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Research Collection
Doctoral Thesis
Aggregation, Coalescence, and Gelation of FunctionalNanoparticles
Author(s): Jaquet, Baptiste P. H.
Publication Date: 2016
Permanent Link: https://doi.org/10.3929/ethz-a-010793615
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Aggregation, Coalescence, and Gelation of Functional Nanoparticles
A thesis submitted to attain the degree of
DOCTOR OF SCIENCES of ETH ZURICH
(Dr. sc. ETH Zurich)
presented by
Baptiste P. H. Jaquet
MSc Chemical and Bioengineering, ETH Zurich
Born on 17.03.1989
Citizen of La Sagne, NE
Accepted on the recommendation of
Prof. Dr. Massimo Morbidelli, examiner
Prof. Dr. Chih-Jen Shih, co-examiner
2016
I
Abstract
The nano-scale world is gaining more interest every day, with exciting developments potentially
allowing significant improvements in many current fields of technology. Nanoparticles, or colloids, are
hence a growing field of research, even though it has been known since close to a century. However,
the size scale in which nanoparticles comes with challenges coming from meso-scale physics together
with nano-scale physics (or chemistry). The industrial production of polymeric products often makes
use of emulsion polymerization, which naturally forms polymeric nanoparticles. These particles can
exhibit a growing number of functionalities, for example the ability of forming films. From an industrial
point of view, it is important to be able to control and understand what factors govern the aggregation
and gelation of the product.
In this work, the specificities of an industrial-type product were studied. It is composed of coalescing
acrylic polymer nanoparticles, on which surfaces poly(acrylic acid) was grown during the
polymerization process.
The effects of the polyelectrolyte layer are studied both experimentally and theoretically over two
chapters of the present work, showing clearly the underlying complexity of working with industrial
products. In order to investigate these effects, aggregation experiments and diverse characterization
techniques were used in order to devise an interaction potential relating the stability of the particles to
the pH and the synthesis recipe of the particles.
The effect of coalescence on stagnant aggregation and shear-induced gelation were studied
experimentally and theoretically over the following two chapters, using population balance equations
in order to understand the interaction between coalescence and stagnant aggregation. In the light of
II
these findings, shear-induced gelation is shown to result from the competition between aggregation,
coalescence and breakup.
Finally, a rather unexpected surfactant adsorption mechanism on polyelectrolyte-covered particles
leading to their aggregation was revealed experimentally and was explained by molecular dynamics
simulations.
III
Résumé
Le monde des échelles de grandeurs `nano` prend de plus en plus d’ampleur dans le temps, avec des
développements excitants pouvant rendre possible des améliorations significatives dans divers
domaines courants de la technologie. Les nanoparticules, ou colloïdes, sont donc un domaine de
recherche en pleine croissance, bien qu’il soit connu depuis près d’un siècle. Malgré cela, l’ordre de
grandeur typique des nanoparticules fait qu’elles viennent avec des problématiques spécifiques à la fois
à la physique ‘méso’ et la ‘nano-‘ physique (ou chimie). Cela fait qu’une compréhension quantitative
des facteurs gouvernant leur physique est loin d’être achevée. La production industrielle de polymères
fait souvent recours à la polymérisation en émulsion, qui par nature génère des nanoparticules de
polymère. Ces nanoparticules peuvent exhiber un nombre de fonctionnalisation constamment croissant,
par exemple la capacité de former des films lorsque la dispersion est séchée. D’un point de vue
industriel, il est essentiel de pouvoir comprendre, prévoir et contrôler quels facteurs vont mener à une
agrégation et une gélation du produit.
Dans le présent travail, les spécificités d’un produit de type industriel sont étudiées. Ce produit est
composé de nanoparticules acryliques pouvant coalescer, sur la surface desquelles des chaines de
poly(acide acrylique) ont été crûes durant le procédé de polymérisation.
Les effets de la couche de polyéléctrolyte sont tout d’abord étudiés sur deux chapitres du présent travail,
à la fois d’un côté expérimental et d’un côté théorique. Ces deux chapitres révèlent clairement la
complexité inhérente aux produits industriels. Afin d’étudier ces effets, des expériences d’agrégation et
de caractérisation variées ont étés appliquées afin de mettre au point un modèle d’interaction liant la
stabilité colloïdale de ces particule au pH et à leur recette de synthèse.
IV
Les effets de la coalescence sur l’agrégation stagnante et induite par cisaillement sont ensuite étudiés
sur les deux chapitres suivants. Tout d’abord, un modèle de bilan de population a été développé afin de
prendre en compte les effets de la coalescence sur l’agrégation stagnante. A la lumière de ce
développement, la gélation induite par cisaillement est expliquée et comprise comme étant le résultat
d’une compétition entre l’agrégation, la coalescence et le cassage des agrégats formés.
Enfin, un mécanisme contre-intuitif d’adsorption inversée de surfactant ionique sur la surface de
particules couvertes de polyéléctrolyte menant à leur agrégation est démontré expérimentalement et
expliqué par simulation de dynamique moléculaire.
V
Acknowledgements
First of all, I want to thank Prof. Massimo Morbidelli for giving me the opportunity to work in his
group. He allowed me to grow as a scientist and as an individual. I enjoyed the possibility of conducting
my research independently and with his confidence. I want to thank the other persons that made this
work possible. Of course, I want to thank Dr. Hua Wu for his support and suggestions, as well as his
expertise.
I want to thank Prof. Chih-Jen Shih for accepting to be my co-examiner, for his availability and
comprehension.
Many thanks to BASF SE, and in particular to Dr. Bernd Reck, Dr Volodymyr Boyko, Dr. Immanuel
Willerich, Dr. Oliver Labisch and Dr. Roelof Balk
A great thanks of course goes to Prof. Marco Lattuada and Prof. Giuseppe Storti. Their contribution to
the model development of the chapter 4 of this thesis cannot be stressed enough.
I want to thank very gratefully Dr. Stefano Lazzari for the work we did together and the support both
scientifically and morally. A great thank also goes to Dr. Tommaso Casalini for his invaluable help in
proving some intuitions are sometimes right. I want to thank Prof. Miroslav Soos for his support and
expertise in the latest stages of this work; you helped me a lot, taking some of your precious time for
me.
I want of course to thank my master, research projects, and “Hilfsassistenten” for their invaluable help
in following my experimental ideas and trusting me. In particular, I want to thank Kevin Maurer,
Antoine Klaue, Luca Colonna, Guido Zichittella, Gabriele Colombo, Daniel Balderas and Marc Siggel
for their help.
VI
Thanks to Bastian Brand, to Alberto Cingolani, and Thomas Villiger for the helpful discussions and
experimental “tips” and explanations. I also want to thank all my friends; in particular David Pfister,
Rushd Khalaf, Lucrèce Nicoud, Marta Owczarz, Vincent Diederich, Bastian Brand, Marco Furlan,
Antoine Klaue, Anna Beltzung, Thomas Villiger, and Fabian Steinebach, and all of the Morbidelli group
for all the good times inside and outside the lab.
Un merci spécial va à mes parents, ainsi qu’à mes amis de Suisse Romande, pour m’avoir soutenu
pendant les bons et les mauvais jours de cette aventure. Merci d’avoir compris que mes absences
régulières ne sont pas le signe que je vous oublie.
Ein grosses Dankeschön geht auch selbstverständlich zu Caroline. Danke, dass du mich unterstützt und
auf mich gewartet hast während der letzten Periode von meinem Doktorat. Und auch Danke, dass du
mit mir die schönen Zeiten des Lebens geniesst.
VII
Table of Contents
Abstract I
Résumé III
Acknowledgements ................................................................................................................................ V
Table of Contents ................................................................................................................................ VII
Chapter 1 Introduction ........................................................................................................................... 1
1.1 Aggregation of industrial functional nanoparticles ....................................................................... 2
1.2 Aggregation, Gelation, and Shear ................................................................................................. 2
1.3 Coalescence ................................................................................................................................... 4
1.4 Scope of the present work ............................................................................................................. 4
Chapter 2 Stabilization of Polymer Colloid Dispersions with pH-sensitive poly(Acrylic Acid) brushes7
2.1 Introduction ................................................................................................................................... 7
2.2 Experimental section ................................................................................................................. 9
2.2.1 The Colloidal Systems ........................................................................................................... 9
2.2.2 Determination of the Fuchs Stability Ratio W ..................................................................... 10
2.3 Results and discussion ................................................................................................................ 14
2.3.1 Particle size and PAA brush conformation behavior ........................................................... 14
2.3.2 Stability of P1 Latex ............................................................................................................ 19
2.3.3 Contributions of PAA Brushes to Colloidal Stability .......................................................... 20
2.4 Concluding remarks .................................................................................................................... 25
Chapter 3 Investigation of the Steric Stabilization of Dispersions with pH-sensitive poly(Acrylic Acid) brushes .......................................................................................................................................... 29
3.1 Introduction ................................................................................................................................. 29
3.2 Materials and Methods ................................................................................................................ 32
3.2.1 The latexes ........................................................................................................................... 32
3.2.2 Characterization of the particles .......................................................................................... 32
3.2.3 Aggregation Experiments .................................................................................................... 33
3.2.4 Simulation of the aggregation curves ................................................................................... 34
3.2.5 Modeling of the interaction potential ................................................................................... 34
3.3 Results and Discussion ............................................................................................................... 35
3.3.1 Characterization of the particles .......................................................................................... 35
3.3.2 Aggregation experiments and the Fuchs stability ratio ........................................................ 38
3.3.3 Steric model ......................................................................................................................... 40
VIII
3.4 Concluding remarks .................................................................................................................... 42
Chapter 4 Interplay between Aggregation and Coalescence of Polymeric Particles: Experimental and Modeling Insights ................................................................................................................................ 45
4.1 Introduction ................................................................................................................................. 45
4.2 Materials and methods ................................................................................................................ 47
4.2.1 Materials .............................................................................................................................. 47
4.2.2 Particles synthesis ................................................................................................................ 47
4.2.3 Light Scattering .................................................................................................................... 48
4.2.4 Differential Scanning Calorimetry ....................................................................................... 49
4.3 Model development .................................................................................................................... 49
4.3.1 Population balance equations ............................................................................................... 49
4.3.2 Calculation of hR t , gR t and S q,t ............................................................ 53
4.4 Numerical Solution ..................................................................................................................... 55
4.5 Results and discussions ............................................................................................................... 56
4.5.1. Particle characterization and experimental conditions ........................................................ 56
4.5.2 Aggregation at room temperature: gR t , hR t and S q,t .............................. 57
4.5.3 Aggregation at higher temperatures ..................................................................................... 67
4.6 Conclusions ................................................................................................................................. 69
Chapter 5 Effects of Coalescence on the Shear-Induced Gelation of Colloids ................................... 73
5.1. Introduction ................................................................................................................................ 73
5.2. Materials and methods ............................................................................................................... 76
5.2.1 Materials .............................................................................................................................. 76
5.2.2 Latex synthesis and characterization .................................................................................... 76
5.2.3 Critical coagulation concentration determination ................................................................ 77
5.2.4 Zeta potential measurement ................................................................................................. 78
5.2.5 Colloidal titrations ............................................................................................................... 78
5.2.6 Couette-flow gelation experiments ...................................................................................... 78
5.2.7 Stirred-Tank DLCA experiments ......................................................................................... 79
5.3.1 Effect of temperature on stagnant aggregation .................................................................... 81
5.3.2 Gelation Curves ................................................................................................................... 83
5.3.3 Effect of salt concentration and temperature on gelation time ............................................. 85
5.3.4 Effect of temperature on cluster breakup ............................................................................. 89
5.4. Conclusion ................................................................................................................................. 95
Chapter 6 The Counter-Intuitive Aggregation of poly(Acrylic Acid)-Grafted Nanoparticles after
Surfactant Addition 97 6.1 Introduction ................................................................................................................................. 97
6.2 Materials and Methods ................................................................................................................ 98
IX
6.2.1 Particles synthesis ................................................................................................................ 98
6.2.2 Particles characterization ..................................................................................................... 98
6.2.3 Aggregation experiments ..................................................................................................... 99
6.2.4 Computational methods ....................................................................................................... 99
6.3 Results and discussion .............................................................................................................. 103
6.3.1 Latex aggregation kinetics ................................................................................................. 103
6.3.3 Molecular dynamics simulations ....................................................................................... 107
6.4 Concluding remarks .................................................................................................................. 110
Chapter 7 Concluding remarks .......................................................................................................... 113
Chapter 8 Appendix ........................................................................................................................... 115
8.1 Experimental proof for the coalescence between particles ....................................................... 115
8.2 - time-evolution of the average fractal dimension .................................................................... 117
8.3 – Derivation of average quantities h,effR x, t and gR x,t ................................................. 120
8.3.1 Calculation of h,effR x,t ................................................................................................ 120
8.3.2 Calculation of gR x,t .................................................................................................... 127
8.4 Derivation of the discretized Equations .................................................................................... 128
8.5 Parameter Values employed in the simulations ........................................................................ 131
8.6 Measured Fractal dimensions .................................................................................................... 134
8.7 Results at higher temperatures .................................................................................................. 135
8.8 Symbol List for chapter 4 .......................................................................................................... 137
X
1
Chapter 1
Introduction
Colloidal and nanoparticles in general play a growingly important role in the current active fields of
research, as well a production. In fact, an important number of commercial polymeric products are used
as dispersions. These cover the fields of paints and coatings, lacquers, adhesives, paper industry, textiles
and leathers[1-3] In order to manufacture these products, emulsion polymerization is commonly used.
Colloidal suspensions are not limited to plastics production, as nanoparticles dispersions nowadays
offer a constantly growing scope of new applications, ranging from photovoltaics [4, 5] to the medical
field [6-8]. Because of the size of colloidal particle (1-1000 nm typically), they are exhibiting physical
and dynamical behaviors common with the macro- scale, as well as with the chemical size scale[9, 10].
These features offer a great deal of complexity and interest, making it uneasy to predict how a colloidal
system will react to an external stimuli, even though colloids have been studied since close to a century
[11, 12].
Great progress has been made in the understanding of colloidal systems since the initial development
of this field [13]. However, most of the theories and mathematical expressions devised in order to
describe the dynamics of dispersions are centered on ideal systems [14]. The use of typical mathematical
models to describe the manufacturing of real-world, complex colloidal dispersions usually is inefficient
at predicting and understanding quantitatively the influences of the control parameters on the dynamics
of the products.
2
1.1 Aggregation of industrial functional nanoparticles
The use of emulsion polymerization to form copolymeric latexes is widely spread in industrial
applications. The quality control of latexes used for high-end applications, such as film-forming
adhesive latex [15], in which optical properties must be closely controlled, is essential. In order to ensure
a reproducibly high quality product, while maximizing productivity, a close control on the aggregation
kinetics is necessary. The use of stabilizers is commonly found in polymer production. However, in the
case of complex polymeric products, where up to ten different monomers are combined and
polymerized together, having a clear understanding of what factors will influence the colloidal stability
is dramatically complex. The main feature of the particles influencing the colloidal stability is evidently
the state of the particles’ surface. Charge density, composition, softness of the interface controls the
dynamics of nanoparticle interactions. In the present work, the case of industrial transparent adhesive
nanoparticle is studied. The industrial production of this high-end commercial product features
significant challenges, coming mainly from two features; the particles are made of a low-Tg copolymer,
which implies that the particles are melting into each other upon contact. On the other hand, in order to
ensure a reasonable commercial production, they exhibit a complex functionalized surface, composed
of a polyelectrolyte “brush-like” layer. This is pH-responsive, and its growth and morphological
properties are poorly understood, as the polymerization method does not allow for an accurate control
of its formation. The challenge here is to clarify the link between the recipe and the morphology, and
then between the morphology and the colloidal behavior.
1.2 Aggregation, Gelation, and Shear
The processes leading to aggregation of colloidal particles are relatively well-known [16]. The
lyophobic particles are stable in suspension as long as a sufficient energy barrier has to be overcome in
order for the surfaces of two particles to contact. The energy allowing to cross the energy barrier can
be supplied by Brownian motion in the case of stagnant aggregation [17], or by an external force field.
The latter case corresponds typically to shear-induced aggregation [18, 19].
When colloidal particles aggregate, whether in stagnant conditions or under the influence of an external
force field, it has been observed that they form fractal structures. Random fractal structures are self-
3
similar (in size scale), and random in nature. They are related to the well-known geometrically
spectacular Mandelbrot fractal set [20], but their random nature do not imply such a degree of organized
geometric self-similarity. The fractal nature of colloidal aggregates imply that the size of an aggregate
is related to its mass by a power-law. This implies that colloidal aggregates are intrinsically porous.
Hence, the volume occupied by the aggregates increases non-linearly to the mass of the aggregates.
This means that along an aggregation process, the occupied volume fraction increases. When the whole
volume is occupied, the aggregates “percolate”, forming a network of aggregated particles, not allowing
the movement of the whole suspension anymore. This process is called gelation, and is macroscopically
recognized as a liquid-solid phase transition of the suspension[21]. Controlled gelation has been
successfully applied in order to form porous monoliths, with applications in chromatography. Gelation
can also be an undesired phenomenon; for example in the industrial production of polymeric
nanoparticles.
When polymeric particles are manufactured industrially or in lab scale, they get stirred, pumped,
filtered, heat-exchanged etc. Many of the unit operations taking place in a typical manufacturing process
imply a significant shear being applied to the latexes. Shear is an external force field allowing to bring
kinetic energy in the dispersion. If the shear rate is sufficient, it may allow nanoparticles which are
stable in stagnant conditions to aggregate. Shear-induced aggregation has been extensively studied in
the last decades, allowing to understand the mechanisms governing it. It has been revealed that the rate
of aggregation under shear is the results of the competition between the shear rate and the inter-particle
interaction potential [22].
Breakup of colloidal aggregates is one of these relevant mechanisms. While only rarely observed under
stagnant aggregation, because the adhesion energy between colloidal particles is typically much higher
than the thermal agitation present in the dispersion, breakup of aggregates caps up the size that
aggregates can physically reach under shear. The breakup rate results of the competition between the
size of the aggregate, on the shear rate (ultimately hydrodynamic stress), and on the adhesion energy of
the clusters [23].
4
Under shear, the evolution of aggregate size is significantly different from what is observed in stagnant
conditions. Where stagnant aggregation typically generates a monomodal size distribution function,
shear-induced aggregation is self-accelerating, generating very large aggregates which cohabit with
primary particles or small clusters. This leads to possible “explosive gelation”, during which the
viscosity of the suspension –after an induction time- sharply rises to infinity, producing a solid gel piece
in small durations.
1.3 Coalescence
This work is focused on film-forming particles. Their physical properties are tuned in order to ensure
that the particles will fuse into each other upon drying of the dispersion. In order to obtain this result, it
is necessary that the particles’ material exhibits a finite viscosity. This means that the glass transition
temperature Tg of the material must be located below the film-forming temperature (mostly room
temperature).
Coalescence has a significant impact on both stagnant and shear-induced aggregation and gelation. In
fact, under stagnant conditions, coalescence will allow the clusters to get more compact, thus increasing
their fractal dimension, and to decrease their occupied volume fraction. This evidently is reflected in
longer gelation times. Similarly under shear conditions, coalescence allows the particles to
interpenetrate. This of course decreases the occupied volume fraction, delaying the gelation. But,
simultaneously, interpenetration of particles reinforce significantly their adhesion energy, thus
reinforcing the clusters, which are then able to grow to larger sizes before breaking apart under shear
stress. The competition between these two effects of coalescence has been reported previously [24], but
never systematically studied.
1.4 Scope of the present work
In the present work, the general case of the aggregation and gelation polyelectrolyte-stabilized, film-
forming nanoparticles is studied. In chapter 2, the initial understanding of the mechanics of poly(acrylic
acid)-covered industrial latex stagnant aggregation is studied and allows to gain qualitative
understanding of the relevant parameters controlling the stabilization of such latexes.
5
In chapter 3, a steric interaction model is used in a tentative to use the aggregation properties in order
to characterize the properties of the polyelectrolyte layer of industrial systems.
In chapter 4, the effect of coalescence on diffusion-limited cluster aggregation is studied, and a
population balance equation model is devised in order to predict the size evolution of clusters formed
by ad-hoc synthesized acrylic particles with different Tg.
In chapter 5, the same particles are used in order to understand the effect of coalescence on shear-
induced aggregation and gelation using rheology.
In chapter 6, a more industrial problem is presented. The presence of surfactant, commonly used during
emulsion polymerization, is shown to have a counterintuitive effect on poly(acrylic acid)-covered
particles. In fact, head-on adsorption of SDS mediated by salt bridges is shown to occur by molecular
dynamics, and to lead to surfactant-induced aggregation in pH ranges in which the particles would not
aggregate without SDS.
6
7
Chapter 2
Stabilization of Polymer Colloid Dispersions with pH-sensitive poly(Acrylic Acid) Brushes
2.1 Introduction
When linear polyelectrolyte (PE) chains are densely grafted on a surface, they form so-called surface
PE brushes [25]. The strong electrostatic interactions among the highly charged chains within the
brushes lead to substantially different properties with respect to surface monolayers of uncharged
macromolecular chains. Thus, great attention has been given recently to these systems [26-33].
Weak (or so-called annealing) PEs such as poly(acrylic acid) (PAA) and poly(methacrylic acid)
(PMAA), when exposed to aqueous media, may undergo conformational changes, depending on the
system pH and salinity. Under alkaline conditions, due to deprotonation of their carboxylic groups, they
are highly charged and hydrophilic, leading to stretching of the backbone and good solubility. Instead,
under strongly acidic conditions, because of protonation of the carboxylic groups, they become weakly
charged and more hydrophobic, with the backbone in a collapsed state, thus less soluble in water.
Accordingly, weak PEs can be used to form brushes on polymeric surfaces, which respond to pH
changes in the environment, leading to potential applications in various areas such as nano-scale
actuators [34], membrane modifiers for pH-controlled permeation [28, 35], charge-driven reversible
polymer and protein adsorption [28, 36-38], nanoparticle stabilization [39], etc.
In this work, we focus on the stability behavior of colloidal particles grafted with PAA brushes. It
is known that weak PEs like PAA and PMAA when grafted on the surface change substantially their
ionization behavior with respect to their un-grafted state in water. Because of the strong electrostatic
8
attraction among the mobile counterions in solution and the opposite charges on the brushes, the
concentration of the mobile counterions in the brush-occupied region increases towards the particle
surface [40, 41]. It follows that the total local charge density (polymer and mobile ions) is substantially
lower in the inner than in the outer region of the brushes. On this process is based, the so-called local
electroneutrality approximation (LEA) [40, 42], which states that the counterion concentration in the
inner region is approximately equal to that of the charged groups on the brushes. As a consequence, the
conformation of the brushes in the inner region is different from that in the outer one [43]. In addition,
experimental observations reveal that the apparent pKa of surface-grafted PMAA differs from that of
PMAA in solution and depends also on both brush thickness and density [44]. In addition to the
electrostatic repulsion arising from the particle charge, grafted hydrophilic polymer chains are known
to stabilize colloidal particles via steric interactions [45]. These steric interactions arise from the mixing
free energy of the polymer chains grafted on the surface [46]; if the dispersant is a good solvent for the
polymer composing the brushes, overlapping two grafted layers (during the aggregation process for
example) will lead to a repulsive effect. The consequence of all the above theoretical and experimental
results is that the contribution of the PAA brushes to the stability of colloidal particles is not a trivial
function of the AA concentration on the particle surface.
In this chapter we investigate experimentally the effect of surface-grafted PAA brushes on the
stability of polymer colloids at different pH values, i.e., under their different ionization and
conformational states. For this, we considered colloidal particles with different thickness of the PAA
brushes and measured their Fuchs stability ratio, W, based on the doublet formation kinetics, at different
pH values, as a function of electrolyte concentration. In the sequel, we first describe and characterize
three types of colloidal particles with different PAA brushes, and measure their W values. Next, we
compare the measured W values of the three colloids, and discuss them with respect to the expected
conformation of the PAA brushes.
9
2.2 Experimental section
2.2.1 The Colloidal Systems
The polymer colloids used in this work are three types of butylacrylate-mehtylmathacrylate-acrylic acid
copolymer latexes, referred to as P1, P2 and P3 in the following, respectively, synthesized ad-hoc by
BASF SE (Ludwigshafen, Germany) through emulsion polymerization, using Na2S2O8 as initiator and
sodium dodecyl sulfate as emulsifier. The characteristics of the three latexes are reported in Table 1. It
is seen that the main difference is the amount of acrylic acid (AA): 0%, 1% and 2% for P1, P2 and P3
latexes, respectively. Thus, particles of P1 latex possess only the sulfate ( 3-OSO ) fixed charges, while
those of P2 and P3 latexes carry both sulfate and carboxylic ( -COO ) charges.
The SDS (sodium dodecyl sulfate) surfactant present in the original latexes, as well as residual
initiator, soluble oligomers and all other possible electrolytes, have been removed completely by ion
exchange resin (Dowex MR-3, Sigma-Aldrich), according to a procedure reported elsewhere [47]. After
cleaning, the conductivity of the mother liquor is <10 μS/cm and its surface tension is close to that of
deionized water, indicating that the colloidal systems are free of SDS and any other electrolyte. The
particle volume fraction for the three latexes after cleaning is about =20%, with pH around 2.7. The
cleaning also leads to replacement of the counterions of the fixed charges (e.g., 3-OSO and -COO )
from Na+ to H+ (the Na+ concentration after cleaning measured by atomic absorption spectrometry is
equal to about 3 ppm).
The particles characterization has been performed using dynamic and static light scattering (DLS
and SLS). The SLS and DLS measurements were carried out using a BI-200SM goniometer system
(Brookhaven Instruments, USA), equipped with a solid-state laser, Ventus LP532 (Laser Quantum,
U.K.) of wavelength 0=532 nm, as the light source, in the scattering angle range of 15 to 150. When
used for DLS, the angle was set to 90. The reported results are averages of at least 2 independent series
of at least 3 measurements. The nominal radius of the particles, Rp, has been evaluated by fitting the
10
measured form factor from the SLS measurements, P(q), defined as ( ) ( ) / (0)P q I q I (where I is the
scattered intensity, q is the magnitude of the scattering wavevector, defined as
0 04 / sin[ / 2]q n with n0, 0 and the refractive index of water, the wavelength of the incident
light, and the scattering angle, respectively), with the Rayleigh-Debye-Gans (RDG) theory for spheres.
The nominal size of the primary particles after cleaning has been obtained by fitting the form factor,
P(q) measured under natural pH and 10 mM NaCl, with RDG theory. All the obtained Rp values are
reported in Table 2.1.
Table 2.1 Characteristics of the three latexes used in this chapter
Latex name Composition (weight ratio)a pH Rp
P1 BA (50) : MMA (50) + 0.7% SDS 5.1 84.1 nm
P2 BA (49.5) : MMA (49.5) : AA (1) + 0.7% SDS 4.9 81.0 nm
P3 BA (49) : MMA (49) : AA (2) + 0.7% SDS 4.2 90.0 nm
a—BA: n-butyl acrylate; MMA: methyl methacrylate; AA: acrylic acid; SDS: sodium dodecyl sulfate
2.2.2 Determination of the Fuchs Stability Ratio W
The contribution of different surface groups to the colloidal stability has been quantified by measuring
the Fuchs stability ratio (W) values under different conditions. To this aim, we have conducted all
experiments for the W measurements at the particle volume fraction =5.0% and different pH, using
NaCl and H2SO4 as the destabilizers.
The principle: The W measurement is based on the observation of the doublet formation kinetics
as discussed in previous work [17, 48]. In particular, at the very initial stage of the aggregation, the
main process is conversion of primary particles to doublets, and therefore the kinetics of primary
11
particle consumption can be approximated as follows:
211,1 1
dNK N
dt (2.1)
where N1 and K1,1 are the primary particle number concentration and doublet formation rate constant,
respectively. Introduction of conversion of primary particles to doublets, )0 1 0( /x N N N , into Eq.
2.1, where N0 (3p3 / [4 ]R ) is the initial number concentration of primary particles, yields:
1,1 01
xK N t
x
(2.2)
Thus, from the measured x values as a function of time, t, the slope of the / (1 )x x vs t line leads to
the estimation of K1,1. Deviations from linear behavior of this line indicate that doublets or larger
clusters also contribute to the aggregation process, and the experiments have to be repeated at lower
conversion values.
From the data above, we can obtain the Fuchs stability ratio, W, based on the following
relationship:
1,1/BW K K (2.3)
where KB (=8kT/[3μl]), with k the Boltzmann constant, T the absolute temperature, and l the viscosity
of the dispersant, is the so-called Smoluchowski rate constant, defining the diffusion-controlled
collision rate between particles of equal size.
It is clear that to obtain K1,1 from Eq. 2.2, we need measure the x value as a function of t. This is
done by taking proper samples of the aggregating system and making SLS measurements. According
to the RDG theory [49], the scattered light intensity I(q) for a system containing only primary particles
12
and doublets can be expressed as:
1 1 2 2
1 2
( ) ( ) 4 ( )
(0)
I q N P q N P q
I N N
(2.4)
where N2 is the number concentration of doublets, and P1(q) and P2(q) are the form factor of the primary
particles and doublets, respectively. Within the RDG theory, the expression for P1(q) is given by:
2
p p p1 3
p
sin( ) ( ) cos( )( ) 9
( )
qR qR qRP q
qR
(2.5)
The expression for P2(q) depends on whether coalescence occurs or not between the two aggregated
particles in the doublet [48]. For the present very soft acrylate copolymer particles (Tg≈15 ̊ C), we have
confirmed experimentally, as shown in the annex 8.1, that coalescence between two particles does
occur. Then, considering two primary particles coalescing and forming a new sphere, P2(q) is given by
21/3 1/3 1/3p p p
2 1/3 3p
sin(2 ) (2 )cos(2 )( ) 9
(2 )
qR qR qRP q
qR
(2.6)
By substituting Eqs 2.5 and 2.6 into Eq. 2.4 and considering that 1 0(1 )N x N= - and 2 0 / 2N xN= ,
we have:
1
( )1 ( )
(0) ( ) 1
I q xF q
I P q x= -
+ (2.7)
where
3 3 3
p p p
p p p
sin( 2 ) (( 2 )cos( 2 )1( ) 2
2 sin( ) ( )cos( )
qR qR qRF q
qR qR qR
(2.8)
Therefore, the plot, 1( ) / [ (0) ( )]I q I P q versus F(q), is a straight line, and its slope yields the value of
the conversion x.
The experimental procedure: We first mix the latex with a proper electrolyte solution so as to reach
13
a desired particle volume fraction, electrolyte concentration and pH (sulfuric acid and sodium hydroxide
were used to tune the pH) to start the aggregation under stagnant conditions. Then, we take samples
from the aggregation system at different times and dilute them immediately in Milli-Q water for
quenching the aggregation and performing the SLS measurements. In addition, for each aggregating
system, pH was measured after the aggregation experiments.
Figure 2.1 Examples of the x/(1-x) vs. t plot for the estimate of K1,1 based on Eq. 2.2, in the case
of P2 latex using H2SO4 as the destabilizer at three concentrations: 0.116 mol/L (), 0.123 mol/L ()
and 0.130 mol/L ().
Fig. 2.1 shows typical plots of / (1 )x x- vs. t at the very initial stage of the aggregation process
for the P2 latex at =5%, using H2SO4 as the destabilizer at Cs =0.116, 0.123 and 0.130 mol/L. They all
are straight lines, whose slopes lead to estimate of the corresponding K1,1=3.19×10-24, 1.30×10-23 and
3.23×10-23 m3/s, and then, based on Eq. 2.3, to the Fuchs stability ratio values of W=3.86×106, 9.51×105,
and 3.82×105, respectively.
14
2.3 Results and discussion
2.3.1 Particle size and PAA brush conformation behavior
To verify the presence of the PAA brushes on the particles of P2 and P3 latexes with 1% and 2%
AA, respectively, we have measured the Rh value of the particles by DLS as a function of pH at a
controlled ionic strength of 25 mM, and the results are shown in Fig. 2.2. It is seen that for P3 latex the
Rh value increases asymptotically as pH increases and the difference between the smallest and largest
Rh values is about 8 nm. These results allow us to conclude that the PAA brushes are indeed attached
to the particle surface of P3 latex. In the case of P2 latex, the Rh value increases with pH only marginally,
about 2 nm, indicating much shorter PAA brushes. Since the thickness of the brushes of P2 latex with
1% AA is only 1/4 of P3 latex with 2% AA, the brush density should be larger for P2 than for P3.
Figure 2.2 The hydrodynamic radius, Rh, of the primary particles as a function of pH, determined by
DLS, for P2 and P3 latexes after cleaning.
To further confirm the presence of the PAA brushes on the particle surface, we have measured the
form factor, P(q) by SLS, at different pH values, and the results are shown in Figs 2.3a and 2.3b for P2
15
and P3 latexes, respectively. The difference in the P(q) curves obtained at different pH is very small; it
follows that the radius of gyration (Rg) of the particles at different pH is also practically the same. Note
that for P3, the Rg value does increase about 1 nm with pH in the given pH range, but this is certainly
incomparable with the Rh variations in Fig. 2.2. Therefore, the almost constant Rg and the significant
increase in Rh with pH together indicate that the growth in Rh results only from a layer of polymer on
the particle surface, instead of swelling of the whole particles. This conclusion arises because from the
definition of Rg: 2 2g i iR M r m , where M is the total mass, im the individual mass and ir is the
distance of im from the center, if the whole particle swells as the Rh variation indicates, due to increase
in ri, Rg must vary proportionally with Rh. Instead, since the mass of the particle is dominated by the
P(BA-MMA), the variation in the inertia by the 1 or 2% mass of PAA is really negligible..
Results of zeta-potential measurements showed no clear trend in the pH range, 5 to 9, where
ionization and conformation variations of the PAA brushes occur. This is due to the displacement of
the slipping plane induced by the transition from the collapsed to stretched state of the brushes [50],
making the zeta-potential measurements far from being a good indication of the surface potential.
Further characterization of the PAA brushes of P3 latex by DLS has been carried out by measuring
the Rh value of the particles as a function of NaCl concentration at each fixed value of pH in the range,
2.2 to 11.0. These experiments show typical behaviors of the annealing PE brushes. As shown in Fig.
2.4a, for pH[2.2,6.0], since most of the AA units are protonated and the brushes are in the collapsed
state, the Rh value is insensitive to the salt concentration. At pH 9.1, the initial slight increase in Rh
(CNaCl<100 mM) could indicate a transition from Osmotic Annealing brush regime (OsmA) to Salted-
brush regime (Sd) (CNaCl>100 mM). In the OsmA regime, AA ionization in the brush interior is
suppressed but increases as the salt concentration increases.
16
Figure 2.3 The form factor of the primary particles, P(q) at various pH values, determined by
SLS, for (a) P2 and (b) P3 latexes after cleaning.
The increased osmotic pressure due to the increased counterion concentration in the brush causes
the brush swelling. This might explain the slight Rh increase in Fig. 2.4a. It must be noted that although
the observed Rh increase is rather small, it is larger than the experimental error. Further increase in CNaCl
leads to the Sd regime, where the behavior of the annealing PE brushes becomes similar to that of
quenched (strong) PE brushes. In this case, the brush thickness (i.e., Rh) decreases with the salt
17
concentration. At pH 11.0 in Fig. 2.4a, due to the higher level of AA ionization, the Sd regime is reached
at a much lower CNaCl value, and we can observe only the decreasing part, which merges with the same
trend of the case at pH 9.1. When the data in Fig. 2.4a are plotted in the form of brush thickness, in
Fig. 2.4b, the slope in the Sd regime is about -1/3, in good agreement with theoretical and experimental
results in the literature [41, 51].
It must be noted that we have also measured Rh at extremely low pH (=1.5) for P3 latex, and we
have reported only one Rh value in Fig. 2.4a at CNaCl=10 mM. Further increase in CNaCl leads to
aggregation of the particles. The only reported Rh value drops below the plateau value observed at
intermediate pH values. This arises mostly due to the fact that apart from the PAA brushes, sulfate
groups ( 3OSO ) coming from the persulfate initiator are also present, which can be protonated at such
a low pH value. This leads to further decrease in the inter- and intra-chain electrostatic repulsion, as
well as the osmotic pressure of the counterions present in the brush, consequently, to further collapse
of the brushes.
For P2 latex, the low pH-responsiveness of its Rh value does not allow to perform the same type
of analysis. It is therefore unsafe to speculate about the difference in the absolute thickness or surface
grafting density of the PE brushes between P2 and P3. Moreover, since the particles were synthetized
through emulsion copolymerization, the acrylic acid content along the chains is unknown and can differ
from P2 to P3.
18
Figure 2.4 Mean hydrodynamic radius (Rh) (a) and the estimated brush thickness () in the salted-
brush regime (b) as a function of NaCl concentration at various pH values for P3 latex. The broken
line represents the theoretically predicted relationship, 1/3C .[41]
19
2.3.2 Stability of P1 Latex
Among the three latexes, P1 is the simplest one possessing only fixed sulfate charges on the particle
surface, without the PAA brushes. The corresponding measured W value at low pH are shown in Fig.
2.5a as a function of the concentration of both NaCl and H2SO4, while the corresponding measured pH
values are shown in Fig. 2.5b.
Figure 2.5 Fuchs stability ratio W (a) and the corresponding pH (b) for P1 latex as a function of the
electrolyte concentration.
As expected, the W value decreases as the NaCl or H2SO4 concentration increases. In addition, as
20
shown in the figure, the electrolyte concentration value where W reaches the value of about one, i.e.,
the CCC (critical coagulant concentration) for fast aggregation, is much smaller for H2SO4 (~0.15
mol/L) than for NaCl (~0.50 mol/L). Note that this difference is not due to the fact that H2SO4 is a
bivalent salt. At such low pH, H2SO4 is only singly deprotonated, and acts as a symmetrical monovalent
electrolyte. To understand the difference in the W values measured above in the presence of NaCl and
H2SO4, one has instead to consider the difference in the association of Na+ and H+ to the surface fixed
charge groups, 3OSO . In particular, the association of H+ with sulfate groups is stronger than that of
Na+ [52]. This leads to a much stronger reduction of the surface charge density in the presence of H+
than in the presence of equal amount of Na+, thus making H2SO4 much more effective in destabilizing
the P1 latex than NaCl.
The above results clearly demonstrate the importance in accounting for the counterion association
with the surface ionizable groups when studying colloidal stability, as also demonstrated in other related
systems[53-55].
2.3.3 Contributions of PAA Brushes to Colloidal Stability
The particles of P2 and P3 latexes, apart from the fixed sulfate charge groups, contain 1% and 2% AA
monomers, respectively, which, as discussed in the previous section, are in the form of PAA brushes
on the particle surface. Clearly, their carboxylic groups, when deprotonated, i.e., COO , contribute
to the surface charges. Fig. 2.6a compares the W values for latexes, P1, P2 and P3, as a function of the
NaCl concentration at pH[1.8-9.0].
21
Figure 2.6 Fuchs stability ratio W (a) for P1 (crossed symbols), P2 (hollow symbols) and P3 (full
symbols) latexes as a function of the NaCl concentration, at pH as reported in (b).
It can be observed that the stability of P1 is weakly dependent on pH. In fact raising the latter to pH=9.1
only makes the P1 latex as stable as the protonated P2 and P3. This weak dependency is clearly reflects
the low pKa of the sulfate groups, which ensures that at pH>3.0, all the chargeable groups of P1 are
ionized and raising the pH more does not affect the particle charge. It is also seen that P1 is slightly less
stable than P2 and P3 at low pH (1.9-2.5).The stabilizing effect of the acrylic acid in this low pH range
can be explained by the fact that the latter is less hydrophobic than MMA or BA, even in its protonated
22
state. Therefore, in the case of P2 and P3, the overlap of their hydrophilic surfaces upon contact leads
to a repulsive interaction. This effect can easily be understood in the scope of polymer solution theories.
It must be noted that the W values for P2 and P3 are basically identical, indicating that increasing AA
(thus carboxylic groups) from 1% to 2% does not affect the colloidal stability in these conditions. The
stabilizing effect of protonated acrylic acid can be even more clearly observed in Fig. 2.7, where the
presence of pAA on the particles’ surface increases shifts the stability curves towards higher electrolyte
concentrations. P1 is slightly less stable than P2 and P3 under acidic conditions.
The situation however changes at higher pH values. The W values for P2 and P3 measured at
various pH values are shown in Figs 2.6a as a function of the concentration of NaCl. At pH around 5.5,
the W curves in Fig. 2.6a for both P2 and P3 move towards larger NaCl concentrations, indicating that
the stability of the latexes increases with respect to that at pH<3. It is therefore evident that now a
significant amount of carboxylic groups have been deprotonated and contribute to the colloidal stability.
In the case of P3 latex, the W values shift much more towards larger NaCl concentrations than those in
the case of P2 latex. This is understandable when one considers that P3 carries 2% AA while P2 only
1%. In this range of pH, located close to the pKa of acrylic acid and its polymer, it is expected that
small differences in the composition of the brushes can lead to a different repulsive effect due for
instance to change in effective pKa. Note that presence of the PE brushes on the particle surface may
contribute to the colloidal stability by steric interactions [45], this point will be covered later.
23
Figure 2.7 Fuchs stability ratio W for P2 (hollow symbols) and P3 (full symbols) latexes as a function
of H2SO4 concentration (a) and the corresponding pH (b).
Finally, as shown again in Fig. 2.6a, at pH~8, i.e., under alkaline conditions, although the W values
for both P2 and P3 continue to increase with respect to the case at pH=5.5, the difference between P2
and P3 is insignificant. This result is somewhat surprising if one considers that at pH=8 and such high
salt concentrations, which ensure an ionization state in the brush close to that in the bulk [41], most of
the carboxylic groups should be deprotonated, and the surface charge should be larger for particles with
24
2% AA than with 1% AA. On the other hand, it is well known from various modeling and simulations
[40, 41] that for the weak PE brushes like PAA, the concentration of the mobile counterions in the
brush-occupied region increases towards the particle surface, because of the strong electrostatic
attraction between the mobile counterions and the opposite charges on the brushes. The consequence is
that the effective charge of the groups (i.e., the AA groups in the present case) is substantially lower in
the inner than in the outer region of the brush zone. In fact, scaling-type models [40, 42] proposed that
the counterion concentration in the inner region is approximately equal to that of the charged groups on
the brushes, leading to the so-called local electroneutrality approximation (LEA).
Steric interactions, which are directly related to the brush thickness and the free energy of mixing
of the polymer chains [56], are expected to play an important role in the stabilization of these lattices.
In fact, at such high ionic strength (Debye length κ-1≈3Å), the electrostatic repulsion arising before the
brushes enter in contact is not likely to stabilize the particles, as noted by Fritz et al. on similar systems
[57]. From Fig. 2.4a, it can be seen that at all the pH and salt concentrations used for aggregation, the
Rh value (thus, the thickness of the brushes) is very similar, not to say identical. This implies that the
only parameter affecting the steric repulsion which changes with the salt concentration is the mixing
free energy of the brushes’ polymer chains. This effect has been investigated in the study of semidilute
polyelectrolyte solutions. It has been determined that it is necessary to account for the monomer-
monomer electrostatic repulsion to explain critical phenomena in polyelectrolyte chains structure[58].
Taking this repulsion into account gives rise to an effective Flory interaction parameter, which depends
on the salt concentration and the ionization state of the polyelectrolyte. This point explains why the
colloidal stabilization, which comes both from the hydrophilic nature of acrylic acid and its charge,
making the polymer chains repel each other when two brushes overlap is affected by the salt
concentration.
Accordingly, since electroneutrality prevails in the inner region, the particle-particle (or brush-
brush) interaction is mainly determined by the AA groups present in the outer region of the brushes.
Now, the difference between the P2 and P3 latexes is the length of the brushes, much longer for the
second, and if the inner part is electroneutral, then the effective repulsion is determined only by the AA
25
groups in the outer region, which are very similar for the two latexes. This would justify the equal
measured W values for the two latexes shown in Fig. 2.6a. Therefore, our above results can be
considered as an experimental confirmation about non-uniform ionization of weak PE brushes, in
agreement with the local electroneutrality approximation.
2.4 Concluding remarks
Poly-acrylic acid (PAA) chains are pH-sensitive polyelectrolytes, whose charging, hydrophobicity,
polarity and conformation change with pH. In this work, we study experimentally the effect of pH on
the stability of polymer colloids with surface-grafted PAA chains (brushes). In particular, we have
prepared three acrylate copolymer colloids using the same recipe, but with 0%, 1% and 2%wt PAA
brushes, respectively. The length of the brushes under the stretched state, measured from the dynamic
light scattering, is about 2 and 13 nm for 1% and 2% PAA, respectively. In the case of 0% PAA, the
particles are stabilized only by the sulfate charges coming from the initiator, while in the latter two
cases they are stabilized by both sulfate and carboxylic charges.
The colloidal stability of each latex has been characterized by measuring the corresponding Fuchs
stability ratio (W) as a function of electrolyte concentration and pH. It is found that at low pH values
(<3), the W values of the particles with 1% and 2% PAA brushes are slightly more stable than the 0%
AA latex. This stabilization certainly comes from the hydrophilic nature of the acrylic acid.
At intermediate pH (~5), the PAA brushes are partially deprotonated, and their contribution to the
colloidal stability is substantial. Moreover, to reach the same W value, one has to use almost a doubled
NaCl concentration in the case of 2% PAA with respect to the case of 1% PAA. This means that at the
intermediate pH the ionization of the carboxylic groups (thus the particle charge) is directly linked to
the total AA groups on the surface. Accordingly, higher particle charge leads to higher stability for P3
than for P2 latex.
Under alkaline conditions (pH>8), one would expect all the carboxylic groups on the brushes to
be fully ionized and participating to stability. On the contrary, we found that the W values are basically
26
the same for particles with 1% PAA and 2% PAA, which implies that the contribution of the ionized
AA in the two cases is practically the same. Such an experimental result confirms the local
electroneutrality approximation (LEA) [40, 42] in the inner region of long brushes, which leads to the
conclusion that only the ionized AA groups in the outer region of long brushes contribute to the particle
stability. Accordingly, the P2 and P3 latexes that differ in the length of the brushes exhibit the same
colloidal stability.
27
28
29
Chapter 3
Investigation of the Steric Stabilization of dispersions with pH-sensitive
poly(acrylic acid) brushes
3.1 Introduction
As seen in the previous chapter, the use of polyelectrolyte brushes to stabilize colloidal systems is an
efficient solution to avoid particle aggregation. The case of weak polyelectrolytes is particularly
interesting, because of the pH-responsiveness of the stabilization [27, 30, 41].
The stabilizing effect arising from polyelectrolyte brushes comes from two main contributions: on one
hand, the charged character of the polyelectrolyte brush allows for an increase in surface charge density,
thus increasing the interparticle repulsion [27, 45, 57]. On the other hand, the hydrophilicity of the
polymer brush gives rise to so-called steric forces. “Steric” stabilization is a well-understood
mechanism. It arises from the free energy of mixing of two lyophilic polymer layers upon overlap, and
contains a purely steric and an elastic contribution. The elastic contribution arises from the physical
compression of the polymer chains. A typical steric interaction potential can be found in the work of
Willenbacher and coworkers [57, 59]:
0 no overlap
24 12 overlap, no brush compression2 2
1
4
USterick TB
U a HSteric LPk T
BU aSteric
k TB
1 12 2 ln compression2 2 4
1
H HL
P L L
(3.1)
30
0 no compression
2 2
3 / ln
2
Elastic
ElasticP
W
U
k TB
U aL
Pk T MB
H H H L
L L
23 /
6 ln 3 1 compression2
H L H
L
(3.2)
Where (3.1) represents the purely steric contribution. (3.2) is the elastic contribution. a is the size of a
monomer unit, ν1 is the size of a solvent molecule (in the present case water), φP is the volume fraction
of the polymer chains, χ is the Flory parameter, quantifying the excess free energy of mixing of polymer
with the water, MW is the molecular weight of the chains, H is the height of the brush, and L is the
surface-to-surface separation.
The industrial synthesis of colloidal polymeric products gives rise to a number of constraints and
objectives. Polymeric products are produced in large quantities, and thus must be efficiently produced,
at high volume fractions, in the shortest reaction time possible. The high volume fraction constraint can
give rise to stability problems. It is therefore commonly decided to include additives in the emulsion
polymerization recipe, in order to modify and improve the stability of the produced particles. A typical
additive for acrylic and acrylic/styrenic adhesives and film-forming polymers are acrylic acid or
methacrylic acid [60, 61]. Because of the time constraint, and in order to obtain a stabilizing effect from
the beginning of the reaction on, the additive is mixed with the monomer mixture and fed during the
reaction time together, using a starved-feed strategy.
Such a synthesis strategy is very efficient a producing stable particles, but also gives rise to a significant
uncertainty about the actual morphology of the particles. In fact, acrylic acid being water-soluble, it will
be initiated by the water-soluble radical initiator before the more hydrophobic co-monomers. The living
chains will then enter the particle phase after having added a few hydrophobic monomeric units,
rendering them water-insoluble [62]. Once the living chain has entered the particle phase, the polymer
chain grows, mostly capturing hydrophobic monomers located in the particles. Therefore, even though
starved emulsion copolymerization is well-known to allow good chain composition control, in this case,
31
the chain formed rather exhibit a kind of “block-like” copolymer structure, effectively synthesizing a
surface-active polymer. The hydrophilic acrylic acid rich extremities of the polymer chains tend to stay
on the surface of the nanoparticles. Therefore, a kind a polyelectrolyte brush is formed, as presented in
the previous chapter. The challenge of this type of particles is to understand the factors influencing the
stabilization of the particles. Namely, what is the link between the recipe (amounts of additive, initiator,
size of the particles) and the colloidal stability of the obtained particles.
The characterization of such nanoparticles is a serious issue. Some analytical techniques are suited to
characterize the size (light scattering) and the charge density (titrations) of the particles. However, no
analytical technique allows to measure directly the radial composition function within nanoparticles. It
is hence rather complicated to make the link between the recipe, the morphology, and the colloidal
stability of these products. We here propose an original method consisting of using the experimentally
obtained colloidal stability in order to gather knowledge about the particles’ structure.
The interaction potential of particles exhibiting electro-steric stabilization contains the coulombic
contribution, which is rather readily modelled and only depends on the surface charge density and the
ionic strength of the bulk, and the steric contribution, which mainly depends on the conformation of the
polymer brush. It was thus decided to investigate the steric contribution to the colloidal stability of
industrially produced poly(MMA-co-BA-co-AA) particles, synthesized with different amounts of
ammonium persulfate as initiator. In order to do so, aggregation experiments were carried out at low
volume fractions, in the presence of a high NaCl concentration, ensuring the absence of interparticle
electrostatic interactions, and varying the pH of the dispersions. The pH affects the protonation state of
the PE chains, thus modifying their effective Flory parameter, governing the steric interactions. The
aggregation kinetics was followed in situ by dynamic light scattering. These kinetic curves were then
fitted using a population balance equation model in order to obtain values of the Fuchs stability
parameter as a function of the pH. The characterization of the latexes using light scattering and colloidal
titrations allowed to gain some quantitative information about the solvent-accessible polymer. Then, a
stability model containing the structural parameters was developed and the unknown parameters were
fitted in order to gain some more structural understanding of the particles.
32
3.2 Materials and Methods
3.2.1 The latexes
All the latexes used in this study were synthesized ad-hoc by BASF SE (Ludwigshafen, Germany).
They were produced by starved emulsion copolymerization of 49% w/w Methyl Methacrylate 49% w/w
Butyl Acrylate, and 2% w/w acrylic acid. The amount of ionic initiator (ammonium persulfate) was
varied between 0.0% w/w (in which case a H2O2/ascorbic acid initiation was used) and 1.2% w/w. All
the latexes were dialyzed extensively against Millipore water until their surface tensions were above 70
mN/m and their conductivities lower than 10 µS/cm, ensuring a complete removal of soluble oligomers
and surfactant used in the polymerization.
Latex name Composition (weight ratio)a
L1 BA (49) : MMA (49) : AA (2) + H2O2/Asc
L2 BA (49) : MMA (49) : AA (2) + 0.2 % APS
L3 BA (49) : MMA (49) : AA (2) + 0.8 % APS
L4 BA (49) : MMA (49) : AA (2) + 1.2 % APS
a: BA=Butyl Acrylate, MMA=Methyl Metchacrylate, AA=Acrylic Acid,
APS=Ammonium Persulfate, Asc=Ascorbic Acid
3.2.2 Characterization of the particles
The latexes were characterized by light scattering and coulometric titrations. The size of the particles
was obtained by static light scattering, using a BI-200SM goniometer system (Brookhaven Instruments,
USA), equipped with a solid-state laser, Ventus LP532 (Laser Quantum, U.K.) of wavelength 0=532
nm, as the light source, at the scattering angle range 15°-150. The particle size was obtained using the
RDG theory.
33
In order to gain as much structural information as possible, the swelling of the polymer brush as a
function of the pH was investigated using the same method as reported in the previous chapter. This
allows for an estimation of the brush thickness at high salt concentrations.
In order to determine the concentration of sulfate groups coming from the persulfate radical initiator
attached to the particles, as well as the amount of solvent-available acrylic acid, titrations were
performed. A suspension of φ=1% particles was prepared, with 10 mM NaCl to provide an ionic
background. The suspension was stripped with wet N2 during 5 min, and thereafter kept under mild wet
N2 flux in the overhead to prevent gas from entering the system. As the particles were extensively ion-
exchanged, their counterions are H+. Hence the concentration of protons in the system is equal to the
concentration of sulfate groups and can be titrated with a base. The titrations were performed with a 10
mM NaOH standard solution (Fixanal®, Fluka) using a Titrino (Metrohm, Switzerland) controlled by a
computer. Before each base addition, the stability of conductivity and pH were verified not to vary more
than 0.02 pH unit and 0.1 µS/cm per minute. This point was shown to be in order to obtain reproducible
and reliable results. This is due to the dynamics of ion exchange when starting with collapsed brushes
[51]. The conductivity and pH of the suspensions were recorded, and the equivalence points were
obtained by the clear kinks in the conductivity curve.
3.2.3 Aggregation Experiments
The dialyzed latexes were diluted to a volume fraction φ = 5·10-4 in order to form stock solutions. Then,
in the vials used for the measurement of the kinetics, the suited amounts of water, H2SO4 and NaCl 4M
were added to reach the desired pH, 1M NaCl and φ = 10-5 after dilution. The stock latex solution was
then added in the prepared salt solution, argon was flushed over the top of the solution, and the vial was
placed in the DLS equipment. The preparation procedure was timed and always was equal to 1 min.
The evolution of the average hydrodynamic radius with time was then followed using a BI-200SM
goniometer system (Brookhaven Instruments, USA), equipped with a solid-state laser, Ventus LP532
(Laser Quantum, U.K.) of wavelength 0=532 nm, as the light source, at the scattering angle 90. The
solutions’ pH were measured after the aggregation using a freshly calibrated pH glass electrode. Some
34
tests were carried and ensured the pH was stable over the aggregation time, varying less than 0.02 pH
units.
3.2.4 Simulation of the aggregation curves
Once the aggregation kinetics were obtained by dynamic light scattering, the Fuchs stability ratio was
obtained by using a PBE approach. The kinetic kernel was already described by Lattuada et al [17]. In
the present case, the Fuchs stability ratio, the fractal dimension and the lambda parameter were fitted
using a least-square algorithm.
3.2.5 Modeling of the interaction potential
In order to get more understanding of the morphology of the particles, the W=f(pH) function obtained
experimentally was compared to a steric interaction potential equation set. From the interaction
potential, the Fuchs stability ratio can be obtained by using eq. (3.3) [53]
22
exp /2 BU l k T
W dlGl
(3.3)
Where U is the total interaction potential, G a hydrodynamic function, and 2L a
la
a dimensionless
center-to-center separation.
Using eqs (3.1) and (3.2), it is possible to devise a steric interaction stabilization model. As observed, a
significant number of parameters are relevant for the application of these works. As discussed
previously, the amount of knowledge on the particles is rather limited. For instance, even if the swelling
of the brushes can be measured experimentally, this does not give access to the thickness of the
collapsed brush, state in which aggregation actually occurs. In the same line, the grafting density or
polymer volume fraction are unknown parameters. Some assumptions were therefore made:
Mw is assumed constant amongst the samples and taken as the molecular weight of a 15 unit
Acrylic Acid chain. This corresponds the length of a pAA chain of about 10 nm, which is the
typical order of magnitude in swelling.
φP is assumed constant amongst the samples and is set to a guessed value of 0.1
35
ρP = 0.2 g/cm3 is assumed constant amongst the samples and chosen such as L = 1 nm for probe
L3 in the fully collapsed state. In fact, knowing the total density of acrylic acid on the surface
from titrations, it is possible to compute the density of the brush from its thickness.
L is computed from the AA – titration results and the chosen ρP. for all samples other than L3.
The Flory parameter χ depends on pH because of the change in hydrophilicity of poly-acrylic
acid with pH. It is assumed to follow a sigmoidal curve between two extreme values:
1 2
2
01 /p
pH pH
, where χ1 and χ2 are the extreme values of the Floy parameter, and
pH0 is the transition pH value, at which half of the acrylic acid should be protonated. It plays
here the role of an “effective pKa”. All of these values are fitted to the experimental results,
using a least-square method.
3.3 Results and Discussion
3.3.1 Characterization of the particles
3.3.1.1 Particle Size and pH-responsiveness of pAA-brush
The swelling behavior of the studied latexes was studied under the same conditions as in the previous
chapter. The use of DLS allows to gain information about the diffusion properties of the particles. The
measurement of the particles by static light scattering allows to reach to the radial mass distribution
function of the particles. This has been shown previously not to be affected by the swelling of the PE
brushes [61]. The particle size obtained by static light scattering, using the RDG theory is reported in
Table 3.2.
36
Table 3.2: Characterization of the particles
Latex name APS [%w/w] Rp [nm]
L1 0.0 92.1
L2 0.2 90.0
L3 0.8 87.1
L4 1.2 93.5
Using the particle size obtained by SLS as a “shell-free” size, it is possible to devise an approximate
“brush thickness” as H PR R . The shell “thicknesses” as a function of and NaCl concentration are
reported in Fig. 3.1
Figure 3.1: Brush Thicknesses for samples L2, L3 and L4 at pH=10.08
It can be observed that the apparent shell thicknesses are very similar at low salt concentrations,
conditions under which the inter-chain electrostatic repulsion swells the PE brush [41]. The change in
swelling with increasing NaCl concentration however depends on the amount of sulfate used in the
synthesis. Namely, more sulfate in the recipe reflects in a more swollen brush at a defined NaCl
10 10000
2
4
6
8
10
12
14
16
1.2 %
0.2 %
(n
m)
NaCl Concentration (mM)
0.8 %
37
concentration. This indicates already that changing the synthesis recipe affects significantly the
morphology and physics of the brush. It is here impossible to speculate about the reason for this effect.
It could come from a higher AA fraction in the brush or sulfate content at the polymer chain ends,
leading to an increased intra-chain repulsion, or more simply to changes in the grafting density of the
brushes. The L1 latex was not investigated in swelling behavior, because it was aggregating at lower
pH ranges.
3.3.1.2 Surface Charges
The surface charges of the diverse latexes were measured by colloidal titrations. This allows to quantify
the surface density of sulfate groups and acrylic acid groups on the surfaces. These results are reported
in Fig. 3.2a) and b), respectively.
Figure 3.2: Fixed sulfate (a) and Carboxy (b) groups on L1, L2, L3 and L4 latexes
0
1
2
3
4
5
6
7
8
0.0 0.5 1.0 1.50
10
20
30
40
50
60
70
80
90
100
b)
Sul
fate
con
cent
rati
on (
mm
ol/K
gP) a)
Car
boxy
l gro
ups
conc
entr
atio
n (m
mol
/KgP
)
% APS
38
One can notice that increasing the APS amount in the synthesis recipe increases significantly the amount
of accessible sulfate groups on the surface of the particles. At the same time, the amount of accessible
acrylic acid decreases slightly with increasing APS amount when APS is used, and is found to be lower
when a redox initiation system is used instead of persulfate.
3.3.2 Aggregation experiments and the Fuchs stability ratio
A typical example of aggregation curve for sample L2 is reported in Fig. 3.3:
Figure 3.3: Aggregation behavior for L2 under different pH conditions. The solid line is the
corresponding PBE simulation result.
It can be observed that at the lower pH range (pH<6.0), the particles follow a fast aggregation regime,
and the aggregation rate does not depend significantly on the pH. This corresponds to the DLCA regime,
where no repulsion is present. Going to higher pH values, it can then be observed that within a limited
pH difference (from pH = 6.12 to pH = 6.17), the aggregation rate decreases significantly. Going higher
with the pH value decreases the aggregation rate even further, and reaches a new lower plateau value.
This fact indicates that a limited extent of repulsion is present, but it does not increase further higher
than the value reached at pH = 7.61. These observations can be easily explained: at low pH, the acrylic
acid is protonated, which of course limits the surface charge to the small amount of sulfate groups
coming from the persulfate initiator. The protonation of acrylic acid also increases its Flory parameter,
or namely makes it less hydrophilic. Less energy is thus needed to overlap two layers of swollen pAA
0
100
200
300
400
500
600
700
800
900
1000
0 1000 2000 3000 4000 5000 6000
Dia
met
er /n
m
Time /s
5.726.056.126.176.427.618.55
39
in water than at high pH. The steric interaction is therefore rather limited. Raising the pH close to the
effective pKa of the pAA brush, some of the AA units will be deprotonated, allowing it to get more
hydrophilic. At even higher pH values, the whole pAA brush is deprotonated, and further increasing the
pH does not affect it anymore. One must be aware that even when the brush is deprotonated, it does not
get swollen, unlike the conditions under which the brush thickness is measured. This is simply due to
the screening of the electrostatic repulsion within the brush by the high NaCl concentration.
Using the PBE model, the kinetics can be used to obtain the Fuchs stability parameter. One example of
fitting is found in Fig. 3.3
Figure 3.4: Evolution of the Fuchs stability ratio W with changing pH at [NaCl] = 1M, as a function
of the pH and APS content.
Summarizing the results in Fig. 3.4, which shows the evolution of the Fuchs stability ratio with pH, a
few trends can be observed. Changing the amount of persulfate initiator in the recipe changes
significantly the pH at which the particles start to be stabilized. It is very clear that increasing the amount
of persulfate gives rise to more pH-resistant particles. Where the sample P2, containing 0.2% APS starts
to get stabilized around pH=5.7, the sample P4, containing 1.2% APS starts to get stabilized around
pH=4.8.
4.0 4.5 5.0 5.5 6.01
10
100
1000
L1 L2 L3 L4
W
pH
40
3.3.3 Steric model
Using the Fuchs stability ratios obtained in the previous section, the parameters needed by the steric
model in order to reproduce the values obtained can be fitted. The first step is to obtain the values of
the Flory parameter for each sample as a function of the pH. These results are presented in Fig. 3.5 .
Figure 3.5 : Flory parameter χ as a function of pH for L1, L2, L3, L4, as fitted using the Steric
stability model.
Here, it can first of all be observed that the value range for the Flory parameter χ is rather unexpectedly
broad. In fact, the Flory parameter is a feature of the polymer composition and is not expected to vary
wildly with a simple change of initiator amount. It must also be recalled that a change of 0.1 in the Flory
parameter corresponds more to a total change in polymer nature than a conformation or a fine change
in the morphology of the chains. This fact hints strongly towards two possible explanations;
remembering that the brushes present on the surface of the particles are not pure poly(acrylic acid), but
rather an AA-rich layer, it could be that the composition of the brush is influenced by the radical
concentration during the polymerization process. Another likely explanation is that, due to the limited
amount of knowledge available about the particle surface morphology, some of the assumptions made
are not realistic. In fact, it would make sense that the polymer’s volume fraction, or molecular weight,
4.5 5.0 5.5 6.0
0.33
0.36
0.39
0.42 L1 L2 L3 L4
pH
41
or density could change by changing the polymerization recipe. This illustrates clearly the limitations
of the modelling approach.
Figure 3.6: Results of the fitting for the Flory parameter at high pH (left) and “effective pKa” (right).
Let us nevertheless investigate the results of assuming that the Flory parameter is a sigmoidal function
of the pH. These results are presented in Figure 3.6. It is observed that the transition pH0 and value of
the Flory parameter at high pH depend significantly on the quantity of persulfate initiator used during
the synthesis. Namely, some clear trends are witnessed: on one hand, the L1 latex seems to be
significantly different from the three other samples. This can be explained by the fact that the whole
initiation process was different. On the other hand, the fitted values of χ1 and pH0 of the three other
latexes seem to evolve on a clear linear trend as a function of APS concentration in the latex. It can be
hypothesized here that the observed linear evolution of these parameters with APS concentration can
be correlated with some changes in the pAA brush morphology. The transition pH corresponds to an
“effective pKa” of the polyelectrolyte brush, which is known to vary both with the length and density
of the brushes [44], it is impossible to conclude whether this change is due to the sulfate presence alone
or to the morphological changes that a modified initiator concentration has on the brushes formed by
emulsion polymerization. Simultaneously, the absolute value of the Flory parameter actually depends
0.0 0.2 0.4 0.6 0.8 1.0 1.20.0
0.2
0.4
0.6
0.8
1.0 1
% APS
4
5
6
7
pH0
42
on the polymer chain composition, which can be locally affected by the APS concentration surrounding
the particles during the synthesis.
3.4 Concluding remarks
In the present chapter, a tentative of quantitative understanding of the stabilization arising from steric
interactions on industrial samples was performed. The aggregation kinetics of a set of polymeric
samples, in which the initiator amount was modified were followed by light scattering at high salt
concentration. A PBE model was used to obtain the Fuchs stability ratio as a function of pH. The Fuchs
stability ratio was then used in order to fit the value of the Flory parameter required in order to
understand how the pH affects the hydrophilicity of poly(acrylic acid) in water. Interestingly, changing
the amount of initiator during the polymerization changes the effective pKa of the pAA brush. This fact
is not trivial to explain, but is probably related to the chain composition change, or a change in the
grafting density within the brush.
The Flory parameter can be accurately described by a sigmoidal function, which corresponds to its
values when fully protonated and fully deprotonated, respectively. The obtained values of Flory
parameters do not compare well with what was measured in the literature using more direct
measurements. The reason for this is probably that the assumptions made in order to simplify the model
are wrong, or that the non-ideal composition of the brushes varies when the amount of initiator used in
the polymerization is changed. Since these parameters cannot be obtained experimentally, no
improvement can be expected. It would be interesting to reproduce the same set of experiments and
apply the same model on sample synthesized in a more controlled way (grafted-on for example).
43
44
45
Chapter 4
Interplay between Aggregation and Coalescence of Polymeric Particles:
Experimental and Modeling Insights
4.1 Introduction
Various industrial processes rely on the handling of colloidal particles, may these be in the form
of suspensions, emulsions or gels. As a matter of fact, colloidal particles applications range from food
to plastics, going through paints, coatings, paper treatment, to more recent niche-fields, such as drug-
delivery.[63]
One of the key features of colloidal particles is their kinetic stability, or the fact that the
dispersed particle phase will sooner or later organize in larger structures (i.e. clusters) and phase-
separate from the continuous phase it is suspended into. This un-avoidable process is regulated by the
interplay of different phenomena occurring, such as aggregation and coalescence. Clearly the
environmental conditions (e.g. temperature, pH, shear rate) and the particles characteristics (e.g. surface
charge, primary particle size), play a key role in determining the rate at which the destabilization process
takes place.[64] Another feature of aggregating colloidal particles is their tendency to form self-similar
(i.e. fractal) structures with fractal dimensions regulated by the conditions in which the destabilization
occurs. The fractal dimension relates the aggregates mass x , with its size R :
fdx R (4.1)
For instance, rather open clusters are formed in fully destabilized systems where the cluster
aggregation is diffusion limited (DLCA), whereas in reaction-limited cluster aggregation (RLCA) more
compact clusters are formed, as not every aggregation event is an effective one due to typically charge-
46
induced particles stability.[65] Note that when aggregation and coalescence occur simultaneously, the
fractal dimension of the formed clusters is typically larger as compared to the non-coalescing cases.
This is due to the fact that coalescence leads to an interpenetration of the particles constituting the
cluster, therefore compacting the cluster itself and increasing its fractal dimension.
After the pioneering work by Ulrich and Subramanian,[66] highlighting how the simultaneous
aggregation and coalescence was indeed shaping the soot cluster growth in flames, several authors dealt
with this topic. Koch and Friedlander[67] introduced a simple, yet insightful and elegant deterministic
model extending Smoluchowski’s[12] approach to account also for coalescence. After that, Xiong and
Pratsinis solved a 2-D population balance equation (PBE) based model accounting for the time-
evolution of aerosol mass and surface area, and successfully compared it to experimental data.[68, 69]
Since then, during the last decades, the picture of aerosol coalescence and aggregation has
become clearer as the characteristic times of metal particles has been quantified and the mechanism of
coalescence unveiled.[70, 71] Moreover, the dynamics of cluster coalescence has been clarified and
expressions describing the time-dependence of the fractal dimension proposed. Another notable
progress in the field of aerosols was the model proposed by Heinson et al., which captures the key
features of the explosive generation of silica nanoparticles.[72] While aerosols have been extensively
investigated in this sense, only few studies are found on polymer particles undergoing aggregation and
coalescence.[73, 74] These works helped clarifying how rubbery (i.e. fully coalescing) particles
organize into clusters,[73] and provided insights on the restructuring of preformed clusters undergoing
coalescence upon temperature increase.[74] In these works particles were typically undergoing full
coalescence 3fd , hence existing light scattering correlations could be used to compare
experimental results (i.e. gyration and hydrodynamic radii) and PBE-based predictions. At the same
time, such correlations cannot be employed for cases where the characteristic times of aggregation and
coalescence are comparable, i.e. when partial coalescence occurs.
In this framework, the aim of this paper is to shed further light on the interplay between
aggregation and coalescence along two main lines, namely i) developing a suitable 1-D PBE based
model accounting for both processes and ii) correlating simulation results with light scattering results,
47
thus extending the existing correlations for rigid clusters to partially coalesced ones. To this end,
polymeric particles with different glass transition temperatures ( gT ) have been prepared and their
aggregation behavior studied in stagnant DLCA conditions. Different temperatures and particles
concentrations have been explored in order to clarify the interplay of the coalescence and aggregation
rate. The developed model, along with the light scattering correlations, was tested against the
experimental data to verify its reliability and potential.
4.2 Materials and methods
4.2.1 Materials
Butyl Acrylate (BA), Methyl methacrylate (MMA), Potassium Peroxydisulfate (KPS) and
Sodium Chloride (NaCl) have been purchased from Sigma-Aldrich, sodium dodecyl sulfate (SDS) by
Apollo Scientific. All chemicals had purities larger than 99% and were used as received. Millipore
(MQ) water stripped for two hours with nitrogen was employed as the continuous phase in the emulsion
polymerization. Ion-exchange resin (Dowex Marathon MR-3 hydrogen and hydroxide form) purchased
from Sigma-Aldrich has been employed as received after the reaction to remove the SDS adsorbed on
the particles surface.
4.2.2 Particles synthesis
The particles were synthesized by starved emulsion polymerization[75] employing the
controlled reaction environment LABMAX©. In particular, the reaction was carried out at 70oC in a 1
L jacketed reactor. The reactor was initially charged with stripped MQ water and SDS and heated up
under nitrogen atmosphere. The initiator (KPS) solution was then added, and the monomer mixture feed
was started. The conversion and particle size were followed by thermogravimetric analysis and dynamic
light scattering (DLS), respectively. The instantaneous conversion always stayed above 99%, ensuring
that starved conditions were achieved. The monomer feed was stopped when the particles reached the
desired size. The reaction mixture was then kept at 70oC during one hour, to ensure complete conversion
48
of the monomer. After the synthesis, the latexes were subjected to several cycles of cleaning using ion
exchange resins until their surface tension was above 71 mN/m (Wilhelmy plate method).
4.2.3 Light Scattering
4.2.3.1 Dynamic Light Scattering
Dynamic light scattering was employed to monitor the particles size throughout the reaction
(data not shown) until the desired particle diameter (200 nm) was reached. All the measurements were
carried out using a BI-200SM goniometer system (Brookhaven Instruments, USA), equipped with a
solid-state laser, Ventus LP532 (Laser Quantum, U.K.) of wavelength 532DLS nm , as the light
source. The temperature was controlled by an external water bath with a precision of 0.1oC. The
measurements were carried out in diluted conditions (occupied volume fraction 51 10 ) and with
10 mM NaCl.
4.2.3.2 Static Light Scattering
Static light scattering was employed to simultaneously follow the particle size (through the
gyration radius gR t ) and the fractal dimension evolution. The employed instrument was a Malvern
Mastersizer 2000, equipped with a laser having 633SALS nm . The structure factor ,S q t was
obtained by dividing the measured scattered light intensity ,I q t by the form factor ,P q t of
the primary particles, obtained by measuring the scattering intensity of the primary particles in non-
aggregating conditions. The radius of gyration gR t has been obtained by fitting the structure factor
in a Guinier plot following the method described in Harshe et al.[23]
49
4.2.4 Differential Scanning Calorimetry
The glass transition temperatures of the samples were determined using a Waters Q200 differential
scanning calorimeter. The samples were heated to 120 oC, cooled down to -50oC and brought back to
120oC. The heating rate was 10oC/min in all cases.
4.3 Model development
4.3.1 Population balance equations
As previously discussed, two internal coordinates are in principle necessary to describe the time
evolution of aggregating-coalescing systems, namely the cluster mass x (i.e. the number of primary
particles in the cluster) and the fractal dimension fd . The corresponding population balance equation
is therefore two dimensional:
3
1 1 1 1 1 1
0 1
3 3
1 2 1 2 1 1 2 2 1 2 1 2
0 0 1 1
, ,, , ( )
, , ( , , , ) , , d d
1( , , , ) , , , , d d d d
2
f
f ff
A
f f f f f
B
f f f f f f
C
f x d tf x d t v x,d
t d
f x d t x x d d f x d t d x
x x d d f x d t f x d t d d x x
(4.2)
1 1 2 2 1 2 1 2( , ) , , ,D D f fx g x x y g x x d d (4.3)
1 1 2 1 2,g x x x x (4.4)
1 2
1 22 1 2 1 2 / /
1 2
ln, , ,
ln fm f fm f
fmf f d d d d
d x xg x x d d
x x
(4.5)
1, 3f f
COAL
v x d dx
(4.6)
01/3 1/3p pCOAL C
p
Rx x x
(4.7)
50
The cluster mass distribution , , tff x d is defined such that , ,f ff x d t dxdd represents the
concentration of clusters consisting of x to x dx primary particles having a fractal dimension
comprised between fd to f fd dd at time t . Note that term A describes the variation in time of the
rate of coalescence , fv x d , and corresponds to a convective or Liouville term.[67] The rate of change
of the fractal dimension is defined in equation (4.6) and depends on the coalescence characteristic time
of the particles (cf. equation (4.7)) as detailed already by Koch and Friedlander.[67] Term B represents
the loss of an x -sized cluster having fractal dimension fd upon aggregation with any other cluster.
Term C instead accounts for the formation of an x -sized cluster with fractal dimension fd starting
from two smaller aggregates which need to satisfy the two constitutive laws (equation (4.4) and
4.5).[76] The integration boundaries are 0; for the cluster mass and 1;3 for the fractal dimension,
accounting for all possible cluster masses and shapes. fmd instead is the fractal dimension “imposed”
by the aggregation regime, i.e. 1.7-1.8 in DLCA and 2.0-2.1 in RLCA conditions. 0, ,p p pR represent
the surface tension, radius and viscosity of a primary particle, respectively. 1 2 1 2, , ,f fx x d d instead
is the aggregation kernel, which will be defined afterwards. All symbols are defined in the Symbol List
in Appendix 8.8.
Despite the solution of multidimensional balances is indeed possible,[77] in this context a
simplification of the balance to 1-D PBEs is desirable, as a fitting with experimental data has to be
performed, requiring the iterative solution of the PBE. To reduce the problem to a 1-D PBE, it is
necessary to multiply the original 2-D PBE (cf. equation (4.2)) with 3
1
fdd and 3
1
f fd dd , while
introducing the following two distributions, along a line already developed in the literature:[78]
3
1
, , ,f fx t f x d t dd (4.8)
3
1
, , ,f f fx t d f x d t dd (4.9)
51
To properly treat the further terms present in the balance (4.2), the following assumption (cf. equation
(4.10)) is made:[78]
, , , ,f D f ff x d t x t d t d x t (4.10)
where
3
1
, 1D f fd d x t (4.11)
The physical meaning of the latter assumption (4.10) is that the clusters of a given size x have the
same fractal dimension ,fd x t . The idea is to obtain two 1-D PBEs, the first one describing the
cluster mass distribution (CMD) in time, while the second one describes their average fractal dimension
evolution in time. As a difference to typically used models, the fractal dimension is time-dependent and
a function of the (average) cluster size.
The resulting 1-D PBEs read:
1 1 1 1 1 1 1 1
0 0
, 1, , , , , ,
2
xd x tx t x x x t dx x x t x t x x x dx
dt
(4.12)
1
1 1 1 1 1 1 1 1
1 1
, 1, 3 ,
1, , , , , , ,
2
fCOAL
x
f
d x tx t d x t
dt x
x t x x x t dx d x t x x x x x t x t dx
(4.13)
where
1/3COAL Cx x (4.14)
1 2 1 21/ , 1/ , 1/ , 1/ ,
1 2 1 2 1 2
2( , )
3f f f fd x t d x t d x t d x tB
c
k Tx x x x x x
(4.15)
,,
,f
x td x t
x t
(4.16)
The Liouville term vanishes in equation (4.12), as this balance accounts solely for the cluster
concentration, disregardful of aggregates shape, whereas it is still present in equation (4.13); further
details are reported in the work by Koch and Friedlander.[67] Note that 1 2,x x is the typical DLCA
52
aggregation kernel, modified in order to account for a time-dependent fractal dimension. Bk is the
Boltzmann constant, T the temperature and c the viscosity of the continuous phase.
Instead of solving the two PBEs (equation (4.12) and (4.13)) to calculate the average fractal
dimension (equation (4.16)), it is possible to differentiate equation (4.16), obtaining an ODE system
describing the time-evolution of the average fractal dimension (derivation details are reported in the
appendix 8.2):
1/3
1, 3
,0 1.75
f fC
f
dd x t d
dt x
d x x
(4.17)
The initial condition selected, ,0 1.75fd x , represents a typical fractal dimension of DLCA
aggregating clusters.[64] It is hence sufficient to solve the PBEs (4.12) coupled with the ODE system
(4.17): this way, one has directly the time evolution of the cluster concentrations while accounting for
a time-dependent fractal dimension. Notably, equation (4.17) is actually analytically solvable and
results in the following expression:
1/3, 3 3 1.75 expf
C
td x t
x
(4.18)
which is very similar to the one derived by Eggersdorfer et al., who studied the fractal dimension time-
evolution of purely coalescing fractal clusters.[79]
To appreciate the features of equation (4.18), parametric simulations with different C values
for differently-sized clusters ( 50,500,5000x ) have been performed (cf. Figure 4.1).
53
Figure 4.1:
Fractal dimension vs time for differently sized clusters (x = 50, 500, 5000) and different C .
A smaller C implies a faster increase in the fractal dimension for a cluster of a given size as can be
seen in Figure 4.1 a), considering 310C s (blue curve) and 410C s (green curve). Comparing
instead the time-evolution of the fd of differently sized-clusters at the same C it is observed how
smaller clusters are able to re-arrange faster towards compact structures, exhibiting larger fd (cf.
Figure 4.1 a) -1c) for 410C s ). At the same time, if the C is large enough as compared to the
process time P considered (e.g. 6 410 300 min 1.8 10C Ps s ) , no coalescence is
observed, no matter how small the cluster size considered (cf. Figure 4. 1a)-1c) for 610C s ).
4.3.2 Calculation of hR t , gR t and S q,t
Once in possession of the time-evolution of the cluster mass distribution, average properties,
such as the hydrodynamic and gyration radii, hR t and gR t , and the average structure factor
54
,S q t have to be evaluated in order to compare them with experimental results obtained from light
scattering. Such average quantities are defined as follows:
2
02
,0
, , ,
, , ,
,
h
h eff
x x t S x q t dx
R tx x t S x q t
dxR x t
(4.19)
2 2
2 0
2
0
, ,
,
g
g
x x t R x t dx
R t
x x t dx
(4.20)
2
0
2
0
, ,
,
S x q t x x dx
S q t
x x dx
(4.21)
While the cluster distribution ,x t is known once the PBEs have been solved (equation (4.12)),
suitable expressions for the hydrodynamic and gyration radii of each cluster, i.e. , ,h effR x t and
,gR x t as well as for the structure factor , ,S x q t are needed. q represents the scattering wave
vector, defined as:
4
sin2
nq
(4.22)
where is the scattering angle, is laser wavelength and n is the refractive index of the continuous
phase. Note that in the present case, clusters may aggregate and coalesce simultaneously and therefore
the usually reported formulas to calculate , ,h effR x t and ,gR x t in the case of rigid spheres cannot
be employed.[80, 81] For the sake of brevity the full derivation of the average quantities gR t ,
hR t and ,S q t , is reported in Appendix 8.3.
55
4.4 Numerical Solution
In the present simplified form, the balance on ,x t would require a significant number of
ordinary differential equations (ODE) to be solved, as the internal coordinate x can go up to roughly
4 510 10 units. In order to reduce the problem size, a discretization method, based on Gaussian basis
functions is employed.[77] The main advantage of the present method is the ease with which it allows
to deal with the convolution integrals. The key idea is to approximate the actual distribution function
with a sum of Gaussian basis functions:
2
1
,G
i i
Ns x x
ii
x t t e
(4.23)
This allows to solve a finite number of ODEs (namely GN ) to obtain the time-dependent coefficients
i t , while the grid positions where the Gaussians are centered ( ix ) are fixed before the integration
start. The parameter is describes the overlapping degree of the Gaussians and is fixed once the
Gaussian centers are defined:[77]
1
1i
i i
sx x
(4.24)
To obtain the discretized balances it is sufficient to plug in the approximations (4.23) in the PBE in
(4.12). Further details are discussed in the Appendix 8.4, whereas the final balance reads:
2 1 1 2
1 1 1
2 W
N N NW W W W
j j jj j jj j j
dC C C C
dt s s s
(4.25)
The vectors and matrixes occurring in equation (4.25) are also defined in Appendix C. The calculations
in the present work have been performed with 90 Gaussians, i.e. 90GN . For numerical stability, the
fixed positions ix were placed on a linear grid (i.e. such as ix i ) up to 15 units, and then spaced
logarithmically up to 43 10 units.
56
4.5 Results and discussions
4.5.1. Particle characterization and experimental conditions
The different nanoparticles used in this study, prepared as described in Section 2, are presented
in Table 4.1, where details about their composition, size, stability and glass transition temperature ( gT )
are found.
Table 4.1: Particle characterization
MMA [% wt] BA [% wt] Diameter [nm] PDI [-] CCC* [M NaCl] gT [oC]
30 70 198 0.027 0.75 -12
50 50 202 0.016 0.65 17
60 40 199 0.017 0.75 36
70 30 198 0.021 0.25 54
*The critical coagulation concentration (CCC) is measured at 25oC
Particles spanning through a quite large range of gT (-12oC to 54 C) were produced. Note that the
reported gT values represent an indication of the temperature interval at which the transition between a
glassy and a rubbery polymer matrix is observed.
The aggregation behavior of the particle systems has been studied in fully destabilized DLCA
conditions, achieved by diluting the latexes in 4 M NaCl water solutions. This specific salt concentration
was chosen as it guaranteed a density match between the continuous phase and the particle phase,
preventing cluster sedimentation to interfere with the experiments. Aggregation experiments were
performed at room temperature in static light scattering (SLS) experiments, whereas in a range between
25-45oC when using dynamic light scattering (DLS). The corresponding viscosities of the continuous
phases (given the 4M NaCl concentration) are reported in Table S2 (Appendix 8.5).[82]
57
4.5.2 Aggregation at room temperature: gR t , hR t and S q,t
Static light scattering (SLS) experiments were performed at 25oC and 4 M of NaCl. Both the
average radius of gyration gR t , and the structure factor ,S q t have been measured in time at
three different occupied volume fractions, namely 5 5 51 10 ,2 10 ,3 10 . In Figure 4.2 a)-c) the
measured gR t vs. time is shown for all concentrations and particle types employed.
Figure 4.2:
gR t for the different particles at 25oC and a) 51 10 , b)
52 10 and c) 53 10 .
Black diamonds: 70% MMA, red squares: 60% MMA, green triangles: 50%MMA, blue circles: 30%
MMA particles, continuous lines: corresponding model predictions. Note that the gR t
predictions for the 70% and 60% MMA overlap. All the parameters values employed to obtain the
model predictions are reported in Table D2 Appendix 8.5.
gR t increases in time for all particle types, although the lower the gT (i.e. the lower the MMA %
and the softer the particles), the smaller is the gR t observed, both in terms of absolute value and
increase rate. This might seem surprising considering that the DLCA characteristic time of (doublet)
aggregation A is composition-independent:
11
31
8C
A DLCApart B partC k TC
(4.26)
a) b) c)
58
As can be seen from equation (4.26), A depends only on the temperature of the system (25oC in this
set of experiments) and the viscosity of the continuous phase C (cf. Table S2, Appendix 8.5) and it is
defined for all systems once the particle number concentration partC is fixed. The smaller gR t
observed for the softer particles (at a given particle concentration) can be rationalized as follows. When
soft particles aggregate into clusters, neighboring particles (in the aggregate) will eventually coalesce
with one another, leading to a more compact cluster with a smaller gR t as compared to equally-
sized aggregates consisting of rigid particles, which are typically more open. This structural difference
affects also the aggregation rate: open clusters have a larger collision radius than compact ones (of the
same mass) and their aggregation probability is therefore larger. Thus, although A is per definition
composition-independent, the softness of the particles in an aggregate indirectly impacts the
aggregation rate by affecting the cluster spatial organization. Deepening whether and to which extent
particles in a cluster undergo coalescence, is therefore key.
A powerful and yet simple way to understand this is based on the comparison of the relevant
characteristic times involved. Besides the already introduced characteristic time of doublet coalescence
C (cf. equation (4.7)) and of doublet aggregation A (cf. equation (4.26)), also the characteristic time
of the entire aggregation process ( P ), defined as the total time for which the system is observed, plays
an important role in this context. Notably, P A in all cases, otherwise no clusters would be formed
if the process time was shorter than the characteristic time of doublet formation. To ease the analysis it
is convenient to introduce the ratio N of the doublet coalescence and aggregation characteristic times:
C
A
N
(4.27)
It is now possible to distinguish three limiting cases:
i) N <<1 - as soon as the primary particles aggregate, they undergo instantaneous coalescence
and the obtained clusters are fully compact all along the aggregation process.
59
ii) 1N - doublet aggregation and coalescence occur at a comparable time scale. Since the
process time P A C the clusters grow while coalescing, hence rather uniformly compact
aggregates are formed. Such uniformity should be granted even though clusters born at different
times exhibit different coalescence extents, as the clustering process is mediated by cluster-
cluster aggregation, which averages out possible local non-homogeneities.
iii) N >>1 - in this latter case, two further sub-cases have to be distinguished:
a) P C - no coalescence occurs as this process is too slow (it lasts even longer than the
full aggregation process); open clusters are formed.
b) P C - coalescence occurs slowly and only after the clusters are formed. The aggregate
“history” or “life” becomes relevant: neighboring particles in a cluster formed at the
beginning of the aggregation process will exhibit a larger degree of coalescence as compared
to particles which only “recently” became neighbors. On average, the clusters will exhibit
(according to the absolute values of P and C ) a rather open structure with coalesced
braches. As compared to case ii) a smaller degree of compactness (i.e a smaller fd ) is
expected to be found, unless P .
Being based on the A and C of doublets, the latter picture is a simplistic one, nevertheless
representing an insightful view into the aggregation-coalescence process.
When fitting the model predictions against the experimental data using as only fitting parameter
C , one obtains the following results: C (i.e. 810C s ) for the 70% and 60% MMA particles,
4300C s for the 50% MMA case and 100C s for the 30% MMA particles. The simulated
gR t are reported in Figure 4.2 a)-c) (continuous lines) for the different particle concentrations; the
good agreement with the experimental data shows the model capability to capture the underlying
physics of the model. Given such C values, it is possible to compare them with
60
150min 9000P s and A (cf. Table 4.2) along the previously described three limiting cases i)-
iii).
Table 4.2 A at 25oC and 45oC, for the different particle systems
A s at 25oC A s at 45oC
1 x 10-5 48 31
2 x 10-5 24 15
3 x 10-5 16 10
It is clear that being C for the 60% and 70% MMA cases, these two types of particles will never
undergo coalescence (cf. case iii a) ) given that C >> A ( N >>1) and C >> P . Note that the
predictions of gR t for these two cases are superimposed and represent a typical DLCA case of
non-coalescing particles (cf. Figure 4.2a)-c)). Such a behavior is consistent with the gT of these particles
systems, which is larger (36oC and 54oC for the 60% and 70% MMA, respectively) than the
experimental one (25oC), thus coalescence is not expected to occur. The 50 % MMA containing
particles instead belong to case iii b) because while C A 1N for any , P C , allowing the
formed clusters to undergo partial coalescence. This observation is supported by the gT of the 50%
MMA particles, which is slightly lower than the experimental one (17oC vs 25oC), thus allowing partial
coalescence to occur. The 30% MMA case on the other hand falls in case ii) as C A 1N ; being
the values very close, a significant extent of coalescence, much larger than the one of the 50% MMA
particles, is expected to occur, especially considering the long process time P A C and the
significant difference in terms of gT vs T (-12oC vs 25oC).
61
To further explore the model potential, the predictions of the hydrodynamic radius hR t
(according to equation (4.19)) of the 60%, 50% and 30% MMA containing particles have been
compared with the corresponding DLS experimental results at 25oC and 51 10 in Figure 4.3.
Figure 4.3: hR t of the 60% MMA particles (red squares), the 50% MMA particles (green
triangles) and the 30% MMA particles (blue circles) and their corresponding simulations (continuous
lines) All the parameters employed to obtain the model predictions are reported in Table S3
(Appendix 8.5).
Note that the C values employed in these simulations were the ones obtained from the fitting of the
gR t data, hence the simulated curves in Figure 4.3 are purely predictive. The reasonable agreement
between simulations and experiments evidences the model reliability and its capability to capturing the
complex physics of the aggregating-coalescing system. When particles are significantly (if not almost
completely) coalescing (30% MMA case), a limited overestimation of the model prediction is observed
for both gR t and hR t . This is probably due to the fact that the equations derived for the
calculation of gR t and hR t have been obtained for the intermediate situation of partially
coalescing particles and not for full coalescing systems.
62
Having explored the model effectiveness in describing size averages, it is now desirable to test
its performance in terms of cluster-structure related quantities and to deepen the interplay between
aggregation and coalescence in determining the clusters spatial organization. As a first step, the average
number of particles constituting the clusters (i.e. the cluster mass), AVEN t , defined as:
0
0
,
,AVE
x x t dx
N t
x t dx
(4.28)
is calculated at 25oC at 5 5 51 10 ,2 10 ,3 10 for the 60%, 50% and 30% MMA containing
particles, employing the previously fitted values of C . The simulation results are reported in Figure
4.4.
Figure 4.4: AVEN t at 25oC for 5 5 51 10 ,2 10 ,3 10 . Red continuous lines: 60% MMA;
Green dashed lines: 50% MMA particles (almost overlapped with the 60% MMA), dotted blue lines:
30% MMA particles. All the model parameters values employed to obtain the model predictions are
reported in Table S3 (Appendix 8.5).
As can be seen from Figure 4.4, the average mass of the clusters is smaller for aggregates consisting of
soft particles. This is due to the decreased reactivity of soft clusters as a result of their higher
compactness and reduced collision radius. Notably though, the observed differences in AVEN (for one
63
set of ) among the different types of particles are relatively small: in the considered time-interval the
largest relative difference between cluster masses is of about 10% (comparing the 30% MMA and the
60% MMA particles). In other words, the difference in reactivity induced by the different softness is
present, but only mildly affects the average cluster mass. From the gR t trends (cf. Figure 4.2) on
the other hand, where the soft 30% MMA particles were showing much smaller radii (by at least 100%)
when compared to the 60% MMA particles, the significant impact of the particle softness on the clusters
morphology can be appreciated. More explicitly, the softness of the aggregating particles strongly
affects the spatial organization of the resulting clusters but only mildly impacts the overall aggregation
kinetics.
Having qualitatively discussed the impact of coalescence on the aggregates structure, it is now
interesting to attempt a more quantitative description. For this reason, the SLS experimental data on the
average structure factor ,S q t (obtained at 25oC, 4 M NaCl and three different occupied volume
fractions, 5 5 51 10 ,2 10 ,3 10 ) have been compared with the corresponding predicted
,S q t (cf. equation (4.21)) at three different time points, namely 60, 90 and 120 minutes. Note that
the model is employed here in “prediction mode”, as the employed C values in the different cases are
those previously fitted against the gR t data set. The results of the comparison are shown in Figure
4.5.
64
Figure 4.5: Experimental (symbols) and predicted (continuous lines) ,S q t at three different times:
60 min (blue squares), 90 min (red triangles), and 120 min (black circles). All the parameters employed
to obtain the model predictions are reported in Table S3 (Appendix 8.5).
From Figure 4.5 it can be seen that the predictions of the model well-describe the experimental
observation in the cases of 60% and 50% MMA containing particles, whereas a poor agreement is found
in the 30 % MMA case. This implies that the model is indeed suitable for describing not only average
sizes, but gives also meaningful insights regarding the structure of clusters undergoing simultaneous
aggregation and coalescence. This holds provided that there is no or only partial coalescence: when
approaching full coalescence the model offers only a qualitative description. As a matter of fact, for the
30% MMA case, the slope of ,S q t vs. q (which represents the fd t of the clusters)[64]
increases in time, suggesting that cluster coalescence is significantly occurring. To further prove this
65
point, the scattered intensity for the 30% MMA clusters has been recalculated assuming fully coalesced
clusters (i.e. spheres) using the Mie theory.[83] The radii of the spheres were calculated as
1/30S pR x R x , while the scattered light intensity from the whole distribution has been computed as:
0
0
, , ,
,
,
x t I x q t dx
I q t
x t dx
(4.29)
Note that ,I q t was then normalized by multiplication with a constant value. In the experimental
conditions considered, the polarization of the light was shown to have no significant effect on the
obtained function (data not shown). In Figure 4.6 the comparison between the experimental and the
simulated ,I q t at 60, 90 and 120 min, is reported for the 30% MMA case for the three
concentrations investigated, corresponding to 5 5 51 10 ,2 10 ,3 10 .
Figure 4.6: Experimental (symbols) and predicted (continuous lines) ,I q t at a) 51 10 , b)
52 10 and c) 53 10 for three different times: 60 min (blue squares), 90 min (red
triangles), and 120 min (black circles) for the 30% MMA particles. All the parameter values
employed to obtain the model predictions are reported in Table S3 (Appendix 8.5).
The reasonable accordance between experimental data and simulations indicates i) that the 30% MMA
particles are significantly coalesced and ii) that such coalescence is not a complete one, as otherwise a
full overlapping would have been observed.
66
The average fractal dimension of the clusters, fd t , was estimated from the experimental
data at 120mint in order to provide a more direct glance at the aggregates spatial organization (cf.
Appendix 8.6, Table S5). Two different ways to calculate the fd t have been used, namely taking
the slope from the double log plot of ,S q t vs q and from 0gR vs I . In some cases it was not
possible to employ the ,S q t data in this sense, as the aggregates under investigation were too
small to get a sufficiently large fractal regime. Notably, apart from the clusters made of 30% MMA
particles, who significantly coalesce exhibiting a 2.3fd t , all other systems have the typical
fd t of non-coalescing clusters in DLCA conditions. For the 60% and 70% MMA particles this is
expected as their gT is much larger than the process temperature and therefore coalescence cannot occur.
On the other hand, the 50% MMA containing particles showed a smaller gR t (as compared to the
60% and 70% cases) and have a gT which is slightly lower than the experimental temperature (17oC vs.
25oC), therefore an effect of coalescence on fd t was expected to be present. This apparent
contradiction can be understood by recalling the different characteristic times for the 50% MMA
particles: 4300C s , 16 48A s (according to the different , cf. Table 4.2) and 9000P s .
While C >> A ( N >>1) and the initially formed clusters are indeed quite open and do not have the
time to coalesce, the total process time is large enough ( P C ) for the aggregates to undergo partial
coalescence, i.e. case iii b). These aggregates are significantly ramified at their “birth” and their
coalescence has to proceed through their branches first, before an extensive change in their structure is
appreciated. Such process is particularly slow and therefore no evident change in the fractal dimension
was appreciated. Interestingly, it turns out that while the fd t is a useful tool to assess and quantify
the extent of coalescence, in some cases measuring also the gR t and hR t is desirable as their
67
time-evolution can be of great help in unravelling structural and spatial information about the
aggregating-coalescing clusters.
4.5.3 Aggregation at higher temperatures
To further test the reliability of the developed model, experiments at higher temperatures (25-
45oC) have been performed with the 60%, 50% and 30% MMA particles. The 70% MMA containing
particles were not considered as their gT is of about 55oC and these particles would therefore remain
“rigid” even at 45oC; higher temperatures were not explored in order to avoid water evaporation which
could bias the experiments. Note that to properly compare the experimental results at different
temperatures, the measured hR t have been plotted against the non-dimensional time, N :
11
8
3B
N part partA
k Ttt C t C
(4.30)
Note that by employing N , the effect of particle concentration and temperature are “filtered out”,
allowing to better appreciate the coalescence effect.
When considering the 30% MMA containing particles, it is worth mentioning that a significant
coalescence was already observed at 25oC, being their gT equal to -12oC. When increasing the
temperature by 10 degrees, the situation was found to be almost unchanged, as shown in Figure S1
(Appendix 8.7), where the hR t of the 30% MMA particles is reported for both 25oC and 35oC
against the non-dimensional time, N .This means that a substantial coalescence (very likely an almost
complete one) occurred in both cases and that for this reason no significant difference in hR t can
be appreciated; in fact in both cases 100C s was employed as fitting parameter.
When studying the 50% MMA containing particles (with a gT of 17oC), a larger coalescence
extent is expected when increasing the temperature above 25oC. Indeed, this is observed when
68
measuring the hR t vs. N at 30oC, 35oC and 45oC: progressively smaller radii and radii increase
are observed (cf. Figure 4.7).
Figure 4.7: hR t vs. time of the 50% MMA particles at 25oC (black squares), 30oC (blue
diamonds), 35oC (green triangles) and 45oC (red circles) along with the corresponding simulations
(continuous lines). All the parameter values employed to obtain the model predictions are reported in
Table S4 (Appendix 8.5).
Increasing the temperature, the softness of the particles increases, causing the clusters to be more
compact and slower aggregating, an effect already discussed in the frame of the composition change
for the gR t and hR t data set at lower temperature. When fitting C against the experimental
hR t data, the following results are obtained: 1100s, 360s, and 210s, for 30oC, 35oC and 45oC,
respectively. The simulation results are reported in Figure 4.7 (continuous lines). Once more the quality
of the prediction is quite good, considering that only one fitting parameter, C , has been used. Given
the optimized values, it is possible to compare them to A (cf. Table 4.2) in the frame of the four limiting
cases identified in paragraph 5.1. At 25oC C A and partial coalescence occurs because of the long
process times, cf. case iii b). When the temperature is raised, C progressively decreases almost
69
approaching a situation where C A . As a result, the coalescence extent is expected to be larger
(indeed the radii are smaller) and the aggregation kinetics slowed down due to the diminished collision
radii. Note that, beyond 35oC the coalescence seems to be almost complete as little difference is
observed between the experimental hR t at 35oC and 45oC. The model slightly overestimates such
predictions of almost fully coalesced systems, as already seen at lower temperatures with the 30% MMA
particles. When plotting the fitted C of the 50% MMA case against 1/T in a semilog plot (cf. Figure
S2, Appendix 8.7), it is possible to appreciate the temperature dependency, already observed in the
literature for both metallic and polymeric particles:[71, 74]
expC
BA
T
(4.31)
A similar analysis (at T = 25, 30, 35, and 45oC) was conducted with the 60% MMA containing
particles but for the sake of brevity it has not been shown. Moreover, as the 60% MMA particles possess
a gT of about 35oC, no real difference between the samples is observed between 25oC and 30oC. Even
beyond this temperatures (i.e. for 35oC and 45oC) only mild differences have been observed
experimentally.
4.6 Conclusions
In the present paper, a deterministic model accounting for the simultaneous aggregation and
coalescence of colloidal particles has been developed. The model is based on 1-D population balance
equations (PBE), whose solution gives access to the particle size distribution as a function of time. The
occurring coalescence is accounted for employing one parameter, the characteristic time of coalescence,
C . Literature correlations,[80] linking the PBE to experimental light-scattering information such as
hR t , gR t and qS t , have been extended in order to account for partial coalescence.
The developed model has then been successfully tested against light scattering data of DLCA
aggregating colloidal particles. In particular, by tuning the available parameter C , it was possible to
70
well-reproduce the observed experimental trends employing polymeric particles exhibiting a broad
range of glass transition temperature values (-12oC - 55oC) in a range of temperatures comprised
between 25oC and 45oC. Limitations of the model were found when the coalescence extent was
significant, whereas for partial and no coalescence a good accordance was observed, both in terms of
average sizes gR t and hR t , as well as for structural parameters, such as qS t and
fd t .
Based on the characteristic times of aggregation ( A ), coalescence ( C ) and the entire
aggregation process ( P ), it was possible to identify three limiting cases, useful to appreciate the extent
of coalescence and the resulting cluster spatial organization. Recalling that P A in all cases, as
otherwise no aggregation would take place, and introducing the ratio of characteristic time of doublet
coalescence over aggregation, /C AN , it is possible to distinguish among the following situations.
i) full, instantaneous coalescence occurs when N <<1 and uniform, compact clusters are obtained all
along the aggregation process; ii) doublet aggregation and coalescence are occurring at a comparable
rate for 1N . Since P A C rather uniform, compact aggregates are formed as the clusters
coalesce while growing; iii a) no coalescence occurs and open clusters are formed when N >>1 and
P C ; iii b) coalescence occurs slowly and only after the clusters are formed when N >>1 and
P C . The initially formed clusters will be rather open; while for large enough P , the aggregates
branches will start to coalesce, hence the cluster “history” becomes of significant importance. For
P the clusters will fully coalesce and become comparable to the situation of case i).
This analysis revealed, among other things, how the fractal dimension of the clusters is not
always enough to appreciate partial coalescence (cf. case iii b)) and that by measuring gR t and
hR t a more complete picture is obtained.
71
The present model might be of help in better characterizing and controlling colloidal processes
where the interplay of aggregation and coalescence regulates the clusters size and structure, which is of
great practical importance for the final applications.
72
73
Chapter 5
Effects of Coalescence on the Shear-Induced Gelation of Colloids
5.1. Introduction
Colloidal systems and nanoparticle suspensions are gaining continuously growing interest with time, as
new applications in the nano-scale are developed almost on daily basis[84-88]. In most of the cases,
these suspensions consist of lyophobic particles dispersed in a continuous phase. The thermodynamic
equilibrium thus consists of two separated phases; one particle-rich and one particle-poor phase. In
order to prevent the phase separation of the suspensions, the particles are commonly rendered kinetically
stable with the help of various strategies allowing to build a repulsive energetic barrier between the
particles, such as the use of charged ligands, surfactants, and steric stabilizers[89]. A typical example
of such colloidal system are polymer nanoparticles produced by emulsion polymerization, a widely
industrially used process[90]. Even though colloid science is a relatively “old” scientific topic, some of
the basic trends of these systems are still unclear and even a qualitative understanding is lacking. One
feature of dispersed colloidal particles is the ability to form fractal aggregates when the stabilization
strategy fails[91, 92]. Colloid instability arises either if the employed surfactant loses its efficacy (e.g.
high electrolyte concentration in the case of ionic surfactants)[89], or when the particle suspensions are
sheared, a condition which is very often met in industrial applications, during pumping or filtration
operations. The formation of aggregates can be either desired, if the aimed product is a powder, but it
may also compromise the quality of the whole product, as in the cases of coatings, paints and adhesives.
74
In either case, aggregation should be tightly controlled. In the most drastic situations, the formation of
particle aggregates can also lead to a process called gelation, when the porous clusters interconnect,
percolate and form one “macro-aggregate” which spans through the whole container[93]. Such a
process may occur both in stagnant, as well as in shear conditions, where in the latter case it is
significantly more abrupt. The process of shear-induced gelation has been a topic of intensive research
in the last years and is rather well understood[18, 19, 22, 94]. If the applied shear rate is high enough,
the added energy can overcome the repulsion barrier keeping the particles apart. The governing rate of
doublet formation in the case of shear-induced aggregation of particles with stabilization has been
developed by Zaccone et al[22], and reads (in the simplified form):
/k 2 Pe1,1 8 Pe m BU Tm
B
Uk Dac e
k T
(5.1)
mU is the value of the interparticle potential at its maximum and 33 / BPe a k T , the Péclet
number, with α being a flow-type related constant, μ is the viscosity, is the shear rate, a is the particle
radius, kB is the Boltzmann constant, D the diffusion coefficient of the particles and T is the temperature..
The first term in the exponential represents the colloidal stabilization, slowing down the aggregation
and competing with the second term, representing the energy brought by the shearing. It can thus be
observed that above a critical size (or shear rate), the shear contribution takes over the stabilization
contribution, allowing for a drastic speeding up of the aggregation. Once small aggregates are formed
and reach the aggregation critical size, they aggregate with each other following an “explosive”
behavior because of the third-power size dependency of Pe
Once the aggregates reach a critical size, the applied stress becomes too large for them to resist, and
breakup occurs[95, 96]. The critical size depends on the topology of the aggregates and on the force
holding the particles together upon contact, which is ultimately related to the surface properties of the
particles. Breakup and re-aggregation typically lead to a compacting of the cluster, since more compact
clusters exhibit a higher average coordination number, which decreases the interface area between water
and the hydrophobic surface. At the same time, increasing the average coordination number increases
the resistance of the clusters to stress.
75
If the material composing the nanoparticles exhibits a non-infinite zero-shear viscosity, coalescence can
occur once particles aggregated[97, 98]. A significant number of important polymeric products actually
undergo coalescence; such as coatings[99] and adhesives[24, 100]. This phenomenon has been
discussed in a previous article[101] for polymer particles and has been shown to have several effects
on the structure of the colloidal aggregates generated in stagnant conditions: reduction in cluster size,
increase in fractal dimension[97].
When combining coalescence and shear-induced aggregation, several effects can be expected: on one
hand, coalescence reduces the volume occupied by the fractal clusters, hence delaying the gelation. On
the other hand, partial coalescence may strengthen the bonds formed between particles, thus allowing
the clusters to grow to larger sizes before breaking, hence reducing the gelation time. The interplay of
these two competing effects makes it hard to predict what will be the effect of changing environmental
and particle parameters (e.g. temperature, particle viscosity, and shear rate) on the stability of the
suspension.
In the present work, we investigate experimentally the effect of (partial) coalescence on the kinetics of
shear-induced gelation. More accurately, we devised an experimental procedure allowing to distinguish
the contributions of aggregation, breakup and coalescence on ad-hoc synthesized polymer latexes. In
particular, we synthetized acrylic polymer colloidal particles with desired size and glass transition
temperature (Tg), and measured their gelation times in a Couette-Flow device as a function of
temperature and salt concentration. We then investigated the structural properties of the aggregates
using stirred-tank breakup experiments and the stagnant colloidal stability using dynamic light
scattering. This allowed to decouple the respective effects of coalescence on cluster breakup,
temperature effect on colloidal stability, and coalescence on the reduction of occupied volume fraction.
The present article investigates the case of hydrophobic acrylic polymeric particles in water, even
though the physical phenomena playing a role here are independent on the nature of the system and can
be found with inorganic particles as well as with other dispersants.
76
5.2. Materials and methods
5.2.1 Materials
All synthesis and analysis chemicals were bought from Sigma-Aldrich, with purities higher than 99%.
The water used was filtered with a Millipore Milli-Q® (MQ) system.
5.2.2 Latex synthesis and characterization
The latexes used during this study have been synthesized by starved feed emulsion polymerization as
described in a previous article[101]. They consist of nanoparticles of random copolymer of methyl
methacrylate (MMA) and butyl acrylate (BA) dispersed in water. The initiator and surfactant used
during the polymerization were potassium persulfate (KPS) and sodium dodecyl sulphate (SDS),
respectively. The latexes were cleaned from surfactant using ion-exchange resin (Dowex® Marathon™
MR-3, Sigma-Aldrich) until their surface tension reached values close to the ones of water (>70 mN/m,
measured with a DCAT 21 and a Wilhelmy plate, Dataphysics, Germany). The ratio of MMA/BA
defines the glass transition temperature of the obtained particles. The particles were characterized in
terms of diameter and polydispersity index by dynamic light scattering using a Zetasizer Nano ZS
(Malvern, UK) at volume fractions of φ = 10-4. To measure the particles glass transition temperature
(Tg), calorimetry was employed, as already reported[101].
An overview of the produced particles in terms of their composition and properties is reported in Table
1.
77
Table 5.1: Particle characterization
MMA [% wt] BA [% wt] Diameter [nm] PDI [-] gT [oC]
30 70 198 0.027 -12
50 50 202 0.016 17
60 40 199 0.017 36
70 30 198 0.021 54
5.2.3 Critical coagulation concentration determination
In order to measure the particles CCC, the stagnant aggregation process has been investigated by
dynamic light scattering (DLS) using a BI-200SM goniometer system (Brookhaven Instruments, USA),
equipped with a solid-state laser, Ventus LP532 (Laser Quantum, U.K.) of wavelength 0=532 nm, as
the light source, with scattering angle set to 90. The sample was thermostated by a water bath (Julabo,
Germany). The particles were first diluted with water to reach a polymer volume fraction φ = 10-4. The
NaCl solution was then diluted to the needed concentration, so that the mixture reaches φ = 10-5 and the
desired test salt concentration after mixing. Both solutions were then tempered in an oven set at the
desired temperature and left for equilibration for 15 minutes. When the desired temperature was
reached, the salt solution was rashly added to the polymer suspension and shortly mixed, in order to
avoid unwanted aggregation due to local higher concentrations of salt or polymer. The time needed to
mix the solutions and to set them in the DLS was measured and always equal to 1 min.
Increasing amounts of salt lead to faster aggregation. Above a certain NaCl concentration, the
aggregation stops accelerating. Such concentration corresponds to the CCC.
78
5.2.4 Zeta potential measurement
In order to investigate the stagnant stability of the studied systems, their zeta potentials have been
measured at the two extreme temperatures of our range (25 and 45oC). Furthermore, to quantitatively
describe the colloidal stability of the used latexes experimentally measured values of zeta potential
(Zetasizer Nano ZS, Malvern, UK) has been compared with those calculated using generalized colloidal
stability model[102]using two different NaCl concentration. The polymer volume fraction for these
experiments was always φ = 5·10-5.
5.2.5 Colloidal titrations
In order to determine the concentration of sulfate groups coming from the persulfate radical initiator
attached to the particles, titrations were performed. A suspension of φ=1% particles was prepared, with
10 mM NaCl to provide an ionic background. The suspension was stripped with wet N2 during 5 min,
and thereafter kept under mild wet N2 flux in the overhead to prevent gas from entering the system. As
the particles were extensively ion-exchanged, their counterions are H+. Hence the concentration of
protons in the system is equal to the concentration of sulfate groups and can be titrated with a base. The
titrations were performed with a 10 mM NaOH standard solution (Fixanal®, Fluka) using a Titrino
(Metrohm, Switzerland) controlled by a computer. The conductivity and pH of the suspensions were
recorded, and the equivalence point was obtained by the clear minimum in the conductivity curve.
5.2.6 Couette-flow gelation experiments
The gelation experiments were carried out in an ARES rheometer (Rheometric Scientific, USA)
equipped with a Couette-Flow device. The radial gap was 0.35 mm, and the gap under the inner cylinder
was set to 3 mm, in order to avoid the contribution coming from the liquid below the bob. The
preparation of the suspensions was the following: first, the diluted salt solution and the polymer latex
were thermostated to the experiment temperature. Then the salt was added to the polymer rashly under
mild stirring. The final dispersion was composed of polymer φ = 5%, NaCl at the desired concentration,
79
and water. The obtained solution was inserted in the thermostated rheometer cup and the bob was
lowered. A solvent trap was used to prevent water evaporation. The used steady shear rate was 4700 s-
1. The viscosity of the suspension was measured over shearing time.
As it is important to ensure that the observed gelation in the rheometer was due to shear, and not simply
to the destabilization of the particles, a part of the mixture was kept in an oven under stagnant conditions
at the same temperature as the experiment. The size of the particles kept under stagnant conditions was
measured after the end of the rheological experiment by DLS (Zetasizer Nano ZS, Malvern, UK), and
always proved the suspension to remain stable, as no change in size was observed. Each shearing
experiment was repeated at least twice to ensure reproducibility. The reported gel time corresponds to
the time-coordinate of the intersection between the viscosity plateau before the onset of gelation, and
the sharp rise in viscosity occurring during the gelation.
5.2.7 Stirred-Tank DLCA experiments
For the breakup experiments, a 2 L stirred-tank reactor was used. It was equipped with a 60 mm Rushton
turbine and four metallic cylindrical baffles. A diluted polymer dispersion φ = 10-5 was inserted in the
tank, taking great care of removing all the air bubbles present in the reactor while filling it up
completely. In fact, the presence of air would lead to adsorption of some particles or aggregates on the
surface of the bubbles and lead to undesirable effects on the aggregation and breakup behaviour. To
ensure that no air would enter the reactor, a magnetic coupling was used to drive the stirrer. For the
same purpose, part of the excess polymer suspension was pumped in a vertical tube (1.5 m high)
connected to the reactor, setting it under light overpressure.
The reactor was equipped with an electrical heating jacket to control its temperature. Once the desired
temperature of the reactor was reached, 60 mL of a 2 M MgCl2 solution was added using a syringe
through a port at the bottom of the reactor, ensuring that the particles are fully destabilized (DLCA
aggregation under shear). The system was then let to equilibrate during one hour. The aggregation and
breakup of the particles was followed by small angle static light scattering (SALS), using a Mastersizer
80
2000 equipped with a flow cell (Malvern, UK). The mean radius of gyration (Rg) and fractal dimension
(Df) of the aggregates were obtained from the structure factor of the aggregates, S q . In particular,
Rg was extracted from a Guinier plot and the slope of the curve in the fractal region was used to extract
Df, using relation fDS q q . In order to measure the size of the aggregates in the reactor, a small
amount (≈10 mL) of suspension was gently diluted in 100 mL of water, and injected in the SALS
instrument. The obtained solution was continuously slowly injected in the cell, in order to prevent
sedimentation of the clusters. Each measurement has been repeated 3 time and the reported values
represent the average values.
81
5.3. Results and Discussion
5.3.1 Effect of temperature on stagnant aggregation
Before discussing the impact of coalescence on shear-induced aggregation, as a reference we first
investigate the particles stability in stagnant conditions. As the shear experiments are performed at
different salt concentrations and temperatures, it is necessary to deepen the effects of these variables on
the intrinsic colloidal stability of the particles employed. Aiming at this, the critical coagulation
concentrations and zeta potentials of the studied particles were measured at the two extreme
temperatures in the studied range, namely 25 and 45oC. The results of these analyses are presented in
Figure 5.1
Figure 5.1 : Critical Coagulation Concentration of NaCl vs particle composition (MMA fraction in
percent) at different temperatures (a) and Zeta Potential vs particle composition at two different salt
concentrations and two temperatures (b)
0.0
0.2
0.4
0.6
0.8
1.0
30 40 50 60 70 80
-70
-60
-50
-40
-30
-20
-10
0(b)
25°C 45°C
CC
C /M
NaC
l
(a)
10 mM, 25°C10 mM, 45°C140 mM, 25°C140 mM, 45°C
Zet
a P
oten
tial
/mV
% MMA
82
As observed, the CCC decreases significantly with increasing temperature for all particle compositions
considered. Conversely, while the zeta potential decreases (in absolute value) with increasing salt
concentration, it does not change significantly with the temperature. This indicates that the ionic
equilibrium of the sulfate groups present on the surface is not influenced by the temperature, and hence
the electrostatic repulsion is constant with temperature. In order to understand more quantitatively the
interaction giving rise to the stability of the colloidal systems, a generalized stability model was
used[102] in order to solve the Poisson-Boltzmann equation, taking into account the counterion
association[103]. The interaction potential was then computed using the DLVO theory, where the total
interaction potential equals the sum of van der Waals attraction and electrostatic repulsion. Let us
analyze as an example the case of the 30% MMA sample. The surface charge was obtained by titration
and is equal to 0.016 mol/KgP. The generalized stability model allows to compute the ionic equilibrium
and the surface potential of the particles. The obtained surface potential at 10 mM NaCl and 25°C is -
76 mV. This compares well with the obtained zeta potential in the same conditions, which is equal to -
73 mV. Increasing the ionic strength to 140 mM gives rise to a surface potential of -29 mV, according
to the model results. This agrees reasonably well with the experimentally obtained -40 mV, considering
the fact that the model is assuming ideal behavior of all involved ions. The so determined surface charge
allows to compute the interaction potential, and more specifically to evaluate the CCC, or the minimum
salt concentration at which the potential barrier value is equal to 0. The CCC computed at 25oC
employing the DLVO theory is equal to 160 mM. This contrasts strongly with the measured CCC
around 800 mM (cf. Figure 5.1), concentration of salt, at which basically no electrostatic repulsion is
present anymore. This strong discrepancy suggests the presence of non-DLVO stabilizing interactions,
which for lyophobic colloidal systems are often associated with hydration forces[104, 105]. It hence
was decided to include a non-DLVO force in the model, in the typical hydration exponential potential
form 20 0
0exphyd
dU aF l l
where d is the surface-to-surface separation The force constant 0F
was fitted in order to match the experimental value of the CCC, whereas the characteristic decay length
83
0l was chosen as the size of a water molecule[89]. It was found that this force constant is equal to
3.1·106 Nm-2 at 25°C. This value is in the same range (107 Nm-2) than previously obtained results[105],
keeping in mind that the surface of the particles in the present study is significantly different.
The effect of temperature on the hydration force is found to be the governing factor influencing the
stability of the particles. In fact, it is found that 0F decreases from 3.1·106 Nm-2 at 25°C to 1.3·106 Nm-
2 at 45°C. Computing the CCC at 45°C using 3.1·106 Nm-2 as force constant results to 850 mM NaCl.
It must be here pointed out that the used stability model takes into account every effect of temperature
on the Poisson-Boltzmann distributions and on van der Waals forces, but not on counterion association.
It is however a conservative approximation, as ion association is an endergonic exothermic
process[103], it will be disfavored by an increase in temperature. This means that the computed surface
potential is lower (in absolute value) than the actual one. Therefore, the particles should become even
more stable by increasing the temperature than what was computed without taking this effect into
account, confirming once more that a non-DLVO repulsive interaction is present, and decreases in
intensity with increasing temperature.
5.3.2 Gelation Curves
The typical process of shear-induced cluster gelation is presented here. It is worth noting, that the
particulate system is stable from a colloidal point of view as long as no shear is applied. When shear is
applied, the energy brought to the system allows the particles to overcome the stabilizing energy barrier
and hence to aggregate. As shown in eq. (5.1), aggregation is strongly accelerated by the shear rate, but
also by the cube of the particle size, in the exponential term of the Arrhenius-type rate constant. This
leads to an explosive behaviour in viscosity, as show in figure 2.
84
Figure 5.2 : Gelation curves for the 50% MMA sample at different NaCl concentrations
An approximate cluster size for the gel point can be derived from the condition where the effective
volume fraction of cluster reaches 0.5, which corresponds roughly to the occupied volume fraction of
randomly packed spheres[106]. The fractal scaling law is well-established and reads:
Df
gf
Ri k D
a
(5.2)
where i is the average number of particles per cluster and fk D a prefactor depending on the
fractal dimension according to empirical relation 2.084.46f fk D D [107]. Considering eq (5.2) it is
possible to obtain the relation between the size of the formed clusters at the gel point and the system
parameters, e.g. phi, size of primary particles and fd
1
3
, 2
fDfc
g gel
k DR a
(5.3)
In the present system, the critical radius of gyration is equal to ,cg gelR =87 μm. This implies that if all
the primary particles are converted into clusters, these clusters must be able to grow at least up to the
aforementioned size in order to allow gelation with primary particles of 100 nm radius and 5% of initial
volume fraction.
0 2000 4000 6000 8000 100000
1
2
3
4
/m
Pa
s
Time /s
140 mM 130 mM 120 mM
85
However, as the aggregates grow the hydrodynamic stress to which they are exposed increases up to
the point when it becomes larger than the aggregate strength, leading to cluster breakup.[108]
It has been observed in this study that no shear-induced gelation is obtained with the MMA/BA
copolymers if T<Tg at φ=5%, as it is the case for the 60% MMA sample at T=25°C and the 70% MMA
sample at all studied temperatures (Tg=36°C and Tg=54°C, respectively). In these cases, it was
impossible to find a salt concentration at which gelation occurred under shear, at least in the 4h shear
experiment performed. It is possible to observe gelation of these samples in these conditions, but only
in stagnant conditions using NaCl concentration in the range of 200-500 mM, depending on the sample
and the temperature. Notably, the formed gels are remarkably weak, since tilting the container where
they are formed is sufficient to break their structures and allow the flowing out of the slurry. This reflects
clearly the effect of breakup on gelation: if these acrylic particles are not coalescing, the bonds formed
between the particles are not strong enough to allow the growth of large enough clusters to reach
gelation under shear stress.
5.3.3 Effect of salt concentration and temperature on gelation time
The gel times (i.e. the time at which the viscosity abruptly spikes) measured under constant shear in the
rheometer for the three different samples at three different salt concentrations and three different
temperatures are reported in Figure 5.3
86
Figure 5.3 : Gel Times vs NaCl concentration for 30% MMA (a), 50% MMA (b), 60% MMA (c) in
the Couette-Flow device at different temperatures
First, let us comment on the effect of NaCl concentration on the gelation time (tgel). It can be observed
that for all the samples at all temperatures, increasing the salt concentration leads to a decreased tgel.
This effect is expected, as an increased salt concentration leads to a decreased electrostatic repulsion
due to the screening of the fixed charges[11, 109]. A decreased repulsion leads to a faster aggregation,
as expressed in eq. (5.1). Nevertheless, it should be observed that the dependency of the gel time on the
0
50
100
150
200
250
300
0
50
100
150
200
250
300
115 120 125 130 135 140 1450
50
100
150
200
250
300
(c)
(b)
25°C35°C45°C
Gel
Tim
e /m
in
(a)
25°C35°C45°C
Gel
Tim
e /m
in
35°C 45°C
Gel
Tim
e /m
in
CNaCl
/mM
87
salt concentrations is a function of the temperature. It is observed that for the “harder” samples (60%
MMA and 50% MMA), the gelation time decreases monotonically with increasing temperature. This
effect arises because of two main reasons; on one hand, as shown in section 3.1, all the used samples
are less stable when the temperature is increased. On the other hand, as shown in our previous
paper[101], increasing the temperature of these samples increases significantly the coalescence rate. An
increase in coalescence rate corresponds to a higher extent of interpenetration between the particles,
hence giving rise to more shear-resistant (i.e. non-breaking) clusters. It is very likely that these two
effects are present and it is non-trivial to suggest when one mechanism prevails over the other one. This
point will be investigated in section 3.4.
When considering material properties, namely the particles with 30% MMA (cf. Table 1) an interesting
phenomenon occurs here (see Figure 3a). It can be seen that increasing the temperature from 25°C to
35°C, the gelation time increases while a further increase to 45°C, results in gel time decreases. This
initial increase in gelation time can be understood by taking into account the second effect of
coalescence, which is the decrease in occupied volume fraction resulting from the compaction of the
clusters. In fact, if full coalescence occurs, the obtained clusters will occupy exactly the same volume
as the particles constituting them, forming a larger sphere. Therefore, if full coalescence occurs, no
gelation is possible, because the clusters cannot percolate without an increase in occupied volume
fraction. It is likely that the initial increase in gelation time observed in the case of the 30% MMA
sample is due to a decrease in coalescence time. It has been reported previously[101] that these soft
particles seem to reach, between 25oC and 35oC under diluted DLCA conditions, an almost fully
coalesced state in comparable times. To evaluate relative importance of involved mechanisms we
perform an analysis of the characteristic times of aggregation, coalescence, and breakup. Following our
previous work,[97, 101] the characteristic coalescence time of clusters can be estimated (is defined) as:
1/30
12
Polcoal
ai i
(5.4)
where pol is the polymer particle viscosity, and 12 the interfacial tension between the phases 1
(particle) and 2 (water).
88
The competition between aggregation and coalescence is not the same in stagnant and shear-induced
aggregation conditions. Namely while stagnant DLCA aggregation tends to slow down with process
time (and cluster size) because of lower diffusivity[110], shear-induced aggregation explosively speeds
up. The small clusters formed at the initial stages of aggregation can hence coalesce, but it requires a
much lower viscosity to allow coalescence on the cluster size range. It therefore makes sense to observe
the effect of decreased coalescence time on gelation time in temperature regions where the full
coalescence was already reached in more diluted conditions. The fact that gelation occurs is also a
strong proof that full coalescence is not occurring, for the reasons stated above. By increasing the
temperature further to 45°C, the gelation time decreases again. This is due to similar effects as those
observed for the other samples, namely a colloidal destabilization or a decrease in breakup rate due to
strengthened bonds between particles.
Let us now discuss the fact that the slope of the tgel vs. NaCl concentration depends on the temperature.
Namely, the influence of salt concentration decreases when the temperature is higher. It seems that the
salt is effective at speeding the gelation up to a certain point, at which the gelation time only depends
very weakly on the salt concentration. It is rather clear that this effect is not due to coalescence; in fact
it is only observed for the 50 % MMA and 60 % MMA samples in the studied temperature range. In
addition, it is not expected that coalescence changes dramatically the effect of salt on aggregation. It is
quite likely that the observed smaller salt effect at larger temperature is due to the already covered non-
DLVO force stabilizing the systems, which is a function of temperature. Since the latexes are stable
under stagnant conditions at the salt concentrations and temperature used, it is certain that some
repulsive energy barrier is present, coming from electrostatics and non-DLVO interactions. This
potential maximum is competing with the hydrodynamic stress, as shown in eq. (5.1), and one can
define a critical Péclet number over which the aggregation rate is almost independent on the interaction
potential[18], the particles aggregate every time they collide. It is hence rather straightforward to
understand that the critical Péclet number for aggregation of the primary particles (or small aggregates)
is reached when enough salt is added or/and when the temperature is high enough. The particles then
aggregate at the same rate, independently on the absolute magnitude of the interaction potential, as long
89
as it is low enough. Adding more salt to the suspension has therefore no more effect. The effect is
however not observed for the softest material (30 % MMA). This is likely due to the fact that the
coalescence rate is much higher for this sample. The occupied volume reduction effect is hence
superimposed on the aggregation/breakup competition.
5.3.4 Effect of temperature on cluster breakup
In order to determine the relative effects of decreased breakup rate and decreased colloidal stability
during the gelations in the rheometer, breakup experiments were carried out in a stirred-tank reactor
under DLCA conditions and relatively low volume fractions. In these conditions, the clusters grow due
to the convection and Brownian motion, capturing all the particles they meet, until their critical breakup
size is reached. They then break, giving rise to smaller fragments or primary particles. This dynamic
equilibrium determines the average cluster size in the system[108]. Since the initial volume fraction is
set constant in all experiments and the particle size is constant, the number of particles is also constant.
Therefore, the aggregation rate is the same in all experimental conditions. The average cluster size is
hence only defined by the breakup rate, which depends on the size of the clusters, but also on the
morphology (fractal dimension), and on the binding strength between the particles. The average size
and fractal dimension of the observed clusters for the 60% and 70% MMA sample at different
temperatures is reported in Figure 5.4.
90
Figure 5.4: Average radius of gyration (a) and fractal dimension for 60% MMA and 70% MMA
samples in the stirred-tank experiments, as a function of temperature.
Looking first at the effect of temperature on the size and fractal dimensions of the clusters of rigid
particles (70% MMA), it can be observed that a slight increase in size is observed between 35°C and
45°C, while the fractal dimension does not show any significant change, with values around 2.8. A
constant fractal dimension indicates that the clusters’ morphologies are not changing, and thus the only
factor governing the critical breakup size is the adhesion force. This moderate increase (1.6 µm to 6.1
µm) can have several origins coming from the effect of temperature on the adhesion of the particles.
Let us analyse two models for adhesion in the literature: according to the well-known Johnson-Kendall-
Roberts (JKR) model, the pull-off force (the force at which the particles are pulled apart) reads
12
3
2cF a [111] The increased cluster size could be explained by an increase in surface energy
25 30 35 40 450
5
10
15
20
25
30
25 30 35 40 452.0
2.2
2.4
2.6
2.8
3.0(b)
60 % MMA 70% MMA
<R
g> /
m
Temperature /°C
(a)
60 % MMA 70% MMA
Df /-
Temperature /°C
91
12 . Such a variation is conceivable, considering the change in colloidal stability observed in the
previous section, which was attributed to a change in non-DLVO interactions. Another popular
adhesion model is the development by Maugis and Pollock[112], allowing for plastic and elastic
deformation. In this case, the pull-off force also depends on the Young modulus (E) of the material. The
pull-off force namely increases with 2
1
E, for both plastic and elastic deformation. It is well known that
the Young modulus decreases significantly when the temperature approaches the glass transition
temperature for amorphous polymers[113]. It is hence likely that the glassy particles become more
elastic while staying under the Tg, (=54°C) allowing the growth of larger clusters.
For the 60% MMA particles, with a lower Tg (=36°C), a more striking effect is observed; the size of
the obtained clusters while T<Tg stays in the same range as for the harder material, it increases
dramatically when the glass transition temperature is reached. The fractal dimension also stays constant
at high values around 2.8, independently on the temperature. This indicates that coalescence only occurs
on the particle scale in these conditions, without being able to significantly influence the morphology
of the clusters. This implies again that the critical breakup size of the aggregates is governed by the
adhesion of the particles only[114]. It must be noted here that all the measurements at T>Tg are
underestimating the size of the obtained aggregates. In fact, it is observed that the formed aggregates
grow to very high sizes (>100 µm) in these conditions, and tend to adhere to the surface of the reactor.
Hence, no real steady-state in size is reached, as some material is taken away from the suspension.
Measurements have been taken after 10 min, 30 min and 60 min, and a decrease in concentration was
systematically observed when T>Tg, while the cluster size stayed constant with time. This is the reason
why no data is reported for the 60% MMA particles above T=39°C as well as for the softer materials,
as most of the material then gets lost in the reactor. This effect actually only accentuates the fact that
the glass transition temperature is the point at which a critical change in binding strength is occurring.
The source of this dramatic change in shear stability can have several sources; on one hand, the surface
energy could be changing, as reported for the hard particles. However, there is no clear reason about
why crossing the glass transition temperature would change the surface energy abruptly. The change in
92
CCC with temperature is also observed for the samples with lower Tg. On the other hand, the most
likely reason why the clusters can withstand the shear at larger sizes is the presence of coalescence,
occurring only at gT T . When coalescence is present, even at rather low extent, the polymer actually
flows from one particle to the other as eq. (5.4) shows. A significantly higher force is thus needed to
take apart the particles, which can now also dissipate some of the stress by viscous relaxation.
Let us now discuss more quantitatively the forces coming into play in the breakup phenomenon. First
of all, the effect of the applied shear has to be consideredError! Reference source not found.. The
maximum shear rate in this experimental system running at 500 rpm is equal to 17300 s-1, evaluated
from the scaling for the maximum dissipation rate proposed by Soos et al [115]. It is hence possible to
compute the hydrodynamic stress 25
8hydF d to which the formed clusters should be able to resist
in order to allow gelation in the rheometer experiments. This hydrodynamic force, corresponding to
,cg gelR , is equal to 283 nN. The hydrodynamic forces to which the clusters withstand in the stirred-tank
experiments are reported in Figure 5.5. As one can see, as long as the temperature stays below the glass
transition temperature of the polymer, the force to which the clusters can withstand stays at least one
order of magnitude below the force to which the critical clusters for gelation in the rheometer are
subjected. This explains very clearly why no gelation can occur below the glass transition temperature
for these clusters. It is however observed that the critical value of 283 nN is not reached in the stirred
tank experiments. This is due to the under-estimation of the cluster size due to the loss of material by
adhesion on the reactor. However, these results show clearly that the criticality in gelation ability is due
to the strengthening of the bonds between the particles constituting the clusters. Moreover, the particle-
particles bonds keep on reinforcing when the coalescence process is eased by increasing the
temperature.
93
Figure 5.5 : hydrodynamic force to which the clusters are subjected as a function of temperature in
the stirred-tank experiments.
Let us now summarize the understanding of the process of gelation under shear in the presence of
coalescence. First of all, if the temperature is set below the Tg of the particles, the binding strength upon
particle contact is too low to allow large clusters to form under shear. This limits the reachable range of
occupied volume fraction, not allowing gelation to occur at the primary particle volume fraction used
in these experiments.
If the temperature is set at (or close to) the Tg, the particles start coalescing. This gives rise to an increase
in contact surface area, which reflects in an increased binding strength. This allows the particles to form
larger clusters upon contact, hence allowing gelation.
If the temperature is above the Tg, three scenarios can be distinguished, depending on the relative
aggregation and coalescence times. Here it must be pointed out that the characteristic coalescence time
is not a parameter that is trivial to define. As already discussed above, the coalescence rate depends on
the size of the coalescing object. Hence small clusters may be coalescing faster than they aggregate, but
large clusters of the same material could well be unaffected by coalescence in their whole structure, but
only in the small size scales. Therefore, their occupied volume fraction and fractal dimension could not
be affected. Keeping the above discussed phenomena in mind, one can define the following cases:
25 30 35 40 450
25
50
75
100 60 % MMA 70 % MMA
Fhy
d /nN
Temperature /°C
94
If τC >> τA, the main effect will be to accelerate the aggregation rate constant and decrease the breakup
rate constant due to colloidal destabilization and increased binding strength, leading to a faster gelation.
If τC ≈ τA, the final effect will depend on the significance of the two processes at elevated temperature.
If the coalescence rate increases faster than the particles are destabilized, the gelation time will increase.
If on the contrary the coalescence rate is only weakly affected by temperature, the gelation will occur
faster due to the aforementioned reasons.
If τC << τA, the formed clusters turn into spheres faster than they grow. This leads to an impossibility for
them to percolate, hence preventing any gelation.
An interesting point is to consider the effect of the parameters that were set fixed in the present article.
Let us for example analyse the effect of changing the initial particle volume fraction. If the volume
fraction is decreased, the rate of aggregation will be decreased, due to the fact that aggregation is a
bimolecular process. This leads to a slowing down of the cluster size increase with time. It means that
each cluster spends more time at one size before aggregating further and growing larger. As eq. (5.4)
shows, the coalescence characteristic time increases with increasing cluster size. This means that at
lower volume fractions and at a defined dimensionless time / At , the clusters will have coalesced
to a higher extent. This allows the clusters to ultimately grow to larger sizes before reaching their critical
breakup rate, possibly allowing a faster gelation.
This discussion explain why it is non-trivial to predict even qualitatively the effect of temperature on
the gelation time of a sheared suspension. It highly depends on the specific interaction potential for the
studied system, on the evolution of coalescence characteristic time with temperature, on the specificities
of adhesion for hard particles, on surface tension, on the volume fraction and the shear rate.
95
5.4. Conclusion
In this article, the intricate effects governing shear-induced gelation of soft colloidal particles have been
studied. To this purpose, same-sized MMA/BA copolymer latex of various glass transition temperatures
have been synthesized by starved emulsion polymerization.
An initial characterization in stagnant conditions revealed how changing the aggregation temperature
of MMA/BA copolymer particles lead to a significant decrease of non-DLVO repulsive forces. A
generalized stability model has been used in order to rationalize the relative effects of electrostatics and
non-DLVO forces in stagnant aggregation.
Using stirred-tank diluted experiments, it has been shown that crossing the Tg of polymer particles
significantly affects the aggregates strength and thus also their breakup rate, even if the temperature
change is relatively small.
Moreover, it has been shown that increasing temperature while above the Tg can lead to two opposite
effects on gelation time: i) a reinforcement of the clusters, decreasing the breakup rate and hence the
gelation time, or ii) such an higher extent of coalescence that the occupied volume fraction decreases,
therefore delaying the gelation. The outcome of a shearing experiment thus depends on the characteristic
times of aggregation and coalescence.
In summary, aggregation, coalescence and cluster breakup all have a significant influence on gelation
time under shear, and are all influenced by temperature, either independently (such as interaction
potential), or depending on one another (such as decreased breakup upon increased coalescence state).
This article identifies independently the main factors influencing these phenomena, allowing to gain a
deeper understanding of the scientifically and industrially relevant process that is colloidal gelation.
96
97
Chapter 6
The Counter-Intuitive Aggregation of poly(Acrylic Acid)-Grafted
Nanoparticles after Surfactant Addition
6.1 Introduction
The production of polymers regularly makes use of emulsion polymerization. This process gives rise to
particles of lyophobic polymer dispersed in a continuous phase, which is often water. The polymer
particles so generated are in a metastable state, with the equilibrium configuration being a phase-
separated system. These lyophobic particles hence have to be stabilized by introducing a repulsive force
between the particles. The nature of this repulsive force can be electrostatic, steric, or linked to the
nature of the surface, such as hydration forces [9, 45, 116, 117].
On an industrial level, the most cost-effective polymerization conditions is to maximize the polymer
volume fraction, while avoiding aggregation. To this purpose, several stabilization mechanisms can be
combined. In this study, acrylic copolymer particles synthesized in semi-batch emulsion polymerization
are studied. The stabilization strategy applied during the synthesis here was double: on one side, a
standard amount of anionic surfactant was used, and on the other side, small amounts of polymerizable,
water-soluble weak acid were added to the monomer feed. More specifically, 0%-2% acrylic acid (AA)
was added to the hydrophobic methyl methacrylate/butyl acrylate mixture. This synthesis method has
already been shown to give rise to a hydrophilic, pH-responsive shell consisting mainly of acrylic acid,
while the more hydrophobic monomers form the core of the particles[61]. The synthesized particles
98
exhibit remarkable colloidal stability when the acrylic acid is deprotonated (pH > 7.0), while a decreased
stability is observed when the pH is decreased below the pKa of the functional monomer.
The goal of this study is to expend the stability study to the case where acrylic acid is used in
combination with sodium dodecyl sulfate (SDS). To this purpose, the aggregation kinetics of several
latexes synthesized ad hoc were followed at different pH values by dynamic light scattering, with
varying amounts of AA and SDS in the particles and the dispersant, respectively. It is observed that
while SDS stabilizes the particles when the pH is low, it destabilizes dramatically the particles when
the AA-shell is deprotonated. In order to understand this phenomenon at the molecular level, molecular
dynamics simulations were performed to identify a “reversed” SDS adsorption on the particle surface.
This adsorption mechanism relies on salt bridges between AA and the sulfate head of the surfactant,
mediated by the high sodium concentration in the bulk.
6.2 Materials and Methods
6.2.1 Particles synthesis
The studied latexes were supplied and synthesized ad hoc by BASF SE. They are copolymers of methyl
methacrylate and butyl acrylate in equal weight fractions, in which desired amounts of Acrylic Acid
were added for stabilizing purposes. The particles were synthesized by semi-batch emulsion
polymerization, using ammonium persulfate (APS) radical initiation system. The latexes were
synthesized in presence of sodium dodecyl sulfate (SDS) as surfactant. The surfactant has subsequently
been removed by extended period of dialysis, which was carried until the surface tension of the mother
liquor was >70 mN/m and the conductivity < 10 µS/m.
6.2.2 Particles characterization
The characterization of the latexes was carried by Static and Dynamic light scattering, and the results
are summarized in Table 1:
99
Name Main Monomers Functional Monomer
[% wt AA] Diameter [nm] PDI [-]
P0 MMA/BA None 161 0.002
P1 MMA/BA 1.0 171 0.005
P2 MMA/BA 2.0 191 0.005
P3 Styrene/BA 2.0 169 0.006
Table 6.1: Characterization of the latexes used
The NaOH, NaCl, HCl, H2SO4 and SDS were all purchased to Sigma-Aldrich with the highest purities
available (>99%) and used without further purification. The water used was filtered with a Millipore
Milli-Q® (MQ) system.
6.2.3 Aggregation experiments
The aggregation process was followed by dynamic light scattering (DLS) using a BI-200SM goniometer
system (Brookhaven Instruments, USA), equipped with a solid-state laser, Ventus LP532 (Laser
Quantum, U.K.) of wavelength 0=532 nm, as the light source, with scattering angle set to 90. The
sample was thermostated by a water bath (Julabo, Germany). The preparation procedure was as follows:
Diluted latex samples were prepared, with a mass fraction φw=5·10-4 in Milli-Q water. The 20 mL vials
were cleaned and dried with nitrogen. Milli-Q water, 2 M NaCl, 2% wt. SDS solution, 0.1 M H2SO4 or
10 mM NaOH solution were added to the vial, after having being freshly filtered through a 0.45 µm
syringe filter. The overhead was then flushed with Argon, in order to avoid pH variations during the
aggregation experiments. Then, the diluted latex was added rashly to the solution, which was
subsequently stirred and set in the light scattering instrument. The overall duration of the procedure of
latex addition and starting of the measurement was measured and always equal to 1 minute.
6.2.4 Computational methods
Atomic charges
Atomic charges for polymethyl methacrylate and polyacrylic acid were computed starting from small
oligomers composed by six repeating units with a syndiotactic configuration; chain ends were saturated
100
with methyl groups. Two different structures were considered for PAA, in order to take into account
the different protonation states due to the pH of the solution. A fully ionized polyacrylic acid (that is,
where all carboxyl groups are present as carboxylate) was employed to mimic high pH values, whereas
a fully undissociated PAA was considered for low pH simulations.
These structures were firstly optimized in vacuo by means of density functional theory (DFT)
calculation at B3LYP/6-31G(d,p) level of theory[118-120]. The obtained geometries were used as input
for a further optimization in implicit water, described through integral equation formalism polarizable
continuum model (IEFPCM)[121, 122] at 300 K, at the same level of theory. Atomic charges were then
fitted starting from electrostatic potentials computed at B3LYP/6-311+G(d,p)[123] level of theory in
implicit water at 300 K, by means of RESP formalism[124, 125]. Charge fitting was performed with a
two-step procedure. In primis, atomic charges were determined by assigning the proper overall charge
value to the considered oligomer, i.e., 0 for PMMA and undissociated PAA, and -6 for the fully ionized
PAA. In the second step, charge equivalence for chemically equivalent atoms was imposed. Oxygen
atoms of carboxylate groups, e.g., have the same partial atomic charge, in order to take into account
charge delocalization due to resonance. The two central units and the ending units of each oligomer
were used for building the polymer chains used in molecular dynamics simulations.
The same charge derivation protocol (structure optimization, electrostatic potential calculation and
atomic charges fitting through RESP formalism) was used for sodium dodecyl sulfate, whose overall
charge is equal to -1. Computations were performed by means of Gaussian 09 software[126].
Creation of the molecular systems
The size of the primary particle is too big to allow an atomistic description of the entire system;
moreover, such analysis would be not meaningful, because the interactions with the surfactant occur
only at the polymer/solvent interface and not in the polymer bulk phase. Since the primary particle size
(100 nm) is much higher than the polyacrylic acid chain length in the fully stretched conformation (10
nm, assumed as the characteristic length at the particle/solvent interface), the system is modeled as a
flat grafted surface of polymethyl methacrylate.
101
The PMMA surface was built as a random coil of 96 polymer chains containing 20 repeating units. Four
polyacrylic acid chains were positioned at 0.1535 nm from the surface and grafted to the closest PMMA
chain by applying a harmonic restraining potential k0(x – x0)2, where k0 and x0 are equal to 2.5363 · 105
KJ mol-1 nm-2 and 0.1535 nm respectively, representative of a chemical bond between sp3 carbon atoms
according to General Amber Force Field (vide infra). Polymer brushes were placed at a distance of 4.1
nm each other, assuming a uniform hexagonal packing. Each polyacrylic acid chain is composed by 40
repeating units, which corresponds to a length of 10 nm in the fully extended conformation and a
monomer density per unit PMMA surface of 4.5 · 10-6 mol m-2, consistently with the experimental
conditions (3.6 · 10-6 mol m-2). Two different systems were created, corresponding to high pH (where
polyacrylic acid chains are fully ionized) and low pH environment (in which undissociated polymer is
grafted on the surface).
a) b)
Figure 6.1: Molecular model for polymethyl methacrylate surface (transparent VdW spheres) grafted
with polyacrylic acid chains (black residues) in explicit water molecules (blue dots) (a); hexagonal
packing of PAA chains (solvent is omitted for the sake of clarity) (b). Each image contains the
simulation box replicated once along x and y.
Each grafted surface (which lies on the xy plane) was solvated with explicit TIP3P water
molecules[127]. No water molecules were added in the xy plane around the PMMA block; in other
words, the base of simulation box coincides with the PMMA random coil. Na+ ions were also added,
102
where needed, in order to assure electroneutrality. This procedure results in a periodic box with an initial
size of 8.2 x 7.1 x 20 nm (where the PMMA thickness is 6.8 nm).
The initial structure is represented in Figure 1, where the simulation box has been replicated along x
and y direction in order to highlight the resulting hexagonal packing when periodic boundary conditions
are employed.
Both systems were equilibrated through 200 ns molecular dynamics simulations in order to reach a
stable arrangement of the brushes on the surface. The attainment of an equilibrated geometry was
checked through time evolution of Solvent Accessible Surface (SAS) and maximum chain height from
the surface related to polyacrylic acid.
Sodium dodecyl sulfate was then added to the periodic box, and additional 500 ns molecular dynamics
simulations were performed for all systems (high and low pH, with and without SDS). According to
experimental conditions, SDS is in large excess with respect to polyacrylic acid; at both pH, the number
of SDS molecules added is equal to the number of PAA monomers (160).
Parameterization and simulation protocol
Polymethyl methacrylate and polyacrylic acid were parameterized according to General Amber Force
Field (GAFF)[128], which proved to be suitable for the simulation of amphiphilic and charged
polymers[129, 130]; ion parameters optimized for TIP3P water molecules were taken from Joung and
Cheatham works[131, 132].
Molecular dynamics simulations were performed by means of GROMACS package, version 5.0.2[133].
Electrostatic long-range interactions were computed through the Particle Mesh Ewald (PME)
method[134], adopting a cut-off value equal to 12 Å; the same value was employed for Lennard-Jones
interactions. Neighbor list was updated every 5 fs, and all covalent bonds involving hydrogen atoms
were restrained through LINCS algorithm[135]. A time step equal to 2 fs, along with Leap-Frog
algorithm, were employed to compute molecular trajectories.
The adopted computational protocol was the following: first of all, energy minimization was applied in
order to remove bad contacts between the solute and the randomly placed solvent molecules.
103
Temperature was raised to 300 K through 200 ps in NVT ensemble; a small harmonic restraint was
applied to the solute in order to avoid wild fluctuations. Density and pressure were equilibrated through
10 ns in NPT ensemble at 300 K and 1 atm. During this phase, a harmonic restrain was applied to
polyacrylic acid chains in order to avoid artifacts. Finally, molecular dynamics simulations were
performed in NPT ensemble at 300 K and 1 atm. Temperature and pressure were controlled through
velocity rescale thermostat[136] and semi-isotropic barostat[137] respectively. Data were collected
every 20 ps.
6.3 Results and discussion
6.3.1 Latex aggregation kinetics
In order to establish a base case, let us investigate the aggregation kinetics of the P0 latex, i.e. the “bare”
particles. This latex is stable after dialysis only thanks to the relatively low charge density of its surface,
coming from the radical initiator used, which leaves an alkyl sulfate group at the end of each polymer
chain. A relatively high concentration of NaCl (0.6 M) is used in order to screen the electrostatic
repulsion between the particles, allowing the acrylic acid containing particles to aggregate. All the
aggregation measurements are performed at low volume fraction (φ=10-5), which allows to follow the
hydrodynamic radius change as a function of time in situ. The results of these experiments are shown
in Figure 6.2a.
104
Figure 6.2: Kinetics of aggregation for P0 (a), P1 (b), P2 (c and d), as a function of pH (a, b, and d)
and of surfactant concentration (c). The surfactant concentration for a,b, and d is always equal to
[SDS] = 0.01 %w/w. The pH value for c) is equal to pH = 9.5
As expected, the change in pH has no effect on the kinetics of aggregation of this low-surface charge
density latex, since no group is to be protonated with changing pH. The presence of SDS allows the
surface to get sufficiently charged and hydrophilic to avoid aggregation. It must be pointed here that
the used SDS concentration, while being very high in terms of surface coverage due to the low polymer
volume fraction, stays significantly below the CMC[138, 139]. The presence of a small amount of
micelles cannot be excluded, but it is observed that this amount is not high enough to induce aggregation
due to depletion forces[140]. The presence of SDS simply stabilizes the particles by adsorption.
Let us now investigate what the effect of surfactant on aggregation kinetics is for the highest amount of
AA-shell (2% Acrylic Acid, P2) at high pH (=9.5). As well as the effect of pH at fixed surfactant
concentration (=0.01% SDS). The experimental conditions are the same as in the P0 latex case. The
results are showed in Figure 6.2c and d, respectively.
0 10 20 30 40 50 60
100
200
300
0 20 40 60 80 10075
100
125
150
175
200
0 20 40 60 80 100 120 140 160
100
200
300
400
500
0 20 40 60 80 100 120 140100
200
300
400
500
pH = 1.08, SDS pH = 8.1, SDS pH = 1.78, no SDS pH = 8.01, no SDS
Rh
/nm
Time /min
a)
d)c)
b) pH = 1.13 pH = 2.92 pH = 6.16 pH = 11.43
Rh
/nm
Time /min
0% SDS 0.002%SDS 0.01% SDS 0.02% SDS
Rh
/nm
Time /min
pH = 1.02 pH = 1.92 pH = 3.24 pH = 9.51 pH = 11.43
Rh
/nm
Time /min
105
Two effects are to be observed here: on one hand, increasing the surfactant concentration at high pH
(>9.0) tends to destabilize the P2 particles. On the other hand, it is remarkable that decreasing the pH
below the pKa of polyacrylic acid [30, 61] has a stabilizing effect on these particles. It has been observed
previously with very similar particles that decreasing the pH below the pKa of acrylic acids tends to
destabilize the particles in absence of surfactant [61]. This effect is expected, since the particles are
stabilized by a hydrophilic charged polyelectrolyte shell. Decreasing the pH below the pKa of the
polyelectrolyte leads to protonation of the charged units, hence decreasing simultaneously the surface
charge (electrostatic repulsion), and the steric stabilization coming from the hydrophilicity of the
polymer. Here, the opposite trend is observed: in the presence of surfactant only, decreasing the pH re-
stabilizes the particles. Decreasing the pH to extreme values (pH=1.02) leads to a slight increase in
aggregation rate. This is due to the protonation of the sulfate groups, generated during the emulsion
polymerization[61].
The stabilizing impact of lower pH is even clearer if the amount of acrylic acid in the polymer is lower.
(Figure 6.2b). For the P1 particles, the aggregation does not occur when the pH is lowered.
These results indicate clearly that the combination of the presence of surfactant, combined with the
deprotonated poly-(acrylic acid) surface and high salt concentration leads to the aggregation. This effect
cannot be due to side effects of the presence of SDS, such as depletion forces, since the experiments
take place below the CMC of SDS. The phenomenon does not happen in the absence of acrylic acid,
and only occurs at high pH, where the solubility of SDS is increased.
In order to investigate this aggregation phenomenon in more details, and more specifically the
mechanism of SDS adsorption on the particles, a fourth latex was synthesized (P3) in which methyl
methacrylate was replaced by styrene. Styrene is significantly more hydrophobic than MMA, and is
able to adsorb a significantly higher amount of SDS than the latter[141]. It is observed in Figure a with
P3 that no destabilization is observed in the presence of SDS, independently on the pH value. This
observation hints strongly towards the fact that the adsorption of SDS on the particle surface is
responsible for the observed destabilizing effect for P1 and P2. More accurately, the amount of
surfactant adsorbed on the particle hydrophobic core modulates the repulsive interactions between the
particles, preventing the destabilizing effect coming from the acrylic acid combination.
106
In order to complete the experimental investigation of the aggregation mechanism, it was decided to
make the P1 latex aggregate as reported in Figure 6.2b. during 60 min at pH=11.43, and then the pH
value was lowered by addition of H2SO4 to pH=3.02 and short mild agitation of the vial using a glass
rod. It was observed that the aggregates formed at high pH redispersed, while the suspension stayed
stable during the following hour. This indicates clearly that the aggregation mechanism of these
particles in the presence of SDS is not related to what is observed in absence of SDS, where increasing
the pH to re-stabilize the particles thanks to the acrylic acid is only able to stop the aggregation process.
Figure 6.3: Aggregation kinetics of P3 at different pH values (a) and P1 (b) with 0.01% w/w SDS
concentration. The pH value for (b) was lowered from 11.43 to 3.02 using sulfuric acid at time = 60
min.
Summarizing this article’s finding, it is observed that the combination of a highly charged
polyelectrolyte layer, a relatively high salt concentration (0.6 M NaCl), and surfactant leads to a
dramatic decrease in colloidal stability. The usual stabilizing components of the DLVO theory, together
with the added contribution of the electro-steric stabilization due to polymer brushes are not able to
explain why this combination leads to destabilization. The presence of surfactant, adsorbing on the
surface due to its hydrophobic tail, should stabilize the particles, and an increased pH value should
deprotonate the polyelectrolyte brush, thus leading to an increased stabilization. The exact opposite
trends are observed.
0 20 40 60 80 100 120
100
125
150
175
200
0 20 40 60 80 100 120 14075
100
125
150
175
200
225b) 1.16
8.95 11.45
Rh
/nm
Time /min
a)
Rh
/nm
Time /min
107
The previously exposed experimental results were not reported so far, and reveal a rather
counterintuitive behavior in the interaction of poly-(acrylic acid) coated particles. In order to investigate
and understand the mechanistic reason for such trends, it was decided to simulate a similar system using
molecular dynamics.
6.3.3 Molecular dynamics simulations
Surface equilibration
First of all, 200 ns simulations have been performed at both low and high pH without SDS, in order to
obtain an equilibrated arrangement for the grafted surface.
At low pH, the undissociated polyacrylic acid chains collapse on the polymethyl methacrylate surface,
because of the attainment of hydrophobic interactions between PAA aliphatic backbone and PMMA,
as shown in Figure 4a. On the other hand, at high pH the ionized polymer chains interact each other
through Na+ bridges between the negatively charged carboxyl groups, thus avoiding the collapse on the
particle surface (Figure 6.4b).
The time evolution of PAA solvent accessible surface is represented in figure 4c and d; in particular,
the values have been normalized with respect to the corresponding values exhibited in the fully stretched
conformation of the polymeric chain, where the SAS is maximum.
At low pH, SAS decrease is mainly due to the hydrophobic fraction of the surface (Figure 6.4c), whose
exposure to the solvent is reduced by the collapse on PMMA surface driven by the hydrophobic
interactions between polymers.
On the contrary, at high pH the hydrophobic SAS fraction remains essentially unaffected, while the
hydrophilic one decreases of about 30% with respect to its initial value (Figure 6.4d). Chain interactions
through Na+ bridges, indeed, reduce the exposure to the solvent of charged carboxyl moieties.
108
a) b)
c) d)
Figure 6:. Representative equilibrated strucures at low pH (a) and high pH (b); Na+ ions are
represented as blue dots, while water is omitted for the sake of clarity; normalized Solvent Accessible
Surface as a function of simulation time at low pH (c) and high pH (d).
SDS effect on stability
As mentioned, after surface equilibration SDS molecules were added in the simulation box and
additional 500 ns were performed for all systems (high and low pH, with and without surfactant) in
order to obtain a meaningful comparison and a better understanding of the surfactant effect. System
equilibration was again verified through the convergence of polyacrylic acid SAS and chain height; the
analysis of molecular trajectories was performed considering the last 100 ns.
Generally speaking, the behavior of SDS molecules can be rationalized by identifying three different
situations. The surfactant can remain free in solution, can be adsorbed on PMMA surface (because of
hydrophobic interactions between the polymer and the aliphatic tail) or can interact with PAA chains.
109
Molecular trajectories showed that the interactions between the surfactant and polymer brushes depend
on PAA protonation state, i.e. on pH.
At low pH, the surfactant is mainly bound to polymer brushes through hydrophobic interactions
between the hydrophobic tail of SDS and PAA backbone. The polar group of the surfactant can interact
with PAA carboxyl moieties through hydrogen bonds, but it remains exposed to the solvent.
Notably, at high pH the binding between polyacrylic acid and SDS occurs through Na+ bridges between
the dissociated carboxyl moieties and the negatively charged sulfate groups. This electrostatic-driven
aggregation can be highlighted by means of radial distribution function between oxygen atoms of PAA
carboxyl groups and sulfur atom of SDS (Figure 6.5).
a) b)
Figure 6.5. Radial distribution function between PAA oxygen atoms and SDS sulphur atom (a);
example of salt bridge between PAA and SDS (b).
Radial distribution function shows that the interaction between the SDS sulfate group and carboxyl
moieties is stronger at high pH thanks to the attainment of salt bridges. As mentioned, at low pH
hydrogen bonds between undissociated carboxyl moieties and the polar head of SDS can occur, but
such interactions are less favored if compared to the electrostatic-driven binding at high pH.
Because of this arrangement, the hydrophobic tail of the surfactant remains exposed to the solvent, thus
increasing the hydrophobicity of the surface and promoting particle aggregation. Such phenomenon can
110
be quantified by computing the overall solvent accessible surface of PAA and the surfactant bound to
the polymer brushes, as highlighted in figure 6.6.
a) b)
Figure 6.6. Fraction of hydrophobic and hydrophilic overall solvent accessible surface (PAA + bound
SDS) at low pH, before and after SDS addition (a); fraction of hydrophobic and hydrophilic overall
solvent accessible surface (PAA + bound SDS) at high pH, before and after SDS addition (b).
While at low pH the addition of the surfactant does not appreciably modify the hydrophilicity of the
exposed surface, at high pH the presence of SDS increases the hydrophobicity of the particle/solvent
interface, thus favoring colloidal destabilization of the latex.
6.4 Concluding remarks
In this chapter, the example of the sometimes counter-intuitive behavior of colloidal particles is shown.
It also demonstrates the efficiency of using colloidal aggregation as a tool in order to investigate the
properties of the surfaces. From the kinetics of aggregation of some industrial latexes, an interesting
effect was observed: adding anionic surfactant, usually used in order to improve the stability of colloidal
systems, brings instability in the system, making particles aggregate in pH ranges where the rest of the
stabilization strategy is the most efficient. This “reversed” adsorption of SDS on the surface, mediated
by salt bridges, also gives rise to the possibility of creating pH-sensitive systems, where reversible
aggregation is observed. Such systems could be of significant interest for the general nanoparticle
technological applications, as well as in simpler polymer production. In fact, if the aggregation is
111
reversible, its presence in a manufacturing process is less likely to lead to a catastrophic failure of the
process.
The present chapter is equally an arguments towards the collaboration between industrial chemistry and
molecular dynamics. A counter-intuitive behavior was observed experimentally, but only simulations
could explain it on the physical level. The possibility of confirming the presence of reversed adsorption
experimentally is relatively small, since probing the surface of nanoparticles would require the use of
small-angle neutron scattering, which is not a readily accessible setup.
112
113
Chapter 7
Concluding remarks
The production and characterization of industrial polymeric latexes gives rise to a multitude of complex
behaviors which, even if rather well understood independently, prevent any educated guessing in their
combined effects. In the present work, the case of copolymeric film-forming particles stabilized with
polyelectrolytes was studied. First, the effect of the polyelectrolyte layer was studied in stagnant
conditions, revealing the pH-responsiveness of the weak polyelectrolyte brush and of its stabilizing
effect, which was shown to act not only on the electrostatic level, but also gives rise to steric
stabilization. In the third chapter, the effect and pH responsiveness of the steric stabilization was
examined while reducing the electrostatic contribution by addition of salt. A steric interaction model
with pH-dependent Flory parameter was devised in order to gain knowledge about the characteristics
of the particles’ surfaces. In the fourth chapter, the coalescing properties of “bare” particles synthesized
to exhibit different glass transitions were simulated using a population balance equation model for the
aggregation in stagnant conditions. In the fifth chapter, the effect of coalescence on shear-induced
aggregation and gelation was studied experimentally and revealed systematically the presence
competition between aggregation, coalescence and breakup in this process. In the last chapter, a
counterintuitive reversed surfactant adsorption mechanism was revealed by molecular dynamics,
leading to aggregation of stable particles covered with polyelectrolyte brushes.
Overall, the present work is a strong argument in favor of using even simplified mathematical models
and simulations in order to tackle complex practical industrial problems. Starting from an intricate
system where several features are expected to play a role, the best strategy is to take each of the effects
114
independently, studying it systematically from the experimental and theoretical point of view, before
coming back to the full complexity of the initial problem.
115
Chapter 8
Appendix
8.1 Experimental proof for the coalescence between particles
One crucial difference between coalescence and non-coalescence systems is that during the aggregation,
the total surface area of the particles in the system decreases for the former and remains constant for the
latter. Then, if an ionic surfactant is adsorbed on the particle surface, when coalescence occurs, the
reduction in the total particle surface area changes the surfactant adsorption equilibrium, and more
surfactant molecules are adsorbed on the particles, leading to increase in the colloidal stability. The
consequence is that one would observe that the doublet formation rate decreases with time for a
coalescence system, while it remains constant for a non-coalescence one.[48, 53]
Therefore, to demonstrate the occurrence of coalescence for our colloidal particles, we have
measured the rate constant for the doublet formation, K1,1 for P3 latex in the presence of SDS, based on
the technique described in the text. The results are shown in Figure 8.1. It is seen that the K1,1 value
decreases monotonically as the time increases, confirming that the total surface area decreases along
the coalescence during the doublet formation, leading to increase in the adsorbed surfactant molecules
on the particle surface, thus the stability.
However, if no surfactant is present in the system, even if coalescence occurs, the K1,1 value does
not change with time. An example is shown in Figure 8.2, for P2 latex in the absence of surfactant. This
arises because even though coalescence may change the density of the fixed charges on the doublet
116
surface, it does not change the fixed charge density on the primary particles, thus without effect on K1,1.
Figure 8.1 Doublet formation rate constant, K1,1, determined from the initial aggregation kinetics at
CNaCl=0.50 mol/L for P3 latex in the presence of surfactant SDS. The solid curve is guiding the eyes.
Figure 8.2 Doublet formation rate constant, K1,1, determined from the initial aggregation kinetics for
P2 latex in the absence of surfactant.
2E‐25
2E‐24
2E‐23
0.0E+0 5.0E+3 1.0E+4 1.5E+4
K1,
1, m
3 /s
t, s
1E-24
1E-23
1E-22
0 500 1000 1500 2000
K1,
1, m
3 /s
t, s
117
8.2 - time-evolution of the average fractal dimension
,,
,f
x td x t
x t
(S1)
3
13
1
, ,,
,,
, ,
f f
f
f
d f x y t ddx td d d
d x tdt dt x t dt
f x y t dd
(S2)
3
13
1
3 3 3 3
1 1 1 123
1
, ,,
, ,
, , , ,, , , ,
, ,
f f ff
f f
f f
f f f f f f f f
f f
d f x d t dddd x t d
dt dtf x d t dd
f x d t f x d td dd f x d t dd d f x d t dd dd
t t
f x d t dd
(S3)
, ,
3 3 3
1 1 13 3 3
1 1 1
, ,,
, , , ,, ,
,
, , , , , ,
, , ,1 1,
, ,
f
x t x t
t t
f f
f f f f f ff
f f f f f f
x t x td x t
ff
f x d t f x d td dd d f x d t dd dd
dd x t t t
dtf x d t dd f x d t dd f x d t dd
dd x t x t xd x t
dt x t t x t
tt
(S4)
118
1 1 1
0
1 1 1 1 1
0
1 1 1
0
1 1
, , ,1 1,
, ,
1, 3 ,
, 1, , , t ,
,
1, , , , ,
2
, , ( , , ) d1
,, 1
( ,2
ff
fCOAL
f
x
f
f
dd x t x t x td x t
dt x t t x t t
x t d x tx
dd x tx t x x x t dx
dt x t
d x t x x x t x x t x t dx
x t x t x x t x
d x tx t
x x x
1 1 1
0
, ) , , dx
t x x t x t x
(S5)
1 1 1
0
1
1 1 1 1 1
0
2
1 1
0
1 1, 3 ,
,
, ,, , ,
,
,1, , , ,
2 ,
,, , ( , ,
,
fCOAL
f
T
xf
T
f
x t d x tx t x
dd x t x tx x t x t dx
dt x t
d x tx x x t x x t x t dx
x t
d x tx t x t x x t
x t
1
3
1 1 1 1 1
0
4
)d
,1( , , ) , , d
2 ,
T
xf
T
x
d x tx x x t x x t x t x
x t
(S6)
Comparing the terms T1 and T3, and T2 and T4 respectively, one gets:
119
1 1 1
0
,
,,1 3 , , ,
, ,
f
f
d x t
d x tx tT T x x t x t dx
x t x t
,x t
1 1 1
0
1 1 1 1 1 1
0 0
, ( , , ) d
1 3 , , , , , , ( , , ) d 0f f
x t x x t x
T T d x t x t x x t dx d x t x t x x t x
(S7)
2 4 1 1 1 1 1
0
1 1 1 1 1
0
,1, , , ,
2 ,
,1( , , ) , , d 0
2 ,
xf
xf
d x tT T x x x t x x t x t dx
x t
d x tx x x t x x t x t x
x t
(S8)
Therefore, considering (S7) and (S8) the final form of equation (S6) reads:
, 1
,fdd x t
dt x t
1,
COAL
x tx
3 ,
, 13 ,
f
ff
COAL
d x t
dd x td x t
dt x
(S9)
120
8.3 – Derivation of average quantities h,effR x, t and gR x,t
8.3.1 Calculation of h,effR x,t
In the following, the Kirkwood-Riseman (KR) theory[80, 142, 143] is extended to the case of
clusters with partially-overlapping primary particles, by using Rotne-Prager’s tensor corrected for
particles overlapping.[144, 145]
Let us start by considering the general formalism of KR theory. The basic idea is to use the
linear relationship between force and velocity, so that for each particle in a cluster one can write:
6i p i iR F u v (S10)
where Fi is the force acting on the ith particle, ui is the ith particle velocity and vi is the fluid velocity at
the center of the ith particle. The fluid velocity is written as a combination of the unperturbed fluid
velocity and of the perturbations due to all other particles in the cluster:
1
'N
i i ij jj ì
v v T F (S11)
The perturbation term is computed using the Kirkwood-Riseman theory. The meaning of Equation
(S11) is that the velocity perturbation on the ith particle due to the jth particle is proportional to the force
applied to the latter. The proportionality term is given by the Oseen tensor, which defines the velocity
profile created by a point-like force located at the center of the jth particle:
2
1
8i j
ijc ij ijR R
R RT I (S12)
where Rij is the distance between the centers of the ith and the jth particles. According to the original
formulation of KR theory, particles are considered as point forces in the calculation of the perturbation
they cause to the surrounding fluid. It is however possible to utilize a more appropriate formulation,
which extends the Oseen tensor to the case where particles size is accounted for. The corresponding
tensor is the Yamakawa (or Rotne-Prager) tensor:[144, 145]
121
2
2 2 2
21 1
8 3i j p i j
ijc ij ij ij ij
R
R R R R
R R R RT I I (S13)
In the case of particles partially interpenetrated, such as those where (partial) coalescence is occurring,
the following form of Rotne-Prager tensor can be used:[146]
1 9 3
16 32 32
ij i jij
p p p ij
R
R R R R
R RT I (S14)
Equation (S14) is only valid when the interparticle distance 2ij pR R . For interparticle distances
larger than the sum of the particle diameter, Equation (S13) should instead be used. At this point, we
will proceed as usual in the case of rigid clusters. We will assume that each particle is subject to the
same average perturbation (mean field approach), and that the particle-particle correlation function can
be used to describe the distribution of particles around the average particle in the cluster. One can
observe that according to Equation (S11) the perturbation that a particle experiences depends on the
relative position of the other particles, and that such perturbation is not spherically symmetric. When
using a mean-field approach, it is instead assumed that, inside a cluster, each particle is surrounded by
all other particles uniformly distributed around it. It is therefore sufficient to compute the average of
Equations (S13) and (S14) over all possible orientations, which leads to a spherically symmetric
perturbation field having the following form:[80]
1
for 26ij ij pang
ij
R RR
T I (S15)
and:
1 1
1 for 26 4
ijij ij pang
p p
RR R
R R
T I (S16)
In this manner, we can write the following equation for the total hydrodynamic force experienced by a
cluster moving with a velocity u in a viscous fluid (obtained after a few algebraic manipulations):
2
0
6
1 4
pNR
g r r h r dr
F u (S17)
122
In Equation (S17), the function h(r) is the following:
11 for 2
4
for 2
pp
pp
rh r r R
R
Rh r r R
r
(S18)
Therefore, the final equation for the cluster hydrodynamic radius is the following:
2
0
,
1 4 , ,
ph
xR tR x t
g r x t r h r dr
(S19)
One can observe that, if there is no overlapping between particles, equation (S19) is identical to the one
developed for fractal clusters, since one only needs to use the second expression for h(r) in Equation
(S18). More explicitly, the integral in Equation (S19) needs to be split in two parts, as follows:
2
2
0 2
,1
1 4 , , 1 4 , ,4
p
p
ph R
pp R
xR tR x t
rg r x t r dr R t g r x t rdr
R t
(S20)
Even the particle-particle correlation function needs to be modified. In fact for DLCA clusters it
becomes:
f
f
,
nn2
3p
3
p,
0 if r ,
if r= ,4·
( , , ) · if r 2 ,
· ·exp if r 2 ,,
D
b
b
d
x
x
d
x t
x t
x
x
t
x t
tt
Nr t
t
ag r x t rR
c rr
R
(S21)
where t is the center-to-center distance of two particles in contact for a given extent of coalescence
(if 2 pt R t , there is no coalescence, otherwise 2 pt R t ) not to be confused with D
which is instead the Dirac delta function. Note that pR t and t are calculated by solving the
following equation system:
123
2
303
2
2 14
4
1 22 2
ADp
p
pp
p p
S tt R t
R t
RR t
t t
R t R t
(S22)
where 0pR is the radius of a single primary particle, while pR t is the radius of a primary particle
embedded in a doublet, undergoing coalescence (assuming that the coalescing particles maintain their
spherical shape). ADS t is the surface area of a coalescing doublet which can be estimated from the
following equation:
/min
min max minCtAD AD
ADC
dS t S t SS t S S S e
dt
(S23)
where minS is the surface area of a fully coalesced doublet, while maxS the one of a non-coalesced
doublet (to any extent). Both pR t and t change in time as a function of the coalescence extent.
t tends to zero as full coalescence implies an overlap of the geometrical centers of the two
coalescing particles. pR t instead is equal to primary particles radius 0pR if no coalescence occurs,
and reaches the value of 1/302 pR at full coalescence (i.e. the radius of a sphere obtained by the complete
fusion of two spheres of radius 0pR ). More details can be found in Jia et al.[74]
Several input parameters are required to employ the , ,g r x t function (cf. equation (S21)),
namely the cut-off length t , the cut-off exponent x and the empirical functions
, , nna x b x N x and ,c x t . As for ,x t , the cut-off length is defined as follows:[81]
1/ ,, 1.45 fd x t
px t R t x (S24)
The empirical functions , , na x b x N x and x , can be estimated instead using the following
empirical equation:[81]
124
m
m
x eF x d
x e z
(S25)
where x is the cluster mass and F x represents the generic function to be estimated (
, ,a x b x x or nnN x ). The values of the empirical parameters of equation (S25) are available
in the literature and are summarized in Table S1.[81]
Table S1
Values of the parameters , ,d e z and m in DLCA condition to be used in equation (S25), in order to
calculate the functions , ,a x b x x and nnN x in equation (S21).
Parameters of the g r function d e z m
nnN 2.0342 1.1477 0.9997 1
a 0.0095 4.1292 0.1997 2
b * 0.6425 6.2352 5.1747 1
2.1976 3.8377 -0.1784 1
* Note: for 7x , 0b
The function ,c x t is obtained by imposing the following equality:
2
0
1 4 , ,x r g r x t dr
(S26)
As a result, one gets:
3 3
3
,
4 11 2
3,
, ,, 241 ,
,
f
b x b x
nn bp
d x
I
p
t
f fNC
a xx N x t t
R t b xc x t
d x t d x t
R
x t t
x t x x t x
(S27)
125
Having clarified the different parameters occurring in the pair correlation function , ,g r x t , it is
possible to continue the integration of equation (S20), which can be re-written as follows:
2 22 3
0 0 2
, , ,
,
1 4 , , , , 4 , ,p P
P
A B C
Ph R R
PP R
I x t I x t I x t
xR tR x t
g r x t r dr g r x t r dr R t g r x t rdrR t
(S28)
Two different cases have to be distinguished, namely i) pt R t and ii) pt R t ,
as the integration boundaries have to be appropriately chosen.
In case i), for pt R t the three integrals become:
3 3nn3
p
,
1,
, 2 , ,, ,
24
2, ,
, ,
· 3
f
x x
d x t
f p f fINC
b
A
Np
C
b
I
bx xt t
t x
d x t R t d x t d x tx t x t t
x
N aI x t
R b
c
t x xR x t x t x
(S29)
f
4 4nn3
p
,
p
1
1, 2
4· 4
, , 1 , 1, 1 2 2, ,
,
,
,
f
b x b x
b
x t
B
df p
d
f fINC INC
I
d R t d d
N x a xx t t t t
R t b x
x t
x t x t
t x t x tc x t x t
R t x x x x
(S30)
p
1,
,
1, 121
,,, ,
,
f
f
df f
C i
x t
x t ncd
x t x tdc x t tx
dtI
x x x tR xt
t
(S31)
126
While for case ii), for pt R t :
3 3nn
3p
1, 2
4· 3
b x b x
A pb x
N x a xI x t R t t
b xR t
(S32)
4 4nn
3p
1, 2
4· 4
b x b x
pbB
a xNx t t R t t
R t bI
x
(S33)
f
22nn3
p
p
1
1, 2 2
4· 2
, ,, 1 1, 21 ,
,
b xb x
pC
dff f
i
b
d nc
N x a xx t t R t
t R t b x
x t x tc x t t
I
R t x x
d dx t
t xx
(S34)
After calculating ,hR x t according to the proper case, the following equation is employed to link it
to the , ,h effR x t , accounting for the rotational diffusivity:
, 12
,,
2 ,1 0.5 1 1
3 ,
hh eff
g
f
R x tR x t
qR x t
d x t
(S35)
To calculate the average hydrodynamic radius, hR t also the scattering structure factor has to be
known. This quantity can be obtained from the following integral:
2
0
1 sin( )( , , ) 1 4 ( , , )
qrS x q t r g r x t dr
qrx
(S36)
Note that only the first term of the particle-particle correlation function, corresponding to the Dirac
delta peak of the nearest neighbors, can be evaluated analytically:
2
0
sin1 sin( )( , , ) 1 4 ( , , )nn
q qrS x q t N x r g r
xx t dr
q qr
(S37)
The rest of the integral of the pair-correlation function in equation (S37) has been obtained by numerical
quadrature.
127
8.3.2 Calculation of gR x,t
The equation for the radius of gyration from the particle-particle correlation function (the same as for
the case of , ,h effR x t , equation (S21)) in the case of a non-coalescing cluster is the following:
2 2 4, 0
4( , ), ,
2g g pR R r g r x t drx
x t
(S38)
In the case of clusters undergoing coalescence, the integral remains the same, while the first term on
the right hand side, which is the radius of gyration of a primary particle, would need to be modified. It
has to become the radius of gyration of the unit element of a cluster undergoing coalescence. This is
not a well-defined object and many approximations need to be introduced. Since the relevance of this
term, especially for large clusters is almost negligible, we will keep it as the non-coalesced primary
particle radius of gyration. Therefore, the integration of equation (S38) using equation (S21) leads to
the following result:
5 52 2, 3
22
2
,
412
2 5
4 2 22· 1 ,
2
1,
, , ,,
,
f
b b
g g p nn bp
d
p f fINC
p
x t
x t t t t tt
x t t x t x tt
a xR R N x
x b x R
c R d d
x x R x x
x t
t x t
(S39)
The function ,c x t appearing in Equation (S39) is defined as for the hydrodynamic radius, cf.
equation (S27). Similarly, the functions , , , nna x b x x N x and ,x t have been previously
discussed (Table S1).
128
8.4 Derivation of the discretized Equations
Let us start to treat the PBE on the cluster concentration ,x t reported in equation (12) (in the main
text). The approximation for the latter balance, stated: 2
1
,G
i i
Ns x x
ii
x t t e
. Note that the GN
Gaussians are placed in the positions ix and have heights i . In order to reduce the problem size, let
us evaluate the function ,x t in the jx positions where the Gaussians are placed:
2
1
,G
i j i
Ns x x
j ii
x t t e
(S40)
Considering that [1... ]Gj N , one can re-write equation (S40) GN times:
21 2
2 211 1 1 2 1 2
2 2 22 21 2 1 2 2 2
2
1 1 21
21 2
1
1
,
,
,
G
i i
N NG G
G
G
i i NG
G
G Gi N iG
Ns x x
i s x xs x x s x xi NN
s x x s xs x x s x xi N
i
N Ns x x
ii
t et e t e t ex t
x t t e t e t e t e
x t
t e
2
2 2 2
1 1 2 2
1 2
NG
N N N N NG G G G G
G
x
s x x s x x s x x
Nt e t e t e
(S41)
Equation (S41) can be manipulated as follows:
22 2
11 1 1 2 1 2
22 2
21 2 1 2 2 2
2 2 2
1 1 2 2
1 1
2 2
,
,
,
N NG G
N NG G
GG N N N N NG G G G G
s x xs x x s x x
s x xs x x s x x
NN s x x s x x s x x
e e ex t tx t te e e
tx te e e
(S42)
Defining then 2expi j i j ix s x x one gets:
129
1 1 2 1 11 1
2 1 2 2 2 2 2
1 2
,
,
,
G
G
GG G G G G
N
N
NN N N N N
C
x x xx t tx t x x x t
tx t x x x
(S43)
A more compact writing of equation (S43) is then:
C (S44)
Where the matrix C , of size G GN N , is the so-called change of base matrix, allowing to “move”
from the cluster-mass space to the Gaussian one.
For further details on the method and to appreciate the versatility of this methodology, literature papers
should be considered,[77, 147, 148] where different applications systems (polymers, colloidal particles,
protein fibrils) have been approached with this technique. The vectors and matrices occurring in the
discretized balance are defined as follows.
The C matrix is reported in equation (S43), whereas the weighted 1 2,W W vectors read:
1
2
1 1
2 2
11
12
1/ , 1/ ,
1 1
1/ , 1/ ,
2 21 2
1/ , 1/ ,
0 0 0 0 0 0
0 0 0 0 0 0
0 0 0 0
f f
f f
f N f NG G
G G
W
W
d x t d x t
d x t d x t
d x t d x t
N N
C W C
C W C
x x
x xW W
x x
(S45)
The elements of the matrix (of size 2N N ) and of the vector size
2 1N ), are defined
as:
130
,
2
, , ,
,
,
exp
i j i jj i
i j i j i j
i j i j
i ji j
i j
s s
x s x x
x x x
s ss
s s
(S46)
The elements of the W vector of size 2 1N are calculated as follows:
1 2,
W WWi j i j
j is s
(S47)
131
8.5 Parameter Values employed in the simulations
Table S2
Viscosities of the continuous phases ( C ) containing 4 M NaCl at different temperatures.
oT C C Pa s
25 31.27 10
30 31.14 10
35 31.04 10
45 48.67 10
132
Table S3
Parameters used for simulations reported in Figure 2-3-4-5-6
Parameter Value
Bk 231.38 10 / KJ
T 298.15 K
C 31.27 10 Pa s
0pR 71.0 10 m
5 5 51.0 10 / 2.0 10 / 3.0 10
n 1.332
90 o
DLS 9532 10 m
C
70% MMA 60% MMA 50% MMA 30% MMA
810 s 34.3 10 s 21.0 10 s
133
Table S4
Parameters used for simulations reported in Figure 7, S1, S2
Parameter Value
Bk 23 11.38 10 KJ
T 298.15 / 303.15 / 308.15 / 313.15 K
C 3 3 3 41.27 10 /1.14 10 /1.04 10 /8.67 10Pa s Pa s Pa s Pa s
0pR 71.0 10 m
51.0 10
n 1.332
90 o
DLS 9532 10 m
C
298.15 K 303.15 K 308.15 K 318.15 K
50 % MMA 34.3 10 s 31.1 10 s 23.6 10 s 22.1 10 s
30 %MMA 21.0 10 s - 21.0 10 s -
134
8.6 Measured Fractal dimensions
Table S5 – fd t measurement of the different particles after 120 min of DLCA aggregation from
two different sources, ,S q t vs q and 0gR vs I
51 10 52 10
53 10
MMA % ,S q t vs q 0gR vs I ,S q t vs q
0gR vs I ,S q t vs q 0gR vs I
30 - 2.45 - 2.33 - 2.98
50 1.77 1.77 1.8 1.78 1.80 1.86
60 1.73 1.86 1.74 1.73 1.78 1.87
70 1.81 1.77 1.80 1.87 1.80 1.72
135
8.7 Results at higher temperatures
Figure S1 hR t vs N of the 30% MMA particles at 25oC (blue squares) and 35oC (red circles)
along with their corresponding simulations (continuous lines, which overlap as full coalescence is
reached already at 25oC). All the parameter values employed to obtain the model predictions are
reported in Table S4 (Appendix 8.5).
N[-]
0 20 40 60
Rh(t
) [n
m]
0
100
200
300
400
136
Figure S2 C versus 1/T for the 50% MMA containing particles, using 343.70 10A s and
42.55 10B and equation (31) of the main text
Note that the term B is of the same order of magnitude as the ones reported in the literature.[71, 74]
Being interested in producing partially coalesced clusters, the present model could be used to calibrate
the coalescence process exploiting the temperature dependence in Figure S2 and equation (31) in the
main text; such dependence is readily available once a few selected experimental points are obtained.
137
8.8 Symbol List for chapter 4
Symbol Meaning
a x Function in equation (S21)
b x Function in equation (S21)
,c x t Function in equation (S21)
C Base of change matrix to “move” from Gaussian to concentration space
d Parameter in equation (S25)
fd Internal coordinate of the distribution function representing the fractal dimension
,fd x t Average fractal dimension of an x -sized cluster at time t
fmd Fractal dimension of the aggregation regime
pD Particle diameter
e Parameter in equation (S25)
, ,ff x d t 2-D particle distribution function
F x Function to calculate , , ,na x b x N x x
jF Drag fore acting on the jth particle in a cluster
, ,g r x t Particle-particle correlation function
1 1 2
2 1 2 1 2
,
, , ,f f
g x x
g x x d d
Constitutive laws
Bk Boltzmann constant
m Parameter in equation (S25)
N Ratio between coalescence and aggregation characteristic time (equation (27) in the main text
nN Average number of nearest neighbor particles
q wave vector
r Radial distance from the center of a reference particle
,m jr Distance between the center of the mth and the jth particles
138
,gR x t radius of gyration of an x-sized cluster
gR t Average radius of gyration of the whole cluster mass distribution
,heffR x t Effective hydrodynamic radius of an x-sized cluster accounting for rotational diffusivity
,hR x t Hydrodynamic radius of an x-sized cluster
hR t Average hydrodynamic radius of the whole cluster mass distribution
0pR Primary particle radius
pR t Radius of a primary particle (embedded in a doublet) undergoing coalescence
is Gaussian overlapping parameter
, ,S x q t Structure factor of an x-sized cluster, measured at a wave vector q and at time t
ADS t Surface area of a coalescing doublet
minS Minimum surface area of a doublet (i.e. fully coalesced doublet)
maxS Maximum surface area of a doublet (i.e. doublet with no coalescence extent)
t Time
T System temperature
jmT Oseen (S12) or Yamakawa (S13) tensor
u Cluster velocity
, fv x d rate of coalescence
'jv Velocity perturbation experienced by the th particle in a cluster
x Internal coordinate in the distribution functions representing the cluster mass
Cluster friction tensor
1 2,W W Weight matrixes for numerical method
z Parameter in equation (S25)
Greek Letters
i Coefficient of the Gaussians used to approximate the distribution function
139
1 2 1 2, , ,f fx x d d
1 2,x x
Aggregation kernels used in the 2-D and 1-D PBEs
Cut-off exponent in the correlation function
, W Vectors employed in the discretized PBE
x Euler gamma function
INC x Euler incomplete gamma function
D Dirac Delta function
c Continuous phase viscosity
p Particle viscosity
Cut-off length in the correlation function
p Particle surface tension
A Characteristic time of doublet aggregation
C Characteristic time of doublet coalescence
COAL x Characteristic time of coalescence of an x-sized cluster
N Non-dimension time
P Characteristic time of the whole aggregation process
Occupied volume fraction
,x t 1-D distribution function obtained as 0th order moment of the 2-D distribution function , ,ff x d t
,x t 1-D distribution function obtained as 1st order moment of the 2-D distribution function , ,ff x d t
140
141
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Baptiste Jaquet
Nationality: Swiss
Born 17.03.89
Curriculum Vitae
Education
10/2011 – 9/2016 Doctor of Science in Chemical and Bioengineering , ETH Zurich
Thesis: Aggregation, Coalescence and Gelation of functional Nanoparticles
9/2010 – 10/2011 Master of Science in Chemical and Bioengineering, ETH Zurich
Thesis: Specific Salt Effect in Protein Aggregation
9/2007 – 7/2010 Bachelor of Science in Chemical and Bioengineering, EPF Lausanne
9/2004 – 7/2007 High School, Nyon
Working experience
10/2011-9/2016 Research Assistant, ETH Zurich, Morbidelli group
Publication List
Arosio, P., et al. (2012). "On the role of salt type and concentration on the stability behavior of a monoclonal antibody solution." Biophysical Chemistry 168–169: 19-27. Gu, S., et al. (2014). "Kinetic Analysis of the Catalytic Reduction of 4-Nitrophenol by Metallic Nanoparticles." The Journal of Physical Chemistry C 118(32): 18618-18625. Jaquet, B., et al. (2013). "Stabilization of polymer colloid dispersions with pH-sensitive poly-acrylic acid brushes." Colloid and Polymer Science 291(7): 1659-1667. Lazzari, S., et al. (2015). "Interplay between Aggregation and Coalescence of Polymeric Particles: Experimental and Modeling Insights." Langmuir 31(34): 9296-9305. Lazzari, S., et al. (2015). "Shear stability of inverse latexes during their polymerization process." AIChE Journal 61(4): 1380-1384. Jaquet, B., et al. (2016) “Effects of Coalescence on Shear-Induced Gelation of Colloids”, Langmuir, submitted