed for analyzing DLS for nonergodic

samples.The simplest scheme is to move the sam-

ple while the data are being collected, thus perform-

ing an ensemble average.3

this is most easily accomplished by rotating or trans-

lating the sample. A second method involves mak-

ing a careful measurement of the correlation function

at a single point and then measuring the average

scattering intensity by rotating or translating the

sample.2

The scattered electric field is expressed as

2.

In addition, the

This

Here the correla-

Several methods

In standard DLS setups,

a sum of a fluctuating E?t? and constant components

Ec.4

The data contain both homodyne and hetero-

dyne terms ??E?t?E*?t?? and ?EcE*?t???, respectively,

and the field correlation function can be extracted

from this data using

?I?t?I?t ? ???

I0

2?f????4N2? 1 ? 2Y?1 ? Y?g1??? ? Y2g1???2, (2)

where Y ? NI0?f????2??I?t?? is the contribution from

the heterodyne signal and must be measured sepa-

rately.To measure the denominator of Y, we sepa-

rately average the static scattering intensity over

many speckles, NI0?f????2.

time-averaged intensity measured during collection

of the correlation function.

speckle averaged intensities, Y can be less than,

equal to, or greater than 1.

Although either of these two methods works well,

for many samples the second technique is preferable,

as it allows more-slowly decaying samples to be stud-

ied.18

The reason for this improvement is that the

critical quantity that must be measured in both cases

is the average scattered intensity, which is used to

properly normalize the correlation function.

tain an accurate value, this intensity must be aver-

aged over a large number of independent speckles; for

example, 1% accuracy requires measurement of 104

independent speckles.In the first technique, the

need to move the sample over this number of speckles

during the course of the measurement sets the min-

imum rotation rate, given the length of the total mea-

surement.This rotation rate in turn limits the

longestmeasurabledecays;thecontributionofdecays

that are slower than the rate of motion between

speckles are obscured by the contribution of the mo-

tion of the sample to the total decay.

the second method relies on a completely indepen-

dent measure of the average static scattering, which

can be done rapidly; then the dynamic measurement

can be done for the full duration of the experiment.

This allows longer decays to be measured, albeit with

reducedaccuracy.Nevertheless,itisstillpossibleto

probe significantly longer decays with the latter

method, and this can be important for studies of vis-

coelastic materials with long relaxation times.

Weusethesecondmethodtocorrectthecorrelation

functions of the agarose gel to account for their opti-

cally nonergodic behavior demonstrated by the vari-

ability in Fig. 4.To test that we can understand this

heterogeneity as a nonergodicity in the language of

Ref. 4, we find that the average static scattering in-

tensity NI0?f????2was measured by averaging mea-

surements made from different regions within the

sample. The individual correlation functions shown

in Fig. 4 were then corrected for the speckle average

using Eq. ?2?.As shown in Fig. 5, the resultant cor-

rected correlation functions and inferred displace-

ments then collapse together as they must, indicating

that we are successfully accounting for the optically

nonergodic behavior. A more stringent test of these

corrections is the comparison of the behavior of this

The numerator is the

For the ratio of time-to-

To ob-

By contrast,

1 July 1999 ? Vol. 38, No. 19 ? APPLIED OPTICS4155

Page 6

sample for scattering at different angles.

this comparison, we invert the data to obtain the

mean-square displacement of the particles; this must

be the same, independent of the angle of the mea-

surement or the particular speckle studied.

raw correlation functions are completely different;

nevertheless, when they are corrected for the speckle

average and inverted to obtain the mean-square dis-

placements, all the ??r2???? overlay for different sam-

To make

The

ple volumes and for different scattering angles, as

shown in Fig. 6. This confirms that we are able to

measure optically nonergodic samples with the mi-

croscope. The shape of the observed mean-square

displacement is consistent with our expectations for a

highly constrained system.

increases with time, whereas at longer times, the

data saturate, reaching a plateau.

The agarose data have a functional form expected

for samples that are strongly elastic.

at short times reflects the high-frequency elastic

modulus dominated by the viscosity of the solvent.

The plateau at longer times reflects the low-

frequency elastic modulus; it is related to the shear

modulus by ??x2? ? kBT??aG,1where ?x2? is the

value of the maximum mean-square displacement in

one direction, a is the radius of the spheres, and G is

the low-frequency elastic shear modulus.

long-time or low-frequency limit of the complete re-

sult that links the Laplace transform of the viscoelas-

tic modulus G?s? to the Laplace transform of the

mean-square displacement ??x ˜2?s??.

ple of agarose, we estimate a shear modulus of 102

dyn?cm2using the measured value of ?x2?.

the modulus experienced by the beads within the

pores in which they are located.

nificantly smaller than the bulk shear modulus mea-

sured for the same gel in a rheometer, which yields a

value of approximately 104dyn?cm2.

that the local environment of the beads does not re-

flect the average properties of the bulk medium,

which in turn suggests that the characteristic length

scale of the pores in the agarose gel are comparable

with or larger in size than the probe beads.

port of this, other measurements of the dynamics of

beads as a function of their size show that the pore

sizes within the agarose are indeed comparable in

size to the beads.19

At short times, ??r2????

The behavior

This is the

For this sam-

This is

However, it is sig-

This suggests

In sup-

5.

The data presented here demonstrate the feasibility

of using a microscope for performing DLS.

main utility of the use of a microscope for DLS is to

probe highly inhomogeneous materials in which the

imaging capability of the optics can be combined with

the scattering to allow particular regions to be

probed. As the sample becomes ever more inhomo-

geneous and these regions become more localized, we

can expect some limitations in the quality of the DLS

data. The first concern will arise from the contribu-

tion of non-Gaussian number fluctuations; these will

become ever more important as the size of the probe

beam is decreased to investigate smaller regions.

However, such effects can be incorporated into the

interpretation of the DLS data if this becomes impor-

tant.20

A second, more subtle problem will arise when the

sample is strongly inhomogeneous and the scattering

is optically nonergodic.The correction to the auto-

correlation function measured at any point requires

knowledge of the average static scattered intensity.

This must be collected by either rotating or translat-

Discussion

The

Fig. 5.

in the text.

placement.

applications of DLS.

Data from Fig. 4 corrected using the procedure described

?a? g1??? and ?b? resultant measure of particle dis-

Nonergodic samples are one of the most challenging

Fig. 6.

and 5?b? at different angles shows that the inferred particle dy-

namics are independent of scattering angle between 20.8° and 55°.

Repeating the measurements and analysis of Figs. 5?a?

4156 APPLIED OPTICS ? Vol. 38, No. 19 ? 1 July 1999

Page 7

ing the sample.

tion is certainly the simplest type of motion to

implement, as most microscope stages are designed

to do this. However, if the sample being studied is

highly inhomogeneous, then translation will likely

move the probe beam out of the region of interest.

Then it will instead be necessary to rotate the sam-

ple to measure the average static intensity.

can be accomplished using a rotation stage mounted

on the microscope. However, even with this means

to determine the average, it may still be difficult to

determine a true ensemble average, as required to

properly interpret the data.

used in obtaining Eq. ?2? is that the medium is

sufficiently spatially homogeneous to allow a mean-

ingful ensemble average to be determined by rotat-

ing or translating the sample.

volume of interest is small, it may not be possible to

determine a true ensemble average; each scattering

direction may be sufficiently unique that a true

average cannot be determined.

likely to be a significant problem only when the

sample is inhomogeneous and when the scattering

volumes of interest are correspondingly small.

Even then it should still be possible to obtain some

average of the scattering intensity, albeit not a com-

plete ensemble average, allowing a correction to be

made even if it is not perfect.

ever, that the limitation placed on obtaining DLS in

that situation is intrinsic to the sample itself, as it

would simply be too small to possess a true ensem-

ble average.

On a microscope, sample transla-

This

The basic assumption

If the scattering

However, this is

We emphasize, how-

6.

The development of a robust scattering microscope

opens the possibility of investigating mechanical and

transport phenomena that were previously inacces-

sible to light scattering.

scale physics requires the simultaneous application

of several techniques to document complex relations.

The microscope is designed for micrometer-scale in-

vestigations and thus forms an excellent platform for

these purposes.The DLS microscope allows us to

include light-scattering results when performing a

wider range of experiments.

a technologically important material used for sepa-

rating macromolecules, it is possible to use the data

presented here in more complete studies of the mi-

croenvironments found in weak agarose gels.

scattering microscope combines the high-quality im-

aging capabilities of a modern optical microscope

with the ability of performing complementary, local-

ized, light-scattering measurements.

We thank Peter Pusey for helpful discussions.

Hinch and G. Daniels provided essential assistance

with microscope customization.

supported by National Science Foundation grant

DMR96-31279.

Conclusions

The study of micrometer-

In the case of agarose,

This

J.

This research was

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