Andreas Silvia

Mike Hawes

DLCA Diffusion Limited Cluster Aggregation DLS Dynamic Light Scattering EBS Extremely Brilliant Source

Keywords:

An dall sa studio, Seán Ó Ríordáin vi with me. Dr. Thomas Zinn taught me about XPCS and assisted me on the beamline with the experiments. Dr. Peter Bösecke helped with early drafts of chapter 2 and along

ACKNOWLEDGEMENTS

Firstly, I would like to thank my thesis supervisor Dr. Theyencheri Narayanan for the experience you brought to this project, in particular your vast knowledge of the literature. You have taught me the high level of precision and perseverance required for the analysis of SAXS data, and your dedication to science, instrumentation and data has been an inspiration.

Dr. Sylvain Prévost, thank you for being my "encadrant". There have been many memorable moments in this thesis, and you have helped me navigate through these while also navigating your joint commitments to the ESRF and ILL. After four years in Grenoble this thesis is the highest mountain I have climbed. Thank you for coming on the expedition with me.

I would like to express my gratitude to the members of the jury for accepting to review this work: Dr. Nadia El-Kissi, Dr. Martine Meireles, Prof. Brigitte Pansu and Prof. Michael Gradzielski.

I would like to thank the EDCSV for their welcome and support. In particular I would like to thank the current director Dr. Guy Royal, and his predecessor Dr. Christelle Breton. Their advice has been a support in rocky moments.

The curved roads this thesis took gave me the opportunity to meet many colleagues from whom I learned a lot. In particular I would like to express my gratitude to Prof. Marguerite Rinaudo. The electrophoretic titration in chapter 5 was your suggestion, and chapter 4 benefited greatly from the samples you kindly provided. Sans vos conseils je n'aurais jamais fini. Je vous remercie pour nos discussions et votre générosité d'esprit.

Prof. Patrick Davidson and Dr. Patrice Rannou accepted to be in my "comité de suivi de thèse", and I thank them for their time and feedback. During my PhD, I had the opportunity to visit Prof. Patrick Davidson and Dr. Ivan Dozov in Orsay, and Prof. Michael Gradzielski in Berlin. They have been perfect hosts, and these short stays were educational.

In my time at ID02 I had the pleasure of working with wonderful colleagues. In particular Dr. Rajeev Dattani who spent many nights collecting data at the instrument

ABSTRACT

The macroscopic properties and long-term stability of a colloidal suspension are controlled by its microstructure and dynamics. Inter-particle interactions determine the microstructure and dynamics of the system to some extent. Colloidal systems are often metastable and so the inter-particle correlations can change with time (and so change the microstructure, and macroscopic properties of the system). As smaller colloidal particles can be used to form larger structures with different properties it is necessary to examine a system across various size scales.

In this PhD thesis a combination of small and ultra-small angle x-ray scattering (U)SAXS and x-ray photon correlation spectroscopy (XPCS) techniques are used to investigate colloidal complexation across both time and space. Spatial scales from nanometers (nm) to microns (µm) and time scales from milliseconds (ms) to months were examined to elucidate the formation pathway and structural evolution of the complexes. To uncover general mechanisms of broad relevance this work uses technical-grade, non-ideal materials. The study is on the complexation of silica nanoparticles and chitosan, a bio-sourced, cationic polysaccharide.

First, a new description for scattering data of polydisperse silica nanoparticles using a fractal model is employed. An investigation into the small-angle scattering of chitosan by neutrons and x-rays, and static light scattering reveals significant differences that are not readily explained. The investigation of nanoparticle and polysaccharide complexes uses kinetic SAXS measurements to study the formation pathways of the complexes, with rapid mixing experiments to access shorter times (stopped-flow apparatus). Combined USAXS and SAXS data are analysed to provide information on the complex structure within a state diagram. Salt was removed from the system using dialysis, and the subsequent behaviour of the system was investigated. It was found that the behaviour of the dialysed system differed to the behaviour of the non-dialysed system over long timescales. In the dialysed system the growth of crystallites of colloidal silica was observed. These results, along with an investigation into the temperature dependency of the system led to some understanding of the microstructure of this type of colloidal complex.

RÉSUMÉ

Les propriétés macroscopiques et la stabilité d'une suspension colloïdale sont déterminées par sa microstructure et le comportement dynamique du système, qui résultent des interactions entre particules. Les systèmes colloïdales sont souvent métastables ; les corrélations entre particules ont donc une dépendance temporelle, affectant la microstructure ainsi que les propriétés macroscopique du système. Les systèmes colloïdaux peuvent être hiérarchiques : de petites particules colloïdales peuvent être utilisées comme blocs pour construire de plus grandes structures ; il faut donc examiner un tel système sur plusieurs échelles, pour caractériser les propriétés des structures de différentes tailles.

Cette thèse exploite la technique de diffusion des rayons-x aux petits angles (DXPA), et aux très voire ultras petits angles (DXTPA, DXUPA), et la spectroscopie de corrélation de photons-x (XPCS) pour étudier la complexation colloïdale sur plusieurs échelles de temps et de dimensions. En examinant l'échelle spatiale de l'ordre de nanomètre au micromètre et l'échelle temporelle de quelques millisecondes à plusieurs mois le mécanisme de formation et l'évolution structurale des complexes peut être élucidé. Afin de relever les mécanismes d'une pertinence générale les matériaux employés dans ce travail sont de qualité technique. Cette étude porte tout particulièrement sur la complexation des nanoparticules de silice avec le chitosan, un polysaccharide cationique d'origine biologique.

Premièrement, une modèle fractal est employé d'une nouvelle façon pour décrire les courbes de diffusion des suspensions de nanoparticules de silice polydisperse. Deuxièmement, les divergences importantes, qui ne sont pas facilement expliquées, entre la diffusion aux petits angles de chitosan par les neutrons et les rayons-x ainsi que par la diffusion statique de la lumière sont révélées. Après avoir caractérisé les matériaux de base, le mécanisme de complexation entre nanoparticules et polysaccharide de charges opposés est étudié par des mesures DXPA cinétiques à l'aide d'un mélangeur rapide pour accéder aux temps courts. Les mesures statiques DXPA et DXUPA sont analysées pour élucider la structure des complexes dans le contexte d'un diagramme d'état. Finalement, la force ionique de la solution est fortement abaissée par dialyse, et le comportement subséquent du système est examiné. Étonnamment, aux temps courts, les dispersions dialysées ne x semblent pas significativement différentes des dispersions à forte force ionique ; aux temps longs, le comportement des échantillons dialysés diffère de celui des dispersions d'origine. Dans le système dialysé des cristallites de silice colloïdale sont observés. Ces résultats ainsi qu'une analyse de la dépendance du système en fonction de la température mettent en évidence la microstructure de ce type de complexation colloïdal.

GETTING STARTED

A colloidal system is any system that contains discontinuities (such as particles) between 1 nm and 1 µm dispersed in an otherwise continuous phase. [START_REF] Jones | [END_REF] Such a definition would indicate that colloid science is esoteric and far removed from daily life. This is not exactly true. One can make a convincing case for colloid science being the science of the everyday. It is common to consume milk for breakfast, milk is a colloid of butterfat globules and protein aggregates in a liquid medium; live in rooms with painted walls, paint is a suspension of pigment in liquid medium; look at liquid crystalline screens; and wear jewellery made of colloidal crystals such as opal. Indeed you have prepared colloids unwittingly if you have whipped egg whites, or cream to form a foam i ; made mayonnaise (a liquid-liquid emulsion); or custard (a gel).

It may then be surprising to know that in his Nobel prize speech in 1991 Pierre-Gilles de Gennes, characterised soft-matter systems using two words: complexity and flexibility. Colloids are considered complex systems because they have many particles which interact. As Anderson notes in an essay entitled "More is different":

The behaviour of large and complex aggregates of elementary particles, it turns out, is not to be understood in terms of a simple extrapolation of the properties of a few particles. 2 Note that there is a hierarchy of size scale present here -small particles collecting into larger aggregates (irregular clusters of individual colloidal particles). In moving from individual particles to aggregates there is a change in behaviour associated with the changing size scale. Aside from the many-body problem of dealing with lots of small particles, there is now the additional problem of secondary structures which exist on a bigger scale with their own individual behaviour. Such systems are familiar: the behaviour of atoms is not that of their i Foam is gas in a liquid or solid medium individual components -electrons, neutrons and protons; a molecule behaves differently to individual atoms. It is also common knowledge that differences in the micro-structure of a material lead to significant differences in the macroscopic properties of that material. Diamond and graphite are both composed of carbon atoms; but differing atomic structures ensure diamond is one of the hardest known materials, and graphite is a soft material commonly used for writing.

Colloidal systems are generally defined by their scale, which is also called the mesoscopic scale: between 1 nm and 1 µm. [START_REF] Jones | [END_REF] They form the bridge between the atomic (≈0.1 nm) and micro/macroscopic worlds. Writing about science at the mesoscale in 2000 Laughlin et al. state:

The success of the sciences of the small and large has been based on some simple general guidelines. One of these guidelines is the expectation that systems possess a unique favoured state and that the important motions of the system can be described as combinations of excitations that involve structures in some sense close to that favoured state.....But these principles are no longer sufficient in the mesoscopic realm. The phenomena of nonexponential dynamics and ageing suggest that many states, each potentially very long lived, can be found for systems with mesoscopic organisation. 3 This quote introduces the link between time and spatial scale. Intuitively we understand that timescales in the mesoscopic realm are different to those in the macroscopic world. Anyone who has left milk in the fridge too long will note that after some weeks it phase separates. This is caused by aggregation and sedimentation. It is accepted intuitively that milk exists in a metastable state -stable for a few weeks (a relatively long time); all the while slowly moving into a new state. Colloidal systems are often out-of-equilibriumthey do not occupy the lowest free energy state. Consequentially systems can be kinetically trapped in some other state, due to a specific sample preparation method for example. This can be useful as it leads to directed assembly, and can be used to engineer a desired state. 4,5 It can also result in additional complexity as the history of each sample requires consideration.

This is what makes colloids exciting to study. They exist across spatial scales and evolve in time. Additionally, an understanding of colloidal systems impacts other areas of science. This is most fully evidenced in the "colloids as (big) atoms" paradigm. 6 Colloids, especially colloidal hard spheres, can be used to study phenomena typically associated with the atomic scale. As colloids are both larger and slower than atoms they are easier to study in laboratory conditions.

Thermal energy and interactions in colloidal systems

Colloid science benefits from having a natural energy scale in the form of k B T , where k B is the Boltzmann constant and T is temperature. Colloids are dominated by thermal or Brownian motion. Particles are considered Brownian if sedimentation can be neglected with respect to thermal motion. In the spherical case this holds if particle radius, R, is less than the sedimentation length, l s e d defined in terms of thermal energy as

l s e d = k B T 4π 3 ∆ρR 3 g (1.1)
where ∆ρ is the density difference between the particle and solvent, and g is acceleration due to gravity. This implies that for the silica particles used in this thesis, which have density ≈1.7 g cm -3 at room temperature (≈300 K), a minimum radius of 0.6 µm would be required for sedimentation to dominate. Therefore the thermal motion of the individual silica particles used in this thesis, which are ≈11 nm in radius, prevents sedimentation. ii Additionally, the interactions between these particles occur on a scale of k B T . The length scale over which an electrostatic interaction occurs in a medium with high dielectric constant (such as water) is given by the Bjerrum length, λ B ,

λ B = e 2 4πε 0 ε r k B T . (1.2)
where e is the electronic charge, ε 0 is the permittivity of vacuum, and ε r is the relative dielectric constant. For example, the Bjerrum length in water at 25 • C is ≈0.7 nm. More specifically the Bjerrum length is the distance at which the Coulomb energy of a particle equals its thermal energy. If similarly charged particles come into contact at a distance much less than the Bjerrum length then they will repel each other. However, long-range Van der Waals forces, which are attractive interactions, are also present between colloidal particles. The macroscopic Van der Waals interaction energy, W V DW , between two spheres, of radius r 1 and r 2 , in the unretarded case follows

W V DW = -A 6l r 1 r 2 r 1 + r 2 (1.3)
ii It is judicious to note that colloidal sedimentation is a highly complex topic. The review of Piazza is recommended for more detail. 7 where l is the surface-to-surface separation between the two particles, and A is the Hamaker constant. 5 Due to this attraction particles need to be prevented from coming close to create a stable suspension. If the particles come close enough to each other the attractive interaction will be many times greater than k B T . This will cause irreversible aggregation.

Mechanisms of colloidal stabilisation

There are two common mechanisms of colloidal stabilisation: charge stabilisation and steric stabilisation. These methods can also be used in combination. Steric stabilisation involves covering colloidal particles with a layer of flexible polymer. When the polymer chains approach each other it is entropically unfavourable for them to interpenetrate which prevents the particles from coming close. 5 As the particles do not come into direct contact they stay stable in solution.

In terms of this thesis charge stabilisation is the more important stabilisation mechanism. This is often synonymous with DLVO theory (named after Derjaguin, Landau, Verwey and Overbeek). 5 Colloidal particles with a surface charge suspended in an electrolyte solution (even pure water can be considered an electrolyte containing H 3 O + and OH -ions) will lead to a distribution of counter-ions and co-ions around the colloidal particles. 8 Counter-ions which have the opposite charge to the colloidal particles will accumulate at their surface while co-ions (which have the same charge as the colloids) will be depleted. This leads to the so-called "double-layer" phenomenon where there are alternate layers of counter-and co-ions. In bulk solution, far from the colloid surfaces the counter-and co-ion concentrations will have a constant average value. When two double layers come into contact a repulsive potential develops; therefore like-charged colloidal particles repel each other. [START_REF] Lekkerkerker | Colloids and the depletion interaction[END_REF] This double-layer repulsion acts over a length known as the Debye screening length λ D λ D = 1 8πλ B n s (1.4) where n s is the salt number density, and λ B is the Bjerrum length described above. [START_REF] Lekkerkerker | Colloids and the depletion interaction[END_REF] It is then clear that increasing salt concentration, n s , decreases inter-particle repulsion length (Debye length) by screening colloidal surface charges. In pure water λ D is ≈1 µm. Calculating the Debye length precisely requires knowing the concentration of ions in solution. iii For a buffer solution of 0.3 mol l -1 acetic acid and 0.2 mol l -1 sodium acetate as used in chapter 5 iv

iii Such calculations are simplifications based on the approximation that λ D = 0.304 × 10 -9 / I nm where I is ionic strength. 5 iv The buffer solution is at pH 4.5 and has pK a of 4.8.

the ion species are essentially due to fully dissociated sodium acetate. In this case λ D will have dropped to ≈0.7 nm. In chapter 6 most of the salt is removed (the buffer is now composed of 0.05 mmol l -1 acetic acid and 0.03 mmol l -1 sodium acetate. The removal of salt causes λ D to increase to ≈46 nm.

The interaction energy of a DLVO system, W D LV O is then the sum of Van der Waals attraction, W V DW , and double-layer repulsion

W D R W D LV O = W V DW + W D R .
(1.5)

At low-salt concentrations W D R is at several k B T and double-layer repulsion dominates.

In high salt conditions W D R is low and W V DW dominates. Particles will aggregate and sediment. [START_REF] Lekkerkerker | Colloids and the depletion interaction[END_REF] In the intermediate case a secondary minimum can appear in the inter-particle potential where particles become weakly flocculated; however the secondary minima is not deep and the particles can be easily re-dispersed. 5 DLVO theory is now more than 70 years old and in this time it has been highly successful. [START_REF] Overbeek | [END_REF] The theory continues to be refined and as such there are numerous review articles describing how the theory has developed. The review by Ninham 11 is recommended as is the review by Hansen and Löwen. 12

Depletion interactions

There is an additional interaction in colloidal systems with two or more components which DLVO theory does not consider. This is called the depletion interaction and is an attractive force arising from a loss in configurational entropy. [START_REF] Lekkerkerker | Colloids and the depletion interaction[END_REF] Consider large noninteracting colloidal spheres mixed with smaller non-adsorbing polymers. If the two spheres approach each other at a certain point there will be a volume between the spheres from which the polymer is excluded. The osmotic pressure on the spheres is then unbalanced. v The excess pressure will bring the spheres closer. This increases the overall volume available to the polymers and so their free energy is minimised. [START_REF] Lekkerkerker | Colloids and the depletion interaction[END_REF] One example of a full treatment of the topic of depletion interactions and colloids is given by the book of Lekkerkerker and Tuinier. [START_REF] Lekkerkerker | Colloids and the depletion interaction[END_REF]

Colloidal structure

The structure of a colloidal system is governed by volume fraction and inter-particle interactions. As we have seen the interaction energy between colloidal particles is on v Osmotic pressure is the "excess pressure required to maintain osmotic equilibrium between a solution and the pure solvent separated by a membrane permeable only to the solvent." [START_REF] Nič | IUPAC Compendium of Chemical Terminology[END_REF] the order of k B T . This means that systems are sensitive to external stimuli (hence the name "soft" matter). Due to the relatively large size of colloidal particles (when compared to atoms or molecules) the dynamics of colloidal systems are slow. Ideally, over long timescales, the system will move to an optimally bound state otherwise it will result in an arrested state such as a glass or gel. This process can take long times (weeks, months, or even years) before the system moves to an equilibrium state or arrested state.

When the states of a colloidal system are dependent on the colloidal volume fraction they can generally be described as gaseous, liquid, crystalline, or amorphous. When the stability of the system is affected a fractal state can result. The structure of aggregated states (such as crystalline or fractal) is exhibited across multiple length scales. Take, for example, a colloidal crystal which is analogous to an atomic crystal and defined as having long-range periodicity in at least one direction. Macroscopically a crystal is observed, like opal. Microscopically a crystal lattice is observed. This crystal lattice is built from colloidal particles. Finally, the colloidal particles are themselves composed of atoms or molecules.

Amorphous states include colloidal glasses and gels which are closely related -the terms are sometimes used interchangeably. Typically the volume fraction of a glass is higher than the volume fraction of a gel (with the exception of Wigner glasses). A gel is defined by IUPAC as "a non-fluid colloidal network or polymer network that is expanded throughout its whole volume by a fluid". [START_REF] Jones | [END_REF] In her 2007 review of colloidal gels Zaccarelli describes the implications of this definition. [START_REF] Zaccarelli | [END_REF] A gel is a low-density state which does not flow, and shows no significant order. At large length scales there may be an associated length arising from the fractal-like structure of the material.

Fractality in colloidal aggregation

Both gels and aggregates can be described using the geometry of fractals. The concept of a fractal originated with Benoît Mandelbrot, a French mathematician, who took time, "to investigate the morphology of the amorphous." [START_REF] Mandelbrot | The Fractal Geometry of Nature[END_REF] His mission was to describe shapes that could be described by scaling laws, such that the degree of their particular irregularities was identical at all scales. We depart from the ideality of the mathematical world and exist in the realm of the experimental and imperfect. As such we define a fractal as an object that is self-similar over many length scales. [START_REF] Herrmann | On Growth and Form[END_REF] It is characterised by a fractal dimension called the Hausdorff-Besicovitch dimension. The concept of fractal dimension is so important to the theory of fractals that Mandelbrot uses it to define what a fractal is. That is a fractal set .... is a set in a metric space, for which Hausdorff Besicovitch dimension D > topological dimension D T

The topological dimension corresponds to the Euclidean geometry with which we are all familiar: cubes are in three dimensions, squares are two-dimensional, lines one-dimensional and a point is zero dimensional. Hausdorff-Besicovitch dimensions correspond to both traditional integer topological dimensions, and the non-integer values existing between them. They can be understood as a degree of roughness, or chaos in the topology of a system (or of objects containing much more information). vi In this thesis the concept of a fractal is invoked throughout as a convenient method of describing a variety of colloidal systems. It is neither a new nor a fashionable idea to use fractals to describe colloidal systems. In pioneering computer simulations, Witten and Sander discovered that aggregates formed by diffusion-limited cluster aggregation (DLCA) grew as "critical objects". [START_REF] Witten | [END_REF] Experiments showed that depending on the repulsive energy barrier between particles aggregates were formed either by DLCA, or by reaction-limited cluster aggregation (RLCA). 20 Both pathways generated aggregates with a characteristic fractal dimension, irrespective of the chemistry of the system. This is understood as an example of universality, and would not be possible without the use of fractality as a concept. 21 Aggregation limited by either diffusion or reaction also highlights the link between structure and interactions. Diffusion limited cluster aggregation occurs when there are negligible repulsive inter-particle interactions, and so the aggregation rate is controlled by the time it takes for particles to diffuse to each other. Reaction-limited cluster aggregation requires a small repulsive barrier between particles. The aggregation rate is limited by the time it takes particles to overcome this repulsive barrier. Recall that inter-particle interactions and thermal fluctuations are a similar order of magnitude in colloidal systems. Note that most colloidal systems are electrostatically stabilised so RLCA is delayed for extended timescales. However, as described in a review of "Fractal-like structures in colloid science" Lazzari et al. 22 note that ... when the complexity of the colloidal system analysed increases; long-range attractive forces, high particle concentrations, polydispersed particles, and patchy particles give rise to significant deviations from the "idealised" picture [DLCA and RLCA] mentioned above. 22 One of the interesting cases described by Lazzari et al. is the work of Kim et al. 23 who found that oppositely charged polystyrene particles had a fractal dimension of 1.2 upon vi A simple example, described by Mandelbrot is that of map-making. When the map of a coastline is scaled more bays and peninsulas are revealed. Each magnification reveals more detail, which would greatly increase the length of a hiker's walk (length of the border between Spain in Portugal was variously 987 and 1214 km long)! [START_REF] Richardson | General systems: Yearbook of the Society for the Advancement of General Systems Theory[END_REF] In using the concept of fractals we accept that a coastline's length contains more information than in a traditional one-dimensional line. aggregation, and that this fractal dimension could be tuned by changing the salt concentration of the system. As such structures are almost linear this gives hope that by tuning the inter-particle interaction a desired structure can be achieved and the challenge of building "designer soft-matter" will be reached.

The investigation of familiar systems using the new tool of fractal dimension was considered exciting, not just in colloid science; but across all domains of physics in the 80s and early 90s, culminating in Kadanoff's article pondering if any physics was to be found within this mathematical eccentricity. 24 Kadanoff warned that ... without organizing principles, the field [of fractals] tends to decay into a zoology of interesting specimens and facile classifications. 24 His point was valid, while some breakthroughs were made (such as the case of colloidal aggregation) overall progress was limited. Interest in colloidal fractals waned over time, and was increasingly faced with questions on the validity of many experimental descriptions of fractality. This is demonstrated in a literature review by Avnir which found that experimental fractality typically spans 0.5 to 2.0 orders of magnitude. 25 At the beginning of this section we required an object to be self-similar over many length scales to be defined as fractal, it would be found that the literature falls short of this definition.

In spite of such criticism which is justified, this thesis invokes frequently the concept of a fractal, and in particular fractal dimension. The reasoning for this is straightforward, and was proposed by Avnir as reason for using the fractal as a descriptor. To quote Avnir:

In many instances, the choice is either to use the limited-range data or to discard it altogether and not have even an approximate picture of the studied object.... Fractal geometry provides a proper language and symbolism for studies of ill-defined geometries. 25

Binary systems

The ultimate goal of much colloidal research is something called "Designer softmatter". 26 In an article entitled "Grand Challenges in Soft Matter Phyics" in 2018 Van der Gucht concluded that the field's ultimate challenge is two-fold ... to find microscopic explanations for macroscopically observed behaviour. The inverse problem -designing a material with a specific desired propertyis obviously much more difficult. Yet, it would be highly desirable.... 26 There are some ongoing attempts to create designer soft-matter. These include creating responsive materials -such as lightweight and flexible body armour; 27 or materials which have selective binding. 28 Such a material has implications for drug design, where a colloidal particle filled with an active ingredient can be delivered to a particular target. While immense progress has been made on such materials, their everyday application remains a challenge. The present task is the first challenge posed by Van der Gucht, "to find microscopic explanations for macroscopically observed behaviour". 26 One simple way of illustrating the link between macroscopic observation and microscopic detail is to create a phase diagram of a system. A phase diagram represents the physical state of a system at thermodynamic equilibrium for a given set of conditions, for example composition and temperature. This provides a map for others to reach the same macroscopic state by engineering microscopic parameters. Perhaps the most famous phase diagram in colloid science was created by Pusey and Van Megen. 29 It describes how a system of sterically stabilised colloidal spheres behave as a fluid before increasing their volume fraction at which point crystallites form and there is fluid-crystal coexistence. 29 Further increasing volume fraction samples become fully crystallised before reaching the highest volume fractions at which time colloidal glass is formed. 29 Such a phase diagram is easily understood and can be translated to other colloidal systems.

Work continued on more complex binary systems, such as colloids and non-adsorbing polymers for example, by Lekkerkerker et al., 30 who showed how depletion interactions could cause complex phase behaviour. By understanding the force driving the phase behaviour (depletion) a phase diagram was generated which provides a blueprint for others looking to understand similar systems. 30 Work continues on understanding systems of charged colloids. DLVO theory shows that background solvent mediates inter-particle interactions. The background solvent cannot be treated as a homogeneous entity. The solvent properties at the colloid interface differ from those of the bulk solution. Inter-particle interactions are modified by the background electrolyte. Additionally, with the addition of depletants, or patches of charge on the particle surface it is possible to create complex potentials with short-range attraction and long-range repulsion. Ideal colloidal systems with such inter-particle potentials have been examined by computer simulation (Monte-Carlo) techniques by the group of Godfrin et al., 31,32 the group of Mani et al., 33 and the group of Mossa et al. 34 amongst others. Their results describe a theoretical phase diagram. At low volume fraction small aggregates are formed in a fluid or monomer phase. As volume fraction increases extended clusters are formed and the system eventually moves into a percolated phase followed by gel formation. This is supported by experimental work by Zhang et al. 35 who additionally found that by controlling short-range attraction in their system the gel growth mechanism could be controlled.

Mani et al., 33 Mossa et al., 34 and Zhang et al. 35 found that under particular conditions elongated colloidal structures resembling rods were present. It would appear that interparticle interactions can be used to create complexes with a particular shape. Such papers also note the existence of theoretically predicted equilibrium cluster phases. 33 Clusters reach a specific size beyond which it becomes energetically unfavourable to grow further. 33 This illustrates that systems of charged colloids interacting through short-range attraction and long-range repulsion have interesting phase behaviour.

This thesis considers a binary system of charged colloids, specifically interactions between oppositely charged polyelectrolyte and colloids. This adds a further level of complexity to inter-particle interaction potentials as multiple species have different geometries and need to be considered differently. This is an experimental thesis and so it simplifies interpretation where necessary. The main inspiration is the work of Shi et al. 36 They examined a system of silica nanoparticles and chitosan, a polyelectrolyte drawing a state diagram which indicated the presence of rod-like complexes. 36 As demonstrated by the theoretical work of Mani et al. 33 and Mossa et al. 34 such structures appear to be a consequence of complex inter-particle interactions. It is therefore interesting to examine the wider behaviour of such a system.

Structure of the thesis

The next chapter, chapter 2 introduces the scattering techniques used in the thesis, along with a description of beamline ID02 where the majority of the experiments were conducted. Following this are two chapters characterising the materials used in the thesis. In order to analyse the complex system examined in later chapters there is a need to understand its component parts.

Chapter 3 examines small silica nanoparticles. Small silica nanoparticles have a large degree of surface roughness, porosity and polydispersity which makes analysis challenging. A paper from the Institute for Reference Materials and Measurements illustrates the challenge of characterising small silica nanoparticles. 37 Four different certified particle size values are published. The values are dependent on the analysis method. 37 There is a need for a simple way of characterising such particles reliably. In chapter 3 the appropriateness of a fractal model called the Hurd and Flower model to describe scattering from small silica nanoparticles is tested. 38 Chapter 4 examines chitosan which is a biologically derived polysaccharide formed from partially deacetylated chitin. Although chitosan is widely used for water purification, and for biomedical applications the solution structure of chitosan is not fully established. Work by Philippova et al. 39 and Novoa-Carballal et al. 40 indicate that chitosan forms aggregates in solution. These aggregates contain hydrophobic domains. Surprisingly these independent studies conclude that the acetyl groups on the chitosan chain do not contribute to the formation of these hydrophobic domains. Their origin remains unknown. The aim of chapter 4 is to use a combination of scattering techniques (SAXS, SANS and SLS) vii to examine the structure of chitosan in solution.

The second half of the thesis examines the complexes formed when silica nanoparticles and chitosan are combined. Chapter 5 continues the work of Shi et al. 36 which indicated that when silica and chitosan are combined rod-like complexes are formed. This is supported by simulation studies of colloids interacting via short-range attraction and long-range repulsion by Mani et al. 33 which indicate that non-equilibrium clusters can occupy an elongated shape which resembles a Bernal spiral. In chapter 5 elongated clusters are identified and an attempt is made at aligning them using shear. More broadly chapter 5 examines the silica-chitosan state diagram and investigates how complexes are formed, how their structure changes within the state diagram, and the final state of the complexes.

Finally, in chapter 6 the effect of removing salt from the system while keeping pH and volume fraction constant is examined. A literature review by De Kruif et al. 41 on systems of protein-polysaccharide indicates that changing the ionic strength of oppositely charged systems should have a large effect on the system. Typically investigations into the effect of ionic strength on such a system are performed by dilution or mixing experiments, 42,43 or by simply preparing samples with differing salt concentration. 44 In chapter 6 the complexes are formed and then salt is removed by dialysis keeping both pH and volume fraction constant. Changes in the complex structure compared to the non-dialysed system are examined. Long-term storage tests are then used to identify the final state of the system. vii Small-angle x-ray scattering (SAXS), small-angle neutron scattering (SANS), static light scattering (SLS) CHAPTER 2

SMALL-ANGLE X-RAY SCATTERING METHOD

Time is important in this work, and so interrogating time's effect on the system is important. There are three ways of doing this: static, kinetic and dynamic measurements. In choosing techniques to study the system we are constrained by both size-and timescales.

The silica nanoparticles used have diameters of a few tens of nanometers. When combined with polyelectrolyte the resulting structures are micron-sized. This thesis characterises the hierarchies of order present, and requires characterisation techniques capable of structural resolution across these three orders of magnitude for turbid samples. Small-angle x-ray scattering, SAXS, and ultra small-angle x-ray scattering, USAXS, can be performed statically on the size-scales required. This means that a sample is measured a single time without any change occurring in the sample.

Alternatively, the sample can be perturbed, and a series of measurements observe how the sample reacts to the perturbation. These are time-resolved, or kinetic measurements. [START_REF] Narayanan | Structure from Diffraction Methods: Inorganic Materials Series[END_REF] Kinetic SAXS and USAXS measurements at ID02 can be performed with ≈ ms temporal resolution to follow structural changes in the sample. In this thesis, such a capability was needed to understand the kinetics of cluster formation and its link to long-term sample stability.

Additionally, the equilibrium dynamics of the system were examined. Dynamic studies involve exposing the system to many fast measurements to follow particle movements. This study requires dynamic resolution in the spatial range from ≈100 nm to ≈1 µm. X-ray photon correlation spectroscopy, XPCS is used to investigate the dynamic behaviour of the system. XPCS is described in chapter 5.

SAXS provides most of the evidence collected in this thesis. This chapter introduces scattering techniques including basic analysis tools and the instrumentation required. k 0 k q θ Fig. 2.1 Schematic defining the incident wavevector k 0 , scattered wavevector k, scattering angle θ and scattering vector q 2.1 Small-angle scattering of particulate systems Small-angle scattering (SAS) of incident radiation from a sample of interest is a technique for characterising microstructure across several orders of magnitude. The concept of small-angle scattering, originating with the work of Debye in 1915, [START_REF] Debye | [END_REF] is that scattered radiation from randomly orientated bodies interferes in the far field. Common sources of incident radiation include x-rays, i neutrons and light. Regardless of the radiation used the underlying principles of scattering remain the same.

SAXS signal originates from electron density fluctuations in the range of nm-µm in a sample of interest. In a scattering experiment the direction of a wave ii of incident radiation, denoted by the wavevector k 0 , changes as a result of interactions between the wave and matter. This gives the scattering vector q, illustrated in fig. 2.1 where q = kk 0 (k denotes the scattered wavevector). Only elastic scattering events are considered in SAXS. The magnitudes of the incident and scattered waves are identical ( |k 0 | = |k|) but their i X-rays, are a form of electromagnetic radiation with high energy and a short wavelength. According to Als-Nielsen 47 the wavelength of an x-ray is on the order of an Ångstrom (10 -10 m) and thus they interact with materials on the atomic scale.

ii Definitions of a wave is given in section A.1 direction is different. This difference can be written as:

q = |q| = 2|k| sin θ 2 = 4π λ sin θ 2 (2.1)
which demonstrates why q is the variable that forms the basis of the following discussion. The structures observed in SAXS are relatively large (from several Å to µm ). Examination of (2.1) shows that the units of q are the inverse of λ. Figure 2.2 plots the inverse of q against θ illustrating why small angles are required to study structures on these scales. The long detector tube (length 34 m) at ID02 enables small angles (≈10 µrad) to be resolved on the detectors. [START_REF] Narayanan | [END_REF] Contrast X-rays are scattered elastically by heterogeneities in electron density. The scattering length is a measure of the strength of the interaction between x-rays and an atom or molecule. Far from the absorption edge scattering length, b , is the product of the number of electrons, Z, and the classical (Thomson) radius of an electron, R e : iii b = r e Z. [START_REF] Feigin | Structure Analysis by Small-Angle X-Ray and Neutron Scattering[END_REF] Experimentally we define a scattering length density, SLD, in units of nm -2 where S LD = ρ s a m p l e N A r e (2.2)

iii The Thomson scattering length, or classical radius of an electron is 2.82 × 10 -6 nm. [START_REF] Als-Nielsen | Elements of Modern X-ray Physics: Second Edition[END_REF] where N A is the Avogadro number, and ρ s a m p l e is the electron density expressed in units of mol nm -3 . Contrast, ∆ρ, is the scattering length density difference between a specimen (particles of volume V and electron density ρ(r)) and the suspending medium, ρ s :

∆ρ = V -1 V [ρ(r) -ρ s ]d r (2.3)
such that the scattering length excess of particle j is then related to its contrast as follows:

b j (q) = V j ∆ρ(r j ) exp(-i q • r j )d 3 r j (2.4)
for samples with poor contrast, ∆ρ ≈ 0, the features of interest become potentially invisible.

In such a case it can be interesting to use neutrons. Neutron contrast comes not from electrons; but from the nucleus of the atom. Therefore x-rays and neutrons provide complementary information. iv

Differential scattering cross-section

A SAXS instrument measures the intensity resulting from interfering scattered waves on the detector. The intensity, I (q), is the square of the amplitude, F (q), of scattered radiation I (q) = F (q)F * (q). This measured scattering intensity is related to a quantity called the differential scattering cross-section, d σ d Ω , per unit solid angle Ω (in dimensions of mm 2 sr -1 ). This relates the number of x-ray photons scattered, I s c , into a detector that subtends a solid angle, ∆Ω, to the number of photons per unit area in the incident x-ray beam, I 0 /A, v and transmission T r vi as follows:

d σ d Ω = I s c I 0 A ∆Ω 1 T r (2.5)
Practically, the calculated value is

d σ d Ω ×1/A (which is dimensionless) i.e.
it is not neceessary to evaluate the beam cross-section.

iv Chapter 4 exploits this complementarity. v A is area. vi Transmission is the ratio of the transmitted beam to the incident beam, T r = I 1 /I 0

Data corrections

Ideally SAS measurements would be comparable between samples, and between scattering instruments. This is possible with correct data treatment. An important correction is background subtraction. In SAXS, samples are typically measured in a quartz capillary. In this thesis colloidal samples are studied -a suspension of colloids in a buffer solution.

A SAXS measurement includes information on the quartz capillary and buffer solution, which need to be removed. An appropriate background should be measured for each sample, preferably in the exact same sample holder. In this case, if the volume fraction of colloids (φ) is negligible, the background would be the buffer solution the colloids are suspended in and container holding the solution. This measurement needs to be subtracted from the sample's scattering curve. The resulting curve can then be considered background subtracted.

This treatment does not remove the dependence on the illuminated volume of sample in the beam. This is achieved by normalising d σ d Ω × 1/A by the sample thickness, giving:

d Σ d Ω = 1 V d σ d Ω (2.6)
The value

d Σ
d Ω is called the specific differential scattering cross-section per unit solid angle, and is expressed in dimensions of reciprocal length. [START_REF]Neutron, X-rays and Light[END_REF] It is proportional to the number density of particles and so can be readily compared between samples. The absolute calibration is implemented by the knowledge of instrument normalisation factors for each detector and electronic gain setting. This can be cross-checked using a calibration standard with known differential scattering cross-section such as water.

Intensity I (q) is often used in place of d Σ d Ω as reduced data still contain some instrumental dependence, in particular on instrumental resolution. In this document, I (q), refers to data which have been corrected and converted to the absolute intensity scale.

Form factor

A scattering curve plots intensity, I (q), against the magnitude of the scattering vector q. A useful first step in analysis is to examine the case of non-interacting (or highly dilute) particles. When inter-particle interactions are neglected I (q) is linked to the contrast (see eq. (2.3)), particle number density, N p , particle volume, V , and a quantity called the normalised form factor, P (q) (where P (0) = 1), as follows:

I (q) = N p (∆ρ) 2 V 2 P (q) (2.7)
The form factor describes the morphology, size, and internal distribution of scattering length densities of particles.

Polydispersity

In the above we have assumed a monodisperse system of identical particles. In realworld systems this is not the case. One of the objectives of the following document is to interpret scattering data from polydisperse systems. A polydisperse system is one where particles have sizes which may vary discreetly or continuously (which can be considered as any experimental system of particles). The polydispersity can be accounted for by introducing a distribution function D N (r ) such that its integral D N (r )d r accounts for particles whose effective size, r , lies in the range r, r + d r . The form factor then becomes:

P (q) = 〈F 2 (q)〉 = ∞ 0 〈F 2 0 (q, r )〉m 2 (r )D N (r )d r (2.8)
where F 2 0 (q, r ) is the normalised form factor of particles of size, r . The factor m(r ) becomes significant for polydisperse particles with inhomogeneous scattering length density e.g. core-shell particles. In this case the volume of the particle and scattering length density cannot be decoupled and so m(r ) is included to account for this. [START_REF] Feigin | Structure Analysis by Small-Angle X-Ray and Neutron Scattering[END_REF] It is common to encounter multimodal populations, for example particles and their aggregates. Each population has a continuous size distribution. Such a situation is described in this thesis.

Scattering from a sphere

The simplest object to examine is a sphere. Scattering from a system of homogeneous spheres is therefore evaluated as

P (q) = 3 s i n(q R) -q Rc os(q R) (q R) 3 2 (2.9)
and a simulation of scattering from such a curve can be seen in fig. 2.3. One observes that the effect of polydispersity is to smear the oscillations observed in the monodisperse case. It is also observed that for a high degree of polydispersity there is an increase of intensity at low-q. This is due to the presence of larger particles. The slope of the curve at high-q is constant for all samples. This is the so-called Porod region described in section 2.1.3. As the particle-solvent interface is identical irrespective of size distribution the slope of q -4 remains constant. The minima of the first oscillation, in fig. 2.3, is located at q R = 4.49, and the distance between the minima of subsequent oscillations tends to π.

Structure factor

Thus far we have discussed the form factor and understood it as being linked to the forward scattering of the system. In more concentrated systems where particles interact, the structure factor becomes important. We can rewrite the intensity, for monodisperse systems, as I (q) = N p (∆ρ) 2 V 2 P (q)S(q) (2.10)

The structure factor S(q) describes the spatial correlations between particles and can be related to thermodynamic quantities such as the compressibility and the pair correlation function g

(2)

N (r 1 , r 2 ) as follows

S(q) = 1 + N p d re iq•r (g (2) N (r ) -1)) (2.11)
where in a system of N particles the pair correlation function describes the correlations of two particles at positions r 1 and r 2 respectively in the presence of the other N -2 particles.

In the limit q → 0, an expression with the compressibility of the system, χ t is obtained:

S(0) = N p k B T χ T (2.12)
This quantity is related to fluctuations in particle number density characterised by mean square deviations from the average 〈N 〉. The implications of this can be seen in the general definition of κ, the compressibility of a solid:

κ = - 1 V ∆V ∆ p (2.13)
where V is the volume of a solid to which a small pressure ∆ p is applied and the deformation of the solid by ∆V is observed. [START_REF] Reif | Statistical physics[END_REF] In this case we are considering an isotropic, homogeneous colloidal solution, not a solid state material. As such our description contains a kind of coarse graining -the pair correlation function of solvent molecules is smeared out and treated as a system-dependent constant; and it is the pair correlation function between colloidal particles we are interested in. [START_REF]Neutron, X-rays and Light[END_REF] It is thus possible to write the compressibility equation for such a case:

N p k B T χ T = 1 + N p d r(g (2) (r ) -1) (2.14)
which links the macroscopic properties of the system, in the form of χ T to the microscopic behaviour in the form of g (2) (r ), which in turn can be linked to the structure factor S(q) and observed scattering I (q). [START_REF] Klein | Light Scattering: Principles and Development[END_REF] These links are most commonly made by solving the Ornstein-Zernike equation for a homogeneous fluid

h(r ) = c(r ) + N p d r ′ c(|r -r ′ |h(r ′ ) (2.15)
which describes two-particle correlation functions separated into two parts: 'direct', described by c(r ), and 'total', contained in the integral. [START_REF] Ornstein | [END_REF] The Ornstein-Zernike equation cannot be solved directly. c(r ) is unknown and therefore closure relations are used to solve the Ornstein-Zernike equation. These closure relations require further approximations. [START_REF] Klein | Light Scattering: Principles and Development[END_REF] In this thesis the mean spherical approximation, c(r ) = -u(r )/k B T (where u(r ) is the interparticle potential), and Percus-Yevick closure relation, c(r ) = g (r ) 1 -exp(-u(r )/k B T ) , are used to describe interactions between approximately spherical particles.

Effect of polydispersity on the structure factor

As can be seen in (2.12) the structure of the system is linked to particle number density fluctuations. In the case of polydisperse particles this is complicated by their broad size distribution. [START_REF] Klein | Light Scattering: Principles and Development[END_REF] I (q) = I e (q)〈N 〉 m α,β=1 F α (q)F β (q)S αβ (q)

(2.16)

where S αβ (q) is known as the Bhatia-Thornton structure factor, and is useful for binary systems (with more components it becomes complicated to express). [START_REF] Klein | Light Scattering: Principles and Development[END_REF] By including a continuous size distribution p(σ) in the Ornstein-Zernike equation, where 〈σ〉 is the mean particle diameter, it is possible to solve the structure factor for polydisperse systems by using the appropriate closure relations. 54

h σ (r ) = c σ (r ) + N p ∞ 0 d σ p(σ) d r ′ c σ (|r -r ′ |)h σ (r ′ ) (2.17)
Note that in this case the expression for compressibility seen in (2.12) ( S(0) = N p k B T χ T ) no longer holds; however the compressibility equation (2.14) can be modified such that the microscopic and macroscopic behaviour can be linked together. [START_REF]Neutron, X-rays and Light[END_REF] Scattering techniques are one of the few ways of gaining information on inter-particle interactions in-situ; extracting this information is not always straightforward. q monodisperse 10% Polydispersity 40% Polydispersity Fig. 2.5 Change in hard sphere structure factor (implemented with the monodisperse approximation I (q) = F 2 (q, r )S(q)) with changing polydispersity. Polydispersity is introduced with a Schulz-Zimm distribution. Volume fraction, φ=0.25. Data are simulated and not in absolute scale.

Data analysis

Elastically scattered x-rays have identical energies; but differing phases. What is detected is I (q) the modulus of the scattering amplitude squared. Information about the phase is therefore lost. This amplitude is related to the electron-density distribution of the scatterer by Fourier transform. In turn the intensity is related by Fourier transform to a quantity called the correlation function γ (r) which can be deconvoluted to retrieve the scattering length density profile of the scatterers. This deconvolution procedure also has a loss of information as illustrated in fig. 2.6. The information loss inherent in retrieving structural knowledge from scattered waves we call here the 'inverse scattering problem'.

When the colloidal system is polydisperse the SAXS model selection is non-trivial. The objective of small-angle scattering studies is to provide a physical explanation for the data. Three approaches to interpreting SAXS data are used in this thesis and will be discussed here: the use of limiting cases, inverse Fourier transform, and direct modelling.

Model independent analysis

In general the first approach to analysing SAXS curves should be to use a series of limiting cases to better understand the spectra. The first is in the large q limit and gives

Autocorrelation

Fourier transform

I (q) A(q) b j (q) γ (r)
Fourier transform Inverse Fourier transform Square Square-root

Deconvolution

Inverse Fourier transform Fig. 2.6 Illustration of the relationships between intensity I (q), amplitude A(q), scattering length b j (q, and the correlation function γ (r).Dotted arrows indicate ambiguous solutions. Adapted from M. Muthig, 2014. 55 the Porod law:

lim q→∞ I (q) = 2π(∆ρ) 2 q 4 S V (2.18)
where S is the total surface area of objects of uniform density, and V the volume in which they are dispersed. This law is only concerned with the interface of objects, as a consequence of the lim q→∞ and so also holds for bicontinuous systems. Observing the slope of q -4 in (2.18) is contingent on having a sharp interface. In the case of a rough interface this slope is instead proportional to q -(6-D s ) where D s is the surface fractal dimension. 56 The second is in the low-q limit. When lim q→0 I (q) this is called the forward scattering. In the case of dilute suspensions a plateau is observed in the low-q limit. This can be used to calculate the radius of gyration, R g , of the scattering object in the Guinier approximation:

lim q→0 I (q) ≈ N p (∆ρ) 2 V 2 e x p - (q R g ) 2 3 (2.19)
This law does not hold in the presence of a structure factor. Thirdly, the invariant, Q, relates the volume fraction, φ, of the scattering objects to the second moment of the intensity: 2 (2.20)

Q = ∞ 0 I (q)q 2 d q = 2π 2 φ(1 -φ)(∆ρ)
as such it can be used to check intensity versus known sample composition, and obtain the sample volume fraction from data in absolute scale.

IFT/ Generalised IFT with S(q) Theoretically, in the case of a non-interacting system where the structure factor can be neglected a Fourier transform of the scattered intensity will compute the pair distance distribution function (PDDF), p(r ). However, due to the phase information loss, noise and necessarily limited q-range an exact calculation is not possible. To solve this problem Glatter developed a regularisation technique to retrieve the most likely PDDF by indirect Fourier transform (IFT). [START_REF] Glatter | Small Angle X-ray Scattering[END_REF] This later became the generalised inverse Fourier transform (GIFT) suitable for systems of interacting particles (i.e. concentrated systems). GIFT makes an ansatz of the structure factor and then retrieves the PDDF by IFT. By assuming a particular shape the PDDF can then be deconvoluted to provide a particle size distribution.

Model dependent analysis

Model dependent analysis is perhaps the most common method of solving I (q). Using prior knowledge of the system a mathematical model is built and the resulting intensity is calculated. Both the form and structure factor are analytically or numerically solved. As for all methods presented here this cannot definitively prove a structure; but it can lead to the most probable structure. To completely understand the data this is the most practical approach.

2.2 Synchrotron small-angle x-ray scattering X-ray characterisation techniques, such as SAXS, continue to be developed and refined today at synchrotrons and x-ray free electron laser (XFEL) facilities. These are large-scale facilities that produce x-ray beams with orders of magnitude more flux (photons) than laboratory sources. Such high-intensity x-ray beams are exploited on instruments, called beamlines, which consist of a source (used to produce x-rays with specific properties), optics to refine the beam, an appropriate sample environment, detector and associated hardware to collect the information desired.

Synchrotrons

The European Synchrotron Radiation Facility (ESRF) where this work was conducted uses a cyclicly accelerating electron beam to produce x-rays. ESRF's electron beam begins with an 100 keV triode gun triggered at either 10 or 1 Hz to generate electrons. The impulse applied to the gun determines the shape and length of bunches of electrons that move around the storage ring. These electron bunches then pass through a pre-buncher and buncher before entering a linear accelerator (linac) which accelerates the electrons from 11 MeV to 200 MeV. [START_REF] Wiedemann | Particle Accelerator Physics[END_REF] A small circular accelerator then increases the energy of these electrons to ≈6 GeV which is the energy of the ESRF storage ring. This energy, E, is chosen to optimise the photon flux on the sample of interest (brightness, B ). [START_REF] Baruchel | Neutron and Synchrotron Radiation for Condensed Matter Studies[END_REF] Both the synchrotron, and storage ring make use of a matrix of electron focusing optics to maintain the desired characteristics (bunch format, electron momentum, bunch/beam shape, energy) of the electron beam. This matrix is composed of a combination of dipole (bending), quadrupole and sextupole magnets, which focus the beam, combined with radio-frequency cavities, to maintain the bunching characteristics and energy of the beam. [START_REF] Baruchel | Neutron and Synchrotron Radiation for Condensed Matter Studies[END_REF]

Small-angle scattering instrument ID02

All the SAXS and XPCS experiments presented in this thesis were performed on the time resolved USAXS beamline, ID02, at ESRF -the European synchrotron. Synchrotron SAXS instruments provide the high photon flux required for high time resolution experiments. This is due to an ≈6 order of magnitude difference in the available flux (number of photons per second) relative to laboratory SAXS instruments enabling fast data collection. Additionally, an instrument such as ID02 provides access to much lower q-values than a laboratory source, down to 10 -3 nm -1 thanks to a high degree of colimation. Therefore, to study the kinetics of hierarchical systems with high temporal and spatial resolution an instrument with the capabilities of ID02 is required. Dynamic measurements, such as XPCS, require a beam with a high degree of coherence which is not possible to obtain with a laboratory source. XPCS is only possible with synchrotron radiation, or x-ray free electron lasers. The main specifications of the beamline will be described, highlighting where the instrumentation enabled specific experiments.

Beamline description

All x-ray instruments consist of a source (used to produce x-rays with specific properties), optics to refine the beam, area to install a sample environment, and detector to collect the information desired.

ID02 uses three undulators (two U21.4 -high flux; and one U35 -wider energy range) as a source. [START_REF] Van Vaerenbergh | AIP Conference Proceedings[END_REF] Undulators look like two piano keyboards with a gap in-between. The keyboard keys are magnets which cause the electron beam (moving through the synchrotron's storage ring) to undulate upon passing through the gap generating a beam of x-rays. By adjusting the gap between the two keyboards the strength of the magnetic field changes, and thus the x-ray energy of the beam emitted by the undulator is changed. [START_REF] Als-Nielsen | Elements of Modern X-ray Physics: Second Edition[END_REF] Primary slits, in the optics hutch, remove excess heat and begin shaping the beam. The resulting beam is monochromatised by a channel-cut, Si-111 monochromator which is liquid nitrogen cooled. A double-mirror set-up, using one partly toroidal and one flat mirror in a horizontal geometry is used to focus the monochromatic beam at the detector position. [START_REF] Van Vaerenbergh | AIP Conference Proceedings[END_REF] A series of slits are used for further collimation, and to curtail the parasitic scattering around the focused beam introduced by various optical elements. The slits are spaced between the monochromator and sample so as to minimise parasitic scattering from the slits themselves. [START_REF] Koch | [END_REF] The instrumentation at the beamline is designed to optimise collimation for high resolution and low background. Increasing collimation means the x-rays are more parallel, practically this means that more of the beam profile has been cut away leaving the collimated central cone.

Attenuators which are inserted when required are also present on the beamline. These are a variety of metallic sheets, each with a different absorption edge used for wavelength calibration. A graphite attenuator is available which broadens the beam. Broadening the beam illuminates a larger sample area with lower x-ray flux-density. This thesis has used broadening the beam to enable measurement of radiation sensitive samples (such as polyelectrolytes in chapter 4).

Specifications of the instrument

Typically SAXS data are collected on a 2D detector. This is useful for observing anisotropy present in the sample as this will result in an anisotropic 2D scattering pattern visible on the detector. In the case of isotropic scattering there is no dependence of the intensity distribution on the azimuthal direction on the detector. To improve statistical accuracy the data are azimuthally integrated to produce a 1D scattering curve which can be interpreted. [START_REF] Stribeck | X-Ray Scattering of Soft Matter[END_REF] All data presented in this document has been corrected for the instrumental parameters using software developed on the beamline and presented by Bösecke et al. [START_REF] Narayanan | [END_REF][START_REF] Boesecke | [END_REF] X-ray detectors Time-resolution is limited by two factors: the available flux, and detector acquisition rate or read-out time. Often detector technology is the limiting factor for time resolution. Hybrid pixel detectors (HPD) have been developed for synchrotron science. 64 ID02 has two such detectors the Pilatus 300 K, and PSI-Eiger 500K (main specifications of detectors used in this thesis are in table 2.1). The Pilatus was used to collect the kinetic Static data presented in this thesis were collected using charge-couple device (CCD) based detectors. These are a mature technology that have been applied to research in the astrophysics, particle physics and synchrotron communities for several decades. 65 A Rayonix CCD detector was used for many of the static curves presented in this work. This has the advantage of a large, gapless active area (17 x 17 cm 2 ) which makes it ideal for studying samples over more than two orders of magnitude in reciprocal space. On those occasions where high resolution was required the FReLoN detector was used due to its small pixel size. The FReLoN additionally benefits from a narrower point-spreadfunction than the Rayonix detector. 66 The normalised point-spread-function is a way of characterising the resolution of a detector. It gives the distribution of the output signal of the detector given a point input signal. 67 The detectors are housed in a wagon which moves on rails inside a 34 m long detector tube under vacuum. [START_REF] Narayanan | [END_REF] The experiments in this thesis would not have been possible without synchrotron radiation. Chapter 3 and chapter 4 benefit from the high resolution offered by a synchrotron SAXS instrument. In chapter 5 complex sample environments exploit the high flux and fast detectors available. Finally, in chapter 5 and chapter 6 the coherence of the synchrotron radiation beam is exploited to perform XPCS measurements.

Detector tube Optics Hutch

Introduction

Colloidal silica is widely used in research, from fundamental to applied, often as a proxy for more complicated or expensive systems. Commercial production of colloidal silica particles dates back to the 1940s. 68 The Stöber process, developed in 1968, provided a straightforward way of producing nearly monodisperse particles for a chosen size. 69 There exists a vast body of research examining the Stöber process, including in-situ studies of the particle formation. [70][71][72][73] One of the mechanisms of particle formation proposes that particles are initially polydisperse and fractal in structure. These initial particles become more dense and form a core, to which concentric layers of secondary particles are added before the structure is terminated with a dense layer of amorphous silica. As the particles are well-described by a spherical model the analysis of Stöber silica often neglects possible inhomogeneities of the particle core.

Unfortunately the Stöber process is only suitable for producing particles that are above fifty nanometers in diameter. Producing monodisperse silica particles below fifty nanometers in diameter, hereafter called silica particles is more challenging. The most common methods use water-in-oil reverse microemulsions or reverse micelles as microreactors for a confined synthesis of silica. [START_REF] Arriagada | The colloid chemistry of silica[END_REF][START_REF] Lin | [END_REF] Unfortunately these methods are not suitable for large-scale production due to relatively low yields. An alternative is to use lysine to produce particles. 76 Lysine improves particle monodispersity but is only suitable for synthesising particles in a small size-range (12 nm to 36 nm).

Not only is the synthesis of silica particles challenging, their characterisation is also challenging. Reference silica particles provided by the Institute for Reference Materials and Measurements illustrate this difficulty. The particles are certified for four different size values, contingent on the sizing method used. 37 The characterisation of silica particles is complicated by the particle porosity, the degree of which is unknown; an irregular particle shape (these are not perfect spheres); and size polydispersity. 37 This thesis focuses on silica particles despite the difficulties in working with them. The size of the silica particles was chosen to match the persistence length of a particular polyelectrolyte -chitosan. In this chapter silica particles of tradename Ludox and produced by W.R. Grace and Co. are characterised. In later chapters these silica particles will be used as building blocks to form more complex silica-polyelectrolyte materials. Understanding the scattering from the more complex system requires understanding the scattering from individual nanoparticles. This will enable separation of the silica form factor from the silicapolyelectrolyte structure factor, enabling understanding of the effect of polyelectrolyte on the system. SAXS studies of silica particles typically approximate the particles as a system of polydisperse spheres. Different methods of quantifying particle polydispersity have been proposed -for example an innovative iterative fitting procedure has been developed and applied. 77 This describes the polydispersity with a multi-modal size distribution of silica particles; but neglects inter-particle interactions or particle porosity. Understanding interparticle interactions is important for describing concentrated systems, and in practical applications for rational design of functional systems. 78 While controlled particle porosity can be a feature, different applications require different types and sizes of pores. Mesoporous silica particles are used as carriers for drug delivery where therapeutics are loaded into pores where they are protected from degradation until required. 79 Silica used in chromatography columns requires pores of diameter ≈20 nm while pores of diameter <2 nm are detrimental and need to be suppressed. 80 Based on the results from SAXS experiments, in this chapter, an alternative model to describe silica particles is proposed. This model, originally developed by Hurd and Flower, accounts for the non-homogeneous nature of the particles -including porosity. 38 Fits from various structure factor models, including the Hayter-Penfold structure factor for macro-ions, are then compared.

Materials

Three different brands of colloidal silica were used: Ludox, Bindzil, and model particles obtained from Polysciences Europe GmbH.

Ludox colloidal dispersions

This work focuses on Ludox silica particles (produced following US patent number: 2,574,902 filed: 1951). Ludox TMA is industrially used both in the investment casting industry, and as a surface modifier: in polishing applications, or as an antislip coating for floors and polishes. 81 The particles are ≈ 22 nm in diameter, and have ζ potential of (-42 ± 3) mV i corresponding to particle surface charge of ≈7.9 × 10 -18 C (≈50 effective charges). ii The electrokinetic or ζ potential is defined by IUPAC as: the potential at the plane where slip with respect to bulk solution is postulated to occur. 84 Therefore ζ potential is a function of particle charge and ionic strength, and gives an indication of colloidal stability. Generally for ζ potential >| 30 |mV a colloidal solution can be considered stable.

Ludox TMA is negatively charged, even in acidic conditions, and is considered stable from pH 4 -7. [START_REF] Materials | Ludox Colloidal Silica in Coatings[END_REF] The manufacturer notes that TMA is deionised to remove sodium ions which ensures particle stability. A small amount of the particle surface is modified with an aluminate compound during the manufacturing process. However, the manufacturer notes that the surface chemistry of the particles is predominantly silica. [START_REF] Materials | Ludox Colloidal Silica in Coatings[END_REF] Other studies using Ludox TMA particles in the literature consider them as unmodified silica particles and generally do not discuss the aluminate surface modification. 82,[START_REF] Schoth | [END_REF][87][88] We consider Ludox TMA as silica particles.

Other silica colloidal particles

Model particles were obtained from Polysciences Europe GmbH. Two different particle sizes were purchased: silica microspheres of diameter 0.1 and ≈ 0.02 µm. This data is presented in section A.2.

The final type of silica used is tradename Bindzil now marketed under a new trade name -Levasil. Bindzil 40/130 was used, it has ≈ 30 nm particle diameter, and ζ potential of (-37 ± 4) mV. iii i The ζ potential was measured using a Malvern Zeta-sizer Nano Z at ≈0.03 %. Samples were dialysed and diluted with Milli-Q water following the methods section of this chapter. The value of (-42 ± 3) mV is in reasonable agreement with the ζ potential value of -56 mV reported by Arriaga et al. 82 ii This is approximated using the formula ψ 0 = z/πε 0 ε r 2r (2 + 2r /λ D ), where ψ 0 is surface charge, z is electronic charge, ε 0 is vacuum permittivity, ε r i relative permittivity, r is particle radius, and λ D is Debye length. 83 iii ζ potential was measured at the same concentration, and in the same manner as Ludox TMA.

Methods

Sample preparation

Silica suspensions for SAXS measurements were prepared as follows. The silica samples were diluted by a factor of 100 using Milli-Q water. Volume fraction, φ, of the original Ludox TMA solution was ≈0.23 (assuming a silica density of 1.7 g cm -3 ) so the samples have φ≈0.0023. Once diluted the samples were dialysed to remove any salt or surfactant present in the original dispersion. Samples were placed in 12 kDa to 15 kDa dialysis tubing, the tubes were sealed and placed in a 1 L bath of weak buffer solution (0.05 mmol L -1 acetic acid and 0.03 mmol L -1 sodium acetate) for 72 h. The buffer solution was changed every 12 h.

To remove any aggregates the diluted and dialysed solutions (hereafter called the samples) were placed in an ultrasonic bath for 1 h. They were then centrifuged at 600 rpm for 10 min, with the expectation that any remaining aggregates would sediment. The supernatant was then removed. This supernatant was then centrifuged again at 600 rpm for 10 min, the stability checked, and a longer centrifugation of 800 rpm for 30 min was conducted. The remaining supernatant was then measured on beamline ID02. To screen interactions 10 mmol L -1 of salt was added to Bindzil 40/130. No salt was added to Ludox TMA.

Silica obtained from Polysciences Europe GmbH was used as purchased, simply diluted to 1 % by weight using Milli-Q water.

SAXS measurements with the peristaltic pump set-up used Ludox TMA as obtained. The Ludox TMA was then diluted in-situ using the peristaltic pump apparatus.

Sample measurement

All SAXS data were collected using beamline ID02 at the ESRF (described in chapter 2).

Flow-through cell A flow-through capillary was used to measure dilute samples with negligible inter-particle interactions. This flow-through cell is composed of a borosilicate capillary, with wall thickness 50 µm and sample thickness is ≈2 mm, mounted in a holder so the capillary can be cleaned between measurements. Sample and its corresponding background, in this case water, were measured in identical conditions enabling precise background subtraction. The capillary was cleaned using sodium hydroxide (NaOH) solution and rinsed with Milli-Q water between every measurement. A background measurement was taken between each silica measurement and compared to the background at the beginning of the experiment.

Peristaltic pump

To study the development of inter-particle interactions a peristaltic pump set-up was used. This set-up enabled a continuous increase in silica particle concentration to be measured efficiently. The set-up is depicted in fig. 3.1. A peristaltic pump was used to draw liquid from a sealed reservoir of Milli-Q water containing a stirrer bar. This caused an equal volume to be drawn from a silica suspension-containing reservoir into the water reservoir, such that the volume of the water reservoir remained constant. The pump was connected to a flow-through cell (as described in the previous paragraph).

As pumping continues the concentration of silica particle in the water-containing-vial increases. Controlling the pumping rate determines the rate of concentration increase and so this method is particularly useful for studying changes in scattering as a function of concentration. The pumping circuit was cleaned using NaOH solution and Milli-Q water between series. Figure 3.2 shows how the scattering invariant increases with pumping time. In eq. (2.20) it was shown that the invariant is proportional to φ(1 -φ). The invariant can therefore be considered a measure of the number of silica particles present. It can be seen in fig. 3.2 the concentration of silica particles increases with time. The invariant, a measure of particle concentration, plotted against time after initial pumping (at constant flow rate) using the peristaltic pump set-up.

Results and discussion

Scattering contrast is the scattering length density difference between silica particles and water. However, determining the precise scattering length density of silica is difficult, even for the ideal case of Stöber silica. 89 This is due to internal porosity of the particles. The pores are inaccessible to solvent or gas meaning that pycnometry iv cannot measure the density of the entire particle. While the density of amorphous silica is 2.2 g cm -3 , the accepted values for the density of Stöber silica are 1.6 g cm -3 to 2.0 g cm -3 . 90,91 As expected the density of silica particles is difficult to determine. There is conflict in published values for the density of Ludox TMA with values of 2.19 g cm -3 , 92 and 2.37 g cm -3 ; 93 however these values are based on evaporation experiments (where density is calculated on basis of liquid that evaporates) and the true density is likely to be lower. Work by Vrij et al. 94 on the density of silica particles of ≈10 nm radius determine particle density of 1.6 g cm -3 to 2.0 g cm -3 .

Scattering length density can be estimated for a dilute system of scatterers by examining the forward scattering I (0) of a scattering curve. Forward scattering is

I (0) = N p ∆ρ 2 V 2 p
where ∆ρ is contrast, N p is number density of particles, and V p is the mean volume of nanoparticles. The scattering length density of silica can then be written in terms of the iv A pycnometer is a device which measures the density of an object based on the volume within a sample chamber from which gas is excluded. volume fraction, φ and I (0) as

ρ S i O 2 = ρ H 2 O + I (0) φV p (3.1)
where ρ H 2 O is the scattering length density of water, 9.41 × 10 -4 nm -2 . Taking I (0) from fig. 3.5 this gives ρ S i O 2 ≈14.3 × 10 -4 nm -2 indicating a silica density of 1.7 g cm -3 which is an agreement with the range published by Vrij et al. 94 Unfortunately there is a logical flaw. Accurately determining the volume fraction requires knowing the particle density. This can be circumvented by conducting contrast matching experiments in which a precise volume fraction is extracted, as performed by Sztucki et al. 91 The calculations show that in all cases there is a favourable contrast between silica colloids and water as observed in fig. 3.3. In section 2.1.1 scattering from a system of spheres was described. The appropriateness of a spherical fit to silica particles is shown in fig. 3.4. It can be seen that for Bindzil particles the fit is reasonable; but there is a considerable deviation for the Ludox particles. In both cases the slope at high-q deviates from the ideal case of smooth particles where a slope of q -4 would be observed (as described in section 2.1.3). At mid-q, called the Kratky Ludox TMA is fit with a polydisperse sphere model using a Schulz-Zimm particle population distribution (R m =10.9 nm, σ = 0.12). Bindzil is fit with a polydisperse sphere model with Gaussian particle population distribution (R m =14.7 nm, σ = 0.19). The Bindzil data have been vertically offset with a factor of 20 for easier viewing. Samples have been diluted by a factor of 100, dialysed, centrifuged and sonicated to remove aggregates and impurities. Bindzil has 100 mM of NaCl added.

region, there is further deviation -it can be seen that the intensity of the fit overestimates the actual intensity. In the case of Ludox the fit deviates in the low-q region. Polydispersity has been described using either a Gaussian or Schulz-Zimm population distribution (details in the Appendix) where R m is the mean value of the particle radius and σ the reduced standard deviation (σ = variance/R m ) . Such discrepancies are further illustrated for the case of Ludox fig. 3.5. A physical explanation for these discrepancies will be demonstrated and then a solution, in the form of the Hurd and Flower model will be presented.

Deviation in the high-q region

There are three possible explanations for the deviation between the fit and data in the high-q region as seen in fig. 3.4 and fig. 3.5. The first explanation is incorrect background subtraction. Error due to incorrect background subtraction is minimised by measuring the silica in a flow-through cell as described in section 3.3. While the data and background approach each other at high-q, as seen in fig. 3.4, this is not sufficient to account for the deviation from q -4 slope.

The scattering of an incident x-ray photon by successive particles is called the multiple scattering. It is undesirable as it causes smearing in the features of a scattering curve. 95 As the silica particles used are small and dilute it is less probable that multiple scattering effects the curves. [START_REF] Narayanan | [END_REF] It is shown in fig. 3.6 that multiple scattering does not affect the scattering curves. By increasing the concentration of scatterers (silica particles) the probability of multiple scattering events is increased. Therefore if there is no increase in the smearing of scattering features when measuring at higher concentrations there is no significant multiple scattering effect. Ludox TMA was measured at three concentrations (φ ≈ 0.0023, 0.0069 and 0.069) in a 1.5 mm quartz capillary. Data are fit using the same parameters, changing only volume fraction in the model. A plot of residuals shows that there is no smearing of features at higher concentrations, which would be expected for a sample with multiple scattering, and so multiple scattering effects can be dismissed.

The second explanation states deviation at high-q is a consequence of instrumental smearing caused by the limited dynamic range of the detector. One method of checking this effect on the high-q region is to remove the intensity in the low-q region using a beamstop. This procedure is followed in fig. 3.7. The yellow points demonstrate where an additional beamstop (12 mm in diameter) has been used to mask the higher intensity. The purple points show the data measured with the standard beamstop (2 mm height). It can be observed that although the readout noise of the detector has some effect on the slope at high-q it is not sufficient to account for the deviation observed in fig. 3.4.

Finally, deviation can come from the sample itself. This can be a consequence of a rough particle surface, or porosity within the particles, or a combination of both of these |I(q)-I fit (q)/ΔI(q)| q /nm -1 Fig. 3.6 Figure 3.6a shows scattering from Ludox TMA for different sample concentrations. Data have been fit using the 'OverlapSph Cut-Off' model implemented in SASfit. 96 Fitting parameters (R m =10.1 nm, σ = 0.11, D f = 2.96) were kept constant. Figure 3.6b shows the fit residuals. There is no change in the fit appropriateness at high-q with respect to concentration indicating multiple scattering can be dismissed.

I(q) /mm -1 q /nm -1 q -4
Fig. 3.7 Scattering from large silica microspheres (1% wt.) of 0.1 µm obtained from Polysciences Europe GmbH.

The data have been measured with an additional beamstop (12 mm in diameter), yellow points, and with a standard beamstop (2 mm height)-magenta points. The larger beamstop masks the high intensity radiation.

There is a small difference visible at high-q between the two measurements caused by readout noise of the CCD detector. effects. Given that error due to experiment has been accounted for, surface roughness and porosity are the most probable origin of the excess scattering observed. This is confirmed by fig. 3.8 which shows a Porod plot of three types of silica. In a Porod plot it is expected that I * q 4 , plotted on the y-axis, gives a flat line for sharp interfaces. As the Polysciences particles are the closest to ideality they form an almost flat line. This indicates a smooth surface with stepwise density profile. The Ludox and Bindzil particles both deviate from flat lines in the high-q region. Either the particle interface in solution is not sharp, or the particles are not compact. Supporting this hypothesis is the literature describing the synthesis of silica particles which indicates that they are both porous and rough. [70][71][72][73] It is further justified by the description of the particle synthesis (following US patent number: 2,574,902 filed: 1951) whereby a continuous influx of nm-sized sodium and silica oxides are boiled. One can assume that this produces dense, fractal aggregates which are subsequently labelled as silica nanoparticles.

Deviation in the mid-q region

As observed in fig. 3.5 the dense sphere model overestimates the intensity in the mid-q region. Figure 3.3 demonstrates that neither incorrect background subtraction nor instrumental smearing can account for this. The most likely explanation is that this is due

I(q)*q 2 /nm -3 q /nm -1
Ludox Bindzil Polysciences to the particles not being ideal spheres. A Kratky plot, fig. 3.9 shows how the particle cross-section is not that of a perfect sphere.

Other explanations include a high-degree of polydispersity, such as the presence of a multimodal population distribution of silica spheres. Inter-particle interactions, such as a structure factor can also disturb this region of a scattering curve, as observed in fig. 2.4. This is why only dilute samples were measured here which do not exhibit a structure factor in their scattering curve. It is in this mid-q region that a combination of polydispersity and strong inter-particle repulsion can cause a loss of intensity. 97 The simplest interaction potential is that of the hard-sphere potential which describes excluded volume effects: a particle cannot occupy space that is already occupied by another particle. 98 In the case of spheres this potential is described by infinite repulsion at the point of contact. One can imagine that for particles with an ill-defined density profile this potential becomes a softer function. In fig. 2.3 it was shown that particle polydispersity also causes features in this part of the scattering curve to be smeared.

Deviation in the low-q region

The difference between the spherical model at low-q and the observed scattering data has three possible sources. It is potentially due to experimental error caused by incorrect Concentration I(q) /mm -1 q /nm -1 Fig. 3.10 Scattering from a concentration series of Ludox TMA. The series was made using the peristaltic pump set-up described in section 3.3. It can be seen that as the suspension becomes more concentrated a strong structure factor peak develops.

background subtraction. This we discount for the same reasons as in section 3.4 -error is minimised by measuring in a flow-through cell.

It is more likely that the deviation arises from particle aggregation. Ludox is negatively charged to prevent particles from sticking together. v Over time it is possible for particles to overcome this repulsive barrier and aggregate. In Ludox this energy barrier is high and so the particles remain stable in solution for long timescales. The number of aggregates is minimised by sonicating and centrifuging the samples before measurement (as described in section 3.3). However, it is impossible to remove aggregates entirely.

Note that in fig. 3.4 the deviation at low-q is different. The dense sphere model overestimates the intensity of Bindzil. This is due to inter-particle repulsion from excluded volume effects. In the case of Ludox there is an excess of scattering with respect to the model. By measuring a concentration series of Ludox TMA, shown in fig. 3.10 a structure factor peak is observed; however the intensity at low-q scales linearly with increasing volume fraction. This is indicative of aggregation.

Inverse Fourier transform (IFT) vi analysis can be used to study aggregation. Here a complete inverse Fourier transform is not possible due to information loss. Instead an indirect Fourier transform is made which includes additional constraints for smoothness. I(q) /mm -1 q /nm -1 Fig. 3.11 Indirect Fourier transform analysis of Ludox followed by deconvolution assuming polydisperse, noninteracting spheres. The probability density function has been fitted with a triple Gaussian size distribution for Ludox (x 1 =11.63 nm, σ 1 = 0.1; x 2 =22.6 nm, σ 2 = 0.15; x 3 =32.9 nm, σ 3 = 0.07). IFT was done using BayesApp. 99 The online tool BayesApp developed by S. Hansen was used to perform an IFT analysis of the data shown in fig. 3.4. 99 Indirect Fourier transform provided the pair distance distribution function, PDDF. A second step analysed the PDDF by assuming it arises from a set of homogeneous, non-interacting spheres to provide a corresponding probability density function, PDF. Both steps in the process: I (q) to PDDF, and PDDF to PDF are mathematically ill-defined. They are constrained to give a physically meaningful result. When extracted the probability density function was fitted with Gaussian size distributions as appropriate. It is clear from fig. 3.11 that there is aggregation present in the Ludox sample given the PDF separates into three clear distributions. This is in agreement with the conclusions of Sen et al. 77 Bindzil, depicted in fig. 3.12, appears to be monomodal and is fitted with a single Gaussian size distribution.

Improving the sphere model

As areas for improvement have been identified it is possible to improve the spherical fit of silica particles. The deviation at high-q due to heterogeneities in the particles can be I(q) /mm -1 q /nm -1 Fig. 3.12 Indirect Fourier transform analysis of Bindzil followed by deconvolution assuming polydisperse, noninteracting spheres. The probability density function has been fitted with a single Gaussian size distribution for Bindzil (R m =17.09 nm, σ = 0.13). The probability density function and original data are shown on the right where it can be seen there is good agreement.

accounted for using an Ornstein-Zernike function given by

I (q) = I 0 1 + q 2 ξ 2 (3.2)
which is used to describe density fluctuations within the particles. [START_REF] Narayanan | Structure from Diffraction Methods: Inorganic Materials Series[END_REF] It uses a correlation length ξ , of density fluctuations, which decays as ≈ (1/r ) exp( -r ξ ) which is appropriate at high-q. Unfortunately the consequence of surface irregularity is morphological deviation from an ideal sphere and the Ornstein-Zernike function cannot represent this behaviour at mid-q. To improve the mid and low-q fit a structure factor can be used to account for inter-particle interactions. The result is shown in fig. 3.13. While an improvement on fig. 3.4 the fit at mid and low-q is still not fully satisfactory. The current analysis disregards the fact that the particles are not perfectly spherical, and often neglects the presence of aggregates. This prevents understanding the structure factor of the system -solving for a structure factor first requires a precise form factor.

This precise form factor is of interest because we rarely study the behaviour of a single particle. It is the collective behaviour of particles that is studied fundamentally and used industrially. A precise effective structure factor can be determined only after q /nm -1 Fig. 3.13 Bindzil silica, with a monomodal size distribution fitted with two independent contributions: a Gaussian distribution of spheres (R m =14.7 nm, σ = 0.19), and an Ornstein-Zernike function to account for high-q deviation (correlation length ξ =1.86 nm and I e x p (0) ≈ 1700× I OZ (0)). A Percus-Yevick hard-sphere structure factor has been added to account for interparticle repulsion with a local monodisperse approximation. The volume fraction of the fit is φ = 0.034. Fitting was done in SASfit. 96 Details of the Ornstein-Zernike function are in eq. (3.2), the Percus-Yevick structure factor and spherical form factor are presented in chapter 2.

implementation of a precise form factor as

S e f f (q) = I (q) N (∆ρ) 2 V 2 p P (q) . (3.3)
It is therefore incumbent to determine a precise form factor to extract the effective structure factor and study inter-particle interactions. As stated in section 3.1 understanding interparticle interactions is essential for understanding the complexation process between silica and chitosan examined in later chapters.

Use of fractal models

Adequately describing polydisperse, highly charged, approximately spherical silica particles with surface roughness and internal porosity is a problem. Our solution is to use the fractal model of Hurd and Flower to describe the particles. 38 This fractal model has the advantage of describing with a single, simple model the roughness of the particle surface seen at high-q (in fig. 3.5); the oscillations at mid-q, and the shoulder also at mid-q. The model deviates at low-q where the structure factor dominates. Given that the use of a fractal model enables an accurate solution of the form factor it becomes a lot easier to construct a physically meaningful structure factor.

Models for spherical fractal aggregates

As highlighted in section 1.2.1 experimentally determined fractals are self-similar over a limited range. When building a model to describe experimental fractals it is incumbent to define such a spatial limit which is done using a cut-off function.

Consider the structure factor for an isotropic system 50

S(q) = 1 + 4πφ ∞ 0 | g (r ) -1|r 2 sin(q r ) q r d r, (3.4)
where φ the particle volume fraction. The radial distribution function (or pair-correlation function, or density-density correlation function), g (r ), is defined as the probability of finding the centre of any particle at distance r from the centre of any other particle. [START_REF] Teixeira | On Growth and Form[END_REF] This definition of g (r ) ensures that

d P (r ) = φ g (r )4πr 2 d r (3.5)
but P (r ) is also linked to the definition of fractal dimension, D f . We define here as

P (r ) = ( r R ) D f (3.6)
which describes n p particles of radius R residing within a sphere of radius r . [START_REF] Teixeira | On Growth and Form[END_REF] R can be considered as the smallest unit size, and r the overall fractal size. Combining these gives

φ| g (r ) -1| = D f 4π 1 R D f r D f -3 h(x) (3.7)
where the cut-off function h(x) has been introduced to limit g(r). [START_REF] Teixeira | On Growth and Form[END_REF] The structure factor can then be written as

S(q) = 1 + D f R D f ∞ 0 r D f -1 h(x) s i n(q r ) q r d r (3.8)
Much work has been done on determining cut-off functions that result in a physically meaningful structure factor. A good cut-off function must be sharp; and it must tend to zero faster than any power law for r >> 1. [START_REF] Jullien | [END_REF] Simulation confirms this experimental assessment. 102 Well-known cut-off functions have been summarised by Soresen and Wang. 103 They generally take the form of an exponential decay, such as that by Teixeira expr ξ which uses a single exponential decay, and correlation length ξ . 104 When eq. (3.8) is evaluated for qξ ≪ 1, D f = 2 using Teixeira's cut-off function the Ornstein-Zernike equation shown in eq. (3.2) is recovered. [START_REF] Teixeira | On Growth and Form[END_REF] The cut-off parameter ξ has the same physical meaning, defining the length scale at which density fluctuations within the material approach the average density of the material.

While these cut-off functions are all useful for describing aggregates they fail to describe materials that are approaching a regular spherical geometry. They are unable to bridge the continuum between fractal and Euclidean geometry in a satisfactory manner.

The model of Hurd and Flower

Here, the fractal model developed by Hurd and Flower in 1988 is implemented. 38 This model was initially used to describe the growth of silica aggregates in a flame; however here it is shown to be effective in describing stable silica particles in solution. It is in the form of cut-off function

h r R :=    4πR 3 3 1 + r 4R 1 - r 2R 2 if r ≤ 2R 0 if r > 2R (3.9)
Equation (3.9) adapts the correlation function of a sphere, radius R, and therefore the spherical form factor is recovered when the fractal dimension, D f = 3 (as illustrated in fig. 3.14). This cut-off function allows for an overlap volume in the integral of the correlation function. It is combined with the pre-factor r D-3 such that

I (q) ≈ r D-3 h r R r 2 sin(q r ) q r d r, (3.10)
this pre-factor accounts for density variation within the cluster or particle. In this case it describes porosity in the silica particles.

Applying the Hurd and Flower model

The Hurd and Flower model has been implemented in SASfit (Breßler and Kohlbrecher, 2015), where it is called the 'OverlapSph Cut-Off'. 96 SASfit was used to perform the fitting on the data shown. Due to the monomodal distribution of Bindzil it can be considered a simpler system, therefore the Hurd and Flower model will first be demonstrated on Bindzil and then Ludox.

Bindzil particles

Bindzil is a polydisperse system with a high degree of porosity and/or surface roughness, as shown in fig. 3.8. It is comprised of a monomodal distribution of silica and can be fit using a single population of fractal aggregates with a Gaussian size distribution as shown in fig. 3.15. To compare fig. 3.15 and fig. 3.13 a plot of residuals is shown in fig. 3.16. The power of the fractal aggregate model is instantly apparent. At high-q the data are well described by both the fractal model and the sphere model with Ornstein-Zernike function. At low-q the sphere model also requires an additional structure factor to account for excluded volume effects. Even so at mid-and low-q the Hurd and Flower model is an improvement over the sphere model combined with Ornstein-Zernike and Percus-Yevick hard-sphere structure factor. The smearing of the density profile provided by the fractal description accounts for the change in excluded volume effects, something that can not be achieved easily with simple structure factors. 105

Ludox particles

As demonstrated throughout this chapter Ludox deviates from an ideal particle in multiple respects. It is polydisperse, has surface roughness and porosity (fig. 3.8), is strongly A fit containing a single population of fractal particles, fig. 3.17 is appropriate at high-q; but fails at low-q as it is unable to account for aggregation. There are two options for describing such aggregation. The first is to add a second population of fractal particles. The second is to use a structure factor with an attractive potential to describe this aggregation and inter-particle interactions. While fitting with two populations of fractal particles works well at low concentrations; this breaks down at higher concentrations as inter-particle interactions dominate. It is therefore necessary to use an appropriate structure factor to describe the change in scattering curve as a function of concentration shown in fig. 3.10.

Modelling the structure factor

Typically charged particles can be modelled using a structure factor appropriate for macro-ions such as those by Hayter-Penfold, 83 Hansen-Hayter, 106 or Grimson. 107 Such structure factors describe purely repulsive inter-particle interactions. At low volume fraction these models work well. Unfortunately, at high volume fraction they fail to describe scattering from Ludox. This is illustrated in fig. 3.18 where the Hurd and Flower model is used at high-q, and the Hayter-Penfold model is used to fit the structure factor peak observed in Ludox TMA at low-q. At low volume fraction there is good agreement 3.17. This is combined with the Hayter-Penfold model and monodisperse approximation for the structure factor peak (RHS=13.9 nm, Z = 65.5, φ = 0.078, T=298 K, n s =0.03 mmol L, ε = 64.2). The mass fractal aggregate fitting parameters are: ξ =207 nm D f = 1.21. The inset shows the fit at low volume fraction with Hayter-Penfold model and monodisperse approximation for structure factor peak (RHS=13.9 nm, Z = 65.5, φ = 0.0012, T=298 K, n s =0.02 mmol L, ε = 64.2). This was combined with a second population of the Hurd and Flower model with a mass fractal structure factor with cut-off ξ =16.6 nm and fractal dimension D f = 2.97.

between the model and data; however there are two contributions one Hurd and Flower model with a Hayter Penfold structure factor (with RMSA approximation), and a second Hurd and Flower contribution with mass fractal aggregate.

Examining the high volume fraction data in fig. 3.18 it is clear why such a composite system is required. It is clear the Hayter-Penfold model cannot describe the data at low-q. To describe the low-q upturn a cluster of five particles arranged in a tetrahedron (tetrahedron double shell model in SASfit) 96 is used. This aggregate is too small to describe the scattering observed. The mass fractal aggregate with Gaussian cut-off function, (implemented in SASfit) 96 which is a larger aggregate, was then used to fit the data. This is in better agreement with the scattering curve.

As shown by using IFT in fig. 3.11 Ludox has aggregates present. These are also particles with a strong negative charge. As seen in fig. 3.18 there is a strong structure factor present. Describing this structure factor is difficult as any structure factor should act on all particles present -both single particles and aggregates. At low volume fraction it is possible to avoid doing this, as the concentration of aggregates is low (for example in fig. 3.18); however at high volume fraction it is not straightforward to solve.

Attempts at modelling the structure factor of Ludox in the literature are limited. Notably there is the work of Kumar et al. 108,109 who use the double-Yukawa potential to describe the structure factor of Ludox HS40 and Ludox LS30 particles; however their work was mostly focused on understanding interactions between the particles and multivalent ions. There is also a thesis recently published by Bareigts 110 who used multi-component model simulations to understand the system. Similarly to Kumar et al. Bareigts was focused on understanding how ions in solution affected the particle structure factor. The simulation results were compared with the structure factor of Ludox HS40 and Ludox TM50. They were found to be in good agreement with SAXS data. 110 Here, as in the work of Kumar et al., the double-Yukawa potential is used with S(q) calculated within the mean spherical approximation. This is because the potential will be used in chapter 5 and so enables continuity throughout the thesis. The model is traditionally used to describe long-range repulsion and short-range attraction; it will not have this physical meaning here. It is used because of its flexibility: the combination of polydisperse system with a strong structure factor and aggregates is challenging to model traditionally. The attractive Yukawa term can describe the aggregates that were visible in fig. 3.11, and the repulsive term can describe the inter-particle repulsion. The reduced potential of such a solution has the form:

V (r ) k B T =    ∞ if 0 < r < 1 -K 1 exp -Z 1 (r -1) r -K 2 exp -Z 2 (r -1) r if r > 1 (3.11)
where r is expressed in units of the diameter of the particle's hardcore, d H S ; T the absolute temperature, k B is the Boltzmann constant; K i the amplitude of potentials 1 and 2; and Z i are interaction ranges (for a single potential Z would be the inverse Debye screening length). 106,111 Solving this analytically is mathematically and computationally challenging. Liu et al. presented an efficient method of computing the analytical solution, in mean spherical approximation. 111 Their MATLAB code was implemented in SASET (Muthig, 2013). 111,112vii The comparison between the double-Yukawa model and combination of Hayter-Penfold and mass fractal aggregate models is shown in fig. 3.19. This demonstrates that the choice of double-Yukawa model is appropriate in describing the shape of the curves.

It is important to note that the double-Yukawa model has no physical significance here beyond its ability to describe the average distance between particles. Having a scalable description of the Ludox structure factor is important for understanding the composite system of Ludox and Chitosan in chapter 5 and chapter 6. The double-Yukawa model with MSA is adequate for this purpose.

Conclusion

Possessing a relatively large surface area, low toxicity, chemical and thermal stability colloidal silica is widely used. 70 Scattering techniques are often used to characterise colloidal silica, typically by fitting with a model of polydisperse homogeneous spheres. Studies on the growth of silica particles during synthesis reveal that silica is composed of aggregates of smaller units with intra-particle and surface porosity. Such inhomogeneities are even more significant in small silica particles (those with diameters of less than 50 nm). This is why current fits of small silica particles fail to describe scattering curves in a completely satisfying manner.

Here we propose a model that accounts for the porosity present in silica using a fractal structure factor developed by Hurd and Flower. Along with providing a significant improvement in the fit of silica the model provides insight into the formation process of silica particles and particle porosity, a parameter which is often important in applications vii Fitting was performed on the concentration series. This is plotted in fig. A.3 of silica. We demonstrate the Hurd and Flower model is also appropriate for use with ideal particles by comparing the model with scattering from ideal spherical particles. When D f = 3 in the Hurd and Flower model the spherical form factor is recovered. This point is developed by fitting Bindzil, industrial silica that has a monomodal distribution. In the case of Ludox particles, we demonstrate that the Hurd and Flower model successfully fits the data. The Hurd and Flower model describes the polydispersity of the particles, roughness of the particle surface, and porosity. Therefore, in cases where a spherical form factor is unable to describe particles in a satisfactory manner the Hurd and Flower model may provide a suitable alternative. In the future it could be considered that a development of the Hurd and Flower model would enable quantification of particle porosity. As porosity is often an important parameter for applications of silica particles this would be useful.

The possibility of extracting the Ludox structure factor is demonstrated by fitting Ludox with a Hayter-Penfold structure factor for macro-ions and a double-Yukawa structure factor. It was decided to use the double-Yukawa structure factor for continuity with the rest of this thesis. Further work on separating contributions from particle polydispersity and inter-particle interactions would be interesting. This could be possible by adapting structure factors developed for calculating inter-particle potential in systems of polydisperse particles such as in the work of Ginoza and Yasutomi. 113 It would be particularly interesting to combine the model developed by Bareigts with the Hurd and Flower form factor, and to examine how this can describe a full concentration series of Ludox. This would be useful to experimentalists and industrialists alike.

Introduction

Chitosan is a linear polysaccharide that is industrially valued due to its use in water purification and biomedical applications. 114 It is formed by partial deacetylation of chitinthe world's second most abundant biopolymer -which is found in crustaceans, insects and fungi. Fully deacetylated chitosan is poly(β-D-glucosamine). The global market for chitosan, chitin and their derivatives was valued at $2.0 billion in 2016. 114 Due, in part to its industrial relevance chitosan has been widely studied in the literature, about 23.500 articles were published on chitosan in the decade 2000-2010. 115 Such studies have demonstrated some of the difficulties in developing products based on chitosan and chitin. Processing chitin is challenging as it is difficult to solubilise. [START_REF] Roy | Solubility of Polysaccharides[END_REF] Additionally, as a biomaterial originating from various organisms the processing method is dependent on the origin of the source material. This results in the quality of commercial chitin being variable which is in turn reflected in chitosan. 114,115,[START_REF] Rinaudo | [END_REF] The chemical and structural formulae of chitin and chitosan are presented in fig. 4.1. Chitosan is generally characterised by three parameters: molar mass, degree of acetylation, and patterning of acetyl groups on the chitosan backbone. 115,[START_REF] Rinaudo | [END_REF] These parameters are related to the chitin source and how the material was processed. Due to the number of possible permutations a variety of chitosans are commercially available; but understanding how parameter variation affects the behaviour of chitosan in an application is difficult. 115 In particular, the structure of chitosan aggregates in solution is incompletely understood. Understanding this aggregation is important, as many industrial applications of chitosan are based on its behaviour in solution (e.g. as a food thickener, in waste treatment, and a drug delivery agent). 39,40 Here we examine the structure, and aggregation of chitosan in solution using small-angle scattering (SAS) techniques. Chitosan with degree of acetylation below 0.5 is soluble under acidic conditions. Upon solubilisation chitosan forms concentration-dependent aggregates which exist in dynamic equilibrium with the chitosan solution. 39 Techniques used to study the aggregates include: light scattering (static and dynamic), 118,119 small-angle x-ray scattering, 120 viscometry, 121 fluorescence spectroscopy, 119,121 rheology, 40 and transmission electron microscopy. 119 It must be noted that there is much contradiction in the literature on chitosan. A review by Boucard et al. notes that the majority of small-angle x-ray scattering, SAXS, studies on chitosan find data that differs from predicted scattering behaviour. 120 In particular hydrophobic domains form within the chitosan aggregates. 40,119 These are formed due to competition between hydrophobic and hydrophilic forces acting on the chain. Independent studies demonstrate that acetyl groups do not contribute to these hydrophobic domains, and that chitosans with no acetyl groups still form hydrophobic domains. 40,119 This has been shown for chitosan samples in a buffer solution of 0.15 M ammonium acetate and 0.2 M acetic acid pH = 4.5; in 0.056 M trifluoroacetic acid with pH from 1.2-1.7; 40 and in 0.3 M acetic acid solution. 119 To explain these experimental data indicating the presence of domains the pearl-necklace model of Dobrynin and Rubenstein was used. 40,122 The pearl-necklace model describes hydrophobic polyelectrolytes as elec-trostatic blobs, or pearls which are connected by chains, or strings. Changing the pearls allows the system to move from a string-controlled to bead-controlled regime. 122,123 It is technically difficult to identify and examine the hydrophobic and hydrophilic domains of a polyelectrolyte in solution. Scattering techniques provide one method of doing so, indeed SAXS has previously been used to examine chitosan aggregates. 120 By using different incident radiation the contrast of a specimen can be varied. Here chitosan is examined by x-ray, neutron and light scattering to investigate its structure in solution.

Materials

Industrial chitosan

Chitosan was purchased from Sigma-Aldrich company. The molecular weight was 100 kDa, and the degree of acetylation was 0.2, as stated by the manufacturer. It was noted that many visible impurities were present when chitosan was dissolved. Steps were then taken to purify the chitosan following an adapted form of the procedure documented in Signini et al. 124 Chitosan was dissolved in 0.3 mol L -1 acetic acid and filtered three times, increasing the grade of filter paper (pore size: 25 µm, 11 µm and 2.5 µm) each time. After filtration the chitosan was placed in 12-15 kDa dialysis tubing and dialysed against 0.2 mol L -1 NaOH for 24 hours followed by dialysis against Milli-Q water for 36 hours with the water changed every 12 hours. When dialysis was complete chitosan was freeze-dried and stored in sealed containers.

Model chitosan

Pre-characterised chitosan was obtained from a collaborator, Professor M. Rinaudo, Université Grenoble Alpes. This was purchased from Primex with tradename ChitoClear, batch TM4778 and product code 42010. The chitosan had a degree of acetylation (DA) of 0.05 determined by nuclear magnetic resonance (NMR) spectroscopy. 125 The molecular weight of the chitosan was 160 kDa, determined by size-exclusion chromatography (SEC) . 125 Given that M w p o l y me r /N u ni t s = (1 -DA)M o g l u c o s a mi ne + DA * M o N -ac e t y l -g l u c o s a mi ne) , there are ≈ 1000 units on average present in each chain.

The intrinsic persistence length of chitosan is reported in the literature as 5 nm to 15 nm. 126 More specifically the literature states that for chitosan without acetyl groups the persistance length is 9 nm. [START_REF] Rinaudo | [END_REF] The persistence length changes slightly with degree of acetylation such that chitosan with DA=0.6 has persistence length of 12.5 nm. [START_REF] Rinaudo | [END_REF] Note that the chitosan persistence length is approximately equal to the radius of the silica particles characterised in chapter 3.

Methods

While SAXS originates from interactions of x-rays with electron density fluctuations in a sample, small-angle neutron scattering (SANS) originates from interactions of neutrons with variations in the scattering length density of nuclei in the sample. Light scattering originates from fluctuations in the dielectric properties (refractive index) within the sample. The underlying principles of light, neutron and x-ray scattering are similar. [START_REF]Neutron, X-rays and Light[END_REF] What differs is the contrast in each technique. Recall, from chapter 2 that contrast, ∆ρ, is the difference in scattering length density of the specimen in vacuum to that of the suspending medium.

For light scattering where dielectric constant, ε, and refractive index of the particles, n have relationship ε = n 2 the following holds: 127

∆ρ S LS = k 2 4π n 2 -n 2 L n 2 0 (4.1)
where n L is the refractive index of the suspending liquid and n 0 is the refractive index of the suspension. Recall eq. ( 2.3) which states that the contrast of particles which have volume, V, is given by:

∆ρ * = ρ(r) * -ρ * s , (4.2)
for scattering by neutrons and x-rays. For x-rays ρ(r) * is the product of electron density, ρ(r ) and the classical electron radius, R e =2.82 fm. For neutrons ρ = i b i /V where b i , the scattering length of neutrons is tabulated for individual nuclei. Importantly, different isotopes have different scattering lengths. Perhaps the most relevant isotopic difference in scattering length by neutrons is that of hydrogen (-3.8 fm), and deuterium (6.6 fm). [START_REF]Neutron, X-rays and Light[END_REF] This difference allows contrast matching experiments. Contrast is varied by changing the ratio of hydrogen and deuterium in solution. This changes contrast from the solvent and highlights different features of the sample under investigation.

Small-angle neutron scattering

SANS measurements were performed on instrument D11 at the ILL. [START_REF] Lindner | [END_REF] Stock solutions (1 %) of chitosan were prepared by dissolving the chitosan in either a D 2 O sodium acetate buffer solution, or acidified D 2 O solution. The acidified D 2 O solution, with a pD of 4.5 (measured pH of 4.8), was produced by diluting 37 % HCl with D 2 O (deuterium oxide, or heavy water). D 2 O sodium acetate buffer solution was made by mixing 0.2 mol L -1 of sodium acetate i in D 2 O and 0.3 mol L -1 of acetic acid ii in D 2 O to achieve a pD of 4.5 (measured pH of 4.8). iii The stock solutions of chitosan were stirred for 24 h to ensure the chitosan was dissolved and then filled into dialysis cells with pore size of 1 kDa. The dialysis cell had a membrane holder to prevent volume changes. Buffer solution was inserted in the second half of the dialysis cell. This was replaced every 3 h for a period of 20 h. Following dialysis the chitosan solutions were diluted using the final dialysed buffer solution to the required concentration.

Small-angle x-ray scattering

SAXS experiments were performed at beamlines ID02 and BM29 at the ESRF. Data collection was challenging as polyelectrolytes are sensitive to x-ray beam damage and have low x-ray contrast. Chitosan beam damage was studied. X-ray exposures of increasing duration were performed and compared to identify the exposure duration at which beam damage occurs. This follows standard protocols for measuring biological macromolecules. 129 Nonetheless obtaining quality data with no beam damage is challenging as observed in fig. 4.2. Five different exposure times were tested: 0.03 s, 0.1 s, 0.3 s, 1 s and 3 s, corresponding to incident photon flux, Φ 0 =9 × 10 10 , 3 × 10 11 , 9 × 10 11 , 3 × 10 12 and 9 × 10 12 photons respectively. Five frames were taken for each exposure time e.g. 5×0.03 s. Beam damage could be examined between frames and between exposure times. Background measurements were also taken in the same manner. The background measurement with exposure time matching that of the spectra was subtracted.

Unfortunately, short exposure times lead to the collection of noisy data. Noisy data are problematic as chitosan and its background have similar intensity. This is shown in fig. 4.3. Finding a compromise between minimising beam damage, and collecting statistically significant data that can be separated from its background was challenging. Short exposure times were used, and multiple frames were averaged together to reduce noise. To improve background subtraction the x-ray beam position on the capillary of the flow-through cell was not changed. Long intervals were then left between x-ray exposures to prevent beam damage. This enabled new material to diffuse into the path of the x-ray i Sodium acetate, Sigma-Aldrich, S2889, anhydrous, for molecular biology, ≥99 % ii Acetic acid, Merck, 27225, puriss.,99.8 % to 100 % iii Note that deuterated samples were prepared at pD 4.5 and pH 4.8; non-deuterated samples were prepared at pH 4.5 beam. For future experiments a better option would be the use of continuous flow during exposure which is possible at ID02.

Measurement procedure

The same samples of chitosan from Sigma-Aldrich were measured by SAXS and SANS on the same day. One aliquot of the sample was measured with neutrons in a cuvette cell on D11, and a second aliquot was taken to beamline ID02 where it was measured using a flow-through cell (details in section 3.3). A third aliquot was kept back and using this SLS data was obtained some weeks later. SLS measurements were performed using an ALV CGS-3 compact goniometer system at the partnership for soft condensed matter (PSCM) laboratories. Model chitosan was measured on different days, initially by SANS in a cuvette cell on D11, and several days later by SAXS in a capillary cell on beamline BM29. 120 They show the characteristic polyelectrolyte peak, indicated with arrows in fig. 4.4, and an upturn at low-q. This correlation peak changes position with changing concentration as is expected for polyelectrolytes. 130 At 1 % the peak is located at 2π/q ma x corresponding to ≈12.5 nm and at 0.3 % the peak is observed at ≈19 nm. As shown in the inset of fig. 4.4 even in linear scale no peak is visible for the lowest chitosan concentration of 0.1 %.

When the chitosan was solubilised in excess salt in the form of a 0.2 mol L -1 sodium acetate and 0.3 mol L -1 acetic acid buffer solution the polyelectrolyte peak was suppressed. This is shown in fig. 4.5. This is expected behaviour -excess salt typically suppresses the correlation peak in polyelectrolyte solutions. 130 Typical polyelectrolyte behaviour is observed in the SAXS data for both chitosan in HCl and chitosan in acetate buffer. The polyelectrolyte peak develops with increasing chitosan concentration and its position is indicated with arrows.

The inset shows the polyelectrolyte peak in linear scale. An attempt was made at determining the chitosan molecular weight from the scattering data. This is done by examining the forward scattering, I (q → 0). [START_REF] Feigin | Structure Analysis by Small-Angle X-Ray and Neutron Scattering[END_REF][START_REF]Neutron, X-rays and Light[END_REF] I (q → 0) = N (∆ρV c h i t o s an ) 2 P (0)S(0)

(4.3)
where N is the number density of chitosan of volume V c h i t o s an . These have contrast ∆ρ. P (0) and S(0) are the chitosan form and structure factors at q = 0, defined in chapter 2, it is assumed they can be neglected (by definition P (0) = 1 and it is assumed that S(0) = 1 corresponding to no interactions between chains). The number density of polymer chains can be written as mas s c h i t o s an N A /M w c h i t o s an V s o l . Where V s o l is the solution volume. Similarly V c h i t o s an can be written as M w c h i t o s an /N A d where d is the chitosan density. It is then possible to rewrite eq. ( 4.3) as

I (0) (∆ρ) 2 = mas s c h i t o s an V s o l M w c h i t o s an N A d 2 . iv (4.4)
Many of these parameters are known: ∆ρ can be found in table 4. . The calculated value is close to neither the mid-q plateau of the scattering at low-q. This discrepancy between observed and theoretical intensity may be attributed to the strong organisation in solution.

Comparing SAXS, SANS and SLS

Both SANS and SLS data were also collected for the chitosan solubilised in D 2 O acidified with HCl and D 2 O buffer solution. In fig. 4.6 results for the sample solubilised in D 2 O acidified with HCl are presented. It can be seen that the SAXS and SANS data differ for all sample concentrations across the entire q-range. It can also be observed that the SANS data do not display any correlation peak. To verify that there is no correlation peak in the SANS data a plot in linear-linear scale was constructed, fig. 4.7, which confirms the correlation peak is absent. It is unexpected that the SANS and SAXS data differ so significantly. It is also unexpected that the correlation peak is not visible iv mm -1 /nm -4 =g/cm 3 ×g mol -1 /mol -1 g 2 cm -6 It can be seen that there are significant differences in the scattering data. All samples were solubilised in D 2 O acidified with HCl. Data at 1.0 % chitosan concentration have been offset by a factor of 10, and data at 0.1 % chitosan concentration have been offset by a factor of 0.1.

in the SANS data. Such a peak has previously been observed when using SANS to study other polyelectrolytes. 131 Nierlich et al. examined the polyelectrolyte peak in solutions of polystyrene sulfonate. They found that for a ratio of salt to polystyrene sulfonate >0.33 the polyelectrolyte peak was visible.

In fig. 4.8 SAXS, SANS and SLS data are presented for chitosan obtained from Sigma-Aldrich dissolved in D 2 O buffer solution. The data are in agreement with those observed in fig. 4.6 and in disagreement with what could be expected. The SAXS and SANS data differ across all sample concentrations and over the entire q-range. As before no correlation peak is observed in the SANS data.

It can be noted that the scattering curves obtained from chitosan in HCl and acetate solution differ. It can also be noted that in both cases the SANS data do not change significantly with respect to the changing sample concentration. No plateau is observed in the SLS data at low-q indicating the presence of large aggregates (>1.5 µm).

Given the surprising nature of the results an equivalent sample in H 2 O buffer solution was prepared and measured at beamline ID02. The SAXS data for this sample was compared with the SAXS for the sample in D 2 O buffer solution, and the results are shown in fig. 4.9. There is no difference in scattering between chitosan solubilised in H 2 O buffer solution or in D 2 O buffer solution as observed by SAXS. The difference at low-q can be ascribed q /nm -1 Fig. 4.8 SAXS and SANS data collected on the same samples of low molecular weight Sigma-Aldrich chitosan in D 2 O buffer solution on the same day (SAXS from ID02). SLS data collected several weeks later on the same samples. It can be seen that there are significant differences in the scattering data, especially in the mid-q region. The blue dashed line highlights the discontinuity in the slope of the SAXS data at 2π/0.05≈125 nm. Data at 1.0 % chitosan concentration have been offset by a factor of 10, and data at 0.1 % chitosan concentration have been offset by a factor of 0.1. to background subtraction which is challenging for such a weakly scattering sample. It appears that substituting D 2 O for H 2 O in the buffer solution does not significantly affect the sample behaviour in solution and has not distorted the results. The SAXS data in fig. 4.8 show a discontinuity in their slope at 2π/0.05≈125 nm. This discontinuity is highlighted in fig. 4.8 using a blue dashed line. It can be noted that there is no such discontinuity present in the SANS data for this q-value. Instead fig. 4.10 shows a discontinuity in the slope of the SANS data corresponding to ≈12 nm. This is illustrated in fig. 4.10 with a dashed grey line and appears independent of the buffer solution.

Model chitosan

It is clear that there are significant differences in scattering data obtained by SAXS, SANS and SLS. It is also clear that the structure of chitosan observed in the scattering data changes depending on whether the chitosan was solubilised in D 2 O acidified with HCl or D 2 O buffer solution.

To better understand the system model chitosan with a known degree of acetylation of 0.05 was measured by SAXS and SANS. The results are shown in fig. 4.10. The SANS curves are almost identical for samples solubilised in HCl or acetate solution; however between SAXS curves there are significant differences. As expected, the HCl SAXS curve the factor of four difference expected in I (q) due to SLD differences between the SAXS and SANS data for the sample solubilised in acetate. The blue dashed lines indicate the factor of eight difference expected in I (q) due to SLD differences between the SAXS and SANS data for the sample solubilised in acidified D 2 O. The calculated differences hold at high-q; but not in the mid and low-q regions. The grey line highlights the discontinuity in SANS data present at ≈12 nm.

has a correlation peak at ≈0.675 nm -1 . This corresponds to the value of ≈0.7 nm -1 found by Boucard et al. 120 for chitosan in HCl with DA = 0.03 . The chitosan in acetate buffer solution shows no obvious correlation peak.

As the molecular weight for the model chitosan is known to be 160 kDa the expected forward scattering can be calculated using eq. (4.3). This gives a value of 2.5 mm -1 for SAXS, and a value of 0.4 mm -1 for SANS. Neither of these values are in agreement with the experimental data as seen in fig. 4.10. Scattering techniques cannot easily determine the molecular weight of chitosan.

Scattering length density

That SAXS and SANS curves of chitosan are different is a result that has not been previously published. However differences in the scattering behaviour, when examined by SAXS and SANS, of another polyelectrolyte, poly(methacrylic acid) have been documented in a series of papers by Pleštil et al. 132,133 Their data were collected over a smaller q-range and not combined with information from light scattering. As a result Pleštil et al. concentrated their efforts on understanding differences in scattering from the correlation peak they observed in both SAXS and SANS. They explained their results as the consequence of scattering length density differences. These differences were caused by condensation of counter-ions on the polymer chain, and a complex hydration shell surrounding the polyelectrolyte in solution. This resulted in three different scattering regions: bulk solution, the hydration shell surrounding the polyelectolyte, and the polyelectrolyte itself.

A difference in scattering behaviour between SAXS and SANS, as observed here, can only be explained by the presence of more than two domains with differing scattering length density. In chapter 3 a one-component system of silica particles in water was examined. Contrast was simply the scattering length density difference between silica particles and water. For such a system the SAXS and SANS data would show scattering curves with an identical form but different intensities due to the different contrast. To explain the chitosan data presented in this chapter there must be a variation in the contrast across chitosan i.e. inhomogeneous contrast. In the introduction to this chapter, section 4.1, it was stated that hydrophobic domains are formed within chitosan aggregates. It was found that acetyl groups do not contribute to these hydrophobic domains. This result is consistent with the scattering data presented here: the difference between SAXS and SANS data persists even for chitosan where the degree of acetylation is known to be small -DA=0.05 (fig. 4.10).

An obvious starting point to understanding these hydrophobic domains is to do as Pleštil et al. 132,133 and examine the effect of counter-ion condensation on scattering length density. This is also consistent with the theory of Manning condensation. 134 Manning theory states that counter-ions occupy two positions in polyelectrolyte solution: bound to the polyelectrolyte chain, and further away from the chain in bulk solution where counterions can be considered as unbound. The position is determined by the condensation threshold given by the ratio of the Bjerrum length v to the distance between charges along the polyelectrolyte chain. [START_REF] Prigogine | Polymeric systems[END_REF] When the condensation threshold is > 1 counter-ions are bound to the polyelectrolyte chain. Here, the parameter is ≈1.3 which is > 1 but not significantly so; therefore the condensed fraction of ions is weak.

Chemsketch (version 2018.2.1 from ACD/Labs) was used to estimate the molecular volume of fully deacetylated chitosan (poly(β-D-glucosamine), DA=0) and molecular volume of chitin (poly(β-D-N-acetyl-glucosamine), DA=1). The contribution of either an acetyl or chloride group to the molecular volume was then estimated, and SLD calculated with the results displayed in table 4.1. It can be seen that the SLD for x-rays does not change when in deuterated solvent, which is consistent with the data in fig. 4.9. This is not true of the SLD of neutrons which changes significantly for chitosan in deuterated and non-deuterated solvent, which is expected. Chitosan contains labile hydrogen atoms that exchange with those from the solvent; therefore chitosan in D 2 O contains both H and D atoms. As x-rays are scattered by electrons the isotopic substitution has no effect on the SLD; however for neutrons, which are scattered by nuclei, the SLD is changed. The change in SLD between chitin and chitosan with counter-ion condensation is unexpected. Recalling eq. (2.10): I (q) = N p (∆ρ) 2 V 2 P (q)S(q) the observed scattering intensity, I (q), is proportional to the contrast squared, (∆ρ) 2 . Even small changes in the SLD of chitosan will have a significant effect on the scattering curves. Examining the contrast of chitosan in DCl gives (∆ρ) 2

x-ray s ≈29.16 × 10 -8 nm -2 and (∆ρ) 2 ne u t r on s ≈3.49 × 10 -8 nm -2 . This means that the I (q) of samples measured by SAXS should be ≈8 times higher than the samples measured by SANS. Looking at fig. 4.6 and fig. 4.10 this prediction is refuted by experiment. While at high-q the SAXS data is larger than the SANS data the difference is not constant, and at low-q the SANS data are larger! In fig. 4.10 dashed blue lines illustrate the factor of eight difference. At high-q the SAXS data have eight times more intensity than the SANS data; however this is not the case at low-q. In the case of acetate buffer (∆ρ) 2

x-ray s ≈21.81 × 10 -8 nm -2 and (∆ρ) 2 ne u t r on s ≈5.84 × 10 -8 nm -2 . The SAXS data should be ≈4 times higher than the SANS data. Again, the difference is not constant at high-q although the SAXS data appear to be more than 4 times higher and at low-q the SANS data are higher. In fig. 4.10 dashed yellow lines illustrate the factor of four difference. The result mirrors that of the HCl curves. At high-q the SAXS data have the expected four times more intensity than the SANS data; however this is not the case at low-q. SLD differences can explain the difference in scattering curves observed at high-q; but they are not sufficient to explain the difference in scattering from aggregates observed at low-q.

Here counter-ion condensation as described by Manning, which describes counterions in either a bound or unbound state, has been considered. This is extended in the three-state model of Kramarenko et al. which adds the third possibility of free, unbound counter-ions inside polyelectrolyte aggregates. [START_REF] Kramarenko | [END_REF] No existing model describes scattering from such a structure and a full description is beyond the current scope of this work. Future SANS experiments that contrast match the chitosan would be required along with considerable computer simulation to fully understand the results presented here.

Correlation peak

The correlation peak, or indeed any peak visible by SAS indicates a structural correlation. This correlation can arise either from inter-chain interactions, or from interaggregate interactions. It is possible to conclude the correlation peak arises from inter-chain interactions by a thought experiment.

This correlation can be examined by assuming the entire volume of the chitosan solution is occupied by cubes with side of distance, d . Each cube will have volume d 3 and have a total of 3d sides per cube. Model chitosan was measured by SAXS at a concentration of 1 % corresponding to 65.4 mmol L -1 . Multiplying with Avogadro's number converts this to 0.036 nm -3 . It is now possible to calculate d = 3 3/0.036=9.1 nm. This corresponds well to the peak position of model chitosan with DA=0.05 which is 9.3 nm as seen in fig. 4.10. vi If the correlation peak was due to inter-aggregate interactions it would have a value less than 9 nm. Note that this model is not intended to mean that chitosan has cubic packing, it simply shows that chitosan fills space in a mesh network.

It can also be considered that the chitosan network is hexagonal instead of cubic. The area of a hexagon is

3 3 2 d 2
, where d is the length of a hexagon side. Assuming the correlation peak position corresponds to the length of the hexagon side d=9.3 nm the hexagon area is 225 nm 2 . The average molecular volume of model chitosan is 0.17 nm 3 . It is known that the length of a chitosan unit is 0.52 nm; 137 therefore each chitosan chain vi A similar calculation can be performed for the chitosan obtained from Sigma-Aldrich. Assuming the manufacturer's estimate of DA=0.25 is correct this gives an approximate molecular weight of 175 corresponding to 57 mmol L -1 and 17 mmol L -1 for the 1 % and 0.3 % samples respectively. This corresponds to predicted correlation peak positions of 9.4 nm and 17.1 nm which are in reasonable agreement with the experimental values of 12.6 nm and 19.3 nm. occupies a surface area of 0.17/0.52 =0.33 nm 2 . Each hexagon contains three chains. The volume fraction of each hexagon which is composed of chitosan is then 3 * 0.33/225 which gives 0.0044. This is in reasonable agreement with the volume fraction for 1 % chitosan of 0.0064. Assuming a hexagonal configuration it can be concluded that the chitosan correlation peak arises from inter-chain interactions, and that chitosan occupies all space.

Overall it appears that chitosan occupies all available space without forming a significant number of very dense clusters. However this contradicts TEM images of chitosan which show discrete aggregates. 119 DLS shows two modes: a fast mode ≈10 nm and a slow mode ≈100 nm. The fast mode is attributed to the polymer chains, and in agreement with the correlation peak position. 39,138 The slow mode is attributed to aggregates. 39,139,140 However there is no clear evidence of structures with a size of ≈100 nm in the scattering data. The low-q scattering indicates the presence of either very large structures >1 µm, or a configuration that occupies all space.

Conclusion

The structure of chitosan in solution was studied by small-angle scattering methods. Scattering curves of chitosan by light, neutrons and x-rays show differing features. The behaviour was demonstrated on chitosan from two different sources, with differing degree of acetylation, solubilised in either acetate buffer, or HCl solution at pH 4.5. One of the chitosan sources is considered "model" chitosan with a degree of acetylation of ≈0.05. The discrepancy in scattering behaviours can only be explained by the presence of several domains with dissimilar scattering length density. This confirms previous work showing that the formation of hydrophobic domains accompanies the aggregation of chitosan in solution, and that this is independent of the degree of acetylation of the chitosan chain. 119 This could mean the hydrophobic domains are caused by the sugar backbone, where one face has a more hydrophobic character than the other. Other materials could exhibit similar behaviour.

An examination of counter-ion condensation on the chitosan backbone reveals that the SLD of chitosan changes in the presence of counter-ions. The simple model of counterion condensation examined here is not sufficient to describe the data. The scattering curves presented here indicate that the domains formed by chitosan in solution are more complex than initially thought. An examination of the correlation peak behaviour of the SAXS curves indicates that chitosan chains occupy all available space. Future experimental and theoretical work is required to understand the system in the form of contrast-matching SANS experiments and computer simulations.

CHAPTER 5 STRUCTURE OF SILICA-CHITOSAN COMPLEXATION

Introduction

The materials used in this thesis were described, and their structure characterised in chapter 3 and chapter 4. Both of these materials are widely used for scientific research as well as industrial applications. However, the aim of this thesis is not to study chitosan and silica individually; but rather to examine how these oppositely charged species interact when combined. Studies examining mixtures of oppositely charged colloidal species have been gaining interest in recent years. This is partly due to fundamental interest: the challenge described by Van der Gucht in chapter 1: "to find microscopic explanations for macroscopically observed behaviour;" 26 requires understanding the microscopic explanations behind the macroscopically observed behaviour in binary systems.

There is interest in such systems from the pharmaceutical industry. Recent work has shown that when nanoparticles are combined with biological fluids nanoparticle-protein complexes are formed. 141,142 It is only nanoparticle-protein complexes that are identified by our bodies, and these determine biological responses such as cellular uptake, bioavailability and toxicity. 141 This has implications for drug delivery and imaging. Given the importance of protein-nanoparticle interactions recent studies have examined two model systems: silica-lysozyme, and silica-bovine serum albumin. 44,143,144i These systems are dependent on the size, shape and charge of both the proteins and nanoparticles. They are also dependent on the pH, ionic strength and temperature of the suspending solution. 144 The parameter space is vast.

In addition to investigation of nanoparticle-protein systems a recent review by Ballauff et al. 145 discusses work on polyelectrolyte and protein systems. 145 Such systems are industrially relevant as polymer chains are often used to prevent proteins from adsorbing to i Both lysozyme and bovine serum albumin are globular proteins. nanoparticles in biological fluids. 145 Therefore, to understand the full lifecycle of nanoparticles in the body it is necessary for a toxicologist to understand not only nanoparticle-protein interaction; but also nanoparticle-polyelectrolyte interaction.

The literature contains many case studies examining oppositely charged systems. For example Gradzielski et al. studied chitosan-surfactant systems and chitosan-microemulsion systems; [146][147][148][149] 130 and the kinetics of complexation of oppositely charged polyelectrolytes was inspected by Takahashi et al. 152 Results from such experimental work can be combined with the results of computer simulations, which use complex inter-particle potentials, such as the work of Godfrin et al. 31 or results from theoretical work by Gurovitch and Sens. 153 The literature referenced above is a small fraction of what has been published. There are many different types of colloidal system resulting in many permutations when combining oppositely charged colloids. Even so there are some similarities which are highlighted in a review, of complex coacervation of proteins and polysaccharides, by De Kruif et al. 41 which found that oppositely charged colloidal systems have a common phase diagram with concentration-dependent gas-liquid phase separation (the liquid phase is either a precipitate or liquid coacervate). Below the phase boundary (coalescence line) complexes are observed as turbidity in solution. 41 Extending the literature search to include other forms of oppositely charged colloidal system it appears that the generalisations of De Kruif can be applied to systems of polyelectrolyte and microemulsion droplets, 148 polyelectrolyte-micelle systems, 154 and polyelectrolyte with nanoparticles. 36 Understanding the structure of these complexes is important. Complex formation is driven by electrostatic attraction and can be controlled by salt concentration -De Kruif plots a phase diagram in terms of conductivity, which is used as a proxy for salt concentration. 41 If the complex structure changes significantly upon administration in a biological fluid this can have serious implications for therapeutic applications. These complexes are not equilibrium structures which means their structure can evolve with time. Such structural flexibility raises the future possibility of controlling inter-particle interactions to obtain a desired structure. Underlining this is a paper by Shi et al. indicating the presence of rod-like complexes in a chitosan-silica system. 36 Rod-like complexes are of interest because they can be aligned in the presence of an external field. This is particularly valuable for optoelectronic applications. 155 Liquid crystalline technology, based on the principle of alignment under field, is now omnipresent -including mobile phone screens. This chapter explores the silica-chitosan system in detail. We begin by reproducing the state diagram of Shi et al. 36 and then use fast mixing experiments to examine how the complexes form. The behaviour of rods is tested by applying an external shear field. Electrophoretic mobility measurements confirm the chitosan-silica complexes are formed by electrostatic interactions. These complexes are examined using a combination of small-angle x-ray scattering (SAXS) and x-ray photon correlation spectroscopy (XPCS) measurements. Storage tests are used to examine the final state of the system.

Materials and methods

This system was inspired by the work of Shi et al. 36 who used Ludox AM and chitosan from Sigma Aldrich in pH 4.5 buffer solution composed of 0.3 mol L -1 acetic acid and 0.2 mol L -1 sodium acetate. 36 Ludox AM particles are negatively charged with a radius of ≈9 nm and are considered stable across a broad pH range. 156 However the Ludox AM particles supplied by Sigma-Aldrich vitrified in solution after several months. Literature from the manufacturer states that Ludox TMA is more stable in solution. [START_REF] Materials | Ludox Colloidal Silica in Coatings[END_REF] This was supported by storage tests which showed that over long timescales Ludox TMA was stable in storage. Due to this stability it was decided to use Ludox TMA instead of Ludox AM.

This system is composed of Ludox TMA and chitosan, of low molecular weight obtained from Sigma Aldrich, in pH 4.5 buffer solution made with 0.3 mol L -1 acetic acid and 0.2 mol L -1 sodium acetate. All materials used here have been characterised in detail. The silica particles were described in chapter 3, and the chitosan was described in chapter 4. These materials were chosen as the silica particle size is thought to correspond approximately to the polyelectrolyte persistence length following work by Shi et al. 156 which indicated the ratio of particle size to persistence length can control the fractal dimension of the complexes formed. 156

Sample preparation

Stock solutions of 0.1 % chitosan in buffer solution were prepared. The desired amount of chitosan stock solution was weighed on a balance and buffer solution added. A vortex mixer was used to combine the chitosan and buffer. While on the vortex mixer the silica dispersion was added using a pipette. This ensured a homogeneous mixture, and consistency between samples. The samples were stored vertically.

Samples made using this method were highly reproducible; however experiments performed in this thesis indicate the mixed solutions are not in thermodynamic equilibrium.

Therefore, it can be expected that changing the preparation method will lead to different results.

Stopped flow device

Kinetic studies are a two-step process first components are mixed, then they are observed. There are two limitations: mixing speed, and observation rate. Stopped-flow mixing experiments study fast kinetics by maximising the mixing speed, and minimising the time between mixing and detection.

The stopped flow apparatus available at beamline ID02 is the Bio-Logic SFM-400, modified with a capillary suitable for use with SAXS/WAXS. This is a four-syringe device, containing three Berger ball mixers and hard stop. The dead time ii varies between 2 and 3 ms at an optimal flow rate of 6.67 ml s -1 . This was determined by the change in transmitted intensity of x-rays (of an energy just above the bromine K-edge) upon mixing KBr with water. 157 The dead-time, combined with the available detectors (normally a Pilatus detector) on ID02, ensure that dynamics on the order of milliseconds can be obtained. A complete description of the set-up, including calibration methods, can be found in Narayanan et al. [START_REF] Narayanan | Advances in Planar Lipid Bilayers and Liposomes[END_REF]

Rheo-SAXS set-up

Rheo-SAXS measures the rheology of a sample, how it flows and deforms, and its scattering simultaneously. This was achieved by using a stress-controlled rheometer (RS6000 Thermo-Haake) with custom polycarbonate shear cell, details in Panine et al. [START_REF] Panine | [END_REF] The response of the sample was observed in both shear controlled, and stress controlled modes. The maximum frequency applied was 10 Hz and the maximum shear stress was 40 Pa.

Electrophoretic titration

A Malvern Zeta-sizer Nano Z was used to measure the electrophoretic mobility of solutions. A comparison was made between the electrophoretic mobility of Ludox TMA and chitosan from Sigma-Aldrich samples solubilised in pH 4.5 buffer solution, and the electrophoretic mobility of a model system of Bindzil 40/130 and chitosan with DA=0.05 samples in low-salt conditions. The chitosan was solubilised with the stoichiometric amount ii Dead-time is the minimum time between initial mixing and initial detection.

Results and discussion

State diagram of the system

A state diagram of the system immediately after sample preparation is presented in fig. 5.1 for comparison with the work of Shi et al. 36 As the diagram is based on observations from a system that is unlikely to be at thermodynamic equilibrium the term "phase diagram" is avoided, and instead the more neutral "state diagram" is used.

Despite replacing Ludox AM with Ludox TMA the state diagram is qualitatively similar to that published by Shi et al., 36 and follows the description of De Kruif et al. 41 Increasing chitosan concentration leads to gas-liquid phase separation. Below the phase boundary the solutions are turbid indicating the presence of complexes. Phase separation occurs in less than five minutes in the phase separated region of the diagram. This phase separation is attributed either to precipitation of complexes, or to complexes forming a liquid coacervate phase. 41

Kinetics of complexation

Stopped-flow mixing experiments were carried out to observe initial cluster formation above and below the coalescence line. An overview of how the fast kinetics relate to the state diagram is presented in fig. 5.2. The first frame was acquired while the syringes were still flushing the mixture into the observation headgroup to allow observation of the mixture at the dead-time (2 ms to 3 ms after mixing). Figure 5.2 shows that most of the kinetics of complex formation occur in the dead time of the stopped flow device. A similarly fast aggregation behaviour was observed by Takahashi et al. when they used stopped flow mixing to examine polyelectrolyte complexes. 152 Complex formation occurs on a sub-millisecond timescale. It can be seen in fig. 5.2 that the kinetics of the system change above and below the coalescence line. The kinetics above the coalescence line are observable on the timescales probed by the experiment. Therefore the kinetics above the phase boundary are examined here.

In fig. 5.3 and fig. 5.4 the kinetics of 0.1 % Ludox TMA and 0.001 % chitosan are examined. The figures show that cluster growth dominates the scattering in the low-q region. The increase in low-q scattering intensity indicates that the number of particles per cluster grows over the experiment duration. However, there is structure compaction visible in the mid-q region. It appears that as the clusters grow they become more tightly packed. The structural compression shown in fig. 5.4 is observed across the biphasic region of the state diagram. This is quantified in fig. 5.5 which shows the kinetics of formation of 1 % Ludox TMA and 0.01 % chitosan. The data are described using a combination of the Hurd and Flower model (explained in chapter 3) and an extension to the Orstein-Zernike function eq. (3.2) which was also described in chapter 3. In chapter 3 the Ornstein-Zernike function was given by:

I (q) = I (0) 1 + q 2 ξ 2 . (5.1)
where ξ is the correlation length of density fluctuations, which decays as ≈ (1/r ) exp ( -r ξ ). This equation is now re-written as an effective structure factor and extended to account for the cluster morphology, using the Fisher-Burford structure function, [START_REF] Narayanan | Structure from Diffraction Methods: Inorganic Materials Series[END_REF] such that:

S e f f (q) = N a (1 + 2 3D f q 2 R 2 g ) D f 2 (5.2)
where N a is the aggregate number of the clusters. Figure 5.5 shows that the eq. ( 5.2) equation describes the data well at low-q, and the Hurd and Flower model describes the data well at high-q. The mid-q region is not fit. There is significant particle compression which increases by time, as seen in the inset of fig. 5.5. It was not possible to fit this region of the scattering curve using straightforward structure factor models, as these models are unable to describe the observed decrease in intensity at mid-q. The missing intensity of such a feature has previously been referred to as a "correlation hole". 97,161,162 It arises from repulsive interactions between silica particles in aggregated systems. 97 Describing the depth of the hole requires a structure factor with high effective volume fraction. Structure factors with such high volume fraction show a sharp correlation peak (structure factor peak). As seen in fig. 5.5 no sharp structure factor peak is observed. It is possible to smear the peak when fitting, by using a high polydispersity; however the polydispersity required is much higher than that used to fit the form factor oscillations. The fitting parameters used in fig. 5.5 are extracted and plotted against time in fig. 5.6. It can be seen that the number of particles per cluster (aggregation number) increases with time, as does R g . As more particles are added to the aggregates the aggregates get bigger. The inset in fig. 5.6 shows the fractal dimension of the clusters grows quickly before reaching an apparent plateau. It can also be seen that most of the kinetics are finished within 10 s. In this time clusters are formed, grow larger, become more compressed, change shape and begin to sediment. The inset shows a plot of residuals from the fit. This can be compared to the non-coalescing region of the state diagram. Increased intensity relative to the form factor of the silica particles indicate that complexes are formed. The kinetics of complex formation finish faster, as shown in fig. 5.2 and their morphology is different. The long-term kinetics of the system do not require a stopped flow device for measurements as they occur on timescales longer than minutes. Clusters in the non-coalescing region of the state diagram are also less compact -there is no intensity loss in the mid-q region. The power-law dependence of the system is not as pronounced as in the phase-separating region of the diagram, the low-q region appears to show a change in slope. This indicates the presence of a hierarchy of structures.

Stopped flow measurements indicate that the monophasic region of the state diagram is particularly interesting for a structural study. Additional measurements revealed a region of the state diagram with a strong q -1 dependence suggesting the presence of rod-like complexes such as those reported by Shi et al. in the Ludox AM-chitosan system. 36 Such power laws have also been observed by Simon et al. 148 in polyelectrolyte-microemulsion drop complexes; and simulation work by Mani et al. 33 that examined colloids interacting via short-range attraction and long-range repulsion identified quasi-linear clusters resembling a Bernal spiral. Given this evidence in the literature for quasi-linear structures in complexes of oppositely charged colloidal systems it was interesting to probe the behaviour of such structures under an external field. If it would be possible to align complexes using an external field this raises the possibility of liquid-crystal-like applications.

Effect of shear

As an initial assessment one of the samples which exhibited the power-law scattering corresponding to a slope of q -1.3 was measured using RheoSAXS. Scattering from this sample, 16 % silica particles and 0.0225 % chitosan, is shown in fig. 5.7. It can be seen that at very low-q there is some deviation from the slope of -1.3; but at intermediate-q the power law dependency holds.

Contrary to expectations, when the sample was sheared, to the maximum rate accessible by the instrument, no anisotropy was visible in the 2D scattering pattern. To further confirm there was no anisotropy present in the data an anisotropy parameter, A H was calculated using the software package SASET by Michael Muthig. This method of determining anisotropy was chosen as it is more precise than principal component analysis (PCA), and it is model independent. 55 Calculating the anisotropy parameter requires excellent statistics and no null or negative values, these conditions are guaranteed by the high intensity statistics of the measurement. A complete mathematical description of the anisotropy calculation can be found in (Muthig,2013). 112 The method takes a ring of intensities on the 2D scattering pattern and uses information theory to calculate the entropy in intensities in this ring. In a perfectly anisotropic 2D scattering ring A H =1 and for an isotropic pattern A H =0. Scattering data was checked for anisotropy by selecting many intensity rings over a range of q-values, this process was repeated for varying values of applied shear stress, τ. As shown in fig. 5.8 no significant alignment, or anisotropy was observed either as a function of shear stress or of q-value. At best there is a slight increase in anisotropy for q values <0.0526 nm -1 (corresponding to lengths >100 nm) and shear stress τ>1 N m -2 . To examine this behaviour more closely the anisotropy parameter, A H , for q-values corresponding to 130 nm to 170 nm, is plotted against shear stress in fig. 5.9. In the plot the frequency of oscillatory shear stress applied is varied. It can be seen that at a frequency of 0.1 Hz the weak anisotropy parameter increases marginally with increasing τ. When the frequency is 0.5 Hz and 1 Hz the anisotropy also increases with τ but the effect is less than at low frequency, even when a higher shear stress is applied. Therefore to observe alignment a low-frequency shear stress needs to be applied, and the sample observed in the correct q-range; however as shown in fig. 5.9 the experimental effect is tiny. A H and applied shear stress τ for small to small angles (as shown in the q-values). In the case of an anisotropic system A H = 1 and for perfect isotropy A H = 0. The sample is 16 % TMA and 0.0225 % CSL and measured in the radial position. Fig. 5.9 Plot showing the anisotropy parameter A H and applied shear stress τ for q-values corresponding to 130 nm to 170 nm. The rheometer frequency was varied, which is shown in the legend. In the case of an anisotropic system A H = 1 and for perfect isotropy A H = 0. The sample is 16 % TMA and 0.0225 % CSL and measured in the radial position.

Additionally there was no significant change in the 1D scattering curves. iii The lack of anisotropy and lack of structural change requires contemplation. iv It is possible that no rod-like structure is present in the sample; however there could be highly dynamic rod-like structures which are transient. Alternatively, some larger structure is present composed of units with a rod-like shape. In this case it is likely that shear would break-up this larger structure, and this fragmentation would be visible in the scattering data. An example of this is the case of colloidal gels. Rheo-SAS studies of colloidal gels, along with numerical studies have found significant anisotropy in the scattering patterns of such samples when sheared. [163][164][165] The visibility of such anisotropy is closely linked to the yield stress of the material. 163,165 Rheo-SAXS experiments are not sufficient on their own to interpret the structures; therefore other experiments were performed.

Charge inversion point

The work of Matusiak et al. 166 demonstrated that silica adsorbs electrostatically to chitosan. This is verified here using an electrophoretic titration. The results of the Ludox TMA and chitosan in pH4.5 buffer solution titration are plotted in fig. 5.10a, along with results obtained for the model system of Bindzil 40/130 (described in chapter 3) and chitosan with a known degree of acetylation of 0.05 (described in chapter 4) in low-salt conditions. Both the model system of Bindzil and chitosan, and the system of Ludox TMA and chitosan show a charge inversion. It can be seen that for Ludox TMA and chitosan this charge inversion occurs at the mass ratio of ≈150. This line is plotted in green on the state diagram, fig. 5.10b. It can be seen that the line corresponds to the boundary between the phase separated and stable region of the diagram. This is reassuring as it verifies the findings of Matusiake et al. 166 that the complexes are formed by electrostatic interactions.

However the charge inversion line does not correspond to the theoretical value that could be expected. From chapter 3 the mass of a single particle of Ludox TMA can be calculated as m p = V p ρ p where, V p = 4/3πR 3 , is the volume of a spherical particle; and ρ=1.7 g cm -3 is the density of the silica particles. In chapter 3 it was found that in low salt conditions each particle has a charge of ≈7.9 × 10 -18 C. Therefore m p /7.9 × 10 -18 =1.2 gC -1 . In chapter 4 the average molecular weight of a single unit of chitosan with a degree of acetylation of ≈0.2 was found to be 170. Each chitosan unit contains a single charge. This gives (170/1.6 × 10 -19 )N A =0.0018 gC -1 . The mass ratio, assuming a maximum of charges, required to achieve charge neutrality is then 1.2 gC -1 divided by 0.0018 gC -1 which gives ≈670. This is ≈4.4 times greater than that measured by electrophoretic titration, and observed in the state diagram, fig. 5.10b. Gurovitch and Sens conducted a theoretical investigation into polyelectrolyte-colloid complexes. 153 They calculated that a charged colloid would attract a polyelectrolyte with a larger and opposite charge 16/5 times greater than the charge on the colloid. 153 They expected overcharging in favour of the polyelectrolyte. In the case examined here where the colloid overcharges they predict a more complex star-like structure with parts of the chain collapsed and parts stretched away. 153 The theory of Gurovitch and Sens simplifies the problem by neglecting the effect of salt or of counter-ions which they indicate can cause overcharging. 153 The effect of salt can be considered on the theoretical charge inversion line which results in a ratio of ≈50 silica particles to neutralise each chitosan chain. v This is 15 times greater than the degree of overcharging suggested by Gurovitch and Sens indicating that the effects of salt and counter-ions are not sufficient to account for the observed overcharging.

It thus appears that the chitosan does not completely cover the silica particles. Instead it appears that the silica particles decorate the chitosan chain in solution, and so only a v Using the formula in chapter 3 with λ D calculated in chapter 1.

percentage of the silica charges are directly neutralised by the chitosan. It is possible that this is the origin of the local rod-like behaviour that is observed in the scattering data. This behaviour is responsible for the phase separation observed in the state diagram. Below the phase boundary complexes are formed; but they are not dense clusters and remain stable in solution. The use of buffer solution additionally reduces the number of silica particles per chitosan chain by screening charges. It appears that phase separation occurs close to charge neutrality, this is supported by the review of De Kruif et al. which indicates that at maximum coacervation complexes should be electrostatically neutral. 41 Therefore, below the phase boundary there must be free silica particles in solution. It is not clear why the complexes below the phase boundary are stable. The structure of these complexes is not clear. To explain the results from rheo-SAXS experiments static SAXS experiments were performed to obtain further structural information.

Static scattering data within the state diagram

Kinetic measurements have shown that complexes above and below the phase separation line are formed at two different rates. Below the phase separation boundary complexes are formed on a sub-millisecond timescale. A rheo-SAXS investigation of a sample with a power law scattering behaviour corresponding to q -1.3 showed little evidence of anisotropy or structural rearrangement on application of shear. Meanwhile, electrophoretic titration demonstrates that complexes are formed by electrostatic interactions. While it is clear that complexes are formed by electrostatic interactions below the phase boundary it is not clear what their structure is, or why they are stable.

The case of 16 % Ludox TMA concentration

Before examining all possible combinations of chitosan and Ludox a case study is performed. The concentration of Ludox TMA is fixed at 16 % and the concentration of chitosan is varied. It can be seen in fig. 5.11 that upon addition of chitosan the scattering changes. Figure 5.11 shows a loss in intensity in the mid-q region as the sample composition approaches the phase boundary. This is caused by compression forcing particles closer together. On crossing the phase boundary line (between 0.05 % and 0.1 % chitosan concentration) there is a significant change in the shape of the scattering curve. This is expected and corresponds to a large structural change: the system passes to large aggregates. As observed in kinetic measurements the large aggregates coalesce and phase-separate almost instantaneously. It is interesting to note that the structures observed close to the phase boundary are large, at least 500 nm. It is unexpected that such large structures are stable q /nm -1 0.0001% CSL 0.0005% CSL 0.001% CSL 0.005% CSL 0.01% CSL 0.05% CSL 0.1% CSL Mid over long periods of time. In chapter 1 it was shown that for the silica particles used here a minimum radius of 600 nm would be required for sedimentation to be significant. Large structures >500 nm could be expected to further aggregate and sediment relatively quickly (as for DLCA, chapter 1). However, the aggregates are not densely packed, and they are electrostatically stabilised in solution preventing barrier to further aggregation.

Electrophoretic titration indicated the complexation process was driven by electrostatic interactions. Then the rheo-SAXS measurements appeared to contradict previous assumptions about the structure. It was decided to use the double-Yukawa model (used to describe the inter-particle interactions in Ludox TMA in chapter 3) to fit the data. The model has the advantage of assuming little besides an isotropic inter-particle potential and has been used by the group of Kumar et al. to examine nanoparticle-protein complexes. 44 The double-Yukawa model can then be used to compare scattering results between these two systems. It can be seen, in fig. 5.12, that the model describes the data well.

In fig. 5.13a the potential corresponding to the fits in fig. 5.12 is plotted as a function of the hard sphere radius. It can be seen that the potential evolves with the addition of chitosan. At low chitosan concentration the potential shows a shallow minima. This minima corresponds to the location of the hard-sphere radius. Then, as the quantity of chitosan is increased the potential depth increases, and the inter-particle potential becomes purely attractive. The values corresponding to the potential depth appear consistent with q /nm -1 0.0001% CSL 0.0005% CSL 0.001% CSL 0.005% CSL 0.01% CSL 0.05% CSL those published. Potentials of such depth were also observed by Kumar et al. in the nanoparticle-protein system. 143 In a Monte Carlo simulation study by Godfrin et al. 32 energy well depths of up to -10.85 k B T were described (in supplementary material). Interestingly, the Bernal spiral structure studied by Mani et al. 33 was found to form at high attraction strengths, with an effective well depth of 10 to 20k B T . The high attraction strengths observed by Mani et al. 33 led to kinetic trapping. However, there is a significant difference between the potential shape observed in fig. 5.13a and those published in the literature. There is no repulsive barrier to prevent the particles from aggregating. After the minima the inter-particle potential returns to 0 k B T . In this case there is no barrier to aggregation and subsequent sedimentation (as described in chapter 1). This is not observed and therefore the potential is not compatible with macroscopic observations of the system (it will also be shown that the potential is not compatible with the system dynamics using XPCS measurements).

The pair correlation function, g (r ), (described in section 2.1.2) can also be extracted from the double-Yukawa fitting parameters and is plotted in fig. 5.13b. It was not possible to calculate g (r ) for the sample with 0.01 % chitosan or 0.05 % chitosan as the fitting parameters were not physically meaningful. The calculation of g (r ) diverged at deep values of the inter-particle potential indicating the Yukawa model is not appropriate close to the phase boundary. When comparing the inter-particle potential with the pair correlation function the lowest chitosan concentration is particularly interesting (16 % silica, 0.0001 % chitosan). The pair correlation function corresponds to that observed from screened silica particles. There is a significant change in g (r ) upon increasing chitosan concentration to 0.0005 %; however this change is not reflected in the plot of interparticle potential, nor in the static scattering data. The potential overlaps until a concentration of 0.005 % is reached, then both g (r ) and V (r ) change along with the static scattering shown in fig. 5.11. Above this chitosan concentration the low-q scattering increases. There is a structural change and the values of V (r ) and g (r ) loose physical meaning.The values of g (r ) correspond to a solid-like transition. Macroscopically this is not observed.

Considering the entire state diagram

Data were considered from within the stable region of the state diagram. The double-Yukawa potential was even less appropriate as the concentration of silica particles decreased. This is illustrated in fig. 5.14 which shows SAXS curves of 1 % TMA with varying chitosan concentration. In the mid-q region there is a lack of agreement between the data and fit. The intensity of the fit is below the intensity of the data. This is because the fit assumes strong attractive interactions to describe the scattering at low-q. These strong attractive interactions correspond to densely packed aggregates in the sample. It is the density of the packing that results in intensity loss in the mid-q region, as seen experimentally in fig. 5.5. The excess of intensity seen here implies the complexes are not very densely packed. Additionally, if densely packed aggregates were present in the sample they would sediment. However, it is clear that aggregates are present in the sample: macroscopically the samples are turbid, and fig. 5.14 shows an excess of intensity at low-q.

Unfortunately the double-Yukawa model is not able to describe these complexes in a meaningful way. The double-Yukawa model is being used here as an ad-hoc solution, where the attractive term of the Yukawa is used to describe scattering from clusters. Electrostatic titration measurements indicate that there is an excess of free silica particles present in solution. It is difficult for the double-Yukawa interaction to describe a heterogeneous system of particles and complexes. Additionally, interactions between these entities are screened by the salt in solution.

In the ideal case of an homogeneous dispersion of particles the number density of the structure factor scales with intensity, and this corresponds to the known sample composition. This introduces a strong constraint in the fitting procedure which can be used to find a good solution. In the case of complexes the main contribution to intensity at mid, and low-q comes from particle-particle correlations within complexes where the local volume fraction within clusters is important. When loose particles (not participating in complexes) are considered this ensures that the structure factor should consider three terms: intra-cluster particles, loose particles, and cross-terms. Constructing such a fit requires strict fitting parameters to obtain a meaningful result. This was not possible here. The double-Yukawa potential was compared around the stable region of the state diagram vi As discussed these values are not representative of the actual inter-particle potential. Their role is to understand patterns around the state diagram. It was found that as the sample composition approaches the phase boundary the depth of the inter-particle potential increases. This is also illustrated in fig. 5.13a. Electrophoretic titration experiments showed that electrostatic interaction lead to complex formation. It could then be expected that the inter-particle potential should be constant for constant mass ratio. This is not the case. Volume fraction is also an important contribution to the inter-particle potential. Increasing the sample volume fraction leads to an increase in excluded volume interactions vii which can be understood if one considers that a large fraction of the particles are not participating in complexes. This is illustrated schematically in fig. 5.15.

vi Plotted in fig. A.6. vii Excluded volume is the volume that a particle occupies rendering it inaccessible to other particles as a result. q /nm -1 0.0001% CSL, 2.88E9 0.0005% CSL, 1.15E8 0.001% CSL, 2.88E7

XPCS

XPCS measurements can provide information on the particle dynamics. If particles become strongly attractive, their dynamics should significantly slow down and eventually arrest. XPCS measurements were performed on samples with a high volume fraction (corresponding to ≥10 % silica in the state diagram fig. 5.1). The ensemble averaged autocorrelation functions g (2) (q, t ) were fit with a stretched exponential decay following the Kohlrausch form g (2) (q, t ) = 1 + βe x p -2(Γ t ) γ (

where t is time, Γ is the relaxation rate, γ is the Kohlrausch exponent and β the speckle contrast is a weakly varying function of q. [START_REF] Madsen | Synchrotron Light Sources and Free-Electron Lasers[END_REF] It is sample independent and largely determined by the coherence of the setup. When the data for a particular configuration has been fitted using (5.3) the β were extracted and averaged for a given q-value. Data were then re-fitted with (5.3) but with β fixed as shown in fig. 5.16. Figure 5.16 shows that the stretched exponential decay describes the data well. In the case of a purely diffusive system the autocorrelation function would be fit with a simple exponential decay. 91 The stretched exponential is indicative of attractive particles whose motion is constrained and is understood to correspond to gel formation.

Γ /s -1 γ q 2 /nm -2 γ Γ D eff =1.8E7 D(q)=1.6E8q 1.3
Fig. 5.17 Kohlrausch exponent, γ , and relaxation rate, Γ , extracted from the stretched exponential fits in fig. 5.16 and plotted against q 2 . The green dashed line is a fit of relaxation rate with Γ = D e f f q 2 in green where D e f f =1.8 × 10 7 nm 2 s -1 . The yellow continuous line is a fit of Γ = D e f f (q)q 2 where D e f f (q) = 1.6 × 10 8 q 1.3 .

The values of the Kohlrausch exponent used to fit the data are plotted in fig. 5.17. It can be seen in fig. 5.17 that for a given sample the value of the Kohlrausch exponent remains relatively constant across the q-range, which is suggestive of gel formation. [START_REF] Krall | [END_REF] The average value of the Kohlrausch exponent is 0.68 which corresponds to values observed in gelling systems by other XPCS experiments in the literature. 169 However, the relaxation rate, Γ , plotted against q in fig. 5.17 behaves as neither a colloidal gel, nor polymer gel. In the case of a colloidal gel an effective diffusion coefficient, D e f f , can be defined such that D e f f = Γ q 2 while for polymer internal motion there is a q 3 behaviour. [START_REF] Krall | [END_REF] In the non-ideal case we can re-write D e f f (q) where there is a q-dependence, and so Γ = D e f f (q)q 2 . Such a case is observed in fig. 5.17 where the exponent has been allowed to vary. In addition to the q 2 dependence expected of colloidal gels there appears to be a q 1.3 dependence. This describes the data well.

Theoretically colloidal particles with short-range attraction and long-range repulsive forces such as those described by the double-Yukawa potential form several types of structure. These can be plotted on a phase diagram described by reduced temperature T * . Godfrin et al. 31 use Monte-Carlo simulation to provide criteria for three different phases: monomer dominated (governed by loose particles); a cluster fluid, where ⪆ 85 % of particles are in clusters of two particles or more; or a percolated state, where ⪆ 93 % of particles are part of a cluster. Percolation is defined as a cluster spanning the entire system size in at least one dimension. Percolated states appear at higher volume fraction, they are a consequence of the overall density of the system forcing it to occupy a particular configuration: at high volume fraction it is difficult to form clusters with a locally higher volume fraction. By combining the results from the XPCS and static measurements it would appear that samples which are labelled as "shallow attraction" in fig. 5.15 exhibit behaviour that could be associated with percolated phase formation. XPCS data indicate the dynamics correspond to gel formation. This implies that the excess scattering in the low-q region which was described by strong attraction with the double-Yukawa model corresponds to the scattering by the gel network. An alternative approach would involve incorporating an extra term for scattering by the gel network, in the spirit of the unified scattering function. 170 In addition it may be necessary to describe the unaggregated silica particles at high salt concentration.

An example of such a fit is shown in fig. 5.18 where the Hurd and Flower model described in chapter 3 is combined with the Fisher-Burford model used in section 5.3.2 and the Percus-Yevick approximation for the square well potential as described by Chen et al. 171 Such a model describes the data well; in the future it would be interesting to extend such a fit to the wider state diagram.

Evolution of the state diagram

Finally, macroscopic observations show that over long timescales a strong gel is observed. This is consistent with the findings from the XPCS experiments. The samples with the shallow attractive potential in fig. A.6 were the first to gel. Initially there is gel-gas coexistence. The gel continues to grow over time until the sample is monophasic. This supports the hypothesis that these samples are forming a percolated phase. Gel formation is observed across the state diagram in the region below the coalescence line. Over long timescales sedimentation changes the local volume fraction and so even samples with individual clusters change to a percolated state. The XPCS experiments, storage tests, static scattering experiments and rheo-SAXS show different aspects of the system. Rheo-SAXS experiments show that the system does not respond significantly to external shear. No significant anisotropy was observed. No significant structural change was observed. The simplest interpretation is that the gel, or percolated structure has a very low yield stress which is destroyed by even simple start-up shear. As the system takes months to enter a state which appears macroscopically gelled it is likely that the microstructure is slow to redevelop. It is possible that smaller complexes are highly dynamic. This would prevent them from being destroyed by external shear. Such a hypothesis would require more detailed experiments, perhaps using improved XPCS to examine the particle dynamics directly which will be possible with the upgraded ESRF-EBS instrument. In a review of the literature Hoffmann et al. 149 note the complexity of the rheology of oppositely charged systems: some polyelectrolyte-surfactant systems see significant changes in viscosity (when compared to the polyelectrolyte viscosity), while others see little effect. The reasons for this are not clear.

Conclusion

This Ludox TMA and chitosan system provides an interesting model to study the behaviour of oppositely charged systems. This chapter began by replicating the state diagram published by Shi et al. 36 and showing this was consistent with the behaviour of other oppositely charged systems. Upon mixing chitosan and silica particles together complexes are formed. There is a phase boundary with a biphasic region, and stable monophasic region. Above the phase boundary complexes grow and sediment to form a coalesced liquid phase, and silica-poor gas phase. Below the phase boundary the complexes are stable for extended periods. The charge inversion point found by electrophoretic titration corresponds to the phase boundary of the state diagram indicating that coacervation is caused by electrostatic complexation.

The formation kinetics of clusters was investigated using rapid mixing experiments. These fast mixing experiments show that coalescence is accompanied by simultaneous cluster growth, compression and structural rearrangement. In contrast the kinetics in the non-coalescing region of the state diagram are faster than the experimental timescale. Storage tests show this is then followed by long-term structural rearrangement leading to gel formation.

Scattering data, and previous work both theoretical and experimental indicated the presence of rod-like structures. An attempt was made to align these rods using shear. This was not possible. Not only was no anisotropy observed; but also no significant structural change was observed. When combined with the results from XPCS measurements it can be concluded that the system corresponds to a gel with very low yield stress.

A double-Yukawa potential was used to fit static scattering data in the "stable" region of the state diagram. Although the fitting parameters found were similar to those obtained by other groups it was found that this potential does not provide physically meaningful information about the system. It also appears that there is a large contribution of free silica particles to the scattering signal which is difficult to separate from the scattering contribution of the clusters. This heterogeneity hinders more quantitative structural analysis. A more suitable structure factor for this system would consider four contri-butions: polyelectrolyte-polyelectrolyte interactions (which are invisible in SAXS and SANS due to poor contrast), particle-particle interactions (which was the aim of this work), particle-polyelectrolyte interactions (which were not considered here, and would require simulation), and finally the contribution from the overall gel structure (which was identified during analysis).

While the system may be locally linear these linear regions are part of much larger structures that then form a gel-like network, or percolated state across the sample. In the future it would be interesting to examine how dynamic these structures are on the scale of individual silica particles, or chitosan units. Additional experiments with improved control over ageing time would facilitate a more quantitative comparison macroscopic observation and microscopic structure and dynamics. This could provide insight to the remaining unanswered questions.

CHAPTER 6

EFFECT OF DIALYSIS ON SILICA-CHITOSAN COM-PLEXES

Introduction

In chapter 5 the silica-chitosan system was examined in detail. It was established that electrostatic interactions are responsible for the silica-chitosan complex formation. Reviews of the literature by De Kruif et al. and investigations by Yaroslavov and Potemkin on electrostatically formed complexes indicate that changing the ionic strength of such a system should dramatically change the system behaviour. 41,42 Understanding how such systems react to changes in ionic strength provides information about complex stability which is useful for future applications as stimuli-responsive materials as such changes are normally reversible. 172 In general to examine how complexes react to changing ionic strength samples are diluted or rapidly mixed. 43 Such a method has the advantage of being close to real-world applications. Here an alternative method of changing ionic strength is employed. Dialysis is used to reduce the ionic strength of the samples. This has the advantage of reducing the salt concentration without significantly changing the sample volume fraction. In dilution or mixing experiments changing sample volume fraction is an inevitable consequence of changing ionic strength. By dialysing the samples studied in chapter 5 it is possible to have a direct comparison of the sample behaviour at low ionic strength. As it seems likely from chapter 5 that the system is not at thermodynamic equilibrium it is expected that the system after dialysis will not be at equilibrium either. It is expected that the result will differ to complexes prepared in the absence of buffer; however this is not examined. This chapter investigates the consequences of dialysing complexes after their formation.

Materials and methods

Sample preparation

Samples were prepared as described in chapter 5.Small silica particles were mixed with chitosan in a pH4.5 buffer solution composed of 0.3 mol L -1 acetic acid and 0.2 mol L -1 sodium acetate. The materials used are described and characterised in detail in chapter 3 and chapter 4.

Dialysis is a straightforward method to remove salt from a sample. It uses a membrane as a sieve to separate smaller species from larger ones. A sample is placed inside the dialysis membrane which has a given pore size. Species smaller than the pore size are forced through the membrane by osmotic pressure. Species larger than the pore size are trapped inside the membrane. Silica particles, and large aggregates are trapped inside the dialysis membrane; however the salt molecules of the buffer are driven out and replaced by the molecules in the reservoir outside the membrane.

All samples described in this chapter have been dialysed against a weak buffer solution after sample preparation. To dialyse the samples were placed in 12 kDa to 15 kDa dialysis tubing and left in a weak buffer solution (0.05 mmol L -1 acetic acid, and 0.03 mmol L -1 sodium acetate) for 72 h. The buffer solution was changed twice a day. Using a conductivity meter (Mettler Toledo SevenCompact) it was found that 72 h was sufficient for the conductivity of the solution to stabilise (see table A.1). A pH meter shows that the sample pH remains constant.

The samples were sealed and stored vertically. Samples were kept in a water bath at 25 • C for the purpose of storage testing. All samples presented here are composed of Ludox TMA, and chitosan of low molecular weight obtained from Sigma Aldrich.

Sample measurements

SAXS and XPCS measurements were performed at beamline ID02 at the ESRF. Samples were measured in capillaries as described in chapter 5.

SAXS Temperature sweep

A Linkam heating stage was used to heat a quartz capillary containing the sample dialysed 16 % Ludox TMA 0.0225 % chitosan to 90 • C. The heating stage increased the temperature by 10 • C increments. A three minute interval allowed the capillary temperature to equilibrate. Three SAXS frames were taken, averaged, background subtracted and normalised.

Microbeam SAXS

A piece of colloidal silica crystallite which grew in the dialysed samples was examined using a microfocused x-ray beam, of wavelength 0.095 nm, on beamline ID13 at the ESRF. An Eiger 4M detector with pixel size of 75x75 µm 2 , and sample-to-detector distance of 0.522 m, was used to acquire the scattering data. The beam was focused to 2x2 µm (full width half maximum, FWHM) and scanned across the sample. The total area of the scan was 600x600 µm 2 , larger than the sample size, with step size corresponding to the size of the beam (2 µm).

The colloidal silica crystallites were carefully extracted using tweezers. An individual crystallite was then glued to the top of a glass needle and mounted in the x-ray beam.

Results and discussion

Macroscopically, there appeared to be no visible change in the sample before and after dialysis. If the sample had phase separated, as seen in chapter 5, this would have been visible by eye. Conductivity measurements of the sample before and after dialysis show that the conductivity changes from mS cm -1 to µS cm -1 range (see table A.1). It is expected that the conductivity drops, which is the goal of dialysis; however it is interesting that the conductivity drop is not accompanied by a visible phase change. This differs to what was observed in protein-polyelectrolyte systems where a phase change accompanied such a drop in conductivity. 41 This was not observed.

Microscopically SAXS experiments show significant differences between the dialysed and non-dialysed samples, as demonstrated in fig. 6.1. It can be seen that after dialysis the scattering features a structure factor peak. The position of this peak corresponds to the peak that would be observed from free silica particles in solution. It can also be seen that there is a drop of intensity in the low-q region between the dialysed and non-dialysed sample. This can be interpreted as a decrease in the number of large complexes present in the sample, and a reduction in the scattering contribution from free silica particles, as there is less screening of strong repulsive interactions. Qualitatively it would appear that the complex structure is not changed by dialysis. Therefore the qualitative effect of dialysis is to reduce the concentration of large complexes and increase the amount of free silica particles in solution. This is a surprising result, as one would expect that decreasing ionic strength leads to an increase in the attractive interaction between oppositely charged q /nm -1 Fig. 6.1 SAXS curves of 16 % Ludox TMA and 0.01 % chitosan. The orange circles show the sample before dialysis, and the blue triangles show the sample after dialysis against a weak buffer of 0.05 mmol L -1 acetic acid, and 0.03 mmol L -1 sodium acetate at pH4.5.

colloids and polyelectrolyte, and therefore an increase in the number of complexes instead of the decrease observed here.

Results from XPCS measurements

XPCS measurements indicate that the dynamics of the system are changed significantly by dialysis. XPCS autocorrelation functions, g (2) (q, t ), were fit with a stretched exponential decay, following the Kohlrausch form, using the procedure described in chapter 5. The fits of the autocorrelation function are shown in fig. 6.2, where it can be seen that the stretched exponential decay describes the data well. However there is a significant difference in the values of the Kohlrausch exponent, which are plotted in fig. 6.3 alongside those of the non-dialysed sample shown in chapter 5. The average value of the Kohlrausch exponent for the dialysed sample is 0.89 which is greater than the value of 0.68 observed for the non-dialysed sample. The increased value of the Kohlrausch exponent indicates the sample dynamics are becoming closer to those observed in arrested states. Such values have been observed in the final transition to a gelled state by other XPCS experiments in the literature. 169 However, this increased value of the Kohlrausch exponent of the dialysed sample matches literature values for supercooled liquids. 173 Instead of a transition to arrested gel phase this could show a transition from colloidal gel to colloidal liquid behaviour. The different colour lines correspond to different values of q where the curves from left to right correspond to increasing q-value 5 × 10 -3 nm -1 to 5 × 10 -2 nm -1 . The values of both the dialysed and non-dialysed samples are constant across the q-range. [START_REF] Krall | [END_REF] The q-range measured here is corresponds to clusters. The dynamics of the free silica particles around the structure factor peak were too fast to be measured with the current instrumental capabilities. This will become possible after the ESRF-EBS upgrade. It will be interesting to examine this with future experiments.

In chapter 5 it was shown that the relaxation rate, Γ , corresponded to neither a colloidal nor a polymer gel but had an intermediate value. It would appear that the dialysed sample behaves in a similar way. Relaxation rates of a dialysed and non-dialysed sample are compared in fig. 6.4. The data are plotted on a linear scale where it can be seen that they diverge. This is further explored in fig. 6.4b where the data are plotted against q 2 . They are fit with Γ = D e f f q 2 ; however it is clear that there is an additional q-dependence. In fig. 6.5 the additional q-depence of the relaxation rate has been explored by fitting with Γ = D e f f q a q 2 . It can be seen that the fits describe the data well.

XPCS measurements indicate the removal of salt has a noticeable impact on the system dynamics. The dynamics of the dialysed sample indicate a change from gel to colloidal liquid state. Multiple studies, such as those by De Kruif et al. and Dubin et al. indicate that decreasing the ionic strength of the system should result in a change of state. 41,174 This trend is observed here using dynamic measurements. not dialysed dialysed Fig. 6.5 Relaxation rate, Γ , extracted from the stretched exponential fits in fig. 5.16 (and fig. 6.2) and plotted against q 2 . The relaxation rate has been fit with Γ = D e f f q a q 2 where a = 0.34 For the non-dialysed sample, which has been multiplied by 10 for clarity, D e f f =1.6 × 10 9 nm s -1 . For the dialysed sample a = 0.29 D e f f =7.1 × 10 7 nm s -1 .

Effect of temperature

To further probe the structure of the sample a temperature sweep was performed. It is expected that temperature has little effect on such binary systems. 41 A Linkam temperature stage was used to increase the temperature from room temperature (20 • C) to 90 • C. The data are plotted in fig. 6.6 which shows a loss of intensity in the low-q region with increasing temperature. This corresponds to a gradual decrease in the size of the largest structures. When the sample is cooled back to room temperature (blue squares) the structures are recovered demonstrating that structure formation is a reversible process. It is therefore clear that the silica-chitosan system behaves differently to the polysaccharide-protein systems examined by de Kruif et al. 41 There is no significant change in either the shape or position of the inter-particle structure factor peak. This is highlighted in the inset of fig. 6.6 where the slight difference in peak height can be attributed to changing scattering length density (SLD) of water (at 22 • C SLD is 9.41 × 10 -4 nm -2 and at 90 • C SLD is 9.09 × 10 -4 nm -2 ). If the structure factor peak was related to interactions within clusters then the change in low-q behaviour would induce a significant change in the height and position of this peak. Additionally, the peak position at q=0.15 nm -1 corresponding to ≈42 nm is too large to correspond to inter-particle distances within a cluster. In fact this distance corresponds to the mean distance between free silica particles in a simple cubic packing at this volume fraction, further suggesting that most particles are free. Therefore, the peak is caused by scattering from free silica particles. That neither the peak height nor peak position change means that the concentration of free silica particles remains approximately constant.

In chapter 5 it was proposed that scattering at low-q for this sample, composition of 16 % Ludox TMA and 0.0225 % chitosan, originates from percolated clusters. Increasing the temperature of the system should destroy the percolated clusters and form a cluster fluid. To investigate this the forward scattering, I (0), was plotted against temperature. Figure 6.7 shows there is a decrease in I (0). The absence of a change in the intensity of the structure factor peak (fig. 6.6) means there is no change in the amount of free silica particles present. Therefore the change in I (0) is a consequence of the break-up of larger structural units. There is a decrease in the number of large structures implying an increase in the number of smaller structures present. This is consistent with a transition between percolated clusters and cluster fluid. To investigate the structural reorganisation quantitatively the form factor and structure factor of silica particles (as calculated in chapter 3 i ) were subtracted from the data shown in fig. 6.6. The results are plotted in fig. 6.8. This highlights the presence of an isosbestic point at slightly lower q than the structure factor peak position (the peak is also visible in fig. 6.6). Isosbestic, or isoscattering points have previously been observed in scattering studies of temperature-induced structural change in gels. 175 Gibaud et al. 175 note that the presence of an isosbestic point in scattering data indicates that the attraction strength is being modified at a constant range. They illustrate this point by showing how the depth of interparticle attraction scales linearly with temperature, while the range of the potential remains constant. 175 Here, at this q-value, it can be assumed that interactions between free silica particles dominate. Fitting shows the range of their inter-particle potential is not likely to change, and so the presence of an isosbestic point is unsurprising. It is surprising that it is visible in a more complex system. Gibaud et al. 175 examined a system of lysozyme at 0.5 mol L -1 of NaCl.

A closer look reveals a slope close to q -1 at high temperature indicating the presence of linear structures. The XPCS data had shown that the sample dynamics indicate a transition from colloidal gel to fluid behaviour. Heating the sample may have accelerated q -2 q -1 Fig. 6.9 Illustration showing the model used provide the fits in fig. 6.10. The spherical particles form cylinders which then form larger aggregates. this transition. An attempt was made at fitting the data with a combined model of long cylinders and long cylinders with a fractal aggregate structure factor. An illustration of this model is shown in fig. 6.9. This is a simplistic description where cylinders are used as a proxy for linear assemblies of spherical particles; a fraction of these linear assemblies form larger aggregates. Figure 6.10 shows a good fit with this simple model. Figure 6.11 and fig. 6.12 show how the fitting parameters change with temperature. The total number density of cylinders (loose cylinders and cylinders contributing to aggregates), N t ot , increase as temperature increases. The overall size of the clusters, given by the cut-off X i decreases with temperature. This is in agreement with the information from the forward scattering, I (0), shown in fig. 6.7; the overall number of aggregates increases while their size decreases. Figure 6.11 shows that the fraction of cylinders that do not contribute to the formation of large aggregates (N s ma l l /N t ot a l ) remains constant at ≈ 1. This is because the concentration of large aggregates is small relative to the concentration of loose cylinders (loose assemblies of particles).

In fig. 6.12 the change in fractal dimension of the large aggregates is plotted against temperature. The fractal dimension of the aggregates is consistent with what is expected from percolated clusters. The cylinder length is also plotted as a function of temperature. The length of cylinders that do not contribute to the larger aggregates, LC 1, increases slightly with temperature; while the length of cylinders that contribute to aggregates, LC 2 appears to decrease slightly.

As the temperature increases beyond 50 • C, the fitting parameters in both fig. 6.11 and fig. 6.12 begin to deviate from trends. This is partly due to a significant reduction in the number of large aggregates making fitting difficult.

When chitosan and silica particles are combined the silica particles decorate the chitosan chain. This gives rise to rod-like structures ≈100 nm in length (at 20 • C). ii These rod-like structures form larger, branched aggregates. The excess silica particles that do not participate in these structures remain in solution and are observed as the structure ii Given the silica particle radius is ≈11 nm this corresponds to ≈ 5 particles per rod. As the temperature increases these large fractal aggregates break apart.

factor peak in fig. 6.6. In the case of the non-dialysed samples examined in chapter 5 it is likely that free silica particles were also present in solution; however the interactions were screened which made separating the contribution of free silica particles from that of complexes challenging.

The biggest barrier to understanding the non-dialysed system studied in chapter 5 was quantifying the heterogeneity of the system. Estimating the amount of free silica particles would enable the scattering signal from the complexes to be confidently isolated. Studying the structure factor of silica particles as a concentration series in high-salt conditions, by extending the work of Bareigts for example, 110 could provide a path to separating the contribution of free silica particles from the scattering data examined in chapter 5. Separating different scattering contributions in this manner is similar to the method used here to analyse the dialysed sample. Understanding the individual materials is necessary to understand the complexes. Understanding the complexes would then allow the state diagram to be mapped onto theoretical phase diagrams of systems with complex interparticle potentials such as those proposed by Godrin et al., 31,32 Mani et al., 33 and Mossa et al. 34 amongst others. In the future it could be possible to formulate a sample with a pre-defined structure.

Effect of time on the state diagram

In chapter 5 it was shown that for long timescales the samples gelled. It was therefore interesting to examine how the dialysed samples changed with respect to time. After a year, crystallites were visible to the naked eye exclusively in the dialysed samples, a photograph of an example crystallite is seen in fig. 6.14.

Figure 6.15 shows that all dialysed samples which were stored for extended timescales had colloidal crystallites visible to the eye after eighteen months. This was an unexpected result. As demonstrated in chapter 3 Ludox TMA is relatively polydisperse which usually inhibits crystal formation -hard spherical colloids cannot crystallise for polydispersity >12 %. 176 However, Cabane et al. used osmotic pressure to make colloidal crystals (including a Laves phase) in solutions of Ludox HS-40. 177 They dialysed Ludox HS-40 against a solution of 5 mmol L -1 NaCl to remove impurities. Polymer was then added to the solution outside the dialysis membranes, which concentrated the silica particles inside the membranes. By concentrating the silica particles slowly fractionation occured -silica particles of different sizes migrated to different crystals to optimise the entire population. For example, the Laves phase (a type of hexagonal packing) contained four large silica particles, and eight smaller particles per unit cell. Additionally crystal-liquid coexistence was observed. Following such a study it was interesting to examine the crystallites observed here in detail.

Investigation of silica colloidal crystals

A crystallite was extracted from one of the dialysed samples and examined using microbeam SAXS on ID13. A mesh scan was performed over the entire crystallite in a single orientation. The crystallite was not rotated in the x-ray beam. In fig. 6.16, an orientation map of a second order Bragg peak is shown. Lighter colours indicate a higher degree of orientation, while black indicates complete isotropy. The dark purple colour indicates an overall directional order in the material. Lighter colours show the presence of colloidal crystalline domains. It can be seen that microcrystals are distributed over the entire material. In section A.4, a second map is presented which shows the location of crystallites for a different Bragg peaks. This plot again shows the presence of an overall order in the large material. This anisotropy is further explored in fig. 6.17a. The average 2D scattering pattern from the entire scan is plotted. Anisotropy is clearly visible. The liquid-like order surrounding the microcrystalline regions has a preferred orientation.

An example of the 2D scattering data from one of these crystalline regions is shown in fig. 6.17b. The first visible order Bragg peaks are visible at q≈0.314 nm -1 , and second visible order Bragg peaks are visible at q≈0.550 nm -1 (both peaks are shown in fig. A.10). The six-fold symmetry indicates a close packed structure. 178,179 Cubic lattice parameters, such as face centred cubic (fcc) and body centred cubic (bcc) can be calculated as

1 d 2 = h 2 + k 2 + l 2 a 2 (6.1)
where hkl are Miller indices, iii d is the spacing between crystallographic planes and a is the lattice parameter. [START_REF] Als-Nielsen | Elements of Modern X-ray Physics: Second Edition[END_REF] In the case of a fcc structure a = 2 2r , where r is the particle radius which is known. None of the possible values for d correspond to the position of the observed diffraction peaks. In the case of a bcc structure it is not possible to calculate lattice parameters exactly as neither the crystallographic plane, nor the lattice parameter is known.

A lattice parameter can be assumed by examining the supplementary material of Cabane et al. 177 and making the ansatz of bcc value closest to the observed diffraction peak position.

iii Miller indices are a form of notation specifiying families of planes within a crystal. 47 The bright domains correspond to orientated crystalline domains. Data were measured at beamline ID13.

However, none of the possible values for d corresponding to this ansatz correspond to the position of the second visible order Bragg peak observed. The same procedure can be followed for the hexagonal Laves phase indexed by Cabane et al. 177 however none of the possible values of d correspond to the second visible order Bragg peak observed in fig. 6.17b. Therefore it is clear that the structures observed by Cabane are not visible here.

There is a final possibility. That the cylindrical structures observed in the temperature study align to form crystalline regions. This would be similar to the hexagonal packing of cylinders observed in the hexagonal phase of liquid crystals. 180 It is known that the Bragg peaks of such crystalline structures appear at 1, 3, 4 relative to the position of the first peak. As the first order peak is at q≈0.314 nm -1 the second order peak should then be at 0.314 × 3 = 0.543. This is almost identical to the position of the observed peak at q≈0.550 nm -1 thereby vindicating the cylindrical model used to fit the data, and illustrating that this structure is essential to the crystal formation.

It must be stated that more data are desirable to properly characterise the crystalline structure. Here we have analysed only one observed structure; however different crystalline regions are observed in the larger crystallite matrix fig. Strikingly, colloidal crystallite growth was observed in all dialysed samples. The samples were stored in the same temperature controlled water bath in the same way as the non-dialysed samples examined in chapter 5. Recall that the non-dialysed samples in chapter 5 formed gels. These dialysed samples avoided the formation of arrested states, such as gels. This indicates that the components can easily associate and dissociate to sample many configurations and find the lowest energy state. 181 In general, colloidal crystallisation is dependent on solution volume fraction. In their seminal 1986 paper Pusey and Van Megen, showed that hard-sphere colloids crystallised for a narrow range of volume fraction (between 0.44 and 0.50). 29 Similarly, colloidal crystallisation using Ludox HS-40, as studied by Cabane et al, 177 occurred for volume fractions between 0.240 and 0.188. As shown in fig. 6.15, the samples showing the crystalline behaviour observed here have volume fractions from 0.03 to 0.1 -significantly lower than those examined by Pusey and Cabane. Additionally, simulation has shown that polydisperse suspensions require high compression, such as the osmotic pressure applied by Cabane, for crystallisation to occur. 182 Aside from the gravity experienced in storage tests no external force was applied to the samples studied here and yet the samples crystallised at low volume fraction. iv It would appear that the colloidal crystallite formation observed here was caused by the dialysis-induced change in inter-particle potential. This is supported by experimental work examining the behaviour of a model polymethylmethacrylate (PMMA) system (445 nm radius)with short-range attraction and long-range repulsion. 35 Zhang et al. 35 found that microcrystalline clusters could be formed at low volume fractions, 0.06 and 0.12, comparable to those used here. Using three-dimensional laser scanning confocal microscopy, it was found that the fractal dimension of their clusters was ≈2.25 which is comparable to the value of ≈2.5 found from fitting the temperature data (plotted in fig. 6.12).

Silica, is one of the four main biominerals along with magnetite, calcium carbonate and calcium phosphate. 183 Biogenic silica is most commonly found in sea creatures such as diatoms, radiolaria, silicoflagellates, and sponges. While sea sponges are commonly known, diatoms and silicoflagellates are types of algae, and radiolaria are zooplankton. These tiny organisms are an essential part of marine ecosystems, as is the silica they use to build their cell walls and microskeletons.

Traditionally crystallisation occurs by nucleation; however in recent years other pathways to crystal formation have been elucidated. These include so-called amorphousto-crystalline pathways, which are commonly observed in biominerals. 184 Multi-step nucleation pathways are common in biomineral formation as the overall energy barrier to nucleation is lower than it would be via a traditional nucleation mechanism. 185 Here we observe some similarities to biomineral formation pathways -the presence of chitosan (and potentially residual salt ions) can be considered, in the framework of Mann, as "chemical control" of the system where specialised molecules promote or inhibit crystal growth. [START_REF] Mann | Biomineralisation: Principles and Concepts in Bioinorganic Materials Chemistry[END_REF] Alternatively, Mann writes that organisms achieve precise structural control by using macromolecules as a template for mineral growth. [START_REF] Mann | Biomineralisation: Principles and Concepts in Bioinorganic Materials Chemistry[END_REF] In the specific case of biogenic silica grown by diatoms amorphous silica uses a scaffold of chitin and chitosan to grow. 187 iv Sedimentation over long timescales causes the local volume fraction to increase; but this is unlikely to reach the theoretical values required. This scaffolding effect would appear to be observed in the system investigated here, as demonstrated by the crystal taking a hexagonally packed cylinder structure. The results of the temperature sweep experiment indicate the system contains a base unit of short cylinders. It is likely that this structure is preferentially chosen due to the silica attaching to the chitosan backbone. These rods attach together to form larger fractal aggregates. It appears that over time the larger aggregates compress and the rods align. This would explain the overall anisotropy observed in the material. Such a multi-step process has many similarities to biomineral formation observed in nature.

When compared to biological systems the system examined here is relatively simple. The path to the final state can be understood by comparing it to biomineral formation. It is clear that understanding how these crystallites grow in a sample which initially appears, both dynamically and structurally, to be the path to gel formation is fundamentally interesting.

Conclusion

The effect of removing salt from chitosan-silica system, which was described in chapter 5, was examined. Dialysing the samples revealed the presence of a strong structure factor peak. There was no longer salt present to screen electrostatic interactions. Upon heating the structure factor peak changed neither intensity nor position, while scattering corresponding to large aggregates at low-q changed significantly. This showed that the structure factor peak originated from scattering by free silica particles. This structure factor peak had been well described by the double-Yukawa model in chapter 3. Quantifying the contribution of free silica particles to the system and isolating the scattering signal originating from complexes was then straightforward.

To examine the structure of the system in more detail one high-concentration sample was heated: 16 % Ludox TMA and 0.0225 % chitosan. Increasing the temperature from 22 • C to 90 • C illustrates that the clusters are approximately cylindrical, and aggregate to form larger fractal objects. This is indicative of a percolated cluster state. These larger fractal objects break apart with increasing temperature; however the number of cylinders remains constant. The cylindrical clusters have a preferred length of ≈100 nm (at 20 • C). The high temperature state therefore resembles a cluster fluid. This indicates that in the future it may be possible to map this systems onto a theoretical phase diagrams.

The XPCS data show the transition from a gel to fluid-like behaviour. In the future it would be interesting to study the dynamics of such a transition in detail as long-term storage tests show the transition from fluid to colloidal crystallites. Significant difference between the XPCS results for the dialysed and non-dialysed samples; resulted in very different final states for the system. The non-dialysed sample became a gel. The non-dialysed sample grew colloidal crystallites. It may be possible to be understand the contribution of the structure and dynamics of the initial states to the final state of the system by examining the internal dynamics of the complexes, or by conducting a systematic investigation into the removal of salt from samples by dialysing the samples against buffer solutions of different ionic strength. Such a study can help to understand the role salt plays in determining the final state of a colloidal system. As these systems are metastable understanding their long-term stability is essential for any application.

The growth of colloidal crystallites in the samples as revealed by long-term storage tests was unexpected. It was shown that the cylinders initially formed in the samples rearrange to form a hexagonal cylindrical phase. As the chitosan-silica system is relatively simple it is possible that there are lessons which can be extrapolated to protein crystallisation problems. v It appears that the colloidal crystallite growth mechanism is similar to the case of biomineral formation. A detailed examination of the crystal growth mechanism has potential to be interesting. It may reveal that microscopic crystallites are formed within a few weeks, such small crystals could be used for serial crystallography experiments. Ultimately, further investigation could lead to understanding how to sample multiple states of a system without falling into the trap of an arrested state.

CHAPTER 7 CONCLUSIONS

This thesis presents an investigation into the structure, dynamics and evolution of a self-assembled colloidal system composed of industrial materials.

Chapter 1 presents a literature review on the topic of colloidal aggregation, and self-assembly including an introduction to the use of fractal models in colloid science. An introduction to small-angle scattering and beamline ID02 is given in chapter 2. The opening scientific chapters (chapter 3 and chapter 4) describe the materials used and challenges involved in analysing their behaviour using scattering techniques. The thesis concludes (chapter 5 and chapter 6) by combining the materials examined in the opening chapters and investigating this binary system.

Conclusions on the materials

Chapter 3 illustrates the challenges involved in describing scattering data from silica particles and proposes a new approach using the Hurd and Flower model. The Hurd and Flower model uses fractal dimension as an additional fitting parameter so that when D f =3 the spherical form factor is recovered. The model was tested on silica particles with a monomodal size distribution (Bindzil 40/30), and was also tested on polydisperse silica particles (Ludox TMA). It was shown that this fractal model was effective at describing the rough surface and porosity of the silica particles along with their polydispersity. In the future it would be interesting to apply this model to scattering from other types of particle. The model is simple to use and is already implemented in a common program for scattering analysis (SASfit) making it easy to apply. It has potential for widespread use.

In chapter 3 the double-Yukawa potential was used to describe the silica structure factor. The choice of double-Yukawa potential was governed by convenience rather than physical meaning. As demonstrated by the recent work of Kumar et al. 108 and Bereigts 110 solving the structure factor of Ludox particles, especially in salt solution, is challenging. This thesis shows the need for a meaningful structure factor that can be easily applied to experimental data. Charged colloidal systems are often studied in salt solution, and so structure factors that can be applied to such data are extremely useful. As an immediate future step it would be interesting to combine the work of Bereigts with the Hurd and Flower model, and testing this model by fitting a concentration series of Ludox (such as the series examined in chapter 3). This code could then be optimised for efficiency and implemented in a common fitting program, such as SASfit, making it easily accessible for the wider SAS community.

In chapter 4 the structure of chitosan in solution was examined by SAXS, SANS and SLS. The scattering curves show different features. It was shown that the scattering length density of chitosan changes significantly in the presence of counter-ions; however the magnitude of this change is not sufficient to describe the scattering data. These results confirm previous work indicating that domains form within chitosan aggregates in solution; however the data indicate these domains are more complex than previously thought. As a derivative of chitin chitosan is widely researched and widely used in industry. Given this, it is astounding that the inconsistency in scattering by SAXS and SANS has not previously been reported. It also raises the question of whether this difference exists for other polyelectrolytes. In the future it would be interesting to conduct neutron contrast matching experiments on chitosan to understand the origin of the curious scattering behaviour. It would also be interesting to measure other polyelectrolytes by SAXS, SANS and SLS to understand if the behaviour is a feature of chitosan, or a more general phenomenon.

Chapter 3 and chapter 4 characterise the materials used in chapter 5 and chapter 6 of this thesis. Both silica particles and chitosan are simple systems which are widely used. This thesis has shown that despite their simplicity there is much still to learn about the structure of chitosan and the structure of silica particles in solution.

Conclusions on the binary system

The binary (chitosan-silica) system is introduced in chapter 5. The chapter begins by replicating the state diagram of Shi et al. 36 which is consistent with other systems of oppositely charge colloids found in the literature. 41 When chitosan and silica particles are mixed complexes are formed. The state diagram shows the presence of a phase boundary. Above the phase boundary samples separate into a coalesced liquid phase and silica-poor gas phase. Just below the phase boundary is a metastable single phase. The complexes are formed by silica particles electrostatically adsorping onto the chitosan chain.

The kinetics of formation of the complexes was studied using in-situ fast mixing experiments in combination with SAXS measurements. These experiments show that the kinetics of complex formation are different above and below the phase boundary. Above the phase boundary kinetic data show complexes are formed in the ≈2 ms between mixing and the first measurement. The overall size of these complexes increases, and they rearrange over time to become more densely packed. This leads to phase separation as the increased density of the clusters causes sedimentation of the coalesced phase.

Below the phase boundary the kinetics of complexation are too fast to be probed by fast mixing experiments. The complexes formed below the phase boundary are stable for extended timescales. Samples indicating the presence of rod-like complexes were identified. One of the original aims of this thesis was to investigate the possibility of manipulating these rod-like complexes. An attempt was made to align samples which appeared to contain rod-like structures using an external shear. Neither significant structural anisotropy nor any significant change in structure were observed under shear. However, the small amount of anisotropy that was observed appears to correspond to the size of rod-like structures identified in dialysed samples in chapter 6 (≈150 nm). It is possible that these rod-like units are present in the non-dialysed sample; but they do not align because their number is small, and they form part of larger clusters. It appears that these larger structures are not broken apart by external shear; although chapter 6 demonstrates they are destroyed by increasing temperature. By simultaneously heating the sample (to destroy large aggregates and increase the number of loose rods, as seen in chapter 6) and shearing the sample it may be possible to increase the degree of alignment; however beyond confirming the structural hypothesis posed here this is unlikely to have any practical application.

In the samples examined without dialysis in chapter 5 it was difficult to quantify the amount of loose silica particles present in the system. The high salt concentration screened electrostatic interactions and prevented the formation of a structure factor peak. Therefore, the double-Yukawa model was chosen to interpret data from around the metastable region of the non-dialysed state diagram presented in chapter 5. The double-Yukawa model assumes little besides an isotropic potential and was previously used by other groups examining similar systems. 44 Unfortunately, in the particular case examined here the double-Yukawa model provided little useful information. Nevertheless, it was possible to qualitatively describe the metastable area of the state diagram. As the phase boundary is approached the concentration of complexes increases. As one moves away from the phase boundary into the metastable region the concentration of loose silica particles increases. As the overall concentration of the system increases excluded volume interactions become more significant, as does structural compression of the complexes.

In chapter 6 dialysing the samples revealed the presence of a strong structure factor peak. There was no longer salt present to screen electrostatic interactions. By heating a sample it was possible to show the structure factor peak originated from scattering by loose silica particles. Quantifying the contribution of loose silica particles to the system and isolating the scattering signal originating from complexes was then straightforward as the silica system without salt had been well described in chapter 3.

A temperature sweep experiment revealed the system self-assembles reversibly. The scattering data was well described by a model of rod-like complexes which join to form fractal aggregates. This assumes that the building unit is not individual silica particles; but silica-chitosan structures. If this assumption is correct it shows similarities with biomineralisation processes.

While it is easier to characterise the structure of samples after dialysis when the salt concentration is low, it is necessary to understand the structure of samples in highsalt conditions. Any biomedical application of oppositely charged complexes, such as understanding how nanoparticle-polyelectrolyte complexes can be used as drug delivery agents (which was discussed in the introduction of chapter 5), requires understanding how the complexes behave in biological fluids which have high ionic strength and in the presence of multi-valent ions. For example the ionic strength of blood is ≈0.15 mol l -1 which is comparable to the ionic strength of the buffer solution used here. 188 Further work is required to understand how to characterise the structure and dynamics of heterogeneous systems with biological relevance.

Similarly further work is required to understand the link between the system dynamics and the final state of the system. In chapter 5 the system dynamics (as measured by XPCS) indicated gel formation. In chapter 6 the dynamics indicated a transition from a gel to a liquid-like state. This was supported by long-term storage tests of the system which showed the formation of a colloidal crystallites with some liquid structure. It would be particularly interesting to examine the dynamics across a wider q-range with further experiments and to understand more about the dynamics of nanoparticle-polyelectrolyte systems.

The final result was the growth of colloidal crystallites in samples which had been dialysed. It appears that the crystals have a hexagonal close packed cylinder structure vindicating the choice of cylinder model used for the temperature sweep study. These rods align to form a crystalline structure. The mechanism of crystal growth was discussed and there appears to be similarity with biomineral formation. Understanding how these colloidal crystallites grow could provide inspiration to the problem of protein crystallisation.

Outlook on the system

This thesis has examined charged colloidal systems, both as isolated materials and as binary systems. It has found a new model to describe scattering from polydisperse and porous nanoparticles in chapter 3. In chapter 4 it found an unusual structure of chitosan in solution. Chapter 5 demonstrated that in this system it is not possible to align rod-like structures formed from oppositely charged colloids and polyelectrolyte, and explanation for why this was not possible was provided. Finally, in chapter 6 a model of the system based on rods joining to form larger fractal aggregates was proposed. It was demonstrated that by removing salt from the system colloidal crystallites grow.

These chapters have traced a journey through the landscape of much of current colloid science. The thesis has focused on fundamentally examining the systems investigated here by using scattering techniques to discern the solution structure, and inter-particle interactions. Chapter 5 described why understanding nanoparticle-polyelectrolyte systems could be relevant to the pharmaceutical industry. This is only one potential application. As the complexes described here change their structure in a reversible manner upon increasing temperature there is potential for them to be adapted as a responsive material. For now such applications remain in the future; however I hope that they provide insight into the "grand challenge" posed by Van der Gucht in chapter 1: understanding the microstructure of colloidal systems is essential for designing materials with specific properties. As a result colloid science will likely remain the science of the everyday. with the particles. It is nevertheless observed in A.1 that despite the polydispersity the Hurd and Flower model fits well.

Distribution functions

Two different distribution functions were used in fitting data presented in chapter 3a Gaussian distribution, and a Schulz-Zimm distribution. They were used as implemented in SASfit (Breßler and Kohlbrecher, 2015). 96 The Gaussian distribution is described as

Gau s s(R m , N , σ, x) = N π 2 σ 1 + e r f ( R m 2σ ) exp -(x -R m ) 2 2σ 2 (A.3)
where R m is the mean of the distribution, corresponding to the mean particle radius; σ is standard deviation from the mean, a measure of particle polydispersity; e r f is the error function; and N is particle number density. While the Schulz-Zimm distribution is described as

SZ(R m , N , k, x) = N R m x r k-1 k k exp(-k x r ) Γ (k) (A.4)
where R m is the distribution mean, and standard deviation σ = R m / k; particle number density is again N , and Γ (k) is the gamma function. Both the Schulz-Zimm and It can be seen that the transmission values of the sample and its buffer (given in the legend of fig. A.2) remain relatively constant. It is possible that the signal may be distorted due to the detector receiving high intensity in certain areas. Figure A.2 illustrates that such distortion is not significant, and that this does not cause the lack of q -4 slope at high-q.

A.2.2 Results and discussion

The concentration series measured using the peristaltic pump was fit using a double-Yukawa model described in chapter 3. A.3 Appendix chapter 5

A.3.1 Results and discussion

Rheo-SAXS

To check for any changes in structure as a function of shear stress data were divided by the 1D scattering curves when stress was applied with the corresponding static scattering curve for a given configuration. Results are plotted in fig. A.5. It can be seen that the change in scattering intensity is less than 2 % as a result of shear.

Double-Yukawa potential

After fitting with the double-Yukawa potential the fitting parameters were extracted and used to plot the corresponding potential. This is shown in fig. A.6. As described in chapter 5 this potential is not a good physical description of the system. Nevertheless it is possible to characterise the curves into four types: repulsive, neutral, a shallow attraction and a strong attraction. 2) and plotted against q 2 and q 3 . The relaxation rate has been fit with f (q) = D e f f Γ a . In the left panel where a = 2 and in the right panel a = 3. The data corresponding to the non-dialysed sample have been offset by a factor of 0.1 for observational ease.

A.4.1 Results and discussion

XPCS

In fig. A.7 a comparison of integer exponents used to fit relaxation rates obtained by XPCS are plotted. It can be seen that neither an exponent of 2 nor of 3 fits the data well. This is why a non-integer exponent of 2.5 was used in fig. 6.5.

Polycrystalline silica

Figure A.8 shows the regions in the mesh scan of the crystallite examined in chapter 6 where a particular second order Bragg peak was observed. The light coloured areas show the presence of crystalline domains. As the overall shape of the material is observed it would appear the material has an overall anisotropy. Crystallites appear to be distributed across the entire area of the material.

In Figure A.9 the maximum projection on the detector observed in the mesh scan is plotted. It can be seen that across the entire mesh scan there is a distribution of lattice parameters. The broadness of the peaks is due to random orientations of the crystallites In fig. A.10 the data from the single frame shown in fig. 6.17b have been integrated azimuthally. The q-values of the first and second visible order peaks are identified. Detector saturation has distorted the integration as the highest intensities are recorded as zeros. This does not prevent identification of the mean q-value. It can be seen that the peaks are broad. This is due to random orientations of the crystallite through the material thickness.

  -Angle X-Ray Scattering WAXS Wide-Angle X-ray Scattering XFEL X-ray Free Electron Laser XPCS X-Ray Photon Correlation Spectroscopy CHAPTER 1

Fig. 2 . 2

 22 Fig. 2.2 Plot of(2.1) where θ is in radians and λ = 1 Å. This illustrates why small-angles are required to study structures on the colloidal (nm to µm) scale.

Fig. 2 . 3

 23 Fig. 2.3 Simulation of the spherical form factor showing the effect of polydispersity, in the form of a a continuous distribution of particle sizes, on the scattering curve. The size-distribution corresponds to a Schulz-Zimm distribution.Particle radius of the monodisperse spheres is 15 nm. Data are simulated and not in absolute scale.

Fig. 2 . 4

 24 Fig. 2.4 Evolution of a hard sphere structure factor as a function of volume fraction for monodisperse spheres. Data are simulated and not in absolute scale.

Fig. 2 . 7

 27 Fig. 2.7 Layout of beamline ID02

Fig. 3 . 1

 31 Fig.3.1 Illustration of the peristaltic pump set-up on ID02. This was used to mix and measure a concentration series of silica particles (Ludox TMA) and water. Silica particles, from a reservoir at 34 %, was mixed progressively with water. This mixture was passed through the capillary where SAXS measurements were taken. A peristaltic pump controlled the pumping rate.

  Fig. 3.2The invariant, a measure of particle concentration, plotted against time after initial pumping (at constant

Fig. 3 . 3

 33 Fig. 3.3 Data (orange diamonds) and background (shown in blue circles) of 0.3 % Ludox TMA measured for different sample-detector distances. Subtracted data is shown in magenta triangles. Transmission values are shown in the legend. There is no difference in transmission for background and sample; indicating scattering probability is low and there is little absorption from the nanoparticles. TMA data measured with different beamstops are shown in the appendix fig. A.2.

4 Fig. 3 . 4

 434 Fig. 3.4 Comparison of Ludox TMA and Bindzil 40/130 particles. Ludox TMA is fit with a polydisperse sphere model

Fig. 3 . 5

 35 Fig. 3.5 Description of deviation of Ludox TMA from a simple spherical fit using a Schulz-Zimm population distribution (R m =10.89 nm, σ = 0.12). The sample is the same as that shown in fig. 3.4.

Fig. 3 . 8

 38 Fig. 3.8 Porod plot showing scattering from large silica particles of 0.1 µm obtained from Polysciences Europe GmbH, from Ludox TMA and from Bindzil 40/130. For clarity the Bindzil data have been offset vertically by a factor of 10, and the Polysciences silica by a factor of 100.In the case of perfectly smooth particles I (q) * q 4 should be flat.

Fig. 3 . 9

 39 Fig. 3.9 Kratky plot showing scattering from large silica microspheres of 0.1 µm obtained from Polysciences Europe GmbH , from Ludox TMA and from Bindzil 40/130. For clarity the Ludox data have been multiplied vertically by a factor of 10, and the Polysciences silica by a factor of 0.1.

Fig. 3 .

 3 Fig.3.14 Simulation of the Hurd and Flower model with fractal dimension, D f = 2, 2.5, 3. This is compared with the standard scattering from a sphere of 15 nm radius (the dashed line). It is observed that the sphere scattering function is recovered for fractal dimensions of 3. Data are simulated and not in absolute scale.

Fig. 3 .

 3 Fig. 3.15 Bindzil, a silica dispersion with a monomodal size distribution fitted with a single model: fractal aggregate with a Gaussian size distribution. The model uses fractal dimension, D f = 2.98 ± 2.1 × 10 -5 , R m =13.1 nm, σ = 0.14. The fractal dimension is close to that of the spherical form factor D f = 3.

Fig. 3 .

 3 Fig. 3.16 Plot of residuals comparing the data shown in figs. 3.4, 3.13 and 3.15. It can be observed that at high, mid and low-q the fractal model provides an improved description of the data relative to the polydisperse sphere fit, and that compared to the combined Gaussian sphere, hard-sphere structure factor and Ornstein-Zernike function the fractal model is more appropriate at low and mid-q.

Fig. 3 .

 3 Fig. 3.17 Description of dialysed Ludox TMA using a single population fractal fit with a Schulz-Zimm distribution. The fitting parameters are: R m =9.1 nm, σ = 0.1, D f = 2.97 ± 3.6E -5

Fig. 3 .

 3 Fig. 3.18 Fit of Ludox TMA with the Hurd and Flower model used as in fig. 3.17. This is combined with the

  Hayter-Penfold and Mass fract. agg.

Fig. 3 .

 3 Fig. 3.19 Single frame from the concentration series plotted in fig. 3.10. The data are fit with the double-Yukawa model which is compared to the combined model of Hayter-Penfold and mass fractal aggregate with Gaussian cut-off function used in fig. 3.18.

Fig. 4 . 1

 41 Fig. 4.1 In fig. 4.1a the chemical and structural formula of fully deacetylated chitosan: poly(β-D-glucosamine). In fig. 4.1b the chemical and structural formula of chitin: poly(β-D-N-acetyl-glucosamine). Chitosan is partially deacetylated chitin.

Fig. 4 . 2

 42 Fig. 4.2 SAXS profiles of 0.1 % Sigma-Aldrich chitosan in 0.2 M sodium acetate, 0.3 mol L -1 acetic acid buffer solution at pH = 4.5. The legend shows the exposure time of the measurement in seconds, and incident photon flux Φ 0 . Shorter measurement times are in dark green, longer measurement times in light green. Data were measured at ID02 and have been averaged and rebinned.

Fig. 4 . 3

 43 Fig. 4.3 SAXS profile, from ID02, of 0.1% Sigma-Aldrich chitosan in acetate buffer solution and the sample background.

4. 4

 4 Results and discussion 4.4.1 Results from SAXS measurements In fig. 4.4 SAXS data for three concentrations, 0.1 %, 0.3 % and 1.0 % of low molecular weight chitosan, from Sigma Aldrich, and solubilised in D 2 O acidified with HCl are presented. The data are consistent with other measurements of chitosan, such as those by Boucard et al.

1 Fig. 4 . 4

 144 Fig. 4.4 SAXS curves, from ID02, of 0.1 %, 0.3 % and 1.0 % Sigma-Aldrich chitosan in D 2 O acidified with HCl.

Fig. 4 . 5

 45 Fig. 4.5 SAXS curves, from ID02, of 0.1 %, 0.3 % and 1.0 % Sigma-Aldrich chitosan in 0.2 mol L -1 sodium acetate and 0.3 mol L -1 acetic acid D 2 O buffer solution.The polyelectrolyte peak is suppressed due to excess salt.

1 ,

 1 and the mass of chitosan and scattering volume are known experimentally. The apparent density, 1.57 g cm -3 , was determined by L. Chiappisi, Institut Laue-Langevin, using a vibrating U-tube densitometer, Anton-Paar DMA 4500 operated at 25 • C. Assuming the manufacturer's molecular weight of 100 kDa for 1 g L -1 of chitosan a forward scattering of 0.16 mm -1 should be observed by SAXS. In fig. 4.4 and fig. 4.5 the observable I (0) values are closer to 4 mm -1

Fig. 4 .

 4 Fig. 4.6 SAXS and SANS data collected on the same samples of low molecular weight Sigma-Aldrich chitosan in acidified D 2 O on the same day (SAXS from ID02). SLS data collected several weeks later on the same samples.

Fig. 4 . 9

 49 Fig. 4.9 SAS curves of 0.1 % Sigma-Aldrich chitosan in 0.2 mol L -1 sodium acetate, 0.3 mol L -1 acetic acid buffer solution at pH = 4.5. SAXS data, collected on ID02, are compared for chitosan in D 2 O and H 2 O acetate buffer solution.

Fig. 4 .

 4 Fig. 4.10 SAS curves of 1 % chitosan with a known degree of acetylation of 0.05. SAXS data were collected at BM29. Diamonds show the chitosan solubilised with the stoichiometric amount of HCl. Squares show chitosan solubilised in 0.2 mol L -1 sodium acetate, 0.3 mol L -1 acetic acid D 2 O buffer solution at pH = 4.5. The data solubilised in acidified D 2 O have been offset by a factor of 10 for easier viewing. The yellow dashed lines indicate the factor of four difference expected in I (q) due to SLD differences between the SAXS and SANS data for the sample solubilised in acetate. The blue dashed lines indicate the factor of eight difference expected in I (q) due to

Fig. 5 . 2

 52 Fig. 5.2 Illustration showing the links between the stopped flow data, macroscopic sample observations after storage tests, and the state diagram.

Fig. 5 . 3

 53 Fig. 5.3 3D graph showing stopped flow data for sample composition 0.1 % Ludox TMA and 0.001 % chitosan. Therefore this is in the phase-separating region of the diagram, as seen in fig. 5.1.

Fig. 5 . 4

 54 Fig. 5.4 2D graph describing the stopped flow data in fig. 5.3 for sample composition 0.1 % Ludox TMA and 0.001 % chitosan. Therefore this is in the phase-separating region of the diagram, as seen in fig. 5.1. Yellow is used to describe the initial times, orange mid times, and green the final spectra as seen in fig. 5.3. The cross hatches show the form factor of the silica particles. R g corresponds to the complex radius of gyration extracted from the Guinier approximation (section 2.1.3). The number of particles per cluster, described by the aggregation number N p , increases. Annotations describe the physical meaning of the spectra.

1 Fig. 5 . 5

 155 Fig. 5.5 Plot of scattering curves showing the kinetics of formation of 1 % Ludox TMA and 0.01 % chitosan. The data are fit, using the dashed green lines, with the Hurd and Flower model described in chapter 3 and eq. (5.2).

Fig. 5 . 6

 56 Fig. 5.6 Plot showing the evolution of the parameters of the Fisher-Burford function, eq. (5.2), with respect to time. The aggregation number, N a , increases exponentially with respect to time, while the overall cluster size, R g increases linearly. The inset shows, D f , which is the fractal dimension of the clusters. This increases before reaching a maximum.

Fig. 5 . 7

 57 Fig.5.7 Scattering data of 16 % silica particles and 0.0225 % chitosan. The sample has a power-law scattering behaviour corresponding to a slope of q -1.3 .

%

  Electrophoretic titration of silica particles against chi-Ludox wt. % (b) Green line corresponds to the electrophoretic titration charge inversion point.

Fig. 5 .

 5 Fig. 5.10 In fig. a the ratio of silica mass to chitosan mass is plotted against their measured electrophoretic mobility. It can be seen that there is a charge inversion point. The position of this point corresponds to the phase boundary in the diagram shown in fig. b. It can be seen that by changing the type of silica particles the position of the charge inversion point changes; however the behaviour is consistent. Figure 5.10b shows the state diagram immediately after sample preparation. The green diagonal line is drawn at the chitosan-silica mass ratio of ≈150 corresponding to the mass ratio at the electrophoretic titration charge inversion point. The light orange and light yellow area (enclosed in a box) was not investigated.

Fig. 5 .

 5 Fig. 5.11 SAXS curves showing how the scattering interactions change with increasing chitosan concentration added to the system. The concentration of Ludox TMA is constant at 16 %. The yellow squares correspond to the sample after phase separation had occured.

Fig. 5 .

 5 Fig. 5.12 The SAXS curves shown in fig. 5.11 are fit with the double-Yukawa model used in chapter 3. The concentration of Ludox TMA is constant at 16 %. Data have been offset to show the quality of the fits.

Fig. 5 .

 5 Fig.5.13 The apparent inter-particle potentials, and pair distribution function, g (r ), plotted as a function of the hard-sphere radius. Both the potentials and g (r ) are calculated using the parameters of the fits presented in fig.5.12.

Fig. 5 .e e p a r t i c l e s C o m p l e x e s 1 Fig. 5 .Fig. 5 .

 5155 Fig. 5.14 SAXS curves of 1 % TMA fit with the double-Yukawa model used in chapter 3. Data have been offset to show the quality of the fits.

168

 168

Fig. 5 .

 5 Fig. 5.18 Plot of 16 % Ludox TMA and 0.05 % chitosan fit with a combination of the Hurd and Flower model, and a Fisher-Burford (S(0)=83, ξ =105, and 2 p=2) and Percus-Yevick structure factor (well width 8 nm, well depth 2 k B T).

  (a) Photograph of sample with gel-gas coexistence. Ludox wt. % State diagram after 18 months.

Fig. 5 .

 5 Fig. 5.19 State diagram after 18 months. The light orange area is from fig. 5.1 and was not measured. The thick grey line indicates which samples were kept in storage tests. It can be seen, in the blue region of the figure, that samples with a high concentration of silica particles gel.

Fig. 6 . 2

 62 Fig. 6.2 Example of fits of equation 5.3 to data for dialysed sample of 16% TMA and 0.0225% chitosan via XPCS.

Fig. 6 . 3 Plot against q 2 Fig. 6 . 4

 63264 Fig. 6.3 Kohlrausch exponent, γ , extracted from the stretched exponential fits in fig. 5.16 (and fig. 6.2) and plotted against q. The green line at γ = 0.89 corresponds to the average value of γ for the dialysed sample. The yellow line at γ = 0.68 corresponds to the average value of γ for the sample which was not dialysed.

Fig. 6 . 6

 66 Fig. 6.6 SAXS curves showing the response of a sample of 16 % Ludox TMA 0.0225 % chitosan to changing temperature. As temperature increases there is a decrease in low-q intensity. This is a reversible change with the low-q intensity recovered on return to room temperature indicated by the blue lines. The inset shows that the position of the structure factor peak does not change with temperature. The scattering length density of water changes with temperature and this accounts for the slight difference in peak intensity with temperature.

Fig. 6 . 7

 67 Fig.6.7 The forward scattering, I (0), extracted from the static curves shown in fig.6.6, is plotted against temperature. It can be seen that there is a dependence of I (0) on temperature. On returning to room temperature the original behaviour is approximately recovered.

Fig. 6 . 8

 68 Fig. 6.8 The inset shows the data presented in fig. 6.6 along with the form and structure factor of free silica particles corresponding to φ = 0.07 and hard sphere radius 14.8 nm. The main plot shows the data after this form and structure factors have been subtracted. The presence of an isosbestic point is labelled, confirming the structures do not contribute to the peak formation.

Fig. 6 .

 6 Fig. 6.10 The subtracted data shown in fig. 6.8 are fit with two spherical form factors, one of which has a fractal structure factor. The data are offset to show the quality of the fits.

Fig. 6 .

 6 Fig.6.11 The fitting parameters from fig.6.10 are extracted and plotted against temperature. The overall cluster number, N t ot , is plotted on the right axis; while the overall cluster size is plotted on the left axis.

Fig. 6 .Fig. 6 .

 66 Fig. 6.12 The fitting parameters from fig. 6.10 are extracted and plotted against temperature. The fractal dimension, D f , is plotted on the left axis; while the cylinder lengths are plotted on the right axis. The length of cylinders that do not contribute to aggregates, LC1, is shown in yellow diamonds; while the length of cylinders that contribute to large aggregates, LC2, are shown in blue squares. The silica particles which form the cylinders have a diameter of ≈22 nm.

Fig. 6 .

 6 Fig. 6.14 Photograph of a sample colloidal crystallite found in a sample of dialysed 5 % Ludox TMA and 0.001 % chitosan. The crystallite is placed on a ruler showing it is ≈1 cm in length.

Fig. 6 .

 6 Fig. 6.15 State diagram for dialysed samples after 18 months. The light orange area is from fig. 5.1 and was not measured. The thick grey line separates which dialysed samples were kept in storage tests. It can be seen that all dialysed samples show colloidal crystallites.

Fig. 6 .

 6 Fig. 6.16 Orientation map of a second order Bragg peak from colloidal crystallite examined by microbeam SAXS.

  6.16 (as shown in fig. A.9). From this study it is clear that the crystalline regions are several tens of µm 2 in size. Properly characterising the size of such regions, and how they change over time would be interesting to examine further.

( a )

 a Average scattering across the entire crystallite. (b) 2D scattering pattern observed in the microcrystalline region of the colloidal crystallite.

Fig. 6 . 17

 617 Fig. 6.17Figure 6.17a shows the average of all frames taken in the scan. It shows the crystallite has an overall

Figure 6 .Fig. 6 .

 66 Fig. 6.17Figure 6.17a shows the average of all frames taken in the scan. It shows the crystallite has an overall anisotropy. In fig. 6.17b a sample 2D scattering pattern observed in a micro-crystalline region of the material is shown. This is a single frame from the scan. Bragg peaks are clearly observed. Data were measured at beamline ID13.

Fig. A. 1

 1 Fig. A.1 Fractal aggregate fit of model silica particles 10 nm in radius. The silica was obtained from Polysciences Europe GmbH.

49 Fig. A. 2 3 .

 4923 Fig. A.2 Data (orange circles) and background (shown in blue squares) of 0.3 % Ludox TMA measured for different distances and with differently sized beamstops. Transmission values, trm, are included in the legend. It can be seen these remain relatively constant. The silica nanoparticles were taken from a bottle that had been open for three years and experienced some fractionation, hence the size distribution is slightly different to that presented in 3.

Figure A. 2

 2 Figure A.2 presents scattering from Ludox TMA with and without background subtraction, for different sample-detector distances, and with differently sized beamstops.It can be seen that the transmission values of the sample and its buffer (given in the legend of fig. A.2) remain relatively constant. It is possible that the signal may be distorted due to the detector receiving high intensity in certain areas. Figure A.2 illustrates that such distortion is not significant, and that this does not cause the lack of q -4 slope at high-q.The concentration series measured using the peristaltic pump was fit using a double-Yukawa model described in chapter 3. In fig. A.3 it can be seen that the model successfully describes the evolution of the main features of as a function of concentration. As the form of the fitting potential does not change with concentration the fitting parameters are constant and are plotted in fig. A.4 as a function of volume fraction.

  Figure A.2 presents scattering from Ludox TMA with and without background subtraction, for different sample-detector distances, and with differently sized beamstops.It can be seen that the transmission values of the sample and its buffer (given in the legend of fig. A.2) remain relatively constant. It is possible that the signal may be distorted due to the detector receiving high intensity in certain areas. Figure A.2 illustrates that such distortion is not significant, and that this does not cause the lack of q -4 slope at high-q.The concentration series measured using the peristaltic pump was fit using a double-Yukawa model described in chapter 3. In fig. A.3 it can be seen that the model successfully describes the evolution of the main features of as a function of concentration. As the form of the fitting potential does not change with concentration the fitting parameters are constant and are plotted in fig. A.4 as a function of volume fraction.

1 Fig. A. 3

 13 Fig. A.3 Three dimensional plot showing the change in scattering from Ludox TMA particles as a function of volume fraction. The data series was made using the peristaltic pump set-up described in section 3.3. The data were fitted with the Hurd and Flower model and a double-Yukawa structure factor with MSA. MATLAB code of Liu et al.111 was implemented in SASET (Muthig, 2013)112 and series fitting was performed.

Fig. A. 4

 4 Fig. A.4 The fitting parameters used in the double-Yukawa structure factor used in fig. A.3 are plotted as a function of volume fraction. It can be seen that the shape of the potential does not change significantly.

Fig. A. 5

 5 Fig. A.5Graph showing the small change in the structure factor of 16 % silica and 0.0225 % chitosan as a function of τ. Data were obtained in rheo-SAXS experiments. The distances quoted in the graph correspond to 2π/q nm, and shear stress in the legend is in units of Pa.

4 Γ /s - 1 q 3 /nm - 3 Fig. A. 7

 41337 Fig. A.7Data obtained from XPCS measurements. The relaxation rate, Γ , has been extracted from the stretched exponential fits in fig.5.16 (and fig.6.2) and plotted against q 2 and q 3 . The relaxation rate has been fit with f (q) = D e f f Γ a . In the left panel where a = 2 and in the right panel a = 3. The data corresponding to the non-dialysed sample have been offset by a factor of 0.1 for observational ease.

Fig. A. 8

 8 Fig. A.8 Image of the polycrystalline silica that was examined with microbeam SAXS. The bright domains correspond to orientated polycrystalline domains. The shape of the overall material can be distinguished. This corresponds to it having an overall anisotropy. Data were measured at beamline ID13 by T. Grünewald.

Fig. A. 9

 9 Fig. A.9 Bragg peaks from the maximum intensity found on a mesh scan of a crystal of silica found in the dialysed samples. The beam size was 2x2µm (FWHM).Data were measured at beamline ID13 by T. Grünewald.

1 Fig. A. 10

 110 Fig. A.10 The data shown in fig. 6.17b have been integrated. The locations of the first and second visible order peaks are identified. The shape of the first peak is distorted due to detector saturation causing the highest intensities to be recorded as zeros.

Table 2 .1 ID02 detectors 48

 2

	Detector	Type	Pixel Size	Acquisition Rate	Active Area
			µm 2	Hz	pixels per side
	FReLoN	CCD	24 x 24	2 (1x1 bin)	2048 x 2048
	Rayonix	CCD	88 x 88	8 (2x2 bin)	1920 x 1920
	MX-170HS
	Pilatus	HPD	172 x 172	400	487 x 619
	300K
	PSI-Eiger	HPD	75 x 75	<22000	1030 x 514
	500K
	data presented in chapter 5, and the PSI-Eiger to collect the XPCS data shown in chapter 5
	and chapter 6.

Table 4 . 1

 41 Chitin and chitosan SLD values

	Name	in	Formula	SLD x-rays	SLD neutrons
		H 2 O/D 2 O
				10 -4 /nm -2	10 -4 /nm -2
	Chitin	H 2 O	C 8 H 13 NO 5	13.66	1.92
	Chitin	D 2 O	C 8 H 10 D 3 NO 5	13.66	3.33
	Chitosan	H 2 O	C 6 H 11 NO 4	14.35	1.85
	Chitosan	D 2 O	C 6 H 7 D 4 NO 4	14.35	4.32
	Chitosan HCl	H 2 O	C 6 H 12 ClNO 4	14.8	1.87
	Chitosan DCl	D 2 O	C 6 H 7 D 5 ClNO 4	14.8	4.49
	Chitosan AcH	H 2 O	C 8 H 15 NO 6	14.07	1.74
	Chitosan AcD	D 2 O	C 8 H 10 D 5 NO 6	14.07	3.94

  Quemeneur et al. examined chitosan and oppositely charged vesicles; 150,151 Yaroslavov et al. examined liposome-polyelectrolyte complexes, 42 and Rinaudo et al. investigated DNA-chitosan complexation,

  Plot showing the anisotropy parameter

	0.886
	0.758
	0.659
	0.489
	0.39
	0.319
	0.178
	0.148
	0.124
	0.1
	0.0764
	0.0526
	0.0289
	0.017
	τ /Nm -2
	Fig. 5.8

v Charge stabilisation is discussed in section 1.1.1. vi See section 2.1.3.

v The Bjerrum length (chapter 1) is defined as λ B = e 2 /4πε 0 ε r k B T where e is the charge of an electron, ε 0 is permittivity of a vacuum, and ε r is dielectric constant of the medium.5

iii A slight change was observed and is plotted in fig. A.5. iv The dialysed samples used in chapter 6 were also examined under shear, and using an electric birefringence set-up where the degree of observed anisotropy was very weak.

i Calculation was made with volume fraction of silica particles in solution, φ = 0.07, and hard sphere radius of 14.8 nm.

v Crystallisation of proteins is required to study their structure using x-ray crystallography; however forming protein crystals is often challenging.

of HCl, diluted to the appropriate condition in Milli-Q water and the silica particles were added.

Small-angle x-ray scattering

All SAXS measurements were performed at beamline ID02 at the ESRF. The beamline is described in section 2.2.2. In addition to stopped flow mixing and rheo-SAXS experiments samples were measured using static SAXS in individual quartz capillaries.

X-ray photon correlation spectroscopy

XPCS measurements were performed at beamline ID02 at the ESRF. The XPCS setup is described in detail in Zinn et al. 160 Data analysis was done using the PyXPCS program developed by Y. Chushkin at the ESRF. Samples were measured in quartz capillaries.

LIST OF TABLES

SUPPORTING INFORMATION A.1 Appendix chapter 2

A plane wave can be described by

where A, is the amplitude of the wave; k is the wave vector; r describes the wave front at time, t; ω is the angular frequency of the wave, and φ is the phase of the wave. [START_REF] Sivia | Elementary Scattering Theory[END_REF] The wave vector, k denotes the direction of propagation of the wave, and its magnitude |k| = k where k is defined as the wavenumber. Waves are periodic in both time and space such that

where λ is the wavelength, T the period of the wave, and ν its frequency.

A.2 Appendix chapter 3 A.2.1 Materials

Silica nanoparticles were obtained from Polysciences Ltd. with a nominal radius of 10 nm to compare with results from Ludox TMA, which is an industrial dispersion. However upon measuring, see fig. A.1, it was found that the particles from Polysciences Ltd. were poorly defined. The particles have a high polydispersity and display a q -x slope at high-q (where x ̸ = 4) indicating a non-ideal morphology. No further work was conducted Supporting Information

A.4 Appendix chapter 6

Table table A.1 shows the change in conductivity per day dialysis. As expected the conductivity drops by two orders of magnitude.