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Understanding the diffusion of a colloid in
a polymer solution
Wim Maeyaert
Nicolas Vander Stichele
Thesis voorgedragen tot het
behalen van de graad van Master
in de ingenieurswetenschappen:
chemische technologie, optie
kunststofverwerking en
productontwerp
Promotoren:
Prof. dr. ir. P. Moldenaers
Prof. dr. ir. J. Vermant
Academiejaar 2011 – 2012
Master in de ingenieurswetenschappen: chemische technologie
Understanding the diffusion of a colloid in
a polymer solution
Wim Maeyaert
Nicolas Vander Stichele
Thesis voorgedragen tot het
behalen van de graad van Master
in de ingenieurswetenschappen:
chemische technologie, optie
kunststofverwerking en
productontwerp
Promotoren:
Prof. dr. ir. P. Moldenaers
Prof. dr. ir. J. Vermant
Assessoren:
Prof. dr. ir. B. Van der Bruggen
Ir. M. Vallerio
Begeleider:
Dr. ir. N. Reddy
Academiejaar 2011 – 2012
c Copyright K.U.Leuven
�
Without written permission of the thesis supervisors and the authors it is forbidden
to reproduce or adapt in any form or by any means any part of this publication.
Requests for obtaining the right to reproduce or utilize parts of this publication
should be addressed to Faculteit Ingenieurswetenschappen, Kasteelpark Arenberg 1
bus 2200, B-3001 Heverlee, +32-16-321350.
A written permission of the thesis supervisors is also required to use the methods,
products, schematics and programs described in this work for industrial or commercial
use, and for submitting this publication in scientific contests.
Zonder voorafgaande schriftelijke toestemming van zowel de promotoren als de
auteurs is overnemen, kopiëren, gebruiken of realiseren van deze uitgave of gedeelten
ervan verboden. Voor aanvragen tot of informatie i.v.m. het overnemen en/of
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Ingenieurswetenschappen, Kasteelpark Arenberg 1 bus 2200, B-3001 Heverlee, +3216-321350.
Voorafgaande schriftelijke toestemming van de promotoren is eveneens vereist voor het
aanwenden van de in deze masterproef beschreven (originele) methoden, producten,
schakelingen en programma’s voor industrieel of commercieel nut en voor de inzending
van deze publicatie ter deelname aan wetenschappelijke prijzen of wedstrijden.
Preface
A master thesis is a group assignment and we would like to use this page to
thank all the people who helped and supported us during the realization of this
scientific work.
“ Genius does what it must, and talent does what it can” Bulwer
Thanks to our promotors, Prof. Dr. Ir. P. Moldenaers and Prof. Dr. Ir. J.
Vermant, for providing us the opportunity to conduct our master thesis under
their supervision. Their constructive comments and support stimulated us to
convert our interest in chemical engineering into scientific thinking. We would
like to thank them for their help during our entire eduction in chemical engineering by providing a unique research and educational environment, together
with all members of the CIT department including our fellow students.
“ Habe einen guten gedanken, man borgt dir zwanzig” Ebner-Eschenbach
We would like to express our in-depth gratitude to Dr. Ir. Naveen, our
mentor during this work. Not only as a researcher but also as a friend Naveen
influenced greatly our final year at CIT. We would like to thank him for all
his guidance, eﬀort and time in making this master thesis better. His explanations about dynamic light scattering, the wonderful data processing software,
as well as his patience in explaining all kinds of problems we encountered, were
invaluable.
i
PREFACE
We would also like to thank Ruth Cardinaels and Jeroen De Wolf for their
support using the rheometer device, and Anja Vananroye for making the lab
the safest place on earth. Special thanks to Denis Rodriguez Fernandez for
providing ellipsoids coated with gold nanorods used for dynamic light scattering experiments in this master thesis and Ward Vanheeswijck for his help and
support with our special Latex compiler and cozy times at the copy room.
“La science se fait non seulement avec l’esprit, mais aussi avec le coeur” L. Pasteur
We greatly appreciate the endless support of our parents during our engineering studies, and the opportunity they gave us to graduate as a chemical
engineer. During those five years they stressed (many times) the importance of
a good education, but more importantly they gave us the right personal values
to develop into young responsible adults. Also special reference to our brothers
and sister for their support, happiness and help during our time at the university
and with this master thesis as a highlight.
A final mention goes to our girlfriends, Elien and Alexandra, for their endless
support and encouraging comments during the performance of this thesis. Even
though they didn’t always follow us in our elaborations, they kept believing in
our decisions.
Wim Maeyaert
Nicolas Vander Stichele
ii
Table of contents
Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
i
Abstract . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
vii
Samenvatting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
ix
List of figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
xi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiii
List of abbreviations and symbols . . . . . . . . . . . . . . . . . . xiv
1 Introduction
1
1.1 Motivation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1
1.2 Context . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
1.3 Aim and structure . . . . . . . . . . . . . . . . . . . . . . . .
3
2 Literature study
5
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
2.2 Viscoelasticity . . . . . . . . . . . . . . . . . . . . . . . . . .
8
2.2.1 Elastic materials . . . . . . . . . . . . . . . . . . . .
8
2.2.2 Viscous materials . . . . . . . . . . . . . . . . . . . .
8
2.2.3 Viscoelastic materials . . . . . . . . . . . . . . . . .
8
2.3 Bulk rheology . . . . . . . . . . . . . . . . . . . . . . . . . . .
9
2.4 Microrheology . . . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.4.1 Colloids . . . . . . . . . . . . . . . . . . . . . . . . .
11
2.4.1.1 Van der Waals forces . . . . . . . . . . . . .
12
2.4.1.2 Electrical double layer . . . . . . . . . . . .
12
2.4.1.3 DLVO . . . . . . . . . . . . . . . . . . . . .
13
2.4.1.4 Eﬀect of polymers on colloidal stability . .
2.4.2 Generalized Stokes-Einstein relationship . . . . . . .
14
15
2.4.2.1 Translational diﬀusion . . . . . . . . . . . .
16
2.4.2.2 Rotational diﬀusion . . . . . . . . . . . . .
18
2.4.3 Active microrheological techniques . . . . . . . . . .
19
2.4.4 Passive microrheological techniques
. . . . . . . . .
19
2.4.4.1 General dynamic light scattering (DLS) . .
20
iii
TABLE OF CONTENTS
2.4.4.2 Diﬀusive wave spectroscopy . . . . . . . . .
20
2.4.4.3 Video based particle tracking . . . . . . . .
20
2.5 State of the art . . . . . . . . . . . . . . . . . . . . . . . . . .
20
3 Materials and methods
23
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . .
23
3.2 Materials . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
3.2.1 Polymers: Polyetheleneglycol or Polyetheleneoxide .
24
3.2.2 Beads . . . . . . . . . . . . . . . . . . . . . . . . . .
25
3.2.2.1 Spheres . . . . . . . . . . . . . . . . . . . .
25
3.2.2.2 Ellipsoids . . . . . . . . . . . . . . . . . . .
25
3.2.3 Extra materials . . . . . . . . . . . . . . . . . . . . .
28
3.3 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
3.3.1 Sample preparation . . . . . . . . . . . . . . . . . .
29
3.3.2 Rheometer . . . . . . . . . . . . . . . . . . . . . . .
31
3.3.2.1 Device . . . . . . . . . . . . . . . . . . . . .
31
3.3.2.2 Maxwell model . . . . . . . . . . . . . . . .
32
3.3.3 Dynmaic light scattering . . . . . . . . . . . . . . . .
33
3.3.3.1 General dynamic light scattering . . . . . .
33
3.3.3.2 Depolarized dynamic light scattering . . . .
36
3.3.3.3 DLS device . . . . . . . . . . . . . . . . . .
39
3.3.4 Comparison between bulk and microrheology . . . .
40
4 Experimental results and discussion
43
4.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.2 Rheology . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
43
4.3 Dynamic Light Scattering of spherical particles and polymers
45
4.3.1 Silica particles in water . . . . . . . . . . . . . . . .
47
4.3.2 Latex particles in water . . . . . . . . . . . . . . . .
48
4.3.3 Polymers in water . . . . . . . . . . . . . . . . . . .
50
4.3.4 Latex particles in polymers . . . . . . . . . . . . . .
54
4.3.5 Comparison between DLS and rheometer . . . . . .
57
4.4 Ellipsoids . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Ellipsoids in water . . . . . . . . . . . . . . . . . . .
59
59
4.4.2 Ellipsoids in polymer . . . . . . . . . . . . . . . . . .
61
4.4.3 Comparison between DLS and rheometer . . . . . .
63
5 Conclusion and future research
65
A Specifications of the geometry
68
iv
TABLE OF CONTENTS
B Results PEG 35 kDa
70
C Matlab code for data processing
74
D Safety and hazard analysis
77
Bibliography
79
v
Abstract
In this scientific work, research is conducted on the use of microrheological
techniques to characterize viscoelastic materials. In traditional rheological experiments, properties of complex fluids such as polymers are investigated using
conventional rheometers. Recently however a new technology, called microrheology, has been developed in this field. In microrheology, rheological properties
of complex fluids are investigated by tracking the movement of microparticles
embedded in them. Microrheology has numerous advantages over bulk rheology
and is becoming more important ever since its discovery.
When adding particles to a fluid, a colloidal solution is formed. Although
most colloids in nature are disklike and rodlike, current technology is based on
spherical colloids. Due to the complex characteristics and the lack of suitable
computational models, only minor research was done on the use of non-spherical
particles in microrheology. Therefore, in this master thesis the use of nonsphercial, ellipsoidal particles will be investigated. While the spherical particles
only yield a translational diﬀusion, ellipsoids also show rotational diﬀusion due
to their shape anisotropy. By tracking this rotational diﬀusion more information
about the viscoelastic behavior of the material can be obtained.
Spherical particles in polyethylene glycol (PEG) are used to accustom to
microrheology and prove the current state of the art in this field. To prove
�
��
the suitability of microrheology, the storage and loss modulus (G and G ) of
the samples are investigated with both dynamic light scattering (DLS) and a
traditional bulk rheometer. A modified, algebraical form of the general Stokes
Einstein relation enables a comparison of both methods. As expected from literature, the use of microrheology with spherical particles was proven an excellent
equivalent for bulk rheology.
vii
ABSTRACT
After proving the value of microrheology for spherical particles, the second part of the thesis focuses on the possibilities of using ellipsoidal beads in
this experimental technique. Regular light scattering measurements, used with
spherical particles, include both translational and rotational diﬀusion in one
value. But by placing a polarizer before the detection optics, scattering of the
translational diﬀusion is suppressed and only rotational diﬀusion is observed.
The aim was to calculate storage and loss modulus from this rotational diﬀusion and compare this result with bulk rheology, in order to prove the suitability
of rotational diﬀusion in microrheology.
But, as the ellipsoids used in this last part are stretched spheres in a prolate
way, they are anisotropic in shape but still have isotropic scattering properties.
Consequently they gave disappointing results with the polarizer, as the rotational diﬀusion was not recognized. To overcome this problem, the ellipsoids
were covered with gold nanorods. The gold nanorods have excellent scattering
properties and as they are attached to the ellipsoids, they scatter the rotational
diﬀusion of the ellipsoids.
Surprisingly however, it was discovered that not the size of the ellipsoids,
but that of the gold rods instead, has to be used in calculations to obtain equivalent results between bulk and microrheology. Observing this results, this would
imply for the first time that the probe size of the bead would diﬀer from the scattering size of the particle. One other explanation is that the nanorods detached
from the ellipsoids due to the polymer solution and were moving freely in the
polymer solution. Either way, for the first time the use of ellipsoids is shown in
determining microrheology properties. Although further investigation is needed,
this idea could considerably enlarge the application scope of microrheology.
viii
Samenvatting
In dit wetenschappelijk werk wordt onderzoek gedaan naar microreologische
technieken om viscoelastische materialen te karakteriseren. In traditionele reologische experimenten worden de eigenschappen van complexe vloeistoﬀen, zoals
polymeren, onderzocht met conventionele reometers. De laatste jaren werden
binnen dit domein echter een aantal nieuwe technieken ontwikkeld, waaronder
microreologie. Hierin worden reologische eigenschappen van complexe vloeistoffen onderzocht via het nauwgezet opvolgen van de beweging van micropartikels
die aan de onderzochte vloeistof werden toegevoegd. Omwille van zijn vele voordelen t.o.v. bulk reologie, blijft microreologie, vooral de laatste jaren, aan
belang winnen.
Door het toevoegen van micropartikels aan een vloeistof, ontstaat een colloïdale oplossing. Hoewel de meeste colloïdale deeltjes in de natuur voorkomen
als schijfjes of staafjes, is de huidige technologie gebaseerd op sferische partikels.
Door de complexe karakteristieken en het ontbreken van gepaste rekenkundige
modellen, is er voorlopig slechts in beperkte mate onderzoek uitgevoerd naar het
gebruik van niet-sferische partikels in microreologie. Desondanks zullen in deze
master thesis de mogelijkheden van ellipsoïde deeltjes in microreologische technieken onderzocht worden. Waar bij sferische partikels enkel hun translationele
diﬀusie kan gemeten worden, laten ellipsoïden, door hun anisotrope vorm, het
ook toe hun rotationele diﬀusie te karakteriseren. Door ook deze rotationele
diﬀusie te observeren, kan er meer informatie over het viscoelastisch gedrag van
het materiaal achterhaald worden.
Sferische deeltjes in polyethyleenglycol (PEG) worden gebruikt om de basisprincipes van microreologie te leren kennen en de huidige ontwikkelingen in dit
domein te toetsen. Om de compatibiliteit tussen bulk en microreologie te bewi-
ix
SAMENVATTING
jzen, worden de opslag en verlies modulus (G’ en G”) van verschillende samples
onderzocht met zowel dynamische licht verstrooiing (DLS) als met een traditionele reometer. Een aangepaste, algebraïsche vorm van de algemene Stokes
Einstein relatie laat toe om de beide resultaten met elkaar te vergelijken. Zoals
beschreven in de literatuur, werd aangetoond dat microreologie met sferische
partikels een geldig alternatief vormt voor bulkreologie.
De ellipsoïden die vervolgens in de initiële experimenten gebruikt werden,
waren uitgerekte sferen. Hierdoor hadden ze, hoewel anisotroop in vorm, isotrope
verstrooiings eigenschappen. Op deze manier waren de resultaten met de polarisator, die de vertstrooiing van de rotationele diﬀusie niet herkende, niet zoals
gehoopt. Als oplossing voor dit probleem werden de originele ellipsoïden bedekt
met kleinere nanostaafjes uit goud. Deze hebben wel uitstekende verstrooiings
eigenschappen, en aangezien ze bevestigd zijn op de ellipsoïden laten ze toe de
rotationele diﬀusie van de ellipsoïden waar te nemen.
Verrassend genoeg bleek uit deze laatste experimenten dat niet de lengte van
de ellipsoïden gebruikt moest worden in de berekeningen om de compatibiliteit
tussen bulk en microreologie aan te tonen. Het is waarlijk door de lengte van
de gouden nanostaafjes mee te nemen in de wiskundige berekeningen, dat deze
compatibiliteit wel werd aangetoond.
Een andere mogelijkheid bestaat erin dat de goud nanostaafjes loskomen van
de ellipsoïden in de polymeeroplossing. Uitgebreider onderzoek is zeker noodzakelijk, maar hoe dan ook wordt in dit werk voor de eerste keer microreologie gebruik makend van ellipsvormige partikels toegepast. Toekomstige experimenten
zullen moeten uitwijzen indien het toepassingsgebied van microreologie hierdoor
aanzienlijk vergroot kan worden.
x
List of Figures
1
Response of a material to a shear strain. . . . . . . . . . . . . . .
2
Typical frequency ranges for diﬀerent measurement devices [8]. .
3
Schematic representation of the rheometer. [8]
4
Stress/strain response of a material to a strain/stress application
. . . . . . . . . .
[11] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
5
7
10
10
Schematic representation of the electro-static potential near a
solid surface in a solution containing ions [13]. . . . . . . . . . . .
13
6
Eﬀect of adding polymers to colloidal solution: steric stabilization
7
and depletion[14]. . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
Eﬀect of adding polymers to a colloidal solution: eﬀect of bridging. 15
8
Evolution of the mean square displacement (MSD) of the particles
as a function of the lag time τ [8] . . . . . . . . . . . . . . . . . .
17
9
Monomer polyethylene glycol (PEG) . . . . . . . . . . . . . . . .
24
10
Scanning electron micrograph (SEM) of silica particles [24] . . .
25
11
Rotation of ellipsoids in a fluid.(a) axisymetric rotation. (b) nonaxisymmetric rotation.
. . . . . . . . . . . . . . . . . . . . . . .
26
12
Oblate and prolate ellipsoid. . . . . . . . . . . . . . . . . . . . . .
27
13
Transmission electron micrograph of polystyrene ellipsoids. . . .
27
14
Transmission electron micrograph of of gold nanorods. . . . . . .
27
15
Schematic representation of prolate ellipsoids covered with gold
nanorods. [Courtesy: Sylvie Van Loon] . . . . . . . . . . . . . . .
16
28
Schematic representation of the critical overlap concentration C∗
[29] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
17
Schematic representation of the radius of gyration. [29]
30
18
Anton-Paar 501 MCR stress-controlled rheometer
. . . . . . . .
32
19
Schematic representation of the cone and plate geometry [12] . .
32
20
Schematic representation of the Maxwell model . . . . . . . . . .
32
�
. . . . .
��
21
Evolution of G and G described by the Maxwell model [31] . .
33
22
Schematic representation of dynamic light scattering (DLS) . . .
34
xi
LIST OF FIGURES
23
Scattering oﬀ a small particle in an ideal solution by incident
light [32] . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
24
Intensity measurement and autocorrelation function measured by
dynamic light scattering [12] . . . . . . . . . . . . . . . . . . . . .
25
36
Schematic representation of the general dynamic light scattering
setup (VV-mode). [35] . . . . . . . . . . . . . . . . . . . . . . . .
26
34
38
Schematic representation of the depolarized dynamic light scattering setup (VH-mode). [35] . . . . . . . . . . . . . . . . . . . .
39
27
The ALV/CGS-3 compact goniometer system. . . . . . . . . . . .
40
28
Bulk rheology: Storage G” and Loss modulus G’ for PEO 1000
kDa at 20 C∗ . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
29
Bulk rheology: Storage modulus G” for PEO 1000 kDa at diﬀerent concentrations. . . . . . . . . . . . . . . . . . . . . . . . . . .
30
44
44
Bulk rheology: Loss modulus G’ for PEO 1000 kDa at diﬀerent
concentrations. . . . . . . . . . . . . . . . . . . . . . . . . . . . .
45
31
DLS: Silica particles in water . . . . . . . . . . . . . . . . . . . .
47
32
Diﬀusion coeﬃcient: Silica particles in water
. . . . . . . . . . .
48
33
DLS: Latex particles in water . . . . . . . . . . . . . . . . . . . .
49
34
Diﬀusion coeﬃcient: Latex particles in water . . . . . . . . . . .
49
35
DLS: Overview diﬀerent molecular weights PEG at 0.5 C∗ . No
particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
51
36
DLS: PEO 1000 kDa at 0.5 C . No particles . . . . . . . . . . . .
51
37
∗
Diﬀusion coeﬃcient: Overview PEG diﬀerent molecular weights
at 0.5 C∗ . No Particles. . . . . . . . . . . . . . . . . . . . . . . .
52
38
Diﬀusion coeﬃcient: PEO 1000 kDa at 0.5C . No particles. . . .
53
39
DLS: PEO 1000 kDa at 20C . No particles . . . . . . . . . . . .
54
40
DLS: PEO 1000 kDa at 5C and 10C . Latex Particles . . . . . .
55
41
DLS: PEO 1000 kDa at 20C∗ . Latex Particles . . . . . . . . . . .
55
42
DLS: Overview PEO 1000 kDa at diﬀerent concentrations. Latex
∗
∗
∗
∗
Particles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
56
43
MSD: PEO 1000 kDa at 20C . Latex Particles . . . . . . . . . .
56
44
DLS-Rheo: PEO 1000 kDa at 20C∗ . Latex Particles . . . . . . .
58
45
DLS-Rheo: PEO 1000 kDa at 10C . Latex Particles . . . . . . .
58
46
DLS: Ellipsoids in water. No polarizer . . . . . . . . . . . . . . .
59
47
Diﬀusion coeﬃcient: Ellipsoids in water. No polarizer . . . . . .
60
48
DLS: Ellipsoids in water. Polarizer . . . . . . . . . . . . . . . . .
60
49
DLS: PEG 1000 kDa at 20C Ellipsoids. No polarizer . . . . . .
62
50
DLS: PEG 1000 kDa at 20C Ellipsoids. Polarizer . . . . . . . .
62
51
DLS-Rheo: PEG 1000 kDa at 20C∗ . Ellipsoids . . . . . . . . . .
64
52
NFPA 704 chloroform and toluene . . . . . . . . . . . . . . . . .
78
∗
∗
∗
∗
xii
List of Tables
1
Classification of colloidal solutions depending on the continuous
medium and the dispersed phase. . . . . . . . . . . . . . . . . . .
2
11
Overview of Rg , C∗ and mesh size at diﬀerent concentrations for
diﬀerent molecular weights Mw . . . . . . . . . . . . . . . . . . . .
31
3
Overview of radius of spheres in water. . . . . . . . . . . . . . . .
50
4
Overview radius of gyration for diﬀerent molecular weights. . . .
53
5
Overview of length of ellipsoids in water. . . . . . . . . . . . . . .
61
xiii
List of abbreviations and
symbols
List of abbreviations
AFM
Atomic Force Microscopy
DDLS
Depolarized Dynamic Light Scattering
DLS
Dynamic Light Scattering
DLVO
Derjaguin, Landau, Verwey & Overbeek
DWS
Diﬀusive Wave Spectroscopy
GSER
Generalized Stokes Einstein Relationship
MCR
Modular Compact Rheometer
MSD
Mean Square Displacement
PEG
Polyethylene Glycol
PEO
Polyehtylene Oxide
VDW
Van der Waals
VV
Vertical - vertical mode
VH
Verical-horizontal mode
xiv
LIST OF ABBREVIATIONS AND SYMBOLS
List of Greek symbols
α
Diﬀusive coeﬃcient
[-]
β
Coherence factor
[-]
γ
Strain
γ˙
Shear rate
[s−1 ]
Γ
Decay rate
[s−1 ]
ΓV V
Decay rate vertical vertical mode
[s−1 ]
ΓV H
Decay rate vertical horizontal mode
[s−1 ]
δ
Phase angle
[rad]
ε
Dielectric constant
[Fm−1 ]
ε0
Dielectric constant vacuum
[Fm−1 ]
ξ
Mesh size
[m]
φ
Electric potential
[V]
κ−1
Debye length
[m]
ψ0
Surface potential
[V]
λ
Wavelength
[m]
σ
Stress
[Pa]
σ0
Stress amplitude
[Pa]
τ
Relaxation time
θ
Angle
[rad]
∆θ
Mean square angular displacement
[rad]
ν
Refraction index
ω
Frequency
[%]
[s]
[-]
[rad/s]
xv
LIST OF ABBREVIATIONS AND SYMBOLS
List of symbols
a
Ah
c∗
D
Dt
Dr
e
E(t)
E
G∗
Radius of the particle
Hamaker constant
Critical overlap concentration
Diﬀusivity coeﬃcient
Diﬀusivity coeﬃcient translational
Diﬀusivity coeﬃcient rotation
Electron charge = 1.60217646.10−19
Electric field Intensity
Elasticity modulus
Complex shear modulus
[m]
[J]
[g/L]
[m2 .s−1 ]
[m2 .s−1 ]
[m2 .s−1 ]
[C]
[-]
[Pa]
[Pa]
�
Storage modulus
[Pa]
��
Loss modulus
[Pa]
Laplace transformation of complex shear
modulus
Intensity auto correlation function (IACF)
(Electric) field auto correlation function
(FACF)
FACF in the vertical-vertical mode
FACF in the vertical-horizontal mode
Intensity field gradient of the scattered light
Intensity field gradient of the incoming light
Ionic strength
Boltzman number=1.3806488.10−23
Eﬀective spring constant
Weight average molecular weight
Avogadro’s number = 6.02214179.1023
Scattering vector
Radius of gyration
Mean square translational displacement
Temperature
[Pa]
G
G
˜
G
g2 (q, t)
g1 (q, t)
g1V V (q, t)
g1V H (q, t)
I
I0
[I]
kB
Ks
Mw
NA
q
Rg
∆r
T
xvi
[-]
[-]
[-]
[-]
[cd]
[cd]
[mol.dm−3 ]
[m2 kg.s.K −1 ]
[-]
[g/mol]
[mol−1 ]
[m−1 ]
[m]
[m]
[K]
Chapter 1
Introduction
1.1 Motivation
Panta rhei (π αντ
´ α ��˜ι) or ’everything flows’ are the famous words of the Greek
philosopher Heraclitus and also form the etymological origin of the word rheology. Rheology is the study of the flow of matter. Many industrially important
substances such as paints, beverages, chocolates, polymers and even flowing
metals during processes have very complex flow behavior. The major concern
of rheology is to measure, describe and establish predictions about the flow behavior of these materials. Rheology is used in everyday life, not only through
the use of consumer goods, e.g. the good mouth-feeling of beverages and chocolates or the ease of application of paint, but also during the processing steps of
many consumer goods.
Although the formal introduction of the word rheology was established in
1929, considerable rheological research has been carried out before. During these
early measurements, many devices were developed to measure the rheological
properties of matter. Traditionally, the characterization of complex materials
is performed using bulk rheology devices such as a rheometer. A rheometer
imposes a specific stress or strain on the material and measures the resulting
strain or stress in the fluid. At present the rheometer is a widely used device to
investigate the rheology of complex materials.
However, more recently, new rheological measurement techniques have been
developed [1, 2]. One method of particular interest is microrheology. In microrheology, micro/nano-scale particles are added to a small sample of the material under investigation. The goal of microrheology is to derive the rheological
properties of the material by tracking the motion of these particles embedded
in them. Microrheology overcomes certain limitations of bulk rheology such as
1
CHAPTER 1. INTRODUCTION
a smaller sample size and a larger range of frequencies and moduli that can
be probed. Additionally, the strains exerted by microrheology are much smaller
than those in bulk rheology, which makes the method useful for fragile materials.
Important advantages also include the possibility of investigating heterogeneity
and a reduced cost [3].
These advantages of microrheology have made this method very popular over
the past decade. The use of particle trackers of diﬀerent size, shape and matter
or the investigation of diﬀerent materials have raised a number of challenges
within this field. A further understanding of microrheology makes it possible
to investigate more materials and enables us to extract information such as the
microrheological properties of the material which supplements bulk rheology. A
major drawbacks of the method however, is that the material under investigation
should be transparent when using dynamic light scattering[3].
1.2 Context
Over the past decades, scientists have investigated the use of spherical particles
in microrheology. For many materials the use of microrheology has been proven
as an equivalent of bulk rheology. Although the concept was originally proven
on synthetic materials, microrheology soon started playing an important role in
the investigation of biological matter. These materials are not available in large
amounts which makes traditional bulk rheology, which requires larger sample
sizes, inappropriate.
More recently, the use of non-spherical particles in microrheological methods gained particular interest. By using non-spherical particles the translational
movement of the particles, as well as the rotation of the beads can be investigated. With the spherical particles used originally, this rotational diﬀusion was
not measurable. A new point of interest has emerged on the use of ellipsoidal
particles. For simple, incompressible, isotropic viscoelastic materials, measurements using these particles provide redundant information which can be used to
check the self-consistency of the measurements by comparing translational and
rotational results for the same material property [4]. Additionally, ellipsoids can
give information about the nature of depletion zones formed in certain materials
as explained in section 2.6.
2
CHAPTER 1. INTRODUCTION
1.3 Aim and structure
The purpose of this research is twofold. First, the equivalence of bulk rheology
and microrheology with spherical particles will be investigated for polyethylene
glycol solutions. In the second part, the use of ellipsoidal particles in microrheology will be investigated. In this section the major concern is to obtain an
equivalence with bulk rheology concerning the rotational diﬀusion of the ellipsoidal particles.
After an extensive literature study in chapter two, the used materials and
methods are further explained in chapter three. The experimental work in
chapter four is divided in two parts. In a first section the results from bulk
rheology experiments are shown. Since bulk rheology is not the main goal of
this scientific work only few experimental work was done in this part.
In the second experimental part, microrheological experiments were carried
out with dynamic light scattering, of which the results are given in section 4.2.
As a way of acquaintance with the device, experiments on silica particles in water
were performed. Later, the beads used for microrheology in this master thesis
were tested in water to fully characterize them. After testing the particles in a
polymer solution the results of microrheology are compared with bulk rheology
for spherical particles.
In the last experimental part the use of ellipsoidal particles in microrheology
is investigated. Firstly translational diﬀusion of the ellipsoids is tracked with a
general dynamic light scattering device. Secondly, to obtain only the scattering
of the rotational diﬀusion of the particles a depolarized dynamic light scattering
(DDLS) device was used.
3
Chapter 2
Literature study
2.1 Introduction
The term rheology was inspired from the Greek ’panta rei’ meaning ’everyting flows’. It was coined by E.G. Bingham, one of the founding fathers
of rheology. In rheology the deformation and flow of materials in response to
applied stress or strain is investigated. Rheology focuses on materials with a
complex molecular structure such as polymer solutions, suspensions, emulsions,
which exhibit a complicated flow behavior. A key diﬀerence between a solid
matter and a liquid material is their contrasting response to an applied shear
strain, which is the amount of deformation perpendicular to a given plane (figure 1). When applying a force, simple Newtonian liquids dissipate the provided
energy through viscous flow, while pure solids store energy and show an elastic
response [5, 6]. Squishy materials on the other hand both store and dissipate energy and are called viscoelastic materials. Rheology reveals both their solid-like
and fluid-like behavior, depending on the time scale used to probe the material.
Figure 1: Response of a material to a shear strain.
5
CHAPTER 2. LITERATURE STUDY
Traditional rheological experiments are performed with mechanical rheometers. However, over the past decades new techniques have been developed to
measure the viscoelastic behavior of complex materials. Microrheology is one
of them and uses embedded micron or nano-sized particles to locally deform
a sample. The goal of microrheology is to derive the rheological properties
of the material from the motion of the colloidal particles embedded within it.
The particles can be either thermally excited (passive rheology) or moved using external forces (active rheology). Microrheological methods enjoy certain
advantages over bulk rheological techniques [3] .
• In microrheology the required sample volume is much smaller (� 1 ml),
which make it possible to study rare and expensive materials, including
biological materials that are diﬃcult to obtain in large quantities.
• The frequency range which can be probed may be orders of magnitude
greater in microrheology than in conventional bulk rheology (figure 2).
• Microrheological techniques exert very low strains which is useful to measure the viscoelastic properties of fragile materials.
• Microrheology allows to measure local anisotropy in inhomogeneous systems.
• Microrheology is a non contact method. If the samples are toxic or hazardous they can be placed in a glass tube and measured using a dynamic
light scattering setup.
Besides the many advantages of microrheology there are some limitations to the
method too [7].
• The most important limitation is that the samples in microrheology most
be transparent to light when using dynamic light scattering device.
• Microrheology is computationally intensive. The raw data coming from
the device need to be transformed in order to deduce the rheological properties of the matter.
• If the probe motion is slow, the time to collect suﬃcient information is
large and therefore this method is also not useful for very stiﬀ or viscous
materials.
6
CHAPTER 2. LITERATURE STUDY
• If the polymer chains in the sample are far apart (large mesh size) very
large probes are necessary to measure the viscoelastic behavior of the
material. If the probes are too large, only part of the probe scatters
light emitted by the laser of the dynamic light scattering device, giving
erroneous results. This will be further explained in chapter 3.
Figure 2: Typical frequency ranges for diﬀerent measurement devices [8].
In order to fully understand the subject the general information about viscoelastic materials will be given in section 2.2. Later bulk rheology experiments
will be explained briefly. To fully understand microrheological methods a brief
introduction is given on colloids. In microrheology, the full frequency dependence of the viscoelastic moduli is obtained from the mean square displacement
of embedded particles using the generalized Stokes-Einstein relation, this is discussed in section 2.4.2. Next, active microrheological methods are listed up. As
these methods require sophisticated instrumentation they will not be discussed
in detail. In the next session the passive measurements will be discussed with
special attention to dynamic light scattering. In section 2.5 the state of the art
about the subject is given.
7
CHAPTER 2. LITERATURE STUDY
2.2 Viscoelasticity
2.2.1 Elastic materials
The behavior of pure solids or elastic materials can be described by Hooke’s
law [9]:
(1)
σ = Eγ
where σ is the stress, E the elasticity modulus and γ is the strain. This law
states that the stress is directly proportional to the strain. All the energy given
to the material while loading is stored in the material. When the load is removed
all the energy is released again.
2.2.2 Viscous materials
For viscous materials Newton’s law describes their behavior:
.
(2)
σ = ηγ
.
where σ is the stress, η the viscosity and γ the shear rate. In this case the
stress is directly proportional to the shear rate. Perfectly viscous materials obey
this law. When a viscous material is loaded all the energy is dissipated. These
are called Newtonian fluids.
2.2.3 Viscoelastic materials
Viscoelastic materials are in between the two extremes (Newtonian fluids and
Hookian solids). The elastic susceptibility of a viscoelastic material is given by
the complex shear modulus G∗ (ω). For an oscillatory shear strain at a frequency
ω, G∗ (ω) determines the stress induced in the material. The real part of the
�
complex modulus Re(G∗ (ω)) = G (ω) is the in phase response of the medium
to the applied strain and is called the elastic or storage modulus [6]. It is a
measure of the elasticity and the storage of energy of the investigated material.
��
The imaginary part of the complex shear modulus Im(G∗ (ω) = G (ω) is the out
of phase response to the applied strain and is called the viscous or loss modulus.
�
It is related to the viscosity of the material and the dissipation of energy. G (ω)
��
and G (ω) are related by the Kramers-Kronig relations [10].
8
CHAPTER 2. LITERATURE STUDY
2.3 Bulk rheology
For bulk rheological measurements a rheometer is used. There are two diﬀerent types of rheometers: drag flows and pressure driven rheometers. The former
applies a shear stress or shear strain between a moving and a fixed solid surface
while the latter generates a shear by a pressure diﬀerence over a closed channel.
Pressure driven rheometers are less frequently used. Of both rheometers many
diﬀerent types exist, which will not be discussed in detail. In a stress/strain
controlled rheometer, after applying an oscillatory stress or strain (figure 4)
the stress/strain response of the material is measured. For an elastic material
the response in completely in phase with the applied stress or strain, while for
an viscous material only an out of phase component is observed. A viscoelastic material has a situation in between (figure 4). The response will have the
same frequency but will be shifted by a phase angle δ. The response will be
decomposed into an in phase component and an out of phase component.
If for example a sinusoidally deformation is applied to a material the strain
γ is given by [11]:
(3)
γ = γ0 sin(ωt)
where γ0 is the amplitude of the strain. The resulting stress σ in the material
is then given by:
(4)
σ = σ0 sin(ωt + δ)
where τ0 is the amplitude of the stress. If the resulting stress is decomposed
into the in phase (sin) and out of phase (cos) component, the stress is given by:
�
��
�
��
σ = σ + σ = σ 0 sin(ωt) + σ cos(ωt)
(5)
It can be derived that the relationship between the phase angle and the in
and out of phase component is:
��
σ
tanδ = 0�
σ0
(6)
Now a modulus can be defined as the ratio of the stress in the material and
the strain applied to it. The decomposition of the stress thus results in the two
dynamic moduli:
�
σ
G = 0
γ0
�
9
(7)
CHAPTER 2. LITERATURE STUDY
��
��
G =
σ0
γ0
(8)
So from equation 6 follows:
��
G
tanδ = �
(9)
G
If the real and imaginary part of the modulus are combined, the complex
modulus G∗ can be defined:
σ0 = |G∗ | γ0
(10)
�
��
or G∗ is a complex number with a real part: G and an imaginary part: G .
�
G∗ = G + iG
��
(11)
During rheology measurements G∗ is measured and later decomposed into
its real and imaginary part, to describe its rheological behavior.
Figure 3: Schematic representation
of the rheometer. [8]
Figure 4: Stress/strain response of
a material to a strain/stress application [11]
10
CHAPTER 2. LITERATURE STUDY
2.4 Microrheology
In order to use microrheological methods, particles must be added to the
investigated material. Because the embedded particles are of microscopic scale
or smaller, a colloidal solution is formed.
2.4.1 Colloids
The name colloids comes from the Greek ’κολλα’ which means ’to stick’.
A colloidal system consists of a dispersed phase and a continuous phase. A
colloidal systems satisfies two important properties. Firstly the behavior of the
system is mainly determined by the thermal fluctuations of the particles due
to the Brownian motion. Secondly the surface properties predominate the bulk
properties of the system. A classification of colloids can be found in Table 1. In
microrheology we mainly deal with suspensions [12].
❤❤❤❤
❤❤❤
❤❤
dispersed phase
❤❤❤❤
❤❤❤
❤❤
SOLID
LIQUID
GAS
SOLID
Solid
Foam
Foam
LIQUID
Suspension
Emulsion
Foam
GAS
Aerosol
Aerosol
Gas
continous phase
Table 1: Classification of colloidal solutions depending on the continuous
medium and the dispersed phase.
The stability of a colloidal system is a key feature in microrheological measurements. Stability is defined as the ability of the system to remain in the
current state. This means that the particles should not stick together or settle
down in the polymer solution but in contrast stay dispersed in the continuous phase. There are two ways of describing the stability of colloidal systems:
thermodynamical and through kinetical stability.
Thermodynamical stability is the state of minimal Gibbs free energy. Normally the energy of colloidal systems is very high and they are therefore thermodynamically unstable. Kinetic stability comes when the transition to a lower
energy state becomes so slow that no aggregation occurs. The tools to influence the stability of a colloidal systems are the interaction forces between the
particles.
There are mainly two forces, attraction and repulsion, acting on colloidal systems. In order to obtain stability a balance between them should be established.
It is very important to understand and to manipulate these forces. The repulsive
forces are electrostatic forces, steric repulsions, electrosteric repulsions and solvatation. Among the attractive forces depletion, hydrogen bonds, hydrophobic
11
CHAPTER 2. LITERATURE STUDY
interactions and Van der Waals forces are observed. To investigate the stability
of a colloidal suspension the overall eﬀect of the diﬀerent forces should be determined. Derjaguin, Landau, Verwey & Overbeek (DLVO) established a theory
which bears their name. The DLVO theory describes the balance between the
attractive Van der Waals forces and the repulsive electrostatic forces.
2.4.1.1 Van der Waals (VDW) forces
To describe the VDW forces between macroscopic bodies one needs to add
the contributions of all the atoms in the solid. This means that the calculation
depends on geometry. Hamaker calculated the total interaction between two
spherical particles. For R1 and R2 >‌> H the potential is given by:
φ=−
Ah R 1 R 2
(
)
6H R1 + R2
(12)
Where R1 is the radius of the first sphere, R2 the radius of the second sphere,
H the distance between the spheres and Ah the Hamaker constant. The Hamaker
constant H is generally a function of density and the nature of the interaction.
The VDW forces depend on the mobility of the electrons and also on geometry.
2.4.1.2 Electrical double layer
The repulsive electro-static forces of the particles come from the electrical
double layer surrounding the particle. This double layer comes from the charged
surface attracting ions of opposite charge. To describe the electrical double layer
diﬀerent models exist. A frequently used model is the Gouy-Chapman model.
This model describes the interaction of a charged surface with the ions in the
solution and the formation of a double layer. For a positively charged surface,
the anions in the solution tend to balance the positive surface charge. The
counter ions are not rigidly held, but tend to diﬀuse into the liquid phase to
the solid surface. This tendency decreases as more ions reach to surround the
surface. The kinetic energy of the counter ions will partly aﬀect the thickness of
the resulting double layer. The Gouy-Chapman model was extended by Stern,
whose model takes into account the finite size of ions and hence cannot approach
the surface closer than a few nanometers. The ions can be adsorbed onto the
surface up to the ’slipping plane’ through a distance known as the Stern Layer
(figure 5). The potential at the slipping plane is known as the zeta potential.
Concerning electrostatic repulsion, the zeta potential is sometimes considered
more significant than the surface potential [12, 13]. Knowledge of this potential
allows to determine the electro-static forces between the particles, which have
a great influence on the stability of the colloidal system.
12
CHAPTER 2. LITERATURE STUDY
Figure 5: Schematic representation of the electro-static potential near a solid
surface in a solution containing ions [13].
2.4.1.3 DLVO
The DLVO theory describes the force between charged surfaces interacting
through a liquid medium. As already stated, it combines the eﬀects of the
VDW attraction force and the electrostatic repulsion force due to the double
layer. The overall potential is a function of the distance from the surface x and
is given by:
φ(x) =
−Ah R
+ 2π�ψ0 exp(−κx)
12x
(13)
where R is the radius of the particle, Ah the Hamaker constant, ψ0 the
standard surface potential, x is the distance between the particles in nm and ε
the dielectric constant.
Also the variable κ appears in formula (13). κ is the inverse of the Debye
length or the distance over which the potential decreased with 66%. The exact
formula of is given by:
κ=
�
2e2 NA
I
εε0 kB T
(14)
κ is given in [nm]. In formula 14, e is the charge of an electron, NA the
Avogadro number, I is the ionic strength, ε is the dielectric constant, ε0 is the
13
CHAPTER 2. LITERATURE STUDY
dielectric constant vacuum, k B the Boltzmann number and T is the temperature
[K].
Summary: In order to determine the potential three factors need to be
determined:
1. the Debye length (κ−1 )
2. the Hamaker constant (Ah )
3. the potential at the surface of the particles (ψ0 )
2.4.1.4 Eﬀect of polymers on colloidal stability
Adding polymers to a colloidal solution influences the stability of the system.
Dissolved polymers can either have a repulsive or an attractive eﬀect on the
solid particles.
Repulsive forces come from the steric interaction between the polymer chains
sticking on the surface of the particles. The stability of the system is determined
by the adsorption and the chemical anchoring of the polymer chains. The stability is less sensitive for changes in pH and ionic strength. The steric interactions
can be controlled by the length of the polymer chains, the density and rigidity
of the polymer chains , and the solvent quality (figure 6).
Attractive forces between the solid particles appear when the polymers do
not interact with the solid surface. Two eﬀects are observed: depletion and
bridging.
Depletion occurs when the osmotic pressure is out of balance. When the
polymer chains are located between the solid spheres, their entropy decreases
and the chains tend to restore their entropy loss by leaving the space between
the particles. Consequently there is an imbalance in osmotic pressure around
the particles by which the particles are pushed towards each other.
Bridging occurs when parts of the polymer chain adsorb on the solid surface.
This results in an attractive force working on large distances (figure 7).
The eﬀects of adding polymer chains to a colloidal solution depends mainly
on the molecular weight, the concentration and the adsorption of polymer on
the solid surface.
14
CHAPTER 2. LITERATURE STUDY
Figure 6: Eﬀect of adding polymers to colloidal solution: steric stabilization
and depletion[14].
Figure 7: Eﬀect of adding polymers to a colloidal solution: eﬀect of bridging.
2.4.2 Generalized Stokes-Einstein relationship (GSER)
In microrheology the stochastic thermal energy of particles, embedded in the
material that needs to be investigated, is used to derive the rheological properties of the material. Particles suspended in a liquid undergo both translational
as well as rotational diﬀusion due to the Brownian motion. Stokes and Einstein derived a method to obtain diﬀusion of a material by tracking Brownian
motion. Their results are combined in the generalized Stokes-Einstein relation
(GSER). Significant attention has been given to microrheological methods using
translational diﬀusion of the colloids so far. The GSER for translational and
15
CHAPTER 2. LITERATURE STUDY
rotational diﬀusion are similar and will be derived further in this section.
2.4.2.1 Translational diﬀusion
First the motion of a sphere in a purely viscous material is considered and afterwards this is generalized for viscoelastic materials. In a purely viscous medium
the particles simply undergo Brownian motion. The dynamics of particle motions are given by the time dependent position correlation function, also known
as the mean square displacement (MSD). This MSD reflects the response of the
material to the stress applied to it by the thermal motion of the beads and is
given by [5]:
�
�
� �
2
∆�x2 (τ ) = |�x(t + τ ) − �x(t)|
(15)
where x is the particle position, τ is the lag time and the bracelets indicate
an average over all times t. From this MSD the diﬀusion coeﬃcient (D) of the
particles can be determined from the diﬀusion equation [8]:
�∆�x(τ )� = 6Dτ
(16)
The diﬀusion coeﬃcient is related to the radius of the particle ’a’ and the
solvent viscosity η of the surrounding fluid via the Stokes-Einstein relationship
[8]:
D=
kB T
6πηa
(17)
where kB is the Boltzmann constant, T the absolute temperature.
On the other hand when a particle is embedded in a purely elastic medium its
movement will be limited and the MSD will reach a maximum value, determined
by the elastic modulus of the material (figure 8).
This derivation can be generalized for more complex materials, exhibiting
both viscous and elastic behavior. As can be seen in figure 8 for simple fluids
the MSD of the particles evolve linearly with time. For more complex materials,
the linear behavior of the MSD disappears and the MSD evolves diﬀerently with
τ .
�
�
∆�x2 (τ ) ∼ τ α
(18)
where is α< 1 and is called the diﬀusive coeﬃcient. In the purely elastic
part the particles are limited in their movement (α =0) and the MSD reach
the plateau. To further derive the GSER, a complex fluid will be modeled
16
CHAPTER 2. LITERATURE STUDY
as an elastic network within a simple Newtonian fluid. After a computational
intensive derivation, the GSER in the Laplace domain can be found in literature
as [5]:
�
G(s)
=
kB T
πas �∆�
r(s)�
(19)
This is the generalized Stokes-Einstein
relationship for translational diﬀusion
�
�
˜
˜
of colloids, where G(s) and ∆r(s) are the complex shear modulus and MSD
in the Laplace domain respectively.
This formula can be explained intuitively. By equating the thermal energy of
� 2 �
the particles, k B T with its elastic energy Ks ∆rmax
/2 an expression of Ks is
obtained. This Ks is the eﬀective spring constant and depends on the elasticity
�
of the matter surrounding the beads. The elastic modulus G of the surrounding
material thus can be expressed as a function of this Ks . The factor that relates
both, is a factor of length: the dimension of the probe. The resulting formula
gives:
G� ∼
kB T
2
�∆rmax
�a
(20)
This simplified explanation shows the main idea behind equation 19.
Figure 8: Evolution of the mean square displacement (MSD) of the particles as
a function of the lag time τ [8]
17
CHAPTER 2. LITERATURE STUDY
2.4.2.2 Rotational diﬀusion
For rotational diﬀusion the derivation is only slightly diﬀerent. The rotational
motion can be described by the mean square angular displacement [15, 16]:
�
∆θ2 (t)
�
(21)
Deriving the viscoelastic modulus for rotational diﬀusion is similar to translational diﬀusion where a is replaced by a3 (or L by L3 for ellipsoids) because
rotational diﬀusion depends on the volume swept by the particle. The rotational
GSER for a sphere is given by:
�
G(s)
=
kB T
�
�
�
4πa3 s ∆θ(s)
(22)
This equation has to be modified with a correction factor for non-spherical
particles, depending on their shape [8].
This result of the generalized Stokes-Einstein relationship is noteworthy: by
observing the time-evolution of the MSD of the particles the frequency dependent viscoelastic response can be obtained. It is important to note that the use
of the GSER requires that the size of the bead is larger than any structural
length scales of the material. For example, in a polymer network, the size of the
beads should be significantly larger than the characteristic mesh size [5]. This
will be explained in detail later in chapter 3.
˜
To compare this with the bulk rheology experiments, G(s)
need to be transformed into the Fourier domain to obtain G∗ (ω) . Generally this can be done
in two ways. The first method is by calculating the inverse unilateral Laplace
transform and then taking the Fourier transform [5]. In this method the real
data are transformed into the complex plane and can cause significant errors in
G∗ (ω) near the frequency extremes [10].
To overcome this problem an alternative method is developed. Here the
complex shear modulus is estimated algebraically by using a local power law to
describe the mean square displacement of the beads in the complex fluid [10].
This is the method used in this master thesis and will be discussed in detail in
chapter 3.
18
CHAPTER 2. LITERATURE STUDY
2.4.3 Active microrheological techniques
In active microrheological methods the probes in the colloidal solution are
agitated by an external force. In a certain way active methods are analogous
to conventional bulk rheology in which an external stress is applied and the
resultant strain is measured. However the way stress is applied is diﬀerent.
Active microrheological methods use embedded particles to deform the material
locally and measure the viscoelastic response of the material. The advantages of
active microrheological methods are mainly twofold. First, they allow to probe
stiﬀ materials, secondly they can measure non-equilibrium behavior, because
large stresses can be applied. As only passive techniques have been used in this
thesis, only the most important methods that exists are listed up [17]:
• optical tweezers measurement
• magnetic manipulation technique
• atomic force microscopy (AFM)
2.4.4 Passive microrheological techniques
In contrast with active microrheological methods in passive methods the embedded particles are not forced to move. In passive methods the thermal energy
of the beads is responsible for their movement. This thermal energy is given
by k B T , where k B is the Boltzman constant and T the absolute temperature.
The only energy input to the beads is their Brownian motion. Thus, in order
to have measurable movement of the beads, the surrounding material should
be soft enough. The displacement of the particle is highly dependent on the
stiﬀness of the material that surrounds it. An embedded particle will only move
when this thermal energy is bigger than the energy needed to deform the sur�
rounding material. For an elastic material the elastic modulus G is related to
the particle size by the equation below [5]:
�
Gy
kB T
= 3
2
a
a
(23)
where y is the particle displacement, a is the radius of the particles or the
probe. From this formula it can be derived that the upper limit of the elastic
modulus that can be mainly determined by the particle size of the beads, by the
possibility to investigate small particle displacements (y) and the temperature.
19
CHAPTER 2. LITERATURE STUDY
2.4.4.1 General dynamic light scattering (DLS)
Th main passive microrheological technique is dynamic light scattering. The
details of this method will be explained in chapter 3 materials and methods.
2.4.4.2 Diﬀusive wave spectroscopy (DWS)
Diﬀusive wave spectroscopy extends the technique of DLS to samples dominated by multiple scattering. The light will hit many particles and will be
scattered many times. Therefore as the individual scatterer move only a small
fraction of the total wavelength of the incident light, an aggregate change of
the total path length by one wavelength will be observed. Therefore DWS is
sensitive at shorter length scales and thus faster time scales than DLS because
these measurements are made in the single scattering limit. A combination of
both techniques makes it possible to probe a large range of frequencies, which
is the key advantage of microrheology [8, 18].
2.4.4.3 Video based particle tracking
Video based particle tracking is a technique to track the movement of embedded solid particles in many types of systems. The technique has certain
advantages but it is especially useful to probe the local structure of the material. This technique provides a direct visualization of possible inhomogeneities
that can be present in the sample while at the same time it tracks about hundred
beads simultaneously [10].
2.5 State of the art
First results about microrheology were published in 1995 by T.G. Mason
and D.A. Weitz. They were the first to test the applicability of the GSER
�
��
equation and the further derivation towards G and G for several distinctly
diﬀerent complex fluids. The first system investigated was a suspension of silica
particles in ethylene glycol with a radius a of 0.21 µm. They measured the mean
square displacement using diﬀusing wave spectroscopy. The bulk rheology was
performed with a strain controlled rheometer using a sample cell with a doublewall couette geometry. Excellent agreement was found between bulk rheology
and DWS.
They performed a second test on a polymer solution at a suﬃcient high
concentration to have an entangled network. They used polyethylene oxide
with a molecular weight of 4.106 g and polystyrene latex spheres with a = 0.21
and a volume fraction of 0.02. Again excellent agreement between micro and
bulk rheology was observed. A third experiment was the study of an emulsion,
20
CHAPTER 2. LITERATURE STUDY
comprised of uniformly sized oil droplets with a radius of 0.53 µm and a volume
fraction of 0.62. Again, very good agreement was obtained. During the following
years the use of spherical particles in microrheology have been investigated in
depth. For example the eﬀects on probe size was explored by Q. Lu and M.J.
Solomon in 2007. Also the limits of the application of microrheology have been
investigated by Todd M. Squires and T.G. Mason.
The use of non-spherical particles in microrheology is a newer development.
With non-spherical particles not only the translation but also the rotation of
the particle can be investigated. In 2003 Z. Cheng and T.G. Mason have established the fundamental principle of rotational diﬀusion microrheology on
anisotropically shaped wax (α-eicosene) microdisks in an aqueous polyethylene
oxide (PEO) solution. They used light streak tracking and compared the results
with those obtained with a strain mechanical rheometer with a concentric cylinder geometry. The rotational diﬀusion measurements of the viscoelastic shear
modulus of the polymer solution did agree well with the experiments with the
mechanical rheometer.
At present, no one has studied microrheology using ellipsoidal particles (prolate) and DLS. This master thesis is the first scientific work to study this system.
Details on this ellipsoids used in this master thesis can be found in chapter 3
materials and methods.
21
Chapter 3
Materials And methods
3.1 Introduction
Using a traditional mechanical rheometer, the bulk viscoelastic properties can
be investigated by applying oscillatory strain and measuring the stress response
of the material. This provides a direct measure of the storage and loss modulus.
In the last decade, other techniques have been developed and improved that
also allow investigating soft materials for local viscoelastic behavior [19, 20, 21].
In this thesis, first microrheological experiments are performed using spherical particles. This is done to get familiar with preparing correct samples and
the measuring methods. In a later stadium it will be investigated using nonspherical particles. Dynamic Light Scattering (DLS) is used to measure the dynamics of the probes suspended in the polymer solution. In order to investigate
the viscoelastic properties of a simple, isotropic uncrosslinked flexible polymer,
experiments are performed on polyethylene glycol (PEG) and polyethylene oxide
(PEO) solutions in semi-dilute regime using particles of varying form. Diﬀerent
molecular weights of polyethylene glycol (PEG) are used in diﬀerent concentrations above the overlap concentration C∗ to obtain many diﬀerent samples with
varying viscoelastic properties. By measuring the thermal or Brownian motions
of the particles, the elastic and viscous moduli of the samples can be obtained.
Using the technique of dynamic light scattering, the dynamics of the probe particles in the polymer sample are determined as a function of time. The data
measured with these microrheology experiments should then have a frequency
overlap with data found with traditional rheology experiments. To calculate
the moduli from the dynamic light scattering experiments a modified algebraic
form of the generalized Stokes-Einstein equation is used as explained in section
3.4.4. The expectation is that for simple uncross linked polymeric systems, this
method should show excellent similarity between bulk and micro-rheology data.
23
CHAPTER 3. MATERIALS AND METHODS
3.2 Materials
3.2.1 Polymers: Polyetheleneglycol (PEG) or Polyetheleneoxide (PEO)
Polyethyleneglycol (PEG), is a hydrophilic polymer or oligomer build with
chains of monomers of ethylene glycol (-CH2 -CH2 -O-) with an hydroxyl group
(-OH) at both ends. PEG exists in diﬀerent chain lengths. Shorter chains are
only a couple of hundreds monomers in length and are liquid at room temperature. The longer chains are up to ten thousands monomers in length and are
solid at room temperature. Polyethyleneglycol with long chains and high molecular weights (over 35000) is also called Polyethyleneoxide (PEO) [22], because
the role of the hydroxyl groups are negligible. In this masterthesis experiments
were performed on long chains with molecular weights varying from 3350 until
1000000 g mol−1 or 3.35 to 1000 kDa. During this thesis, the molecular weight
will often be mentioned in Dalton [Da]. It is defined as one twelfth of the rest
mass of an unbound neutral atom of carbon-12 in its electronic and nuclear
ground state. One Dalton is approximately equal to the mass of one proton or
one neutron and can be used as an equivalence for 1 g mol−1 [23]. The choice for
PEG was obvious for diﬀerent reasons. First of all it is a safe and easy polymer
to work with. Also, it has good viscoelastic properties and it is rather easily
soluble in water. Another important fact is that it was available in diﬀerent
molecular weights in the lab. Applications of PEG range from being a half fabricate in the production of polyurethane to being an ingredient in suppository
pills. It can also be used to conserve wood in the domain of archeology.
Figure 9: Monomer polyethylene glycol (PEG)
24
CHAPTER 3. MATERIALS AND METHODS
3.2.2 Beads
3.2.2.1 Spheres
In order to test the DLS, in the first range of experiments a solution of spherical
silica particles in a water was used. These particles, with a theoretical diameter
of 30 nm, are typically amorphous and nonporous (figure 10). Their scattering
intensity is strong and overshadows any scattering from dust particles. This is
an advantage because of the clean data and easy fitting.
In a second phase, spherical surfactant free sulfate latex particles were used
with a diameter of 210 nm. These are typically made of polystyrene. The
diameter of the particles is suﬃciently small to ensure that the particles are
small enough to have colloidal properties. Also important is that this diameter
is big enough to ensure that the particles are larger than the mesh size of the
polymer. This mesh size, which depends on the polymer concentration, is the
average size of the space between the polymer coils. Particles should be bigger
than this mesh size to probe the material, if not they would just diﬀuse through
the material without probing the matrix . Calculations to prove this are shown
in paragraph 3.3.1.
Figure 10: Scanning electron micrograph (SEM) of silica particles [24]
3.2.2.2 Ellipsoids
So far, very few research has been done on the use of ellipsoidal particles to compare bulk rheology with microrheology. This will be the subject
of this master thesis. Ellipsoids have three main directions to move in a fluid.
They can translate, rotate about its axis of symmetry or rotate about its nonaxisymmetric rotation (figure 11). The motivation for using ellipsoidal particles
in microrheology for the investigation of viscoelastic materials is twofold [4].
25
CHAPTER 3. MATERIALS AND METHODS
First, for simple, incompressible, isotropic linear viscoelastic materials, investigating both translational and rotational diﬀusion gives redundant information. This information can be used as a self-consistency check between G∗ (ω)
translation and rotational diﬀusion. If in contrast, the moduli measured with
translational diﬀusion diﬀer from those measured with rotational diﬀusion, extra information can be extracted from the coupling of the investigated material
and the probes or from the anisotropy of the material.
Second, some materials are known to form depletion zones. These materials
show regions of lower polymer concentration around particles. As the ellipsoid
will rotate freely about its axis of symmetry, the lack of polymer macromolecules
around the particle aﬀects its axisymetric rotation much more significant than
its translational or non-axisymmetric diﬀusion [4]. Quantitative measurements
of the axisymetric rotation compared to the translational or non-axisymetric
diﬀusion will give information about the nature and extent of the depletion
zone.
Figure 11: Rotation of ellipsoids in a fluid.(a) axisymetric rotation. (b) nonaxisymmetric rotation.
• General ellipsoids
The ellipsoidal particles were made of polystyrene. They are made by stretching
spheres in one direction, which makes them prolate (figure 12). The beads used,
have a theoretical length of approximately 650 nm which is bigger than the
spherical ones and an aspect ratio P of around 3.9. This means that the length
of the ellipsoids was 3.9 times longer than the width. These dimensions proved
suﬃcient to match the mesh size of the polymers (figure 13).
26
CHAPTER 3. MATERIALS AND METHODS
• Ellipsoids covered with gold
The general ellipsoids used are anisotropic in shape. To observe only their
rotational diﬀusion, depolarized setup of the DLS was used, as will be explained
in section 3.3.3. But although they are anisotropic in shape they still have
isotropic scattering properties. As a result the polarization of the scattered light
from the sample is the same as the polarization direction of the incoming light,
giving disappointing results when using the polarizer. This is why the ellipsoids
were covered with gold nanorods (figure 14), which have excellent, anisotropic
scattering properties and make it possible to observe the rotational diﬀusion of
the ellipsoids with the polarizer. In figure 15 a schematic representation of the
ellipsoids covered with gold nanorods is given.
Figure 12: Two diﬀerent kinds of stretching spheres are possible to obtain ellipsoids. The results are called oblate ellipsoids or prolate ellipsoids. In this
master thesis prolates were used. [25]
Figure 13: Transmission electron
micrograph of polystyrene ellipsoids.
Figure 14: Transmission electron
micrograph of of gold nanorods.
27
CHAPTER 3. MATERIALS AND METHODS
Figure 15: Schematic representation of prolate ellipsoids covered with gold
nanorods. [Courtesy: Sylvie Van Loon]
3.2.3 Extra materials
• Water
Of course water was necessary during the complete duration of the thesis. To
clean the tools for example, regular tap water was used to wash the soap oﬀ and
then normal demineralised water was used to make everything clean. Even a
third check was done with ultra pure water. The ultra pure water was used to
make the samples and to dissolve the polymers. It is obtained out of a Sartorius
Stedium Arium 611DI system and has a conductivity of 0, 055 µS/cm.
• Sodium Chloride
In order to obtain a constant level of ionic strength (electro-static double layer)
and to reduce possible eﬀects of CO2 absorption in the samples, a 25 millimolar
solution of Sodium Chloride (NaCl) in water was used to dissolve the polymers.
• Chloroform & Aluminum foil
To prevent bacterial growth in the samples a drop of chloroform was added
in the main polymer solution. All samples were covered with alumina foil to
prevent degradation from the light.
28
CHAPTER 3. MATERIALS AND METHODS
3.3 Methods
3.3.1 Sample preparation
In order to get adequate and correct results, all the samples were created
with the greatest care and patience. The first thing was to triple clean every
tool used in the process. This was essential to avoid dust particles to interfere
because dust particles also scatter light, which would result in bad experimental
measurements. For the diﬀerent molecular weights of the PEG used, diﬀerent
concentration samples were made ranging from 0.1 to 20 times the overlap concentration C∗ . This overlap concentration indicates the point where the solution
begins to exhibit viscoelasticity due to the entanglements of the polymer coils.
At this concentration the neighboring polymer coils are overlapping which each
other (figure 16). The overlap concentration C∗ was calculated using the following equation [26]:
C∗ =
Mw
4
3
3 N a Rg
(24)
Where Na is Avogrado’s number and Rg the radius of gyration of the polymer. The radius of gyration is given by an empirical relation [27]:
Rg = 0.21Mw(0.58±0.031)
(25)
The squared radius of gyration gives the average squared distance between
monomer units and the center of mass of the polymer coil. About 92% of the
polymer mass are within the radius of gyration (figure 17) [28].
Figure 16: Schematic representation of the critical overlap concentration C∗ [29]
29
CHAPTER 3. MATERIALS AND METHODS
Figure 17: Schematic representation of the radius of gyration. [29]
In first instance 4 molecular weights of PEG were used: 3.35 kDa, 8 kDa,
20 kDa and 35 kDa. Later on PEO of molecular weight 1000 kDa was used to
obtain higher viscoelastic samples. All molecular weight samples were produced
as follows: A 100 ml of de-ionized water was mixed with 0.145 g of N aCl to
obtain a 25 millimolar N aCl − H2 O solution. This salt was added to ensure
a constant ionic strength in the solution. For every molecular weight the right
amount was measured to prepare a 100 ml 20 C∗ stock solution. The polymers
were added iteratively into the stock solution while mixing it very gently on a
magnetic stirrer at a temperature of 40° C to homogenize. After adding a drop
of chloroform to prevent bacterial growth in the polymer, the stock solutions
were covered in alumina foil to prevent degradation by light and it was put on a
rolling shaker for a couple of days. When the polymer was completely dissolved,
the 20 C∗ stock solution was used to create less concentrated samples. The right
proportions of the stock solution were mixed with more saltwater to obtain 10
ml samples of diﬀerent concentrations. In these smaller samples tiny drops of
the beads were added, and the sample was thoroughly shaken by a vortex mixer.
It was very important to make sure that the bead diameter was bigger than the
mesh size of the polymer sample as explained earlier. The mesh size is normally
only a few nanometers and can be calculated using the following equation [26]:
ξ = Rg (C ∗ /C)0.75
(26)
For both the spherical particle of 210 nm diameter and the ellipsoidal particle
of 650 nm this was satisfactory. Another important aspect was to inject a high
enough bead concentration to ensure the bead scattering to be dominant over
the polymer scattering. However a too high concentration would induce multiple
scattering instead of the desired single scattering. A good balance between these
30
CHAPTER 3. MATERIALS AND METHODS
was necessary to have a good scattering sample.
The final step to get the samples ready for the DLS measurements was to
add them in glass tubes of 1 cm diameter. These were carefully washed with
ethanol and deionized water. For the rheometer experiments no extra procedure
was necessary and the same samples were used as the ones for DLS. The beads
did not eﬀect the rheology data.
Mw [g/mol]
3350
8000
20000
35000
100000
Rg [m]
2.44084E-09
4.05452E-09
6.91734E-09
9.58584E-09
6.76766E-08
C∗ [g/ml]
0.091324338
0.047580685
0.023953756
0.015752088
0.001278925
ξ[m] at C = C∗
2.44084E-09
4.05452E-09
6.91734E-09
9.58584E-09
6.76766E-08
ξ[m] C = 20 C∗
1.29811E-10
2.15632E-10
3.67884E-10
5.09803E-10
3.59924E-09
Table 2: Overview of Rg , C∗ and mesh size at diﬀerent concentrations for different molecular weights Mw .
3.3.2 Rheometer
3.3.2.1 Device
The bulk rheology experiments were performed on an Anton-Paar MCR 501
stress controlled rheometer (figure 18). A cone and plate geometry (CP50-1/Ti)
with a diameter of 49.94 mm and a cone angle of 0.993° (figure 19). A solvent
trap with evaporation blocker was used during all the experiments. Additionally
water was put around the loaded sample to further prevent evaporation. This
was done with the greatest care so as to prevent the water from mixing with
the polymer sample. More information about the geometry can be found in
appendix A.
The instrument has a fully automatic tool recognition system and a T ruGapT M
system to monitor and control the real gap, eliminating errors from thermal expansion and normal force during experiments. Temperature control is achieved
by a Peltier element and all experiments were done at 25°C. The gap between
the cone and plate geometry was fixed at 0.048 mm and a sample of 0.57 ml
was loaded. The sample material that bulged out from in between the cone and
the plate was scraped away. Measurements were controlled and analyzed using
Physica RheoPlus software.
All samples were pre-sheared for 100 seconds at a shear rate of 10 s−1 . Afterwards a strain sweep was performed in order to determine the linear viscoelastic
region. All subsequent frequency sweep measurements are performed at strain
within the linear viscoelastic regime. Experiments are repeated twice to check
their reproducibility.
31
CHAPTER 3. MATERIALS AND METHODS
Figure 18: Anton-Paar 501 MCR
stress-controlled rheometer
Figure 19: Schematic representation of the cone and plate geometry
[12]
3.3.2.2 Maxwell model
Maxwell derived a mathematical model to describe viscoelastic behavior of
materials. The model describes the dual nature of a viscoelastic fluid starting
from simple phenomenological tools. A Hookean spring represents a perfect
solid material, while a dashpot will be the model for a Newtonian fluid. In the
Maxwell model both elements are placed in series (figure 20).
Figure 20: Schematic representation of the Maxwell model
Mathematically G(t) is represented in the model as [30]:
G(t) = G0 exp(
−t
)
τ
(27)
where τ is the relaxation time or the time where 66% of the stress in the
material is released. The storage and loss modulus can be defined as:
32
CHAPTER 3. MATERIALS AND METHODS
�
G (ω) =
��
G (ω) =
�
G0 ω 2 τ 2
∝ ω2
1 + ω2 τ 2
(28)
G0 ωτ
∝ω
1 + ω2 τ 2
(29)
��
The evolution of G and G is given in figure 21.
Since PEO is a viscoelastic material measured in the linear viscoelastic
regime, it should look like the Maxwell model. The slopes of the moduli in
figure 21 are logically explained by the proportional relation of the moduli to
the frequency ω in equations 28 and 29.
�
��
Figure 21: Evolution of G and G described by the Maxwell model [31]
3.3.3 Dynamic light scattering (DLS)
3.3.3.1 General dynamic light scattering for spherical particles
The main passive microrheological technique is dynamic light scattering. A DLS
experiment is based on the elastic scattering of light by particles undergoing
Brownian motion in a suspension. A laser beam hits on a sample and the light
is scattered by the particles into a detector placed at a certain angle θ with
respect to the incoming beam (figure 22). According to the Rayleigh theory,
the incident light and its electrical field will distort the electron distribution
in the molecule or particle and induce an oscillating dipole that will re-radiate
light, as illustrated in Figure 23.
33
CHAPTER 3. MATERIALS AND METHODS
Figure 22: Schematic representation of dynamic light scattering (DLS)
Figure 23: Scattering oﬀ a small particle in an ideal solution by incident light
[32]
As the particles move and rearrange in the sample, the intensity of the light
I(t) that reaches the detector fluctuates in time. From this constantly changing
signal I(t) the autocorrelation function of the intensity of the light, g2 (q, t),
is calculated. This function states how the intensity of the light at time t is
correlated to the intensity of the light at time t + τ , and is given by [33, 34] :
g2 (q, τ ) =
�I(t)I(t + τ )�
�I(t)�
2
(30)
where the brackets�� indicate an average over time. This means the intensity
at time t + τ is compared with the initial intensity at time t. For small τ , the
particles are close to their initial position and the correlation between I(t) and
I(t + τ ) is very strong. For larger τ the particles had time to move a lot and
the correlation will fall down (figure 24). The autocorrelation function of the
intensity is hence a direct measure of the Brownian motion of the particles [35].
34
CHAPTER 3. MATERIALS AND METHODS
Although the intensity of the light is measured, one is interested in the
electric field intensity E(t), but this electric field intensity is too diﬃcult to
measure. Using the Siegert relation a correlation between the electric field
intensity correlation function g1 (τ ) and the correlation function for the intensity
of the light is given by [5, 35]:
g2 (q, τ ) = 1 + β |g1 |
2
(31)
where β is the coherence factor, depending on the laser beam and instrumentation optics. For idealized conditions of perfect coherence β=1. In practice
however β is slightly lower than 1.
If all the particles are statistically independent, and moving randomly due
to thermal impulses only, then [5, 35]
g1 (q, τ ) = exp{−Γτ }
(32)
where Γ is the relaxation rate and τ the correlation time. For optically
isotropic, monodisperse spheres Γ is related by the translational diﬀusion of the
particles by [35]:
Γ = Dt q 2
(33)
where Dt is the translational diﬀusion coeﬃcient and q is the scattering wave
vector given by:
q=
4πν
θ
sin( )
λ
2
(34)
where ν is the refraction index of the sample and λ is the wavelength of the
laser in vacuum.
The translational diﬀusion coeﬃcient can be related to the mean square
displacement < ∆r2 (τ ) > of the particles by:
Dt =
< ∆r2 (τ ) >
6t
(35)
If formula 35 is combined with 32, the electric field autocorrelation function
is given by:
g1 (τ, q) = exp(
−q 2 < ∆r2 (τ ) >
)
6
(36)
Formula 36 couples the electric field autocorrelation function to the MSD
of the spherical particles. This result is of major concern in this master thesis
because from this MSD the complex shear modulus can be obtained by use of
the GSER as explained in chapter 2.
35
CHAPTER 3. MATERIALS AND METHODS
Figure 24: Intensity measurement and autocorrelation function measured by
dynamic light scattering [12]
3.3.3.2 Depolarized dynamic light scattering for ellipsoidal particles
In order to investigate the rotational diﬀusion of the ellipsoids an depolarized
dynamic light scattering (DDLS) device is used because that polarization direction of the incoming light is diﬀerent from the polarization direction of the
scattered light from the sample. This may occur for several reasons, e.g. if the
particles exhibit a suﬃciently large shape and/or optical anisotropy.
In standard DLS experiments the polarization of the incoming beam is normal to the plane defined by the wave vectors of the incident and the scattered
beam respectively. The intensity of the scattered light is usually measured without the use of a polarizer (VV-mode), as seen in figure 25.
In a DDLS experiment however, a polarizer is used that is oriented 90° with
respect to the polarization of the scattered beam. (VH-mode), as given in figure
26. The correlation function contains information about both translational and
rotational diﬀusion. But as the polarizer is set at 90° only the light scattered
whose polarization direction has been changed is measured. Consequently the
coupling between translational and rotational diﬀusion is measured.
In general DLS measurements (VV-mode) the decay rate is given by [38]:
ΓV V = D t q 2
(37)
which is analog to equation 33. However in the VH-mode the decay mode is
given by:
ΓV H = Dt q 2 + 6Dr
36
(38)
CHAPTER 3. MATERIALS AND METHODS
As the translational diﬀusion cannot be completely blocked, the rotational
diﬀusion is still present in equation 38. Dr represents the rotational diﬀusion coeﬃcient. Consequently, analog to equation 32, the electric field autocorrelation
functions are given by:
g1V V = exp(−Dt q 2 τ )
(39)
g1V H = exp[−(Dt q 2 + 6Dr )τ ]
(40)
Where equation 39 is again completely analog to equation 32. As this technique was used to calculate the diﬀusivity and radius of the prolate ellipsoids,
formulas to derive the diﬀusion coeﬃcient for prolate ellipsoids are given in
equation 41 and 42. For thin rods with a length of L and an aspect ratio (P )
in a fluid of viscosity η and temperature T formulas where derived by Boersma
and Brenner [39, 40], but for ellipsoids with small aspect ratios, as the one used
in this master thesis, correction factors Cr and Ct are added. They are required
to account for the relative importance of end eﬀects due to the finite rod length
[41].
Dt =
kB T (ln(P ) + Ct )
3πηL
(41)
Dr =
3kB T (ln(P ) + Cr )
3πηL3
(42)
Empirical expressions to relate Ct and Cr to the hydrodynamic dimensions
are given by [41]:
Cr = 0.312 + 0.565P −1 − 0.100P −2
(43)
Ct = −0.662 + 0.917P −1 − 0.050P −2
(44)
In order to couple the electric field autocorrelation function to the mean
square angular displacement of the probes, the electric field autocorrelation
function of the VV-mode and the VH-mode are equated to obtain:
g1V V
exp(−Dt q 2 τ )
=
= exp(6Dr τ )
V
H
exp[−(Dt q 2 + 6Dr )τ ]
g1
(45)
Additionally, equivalent to equation 35, the rotational diﬀusion coeﬃcient is
given by:
Dr =
�
∆θ2
2
37
�
(46)
CHAPTER 3. MATERIALS AND METHODS
Combining equation 45 and 46 and taking natural logarithm on both sides
gives:
ln(
�
�
g1V V
) = 3 ∆θ2 τ
V
H
g1
(47)
From formula 47 the mean square angular displacement can be obtained from
the electric field autocorrelation function. By use of the GSER for rotational
diﬀusion, as explained in chapter 2, the complex shear modulus can be derived.
Figure 25: Schematic representation of the general dynamic light scattering
setup (VV-mode). [35]
38
CHAPTER 3. MATERIALS AND METHODS
Figure 26: Schematic representation of the depolarized dynamic light scattering
setup (VH-mode). [35]
3.3.3.3 DLS device
The DLS set-up used in this master thesis consists of a goniometer that holds the
sample, a coherent light source and detection optics that measure the intensity of
the scattered light as a function of time. By placing the system on a Newport air
damped table everything is stabilized and mechanical vibrations are reduced.
The light source is a continuous wave (CW) laser that emits red light with
an operational wavelength of 632,8 nm at a power of 35mW. This wavelength
ensures enough distance from the absorption lines of the sample to prevent local
heating. The compact goniometer system (ALV/CGS-3) defines the scattering
geometry and also holds the sample. It consists of an arm that holds the detector
and a vat with a sample holder (figure 27). The detector is placed on a movable
arm that can rotate about the center of the vat from 15 to 150 degrees. The vat
is filled with toluene and the sample is positioned such that the laser beam passes
through the center. Toluene is used because it reduces undesirable reflections
and refractions as it has a similar index of refraction as the glass of the sample.
The vat is connected with a fluidsbath to control the temperature of the sample.
The duration of the experiments varied depending on the concentration. The
higher the polymer concentration, the longer the experiments were executed. In
water the correlation function went to zero within 30 seconds per angle while for
the most concentrated samples experiments took over 10000 seconds per angle.
39
CHAPTER 3. MATERIALS AND METHODS
Figure 27: The ALV/CGS-3 compact goniometer system.
3.3.4 Comparison between bulk and microrheology
From the raw data obtained from microrheology, the MSD was calculated as
explained earlier. This MSD is plugged in the general Stokes Einstein relation
(GSER) as deduced in chapter 2 to obtain G(t) for translational or rotational
diﬀusion. In order to compare with the bulk rheology, these data need to be
transformed from the time domain to the Fourier domain G∗ (ω) and be decou�
��
pled into G and G .
Generally two methods were developed to do the transformation from the
time to the frequency domain. The first methods takes the inverse unilateral
Laplace transformation and then afterwards takes the Fourier transform [5]. In
practice however, the numerical calculation of this method can give significant
errors in G∗ (ω). Therefore an alternative method was developed. In this method
a local power law is used to describe the MSD of the beads in the fluid. With
this local power law the complex shear modulus can be estimated algebraically.
In this method no numerical transformations are used, thus avoiding the significant errors in G∗ (ω). The power law is determined from the logarithmic time
derivative of the MSD [8].
For spherical particles the GSER for translational diﬀusion in the Fourier
domain is given by:
G∗ (ω) =
kB T
πaiω �∆r2 (τ )�
(48)
Now, an expression needs to be found for the storage and the loss modulus.
� 2 �
�
��
If ∆r (τ ) can be written in a local power law form, G (ω) and G (ω) are
40
CHAPTER 3. MATERIALS AND METHODS
given by [8]:
�
(49)
��
(50)
G (ω) = G(ω) cos(πα(ω)/2)
G (ω) = G(ω) sin(πα(ω)/2)
where:
kB T
(51)
πa �∆r2 (1/ω)� Γ(1 + α(ω))
�
�
�
�
In equation 51, the local power law α(ω) is given by �(∂ln ∆r2 (τ ) /∂lnτ )�τ =1/ω
G(ω) =
and Γ represents the gamma function. The gamma function comes from the
�
�
Fourier transformation of the local power law. ∆r2 (1/ω) represents the mag�
�
nitude of ∆r2 (τ ) evaluated at τ = 1/ω. The DLS data was converted into
actual microrheology data using Matlab. The Matlab code, with the exact
calculations can be found in Appendix C.
For ellipsoidal particles the rotational diﬀusion is tracked, and the GSER for
rotational diﬀusion needs to be used.
G∗ (ω) =
�
kB T
4πa3 iω �∆θ2 (ω)�
��
(52)
The derivation of G and G is analog as in the translational case. The Matlab
code, with the exact calculations can be found in Appendix C.
41
Chapter 4
Experimental results and discussion
4.1 Overview
In this chapter the experimental results will be presented and analyzed. First the
bulk rheology results will be shown. The focus in the rheology experiments lies
�
��
in the range of angular frequencies where crossover between G and G occurs.
�
��
Although not all samples will show the perfect crossing of G and G , the trend is
definitively there. Of course a considerably part of this chapter will focus on the
DLS results. In a structured way the results will handle the diﬀerent particles
in water, the polymers without particles and the various particles in polymers
of diﬀerent molecular weights and concentrations. An important sample that
will come back throughout the whole chapter is the PEG 1000 kDa at a 20
C∗ concentration. In the last but most important part, the results from bulk
rheology will be compared with microrheology from DLS. A special attention
will be given to the work with the ellipsoids and the unique results obtained.
4.2 Rheology
After investigating many diﬀerent concentrations and molecular weights of PEG
(or PEO), the PEO sample of 1000 kDa with a concentration of 20 C∗ was chosen
to compare bulk rheology with microrheology. The reason here fore was that it
showed suﬃcient viscoelastic behavior.
�
��
In figure 28 the evolution of the storage (G ) and loss (G ) modulus as a
function of angular frequency (ω), measured with the stress controlled rheome-
43
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
ter, is shown. From this result it can be seen that at lower frequencies viscous
��
�
response is dominant as G � G . At higher frequencies the elastic response also
�
��
becomes significant and eventually there will be crossover between G and G .
��
If the frequency goes further, the elastic modulus G will be more significant.
As one can see, the slope of the moduli are as expected with the Maxwell model.
�
Figure 28: Bulk rheology: Storage modulus (G ) with open symbols and loss
��
modulus (G ) with filled symbols, as a function of angular frequency for 1000
kDa PEO at 20 C∗ .
�
Figure 29: Bulk rheology: Storage modulus (G ) for diﬀerent concentrations
from 5 C∗ to 20 C∗ , as a function of angular frequency for 1000 kDa PEO.
44
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
��
Figure 30: Bulk rheology: Loss modulus (G ) for diﬀerent concentrations from
5 C∗ to 20 C∗ , as a function of angular frequency for 1000 kDa PEO.
In order to investigate the reliability of the measurements, the results of PEO
1000 kDa with a concentration of 20C∗ are compared with samples of 15C∗ ,
10C∗ and 5C∗ . Because the samples of the lower concentrations become too low
viscous, the viscoelastic behavior disappears. The lower limits of the rheometer
are reached and the measurements for the lower polymer concentrations become
less accurate. The fact that the date of the 5C∗ and 10C∗ are almost exactly
the same, proves this statement. Another derivation can be made out of figures
29 and 30. The more viscous the sample, the lower the angular frequency at
which cross over happens. This is normal as this frequency is the inverse of the
relaxation time of the sample. The higher the viscosity, the higher the relaxation
time and thus the earlier the cross over. On the other hand the moduli of the
cross over are getting higher for more viscous samples. To conclude, more
viscous samples have higher moduli at cross overs and the cross over frequency
is earlier.
4.3 Dynamic Light Scattering of spherical particles
and polymers
The first experiments were done with the purpose of getting to know the DLS
system and to getting familiar with the outputs and the computational intensive
data processing. To avoid complications in the beginning, the first experiments
were done in water. Besides gaining experience with the methodology the second
goal of the experiments was to gain an insight in the formulas to calculate
45
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
diﬀusion, viscosity and radius. The results of the DLS measurements come in
two columns. The first column represents the time τ, in milliseconds and the
second contains the values of the normalized intensity auto correlation minus
one, g2 − 1. In the results, shown in the following figures, these outputs were
plotted and fitted using a stretched exponential. This stretched exponential or
Kohlrausch-Williams-Watts (KWW) function is given as [36, 37]:
g2 − 1 = A ∗ (e− )
(53)
were β indicates the stretched nature of the function. It is usually between 0.9
and 1 for monodisperse particles. A represents the amplitude but is usually set
to 1. Γ is the inverse of the relaxation time in seconds which denotes the average
time, per angle, for which the correlation is lost. The relaxation time for each
angle can be obtained from the fitting of equation 53 or in the graphs as the
value on the x-axis for which the specific curve have fallen 66%. τ is the time
in milliseconds coming out of the DLS. From the Γ obtained out of the fit, it is
possible to calculate the diﬀusion coeﬃcient D. The relation used here is
Γ = 2Dq 2
(54)
By determining the diﬀusion coeﬃcient D, it is only a small step further to
calculate the experimental diameter of the particles using the Stokes Einstein
relation [8]:
η=
kB T
6πDa
(55)
where η is the viscosity of the sample, kB is the Boltzmann constant, T is the
temperature and a is the radius of the particle. One should also realize that
the radius calculated here, includes the Debye length κ−1 . As the samples used
here are simple water samples the approximation form can be used [12]:
0.304
κ−1 [nm] = �
I(M )
I is the ionic strength, which in this case is 0.025 mol/l.
46
(56)
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
4.3.1 Silica particles in water
As a matter of testing, the first experiments were done with spherical silica
particles with a diameter of 30 nm. Silica is a wonderful material for dynamic
light scattering as it scatters very good. All the experiments were done for
scattering angles 30°-150°, but to have a better visualization only three angles
are presented in figure 31.
Figure 31: DLS: The normalized intensity auto correlation function g2 − 1 is
plotted as a function of time for spherical silica particles with diameter 30 nm
in water. Only 3 diﬀerent angles 50°, 90° and 150° are presented.
As expected the exponential decay, which represents the correlation function,
occurs smoothly as the particles are monodisperse. As the particles are almost
not hindered in the water, they can move freely and the correlation function
goes to zero. The experiment took only 30 seconds per angle which indicates
that the correlation was lost rapidly. The relaxation times are between 0.1 and
1 millisecond.
For the next calculations all the angles are taken into account. Plotting
Γ values for each angle as a function of q 2 shows the expected linear relation
(figure 32). By linear plotting, it was possible to obtain the slope, which gave
the diﬀusion coeﬃcient. In this case D is 1.5 10−11 [m2 /s] which is high as
expected. The calculations to verify the particle size are given in Table 3.
47
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
Figure 32: Diﬀusion coeﬃcient: The inverse relaxation time Γ in inverse seconds
(dots) is plotted as a function of the scattering vector for the silica particles in
water sample. Using a linear fit, the slope is obtained which is 2 times the
diﬀusion coeﬃcient.
4.3.2 Latex particles in water
The particles used now are surfactant free white sulfate latex particles with an
apparent diameter of 210 nm. It are these particles that will be put in the
polymer samples in a later stage. Again only the 50°, 90° and 150° angles are
represented (figure 33). As with the silica particles the curves are smooth and
rapidly going down and the fit covers the data nicely. Experiments were again
at 30 seconds per angle.
48
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
Figure 33: DLS: The normalized intensity auto correlation function g2 − 1 is
plotted as a function of time for spherical Latex particles with diameter 210 nm
in water. Only 3 diﬀerent angles 50°, 90° and 150° are presented.
As the graphic shows, the relaxation times are around 1 millisecond. This
indicates that the correlation was a bit longer here than with the silica particles.
This is as expected as the particles are bigger here. Again D is obtained by a
linear plot of the inverse relaxation time Γ and the scattering vector q (figure
34). The diﬀusion coeﬃcient, which is 2.27 10−12 [m2 /s], is a bit smaller than
with the silica particles, which is also logical.
Figure 34: Diﬀusion coeﬃcient: The inverse relaxation time Γ in inverse seconds
(dots) is plotted as a function of the scattering vector for the Latex particles
in water sample. Using a linear fit, the slope is obtained which is 2 times the
diﬀusion coeﬃcient.
49
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
Now that all the data are given, an overview in table form shows the results.
The viscosity of water is used with a value of 0.89 mPa.s. The experimental
radii are calculated and when the Debye Length is subtracted from the these
and the result is multiplied by two, the adjusted diameter is found. As one can
see, the errors are negligible and the results are very close to the sizes provided
by the manufacturers.
Form + Size
Diﬀusivity [m2 /s]
Radius a [m]
Debye Length κ−1 [m]
Experimental Diameter [nm]
Absolute Error [nm]
Relative Error [%]
Spherical Silica 30 nm
1.49E-11
1.64E-08
1.9 E-09
29.08
0.92
3.07
Spherical Latex 210 nm
2.27E-12
1.07E-07
1.9 E-09
211.90
1.90
0.90
Table 3: Overview of radius of spheres in water.
4.3.3 Polymers in water
The next series of experiments were polymers in water. The first experiments
were performed for the four lower molecular weights (3.35, 8, 20 and 35 kDa) for
a very dilute concentration of only 0.5C∗ . The goal of this series of experiments
is to characterize the polymers and to check the theoretical radius of gyration
Rg , calculated in chapter 3, with the one obtained out of the measurements.
Figure 35 shows the intensity autocorrelation function of tau for 4 diﬀerent Mw
PEG in water at 0.5 C∗ , and figure 36 shows the same relation for PEO 1000
kDa at 0.5C∗ . Immediately one can observe that the average relaxation times
for these samples are very short. With values around 10−2 ms the relaxation
times are significantly lower then these of the particles in water. The polymer
coils are thus very small and diﬀuse very fast. One can also see the small but
certain evolution of increasing relaxation time for higher molecular weights. All
the experiments were performed during 30 seconds per angle. The smaller the
angle, the harder it is to obtain correct results. This is because at these angles
even the tiniest dust particles scatter hard. For this reason, the 50° angle in
some measurements was not of suﬃcient quality to present so the 60° angle is
shown instead.
50
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
Figure 35: DLS: The normalized intensity auto correlation function g2 − 1 is
plotted as a function of time for PEO 1000 kDa for the diﬀerent molecular
weights (3.35, 8, 20 and 35 kDa) at a concentration of 0.5 C∗ . Only 3 diﬀerent
angles per sample are presented. No particles are present in this sample, just
polymer.
Figure 36: DLS: The normalized intensity auto correlation function g2 − 1 is
plotted as a function of time for PEO 1000 kDa at 0.5 C∗ in water. Only 3
diﬀerent angles 50°, 90° and 150° are presented. No particles are present in this
sample, just polymer.
The graphs of the polymers in water are obviously less smooth than those
51
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
of the particles in water. This is because the polymer coils scatter less than the
solid particles. The coils are not in a solid predefined state and their form and
measurable size is constantly changing. In order to obtain the radius of gyration
of the polymer, the diﬀusivity was also calculated for these samples. Figure 37
shows the inverse relaxation time Γ as a function of the scattering angle q2 for
4 diﬀerent Mw PEG in water at 0.5 C∗ , and figure 38 shows the same relation
for PEO 1000 kDa at 0.5C∗ .The values of the diﬀusivity coeﬃcients make sense
as they are decreasing with higher concentration. It can be remarked that the
data of the 1000 kDa sample is less good then the others.
Figure 37: Diﬀusion Coeﬃcient: The inverse relaxation time Γ in inverse seconds
is plotted as a function of the scattering vector for the diﬀerent molecular weights
of PEG (3.35, 8, 20 and 35 kDa) at a concentration of 0.5 C∗ . Using a linear fit,
the slope is obtained which is 2 times the diﬀusion coeﬃcient. All the diﬀusivity
coeﬃcients are in table 4.
52
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
Figure 38: Diﬀusion Coeﬃcient: The inverse relaxation time Γ in inverse seconds is plotted as a function of the scattering vector for PEO 1000 kDa at a
concentration of 0.5 C∗ . Using a linear fit, the slope is obtained which is 2 times
the diﬀusion coeﬃcient.
Again an overview in table form is given, and the experimental radius of gyration, obtained out of the diﬀusivity, is compared with the theoretical Rg from
chapter 3. Normally the experimental and theoretical radius of gyration should
be the same, but the results show consistently lower values for the experimental
values. A feasible explanation for this, is that even the 0.5 C∗ concentration was
not dilute enough, meaning that the diﬀerent polymer coils were sensing each
other. This oﬀ course limits the full expansion of the coil, and also explains the
lower Rg as compared to the theory.
Mw [kDa]
Diﬀusivity [m2 /s]
Experimental Rg [m]
Theoretical Rg [m]
3.35
1.47E-10
1.67E-09
2.44E-09
8
9.91E-11
2.47E-09
4.05E-09
20
6.07E-11
4.04E-09
6.91E-09
35
4.39E-11
5.58E-09
9.58E-09
1000
3.41E-11
7.19E-09
6.76E-08
Table 4: Overview radius of gyration for diﬀerent molecular weights.
Figure 39 shows another sample of the 1000 kDa polymer. Unlike the previous ones, this sample is at a much higher concentration of 20 C∗ . Because of
the higher concentration, the coil’s movements are more diﬃcult which is shown
in the elevated relaxation time of 1000 ms. This is also proven by the fact that
this experiment took 10000 seconds per angle to obtain good correlation data.
53
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
Figure 39: DLS: The normalized intensity auto correlation function g2 − 1 is
plotted as a function of time for PEO 1000 kDa at 20C∗ in water for angles 60°
and 90°. No particles are present in this sample, just polymer.
4.3.4 Latex particles in polymers
After the previous series of experiments the characterization of the polymers and the particles was done. In order to save experimentation time and to
investigate thoroughly enough, it seemed appropriate now to limit the future
measurements to only 2 molecular weights. Because the rheology data showed
diﬃculties with the lower molecular weights, even at higher concentrations they
were not suﬃciently viscous, the 3.35, 8 and 20 kDa samples were not further
investigated. Moreover, to keep this section structured and concise, the results
of the PEG 35 kDa samples are shown in appendix B. In the next sessions all
the attention will go to the 1000 kDa molecular weights.
First the separate results are shown in figures 40 and 41 for the 5 C∗ , 10C∗
and 20 C∗ concentrations at diﬀerent angles. After, they are summarized in figure 42, which compares the diﬀerent concentrations for the angle 90°, including
a 2C∗ sample. Obviously the increasing relaxation time is proof of the more
diﬃcult movements of the embedded particles.
54
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
Figure 40: DLS: The normalized intensity auto correlation function g2 − 1 is
plotted as a function of time for PEO 1000 kDa at a concentration of 5C∗ and
10C∗ . In every sample 3 diﬀerent angles are plotted 50°, 90° and 150°. The
particles in the sample are spherical Latex particles with 210 nm diameter.
Figure 41: DLS: The normalized intensity auto correlation function g2 − 1 is
plotted as a function of time for PEO 1000 kDa at a concentration of 20C∗ . Only
3 diﬀerent angles are plotted 50°, 90° and 150°. The particles in the sample are
spherical Latex particles with 210 nm diameter.
55
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
Figure 42: DLS: The normalized intensity auto correlation function g2 − 1 is
plotted as a function of time for PEO 1000 kDa at diﬀerent concentrations from
2C∗ to 20C∗ . Only one angle, 90° is presented. The particles in the sample are
spherical Latex particles with 210 nm diameter.
Figure 43 shows the actual MSD as a function of time for the 1000 kDa
20C∗ sample. As one can see, the MSD grows slowly at lower times and faster
at higher times.
Figure 43: MSD: The Mean Square Displacement < ∆r(τ )2 > in [µm2 ]is plotted
as a function of time in seconds for the PEG 1000 kDa sample at a concentration
of 20 C∗ .
56
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
4.3.5 Comparison between DLS and rheometer
After collecting all the necessary data in the previous parts, the main goal of
the thesis is presented here: The comparison of the microrheology DLS results
with the bulk rheology data. The Matlab code, created for the conversion, can
be found in Appendix C. The formulas presented in paragraph 3.3.3 were used
to do the conversion and are shown again for clarity.
�
(57)
��
(58)
G (ω) = G(ω) cos(πα(ω)/2)
G (ω) = G(ω) sin(πα(ω)/2)
where:
kB T
(59)
πa �∆r2 (1/ω)� Γ(1 + α(ω))
�
�
�
�
In equation 59, the local power law α(ω) is given by �(∂ln ∆r2 (τ ) /∂lnτ )�τ =1/ω
G(ω) =
and Γ represents the gamma function (nothing to do with inverse relaxation
time). The gamma function comes from the Fourier transformation of the lo�
�
�
�
cal power law. ∆r2 (1/ω) represents the magnitude of ∆r2 (τ ) evaluated at
τ = 1/ω. Attention should be paid to the units of the frequency as the rheology data are in rad/s and the DLS converted data are originally in Hertz.
Multiplying the DLS frequency by 2*π solves this problem.
The result presented in figure 44 is for 20C∗ of the 1000 kDa polymer. For
this sample good rheological data were obtained and the conversion also succeeded perfectly. The only adjustment to the original formulas presented above
is a multiplication by a factor 2 in the G(ω) formula. A correction factor of 2
is also used in literature [8]. For the lower frequencies the DLS data are not
shown because of too many irregularities. However, the DLS results fit the
rheology data perfectly so that this result can be considered as the proof that
micro-rheology is a valuable methodology.
57
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
�
��
Figure 44: DLS-Rheo: Storage modulus (G ) and loss modulus (G ) are plotted,
as a function of angular frequency for 1000 kDa PEO at 20 C∗ for both the bulk
rheology data and the DLS microrheology data. The particles in the sample are
spherical Latex particles with 210 nm diameter.
Also for the 10C∗ sample of 1000 kDa, the bulk rheology data fit the DLS microrheology data nicely. The correction factor of 2, however, was also necessary
to obtain the fit presented in figure 45.
�
��
Figure 45: DLS-Rheo: Storage modulus (G ) and loss modulus (G ) are plotted,
as a function of angular frequency for 1000 kDa PEO at 10 C∗ for both the bulk
rheology data and the DLS microrheology data. The particles in the sample are
spherical Latex particles with 210 nm diameter.
For the PEG 35 kDa, the conversion was also done and fitted, but due to a
lack of acceptable rheology data (not viscous enough) the results are presented
58
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
in appendix B.
4.4 Ellipsoids
The second part of the results focuses on the ellipsoids. In contrast to the
spherical particles, the measurements with ellipsoids needed to be performed
twice. Once without a polarizer to obtain translational diﬀusion, and once
with a polarizer to obtain both translational and rotational diﬀusion. After
proving the possibilities for microrheology for spheres, this section will show the
possibilities of ellipsoids.
4.4.1 Ellipsoids in water
In the first measurements, the experiments were done with ’simple’ ellipsoidal particles. However, the results of these particles in water were very disappointing, as no rotational diﬀusion was measured with the polarizer. A solution
for this problem was to cover the ellipsoids (length 650 nm) with smaller gold
nanorods (lenght 60 nm) as they would increase the scattering, but have the
same movements as the big ellipsoids. The first experiments with these goldnanorods-covered-ellipsoids was in water and produced excellent results. The
measurement without the polarizer (figure 46) was analyzed using the same
methodology as spheres in water. By plotting the inverse relaxation time Γ, obtained out of the stretched exponential KWW fit, as a function of the scattering
vector q2 and doing a linear fit, the slope resulted in two times the diﬀusivity
coeﬃcient D.
Figure 46: DLS: The normalized intensity auto correlation function g2 − 1 is
plotted as a function of time for polystyrene ellipsoids with a length of 650 nm,
covered with gold nanorods with a length of 60 nm, in water. Only 3 diﬀerent
angles are plotted 50°, 90° and 150°. No polarizer is used.
59
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
Figure 47: Diﬀusion coeﬃcient: The inverse relaxation time Γ in seconds (dots)
is plotted in function of the scattering vector for polystyrene ellipsoids with a
length of 650 nm, covered with gold nanorods with a length of 60 nm, in water.
Using a linear fit, the slope is obtained which is 2 times the diﬀusion coeﬃcient.
The same sample was also measured without polarizer. It was immediately
obvious that this experiment had to run longer. As the polarizer blocks much of
the light, the intensity measured was far below the normal conditions. However,
the results are expected to be in the same range as the previous ones.
Figure 48: DLS: The normalized intensity auto correlation function g2 − 1 is
plotted as a function of time for polystyrene ellipsoids with a length of 650 nm,
covered with gold nanorods with a length of 60 nm, in water. Only 3 diﬀerent
angles are plotted 50°, 90° and 150°. Polarizer.
60
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
Again an analysis is made of the samples. The viscosity of water is used
with a value of 0.89 mPa.s and the Debye length is again subtracted from the
obtained radius. After multiplying this by 2 the wanted diameter (or length in
this case) is found. The theoretical length of the ellipsoids is very close to what
was measured and calculated with the DLS data and the errors are negligible.
Form
Name/ TEM size
Diﬀusivity [m2 /s]
Radius a [m]
Debye Length κ−1 [m]
Experimental Diameter [nm]
Absolute Error [nm]
Relative Error [%]
Ellipsoid
Polystyrene 650nm + gold 60 nm
7.43E-13
3.30E-07
1.9 E-09
656.59
6.59
1.01
Table 5: Overview of length of ellipsoids in water.
4.4.2 Ellipsoids in polymer
Ellipsoids covered with gold nanorods were mixed in the polymer sample.
To limit the experimentation times, for high concentrations more then 10 000
seconds per angle were necessary, ellipsoids were only added in the 20 C∗ sample
of the 1000 kDa. Both experiments, with and without polarizer, were executed.
Again the obtained data was good and the fit was decent. The results are shown
in figures 49 and 50.
61
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
Figure 49: DLS: The normalized intensity auto correlation function g2 − 1 is
plotted as a function of time for polystyrene ellipsoids with a length of 650
nm, covered with gold nanorods with a length of 60 nm, in PEG 1000 kDa at
concentration 20 C∗ . Only 3 diﬀerent angles are plotted 50°, 90° and 150°. No
polarizer is used.
Figure 50: DLS: The normalized intensity auto correlation function g2 − 1 is
plotted as a function of time for polystyrene ellipsoids with a length of 650
nm, covered with gold nanorods with a length of 60 nm, in PEG 1000 kDa at
concentration 20 C∗ . Only 3 diﬀerent angles are plotted 50°, 90° and 150°. A
polarizer is used.
62
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
4.4.3 Comparison between DLS and rheometer
In order to compare the microrheology of the ellipsoids with the obtained rheology new conversions were made. Not only translational diﬀusion, but also
rotational diﬀusion had to be taken into account. For translational diﬀusion,
the formulas were the same as with the spheres, but for the rotational diﬀusion
new formulas were necessary to obtain the comparison. The Matlab code is
presented in appendix C. A revision of the necessary equations for rotational
diﬀusion, explained in paragraph 3.3.3.2, shows the way of working:
In order to couple the electric field autocorrelation function to the mean
square angular displacement of the probes, the electrical autocorrelation function of the VV-mode and the VH-mode, obtained by DLS, are divided to obtain:
g1V V
exp(−Dt q 2 τ )
=
= exp(6Dr τ )
V
H
exp[−(Dt q 2 + 6Dr )τ ]
g1
(60)
The rotational diﬀusion coeﬃcient is given by:
Dr =
�
∆θ2
2
�
(61)
Combining both results and taking the natural logarithm, the following relation is obtained. :
ln(
�
�
g1V V
) = 3 ∆θ2 τ
V
H
g1
(62)
Unlike the translational diﬀusion, the rotational diﬀusion is independent of
the scattering angle and analogously to formula 59 the following equation was
used:
G∗ (ω) =
kB T
4πa3 iω �∆θ2 (ω)�
(63)
In figure 51 the fit is shown, both for the translational and rotational diﬀusion, with the bulk rheology data of the sample. The rotational diﬀusion was
not optimally obtained but the fit is still good. As the PEO 1000 kDa sample
of 20C∗ was made several months before the ellipsoids were investigated, the
original rheology could not be trusted and the sample was again measured with
the rheometer after the DLS experiments. As expected, the polymer degraded
already and was comparable with a 15C∗ sample instead of the original 20C∗ .
Initially the fitting of the DLS data on the bulk rheology was completely wrong.
After thorough consideration, it was discovered that this was due to the size of
the ellipsoids. Surprisingly the data should not be fitted with the length of the
ellipsoids, but with the length of the gold nanorods. This is a remarkable dis-
63
CHAPTER 4. EXPERIMENTAL RESULTS AND DISCUSSION
covery as this has never been discovered before. There are two possible ways to
look at this result. The first explanation is simply that the gold nanorods have
come oﬀ the original big ellipsoids and that they are responsible for the scattering. Due to a lack of time, multiple experiments were impossible to actually
confirm or deny this hypothesis. A second way of looking at the result is that
this might actually be a major braketrough in microrheology. What happens is
that although the big ellipsoids, who satisfy the mesh size, are responsible for
the movements, it is the gold nanorods that are responsible for the scattering.
If this is the case, major opportunities present themselves, like measuring viscoelastic properties of gels and other materials with big mesh sizes. However,
as it is unknown what actually happens in the polymer sample, no definitive
conclusion can be drawn. More investigation is definitely necessary.
�
��
Figure 51: DLS-Rheo: Storage modulus (G ) and loss modulus (G ) are plotted,
as a function of angular frequency for 1000 kDa PEO at 20C ∗ (theoretically)
for both the bulk rheology data from the rheometer as the DLS microrheology
data. The particles in the sample are polystyrene ellipsoids with a length of 650
nm, covered with gold nanorods with a length of 60 nm.
64
Chapter 5
Conclusion and future research
In this scientific work, research was conducted on the use of microrheological
techniques to characterize viscoelastic materials. Current state of the art in
microrheology focuses on the use of spherical particles embedded in the fluid
under investigation. However this master thesis focused on the use of ellipsoidal
particles to deduce the viscoelastic behavior of the material. After performing
traditional bulk rheology measurements using a rheometer device, the obtained
results were compared with microrheology in order to prove their equivalence.
The polymer used during the experimental work was polyethylene glycol (PEG),
a hydrophilic polymer with hydroxyl groups at both ends.
In the first experimental part of the thesis traditional bulk rheology experiments were performed. Although this method is well established some diﬃculties were encountered working with low viscous samples such as PEG of lower
molecular weight and small concentrations. Due to the low viscoelasticity of
these samples, the rheometer was pushed to its limits and the data were not
reproducible. Therefore, the decision was made to look at PEG with higher
molecular weights, and a standard sample was made at suﬃciently high concentration. A PEO 1000 kDa sample at 20C∗ became the standard throughout the
thesis and provided steady and acceptable data with a visualization of the cross�
��
over of storage (G ) and loss modulus (G ). To obtain even better data, higher
concentrations were also investigated. But due to their increasing turbidity, the
samples became useless in dynamic light scattering experiments.
During this thesis the dynamic light scattering device was the main microrheological technique. Prior experiments with spherical silica particles embedded
in water, allowed to familiarize with the concept of dynamic light scattering and
the computationally intensive data processing. Afterwards, diﬀerent molecular
weights and concentrations of PEG and PEO were investigated using embedded
65
CHAPTER 5. CONCLUSION AND FUTURE RESEARCH
spherical particles. Most experiments were performed with a multiangle setup,
covering 13 diﬀerent angles. Consequently, the duration of the experiments for
more concentrated samples covered several days, as the diﬀusion coeﬃcient of
the colloids diminished. The results obtained for spherical particles in polymers were of high quality and most of the results were fitted perfectly using a
stretched exponential fit. Afterwards, the mean square displacement obtained
from microrheology was converted into actual viscoelastic moduli, in order to
compare the DLS results with the traditional bulk rheology data. In line with
what literature describes, microrheology using spherical particles was confirmed
as a valuable alternative for bulk rheology.
After this first set of experiments the use of ellipsoids in microrheology was
investigated. By using a polarizer, the scattering from translational diﬀusion
was suppressed and mainly rotational diﬀusion was observed. However due to
the optical isotropy of the ellipsoidal beads no light scattering was perceived. In
order to overcome this problem the ellipsoids were covered with gold nanorods,
which have excellent scattering properties. This led to proper results with the
dynamic light scattering device. However, to obtain appropriate resemblance
between bulk and microrheology for rotational diﬀusion, initial conversion, using
the size of the big ellipsoids was oﬀ by several decades.
Surprisingly however, using the size of the gold nanorods instead, resulted
into a perfect fit with the bulk rheology data. This raised questions which can
be answered with two hypotheses.
A first explanation is that the gold nanorods detached from the original
ellipsoids and moved freely at their own diﬀusion rate. Although the results in
water do not show any indication of this, time constraints made it impossible
to actually confirm or deny this hypothesis with additional experiments.
A second way of looking at the result is considering this to be a new way of
looking into microrheology. Although the big ellipsoids satisfy the mesh size of
the polymer and probe the material, it are the gold nanorods that are responsible for the scattering and it is their length that matters in the mathematical
conversion between bulk rheology and microrheology. If this is the case, major
opportunities would emerge, such as measuring viscoelastic properties of gels
and other materials with big mesh sizes. However, as it is currently unknown
what actually happens in the polymer sample, no definitive conclusion can be
drawn yet.
The aforementioned conclusions make it clear that this research is far from
finalized, and that in the best case only the top of the iceberg has been exposed.
Further investigation will need to clarify this.
66
l
Appendices
Appendix A
Specifications of the geometry
l
68
APPENDIX A. SPECIFICATIONS OF THE GEOMETRY
69
Appendix B
Latex particles in PEG 35 kDa
Diﬀerent concentrations of the 35 kDa molecular weight are shown. First the
results are shown for the 5 C∗ , 15C∗ and 25 C∗ at diﬀerent angles and to
summarize a graph is shown that compares these concentrations for the angle
of 90°, including a 0.1 C∗ . Obviously the increasing relaxation time is proof of
the more diﬃcult movements of the embedded particles. The measuring time
also went up to 10 000 seconds per angle during these diﬀerent measurements.
The long time per angle made the various experiments very time consuming.
PEG 35 kDa concentration at 5C ∗ . Latex particles.
70
APPENDIX B. Results PEG 35 kDa
PEG 35 kDa concentration at15C ∗ Latex particles.
PEG 35 kDa concentration at25C ∗ Latex particles.
PEG 35 kDa overview diﬀerent concentrations. Latex particles
71
APPENDIX B. Results PEG 35 kDa
Conversion DLS and rheometer PEG 35 kDa
Because for PEG 35 kDa the rheology data were less successful, the conversion
of the DLS data was done without comparing to bulk rheology.
Diﬀerent concentrations are presented in increasing concentration. This way
�
��
one can see the evolution of the cross-over between G (ω)and G (ω) appearing
at lower angular frequency and higher values for the moduli. This matches the
expectations perfectly as this cross-over represents the inverse of the relaxation
time. In other words, the more concentrated, the higher the relaxation time and
the earlier the cross-over point.
DLS-Rheo Comparison: PEG 35 kDa concentration 5C ∗ Latex particles
DLS-Rheo Comparison: PEG 35 kDa concentration 15C ∗ Latex particles
72
APPENDIX B. Results PEG 35 kDa
DLS-Rheo Comparison: PEG 35 kDa concentration 25C ∗ Latex particles
73
Appendix C
Matlab code for data processing: Translational
Diﬀusion
74
APPENDIX C. MATLAB CODE FOR DATA PROCESSING
75
APPENDIX C. MATLAB CODE FOR DATA PROCESSING
Matlab Code for data processing: Rotational Diffusion
76
Appendix D
Safety and hazard analysis
Working in a chemical lab always entails hazardous situations which should be
carefully analyzed prior to the actual experimental work. Although the hazard
in this master thesis were relatively limited, the potential threats are listed up.
• Chloroform
In order to prevent bacterial growth, a few drops of chloroform were added to
our samples. Chloroform is a colorless, dense organic compound with structured
formula CHCl3 .
Chloroform vapors depress the central nervous system (used as anesthetic).
It is dangerous at approximately 500 ppm according to the U.S. National Institue
for occupational Safety and Healt.Breathing about 900 ppm for a short time
can cause dizziness, fatigue, and headache. Chronic chloroform exposure can
damage the liver [42]. The NFPA 704 code fo chloroform is given in figure 52.
• Laser beam of DLS
The laser beam from the DLS is a continuous wave (CW) laser that emits red
light with an operational wavelength of 632,8 nm at a power of 35mW. This
laser beam is classified as a class 3B laser, which means the laser is hazardous
if the eye is exposed directly, but diﬀuse reflections such as those from paper or
other matte surfaces are not harmful. Protective eye wear is typically required
where direct viewing of a class 3B laser beam may occur. Class 3B lasers must
be equipped with a safety interlock.
77
APPENDIX D. SAFETY AND HAZARD ANALYSIS
• Toluene
The DLS samples were placed in a toluene bath. Toluene is used because it
reduces undesirable reflections and refractions as it has a similar index of refraction as the glass of the sample. Toluene should not be inhaled due to its
health eﬀects. Low to moderate levels can cause tiredness and confusion or loss
of appetite, even hearing and color vision loss have been observed. These symptoms usually disappear when exposure is stopped. Inhaling high levels of toluene
in a short time may cause nausea, or sleepiness. Additionally it can cause unconsciousness, and even death[43]. In figure 52 the NFPA code of toluene is
given. As can be seen from the figure toluene has intense health hazards and
high flammability risks.
Figure 52: NFPA 704 of chloroform and toluene. In this fire diamond every
thread is given a score from 0 to 4, ranging from no hazard to severe risk. Blue
stands for health risks, red for flammability, yellow for reactivity and white for
special codes for unique hazards [44].
78
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