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Light Scattering Probes of Viscoelastic Fluids and SolidsFrank Scheffolda; Peter Schurtenbergera

a Department of Physics, University of Fribourg, Fribourg, Switzerland

Online publication date: 06 November 2003

To cite this Article Scheffold, Frank and Schurtenberger, Peter(2003) 'Light Scattering Probes of Viscoelastic Fluids andSolids', Soft Materials, 1: 2, 139 — 165To link to this Article: DOI: 10.1081/SMTS-120022461URL: http://dx.doi.org/10.1081/SMTS-120022461

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SOFT MATERIALS

Vol. 1, No. 2, pp. 139–165, 2003

REVIEW

Light Scattering Probes of Viscoelastic Fluids and Solids

Frank Scheffold* and Peter Schurtenberger

Department of Physics, University of Fribourg, Fribourg, Switzerland

ABSTRACT

In this article, we discuss recent advances in static and dynamic light scattering

applied to soft materials. Special emphasis is given to light scattering methods that

allow access to turbid and solid-like systems, such as colloidal suspensions,

emulsions, glasses, or gels. Based on a combination of single- and multispeckle

detection schemes, it is now possible to cover an extended range of relaxation times

from a few nanoseconds to minutes or hours and length scales below 1 nm up to

several microns. The corresponding elastic properties of viscoelastic fluids or solid

materials range roughly from below 1 Pa to several 100 kPa. Different applications are

discussed such as light scattering from suspensions of highly charged colloidal

particles, colloid and protein gels, as well as dense surfactant solutions.

Key Words: Dynamic light scattering; Photon correlation spectroscopy; Diffusing

wave spectroscopy; Microrheology; Colloids; Biopolymers; Micelles.

*Correspondence: Frank Scheffold, Department of Physics, University of Fribourg, CH-1700

Fribourg, Switzerland; E-mail: [email protected].

139

DOI: 10.1081/SMTS-120022461 1539-445X (Print); 1539-4468 (Online)

Copyright D 2003 by Marcel Dekker, Inc. www.dekker.com

Downloaded At: 17:20 15 January 2011

INTRODUCTION

To characterize the structural and dynamic properties of soft materials, information

on the relevant mesoscopic length scales is required. Such information is often ob-

tained from traditional photon correlation spectroscopy (PCS) techniques, such as static

and dynamic light scattering (SLS/DLS) in the single-scattering regime.[1 – 8] In dense

systems, such as colloidal suspensions and gels, however, these powerful techniques

frequently fail due to strong multiple scattering of light. Furthermore, many (dense)

soft materials are viscoelastic solids, meaning that they are nonergodic. Ergodicity,

however, is an important condition for the applicability of PCS.

We will review some of the recent advances in light scattering from turbid fluid-

and solid-like media. After a brief introduction to dynamic light scattering, we will

show how multiple scattering suppression schemes are used to study moderately turbid

systems. These techniques provide the same kind of information as conventional SLS/

DLS, even in the presence of significant multiple scattering. For even more turbid

samples, diffusing wave spectroscopy (DWS) can be used in order to study the internal

dynamics by measuring the intensity fluctuations of the diffusively transmitted light.

Because in the case of DWS the light is scattered from a large number of scatterers,

each individual one must move only a small fraction of a wavelength for the

cumulative change in optical path length to be a full wavelength. Therefore, DWS can

probe motion on very short length scales from less than 1 nm to about 50 nm.[9 – 13] In

the second part of this article, we address in detail how to study the internal dynamics

of solid-like nonergodic materials in the single and multiple scattering regimes. Finally,

we will illustrate our considerations with a number of recent experiments on soft

complex materials, such as correlated colloidal suspensions and gels. Most of our

examples were taken from studies on colloidal particle assemblies, where multiple

scattering and nonergodicity frequently play an important role. However, our con-

clusions can easily be applied to any kind of soft material, provided the relevant

length and time scales are accessible to light scattering.

PHOTON CORRELATION SPECTROSCOPY

Coherent laser light scattered from a rough or disordered material shows the

typical random speckle pattern. If the material under investigation is soft, the individual

speckles fluctuate due to the microscopic motion in the medium.[5] Photon correlation

spectroscopy analyzes these temporal fluctuations with a time resolution down to a few

nanoseconds.[6,14,15] Based on the experimentally obtained intensity autocorrelation

function (ICF), local dynamic properties on length scales from one to a few hundred

nanometers can be studied. Many different methods and geometries are employed for

this kind of experiments. A selection of current experimental techniques will be

discussed in this section.

Dynamic Light Scattering (DLS) from Weakly Scattering Materials

In a dynamic (single) light scattering (DLS) experiment, laser light (wave vector k)

is incident on a scattering cell of dimension L containing a weakly scattering medium.

140 Scheffold and Schurtenberger


This means that in the case of conventional DLS, the scattering mean free path l � L

and, therefore, the transmission in line of sight has to be high, exp(� L/l ) > 95%. In

the most simple case, we can consider a system of N independent point scatterers

illuminated with laser light. The scattered light is then detected at a certain angle y;

hence, incoming and scattered wave vector (with |k| = |k’|, elastic scattering) define the

scattering vector q = k � k’’’, |q| = 2|k| sin(y/2)) (Figure 1). Each scatterer at a position

ri feels the same incident field amplitude.a

EðriÞ ¼ Eiei½kðr0þriÞ�ot� ¼ ðE0

ie�iotÞeikri ð1Þ

This field amplitude induces a dipole of the same frequency emitting a spherical wave.

Hence, the field amplitude at the detector position is given by the following:

Es / eikrieijkjjr�rij

jr � rijð2Þ

In the far field (|r| � |ri|):

jr � rij ffi r � rik000

jkj þ ð3Þ

and we obtain the following expression for the field amplitude at the detector position:

Esðq; tÞ / eikr

r

XN

i¼1

eiðk�k000ÞriðtÞ ¼ eikr

r

XN

i¼1

eiqriðtÞ ð4Þ

To experimentally characterize the internal dynamics, the intensity autocorrelation

function (ICF) is introduced:

G2ðq; tÞ ¼ hIðq; 0ÞIðq; tÞiT ð5Þ

aThis approximation for weak scattering is equivalent to the first Born approximation. Within the

frame of the Raleigh Gans Debeye (RGD) formalism, it can easily be expanded to scatterers of any

size a and shape, as long as the scattering contrast is weak (ka|n1/n2 � 1| � 1, where n1/n2 denotes

the index contrast).[16,17]

Figure 1. Static and dynamic light scattering (SLS/DLS) scheme. A weakly scattering sample is

illuminated with a laser beam. The scattered light intensity and its fluctuations are recorded with a

single-photon detector at an adjustable angle y. The scattering vector q = k � k’’’’ is defined as the

difference between incoming and scattered wave vector.

Light Scattering Probes 141


For ergodic (fluid-like) systems, the normalized ICF can be directly related to the

normalized field autocorrelation function, g1(t), via the Siegert relation:

g2ðq; tÞ ¼ hIðq; 0ÞIðq; tÞiT

hIð0Þi2T

¼ 1 þ bjg1ðq; tÞj2 ð6Þ

g1ðq; tÞ ¼ hEðq; 0ÞEðq; tÞiT

hjEðq; 0Þj2iT

ð7Þ

The intercept b depends primarily on the detection optics and takes an ideal value

of 1 that can almost be reached when using single-mode fibers for the detection.[18] For

randomly moving scatterers, the individual particle mean squared displacement Dr(t) is

a random Gaussian variable and Eq. 7 is readily evaluated, assuming that the cross

terms i 6¼ j vanish for uncorrelated scatterers:

g1ðq; tÞ ¼ hEðq; 0ÞE*ðq; tÞihjEðq; 0Þj2i

/X

i;j

�eiq½rið0Þ�rjðtÞ�

�/ he�iqDrðtÞi

¼ e�q2hDrðtÞ2i=6 ð8Þ

Combined with the Stoke–Einstein relation (D0 = kT/6pZa) for freely diffusing spheres

[Dr(t)2 = 6D0t],[19] this expressions serves as the basis for the size analysis (particle

radius a) in the submicron range, one of the most prominent applications of PCS.[8]

g1ðq; tÞ ¼ e�D0q2t ð9Þ

For interacting and therefore corrrelated scatterers (spherical particles of size a) the

scattered intensity factorizes: I(q) = P(q) � S(q). Here, P(q) = k02 (dssc/dO) is the Mie

scattering function of the individual particle.[16,17] The intermediate scattering function

Sðq; tÞ ¼ SðqÞ � g1ðq; tÞ ð10Þ

then comprises information about structural correlations S(q) and temporal fluctuations

via g1(q, t).

3DDLS and TCDLS—Multiple Scattering Suppression Techniques

While we have seen that SLS and DLS provide interesting insight into the

structural and dynamic properties of soft materials, their application to many systems of

scientific and industrial relevance has often be considered as too complicated, due

to the very strong multiple scattering frequently encountered. The interpretation of a

SLS/DLS experiment becomes exceedingly difficult for systems with nonnegligible

contributions from multiple scattering. Particularly for larger particles with high

scattering contrast, this limits the technique to very low particle concentrations, and a

large variety of systems are, therefore, excluded from investigations with dynamic light

scattering. However, Schatzel demonstrated that it is possible to suppress contributions



from multiple scattering from the measured photon correlation data.[20] During the last

few years, a number of different theoretical and experimental approaches to this

problem appeared.[21] The general idea is to isolate singly scattered light and suppress

undesired contributions from multiple scattering in a dynamic light scattering

experiment (for details, see Ref. [20]). This can be achieved by performing two

scattering experiments simultaneously on the same scattering volume (with two laser

beams, initial wave vectors ki1 and ki2, and two detectors positioned at final wave

vectors kf1 and kf 2) and cross-correlating the signals seen by the two detectors. If both

experiments then have the same scattering vector q (i.e., in magnitude as well as in

direction), but use different scattering geometries, and in the absence of multiple

scattering, each detector sees the same spatial Fourier component of the sample. The

corresponding relations between the autocorrelation function g1(q, t) and the measured

intensity autocorrelation [G11(1)(t)] and cross-correlation [G12

(1)(t)] function, where (1) in-

dicates singly scattered photons, can then be written as

Gð1Þ11 ðtÞ ¼ I

ð1Þ21 ð1 þ b11jg1ðq; tÞj2Þ ð11Þ

and

Gð1Þ12 ðtÞ ¼ I

ð1Þ1 I

ð1Þ2 ð1 þ b12jg1ðq; tÞj2Þ ð12Þ

In the autocorrelation experiment, the intercept b11 depends primarily on the detection

optics and takes an ideal value of 1 that can be almost achieved when using single-

mode fibers for the detection, whereas for the case of the cross-correlation experiment,

b12 is, in addition, reduced due to phase mismatch dq = |dq| (describes the match of the

scattering vectors q1 and q1) and misalignment dx = |dx| (the spatial match of the

scattering volumes). For single scattering, both auto- and cross-correlation, therefore,

yield the same information. However, in the case of multiple scattering, the situation

changes. In the autocorrelation experiment, the multiply scattered photons contribute to

G11(1) (t) as well and make an interpretation of the data and a deduction of g1(q, t) very

difficult, if not impossible. However, in the cross-correlation experiment outlined

above, only singly scattered light will produce correlated intensity fluctuations on both

detectors. In contrast, multiply scattered light will result in uncorrelated fluctuations

that contribute to the background only due to the fact that it has been scattered in a

succession of different q-vectors, and the contributions from multiple scattering to the

signal are suppressed by a factor of order (Rdkj)�1, where R is the beam waist, and

dkj�1 denotes the magnitude of the smallest of the two wave vector combinations

ki2 � ki1 and ki2 + ki1.[20] This leads to the following expression for G12(1) (t):

Gð1Þ12 ðtÞ � I1I2 þ b12I

ð1Þ1 I

ð1Þ2 jg1ðq; tÞj2 ð13Þ

where Ij is the average intensity summed over all contributions (singly and multiply

scattered photons) measured at detector j, and Ij(1) contains the singly scattered light

only. Such a cross-correlation experiment thus provides us with g1(q, t) even for turbid

suspensions, and multiple scattering will be visible in the decreasing intercept only.

Several different proposals for an experimental realization of this task were

presented in the past. Schatzel[20] and his collaborators successfully implemented the



two-color method (TCDLS).[22] While this method has been demonstrated to work very

well by several groups (see Ref. [21] and references therein), it is technically extremely

demanding and requires sophisticated alignment procedures. A particularly interesting

scheme is the so-called three-dimensional cross-correlation experiment (3DDLS), in

which the two incident and the two detected light paths are placed at an angle d/2

above and below the plane of symmetry of the scattering experiment.[20,21,23 – 25] The

initial (ki1, ki2) and final (kf11, kf1) wave–vector pairs are rotated by some angle about

the common scattering vector q = q1 = q2 for the two scattering processes 1-1 and 2-2,

whereas the two other scattering processes 1-2 and 2-1 detected in this experiment have

different scattering vectors. These additional scattering processes will, therefore,

contribute to the background only, which means that the maximum intercept for

3DDLS is only one quarter of the value obtained for autocorrelation or other cross-

correlation schemes, such as the two-color method, i.e., b12, ideal = 0.25.[20]

A schematic layout of a 3DDLS experiment is shown in Figure 2. A laser beam is

split into two parallel beams that are then focused onto the scattering cell by a lens.

The lens has to be chosen in order to allow for a sufficiently large angle d required for

an efficient suppression of contributions from multiple scattering.[25] An identical lens

is used for the detection side, and the scattered light with wave vectors ki1 and ki2 is

collected using single-mode fibers. For turbid colloidal suspensions, due to the strong

attenuation of the singly scattered intensity that essentially decays like I (1) � exp(�s/l),

the size of the scattering cell becomes important (s is the optical path across the cell

for singly scattered light, and l is the scattering mean free path, which may easily be

less than 1 mm for dense colloidal suspensions). In order to avoid a strong reduction of

the singly scattered light and the subsequent reduction in signal (see Eq. 8), the light

path should be of the order of l. In principle, this could be achieved best by using flat

cells. However, because these cells will then be used in general with beams at

nonnormal incidence angles, this will lead to considerable beam deflection and

Figure 2. Schematic view of a 3DDLS setup. Side-view) Two parallel beams are focused onto the

scattering cell (vat) by a lens. An identical lens is used for the detection side, and the scattered light

is collected using single-mode fibers (SMF). Top-view) A schematic description of the y�2ygeometry for two different scattering angles, angles a) q = 90� and b) q= 30�, that can be used to

work with rectangular scattering cells to minimize the optical path length.



displacement. This problem can be overcome by implementing a different approach, as

outlined in Figure 2. One can use square cells with 10 mm path length that are of

superior optical quality for experiments at low scattering angles, and position them

such that the scattering volume is located in a corner of the cell to have short optical

path lengths. For experiments at scattering angles different from 90�, we can then turn

the cell in a so-called y � 2y geometry, where the sample is rotated by half the rotation

angle (90� � y) in order to recover a symmetrical situation in which the displacement

of the incident and scattered beam almost cancel.[25]

A 3DDLS experiment cannot only be used to determine the dynamic structure

factor for turbid solutions, but one can also perform a static light scattering experiment,

where the 3DDLS scheme allows to correct for multiple scattered light.[21,25] The basis

for this is the fact that the intercept of the cross-correlation function depends on the

ratio between the singly scattered light I(1) (q) and the total scattered intensity I(q)

(Eq. 8). The mean intensity measured in a static light scattering experiment contains

contributions from singly scattered and from different orders of multiply scattered light.

To extract the single scattering contribution I(1) (q), I(q) has to be corrected with the

reduced intercept b12/b12(1) of the cross-correlation function, where b12

(1) denotes the

intercept measured in the limit of single scattering only, which can be measured

independently in a dynamic light scattering experiment with a highly diluted sample.

The single scattering contribution to the total intensity observed at detectors 1 and 2

can then be obtained from

Ið1ÞðqÞ ¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiIð1Þ1 ðqÞIð1Þ2 ðqÞ

q¼

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiI1ðqÞI2ðqÞðb12=b

ð1Þ12 Þ

qð14Þ

When doing static light scattering measurements with rectangular cells on concentrated

suspensions, where the mean free path length l is of the order of the path length s in the

cell, a transmission correction of the scattering intensity with the scattering angle y is

necessary, because s varies with y (see Figure 2).

Several research groups[21] demonstrated the feasibility of such an experiment and

clearly showed that static and dynamic light scattering need not be restricted to dilute

suspensions of small particles, but can be used to successfully characterize turbid

suspensions with transmission values as low as T � 0.01 (which for square cells and

y � 2y geometry allows for a scattering mean free path of approximately 50 mm or

more). This opens a completely new field of suspension characterization using static

and dynamic light scattering experiments combined with novel cross-correlation

schemes in numerous areas of soft condensed matter research and technology.

Diffusing Wave Spectroscopy—DWS

In conventional DLS (Figure 3), the sample has to be almost transparent (and,

hence, often highly diluted). As we have seen previously, 3DDLS allows some of

the shortcomings of DLS to be overcome. However, the signal strength decreases

roughly exponentially along with increasing turbidity (or decreasing scattering

length l), and hence, multiple scattering suppression is best applied to moderately

turbid samples.

Diffusing wave spectroscopy (DWS) extends the previously described single light

scattering techniques (DLS, TCDLS, 3DDLS) to strong multiple scattering, by treating



the transport of light as a random walk (Figure 4). Over the last 15 years, the DWS

technique has been applied to study and characterize a variety of optically dense

complex systems such as colloidal dispersions and gels,[26 – 30] ceramic slurries and

green bodies,[31] biopolymers and gels (yogurt and cheese),[32,33] granular media,[34]

and foams.[35a,b]

To derive the autocorrelation function for diffuse transport, we first consider an

individual multiple scattering path with n scattering events.[11,30] The normalized field

autocorrelation function of a path of length s = nl is given by

gn1ðtÞ ¼

Z 2k0

0

qPðqÞSðq; tÞdq

� �n Z 2k0

0

qPðqÞSðqÞdq

� �n

ð15Þ

with the normalized field autocorrelation function given by

g1ðtÞ /X1n¼1

PðnÞgn1ðtÞ ð16Þ

In many cases, the intermediate scattering function at sufficiently short times is

dominated by a diffusion-type process and can be written the following way:

Sðq; tÞ ¼ SðqÞe�aðtÞbðqÞq2 ð17Þ

In the following, some typical cases are listed:

. For an uncorrelated suspension S(q) � 1 of identical particles in a simple liquid:

aðtÞ ¼ D0t; bðqÞ ¼ 1

. For an uncorrelated (dilute) suspension S(q) � 1 of identical particles suspended

in a viscoelastic fluid or solid:

aðtÞ ¼ hDr2ðtÞi=6; bðqÞ ¼ 1

Figure 3. In diffusing wave spectroscopy, light is scattered along a path of n steps (scattering

events) with the probability P(n). From the cumulative phase shift, the correlation function g1(t) for

a given path length distribution is obtained.



. For a correlated suspension S(q) 6¼ 1 of identical particles:

aðtÞ ¼ D0t; bðqÞ ¼ HðqÞ=SðqÞ

. For a fractal gel of identical particles in a simple fluid:

aðtÞ ¼ d2ð1 � e�ðt=tcÞp

Þ=6; bðqÞ ¼ 1

The time-dependent parameter a(t) characterizes the self-motion of the individual

particles as expressed by their mean squared displacement. For small particles and

strongly correlated suspensions, collective motion is detected by light scattering. b(q)

takes account for this influence in the examples given above.

Due to the typically large number of scattering events n along a multiple scattering

path, for small values of 1 � S(q, t)/S(q), a full decay of the DWS correlation function

g1n(t) will be observed and, therefore,

Sðq; tÞ ffi SðqÞ½1 � q2aðtÞbðqÞ� ð18Þ

hence,

gn1ðtÞ ¼ 1 �

R 2k0

0q3PðqÞSðqÞ½aðtÞbðqÞ�dqR 2k0

0qPðqÞSðqÞdq

" #n

ð19Þ

Figure 4. DWS setup: An intense laser beam (Verdi from Coherent) is scattered from a turbid

sample contained in a temperature controlled water bath. Two-cell DWS (TCDWS): Light

transmitted diffusively from the sample cell is imaged via a lens onto a second cell (containing a

highly viscous colloidal suspension of moderate optical density). Subsequently, the light is detected

with a single-mode fiber (SMF) and analyzed digitally (correlator and PC). Multispeckle DWS

(MSDWS): In backscattering, the fluctuations of the scattered light are analyzed with a CCD camera

(or a SMF). MSDWS can also be applied in transmission geometry, in parallel to TCDWS, by

placing a beam splitter between the sample and the lens.



which we can write again as an exponential (with e�x ffi 1 � x; x � 1):

gn1ðtÞ ¼ exp �n

R 2k0

0q3PðqÞSðqÞ½aðtÞbðqÞ�dqR 2k0

0qPðqÞSðqÞdq

( )ð20Þ

We find

gs1ðtÞ ¼ exp �2k2

0

s

l*aðtÞ ½b S�½S�

� �ð21Þ

using n = s/l* � l*/l and the well-known relation between l and l*:[26,27]

l*

l¼ 2k2

0

R 2k0

0PðqÞSðqÞqdqR 2k0

0PðqÞSðqÞq3dq

ð22Þ

The transport mean free path l* is the effective step length of the random walk. (Note

that l* directly determines the transmission coefficient for diffuse light: T = Itransmitted/

Iincident ffi l*/L.) The brackets [ ] in Eq. 21 denote the angular average of the dynamic

and static quantities accessible by DWS:[36,37]

½X� ¼Z 2k0

0

q3PðqÞXðqÞdq

Z 2k0

0

q3PðqÞdq ð23Þ

Replacing the sum by an integral, we obtain the following expressions for the field

autocorrelation function.b We note again that for an uncorrelated suspension S(q) � 1

of scatterers [b S]/[S] = 1.

g1ðtÞ ¼Z 1

0

PðsÞ exp �2k20

s

l*aðtÞ ½b S�½S�

� �ds ð24Þ

For transmission through a slab (plane wave limit):

g1ðtÞ ffiðL=l* þ 4=3Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6k2

0aðtÞ½b S�½S�

qsinh ðL=l* þ 4=3Þ

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6k2


qh i ð25Þ

bAbsorption can be taken into account by substituting 2k02a(t)([b S]/[S]) (s/l*) ! 2k0

2a(t) ([b S]/[S])

(s/l*) + (s/3la), where la is the microscopic absorption length along the multiple scattering paths.

For nonabsorbing scatterers, la is the absorption length of the pure solvent.



For backscattering from a semi-infinite medium (L/l*!1, plane wave limit):

g1ðtÞ ¼ exp �g

ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi6k2


s !ð26Þ

DWS is able to provide information about the local dynamics of particle

dispersions, similar to DLS and 3DDLS, however, without any restrictions on particle

concentration and turbidity. Due to the typically large number of scattering events,

DWS is sensitive to small local displacements in the nanometer range. Therefore, in

principle, DWS is well suited for studying solid materials such as colloid or polymer

gels, provided the scattering signal is properly analyzed. In the following, we will

describe how to do such an analysis in order to access the dynamics of viscoelastic

solid materials with PCS.

NONERGODICITY IN LIGHT SCATTERING

In solid-like media, the scatterers are localized near fixed average positions,

probing only a small fraction of their possible spatial configurations by thermal motion.

As a consequence, the measured time-averaged quantities (such as the scattered

intensity or its autocorrelation function) differ from the ensemble-averaged

ones.[38 – 42,48] Experimentally, one finds that a series of (time-averaged) measurements

on a given sample yields a set of different results, each being of limited use for the

characterization of the medium (see inset Figure 11). A comprehensive analysis of

nonergodicity in dynamic (single) light scattering was given by Pusey and van Megen

in 1989.[38] In the following, we will briefly summarize their main results. The authors

consider a viscoelastic solid material, as shown in Figure 5, where the motion of the

scatterers is restricted in space (as it is in the case, e.g., for a polymeric or a colloidal

gel). Light scattered from such a nonergodic system comprises a fluctuating part IF and

Figure 5. Left: In a viscoelastic solid (as shown for a colloidal network), the individual particle

motion is spatially constrained, leading to a plateau value in hDr2(t)i. Right: Due to the constrained

motion, the (properly ensemble averaged) correlation function g1(t) does not decay to zero.



a time-independent part IC. The time average now depends on the exact configuration

of the system: hIiT = hIFiT + IC. For a given configuration, the (time-averaged) ICF is

given as follows:

g2ðtÞT ¼ hIð0ÞIðtÞiT

hIi2T

¼ 1 þ hIi2E

hIi2T

½g1ðtÞ2 � g1ð1Þ2� þ 2hIiE

hIiT

1 � hIiE

hIiT

� �

� ½g1ðtÞ � g1ð1Þ� ð27Þ

The measured ICF still decays to one for large t, however, never attains a value of 2 at

t = 0, except for in the limiting case of an ergodic medium, where g1(1)2 = 0.

Illustrated in Figure 5 is how the properly ensemble averaged field correlation decays

for a viscoelastic solid. If hIiT = hIiE, the second term in Eq. 27 is zero, and

consequently, the measured correlation function is simply 1 + g1(t)2 shifted by its

plateau value g1(1)2. However, in general, the second term in Eq. 27 leads to a

complex relation between the measured ICF and g1(t)2 that cannot be solved without

prior knowledge of hIiE/hIiT.

The Pusey van Megen Method

Pusey and van Megen[38] proposed to use their Eq. 27 to obtain g1(t) from a

combination of static and dynamic experiments. The idea is to measure the time-

averaged ICF (as obtained from a digitial correlator) for a single sample orientation,

and to subsequently rotate/translate the sample rapidly in order to obtain the ensemble

averaged intensity hIiE for a given scattering angle.[38 – 40,42] This approach can be

applied to DLS and DWS.[43] However, it implies that the measured intensity is

constant over the measurement time (which is not always easy to achieve with the

required precision, in particular for streched decays or additional slow relaxations).

For multiple scattering suppression techniques, such as TCDLS or 3DDLS, the

situation is more complex, because the cross correlated intensity correlation function is

analyzed. Here, the nonfluctuating part of the scattered intensity I(kf1) 6¼ I(kf 2) is not

the same for the two cross-correlated signals at directions kf1, kf 2, hence, adding

significant complexity to the application of the Pusey van Megen method for these

systems. Even if a similar analysis should be possible, one would still suffer a signi-

ficantly decreased signal-to-noise ratio due to the nonequal count rates at both

detectors.c

Rotation and Translation

The most direct method of performing ensemble averaging of scattered light for

nonergodic samples is based on the idea of collecting light scattered by different

cWe note that it seems feasible, though tedious, to manually adjust the cell position in order to have

the same signal count rate on both detectors. Under that premise, we assume that the Pusey–van

Megen method can also be applied to 3DDLS, though the details of such an approach have not yet

been worked out.



parts of the sample, thus performing the ‘‘real’’ ensemble averaging.[41] Experiment-

ally, the sample is slowly moved or rotated, while the ICF is collected. Obviously,

this leads to an additional decay of g2(t), which becomes increasingly rapid with

increase of the translation velocity or the rotation frequency.[41] The rotation/

translation method can be extended to 3DDLS/TCDLS as well as to DWS without

any particular difficulties.[44 – 47,72] It has, however, an important disadvantage of

experimental complexity (precise translation or rotation of the sample is required).

Also, it has been claimed (based on exprimental observations) that this method is less

efficient than, e.g., the method of Pusey and van Megen described in the previous

section.[42]

Two-Cell Technique

To overcome the problem of nonergodicity in DWS, a system of two independent

glass cells [thickness L1, L2] can be used.[29,48 – 50] The first cell contains the sample

to be investigated, which can be a stable ergodic or an arrested nonergodic sample.

The second cell, which serves to properly average the signal of the first cell only,

contains an ergodic system with very slow internal dynamics and moderate turbidity

(Figure 6).d

Figure 6. The two-cell setup. The first cell is filled with a liquid or solid-like sample, and the

second cell is filled with an ergodic system (e.g., polystyrene latex spheres in glycerol). After the

first cell, the time average of the transmitted light intensity depends on the position at the interface.

Adding a second ergodic cell leads to an efficient average of the static speckle pattern after L2,

without obscuring the time fluctuations that stem from the scatterer motion in the first cell (for

details, see text).

dIf colloidal particles with ka > 1 are used, typical values L2 ffi (1 � 2)l* are sufficient to suppress

transmission of unscattered light (because l* � l).



Sandwich Geometry

The original setup, as described in Refs. [29,48], is based on a sandwich of two

glass cells of equal dimensions. Light is incident on the first cell L1. The transmitted

speckles at the back of L1 are then diffusively broadened by the presence of the second

cell to a size L22. Because the second layer is ergodic, a two-dimensional average over

all speckle spots within the same area L22 is obtained. The correlation function

g2(t) � 1 of the two-cell setup can be expressed through the joint distribution function

P2(s1, s2) of path segments s1, s2 in the cells:

g2ðtÞ � 1 ¼Z 1

0

ds1

Z 1

0

ds2P2ðs1; s2Þ exp � 1

3k2

1hDr21ðtÞi

s1

l1*

� �� exp � 1

3k2

2hDr22ðtÞi

s2

l2*

� ��2 ð28Þ

In the case that both layers decouple (i.e., no loops of the scattering paths between the

two layers), P2(s1, s2) reduces to a product of two terms P1(s1)P1(s2) and, therefore,

g2(t) � 1 = [g2(t, L1) � 1] [g2(t, L2) � 1]. (A general theoretical treatment of the two-

cell technique is given in Refs. [48,51].) One can now directly determine the con-

tribution of the first cell by dividing the signal of the two-cell setup g2(t) � 1 with the

separately measured autocorrelation function g2(t, L2) � 1 of the ergodic system in the

second cell:

g2ðt; L1Þ � 1 ¼ ½g2ðtÞ � 1�=½g2ðt; L2Þ � 1� ð29Þ

Separated Cells

Recently, a slightly modified version of the two-cell method was installed by

several groups (J. L. Harden, unpublished).[50,52,53] In this version, the two cells are

separated by a distance of several centimeters, thereby allowing additional optical

elements (such as a beam splitter) to be introduced between both cells. The speckle

pattern at the surface of cell L1 is imaged via a lens onto cell L2, as shown

schematically in Figure 4 (and explained in detail in Ref. [50]). In addition to being

more flexible, this setup largely simplifies the decoupling of light propagation in

both cells.

Mechanical Translation

Finally, it is worthwhile to mention that efficient averaging can also be achieved

by mechanical translation of a solid second cell (such as a glass frit).[52] Obviously,

such a setup lacks some of the simplicity of the original two-cell setup. However, if

stability with respect to drifts and temperature is required over an extended duration of

hours or days, for example for the study of colloidal aging, this can be the method of

choice. Also, the decay time of g2(t, L2) can be chosen at will and be adapted to the

time evolution of the system under study.



Multiple Runs

Another practical way to obtain ensemble-averaged correlation functions is

to record a sufficient number of statistically independent time-averaged ICFs. From

Eq. 27 with hhIiE/hIiTi = 1:

hg2ðtÞTiE � 1 ¼ limN!1

1

N

XN

n¼1

hg2ðtÞiT ¼ g1ðtÞ2|fflffl{zfflffl}g2ðtÞ�1

� g1ð1Þ2|fflfflfflffl{zfflfflfflffl}g2ð1Þ�1

ð30Þ

This means that the average of (statistically independent) experimental ICSs yields

the correct correlation function g2(t) � 1 shifted by its plateau value g2(1) � 1.

This result is physically intuitive, because the loss of ergodicity also implies a

lower intercept value. It is, therefore, always a strong indication of nonergodicity

when the measured intercept value decreases before the previously mentioned re-

producibility issues become evident. To ensure statistically independent measure-

ments, it is again possible to measure different system configurations, e.g., by moving

the sample.

Sometimes the arrested system can help fulfill this requirement. In many cases,

soft solid materials only appear nonergodic but still show a very slow relaxation

process, such as aging. While in DLS aging leads to a slow decay on a time scale tA

of hours or days, it can be substantially faster in DWS, because the latter technique

is more sensitive to subtle changes in the system configuration. Hence, if tA is

substantially larger than the time window of interest (e.g., a time window from 10 ns to

1 ms like in Figure 11) but of the order (or slightly larger) compared to the

measurement time (typically a few minutes), it is possible to get a reasonable ensemble

average hg2(t)TiE from a repeated measurement of g2(t)T.[54] In order to make this

method work in practice, one has to adjust the measurement time to tA. If the

measurement time chosen is too long, the intensity IC(q) is no longer constant over the

time of the measurement, and the digital correlator picks up the slow fluctuations

leading to sudden jumps in the experimentally observed correlation function. While this

method is certainly less accurate compared to the previous ones, it nevertheless can

rapidly provide information about a new material under study.

Multispeckle Analysis

Statistically independent correlation functions are most easily obtained using a

CCD camera (see Figure 4). Instead of analyzing the fluctuations of the intensity at a

single spatial position (one speckle spot), one analyzes a large area of the intensity

pattern of the scattered light (hence, multispeckle) using a CCD camera. Such an

approach was first used in dynamic light scattering to replace a standard detector in a

goniometer SLS/DLS setup.[55] Later, it was demonstrated that, particularly for low-

angle SLS/DLS, an area CCD detector has tremendous advantages for the study of slow

relaxation processes.[56] Multispeckle DWS (MSDWS) was reported in 1993;[57] it is

relatively simple to set up due to the weak angular dependence of the diffuse light, and

furthermore, it allows efficient spatial averaging due to the large number of

uncorrelated speckle spots available.[50,55,56,58,59] The main advantage of the CCD



camera as a detector is the access it provides to nonergodic samples and the

significantly improved data acquisition time, because a large number of scattering

experiments are performed simultaneously. In standard DWS or DLS measurements,

the data acquisition time has to be several orders of magnitude larger than the typical

relaxation time of the correlation function g2(t) � 1, a restriction that does not apply to

multispeckle-DWS/DLS (MSDLS, MSDWS). The main drawback of camera-based

DLS/DWS is the limited time resolution of CCD cameras. Typically, correlation times t

down to a few ms can be accessed (as compared to 10 ns with a standard photo-

multiplier-digital correlator setup), which is a particular problem for DWS, where fast

relaxation processes are frequently found. However, a combination of two-cell DWS

and multispeckle DWS, as shown in Figure 4, turns out to be a perfect combination for

overcoming most of the commonly encountered experimental limitations.[50,59] Using

both techniques allows a range of 10 ns to at least 10,000 seconds to be covered (the

latter only limited by the waiting time), hence, more than 12 orders of magnitude in

correlation time.[50,53,57,59] It is worthwhile to note that it seems impossible to combine

the advantages of MSDLS with current multiple scattering suppression schemes,

because a well-defined scattering geometry is necessary (as explained before) for

obtaining a nonzero cross-correlation signal.

APPLICATIONS

In the following examples, we will illustrate how the previously discussed

techniques can be applied to the study of fluid and solid-like soft materials.

Positional Correlations and Dynamics inSuspensions of Charged Spheres

In the first example (Figures 7 and 8), a 3DDLS measurement of two moderately

concentrated colloidal suspensions in a water/ethanol mixture is shown (particle radius

a = 58.7 nm).[30,60] In this concentration regime, the samples already appear strongly

turbid, which makes the application of traditional static light scattering impossible. The

3DDLS allows quantitative information about the mesoscopic interparticle interactions

and dynamics in this regime to be obtained. In the classical DLVO theory for charged

particles, the (kinetic) stability is explained by assuming that a repulsive Debye–

Huckel-type interaction superimposes to the attractive van der Waals interaction,

resulting in a net repulsion or attraction depending on the screening by the surrounding

electrolyte.[1 – 3] In a fully deionized suspension, repulsive forces are therefore strong,

leading to short-range order (or even crystallization) in the particle positions, as

characterized by the radial pair correlation function g(r).

The particle dynamics in dense colloidal suspensions is influenced by hydro-

dynamic interactions (HI), particularly if strong repulsive forces are present.[61] HI are

mediated by the solvent in which the particles are suspended. They are specific to

colloidal suspensions and do not have an analog in atomic physics (such as direct

interactions[7]). In dynamic light scattering, Eq. 9, HI can be taken into account by

introducing an effective q-dependent diffusion constant: Deff(q) = D0H(q)/S(q), and

H(q) can be extracted from the measured static and dynamic structure factor.[5,30]



Excellent agreement between experimental and theoretical calculations is found for

concentrations up to about F ffi 1% (theory: pairwise additivity approximation

(PA)[61]). HI in more concentrated systems deviate from this model,[30,62] but a

detailed understanding of this behavior is still lacking.

Figure 8. Left: Static structure factors S(qa) of deionized suspensions of polystyrene latex

particles (volume fraction F, radius a = 58.7 nm) measured with 3DDLS at 25�C. Solid lines: Best-

fit polydisperse HNC calculations using a Yukawa interaction potential with adjustable volume

fraction and effective charge Zeff.[30,61] Right: Hydrodynamic function of charged sphere

suspensions determined by 3DDLS. Solid lines: Pairwise additivity approximation (PA).

Figure 7. Suspensions of polystyrene latex particles (radius a = 58.7 nm) at different volume

fractions (from left to right): F � 0.01%. Deionized suspensions at F = 0.13%, 0.42%, and 1.05%

(from left to right).



Colloidal Aggregates and Gels

Aggregation and gelation (Figure 9) in complex fluids has been, for a long time, a

field of intense research, where fundamental as well as applied questions are equally

important. Applications of gels and sol–gel processing include such different areas as

ceramics processing, cosmetics and consumer products, food technology, to name only

a few. Gels are formed by chemical or physical reactions of small subunits (molecules,

polymers, or colloids) that can be reversible or irreversible.

The macroscopic features that bring together such different materials are based on

the microstructural properties of all gels, which can be described as random networks

built by aggregation of individual subunits. Starting from a solution of the subunits, the

system is destabilized, which leads to aggregation, cluster formation, and gelation. At

the gel point, a liquid–solid transition is observed that can be characterized by the

appearance of a storage modulus in rheological measurements. In dilute colloidal gel

networks, the individual clusters show a fractal structure leading to a power law

dependence of the structure factor S(q) / q�df for 1/a < 1/q < Rc. The fractal di-

mension is a measure for the compactness of the individual cluster—the higher the

df, the more compact the clusters. For diffusion-limited cluster aggregation (DLCA)

df ffi 1.8, while for reaction-limited cluster aggregation (RLCA), df ffi 2.1 is

expected.[63 – 68] In the most simple picture, the individual clusters grow at the same

rate until they fill the whole accessible volume, with a critical cluster radius given by

Rc � aF�1=ð3�df Þ ð31Þ

Under highly dilute conditions, the structure and dynamics of colloidal gels can be

analyzed using single-light scattering.[69,70] However, with colloidal suspensions or

polymer and biopolymer solutions, the limit of single scattering is quickly reached,

and standard SLS and DLS experiments cannot be used to follow aggregation and

gelation and to characterize the gel structure. The use of cross-correlation techniques

such as 3DDLS can help to expand this regime considerably. This is illustrated with

a recent study of the heat-induced aggregation and gelation of the globular protein

b-lactoglobulin.[72] b-lactoglobulin, the major component of whey, with a molar mass

of 18,600 g/mol and a radius of 2 nm, is one of the most intensively studied

globular proteins. Globular proteins denature if heated to sufficiently high tem-

peratures, which generally leads to aggregation and the formation of a gel, if the

Figure 9. In aggregation and gelation, attractive interparticle interactions lead to the growth of

individual (fractal) clusters until they fill the whole accessible volume at Rc � aF�1/(3�df).



protein concentration is high enough.[71] The structure of these gels depends on

external conditions such as pH, ionic strength, protein concentration, and tem-

perature. For b-lactoglobulin, the aggregation process can be quenched by rapidly

lowering the temperature. At room temperature, the aggregation is very slow, and if

the system is diluted, it remains stable, as no further aggregation or break-up of the

aggregates is observed for a period of months. It thus becomes possible to

characterize the diluted aggregates using scattering techniques at different stages of

the aggregation process up to the gel point. However, it is not possible to study

more concentrated systems and the gels with standard light scattering techniques,

because these systems are too turbid. While neutron and x-ray scattering can be used

for turbid systems and applied correspondingly, they probe the gel structure only on

length scales smaller than about 50 nm. Using 3DDLS, the effect of multiple

scattering on the scattered light intensity can now be effectively corrected, even if

the transmission is only 1%. It was thus possible to determine the structure factor of

aqueous solutions of the globular protein b-lactoglobulin as a function of heating

time.[72] The total scattering intensity and the intensity cross-correlation function

(G2(t)) were determined over a range of scattering angles between 10 and 150

degrees. For the gelled samples, a fraction of the concentration fluctuations relaxes

very slowly or is stationary on the time scale of the experiment, i.e., the samples are

nonergodic. As the calculation of the singly scattered intensity I (1)(q) using Eq. 8

requires the correct intercept b12 for every scattering angle, the samples were slowly

rotated during the measurement so that the rotation velocity determines the terminal

relaxation of the correlation functions. This guarantees that the correct intercept is

obtained, and the correction for multiple scattering is possible even for the non-

ergodic gels. The measurements clearly revealed that the structure factor of the

aggregated and the gelled proteins can be described by an Ornstein–Zernike equation

of the following form:

SðqÞ ¼ IðqÞIð0Þ ¼ 1

1 þ ðqxÞ2ð32Þ

where x is the correlation length of the concentration fluctuations in the system.

Shown in Figure 10A is the q-dependence of the scattered intensity corrected for

multiple scattering for a solution of b-lactoglobulin with a concentration of 40 mg/

mL at different heating times at a temperature of 76�C. The inset, where the ratio

b12/b12(1) of the measured to the ideal intercept as a function of q is shown for the

different individual measurements, demonstrates the effect of multiple scattering on

the original scattering data.

It is possible to generate a master curve from the data before and after the gel point

by plotting S(q) vs. qx, which demonstrates that the local structure of the gel fraction is

the same as that of the individual aggregates. This is shown in Figure 10B.

However, the use of the 3DDLS technique also has its limits, as measurements for

b-lactoglobulin gels were only possible at concentrations below approximately 80 mg/

mL. This restriction becomes even more important for colloidal gels, where the

scattering cross section of the primary particles is much larger, and the limit of the

3DDLS technique to provide a measurement of the single-scattering S(q) and S(q, t) is

reached at much lower particle concentrations.



For concentrated colloidal dispersions, DWS is able to provide information about

the dynamics during aggregation and gelation.[29] Shown in Figure 11 is the measured

autocorrelation function as a function of time after a fluid–solid sol–gel transition. A

suspension of monodisperse polystyrene latex spheres is destabilized by increasing the

solvent ionic strength with a catalytic reaction. Thereby, the electrostatic repulsion of

the double layer is reduced, and the particles aggregate due to van der Waals attraction.

At early stages, clusters form due to particle aggregation, and the decay of the

correlation function shifts to higher correlation times due to the slower motion of the

clusters (data not shown). Gelation occurs when a single cluster fills the entire sample

volume. After the sol–gel transition, we observe that the correlation function g2(t) � 1

does not decay to zero but remains finite. At the gel time, the time dependence of the

mean-squared displacement changes qualitatively from diffusion to arrested motion

well described by:

hDr2ðtÞi ¼ d2½1 � e�ðt=tcÞp

� ð33Þ

At short time intervals, this expression reduces to a power law Dr2(t) / tp, and the

exponent for diffusion p = 1 drops rapidly at the gel point and takes a value of p ffi 0.7

Figure 10. (A) q-dependence of the scattered intensity corrected for multiple scattering for a

solution of b-lactoglobulin with a concentration of 40 mg/mL at different heating times at a

temperature of 76�C. Heating times (top to bottom): 8 h, 4 h, 2 h, 0.5 h, 0.25 h. The solid lines are

fits of the data to Eq. 32. Inset: Reduced intercepts b12/b12(1) for the measurements shown in A. (B)

S(q) vs. qx, where S(q) = I(q)/I(0) for all data sets shown in Figure 10A. The solid line is Eq. 32.

The solid symbols are from measurements with a protein concentration of 20 mg/mL after heating

times of 1 h (circles), 2 h (squares), 4 h (triangles up), and 9 h (triangles down). (Data from

Ref. [72])



for all t > tgel. Similar features of the fluid–solid transition have been observed for

many colloidal systems, such as in ceramic processing of aluminum oxide powders or

in yogurt and cheese making.[31,33,73]

It is possible to directly link the results from DWS to the macroscopic storage

modulus, taking advantage of a recent model developed by Krall and Weitz. Once

the gel spans the whole sample, the signal is dominated by a broad distribution of

elastic gel modes. For the case of fractal gels, the storage modulus is given by

G0’ = G’(o ffi 0) = 6pZ/tc.[70]

DWS-Based Tracer Microrheology

The underlying idea of optical microrheology is to study the thermal responses of

small (colloidal) particles embedded in a material.[74 – 80] Thereby, it is possible to

obtain quantitative information about the loss and storage moduli, G’(o) and G@(o),

over an extended range of frequencies.[81,82] This technique was introduced some years

ago when Mason and Weitz suggested a quantitative relation between the tracer mean-

squared displacement and the complex shear modulus (Refs. [77–79]):e

~GðsÞ ¼ s

6pa

6kBT

s2hD~r2ðsÞi

� �ð34Þ

One of the most popular techniques with which to study the thermal motion of the

tracer particles is diffusing wave spectroscopy (DWS), because it allows relatively easy

Figure 11. Sol–gel transition in dense colloidal suspensions (298 nm suspension at 20% vol.

fract.). (a) Two-cell correlation function g2(t, L1) � 1 in the gel state after t = 108, 256, 734, 4683,

14400 min. The solid lines have been recalculated from the fit to the data. The raw data g2(t) � 1 is

also displayed [dashed lines]. Inset: Repeated (time averaged) measurements of g2(t) � 1 in the gel

state (single layer). (b) Mean-squared displacement hDr2(t)i at different stages t = 0, 82.5, 88, 132

min of the gelation process (full symbols before the gel point and open symbols after the gel point).

Solid lines: Subdiffusive motion (fit with Eq. 33) (Data from Ref. [29]).

eRecently, a rigorous theoretical derivation (albeit certain constraints) for this generalized Stokes–

Einstein relation was reported in Refs. [83,84].



access to small particle displacements over an extended range of distances from less

than 1 nm to about 50 nm. Thereby, elastic moduli from less than 1 Pa to 1 MPa can

be accessed in a two-cell or multispeckle DWS experiment (Refs. [31,85,86]).f Shown

in Figure 12 are some recent measurements, where tracer particles (polystyrene spheres,

diameter 720 nm and 1500 nm) were introduced in an otherwise transparent matrix

consisting of a concentrated surfactant solution (aqueous solution of the surfactant

molecule hexa-ethylene glycol mono n-hexadecyl ether (C16E6) at 100 mg/ml. Under

the chosen conditions, these surfactants form giant, polymer-like micelles, which results

in a surprisingly strong viscoelastic liquid.[59,82,87] Good agreement between classical

rheology and DWS-based microrheology is found, with a dramatically increased

frequency range for the latter technique. For a quantitative comparison, a scaling factor

of 3/2 had to be introduced. The origin of this factor is not yet well understood but

likely reflects the imperfect coupling of the tracer sphere to the medium (for details,

see Ref. [59]).

Summary and Outlook

In the past, light scattering probes have been major tools for physicists and

chemists to use to access local nano- and mesoscopic properties of complex fluids.

fDWS can detect particle displacements in the subnanometer range with visible light. For (the

limting case of) L/l* = 150, a displacement offfiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffihDr2ðtÞi

p� 1 nm leads to a detectable decay of

g2(t) � 1 � 0.99. For a purely elastic solid material, we find from Eq. 34, G = kT/(pad2) (with

hDr2(s)i = d2 and ~1 ¼ 1=s). Hence, a 1 mm tracer particle and a still detectable 0.1 nm displacement

corresponds to a measureable elastic modulus of ca. 1 MPa.

Figure 12. Left: Intensity autocorrelation function g2(t) � 1 in transmission geometry (L = 2

mm, l* = 0.149 mm. Inset: CCD-camera-based multispeckle DWS in backscattering geometry,

L = 5 mm). Right: Classical rheometry and optical microrheology of giant micellar solutions

(c = 100 mg/mL) at T = 28�C. Microrheology data were multiplied by a scaling factor of 3/2, as

discussed in the text. (Data from Ref. [59]).



With the recent developments summarized in this article, it has now become possible to

study a new class of solid-like and turbid materials. Currently, these techniques are

mainly used by specialists. However, in the near future, we expect them to spread

rapidly in academia and applied research. The first instruments of this generation are

already commercially available,[88] and applications in industrial environments are

underway.[89]

In this article, we tried to give a comprehensive guideline to the different methods

while using a minimum of mathematical expressions. Furthermore, some examples

were given on how to perform light scattering on solid-like and turbid materials. Other

important systems of this class that remain unmentioned include polymer solutions and

gels, colloidal glasses, emulsions, ceramic slurries and green bodies, pastes, and all

kinds of soft viscoelastic fluids or solids accessible to DWS-based tracer micro-

rheology. In summary, we can say that it is now possible to access any kind of soft

material using light scattering techniques, provided the system is homogeneous on

macroscopic length scales.

ACKNOWLEDGEMENTS

We would like to thank our colleagues and collaborators for their contributions to

the different projects reviewed in this article, namely Claus Urban, Sara Romer,

Sergey Skipetrov, Luis Rojas, Frederic Cardinaux, Luca Cipelletti, Taco Nicolai and

Ronny Vavrin. We gratefully acknowledge financial support from the Swiss National

Science foundation.

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Received January 14, 2003

Accepted March 4, 2003



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