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
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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
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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
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
120.5820, 290.5850, 300.6480.
Dynamic light scattering ?DLS? is a well-developed
and general technique for studying the dynamics of
small objects.Traditional applications have focused
recent developments suggest an alternate interpre-
tation of the data in which the dynamics of the par-
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
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
Essential to this is the
Fortunately, the chal-
They also moti-
This would complement
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
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@
Received 7 January 1999; revised manuscript received 19 March
© 1999 Optical Society of America
1 July 1999 ? Vol. 38, No. 19 ? APPLIED OPTICS4151
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
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
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
We take advan-
We also collect the
In addition, by
This is impor-
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.
To obtain a colli-
In detail, a
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-
That is, ds?
As the illu-
?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
4152 APPLIED OPTICS ? Vol. 38, No. 19 ? 1 July 1999
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
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
The condenser mag-
Note that, without this
We can easily manipu-
Using a phase telescope
The image of the BFP
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-
To allow the
Thus, in ad-
To demonstrate the utility of this arrangement for
DLS, we first investigate the scattering from a stan-
dard sample of polystyrene spheres.
the diffusion so that the measured correlation func-
tion decays on time scales similar to those of the
agarose sample discussed below.
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
As a solvent,
The spheres are
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
In Fig. 3
1 July 1999 ? Vol. 38, No. 19 ? APPLIED OPTICS4153
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
The particles are
This is illus-
By themselves, these
data can be corrected to account for the static com-
ponent of the scattered intensity.
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-
The first problem that
By contrast, for a
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
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-
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
4154 APPLIED OPTICS ? Vol. 38, No. 19 ? 1 July 1999
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
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
By contrast, for more-
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 ?
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
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
The scattered electric field is expressed as
In addition, the
Here the correla-
In standard DLS setups,
a sum of a fluctuating E?t? and constant components
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 ? ???
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-
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
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
probe significantly longer decays with the latter
method, and this can be important for studies of vis-
coelastic materials with long relaxation times.
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-
1 July 1999 ? Vol. 38, No. 19 ? APPLIED OPTICS4155
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-
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????
This is the
For this sam-
However, it is sig-
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-
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-
in the text.
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
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
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-
On a microscope, sample transla-
The basic assumption
If the scattering
However, this is
We emphasize, how-
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
The study of micrometer-
In the case of agarose,
This research was
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1 July 1999 ? Vol. 38, No. 19 ? APPLIED OPTICS4157