optical methods for the investigation of application

POLITECNICO DI MILANO

Dipartimento di Ingegneria Nucleare

Optical methods for the investigation of

application-oriented complex fluids

Emanuele Vignati

Ph.D. Thesis

Radiation Science & Technology – XVII Course

Tutor: Prof. Roberto Piazza

Supervisor: Prof. Roberto Piazza

Coordinator: Prof. Marzio Marseguerra

2002—2005

Coordinator: Prof. Marzio Mareseguerra

Supervisor: Prof. Roberto Piazza

Tutor: Prof. Roberto Piazza

Ph.D. Student: Emanuele Vignati

The road goes ever on and onDown from the door where it began.

Now far ahead the Road has gone,And I must follow, if I can,

pursuing it with eager feet,Until it joins some larger way

Where many paths and errands meet.And whither then? I cannot say.

J.R.R. Tolkien

Contents

1 Abstract 9

2 Optical microscopy 13

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2.2 Image formation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2.3 Fluorescence microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.4 Phase contrast microscopy . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.5 Polarised light microscopy . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.6 Differential interference contrast . . . . . . . . . . . . . . . . . . . . . . 28

2.7 Digital Imaging . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35

3 Light Scattering Techniques 36

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.2 Fluctuation Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Particulate Approach . . . . . . . . . . . . . . . . . . . . . . . . . . . . 39

3.3.1 Rayleigh scattering . . . . . . . . . . . . . . . . . . . . . . . . . 40

3.3.2 Rayleigh-Debye-Gans scattering . . . . . . . . . . . . . . . . . . 41

3.3.3 Mie Scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.4 Scattering from interacting particles . . . . . . . . . . . . . . . . . . . . 45

3.5 Scattering by fractal aggregates . . . . . . . . . . . . . . . . . . . . . . . 46

3.6 Statistics of the Scattered Field . . . . . . . . . . . . . . . . . . . . . . . 48

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

4 Interfacial tension apparatus 52

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

4.2 Young and Laplace’s equation . . . . . . . . . . . . . . . . . . . . . . . 53

4.3 Interfacial tension and measurement method . . . . . . . . . . . . . . . 54

4.4 Micropipettes Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

4.5 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.6 Calibrating measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.7 Surfactant adsorption . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

5 Medium angle light scattering apparatus 64

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

5.2 Experimental setup . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 66

7

CONTENTS CONTENTS

5.3 Calibration measurements . . . . . . . . . . . . . . . . . . . . . . . . . . 735.4 Scattering measurements on polystyrene particles . . . . . . . . . . . . 76References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79

6 Pickering Emulsions 806.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 816.2 Solids-stabilised emulsions . . . . . . . . . . . . . . . . . . . . . . . . . . 826.3 Preparation of silica colloids . . . . . . . . . . . . . . . . . . . . . . . . 846.4 Macroscopic stability . . . . . . . . . . . . . . . . . . . . . . . . . . . . 876.5 Emulsions interfacial tension measurements . . . . . . . . . . . . . . . . 896.6 Effects of particles’ surface roughness. . . . . . . . . . . . . . . . . . . . 906.7 Surface diffusion effects . . . . . . . . . . . . . . . . . . . . . . . . . . . 94References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

7 Gelation of waxy crude oils 1007.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1017.2 Introduction to colloidal aggregation . . . . . . . . . . . . . . . . . . . . 1027.3 Waxy crude oil microscopy measurements . . . . . . . . . . . . . . . . . 1077.4 Characterisation of the model system . . . . . . . . . . . . . . . . . . . 1097.5 Colloidal gelation induced by depletion forces . . . . . . . . . . . . . . . 114References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

8

Abstract

Main aim of this work is showing that many basic questions concerning complex flu-ids of applicative interest can be profitably tackled by exploiting innovative opticalinvestigation methods.

By “complex fluid” here we mean any mixture containing either dispersed parti-cles, or species having large molecular weight, or mesoscopic aggregates spontaneouslyformed by peculiar molecular components like amphiphilic molecules. Therefore, com-plex fluids include for instance dispersion of particles in the colloid size range, polymersolutions (including biological polyelectrolytes), and micellar, emulsions, lyotropic liq-uid crystal, vesicle phases formed by surfactants in solution. From the structuralpoint of view, all these systems display organisation on mesoscopic length scales, suit-able to be investigated by electromagnetic radiation in the optical range. In addition,they often display complex dynamic properties, and a rheological behaviour that isintermediate between simple liquids and solids. Indeed, albeit missing any long-rangecrystalline order (hence belonging the general class of fluid structures) the may formsolid-like phases with a finite yield modulus like gels and glasses.

In particular, in this thesis we shall deal with colloidal suspensions, colloidal gels,and emulsions. Colloids are solid particle, with a size ranging between several nanome-tres to few micrometers, stabilised by repulsive (for instance, electrostatic or steric)interparticle forces which prevent particle coagulation due to dispersion forces. Bylowering the latter repulsion barrier, the particle may form large but tenuous aggre-gates, quite often with a fractal morphology, that form the basic scaffolding networkof a colloidal gel that, albeit being often dilute in terms of particle volume fraction,display a solid-like rheology. By emulsion, we mean any dispersion of a liquid intoan immiscible continuous phase (like water in oil, or viceversa) in the form of meso-scopic droplets displaying long–term stability: because of the large surface energy costinvolved in creating the dispersed phase, emulsion formulation necessarily requires anadditional stabilising species, which is usually a surfactant.

One of the most interesting feature of complex fluids is that the effective interactionsbetween the macromolecular components can easily be ‘tuned’: as a consequence, in thelast decades they have been used as ‘model systems’ for studying fundamental problemsin condensed matter and statistical physics. Besides their interest for basic research,complex fluids play a major role in a large number of technological sectors ranging frommining to industrial processing, from the consumer market to high–tech, including forinstance oil recovery and transportation, detergency, painting and coating, agriculture,cosmetics, pharmaceutics, and the food industry.

9

Abstract

In this thesis we shall mainly deal with two specific problems related to the oilindustry, namely:

a) The origin, stabilisation mechanism, and morphology of emulsions stabilised bysolid particles

b) The formation and structure of gel phases induced at low temperature by thecrystallisation of the paraffin component of crude oils.

Emulsions stabilised by solid particlesAlthough surface active agents are usually required to form stable emulsions, since thebegin of last century it is known, thanks to the works of Pickering and Ramsden, thatsolid particles in the colloidal range can also promote emulsion stabilisation by beingtrapped at a droplet interface and forming a steric barrier to coalescence: dispersedsystems of this kind are called Pickering Emulsions. Stable water/crude oil emulsionsnaturally occur in many steps of crude oils processing as extraction, waste water treat-ments, transportation, and refining. Spontaneous emulsification, often leading to anincrease of several order of magnitude of the oil viscosity, may give rise to arrest inextraction. Moreover, emulsified water causes severe corrosion problems to the down-stream units. Clays, asphaltenes, and waxes are naturally dispersed in crude oils andit is believed they are the primary agents for emulsion stability. Although there hasbeen a considerable amount of studies concerning crude oil emulsions characterisation,physical understanding of the microscopic mechanisms leading to emulsion stabilisationis still partly lacking.

We shall show that many of the open problems related to Pickering emulsion stabil-isation and structure can be profitably studied by investigating model systems, wherethe stabilising agent are custom–made spherical, monodisperse silica particles, andusing accurate video microscopy and micromanipulation methods. The particles wehave synthesized have indeed an internal fluorescent core, which makes the emulsionsthey stabilise particularly suitable to be observed by fluorescence microscopy. In ad-dition, the particle surface properties can be modified to investigate the effects ofparticle wetting on emulsion formation and stability. We have used two kinds of par-ticles, differing primary in their surface structure. The first one are spherical colloidswith a smooth surface, while particles of the second batch display noticeable surface‘roughness’, mimicking therefore an important, and so far neglected, feature of natu-ral colloids. By means of optical microscopy, we have investigated the morphology ofthe surface layers of adsorbed particles and probed the dynamic properties of trappedparticles at the droplet’s interface. In order to quantify the effect of particle interfacialtrapping on the interfacial tension γ of a single droplet, we have also developed a novelmicro-tensiometer, which allows measuring γ by detecting the negative pressure neededto suck the droplet into a custom-made glass micropipette.Our main findings can be summarised as follows:

• Particle trapping at the water/oil interface does not lead to appreciable changesin oil/water droplets interfacial tension. This is, to our knowledge, the firstexperimental evidence that particles adsorption doesn’t lower emulsion interfacialtension.

• Surface-roughness appreciably lowers the particle emulsifying power, while smoothparticles generally stabilise droplets by forming a regular, compact layer, the in-

10

Abstract

terfacial organisation of ‘rough’ particles displays a rich morphology, sometimesmarked by ‘colloidal lumps’ suggesting surface-mediated attractive forces of cap-illary origin.

• No straightforward relation exists between the degree of droplets surface-coverageand macroscopic emulsion stability. Long-term emulsion stability is observedfor very limited particles adhesion, or conversely, rapid coalescence of emulsionscomposed by densely-covered droplets may take place.

• Trapped particles exhibit vigorous Brownian motion, with a surface diffusioncoefficient that, on poorly-covered droplets, basically has the same value as inthe surrounding bulk phase. We have found experimental evidences suggestingthat particles redistribution on droplets may play a role in stabilising dropletswith low or inhomogeneous particles coverage.

Gelation induced by paraffin in crude oilsA second context in which crude oil bring forth its complex fluid behaviour is itsgelation induced by phase separation of the paraffin component at low temperature.Although crude oil is in a fluid phase when spilled from the reservoir, at sufficiently lowtemperature it undergoes a phase separation, where paraffins start to crystallise andseparate from the continuous organic phase. Aggregated wax solids can accumulateupon pipe walls seriously hindering oil flow. Much greater problems arise if the flow isarrested for pumping breakdown: if the oil temperature is sufficiently low, as it happensfor instance in submarine pipeline transportation, aggregation of the solid phase caninduces gelation of the whole system. Restarting crude oil flow can be a very seriouschallenge, since it requires huge pressurisation of the pipeline in order to break the geland restore the flow condition.

Differences in crude oil compositions, in distribution of paraffins molecular weightand structure (linear or branched) between crude oils leads to wide inhomogeneity inphase behaviour. For example, some crude oils gel at temperature above 30 degreewhile others do at temperature below −20 degree.

Gelled crude oil show many similarities with attractive colloidal gels. Both ofthem present the same dependency of viscosity against volume fraction and are ableto partially restore their structure after shearing. These similarities lead to envisiongelation of crude oil as a colloidal aggregation process of the wax crystallites, and to tryand exploit this similarity in order to find specific conditions leading to the formationof a weaker gel. For instance, a promising feature of colloidal gels is their sensitivityto shear during formation. This could be exploited for reducing the pressure requiredto restart flow in a plugged pipelines.

Using polarisation microscopy and light scattering, we have studied the gelationprocess both in crude oils and in a model system, whose phase behaviour and rheo-logical properties resemble those of natural oils, with the aim of characterising the gelmorphology and assessing the influence of the cooling rate on gelation. Colloidal gelsdisplay structural organisation on very large length scales, and therefore preferentiallyscatter light at low angle. This specific requirement led us to set up a custom lightscattering apparatus based on a new optical detection system1, which exploits an in-

1This setup is deeply based on the work performed by Luca Cippelletti in Montpellier, still unpub-

lished at the time I’m writing this thesis

11

Abstract

verted telescope system lens to demagnify the scattering angle and improve angulardetection. Moreover, we have designed and built custom cells, both the microscope andthe light scattering apparatus, ensuring careful control of the sample thermal cycle.

Our measurements, together with rheological measurements performed in Enitec-nologie, confirm that the structure and dynamics of crude oils at low temperature bearstrong resemblances to attractive colloidal gels. For the specific crude oil we studied, wefound that aggregation is favoured by lowering temperature and decreasing the coolingrate. Sensitivity on the thermal history, and in particular on the cooling rate, may beexpected for colloidal gels. The gel strength is indeed heavily dependent on the effec-tive volume fraction of the dispersed phase and on the size of the aggregating colloidalparticles. The latter results from a competition between the kinetics of paraffin crystalgrowth and that of crystallite aggregation via dispersion forces in a space–spanningdisordered network. Since lower cooling rates leads to the formation of larger paraffincrystallites, slower thermal cycles may be expected to promote gelation. Light scatter-ing measurements and microscopy observation on the model system establishes stronganalogies with the diffusion limited cluster aggregation gels.

Thesis outlineThis thesis is organised as follow.

Chapter 2 and chapter 3 introduce the experimental methods used for this work: basicand advance techniques in transmitted light microscopy are presented, fundamentalconcepts of scattering by particles dispersions are provided, and the particular case oflight scattered by a gel is discussed.

Chapter 4 and chapter 5 specifically deal with the apparatuses we designed and built.Chapter 4 describes our custom apparatus for measuring the interfacial tension of amicrometric droplet using a micropipette method. Calibration measurements on purefluid and on a surfactant solution are discussed. Chapter 5 presents the apparatus wedeveloped for measuring low-to-medium angle light scattering. Calibration measure-ments on three different pinholes and two colloidal systems are presented.

In chapter 6 we report our measurements on Pickering emulsions and in chapter 7 weaccount for gelation of crude oils experiments.

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Optical microscopy

Abstract

We deal with optical microscopy, with special emphasis for transmittedlight in Kohler configuration. Image magnification and resolution is de-scribed from an optical point of view and each optical component partakingimage formation is presented. Fluorescence microscopy is introduced. Ad-vanced contrast enhancing techniques, that are phase contrast, polarisedlight microscopy, and differential interference contrast, are described. Atthe end of the chapter, digital imaging methods are briefly introduced.

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2.1 Introduction Optical microscopy

(a) Anton von Leeuwenhoek’s simple micro-

scope(b) Hooke compound microscope

Figure 2.1: Simple and compound microscopy

2.1 Introduction

Photomicrography, the use of photography to capture images in a microscope, hasbeen an useful tool to scientists since a long time. For many years, the biological andmedical sciences have relied heavily on microscopy to solve problems related to theoverall morphological features of specimens as well as a quantitative tool for recordingspecific optical features and data[1].

More recently, microscopy has experienced an explosive growth as a tool in thephysical and materials sciences as well as in the semiconductor industry, due to theneed to observe surface features of new high-tech materials and integrated circuits.Microscopy is also becoming an important tool for forensic scientists who are constantlyexamining hairs, fibers, clothing, blood stains, bullets, and other items associated withcrimes.

The first simple microscopes were developed in the XV century. These were asingle1convex lens through which the specimen could be focused on the observer’s eye.In the 1600’s, Anton von Leeuwenhoek (see figure 2.1) was able to see some largerbacteria.

Around the beginning of the 1600’s, the first compound microscope was developed(see figure 2.1). It consisted of an objective close to the specimen and of an eyepiececlose to the observer’s eye: in this way it was possible to perform a two-staged magnifi-cation. These microscopes suffered from chromatic and spherical aberration more thansimple microscopes being composed of multiple lenses. The XIX century saw the firstchromatically corrected microscopes: they were built with lenses of different colourdispersion. By the end of the 19th century there was a high degree of competitionamong microscope manufactures along with a great improvement in the mechanicaland optical quality of compound microscopes (see figure 2.2).

1They are called simple microscopes because the specimen’s image is formed by only one lens. In

contrast, if the microscope forms the image through the objective and the eyepiece, we refer to it as a

compound microscope

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2.2 Image formation Optical microscopy

(a) Powell and Lealand N 1 micro-

scope, 1850(b) Olympus Provis AX 70

Figure 2.2: Past and present microscopes

Modern microscopes (see figure 2.2) far exceed the quality of those prior to themid 1900’s. The advances in glass formulation and in synthetic anti-glare lens coatingsallow a very good correction for optical aberration. Integrated circuit technology hasallowed manufactures to produce “smart” microscopes.

In this chapter, we will depict microscope functioning and we will describe pivotaltechniques greatly improving microscopy: polarising microscopy, differential interfacemicroscopy, phase contrast and fluorescence microscopy. At the end, we will illustratesome digital imaging techniques.

2.2 Image formation

Microscopes are instruments designed to produce magnified images of a small specimenon a detector as human eye or a camera.

The human eye receptors are of two kinds: cone cells used for sensing colour androd cells used for distinguishing levels of intensity. These cells are located on the retina;the iris, the cornea and the crystalline lens are the ‘mechanisms’ for admitting lightand focusing it on the retina.

In the optical camera, the image is formed on films. Common colour print films aremade of the superposition of twenty or more layers on a cellulose base. Only few of theselayers, which are made of sub-micron sized grains of silver-halide crystals dispersed in aspecial edible gelatin, actually contribute to form the image. The unmodified grains areonly sensitive to the blue portion of the spectrum, however adsorbing specific organicmolecules to their surface make them reactive to blue, green, and red light. Thesemolecules act as spectral sensitisers.

Digital cameras consist of a dense matrix of photodiodes that store electric charge as

15


light impinges on them[2]. Digital cameras are sensitive to light intensity and colours.We could think to each one of these detectors as an array of receptors sensitive to

varying degrees of light intensity and different colours. Therefore the magnified imagemust satisfy these requirements to be seen clearly by the detectors:

• It must spread on the detector at a sufficient visual angle. Unless the light fallson non-adjacent receptors (natural cells, photodiodes, silver grain) close-lyingdetails are not separated.

• There must be sufficient contrast between adjacent details and/or background torender the image visible.

There are two possible configuration for optical microscopy:

(a) Brightfield transmit-

ted light(b) Brightfield reflected light

Figure 2.3: Kohler illumination for transmitted and reflected microscopy. The left

image shows forming light path and conjugates planes in transmitted light microscopy.

transmitted light The specimen is sitting between the objective and the condenserlens as in figure 2.3. Illuminating light, formed by the condenser lens, falls onthe specimen and it is collected by the objective.

reflected light Objective is used both as matching well-corrected condenser and asimage forming lens. Thus the light passes twice from it, see figure 2.3.

It is useful to understand how each microscope component takes part in the processof image formation. In the following we will refer to the transmitted light configuration.

The custom microscope optical train consists of an illuminator, a substage con-denser, a specimen, an objective, an eyepiece, and a detector, which is either someform of camera or the observer’s eye (see figure 2.4).

The most common light source is a built-in incandescent tungsten-halogen bulb.Collector and condenser lenses shape the impinging light into a cone which illuminates

16


Figure 2.4: Modern microscope light train

the specimen. Image forming light rays go through the objective, the tube lens, andthe eyepiece to form the specimen’s image on the observer’s eye. To capture the imagewith a camera, an additional special positive lens is needed to form the specimen’simage on the ccd surface or on the chemical emulsion layered onto the film base.

Specimen illumination

Illumination of the specimen is an important variable in achieving high-quality imagesand for viewing thick specimen. It should be bright, glare-free and evenly dispersed inthe field of view. Microscope’s manufacturers suggest to adjust the optical elements inthe Kohler illumination2 configuration to optimise the illumination.

This arrangement establishes two sets of conjugates planes. The ones in the pathof the illuminating rays are:

• the lamp filament;

• the condenser aperture diaphragm;

• the back focal plane of the objective;

• the eyepoint3 of the eyepiece.

The other set of conjugates planes are related to image formation light path and consistof:

• the field diaphragm;

• the focused specimen;

2This procedure was developed by August Kohler, 1866–19483The eyepoint is the point where the observer places the front of the eye during observation

17


• the intermediate image plane;

• the eye’s retina or the film plate of the camera.

This pair of conjugate plane presents many advantages[3]. The field diaphragmbehaves like the virtual source of light. It influences the amount of excess light con-trolling the width of the bundle of light rays reaching the condenser. Actually, if thefield diaphragm is too opened the scattered light originating by the specimen and thereflected light from optical surfaces could degrade image quality. Oppositely, if thefield diaphragm is closed too much the field of view will be restreined.

The condenser aperture diaphragm is responsible for controlling the angle of theilluminating light cone (its numerical aperture NA). The condenser and the field di-aphragms control the shape of the illuminating cone independently.

Substage condenser

Figure 2.5: Abbe condenser

The substage condenser gathers light from the microscope light source and concen-trates it into a cone of light that illuminates the specimen with an uniform intensityover the entire viewfield. It is crucial that the condenser light cone is properly adjustedto optimise the intensity and angle of the light entering the objective front lens.

The light from the microscopes illumination source passes through the condenseraperture diaphragm, located at the base of the condenser, and is concentrated byinternal lens elements, which then project light through the specimen in parallel bundlesfrom every azimuth, see figure 2.5. The size and numerical aperture of the light coneis determined by adjustment of the aperture diaphragm. After passing through thespecimen, the light diverges into an inverted cone with the proper angle to fill the frontlens of the objective.

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Care must be taken in order to guarantee that the condenser aperture is openedto the correct position with respect to the objective numerical aperture. When thecondenser aperture diaphragm is opened too widely, stray light generated by refractionof oblique light rays from the specimen can cause glare and lower the overall contrast.On the other hand, when the aperture is excessively close , the illumination cone isinsufficient to provide adequate resolution and the image is distorted due to refractionand diffraction from the specimen.

Specimen imaging

Specimen modifies both amplitude and phase of impinging light by means of a combi-nation of refraction and absorption processes.

The objective collects exiting lights and focuses it to infinity. A second lens, setinto the microscope, reforms the image on the intermediate image plane. This secondlens, referred as tube lens, is a positive lens with a fixed focal length that depends uponmanufacturer and model (180mm for Olympus microscopes).

Modern microscopes adopt this imaging system with an infinity–corrected objec-tives, instead older microscopes mounted objectives that formed directly the interme-diate image plane. The former allows to introduce auxiliary components (interferencecontrast prism, polarisers, fluorescence cubes, . . . ) without altering the tube lengthand introducing only minimal effects on focus and aberration corrections.

The intermediate image plane is formed at a distance from the eyepiece shorterthan its focal length, in order to produce a virtual image as if it were at a distance of250mm from the eye.

As cameras can detect only real images, this requires a particular setup to mounta camera after the eyepiece. A special positive lens, the projection lens, is placedafter the eyepiece to focus the image on the ccd array (or photographic emulsion).Modern microscopes usually mount a beamsplitter after the intermediate image plane,see image 2.4, in order to allow to use the eyepiece and the camera at the same time.

Image magnification

Microscope magnification depends upon the objective, the tube lens, the eyepieceand/or on additional lenses (the projection lens for the camera). The overall magnifi-cation should be measured with a stage micrometer but, with a hand waving argument,it could be simply taken as the product of the objective and eyepiece magnification.

Let’s consider an object of size lobj at a distance a from a lens of focal length f .The object’s image, of size limg, is formed at a distance b according to the lens law:

1

a+

1

b=

1

f(2.1)

The lateral magnification M is the ratio of the linear size of the image to the linearsize of the object:

M =limg

lobj=b

a.

The intermediate image plane magnification is the ratio of the tube lens focal lengthand the objective focal length:

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Mint =ftl

fobj

Mint is the magnification inscribed on the objective barrel. The eyepiece magnificationdepend upon its focal length:

Meye =250

feye

Meye is the magnification inscribed on the eyepiece rim.

When looking through the eyepiece, the total magnification is the product of the :

M = MintMeye =ftl

fobj

250

feye

Image resolution and contrast

Some of the light impinging on the specimen passes through or around it: this iscalled direct or undeviated light. Some of the light is diffracted by the specimen andis rendered one-half wavelength out of phase in respect to the undeviated light.

Thanks to the set of conjugate planes in Kohler illumination, the objective spreadsevenly the undeviated light and focuses the diffracted light at various localised placesacross the image plane. The diffracted light causes destructive interference with theundeviated light, forming a pattern of light and dark that we recognise as an image ofthe specimen[3, 4, 5].

The local difference in light intensity between the image and the adjacent back-ground related to the intensity background, is called image contrast C:

C(x,y) =Is(x,y) − Ib

Ib

The contrast is the ability of a detail to stand out against the background or otheradjacent details.

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Abbe theory for resolution Image resolution’s limitation stems from the objectivefinite pupil and its inability to accept all the image spatial frequencies.

If an image is decomposed in its frequency components (for example with Fouriertransform algorithm), its zeroth frequency will be associated to the average imageintensity (the background) and higher frequencies will be associated to finer details. Ifsome of the higher frequencies were eliminated, the image could be reconstructed (withan inverse transform algorithm), but it would lack sharp contours and crisp edges.

This is similar to what happens inside a microscope. Fourier optics states that aconverging lens has the ability to perform the spatial Fourier transform of the fieldamplitude distribution impinging on it. Let’s consider an object located at the frontfocal plane of the converging lens with focal distance f and an amplitude transmittancetA(x,y). Let’s the object be uniformly illuminated by a normally incident plane waveof amplitude A and wavelength λ. The amplitude distribution in the generic point(u,v) at the back focal is (see [4]):

Uf (u,v) =A

iλfF [tA(x,y)]

where u = fxλf , v = fyλf , fx and fy are the spatial frequencies. Calculating theintensity, we obtain:

If (fx,fy) =

(A

λf

)2

|F [tA(x,y)]|2

that relates the intensity distribution to the object power spectrum.

Considering that the specimen is located in the object front focal plane, the intensitydistribution in the back focal plane depends on the specimen power spectrum. Theback focal plane of a microscope objective lays inside the objective itself. This spatiallimitation acts as a low-pass filter and introduces a cut-off frequency.

A handwaving estimateoften useful to calculate the resolution of a microscope is:

d = 1.22λ

NAobj + NAcond

being d the smallest resolvable distance between two object, λ the illumination wave-length, and NA the numerical aperture 4.

Optical Transfer Function Frequency analysis and linear system theory have beenapplied to optical systems only in the last fifty years. However they are of paramountimportance to understand and improve optical components quality.

Any optical system that images a real object, in the plane (s,t), could be describedas a linear operator T [·] that transforms the object light intensity distribution o(s,t) ina spatial magnified, two dimensional intensity distribution f(y,x) = T [o(s,t)][4].

The object plane is considered as a series of closed arranged points that are trans-formed individually:

f(x,y) =

∫∫o(s,t)T [δ(Ms− x,Mt− y)] dsdt

4NA = n sin(ν), where n is the refraction index of the medium between the objective and the

coverslip and ν is one-half the angular aperture

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2.3 Fluorescence microscopy Optical microscopy

where M is the lateral magnification of the system and h(x,y; s,t) = T [δ] is the pointspread function which equals the Fraunhofer diffraction pattern of the exit pupil.

In order to achieve space invariance in the imaging system, it is necessary to removethe effects of magnification M . This can be done by defining reduced coordinates inthe object space:

s = Ms t = Mt

Assuming the system is space invariant: h(x,y; s,t) = h(x − s,y − t), in real spacewe have:

f(x,y) =

∫∫h(x− s,y − t)o(x,y)dsdt

Using the convolution theorem, it can be written as:

F (u,v) = O(u,v)H(u,v)

where F (u,v) = F [f(x,y)], O(u,v) = F [o(x,y)] and H(u,v) = F [h(x,y)] is the OpticalTransfer Function (OTF). Thus it is possible to fully characterise the optical systemproperties by H(u,v) only.

A microscope exit pupil has always circular shape, thus the point spread functionin a diffraction-limited system is the Airy function:

h(x,y) = Ai(d) =

(J1(d)

d

)2

d =2π

λ

√x2 + y2NA

where J1() is the Bessel function of the first order and λ is the illuminating wave-length. If the pupil area is equal to w and the distance image-exit pupils zi, the OTF is:

OTF (ν) =2

π

acos

(ν

ν0

)− ν

ν0

√

1 −(ν

ν0

)2

where ν =√u2 + v2 and v = w

2λziis the cutoff frequency of the system. Note in the

previous figure that all the spatial frequencies lesser than ν0 are passed by the system,but with different amplitude.

2.3 Fluorescence microscopy

Fluorescence microscopy is a rapidly expanding and invaluable tool of investigation[6, 7]. The use of fluorochromes made possible to identify cells and sub-microscopiccellular components and entities with a high degree of specificity amidst non-fluorescingmaterial. More than this, a fluorescence microscope can reveal the presence of fluo-rescing material with exquisite sensitivity. An extremely small number of fluorescingmolecules (as few as 50 molecules per cubic micron) can be detected. In a given sam-ple, through the use of multiple staining, different probes will reveal the presence ofdifferent target molecules. Although a fluorescence microscope cannot provide spatial

22

2.3 Fluorescence microscopy Optical microscopy

resolution below the diffraction limit of the respective objects, the presence of fluoresc-ing molecules below such limits is made visible. New techniques as confocal microscopy,multi-photon microscopy and FRAP (Fluorescence After Photobleaching) are based onfluorescence.

Fluorescence microscopy is an excellent method to specimen where fluorescence ispresent, either in its natural form (primary or autofluorescence) or when the sampleis treated with chemical agents, or fluorochrome, capable of fluorescing (secondaryfluorescence). Fluorescing specimen, irradiated by excitation light, re-emit light at alonger wavelength. If this much weaker light is separated from the excitation one, itcan be used to form an image where fluorescing areas shine against a dark background.

Figure 2.6: Fluorescence Images of African Green Monkey Kidney (CV-1) Cells, image

taken from “Molecular expression website”: http://micro.magnet.fsu.edu/

Light source As fluorochromes present a very low quantum yield, powerful lightsources are needed to generate an emission capable of detection. Moreover, lightsource’s emission spectrum should be matched to the fluorochrome’s absorption spec-trum. For these reasons, a standard incandescent tungsten-based lamp is not suitablefor fluorescent microscopy. Instead an arc (burner) lamp, filled with high pressure gas(mercury or xenon), is chosen.

(a) Tungsten (b) Mercury

Figure 2.7: Tungsten and mercury emission spectra

23

2.4 Phase contrast microscopy Optical microscopy

Sample illumination The most spread configuration is the episcopic illumination:the illuminator is designed such that light, passing through the objective, falls ontothe specimen, see figure 2.8. The objective is used first as a condenser, when lighttravels from the source to the specimen, and then as a standard objective to collectlight emitted by the specimen, as in reflected light microscopy. This setup has severaladvantages:

1. the illuminated area is always the observed area;

2. the system is always well-corrected: the ‘condenser’ is matched with the ‘objec-tive’ and is in correct alignment;

3. it is possible to combine epi-fluorescence with transmitted light phase contrastor DIC.

As light travels through the objective twice, it is of paramount importance tohave objectives of highest quality (good chromatic and spherical correction). Anotherconsequence is that objective numerical aperture affects more than just in transmittedillumination. Larger numerical apertures yield to brighter images, since the imageintensity scales as:

I =NA4

M2

where NA is the numerical aperture and M is the objective magnification. Oil-immersion objectives are preferable because they minimise reflection at the coverslideinterface.

Fluorescence cube Since fluorescence emitted light is much weaker than excitationlight, it is necessary to prevent the later to reach the detector. This feature is carriedout by the fluorescence cube, see figure 2.8. A filter cube is composed by an excitationfilter, a dichroic mirror, and a barrier filter: all of them are interference filters. Thefirst allows only the wavelengths selected by the illuminator to pass through the waytowards the specimen. The dichroic mirror has high reflectivity for shorter wavelengthsand high transmission for longer wavelengths. It is positioned between the excitationand the barrier filter, oriented at 45 degrees to the light passing through them (seefigure 2.8). All the three filters are normally housed into a singled cube. As the filtercube selects both the exciting and fluorescence wavelength, it must be matched withthe fluorochrome type in the specimen.

2.4 Phase contrast microscopy

Phase contrast microscopy is used primarily to detect ‘phase’ specimen: objects whichslightly retard the phase of diffracted light. As light rays pass through areas of differentoptical path they may be retarded in phase by up to 1/4λ but will remain unchangedin amplitude. Diffracted light arrives at the image plane slightly out of step with theundeviated light. The resulting image lacks of contrats and specimen’s details arealmost invisible.

In the early 1950s Zernike5 discovered a method by which phase differences canbe transformed into amplitude differences[8, 9]. Zernike invented what is known as

5Fritz Zernike (1888–1966) received a Nobel prize in 1953 for his discovery of phase contrast

24


(a) Filter Cube (b) Epi-illumination

Figure 2.8: Fluorescence filter cube and light path

positive or dark phase contrast. An alternative method, negative or bright phasecontrast, was subsequently developed and has supplanted Zernike’s original approach.In positive phase contrast the object appears darker than the surrounding background.In negative phase contrast the object appears brighter than the background[10].

There are a number of techniques to view such samples, like the central dark back-ground method or the schlieren method, but the observed intensity is not linearly relatedto the phase shift. Phase contrast enhances image contrast by linearly transformingphase differences into amplitude differences[11]. Therefore light intensity is strictlyrelated to thickness variation of the specimen.

(a) positive phase contrast (b) negative phase contrast

Figure 2.9: Horsefly is the common name for many of the stout flies in the family Taban-

idae, image taken from “Molecular expression website”: http://micro.magnet.fsu.edu/

25


Phase contrast theory Zernike’s idea is to separate the zeroth order (undeviated)light from the diffracted light at the real focal plane of the objective. The unscatteredlight is brought to focus in the back focal plane of the objective. On the contrarythe diffracted light, generated at the specimen plane, is spread over the entire rearfocal plane. So it is possible to insert a phase object only on the zeroth order lightpath. If the phase plate slows down the undeviated light by λ/4, both light pathscome at the image plane in phase and constructive interference takes place: this isnegative phase contrast. If the phase plate speeds up the zeroth order light by λ/4,destructive interference takes place in the image plane: what we have is now positivephase contrast.

Let’s consider a phase object which optical amplitude transmittance is t(x,y) =exp(jΦ(x,y)), where Φ(x,y) = 2π

λ n(x,y)h(x,y) is the phase shift, n(x,y) is the refraction

index and h(x,y) is the object thickness. An electrical field E(0) = E0 exp[j(kx− ωt)]impinges on the object and it is diffracted, generating the field E(1):

E(1) = t(x,y)E(0) = E0 exp[j(kx− ωt)] exp(jΦ)

At the image plane the interference of the two fields gives:

I(x,y) = |E(1) + E(0)|2 = 2I0[1 + cos(Φ)]

For a small phase shift, the intensity is constant.If the field E(0) travels through a phase plate, it becomes:

E(2) = E0 exp[j(kx− ωt)] exp(jα)

where α = ±π/2 for respectively destructive or constructive interference. The intensityat the image plane is:

I(x,y) = |E(2) + E(1)|2 = E20 | exp(iΦ) + exp(iα)| = 4I0 cos2

(α− Φ

2

)

which can be approximated to the first order on Φ:

I(x,y) = 4I0

(cos2

(α2

)+

1

2sin(α)Φ(x,y)

)

It can be seen that the intensity is linearly related to the phase shift of the object, thusthe information about its optical path is not lost.

Phase contrast setup The accessories needed for phase contrast microscopy are asubstage condenser equipped with a ring annulus and a phase contrast objective, whichcan be used also for ordinary brightfield microscopy.

The ring annulus is placed at the front focal plane of the condenser, conjugatedwith the rear focal plane of the objective. The specimen is illuminated by a hollowcone of light, and the undeviated light is focused at the rear focal plane in the shapeof a ring of light. The phase contrast objective has a phase ring in its back focal plane,that overlaps the image of the condenser ring annulus. The phase ring is composedby a absorbing metallic layer and a phase retarding material. Since undeviated lightis brighter than the diffracted light, the absorbing layer balances the two intensitiesgiving a better contrast. The phase retarding layer could be optically thinner or thickerthan the rest of the plate in order to speed up or slow down the undeviated light.

The phase ring thickness is calculated assuming monochromatic green light, so thespecimen is best viewed when a green filter is used.

26

2.5 Polarised light microscopy Optical microscopy

(a) Phase contrast setup

Figure 2.10: Phase contrast light path

2.5 Polarised light microscopy

This technique exploits optical properties of anisotropic materials to reveal informa-tion about structure and composition[12]. Anisotropic materials have properties thatdepend on the vibrational plane’s orientation of the light passing through them.

Two polarisers are inserted in the microscope optical path: they transmit lightvibrating in a single plane. The first polariser selects the vibrational direction oflight impinging on the specimen; the second one, also called analyser, recombines thepolarised light coming out from the sample.

Polarising microscopy can provide informations about composition, three-dimensio-nal structure, thermal history of the sample, stress which the specimen was subjectedunder formation and properties of liquid crystals.

Polarising microscopy could be exploited in two different ways. If the polariser di-rection is right-angled to the analyser’s one the undeviated light and sample’s isotropicpart will remain in extinction. The only light in the image is the one depolarised bythe anisotropic part of the sample.

If an anisotropic specimen is brought into focus and rotated around the optical path,it will present changes in brightness and/or in colours. When polarised light impingeson a birefringent material, it is resolved into two components that are polarised inperpendiculars directions and travel through the material with different velocities. Thetwo components, emerging from the specimen, are out of phase but don’t interfere sincethey have orthogonal polarisations. The retardation Γ, which depends upon sample’sbirefringence B and thickness t, could be quantified by:

Γ = t ·B = t|nhigh − nlow|

where nhigh and nlow are respectively the higher and the lower refraction indexes. Theanalyser recombines the two components and interference takes place. The interference

27

2.6 Differential interference contrast Optical microscopy

(a) (b)

Figure 2.11: (a) View between crossed polarisers of shear-induced birefringence textures

in PTFE ‘fibrillar’ particles. (b) Optical light path in polarised microscopy

pattern depends upon retardation and wavelength. The image’s colours are due toresidual complementary colours, in other words white light minus the wavelengthswhich interfere destructively. It is possible to calculate sample thickness and materialbirefringence from polarisation colours and intensity variations

2.6 Differential interference contrast

Differential interference contrast, or DIC, is a phase imaging technique that forms animage of the gradient of refractive index in the sample, for both high and low spatialfrequencies[13]. Those regions of the specimen where the optical paths increase along areference direction appear brighter (or darker), while regions where the path differencesdecrease appear in reverse contrast. As the gradient of optical path difference growssteeper, image contrast is dramatically increased. This technique essentially acts as ahigh-pass filter that emphasises edges and lines. It gives good rejection of out-of-focusinterference thus offering an excellent mechanism for rendering contrast in transparentspecimens. Since it doesn’t use any blocking filter, it is possible to take advantage offull numerical aperture. Moreover, the image doesn’t suffer from halos (as sometimesin phase contrast) and optical staining could be exploited by means of phase retarderplate.

Unfortunately, the shadow-cast appearance of DIC image is just a qualitative indi-cator, although a good one, of the real structure geometry. It is not possible to extractquantitative information about refractive index or specimen thickness, as in polarisedmicroscopy.

Differential Interference Contrast optical configuration Four optical compo-nents are needed for DIC microscopy, see image 2.12: a polariser, an analyser, a con-denser Wollaston or Nomarski prism and an objective Wollaston or Normaski prism.

28

2.6 Differential interference contrast Optical microscopy

Figure 2.12: (a) Differential interference contrast optical components (b) DIC image

of polypropylene fibers, image taken from http://micro.magnet.fsu.edu/

The polariser and the analyser, in a crossed configuration, give the same result as po-larised light microscopy. The first Nomarski prism, located at the front focal plane ofthe condenser, splits the polarised light emerging from the polariser into two orthogonalpolarised components, slightly spatial separated. The second Nomarski prism, locatedat (or near) the objective back focal plane, recombines the two sheared wavefronts ina polarised beam.

Nomarski and Wollastone Prism If a polarised light beam impinges on an uniassicbirefringent crystal, it is divided into two orthogonally polarised wavefronts whichexperience different refraction indexes: the ordinary wave, vibrating perpendicular tothe crystal’s optical axis, sees a refraction index no, the extraordinary wave, vibratingparallel to the optical axis, perceives a refraction index ne. If ne > no the crystal iscalled positive, otherwise it is called negative.

Wollaston and Nomarski prisms are composed of two wedge-shaped slabs of positiveuniassic high-grade optical quartz. The two wedges are prepared with orthogonaloptical axis, and are cemented along the hypotenuse.

If linearly polarised light impinges on a Wollaston prism, it will be separated intotwo right-angled waves: the ordinary one and the extraordinary one. As the plate ismounted with its optical axis oriented at 45-degree to the microscope’s optical path, thetwo components present the same amplitude. Since the two crystals have orthogonalaxis, both waves experience refraction at the wedges interface: the ordinary wave passesfrom a medium with refraction index no to a medium with ne, the opposite happens tothe extraordinary wave (see figure 2.13). The prism introduces a shear angle betweenthe two components.

As the Wollaston prism is located at the condenser’s front focal plane, rays emergingfrom it are collimated by the condenser lens, travel parallel through the specimen andare focused by the objective at its back focal plane. A second Wollaston prism, locatedin this plane, removes the shear between the two components.

29

2.7 Digital Imaging Optical microscopy

(a) Wollaston prism (b) Nomarski prism (c)

Figure 2.13: Light propagation in a Wollastone (a) and Nomarski (b) prism. (c) Un-

deviated light is blocked by the Analise, diffracted light passes through it

Since the back focal plane is usually inside the objective, it is difficult to adapt theprism for a standard objective. It is generally preferred to adopt a Nomarski prism. Itconsist of two wedges, assembled like a Wollaston prism, of which one has the opticalaxis orientated obliquely with respect of the prism’s surface, see figure 2.13. The sheartakes place at the air-quartz interface and the refraction at the wedge’s interface causingthe waves to crossover outside the prism. Consequently, the Nomarski prism doesn’trequire to be physically located in the objective back focal plane.

Image formation After emerging from the condenser Normaski prism, the twosheared coherent light components travel parallel to each other, separated by a sheardistance which is less than the lateral resolution of the objective6.

Undeviated wavefront pairs go through the same optical paths, therefore the ob-jective Nomarski prism reverses the action of the condenser filter. The wavefronts arerecombined in a linearly polarised light, whose vibrating direction is parallel to thepolariser transmission axis. Since the analyser blocks that kind of polarised light, theundeviated light doesn’t take part in image formation, see figure 2.13.

If the paired wavefronts encounter a phase gradient in the specimen, the waves willtravel along different optical paths. The second prism recombines them in an ellipticallypolarised light, which has a component parallel to analyser transmission axis and cangenerate intensity in the image plane. All the waves exiting from the analyser have thesame polarisation, thus interfere at the image plane creating an image of the specimen.

2.7 Digital Imaging

A digital image, like the ones created by CCD, is represented by a matrix I(i,j) ofinteger numbers obtained by sampling the 2-D intensity distribution of a real image.

6Since the shear distance, which depends upon condenser Normaski prism geometry, should be

comparable with the objective resolution, it is necessary to employ different prisms for objectives of

various magnification

30


Matrix values are positive integer numbers ranging from zero to N = 2d − 1, wherethe bit depth d is the number of bits used to store information about each pixel. Thebit depth is usually 8-bit for grayscale imaging, however it could be higher if intensitymeasurement are needed. It is possible to employ mathematical operators on the imagematrix to improve its quality, to correct optical mistakes, and unravel hidden details.

In this section I will introduce some of techniques frequently used in my microscopyexperiments. Although this will not exhaust the subject it will provide some hints onthe capability of digital imaging and filtering.

Point-by-point process The simplest operation is the point-by-point process inwhich single point values I(i,j) are transformed separately by an operator T (·). Theimage pixels are modified depending only on each pixel value, without influence ofneighbour information. A typical application is histogram optimisation for the proba-bility density function of intensities. The probability density function is usually a veryirregular function. It may happen that the lower values or the higher ones are not useddue to exposition mistakes, diminishing the effective bit depth. For example if intensi-ties i are imin ≤ i ≤ imax, applying the linear operator T (i) = N i−imin

imax−iminwill raise the

bit depth to its original value, enhancing image visualisation. Moreover, let’s supposenow that information is evenly spread across all intensities, thus it would be useful toequalise its distribution. This is done by applying the operator T [i] = NP (i) to eachintensity value i of the image, which has intensity probability distribution function P ,as in figure 2.14.

Convolution process If the operator T (·) modifies single pixel values by taking inaccount neighboor pixels, it performs a convolution process. The transformed imageI ′(i,j) is: I ′ = h(i,j) I(i,j) where h(i,j) is the filter function. The same operationcould be accomplished in the Fourier space, reducing often computational time becauseit becomes a point-by-point process. Thus filters can be indicated either with the filterfunction h(i,j) or with its Fourier transform H(k,l) = F [h(i,j)]. Some of the mostused linear filters are the low pass, the high pass, and the Laplacian.

Low pass filters allow to pass low frequencies while attenuating or blocking higherones. they are often used for removing noise from images, although this may flattendetails as well, as in figure 2.15.

H(k,l) =

1 k2 + l2 = w2

0

0 else=⇒ h(i,j) =

J1(r/w0)

r/w0

To avoid ringing effects, it is better to use a Gaussian low pass filter as:

H(k,l) = exp

(−k

2 + l2

w20

)h(i,j) =

π

w20

exp[−π2w20(i

2 + j2)]

which attenuates high frequencies without removing them. A similar effect is obtainedwith the median filter, in which a pixel is substituted by averaging it with its nearestneighbours.

High pass filters are used to enhance edges, see figure 2.15. They remove (orattenuate) lower frequencies passing the higher ones. As for a low pass filter, it isbetter to use a Gaussian filter to avoid ringing artifacts:

H(k,l) = 1 − exp

(−k

2 + l2

w20

)

31


(a) original image (b) equalised image

Figure 2.14: Example of image equalisation. The probability distribution function is

plotted in the graphs insets

Gaussian low pass and high pass are often used together in the Difference Of Gaussianfilter, or DoG, which acts as a bandpass.

Another method to enhance edges involves differentiation. It is known that differ-entiation for discrete function I(i,j) is:

∂I(i,j)

∂i= I(i+ 1,j) − I(i− 1,j) = [−1 0 1] I(i,j)

∂2I(i,j)

∂i2= I(i+ 1,j) − 2I(i,j) + I(i− 1,j) = [1 − 2 1] I(i,j)

So that the Laplacian operator is:

∇2I(i,j) =

0 1 01 −4 10 1 0

I(i,j)

To enhance all edges I(i,j) −∇2I(i,j) is calculated applying the operator:

∇2I(i,j) =

0 −1 0−1 5 −10 −1 0

I(i,j)

32


In figure 2.15 it is possible to see the effect of Laplacian and edge enhance filters.All the filters presented until now are linear, however there are nonlinear filters that

could be useful, like the median or the binary threshold which sets to zero all valuesbelow a fixed one and sets the others to N.

The median filter is very useful to remove noise like a low pass filter, although itpreserves edges. This filter substitutes the central pixel with the median taken over itssurrounding ones.

33


(a) Original image (b) Gaussian low pass filter, w0 = 34 pixels

(c) High pass filter, w0 = 34 pixels (d) Gaussian high pass filter, w0 = 34 pixels

(e) Laplace filter (f) Laplace edge enhancing filter

Figure 2.15: Convolution filters

34

REFERENCES REFERENCES

References

[1] B. Herman and J. Lemasters, editors. Optical Microscopy Emerging Methods andApplications (Academic Press) (1993).

[2] S. I. K. R. Spring. Video Microscopy, the fundamentals (Plenum Press) (1997).

[3] S. Inoue and R. Oldenbourg. Handbook of Optics, vol. 2 (Mc Graw–Hill) (1995).

[4] J. W. Goodman. Introduction to Fourier Optics (Mc Graw–Hill) (1996).

[5] P. J. Evennett and S. Bradbury. Contrast Techniques in light microscopy (BIOSScientific Publishers) (1996).

[6] F. W. D. Rost, editor. Fluorescence Microscopy (Cambridge University Press)(1992).

[7] M. Abramowitz. Fluorescence Microscopy: the essentials (Olympus America)(1993).

[8] F. Zernike. “Phase-contrast, a new method for microscopic observation of trans-parent objects. Part I.” Physica, 9, (1942), 686–698.

[9] F. Zernike. “Phase-contrast, a new method for microscopic observation of trans-parent objects. Part II.” Physica, 9, (1942), 974–986.

[10] M. Davidson and M. Abramowitz. “Optical microscopy.” In J. Hornak, edi-tor, “Encyclopedia of Imaging Science and Technology,” vol. 2, pages 1106–1141(Wiley–Intescience) (2002).

[11] S. E. Ruzin, editor. Plant microtechnique and microscopy (Oxford university press)(1999).

[12] P. C. Robinson and S. Bradbury. Qualitative Polarized-light microscopy (Oxfordscience publications) (1992).

[13] G. Nomarski. “Microinterferometre differentiel a ondes polarisees.” J. Phys. Ra-dium, 16, (1955), 9s–11s.

35

Light Scattering Techniques

Abstract

Light scattering theory is discussed. First, we introduce the fluctuationsapproach, in which scattered light arises from dielectric constant fluctua-tions in a medium. Then, our attention is focused on static light scatter-ing from colloidal particles, describing both general theory and the specialcases of Rayleigh, Rayleigh-Debye-Gans, and Mie regimes. Light scatter-ing from colloidal aggregates is pointed out, showing that intensity distri-bution contains information of paramount importance for describing thiskind of system. At the end we introduce dynamic light scattering.

36

3.1 Introduction Light Scattering Techniques

3.1 Introduction

The interaction between an electromagnetic wave and a medium yields scattered elec-tromagnetic radiation which propagates in a direction different from the originatingone. Light scattering at the basis of every processes in which light propagates in anon-rectilinear direction. Reflection and refraction arises from the superposition of theincident field and the one scattered in the forward direction by the molecules of themedium. Diffraction can be interpreted as a particular case of diffraction from objectswhose dimensions are larger than the wavelength of light.

Light scattering can be exploited as a probe for studying both the spatial structureand the dynamics of many materials. For a generic scattering process, both the wavevector and the frequency of scattered radiation can be different form the incident ones.Let’s call ki, ωi the incident wave vector and frequency, ks ωs the scattered wave vectorand frequency, and

q = ki − ks ∆ω = ωi − ωs

We will see that light scattered at q depends on index refraction fluctuation with ascalar dimension of

d =2π

q

and the frequency variation depends on dynamics with the characteristic time τ ∼∆ω−1.

3.2 Fluctuation Approach

It is well known from the Lorentz model[1] that, if any particle in space is subjectedto an electric field of strength E, its constituent electrons become subjected to a forcein one direction and its constituent nuclei to a force in the opposite direction. Thus adipole moment is induced in the particle, and, if the particle is optically isotropic, itwill be parallel in direction to the electric field. The magnitude of the dipole momentp is proportional to the electric field strength where the proportionality constant α, isknown as the polarizability of the particle p = αE. The light beam impinging on a gas,on a liquid, or generally on a dielectric medium, will induce an oscillating dipole in anyparticle on its path. Such oscillating dipoles are themselves sources of electromagneticradiation being in practice small elementary antennas: this emitted radiation is whatwe will refer as scattered radiation. However it is easy to show that if we consider anhomogeneous medium we do not observe any scattered light, since it is always possibleto find two volume elements which scattering contributions are mutually extinguishedin the detector plane.

Fluctuations are therefore required in order to observe scattering. Let us considera fluctuating dielectric medium whose local dielectric constant can be expressed as:

ε(r,t) = 〈ε〉 + δε(r,t) (3.1)

Although the medium is considered to be isotropic, local anisotropy may arise from thepresence of fluctuations, thus they should be considered more generally as symmetrictensors δε. A common geometry for scattering experiments is the following: an incidentbeam propagates horizontally and is linearly polarized on the vertical direction. θ isthe scattering angle formed between the scattered and the incident wave vectors ks

37

3.2 Fluctuation Approach Light Scattering Techniques

Figure 3.1: Sketch of wave vectors ki and ks and scattering wave vector q

and ki, and it is related to the modulus of the scattering wave vector q = ks − ki

through q = 2ki sin(θ/2), see figure 3.1. We have implicitly assumed elastic scattering,so |ki| = |ks|. If the incident fields impinging on the sample is given as:

Ei(r,t) = niE0 exp [i(ki · r − ωt)]

where ni is a unit vector in the direction of the incident field, the expression of thefield Es at distance R in the far field, will be given by [2]:

Es(R,t) =E0

4πεRei(ksR−ωt) ks ×

ks ×

[ ∫

Vs

dV (δε(r,t) · ni) eiq·r

]

where r denotes the position in the scattering volume Vs with respect to the origin ofthe coordinate system. Introducing the Fourier decomposition of wave vector q of thefluctuations:

δε(q,t) =

∫dV δε(r,t) exp[ iq · r]

it is possible to calculate the polarized scattered field in the direction nf :

Es(R,t) = − k2sE0

4πεRδεif (q,t) exp[i (ksR− ωt)] (3.2)

where δεif (q,t) ≡ nf · δε(q,t) · ni. The polarization of the scattered field is determinedby δεif . If fluctuations are isotropic, the incident and scattered fields have the samepolarization direction.

Scattered intensity at (ks,ωs,nf ) from incident component (ki,ωi,ni) is:

Iif (q, ωs, R) =I0k

4s

32π3R2ε20

∫ ∞

−∞dt〈δε∗if (q,0)δεif (q,t)〉 exp[i(ωs − ωi)t]

Anelastic scattering rises only from time-depending fluctuations. Rather, if dielectricfluctuations are time-indipendent the scattering is elastic.

38

3.3 Particulate Approach Light Scattering Techniques

The scattered intensity at wave vector q is:

Iif (q) =I0k

4s

32π3R2ε20〈|δεif (q)|2〉 (3.3)

I(q) is proportional to the fluctuations of dielectric constant of similar wave vector,meaning that light scattering probes fluctuations of the refractive index with the samespatial frequency.

Fluctuation in a gas

Let consider N molecules of polarizability α in a volume V . The polarization vectorP is:

P =N

VαEi =⇒ D =

(ε0 + ρα·

)Ei =⇒ ε = ε0 +

N

Vα

Looking at eq. 3.1, we can connect the dielectric constant’s fluctuations with moleculespolarizability:

δε(r) = αρ(r) = α1

V

∑

i

δ(r − ri) δε(q,t) = α∑

i

exp[iq · ri(t)]

where ρ(r) is the molecular local density. Thus for a diluted gas eq. 3.3 becomes:

Iif (q) =I0k

4s

32π3R2ε20〈|αif |2〉S(q) (3.4)

The static structure factor is:

S(q) = 〈∑

i,j

exp[iq · (ri − rj)]〉 = 〈|δρ(q)|2〉 (3.5)

If we assume scalar polarizability and that spatial correlation between molecules in adiluted gas are neglible S(q) = N and eq. 3.4 becomes:

Iif (q) =I0k

4s

32π3R2ε20α2(ni · nf )2N (3.6)

3.3 Particulate Approach

We are interested in calculating the scattering amplitudes provided local fluctuationsof dielectric constant arise from particles dispersed in a solvent. We assume that:

1. The intensity of light scattered by the solvent is negligible compared to the onescattered by the particles.

2. Just a small fraction of the incident light is scattered by the particles so thatonly single scattering has to be considered.

3. The refractive index profile np of individual particles has orientational symmetry,meaning so that it only depends on the distance r from the center of the particles.

39


Let a fixed non-absorbing particle, dispersed in a solvent with refraction index ns,be illuminated by a planar scalar wave from the z-direction:

Ei(z) = E0 exp[i(kz − ωt)]

The scattered wave in the far field is[3]:

Es(r) = S(θ,φ)exp[ik(r − z)]

ikrEi

where S(θ,φ) is the scattering function. Since the scattered field is not usually inphase with the incident field, the scattering function is a complex one: S(θ,φ) =F (θ,φ) exp[ iΦ(θ,φ)]. Scattered intensity is:

Is(r) =F 2(θ,φ)

k2r2I0

If now consider a collection of N non-interacting particles, the total field Es(r) in apoint P is the incoherent superposition of each scattered field E i

s this because particlesare randomly distributed. The total scattered field is thus proportional to

√N and the

intensity is proportional to N , as for scattering in a diluted gas.

The previous equations could be generalized for introducing polarization effects:

(E⊥

s

E||s

)=

(S1 S2

S3 S4

)exp[ik(r − z)]

ikr

(E⊥

i

E||i

)Is =

ε0c

2

[|E⊥

s |2 + |E||s |2]

(3.7)

Along with the scattering function, we define S as the scattering matrix. Since thescattering matrix is related to the interactions between the incident field and the parti-cle’s volume elements, it depends only on the scatterer’s geometry and on palarizationsymmetries.

Let’s define the scattering cross section σs as the ratio between the total scatteredpower P =

∫dΩr2Is and the incident intensity:

σs =1

k2

∫dΩF 2(θ,φ) (3.8)

σs is equal to the extinction cross section for non-absorping particles, instead σext =σs+σabs

1 for absorping particels. Extinction efficiency Q is defined as the ratio betweenthe extinction cross section and the geometrical area of the particles. The relationshipbetween extinction cross section and scattering matrix is stated by the optical theorem:

σext =4π

k2Re[S(0)] (3.9)

3.3.1 Rayleigh scattering

If a scatterer is small compared to the wavelenght both in the particle and in thesolvent, it will see an homogeneous electric field and it will behave as an oscillatingdipole.

1We define absorption cross section as σabs =

R

dΩr2Iabs

I0

40


Refering to eq. 3.7, particle dimension should satisfy this condition:

k a << 1 =⇒ λ

2πnp<< a

since np is usually bigger than the refraction index ns of the continuous medium.For a spherical particle of radius a and scalar polarizability α, the scattering matrix

is[3]:

S(θ) =ik3α

4πεm

(1 00 cos θ

)(3.10)

Scattered intensity for unpolarized incident lihgt is:

I(q) =|α|2k4

(4πεmr)2(1 + cos2 θ)I0 α = 4πεm

(n2

p − n2s

n2p + 2n2

s

)a3 (3.11)

3.3.2 Rayleigh-Debye-Gans scattering

Let’s assume now that each particle’s element gives Rayleigh scattering indipendentlyof other elements. The total scattered field in one direction is given by the interferenceof each element’s field. It is essential that the internal field is very close to the externalone, in order that each element perceives the same incident field. Two conditions mustbe satisfied:

|np − ns| << 12π

λ|np − ns|a << 1

The first is needed for having equal amplitude, the second states that optical pathsdifferencies are negligible. Each volume element dV is a Rayleigh scatterer with thescattering matrix in eq. 3.10, where polarizability is:

dα = ε0(n2p(r) − n2

s)dV ≈ 2ε0ns(np(r) − ns)dV

The overall scattering matrix is:

S(θ) =insk

3V

2πF (q)

[1 00 cos θ

]F (q) =

1

V

∫

Vd3r(np(r) − ns) exp[iq · r] (3.12)

where F (q) arises from the differences in scattered field’s optical path for distinctelement volumes. It should be pointed out that F (q) is the Fourier decomposition ofwave vector q of the refraction index distribution inside the particle.

When incident light is unpolarized, the scattered intensity is therefore:

I(q) =V 2k4

(2πr)2[F (0)]2 P (q)(1 + cos θ)I0 (3.13)

where the form factor P (q) = |F (q)/F (0)|2 has been introduced.F (q) for an homogeneous sphere of radius a is related to the spherical Bessel func-

tion J3/2[3]:

F (q) = 3

√π

2

∆n

(qa)3/2J3/2(qa) =

3∆n

(qa)3[sin(qa) − qa cos(qa)]

In figure 3.2 are plotted some form factors of spherical colloids at different radii.

41


(a) normal plot (b) semilog plot

Figure 3.2: Rayleigh-Debye-Gans form factor for spherical colloids. np = 1.59, nw =

1.333, λ = 632.8nm

(a) normal plot (b) semilog plot

Figure 3.3: Mie and Rayleigh-Debye-Gans form factors for spherical colloids of various

sizes. Solid lines are form factors from RDG theory, scattered points are calculated by

‘Mie’ software. np = 1.59, nw = 1.333, λ = 632.8 nm

3.3.3 Mie Scattering

If the conditions for the previous scattering regimes are invalid, it will be very difficultto calculate the scattering matrix coefficients because each volume element sees a fieldwith a different amplitude. This leads to consider Maxwell equation for internal andexternal fields.

42


Mie[4] calculated the analytical expression for the field scattered by isotropic spher-ical particles of any size. I will only report the final expression.

Let’s call x = ka = 2πnsa/λ and m = np/ns. The scattering matrix is:

S =

−∑n

2n+ 1

n(n+ 1)(anπn + bnτn) 0

0 −∑n

2n+ 1

n(n+ 1)(anτn + bnπn)

(3.14)

where the functions πn and τn are related to the angular dependance of scattered light:

πn =P 1

n(cos θ)

sin θ

τn =d

dθ[P 1

n(cos θ)]

(3.15)

P 1n are the Legendre function of the first order.

The coefficients an = f(x,m) and bn = f(x,m) are the weights for the angulardependance functions:

an =mψn(mx)ψ′

n(x) − ψn(x)ψ′n(mx)

mψn(mx)ξ′n(x) − ξn(x)ψ′n(mx)

bn =ψn(mx)ψ′

n(x) −mψn(x)ψ′n(mx)

ψn(mx)ξ′n(x) −mξn(x)ψ′n(mx)

(3.16)

ψn(ρ) and ξn(ρ) are the Riccati-Bessel functions, which are related to the first orderBessel spherical functions jn(ρ), and to the ones of the third order (Henkel function)

h(1)n (ρ). The latter are complex functions.

ψn(ρ) = ρjn(ρ)

ξn(ρ) = ρh(1)n (ρ)

(3.17)

It is possible to implement a procedure for calculating matrix elements on a com-puter, taking advantage of two useful properties. an and bn fall to zero very rapidly,thus the sum in eq. 3.14 could be arrested at nmax = x+ 4x

1

3 + 2; it exists a ricorsivedependance between the angular functions:

πn =2n− 1

n− 1cos θπn−1 −

n

n− 1πn−2

τn = n cos θπn − (n+ 1)πn−1

π0 = 0

π1 = 1

In figure 3.3 are plotted some form factors of spherical colloids at different radii,calculated with the ‘Mie’ software I have developed in Labwindows-CVI with GnuScientific Library.

In the sum of eq. 3.16, the terms which nullify the denominators will overcomethe others. These quasi-resonances2 give rise to the so called fine ripple structure[5]behaviour of the extinction efficiency, see figure 3.4. In the figure, extinction coefficiencyvalues are obtained starting from calculations of S(0) with Mie software, in agreementwith the optical theorem.

2It is incorrect to speak of resonances because the denominators of eq. 3.16 go to zero only for

imaginary refraction indexes. Thus it requires particles which only absorb light. However, some terms

are bigger than others thus extinction efficiency shows peaked features.

43


(a) (b)

(c) (d)

Figure 3.4: Behaviour of the extinction cross section at the fixed wavelenght λ = 560nm

for polistirene particles (np = 1.59) in water. (b) Logarithmic plot of Qext, which scales

as ∼ x−4 for small x, as predicted by Rayleigh scattering. In figure (c) and (d) is

possible to see the fine structure and the resonances.

44

3.4 Scattering from interacting particles Light Scattering Techniques

3.4 Scattering from interacting particles

We are interested in calculating the instantaneous amplitude of field of the light scat-tered by an assembly of N particles. Let’s assume, for simplicity, that particles are

isotropic and consider E(⊥)i and E

(⊥)s only. The total field is the superposition of each

particle scattered field:

Es(q) =N∑

i

E0Si(θ)exp[iq · ri(t)]

ikri(t)

where ri(t) is the position of the particle i at time t. Let’s rewrite the previous equationas:

Es(q,t) = E0

N∑

i=1

bi(q) exp[iq · ri(t)] bi(q) = S1(q)

The instantaneous intensity of the scattered light is therefore:

Is(q,t) = |Es(q,t)|2

with an average value given by:

Is(q) = I0

N∑

i,j=1

〈bi(q)bj(q) exp [iq · (ri − rj)]〉 (3.18)

with an ensemble average on the right side of the previous equation.

For monodisperse particles bi(q) = b(q), the average scattered intensity can bewritten as:

Is(q) = N [b(0)] 2 P (q)S(q) (3.19)

where P (q) is the single particle form factor introduced in eq. 3.5:

P (q) ≡∣∣∣∣b(q)

b(0)

∣∣∣∣2

(3.20)

and S(q) is the structure factor, which reflects correlation of particle positions, affectedfrom inter–particle mutual interaction:

S(q) ≡ N−1N∑

i,j=1

〈 exp [iq · (ri − rj)] 〉

It is possible to relate S(q) with the well known pair distribution function g(r), whichcounts the number of molecules at distance (r,r + dr) from a fixed particle:

S(q) = 1 + ρ

∫d3r exp(iq · r)g(r) = 1 + F [g(r)] (3.21)

ρg(r) =1

N

∑

i6=j

δ(r − ri + rj) (3.22)

45

3.5 Scattering by fractal aggregates Light Scattering Techniques

3.5 Scattering by fractal aggregates

Weitz and Oliveria[6] show that colloidal aggregation leads to fine branched structuresthat show fractal features and auto-similarity properties. The latter can be measuredcounting the number n(R) of colloidal particles with radius R0 inside a fractal objectin a spherical volume of radius R:

n(R) =

(R

R0

)df

(3.23)

The adimensional quantity df is the fractal dimension, which is a fractional number forfractal objects. The Gyration radius is another useful quantity for describing colloidalaggregates. It is defined as:

Rg =

∑imir

2i∑

imi= R0n

1

df (3.24)

where n is the number of monomers of radiusR0 and massM0 which form the aggregate.From this equation is possible to calculate aggregate’s mass:

M = M0

(Rg

R0

)df

Texeira[7] showed that the pair correlation function for a real fractal object is:

g(r) ∼ r−α exp

(− r

Rc

)(3.25)

where the exponential introduces a cutoff lenght which is due to the finite extension ofa real system. The pair correlation function is indeed related to n(r) by:

n(R) ∼∫ R

0d3rρg(r) ∼ R3−α α = 3 − df

Since aggregating particles have usually small radii, they are in the Rayleigh-Debye-Gans scattering regime. It is interesting to point out that if the aggregating particleof radius R0 satisfy RDG criteria 2π

λ |np − ns|R0 << 1, then the aggregates satisfy thesame criteria too. Indeed, refraction index of an aggragate with gyration radius Rg is:

naggr(Rg) = φV (Rg)np + (1 − φV )ns

where φV (Rg) =

(R0

Rg

)3−df

is the volume fraction occupied by the single particles.

Since df ≈ 2, the bigger the gyration radius grows the smaller the refraction indexmismatch becomes.

The scattered light intensity obeys to eq. 3.19, in which P (q) could be associatedto the form factor of single colloids and S(q) to the structure factor of aggregates,thus P (q) and S(q) refer to objects with very different lenght scales. This allow us tomeasure S(q) from light scattered intensity distribution, neglecting P (q). The structurefactor of an aggregate is the Fourier trasform of eq. 3.25[7]:

S(q) ' 1

(qa)df

dfΓ(df − 1)

(1 + (qRg)−2)df−1

2

sin[(df − 1) arctan(qRg)]

46

3.5 Scattering by fractal aggregates Light Scattering Techniques

This is usually simplified with the Fisher–Burford function[8, 9]:

S(q) ∼ 2(Rg/a)

df

(1 +

2q2R2g

3df

) df

2

(3.26)

The two asymptotic behaviours of the Fisher–Burford function are:

qRg << 1 −→ S(q) ' S(0) qRg >> 1 −→ S(q) ' (qRg)−df (3.27)

Thus, I(q) is constant for low q and its intercept is related to the average aggregates’mass[10]. At higher q, the scattered intensity scales as q−df . The interpolation of thetwo asymptotic function gives the average gyration radius < Rg > as in figure 3.5

Figure 3.5: Scattered intensity from a colloidal aggregation system. Two regions are

cleary visible: at low q intensity is constant, at higher q it scales as q−df .

47

3.6 Statistics of the Scattered Field Light Scattering Techniques

3.6 Statistics of the Scattered Field

In section 3.2 we stated that the scattered field arises from the contribution of eachparticle in the scattering volume. If particles positions are independent, every contri-bution is added with a random phase, though with the same amplitude. The scatteredfield is therefore to be considered in terms of a fluctuating random variable whose timedependence is determined by the instantaneous configuration of the scatterers. It canbe expressed as the result of a fixed steps random walk in the complex plane, leadingto a gaussian variable with a zero mean square value.

It is known that a monochromatic electromagnetic plane wave is perfectly coherent,meaning that the value of the electric field associated can be exactly predicted inevery point of the wave front at every time of the propagation. Since no wave isperfectly monochromatic, such predictability is restricted to time intervals known asthe coherence time τc, and similarly to coherence lengthscales `c = cτc. In many lightscattering circumstances we are interested to “measure” the coherence associated tothe scattered field. The expression

G1(t,τ) = 〈E∗(t)E(t+ τ) 〉 (3.28)

gives a quantitative estimate of the (first order) time coherence of the electric field,where the product E∗(t)E(t+ τ) is averaged over an ensemble of different systems. Inthe (usually fulfilled) ergodic approximation it is possible to repeat the average manytimes on the same system. Accordingly to Glauber frameworks [11], let us consider apoint-like light source and define the time coherence degree of first order, of the emittedfield E(t):

g1(τ) ≡ 〈E∗(0)E(τ) 〉〈E∗(τ)E(τ)〉 (3.29)

The case of a monochromatic light source with bandwidth ∆ω gives thus a time cor-relation function g1(τ) decaying to zero over the coherence time τc = 2π/∆ω. Whiletime coherence is related to the emission mechanism of the light source, the spatialcoherence depends solely from the source geometry: let us consider the linear source Lof Fig. 3.6 and let us consider the observation P at distance R in the far field, so thatRλ/L2 1

Figure 3.6: Coherence length generated from a linear light source.

48


The phase difference between P and P ′ is

∆φ =2π

λ(AP ′ −AP −BP ′ +BP ) =

2π

λ

∆PL sin θ

R(3.30)

The displacement ∆P causing a phase shift ∆φ = π is :

∆P =λR

2L sin θ(3.31)

defining a coherence angle θc = ∆P/R we may observe that θc is essentially the lightsource diffraction aperture as viewed in P . Therefore if we consider a light sourceL = 1 mm emitting at λ = 0.5µm we can observe at R = 1 m a coherence areaAc ∼ 0.25 mm2 persisting for a time scale of τc. In other words placing a screen at Rwe do not observe a uniform distribution of light intensity, but light and dark spotsknown as ‘speckle pattern’ evolving in time over the coherence time τc.

When a laser light is scattered by an amorphous assembly of particles the instanta-neous far–filed pattern of scattered radiation constitutes indeed a random diffraction or

‘speckle’ pattern. Each speckle subtends a solid angle of the order of (λ/V1/3s )2 where

λ is the light wavelength and Vs is the scattering volume. The instantaneous intensityof a speckle is given by |Is(q,t)| = |Es(q,t)|2. The correlation function of the scatteredfield is:

〈E (q,0)E∗ (q,τ)〉〈I (q)〉 =

F (q,τ)

S(q)≡ g1 (q,τ) (3.32)

where F (q, τ) is the coherent intermediate scattering function:

F (q,τ) =[N b2(q)

]−1 ∑

i,j

〈 bi bj exp ( iq · [ri(0) − rj(τ)] ) 〉 (3.33)

In a dilute enough suspension interactions can be neglected (S(q) ≈ 1) and the positionsof different particles are uncorrelated. The cross terms (i 6= j) average to zero andthe intermediate scattering function is in practice a measure of the field correlationfunction:

g1 (q,t) ' F (q,τ) =[Nb2(q)

]−1∑

i

⟨b2i exp ( iq · [ri(0) − ri(τ)] )

⟩

The displacement [ri(0) − ri(τ)] is a gaussian variable with a mean square value givenby [12]:

〈[ri(0) − ri(τ)]〉 = 6D0,i τ

D0,i is the free particle diffusion coefficient of a particle of radius Ri:

D0,i =kBT

6πη0Ri(3.34)

where η0 is the shear viscosity of the suspension medium.Thus considering monodisperse particles in brownian motion, g1 becomes:

g1 (q,τ) =[Nb2(q)

]−1N∑

i=1

bi exp[−D0q

2τ]

(3.35)

49


Since the (3.35) has an exponential decay, experimental results are often plotted asln (g1(q, τ) against τ or q2 τ . A useful quantity is the initial slope Γ(q) of this plot,that is the first cumulant of (g1(q, τ):

Γ(q) = limτ→0

d

dτln[(g1(q, τ)

](3.36)

Figure 3.7: Time correlation function for a dilute colloidal system in brownian mo-

tion. Fitting experimental data with a two parameters exponential leads to diffusion

coefficient extimation (first cumulant) and particle size distribution variance(second

cumulant)[13]

50


References

[1] M. Kerker. The Scattering of Light and other electromagnetic radiation (AcademicPress, New York) (1969).

[2] B. Chu. Molecular Forces (Wiley, New York) (1967).

[3] H. C. V. de Hulst. Light Scattering by Small Particles (Dover, New York) (1981).

[4] G. Mie. “Beitrage zur optik truber medien speziel kolloidaler metallosungen.”Annal Physics, 25, (1908), 377–445.

[5] C. F. Bohren and D. R. Huffman. Absorption and scattering of light by smallparticles. (John Wiley & Sons) (1980).

[6] D. A. Weitz and M. Oliveria. “Fractal Structures Formed by Kinetic Aggregationof Aqueous Gold Colloids.” Physical review letters, 52, (1984), 1433–1436.

[7] J. Texeira. “Experimental methods for studying fractal aggregates.” In H. Stanley,editor, “On growth and form,” (martinus Nijhoff) (1986).

[8] G. Dietler, C. Aubert, D. S. Cannell and P. Wiltzius. “Gelation of ColloidalSilica.” Physical Review Letters, 57, (1987), 3117–3120.

[9] M. E. Fisher and R. J. Burford. “Theory of Critical-Point Scattering and Corre-lations. I. The Ising Model.” Physical Review , 156, (1967), 583–622.

[10] M. Carpineti. Dinamica “spinodale” e transizione tra cinetiche universali in pro-cessi di aggregazione colloidle. Ph.D. thesis, Universita degli studi di Milano(1994).

[11] R. J. Glauber. “The Quantum Theory of Optical Coherence.” Phys. Rev., 130,(1963), 2529–2539.

[12] B. J. Berne and R. Pecora. Dynamic Light Scattering (New York: Wiley) (1976).

[13] D. E. Koppel. “Analysis of macromolecular polydispersity in intensity correlationspectroscopy: the method of cumulants.” J. Chem. Phys., 57, (1972), 4814–4820.

51

Interfacial tension apparatus

Abstract

After introduce basic concepts of interfacial tension for a small droplet ina liquid/gas, the equation of Young and Laplace is introduced. Yeung’s‘micropipette method’, a new technique to measure interfacial tension formesoscopic droplets is discussed together with its importance for emulsionsmeasurements. Large emphasis is devoted to the development of a similarapparatus and to the description of the crucial components.

52

4.1 Introduction Interfacial tension apparatus

4.1 Introduction

It happens often, in our everyday life, to see a water droplet falling on a table: once onthe surface of the table, the droplet either keeps its oval shape or it stretches out be-coming a thin film. The fate of the falling droplet depends on the interactions betweenthe molecules of water and the ones on the surface of the table. A single moleculeinside a fluid experiences an isotropic distribution of similar molecules that resultingin no overall forces acting on it. If the molecule is brought at the interface betweentwo different substances, it will experience an anisotropic distribution of matter orig-inating a net force. For example, a water molecule at the liquid-vapour interface seesmore molecules inside the liquid than in the vapour phase, thus it feels a force directedtowards the liquid.

The surface tension is defined as the energy necessary to bring a molecule from thebulk phase to the surface of a fluid at equilibrium with its vapour. When the vapour issubstituted with another immiscible fluid it is called interfacial tension. The interfacialtension is a physical quantity of paramount importance in any application where twoimmiscible fluids are in contact or in the presence of surfactant agents. It is crucial,for instance, in order to understand phenomena such as the adsorption of stabilisers,the drainage of thin films and the formation of new emulsion drops[1, 2, 3].

This physical quantity was not directly measured on emulsion droplets until someyears ago but was instead inferred from measurements that involved samples sizingmillimetres or more. Some examples of such bulk techniques are the Wilhelm plate,the Du Nouy ring, the dynamic drop volume method, and the drop shape method[4, 5].However such an extrapolation is incorrect. Since almost all interesting systems containsurfactants or partitioning agents, any difference in the agents distribution between thebulk and the interfacial phases is leading towards distinct interfacial tension values. Ithas been demonstrated[6] that partitioning effects depend upon the physical interfacialarea beetwen the two system in the presence of surfactants. So emulsion’s interfacialtension can be measured only with tensiometric techniques acting on the micrometerscale.

In this chapter I will describe the development of an experimental apparatus, orig-inally conceived by Yeung[6, 7, 8], suitable for measuring the interfacial tension of asingle emulsion droplet. This technique, borrowed from the field of Biophysics [9], in-volves capturing single droplets with a glass micropipette and measuring the pressureneeded to suck in the droplet. Moreover, it allows to perform simple but meaningfulmechanical experiments on micrometric droplets or vesicles[10, 11, 12].

4.2 Young and Laplace’s equation

Let’s consider a soap bubble in air: in the absence of any external field, such as gravity,it assumes a spherical form to minimise the surface area. If the bubble radius is r andthe surface tension is σ, the total surface free energy is E = −− 4πr2σ. If we assumethat the bubble undergoes a surface reduction, its radius diminishing by dr, then thechange in the free energy is dE = 8πrσdr. As any reduction in the surface reduces thefree energy, the bubble will vanish unless there is a pressure difference ∆P , across thefilm which balances the free energy reduction.

Let’s assume now a generic curved surface as in figure 4.1 with two radii of curvature

53

4.3 Interfacial tension and measurement method Interfacial tension apparatus

Figure 4.1: An arbitrarily curved surface

R1 and R2. If the surface is displaced by a small distance outward, the change in thearea will be:

dA = (x+ dx)(y + dy) − xy = xdy + ydx

The energy spent to broaden the surface is:

dE = σdA

The pressure across the surface acts on an area xy and trough a distance dz. Thecorresponding work is therefore:

dW = ∆Pxydz

From figure 4.1, it could be seen that:

x+ dx

R1 + dz=

x

R1=⇒ dx =

ydz

R1

y + dy

R2 + dz=

y

R2=⇒ dy =

xdz

R2

The surface is at mechanical equilibrium when the surface free energy variation equalsthe work made by the pressure, thus giving the equation of Yeoung and Laplace:

∆P = σ

(1

R1+

1

R2

)(4.1)

In the case of a spherical bubble, R1 is equal to R2, thus:

∆P =2σ

R(4.2)

4.3 Interfacial tension dependence on geometrical attri-

butes of the system

If we consider a generic oil–water system, its geometrical attributes are the oil volumefraction Voil, the water volume fraction VH2O, and the interfacial area A. Let’s assume

54

4.3 Interfacial tension and measurement method Interfacial tension apparatus

that a surfactant is dispersed only in the oil phase with an initial concentration C0. Atequilibrium, the surfactant will be partitioned between the two bulk phases and thesurface. The mass balance of the surfactant is:

Voil(Coil − C0) + VH2O(CH2O) + ΓA = 0 (4.3)

where Coil and CH2O are the surfactant concentrations at equilibrium, in water and oilrespectively, and Γ is the interface concentration (mol/m2), also called surface excessconcentration.

The three concentrations are interrelated through a chemical potential balance forsurfactants in the bulk phases and the surface. Assuming an initial concentrationbelow the CMC1, the surfactant concentrations in the two bulk phases are related tothe partition coefficient Kp. Chemical balance leads to:

Kp =CH2O

Coil=vH2O

voilexp

(−µ

(0)H2O − µ

(0)oil

KBT

)(4.4)

where vH2O and voil are the molar volumes and µ(0)H2O and µ

(0)oil are the standard chemical

potentials.Assuming a monomolecular adsorption at the interface and non–interacting sur-

factant molecules, chemical equilibrium betwen the oil phase and the surface leads toLangmuir isothermal adsorption:

Γ

Γ∞=

KlCoil

1 +KlCoil(4.5)

where Γ∞ is the saturation interfacial coverage andKl is the affinity coefficient, which isthe ratio of adsorption and desorption rates Kl = Ka

Kd. The surface excess concentration

Γ is related to the interfacial tension by the Gibbs equation[4]:

Γ = −Coil

RT

dγ

dCoil(4.6)

which leads to the Szyszkowski[13] equation for the interfacial tension:

γ = γ0 −KBTΓ∞ ln(1 +KlCoil) (4.7)

Surfactant concentration in the oil phase is calculated using equations 4.3–4.5, andthus is[6]:

2KlCoil =

√[1 + φ2(1 − C)2 + 2φ(1 + C

]−[1 + φ(1 − C)

](4.8)

where the two dimensionless quantities φ and C are defined as:

φ =KlΓ∞

(Voil/A) +Kp(VH2O/A), C =

C0

Γ∞

Voil

A

1CMC stands for critical micellar concentration. It is the concentration at which surfactant

molecules begin to form aggregates, also called micelles. The molecules in the micelle are all orientated

to screen the group that hates he solvent.

55

4.4 Micropipettes Method Interfacial tension apparatus

Substituting equation 4.8 in equation 4.7 shows that the interfacial tension dependsupon the geometrical attributes Voil/A and VH2O/A. Passing from ‘bulk’ sample toemulsions changes the interfacial area A of several order of magnitude: so it is incor-rect to extrapolate the interfacial tension for an emulsion from a measurement on abulk sample. Experimental data, taken by Yeung[6], show that the difference betweendifferent method could be greater than the measured value.

4.4 Micropipettes Method

This method allows to perform mechanical test on a micrometric droplet to measureits interfacial tension. A capillary micropipette, partially filled with the continuousfluid, is brought into contact with a disperse droplet using micro-manipulators, anda controlled negative pressure ∆P is progressively applied to the pipette until thedroplet is sucked in. The whole procedure is monitored by visualisation under aninverted microscope, simultaneously allowing for measurement of the droplet radius Rand the negative pressure applied.

(a) (b)

Figure 4.2: (a) Pressure balance on the droplet during aspiration. (b)The dispersed

fluid wets the micropipette with a contact angle θ

Let’s consider the droplet of radius Rd in figure 4.2 during aspiration inside themicropipette of radius Rp and calculate the pressure balance. The pressure inside thecapillary, P3 = PM +Pa, is given by the sum of the ambient pressure Pa and the suctionpressure. P1 = 2γ

Rdand P2 = 2γ

R are the Laplace pressures inside the droplet and R isthe radius of curvature of the meniscus inside the capillary.

Rising the suction pressure makes the meniscus radius grow until it matches thecapillary size at the critical pressure Pcr; for higher pressures the whole droplet ispulled in the micropipette. If the suction pressure is changed slowly we can considerthe system as always in equilibrium. Calculating the pressure balance when Pcr isapplied, leads to:

2γ

Rd− 2γ

Rp+ Pcr = 0 =⇒ γ =

RpPcr

2(1 − Rp

Rd

) (4.9)

Being based on threshold detection, the micropipette method is quite precise. Further-more, multiple measurements on many droplets, which can be performed in a relatively

56

4.5 Experimental setup Interfacial tension apparatus

short time span, improve the statistical accuracy.

The previous equation is valid under two conditions. It has been assumed thatthe pressure at the droplet level is equal to atmospheric pressure, thus neglecting thedependence from the height of the continuous fluid. This is correct provided thatthe micropipette is kept slightly under the continuous fluid’s surface. The secondassumption is that the dispersed fluid doesn’t wet the capillary. If the fluid wets orpartially wets the micropipette, as in figure 4.2, the equation 4.9 is incomplete since itneglets the surface forces arising at the liquid-glass interface. In this case, the pressureP2 is[4]:

P2 =2γ cos θ

Rp

It would thus be necessary to know the contact angle to measure the interfacial tension.To overcome this problem it is necessary to treat chemically the capillary in order toavoid wettability.

4.5 Experimental setup

Figure 4.3: The micropipettes apparatus in our lab

The apparatus I have developed is mounted on the inverted microscope OlympusIX70, equipped with the long distance condenser Olympus U-UDCB-2. Two micro-capillaries are mounted on two holders and controlled by manual micro-manipulators(Narishige MN-151) fixed to the microscope stand. One of the two pipettes is used toform droplets of an immiscible liquid with a controlled volume (down to a few tens ofpicoliters) in a continuous phase contained in a Petri dish using a digital micro-injector(Narishige IM300). The pre-formed drop is transferred from the first to the secondmicropipette by applying negative pressure ∆P on the latter. This is accomplished

57

4.5 Experimental setup Interfacial tension apparatus

using a simple syringe, acting as air buffer, connected to the suction pipette via PTFEtubing. ∆P is measured with a precision miniaturised strain-gauge (Druck PDCR42)hermetically jointed to the pipette holder and the pressure signal is passed to a digitalprocessing unit, calibrated in the ±1 bar pressure range and connected to a PC via aserial port (protocol RS232).

The whole procedure is controlled by visualisation under the inverted microscope,simultaneously allowing for measurement of the droplet radius and the negative pres-sure applied. Droplet is photographed with a CCD (Olympus DP50); the image isanalysed with the software Image ProPlus (Media Cybernetics).

(a) 5µm (b) 15µm

Figure 4.4: Micropipettes SEM images

Micropipettes Expression 4.9 is based on a well-defined micropipette geometry,which must have a flat, sharp-cut circular edge, unlike those commonly used in micro-biology for cell-holding or injection. Obtaining micropipettes with precise, controlled,and reproducible geometry using laboratory pipette pullers and forges is far from easy.Increasing application of micropipettes in microbiology has however lead producers tooffer custom-made micropipettes for specific needs. Our micropipettes, see figure 4.4are custom-made by Eppendorf (Germany) and it have a terminal part with an internaldiameter of either 5 or 15 µm. They are bent to an angle of 30 allowing the pipettetip to lie parallel to the microscope table.

Micropipettes must be chemically treated to avoid wetting by the dispersed fluid.To accomplish that, in the case of measures of the interfacial tension of oil dropletsin water (o/w), the pipette is filled with a HCl solution (φV = 10%) for ten minutesand the rinsed with distilled water, whilst for water-in-oil (w/o) emulsions we silanizethe capillaries keeping the glass in a solution of dichlorodimethylsilane (C2H6Cl2Si) incyclohexane (φV = 5%) for thirty minutes and then rinsing with cyclohexane.

Pressure sensor PDCR42 The pressure sensor PDCR42 (Druck) shown in fig-ure 4.5 measures the relative pressure applied on it. It exploits a piezoresistive siliconcrystal which undergoes a resistance change when its sensitive area is deformed bypressure and to measure the strain the piezo is integrated in a Wheatstone bridge cir-cuit. Refering to the picture 4.5, R1 is the piezo’s resistance to be measured whilethe other three are well known and are equal to R1 when ambient pressure is applied.

58

4.6 Calibrating measurements Interfacial tension apparatus

(a) Pressure sensor (b) Sensor section (c) Wheatstone Bridge

Figure 4.5: Pressure Sensor PDCR42

Calculation of the voltage V0 when voltage bias VB is applied leads to:

V0 =R1

R4− R2

R3(1 + R1

R4

)(1 + R2

R3

)

If all the resistances are equal, V0 is be zero and the bridge is said to be balanced.However if R1 is shifted by a change in the piezo’s strain, a voltage will appear acrossthe middle of the bridge. Thus gauging V0 allows us to know the piezo’s resistance andthe pressure applied on it.

The sensor is hermetically jointed to a custom chamber welded to the pipette holder,in order to measure the pressure inside the micropipette, see figure 4.5.

4.6 Calibrating measurements

The validity of our apparatus can be established measuring the interfacial tensionbetween water and several non polar liquids. Since pure liquids are involved, theinterfacial tension is a property of the two fluids independent from the measurementmethod adopted. We measured the interfacial tension between water and octanol(Aldrich), toluene (Aldrich), hexane (Aldrich) and iso-octane (Aldrich). The nonpolarliquids are used as received.

Using the micro-injector connected to the 5 µm internal diameter micropipette,micrometer-sized oil droplets are created inside a Petri dish filled with distilled water.A 15 µm internal diameter micropipette, previously acid-cleaned and connected tothe pressure sensor, captures the droplet and performs the measurements. Table 4.1shows the comparison between experimental data and the corresponding literaturedata[14], the standard deviation are calculated on measurements on ten droplet. Theexperimental data are in good agreement with the literature.

4.7 Surfactant adsorption

We have measured the interfacial tension of hexane dispersed in a solution of thesurfactant triton-X100 (Sigma) in water at different concentration. Solutions of distilled

59

4.7 Surfactant adsorption Interfacial tension apparatus

(a) (b) (c)

Figure 4.6: (a) The droplet is captured by the capillary. (b) Suction pressure is being

applied. (c) Critical pressure is applied.

Nonpolar liquid Literature value (mN/m) Experimental data (mN/m)

Octanol 8.52 8.32±0.34

Toluene 36.1 35.84±0.35

Hexane 51.3 51.87±0.69

Iso-octane 51.8 51.75±0.28

Table 4.1: Interfacial tension of non polar liquids and water in absence of surfactants

at room water

water and triton at different concentrations are prepared and transferred in a Petri dish.Hexane droplets are made with the aid of a micropipette and the micro-injector, asdescribed in the previous section. Droplets’ interfacial tension is measured with themicropipettes method, using a capillary with 15 µm internal diameter. Experimentaldata are shown in figure 4.7 and in table 4.2.

While the surfactant concentration is below CMC, the system could be described interm of dynamical adsorption and desorption at the interface, thus interfacial tensioncan be inferred from Szyszkowski equation:

γ = γ0 −KBTΓ∞ ln(1 +KC)

Thus knowing γ0 from literature and fitting experimental data with the previous equa-tion gives the surface excess concentration and the affinity coefficient K. We obtainK = 3.33 1010cm3/mol and Γ∞ = 2.31 10−6mol/m2.

While the surfactant concentration is above CMC, the interfacial tension of thedroplets is constant. In this second region the experimental data can be fitted witha constant. The two fitting curve are intersecting at CMC concentration: cCMC =0.19mM .

60


Figure 4.7: Interfacial tension of a surfactant stabilised emulsion of hexane in water

measured with the micropipettes method.

61


c (mol/l) γ (mN/m) c (mol/l) γ (mN/m)

0 51.8±0.99 10−4 4.10±0.74

10−6 32.0±2.7 2 · 10−4 1.60±0.18

3 · 10−6 26.9±0.69 3 · 10−4 1.64±0.16

5 · 10−6 22.9±1.7 5 · 10−4 1.69±0.24

10−5 19.3±1.5 7 · 10−4 1.58±0.12

3 · 10−5 11.00±0.49 10−3 1.61±0.04

5 · 10−5 8.72±0.49

Table 4.2: Micropipette method measurements of hexane-water interfacial tension in

the presence of the surfactant Triton-X100 at different concentrations

62


References

[1] E. Ruckenstein. “Interfacial free energy, surface excess, and stability of macroemul-sions.” Langmuir , 4, (1988), 1318–1321.

[2] Tadros and Vincent. “Emulsion stability.” In P. Becher, editor, “Encyclopedia ofEmulsion Technology,” vol. 1, chap. 3 (Marcel Dekker) (1983).

[3] Overbeek, Lekkerkerker, Verhoeckx and D. Bruyn. “On understanding microemul-sions : II. Thermodynamics of droplet-type microemulsions.” Journal of ColloidInterface Science, 119, 2, (1987), 422–441.

[4] A. T. Adamson. Physical Chemistry of Surfaces, chap. 2 (John Wiley & Sons), 5ed. (1990).

[5] F. Ravera, M. Ferrari and L. Liggeri. “Adsorption and partitioning of surfactantsin liquid–liquid systems.” Advances in Colloid and Interface Science, 88, (2000),129–177.

[6] Yeung, Dabros and Masliyah. “Does Equilibrium Interfacial Tension Depend onMethod of Measurement?” Journal of Colloid Interface Science, 208, 1, (1998),241–247.

[7] K. Moran, A. Yeung and J. Masliyah. “Measuring interfacial tensions ofmicrometer-sized droplets: a novel micromechanical technique.” Langmuir , 15,(1999), 8497–8504.

[8] J. L. Drury and M. Dembo. “Hydrodynamics of Micropipette Aspiration.” Bio-phys. J., 76, (1999), 110–128.

[9] E. Evans and R. Skalak. “Mechanics and thermodynamics of biomembranes: part2.” CRC critical reviews in bioengineering , 3, (1979), 331–418.

[10] A. Yeung, T. Dabros, J. Masliyah and J. Czarnecki. “Micropipette: a new tech-nique in emulsion research.” Colloids Surf. A, 174, (2000), 169–181.

[11] A. Yeung, T. Dabros, J. Masliyah and J. Czarnecki. “On the interfacial propertiesof micrometre-sized water droplets in crude oil.” Proceedings of the Royal SocietyA, 455, 1990, (1999), 3709–3723.

[12] V. Heinrich and W. Rawicz. “Automated, High-Resolution Micropipet AspirationReveals New Insight into the Physical Properties of Fluid Membranes.” Langmuir ,21, 5, (2005), 1962–1971.

[13] C. H. Chang, Franses and I. Elias. “Adsorption dynamics of surfactants at theair/water interface: a critical review of mathematical models, data, and mecha-nisms.” Colloids and Surfaces A, 100, (1995), 1–45.

[14] R. C. Weast, editor. CRC Handbook of chemistry and physics (CRC press), 69ed. (1998).

63

Medium angle light scattering

apparatus

Abstract

We present here our custom developed apparatus for measuring scatteredlight at low-medium angles with a charged coupled device (CCD) camera.We discuss in details all the experimental aspects of the apparatus, de-scribing the laser source, the optical detection system, the CCD camera,the data processing and the acquisition/processing software. The calibra-tion of linearity, dynamic range and accuracy of the instrument is carriedout by using pinholes. At the end of the chapter we show the measuredform factor of polystyrene spheres dispersions in water.

64

5.1 Introduction Medium angle light scattering apparatus

5.1 Introduction

We have seen in chapter 3 that the scattered light intensity I(q) in the far field isrelated to the local fluctuations of dielectric constant ε with spacial frequency q. Forelastic scattering the wave vector q depends only on the wavelength λ, the scatteringangle θ and the medium refraction index ns:

q =4π

λns sin

(θ

2

)

Scattered angles distribution is isotropic only for Rayleigh scatterers, whereas the scat-tering cone is usually very narrow for samples that present inhomogeneities on micronlenght scales or larger. However, there are a variety of interesting systems in this sizerange, as the ones encountered in colloidal aggregation, polymer blends, gel formation,and phase transition in critical systems. The measurement of scattered light at lowwave vectors is usually known as the small angle light scattering technique.

Static light scattering is usually measured with a phototube mounted on a go-niometer, in order to detect scattered intensity at different angles. Although the pho-totube allows exquisite sensitivity, its geometrical volume prevents measurements below15 −−20 degrees.

The simplest experimental method to measure low angle scattered light is to forwardtransmitted and scattered light on a screen in which a small hole allows the former topass[1]. A CCD images the screen measuring the intensity distribution. This setup isvery simple since it doesn’t need any lens, but it has a serious drawback: most of thelight is lost on the screen since it is too low to be acquired by the CCD.

A better setup, developed by Ferri[2, 3], applies two lenses for impinging the scat-tered light directly on the CCD whilst the transmitted beam is reflected away of theoptical path by a beam stopper (see figure 5.1). Considering a geometrical optics point

Figure 5.1: The lens closer to the CCD conjugates the focal plane of the other lens

with the CCD plane. The transmitted beam is blocked by the beam stopper. The light

scattered at similar angles is focused on the CCD at equal distance from the optical

axis

of view, the lens (LA) closer to the sample focuses the scattered beam at angles θ in

65

5.2 Experimental setup Medium angle light scattering apparatus

a very narrow ring of radius r = fA tan θ, where fA is the focal lenght of the lens LA.Therefore this lens relates the angular distribution oflight intensity to the radial distri-bution in the focal plane. This remains valid also from the Fourier optics point of view,because the intensity distribution in the focal plane is related to the energy spectrumof the scattered light[4]. However, the light intensity can’t be measured by a CCDin this plane because the transmitted beam is several orders of magnitude strongerthan the scattered one and it raises blooming problems. In order to remove only thetransmitted light without affecting the scattered intensity distribution, a beam stopperis inserted at the intersection between the optical axis and the focal plane of the lensLA. The intensity distribution on the focal plane is imaged on the CCD’s sensor planeby means of the second lens LB. The distances a and b are fixed according to the lens’law, so that 1/a+ 1/b = 1/fb. Our laboratory provides a similar apparatus, developedby Luca Ghiringhelli[5], which measures the light scattered at scattering wave vectorsin the 80–104 cm−1 range.

In this chapter I will describe the custom apparatus I have developed to detectscattered light in the intermediate range between the ones probed by the small angleapparatus and the ones measured by the standard phototube configuration. The ap-paratus exploits three lenses, where one is the Fourier lens which plays the role of LA,and the other two form an inverted telescope system, which demagnifies the scatteringangle. The optical detection system of this apparatus is fully based on the work ofLuca Cippelletti, still unpublished.

5.2 Experimental setup

The general layout is decribed in pictures 5.2 and 5.3. The key-feature is to exploitan two lenses-inverted telescope system as an angle demagnifier, in order to increaseangle acceptance on the CCD sensor. The optical elements, except the CCD and thebeam stopper, are mounted on an X26 Newport rail system which, togheter with anX26 optical mount, are designed to mantain the optical axis at 40 mm above the rail.

Light source The light source is a 17.5 mW single-mode fiber-coupled (NA = 0.11)diode laser FiberMaxTM(Bluesky research), with λ = 645−−665 nm and λ = 658 nm.A fiber’s FC connector is mounted on a fiber collimator (OZ optics) with 6 mm of focallenght, so that the collimated beam diameter at 1/e2 is:

d = 2NA · f = 1.32 mm

and the diverging angle is slighty less than 1 mrad. A fiber collimator is inserted in agimbal mount (Newport) to allow precise alignment.

A laser diode driver MPL250 (Wavelength electronics) drives and controls the laserdiode, both in constant power and constant current mode. The driver’s voltage signalsare controlled by a computer via an analog-digital (A/D) multifunction I/O board(National Instrument) and a 12 turn trimpot adjusts the laser diode output power.The diode is kept in thermal contact with a copper heat exchanger in which waterfrom a thermal bath flows. This solution is necessary because the diode’s temperaturecould rise considerely on use (up to ten degrees in several hours), changing the outputpower and the laser wavelenght.

66


(a)

(b) (c)

Figure 5.2: (a)Technical draw of medium angle light scattering apparatus. (b) Detail

of the fiber output. (c) Detail of the beam stopper.

Since the output beam polarization depends strongly on the fiber’s geometricalconfiguration, it is necessary to insert a vertical oriented polarizator on the opticalpath to ensure a stable polarized beam. This is strictly necessary because scatteredintensity depends on incident field polarization, see chapter 3.

The output beam is attenuated by a slighty tilted 0.1-density filter which, actingas a beam sampler, reflects part of the incident light on a photodiode connected to theA/D board. The density filter is located after the polarizator, in order to measure thepower inpinging on the sample.

67


Figure 5.3: The medium angle light scattering apparatus.

Detection optics Light scattered by the sample at angle θ is collected by the lensL1 (Newport, aspheric condenser f1 = 17 mm, Φ1 = 25.2 mm) and focused on its focalplane at distance d1 = f1 tan θ from the optical axis as an ideally diffraction limitedspot. The lens L2 (Melles griot, aspheric condenser f2 = 53 mm, Φ2 = 68 mm) islocated so that its front focal plane overlaps with the back focal plane of L1. Lightspots at distance d1 on front focal plane are focalised to infinity as parallel bundleswith angle θ1 = atan(d/f2). Thus lenses L1 and L2 togheter transform the scatteredlight at θ in a parallel bundle at

θ1 = atan

(f1 tan θ

f2

)

which for a small angle can be approximated as:

θ1 =f1

f2θ

These lenses demagnify the scattered angle, increasing the angle acceptance at theCCD, see figure 5.4. We choose lenses L1 and L2 between the availabe ones, tryingto maximize the ratio f1/f2 and to reduce vignetting, which arises when L2 doesn’tgather all the optical rays exiting L1. The sample cell is positioned against the lens L1

to maximize the acceptance angle.

The transmitted beam is removed from the optical axis by a beam stopper posi-tioned by a 3D-microtranslator in the center of the L1 focal plane. The beam stopperis a needle (Φ=0.7 mm) with a planar reflectant surface oriented at 45 degree. Light re-flected by the needle, impinges on a second photodiode, connected to the A/D board,which monitors the transmitted light. We calculate beam attenuation as the ratiobetween transmitted and incident light.

The Fourier lens L3 (Newport, achromatic doublet f3 = 30 mm, Φ3 = 25.2 mm)acts in a similar way as L1 with the CCD sensor in its focal plane, see figure 5.4. Theintensity distribution on the CCD is proportional to the energy spectrum of light atlens input. Thus, the optical system creates a relationship between light scattered by

68


(a) Inverted telescope (b) Fourier lens

Figure 5.4: Inverted telescope lens system and Fourier lens

the sample at angle θ and light intensity on the CCD at distance d from the opticalaxis:

d =f3 · f1

f2tan θ (5.1)

It is useful to point out that θ is the scatterering angle in air, which depends onscattering angle in the medium through Snell’s law:

θs = asin

(sin θ

ns

)

The distance l between L2 and L3 is an important parameter to set to reducevignetting. The distance d is set in the following way. A uniform scatterer1 (‘magic’scotch) is located on the sample’s plane. Light scattered by the sample is shaped on thefocal plane of L3 as a circle of uniform intensity. The radius depends on the maximumangle which the lens L3 is able to accept, thus it is maximum at the distance l whichminimizes vignetting.

The scattered light is measured only in the quarter plane above the beam stopperto maximize the CCD sensor’s area usage.

To reduce stray light, the sample cell is slighty tilted to prevent beam reflectionsfrom falling on the CCD, and the instrument’s room is kept dark during measurements.

CCD The CCD camera is an interline transfer Olympus DP50 digital camera, whichis actually a Pixera Penguin 600 CL model. The pixel matrix is rectangular with1392x1040 pixels, 4.65 µm in size. Raw analog pixel data are digitalized by its own 16-bit frame grabber, resulting in a nominal S/N ratio of 62 dB and a nominal dynamicalrange of 60 dB. An internal Peltier modulus cools the CCD sensor 20C below ambienttemperature in 2 minutes. An electronic shutter controls the exposition time in the0.1 ms–64 s range. The CCD is mounted on a xyz-microtranslator to position itsleft-lower pixel-matrix corner on the optical axis.

The CCD can acquire images at different resolution: 640x480, 1392x1040, and 2776x

2074. The latter is achieved using a Pixera light-swing opto-mechanical method, called DiRact-

orTM. Unfortunately, the CCD’s acquisition time is quite long as it takes up to 20 s to acquire

an image, thus we usually choose to acquire only 1392x1040 grayscale images at 16-bit depth.

1We define as uniform scatterer a system whose scattering matrix doesn’t depend on the scattering

angle

69


Data analysis Light scattered by the sample at angles θs is mapped by L3 in a ring on the

CCD sensor. By using Eq. 5.1 we can calculate the correspondence between the wave-vector q

associated to scattering angle θs:

q =4π

λns sin

1

2asin

[1

nssin

(atan

(d · f2f1 · f3

))](5.2)

for small angles, such that θ ' sin θ ' tan θ, it can be approximated by:

q =2π

λ

d · f2f1 · f3

Note that refraction effects on the scattering angle exiting the sample’s cell exactly compensate

laser wavelenght reduction in the medium.

The images are processed dividing the pixel matrix in concentric rings centered on the

optical axis, in order to establish a relationship between ring distance and q values.

In order to measure the scattered intensity we need to acquire and process three images,

where each one is the average of several (usually 10) snaps at fixed exposition time. The first

image is taken without any light falling on the CCD. This dark image measures the CCD dark

current. The second image is taken while the laser passes through the cell filled only with the

system solvent. This blank image takes in account for stray light. Finally the third image, which

is the sample image, is the measurement of the light scattered by the sample. We subtract the

dark image from both the sample and blank images, eliminating so the electronic noise, and

we average the intensity values on each ring, obtaining the blank raw data and the sample raw

data, which are the superposition of the scattered intensity and of the stray light attenuated

by the sample. Since both the scattered and stray light are attenuated by the same factor

when passing through the sample, the blank raw data are reduced by the beam attenuation,

as measured by the two photodiodes. Finally we obtain the scattered intensity raw data by

subtracting the blank data from the raw data.

The scattered intensity measured on the CCD is related to the sample scattered intensity:

ICCD(d) = I(θ)∆Ω sin2 Φ (5.3)

where ∆Ω is the correction due to projected solid angle, sin2 Φ is the dipole term which account

for polarization effects and Φ is the angle between the laser polarization and the direction of

observation, see fig 5.5.

The solid angle associated with each pixel at distance R from the sample is:

∆Ω =Apix

R2k · n = Apix

f

(f2 + d2)3/2

where the factor cos θ arises from the projection of a pixel’s area on the scattered wave number.

The term sin2 Φ in eq. 5.3 varies from pixel to pixel inside each ring. If we refer to a pixel

which coordinates are (x,y), we can write:

sin2 Φ =x2 + f2

d2 + f2

The intensity measured on each ring is thus divided by these correction factors in order to

calculate the intensity scattered at q by the sample:

I(θ) =ICCD(x,y)

Apix

R2k · nx

2 + f2

d2 + f2

(5.4)

70


Figure 5.5: Scattered light at wave vector k is measured by a pixel with (x,y) matrix

coordinates. θ is the scattering angle, between axis z and k, Φ is the angle between

axis y and the projection of k on the plane xy. The pixel drawn is at distance R from

the scatterer, and at distance d from the intersection between the sensor plane and the

optical axis. Versor n is ortogonal at pixel area.

Software acquisition and data analysis A custom software, developed in Labwin-

dows CVI (National Instruments), acquires and processes the images. A CCD interface is

carried out by means of the Pixera dynamic link library PixSDK, which provides setting pro-

cedures for exposition time, image resolution, pixel depth and acquisition procedures. Our

software allows for storing the three images needed for a single measurements and for process-

ing the images as described above. It makes dark image subtraction, raw data processing and

applies correcting factors. It allows for saving radial intensity distribution as a function of the

distance from the optical axis in a text file, which we plot using Easy Plot software.

In the following, I will point out only two main features of this software. The CCD stores

the acquired images in teh PC’s memory in a structure similar (but not equal) to DIB format

and it returns only a handle to the memory area. We found that the structure has an header

in which the image information are stored and an area to store pixel values. The header is

defined as the Windows bitmap header:

typedef struct tagBITMAPINFOHEADER

DWORD biSize;

LONG biWidth;

71


LONG biHeight;

WORD biPlanes;

WORD biBitCount;

DWORD biCompression;

DWORD biSizeImage;

LONG biXPelsPerMeter;

LONG biYPelsPerMeter;

DWORD biClrUsed;

DWORD biClrImportant;

BITMAPINFOHEADER

Pixel data are stored in a matrix with many rows as the pixels and three columns (one for each

RGB colour):

unsigned short int intchannel[1447680][3]

We define a new structure as the union of these two structures, in which we cast 2 the handle

returned by the CCD. In this way we are able to access pixel data, which we store in a simpler

unsigned short integer array dynamically allocated as a grayscale image. The grayscale value

is the average of the colour channels.

We adopt the following procedure to average each ring data. Ring arrangements are defined

in a text file so that they could be easily changed to be optimised depending on the sample being

studied. The optical axis is determined by removing the beam stopper and shining the strongly

attenuated laser beam directly on the CCD. In this way we determine the pixel matching the

optical axis. Using ring parameters and optical axis position, the software assignes each pixel

to its corresponding ring on the basis of its distance from the optical axis and it stores each

ring data in this dynamically allocated structure:

typedef struct

unsigned long int point;

unsigned short int *row;

unsigned short int *column;

sparse_matrix;

The first field gives the number of pixels in the ring while the other two fields hold pixels

coordinates. This structure is defined in a similar way as a sparse matrix, in which only

non-zero elements are kept. All Sparse matrix structures are stored in a new array. This

construction has three advantages:

1. We need to follow this long procedure only once, since we could save optical axis coordi-

nates, ring parameters (minimum and maximum radii), and the array of sparse matrix

in a second file B. This file contains all the data needed to recover rings arrangment

until optical configuration is unchanged. Loading this small file3 in memory is almost

instantaneous.

2. It is possible to save different arrangments and to switch easily and quickly among them.

3. Averaging intensity on each ring requires few seconds since pixels coordinates are already

stored in the corresponding sparse matrix structure.

2The operation of cast is to assign a variable changing its type, for example from float to double3File size is usuall 1–3 Mb

72

5.3 Calibration measurements Medium angle light scattering apparatus

5.3 Calibration measurements

We measure the diffraction pattern of pinholes with nominal diameter d = 20µm and d = 30µm

by centering the pinhole at the cell plane, with the beams stopper out of the beam path.

Measurements are carried out by acquiring the diffracted light images and subtracting the dark

image from it. There is obviously no need of blank images. The pinhole’s diffraction pattern

is related to Airy function[4]:

I(r) =

(A

λz

)2

Ai(kdr/2z) =

[2J1(kdr/2z

kdr/2z

]2

where r is the distance on the detector plane, z is the distance of the detector from the pinhole,

and J1 is the first order Bessel function. Considering our optical setup, we expect to measure

an intensity profile in agreement with an Airy function with z = f3, see figure 5.7–b.

Figure 5.6: First maximum intensity of light diffracted from a 20 µm diameter pinhole

at variance with exposure time.

We first check the linearity of the CCD with respect to the energy falling onto the CCD’s

sensor, by plotting the intensity of the first maximum versus CCD’s sensor exposure time. The

intensity values are reported in a 16-bit integer format, as digitalized by the A/D converter.

We found that the CCD’s response is linear until Ilin = 3 · 104, as reported in figure 5.6. Thus

we must always set a suitable exposition time in order to have maximum intensity detected

well below Ilin.

We carry out the angular calibration of the system by comparing the angular positions of

secondary maxima and minima with the corresponding theoretical value. Eq. 5.2 refers to a

sample in solution, so for a pinhole we should write it in the following way:

q =4π

λsin

[1

2atan(d ·M)

]M =

f2f1f3

(5.5)

Using nominal the focal lenght we find M = 0.1039 mm−1. However we choose to measure it.

If we approximate the previous equation for small angles, we can write:

q =2π

λMd

73


(a) Experimental data (b) Theorical data

Figure 5.7: Theoretical and experimental data of light diffracted from a 20 µm diam-

eter pinhole. By comparing position of the first minimum and maximum we measure

the actual parameter M in eq. 5.5. Experimental data are taken with 10 ms exposure

time and 3 mW laser output.

thus it is possible to calculate M from the comparison of theorical and experimental positions

of the first intensity minimum and the first secondary maximum, see figure 5.7. We find

Mmin = 0.09239 and Mmax = 0.09351, by averaging them it results M = 0.09295 which is

slighty different from the nominal value Mnom = 0.1.

Figure 5.8: Calibration curve used to correct nominal q-value associated to each ring.

Experimental data are taken with 10 ms exposure time and 3 mW laser output.

For a better calibration we compare the measured q values for the first four secondary

maxima with the theoretical ones for both a 20 µm and 30µm diameter pinhole, see figure 5.8.

By interpolating them we get a q-calibration curve to correct the nominal q-value associated

74


with each ring. We find:

q = 1.04 · qnom − 302 (5.6)

Figure 5.9: Theoretical and experimental data of light diffracted from a 20 µm diam-

eter pinhole.

Figure 5.10: Theoretical and experimental data of light diffracted from a 30 µm

diameter pinhole.

Finally, in figure 5.9 we report the diffraction pattern of both 20 µm and 30 µm diameter

75

5.4 Scattering measurements on polystyrene particles Medium angle light scattering apparatus

pinholes as a function of the scattering wave vector q on a log-log plot. Data are taken from two

images associated with different exposure times. The light intensity of the primary maximum

is measured with 0.1 ms exposition time, while other data are measured with 15 ms exposure

time. We can see that experimental data fit the theory quite well, with 5 secondary maxima

clearly resolved for the 20 µm pinhole. We can also guess the dynamical range of our apparatus

from the figure. Observing that points start to become noisy at about 3 decades below the

main peak, we are able state that our apparatus has ∼ 3 dB dynamical range.

5.4 Scattering measurements on polystyrene particles

Measuring diffraction from a pinhole is an easier task than measuring light scattering from

a solution, because the former is a static system in which there is no need for optical noise

subtraction as in the latter case. To test our apparatus with a ‘real’ system, we measure static

light scattering from a solution of polystyrene particles in water. In this measurements we

apply both the beam stopper and the subtraction of the blank raw data.

Figure 5.11: Mie theory and experimental data of light scattered from monodisperse

polystyrene beads with 7 µm diameter.

We use particles with 7 µm, 20 µm, and 40 µm (Polyscience). In figures 5.11,5.12, 5.13 we

report our measurements. The light intensity of the primary maximum is measured with a 0.1

ms exposure time, while other data are measured with a 5 ms exposure time. Measurements

on the 20 and 40 µm systems are in a acceptable agreement with data calculated with Mie

theory for strictly monodisperse particles. Measurements on the 7 µm system are instead not

so good, probably because particles aggregation.

Measurements on polystyrene particles are noisier than pinhole’s ones. Two different ele-

ments leads to this effect:

1. the beam stopper,

2. blank image subtraction

76



polystyrene beads with 20µm diameter.


polystyrene beads with 40µm diameter.

77


The first cause arises because lens L1 doesn’t focalise incident beam in a diffraction limited

spot, but in a much wider one which is difficult to remove without affecting intensity at low q.

Moreover, the beam stopper augments stray light introducing reflections.

Blank image subtraction requires that laser output power doesn’t fluctuate. The laser

applied in this apparatus, however doesn’t show such good stability although it is controlled in

constant power. Measuring the output beam with a photodiode shows power fluctuations up

to ' 5%.

78


References

[1] N. Pusey, A. D. Pirie and W. C. K. Poon. “Dynamics of colloid-polymer mixtures.” Physica

A, 201, (1993), 322–331.

[2] F. Ferri. “Use of a charge coupled device camera for low-angle elastic light scattering.”

Review of Scientific Instruments, 68, (1997), 2265–2274.

[3] R. H. Tromp, A. R. Rennie and R. A. L. Jones. “Kinetics of the Simultaneous Phase

Separation and Gelation in Solutions of Dextran and Gelatin.” Macromolecules, 28, (1995),

4129–4138.

[4] J. W. Goodman. Introduction to Fourier Optics (Mc Graw–Hill) (1996).

[5] L. Ghiringhelli. Realizzazione di un apparato per la misura di diffusione di luce a basso

angolo. Master’s thesis, Politecnico di Milano (1998–1999).

79

Pickering Emulsions

Abstract

Emulsion stability does not necessarily require surface-active agents, but can also

be efficiently promoted by dispersed particles in the colloidal size-range. This

systems are known as Pickering emulsions . We have explored some basic fea-

tures of Pickering emulsions using ‘model’ silica particles with a fluorescent core

as stabilising agent. This allows the interfacial adhesion processes to be de-

tected and quantified by video-microscopy. We studied two colloidal systems,

differing primarily in the particle-surface structure. The first one is composed of

monodisperse, smooth-surfaced spherical colloids. Particles of the second batch,

although still monodisperse, display noticeable surface ‘roughness’. In particu-

lar, we show that particle trapping at the water/oil interface does not lead to

appreciable changes in O/W droplet interfacial tension, confirming therefore the

steric origin of Pickering emulsions’ stabilisation. We also found that surface

roughness appreciably lowers particles emulsifying power, and that no straight-

forward relation exists between the degree of droplet’s surface-coverage and macro-

scopic emulsion stability. Finally, we studied surface diffusion of trapped particles

and suggested that particles redistribution on droplets plays a role in stabilising

droplets with low or inhomogeneous particle coverage.

80

6.1 Introduction Pickering Emulsions

6.1 Introduction

Emulsions are of enormous practical interest for their widespread occurrence in processes re-

lated to food, cosmetics, and pharmaceutical industries. Emulsions are dispersions of one

liquid in another liquid, in the form of micrometric droplets immersed in the continuous phase.

Most commonly they present themselves as oil-in-water (o/w) or water-in-oil (w/o). Although

emulsions present a huge interfacial area, which is associated with a positive free energy, it is

possible to prepare emulsions with a long term stability by means of an emulsifier and provid-

ing mechanical energy to break up the mixture. There are many chemical agents that can act

as emulsifiers: some examples are surfactants, proteins and amphiphilic polymers.

An emulsified system can separate through to two different processes: coalescence and Ost-

wald ripening. The system undergoes coalescence when the liquid film separating two droplets

disappears, contact is achieved and their contents flow together to form a new larger droplet.

This process consists of three phases: first, one droplet approaches another one in the contin-

uous liquid; second, a thin liquid film formes between the two droplets and it begins to drain;

third, as the film’s thickness becomes sufficiently small it ruptures and the two droplets form

a bigger one. Coalescence can be enhanced by droplets flocculation and by creamming. Emul-

sion droplets interact via hydrodynamic and surfaces forces: when the overall attractive force

between droplets is large enough to overcome thermal agitation, flocculation occurs. Emulsion

droplets become connected either in flocs, which are discrete aggregates, or in a single network

structure. The system undergoes a creaming process when there is a difference between the

dispersed and the continuous phase densities. Under the influence of gravity, separation occurs

with the denser phase sitting at the bottom and with the less dense phase on top.

In Ostwald ripening processes, the dispersed phase is transported through the continuous

phase from smaller to larger droplets, without droplets contact. Since the latter have a lower

surface to volume ratio than the former, this process occurs with a net reduction in interfacial

energy. However, the dispersed phase must be significantly soluble in the continuous phase.

Neglecting this last process, an emulsion will have a long term stability if the droplets are

prevented from coming close to each other and if coalescence is inhibited.

Griffin[1] introduces the hydrophilic-lipophilic balance (HLB) concept to describe the abil-

ity of emulsifiers to stabilise either o/w or w/o emulsions. For example, rather hydrophobic

emulsifiers, which have HLB number below 6, usually stabilise w/o emulsions, whereas more

hydrophilic emulsifiers, with a HLB number above 10, are suitable to stabilise o/w emulsions.

HLB number can be inferred by the emulsifier’s molecular composition and geometry. In fact,

what determines the tendency of emulsifier’s monolayers to bend towards water or oil, is the

packing parameter of the emulsifiers at the oil/water interface, which in turn selects the kind of

emulsion stabilisation. The packing parameter is essentially related to the molecular geometry

of the system, hydrated by water on one side and solvated by oil on the other.

Starting from the original observations by Ramsden[2] and the seminal studies by Pick-

ering[3], evidence has been gathered that emulsion stability does not necessarily require am-

phiphilic surfactants to reduce the interfacial tension, but can also be efficiently promoted by

dispersed particles in the colloidal size-range: this kind of emulsions are known as Pickering

emulsions.

Our aim is to develop a model system to study Pickering emulsions properties. We chose

to use emulsions of hydrocarbon in water stabilised by silica monodispersed solid particles,

synthesised with the Stober method, containing a fluorescent core which allows the interfacial

adhesion processes to be detected and quantified by video-microscopy.

We shall examine two colloidal systems, differing primarily in the particle-surface structure.

The first one is composed of monodisperse, smooth-surfaced spherical colloids. Particles of the

81

6.2 Solids-stabilised emulsions Pickering Emulsions

second batch, although still monodisperse, display noticeable surface ‘roughness’, mimicking

therefore an important, and so far neglected, feature of natural colloids. In particular, we will

show that:

• Particle trapping at the water/oil interface does not lead to appreciable changes in O/W

droplets interfacial tension.

• Surface-roughness appreciably lowers particles’ emulsifying power. The interfacial or-

ganisation of ‘rough’ trapped particles displays a rich morphology, sometimes marked by

‘colloidal lumps’ suggesting surface-mediated attractive forces of capillary origin.

• No straightforward relation exists between the degree of droplets surface-coverage and

macroscopic emulsion stability. Long-term emulsion stability is observed for very limited

particles adhesion, or conversely, rapid coalescence of emulsions composed by densely-

covered droplets may take place.

• Trapped particles exhibit vigorous brownian motion, with a surface diffusion coefficient

that, on poorly-covered droplets, basically has the same value as in the surrounding bulk

phase. Particles redistribution on droplets may play a role in stabilising droplets with

low or inhomogeneous particles coverage.

6.2 Solids-stabilised emulsions

Probably the most important system of solid-stabilised emulsions are water in oil emulsions,

stabilised by asphaltenes1, fine clays less than one micron in diameter and waxes. These

susbtances are often encountered during the extraction of bitumen from oil sands, crude oil

dewatering, the separation of fines from shale oils and the separation of oils from waste waters.

Diluted bitumen obtained from bitumen froth contains commonly 0.5 % of fine solids and 3 %

of micron-sized emulsified water droplets. This water causes serious corrosion problems to the

downstream units due to the chlorides dissolved in it. Emulsions change spill oil properties to a

large degree. As a matter of fact, stable emulsions can contain up to 60%–80% representing an

expansion in volume of spilled material from 3–5 times the original volume. Most significantly,

oil viscosity increases of three orders of magnitude: from ' 0.1 Pa s ' 1000 Pa s. These are

just some important examples of the crucial role played by Pickering emulsions in oil industry

processes.

Until a few years ago, most experimental studies on ‘Pickering emulsions’ aimed at pro-

viding a macroscopic, semi-quantitative description of the stability diagram of emulsions sta-

bilised by naturally-occurring, but not so well-characterised, colloidal particles. In spite of

their limitations, these studies have allowed some general conclusions to be drawn on the

colloid-stabilisation mechanism[4]. First of all, particle wetting properties are of paramount

importance: hydrophilic colloids tend to stabilise continuous water (o/w) emulsions, while

water-in-oil (w/o) emulsions are better stabilised by oil-wettable particles. Furthermore, the

stabilising colloids tend to be small compared to the emulsion droplets, and stabilisation is more

readily obtained using marginally stable dispersions within respect to salt-induced coagulation.

Attention has more recently shifted to simpler emulsions, stabilised by model colloids, with

relatively well-characterised size and shape distribution and controlled surface properties[5].

1Asphaltenes are polyaromatic compounds of large molecular weight found in oil defined by the

amount that precipitates when the oil is dissolved in hexane or pentane. They contain a large variety

of chemical species with functional groups including acids and bases. Some of these groups are hy-

drophobic whereas polyaromatic skeleton is more polar and so hydrophilic, thus asphaltenes are surface

active

82

6.2 Solids-stabilised emulsions Pickering Emulsions

In particular, silica particles have been investigated at length, because their surface properties

resemble natural inorganic colloids like clays. Detailed studies of emulsion stability for Ludox or

fumed silica (Aerosil) have been made by Midmore[6] and Binks et. al.[7]. Optical observation

of particle adhesion at the droplet interface has however been clearly obtained in beautiful

experiments on latex-stabilised emulsions by Velev et. al.[8], who also described careful particle

synthesis and emulsions preparation methods yielding close-packed particle assemblies on the

droplet surface. A similar protocol, followed by solvent exchange of the continuous phase, has

been employed to obtain colloidal micro-capsules with selective permeability (‘colloidosomes’)

that could possibly be used for cellular immuno-isolation [9].

(a) (b) (c)

Figure 6.1: (a) Position of a small spherical particle at a planar α/β interface for a

contact angle, measured through α phase. (b) Corresponding position of particles at

a curved interface. (c) Sketch of a particle at the interface between liquids alpha and

beta.

In order to understand how solid particles can stabilise emulsions, let’s consider a particle of

radius a adsorbed at a fluid (α)-fluid (β) interface. As depicted in fig 6.1, the particle’s surface

fraction exposed to each fluid phase depends on the interfacial tension γαβ and on particle’s

contact angle θ, which in turn relies on the interfacial tensions γαβ , γαp, and γβp:

cos(θαβ) =γαp − γβp

γαβ(6.1)

To entrap the particle in figure 6.2 it is necessary to create a cavity in the fluids, but forming

the surfaces A1 and A2 requires energy. The former surface is the separation of fluid β and

the particle, the latter separates fluid alpha from the particle. However the entrapped particle

removes the surface A3, which is the fluid-fluid area separation, freeing therefore energy. The

energy balance is thus:

E = γαpA1 + γβpA2 − γαβA3

83

6.3 Preparation of silica colloids Pickering Emulsions

Simple geometrical considerations lead to:

A1 = 2πa2(1 + cos θ) A3 = πa2(1 − cos2 θ)

thus

E = 2πa2(1 + cos θ)(γβp − γαp) + πa2(1 − cos2 θ)γαβ

Using eq. 6.1 in the previous equation leads to:

E = πa2γαβ(1 − | cos θ|)2 (6.2)

The variation of energy E at variance with contact angle is reported in figure 6.2 for isoc-

tane/water system and silica particles with diameter a = 500 nm. Particles adsorption reduces

system free energy and for contact angles θ ∼ 90 the entrapping energy is much higher than

the thermal energy kBT at room temperature: this leads to a strongly entrapment of particles

at the interface. Coalescence is indeed hindered, because the two droplets should first remove

the entrapped particles by lateral displacement before they can coalesce, but thermal energies

are insufficient for particles desorption.

(a) (b)

Figure 6.2: (a) Variation of energy attachment E relative to kBT of a spherical particle

of radius a at a planar isooctane-water interface. a = 500nm, γow = 50.8mN/m T = 298

K. (b) Semilog plot of (a)

6.3 Preparation of silica colloids

Silica dispersions in alcohols can be made by ammonia-induced hydrolysis and condensation of

orthosilicate tetraethoxysilane2 (TES) using the Stober synthesis method, yielding monodis-

perse spherical particles with a size covering most of the colloidal range [10]. An ingenuous

modification of Stober reaction, due to van Blaaderen and Vrij [11], allows to obtain particles

with a pure silica surface and a fluorescent core, which is obtained by linking dyes to the organo-

silicates 3-Amminopropyl-triethoxysilane (APS). Interfacial adsorption of dyed particles can be

2C8H20O4Si

84


easily detected even when particle size is below the optical resolution limit, just by monitoring

local epi-fluorescence intensity.

Particles’ size can be finely tuned changing free TES amount in solution and colloidal surface

charge. The latter is modified by either ammonia and water concentration. Siloxane oligomers

aggregation occurs during the initial reaction phases and it finishes when the aggregates are

large enough to become stable colloidal particles. At this point the number of particles becomes

fixed and any subsequent growth proceeds by monomer addition of remaining TES. Weakening

colloidal stability leads to bigger particles because the number of stable olygomers diminishes.

Ammonia concentration plays two contrasting roles in influencing colloidal surface charge:

• it changes pH of solution towards basic values. This negatively charges silica particles[12]

thus increasing their stability against aggregation;

• it screens electrostatic repulsion between particles diminishing stability;

educes Imhof et al.[13] showed that also APS concentration affects the final diameter size,

because it diminishes colloidal stability introducing a positive charged group (R − NH+3 ) that

reduces the overall surface’s negative charge.

Materials and methods Pure silica particles nucleation and growth are accomplished in

degassed ethanol/distilled water/ammonia solutions by adding TES, and letting the reaction

proceed for four hours under mild stirring. It is possible to add more TES at this stage to make

the particles larger. We report particle size and chemical quantities in table 6.1. All chemical

products are from Fluka.

Fluorescent particles are obtained with the following procedure. Fluorescein isothiocyanate3

(or equivalently, rhodamine B isothiocyanate[14]) and freshly-distilled 3-Amminopropyl-trie-

thoxysilane are first made to react for a day under nitrogen and dark conditions. The resulting

slurry is re-dispersed in anhydrous ethanol and then micro-filtered. Nucleation and growth of

particle cores are made in degassed ethanol/water/ammonia mixtures by adding small volume

fractions of coupled dye and tetraethoxysilane, and letting the reaction proceed for some hours

under mild stirring. The fluorescent cores are then coated with pure silica shells by progres-

sively adding TES in small steps under fast stirring. We report particle size and chemical

quantities in table 6.2. All chemical products are from Fluka.

The particles are transferred to water by extensive dialysis4 or centrifugation, and then

precipitated by carefully adding acetone. The solvent is eventually removed by a rotating

evaporator, yielding a finely-divided powder that can be easily re-dispersed in water.

We measured particle size and polydispersity with dynamic light scattering (DLS) and

scanning electron microscopy (SEM).

d (nm) EtOH (φvol) NH3OH 25% (φvol) TES(∗) (φvol) TES(∗∗) (ml)

171±24 87.7% 7% 5.3% 0

200±48 85% 9.5% 5.5% 0

185±50 88.9% 9.5% 1.6% 4

200±38 87% 9% 4% 3

Table 6.1: Pure silica particles synthesys data. Volume fractions data refer to an

overall volume of 20 ml. Particle sizes are measured by DLS. (*) TES quantities

applied initially. (**) TES quantities added in a second time.

3isomer I4Membrane dialysis has 12000 Molecular Weight Cut-Off.

85


Core formation

EtOH APS FITC

PA 5.600 ml 0.777 ml 144 mg

PB 19.017 ml 2.639 ml 489 mg

Shell formation

EtOH NH3OH 25% TES(∗) APS+FITC TES(∗∗)

PA 82.2 ml 9.42 ml 3.66 ml 0.96 ml 6 ml

PB 857.85 ml 98 ml 38.15 ml 10 ml 125 ml

Table 6.2: Fluorescent silica particles synthesys data. (*) TES quantities applied

initially. (**) TES quantities added in a second time.

Figure 6.3: DLS field correlation functions g1(t) and SEM images for PA (left image

and full dots) and B (right picture, open dots).

Particle characterisation The two different colloidal systems that we shall mainly dis-

cuss in what follows, which we will refer to as PA and PB, have been obtained using an identical

core-polymerisation procedure, yielding a fluorescent core diameter dC ≈ 300 nm, but a dif-

ferent total amount of TES in the shell-formation step. Scanning electron-microscopy (SEM)

pictures and dynamic light scattering (DLS) correlation functions for PA and B are shown

in Fig. 6.3. Electron microscopy shows that PA are spheres with pretty uniform size and

a smooth surface. Two-cumulant fits of DLS correlation functions yield a particle diameter

dA = 0.51 ± 0.04 µm, in good agreement with direct SEM visualisation. DLS particle size for

86

6.4 Macroscopic stability Pickering Emulsions

System d (µm) Surface tsil (min)

PA 0.51 smooth 0

P silA 0.51 smooth 15

PB 0.77 rough 0

PB,5 0.77 rough 5

PB,10 0.77 rough 10

PB,15 0.77 rough 15

PB,35 0.77 rough 35

PB,50 0.77 rough 50

Table 6.3: Size (diameter d by DLS), surface features, and exposure time tsil to HMS

of silica particles systems used for emulsion stabilisation.

colloids B is found to be about 50 % larger (dB = 0.77±0.05 µm). However, SEM pictures show

that these particles, albeit still uniform in size, display a rather ‘rough’ surface, characterized

by a high number of little lumps adhering to a large spherical core, which is only slightly larger

than PA. A possible explanation for this curious morphology is the occurrence, during particle

synthesis, of secondary nucleation processes, known to happen frequently while growing very

large silica particles. Secondary nucleation seems to be avoided or strongly reduced by progres-

sive dilution of the suspension between successive additions of TES. Indeed, using this method,

we managed to grow monodisperse colloids (particles PC) with a diameter larger than 1 µm

and still retaining a smooth surface. Specific features of the latter colloids will be discussed in

connection with emulsion-forming properties of PB.

Surface treatments Attempts to modify the particles’ wetting properties are performed

in these ways:

1. we promote physisorbing of short-chain alcohols on particles’ surface, adding alcohols in

the colloidal dispersions. This leads to an increased surface hydrophobicity and reduces

particle’s contact angle with the disperse phase;

2. we expose dried particles powder to dilute solutions of hexamethyldisilazane (HMS) in

n-hexane, controlling the silanization extent by quenching the reaction after agiven time

adding dilute water-in-acetone solutions. Table 6.3 summarises the principal features of

the specific particles systems used for emulsion stabilisation.

3. we promote physisorbing of a non-ionic surfactant (Triton X100) on particles’ surface.

6.4 Macroscopic stability

Emulsions preparation The particle-stabilised emulsions we have studied were prepared

by adding to aqueous suspensions of PA,PB or PC either isooctane or octanol. The former

solvent has a high interfacial tension γ = 50.8 mN/m with water, typical of hydrocarbon

liquids. Conversely, water’s interfacial tension with long-chain insoluble alcohols like octanol,

is much lower (γ = 8.5 mN/m) and therefore emulsification is easier. The emulsification was

obtained using an Ultra-Turrex homogeniser with a 8 mm rotating head, typically mixing at

20,000 r.p.m. for 2 minutes.

87

6.4 Macroscopic stability Pickering Emulsions

(a) Surfactants (b) Alcohols (c) Silanization

Figure 6.4: Fraction of emulsified isooctane in water compared to the total volume

after three days. (a) Emulsions prepared by adding 80/20 isooctane/water mixtures

and stabilised by both surfactants and PC+surfactants. (b) Emulsions prepared by

adding 1/1 isooctane/water mixtures and 0.1% w/w of particles PC , surface-modified

with alcohols. (c) Emulsions prepared by adding 75/25 isooctane/water mixtures and

stabilized by silanized particles PsilA .

Surfactant Physisorbing The synergy of simple surfactants with Ludox silica colloids,

leading to enhanced emulsification power, has been previously observed [15]. We came across

this peculiar behaviour also studying emulsions-forming in the presence of the nonionic deter-

gent Triton X100. For 20% water / 80% isooctane mixtures without colloids, we observed a

triggering of the O/W emulsification for surfactant concentrations exceeding cT ≈ 2.5 × 10−3,

while oil is fully emulsified for cT ≈ 5× 10−3. Conversely, in the presence of PC at ΦP ≈ 10−3,

the threshold value for cT is reduced, and more than half of the isooctane gets emulsified in

conditions where emulsification by Triton alone is still negligible, as reported in figure 6.4.

Non-monotonic effects of alcohol addition Alcohols interact strongly with surface

silanol groups. The steric stabilisation of silica particles commonly exploits chemical attach-

ment of long-chain alcohols to silica, making particles’ surface hydrophobic and allowing to

disperse silica colloids in non-polar solvents[16]. This reaction requires very dry conditions to

avoid competition with water. However, even in aqueous solutions, residual preferential adsorp-

tion of alcohols on silica may be expected [12], leading to an increased particles hydrophobicity.

Emulsions were prepared from equal volumes of isooctane and aqueous dispersions of PC at

fixed particles’ concentration cP ≡ 0.1% by weight and variable amounts of short-chain alco-

hols. Figure 6.4 shows the fraction of emulsified over total volume after three days as a function

of alcohol concentration calc. We can summarize our main experimental findings as follows:

Ethanol Addition of a few percent ethanol leads to the formation of a limited amount of

emulsified mixture, which however collapses on a time-scale of the order of 1-2 days.

Conversely, total and long-term stable emulsification is obtained only in the presence of

calc ≥ 10%.

Propanol Partial emulsification takes place even for very low propanol contents (calc ≈ 0.5%).

The emulsions are relatively stable, displaying however slow coalescence on time-scales

of a few days. The amount of emulsified volume shows however a very peculiar trend

88

6.5 Emulsions interfacial tension measurements Pickering Emulsions

as function of propanol concentration calc, increasing up to about 60 % of the total

volume for calc ≈ 1%, but then considerably diminishing at larger calc and vanishing for

calc & 15%

Butanol Addition of a minimal amount of butanol yields very stable emulsion. A non-

monotonic trend of the emulsified volume on alcohol content is still observed.

Effects of particles silanization Particles PsilA display intermediate wetting properties,

and do stabilise O/W isooctane emulsions. The emulsified fraction of the total volume rapidly

grows by increasing particles PsilA concentration, approaching to saturation for a colloidal con-

centration of about 10 g/l, this corresponding to a particles’ volume fraction ΦP ≈ 5 × 10−3,

as shown in figure 6.4. The volume fraction ΦO of isooctane within the emulsified O/W phase,

estimated from meniscus locations, stays approximately constant at ΦO ≈ 0.8, a value fairly

larger than the one for random close-packing for hard spheres suggesting so appreciable droplets

deformation.

6.5 Emulsions interfacial tension measurements

Particles’ surface adhesion is very strong. Direct microscope observations show that, once freely-

diffusing particles get trapped, they stick to the interface for an indefinite time, performing

a surface brownian motion, as we shall discuss at length later. Moreover, particles remain

constrained at the droplet interface even when small amounts of the emulsified phase are

sampled and extensively diluted in pure water.

(a) (b)

Figure 6.5: (a) Interfacial tension of isooctane droplets stabilised by PsilA versus par-

ticles concentration. Inset: fluorescence image of a particle-coated droplet just before

suction into a micropipette. (b) Interfacial tension of octanol droplets stabilised by

silica particles PB at 4 mg/ml concentration.

Dilute droplets suspensions, prepared in the aforementioned way, were used to measure

by micropipette tensiometry the interfacial tension of isooctane O/W emulsions stabilised by

89

6.6 Effects of particles’ surface roughness. Pickering Emulsions

particles PsilA . Fig. 6.5 displays average results and standard deviation for measurements on

10-15 droplets as a function of particles’ concentration. The microscope image in the inset

captures the droplet-pipette geometry just before suction. Notice that the meniscus of the oil

plugged in the pipette is very close to being hemispherical, confirming that the micropipette

is preferentially wetted by the continuous phase. Results show that γ stays approximately

constant within the whole investigated particles range, and essentially coincides with the value

obtained for isooctane droplets injected in pure water. This finding is the first experimental

demonstration that IFT reduction cannot be the operative stabilisation mechanism in Pickering

emulsions, leaving steric hindrance or surface rheology effects as most probable candidates to

play such a role. The observed slight apparent increase of γ with cP may be possibly due to

partial rigidity of the emulsion ‘skin’. This effect can be detected experimentally by observing an

anomalous droplet geometry during suction. For most of the droplets we measured, deviations

from the hemi-spherical shape were small. Yet, for a small number of droplets showing a very

strong particles surface ‘crowding’, ‘skin effects’ were conversely very pronounced, up to the

point that partial ‘peeling’ of the surface layer, leading to the formation of a toroidal ‘particles

ring’ around the pipette tip, was noticed.

Before the following discussion about effects of particles roughness and surface hydropho-

bicity, it is useful to compare the results about the interfacial properties of octanol in water

emulsions stabilised by different kinds of particles. In figure 6.5 the values for γ are obtained us-

ing the micropipette method. It is to be noticed that the interfacial tension of octanol droplets

in water, prepared by direct micro-injection, is found to be γ = 8.42 ± 0.34 mN/m. The

effects of particles interfacial adsorption on γ have been theoretically investigated by Levine

et.al. [17, 18]. According to their model, a close-packed layer of adsorbed particles with suffi-

ciently large contact angles θ (which means, in our case, more hydrophobic colloids), leads to

a reduction of γ, reaching a maximum of about 50% for θ = 90. This effect is experimentally

observed when slightly hydrophobic silica particles are made adsorbing on carbon tetrachloride

sessile droplets over long time scales (at least one hour) [18]. The reason why no similar effect

is found in our measurements is probably that, as better discussed in the following, the degree

of surface coverage for rough silanized particles is extremely low, implying very slow adsorption

kinetics.

6.6 Effects of particles’ surface roughness.

Native colloids differ in many aspects from ‘model’ spherical particles. In addition to their

intrinsic polydispersity, another factor potentially capable of influencing interfacial adhesion is

represented by the highly irregular surfaces displayed by most environmental particles. We have

explored particles shape effects by studying Pickering emulsions stabilised by PB, which display

noticeable surface roughness while still retaining low size polydispersity. Both macroscopic

stability measurements and direct visualisation of the adsorbed particles layer reveal remarkable

differences with the behaviour of PA.

On the basis of the observations made with PA, dried particles were brought to react with

HMS, prepared at the same concentration used for system Asil, for short times ranging in five

steps from 5 to 50 minutes. As shown in table 6.3, particles B exposed to HMS for X minutes

will be indexed as PB,X.

Visualisation methods In order to visualise colloids adhesion to droplets interface, emul-

sions are sampled, diluted in the continuous phase, and tightly sealed into 50 × 4 × 0.2 mm

rectangular glass-capillaries (VitroCom, USA), with a controlled glass thickness of 0.18 mm,

allowing to use high N.A. objectives with limited optical aberration while avoiding droplet

90


squeezing. Samples were observed using an Olympus IX70 inverted microscope and visualised

either using a standard CCD or, for high resolution fluorescence images, an Olympus DP50

digital camera. The images were processed using ImagePro (Media Cybernetics, USA) video

acquisition/elaboration software.

Emulsifying power and colloidal layer morphologies As a general feature, PB

shows a quite reduced propensity for stabilising O/W emulsions rather than PA. At variance

with the latter, in the absence of surface treatment, the former failed to emulsify not only

isooctane, but also octanol. Sufficiently long silanization did lead to the formation of stable

octanol emulsions, with specific surface morphologies we shall shortly describe. Yet, non-polar

hydrocarbons, like isooctane, could not appreciably be emulsified even by PB,50. There are

two possible reasons for this partial emulsification failure, since PB are not only rougher, but

also larger than PA. Preliminary emulsions stability studies, made with monodisperse silica

particles with diameters ranging between 200 − 400 nm, showed indeed that smaller particles

are more efficient in stabilising emulsions. However, this does not seem to be the case, since

isoctane emulsions stabilised by PC, which are larger than PB, can be easily obtained (see

Fig. 6.6. Therefore, we regard it fair saying that surface roughness has a negative impact on

emulsification power, a reasonable explanation being that a reduced surface contact consid-

erably reduce the strong interfacial potential well, keeping smooth particles at the interface.

The following is a short survey of the observations made on the stability and surface mor-

phology of emulsions prepared by mixing octanol with aqueous suspensions of silanized PB in

1/1 volume ratio.

• Weakly-silanized particles (tsil < 10 mins) fail to stabilise O/W emulsions, giving only

very large droplets which totally collapse in a few hours. Emulsifying power increases

with tsil. The fraction of emulsified octanol reaches about 30 % for tsil = 10 mins, and

levels off at about 50 % for tsil ≥ 10 mins.

• Droplets surface coverage shows a more complex and partially reversed trend. Particles

exposed to HMS for 10 minutes yield an average 10 % droplet area coverage, although

examples of a much higher particles’ surface density are not uncommon (see Fig. 6.7A).

The surface coverage for tsil = 15−35 mins depends on the droplet size: smaller droplets

are much more densely covered (with a fractional surface coverage reaching up to 50

%) than larger ones (where typically only 5% of the surface is covered with colloids).

The most surprising finding, however, concerns particles exposed to HMS for the longest

period of time(tsil = 50 mins). Most droplets surfaces show indeed a very limited number

of trapped particles (Fig. 6.7C), with an average surface coverage of the order of only

5% that, reversely with what is found for PB,15 and PB,35, does not depend on droplets

size. This value is far below the minimal surface coverage of about 30%) estimated by

Midmore for stabilization of W/O emulsions by flocculated silica colloids[6].

• Surface layer morphology displays noticeable changes as a function of particles’ silaniza-

tion degree. For tsil = 15 − 35 mins (Fig. 6.7B) particles form close-packed ‘lumps’

(particle-clustering was sometimes observed also for PA, but to a far smaller extent).

Conversely, for tsil = 50 mins, the small number of adsorbed particles shows an appar-

ently random distribution.

These unexpected findings lead to two conclusions:

1. emulsions can be stabilised considerably even for very low particle trapping;

91


Figure 6.6: Fluorescent microscopy image of an isoctane emulsion droplet stabilised

by particles PC .

2. the surface particles layer can be highly inhomogeneous.

We defer the former aspect to the last section, after a discussion on particles’ surface mobility.

Here we only point out that the observed low degree of particles coverage must be a kinetic

effect. It is true that surface roughness leads to a smaller contact angle of the trapped particles,

corresponding to a lower effective degree of particles hydrophobicity. This has been phenomeno-

logically quantified by Cassie and Baxter [19] by introducing an apparent (averaged) contact

angle taking into account the fraction f of the particles’ surface that effectively contacts the

interface: low values of f might prevent even very hydrophobic particles to be trapped. Yet,

adsorption is to some degree observed. Since the interfacial trapping energy is much higher

than kBT , so that adhesion is practically irreversible, there is nothing preventing a progressive

increase of surface coverage until random or ordered (but necessarily defective, see [20]) close-

packing is reached. As we previously pointed out, equilibration of (smooth) particles’ surface

layers requires a few hours [17, 18]. However, our measurements have generally been performed

after a much longer emulsion equilibration time. It is therefore very likely that surface rough-

ness strongly affects the adsorption kinetics, to the point that particles trapping can take place

only because of the strong shear forces associated with the initial mixing process.

The second observation suggests the existence of attractive interactions between trapped

particles. Examples of apparent attractive forces in two-dimensional colloidal layers have been

92


(a) (b)

(c) (d)

Figure 6.7: Examples of surface morphology for octanol emulsions stabilised by par-

ticles PB silanized for different times. (a) Dense surface layer of PB on a droplet of an

unstable emulsion (tsil = 10 mins); (b) ‘Colloidal lumps’ (tsil = 15 mins); (c) Stable

emulsion showing very weak interfacial trapping (tsil = 50 mins). (d) Comparison with

octanol droplets stabilised by PC , showing no particle-clustering.

previously reported [21]. Recently, Stamov et.al. [22] has proposed a mechanism for attraction

between irregular-shaped particles at an interface based on pinning and concomitant non-

uniform wetting, causing an irregular shape of the particles meniscus. Attractive forces stem-

ming from this effect are strong and very long-ranged. The fact that particle-clustering is

much more accentuated for rough rather than smooth particles suggests that, in our case, this

capillary effect may actually be the leading mechanism for particle segregation.

93

6.7 Surface diffusion effects Pickering Emulsions

The observed surface morphology highlights a fundamental difference between Pickering

and simple emulsions. Perturbations of the surfactant homogeneous interfacial layer, due for

instance to film stretching, lead to strong interfacial tension gradients tending to restore the

original configuration. This ‘self-healing’ mechanism of the interfacial film is actually the main

source of surfactant emulsions stabilisation. Conversely, we have seen that wild fluctuations in

the surface distribution of adsorbed rough particles occur spontaneously, without prejudicing

the emulsion stability. A full understanding of the mechanisms of Pickering emulsions stabili-

sations requires therefore abandoning a continuous elastic model of the surface film, and taking

into full consideration film-mediated interparticle interactions.

6.7 Surface diffusion effects

Densely-packed layers are commonly supposed to hinder coalescence since merging of droplets

requires lateral displacement of the densely-packed, weakly mobile particle layers. As we shall

see, surface particles ‘reallocation’ effects may partly account for droplets stability in the pres-

ence of a very limited particle interfacial adhesion. Trapped-particles mobility is expected

therefore to be an important parameter to be evaluated. For emulsions stabilised by silanized

PB, colloidal surface motion can be profitably studied on droplets that, albeit stable, display

limited coverage. Particles thermal motion was indeed evident in all observed morphologies,

however for densely packed structures, where particles are trapped within neighbour cages,

this motion can rather be described as a localised vibration. Conversely, for emulsion droplets

stabilised by silanized PB, free particles surface diffusion can be observed.

Particle-tracking was performed by recording single particle positions as a function of time

from digital movies, with minimal time resolution imposed by the imaging CCD 0.1 s frame

rate (slightly longer for very long movies, due to data-transfer dead time).

The inset of Fig. 6.8 shows two examples of brownian trajectories performed by PB,15

trapped on a very poorly covered droplet, so that interparticle collision is negligible. The main

body shows that the mean square displacement < r2 >, averaged over about 20 trajectories,

is linear in time as expected for brownian motion. Recalling that for 2-dimensional brownian

motion is:

< r2(t) >= 4DSt

the surface self-diffusion coefficient is

DS ≈ 5 × 10−9cm2s−1

Since DLS correlation functions from PB yield a bulk translational diffusion coefficient

D = 5.9 × 10−9cm2s−1

this result implies that surface and bulk diffusion take place essentially at the same rate. Squares

in Fig 6.8, which conversely refer to particle motion on a relatively ‘crowded’ droplet (fraction

of surface coverage f ≈ 0.5), are consistent with

DS ≈ 1.7 × 10−9cm2s−1

showing that surface self-diffusion is hindered by interparticle interactions. Notice, however,

that DS is reduced by only a factor of three, which is too small to argue in favour of stabilisation

arising from a viscous, rigid particle layer [4]. The latter mechanisms could, of course, be

important at higher surface coverage.

An important note of caution concerns gravity effects. The gravitation length

lg = kBT/mg

94


Figure 6.8: Mean square displacement for particles PB diffusing on a weakly-covered

and a ‘crowded’ droplet. Inset: Brownian 2-D paths for two particles (tics on the axes

are spaced by 10 µm).

for PB is, depending on surface roughness, around 1−2 µm (or, equivalently, their Peclet number

Pe = kBT/mga, where a is the particle radius, is of order one). This means that sedimentation

over typical droplet length scale is relevant. We noticed indeed that, on poorly-covered droplets,

particles progressively concentrate towards the droplet bottom (brownian motion in ‘dilute’

conditions presented in Fig. 6.8 was in fact studied for a particle initially sitting near the

droplet lower pole). Time scale for sedimentation on 20 − 40 µm diameter droplets is of

the order of few minutes, consistent with a Stokes sedimentation velocity vS = DS/lg ≈0.3−0.5 µm/s. Particles therefore move in an potential well, and for large displacement should

not behave as free diffusers, but rather as brownian harmonic oscillators. This means that the

probability of a particle, initially sitting at the bottom of a droplet of radius R, undergoing a

given lateral displacement r decays to 1/e when r = rg ≈√

2Rlg. For droplets with R ≈ 20 µm,

corresponding to those used in our diffusion measurements, we have rg ≈ 6 − 10µm, which is

slightly beyond the maximum displacement we followed. However, for smaller droplets we

did observe noticeable deviation from a linear time-dependence and partial ‘saturation’ of

< r2(t) >.

95


Figure 6.9: Fluorescent microscopy image of isooctane/water emulsion droplet sta-

bilised by PC. Brighter regions indicate trapped-particle location.

Hindering of coalescence by static steric hindrance is generally supposed to require densely-

packed, and rather uniformly spread, particles [4]. However we observe stable emulsion droplets

presenting particle surface coverage as low as 5% or confined into localized lumps. Observations

suggest that, when two droplets are in contact, trapped-particle distribution on their surfaces

is not random as reported in fig. 6.9. In quite a few situations, we have indeed captured

images of droplet pairs where fluorescence intensity is much higher near the contact region,

bespeaking enhanced local colloid concentration either in the continuous phase or within the

droplet interfaces. High-resolution images suggest that particles within the contact region

can sometimes form a monolayer, suggesting simultaneous adhesion of a given particle to two

distinct droplets. This ‘bridging’ mechanism may help to keep the droplets at a finite distance,

hindering inter-droplet film drainage.

We have also collected evidence of peculiar particle reallocation effects associated with

droplet closeness, as shown, for instance, in the time-frame of Fig. 6.10. The four bright-field

images portray sequential detachment and departure of a small, partially covered droplet from

a larger one. The four pictures have been tightly focused with artificial contrast enhancement

in order to spot particle-covered regions on the small droplet (particles on the larger droplet are

visible only as a rim in the plane of focus). They show that the particle-covered area undergoes

noticeable morphology changes as the droplets drift apart: colloids on the small droplet are

mostly localised around the contact region when drops are touching, but progressively spread

out over the entire surface of the small droplet as the droplets separate. Colloid redistribution

may therefore play a role in stabilising droplets against coalescence.

96


Figure 6.10: Time-sequence (from A to D, in order) of detachment and drifting apart

of two octanol droplets stabilised by PA. Brighter regions on the smaller drop indicate

trapped-particle location.

97


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fluorescent, monodisperse silica spheres.” Langmuir , 8, (1992), 2921–2931.

[12] Iler. The colloid chemistry of silica and silicates (Corner University press) (1955).

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troscopy of fluorescein (fitc) dyed colloidal silica spheres.” J. Physical Chemistry B , 103,

(1999), 1408–1415.

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Spheres: Synthesis, Characterization, and Fluorescence Confocal Scanning Laser Mi-

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of O/W emulsions.” Colloids Surf. A, 145, (1998), 133–143.

[16] M. Fuji, T. Takei, T. Watanabe and M. Chikazawa. “Wettability of fine silica powder

surfaces modified with several normal alcohols.” Colloids ans surfaces A, 154, (1999),

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99

Gelation of waxy crude oils

Abstract

Some crude oils are characterised by a greater paraffin component fraction. These

are known as waxy crude oils. We deal with gelation mechanisms in these oils

which present some interesting similarities with colloidal gels. A basic introduc-

tion on interaction in a colloidal dispersion is given, for a better comprehension

of the forces leading to particles aggregation and gelation. We present our charac-

terisation of a particular crude oil, rich in the paraffin components, with polarised

light microscopy technique, showing that aggregation is favoured by lowering tem-

perature and decreasing the cooling rate. We study a model system, whose phase

behaviour and rheological properties resemble those of natural oils, with the aim

of characterising the gel morphology. At the end of the chapter, we present light

scattering measurements on a colloidal gel induced by depletion forces.

100

7.1 Introduction Gelation of waxy crude oils

7.1 Introduction

The phase separation of the paraffin component from both crude and distilled oils leads to seri-

ous issues for petroleum industry[1, 2]. Although crude oil is in a fluid phase in the reservoir, it

could undergoes a phase separation at ambient temperature, where paraffins start to crystallise

and separate from the continuous phase. Wax solids can accumulate upon pipes walls inducing

loss of flow. Much greater problems arise if the flow is arrested for pumping breakdown: if the

oil temperature is sufficiently low, as it happens for instance in submarine pipeline transporta-

tion, aggregation of the solid phase can induces gelation of the whole system. Restarting crude

oil flow can be a very serious challenge, since it requires huge pressurisation of the pipeline in

order to break the gel and restore the flow condition. It has been observed that gelation can

be induced when wax solid fraction as low as 1–6% has separated from solution[3, 4, 5].

The phase stability of the paraffin component in an organic solvent could be fully described

in term of the cloud point temperature and the pour point temperature. The former is defined

as the temperature at which paraffins start to crystallise upon cooling, whereas the latter is

the temperature at which the system undergoes gel transition, which triggers the onset of a

visco-elastic behaviour. The re-start problem in pipelines is indeed due to the visco-elastic

behaviour of the gelled system. It has been observed[6, 7] that below the pour point crude oils

present a yield point1 which further increases when the temperature diminishes. The rheological

behaviour of the crude oil studied in this work is reported in figure 7.12. While the system

exhibits a Newtonian behaviour at 40 degree, it presents a critical stress σ below 30 degree.

Raising the stress applied from σ < σ to σ > σ cause a drop in the crude oil’s viscosity of

several order of magnitude.

Figure 7.1: Flow curve for oil A7.1 at different temperatures.

Crude oils present an high dishomogeneity in the phase stability of the paraffin component.

Some crude oils gel at temperature above 30 degree, while others present a pour point at

temperature below −20 degree. This feature can be explained by the differences in paraf-

fin composition both in molecular weight and in molecular structure (n-paraffin, iso-paraffin,

and branched paraffin),in wax volume fraction, thus more generally by the physic-chemical

1The yield point is defined as the minimum pressure required for making the system flown. Being

a pressure, it is measured in Pascal2All rheological data are provided by Enitecnologie

101

7.2 Introduction to colloidal aggregation Gelation of waxy crude oils

differences between crude oils.

Gelation processes is due to interaction between wax crystals, although a comprehension

of the physical mechanisms is still partly lacking. Some hypothesis are qualitatively related to

colloidal aggregation models[7, 8] since they refer to attractive interparticles forces. However,

other models compare gelled crude oils to polymer gels[9].

Our activity on this topic is inserted in a project of EniTecnologie, focused on further

exploring waxy crude oils properties in terms of colloidal aggregation processes, for identifying

exploitable features for inducing the formation of a gel with weaker elastic properties. For

example, a promising property of colloidal gels is their sensitivity to shear during formation.

This could be exploited for reducing the pressure required to restart the flow in a plugged

pipeline.

We characterise, by means of optical methods, the morphological properties of a real waxy

crude oil, of a model system with a similar rheological behaviour of the crude oil, and of a

reversible colloidal gel. Experimental structural parameters as the fractal dimension and the

gyration radius can be used in rheological models, which are outside the aim of this thesis, for

calculating mechanical properties these systems.

7.2 Introduction to colloidal aggregation

Polymer gels are usually formed by linking of polymer chains in a network which fills the entire

volume. Molecules bind in covalent bonds, which are greatly stable compared to thermal energy.

Colloidal gels are instead formed by particles aggregation under the actions of interparticles

forces, weaker than covalent bonds. We will see in the following the nature of these forces.

In order to treat a colloidal aggregation process we should keep in mind we are dealing with

a multicomponent system composed by particles dispersed in a solvent. However it is possible to

deal with one-component system of particles in which only the pairwise interparticle forces are

considered[10], neglecting the fluid-particles interaction. The physical idea is the coarse graining

procedure, which shows that the liquid suspension medium can be regarded as a continuum

described only in terms of macroscopic quantities such as density or dielectric constant. In this

framework the excess equilibrium properties of a single component in the system are therefore

derivable in terms of the potential of the mean force UN :

UN (r1, ..., rN ) =∑

i<j

U2(ri, rj) (7.1)

where U2(ri,rj) is the pairwise potential between particles i,j. Moreover for spherical particles

we may assume:

UN (r1, ..., rN ) =∑

i<j

U2(|ri − rj |)) (7.2)

Thus in the coarse graining procedure, UN depends only on inter-particle pair interactions

neglecting both solvent-particles and many-particles interactions. This approximation is usually

quite accurate for molecular systems[11].

We will consider some basic mechanism responsible for the net interaction between colloidal

particles.

Van der Waals dispersion forces It is known from classic electrodynamic that [12] two

electromagnetic fluctuating dipoles experience an effective attractive force growing considerably

as their distance r vanishes. The forces in question are the dispersion, or London-Van der

Waals forces, scaling as r−6, or r−7 if we are keeping account for relativistic retardation effects.

Between two spherical colloidal particles of radius R the dispersion forces arise on integration

102


over all pairs of contributes from the volume elements in the particles, as shown in Fig. 7.2.

Considering that dispersion forces decay so much quickly as the interparticle distance increases,

we may assume that only the nearest volume element of distinct particle will appreciably

contribute to the integral of all the contributes. The resulting expression, given as a function

Figure 7.2: Van der Waals dispersion forces: the interparticle force is obtained inte-

grating over all the fluctuating elementary dipoles.

of the centre-to-centre distance r is[13]:

UA(r) = −A6

[2R2

r2 − 4R2+

2R2

r2+ ln

(1 − 4R2

r2

) ](7.3)

The Hamaker constant[14] A is of the order of a few kBT and is determined essentially by the

frequency-dependent polarizability mismatch between the particles and suspension medium.

The main consequence of the dispersion forces is the formation of a minimum in the in-

terparticle potential, close to contact (r = 2R), as deep as many kBT , capable to aggregate

irreversibly the colloidal particles. The suspensions can realistically be stabilised against pre-

cipitation only counteracting the Van der Waals strong attraction with a repulsive barrier in

U(r). This mechanism is called colloidal stabilisation: in aqueous suspensions this is commonly

an electrostatic repulsion, whereas for suspensions dispersed in organic solvents this is steric

repulsion. In the former stabilisation particles are surrounded by a cloud of charges, in the

latter one particles are coated with layers of polymers or surfactants.

DLVO potential Most particles become electrically charged when they are dispersed in a

polar solvent as water, because of the dissolution of ionizable groups at the particle’s surface.

Therefore on the surface of the colloids there is a charge Zq, being Z the number of ionised

groups and q their unit charge. The freed counter-ions form an electrical double layer around the

particle which moves together with it. When two particles come so close to overlap the double

layers, it arises a repulsive force that prevents aggregation and keeps distinct the particles.

The spatial distribution of all the ions around colloidal particles can be determined on the

basis of the Poisson-Boltzmann equation. This means that the electrostatic Poisson equation

must be solved assuming the spatial distribution of the counterions is given by an equilibrium

Boltzmann distribution. This is clearly an approximation justified by the observation that

thermal agitation of the counterions is rapid enough that the distribution of macroions can be

considered, at any time scale of practical interest, as the equilibrium one. In this picture we

don’t care about possible correlation between (small) ions meaning that in this sense we are

dealing with a mean field approach. The solution was given in 1948 by Verway and Overbeek in

their extensive monograph[15] leading to the so called DLVO interaction energy of an isolated

pair of macroions immersed in a bath of electrolytes. The result is an electrical double layer

103


Figure 7.3: Charged particles (macroions) are surrounded by an opposite charged cloud

of counter–ions.

surrounding the particles, composed from counterions and the ions of any electrolyte maybe

present in the suspension[16, 17]. The spatial extent of the double layer is of the order of the

Debye-Huckel length κ−1 = λDH :

λDH =

(1000e2NA

εkBT

∑

i

z2iMi

)−

1

2

(7.4)

Where the concentrations Mi of different species of ions which charge is zi are expressed in

mol l−1. Two particles approaching one another experience an electrostatic repulsion screened

from the surrounding cloud of opposite charges. The screened Coulomb or Yukawa poten-

tial provides a useful analytic expression of the effective interaction between pairs of charged

macroions of radius R:

UY (r) =q20

ε r(1 + κR)2exp[−κ (r − 2R)] =

q2eε r

exp(−κr) (7.5)

being ε the dielectric constant of the liquid, κ = λ−1DH the effective screening parameter, and qe

the effective charge of the macroion usually smaller than bare charge of the colloid q0:

qe =q0

1 + κRexp(−κR) (7.6)

The presence of added salts clearly reduce the spatial extent of λDH and therefore the amplitude

and the range of the repulsive potential. The DLVO potential consists in practise of two

contributions:

UDLV O = UA + UY (7.7)

UA, UY being the Van der Waals (Eq. 7.3)and the screened Coulomb (Eq. 7.5)potential re-

spectively. Since the infinitely deep Van der Waals attraction is hindered by a finite wall of a

few kBT , the system is indeed in a metastable state. The long-time stability of the suspension

can be assured by the rigorous absence of added salts.

Sterical stabilisation Colloidal suspension fully stable on a thermodynamic point of view

are achievable only coating the surface of the particles with layers of additives. They include

certain water-loving polymers and surfactants with water-loving chains. Coagulation of two

104


approaching coated particles is fully counteracted by the compression-induced repulsion of the

interpenetrating adsorbed layers.

Depletion Forces Another source of interactions between particles of colloidal suspensions

arises from the addition of non–adsorbing polymers (or surfactants). It has been shown that

([18], [19]) added polymers give rise to an (attractive) force between particles depending on

both the polymer size and concentration. When the inter–particle distance is sufficiently small,

roughly of the order of the added polymers, they experience a force corresponding to the nega-

tive of the osmotic pressure of the additive. This is first stated by Asakura and Osawa (1958)

and subsequently but independently by Vrij (1976). In principle the full mechanism appears

rather complicated since involves the statistical mechanics of the mixed polymer-particles. Let

us make use of some important simplifications:

• Additives are assumed to behave as non–interacting spherical particles of radius ρ .

• Additives interact with particles just through excluded volume.

• Colloidal particles themselves interact through excluded volume.

We shall call ‘depletion zone’ the spherical crust of radius ρ around the colloidal particles (cf.

Fig.7.4). When two particles of radius R approach until the centre-to-centre distance is smaller

than 2(R + ρ) polymers will be ejected from the overlap volume. The basic concept is that

Figure 7.4: Depletion forces are induced by the addition of a non–adsorbing system

(polymers, micelles, small colloidal particles).

polymers exclusion from the depletion zone causes a local imbalance in the isotropic osmotic

pressure Πpol exerted by the polymers against the colloidal particles. The restoration of the

equilibrium results into effective attractions between the particles. Indeed they are forced to

approach more and more by the excess of the osmotic pressure of the additives. Both the

spatial range and the depth of the resulting interaction potential can be tuned varying the

polymer molecular weight and concentration respectively. In principle the role of the polymers

105

optical methods for the investigation of application

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