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Research Collection
Doctoral Thesis
Development of fiber optic based dynamic ligth scattering for a
characterization of turbid suspension
Author(s):
Urban, Claus
Publication Date:
1999
Permanent Link:
https://doi.org/10.3929/ethz-a-003810850
Rights / License:
In Copyright - Non-Commercial Use Permitted
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ETH Library
Diss. ETH No. 13067
Development of Fiber Optic Based
Dynamic Light Scattering for a
Characterization of Turbid Suspensions
A dissertation submitted to the
Swiss Federal Institute
for the
Doctor
of
of
Technology Zurich
degree
of
Natural Science
presented by
Claus Urban
Dipl. Phys., University
born
on
Prof. Dr. P.
1965
Germany
the recommendation of
Schurtenberger,
Prof. Dr. L.
Dr. H. J.
Karlsruhe, Germany
May 14,
citizen of
accepted
of
Gauckler,
Watzke,
examiner
co-examiner
co-examiner
Zürich 1999
i
CONTENTS
Contents
iii
1
Summary
2
Zusammenfassung
3
Introduction
1
4
Light Scattering Theory
4
Light Scattering
9
5
4.1
Static
4.2
Dynamic Light Scattering
15
4.3
Cross-Correlation
18
4.4
3d Cross-Correlation Scheme
20
7
of the Cross-Correlation Function
4.4.1
Intercept
4.4.2
Overlap Volume
4.4.3
Goniometer
4.4.4
Correction for Static
Experimental
5.1
6
v
Detailed
Angle
22
and
Scattering Angle
Light Scattering Experiments
22
24
28
Realization
Description
20
of the
Opto-Mechanical Components
30
30
5.1.1
Producing Parallel
5.1.2
Lenses
31
5.1.3
Detection Unit
32
5.1.4
Alignment
Beams
of the Instrument
Characterization of Turbid Model
Suspensions
32
35
6.1
Materials and methods
35
6.2
Results and discussion
35
Static Structure Factor
45
7.1
Introduction
45
7.2
Materials and Methods
50
7.3
7.2.1
Experimental Set-Up
50
7.2.2
Methods
50
7.2.3
Materials
51
Results and Discussion
52
CONTENTS
ii
8
Influence of Dilution
the Particle Sizes in Milk
57
8.1
Introduction
57
8.2
Materials and Methods
58
8.3
9
on
8.2.1
Experimental Set-Up
58
8.2.2
Methods
58
8.2.3
Multi
8.2.4
Methods
60
8.2.5
Materials
60
Angle
Laplace Transformation
Inverse
58
61
Results and Discussion
8.3.1
Full-Cream Cow Milk
61
8.3.2
Skimmed Cow Milk
63
8.3.3
Emulsion Based
on
Vegetable
Protein Stabilized Emulsions
Oil
66
69
9.1
Introduction
69
9.2
Materials and Methods
69
9.3
9.2.1
Experimental Set-Up
69
9.2.2
Materials
70
9.2.3
Methods
70
Results and Discussion
72
9.3.1
Size Distribution
72
9.3.2
Interactions
74
10 Outlook
79
Danksagung
85
Lebenslauf
87
iii
Summary
1
Dynamic light scattering (DLS)
is
one
popular experimental techniques
of the most
application
its
However,
in the characterization of colloidal systems.
tems of industrial relevance has often been considered to be too
of
DLS
a
contributions from
investigations
of many
high dilution,
with emulsions and
the
industrially
ments with
a
can
so-called
be
cross
to other
to be
dispersions
ing
volume and identical
signals
obtained
multiple scattering
are
respect
experiments
optical
and
scattering
an
matching,
a
the
only,
this instrument two
the
alignment
of such
use
We
and the
were
highly
high
an
particle sizing
emulsions to the determination of the
that
can
not be
Experiments
optically
in
we
light scattering
used state of the art
able to construct
turbid
challenging
a
a
suspensions.
stability
it
laser
can
be used
in fundamental research
a
of concentrated model
light
Due to its
as
well
wide field of
industrially relevant systems
dynamics
such
as
suspensions
matched.
with well defined model
turbid colloidal systems
scatter¬
instrument for inves¬
of the two
inherent
dynamic light scattering experiments
from
same
and all contributions from
Therefore
high stability.
instrument for measurements with
applications ranging
where the
so-called 3d cross-correlation
in industrial laboratories for routine measurements. This opens up
new
turbid
important improve¬
industrial environment represents
opto-mechanical components.
easy
an
contrast
required precise matching
and the necessary
for static and
as
in
or
technique highly
samples. With
events
suppressed. However,
to both the
compact design, the
problem
Then the cross-correlation of the two
vectors.
single scattering
tigations of complex systems
task with
solution to this
performed simultaneously having
scattering
contains
working
With this
have constructed
we
instrument for the characterization of turbid
are
require
or
dynamic light scattering experi¬
in
dilution
as
possible
not
for artefacts when
elegant
A very
light
are
changed.
these considerations
light scattering experiments
very low concentrations. Therefore
technique.
such
high scattering
sizes with
in their natural state, which is
methods,
sample composition need
on
larger particle
high probability
scattered
correlation
investigated
compared
Based
non-negligible
self-associating systems.
suppressing multiply
ment
difficult for systems with
relevant
a
due
interpretation
technique to
which results in
consists of
systems
For
multiple scattering.
immediately limits
contrast this
very
exceedingly
becomes
experiment
complicated
The
strong multiple scattering in undiluted solutions.
to the very
to many sys¬
can
successfully
systems demonstrate clearly that highly
be characterized
using
the 3d
instrument,
1
IV
and that the
factors
can
correlation
relevance.
particle
correctly
be
technique
can
possible
particle
to
an
tor and
which
ment
access
on
determined.
In
applicable
unstable
represents
a
possibilities
in the
dynamic
step
complex
a
against
dilution.
can
samples.
investigation
protein
be
These
particle
stability
for
a
of
state,
or
stabilized emulsions
obtained,
when
experiments
size
at least
a
we
particle
a
it is
dilution
then show
form fac¬
investigating systems
show that the 3d instru¬
quantitative characterization
technique
opens up
new
distributions, equilibrium properties,
complex particle suspensions.
variety
changes
Using the 3d instrument, however,
We thus believe that this
of
cross-
'realistic' systems of industrial
dilution could lead to dramatic
major improvement and allows for
importance
show that the 3d
we
interactions via measurements of the
behaviour and
of considerable
next
form factors and static structure
in their 'natural'
with
the static structure factor
are
to
size distribution.
experiments
particle
a
of milk show that
particle
of undisturbed 'native'
the
the
investigate systems
be low. On further
how
distribution,
is also
Investigations
in the measured
often
size
SUMMARY
of industrial
applications.
This could be
V
Zusammenfassung
2
Die
dynamische Lichtstreuung (DLS)
niken
Charakterisierung
zur
kolloidalen
von
einer
komplexen Flüssigkeit
Streuvektors
q"1.
hauptsächlich
Lichtstreuung
Im Falle kolloidaler Teilchen werden die Intensitätsfluktuationen
durch die diffusive
der Korrelation
von
der Teilchen verursacht. Basierend auf
Bewegung
Diffusionskoeffizient und
als eine sehr
grössenbestimmung
praktische
eine breite
und
zerstörungsfreie
Anwendung.
einigen
zu
vielfach
zur
stark wechselwirkenden kolloidalen
laubt
wird die
dynamischen Verhaltens
des
Untersuchung
Methode
Die Technik ist
geeignet. Weiterhin
Mikrometern
findet die
Teilchengrösse
kolloidalen Teilchen in einem weiten Grössenbereich
von
Suspensionen
zur
von
von
pliziert
betrachtet. Die
mit nicht
schwierig.
vernachlässigbaren Beiträgen
niedrige
Industrie relevante
hohen
was
z.
zu
ses
Problems besteht in der
von
nik. Mit dieser Technik können
Zustand untersucht
Methoden,
z.
werden,
was
Verdünnung
B.
und sie
er¬
Systeme aufgrund
Systemen
als
wird für
zu
der
kom¬
Systeme
grössere Teilchen
mit einem gros¬
Konzentrationen. Deshalb können viele für die
werden oder sie bedürfen einer sehr
Artefakten führt. Eine sehr
Unterdrückung
dynamischen Lichtsreuexperiment
und/oder
verfolgen. Jedoch wur¬
B. bei Emulsionen und selbstassoziierenden
einer hohen Wahrscheinlichkeit
Nanometern
mehrfach gestreuten Lichts ausserordentlich
Dispersionen nicht untersucht
Verdünnung,
wenigen
DLS-Experimentes
Dies limitiert die Technik besonders für
Streukontrast auf sehr
sen
eines
Charakterisierung
verwendet,
und Gläsern
des Lichts in unverdünnten
Interpretation
Teilchen-
zur
konzentrierten
de die Anwendbarkeit der DLS auf viele industriell relevante
Mehrfachstreuung
dynamische
dynamische Lichtstreuung
beispielsweise Aggregations- und Gelierungsprozesse
sehr starken
zur
des inversen
Längenskala
und untersucht diese Fluktuationen auf der
Verfügung
Tech¬
DLS stellt ein Mass für die
Systemen.
Brechungsindex
Zeitskala der Fluktuationen im
bis
populärsten experimentellen
ist eine der
des mehrfach
Systemen
elegante Lösung
gestreuten
zu
die¬
Lichts in einem
mittels einer sogenannten Kreuzkorrelationstech¬
optisch
sehr trübe
eine bedeutende
oder
Systeme
in ihrem natürlichen
Verbesserung gegenüber
Kontastvariation, bietet,
anderen
bei denen die Zusam¬
mensetzung der Probe verändert wird.
Auf diesen
Überlegungen
tionsinstrument
sem
das
zur
basierend haben wir ein
Charakterisierung
Instrument werden zwei
gleiche Streuvolumen
lation der beiden
Signale
von
optisch
sogenanntes 3d Kreuzkorrela¬
trüben Proben
gebaut.
Mit die¬
Lichtstreuexperimente gleichzeitig durchgeführt,
die
und identische Streuvektoren besitzen. Eine Kreuzkorre¬
enthält dann
nur
Beiträge
von
Einfachstreuprozessen,
alle
ZUSAMMENFASSUNG
2
VI
Beiträge
Mehrfachstreuereignissen
von
chen Instruments
Untersuchung
zur
Umfeld stellt eine herausfordernde
Überlagerung
bei der
der beiden
werden unterdrückt. Die
komplexen Systemen
von
Aufgabe
Stabilität dar. Deshalb verwendeten wir modernste
Damit
Komponenten.
Messungen
baus,
an
wir in der
waren
sehr trüben
in der
dynamische Lichtstreuexperimente
strielaboratorien für
Routinemessungen
Anwendungen ausgehend
neuer
strierelevanten
konzentrierter
zu
und die
optische
Systemen
grosse
optomechanische
ein Laser-Lichtstreuinstrument für
bauen.
Wegen
seines
Auf¬
kompakten
es
für statische und
Grundlagenforschung
als auch in Indu¬
benutzt werden. Dies eröffnet ein weites
von
Teilchengrössenbestimmungen
wie Emulsionen bis hin
Modellsuspensionen,
Präzision
nötige
notwendige
und
Justage und der hohen Stabilität kann
der einfachen
Gebiet
Lage,
Suspensionen
in einem industriellen
im Hinblick auf die
Lichtstreuexperimente
eines sol¬
Nutzung
zu
bei denen der
der
der
Dynamik
Kontrast nicht
angepasst
Bestimmung
optische
in indu¬
werden kann.
mit
Experimente
Verwendung
lich der
gut definierten Modellsuspensionen zeigen deutlich, dass
des 3d-Instrumentes sehr trübe kolloidale
Teilchengrössenverteilung
unter
Systeme erfolgreich hinsicht¬
charakterisiert werden
können;
ebenfalls können
Formfaktoren und statische Strukturfaktoren bestimmt werden. In einem nächsten
Schritt
le
zeigen wir,
Systeme
von
zeigen, dass
dass die 3d Kreuzkorrelationstechnik auch auf
industrieller
eine
Bedeutung
Verdünnung
oft die
Möglichkeit, Systeme
dest kann eine
an
dramatischen
zu
Teilchengrössenverteilung führen
anwendbar ist.
kann.
Dagegen
durch Proteine stabilisierten Emulsionen
stemen durch
turfaktors
Messungen
Wechselwirkungen
gen,
werden. Mittels weiterer
zeigen
wir
dann,
Verbesserung
in der
Lichtsreuung
Sy¬
des
dynamischen Verhaltens
komplexer Suspensionen möglich.
darstellt und
unveränderter natürlicher Proben erlaubt. Da¬
Gleichgewichtseigenschaften,
grosser
Experimente
Experimente zeigen,
Untersuchung
von
zumin¬
wie in unstabilen
zur
wendungen
in der gemessenen
untersuchen,
zu
Möglichkeiten
neue
Milch
bietet sich mit dem 3d-Instrument
untersucht werden können. Diese
quantitative Charakterisierung
durch werden
Veränderungen
von
rea¬
des Formfaktors der Teilchen und des statischen Struk¬
dass das 3d-Instrument eine grosse
eine
Untersuchungen
in ihrem natürlichen Zustand
Verdünnung gering gehalten
komplexe
von
Teilchengrössenverteilunund der Stabilität
Dies ist auch für eine Vielzahl industrieller An¬
Bedeutung.
1
Introduction
3
Dynamic light scattering (DLS)
is
popular experimental techniques
of the most
one
suspensions. DLS provides
in the characterization of colloidal
scale for fluctuations in the index of refraction of
these fluctuations
In the
the
on
particles intensity fluctuations
of colloidal
case
particles. Based
the diffusive motion of the
by
sion coefficient and
DLS is
particle size,
non-destructive method for
particles
to several micrometers.
Moreover,
dynamic
sions and
over a
However, its application
solutions. The
used
to many
complicated
interpretation
a
as
frequently
caused
very convenient and
a
is suitable for the char¬
few nanometers
a
used for
and/or strongly interacting
colloidal suspen¬
aggregation
as
DLS
and
multiple scattering
gelation.
con¬
in undiluted
exceedingly
becomes
experiment
of
investigations
systems of industrial relevance has often be
due to the very strong
of
probes
predominantly
wide range of sizes from
and it allows to follow processes such
glasses,
and it
scattering vector, q"1.
are
technique
The
of the time
the correlation between diffu¬
on
widely
DLS is also
behaviour of concentrated
sidered to be too
now
particle sizing.
acterization of colloidal
the
complex fluid,
a
scale of the inverse of the
length
a measure
difficult for
systems with non-negligible contributions from multiple scattering. Particularly for
larger particles
with
concentrations, and
gations
very
already
called
are
been
of systems
implemented
propagation
'diffusing
features such
at
large variety
a
large
wave
as
range of
imental
generally
particles
spatial length
scales.
A very
contributions from
to this
a
ume
problem [2, 3, 4].
scattered
scattering
light
will
can
be used in order to
due to the inherent
this
multiple scattering
so-
important
averaging
interesting opposite approach
The
details
vectors but
produce
problem,
works in the limit of
not allow to determine
over
aims
from the measured
using
idea is to isolate
general
see
[2]).
scattering experiments simultaneously
with identical
singly
products,
to this
number of different theoretical and exper¬
dynamic light scattering experiment (for
performing
approach
and suppress undesired contributions from
two
to very low
sample [1]. Unfortunately,
the
of the
approaches
light
across
does
There exist
recent
diffusion model
polydispersity
correlation data.
scattered
light
spectroscopy'
sufficiently suppressing
photon
of the
a
technique
therefore excluded from investi¬
in commercial
strong multiple scattering, where
describe the
a
contrast this limits the
dynamic light scattering. One quite
with
which has
high scattering
multiple scattering
This
on
the
different
correlated fluctuations
singly
can
same
be achieved
scattering
geometries.
on
in
Then
both detectors. In
a
by
vol¬
only
con¬
trast, multiply scattered light will result in uncorrelated fluctuations that contribute
INTRODUCTION
3
2
to the
due to the fact that it has been scattered in
background only
a
succession of
different q-vectors.
Several different
in the past. In his
presented
for
proposals
pioneering
pressing multiple scattering by
a
turbid
a
sample
acterization of colloidal
scattering angle
a
single scattering angle
article Phillies outlined the
of cross-correlation
means
[3]. However,
used
were
discussed
subsequently
and
requires
very
initial
plane
(kn, A^)
about the
1-1 and
(&/i, kf2)
scattering
have different
for the 3d
ßi2,ideal
=
or
0.25
experiment
[2].
one
of such
an
successfully
new
[6, 7],
field of
and
experiment
characterize
suspension
placed
pairs
at
The outline of this thesis is
of laser
on
are
rotated
terized with respect to size
above
some
scattering
angle
processes
processes 1-2 and 2-1 detected in this
These additional
clearly
extremely
which
scattering
means
processes
that the maximum
the two-colour
as
have very
shown that
turbid
characterization
recently
auto¬
method,
i.
e.
demonstrated
dynamic light
scatter¬
suspensions. This
it
can
opens up
a
using dynamic light scattering
numerous
areas
of
technology.
as
and static
model systems. It is shown that
by
suspensions of small particles, but that
light scattering
dynamic
in which
quarter of the value obtained for
follows:
the
after
an
experimental
correlation instrument is described in detail.
demonstrated
A par¬
angle 6/2
an
combined with novel cross-correlation schemes in
soft condensed matter research and
technically
it is
experiment,
ç2 for the two
=
[8, 9, 10]
Several research groups
need not be restricted to dilute
background
are
other cross-correlation schemes such
ing
experiments
vectors.
only
feasibility
completely
qi
=
background only,
the
be used to
vector
scattering
is
While this method
[5].
groups
group in
scattering experiment (see figure 1). The
wave
scattering
will therefore contribute to the
correlation
by Schätzel [2], whose
by several
light paths
vector q
two other
2-2, whereas the
intercept
not restricted to
sophisticated alignment procedures.
of symmetry of the
and final
common
experiment
of this method for the char¬
scheme is the so-called 3d cross-correlation
the two incident and the two detected
and below the
laser
counterpropagating
the two-colour method
has been demonstrated to work very well
ticularly interesting
use
sup¬
and described
techniques
only. Several other experimental lay-outs
particular successfully implemented
extremely demanding
the
possibility of
is limited due to the fact that it is restricted to
suspensions
of 90°
a
was
realization of this task have been
in which cross-correlation of
relatively simple experiment
beams in
experimental
an
The
overview of the theoretical
realization of the 3d
feasibility
of the instrument is
light scattering experiments
optically
distribution,
turbid
as
well
with well defined
particle suspensions
as
cross-
can
be charac¬
the determination of the
particle
3
Figure
(left)
1: Schematic
description of
and 3d cross-correlation
form factor and static structure
interactions. In
a
next
the
wave
distribution,
can
arrangement
in auto-correlation
(right) experiments.
factor,
where the latter
provides
an access
step then the applicability of the 3d instrument
systems of substantial industrial importance
the 3d instrument
vector
help
interaction
were
investigated,
to
particle
to realistic
and it is shown how
in the characterization of such systems in terms of size
potential
and
stability.
Light Scattering Theory
4
Scattering experiments
continue to be used
condensed matter research
of
ure
well
as
impinges
2).
With
a
on a
information about the
tered beam with
sample properties
polarization,
to
respect
experiments
vector q
through
=
are
of the
the relation E
particles
microscopic
its
For
commonly
as
wavelength
local structural
or
as
light
x-ray
properties
determined
by
with matter
obtained with quantum field
namic
laser
theory,
depends
differ
theory
which is used for the
light scattering
field in the form of
a
in the
plane
response is
on
system.
The deformation of the
tensor. In
a
through
the
the
reciprocal
of the
scattering
is related to
scattering
small
in
can
on
on
and k
the
or
and neutron
polymers,
=
the
higher
with
scattering
such
while laser
as
are
micelles
light
A
27r/A.
investigated
be
particles
wavelength
a
Wirkungsquantum
A of the radiation
or
scatter¬
systems [14, 15].
the electronic structure of the mate¬
only
most
cases
little from the classical
description
a
=
of the theoretical
an
incident
the results
electrodyof
background
electromagnetic
E0et(k'r-a,'i)
polarization
charge
(1)
of the
distribution
polarizability a(u>)
of the
microscopic picture the mechanisms
Dependent
and
on,
wave
impinges
is described
so
scattering
following chapters. When
El(r,t)
matter, the
and
scat¬
be resolved with
quantum mechanical properties. In
its
function
by analysing the
is often the method of choice in the characterization of colloidal
Interaction of
a
as
(see fig¬
can
particle
a
investigate crystal structures,
used to
microemulsions,
magnitude
example
l
beam. Additional
wavelength, spin
the smaller the structures that
particles,
principle
the nature of the interaction between the
where % is Planck's
that the smaller the
E of the
on
energy,
(soft)
radiation,
or
structures
is measured
be obtained
can
The energy E of
hk,
=
scattering experiment.
1
by
the structures that
order of
same
47r/Asin(0/2).
means
Energy
depends
sample. Basically
beam and the
rial
incident beam of
an
tool both in
powerful
resulting scattering intensity
the information obtained also
This
is
very
a
characterization. The basic
sample
in
and is scattered
sample
detector the
as
scattering angle 0 between the incident and the scattered
of the
ing
as
scattering experiment [11, 12, 13]
a
which
a
LIGHT SCATTERING THEORY
4
4
of
by
charge
the
distribution of the
electromagnetic
field
system, which in general is
polarization
the kind of beam these structures could be atoms,
of
a non
a
conducting
nuclei, electrons, particles,
5
beam stop
Figure
Sketch
2:
emitted
particles
measured
material
field;
as
are
with
of
by
a
alignment
ot(uj)
the
by
a
The
sample.
scattering angle 0.
increasing frequency
complex
part is the
the
of permanent
higher frequencies,
even
is scattered
source
function of
a
An incident beam
scattering experiment.
a
into
come
electromagnetic
to follow the
dielectric constant
e(u>)
[16].
resonance
=
2,
where the real
square of the index of refraction
imaginary part
polarization
due to the
celerated
mechanism
describes the
can
(2)
electromagnetic field.
charges
3.
subregions
which
are
Then each of the atoms in
field. In the far field
geneous
(and
2According
an
ideal gas
infinite in
to
through
3These subregions
a
=
small
a
are
eoV/N(n2
a
force
a
for each
[17]
—
the
on
the
charges
these
can
then be divided
wavelength
essentially
the
a
of the
light
incident
same
superposition
Due to the fact that the medium is homo¬
subregion
polarizability
a
another
subregion
can
be found
is related to the refractive index
1)
homogeneous
on
ac¬
perfectly homogenous
region
to the cube of the
sees
acting
The
sample.
electrodynamics,
we now assume
subregion
the
total scattered electric field is then
space),
still
by
the illuminated
subregion.
Clausius-Mosotti
If
compared
picture the
of the scattered fields of each
light through
As known from classical
by light,
small
caused
as
light [18].
then radiate
of
absorption
be viewed
medium which is illuminated
into
at
polarizability
With the
e' + ie" is determined
and,
n
and the
is
text.
see
this process relaxes and oscillations of ions
electrons
or
scattering intensity
For details
dipoles, which try
radiation
of
the
length
scale of the
wavelength
of the
light
n
of
LIGHT SCATTERING THEORY
4
6
visible in
a
homogeneous medium,
in the medium
homogeneous
or
on
medium
dielectric constant
fluctuations take
constantly
surfaces.
e
is, according
or
place,
scattering
that
reason
the index of refraction
n
because in
in motion. Therefore
scattering intensity
light
in the medium
will fluctuate in
e or n
the detector
by
seen
at
inhomogeneities
nevertheless is scattered
liquid, for example, the
a
beam is
primary
by
a
result of local fluctuations of the
a
to
only
occurs
light
Einstein,
cancellation of the scattered
finite
The
and
the
Only
that cancels radiation due to destructive interference.
a
atoms
Such local
[19].
or
molecules
given subregion
no
longer
takes
are
and total
and
place
a
exists. If
e(r,i)
=
e0I +
(3)
<fe(r,t)
is the local dielectric constant of the medium where
5e(r, t)
describes the fluctuations
of the dielectric constant and I is the second rank unit tensor, the
resulting
scattered
electric field is
Es(R,t)
The
subscript
=
-r^T-e**'* /
vector, ki and kf
or
small,
or
so
are
the incident and final
the
k%
~
the
over
simplified
Es(R,t)
=
(k,
(<fe(r, t) ni)]]
x
scattering volume,
the
q
cross
light
electromagnetic
of the
wavelength
products
(4)
R is the
ki—kf is the scattering
=
vectors of the
of the
change
x
detector,
wave
out the
kf. Working
the scattered electric field is
is
polarization
quasielastic scattering
that
integral
volume and the
scattering
describing
unit vectors
For elastic
d3re<^-^[nf [kf
V describes that the
distance between the
are
Jv
47rrieo
in
and rij and
wave
of the
(4)
the
nf
(see fig 3).
light
is
zero
equation
for
to
-^e>kfR-^8etf(q,t)
(5)
where
£e,/(q, t)
is the
component of the dielectric
incident and final
Equation (5)
polarization
describes the
=
generally,
a
however,
two
or
more
Se(q, t)
(6)
m
constant fluctuation tensor in direction of the
of the
light.
through
local
the refractive index of the medium.
Col¬
scattering
fluctuations of the dielectric constant
loidal systems,
nf
or
often consist of
phase system
with sizes in the range of nanometers
or
of
electromagnetic
particles dispersed
where the different
micrometers.
waves
in
a
phases
Examples
solvent,
or
more
form structures
are
suspensions
of
7
Figure
3:
described
by
vector
k; and the direction of polarization
rn
An incident
scattered
solid
wave
can
plane
wave
then described
field
is scattered in
Ef, kf and polarization
via
particles, emulsions, aerosols
the electric
gels.
or
vector
(x,y,z)
these structures.
incoming
is scattered
wave
A"
Assuming
electric field is described
only
particles
Because the different
in the
wave
0.
The
=
phases normally
light
is scattered
volume and that the
scattering
before it reaches the
once
the
rif.
have different refractive indices it is clear that in colloidal systems
by
Ei,
detector,
the scattered
by
N
Es(q)
(7)
EUqyqri
=
1=1
where q is the
at the
particle
scattering
position
n.
For
1.
is the
at
The
scattering amplitude
larger particles
experiment
Is{q)
=
volume
the
scatter
light
or
where all
the
particle
particles
are
is
given by
a.
qa,
ji(x)
strongly
spherical
is the
proportional
more
intensity Is(q)
\Es(q)\2- Is{q)
is
is
to the
Bessel function of index
particle
than smaller
volume and therefore
ones.
In
a
light scattering
is measured rather than the electric field
consequently proportional
diameter to the power of six.
identical
of the i-th
^3j1(qal/2)
~
diameter and
particle
scattering amplitude
is the
spherical particles bt(q)
Hq)
where
bt(q)
vector and
(bt(q)
=
Is(q)
b(q)),
=
it is
Na6a(q)
Es(q)
to the square of the
For the
with
particle
monodisperse
case,
given through
(9)
Here N denotes the number of the
scattering
section.
cross
differential
scattering
da(q)/dQ
section
dcr(q)/diï
cross
of the scattered
0. For
a
is
scattering intensity
The
wavelength
polarizability
particle
In
proportional
is scattered
a,
on
to
more
the other
the
more
on
the
scattering
vector q,
.
9
strongly
is
_
(io)
section and therefore also the
cross
than
light
with
directly related
a
with
light
that
means
a
longer wavelength4.
to the molar
of the
mass
M.
often the so-called
excess
Rayleigh
ratio
AR(q)
[20].
and the pure
between the
=
W^M
Vsdü
AIs(q)
R2
Io
Vs
denotes the difference between the
Als(q)
solvent, Io
scattering
is not known
is the
the measured
intensity Iref{q)
Then
AR(q)
For
suspension
is
a
Normally the scattering
to exclude unwanted effects of
reference
R is the distance
to obtain absolute values of the
is
misalignment
ratio
volume
scattering
inten¬
of the instrument
finally normalized with
sample where the Rayleigh
the scattered
ARref(q)
is known.
given by
of
monodisperse particles AR(q)
AR(q)
4This is the
intensity scattered by the suspension
volume V^s and the detector.
scattering intensity, Als(q)
of
:iu
intensity of the incident beam and
exactly. Nevertheless,
sity and furthermore
a
even
important parameter, which
A-4 and to a2. This
AÄ(9)
Here
or
^r«
hand,
light scattering experiments
is used
o~(q)
scatterer and in the far field detection
point
to note that the differential
important
shorter
is the
intensity
167T2
=
It is
section
cross
is the total
a(q)
and
a
expressed through
be
can
with diameter
scattering
respectively the scattering angle
limit
particles
The total
dependence
determines the
on
LIGHT SCATTERING THEORY
4
8
reason
=
can
KcMP{q)S{q)
for the blue colour of the
sky.
be written
as
(13)
9
Light Scattering
Static
4.1
with
K
47T
--=
2
n
2{dn/dcy
—Ar
.—
.
NAX4n
c
Static
particle
==
dn/dc
4.1
refractive index of the solvent
=
-
( mass/volume)
concentration
refractive index increment
=
-
particle
of the
M
==
molar
m
-=
particle
S(q)
-=
static structure factor
mass
form factor
Particle Form Factor and Static
Light Scattering:
Structure Factor
In
static
a
light scattering (SLS) experiment
function of the
measured
as
through
variation of the
cross
a
a
section
which
<r(ç)
static structure factor
principle
usually
This is
effects
a
S(q),
done with
condition
S(q)
=
on
highly
the
dent
size
1 for all q,
between the
the radius of
of the
particle
shape
of the
P(q)
we
form factor
particles,
diluted
samples
particles. S(q)
is known.
us
can
P(q),
and the
particles.
(14)
obtain the size and form of the
S(q) provides
gyration Rg
scattering
Na6P(q)S(q)
=
Is(q) directly yields
scattering intensity if P(q)
or
product
to the
particles.
to avoid influence of interaction
angular dependent scattering intensity.
tion the static structure factor
potential V(r)
a
proportional
which reflects interactions between the
determination of
through S(q)
is
is
is
available
experimentally
vector q, which is
information about size and
Is(q)
Therefore from
intensity of the scattered light
scattering angle 0. Is{q)
which in
essentially yields
scattering
the
the
particle
Under these
form factor. In addi¬
with information of the interaction
be obtained from the
It is also
and interaction
possible
to
angular depen¬
get both, particle
potential V(r) from the angular
dependent scattering intensity.
The
the
scattering intensity Is(q)
particle
form factor
P{q),
the local structure based
illustration of the
4.
Like the
on
therefore is
product of
a
single particle property,
and the static structure factor
S(q)
which describes
the interactions between the individual
physical background
scattering by
a
of the
particles form
local fluctuations in
a
can
be
particles.
A
given with figure
homogeneous medium,
a
particle
LIGHT SCATTERING THEORY
4
10
q
Figure
Scattering of
4:
For details
can
the
a
particle large compared
be divided into volume elements with dimensions that
A of the
wavelength
light
scattered
by
forward direction
0 and shows
compared
a
=
the
0),
i.
form factor
For
a
the
particle
of the
the
are
same
is the
small
compared
superposition
of the
particle.
e.
P(q)
=
Is[q)/Is{q
=
0),
the
scattering
effects in
decreases from
unity
with dimensions
particles
A don't show these extrema in the
seen
by
all
subregions
particle
of the
form
particle
and therefore also interference effects.
particle
sizes it is
wavelength A, particle
P(q).
form factor
light and/or
can
for
a
be calculated
22, 23, 24, 25].
particle
at
which is characteristic for the size and also
It should be mentioned that
wavelength
to the
given by
which is
If the
now
necessary to find
a
relation between
radius r, refractive index ratio
particles
small
are
small contrast between the
compared
particles
m
=
to the
np/ns
The
is divided
and the solvent
(15)
using the Rayleigh-Gans-Debye (RGD) approximation [21,
theory
into,
and
wavelength
qr(m-l)<l
P(q)
to
incident electric field.
scattering intensity without interference
angular dependence,
evaluation of
the variables
see
P(q),
factor because the difference in the electric field
negligible,
light.
from the volume elements to the detector have to be taken into
Is(q
for the form of the
are
the
of
A
all volume elements where interference effects due to the different
intensity Is(q) divided by
small
consequently
and
resulting particle
The
account.
light
intensity Is(q) from the particle then
optical path lengths
=
wavelength
text.
see
The scattered
q
to the
(l/jim)
sees
assumes
the
that each of the small volume
same
incident
light
wave.
This
elements,
implies
the
that the
of
phase
particle
can
a
11
Light Scattering
Static
4.1
transversing
wave
the
as
it would be if the
be calculated from
P(q)
where V is the
in
same
Rayleigh-Gans-Debye approximation generally P(q)
not there. In the
were
is almost the
particle
particle
equation (16)
be
can
qRg
,
calculated, and
this leads to
g
equation (16)
< 1
integral
the
monodisperse spheres
of
case
77\?fsin(^r)
=
(16)
e'qr dV
v J
volume. For the
P^)
In the limit of
=
~
ir
cos(?r)]2
given through
is
the
(17)
'Guinier-approximation'
[26]
P(q)
and with the
expansion of
the
exponential
P(q)
With the
low
equations (18)
and
(19)
AR(q)
from the
to q
—
gradient
0 the molar
of the
1-^S
=
from
mass
an
+
P(q)
M of the
<C 1 this
qRg
yields
(19)
..-
be calculated in the limit of
can
extrapolation
the radius of
curve
(18)
function for
the value of
scattering angles. Therefore,
ratio
^
=
Rayleigh
of the measured
particles (limg_>0 AR(q)
gyration Rg
of the
~
particles
M)
and
can
be
obtained 5.
For
large
larger particles,
values of
m
the
but still in the range of the
RGD-approximation breaks
cident electric field in the
of the
particle
particles
are
particle,
by
caused
and that of the solvent. A
spheres,
exists with the Mie
satisfy specific boundary
though
Mie
scattering
Debye solution,
and it is also
is much
it describes the
highly
particle
spherical
latex
is taken into account.
particle
dependence
sensitive to the
particles having
a
of
P{q)
particle shape.
theory
5(0)
=
1
if the
The inci¬
and the scattered electric field
on
A
with
the
particle. However,
Rayleigh-Gans-
particle
size very well
comparison of
the
experimental data
size in the range of Mie
Dand in the limit of low concentrations where
problem,
where the difference
difficult to calculate than the
form factors calculated with RGD and Mie
with
to this
conditions at the surface of the
more
light, and/or
the difference in the index of refraction
scattering theory [27],
dent electric field outside and that inside the
the
down due to distortions of the in¬
complete solution
in the electric field inside and outside the
must
wavelength of
scattering
particle
measured
is shown in
Comparison of the particle form factor P(q)
Figure
5:
theory
with
m
LIGHT SCATTERING THEORY
4
12
=
experimental
1.2. For details
5. The Mie
figure
see
theory
data
(suspension of
third
if the condition
describes the
qr{rn
—
the
1)
regime of very large particles,
rather than scattering
[28].
In
wave
with the
cross
experimental
depth
<C 1 is
qr ^>
and the
section
or
data very
position
the
process
can
shifted to lower
increasing particle size,
While
factor
P(q)
S(q)
is
a
nm
and
with
calculating particle
describes the
while the RGD
of the minima in
wave
approximated by
be
to
a
sizes in small
intraparticle
shape
grating
so
of the
a
occurs
hardly
can
interaction
particle
no
in¬
can
be
the diffraction pattern is
that Fraunhofer diffraction is
angle light scattering
interference
result of interference effects of the
P(q).
also very well. In
aperture of the particle. Consequently
regime. Similar
used for
274
=
well,
electromagnetic
obtained in the Fraunhofer
mainly
r
1, Fraunhofer diffraction of the light
formation about the internal structure and the overall
angles
with
satisfied, RGD works
large particles
penetrate and therefore the scattering
of the
spheres
latex
text.
approximation fails by calculating
However,
calculated with RGD and Mie
effects,
light
instruments.
the static structure
scattered
by
different par-
13
Light Scattering
Static
4.1
light
tides. Each
particle
if there is
regular arrangement
there exist fixed
particles,
a
particles, taking
S(q)
given
is
of the
i.
an
e.
means
system
form
they
structure is the
a
particles
by
S(q),
particles.
a
lattice in
less strong than in
neighbours.
nearest
correlation function
pair
(20)
particles,
Correlation between particles
1 for all q.
=
a
crystal. However,
crystal
a
and
expression for
An
the
consequently
the local
which is related to the static structure
g(r)
of
figure
particles
6
finding
examples
a
distance
as
in
an
almost
and
g{r)
are
not
to oscillations of the
perfectly regular
structure factor shows
interaction
is shown in
crystals
potential
(B).
a
particle
sions, however,
long-range
narrow
potentials compared
relatively
that show
between the
shows
at
particles
An
is
pronounced
crystal
static structure
The reason, while the
peaks
in
do not form
particles
particles
and the rela¬
crystals formed by atoms,
the static
Nevertheless, colloidal suspensions
Bragg peaks depending
distances goes
a more
interaction
to
peaks.
particles.
a
is the finite temperature, which leads
on
example for
only
mainly the peak for the
larger
show
broad
shown. In
pair correlation function
and the
particles.
the
i.
perfect lattice (A). The
Because in colloidal systems
The order of the
consequently S(q)
finding
are
lattices due to the Brownian motion of the
weak interaction
also form
infinitesimally
particles.
potentials
Bragg peaks
reflects the fixed distances between the
particle
from the
r
for different interaction
arranged
are
at
(21)
particles. g(r) directly describes
of the
density
particle j
a
pJ<Pre"*\g{r)-l]
l +
=
factor then shows the characteristic
can
scattered
through
probability
tively
light
ensemble average. For uncorrelated
are
some
where p denotes the number
S(q)
the
correlations between
spatial
local structure similar to
S{q)
the
superposition of
the
or
an
particles, S(q)
ideal gas of
the order is in the range of
In
by different
scattered
light
jj £ (e^-1^)
=
bracket denotes
interactions between the
factor
due to interactions between the
between the
A
But
with
angular
that
particles
into account interference reveals the static structure factor
%)
where the
angular dependence given by P(q)-
an
path lengths.
of the order in the
measure
with
phase differences
due to the different
particles
all
a
scatters
a
the concentration and
hard
sphere suspension
in the range of nearest
nearest
neighbours,
quickly to unity. Charge
the
neighbours,
probability
stabilized suspen¬
order at similar concentrations due to the
potential (C). Dependent
on
the
strength
of the interaction
LIGHT SCATTERING THEORY
4
14
S(q)
A 9
O—Ç^^^^D
o
O-O^y-O-O
il
il
9—T^TT^^YY
M
TTT^YtYY
0—0—6—O—C^<^<)
9—9—9—9—9~
B
o
C Q
Figure
6:
Structure, pair
correlation
function g(r)
and static structure
factor S(q)
for different systems.
potential
and
g(r)
the
particles
show
more
can
order similar to
a
pronounced peaks than
The static structure factor
S(q)
can
lattice,
in
a
with the consequence that
hard
core
S(q)
suspension.
be calculated from 21.
For
monodisperse
15
Dynamic Light Scattering
4.2
scattering angles
and in the limit of low
particles
compressibility (dli/dc^1
suspension by
of the
S(0)
In
virial
a
expansion
S(0)
of
0 it is related to the osmotic
—y
q
^kBT (dlL/dc)-1
(22)
2A2Mc +
(23)
=
as
5(0)
-
•
•
•
is related to the interaction
A2
the second virial coefficient
1
=
pair potential V(r)
through
A2
Combining equation
^^/[l_e-^lr2^
=
13 with the
in terms of the second virial coefficient
and the concentration
is
c
can
given through
a
in the limit of q
particles
As described before in
+
equation
in the dielectric constant
fact, that
from the
in
a
solvent, light
liquid
light
particles
is not
a
scattered
in the
is
As
to
an
potential
ratio
AR(q)
(25)
2A2c
can
mainly scattered by
by
different
is the
general
intensity
at two times
cause
particles.
the
particles
are
particles change randomly
of the
an
fluctuations,
analysis
of the
But if
constantly
in
suspended
a
so-called
r, the
r
or
of
and also the number of
a
fluctuation of
diffusion of the
particles
information about the diffusion
in terms of
intensity fluctuations
to
looks like statistical noise.
separated by
Particles
Both effects lead to
[14, 20]. Referring
have different values.
Due to the
different refractive index
a
Because the Brownian motion
time correlation function
intensity
is due to local fluctuations
light
scattering experiment the phase relations
a
volume fluctuates.
be obtained from
the scattered
of
the index of refraction of the medium.
result in
a
scattering
suspension
process
or
scattering
5
static system, rather the
scattering intensity.
in the
a
and the interaction
suspension the particles usually have
Brownian motion.
the
yields
Dynamic Light Scattering
4.2
the
0
measurement.
AR(q)
a
—>
obtained, if the Rayleigh
be
J\c
in
particle form factor (equation
of the
(equation 23)
from which the size and form of the
expression
(24)
J
L
approximations
and the static structure factor
19)
J
Ml
figure
7 the time
Nevertheless,
if
dependence
one
looks at the
intensity values I(t) and I(t
is very small
compared
of
+
r)
do in
to the characteristic
LIGHT SCATTERING THEORY
4
16
<I2>
+
<I>
<I>'
Time
Figure
7:
function.
Fluctuations in the scattered
For details
text.
see
fluctuations, I(t)
time of the
intensity and the intensity auto-correlation
and Ht +
r)
correlated, while with increasing separation
A
measure
to
ergodic systems
so
that it
can
relation to the
starts with
a
a
+
r)>
as
to
=
maximum value of
decay
[29].
0
temporal fluctuations
in
(I2)
are
tric fields the
described
a
DLS
experimentally
by
=
a
of
the
starting point
intensity correlation function
decays
experiment
shown in
in the
figure
course
and its
7. The correlation
of time to
a
the fluctuations in the
intensities
are
available property is the
(I)2
with
(27)
measured rather than elec¬
intensity correlation function
(28)
=
l +
through
the
Siegert
relation
|5r1(ç,r)|2
Gaussian distribution of the
monodisperse spherical particles
amplitude
<M«,0P>
is related to the field correlation function
assumption
on
{Es(q,t)E*s{q,t + T))
(\Mq,m
52(ç,r)
of
The
independent
(26)
normalized field correlation function
*(*'t)
under the
the correlation will decrease.
time ra.
gi(q,r)
Due to the fact that in
somehow
/T /(*)/(* + r) dt
intensity is
and
are
rT
dynamic light scattering experiment (DLS)
of the electric field
g2{q, t)
11
lim
=
r
close, they
intensity auto-correlation function.
the infinite time average is
be chosen
the characteristic
In
in time
of the correlation is the so-called
(I(t0)I(t0
For
will be very
(29)
intensity profile.
the field auto-correlation function
can
In the
case
be written
17
Dynamic Light Scattering
4.2
as
9i(q,T)
=
S(q,T)^
(30)
where
N
JV
i
%>^)
iŒ&A^M0)-r'(T)]}
=
ly\° I
H(q)
denotes the dynamic structure factor,
in the
suspension and S(q) is the static
6
interactions
in the system the
function with respect to the
lag
describes the
hydrodynamic interactions
structure factor.
dynamic
time
(31)
1,3 = 1
In the limit of
structure factor is
a
negligible
single exponential
r.
gi(q,T)~e-D^
(32)
where Dc is the collective diffusion coefficient of the
q is the
scattering
For the
case
of
particles
polydisperse suspensions
in the
inverse
tion
G(T)
Laplace
of the
transformation is
present in
a
more
high
suspension and
a sum
on
In
of exponentials for each
particle
principle
gi(q,r)
is not anymore
it
can
an
ill-conditioned
function,
Therefore it is
statistical accuracy when
evaluation of
particle
and
30
by
problem,
using
integral
over
sin¬
species
be done
by performing
in order to obtain the size distribu¬
e.
any statistical
will lead to different
important
size distributions.
an
i.
a
size from the correla¬
particle suspension [30, 31, 32]. Unfortunately
measured correlation
equation
particles g\ (q, r)
it is rather
complicated.
transformation
the size distribution.
with
of
The evaluation of the
suspension.
tion function then is
an
in the
vector.
gle exponential function of the time,
of
particles
to
the inverse
Replacing
measure
Laplace
Laplace
noise, which
possible
is
ever
results for
correlation functions
transformation for the
the discrete
the size distribution
the inverse
sum
gi(q,r)
in
can
equation
be
31
expressed
through
91
This equation
can
{q>T)
be written
_
fN{ryP{qr)e-D^rdr
~
f0°° N(ryP(qr)
dr
^
as
9l(q,r)= /
Jo
G(T)e-TrdT
(34)
with
N{ryP{gr)
Io°° N(r)reP(qr)
eThen
H(q)
=
1 and also
%)
=
1.
dr
{
}
and T
=
expanded
Doq2
is the
decay
From
a
around the average
so-called cumulant
the coefficient of variation
For
rate.
decay
=
be
can
a
i +
analysis [33]
of
-(r)V
gi(q,r)
(36)
+
decay
the average
rate
(F)
and
be obtained.
a can
Cross-Correlation
4.3
The evaluation of data from
described
above,
is based
on
static
a
interpretation
scattered
suppress
light
general
contributions from
see
[2]).
simultaneously
vectors
the
same
kn and fc;2, and
seen
(Gii (r))
tered
have the
photons,
same
can
GffCr)
GSM
where I\ and I2
are
=
=
vector q.
light
by performing
volume
(with
two
multiply
However, while
exist
a
if multi¬
technique
to
scattering experiments
two laser
at final
wave
beams, initial
vectors
and
wave
kf\
kf2)
two detectors.
+
scattering
(37)
r))
vector q, but
multiple scattering,
S(q, r)
the
use
different scatter¬
corresponding
relations
and the measured auto-correlation
where
(1)
indicates
singly
scat¬
as
li1)2{l+ß\S{q,r)\2)
/^/^(l+^l^r)!2)
-Sq2R2
=
an
occurs,
and suppress undesired
seen
(38)
by detectors
of the cross-correlation function
ßu
scat¬
dynamic light scattering experiment (for
the average scattered intensities
respectively, and the intercept
light
extremely difficult
is
scattered
(Gi2 (r)) function,
then be written
only singly
as
so-called cross-correlation exper¬
a
(h(t)I2(t
structure factor
and cross-correlation
a
by the
=
and in the absence of
dynamic
S(q, r)
positioned
two detectors
experiments then
between the
in
scattering
G12(r)
ing geometries,
singly
be achieved
cross-correlating the signals
If both
scattering
contributions with
multiple scattering
can
of the
scattering intensity, there
idea is to isolate
This
on
structure factor
contributes to the
multiple scattering
iment. The
details
dynamic
is
due to the fact that
impossible
has lost its close relation to the
the calculation of the
light
the laser
multiple scattering
If
of the data is difficult if not
light
scattered
dynamic light scattering experiment,
or
assumption that
the
tered before it reaches the detector.
and
gi(q,r)
(T)
rate
e
-
size distributions
narrow
P-{T)r
9i(q,r)
ply
LIGHT SCATTERING THEORY
4
18
-Sx2
ße-^-e—
.
G12 (r)
is
1 and
2,
given by
(39)
19
Cross-Correlation
4.3
intercept ß detected in the auto-correlation experiment primarily depends
The
due to
intercept
the additional terms describe the reduction of the
optics, and
the detection
on
vectors gi and
q2)
volumes).
single scattering,
same
For
In the auto-correlation
Gtii(t)
very difficult if not
above
scattered
multiply
denotes the
This leads to the
Gu(t)
where
I3
scattered
is the average
photons)
only. Such
structure factor
visible in the
S(q, r)
even
experiments
angle
tors,
as
outlined
experiment
on
will result in uncorrelated
light
due to the fact that it has been
factor of order
a
wave
(RSk^)-1,where
vector combinations
following expression
for
kt2
experiment
l)j
'
contains the
should thus
kti
G\2{t)
all contributions
over
5k 3
—
ß^I^lSiq^y
summed
experiments
can
be realized
In the Phillies
provide
of 0
light
=
90°
(figure
from the wrong
8
A).
For
(40)
(singly
singly
us
and
multiply
scattered
with the
light
dynamic
will be
multiple scattering
be decorrelated
using
(figure
a
8
B).
experiment
[3]
must be
signals
geometries
scattered
light
light
with
a
prevented
two
variable
(figure
8
single
scattering
experiments use different
from the wrong
from the wrong
C).
a
to enter the detec¬
experiments
experiments
geometry where they produce different scattering
This is realized in the 3d set-up
for the
from both
Therefore it is limited to
set-up [5]. Here the
The
different
experiments
and interference filters block the
front of the detectors
using
set-up
is measured with both detectors.
it is done in the two colour
wavelengths
S(q,r)
deduction of
decreasing intercept only.
scattering angle
either
scattered
changes.
contribute to
photons
a
the
yield
intensity fluctuations
correlated
for turbid suspensions, and
scattering experiments.
two
scattering
the situation
in the cross-correlation
measured at detector j, and
Cross-correlation
[2].
intensity
cross-correlation
a
match of the
scattering
q-vectors, and the contributions from multiple
hh +
«
scattered
of the data and
of the smallest of the two
magnitude
kfi [2].
multiply
suppressed by
are
spatial
multiple scattering,
background only
succession of different
signal
is the
produce
will
light
In contrast,
to the
scattering
and k%i +
a
[<£x[
interpretation
fluctuations that contribute to the
scattered in
the
impossible. However,
only singly
both detectors.
an
of
case
experiment
well and make
as
=
describes the match of the
both auto- and cross-correlation therefore
in the
However,
\Sq\
=
misalignment [Sx
and
information.
(Sq
mismatch
phase
can
in
also
vectors
LIGHT SCATTERING THEORY
4
20
counterpropagating beams
90°
fixed scattering angle 6
=
qi
=
-02
wavelengths A,1 * X2
variable scattering angle, but
difference angle 8q depends on 0
two different
scattering angle 6
3-dimensional scattering geometry
variable
\
11
fl
Figure
8:
An outline
schemes. A:
4.4.1
colour, C:
B: Two
Phillies,
vector
arrangement for different cross-correlation
coding
3D
Intercept
of the Cross-Correlation Function
intercept ß of the correlation function
of the
can
dynamic light scattering experiment,
function in the limit of
for
wave
3d Cross-Correlation Scheme
4.4
The
of the
auto-correlation,
i
=
r
0 and
—>
1, j
The theoretical maximum of
=
ß
is
r
—>
be viewed
i.
oo,
e.
=
1 for
an
the
signal-to-noise
ratio
the difference in the correlation
<7jj(0)
—
2 for cross-correlation
ßu
as
gij(oo)
=
ß (i
=
j
=
1
experiments, respectively).
auto-correlation
experiment, while
3d Cross-Correlation Scheme
4,4
in
cross-correlation
a
[2, 14]0.
The
experiment ß depends
for this
reason
21
two detectors 'sees' the
light
that in
is,
the geometry of the beam
on
cross-correlation
a
G12(r)
=
each of the
from both incident beams and therefore also from the
experiment. Thus the cross-correlation function
wrong
experiment
paths
{I{(0)P2(t))
<iï(0)i?(r)>
+
be written
can
(iî'(O)iî(r))
+
as
(41)
<iï'(0)i?(r)>
+
where Arabic numbers denote the detector and Roman numbers denote the incident
beam.
experiment the undesired contributions from the
In the 3d cross-correlation
'wrong'
incident beams
duce different
only
are
scattering
the second term
intercept
(r))
rates, the latter
match of the
lower
is
ß\2
intercept
can
experiment
obtainable
of the
instrument,
non-linearities
volumes
scattering
41
.
\ßM\g?2\2
to be considered.
points need
pro¬
In eq.
Gi2(t)
± +
theoretically
they
or
in the
Sx
5q2R2
0.257.
=
take
place
due to detec¬
and due to restrictions in
high
count
Any misalignment, either
in the
mainly
take effect at
scattering
vectors will lead to
a
intercept described by
give
0.9/?0)
an
for
scattering
not differ
7In the
488
therefore decorrelated.
=
ß
To
are
but
photomultipliers,
3(/x)(l2) + (iï(0)ff(r))
{ii + r?M + i?)
[34], misalignment
two
the
=
overlap volume. While detector
spatial
q2 and
for the 3d cross-correlation
tor non-linearities
by
correlated contributions to
gives
A reduction of the
the
^
vectors q-y
Q
9l2{
The
still detected
nm
example
a
of the
frequently
=
precision
(44)
ßoe-Re'^~
in
alignment required
used value for the beam radius of R
volumes should be closer than 5 pm and the two
more
than
two colour
and A2
=
514
5q
=
0.005
from
an
Ar+-laser)
therefore for the cross-correlation function
M
„
g"{)
we
-
~
=
intercept
of
ß
=
intercept
=
100 pm the two
scattering
are
two laser beams different in
used.
ß
=
vectors should
1.
get
(see
eq.
=
'wrong'
incident laser beam and
41)
(Ji(°)W)
IwfT
l +
wavelength (Ai
Narrow bandwidth interference niters in
front of the detectors block the undesired contributions from the
with the maximum
of
an
//m-1.
experiment described above
nm
for
/?Mbf2|2
(
}
LIGHT SCATTERING THEORY
4
22
4.4.2
Volume
Overlap
Because in
cross-correlation
a
volume
seen
will decrease
ßxi
one
incident beam
9 shows the
consequently. Figure
by
be
can
angle 5zi
of the two beams it is
distance to the center. The
of the
projection
If the diameters of the detection beams
dent
the
beams, there
overlap volume of the
of the
projection
approximation
a
regions detected
exist
a
volume at
l/sin0 larger
of the cross-correlation function
are
the
volume
Because of
intensities
scattering
two incident beams and
/;
equation
the
are
(including
~Wm
~
those
is
a
decreases for
regions
0
=
are
=
90°.
=
are
outside
10°.
It is to first
volume at 0
=
incident beam. The
90°, but
intercept
(45)
w>
from the
intensities
are
originating
illuminated
ß\2
e.
by
overlap
scattering angle,
i.
overlap
as
from the total
one
beam
only).
volume and the
above,
the
Therefore
ßn
discussed
ß\2(Q).
=
volume of the
scattering angles and has its maximum value
at
90°, and this effect is stronger for larger values of the difference angle 5^.
Goniometer
can
of the
be determined
Angle
In the 3d cross-correlation
plane
~
which
the
of 0
9
as
scattering angle,
and for small
Experimentally /?i2(0)
4.4.3
on
figure
indicates the
region
of 6
45 and the fact that the ratio of the
function of the
large
one
originating
scattering
scattering volume Vov/Vsv depends
intercept
shaded
scattering
only by
volume of
in
photomultipliers that
scattering angle
than the
increasing
than the diameters of the inci¬
brighter
be written
can
ßl2
scattering
a
of the volume is illuminated
larger part
where I?v
The
region
scattering angle
a
larger
from the
two beams.
scattering
factor
are
of the
is marked. Due
common
The dark shaded
volume at
scattering
detectors,
the
by
smaller with
is the
scattering volume, however,
the incident beams and the detector beams.
reflects the
seen
overlap
getting
scat¬
both incident beams.
geometrical arrangement
two incident laser beams. The volume where the two beams
to the difference
from incident
intensity originating
must be illuminated
by the detectors
regions where only
If there exist
the
intensity originating from incident beam 2, the
beam 1 is correlated with the
tering
experiment
goniometer,
and
QGom
are
a
highly
diluted
sample.
Scattering Angle
experiment the plane
and the two
not identical. Therefore the
denoted with
by measuring
of
symmetry, which is defined by the
scattering planes
scattering angle
of the individual
0 and the
not identical. The difference
angle
depends
on
of the
experiments
goniometer,
the difference
angle
23
3d Cross-Correlation Scheme
4.4
Figure
9:
tering angles of 0
(dark grey)
90°
=
and scattering volume
(fat lines)
volume
Overlap
and 0
(slightly grey).
10°
=
(shaded areas) for
scat¬
For details
see
text.
53d of the experiment and is also
a
sketch of the
scattering- and
vector
wave
difference
a
function of the
angle
given through the trigonometric
are
.
CD
0
=
—
OC
2
OA
5
cos2
expression
for the
•
=
m
projection of the triangle OCD
symmetry, the distances AB and CD
results in the
(46)
OA
sin
OAB is the
functions
AB
®Gon
triangle
10 shows
arrangement of one scattering experiment. Goniometer-,
sm
Because the
goniometer angle. Figure
are
identical.
Combining
on
the
the
plane of
equations
in 47
scattering angle
&
0
®Gon
/,7x
sin—= cos-sin
2
In the limit of low
at
&Gon
=
from ©Gon
—
Aq/q
<
scattering angle
is shown in
5° to
scattering angle,
vector of
goniometer angle ©oon
180° the
goniometer angle
2
©Gon
=
~*
=
0, ©Gon
©Gon
11. In the actual
—
5.
and 0
are
©Gon
=
angular
125° results in
a
identical,
while
The relation of 0 to the
range of the instrument
125° the maximum difference between
located at
2%.
figure
is 0
K 47 J
2
goniometer-
deviation of the
and
scattering
24
LIGHT SCATTERING THEORY
4
Figure
10:
Sketch of the
explain the relation of
difference angle 5.
the
vector
wave
arrangement of
scattering angle 0
The shaded
plane
is the
to the
scattering experiment
one
to
goniometer angle ©Gon and the
plane of symmetry
or
the
goniometer
plane of the experiment.
180
160
-,
1
1
1
1
j
1
^—
-_
—,^—
140
—><^—
120
100
~7*^^
-_
S^
—?<£-
CD
80
—^^—
60
40
—,^—
-_
20
0
1
1
~/>
—
^—i—i
0
1—i—i
20
1—i—i——i—i—i
40
60
1—i—i
1—i—i
100
80
1—i—i
1—i—i
120
140
1—i—i—
160
180
Gon
Figure
for
a
4.4.4
11:
Relation between the scattering
difference angle of 5
=
angle © and the goniometer angle ©Gon
14°.
Correction for Static
Light Scattering Experiments
With the 3d cross-correlation instrument also static
in turbid
samples
can
light scattering
be done. The undesired contributions from
measurements
multiply
scattered
4.4
3d Cross-Correlation Scheme
light
to the measured scattered
Is is determined
intensity Is
cross-correlation
dynamic
bined static and
25
dynamic
by performing
a
com¬
In the static measurement
experiment.
function of q, while the
as a
be excluded
can
gives
measurement
the
part of
Is which is only singly scattered via the intercept ß\2 of the cross-correlation func¬
tion
In the measurement the reduced
(see equation 40).
ß\2
where
is the
intercept measured
contributions. This
ing
volume and
of the
can
automatically
a
ß\2(ß\2
is
multiple
scatter¬
as a
an
used,
by the overlap
corrects for the effects caused
single scattering intensity
The
without
sample
diluted
of the instrument, which lead to
misalignment
intercept.
with
intercept
angular dependence
function of the
scattering
vector
then be calculated from
i{1)(q)
yji[%)ii%)W(„\
=
%h{q)h{q)
-
(48)
\ß(i)
12
If
rectangular
cells
together
used in the
experiment
sample cell,
the measured
with the 0
in order to
measure
sample,
angle.
As
a
decrease at
because the
result,
surface of the
to
90°,
53d,
the
i.
e.
—
=
given by
T
(90°
I/Io,
The surfaces
indexes for water
—
©Gon)/2. Neglecting
of the
incoming light
for this
case
inside the
effects, namely for
turbidity
cell and for the
vary with the
and the total
is
the effect caused
parallel
water-glass
nw
and
and
glass
ng
angles
equations
to the
scattering
scattering intensity
goniometer angle ©Gon
by the
sample
the transmission T
with their
difference
angle
cells surface. From
through
light intensity for
glass-water
an
the
sample
incident
corresponding
cell is
intensity
refractive
result in transmission coefficients of
(2 sin 9q
6w
nQ
cos
9g
nw
cos
9w \ sm($w + @g)
cos
Ow
nGcos9G
these
light
where i" is the transmitted
nw
ing
scattering
light paths s(©)
scattered
_
A scheme of the
the
cell under half of the difference of the
equations [28]
=
single
light paths s(©)
are
20 set-up of the instrument the incident laser beams hit the
sample
polarization
the Fresnel
Io-
9
both the
of the
on
chapter 5,
different from 90°.
angles
Due to the 0
length
with short
to be corrected for two
intensity has
Fresnel refraction for non-incident beams
of the
20 set-up, described in
—
cos
(2s'm9w
\sin(0G
cos
+
I
(49)
/
9g\
9w) J
used for this calculation is shown in
with the Snellius law sin
9q/ sin 9w
=
figure
nw/na
12.
Combin¬
the transmission
Sketch
12:
Figure
refractive index
mission
through
through
a
of
sample
glass plate
Indicated
nw
the
a
sample
where na
=
1-54 for
=
sample,
s~l
used
angles
256
the
sin8(0G
can
for
index
by
îîg
in water with the
the calculation
a
of
the trans¬
non-incident beam
Fresnel refraction is
sin4 9q cos4 9w sin4 9w cos4 9g
nw
=
+
(50)
9W)
1-33 for water.
inside the
optical path length
which
hx(Io/I),
the
refractive
cell with Fresnel equations for
quartz and
If s{9) denotes the
of the
are
with the
cell filled with water caused
T
t
LIGHT SCATTERING THEORY
4
26
be calculated from
angular dependent
a
sample
cell and
r
is the
turbidity
transmission measurement
correction for the
sample turbidity
is
using
given
by
TT(9)
=
e~TS^
(51)
The transmission corrections for non-incident beams
correction)
the
and for the
turbidity
r
=
sample turbidity
1 mm-1 and
a
are
path length
shown in
of
so
=
1
on
the
figure
mm.
sample
13 for
a
cell
(Fresnel
sample
with
4.4
3d Cross-Correlation Scheme
1
27
-
~~~
-
Fresnel
0.8
-
turbidity
-
e
_o
Ü
g
0.6
-
o
o
Ö
o
0.4
-
§
^^~^-~^~~
-
0.2
0
—
-
-
i
i
.
|
20
i
i
i
|
i
i
40
i
|
i
.
60
i
80
Goniometer
angle
6
Figure
13:
Correction for the transmitted intensity
placed
in
©
a
—
2©-goniometer. Shown
for sample turbidity.
are
100
120
140
Gon
through
a
rectangular sample
cell
the Fresnel correction and the correction
Realization
Experimental
5
A schematic
layout
experimental set-up
of the
with turbid colloidal model
argon-ion
the
light
values
then
7,
EXPERIMENTAL REALIZATION
5
28
laser
8 and 9. A
in order to
300-8) operating
(Spectra Physics,
the
as
light
bring
the
illuminating
convex, focal
which
are
length
at
a
is much
sThis
described
m a
=
higher,
is
127) operating
632.8 nm8
for details
chapter
(Dantec
mode fiber
then focused onto the
even
A0
at
a
sufficiently large
In
&
Spindler
5
[2]).
higher
our
current
by
is used
The
experiments described
scattering
in
set-up
chapters chapters 7,
angle
R
=
50
we
=
14: An outline
of
the
14°,
factor of
and 63,1
an
index match
For the
—
experiments
18° which results
10-3
photomultiplier
have
order contributions
8 and 9
ran
This
efficient
scattering volume and 5
cell is immersed in
were
an
split
the lens
Hoyer).
a
nm
chapters
in
required for
an
nm as
633
=
60x30)
cell
scattering
90 mm, diameter 63 mm,
6 beam waist and difference
suppression factor of 4
Figure
to
DAN
(The efficiency for suppressing higher
see
488
=
suppression of double scattering by approximately
valid for the
m
wavelength of Ao
experiments described
chosen values of R < 50 pm. for the beam waist at the
7 TO-3 for A0
were
experiments
obtained with
sample turbidity
suppression of contributions from multiple scattering.
a
6
14. The
beam to the instrument. The laser beam is
lens has been chosen in order to allow for
which results in
figure
chapter
model
for the
source
polarization preserving single
parallel beams,
(planar
L\
HeNe laser
a
implemented
was
Innova
described in
To extend the measurable lange of
source.
(eq. 10)
into two
(Coherent
suspensions
is shown in
optical
fibers
experimental set-up for 3d cross-correlation.
29
vat
(ALV 88/ALV,
Water is used
heat
without flat entrance and exit
the index
as
exchanger coil,
guided by
is collected and
(Fi, F2,
OZ
side, and
a
Optics) [35, 36].
fiber connectors to
a
the scattered
combination of
light
with
GRIN lens and
a
home-built
housing
the
for
a
scale
than 1
mm
important.
across
(photon transport
principle
In order to avoid
free
mean
signal,
the
general
length
and
that
are
of
we
half the rotation
the
outlined in
a
by using
figure
displacement
For
so-called 0
angle (90°
—
0)
is
s
be less
cell becomes
should be of the order of L.
flat cells.
However,
displacement.
15.
We
use
experiments
—
at
in
recover a
a
combination of
motorized rotational stage above the
a x
—
of the cell to
is rotated
can
by
situation in which
This 0
y translation
sample position, which
path
different from
sample
of the incident and scattered beam almost cancel.
geometry is implemented by using
mm
scattering angles,
scattering angles
symmetrical
this will
implemented
a corner
20 geometry, where the
in order to
angles,
square cells with 10
at low
In
since these cells
We therefore
superior optical quality for experiments
turn the cell in
where
easily
may
with beams at non-normal incidence
optical path lengths.
(ALV
characteristic
a
scattering
position them such that the scattering volume is located
have short
90°
approach
exp(—s/L),
path length), which
light path
lead to considerable beam deflection and
different
~
correlator
attenuation of the
strong
and L is
light
H5783P-
strong reduction of the singly scattered light and
a
this could be achieved best
will then be used in
a
due to the
scattered
singly
digital
a
for dense colloidal suspensions, the size of the
in
fc/2
amplifier/discriminators
two
and fed into
suspensions,
the cell for
subsequent reduction
is
mode fiber
(PM;
tube
photomultiplier
intensity which essentially decays like 1^
optical path
length
the
For turbid colloidal
scattered
single
a
(L2)
and
kf\
vectors
wave
a
The end of each fiber is connected via standard SMA
(C3866 photon counting unit, Hamamatsu)
singly
An identical lens
thermostat.
a
01, Hamamatsu), and the PM signal is processed by
5000/E, ALV).
mm).
fluid. The vat temperature is controlled via
matching
which is connected to
used for the detection
outer diameter 85
windows,
—
20
stage and
a
be controlled from
within the ALV correlator software and allows for automated measurements of the
angular dependence
of the static and
dynamic
structure factor.
EXPERIMENTAL REALIZATION
5
30
•
K
1
/
„
90°/
90°-e
Figure
15:
Description of the 0
—
20 geometry used in order to work with
rectangular
scattering cells and minimized optical path length.
5.1
Description
Detailed
of the
Opto-Mechanical Compo¬
nents
Producing
5.1.1
Using
a
compared
more
mode
single
to
a
Parallel Beams
optical fiber for
'direct' laser beam. The
precisely,
the
Gaussian
as
only
filter. Out of the fiber
intensity profile. Furthermore
the laser beam must be
whole
light scattering
lasers
working
a
single
coupled
instrument. As
at different
a
to the
always
only guides
instrument,
comes
exchange
significant advantages
a
as
can
single
light
it also works
of the laser is easy to
mode
be used.
the
or,
as
beam with the desired
into the fiber without
long
wavelengths
collimator is included which allows
an
has
mode fiber not
TEMoo-mode from the laser
high quality spatial
a
light
the incident laser
perform
re-alignment
operation
is
of the
fulfilled,
even
In the used DANTEC fiber
variation of the focal
point from negative
Detailed
5.1
of the
Description
values to
approximately
the beam
propagation
200
w(z)
radius is
The
through ©div
and
\
divergence 0^
\/(nw0).
=
divergence angle
resulting parallel
is small
from the
of ©jw
a
distance
5.1.2
Lenses
beams
Wo
difference in
that the beam radii
are
mounted
nearly
scattering
the
optical
volume. Therefore
axis. This system
infinity.
a
focal
planoconvex singlets
length
of
/
90
=
optical
error
optical axis,
a
the two
distances
of
generation
rotation and tilt.
prism tables, which allow
are
an
used to generate the difference
are
But,
even
because the
spherical
located in
They
used.
optical axis,
distance of 1.5
vat, filled with
cm
from the
water with
calculated to 53d
=
14°.
position
on
a
is
set-up
optical
a
are
infinity
planoconvex
a
focal
common
to
image
optical
on
also
In the 3d
lens.
mounted with
to the
an
points
clear aperture of 60
x-y-z translation
a
parallel
mm
two axis lens
a
stage. The main
axis is
spherical
of the two beams is fixed with respect to the
abberation is not
a
with
arranged
object
of the lens system for beams
For the 3d instrument
chosen,
axis.
issue in the 3d
an
where the two
Taking
refractive index of
From Gaussian beam
n
set-up.
parallel
beams have
into account the index
=
1.33 the difference
propagation
the
place
a
matching
angle
can
be
of the beam
given trough
z'
=
(53)
7-TT2
z2 +
TT-Wf
where zf is the distance of the beam waist from the focal
image
equal
identical at
made of BK7 with
mm are
which allows to tilt the
abberation.
they
projects
holder,
waist is
nm
divergence
3 cm, the
and the mirror for the
The lens of best form for this task is
instrument two
and
632.8
=
splitting
due to the beam
are
given
This results in
is used.
path length of
splitter
on
mm
A is
wavelength
between the incident and the detection beams and to focus the beams at
the
in
from the beam waist where the
HeNe-laser with A
1
=
The transmission and detection lenses L\ and L2
angle 5^
z
a
Although
volume. The beam
scattering
parallel
a
adjusted,
(52)
of the laser beam with the
10-4.
•
beams have
enough
the two
2
laser beam is
w0
In the actual set-up
=
oo)
—)
(
1 +
beam waist at the exit of the fiber
a
angle
Wq
=
—>
by
then is described
denotes the beam radius at
wQ.
(/
A collimated
mm.
w(z)
where
31
Opto-Mechanical Components
side and
z
accordingly
for the
object side, /
point
is the focal
of the lens at the
length
and w0 the
beam waist before
with
a
short focal
focussing.
length,
calculated
For
in the 3d set-up, z'
as
i.
0,
=
/
=
90 mm,
then is calculated to be Wf
Wf < 100 pm
using
a
w0
36 pm.
=
pinhole,
With the detection unit
the beam
paths
formed
lenses. The lens and
The radius at the beam waist Wf
=
1
and
mm
z
26
=
Experimentally
different
be
estimate if
rough
a
through.
a
symmetric arrangement of
the detection
by
optical
single
optics has
mode
scattering angle
incoming
the
laser beams and
to be fulfilled.
fibers
optical
It consists of
with
equipped
grin
fibers mount allow tilt and translation in x-y-z direction
To allow measurements at
the whole detection unit is mounted
on
the
arm
of the
goniometer.
main
Alignment
5.1.4
of the Instrument
Due to the fundamental
light scattering
principle
for the two individual
placed
with
an
identical
alignment
leads
strongly depends
immediately to
investigated
a
on
the
of the
and of the statistics of the
statistical accuracy
In this
particular
experiment,
compared
to
a
impor¬
scattering
vectors
scattering
volume
i.
e.
on
each other at the
symmetrically
scattering
a
any mis¬
to the
volume
instrument. For the detection of the scattered
points
at the
plane
of the
placed
light
an
on
sample
partially
prolonged
realization of the 3d instrument described
and to direct the two beams
ratio of the
alignment
(nearly) perfect aligned
used to focus the two laser beams with the focal
cross
an
limitation of the maximum turbidity of the
instrument the duration of the measurement has to be
same
is
signal-to-noise
quality
alignment
the
implemented
is
It should guarantee identical
scattering experiments
cross-correlation function
aligned
technique
the rotational axis of the instrument. As the
on
technique
of the cross-correlation
instrument where this
tant and often also difficult issue.
they
can
The beam diameter
cm.
obtain
we
for the lens and in y-z direction for the fiber mounts.
the
the waist of the focussed
e.
where both beams have to pass
lens L2 identical to L\ and two
to be
lens
Detection Unit
5.1.3
a
a
using
In the present set-up
of
by
well collimated laser beam focussed
a
point of the lens.
beam is at the focal
a
EXPERIMENTAL REALIZATION
5
32
mis¬
to obtain
one.
above,
a
lens is
scattering volume,
goniometer
so
that
the rotational axis of the
identical lens is used
together
Detailed
5.1
with two
a common
laser beams. This
scattering
plane
of the
goniometer,
that the bisector of the
must
The
at
alignment procedure
justment part,
a
flow chart
goal
order to
part
are
ensure
to
that
they
generate
the instrument
two
the
can
vectors and of
paths
of
zero
of the two
goniometer
plane perpendicular
the
degree
a
laser beams.
mechanical and
respect
parallel laser beams which
an
optical
is shown in
of the
housing,
to the main
and
placing
figure
beams, respectively,
are
ad¬
16.
the index
goniometer
in
optical
the rotational axias of
the two lenses in such
in the middle of the two
the axis without the lenses.
cross
of the
plane
the detection beam
illuminating
alignment
with
in the
lays
alignment procedure
are
course
the rotational axis of the instrument
be divided into
the
a
the
is thus to focus
have identical rotation axis. The main steps of the
two incident and detection
lays
paths symmetrically to
between the two beams
describing
perpendicular,
rotational axis which
on
33
alignment procedure
scattering angle
a
and the second
the beam
beams, which define
point
The main steps of the mechanical part
matching vat,
of the
laser
on one
angle
completely overlay
pick up
guarantees identical scattering
incoming
goniometer. Furthermore,
paths
that
volume. The
with the first lens the two
such,
Opto-Mechanical Components
single mode optical fibers
illuminating
to the
of the
Description
crossing
a
in that
points where
way that the
point
the beam
on
paths
the
hit
5
34
Centering of
Centering
Mounting
Hlliill
goniometer
of the second
the Dantec fiber and
passing
housing
of the index match vat
111
Centering
the
EXPERIMENTAL REALIZATION
adjustment
a
(separately)
collimated laser beam
goniometer
parallel
beams
of the detection fibers and
Mounting
maximum transmitted
Mounting
of
the rotational axis of the
the laser beam into two
Splitting
unit
and
centering
Mounting
and
adjusting
intensity
of the second
adjusting
goniometer unit
the lenses
^^^11
Fine
highly
tuning with
diluted
Figure
16:
sample
a
to
Survey of
measurement with
achieve maximum
the
a
intercept
alignment procedure.
35
Characterization of Turbid Colloidal Suspen¬
6
Com¬
Using Light Scattering Techniques
sions
bined with Cross-Correlation Methods
Materials and methods
6.1
Measurements
(ACF)
were
(CCF)
correlation functions
weighted
individually analysed using
were
inverse
Laplace
transform
size distributions
CCF determination
intercept ß\2
of the
was
suspensions
The
(i.e.,
for each
also
were
investigate
were
made for
scattering angles
measurements for 10 seconds
were
cylindrical quartz
were
15°
cells with 8
latex
at
a
150° at
<
averaged
performed
polystyrene
0
<
for each
mm
correlator).
and the value
by
means
of
Measurements
increment of
an
The
angle.
inner diameter
temperature of T
=
1°, and
suspensions
(Hellma,
were
10
were
540.110-
25.0 ± 0.01°C.
used in these measurements
particles
Polybead polystyrene microspheres)
We have
performed
measurements with
in order to demonstrate the
multiple scattering
suspensions
R2
of the
obtained from
(specifi¬
Polyscience.
Results and discussion
6.2
from
each),
course
For
goniometer system (ALV/DLS/SLS-5000F single mode
compact goniometer system with ALV-5000 fast
cations:
s
in the
at very low concentrations
fiber
Materials: The
resulting intensity
obtained from the cumulant fit.
commercial
Measurements
and
scattering angle [20].
10 measurements of 120
using
inserted into
The
intensity measured
the
SLS
a
analysis [33]
second order cumulant
averaged
then
were
used
was
a
algorithm (CONTIN) [31, 32].
light scattering (SLS) experiments
static
QS).
intensity auto-
Measurement durations
taken.
were
10
60 seconds for ACF and 120 seconds for CCF determination. The correlation
functions
an
cross-
or
scattering angles 0, and
different
performed at
were
=
274 nm,
static and
of the
of
bimodal
of the
in turbid colloidal
highly monodisperse
which
respectively,
we
latex
both
suspensions
were
ratio
dispersion
technique
particles
carefully
obtained with these
of
We have
with radii of R±
=
8.77)
that the two
prepared
~
23
nm
two
and
characterized by
and
individually
The
corresponding q-dependence
dispersions
is shown in 17. The two
then mixed and the relative concentrations
Ci/c2
polystyrene spheres
to suppress contributions
suspensions.
dynamic light scattering experiments.
scattering intensity
(weight
capability
a
were
particle components equally
adjusted
such
contribute to
CHARACTERIZATION OF TURBID MODEL SUSPENSIONS
6
36
10000
1000
§,
100
)—i
10
-
1
-
50
0
Figure
17: Results
with radii
of R±
from sample
=
23
nm
0
of two different polystyrene spheres
characterization
and R2
150
100
angle
=
274
Results
nm.
from
light scattering
static
experiments with the individual and almost monodisperse particle suspensions (E:
R
=
23 nm; G: R
function of
solutions
weight
the
were
ratio
the total
as
=
nm).
274
magnitude of
the
Shown
scattering
used in the
with
pronounced particle
principle
to the
I(q)
varies
intensity weighted
size distribution.
dynamic
can
0.1
measurements the
mmol/1.
The
sample
considerably
were
For this
particles
and
with the
highest
resulting Debye length
as
a
The stock
the
same
on
of /c_1
in the accessible
scattering angle,
scattering angles
have different contributions
particular
can
concentration ratio
be calculated from the
Direct and
structure factor
hydrodynamic
depend
the salt content of the
concentration had
=
28
48°. Due
particle sizing experiment
particles.
dynamic
with
done at four
structure factor
also influence the
size and concentration of the
larger spheres
suspension
known concentrations and form factors of the
interactions which
to
scattering angle of
a
classical
a
Measurements
of the
adjusted
were
I(q)
mixtures.
form factor of the
particle components
of the mixture the theoretical
our
{Attu/Xq) sin(0/2).
=
of the mixture at
should be measurable in
dynamic light scattering.
where the two
vector q
experiments with
q-range, their relative contribution to
which in
the time average intensities
diluted and the relative concentrations
scattering intensity I(q)
to the very
are
nm
together
a
on
sample.
the
In
salt content of
with the chosen
37
Results and discussion
6.2
particle
radii and volume fractions
interactions
on
S(q, r)
neglected,
be
can
S{q'T)=
A2(q,r)
where
r6P(q,r)
~
and diffusion coefficient of
a
S(q, r)
and
can
hydrodynamic
by
thus be described
(
r*(r)A»(,,r)*-
D0(r)
and
that effects of direct and
ensure
kT/6irnr represent
=
particle with radius
r,
scattering intensity
the
respectively,
'
and
n
is the
viscosity
of the solvent.
At 0
34° almost
=
only
large spheres
the
intensity corresponds
size distribution and their contribution to the total
proximately 90%,
at 0
=
approximately 30%,
weighted
when
0.95)
=
transformation
titatively
=
86° the small
auto-correlation measurements
using Contin [31, 32]
required
the
in order to
appropriate relative amplitudes
shown in
as
figure
total latex concentration of 0.21%
a
tion is still
has
a
is
the mixture is
pronounced
very
considerably,
decay
caused
by
scattering angle
the inverse
perform
are
Laplace
in fact able to quan¬
intensity weighted
in the
for
higher particle
size
an
concentrations.
undiluted mixture
by weight. Although the colloid
we
influenced
only slightly
ments. As shown in
istic
populations
concentra¬
relatively low and the collective diffusion coefficient measured by dynamic
light scattering
actions,
intensity
suspensions (transmis¬
This is demonstrated with auto-correlation measurements for
at
ones
18.
completely changes
situation
distinct
for each
dilute
on
to obtain the size distribution. We
distribution from Contin
However, the
two
dynamic light scattering experiment
a
and
recover
corresponding
The
while
and the small
dominate the
particles
to ap¬
contribution,
same
approximately 70%
contribute
and at 0
visible in
performing
sion T
components have the
48° both
size distribution with 85%.
clearly
are
=
large spheres
the
61°,
at 0
intensity weighted
visible in the
are
completely
now
effect
on
much smaller
in the
now.
sample 9,
auto-correlation
9This has been verified using
concentrated
length
sample
transmission of T
=
inter¬
0.05. This
measure¬
20, the intercept of the auto-correlation function has decreased
are
photon path length
of the coherence
a
interparticle
the results obtained from auto-correlation
the fact that the coherence
an
turbid with
observe strong distortions at short
times
measured in
contributions from
by
an
from 2-3
is then the
length
same as
intercept
is
mainly
of the laser is shorter than the average
the shift of the characteristic correlation times
Ar+ laser with
to
times r, and the character¬
While the decrease in
experiment
cm
lag
is
an
a
genuine
étalon
approximately
for the diluted
10
effect of
installed,
m.
sample
The
multiple scattering
which leads to
intercept
shown in
figure
an
increase
measured for the
20.
38
6
CHARACTERIZATION OF TURBID MODEL SUSPENSIONS
10
1000
100
radius
10
tion
18:
of
of
the
(dashed line,
radii
Contin to the auto-correlation
=
(dotted line)
34°; (B): ©
[7, 8].
This for
=
=
on
functions
large particles
also evident from
or
such
figure
18
A-D,
=
scattering,
a
decay
times is
and any attempt to
measurement would
utterly
from
of
the individ¬
an
droplets
(solid line)
to
recover
recover a
and
radii in
the correct distribution
particle sizing experiments
to low concentrations
only.
which show the result from Contin when
completely
application
86°.
the auto-correlation function measured with the turbid
distribution of
distribu¬
for different scattering angles ©.(A):
61°; (D): Q
emulsion
the measurement
measured with dilute
particle hydrodynamic
as
1000
(nm)
intensity-weighted
with the results obtained
example severely limits the ability
of diffusion coefficients
with
based
bimodal mixtures
48°; (C): 0
calculated
theoretically
monodisperse suspensions)
concentrated
0
radius
A comparison
1000
(nm)
100
1000
(nm)
of hydrodynamic
ual dilute
radius
100
radius
Figure
100
(nm)
suspension.
applied
The
distorted due to the effects of
correct
particle
This is
to
resulting
multiple
size distribution from such
fail.
In contrast to the results obtained from auto-correlation
experiments, the
use
39
Results and discussion
6.2
19:
tion
of hydrodynamic
tion
of Contin
of
radii
the
theoretically
(dashed line)
to the cross-correlation
and concentrated bimodal mixtures
0
=
of
a
34°; (B): 0
=
48°; (C): 0
multiple scattering.
Contin have been
intrinsically
of the
and the actual
œ
plotted.
0.17 for
our
a
us
intercept
mismatch and
instrument,
actual distribution of
from
61°; (D): ©
lower value for the
phase
decay
from
an
=
distribu¬
applica¬
measured with the 3d instrument
functions
corresponding
The
intensity-weighted
with the results obtained
This is shown in
correlation function and the
ßu
=
calculated
(nm)
(solid line) for different scattering angles
3d cross-correlation set-up allows
from
1000
radius
(nm)
A comparison
(nm)
100
1000
100
Figure
radius
(nm)
radius
1000
100
1000
100
radius
0.
(A):
86°
to very
figures
efficiently
20 and 19
distribution of
suppress any effects
where the
A-D,
hydrodynamic
cross-
radii from
is much lower due to the combined effect
ideally aligned
misalignment,
and the effect of
times remains
instrument
which results in
multiple scattering
unchanged
comparison of the auto-correlation function
cross-correlation function obtained from the turbid
as
is
ßi2,ideai
an
40.
=
intercept of
However,
already clearly
from the dilute
0.25
sample
the
visible
and the
sample (see figure 20). Figures
6
40
CHARACTERIZATION OF TURBID MODEL SUSPENSIONS
1.07
01
O.OOOl
10000
100
0.01
lag tine T (ms)
Figure
from
Results
20:
auto-
and cross-correlation
experiments with dilute and
different polystyrene spheres
with radii
of Ri
concentrated mixtures
of
and R2
comparison of the measured auto-correlation function with
dilute
nm.
(dashed line)
A
and concentrated
A-D demonstrate that the
use
of
determine the correct distribution of
highly
(dotted line)
obtained with the 3d instrument
function
19
274
=
two
turbid
from
3d cross-correlation scheme allows
a
decay
times
(or hydrodynamic radii)
light
also
we
allows
to isolate the
us
single scattering dynamic
can
go
one
the differential
scattering
cross
single scattering particle
suspensions.
function
section
can
on
da(q)/dCl,
form factor
or
the ratio between the
intensity I(q) (eqn. 40).
to
even
for
multiply
structure factor
highly
turbid suspen¬
The
singly
which
with the
can
mean
intercept
scattered
intensity
scattering experiment
ent orders of
scattered
light.
us
q-dependence
of
be used to determine
the static structure factor of colloidal
contains contributions from
multiply
for
even
provide
The basis for this is the fact that the
depends
scattered
us
step further: combined static and dynamic light scattering
experiments using the 3d spectrometer
the
a
suspensions.
and therefore to determine correct size distributions
sions,
nm
(solid line).
While the measurements above show clearly that the suppression of
scattered
23
mixture and the cross-correlation
concentrated mixture
a
=
singly
To extract the
of the cross-correlation
light
I^(ç)
measured in
and the total
a
static
light
scattered and from differ¬
single scattering
contribution
Results and discussion
6.2
I^(q), I(q)
has to be corrected with the reduced
ß\2
correlation function where
scattering only,
experiment
total
which
with
diluted
highly
a
intensity observed
system for
R
of the
q-dependence
274
—
dilute
a
particle
form factor
typical
for
has
=
I(q)/I(0)
=
with the
experimental data.
a
contribution from
where the
scattering already leads
average
particle
suspensions
becomes
21 A that
to
more
singly
to
multiply
contribution from
a
looking
effects. The
in
experiments.
figure
This
light, whereby
decrease in the
in
P(q),
multiple
polydispersity
function for
at the reduced
and the
polydisperse
a
21
intercept
B, which
we
value of 1
means
intercept corresponds
I(q).
ßi2/ß[2
the ratio
that there is
to
a
an
no
increasing
ßu/ßu
>
12 % transmission is due to
an
The fact that
by
as
obtained from
quantity directly yields
artefact
intercept
multiple scattering
important role of multiple scattering
points of the sample with
influence the
12% and
=
suspensions, the minima and
(Mie) scattering
The
multiply scattered light
caused
particle concentrations, where
wrong numbers for the
a
where
light scattering
the presence of small amounts of
1 for the first two data
mainly
2.9% results in
=
lower than in forward direction.
magnitude
suspensions shown
multiple scattering, and
cr
becomes dominant close the minima in
data.
obvious when
scattered
of
are
Mie-
using
higher concentrations,
For these turbid
even
completely
simultaneous cross-correlation
A calculation
illustrated with static
smeared out due to
experimental
function of q for both
of
A).
size in any attempt to fit
to the
even
21
is several orders of
figure
with radius
leads to low transmission values of T
multiple scattering
intensity
It is obvious from
are
particles
However, the scattering data
the 3d instrument for two different
maxima of the form factor
latex
given polydispersity
very different appearance when measured at
1.5%, respectively (see fig.
goniometer
shows distinct minima and maxima that
good agreement
significant multiple scattering
classical
a
on
A. In the accessible q-range the normalized
21
multiple scattering becomes important. This is
T
measured
with
on
to the
Pu12
relatively monodisperse spherical particles.
data obtained
single
dynamic light scattering
a
\
particles
a
in the limit of
=
highly monodisperse
of
figure
cross-
thus be obtained from
for these latex
theory
very
P(q)
can
\Zli%)li%)
suspension
in
°f tne
sample. The single scattering contribution
scattering intensity
is shown in
nm
independently
at detectors 1 and 2
=
ßi2/ß\2
intercept
intercept measured
denotes the
be measured
can
I{l\q)
The
41
dust effects at very low
we
obtain
scattering angles,
which
can
of the cross-correlation function. This is also evident when
CHARACTERIZATION OF TURBID MODEL SUSPENSIONS
6
42
looking
which
at the standard deviations for the different values also
are
at very low
largest
and which
are
general larger
in
contributions from dust
latex
scattering angles
are
not
for the
Figure
in
factor,
concentration,
where
with the lower
dominated
the contribution from the
by
particles.
A combination of the total scattered
ßi2/'ß\2 together
scattered
with
I^(q),
intensity
which is
allows for
directly
48
equation
intensity I(q) and the reduced intercept
a
calculation of the
single
figure
21 C.
for both concentrations in
plotted
P(q)
The thus obtained corrected intensities agree very well with the calculated
singly
These results
light.
scattered
particle
form factor
even
under
clear that such
course
the correction of
sizing, but allows for
is less than
strong q-dependence and the scattered intensity
The
singly
ability
to suppress
scattered
light
measurements. In
scattered
colloids,
By this
has
one
generally
tries to
as
important
some
particular
intensity such
one
multiple scattering
large particles
that
from
and
However, if
one
can
successfully
we can
particles completely
correct for
choose
as
0
=
and
a
q-dependence
of
light scattering
of the
strongly interacting
does not
cover
the
regions
with
to contributions
from cuvettes and vat, dust
exceedingly
multiple scattering,
dominates any
background
perform
5°, which would be impossible for
scattering cells
a
etc.),
difficult.
these aspects lose
experimental conditions where the scattering
in fact been able to benefit from this and
low
shows
system is only weakly scattering.
measurements at low values of q become
importance, and
from the
of
highly susceptible
background (solvent, reflexes, stray light
consequently
their
measurements
S(q)
suppressed significantly [37].
suspensions
or
particle
in
pronounced q-dependence
multiple scattering
intensity. This makes these
low
where
consequences for static
aims for conditions where the
ensure
be
It
contributions to the static
and determine the
for systems with very
for
applications
suspensions,
can
P(q)
to the minimum in
is not restricted to
colloidal
single
evaluate the
can
for
10% of the total intensity.
multiple scattering
strongly interacting
structure factor in
we
circumstances, where close
procedure
a
that
demonstrate,
single scattering
the contribution from
is of
B,
21
and in the minima of the form
sample
completely
given
a
contamination. We have
SLS measurements at
conventional
angles
as
approach using normal
goniometer system.
Our measurements show that the
dynamic light scattering experiments
multiple light scattering.
use
of the 3d cross-correlation
allows to
The current value of
correlation function achieved with
our
effectively
ß12
~
intercept
high enough
correlation functions of sufficient accuracy in order to
in
suppress contributions of
0.2 for the
instrument is
technique
in the
that
we
successfully apply
cross-
obtain
inverse
43
Results and discussion
6.2
B
1.4
'I-
1.2
I
1
a
0.8
<4
0.6
°£
0.4
Ï
I
0.2
0
0
10
5
15
q
30
25
20
.+.
4_
-0.2
0.0001
10
35
20
15
(1/pm)
q
25
30
35
(1/pm)
1 T~
0.1
B
-
0.01
0.001
0.0001
Figure
Results
21:
from
trated turbid monomodal
(A):
lute
=
light scattering experiments with dilute and
particle suspensions (R
Normalized scattering intensity
suspension
concentrated
T
static
12%;
measured
on
a
concentration
on
0.05%,
T
=
for polydisperse spheres following
274
and
ßu/ßu
polydispersity
obtained
from
a
=
T
=
12%;
;
1.6%).
as
a
a
a
cross-correlation
function
0.05%,
intensity P(q)
=
I(q)/I(0)
as
a
2.9%).
function of
(E:
T
=
q
0.95,).,
concentration
a
for
di¬
and
0.01%,
theoretical cal¬
Schulz distribution with average radius
suspensions
concentration
(•:
The reduced intercept
measurements
as
T
(B):
a
=
using the 3d
function of q. (•:
1.5%,)
The
standard deviations obtained from 10 individual measurements.
normalized
=
Also shown is
2.9% using Mie theory.
ment with dilute and concentrated
0.01%,
I(q)/I(Q)
—
the 3d instrument
culation
nm
274 nm,
commercial goniometer system
suspensions measured
:
P(q)
=
concen¬
function of
q,
error
concentration
bars
(C): Single
where
instru¬
given
are
scattered
multiple
scatter¬
ing contributions have been corrected for by using figure 21 B and equation 48, for
concentrated
T
=
12%;
suspensions measured
:
concentration
on
0.05%,
T
the 3d instrument
=
culation
for polydisperse spheres following
274
and
nm
polydispersity
o
=
1.5%,).
a
(•:
concentration
Also shown is
a
0.01%,
theoretical cal¬
Schulz distribution with average radius
2.9% using Mie theory.
6
44
Laplace
CHARACTERIZATION OF TURBID MODEL SUSPENSIONS
transformation programs such
60-300 seconds. This opens up
sizing
in
industrially
a
as
Contin after measurements of
wide field of
relevant systems such
new
as
applications ranging
we can
directly
use
intercept of the cross-correlation function and the
multiple scattering. Therefore,
but also for
systematic SLS
can
the relation between the
correct the measured values of the scattered
our
not be
optically
experimentally
amount of
particle
matched.
determined
multiple scattering
and
intensity I(q) for contributions from
3d spectrometer
studies of turbid
from
emulsions to the determination of the
dynamics of concentrated model suspensions that
Moreover,
typically
can
not
suspensions.
only
be used for
DLS,
45
Colloidal
Suspensions
Introduction
7.1
properties of
Structural
colloidal systems
are
interactions, which
the
Understanding
ence.
a
Strongly Interacting
Static Structure Factor of
7
of considerable interest in colloid sci¬
cause
relationship between microscopic characteristics
these structures,
and
loidal systems which is also of industrial relevance
light scattering experiments
of the structural
standing
analogy
an
The colloidal
enters
such
as
as
to
simple liquids,
particles
then
i.
play the
e.
dielectric constant, index of
the
by treating
by
the
tools used to
has been achieved
suspension
molecules,
corresponding
refraction,
theory and
state
experimental
suspensions
role of the fluid
continuum characterized
a
[38, 39]. Liquid
considerable progress in the under¬
of colloidal
properties
to find
help
macroscopic properties of col¬
often the theoretical and
suspensions of interacting particles. A
describe
making
are
can
or
ionic
as a
'macrofluid'.
and the solvent
continuum
strength (or
by
only
parameters
Deb ye
length)
[40, 41].
The structure of colloidal
particles
can
which is the
is
be
suspensions arising
probability
finding
S(q)
which
can
sity Is{q)
q is
be measured in
measured
as
given by equation
size, shape
tor
S(q)
a
a
=
1 +
n
J \g{r)
function of the
S(q)
14. The
can
particle
also be written
of the correlations between the
for uncorrelated
exhibit
particles S(q)
a
It
(57)
P(q)
0
or
The
the
scattering
scattering
inten¬
vector
contains information about
particles,
and the static structure fac¬
based
on
the interactions between the
^X>*s(r,"r')>
(58)
an
ensemble average.
particles
=
r.
as
=
brackets denote
the distance
1] e~tqr d3r
form factor
suspension
%)
angular
-
g(r),
function
via
scattering angle
and the internal structure of the
particles. S(q)
particles
particles separated by
light scattering experiment.
static
reflects the order in the
where the
pair
a
of
related to the static structure factor
directly
pair correlation
described with the
quantitatively
of
from the interactions between the
i and
1 for all q.
j
Equation
located at
Suspensions
58 is
positions
of
rt
a measure
and r3, and
charged monodisperse
characteristic form for the static structure
factor,
which
closely
STATIC STRUCTURE FACTOR
7
46
S(q)
resembles
S(0),
small value
In the limit of q
simple liquids.
found for
related to the isothermal
directly
which is
0 it starts from
—>
a
of the
compressibility x
suspension through
S(q
S(q)
then increases with
lations
Since the
d.
of the
suspension
of the main
position
peak
minima and maxima in
which
reciprocal
in
separated by
correspond
space
a
position
experimentally
the
i.
e.
the
to
to
e.
peak
a
in
g(r).
potential, analogies
particle suspension
to the
is treated
as
The
the
a
^/Vp/<&,
higher
of
finding
a
order
one
S(q)
in
particles
two
relation between
correlation function
pair
theory of simple liquids
a
the
value of
peak positions
In order to find
factor,
=
approach
and
probability
increased
an
d
values of q the
particle gas10.
ideal
an
determined static structure
and the interaction
made,
r, i.
pronounced
become less
to the value of
corresponds
distance
interparticle
mean
4n/3a3 through
=
higher
scales with $s At
S(q)
of the
corre¬
be calculated from the volume fraction <&
can
volume Vp
particle
and the
(59)
nkBTX
reciprocal
at the
distance
interparticle
=
the
q and has its main maximum where
increasing
strongest, which is
are
0)
=
'macrofluid'. In
are
often
the
simple liquids
Ornstein-Zernike equation
h(r)
is used to
g(r)
—
provide
1, which
this link
split
is
particles trough c(r),
particles.
an
Since
c{r)
=
[42].
into
f h(\r
n
-
s\)c(s) d3s
It consists of the total correlation function
sum
60 contains
of indirect correlations mediated
no
A
simple
approximation (MSA) [41]
the interaction
closure relation is for
ß
=
potential U{r)
l/(kßT)
the Perçus-Yevick
is the
is
(PY)
and the
leads to qmax
~
2n/d.
neighbour
a
two
mean
pair
spherical
weakly
c{r)
in¬
and
given through
reciprocal
10For particles that interact with
related to the nearest
the
example
which has often be used to describe relative
c(r)
where
to the
potential
—
potential
The relation between the direct correlation function
teracting particles.
h{r)
the other
by
information about the interaction
additional closure relation is needed to link the interaction
correlation function.
(60)
part describing the direct correlations between
a
and the
equation
+
=
thermal energy. On
hypernatted
chain
short range hard
shell of
(61)
-ßU{r)
particles
a more
(HNC) [41]
sophisticated level,
closure relations
sphere potential, the main peak
around
a
primary particle,
i.
e.
at d
=
of
are
S(q)
is
2a, which
47
Introduction
7.1
applied
often
to describe the structural
it is known that the PY closure relation leads to
sphere potentials
the
Rogers-Young
h(r)
aims at
of
properties
good
and the HNC closure relation for
results for short
long ranged
Because
ranged
Yukawa
hard
potentials,
[43]
closure
=
particle suspensions.
e-
-1 +
x
(l + ^{f(r)[h(r)-c(r)]}-l\
fin
V
from both closures
taking advantage
J
PY and HNC with
by combining
a
mixing function
f(r)
where
is the
a
can
be obtained
a
the isothermal
fluctuation and the virial route.
(63)
e~ar
a
—>
While the fluctuation
xP
the virial
compressibility
where the pressure P is
can
1
3
Because these closure relations
generally
are
irßn
/•«>
/
compressibility
a
v^a the
directly
is
0
(64)
equation
of state
z_,_^dU
dr
approximations the compressibilities calculated
not identical. In the
guarantees thermodynamic consistency,
vides very accurate results for the
(66)
g(r)—— dr
r
jo
is chosen in such way, that Xt
a
—>
Xt anc^ Xt
n
2_a_
1
n
parameter
mixing parameter
given through
=
are
The
closure
-5(0)
be calculated from the
ßP
via both routes
=
Rogers-Young
compressibilities
the static structure factor in the limit of q
given by
0 the
the HNC closure.
—> oo to
by calculating
-
In the limit
mixing parameter.
reduces to the PY and for
1
=
=
Rogers-Young
Xt
at least for the
in
closure the
mixing
this way the RY closure
compressibility,
and it pro¬
pair correlation function g(r) when compared
to
'exact' Monte Carlo simulations.
Light scattering experiments
with
polydispersity
tation of the
account
[44].
generally
is
in
with 'real'
size, charge
light scattering
data
or
particle suspensions
refractive index of the
consequently
often have to deal
particles. An interpre¬
has to take this
polydipersity
In the theoretical calculation the continuous distribution of
replaced by
a
histogram
which then results in
a
mixture of
into
particles
particles
of p
consisting
components. Sizes aß and number densities raM of the components
histogram
chosen such that the first two moments of the
are
distribution have the
a
STATIC STRUCTURE FACTOR
7
48
set of
coupled
same
values. The Ornstein-Zernike
cßV{r)
=
+
n^2 (
which have to be solved with the
polydisperse sample
a
and
particle components,
form
—
) / Kx(r)c\„(\r
is
now
given by
scattering intensity
the
equation
14
can
na*
=
be written in
n|) d3r1
-
generalized Rogers-Young
Is(q)
where a6 and
equation
integral equations
/V(r)
With
and the continuous
closure relations.
weighted
is the
a
(67)
of the
sum
modified form
as
(68)
P(q) S{q)
represents intensity-weighted average particle volume and particle
P(q)
factor, respectively,
EU Nß(a)a*
S(q)
and
no
the
the effective structure factor
longer simply
interparticle
factors into
taking
single particle quantities P(q)
S(q) only.
correlations
s(q)
into account the size distribution.
=
wt
and
a
Is{q)
part describing
The effective static structure factor
S BMBÀq)SM
()
p{q) ß,u=i
where
denotes the
Bß
factors, depends
factors
now
defined
are
scattering amplitudes
also
on
and
SßU
single particle properties.
the
The
partial
partial
static structure
static structure
as
^(«)
Because of the different
=
iEE^q(Rj Ri)
n
size, charge
(7i)
j=i k=i
or
refractive index of the
particles
static structure factors have their maximum at different values of q.
quence the
peak
in
S(q)
is shifted and reduced and
washed out. Furthermore the value of
[45],
thus
In this
5(0)
higher
increases with
the
As
a
partial
conse¬
order oscillations
are
increasing polydispersity
5(0) ^ nkBTx.
chapter
we
demonstrate that with the 3d cross-correlation instrument the
static structure factor
S(q)
can
be measured in
particle suspensions
that exhibit
a
49
Introduction
7.1
high turbidity.
which
This
provides
previously
were
direct
a
excluded from these
Is(q)-
contributions to the measured
which
polystyrene spheres,
matched, which
neglected
be
when
the
While
existing
for their
that
values of q.
is
an
are
typical example
A
finding.
and
experimental
increased forward
known
precisely.
aggregates.
To first
uncorrelated gas of
contribution to the
values of q, where
contribution to
S(q)
can
to
as
Another
S(q)
intensity
S(q)
too
we
Particular
origin
of the
a
could then lead to
values,
fractions,
high
values for
have
performed
care was
careful
a
preparation
charged particles
measurably
S(q)
to
exist for this
at low
discrepancy
is known to result
we
believe that this
performed
where the size distribution
to
particles
as
a
small number of
something
an
ones.
Their
increase in
Is(q)
at low
would be small and their
for deionized
interacting (i.
could contribute
like
of small
liquid
significant
non
suspensions
e.
multiple
highly screened)
significantly
and thus lead
5(c).
a
systematic study of the static
our
multiple scattering
to the
recently
the presence of small amounts of
volume fraction with
given
too low at
would be the
are
results have been
correlated
is
possible
correspond
would
and
polydispersity
generally
could be the presence of
highly
if
data
several
Nevertheless
published
reveals
peak position
all, polydispersity
particle suspensions
in
safely
experimental
between
would contribute
for the small 'monomeric'
particle
are
There
22.
Is{q) considerably suppressed. Finally,
function of
a
figure
in
data. First of
reason
big particles
to determine the amount of
and
given
approximation they
at these low volume
apparently
a
is
some
can
suspensions
the
S(q) quite accurately
which would not be measurable for
Therefore
as
reproducing
scattering intensity [44].
be very low at low q
scattering,
particles
of
However, several other possible explanations
theory
optically
particle size,
the
on
and
predictions
particularly exciting possibility
with very well characterized model
was
capable
although relatively weak,
unlikely explanation,
an
between theoretical
attractive interactions of electrostatic
proposed
in
lower, depending
structure factors in deionized
on
integral equations
low values of q. A
between
data
or
effects
multiple scattering
taken into account, the theoretical values for
properly
[47, 48],
model systems, cannot be
as
charged
of
suspensions
aqueous
instruments.
of the minima and maxima in
height
reasons
light scattering
discrepancy
characteristic
[46, 37].
used
IO-3
to volume fractions of
A survey of the
a
widely
are
example
limits the concentration where
standard
using
For
multiple scattering
due to
investigations
for systems
potential
to the interaction
access
precise
in which all
3d-instrument,
and
structure factor
which allows
subsequently
determination of the
possible precautions
us
correct for it.
sample polydispersity
have been made to prevent
7
50
STATIC STR UCTURE FACTOR
o
u
a
u
-i—»
o
GO
0
0
2
1
Scattering
Figure
sult
Measured static structure
22:
from
deviation
Rogers-Young
a
from theory
and
aggregation
7.2.1
in
factor 5(c) compared
(from [37]).
at low q. For details
remove
all
see
experimental
predicted
re¬
data show
a
text.
existing large particles.
Experimental Set-Up
experiments
chapter
7.2.2
were
performed using
the 3d cross-correlation instrument described
5.
Methods
Measurements
were
done with
suspensions
of latex
spheres (diameter: (110±10) nm)
with volume fractions in the range from 0.05% to 1%
the
The
with the
Materials and Methods
7.2
The
calculation
Vector
by volume.
suspensions cristallize when fully deionized due
interactions of the
highly charged particles,
a
to the
As it
was
long-range
found that
electrostatic
mixture of ethanol and water
was
used
as
51
Materials and Methods
7.2
the solvent. A volume ratio of 70% ethanol and 30% water prevents cristallization
in the
investigated
particle
decreases and the
is thus lower. The
cells with
cylindrical
and
deionization the
fully
an
Combined static and
tion
were
kept
exchanger
resin
s
analysis
fit. The ten
and 120 s,
by
means
tem
an
then
were
diluted aqueous
of
dynamic
10° to 0
=
on
were
=
with 2160
7.2.3
125° at
analysed using
a
each
and
second
a
intercept ßi2 of the cross-correlation
scattering intensity for multiple scattering
of the latex
suspensions
and static
size distribution
transmission electron
were
scattering angle
light scattering
particles
with
a
were
also characterized
commercial
goniometer
(ALV/DLS/SLS-5000F single mode fiber compact goniometer system).
particle
sec¬
cross
scattering angle,
the
in
averaged.
was
a
5
mM/1
NaCl
was
particle
Laplace
size distribution
microscopy. The micrographs
were
sys¬
To avoid
added to the solvent.
calculated with the inverse
tion routine Contin. Furthermore the
The
performed
were
the diffusion coefficient is calculated from the cumulant
influence of electrostatic interactions
The
equal
measurement.
a
done at each
depending
10 mm,
x
correlation functions
range from 0
were
mm
single scattering
corresponding
in order to obtain the
Additionally
runs
to
10
in contact with
days
(Dowex) prior
angular
function needed for the correction of the
contributions.
particles
In order to achieve
Hellma).
least 10
concentration. The correlation functions
order cumulant
Highly
(rectangular cells,
per
(Versapor
0.8 pm filter membrane
and to obtain the
measurements
duration between 20
sample
at
intensities and the
temperature of 25°C. Ten
the
cells
a
dynamic light scattering experiments
measured at intervals of 1° in the
a
using
sample
multiple scattering
o~(q). Scattering
having
filtered
inner diameter of 8 mm,
samples
volumes of mixed bed ion
order to suppress
were
into the quartz
charges
groups, the number of
of electrostatic interactions between the
strength
suspensions
acrodisc, Gelman)
charge
of the surface
equilibrium
dissociation
the association-
changes
concentration range, because the alcohol
was
calibrated
transforma¬
measured with
using
a
Pt-grid
lines/mm.
Materials
suspension
experiments
was
surface which
of 8.3 %
by
of sulfate latex
particles
with the diameter of 110±10
nm
used for the
obtained from UDC. It is stabilized with 2500 sulfate groups
are
fully protonated
volume.
at
pH
7. The stock solution has
a
on
the
concentration
STATIC STRUCTURE FACTOR
7
52
1.2-r
1-
0.8-
©
0.6-
0.4-
0.2-
0
1—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—|—i—i—i—i—
(1/nm)
q
Figure
23: Measured
the Mie-calculation
a
(circles)
radius
(line) particle form factor P(q).
and calculated
of
r
59
=
nm
0.025
0.02
0.015
0.01
0.005
0
was
From
obtained.
Results and Discussion
7.3
The latex
particles
were
transmission electron
highly
dilute
70/30)
with
Mie-theory.
was
as
the solvent
are
A fit to the
static and
figure
compared
23
in
an
particle
This behaviour could be due to
a
and to
a
=
(87.2
±
radius of 59 nm,
7.3)
nm
combination of the effect of the
shrinking
of the
(volume
transmission electron
experiments. However, from
r
mixture
a
form factor calculated
particle
a
and
data measured with
ethanol/water
data reveals
too small values with
sample preparation
dynamic light scattering
experimental
with the
experimental
also found from DLS
microscopy considerably
in the
In
microscopy.
by
suspension (0.001 % by volume)
ratio
which
characterized
particles
were
obtained.
drying
process
in the presence of the
electron beam and is also described in the literature.
Static structure factors
0.04
was
%
to 1
were
measured for
% by volume. As solvent
used and the
suspensions with
suspension
water
low volume fractions of
as
was
a
fully
particle
mixture of 70
deionized
concentrations
parts ethanol
using
an
ion
the solvent show the formation of
approximately
0.1
ranging
to 30
parts
exchanger.
a
colloidal
from
water
Deionized
crystal
at
% for these particles. These cristallized
superimposed
on
the
structure
ethanol/water
mixture
measured
as
light scattering experiments
scattering.
A normalization of
factor then
provides
performed solving
Results
up to
polydispersity
figure
were
good agreement
from
intensities
were
Scattering
turbidity11
corrected for
the measured
by
5(c).
dynamic
and
multiple
form
particle
Theoretical calculations
were
equation with the Rogers-Young closure
of the latex
particles (r
=
59 nm,
a
—
%).
8.6
24 for different volume fractions. The results demonstrate
S(q)
not accessible.
previously
Using
suspensions
with combined static and
and division
particle suspensions
be measured in
can
a
This allows
high turbidity.
interaction effects in colloidal
important information about
to obtain
than 1%.
volume concentrations that exhibit
high
which
higher
Is(q)
that the static structure factor
clearly
the alcohol prevents the
the Ornstein-Zernike
shown in
are
re-cristallization
the static structure factor
into account the
taking
Bragg peaks disappear,
subsequently
and
are
be shear melted
scattering angle
function of the
a
which
too short time for the measurement.
a
cristallization up to volume fractions
I(q)
can
crystal
While the
peak.
solvent,
the
as
in the
high intensity
a
and the
suspensions
after about 10 s, which is
place
takes
liquid
the
by simply shaking
with
Bragg-peaks
show
samples, however,
an
53
Results and Discussion
7.3
data
experimental
The
with theoretical results obtained from
solving
are
now
suspensions
quantitatively
in
the Ornstein-Zernike
equation with the Rogers-Young closure relation.
For
larger
higher
concentrations the
values in q. The values of the effective
lation vary between 160-250
data at
higher
knowledge
of the
our
data
per
particle. However,
are
not
we
in the accessible q range in order to
determines the effective
investigate
(SANS),
the
where
we
is
highest
i.
e.
is
an
have both
overlap
and
1%)
when
are
shown in
higher values,
using rectangular
figure
cells
25.
The
as
we
lack
a
normalize the
the
5(c)
particle
data for these
symbols
the
very
precise
peak
experimen¬
peak
We therefore
plan
neutron
scattering
data at low q and
has reached its
can
asymptotic
form factor become
high
in
of the main
height
small-angle
light scattering
where
calcu¬
ability to analyse
adjustable parameter.
with
light scattering
Rogers-Young
in the normalization of the
unambiguously
limit of one, and where the oscillations in the
Nevertheless the
our
is shifted towards
would need at least the second
concentrations also with
extend the q range to much
visible.
problems
particularly important
charge,
S(q)
in
absolute scale and
on
particle concentration,
tally determined 5(c). This
to
charges
peak
used in the
charge
concentrations is limited due to
scattering data. Since
S(q)
of the main
position
concentrations
represent the measured
clearly
(0.5%
scattering
particle
intensities while the line represents the
experimental
The
culations
peak
the
using
occurs
Rogers-Young
and test for the
experimental
calculations and
improved
However,
tering
it is
can
out that
the main
provide
ical results for
us
5(c)
with
a
first
peak
colloidal
theories to
even
has moved to
theory,
Nevertheless,
study
to
dispersions,
improve
our
of
improved
too
the
theory and
sample preparation
well
as
as
software dust filter.
under these conditions
measurably
by
fluctuations of the in¬
multiple scat¬
of
use
high intensity
a
classical
at low q. At
values of q in
sufficiently high
we
we
still observe
need additional
a
our
data shown in
structural and
and to
understanding
figure
24
and test the
of colloid
=
experiments and
clearly
dynamic properties
investigate
c
0.11% is in
characteristic deviation for
effort in order to reach any conclusions about the
improved capability
acting
a
limited
are
comparison between experimental data and theoret¬
0.23%. However, it is clear that
for this deviation.
we
in the limit of small q values. While the data at
reasonable agreement with the
experimental
of
main
between theoretical
at low values of q, and the
completely neglected
higher concentration,
order to
enormous
use
with the cal¬
determine the asymp¬
systematic comparison
a
scale)12.
concentrations, the
unambiguously
cause
with the
would result in
able
which
arbitrary
a
good agreement
Under these conditions
data.
point
to
(on
previously existing discrepancies
protocol
measurement
not be
in
are
additional effort both in
an
important
form factor
closure relation. At low
light scattering set-up
=
24
at values of 0 < 16°. It is clear that
experiment will require
an
figure
particles,
influence of dust
enormous
tensity
data shown in
at too low values of q in order to
5(0)
totic value
c
STATIC STRUCTURE FACTOR
7
54
stability,
a
consider¬
possible
reasons
demonstrate
of
strongly
applicability
our
inter¬
of modern
interactions and structure
evaluation.
12We also plan
to
change
absolute scale and therefore
the present set-up for the 3d instrument to allow measurements
a
normalization of
scattering
data.
on
7.3
Results and Discussion
55
0.025
0.025
0.025
0.01
0.015
q
Figure
at
24:
different
Static structure
factors
concentrations. The
calculations.
0.025
0.02
(1/nm)
measured with latex
experimental data
are
suspensions
compared
with
(r
=
59
nm)
Rogers-Young
56
7
STATIC STRUCTURE FACTOR
200
c
150-1
=
0.5%
100
50-1
(®Bmnnmni
0-
0.005
o
0.01
0.015
0.02
0.025
0.03
0.025
0.03
250
0.015
q
Figure
latex
25:
Scattering intensity
suspensions (r
=
59
nm)
and
(1/nm)
particle form factor (on
a
with 0.5% and 1% concentrations.
arbitrary scale for
57
Influence of Dilution
8
on
the Particle Sizes in
Milk
Introduction
8.1
Dynamic light scattering is
It is
a
a
powerful tool
size distribution in colloidal
of interaction
However,
many
as
suspensions, and
characterization of colloidal systems.
systems, often of industrial relevance, could
experiment,
suppressed,
in the
and the measurable range of
samples
performance
tested
was
non-negligible turbidity,
investigated
with
dynamic
multiple scattered light
be extended to
can
concentration. The construction of the instrument
and its
stability.
and
so-called 3d cross-correlation
a
contributions from
disturbing
was
are
high turbidity
described before in detail
measurements with well defined model
using
particle
used for investi¬
dynamics
with
not be
past. We therefore developed
where the
frequently
colloidal
samples
of the
investigations
it is also
potentials (see chapter 7),
the method is known to fail for
light scattering
or
a
convenient and fast method for
non-destructive,
gations
for
systems
(see chapter 6).
Here
nique
to
we
make
now
methods
tering effects due
of
sample.
can
to the
practical
of
in
many other
of the concentration
use
for
the
or
light scattering techniques
undesired
multiple
avoided with
are
systems such
of the
significant change
a
as
for
a
to
much
overcome
larger
this
range of
a
light
scat¬
dilution
suspensions
or
self
sample properties
as¬
upon
It is clear that the
composition of the sample.
help
to their
is in the colloidal range,
Frequently
problem
respect
systems, like for example emulsions
3d cross-correlation instrument could
to
particles
mostly high sample turbidity
a
relevance. In industrial
technological
to be characterized with
be used.
principle
react with
sociating systems,
and
If the size of the
While this often is not
inorganic particles,
changes
step and demonstrate the applicability of the tech¬
systems often need
size distribution.
scattering
of the
next
complex suspensions
processes, 'real world'
particle
a
problem
samples
and enable
us
than
previously
great
interest in
considered.
As
a
the food
milk is
'real world' system
industry
given
in table 1. Milk
the components
choose
because many
trimodal distribution of
particle
we
are
also
milk,
products
can
which is
are
based
be characterized
whey proteins,
polydisperse
a
on
as
a
subject
of
it. The
composition
colloidal
suspension
casein micelles and milk fat in
in size
[49].
of
cow
with
a
water, where
In this article the behaviour of the
size distribution of milk under dilution is described. A characterization of
INFLUENCE OF DILUTION ON THE PARTICLE SIZES IN MILK
8
58
cont.
a
whey proteins
casein micelles
milk fat
0.5
5
100
1000
radius
nm
mass
%
92
0.75
2.5
4
volume
%
83
1
11
5
Table 1:
such
phase
Composition of cow milk,
complex system
also
Laplace
ments
Therefore
this
also
as
content in the measured
reduced due to contributions from multi¬
considerable task with respect to the data
a
in order to
even
developed
signal
fractions (from [49]).
a
recover
size distributions from DLS
under ideal conditions of
new
experi¬
high signal-to-noise
analysis procedure
data
analysis.
ratio.
which is described in
well.
Experimental Set-Up
experiments
chapter
were
performed using
the 3d cross-correlation instrument described
5.
8.2.2
Methods
8.2.3
Multi
A
and volume
Materials and Methods
8.2.1
in
we
chapter
8.2
The
require
difficult to handle
are
significantly
represents
inversions
of particles
under conditions where the
correlation function will be
ple scattering
size
Angle
Laplace
Inverse
Transformation
dynamic light scattering experiment probes
scattered
light
which
are
the calculation of the
is
performed
is
an
leads to
a
mainly caused by the diffusive
particle
size distribution
an
intensity of the
motion of the
inverse
Laplace
function,
typically
problem,
which is
this
ever
infinite set of
means
that any noise
present in
a
e(r, T)
particles.
For
transformation
the measured correlation function. The inversion of
ill-conditioned
correlation
G{T)
on
the fluctuations in the
on
g(r), however,
the measured
dynamic light scattering experiment,
possible results for the calculated
size distribution
that fit
/i\
9W{t)=
f
max
e-rTG(r)dr
+
e(T,r)
(72)
59
Materials and Methods
8.2
within the noise level. In
general
because any variation in
g(r)
of
a
polydisperse sample
G(T).
the
two
even
function
a
peaks
depends
the
on
performed with
noise
height
of the
level,
and in
a
experiment
duration of
a
from each size
factors,
in
as
are
distribution, taking
form factors of each
contribution of
particle species
a
amplitude in the
appropriate particle
but there
are
it is
are
usually
different
be obtained in
practical
experiment
limits for the
simultaneous fit of
angles improves
particle species
a
an
a
an
inverse
to the fact that the
varies with the
species
can
at different
the
Laplace
particle
(see figure 18). Consequently
the
scattering
be recovered best
=
P(q, r)
a
Mie calculation
[27, 50]
^P^f^
additional
m
—
(73)
particles,
np/ns
which
of the
input parameter
by taking advantage
angles.
to obtain
and
no
longer depends
particles (rip)
to the
for the Mie calculation. A
set of correlation functions measured at different
the result
form
poly disperse sample
calculated with
intensity
by performing
form factors
as
corresponding particle
Ig. The angular dependence of the size distribution
then is the number distribution of
is needed
scattering
intensities calculated
scattering
size distribution for this
Therefore the refractive index ratio
(ns)
often
the fact that in
on
to the scattered
GN^
solvent
can
measurement decreases the noise level
scattering angle due
be taken into account
GN(r)
as
particle species
at its maximum contribution to
can
that
auto-correlation
into account the
This is based
varies with the
transformation,
GI(q1 r)
G(T)
an
means
experiment ß\2 decreases
set of correlation functions measured at different
a
reproduced.
and the
intercept
accuracy of the calculated size distribution consists
intensity weighted size distribution,
q.
good
factor of
a
the correlation
on
smaller
a
cross-correlation
0 with the additional constraint that the
angles
on
Because the noise
prolonged
resolution,
improves the
simultaneous fit of
where
not be obtained better than
experiment.
an
A method which
the
as
decreasing resolution,
with
even
in
inversion
quite precisely using
intercept ß, where
a
(main) peaks
the
sample having negligible turbidity and with comparable duration
and therefore increases the
angle,
[30, 32]
the resolution in
is not
of the measurement. Of course,
a
normally
can
increasing sample turbidity,
cross-correlation
in
Contin
or
under close to ideal conditions.
relatively higher
with
a
of these
amplitude
position of
While the
often be recovered
can
exponential sampling
methods like
G(V),
noise level restricts the resolution of
smaller than the noise contribution cannot taken into
account for the calculation of
G(T)
higher
a
scattering
of the different contributions of the
The constraint of the
same
angular dependence
of
60
INFL UENCE OF DIL UTION ON THE PARTICLE SIZES IN MILK
8
the
of the size
to the
results of this method
be
can
comparable
with dimensions
expected
or
diluted
prior
or
in
a
a
cell
rectangular
rectangular
the
on
correlation
were
particles
given elsewhere [51].)
covering
the whole
x
10
mm)
of the
for the
done within
a
more
range and
P(q,r).
a
to
1:1000)
particles
were
was
were
in 0
recover
averaged
s
s
>
400 pm,
and 300 s,
Ten
were
ratio
cross-
were
measurements,
used for the multi
the size distribution of
for each dilution step.
used for the calculation of the
performed
For
samples.
high signal-to-noise
a
duration between 60
equally spaced
All measurements
sample
time of 5 hours after dilution. Three
transformation in order to
1.5 of the
=
turbid
adjusted
was
were
inserted either
were
dilutions 1:100 and
other,
to obtain
sample,
The individual results then
refractive index nv
form factors
mm
path length inside the cell
angular
Laplace
particles.
(for
mm
then
range 30° < 0 < 120° at intervals of 5°.
angular
inverse
inner diameter of 8
each measured for
functions,
an
8.2.5
will be
of
best
light. (Details
A of the
wavelength
samples
turbidity
taken in
the
experiments with suspensions
for
1:100 and 1:1000. The
a
(10
cells the
Measurements
angle
approximately A/2,
radius of
measurement with deionized water to volume ratios milk to
(or undiluted), 1:10,
depending
ß\2.
to
round quartz cell with
a
pronounced
a
types of milk used for the experiments
For the dilution series the different
in
a
form factor possesses
Methods
8.2.4
of 1:1
than
than the
larger
performance
and its
algorithm
particle
particles larger
for
at their maximum contribution
single particle species
Because the
scattering intensity.
angular dependence
of the
to about the maximum
distribution, and particularly the peak heights,
resolution obtainable for every
the
size distribution increases the resolution
intensity calculated from each individual
at
a
A
particle
temperature of 25°C.
Materials
Measurements
were
performed using
Customary, homogenized
full-cream
volume and skimmed
milk with
In addition
a
The emulsion
cow
protein-stabilized
was
a
three kinds of milk different in
cow
milk with
a
milk fat content of 3.8%
milk fat content of
emulsion based
obtained from Dr. M.
on
composition.
0.3% by volume
vegetable
oil
was
Leser, Nestec, Switzerland.
were
by
used.
characterized.
Results and Discussion
8.3
scattering effects
the
hardly
auto-correlation
completely
be
can
in the
seen
experiment performed with
particle
wrong
size
multiply
measurement of the size distributions
to have
interactions
coefficients of the
particles
a
size distributions of full-cream
particle
visible for both
a
cases.
would fail
badly
and
an
yield
experiment
This results in
light.
M/l [49], long
a
correct
range electrostatic
known to influence the diffusion
are
cow
figure
The undiluted milk shows
milk measured with
(see
30).
eq.
are
dilution. From
a
in table
1,
it
measured,
comparison
i.
e.
we
peak
a
the casein micelles and the second
possible,
classification is not
to volume fractions
small to the
large particle
in table 1
(1:1000)
The
on a
using
given
the ratio is
method,
fit
when
compared
concentrated
in
Mie
theory,
peak positions
fraction
are
the
samples
a
and 200
radius of 80
nm
for the milk
nm
particle components given
in the size distribution represents
droplets.
are
both have
a
resulting
in reasonable
a
clear
we
know
However,
too close and
broad size distribution
intensity weighted
size distribution
volume ratios of 1.6:1 for the
agreement with the predictions
globules.
In the diluted
sample
1.1:1.
milk,
cannot
nm
is very small
be recovered
because their contribution to the
to the
nm
the milk fat
conversion of the
whose radius of 5
whey proteins
angle
peak
populations
dilution of
shift to smaller values in size upon
for casein micelles and fat
only
particle components
peak
because the
from the literature that these two
based
a
located at
with the sizes of the different
reasonable that the first
seems
[49]. Nevertheless,
observe
a
26. A bimodal size distribution is
shoulder up to radii of 1000 nm, while radii of 65
diluted 1:1000
given
While
size distribution is calculated
1:1000 and with undiluted milk is shown in
with
light.
in undiluted milk. Because milk is known
which otherwise
from which
of the incident laser
the 3d cross-correlation
scattered
multiple
of
signs
Full-Cream Cow Milk
8.3.1
The
neglected,
be
can
even
1000 for the kinds
scattered
samples
salt content of about 0.1
relatively high
a
these
distribution,
to suppress the
successfully allows
multiply
of
background
paths
the
samples
:
show clear
samples
concentrated
higher
dilution of 1
highest
at the
with the undiluted
Especially
scattering.
beams
only negligible
are
investigated,
of milk
stages of dilution. Multiple
measurements of milk at different
performed
We have
a
61
R,esults and Discussion
8.3
large
even
compared
with the
to the other
improved
scattering intensity
casein micelles and emulsion
droplets.
multi
is too small
While the
cannot be measured with conventional DLS due to the
more
multiple
INFLUENCE OF DILUTION ON THE PARTICLE SIZES IN MILK
8
62
radius
Figure
Particle size distribution
26:
dilution. For details
of the
scattering
a
conventional
measured
the
sample
with
size distribution is
instrument,
cow
different steps of
milk at
dilution 1:1000
a
dynamic light scattering
particle
correlation
of full-cream
text.
see
light,
(nm)
instrument
exactly
which supports the
the
was
also measured with
(ALV/DLS/SLS-5000F).
same as
measured with the 3d
performance
The
cross-
of the 3d cross-correlation
instrument.
Undiluted milk with
presence of
large
fat
at the limit of the
it is still
a
total
droplets
capability
measurable,
the
particle
volume fraction of about 16 % and the
with radii up to 1 pm represent
of
our
sample
a
which is
design13.
3d instrument in its present
intercept of the cross-correlation function of about
clearly limits the obtainable resolution of the
size distribution due to the low
to-noise ratio when
sample.
13This limit
regime.
can
be
compared
pushed
While
to
even
with
a
dilute
higher turbidity using
a
Nevertheless
laser
wavelength
we
0.02
signal-
believe it is
in the
near
infrared
63
Results and Discussion
8.3
quite remarkable
than 15 % and
that such
containing
a
diluted
An
as
can
be
more
correctly
resulting quality
of the
obtained with
highly
usually
samples.
of the behaviour of the full-cream milk upon dilution
explanation
found with results of
are
that the
comparable quality
obtained size distribution is of
volume fraction of
a
large particles
amount of
significant
dynamic light scattering, and
characterized with
with
polydisperse sample
a
known to consist of sub-micelles with radii between 5
for the compact micelle is between 50
reported
example due
is removed for
nm
and 140
nm
to dilution with pure water
or
nm
and 20
nm.
The size
in radius. If calcium
dialysis,
[52, 53, 54, 55, 56].
and form small submicelles
disintegrate
Casein micelles
measurements with casein micelles.
previous
be
can
the casein micelles
For this
reason
the size
distribution measured with the dilution 1:1000 is shifted to smaller values. Because
the submicelles
large
fat
small in
are
droplets.
size,
it is difficult to
them in the presence of the
see
Therefore the volume concentration of the casein
peak
in the size
distribution appears to be reduced upon dilution. A better characterization of the
behaviour of casein micelles upon dilution
with skimmed
cow
can
in fact be obtained from
milk.
Skimmed Cow Milk
8.3.2
In the skimmed milk almost all of the milkfat is removed and
of 0.3 %
by
figure
shown in
500
peak
nm.
volume remains. Skim milk is
in water. Measurements of
suspended
main
at
a
In the undiluted
27.
radius of 140
While the latter
one
nm
in skimmed milk.
an
by
volume
the contribution of the
seen
in
a
interesting
the main
peak
appears at
a
on
we
case
a
radius of 22
nm.
the
particle
smaller
one
which
peak
obtain
a
whey proteins
cow
milk
are
size distribution shows
located at
nm
are
and
area
a
a
radius of about
to the
particle
can
be
particle component
assuming
casein micelle
a
volume content of the milkfat of 0.5 %.
scattering intensity
considerably
explained
with
a
is too small
milk, however,
size distribution.
radius decreases
This
the main
Dilution of the skimmed
the measured
located at 140
and casein micelles
mainly whey proteins
micelles,
scattering experiment.
effect
small amount
a
dilution series with skimmed
a
Calculated from the
concentration of 11 %
Again,
and
only
represents the residual fat, the main peak describes
the size distribution of the casein
to be
experiments
The
and
a
shows
amplitude
second
disintegration
of
peak
of the
casein micelles into small submicelles upon dilution.
Since the size distribution is normalized to
a
total
area
of
1, the peak
areas
64
INFL UENCE OF DIL UTION ON THE PARTICLE SIZES IN MILK
8
i
dilution: undiluted
I
I
m_
i
m
-E32L
10000
1000
100
10
BE
gg
1:10
??
^
I
3
si
^
a
^
o
-1^3?
10
f
t
r n
1
1—r-
1000
100
10000
I
"o
>
10000
100
radius
Figure
27: Particle size distribution
micelles with
details
see
a
text.
residual
milkfat
1000
of skimmed
content
of
10000
(nm)
cow
0.3 % at
milk
consisting mainly of
different steps of
casein
dilution.
For
r-t-
CD
c-f.
CD
ta
05
m
=—i
to
Ci
re
to
3
o
o
s
s
o
Ci
.
CS-
S
o-
co.
~3
Co
to
CD
S-
CD
to
CD
=—i
Ci
as
CO.
c*.
cd
to
ri
CD
-1
OO
cm'
^^
o
o
o
o.
o
o
o
oo
^1
S
M
^
o-
o
o
o
o
oo
X-
^^^^^^^^SSSS^^^
|
|
volume distribution (a. u.)
OS
Ol
Ü
O
t~*.
to
to
O
C
to*
Ö
b
cl
EU
c
Co
0)
to
to
00
INFLUENCE OF DILUTION ON THE PARTICLE SIZES IN MILK
8
66
particle
to the volume concentrations of the different
directly correspond
disintegration
More than half of the volume fraction of the casein micelles shows
The volume ratios
into the smaller submicelles upon dilution.
dilutions of 1:10 and 1:100 and 1.3:1 for the
Furthermore the form of the
of the
peak
to smaller values in
continuously shifted
found in the
has been found upon
for these systems
a
where
[57, 58],
a
a
Although good
stability of undiluted milk
Emulsion Based
8.3.3
results
is not
on
Particle size distributions of
given
figure
in
radius of 190
a
looks like the
done with
diluted
small
a
sample
of casein micelles
use
dialysis
the reasonable
guaranteed
Vegetable
light scattering
time and the
long
high
at
of
dilution
temperature
[56].
Oil
case
are
emulsions based
on
vegetable
the distribution shows
a
oil
are
maximum at
and down to about 20
nm
nm
in
angle light scattering
show
a
bimodal distribution of
and 6 pm.
nm
instrument
In
figure
intensity weighted,
particles.
From
a
this
comparison
means
Previous measurements
(Mastersizer, Malvern)
particle
a
However,
the
peaks
that
sample.
they
are
a
diluted
sample
and
cross-
The size distributions in
figure
the
larger
strongly weighted by
of the cross-correlation measurement and the auto¬
these measurements
particle
a
30 the measurement with the Mastersizer
dilution of the emulsion reduces the size of the
made in order to avoid
with
sizes with its main
correlation measurement with the data obtained from the Mastersizer it is
visible that
but it
clearly understood,
dissolved upon dilution.
correlation measurements with the undiluted
change
or
dialysed ultracentrifugate
with auto-correlation measurements with
compared
ples
with
size distribution of the casein
obtained with
protein stabilized
larger particles
at radii of 150
are
successfully
to
The mechanism of this is not yet
to smaller values.
30
good agreement
disintegration
particle
with shoulders up to 1000
nm
disintegration
If the emulsion is diluted the maximum of the size distribution is shifted
radius.
is
were
For the undiluted
29.
This is in
radius is
considerable effort has been made in the past to find
dialysed whey,
skimmed milk and
a
prerequisite
suitable diluent which does not affect the
micelles.
the
nm
removal of calcium from the solvent either due to
As dilution has been
[56].
dilution
a
at 140
size, which again supports
literature,
1.2:1 for the
are
with the dilution of 1:1000.
remaining particle class
(see figure 28).
of the casein micelles upon dilution
previous reports
sample
classes.
clearly
demonstrate
multiple scattering
size distribution
drastically.
in
a
again
clearly
particles.
that
a
dilution of
sam¬
light scattering experiment
can
8.3
67
Results and Discussion
10000
10000
(nm)
radius
Figure
29:
Particle size distribution
etable oil. For details
see
of
a
protein stabilized emulsion based
particle
librium
a
represents
a
major
im¬
quantitative characterization of the undisturbed 'native'
size distribution in natural milk
believe that this
veg¬
text.
It is obvious that the 3d cross-correlation instrument
provement and allows for
on
technique opens
properties, dynamic
up
new
as
as
in artificial emulsions. We thus
possibilities
behaviour and
that could be of considerable
well
importance
stability
for
a
in the
of
investigation
of the
equi¬
complex particle suspensions
variety of industrial applications.
68
8
INFLUENCE OF DILUTION ON THE PARTICLE SIZES IN MILK
0.2-
0.15-
—
master sizer
auto-correlation
3
0.1-
0.05-
/
-f^
10
y
v
7J
\
'"I
l
100
10000
1000
100000
1000000
0.2master sizer
0.15-
cross-correlation
3
0.1
0.05-
10
100
1000
radius
Figure
30:
sion based
Intensity weighted particle
on
vegetable
oil. For details
size distribution
see
100000
10000
1000000
(nm)
text.
of
a
protein stabilized emul¬
69
Protein Stabilized Emulsions
9
i
i
Introduction
9.1
In
chapters
6 and 7
high turbidity by
a
we
have shown that
of
means
particle
3d cross-correlation instrument.
is also
applicable
been shown that
due to
We
dilution,
in the
Chapter
emulsions, which
8 then demonstrates that this
composition
with respect to size
are
potential
of such
technique
It has
undiluted milk.
as
complex systems,
particle
with
example
for
size distribution.
step further and investigate the possibilities in characterizing
of
distribution, stability
As model systems
light scattering.
of
means
size distribution and interaction
leads to artificial results in the measured
complex systems
by
able to characterize model systems of
real world systems such
complex
changes
even one
go
now
to
we are
great importance
in
and interaction
whey protein stabilized
choose
we
industry
as
potential
well
as
in
general
research
[59, 60].
Emulsions
various industrial
name
are
of
gels
are
a
only
a
widely
occur
for
fields,
in nature, and
particular importance
as
one
phase
surfaces,
emulsions
are
are
requires
be studied.
a
During
From
liquid dispersions
liquids
in the continuous
phase.
unstable. The two
the emulsification process
as
for
phases separate
example detergents
proteins
or
the
stability
layer
of
with
proteins
is adsorbed
on
the
To
proteins
droplet
droplets.
surface which increases
of the emulsion due to steric and electrostatic stabilization.
Materials and Methods
9.2
9.2.1
The
a
new
lower the surface tension and
therefore reduce the amount of energy needed for the formation of the
Furthermore
to
immiscible
Basically two
lot of energy for the formation of the
such
in
industry, to
by van der Waals interaction between the droplets.
phase separation, stabilizing agents
added.
[61].
and in food
products
stabilized emulsions which
droplets dispersed
therrnodynamically
time due to coalescence driven
avoid
can
builds small
Since the emulsification process
whey protein
on
food model systems
wide range of microstfuctures
mixed where
the basis of many
are
example paints, pharmaceuticals
chapter we focus
few. In this
they
Experimental Sef-Up
light scattering experiments
strument described in 5.
were
performed using the 3d
cross-correlation in¬
PROTEIN STABILIZED EMULSIONS
9
70
9.2.2
Materials
For the
preparation
Isolates,
Sueur, MN, USA)
Le
content
of the emulsions
89
was
the
as
stabilizing protein.
The total
% consisting of the three components /?-lactoglobulin
(15 %)
lactalbumin
used
was
whey protein BIPRO (Le Sueur
commercial
a
and
(5 %).
albumin
serum
(79 %),
Less than 10 % of the
(Morgia)
denaturated. As the oil weiused soya oil
with
density of
a
protein
915
a-
protein
is
kg/m3
at
20°C.
Methods
9.2.3
Preparation
of the Protein Solution
without further modification
large particles,
which
aqueous solution of the
an
probably
BIPRO
purified prior
was
solution of 10% BIPRO
shifted with 0.1 N HCl
Because the
individual
urated
proteins, only
at 5000 g in order to
which
salt
removed with
of the
was now
purified protein
magnetic agitation
adjusted using
Emulsion
weight)
a
the
the
mixture
clear,
with the
the three
an
protein
excess
Then the
to
pH
was
points of the
but most of the denat¬
centrifuged
adjusted
the
protein components.
in
an
ultracentrifuge
pH of the remaining
7 with 0.1 N NaOH. The
of deionized water and the
preparation of the protein solution
3 %
so¬
by weight
dissolved in deionized water under moderate
was
Small deviations from
Emulsions
protein solution.
homogenization
with
weie
pH 7
of the solution
piepared by mixing
The mixture then
immediately
homogenizer (Modell H5000)
were
afterwards
was
(20 % by
prehomogenized
homogenized using
head pressure of 80 bar.
a
soya oil
a
with
laboratory
Two passages
were
process.
Light Scattering Experiments
of
was
an
0.1 N HCl.
commercial mixer and
means
was
emulsion,
of the solution
pH
match the isoelectric
larger aggiegates.
for 1-2 hours.
Preparation
done for the
exactly
dialysis against
lyophylized. For
was
points of
The solution then
almost
a
not
and the
prepared,
small amount of the native
a
contains
of the emulsion. An aqueous
preparation
the isoelectric
remove
solution,
lution
was
pH of the solution did
protein aggregates.
was
to the
by1 weight
near
Bipro protein also
the measured size distribution of the
particles
on
studies had shown that
aggregates of denaturated protein. To avoid
are
influence from these
protein
Light scattering
dynamic light scattering
Particle size distributions
at
a
were
determined
by
temperature of 25°C. Multiple scattering
71
Materials and Methods
9.2
effects due to the
high turbidity
technique.
correlation
of the emulsions
Measurements
were
were
suppressed using
duration each
were
used. Three
was
taken,
s
>
inside the
were
Investigations
a
scattered
multiply
(Hellma)
angle
[51].
8.2.3 and
chapter
reasonable
a
adjusted
were
to values of
the whole range of measured
inverse
The
dynamic light scattering
transformation.
Laplace
size distributions
resulting
a
product
of the
P(q) S(q).
~
While the
differential
particle
scribes the interactions between the
section
cross
form factor
droplets,
da(q)jdVl
depends
cross-
general
the
on
is obtained
factor,
i.
e.
size, shape
the static structure factor de¬
particles. Basically P(q)
which in
particle interactions,
the
intercept of
part of Is which is singly scattered.
the
and the internal structure of the emulsion
the absence of
The measured
measurements.
form factor and the static structure
particle
performed
were
contains contributions of different orders
j directly gives
single scattering
After correction the
potential
is corrected with the reduced
light,
which
function,
correlation
do~(q)/dQ,
cell
multi
scattering intensity Is, which
which is
s
highly concentrated samples
of the emulsions in terms of interaction
with combined static and
of
to 300
averaged.
then
average
with the
functions, covering
analysed using
The routine is described in
were
even
rectangular sample
200 pm. Each set of correlation
scattering angles,
In order to obtain
scattering angles 0.
of the cross-correlation function
light paths
the
s
where the duration of the measurements increases with
the concentration, at different
intercept
of
As solvent for the
by weight.
correlation functions of 120
cross-
cross-
protein stabilized emulsions
done with
different concentrations from 0.01 % to 10 % oil content
dilution deionized water
the 3d
can
be measured in
is fulfilled with
highly
diluted
i
samples. But,
tions
concentrations
hereby,
strength
lasting
total
known to
are
at least
as
s were
a
electrostatic,
done in the
and the
Another
with the
DLS results
are
of direct and
reduced
range from 0
intercept
possibility
=
by
20° to 0
=
120° every 2°. The
to obtain the differential
a
calculation of
In this
step
we
profit
interaction effects due to
interactions.
the ionic
of the cross-correlation function
consists in
according polydispersity.
hydrodynamic
change
Three measurements, each
P(q)
dynamic light scattering experiment
much less affected
a
Particle interac¬
subsequently by increasing
by adding NaCl.
scattering angle
distribution measured with,the
theory
are
angular
function of the
section.
react upon dilution with
the measurement of the structure factor.
scattering intensity
as
possibly
form factors need to be determined also at the
of the emulsion to 50 mM
30
analysed
cross
emulsions
distribution, p'article
of their size
same
as
were
scattering
from the size
and
using
Mie
from the fact that the
a
partial
cancellation
PROTEIN STABILIZED EMULSIONS
9
72
Results and Discussion
9.3
Size Distribution
9.3.1
Size distributions of the emulsion
tions diluted from
The results
weight.
and 1.5 %
by
mass were
from 90
droplets
nm
up to 700
highest
two
a
nm
radius of
r
peak
is
now
sizes remains.
sions
are
to
180
significant changes,
auto-correlation experiments
completely
expected,
as
even
highest
the
dilution
measured, |a
radii,
a
where the
position
of the
large particle
shoulder to
at
high
be orders of
diluted
sample
for such
required
magnitude higher
Although
the dilution
measure
As for 'normal'
dilutions.
with 0.01 % oil is still
experiments
would lead to
However, with cross-correlation
if the 'natural' concentration cannot be
even
small
% and 5 %
(10
to their size distribution.
respect
wrong results for the size distribution.
experiments,
%,
A broad distribution
nm.
the undiluted emulsion with 20 % oil content is too turbid to
too turbid to be
only
1
experiments show clearly that whey protein stabilized emul¬
not stable upon dilution with
series shows
by
in radius is visible. Further dilution of the
located at radii lower than 100 nm, and
These
%,
oil content of 5
an
concentrations
=
emulsion then shifts the size distribution to smaller
main
content
measured. The dilution series first shows
maximum located at
peak
protein
A maximum concentration of 10 % oil
31.
distribution for the
in the size
with the
of oil
figure
shown in
are
with 20 % oil and 3 %
present set-up. Further diluted samples
0.1 % and 0.01 %
changes
prepared
emulsion
measured for different concentra¬
were
content could be measured with the 3d cross-correlation instru¬
protein
ment in the
oil)
an
droplets
the
measured,
concen¬
than for auto-correlation measurements,
tration
can
and
therefore expect the measured size distribution to be close to the 'real' size
we
distribution of the undiluted
indicate that the
series, which
dilution
first before it starts to
Without
aiming
sample. This assumption
change
higher
at
to describe the
distribution which needs further
which in
an
principle
can
important value
sions. In
droplet
with
size
previous
for
distributions,
the
studies the surface
diluted
a
samples.
as
of this effect of the dilution
investigations,
example in
on
quantities
area was
cumulant fit of
such
as
the
obtained from
a
coverage and
typically
findings,
area
is needed
as
stability of emul¬
calculated from
it is worth
droplets
surface area,
droplet
dynamic light scattering
our
the size
on
the modification of the oil
study of protein
In view of
the
dilution.
be calculated from DLS results. The surface
size obtained from
highly
clearly supported by
size distribution remains constant
droplet
reason
upon dilution has serious consequences
is
average
an
data measured
pointing
light scattering measurement,
are
out that
intensity
9.3
73
Results and Discussion
3
a>
S
_3
"o
!»
100
10000
1000
radius (nm)
Figure
31:
Particle size distributions
stabilized with 3 %
see
text.
whey protein
of
in water
an
emulsion
consisting of
for different steps of
20 % soya oil
dilution. For details
weighted.
as
PROTEIN STABILIZED EMULSIONS
9
74
This
means
particlel
function of the
a
dominate the size
size distribution reflects the scattered
[the
scattering
fit,
is therefore
distribution,
weighted by
investigated,
a
which due to the
ume
theory.
angle
while such
analysis
an
tional auto-correlation measurements
important point
the emulsion.
to consider is the
From
a
on
can
particles.
samples
diluted
face
allow
the
Such
area.
It is thus
where
one can
experiments
only
hope
from
a
The effect of dilution
a
average
particle
highly
diluted
at the
same
on
This
the vol¬
yields
goniometer system,
that the
can
be
the second
specific
surface
area
surface to volume ratio of
larger
slightly
or
reasonable estimate of the total
sur¬
with cross-correlation schemes that
the size distribution of the
form
sample,
facto^ changes
it is
as
measurements of
concentration a's
suppressed.
are
sample by adding
care
particle
dilution, P(q)
done. Because
S(q)
Is(q)
and
interactions.
~
emulsion has
Because the
can't be measured with
P(q)
a
then has to be measured
P(q)S(q), particle
Two routes for the measurement of the
concentrations
although
with
protein stabilized
normally
interactions have
particle
form factor at
investigated.
First electrostatic interactions
of the
be
scattering angles.
serious consequences for the determination of
higher
can
Interactions
9.3.2
to be
strongly
P(q,r)
where
measurement with undiluted
to obtain
size distribution at different
as
by calculating
be obtained
effectively suppressing multiple scattering and performing
droplet
such
polydisperse samples
of
we see
only possible
are
cumulant
in the size distribution upon dilution of
change
increases with dilution. This is caused due to the
the smaller
on
be used in combination with tradi¬
classical
a
distribution
histogram
area
also
Laplace transformation used
inverse
distribution, from which also the surface
However,
can
clearly improved approach.
a
depends
would therefore
area
l/(r6P(q,r))Gi(q,r)
=
The multi
form factor
polydisperse samples,
For
Better results
area.
experiments represents
obtained.
particle
example obtained from the
for
as
intensity Is
r6P(q), larger particles strongly
~
calculation of the surface
density distribution Gn(r)
calculated with Mie
in these
size,
large particles.
the
underestimate the total surface
the number
Is(q)
size. Because
vector. An average
the
the emulsion
that
can
be screened with
salt. The emulsion is then
has to be taken
as
higher
(0.01%,
increased ionic
expected
the size distribution may
weaker electrostatic stabilization at
at three different concentrations
an
to be
change
hard-sphere like,
in response to the
salt content. Emulsions
0.1% and 1%
oil)
strength
were
measured
with deionized water and
9.3
Results and Discussion
with
a
mM/1
50
solution
the ratio of the
as
both intensities
figure
no
corrected for
were
structure in the
of the Ornstein Zernike
0.4 from
=
fitting
a
was
charge
factor, S(q)
periment.
Due to
P(q)
partial1
a
Assuming
a
coefficient of the
For the calculation of the
tribution
can
form factor
In
can
P(q)
33
of the
particles and,
for
particle
than
a
conditions for the
transformation
DLS
a
much less affected
are
ex¬
by
in¬
the measured collective
where D0 is the diffusion
distribution,
histogram. Using Mie-theory
is1
quite
or
the continuous
the average
particle
clearly required
In
general
factor of itwo from
signal-to-noise
provides improved
also
a
on
are
in
is very sensitive to the size
the relative volume fractions of
attempt
to
quantitatively
scattering amplitudes
Laplace
ratio.
Is(q)
to
precise method for the determination
in any
relative
compared
Calculated and measured data
remarkable since
size classes. Therefore
from DLS data.
not better
concentrations
hydrodynamic interactions,
1-4$),
Schultz
a
polydisperse systems,
of the size distribution is
Is(q)
higher
unity.
form factor either the measured size dis¬
scattering intensity.
agreement. This
the individual
agreement with
calculated from the size distribution is shown
the measured normalized
excellent
a
+
values
then be calculated.
can
figure
particle
example by
replaced by
be
form factor at
D0(l
=
60 nm,
=
interactions and $ is the volume fraction of the
particles without
be described for
(r
small deviations from
hard-sphere like behaviour
particles.
distribution
only
DLS experiment
diffusion coefficient Do behaves like Dq
closure
Rogers-Young
experimental data), typical
shows
particle
for the
S(q) theoretically
the
using
cancellation of direct and
a
that there is
from the size distribution measured in
measured size distributions j with
teraction effects.
means
area, and 0.5 mM salt content. In
The second route to determine the
calculation of
1, that
=
measured size distribution
Schultz distribution to the
the measured static structure
a
calculated
equation from the
of 250 nm2 for the surface
consists in
S(q)
find
have also tried to estimate
S(q)
with 1 % oil content.
is shown in
multiple scattering and turbidity,
we
where
due to electrostatic interactions. In addition to
sample
experimental approach, 'we
sample
salt,
intensities measured without and with
scattering
calculated
S(q),
The static structure factor
solvent.
as
In the accessible q-range
32.
regular
this
cr
75
inversion
However,
results for the
methods,
the multi
particle
can
even
angle
calculate
be obtained
under ideal
inverse
Laplace
size distribution which
are
the basis for these calculations.
This
size
now
allows
distributions,
colloidal
us
for the first time to obtain
interaction effects and
suspensions such
as
stability
emulsions and thus
a
of
consistent
highly
perform
picture
of
concentrated
for
example
particle
complex
an
in-situ
76
PROTEIN STABILIZED EMULSIONS
9
J.. J-
A
1-
A A
A
AA
A
"Aa
A
(',A
-,
,
„A.»
,
,aa.A(.A
/W
^-I.uujaav'^—ft^tfWfti^tr'
.
A
0.5-
0-
1
'
'
1
'
'
'
'
'
1
'
'
'
1
'
,
.
,
1.5
0.005
0.015
0.01
q
Figure
32: Static structure
actions. A: 0.01
investigation
wide
of
%,
o;
factor determination
0.1\%,
stability
and
D; 1
0.02
0.025
(1/nm)
via
screening of electrostatic
inter¬
% oil
aggregation
under realistic conditions relevant for
variety of commercially important systems.
a
Results and Discussion]
9.3
77
1
0.1-d
0 01
0.001
0 0001
0
0.005
0 01
0 005
0 01
q
Figure
33:
Normalized
tribution, symbols: measured) for different
:
%.
0 02
0 025
0 015
0.02
0 025
(1/nm)
scattering intensity (solid
i
1
0 015
line: calculated
from
oil concentrations: A: 0.01
the size dis-
%,
o:
0.1
%,
78
9
PROTEIN STABILIZED EMULSIONS
79
Outlook
10
high turbidity
tems of
mation
a
DLS
of the
quantitatively
be recovered with such
can
multiple
can
using cross-correlation techniques colloidal
that
clearly
This work demonstrates
experiment
total scattered
also SLS measurements
can
intensity.
It is
compared
when
Therefore it
Another
tance of
commercially
to
be used also in
can
amounts of
multiply
example
scattering angles
factor have
scattering
scattered
minimum,
a
and
multiple scattering
can
also
where the
using the
direct
high stability
in
dependence
scattered to
single
and is
relatively
easy to
instruments.
light scattering
is the clear demonstration of the
with
can
be
particle
influence
strongly
instrument,
be avoided
is
industrial environment.
light
With the 3d
data.
a
working
when
implemented
technique
the ratio of
available normal
important result of this thesis
multiple scattering
at
an
on
strong suppression of
shown that the 3d cross-correlation in¬
particularly
strument constructed in this work possesses
align
be done
of the cross-correlation function
intercept
due to the
experiments
the cross-correlation
light. Although
scattered
A maximum of infor¬
investigated.
be
sys¬
on
quite
present under these
form factor
the other
Significant
conditions,
for
the static structure
or
quantitative analysis
a
of the
light
such influences of
hand,
it is clear that the instrument is
easily. Finally,
not restricted to 3d cross-correlation
dilute systems.
impor¬
experiments but
can
also be used in normal
auto-correlation mode.
Based
on
the
experience
instruments has
design.
As
infrared
regime
a
now
new
emerged,
will be used
samples
spectrometers will
which will have
as
to
the
even
open
light
generation
an
dynamic properties
systems that
be studied with
source, thus
and
of 3d cross-correlation
even more
higher turbidity
new
of the static and
can
a new
feature modern diode lasers with the
range of measurable
that these
with this work
compact and robust
wavelength
considerably extending
or
near
the
concentration. We believe
exciting possibilities
of colloidal
in the
for
an
investigation
systems, and extend the
optical techniques significantly.
range of
10
OUTLOOK
81
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1997.
Danksagung
Ich möchte mich bei all denen
grösste Dank gebührt
zum
die
Schurtenberger
zu
dieser Arbeit
für seine
beigetragen
haben. Der
beispielhafte Betreuung.
Sein
seine exzellenten Ideen und die stete Diskussionsbereitschaft
persönlicher Einsatz,
haben
Peter
bedanken,
beigetragen.
der Arbeit entscheidend
Gelingen
Bedanken möchte ich
Anregungen und sein Engagement für dieses
Übernahme des Korrefereates. Ebenso danke ich
mich auch bei Heribert Watzke für seine
Projekt
bei der Nestlé sowie für die
Ludwig
Gauckler für seine spontane
Bereitschaft, das Korreferat
zu
übernehmen.
Bei meinen Mitstreitern während der Dissertation mochte ich mich besonders
bedanken. Götz Jerke hat mir den
Einstieg
in die Welt der Kolloide und der Licht¬
streuung erleichtert. Luigi Cannavaciuolo konnte oft
puter wieder einmal nicht das
taten
was
genug
weiterhelfen,
wenn
Com¬
sie sollten. Auch Cornelia Sommer und Sara
Romer dürfen auf keinen Fall unerwähnt bleiben. Durch ihrer aller stete Hilfs- und
Diskussionsbereitschaft wurde vieles einfacher.
Mein Dank
te
auch
geht
Zusammenarbeit,
ebenso
terstützung konnte ich
abwechslungsreiche
an
Thomas Gisler für seine wertvollen
an
Tips
und die gu¬
Jerome Crassous und Martin Leser. Auf ihre Un¬
stets zählen. Mit Sabine
Hilger
und interessante Feldwoche im
verbindet mich eine besonders
bei der die
Tessin,
Lichtstreuung
auch einmal die freie Natur erleben durfte.
Rolf Weber und Horst Seinecke bin ich
der Konstruktion und dem Bau der
Probleme
ner
nicht
waren
der Suche nach dem
für
dagegen
wenn
Leitfähigkeitsmessgerät, Gefriertrockner
gedankt.
Technologie
aufgehoben;
Elektronische
einmal ein Rech¬
konnten Harald Lehmann und Karel Zeemann helfen. Bei
mehr konnte ich mich auf die Hilfe
ler Geduld sei
Lichtstreuapparatur verpflichtet.
bei Max Wohlwend gut
funktionierte,
besonderem Dank für die Hilfe bei
zu
von
Eveline
Blöchliger
und
dergleichen Dinge
immer verlassen. Ihrer al¬
Diese Arbeit wurde durch die Nestec SA und die Kommision
und Innovation
(KTI)
finanziell
gefördert.
Zum Abschluss möchte ich mich bei meinen Eltern bedanken. Ohne ihre Un¬
terstützung
hätte ich nicht studieren können.
Lebenslauf
Name:
Claus Urban
Geburtstag:
14. Mai 1965
Geburtsort:
Karlsruhe, Deutschland
Schulbildung:
1971
1976
-
-
Hauptschule
in Karlsruhe
1976
Grund- und
1982
Realschule in Karlsruhe mit dem Abschluss
der Mittleren Reife
1985
-
Technische Oberschule in Karlsruhe
1987
Abschluss:
Beruflicher
1982
-
Fachgebundene Hochschulreife
Werdegang:
1985
Ausbildung
zum
Physiklaboranten
Bundesforschungsanstalt
an
Ernährung
für
der
in
Karlsruhe
1988
-
1995
Studium der
-
09/1994
der Universität Karls¬
(TH)
ruhe
03/1993
Physik an
Diplomarbeit: Optimierung
genschaften
des
der Materialei¬
Hochtemperatursupraleiters
YBa2Cu307-x hinsichtlich der Verwendung
in
1991
-
1995
magnetischen Lagern
Technische Dokumentation im
Reinisch
-
Ingenieurbüro
Technische Dokumentation
in
Bretten
11/1995
-
03/1999
Studium
mere
der
schule
zum
Doktorat
am
Eidgenössischen
(ETH)
for
Optic
a
Poly¬
Technischen Hoch¬
Zürich
Thema der Dissertation:
ber
Institut für
Based
Development
of Fi¬
Dynamic Light Scattering
Characterization of Turbid
Suspensions