A small angle photon correlation imaging instrument for the study of thermoreversible gels

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Politecnico di Milano

School of Industrial and Information Engineering Master Degree Programme of Engineering Physics

A SMALL ANGLE PHOTON CORRELATION

IMAGING INSTRUMENT FOR THE STUDY OF

THERMOREVERSIBLE GELS

Supervisor: Dr. Stefano Buzzaccaro

Co-supervisor: Prof. Roberto Piazza

Author:

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“a chi ha mano quadri e cuori dove regnano denara e bastoni”

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Abstract

Colloidal systems became one of the most suitable tools for research on statistical physics, thanks to the rather easiness of tuning the inter-particle interactions, coupled to the fact that size of the colloidal particles is suitable to use light scattering techniques. Unfortunately, in many case, these materials display a complex behaviour, characterized by spatially and temporally heterogeneous dynamics, posing many challenges to the study of these materials. To that end, Dynamic Light Scattering (DLS) and Optical Microscopy (OM), two of the favourite optical methods for investigating soft materials, suffer from some drawbacks. DLS provides us with detailed information on the microscopic dynamics of particles and structures down to the nanoscale, but lacks spatial resolution. Per contra, OM methods allow us to resolve spatially heterogeneous structures, but the spatial scales they probe are bounded from below by the resolution limit and, from above, by the limited field of view of the microscope. To overcome these limitations, this thesis work concerns the development and calibration of a small angle Photon Correlation Imaging (PCI) instrument. PCI is an optical technique, that, combining the correlation methods of Dynamic Light Scattering (DLS) and the spatials resolution of an imaging technique, provides both temporally and spatially resolved analysis of the inspected systems. The key point of the PCI is the decoupling of the length scale over which the dynamics is probed from the size of the field of view. We pay particular attention in the design of a custom thermalizer for precise temperature control of the sample, allowing us to study thermo-sensitive material. After the calibration and testing of the setup, we investigate the settling and restructuring dynamics of colloidal gels generated by short-ranged interactions, whose strength can be tuned by changing the temperature of the sample. Our preliminary study shows that a survey of the kinetics of gel settling and restructuring

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highlights several distinctive features that allow us to relate three different settling scenarios with the strength of the inter-particles interactions.

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Abstract

I sistemi colloidali sono diventati uno degli strumenti più adatti per la ricerca sulla fisica statistica, grazie alla possibilità di regolare facilmente le loro interazioni e all’idoneità delle scale spaziali per le tecniche di scattering della luce. Purtroppo, in molti casi, questi materiali mostrano processi complessi, caratterizzati da dinamiche spazialmente e temporalmente eterogenee, rendendo il loro studio una ardua sfida. Perciò, lo Scatterind Dinamico di Luce (DLS) e la Microscopia Ottica (OM), due tra i metodi più usati per studiare la materia soffice, hanno qualche limite. Il DLS fornisce informazioni dettagliate sulla dinamica microscopica delle particelle e delle strutture fino alla nanoscala, ma manca di risoluzione spaziale. Al contrario, i metodi di OM permettono di risolvere strutture spazialmente eterogenee, ma le scale spaziali di indagine sono limitate dalla risoluzione e dal campo visivo del microscopio. Per superare queste limitazioni, questo lavoro di tesi riguarda lo sviluppo e la calibrazione di uno strumento di Photon Correlation Imaging a basso angolo. La PCI è una tecnica che opera nel Near Field, che, combinando i metodi di correlazione dello Scattering Dinamico di Luce (DLS) alla risoluzione spaziale di una tecnica di imaging, fornisce un’analisi dei sistemi osservati con risoluzione sia spaziale che temporale. Il punto chiave della PCI è il disaccoppiamento delle scale spaziali delle dinamiche osservate dalle dimensioni del campo visivo. Un aspetto importante è stata la progettazione di un termalizzatore per controllare la temperatura del campione, che ci ha permesso di studiare materiali termo-sensibili. Dopo i test e la calibrazione, abbiamo studiato la deposizione e la ristrutturazione di gel colloidali generati da interazioni a corto raggio, la cui forza può essere regolata variando la temperatura. Gli studi preliminari hanno mostrato che un indagine sulla cinetica di sedimentazione e ristrutturazione evidenzia caratteristiche peculiari che ci permettono di relazionare tre diversi scenari

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Introduction

Soft matter gradually gained importance since last century because of its possible application in material science, biology and statistical physics. Its matters of study are various, from colloidal particles, that have been incredibly helpfull in investigation on diffusion processes and sedimentation, to polymers, foams, gels, biological materials, which in turn, constitute a structural component of living cells. Since the first years of 1900, J. B. Perrin’s experiments with soft matter, gave him the possibility to determine Avogadro constant and electron’s charge. Furthermore he obtained the experimental confirmation of the equipartition theorem from the concentration profile, induced by sedimentation, through the barometric law of perfect gas at equilibrium. The progress in nanotechologies rapidly enlarged the possibilities for soft matter to be the object of statistical physics studies, making available systems with characteristics never thought possible to be artificially reachable before. For example for the theory of diffusion, the use of particles with well known shapes allowed to determine the size distribution profile of colloids suspended in solution [1][2]; or, more in general, for fluids mechanics and rheology, where it increased and confirmed the power of its mathematical treatment [3]. Furthermore by acting on the solvent, scientists were able to adjust the effective potential between particles, making colloids model systems of particles interacting through determined and controllable forces. This kind of intervent on the solvent includes the addition of depletant agents, like surfactants, allowing to obtain almost ideal sytems of rigid sphere, whose attraction or repultion can be tuned, or a direct action on the salinity or temperature of the sample.

Colloids science has always used many different techniques, but with the invention of laser and its widespread commercialization, optical tecniques became more and more interesting and exploited. Coherence properties of

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laser sources allowed the application of correlation techniques to the typical speckle patter generated by diffracting object. Such techniques include Dynamic Light Scattering (DLS), Time Resolved Correlation (TRC) and the youngest Photon Correlation Imaging (PCI), that being a method operating in near field, and exploiting technologies of multipixels detector, allows to resolve both spatially and temporally the dynamics of the samples object of study.

All these techniques operate on the autocorrelation of the light scattered by the sample, but while a DLS experiment does not allow to obtain a temporally resolved set of data, thus being useless for non ergodic system, or for systems whose dynamics evolves during time, TRC fulfills this aspect. The next step in the evolution of correlation techniques was PCI, where each region of the speckle pattern contains the informations of a precise region in the sample, and the temporal resolution is guaranteed in the same way of TRC. It is immediately clear how PCI can be a powerful imaging technique for systems that undergo structural rearrangements and changes in the dynamics. Among this kind of systems, gels represent one of the most curious, because they never show a completely arrested dynamics, but rather an intermittent and spatially heterogeneous one. A peculiarity of these systems is the occurrence of localized relaxation events triggering, in other regions, rearrangement events that change the overall structure. When these events are characterized by sufficiently slow dynamics, there are no severe technical limitations in the experimental setup, and the original wide-angle scattering PCI can be exploited. But if we want to study, for example, the initial collapsing stage of a gel into fast settling clusters [4], the charateristic times of decorrelation of the speckle pattern make these events detectable only collecting the light scattered at small q-vectors. For this reason the main focus of this thesis work has been the realization of small angle PCI apparatus. Collecting the light at such angles, it is possible to obtain the amplitude of the correlation function for fluctuations with bigger length scale Λ = 2π/q [5].

In the first part of this thesis I will treat the fundaments of the physics of soft matter, the potentials and the phase transition events. Thereafter, I will linger on the optical techniques and the correlation methods available to investigate such systems, their evolutions, their benefits and their limitations. In the second part I will explain in detail the Small Angle PCI

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apparatus that I built, the choice of each optical elements, the thermalization of the samples, and finally the measurement run, the problems raised during the calibration, the possible solutions and the results obtained.

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Part I

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Chapter 1

Statistical Mechanics of

Colloidal Systems

1.1 What is a colloidal suspension?

Colloidal solutions are basically binary systems, in which colloidal particles are dispersed in a solvent. Most of the times the latter is pure water, where, eventually, in some specific experiments a solute is added. Unlike a “normal” solution, whose solute and solvent constitute only one phase, a colloid is composed by a dispersed phase (the suspended particles) and a continuous phase (the medium of suspension), that is the solvent. Let’s explain briefly which is the role of solvent and solute, and why the latter differs from a colloid. The solvent quite often is water, or in general the substance that dissolves the eventual solute. The solute represents all matter that chemically reacts with the solvent, thus changing its physical properties. It consists in all the “non colloidal” components present in the solution (ions, controions, surfactants molecules, etc). Colloidal particles are then all that stuff that is suspended in the solution, without chemically reacting with it, but only subjected to the mechanics of collision and deposition. The characterizing dimensions of colloidal particles are mesoscopic, term that, related to our case, means that they are small enough to make their dynamic to be mainly dominated by brownian motion, but sufficiently big, with respect to solvent and solute molecules, that their shape, surface and charge distribution can be treated as macroscopic quantities. Quantitatively, dimensions of colloidal particles are generally from the order of 50 nm to few

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µm. Some examples of colloids are solutions of polymer chains, biological macromolecules, and even small unicellular organism that behave as colloidal particles, and are subjected to brownian bombardment. In physics this kind of solution are often used as model systems, largely studied to validate statistical mechanics theories and physics of liquids.

1.2 Statistical mechanics

Since a colloidal suspension is a two-component mixture made of Np

particles with mass mp and Ns molecules of solvent with mass ms, it is

possible to write the potential energy of the system as:

U (R, r) = Upp(R) + Uss(r) + Ups(R, r) (1.1)

where R and r represent, respectively, colloidal particles and solvent coordinates. Upp(R), Uss(r), Ups(R, r) are instead the direct interactions

It’s worth to linger a while on the the term Ups(R, r) and to the possiblity

to act on it through the suitable modification of the solvent. The interaction between particles and solvent, exploited by such term, has the effect of generating, around the colloidal particles, a layer of molecules which physical properties differs from the farther ones. The primary consequence of this phenomenon is that the presence of the solvent actually modifies the physical behaviour of the particles, in particular for what concerns the the mutual interaction of the suspension particles Upp. Let’s remember

that colloidal particles don’t chemically react with the solvent, that’s why specific solute it’s often used. This consideration allows to highlight one of them most interesting properties of the colloids: modifying solvent’s properties and nature makes possible to act directly on Ups, thus modifying

the interaction potential of the particles. Another observation on Ups is

that the range δ of the interactions between particles and solvent is usually much shorter than the particles size a. When two particles are so close that the layers of solvent near to them (and then with different properties from

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the farther ones) overlaps, the solvent interactions modifies the interaction term Upp between the colloids. With these considerations it seems natural

to move towards the definition of an effective potential W (R) that depends only on the colloidal particles coordinates. This kind of potentials needs a mapping from the multi-variable potential in Eq. (1.1) to a single variable equation, and in turn this process needs to draw a complete model of the physical properties of these particles. So the full Hamiltonian of the system can be written as:

H(R, r, P, p) = T (P, p) + U (R, r)

= Ts(P) + Tp(p) + Upp(R) + Uss(r) + Ups(R, r) (1.2)

where Ts(P) and Tp(p) represent respectively the kinetic energy of the

particles and of the solvent molecules. The canonical partition function can then be written as

Z = e−βF = ApAs

Z

d(Np)

Z

d(Ns)e−β[Upp(R)+Uss(r)+Ups(R,r)] (1.3)

where      Ap = 1 Np!Λ3Npp = Z ig Np VNp As = _N 1 s!Λ3Nss = Z_Nsig VNs (1.4)

and Λp , Λs are the thermal lenght

Λp = h p2πmpkBT ; Λs= h √ 2πmskBT (1.5)

It is so possible to define the effective potential W (R) through the equation

Z = e−βF0 _{= A}

p

Z

d(Np) e−βW (R), (1.6)

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Boltzman factor of Upp(R) + Ups(R, r) is made over all configurations of

the solvent molecules, and all with the colloidal particles fixed, makes the effective potential result in

W (R) = −kBT ln

D

e−β[Upp(R)+Ups(R,r)]

E

(1.7)

It’s worth noticing that the effective potential yields a full description of the structural properties of a suspension, but only in static condition. For it to be valid also in dynamic conditions, we have to make the assumption that the solvent restructures very quickly, in response to any changes of the colloidal particle configuration. Furthermore it is also possible, as first approximation, to consider W (R) as a sum of pair potentials, involving only two particles: something like W (R) =P

i6=jwij(Ri− Rj).

1.3 Effective potential for colloidal solutions

Since a colloidal particle is composed by many molecules, we can reasonably assert that Upp is the result of all the molecular interaction contributions.

This means that the amplitude of the effective interaction potential is much bigger than the single molecule’s one. Nevertheless it is possible to demonstrate how the kinetic energy is in the order of kBT [6]. These

considerations bring to important consequences on the behaviour of colloidal suspensions. In fact, in the case of molecules we typically have kBT ≈ |W0|

where W0 is the minimum of the molecular interaction potential. In

the case of colloidal particles, in the same temperature conditions, the quantity kBT is strictly less then |W0|. All this analysis can be more

clearly represented in Fig. 1.1a and 1.1b, where the former reproduce the situation in which the colloidal interaction potential is only attractive, excluding the region R < r0 where the impossibility of compenetration

rules. Since, as we said, the thermal energy is much smaller than W0, it

will only be sufficient to make the particles oscillate around the minimum, and the particles substantially stick together irreversibly. In order to avoid this it is possible to transform the potential shape, and make it similar to the one illustrated in Fig. 1.1b, where a relative maximum is present at rmax > rmin. If ∆E > kBT , where ∆E = W (rmax) − W (r → ∞), then

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Figure 1.1: a) qualitative trend for a Van der Waals potential for molecules; b) potential for colloidal particles, note the presence of a maximum that stabilezes the particles at bigger distance. Source: Ref. [7]

the potential barrier, with amplitude ∆E created by the maximum. In this way the two particles do not stick irreversibly together. This kind of suspension is defined as stable. Such a condition can be obtained by using particles with a repulsive term, present for example in electrostically charged particles, or in solution in which surfactants or polymers are added

1_{. Remebering what we said at the beginning of this section, that is that the}

interaction term Uppin the potential 1.1 is the result of all the contributions

of the molecules constituting the particle, we can also assert that, besides the intensity, the range is modified too. Expressing the potential in the form:

u(r) = −C

rn (1.8)

where C is an arbitrary constant, and assuming a couple interaction as first approximation, we can calculate the interaction energy U between the particle and an infinite half-space, as

U (d) = −2πρC Z ∞ d dz Z ∞ 0 dR R (z2_{+ R}2₎n/2 = 2πρC (n − 2)(n − 3) 1 dn−3 (1.9)

where ρ is the density of moleules in the half-space. The energy expressed in Eq. 1.12 is the sum of all the terms associated to the interaction between

1

The stabilization through adding surfactants can be obtained if the concentration is not enough to generate depletion interactions (see Sec. 1.5)

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each molecul and the half-space. Such an integral can be solved considering that all the molecules on a plane of the half-space distant z from the single molecule and disposed on a circumference with radius R, centered on the line joining the particle and the plane, are distant r = √R2_{+ z}2_{. The}

effect of the collective interactions thus makes the interaction range to decay more slowly. This result can be also extended to the case of two infinite half-spaces of molecules separated by a distance d and each with an uniform density equal to ρ1 and ρ2. Integrating on all the volume of a

cylinder with base equal to ∆S with its axis parallel to the perpendicular line of the plan of the half-space, the energy is:

U (d) = − 2ρ1ρ2C∆S (n − 2)(n − 3) Z ∞ 0 dz (z + d)n−3 = 2ρ1ρ2C∆S (n − 2)(n − 3)(n − 4) 1 dn−4 (1.10)

and since E(d) = U (d)/∆S the energy per unit surface can be expressed as

E(d) = − 2ρ1ρ2C (n − 2)(n − 3)(n − 4)

1

dn−4. (1.11)

Such relation allows to use the Derjaguin approximation for the force between to particles. Derjaguin approximation, valid when the range of interaction dint is much smaller than R1 and R2, considers the couple of

particles interacting as two half-spaces. Since the necessary work to bring the particle to distance d from infinite is proportional to d, the force of interaction between two particles is

F (d) = 2π R1R2 R1+ R2

E(d). (1.12)

and the spatial trend is the same, namely d−(n−4). Since the interaction range of a typical colloidal particle, although much bigger than the molecular one, is however quite small with respect to its radius, such approximation is generally valid.

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Figure 1.2: Derjaguin approximation for big spheres at small distance is valid when d R1, R2. Source: Ref [8]

1.4 Van der Waals interactions

Considering an atom from a classic point of view, it has dielectric dipole moment equal to zero and thus we expect no interactions characterized by an electrostatic or electromagnetic nature. Similarly, quantum mechanics reaches the same result, but also demonstrates how the quadratic mean value is different from zero. This purely quantistic result is the origin of the Van der Waals force, and it is possible to consider the atom as having a dielectric dipole instantaneously assuming a value of the order of ea0 where

e is the electron charge and a0 is the Bohr radius

a0=

e2

8πε0~ω (1.13)

where ~ω is the hydrogen ionization energy (Rydberg energy). At distance r from such dipole, an other atom feels an electric field

E(r) = p1 4πε0r3

= ea0 4πε0r3

. (1.14)

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allowing to write the energy of interaction between these two dipole forces as u(r) = p1p2 4πε0r3 = αe 2_a2 0 (4πε0)2r6 = ~ω 8π2_ε2 0 α2 r6 (1.15)

where α = 4πε0a3 is the vacuum polarizability of the hydrogen atom. For

two different atoms the interaction energy in Eq. 1.15 can be generalized as

uAB = −

3αAαBIAIB

4(IA+ IB)

r−6 (1.16)

where we indicated the respective ionization energies of the two atoms A and B as IA = ~ωA and IB = ~ωB. Since we considered only a single

oscillating frequency, neglecting all the possible atomic transitions, the previous model results highly simplified2. The usefulness of this approach can be found in the results of the model for what concernes the nature of these interactions, beacause Van der Waals forces show the typical trend proportional to r−6. The order of magnitude of the thermal energy kBT for such interaction is recovered at distance ∼ 0.1 nm. Using the

Derjaguin approximation, that has been introduced in the previous section, and assuming that the interaction between two atoms or molecules is not affected by the presence of a third one, one can express the force, and then the interaction energy, between two macroscopic particles as

U12= − A12 π2 Z V1 dV Z V2 dV0 1 |r − r0_|6 (1.17)

where ρ1 , ρ2, V1 and V2 are the densities and volumes of the particles and

A12 = π2C12ρ1ρ2 is the Hamaker constant [9]. It is important to observe

that if r σ, the result of the integral is:

UA(r) = − A12 36 σ r 6 (1.18)

where σ is the diameter of the particles. This expression shows how the

2_{Through the perturbation theory these contributions can be taken into account, but}

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asymptotic behavior for large distances is still r−6. Whereas, when the surfaces separation d = r − σ is in the order of σ, such expression becomes:

UA(d) ≈ −

A12 σ

24 d (1.19)

which results in agreement with the Derjaguin approximation. It is however important to take into account the presence of a medium, beacause until now the model assumes the force to act in vacuum. Nevertheless considering a solvent in which the particles are immersed, through the Lifshitz theory of Van der Waals force [10] it is possible to reach the conclusion that the presence of a medium only affects the Hamaker constant and not the dependence on distance, which remains proportional to r−6.

1.5 Depletion interactions

One of the most peculiar phenomena appearing in colloidal solution are depletion interactions. They are not associated to the presence of the colloidal particles alone, but they rise from the presence of another component in the solution, sized in the colloidal range [11]. These other particles should be much larger than simple molecules, but still rather small compared to the colloidal particles we are considering. The candidates for this range of sizes are polymers (tens of nanometers), surfactants (few nanometers) or proteins. The most investigated situation, used to describe depletion interaction, is the mixture with polymers, supposing them to not adsorb to the colloid particle surface. An observed behaviour of these particles is the induction of attractive forces between particles, which strenght depends on both the polymer size and concentration, and give rise to rich phase characteristics of the colloidal dispersion.

Considering the schematic drawing in the Fig. 1.3, where the big particles are the colloids and the polymer coils are represented as the smaller spherical objects. In order to get a quantitative description, it is uefull to make some simplifying hypotheses:

• Colloidal particles interact as hard spheres. That is, the direct interparticle potential Upp is just due to excluded volume.

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Figure 1.3: The colloidal particles have radius R and the polymer coils are represented as spherical objects of radius r < R. The overlapping region Vol makes impossible for the polymers to stay between the colloids, thus arising the resulting attraction of the two. Source: Ref [8]

• Polymers do not interact at all with themselves, meaning that if there were only polymers, they would behave as an ideal suspension. • However, the polymer coils do interact with the colloidal particles,

behaving as hard spheres of radius r.

Associating to each colloidal particle a spherical shell with internal radius R and external radius R + r where the “small” ones cannot fully enter, because of their dimensions. Such a zone will be called the “depletion zone”. The polymer solution has an osmotic pressure Π = ρpolkBT ,

where ρpol is the polymer number density. Such an isotropic pressure

will be applied to any direction, but, if the center-to-center distance of two colloidal particles is less than 2(R + r), it means that their depletion zones overlap, and there is a region Vol in the gap between the

two colloidal particles where no polymer coils can enter. The osmotic pressure will result unbalanced, and the two particles are pushed against each other. This resulting attractive interaction can be modeled as a potential that is the integral of this force up to the particle distance d of the force produced by the osmotic pressure Π on the projected area A of the overlap volume perpendicular to the line connecting the particle centers.

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U (d) =          ∞ if d < 2R −ΠV_ol(d) if 2R < d < 2(R + r) 0 if d > 2(R + r) (1.20)

The last uncovered task is evaluating the overlap volume as a function of the inter-particle distance D, which, introducing the size ratio δ = r/R, can be written as:

Vol(d) = vp(1 + δ)3 1 −3 4 δ 1 + δ + 1 16R3 d3 (1 + δ)3 (1.21)

This effective potential, generated by the presence of the polymer, can be tuned by varying the radius r of the polymer itself, whereas a change in the density ρpol changes Π. Moreover it is worth noticing that the range of

such potential is generally smaller than the radius of the colloidal particles. Depletion effect then causes short-range attractive potentials that are not usually found in molecular systems. This kind of attraction can also be seen as the maximization of entropy for the polymer particles: since the number of polymers is much larger than the particles, we can consider the system as dominated by the entropy of the polymer. Being presence of the colloidal particles a limitation for the available volume to the polymer, the latter will tend to stick the colloidal particle together, so that their depletion zones overlap, and the total excluded volumes decreases, allowing the polymer coils to gain possible states, and then to increase the entropy of the system.

1.5.1 Phase Diagram and gelation driven by depletion

interaction

As said before, depletion forces produce a rich variety of phase diagrams. Because of the attractive short range potential their phase diagrams show the coexistence of a “colloidal liquid” and a “colloidal gas”. All the characteristics of each diagram depend on the size ratio δ = r/R between the small and the large particles. The phase diagrams in Fig. 1.4 are shown for different values of δ. Note that φr_dep is the depletant volume fraction considering a reservoir which is in equilibrium with the sample through an osmotic membrane. We will refer to three particular values of δ as example

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Figure 1.4: From left to right the phase diagram for δ = 0.6, δ = 0.4, δ = 0.1 are showed. Φp is the particle volume fraction. F, S, L and G indicate respetively the state of fluid, solid, liquid and gas. cp is the critical point, tp is the triple point. Source: Ref [12].

to display different peculiarities in the phase diagrams.

δ = 0.6 Referring to the left figure in Fig. 1.4 we can notice a similarity between the phase diagram and the P - T diagram for a common molecular substance. When the temperature is low, only a single fluid F and a solid phase S can exist. Above the critical temperature a G/L phase separation is osbserved. Above the triple point, where gas, liquid and solid states coexist, we can only have G + S.

δ = 0.4 Referring for the central diagram in Fig. 1.4, for such values of δ the G + L coexistence region disappears, becoming metastable and sinking into the F + S region. This means that the system does not presents a triple point and it appears in a single stable fluid phase. Since the process of nucleation of the crystal is very hard in the presence of a short-range attraction, it is possible to enter the fluid/solid coexistence region without nucleating the crystal, thus making possible to distinguish a G + L region of coexistence near the critical point, caused by critical fluctuations. Such fluctuations have been shown to increase crystal nucleation when associated to a metastable G/L transition. It is thermodynamic instability that rules this phenomenon and brings to a phase separation. However this separation is characterized by a fractal structure which permeates the two phases that can successively separate because of gravity, if there is an appreciable difference in density between the solvent and the

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particles.

δ = 0.1 Always in Fig. 1.4, on the right, we now find the diagram for lower values of δ. Here the G + L region sinks even further and becomes flat. The peculiarity of this situation can be found in the process of gelification, that happens when, entering the metastable coexistence, G and L begin to separate, but when the particles generate the fractal structure, are no longer free to move because of the short range of interaction and they remain in a “frozen” disordered state, which is called fractal gel (or an “attractive” glass). Arresting the phase separation, what actually happens is a sol-gel transition [13]. 1.5.2 Gelation driven by critical Casimir forces

The Casimir effect is a well-known effect in quantum physics. If the quantum fluctuations of the electromagnetic field are confined between two parallel conducting and uncharged surfaces, the boundary conditions imply a reduction of the electromagnetic fluctuation spectrum between the plates, resulting in the attraction of the two plates. An analogue effect for binary fluids was first conceived by Fisher and de Gennes, when they proved that an attractive force arises, due to the confinement of concentration fluctuations having range of the order of the correlation length of the liquid [14]. This effect becomes significant near the critical point of the solvent mixture, where the solvent correlation length diverges and becomes comparable to the surfaces separation distance. Specifically, when colloidal particles are dispersed in a binary liquid mixture displaying a miscibility gap, a reversible aggregation is exhibited when the system is close to the mixture coexistence curve. These effects have been ascribed to the confinement of concentration fluctuations within the thin fluid film separating the surfaces of two particles, yielding an effective short range interaction that is similar to the critical Casimir effect [15] [16]. In the critical Casimir effect, boundary conditions are set by the surface adsorbing properties that are controlled by adding surfactants, making this force tunable and usable for practical applications [17].

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1.6 Gels

Gels, in their most general meaning, are systems in which a small quantity of solid component, that we shall call gelling agent, is dispersed in a fluid phase, and determines the phase of the entire system. Gels are dilute cross-linked system, exhibiting no flow when in the steady-state. It is the crosslinking within the fluid that gives a gel its three-dimensional structure that spans the volume of the medium and entraps the liquid. Thanks to different kind of interaction, solid component is actually able to create a percolating structure that crosses the entire solution, bestowing solidity to the system, even though the system itself is mainly composed by liquid. This process is known with the name of gelation, it can be induced by peculiar condition of temperature, pH or salt concentration. In the absence of perturbing forces like gravity that make large clusters sediment fast to the bottom of the cell, the cluster size grows in time until a single “percolating” cluster fills the whole volume. What forms is then a hydrogel, which is a disorder but solid structure, although it may contain only little solid material (the particle volume fraction can be less than 1% ). If the solvent is then extracted via special methods (such as a “supercritical extraction”), we are then left with a tenuous solid, mostly made of air, but which can be still very rigid and has a huge internal surface: these materials, which are called aerogels, are the lightest solids ever made and have many technological applications. Gelation, then, consists in creation of a network by cross-links. This internal network structure may result from physical bonds (physical gels) or chemical bonds (chemical gels). The main difference between the former and the latter is that, in chemical gels, the links between the gelling agent particles are characterized by covalent bonds having typical energy greater then kBT .

This implies that, unless providing external energy, the phenomenon is irreversible. Physical gels, on the opposite, are characterized by reversible cross-links, where the typical energy is at the order of the thermal energy. The fact that, in physical gels, the particles are not frozen in their position, implies that the gel is not in a specific configuration that does not evolve in time. This peculiar property of physical gels allows us to study for example the trend of the time correlation of the system structure.

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1.7 Brownian motion

This section will discuss the peculiar aspects of this kind of motion, its fundamental theoretical models, and in particular the deductible information from the experimental setup that represented one of the purposes of this thesis. Brownian motion, named from its discoverer Robert Brown, is defined as the random movement of a particle in a fluid caused by the chaotic collisions between such particle and the surrounding solvent molecules. The particle is hitted from all directions and therefore, the time-averaged momentum transfer h∆pi = 0. However, at each instant ∆p(t) is a non-zero fluctuating quantity, so the particle trajectory can be viewed in simplified statistical description which is often called a Random Walk (RW). Considering the simplest one dimensional model of a discrete motion, it can be demonstrated [18] that the probability P (k; N, p) to make k steps along the positive direction of the X axis on a total of N (then N − k steps to the left, each one with probability 1 − p) is a binomial distribution

P (k; N, p) =N k

pk(1 − p)N −k (1.22)

In the limit N → ∞, the discrete values of the binomial distribution in Eq. 1.22 tends to a continuous Gaussian distribution:

p(k) = 1 σ√2πexp −(k − hki) 2 2σ2 (1.23)

with expectation value hki. The variance σ for the position is equal to those of the binomial, namely:

σx= 2Lσk= 2L

p

p(1 − p)N (1.24)

The convergence is particularly fast for p = 1/2, but in any case this limit distribution is reached for any p if N is large enough. Evaluating this situation in time, calling τ the time of a single step so that the number of steps in a time t is N = τ /τ and the distribution of positions at time t can be written in terms of the diffusion coefficient D = hx2i/2t, as

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p(x, t) = 1 2√πDtexp − x 2 4Dt (1.25)

This mathematical model of the brownian motion is then useful to describe the diffusion processes and can be extended to a macroscopic approach for mass diffusion. Considering a large number N of particles, the fraction of them that is found between x and x + dx at time t is:

n(x, t)dx = N P (x, t) (1.26)

and it is ruled by the generalized diffusion equation ∂n(x, t) ∂t = (1 − 2p) L τ ∂n(x, t) ∂x + D ∂2n(x, t) ∂x2 . (1.27)

Note that the first term on the left side gives the sign to the direction of the progressive drift, which is positive for p > 1/2 and negative for p < 1/2. For this reason (1 − 2p)L/τ can be defined as a “drift speed” of the particle under the action of an external force. Taking the example of gravity, one can investigate the process of sedimentation, argument of the next section, under alternative methods, but we will mostly focus on the perspective that contemplates gravity as an osmotic membrane.

1.8 Colloid sedimentation

Particles settling in solution has an important role in many physical, geological and environmental processes, just think about forced sedimentation, through centrifugation, which has various applications as a separation tool in chemical industry, or sample analysis in biology and medicine. Furthermore sedimentation had also revealed as one of the most usefull means for developping statistical mechanics. Althoug the similarity with a gas subjected to gravity, colloidal dispersions show substantial differencies in term of gravitational lenghts, due to the enormous difference of mass between particles and solvent, so with the time they settle under their own weight to the bottom of the container.

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Taking a spherical particle of radius a, made of a material with density d, bigger than the density of the solvent, it will be subjected to the weight mg = dvpg, where vp is the volume of the particle that is responsible

for the upward Archimede’s force mg = dfvpg that limits it settling speed to

vs= Fg f = 9∆dg 2η a 2 _(1.28)

where Fg is the net gravity force on the particle resulting as

Fg = (d − df)vgg =

4 3π∆da

3 _(1.29)

and f is a friction coefficient that, for a sphere, is f = 6πηa, where η is the coefficient of (dynamic) viscosity. The velocity vsis called Stokes speed and

in its expression it is immediately evindent how large and dense particles sediment faster. Another possibility to look at the settling of a colloid is to consider the gravitational force as a way to “confine” the particles close to the cell bottom and then what opposes to this tendency is the concentration of the particles, whose profile, acting as an osmotic membrane, generates a force that pushes the particles up against gravity. When this “osmotic force” becomes so large that it exactly balances gravity, a sedimentation equilibrium is reached. In order to find a valid criterium to state that a particle may be regarded as colloidal we can consider the probability that a particle is found at a height z from the cell bottom as the Boltzmann factor of the gravitational energy, P (z) = exp(−βm∗g), where m∗ = (4/3)π∆dg is an effective mass obtained considering the Archimede’s force. The particle volume fraction profiles is then

Φ(z) = Φ0exp(−z/lg) (1.30)

where Φ0 is the suspension volume fraction at the cell bottom, and

lg = kBT /m∗g (1.31)

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to be regarded as colloidal is having lg a. With simple calculations [19]

is possible to demonstrate that if we able to measure the local particle concentration in a sedimentation process at equilibrium, we can also obtain the local value of the osmotic pressure by a numerical integration of the experimental profile, namely

Π(z) = ∆dg Z ∞

z

Φ(z0)dz0 (1.32)

and having both φ(z) we get Π(φ), that is the full equation of state of the colloidal suspension.

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Chapter 2

Optical Techniques

In this chapter all the fundamental of the optical techniques used for the signal generation and measurements will be discussed in detail. In first sections the importance of the concept of coherence and all the physical quantities related to it will be treated. After that we will linger on the technique called Dynamic Light Scattering [20] as the ground to develop further variation such as Time Resolved Correlation, useful to get temporal information about the evolution of the concerned quanties, or Photon Correlation Imaging, that allows to reach spatial resolution of the sample dynamics, being a technique working in Near Field. Nevertheless, it is appropriate to start from the principles of scattering events, in order to fully understand why the former techniques represent a powerful intrument for colloid science.

2.1 Temporal and spatial coherence

All we said about the interaction between radiation and scattering particles we want to study, must be better scrutinized, taking in consideration the properties of light source and the peculiar characteristics of the scattered and detected light. In particular we have to debate about coherence. In general with coherence we mean all properties of the correlation between physical quantities of a single wave, several waves or wave packets. Temporal coherence describes the correlation between waves observed at different moments in time. Spatial coherence describes the correlation (or predictable relationship) between waves at different points in space, either lateral or

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longitudinal. In quantum optics, in order to characterize the coherence properties, together with statistical properties of an electromagnetic source, correlation functions are employed [21]. The degree of coherence is the normalized correlation of electric fields. It will be deeply explained that the term g1(τ ), it is useful for quantifying the coherence between two

electric fields, in the same way Michelson intended it in his linear optical interferometer. Whereas g2(τ ), typically is used to find the statistical

character of intensity fluctuations. Since g1(τ ) refers to the amplitude

correlation and g2(τ ) to intensity correlation, the latter will be more

useful, considering the technological limitation with respect to the physical properties of the phenomena we shall investigate.

2.1.1 Temporal Coherence

Considering a generally time-varying real field uR(t) and exploiting the Fourier TransformF [uR] = ˜uR(ω) one can discuss temporal fluctuation in frequency domain, where it is basically related to non-monochromaticity. Introducing the associated analytical signal:

u(t) = 1 π Z ∞ 0 dω ˜uR(ω) e−iωt= 2 Z ∞ 0 dν ˜uR(ν) e−i2πν, (2.1)

which is then a complex quantity obtained by suppressing the negative frequency components of uR(t) and doubling the amplitude of the positive ones. For a narrow-band signal, having a spectrum centered on ω0 of

width ∆ω ω0 , we can write uR(t) = A(t) cos[ω0t − φ(t)], hence

u(t) = U (t)e−iω0t_{, where U (t) = A(t)e}iφ(t) _{is called the complex envelope.}

It is well known that any signal with finite bandwidth must display temporal fluctuations. In other words this envelope U (t) of a signal does not significantly change in time on time scales much shorter than a time τc = 2π/∆ω, that we will call coherence time, and to which we can

associate a coherence length lc = cτc. For this purpose, let us introduce

the time correlation function of the analytic signal, or self-coherence function

Γ(τ ) = hu∗(t)u(t + τ )it, (2.2)

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not depend explicitly on t (the process is stationary). Normalizing Γ(τ ) to is initial value Γ(0) = h|u(t)|2it = I, we get the coherence of first order

(usually simply named “field correlation function”)

g1(τ ) =

hu∗(t)u(t + τ )it

I (2.3)

The Wiener-Kintchine (WK) theorem, provides the fundamental link between the time and frequency description stating that Γ(τ ) and Pu(ω) are

Fourier transform pairs, where Pu(ω) is defined as the power spectral density

P_uR(ω) = lim T →∞ 1 T Z T −T dt uR(t) eiωt (2.4)

From the definition 2.1 it can be easily shown that the power spectrum Pu(ω) of the complex analytic signal is just 4PuR(ω) for ω ≥ 0 , and 0

otherwise. The following form of the WK theorem    P (ω) =F [g1(τ )] = R∞ −∞dτ g1(τ ) eiωτ g1(τ ) =F−1[P (ω)] = R∞ −∞dω P (ω) e −iωτ (2.5)

will be particularly usefll for our purposes. The degree of temporal coherence is strongly related to the signals detected in classical interferometric measurements, such as those obtained with a Michelson interferometer [21]. Qualitatively, the beams propagating in the two arms of the interferometer can interfere only if the difference ∆l between the optical paths is smaller than the coherence length of the source lc . Quantitatively, one finds that

the time dependence of the detected intensity is given by

I = I0{1 + Re[g1(∆t)]} (2.6)

with ∆t = ∆l/c, which is then proportional to the real part of the time correlation function, evaluated at the delay ∆t. In order to treat about a medium containing scatterers we can briefly discuss the temporal properties of a narrowband thermal source, described as a collection of many microscopic independent emitters radiating at the same frequency ω0. We

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can also consider collisions between these emitters as the reason of abrupt phase jumps.

As an important example for what follows, we briefly describe the temporal properties of a narrowband thermal source, defined as a collection of many microscopic independent emitters, such as a collection of thermally excited atoms, all radiating at the same frequency ω0 , but undergoing

collisions that induce abrupt phase jumps. With a large number N of identical emitters, we have therefore a joint Gaussian statistics that can be written as: pA(A) = A σ2exp − A 2 2σ2 (A ≥ 0) (2.7)

Since a photodetector does not respond to the instantaneous optical intensity associated to the signal, because the oscillations have frequencies too high for common technology, what we get measuring is the value averaged over many optical cycles that, for a narrowband signal, is Irad = (ε0c/2)A2, where ε0 is the vacuum permittivity and c the speed of

light. So changing variable, we get

PI(I) = 1 2σ2exp − I 2σ2 = 1 hIiexp − I hIi (2.8)

Where I1 is the intensity and it is possible to see that it has an exponential probability density, with a decay constant given by its average value hIi.

These kinds of probability distribution for the field and intensity apply to a gaussian distribution characterizing any “random” optical source. However, the spectrum and the time–correlation function depend on the physical origin of the frequency broadening. Indeed, for independent emitters, we have hui(0)uj(t)i = 0 for i 6= j. Hence:

Γ(τ ) = hU∗(0)U (τ )i =

N

X

i=1

hui(0)ui(τ )i = N hu(0)u(τ )i (2.9)

The field correlation function of the system coincides therefore with the

1_{Following a common convention, rather than the “radiometric” intensity I}

rad , we

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correlation function for a single emitter, g1(τ ) ≡ gi1(τ ), which is determined

by a specific physical mechanism.

We can conclude that if each wave is modelled by a vector, then it can be seen that if a number of vectors with random angles are added together, the length of the resulting vector can be anything from zero to the sum of the individual vector lengths, a 2-dimensional random walk. In the limit of many interfering waves the distribution of intensities (which go as the square of the vector’s length) becomes exponential and that these probability distributions for the field and intensity apply for instance to a spectral lamp, but also, as we shall see, to a medium containing scatterers. As a matter of fact, a gaussian distribution for the field characterizes any “random” optical source. However, the spectrum and the time–correlation function depend on the physical origin of the frequency broadening.

2.1.2 Spatial Coherence

Suppose an electromagnetic monocromatic source, with extension D, made of independent emitters (so that it’s spatially δ-correlated), and consider the field thus produced in two differen points, P1 and P2 distant d from

each other, both placed at distance z from the source. We’ll now exploit the case in which z is really big or really small with respect to other lenghts in the system. If points P1 and P2 are really close (i.e. d z), all the

points of the source basically are almost at the same distance: considering the two extremes U and V , whose path difference is maximum, we have rU ' rV and r0_U ' r0_V. This implies that the phase difference of the field on

P1 and P2 is so small that any intensity or phase fluctuation of the source

(for example of thermal nature, or in our case due to the movement of the emitters) will entails in strongly correlated fluctuations of fields in P1 and

P2. This is the case in which the lack of coherence of the sigle emitters

that compose the source results as irrelevant on field coherence, provided that the path difference between them is negligible. In quantitative terms, it is possible to determine the limits of these path difference so to link the wavelength of the source to its dimensions, obtaining a realtion for the coherence area. The fields in P1 and P2 will remain reasonably correlated

until the phase differences of the the various emitter will also remain similar. The difference oh optical path ∆rn between the n-th emitter and

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emitted radiation. Namely

∆rU − ∆rV λ (2.10)

where ∆rU and ∆rV are the lenght previously defined. From this condition,

exploiting the paraxial approximation, namely, for small propagation angles with respect to the optical axis, which is the condition required for the Fresnel approximation in diffraction to hold, in the scheme in the Figure 2.1, we have that

∆rU− ∆rV ' d · ϑ ' d ·

D

z (2.11)

Expliciting this relation with the coherence condition in Eq 2.10 we obtain dD

z λ (2.12)

This result allow to define, for a finite dimension source D, a coherence area Ac' λz D 2 (2.13)

of the source on the observation plane at distance z. The dependence on the distance z let us to define a “coherence cone” with solid angle ∆Ω ' (λ/D)2 which corresponds to an angulare aperture 2α ' λ/D, exalctly what we get from the diffraction pattern of a single slit in Fraunhofer optics [22]. This Far Field optics is infact obatined in conditions of distances z called Fraunhofer region. In further sections we shall see more in details all the possible conditions of distances between source and observation planes and what they involve.

Considering a quasi-monochromatic source, so that all delays in propagation are much shorter than τc , we call mutual intensity the spatial correlation

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Figure 2.1: Coherence propagation of a bidimensional example source. If z is much grater than d, the paraxial approximation is valid.

J12= J (r1, r2) = hu∗(r1, t)u(r2, t)i = hU (r1, t)U (r2, t)i, (2.14)

which, when r1 = r2 = r, becomes just the intensity I(r) in r. The

normalized mutual intensity is called degree of spatial coherence

µ = J (r√1, r2) I1I2

(2.15)

An extremely interesting result about spatial coherence comes from considering how J12 propagates from a given surface, where it is known,

to another surface. The general problem is rather complicated, but it considerably simplifies if the first surface is actually a planar source Σ that can be considered as fully spatially incoherent, by which we mean that, over Σ,

J (ρ1, ρ2) = I(ρ1)δ(ρ1− ρ2) (2.16)

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observation plane further down the propagation axis, one indeed obtains in the paraxial approximation

J (r1, r2) = e−iψ (λz)2 Z Σ dρ I0(ρ) exp i2π λzρ · r , (2.17) where ψ = π[r2

1 + r22]/λz . Hence, apart from a scaling and phase factor,

the mutual intensity is the Fourier transform of the intensity distribution across the source. Eq. 2.17 is the Van Cittert-Zernike (VCZ) theorem, arguably the most important result in statistical optics. By means of the VCZ theorem, it can be shown that the coherence area is quantitatively given by

Ac= (λz)2

R |I(x, y)|2_dxdy

|R I(x, y)dxdy|2 = (λz)2

D hI2_i

hIi2 (2.18)

where D is the area of the source. For an incoherent source with uniform intensity (which may be an incoherently and uniformly illuminated sample) so that hI2i = hIi2, Ac = (λz)

2

D consistently with our qualitative approach.

The coherence area basically yields the size of the speckles produced by a source or a diffuser around each point P on the screen. Since the field in P is a random sum of the contributions coming from all points on the source, which are independent emitters, the total amplitude has a Gaussian statistics. The distribution of the speckle intensity (namely, the distribution of the intensity at different points on the screen) is hence exponential, so there are many more “dark” speckles than “bright” speckles. What is more important, according to the VCZ theorem the “granularity” of the speckle pattern should depend only on the geometry of the source, and not on its physical nature. We shall later see that this is not always necessarily true (for example approaching the source to the observation plane).

Intensity correlation

Scattering techniques used in this thesis project, however, usually probe intensity correlations, which are described by means of the normalized time–correlation function

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g2(τ ) =

hI(t)I(t + τ )it

hI(t)i2 t

= hu

∗_(t)u∗_{(t + τ )u(t + τ )u(t)i} t

hu∗_(t)u(t)i2 t

(2.19)

Note that, for τ → ∞, g2(τ ) → 1, whereas g1(τ ) → 0. While for an ideal

monochromatic source g2(τ ) = 1 for all values of τ , for a random source,

we should evaluate the average value of the product between the intensity and its corresponding delayed term:

hI(t)I(t + τ )i =

N

X

i,j=1

hu∗_i(t)u∗_j(t + τ )ui(t + τ )uj(t)i (2.20)

For a very large number N of emitters, yelds the important Siegert relation [23] :

g2(τ ) = 1 + |g1(τ )|2 (2.21)

Hence, for a random source, g1(τ ) provides all the informations about

correlation and g2(τ ) can be directly obtained from it. Nevertheless, the

asymptotic behavior differences between g2(τ ) and g1(τ ) guarantees, as we

shall see, a crucial advantage for intensity correlation techniques.

2.2 Scattering and speckle field

Models of light scattering can be divided into three domains based on a dimensionless size parameter, α which is defined as:

α = πDb/λ (2.22)

where πDb is the circumference of a particle and λ is the wavelength of

incident radiation. Based on the value of α, these domains are:

α 1 Rayleigh scattering (small particle compared to wavelength of light);

α = 1 Mie scattering (spherical particle about the same size as wavelength of light);

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Rayleigh scattering is a process in which electromagnetic radiation (including light) is scattered by a small spherical volume of variant refractive index, such as a particle, bubble, droplet, or even a density fluctuation. In the Mie regime, the shape of the scattering center becomes much more significant and the theory only applies well to spheres and, with some adaptations, to particular shapes as spheroids and ellipsoids. Since the particles we’re going to study have dimensions between 0.1 − 1µm, using a laser emitting in the green visible light, our reference will be Mie Scattering. When a surface is illuminated by a light wave, according to diffraction theory, each point on an illuminated surface acts as a source of secondary spherical waves. In any point of the space, the field is the sum of waves which have been scattered from each point on the illuminated surface.

2.2.1 Structure factor

For what concerns a typical scattering experiment, an example of geometry is shown in Figure 2.2, where, on the left side is represented the incident radiation with wave-vector ki and the radiation scattered by the sample

is represented with wave-vector ks. The respective frequencies are ωi and

ωs that in general are differnet since the scattering process can cause a

change in terms of energy. In our application we will deal with elastic or quasi-elastic scattering events, so that in the process the total energy of the system is conserved, or, in terms if frequencies ωi ' ωs. Since this

condition directly implies |ki| = |ks|, the scattering vector q is related to

the incident and scattered radiation wave-vectors as q = ki − ks, which

quantifies the transferred momentum ~q. From a pure gemetrical approach is then possible to relate the magnitude of the scattering vector to the scattering angle ϑ by q = 2kisin ϑ 2 (2.23)

where ki = |ki| = 2π/λ, and λ is the wavelenght of the radiation in the

scattering medium, usually defined as the ratio between the wavelegnth in vaccum and the refractive index of the material.

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Figure 2.2: Geometry of a scattering experiment. In the small figure on the left, the phase delay generated by a scattering event is showed. Source: Ref. [19]

Each one of the enormous number of scatterers that compose the inspected sample, once stimulated by the incident radiation, generates a spherical wave, whose position rj is measured from a arbitrary origin. In a

colloidal dispersion each particle behaves as a radiating dipole that emittes what we call the scattered field Es. This scattered field, can be written as

Ej = E0eiki·rj

eiks·(R−rj)

|R − rj|

(2.24)

where R = |R| is the distance in which the field is revealed, along the direction ks, the index j just denote that the field is the one emitted by

the particle placed in rj. The first exponential term is the phase delay of

the incident field E0 in rj (and ki· rj is the segment AP in the Figure 2.2).

The second exponential is the spherical wave centered in rj, where |R − rj|

is equal to the segment OB. With the condition of having R rj we can

obtain Ej(q) ' E0ei(ki−ks)·rj eiksR R = E0e iq·rje iksR R (2.25)

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which is a spherical wave centered in the origin, for which the scattering wave-vector q and the position rj of the single scatterer modulates, through

a pure phase factor, the amplitude of the field. Summing the contributions of all the scatterers along the direction of ks we got the total field

Es(q) = E0 eiksR R X j eiq·rj _(2.26)

Since the amplitude is impossible to measure with any optical detector, what is actually revealed is the intensity Is(q) ∝ |Es(q)|2 resulting from

the average over a huge number of oscillations. If the sample is not a gas of point-like molecules, particularly if we manage with solutions or gels, it is possible to show that the scattered intensity is directly related to the component at wave-vector q of the Fourier transform of g(r) with the expression Is(q) = I0N 1 + n Z d3r g(r)exp(iq · r) (2.27)

where the second term contributes only for q = 0. Subracting this contribution using h(r) = g(r) − 1 we can obtain a really important quantity for investigating the structure of materials: the structure factor

S(q) = Is(q) Isig

= 1 + n Z

d3r h(r)exp(iq · r) (2.28)

where Isig is the intensity scattered by an ideal gas. With scattering

experiment is then possible, passing through the structure factor, to obtain the pair distribution function exploiting a Fourier transform of the sum on Dirac δ given by Is(k) = I0 Z d3r eik·r X i,j δ[r − (rj− ri)] (2.29)

valid for an ergodic system [19], that is a system in which the temporal average coincides with the average made in all the possible configurations.

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2.2.2 What is a speckle?

Objects illuminated by light from a highly coherent cw laser are readily observed to acquire a peculiar granular appearance. A speckle pattern is an intensity pattern produced by the mutual interference of a set of wavefronts [24]. The speckle effect is a result of the interference of many waves of the same frequency, having different phases and amplitudes, which add together to give a resultant wave whose amplitude, and therefore intensity, varies randomly. It appears chaotic and unordered, and is best described quantitatively by the methods of probability and statistics. Thus the speckle pattern consists of a multitude of bright spots where the interference has been highly constructive, dark spots where the interference has been highly destructive, and irradiance levels in between these extremes. Suppose we illuminate with a laser beam a light diffuser, for instance a window made of ground glass: then, a complex figure made of many irregular spots forms on a screen placed beyond the diffuser, which is what we call a speckle pattern. If we insert a lens and enlarge the beam spot on the diffuser, the speckle size reduces. Conversely, if we move the diffuser towards the lens focus plane, the speckle pattern becomes much coarser. Hence, the speckle size depends on the extension of the illuminated region on the diffuser. Furthermore it is worth to say that, considering a detector with a finite area, on each point of such a surface, a value of intensity, depending on how the contributes of each particle diffused field sum with each other, is detected, but the speckle pattern generated has different characteristic depending on the setup chosen for the light path. In particular, as we shall see in further sections, without specific adjustments, each speckle is formed by the contributions of the field coming from all the illuminated area, thus not allowing a spatial resolution of the dynamics.

2.3 Dynamic Light Scattering

The most popular optical correlation technique in colloid science is Dynamic Light Scattering. This kind of scattering allows to investigate on systems for which the variation in frequency of diffused light with respect to the incident one is only due to the motion of the scatterers. For this reason it occurs a phase modulation of the diffused light, corresponding to small

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broadening ∆ω ω of the spectral bandwidth 2. Since we can classically think to variation in frequency to depend on the phase oscillations of each particle’s contribution that is detected by the revealer. In other words, we can say that particles diffuse light always at the same frequency, but with a phase that varies in time due to their motion. The fact that the frequency shift is quite small allows to assume that the scattering process is quasi-elastic, namely that the photon energy is conserved after collisions. Under this hypotesis the wave-vector of the incoming radiation changes direction but not its modulus. This condition bring us to consider temporal correlation of the intensity fluctuations of the diffused light, operating in time domain instead of frequency domain. In the far-field situation of DLS, each particle in the scattering volume contributes to the diffused field: if particles positions are indipendent, each contribute sum with a random phase. From this perspective, the electric field can be written as a random variable whose time dependence is determined by the instantaneous spatial configuration of all the scatterers. At distance R the scattered field is then

Es(q, t) = E0 exp[i(kR − ωt)] R N X j=1 b(q, t)exp[−q · rj] (2.30)

where k is the modulus of the wave-vector (equal for scattered and incident waves), E0 is the incident field at frequency ω, rj indicates the positions

of the single scatterers and b(q, t) is a scattering lenght that is the result of each single diffuser properties. In the sum we have dephasing terms due to different position, changing in time, of the particles. We also know that hE(q, t)i = 0, but hI(q, t)i = h|E(q, t)|2_{i ∝ N . We define an electric field}

normalized temporal correlation function as

g₁c(q, τ ) = hE(q, t)Es(q, t + τ )i

hI(q, t)i (2.31)

If we consider a system composed by independent particles, we can assume that the mixed terms coming from the product of the electric field diffused

2_{Measuring a spectral broadening that is much smaller than the source intrinsic}

bandwidth is of course extremely challenging: as a matter of fact, it is totally out of question for any spectroscopic method relying on field correlations. Yet, things change dramatically if we consider intensity correlations. This is probability easier to see in the time domain.

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by different particles are null. Since we also assumed that the particles are identical, and that the scattering amplitudes do not depend on time, we can write the correlation function as

g₁c(q, τ ) = hE(q, t)Es(q, t + τ )i

|E_s(0)|2 = F (q, τ )e

−iωτ _(2.32)

where we defined the intermediate scattering function (ISF) as

F (q, τ ) = X i,j e−iq·[ri(0)−rj(τ )] (2.33)

which is nothing but the FT (in frequency) of the dynamic structure factor S(q, ω). So, neglecting interactions, amounts, of course, to assume that the position of different particles are uncorrelated, g1(q, τ ) can be written as

gc₁(q, τ ) = hexp{−iq · [r(0) − r(τ )]}i = hexp{−iq · ∆r}i (2.34)

where ∆r(τ ) = r(τ ) − r(0). If we neglect fluctuations due to a variation in the number of scatterers N , but considering the temporal dependence of their diffused field because of particles relative position changes, we can visualize them as fluctuation with respect to an average value, whose dynamics depends on time. We shall also consider that the source field, even if coming from a stabilized laser, is subjected to temporal fluctuations. We can then introduce a normalized temporal correlation function for the impinging field gs₁(q, τ ). Since amplitude fluctuations coming from source and those due to particle motion are completely decorrelated, we can write temporal correlation revealed by the detector as

g1(q, τ ) = gc1(q, τ )g1s(q, τ ) (2.35)

Since g₁s(τ ) decays to zero on the correlation time τc of the source, which is

far shorter than the Brownian correlation time, there is no way to follow the decay of g₁c . Consider however the intensity correlation function. Again, we can write

(50)

g2(q, τ ) = g2c(q, τ )g2s(q, τ ) (2.36)

Yet, in this case, for τ τc , g2s(q, τ ) decays to one, and we have:

g2(q, τ ) −−−→ tτc

g₂c(q, τ ) (2.37)

which is exactly what we want to measure. In other words, we actually want to avoid using a source with a very long coherence time, for we need τc

to be much shorter than the physical fluctuation time of the sample. Let’s now consider a situation in which the characteristic detection time is much less than the characteristic time of the dynamics of our sample, explicitely the necessary time for the sample to change configuration. In this way it is like we can take many picture of the motion of the emitters, thus making possible to assert that each speckle is generated by a specific configuration of the particles, and not by an average on many sequential configuration. For ergodic systems, however, in a time much longer than the characteristic time of the motion, the configurations explored by the system will be enough to consider the temporal average to coincide to the the average on the configurations. As previously mentioned, choosing a value for the scattering vector k implies to get a defined investigation spatial scale. This means that the speckle at time ti will be correlated with the speckle in the

same position, at time tj only if in ∆ti,j, the emitters, that are sources of

the light formnig those speckles, moved of a distance comparable to 2πk−1. It is then possible to obtain informations on the dynamics that characterizes the sample on spatial scales of the order of 2πk−1 simply calculating the autocorrelation index of each speckle in times t an t + τ , where we mean τ as the correlation delay. We have then available an index of correlation, depending on the delay τ , for each instant of the carried out measurement. The index concerned is g2(τ ) expressed in the form of Eq 2.42. The goal

of DLS is to obtain such a function. In case of purely Brownian motion of the particles dispersed in solution, it is possible to demonstrate [25][1] that

g2(τ ) − 1 = e−Dk

2_τ