3.2 Colloidal systems - Treccani, il portale del sapere

167VOLUME V / INSTRUMENTS

3.2.1 Colloid and surface science issuesin the petroleum industry

Colloid science, by convention, deals with small units ofmatter (gas, liquid, or solid) characterized by having atleast one dimension between 0.001 and 1 mm in length. Itis particularly interested in the behaviour of these colloidalobjects when they are dispersed in a second medium,hence colloid science generally deals with multiphasesystems. Typical colloidal systems are represented bysolids dispersed in liquids (dispersions), liquids dispersedin liquids (emulsions), liquids or solids dispersed in gas(aerosols) and gases dispersed in liquids (foams). Thisdefinition acquires its significance because of theimportance of surface-mediated effects upon thebehaviour of the colloid object and, in many cases, thesystem as a whole, when the dispersed phase is small insize (Evans and Wennerström, 1994). In fact, the ratio ofthe surface area to volume increases rapidly as the size ofthe dispersed particle or droplet decreases; for a sphere ofradius r, the ratio of surface area to volume varies as 3/r,so that for a sphere 1 mm in diameter, this ratio is 1,000times greater than that of a sphere 1 mm in diameter.

For dispersed phases of the order of a few mm or less,electrostatic and van-der-Waals forces and Brownianmotion play a fundamental role in their macroscopicphysical behaviour, which includes such processes asaggregation (or flocculation), coalescence, gelation, etc.These processes are illustrated in Fig. 1 for some typicalcolloidal systems together with typical surface-mediatedprocesses such as adsorption, which is important in thestabilization of colloidal solid dispersions and in thereduction of the interfacial tension in liquid-liquid andliquid-gas systems.

In a general sense, the lower limit for the colloidaldomain can be associated with the passage in scale fromsystems with definable interfaces to single molecules. Asalready suggested, the upper limit is defined by the sizescale at which interfacial phenomena strongly impactupon the behaviour of the dispersed phase. Nevertheless,there is some fuzziness at both ends of the scale. In fact,colloid scientists often include solutions of polymermolecules within the scope of their field; while these

3.2

Colloidal systems

A

C

D

B

air

water

micellesmonomers

adsorbed film

Fig. 1. Some typical colloidal systems and their behaviour. A, self-association of surfactant molecules (micellization) and their migration to air-water interfaces; B, formation of emulsions (droplets of one liquid dispersed inside a secondliquid) favoured by surface active agents (surfactants) which lower the energy of the interface; C, stabilization of colloidal particles by repulsive electric charges; D, stabilization of colloidal particles by adsorption of dispersants that inhibit approach of two particles by theirsteric repulsion or by their electric charge.

molecules, in solution, may have physical dimensions upto 1 mm or greater, there is no recognizable interface assuch. To deal with the case of macromolecules and otherdissolved species, colloid science has found it useful tointroduce a distinction between lyophilic (solvent loving,i.e. dissolved) and lyophobic colloids (solvent hating, ordispersed). A polymer that separates from solution passesfrom the lyophylic to lyophobic colloidal state. At theupper end of the size scale, colloid science readily acceptssurface phenomena in general within its ‘scope of work’.Obviously, the processes determining surface charge andproperties (adsorption, etc.) are the same as those ofinterest in determining colloid behaviour, and the tools,methods and results of colloid science find richapplications in all systems where surface properties are ofsignificance.

Colloidal issues and surface phenomena are pervasivein the production, transport and processing of oil and gas.They are the cause of operational problems and play afundamental role in determining the ultimate recovery ofoil from the reservoir. They are also central to thetechnologies used to drill wells, to optimize wellproductivity, and to assure the flow of oil and gas throughpipelines. The ‘colloidal dimension’, in fact, ischaracteristic of the porous reservoir rock itself, which iscomprised of mineral phases that may have an extensionof tens of mm or less and whose microscopic pores areoccupied by two or more immiscible phases (water and oiland/or gas). Furthermore, the wetting state, or wettability,of the rock, which determines the rate and ultimate levelof oil recovery, is a complex function of the type anddistribution of mineral phases, the nature of the brine andcrude oil, and the physical conditions and geological andproduction history of the reservoir. The oils and brinesproduced from the reservoir themselves may be socomposed that the change in thermodynamic conditionsthey experience during production provokes the nucleationand separation of colloidal solid organic or mineral phaseswithin the well or surface facilities, or during transportthrough undersea pipelines.

The importance of these issues for the near-, medium-,and long-term profitability of the oil industry generates acontinuing demand for research and technologicaldevelopment carried out within the oil and servicecompanies, universities, and research institutes.

Table 1 reports, within the broadly identifiable stagesof oil and gas production, transport and processing, someof the areas in which colloidal and surface phenomena andtechnologies based on colloidal systems are important.This will be a useful reference for the following part ofthis text, in which several technological areas rich incolloidal issues are discussed.

Organic solid phase separation phenomena in oil production and processing

Petroleum is a thermodynamically complex mixture ofmany thousands of organic compounds and, whenproduced and utilized, it undergoes a series of phasechanges as a function of changes in temperature, pressureand composition. These changes, which can include theloss of volatile fractions and the formation of solid organicfractions (waxes and asphaltenes), can occur at any point

from the reservoir to the refinery, depending on the natureof the crude oil and the physical environment to which it isexposed during transport, storage and processing. Phaseseparation and deposition of organic solids poses a majorissue for crude oil transport offshore, in particular,because of the limited access to the pipelines and the highcost of intervention.

In general terms, crude oils are exposed to highpressure (P) and moderate to high temperature (T ) in thereservoir, moderate to low T and P during transport andstorage, and high T and low P in the refinery. Changes inT and P, loss of light fractions and chemicaltransformations at high T within the refinery may favouror disfavour the formation of organic solid phases. In thissection, we will examine the characteristics and some ofthe problems associated with two of the heavier petroleumfractions, asphaltenes and waxes, whose phase separationin oil processing facilities and pipelines is a major issuefor oil operators. These crude oil components display arich range of colloidal and surface-mediated behaviours,including nucleation and growth of the solid phase and theformation of deposits on pipeline walls. For both,aggregation processes are important, leading to gelation ofthe bulk oil in the case of waxes. In addition, asphaltenesare surface active compounds (surfactants) and both, inthe solid phase, tend to stabilize emulsions as discussed inthe section dedicated to Pickering emulsions.

AsphaltenesAsphaltenes are the heaviest, most polar fraction of

crude oil. They are defined by a solubility criterion ratherthan molecular structure and are formed of a largenumber of compounds which vary from oil to oil. Bydefinition, then, asphaltenes are the fraction of oilinsoluble in n-paraffins (in general, n-C5 or n-C7) butsoluble in benzene (or toluene; ASTM, 2000, 2001a,b).As would be expected, the quantity and the quality (i.e.composition) of the asphaltene fraction depends on the n-paraffin employed and on the procedure employed toseparate the asphaltenes from the solvent mixture (Ciminoet al., 1995). In general, they are comprised ofpolyaromatic sheets with alkyl side chains; heteroatomssuch as O, N, S and metals (V, Ni, Fe) are also present.The following range of molecular properties is found (Yenand Chilingarian, 1994): a) H/C atom ratio from 0.8to1.4; b) average monomer molecular weight from 500 to3,000 u; c) S from 0.5 to 10 weight %; d ) N from 0.6 to2.6 weight %; e) O from 0.3 to 4.8 weight %. Weemphasize the fact that, in chemical terms, the asphaltenefraction of a crude oil is comprised of a mixture ofcomplex molecules.

A remarkable characteristic of the asphaltenes is theirtendency to aggregate reversibly in hydrocarbon media orcrude oil, even at very low concentration. This behaviourleads to artificially high average molecular weights(1,500-100,000 u) as determined by osmometric and manyother methods; current best estimates place the averagemolecular weight of asphaltene monomers equal to or lessthan about 1,200 u.

Asphaltene phase separation is an important issue inoil production. It leads to the formation of heavy organicdeposits from the reservoir to the oil centre, reducing orblocking entirely the flow of oil into the well (by clogging

SURFACES AND DISPERSE SYSTEMS

168 ENCYCLOPAEDIA OF HYDROCARBONS

COLLOIDAL SYSTEMS


Table 1. Some colloidal issues in oil and gas production, and transport

Technological area Colloidal aspects

Exploration and appraisal

Petrophysical evaluation of reservoir cores Wettability of the rock; capillary pressure; relative permeability

Crude oil, brine phase behaviour in production lines

Physical state of heavy components in the oil; thermodynamics, kinetics of nucleation and growth of solid or gas phases

Basin modelling Diagenetic phenomena, including nucleation events

Production

Drilling fluidsClay dispersions in water, oil; mechanism of destabilization of shales by aqueous drilling fluids; fluid invasion/filtration properties in porous media

Cements Permeability to gas and gas-blocking additives

Well productivity• acidizing Dissolution, nucleation/precipitation of mineral phases

• corrosion inhibition Inhibitor adsorption onto pipe walls; pitting

• fracturingRheology of concentrated polymer solutions and gels, and their filtrationbehaviour in porous media

• sand controlCapillary forces holding unconsolidated sands in place; chemical consolidation of sands

• scale inhibitionThermodynamics, kinetics of nucleation; threshold inhibition mechanisms;inhibitor retention/release in reservoir rock

Produced fluids management• oil-water separations

Coalescence, membrane-based processes; solid-stabilized emulsions; emulsion-breaking additives

• water re-injection Impact of dispersed solids and oil on injectivity/filtration behaviour

• reduction of water in wellPolymer solution and gel rheology, cross-linking; propagation, injectivity, gel stability in porous media

Transport

Flow assuranceThermodynamics, kinetics of nucleation/precipitation of asphaltenes, wax,hydrates; inhibition/remediation technologies

Heavy oil transport as aqueous emulsion Emulsion formation and rheology; stabilization and destabilization

Multiphase flow Emulsion formation, non Newtonian rheology

pores of the reservoir rock), flow through pipelines, etc.For this reason, considerable effort has been dedicated togaining a better understanding of the factors governingasphaltene stability and the development of predictivemodels. An early physical model of asphaltenes in oil,proposed by Frederik Nellensteyn in 1938, described theasphaltenes as solids insoluble in the oil, but dispersed (orpeptized) by another oil fraction, i.e. resins, adsorbed ontheir surface (the resins are compounds chemicallyintermediate between the oil and asphaltenes). This pictureconstitutes the so-called lyophobic-colloid model, inwhich solid formation is believed to be the result of thedesorption of resin molecules from the surface of theasphaltene aggregates.

A competing theory considers asphaltenes and oil toconstitute a true solution and attributes the colloidalbehaviour (association and deposition), in analogy withthe polymer solution theory, as arising from the largedifference in dimensions between asphaltenes and theoil matrix (Hirschberg et al., 1984). In this lyophiliccolloid model, de-stabilization is the result of areduction in the solvating power of the hydrocarbonmatrix, brought about by a change in pressure or theaddition of some compounds. According to this viewthen, the asphaltenes are initially lyophilic colloids inthe oil, bearing some analogy to macromolecularsolutions, only to become lyophobic colloidal solids,upon phase separation.

The differences between these models are not onlysemantic, but lead to different predictions of asphaltenebehaviour under different conditions. This means that theycan be subjected to direct experimental test, as indeed theyhave been over the past decade or so (Cimino et al., 1995).Evidence in support of this view includes the following:• The lyophobic-colloid model is based on the

assumption that asphaltenes are inherently insoluble inoil, though common laboratory and field experiencesdemonstrate this to be false (Porte et al., 2003).

• The assumption that the resins adsorb onto the surfaceof the asphaltene aggregates is falsified by directexperimental test (Espinat and Ravey, 1993; Cimino etal., 1995).

• Asphaltene fractions form aggregates of the samedimensions – as revealed by SAXS (Small Angle X-ray Scattering) and SANS (Small Angle NeutronScattering) studies – in toluene, in the absence of theresin component, as they do within the crude oil. Thusthe resin fraction does not govern asphaltene solutionbehaviour.

• The same flocculation behaviour is observed both forasphaltenes in crude oil and for model mixtures (madeup of previously ‘purified’ asphaltenes, then dissolvedin solvents). This implies that the stabilization/de-stabilization mechanisms result from the asphaltenesthemselves.Notwithstanding the strong evidence against the

lyophobic model, and in spite of the fact that the lyophiliccolloid model is implicit in many of the predictive modelsand tools employed by the industry, technicians working inthis sector have shown little propensity thus far to drop theold convention.

Asphaltene deposition in petroleum production and processing

As noted above, the main problems in oil productioncaused by asphaltenes are connected with the formation ofsolid deposits. Interestingly, the risk and severity ofasphaltene deposition is inversely correlated withasphaltene content, relatively light oils with lowasphaltene content (�4 weight %) being the mosttroublesome. Another characteristic is that deposition ismuch more sensitive to depressurization of the oil than toa drop in temperature. In many instances, asphaltenedeposition is observed within the production string onaccount of the pressure drop experienced by fluidsentering from the reservoir; on the other hand, themaximum deposition is, in general, found close to thebubble pressure. This behaviour is explained by thedecrease in density of the oil as pressure declines to thebubble point, which reduces its solvent power towards theasphaltenes (Correra et al., 2005). Depending on the fluiddynamic conditions, the precipitated asphaltenes maybuild up locally, choking off fluid flow, or they may betransported with the fluids leading to the formation ofstable oil/water emulsions or sludges further downstream.Asphaltenes also generate problems in the refinery. Theirprecipitation can be provoked, for example, by theblending of different streams, where one of the streams ishighly paraffinic. In this case, the asphaltene depositgenerally forms in storage tanks, requiring periodicwashing operations. However, they can also stabilize

emulsions and foams, and cause fouling in refinery plants.Last but not least, deposits of de-stabilized asphaltenes inhot points of refinery plants may undergo chemicalreactions, with coke formation and related foulingproblems.

Asphaltene phase behaviour modellingOwing to their large potential impact on oil

production, operators need to be able to predict whether acertain oil will generate asphaltene deposits, at what depthwithin the well, and at what rate. While several practicaland semi-empirical approaches have been developed, acorrect physical model of the phenomenon is compulsoryin order to generate a detailed prediction of depositionrisk. Such a model has been developed based on the Flory-Huggins model of polymer solution behaviour. As inputdata, it employs results of titration experiments on stocktank oil. In these tests, precipitant (a paraffin) is added tothe oil (eventually pre-mixed with toluene) in smallincrements, while the sample is monitored betweenconsecutive precipitant additions. When asphalteneprecipitation takes place, the observed variable (e.g. lighttransmittance) changes value sharply, defining thecomposition at the onset of asphaltene precipitation. Inthis way, it is possibile to define the solubilitycharacteristics of the asphaltene fraction; on the otherhand, information on the oil characteristics are availablethrough the standard PVT (pressure, volume, temperature)kind. The model employs this information to generate aphase envelope, i.e. to define the zone of asphalteneinstability in the T,P plane (Correra, 2004a).

WaxesWaxes are straight- or branched-chain paraffins and

are found in oils in varying amounts and carbon numbersup to C70. Wax solubility depends upon its chemicalcomposition, the composition of the oil, pressure and,above all, temperature. For a given composition andpressure, wax solids separate as the temperature is loweredbelow a threshold (the cloud point, or Wax AppearanceTemperature, WAT; Létoffé et al., 1995). For this reason,as the temperature is lowered, the viscosity can increase bymany orders of magnitude. Further, while oils are usuallyNewtonian fluids above the WAT, the appearance andgrowth of wax crystals in the oil produces increasinglynon-Newtonian behaviour. For many oils, below acharacteristic temperature (the pour point), completegelation takes place. Typically, as little as 1-5% wax solidsare present at the pour point (Visintin et al., 2005).

Wax related production problemsWax deposition is perhaps the most significant flow

assurance problem for offshore production: the USMinerals Management Service (Alvarado, 2003) reportedin 2002 that all of the 51 severe plugs in the US Gulf ofMexico between 1992-2002 were wax-related. Waxdeposition generally happens in pipelines or sea lineswhen the internal wall temperature falls below the WAT; ifnot adequately contrasted with additives or piggingoperations, the wax deposit can grow until it causes asevere loss in productivity. Frequently, wax deposits arethe cause of pigging devices that are blocked within apipeline.



Assessment of the conditions for solids deposition isrelatively easier for waxes compared to asphaltenes: thephenomenon depends primarily upon temperature.Differential Scanning Calorimetry (DSC) is a useful toolfor identifying the WAT and for determining the amountof wax that will solidify at each temperature. However, forpredicting the rate and severity of wax deposition in apipeline, a fluid dynamic model is required.

Wax depositionAccording to the traditional view, wax deposition is

governed by the radial mass (wax) flux to the cold pipewall. Various mechanisms of radial transport aredescribed in literature, such as molecular diffusion,shear dispersion, Brownian diffusion, gravity settlingand others (Borghi et al., 2005). In general,commercially available software models give a strongpredominance to molecular diffusion to the cold wall,where the wax components change phase and to whichthey adhere. This change of phase generates a radialconcentration gradient of wax molecules which diffuseto the wall according to Fick’s law of diffusion, leadingto growth of the deposit (Fasano et al., 2004).

Evaluations of existing software seem tounderestimate deposit growth, so that not even a reliableestimation tool is available at present. This considerationhas led various research groups to look for alternativemodels of wax deposit formation and growth. The mostimportant findings of this research can be summarized asfollows:• Wax solids aggregate to form a gel-like, percolating

structure even at very low ax concentrations and at alltemperatures below the WAT (Vignati et al., 2005).Thus the role of radial transport is perhapsoverestimated, as axial mass transport is already ableto furnish enough waxes to allow deposit formation(Correra, 2004b).

• For the same reason, the kinetics of deposit growthwill be dominated by heat transfer. When the pipelinewall temperature falls below the WAT, a gelation frontbuilds up from the wall; this process is very fast(Merino-Garcia and Correra, 2006).

• Two mechanisms can limit deposit growth. First, thegel front will stop growing if heat transfer conditionsare such that the WAT is attained on the front (Bidmusand Mehrotra, 2004). Second, a sufficiently high shearstress at the wall will hinder wax adhesion, in whichcase wax solids will not be deposited, but will insteadbe carried through the bulk in the form of particles.

• Formation of the gel-like deposit is followed by aso-called ‘ageing’ process in which the deposit layermaintains a constant thickness, while its compositionchanges over time, becoming more concentrated inheavy waxes. In this phase, radial transport of waxmolecules into the gel is dominant (Venkatesan et al.,2005).On the basis of these observations, it is not strange

that present simulations are unreliable: they predictdeposit thickness on the basis of a mechanism (radialtransport) different from the correct one (gelation, axialtransport). Progress towards a more reliable predictionof deposit growth is also hindered by the lack ofsuitable field data.

Wax pluggingAs noted above, gelation of the bulk oil takes place

when the temperature falls below the pour point, asituation that may occur during un-programmed shut-down of a pipeline, particularly offshore. Re-start of flowin the presence of a gel plug is possible only if sufficientpressure can be placed on the pipeline. Currentunderstanding of crude oil gelation below the pour point isthat gelation is the result of the formation of a percolating,fractal network of wax crystals held together by weakattractive forces. The processes involved in waxcrystallization and oil gelation are illustrated in Fig. 2,which also shows a photomicrograph of wax crystalsformed in a crude oil. Crude oil gels are thus consideredto be members of the family of gels comprised of weakly-attractive, dispersed colloidal particles, such as thoseformed from silica (Visintin et al., 2005). The rheologicalbehaviour of crude oil gels provides evidence in support ofthis mechanism. Strong shearing of the gelled oil reducesthe viscosity to a level consistent with that of a dispersionof solid particles. As shown in Fig. 3, following return toquiescent conditions, the sheared crude oil gel re-heals toa significant degree. This behaviour is indicative ofreversible, attractive interactions between the solid waxparticles, and is expected from the colloidal particle gelmodel.

To better understand the behaviour of crude oil gels,and to facilitate their comparison with other gels formedby dispersed colloidal particles, fractal geometry has beenused to model their internal structure. Dispersions ofweakly attractive particles display rheological propertieswhich are strongly dependent on the modality of particleaggregation. The use of fractal geometry can simplify thedescription of the structures formed during aggregationbecause a small number of parameters provides asatisfactory representation of these highly irregularobjects. The fractal dimension Df , for example, provides ameasure of the compactness of the aggregate. Values of Df obtained for particle aggregates in the dilute regimerange between 1.8 and 2.1, respectively, passing from theso-called diffusion-limited cluster-cluster aggregation tothe so-called reaction-limited cluster-cluster aggregation,delineate two ways in which structures may build up fromcolloidal particles. Diffusion-limited cluster aggregationrefers to the case where all collisions between particlesdiffusing in space lead to cluster formation and is a goodrepresentation of colloidal particle gelation where theinteraction is strong and short-ranged. In reaction-limitedcluster aggregation, most collisions between particles donot produce a bond; this is a good representation ofcolloidal particle gelation where there is a repulsivebarrier against aggregation. Df is greater in the latter casebecause not all collisions form links, and many structuresare explored before adopting a stable configuration.

The fractal dimension, 1.8, found by light scatteringfor wax aggregates in a ‘model’ oil just below the WAT isconsistent with diffusion-limited cluster-clusteraggregation, while that determined from rheologicalproperties below the Pour Point ranges from 2.1 to as highas 2.7 for the model oil and several crude oils. Theseresults are accounted for by considering that in waxycrude oils the development of the solid fraction isgoverned by temperature: the fraction of wax solids

COLLOIDAL SYSTEMS


increases as the temperature decreases below the WAT.Thus, wax crystallization proceeds in the presence of aweak percolating network, leading to the overlapping ofaggregates and the formation of interpenetrating clusters.This mechanism suggests the possibility to employmechanical forces or chemical additives in order tomodify the interaction between wax crystals, and therebyreduce the tendency for wax deposits to build-up or reducethe energy to re-start flow in a gelled pipeline.

Oil-water emulsionsWith rare exceptions, and much to the frustration of

petroleum companies, water (reservoir brine) is co-producedfrom the reservoir together with oil and gas. Although theamount of water produced may be very low initially (less

than a few percent of total fluids produced), waterproduction increases as the field matures and usuallydominates fluid production late in the field life. Passage ofthe produced oil and water through pumps and valves orother high shear conditions favours formation of fineemulsions which can be stabilized by surface activecomponents present in the crude oil, chemical additivesemployed for other purposes (e.g. corrosion inhibition), andorganic and inorganic solids.

Oil-water separation is accomplished with ad hoc gravityseparators. The oil generally needs further de-salting andde-watering in order to achieve commercial quality. Removalof oil from water is also required before the latter can bedisposed of into superficial bodies or re-injected into thereservoir. Today this is often accomplished with



aggregation

lamellae, 1-2 nmthick each

gelation

sheet-likeparticles

crystal bundles

0.1-5 mm

2-10 mm

Fig. 2. Processes involved in waxy crude oil gelation; insert photomicrograph shows colloidal wax aggregates in a crude oil.

shear rate (0.01 s�1) applied for 60 seconds

stor

age

mod

ulus

(Pa

)

10,000

100

1,000

time (min)0 10 20 30 40 50

Fig. 3. Plot of the storagemodulus of a waxy crude oilgel illustrating partialrecovery of gel strengthfollowing shear stress for 60 seconds at 0.01 s�1.

hydrocyclones which can reduce the level of residualdispersed oil to 10-30 mg/l.

Oil-water separations often present considerablechallenges. One example is given by heavy crude oils, whichhave densities close to that of water and are extremelyviscous when cooled to surface conditions; as predicted byStoke’s law, these factors render gravity-based separationprocesses relatively less effective and hinder coalescence.Further, the presence of surface active components in thecrude oil, together with the tendency to deposit organicsolids, favours the formation of solid-stabilized emulsionswhich can be extremely persistent. These may accumulate atthe oil-water interface in the separator units reducing theirefficiency, making it difficult to achieve the required purityof the oil for sale or the brine for disposal. Chemicaladditives are widely used to manage emulsion formation andstability. Since every oil is unique, widely differentiateddemulsifier chemistries have been developed; theirapplication is very much an art.

In the following the stabilization of emulsions in thepresence of colloidal solids, an area of very active academicresearch today, will be examined. The formation ofemulsions is not always undesired in the upstream industry.In fact, ultra-heavy oil-water emulsions are intentionallygenerated in some cases in order to facilitate transport andimprove the commercial value of the crude oil, as discussedbelow. Finally, an emerging nano-technology based on theformulation of nano-emulsions which may, one day, aid inthe deployment of chemical additives will be described.

Solids-stabilized Pickering emulsionsEmulsions stabilized by colloidal solids were first

reported by Walter Ramsden (1903) and Spencer U.Pickering (1907) and have come to be known by the name ofthe latter. Though their role in many industrial and petroleumprocesses has long been recognized, the scientific study ofthese emulsions was only sporadic until the turn of the 21st

century, when the scientific interest forspontaneously-organizing systems provoked a literalexplosion of publications in this area (Binks, 2002).

Pickering emulsions are generated when droplets areformed in the presence of colloidal particles of appropriatesurface wetting properties. As a general rule, Pickeringemulsions are stabilized by particles of intermediatewetness, and the preferred internal phase (the dispersed one)is that which is less wetting to the solid phase (i.e.predominantly water-wet solids favour formation ofoil-in-water emulsions). The driving force for trapping ofsolids at the interface is the reduction of the total surfacearea. When a particle of radius a moves at the oil-waterinterface (Fig. 4), it removes the interfacial area A1 from thetwo liquid phases (ow, oil-water) at the cost of creating thesolid-water (sw) area A2 and the solid-oil (so) area A3. Thesystem free energy variation is:

where g is the interfacial tension and ÿ is the contact angle.Since the free energy after particle adsorption decreases, theprocess is a spontaneous one. Once at the surface, theparticles are bound very tightly; the energy of trapping is afunction of wetting and is many times the thermal agitationenergy kT (where k is the Boltzmann constant and T, the

thermodynamical temperature). Hence, trapping isessentially irreversible.

Another general characteristic is that a greater amount ofsolids present during emulsion formation favour thestabilization of more, and hence smaller, emulsion droplets.Many microscopy studies have shown dense and sometimesclose-packed surface coverage by the colloids. These andother observations led to the formulation of the generalhypothesis that emulsion stabilization by solids derives fromthe rigidity of the particle layers at the surface and from theenergy that would be required to reorganize the surfacebefore coalescence of the droplets could occur.

These and other hypotheses have been examinedexperimentally by EniTecnologie with submicron fluorescentcolloidal silica particles which were prepared and used toexamine surface coverage and particle mobility on thesurface (Vignati et al., 2003). Also, by using a micropipetteto measure the surface tension of single particle-stabilizeddrops, it was demonstrated directly that the particle coveragehad no influence on interfacial tension. These studiesdiscovered a much richer range of behaviours than hadpreviously been recognized. In particular:• No straightforward relation exists between the degree of

droplet surface coverage and macroscopic emulsionstability. Long-term stability was observed for someemulsions with very few adhering particles; conversely,some emulsions comprised of densely covered dropletswere found to undergo rapid coalescence.

• Particle redistribution on the droplet surface may play arole in stabilizing droplets with low or inhomogeneousparticle coverage.

• Surface roughness appreciably lowers particleemulsifying power. Interfacial organization of trappedrough particles displayed a rich morphology, sometimesmarked by ‘colloidal lumps’, suggestingsurface-mediated attractive forces of capillary origin.

• Trapped particles exhibit vigorous Brownian motion,with a surface diffusion coefficient that, on poorlycovered droplets, has essentially the same value as in thesurrounding bulk phase.On-going studies in many laboratories are exploring

these and other behaviours not previously recognized.

A aow ow− ≈ − −( )γ π γ12 2

1 cosÿ

∆G A Asw so= + −γ γ2 3

COLLOIDAL SYSTEMS


oil

water

A2

A3

A1

Fig. 4. Diagram showing reduction in total surface area when a solid particle migrates to a liquid-liquid surface. A1 indicates oil-water interfacial area eliminated by the particle at the interface, and A2 and A3indicate particle surface exposed to the water and oil phase,respectively.

Simultaneously, efforts to develop new applications ofPickering emulsions have emerged. These include the use ofsolids as non-surfactant emulsifying agents (Binks, 2002),the use of solid-stabilized emulsions for drug or reagentdelivery (Dinsmore et al., 2002), and the use of magneticcolloidal particles to generate drops that can be manipulatedby an external magnetic field (Melle et al., 2005).

Returning to petroleum systems, emulsion stabilizingsolids found in petroleum emulsions can be inorganic (clays,scale, silica) or organic (waxes, asphaltenes). The latter cangenerate elastic films which are very resistant to breaking, asshown for oil droplets in produced water in Fig. 5. The crudeoil in the figure is known to deposit both wax andasphaltenes under ambient conditions, and many disperseddroplets have irregular surfaces indicative of the presence ofa mechanically resistant film. Basic studies underway maysuggest new chemical strategies for de-sorbing the colloidalsolids, for reducing their tendency to migrate to the surface,or for hindering the development of organic films on thesurface of the drops.

Oil-in-water emulsions for transport of ultra heavy oilsThe formation of oil-in-water emulsions has been

exploited industrially for the transport of viscous, heavyoils. This application is based on the fact that therheology of an emulsion formed from two liquids ofradically different viscosity is, to a first approximation,determined by that of the external, or continuous, phase.Thus, where an ultra-heavy oil might have a viscosity of10,000 cP or more, when emulsified in water, the mixturewill have a viscosity some orders of magnitude lower. Fig. 6 shows how the viscosity of a crude-oil-in-wateremulsion varies as a function of the fraction of theemulsified heavy oil phase for an Italian oil whoseviscosity is 13,000 cP; note that even with 70% of heavyoil in the emulsions, with water as the external phase, theviscosity is roughly two orders of magnitude lower thanthat of the crude oil itself.

The economics of using water-external emulsions totransport heavy crude oil become more attractive as thefraction of oil transported increases. Beyond a certain oilfraction, however, there is a strong tendency for the emulsionto invert to a water-in-oil emulsion, with disastrousconsequences for the viscosity. Technologically, thechallenges to developing emulsion transport processes arethose of:• Assuring that water remains as the external phase, in

emulsions containing at least 70 volume % of oil, overthe entire range of possible temperature excursions andshear forces that will be encountered.

• Controlling the tendencies of the emulsion to sedimentand to undergo structuring during prolonged storage, orin the event that flow in the pipeline must be interruptedfor an extended period.

• Assuring effective de-emulsification with recovery ofwater-free oil at the refinery.Beginning in the early 1980s, Petróleos De Venezuela

Sociedad Anónima (PDVSA) took on these challenges,developing an integrated process for fluidizing theultra-heavy crude oil of the Orinoco Belt in the form of anaqueous emulsion. This process, called Orimulsion,generates a 70% oil in water emulsion which is transportedvia ship worldwide where it is burned in thermoelectric

power plants. Present production is of the order of 4 milliontonnes/y of emulsion (Marruffo et al., 1998).

Fig. 7 outlines the basic elements of the Orimulsionprocess (Salager et al., 2001). The crude oil is diluted withlight hydrocarbon in the wellbore to facilitate its transport tothe surface where it is dehydrated and desalted ingravitational and electrostatic separators in order to removereservoir brine. The solvent is recovered by flash distillationand recycled to the wellbore. The product (Orimulsion) isprepared by emulsification of the heavy crude with anaqueous ethoxylated nonylphenol solution to generate an85% oil phase-in-water emulsion. Mixing is done under mildconditions to avoid generating so-called multiple emulsions,in which the internal oil phase contains, in its turn, dispersedwater droplets. Next, this concentrated emulsion is diluted to70% oil content by adding additional water. The average sizeof the dispersed oil droplets is 8-15 mm and the apparentviscosity is 350-750 mPa�s (measured at 30°C and 100 s�1).

Eni has developed a similar process based on the use ofdispersing agents rather than surfactants (Bertero et al.,1994). The two differ in that surfactants lower the interfacialtension, making it easier to create new, or to deform, oil-water interfaces. Dispersing agents, on the other hand,stabilize the dispersed phase by adsorbing at the interfaceand creating an electrostatic or steric barrier against closedrop-drop contact; they may act only weakly on theinterfacial tension. Experiments showed that the use of thedispersing agents generated oil-in-water mixturesparticularly resistant to inversion, thus removing a major riskfrom industrial applications. The additive was pilot-tested inthe Gela oilfield in Sicily, where the emulsion was generateddirectly within the wellbore, using the mixing energy of therod pump or down-hole jet pump. The strong reduction inviscosity, compared to the use of conventional diluent,resulted in a four-fold increase in well productivity.

Nano-emulsionsThe so-called nano-emulsions (or mini-emulsions)

represent a type of emulsion of rapidly growing scientificand industrial interest, and for which many potentialapplications to the petroleum industry can be foreseen.



Fig. 5. Dispersed oil drops in reservoir brine with irregular surfaces indicative of the presence of a solid film at the oil-brine interface.

These emulsions are characterized by internal or dispersedphase droplet sizes smaller than 1 mm. One characteristic ofthese emulsions, when the drop size is less than thewavelength of visible light (i.e. less than about 0.2 mm), isthat they are optically translucent or transparent.

At the submicron size scale, buoyancy forces are lesssignificant than Brownian motion, eliminating any tendencyfor sedimentation or creaming of the internal phase. On theother hand, the small droplet size and large surface areamakes these emulsions highly unstable thermodynamically.Ostwald ripening (Taylor and Ottewill, 1994) describes theprocess whereby the smaller drops in a mixture shrink at theexpense of the growth of the larger drops, a process whichreduces the total surface area and, hence, the total energy ofthe system. The rate of transfer of the internal phase betweendroplets depends upon its solubility in the external phase.

Several factors can reduce or eliminate this de-stabilizationmechanism: • Extremely low solubility of the internal phase in the

external phase.• The presence of a solute (insoluble in the external phase)

in the internal phase, in which case the driving force fortransfer of the internal phase to the larger droplets willeventually be countered by the increase in osmoticpressure of the smaller droplets as the soluteconcentration increases.

• A tight, monodisperse dispersion of droplet sizes.Provided the Ostwald ripening mechanism can be

controlled, the lack of sedimentation/creaming andextremely small size of the dispersed droplets makesnano-emulsions a system of considerable interest for a broadrange of applications in the pharmaceutical, cosmetics, food

COLLOIDAL SYSTEMS


anhydrous oil viscosity

visc

osit

y (c

P)

100,000

1

10,000

1,000

100

10

volume fraction of oil0.20 0.40 0.60 0.80 1.00

Fig. 6. Viscosity of heavycrude oil in water emulsionsas a function of volumefraction of dispersed oilmeasured at a velocitygradient of 100 s�1

(Bertero et al., 1994); dashed line indicatesviscosity of the anhydrouscrude oil.

steaminjection

diluentinjection

gas

water

recycled diluent

pipelining

embarking

bitumen storage

emulsification plantOrimulsion

storage

downhole

extractionof dilutedbitumen

flashseparatorheating separator dehydration

surfactantwater

additives

Fig. 7. Basic elements of the Orimulsion process (Salager et al., 2001).

and other industries. EniTecnologie is pioneering thedevelopment of nano-emulsion applications to the petroleumindustry (Del Gaudio et al., 2006).

There are two general routes to formation ofnano-emulsions. The first is based on mechanical mixing atvery high energy (e.g. ultrasonication; Landfester et al.,1999) which, in the presence of suitable surfactants, canprovide emulsions with average droplet size somewhat below1 mm, although the particle size distribution can be far frommonodisperse. Such nano-emulsions therefore rarely haveprolonged stability, which limits the range of their potentialapplications. The second route exploits techniques whichmake it possible to generate nano-emulsions, using modestenergetic content, based on self-emulsification or phaseinversion. For example, phase inversion can be obtained bydiluting a concentrated emulsion (Forgiarini et al., 2001), orby temperature change (Phase Inversion Temperature, PIT,method; Shinoda and Saito, 1968). Many authors attributenano-emulsion formation to the presence of a liquidcrystalline phase during phase inversion (Izquierdo et al.,2005), although nano-emulsions have also been preparedfrom systems which avoid liquid crystalline phases entirely(Pons et al., 2003).

To date, there has been a lack of general routes tonano-emulsions. The PIT approach, for example, can beemployed only for a limited class of surfactants and thestability of the nano-emulsions produced is extremelysensitive to temperature. In recent years, increasing attentionhas been given to understanding the mechanism of theprocesses involved in the different methods ofnano-emulsion preparation, in the interest of developingsimpler, more reliable and general routes.

The low energy process developed in EniTecnologie isbased on carrying out the phase inversion in a systemcharacterized by extremely low interfacial tension, andgenerates monodisperse nano-emulsions in the presence ofonly very gentle mixing. The process reproducibly providesnano-emulsions characterized by extremely small dropletsthat are extremely stable to storage (over 6 months at roomtemperature), to centrifugation (stable to 5,000 g for 15minutes) and to temperature (formulations resist for an houror more up to 100°C).

This method allows the preparation of nano-emulsionswith up to 35% of dispersed phase which can contain 10% ormore of dissolved additives. For example, water-in-oilnano-emulsions have been prepared with 20% of waterphase containing 15% by weight of a typical oilfield scaleinhibitor (phosphino-policarboxylic acid). Because of thesmall size of the dispersed phase, these compositions passreadily through reservoir rock without filtration. Hence, theyrepresent a vehicle for introducing water-soluble chemicaladditives into the reservoir, without having to introduce largevolumes of water. This is important because it reduces thetime for a treated well to recover its original oil productionrate following treatment. It is also of interest for treatment ofreservoirs containing water-sensitive mineral phases becausethe water-in-oil nano-emulsion can deliver the water solubleadditive in a hydrocarbon continuous phase.

Another important potential application is for chemicaladditive delivery into flow lines, e.g. to control against solidsdeposition, corrosion, etc. In many situations, it is necessaryto dose the production stream with more than one additive.Where these additives are incompatible, either because of

their oil/water solubilities or because they undergo chemicalreaction (e.g. acid/base), this poses considerable operationaland economic difficulties. In many cases, facilities forchemical storage are limited (e.g. offshore) or there may beonly a single chemical feed line available. Nano-emulsionsappear to offer an ideal solution to this problem. In fact, theextremely limited interaction of the dispersed andcontinuous phase make it possible to load even chemicallyincompatible additives into the two phases withoutcompromising the excellent stability and resistance tocreaming/sedimentation and shearing of the nano-emulsions.Nano-emulsions are in a good position to be one of the firstnanotechnologies to find application in the oil industry.

References

Alvarado A. (2003) View of flow assurance issues, in: Flow assurance.A holistic approach, Houston (TX), 15-16 May.

ASTM (American Society for Testing and Materials) (2000) Standardtest method for determination of asphaltenes (heptane insolubles)in crude petroleum and petroleum products - IP 143/01, ASTMD-6560.

ASTM (American Society for Testing and Materials) (2001a) Book ofstandards, v.5.01, Standard test method for insolubles in usedlubricating oils, ASTM D-893-97.

ASTM (American Society for Testing and Materials) (2001b) Book ofstandards, v.5.01, Standard test method for characteristic groupsin rubber extender and processing oils and other petroleum-derivedoils by the clay-gel absorption chromatographic method, ASTMD2007-01.

Bertero L. et al. (1994) An innovative wav to produce and transportheavv oil through dispersion in water: laboratory study and fieldtest results, in: Proceedings of the Society of Petroleum Engineers69th annual technical conference and exhibition, New Orleans(LA), 25-28 September, SPE 28543.

Bidmus H.O., Mehrotra A.K. (2004) Heat-transfer analogy for waxdeposition from paraffinic mixtures, «Industrial & EngineeringChemistry Research», 43, 791-803.

Binks B.P. (2002) Particles as surfactants-similarities and differences,«Current Opinion in Colloid & Interface Science», 7, 21-41.

Borghi G.P. et al. (2005) In-depth investigation on wax depositionmechanisms, in: Proceedings of the Offshore Mediterraneanconference and exhibition, Ravenna (Italy), 16-18 March.

Cimino R. et al. (1995) Solubility and phase behaviour of asphaltenesin hydrocarbon media, in: Sheu E.Y., Mullins O.C. (editors)Asphaltenes. Fundamentals and applications, New York, PlenumPress, 97-130.

Correra S. (2004a) Stepwise construction of an asphaltene precipitationmodel, «Petroleum Science and Technology», 22, 943-959.

Correra S. (2004b) A new perspective on wax deposition, in:Proceedings of the International conference on heavy organicdeposition, San José del Cabo (CA), 14-18 November.

Correra S. et al. (2005) Estimation of the solvent power of crude oilsmeasurements, «Industrial and Engineering Chemistry Research»,44, 9307-9315.

Del Gaudio L. et al. (2006) A low energy method for water-in-oilnano-emulsion preparation, in: Proceedings of the World congresson emulsions, Lyon (France), 3-6 October.

Dinsmore A.D. et al. (2002) Colloidosomes: selectively permeablecapsules composed of colloidal particles, «Science», 298, 1006-1009.

Espinat D., Ravey J.C. (1993) Colloidal structure of asphaltenesolutions and heavy-oil fractions studied by small-angle neutronand X-ray scattering, in: Proceedings of the Society of Petroleum



Engineers international symposium on oilfield chemistry, NewOrleans (LA), 2-5 March, SPE 25187.

Evans D.F., Wennerström H. (1994) The colloidal domain: wherephysics, chemistry, biology, and technology meet, Cambridge,VCH.

Fasano A. et al. (2004) Mathematical models for waxy crude oils,«Meccanica», 39, 441-482.

Forgiarini A. et al. (2001) Formation of nano-emulsions by low-energy emulsifications methods at constant temperature,«Langmuir», 17, 2076-2083.

Hirschberg A. et al. (1984) Influence of temperature and pressure onasphaltene flocculation, «Society of Petroleum Engineers Journal»,24, 283-293.

Izquierdo P. et al. (2005) The influence of surfactant mixing ratio onnano-emulsion formation by the pit method, «Journal of Colloidand Interface Science», 285, 388-394.

Landfester K. et al. (1999) Formulation and stability mechanism ofpolymerizable miniemulsions, «Macromolecules», 32, 5222-5228.

Létoffé J.-M. et al. (1995) Crude oils: characterization of waxesprecipitated on cooling by d.s.c. and thermomicroscopy, «Fuel»,74, 810-817.

Marruffo F. et al. (1998) Orimulsion. A clean and abundant energysource, in: Proceedings of the 17th congress of the World EnergyCouncil, Houston (TX), 13-18 September.

Melle S. et al. (2005) Pickering emulsions with controllable stability,«Langmuir», 21, 2158-2162.

Merino-Garcia D., Correra S. (2006) Kinetics of waxy gel formationfrom batch experiments, in: Proceedings of the 7th Internationalconference on petroleum phase behaviour and fouling, Asheville(NC), 25-29 June.

Nellensteyn F.I. (1938) The colloidal structure of bitumens, in:Dunstan A.E. (managing editor) The science of petroleum. Acomprehensive treatise of the principles and practice of theproduction, refining, transport and distribution of mineral oil,London, Oxford University Press, 6v.; v.IV.

Pickering S.U. (1907) Emulsions, «Journal of the Chemical Society»,91, 2001-2021.

Pons R. et al. (2003) Formation and properties of miniemulsions formedby microemulsions dilution, «Advances in Colloid and InterfaceScience», 106, 129-146.

Porte G. et al. (2003) Reversible description of asphaltene colloidalassociation and precipitation, «Langmuir», 19, 40-47.

Ramsden W. (1903) Separation of solids in the surface-layers ofsolutions and ‘suspensions’(obsservations on surface-membranes,bubbles, emulsions, and mechanical coagulation). Preliminaryaccount, in: «Proceedings of the Royal Society of London», 72,156-164.

Salager J.L. et al. (2001) Heavy hydrocarbon emulsions, in: SjöblomJ. (edited by) Encyclopedic handbook of emulsion technology, NewYork, Marcel Dekker.

Shinoda K., Saito H. (1968) The effect of temperature on the phaseequilibria and the types of dispersions of the ternary systemcomposed of water, cyclohexane and nonionic surfactant, «Journalof Colloid and Interface Science», 26, 70-74.

Taylor P., Ottewill R.H. (1994) The formation and ageing ratesof oil-in-water miniemulsions, «Colloids and Surfaces. A:Physicochemical and Engineering Aspects», 88, 303-316.

Venkatesan R. et al. (2005) The strength of paraffin gels formedunder static and flow conditions, «Chemical Engineering Science»,60, 3587-3598.

Vignati E. et al. (2003) Pickering emulsions: interfacial tension,colloid layer, morphology, and trapped-particle motion,«Langmuir», 19, 6550-6656.

Vignati E. et al. (2005) Wax crystallization and aggregation in amodel crude oil, «Journal of Physics. Condensed Matter», 17,3651-3660.

Visintin R.F.G. et al. (2005) Reological behavior and structuralinterpretation of waxy crude oil gels, «Langmuir», 21, 6240-6249.

Yen T.F., Chilingarian G.V. (edited by) (1994) Asphaltenes andasphalts, Amsterdam, Elsevier.

Thomas Lockart

Eni - Divisione E&PSan Donato Milanese, Milano, Italy

Sebastiano Correra

Eni - Divisione E&PSan Donato Milanese, Milano, Italy

COLLOIDAL SYSTEMS



3.2.2 Polymer colloids and their interfaces

Colloidal dispersions of polymeric materials

Colloids and their relevanceThe dispersed phase of a colloidal system is made of

particles so small that their dominant motion is diffusive,the so-called Brownian motion, which results from thecollisions between the particles and the moleculesforming the continuous phase. The upper limit size ofsuch particles is typically of the order of 1 mm and canbe defined by excluding gravitational and inertial forcesfrom significantly affecting the particle motion. BesidesBrownian motion, forces of various origin, such aselectrostatic, dispersion and viscous, contribute todetermine the behaviour displayed by colloidaldispersions and to their peculiar properties. The complexinterplay between these forces determines the evolutionof a specific colloidal dispersion, which can occur intime scales ranging from seconds up to years. Thismakes their handling so challenging and, at the sametime, so interesting.

As discussed in paragraph 3.2.1, colloids play afundamental role in the oil industry during both theupstream and the downstream processes. These colloidscan be of various nature such as asphaltenes, water orwaxes dispersed in oil or oil dispersions in water.However, colloids are present in many more systemsthan these: they might be prepared from a variety ofinorganic or organic materials like gold, silica, carbonblack, emulsion droplets and polymers or exist naturallyas clay and proteins. They can exhibit various shapes asrods or platelets, although here the focus is on sphericalparticles.

In this paragraph we intend to give an overview ofthe behaviour of colloidal dispersions in terms of theirstability, morphology and kinetics of aggregation andbreakage. Most of the notions that we will bediscussing are of general validity for all colloids, butdue to the extremely large variety of these systems it isconvenient for illustrative purposes to focus on the caseof polymer colloids. Their properties are in fact theclosest to the dispersion of solid particles in liquidphases which are typically encountered in the oilindustry. Moreover, polymer colloids have been deeplyinvestigated in the literature where a number ofexciting theoretical and experimental results have beenreported, some of which are discussed in thisparagraph.

Polymers can be synthesized directly fromhydrocarbons and offer a wide variety of properties

which make them suitable for various everydayapplications. This has driven the desire to producepolymers with tailored properties and has resulted in thedevelopment of many polymerization processes whichare widely used in industry to produce materialsdedicated to various market sectors. In some of the mostefficient processes the polymer material at some stage isin the colloidal state. Controlling the colloidalcharacteristics of these systems is crucial to determiningthe final properties of the produced material. In someapplications the colloidal state has to be preserved andthe material is used directly in this form, whereas inothers it is an intermediate state to be later transformedinto a polymer powder. In all of the above cases it isuseful to first briefly describe how these polymercolloids are manufactured and then to identify thefactors that affect their colloidal behaviour andconsequently their post-processing.

Processes for producing polymer colloidsDispersions of polymeric particles, usually referred

to as latexes, can be prepared by several heterogeneouspolymerization processes. Among the advantages ofthese processes compared to the correspondinghomogeneous processes are the easy temperaturecontrol, the uniformity of the product and the very highfinal conversion. In most cases no organic solvent isneeded and no stripping to remove residual monomer isrequired.

Initially, the monomer is dispersed in the form ofdroplets in the liquid, typically an aqueous phase understirring, and is stabilized by suitable surface activemolecules or surfactants.

The four main heterogeneous processes, namelydispersion, precipitation, suspension and emulsionpolymerization, are introduced here and their mainfeatures are briefly discussed.

Dispersion polymerizationBoth the monomer and the initiator are soluble in

the liquid phase and therefore the polymerizationstarts under homogeneous conditions. The polymerformed, however, is not soluble in the aqueous phaseand nucleates into a separate phase, thus leading tothe formation of polymer particles. The initiator andthe monomers are soluble in the polymer and swellthe particles, which become the main locus ofpolymerization and result in a dispersion of particlesof about 0.1-10 mm in diameter. The particles have tobe stabilized against coagulation during thepolymerization and this is typically achieved throughnon-ionic surfactants, i.e. through stericstabilization.

Precipitation polymerizationPrecipitation polymerization starts homogeneously,

as does dispersion polymerization, and both monomerand initiator are soluble in the liquid phase, but not in thepolymer phase. The polymer phase itself is insoluble inthe liquid phase and precipitates out of the latter,nucleating and coagulating usually in an uncontrolledmanner, resulting in irregular and polydisperse particles.

Suspension polymerizationIn this case the monomer is dispersed in water in the

form of micron-sized droplets stabilized by a surfactant.The actual droplet size is mainly determined by thestirring speed, and uniform shear rates in the reactorallow the production of stable and monodisperse dropletpopulations, typically larger than 1 mm. Addition of amonomer soluble initiator forces the polymerization totake place inside the monomer droplets which thereforeact as small bulk reactors. The reaction is completedonce the monomer droplets are completely convertedinto polymer. The solubility of the polymer in its ownmonomer determines the final morphology and surfacecharacteristics of the particles. If the solubility is poor,as in the case of poly(vinyl chloride), the polymerprecipitates in the monomer droplets and irregular,porous particles are produced. On the other hand, in thecase of good solubility, e.g. polystyrene, the polymerswells in its monomer and the final particle morphologyis homogeneous with a smooth external surface.

Emulsion polymerizationThis is the polymerization process currently most

common in industrial applications. The monomer isdispersed in the liquid (aqueous) phase in the form ofdroplets (1-10 mm) stabilized by surfactants, while theinitiator is soluble in the aqueous phase. Thepolymerization initiates consequently in the aqueousphase, where the growing oligomers rapidly become toohydrophobic and tend to nucleate as a separate phase.This can occur through a classical homogeneousnucleation process or, in the presence of surfactantmicelles, through micellar nucleation, and leads to theformation of surfactant stabilized primary particles a fewnanometres in size. These grow by polymerization of themonomers which are transported through the aqueousphase from the large monomer droplets, which act asmonomer reservoirs, until they are completely depleted.The particle sizes typically produced by emulsionpolymerization are in the range of 50-500 nm. Emulsionpolymerization can also be performed withoutemulsifiers or surfactants. In this case the oligomersnucleate in the aqueous phase and aggregate to formlarger particles (homogeneous nucleation). The watersoluble fragment of the initiator, which is an end groupof the oligomeric chain, tends to remain on the externalsurface of the particles thus providing stabilizationagainst further coagulation. In this case the final particlesizes are in the range of 100-500 nm. A disadvantage ofsurfactant-free emulsion polymerization is the smallervolume fraction of monomer at which the process can beperformed. Another interesting possibility is theso-called miniemulsion polymerization where the initialmonomer droplets have sizes in the range of 100 nm and

are stabilized thermodynamically through the addition ofa small amount of water-insoluble molecules in additionto the surfactants. The polymerization is started directlyin the monomer droplets through the addition of amonomer soluble initiator and in this respect it is similarto suspension polymerization. However, the reactiontimes are faster than in suspension polymerization and itis also possible to polymerize extremely hydrophobicmonomers, which do not undergo normal emulsionpolymerization.

Processing of polymeric colloidsOne key issue in determining the performance of a

polymer colloid both during its preparation, as discussedabove, and during its post-processing prior toapplications, is its stability against aggregation. Amongthe most important factors for colloidal stability is thechemical composition of the particle surface. Thespecific ways adopted to prevent the particles fromaggregating include the addition of ionic and non-ionicsurfactants, the self-assembly of charged polymer-chaintails originating from the initiator, or even the grafting ofstabilizers on the surface. In addition, the solubility ofthe various monomers, oligomers and polymer chains inthe aqueous phase, which might adsorb onto the surfaceof the particles, strongly affects the surface compositionand thus the stability of the particles. Different degreesof stability against aggregation are required dependingupon the specific applications the polymer colloid isdesigned for, as is discussed in the following, using twoexamples where specific stability requirements suggestdifferent stabilization mechanisms. These mechanismsare discussed later in detail in the section on colloidalstability. Two examples of polymeric colloid processingare reported below.

Film formation: paper coatingThe polymer or copolymer latexes prepared for paper

coating applications make use of styrene, butadiene andacrylate monomers and various modifiers to improveperformance and stability. In order to understand whystability is important for coating applications let usconsider the coating process itself. The paper to be coatedis prepared on long paper webs which during coating passwith high velocity through several narrow gaps andblades exposing the material to high shear rates. The latexis spread and coated onto the paper by these blades. Inorder to prevent aggregation and the build up of polymermaterial close to the blades, the latex has to be stable atshear rates corresponding to velocity gradients of up to104-106 s�1. In order to provide stability under suchextreme conditions, carefully calibrated combinations ofstabilization mechanisms provided by electrostatic andsteric surfactants are applied.

Aggregation of latexes for extrusion Another application area for the polymer materials

prepared by emulsion polymerization is extrusion andtailor shaping; however, in order to feed the polymer intoan extruder, the small colloidal particles must first beseparated from the liquid phase. The easiest way to dothis is to aggregate them into larger clusters of sizes inthe range of 50 mm-1 mm, which can then be recovered



by sedimentation, filtration or centrifugation. Suchaggregation processes are conducted in largemechanically stirred vessels where the latex is mixedwith a suitable coagulation agent. Consequently, thelatex has to be stable enough not to coagulate in thepolymerization reactor, but it cannot be too stable or itsaggregation in the subsequent coagulator would not bepossible. These constraints necessitate the use ofelectrostatic stabilization. This provides a sufficientcolloidal stability in the polymerization process whichcan then be significantly decreased by properly adjustingthe electrolyte concentration in the coagulator.

Colloidal stability: kinetic versus thermodynamic effects

Lyophilic versus lyophobic colloidsLyophilic and lyophobic colloids (which, if the

solvent is water, are called hydrophilic and hydrophobic,respectively) exhibit different colloidal stability which isof thermodynamic nature in the first case and of kineticnature in the second. A simple experiment illustrates thepoint: after aggregating a lyophobic latex and separatingthe polymer bulk, it is almost impossible to re-dispersethe material in the solvent. This is, instead, possible witha lyophilic material, like gelatin or protein crystals,under suitable conditions.

The surface of lyophilic colloids is solvated by thesolvent (water) and this provides a negative total freeenergy change (negative interfacial surface tension) inthe system upon dispersion, resulting in athermodynamically stable state. Lyophilic colloids canbe seen as complex macromolecules, all parts of whichclosely interact with the solvent, and hydrationfrequently means a direct adsorption of water moleculesat the polar groups throughout the entire macromolecule.It is, however, possible to modify the solvency of thelyophilic sites so as to force the colloid out ofequilibrium to undergo phase separation.

In lyophobic colloids instead, the total free energychange in the system upon dispersion is positive(positive interfacial surface tension) and even in verysmall particles most of the material remains in a ‘bulk’state which does not interact with the solvent molecules.

Because lyophobic colloids thermodynamicallyfavour the bulk state, they are intrinsically unstable andalways evolve towards this state. All we can do is toretard this process, by introducing an additional energybarrier against particle aggregation.

Kinetic stabilityAs previously mentioned, thermal energy (kBT,

where kB is the Boltzmann constant) in the form ofBrownian motion induces collisions between particlesleading to aggregation. Kinetic stabilization can beachieved by building an energy barrier sufficiently largewith respect to the thermal energy, as schematicallyshown in Fig. 1.

Two main mechanisms of stabilization are commonlyused. In the first, the particle surface is chargedelectrically, thus generating repulsive electrostatic forces.The second mechanism involves the addition of non-ionic polymer layers onto the particle surface which have

some affinity to the liquid and the colloidal phase andprevent the particles from closely approaching eachother, thus resulting in steric stabilization. Thecombination of electrostatic and steric repulsiveinteractions often provides stability degrees superior tothose achievable using only one of them.

Interparticle forces and potentials

Dispersion forcesDispersion forces are strongly attractive. The

molecules that constitute polymer particles typically donot exhibit permanent dipoles. However, fluctuatingdipoles are present due to the electromagneticbackground radiation and their strength depends on themolecular polarizability. These interact with the dipolesinduced in the surrounding molecules leading to shortrange attractive forces, usually referred to as van derWaals’ forces, while molecular overlap is prevented bystrong repulsion on the length scales of the electronclouds of atoms. The dispersion potential (Vdispersion) thatgoverns the attractions between two colloidal particlescan be computed by considering the interaction betweeneach molecule in one particle with every other one in thesecond particle, while neglecting the effect of thesurrounding molecules, through the following volumeintegrals:

where A is the Hamaker constant accounting for theeffect of the material properties, while the geometry isaccounted for by the position vectors, x, and the interparticle distance, r. The expression given above for theHamaker constant applies to particles immersed in afluid with different polarizability, where N is the numberof molecules contributing to the potential and a(ixn)their polarizability, while N� and a�(ixn) are the samequantities but referring to the surrounding fluid.

V A d dr

r

dispersion VV= −

= −( ) ⋅∫∫π 2

1 2

6

2

1 2 1

21

x x

x x x −−( )= ( ) − ( )

=

∞

∑

x2

2

0

3

8A k T N i N iB n n

n

α ξ α ξ

COLLOIDAL SYSTEMS


G

aggregation state

colloid

bulk

kineticallystable

unstable

Fig. 1. The addition of an energy barrier (G is the Gibbs free energy) retards the aggregation of the colloid into the thermodynamically preferred bulk state.The barrier can be realized through electrostatic or stericinteractions.

The two most common geometries for which exactanalytic expressions for the dispersion potential havebeen obtained are the semi-infinite parallel plategeometry, which takes the form:

and the one of two identical spherical particles of radius a,which is given by:

Although these simple equations highlight therelatively long-range character of the attraction betweenthe particles, the accuracy of this treatment is limited bythe assumption of simple additive interactions amongmolecules, which does not account for the very shortrange repulsion. Better results are obtained using thecontinuum theory which includes the many-bodyinteractions among polarized molecules. This theoryrelies on the frequency-dependent dielectric permittivityas the relevant material property and is based onMaxwell’s equations. The solutions for geometries andsituations of interest in colloid science, although rathercomplicated, are available. In the following, the twoanalytical expressions above are often referred to, sincetheir simplicity allows for analytical treatment whileretaining the qualitative features of the more accurateexpressions.

Electrostatic forcesElectrostatic forces between two spherical particles

are typically repulsive and for suitable conditions canbalance the attractive dispersion forces, providing astable colloidal dispersion. The origin of the charges onthe colloidal particles responsible for the electrostaticforces can be diverse. These can be due to chargeablesurface groups of the material itself, as for example inthe case of silica, to some chargeable polymer chainend-groups originated during the polymerization processor to ionic surfactants adsorbed on the surface.

When considering the electrolytic solutionsurrounding a charged surface, two regions can bedistinguished, depending upon the mobility of thecounter-ions in the vicinity of the surface. In the innercompact layer, a few nanometres deep, the counter-ionsare bonded to the surface charges and cannot move. Atlarger distances the diffusive thermal motion of thecounter-ions prevails up to the point where the effect ofthe presence of the charged surface vanishes and localelectroneutrality is achieved. This outer layer, usuallyreferred to as the diffuse layer, together with the innerone, constitutes the electric double layer whichcharacterizes the behaviour of charged surfaces inelectrolytic solutions. The concentration of counter-ionsand, therefore, the electric potential in the double layercan be described by proper balances of mass and charge.

Taking carboxylated surfaces as an example, thesurface charge, s, is related to the surface potential, ψs,by the following relation, which includes the dissociationconstant, K, of the carboxylic group:

where Gtot is the total density of chargeable sites, i.e.carboxylic groups in this case, e the elementary chargeand aH the concentration of the counter ion in thesolution. This equation describes the interrelationbetween the charge and the potential in the inner layer.

The ions of concentration, n, in the diffusive part ofthe double layer are experiencing thermal motion andfollow the Boltzmann distribution:

where nb is the bulk ion concentration and z its valence.Substituting this relation in the Poisson equation for thediffusive layer, the nonlinear Poisson-Boltzmann equation isobtained:

(where ere0 is the permittivity), which is often used as anapproximation of the entire double layer, i.e. the Gouy-Chapman approximation, where the existence of the innerlayer is ignored. Since no analytical solution exists for thisequation, various approximate solutions have beendeveloped and a suitable one accurate within 5% for ka�0.5(see below), is given by:

It is seen that the key dimensionless parameterdetermining the surface charge in the diffusive part of thedouble layer is the parameter k, whose converse usuallyreferred to as the Debye length, represents the distancewhere the electrolyte concentration is significantly affectedby the presence of the charged surface, usually referred to asthe depth of the diffusive layer, and is given by:

with I��k zk2nk �2, I being the ionic strength of the solution

and NA Avogadro’s number. When two charged bodies approach each other the

corresponding electrical double layers overlap, leading toa repulsive interaction potential. This can be computedfrom the solution of the double layer model describedabove. For example, in the case of two equal spheres ofradius, a, the repulsive potential as a function of thedistance between the two spheres, r, can be computedfrom the following approximate expression valid for lowsurface potentials:

This equation underlines the strong dependence ofthe electrostatic interaction potential on the ionicstrength of the solution through the Debye length, k�1.For very low ionic bulk concentrations, depending also

V r a relectrostatic r s( ) ln exp= + −( ) 2 10

π ε ε ψ k

k =2 2

0

N e Ik T

A

r Bε ε

σε ε

ψ

= − ⋅

( ) +

2

sinh 22

0r B

s B

k Tez

e k Ta

k

k / taanh /e k Ts Bψ 4( )

1 2sinh2

0r r

rr

eznez k Tb

rB

∂∂

∂∂

= ( ) ψ

ε εψ /

n n ez k Tb s B= − ( ) exp /ψ

σψ

= −+ −

e

aek T

K

tot

Hs

B

Γ

1 exp

A ar a

ar

r aln= −

−+ +

−6

2

4

2 42

2 2

2

2

2

22

2r

V rdispersion( ) =

V r = Ardispersion( )

12 2−

π



on the strength of the surface potential (charge),dispersions of charged particles can be kinetically stableover years. However, as the ionic strength of the solutionincreases, the Debye length decreases, which means thatthe presence of the charged surface is felt at shorterdistances. This is often referred to as double layercompression and is typically responsible for theaggregation of electrostatically stabilized colloid.

Forces due to soluble macromolecules (surfactants)Soluble macromolecules are often present in

colloidal polymeric dispersions (latexes). A few solubleoligomers, depending on the monomer solubility, areproduced during the polymerization and eventuallyadsorb on the surface of the polymer particles.Surfactants are often added to the latex in order toimprove its stability. The effect that such solublepolymers have on the stability of the dispersion dependson their physico-chemical characteristics and on theirpartitioning between the liquid phase and the polymersurface.

Ionic surfactants are one large class of solubleoligomers or polymers, which adsorb on the latexsurface with their hydrophobic tails and improvestability through the electrostatic interactions caused bytheir charged heads which stick out into the solution.This mechanism is generally repulsive, as was discussedabove in the context of electrostatic forces.

On the other hand, non-ionic surfactants can eitherexhibit repulsive or attractive interactions. Somemacromolecules do not adsorb on the latex surface butremain in the liquid phase and generally lead to weak ormoderately strong attractive interactions between theparticles. Others adsorb on the latex surface eitherirreversibly or at least partially reversibly. Irreversibleadsorption can be obtained either by grafting on thesurface by covalent bonds or using, for example, acopolymer whose lyophobic tail sticks firmly to thepolymer particle while the lyophilic tail remains in thesolution. If the lyophobic tail is chosen properly, thisadsorption can be considered permanent. In most casesnon-ionic surfactants adsorb at random points along theirchain onto the surface, whereas other sections stillremain in solution for entropic reasons. In all thesesystems, the coverage of the surface and the interactionof the macromolecules with the solvent determinewhether the interparticle forces are attractive orrepulsive. Here two specific cases will be discussed indetail.

Depletion forces. Non-adsorbing macromoleculesand colloidal particles in dispersion tend to achieve ahomogeneous spatial distribution in order to minimizeconcentration gradients. The spatial size andconfiguration of a macromolecule coil is given by itsradius of gyration, Rg, whereas the size of a sphericalcolloidal particle is simply its radius, R. In most casesthe colloidal particle is significantly larger than themacromolecule. Therefore, around each colloidalparticle we can identify a shell roughly equal to Rg inwidth, where the centre of mass of a macromolecule insolution cannot enter. If two colloidal particles approacheach other to within 2Rg, as frequently occurs due totheir Brownian motion, the macromolecules are

excluded from this region. This causes an increase in theosmotic pressure of the macromolecules which are nowconfined in a smaller volume. The excluded volume can,however, be minimized by bringing all the colloidalparticles into closer contact. The net result is that theexcluded macromolecules create an effective attraction,referred to as depletion force, among the particles, andthe range of this attractive force depends on the size ratio(Rg/R) corresponding to the excluded volume. As a firstapproximation, we can relate the potential correspondingto these depletion forces to the number density ofmacromolecules per free volume of the dispersion, cm

F asfollows:

Values of this attractive potential are typically in therange of 1-5 kBT. This is, therefore, a relatively weakinteraction which, in the presence of dispersion andelectrostatic forces, does not generally play a significantrole. Preparing colloids that exhibit depletion attractionsas the only interaction forces has, however, alloweddetermining phenomena, such as phase separation,which are discussed below.

Steric stabilization. The interaction between twoapproaching surfaces covered with polymer coils isregulated by the increasing concentration of the coils inthe gap, which gives rise to an increased osmoticpressure. This exerts a normal force over the surface,resulting in a net interparticle repulsion. Several factorsinfluence these interactions as discussed in thefollowing.

The spatial structure of a flexible macromolecule isdetermined by its interactions with the solvent and canbe characterized in a simple way through its pairinteraction excluded volume, v, which means that theysimply cannot overlap. At the theta state, v�0, thedispersion attractions between different segments of thecoil are balanced by the excluded volume and thepolymer behaves almost ideally. In good solvents, forwhich v�0, the configurations available to the coil arereduced by the excluded volume and the polymer swellsin order to reduce its increased free energy. This is,however, limited by the chains elasticity, resulting in theequilibrium swollen state. In poor solvents, i.e. v�0, thecoils collapse and occupy less volume, and in thelimiting case reach the polymer bulk density.

The colloidal stability realized by macromoleculesgrafted onto the polymer surface or by copolymers witha strongly lyophobic branch depends mainly on thethickness, �, of the polymer layer, the excluded volume,v, and the surface coverage, Fp. The free polymerbranches extend into the solution and in theta and goodsolvents they increase their available volume bystretching out in the solution. At low coverage, Fp�1,the polymer layers can interpenetrate until the surfacesstart to compress the polymer chains at separations onthe order of one polymer layer, �. At larger coverage,1�Fp�3, this repulsion already occurs at largerseparations, approximately 2�. In poor solvents at lowcoverage, Fp�1, the chains can exhibit an attractiveminimum which, however, can disappear upon increaseof the coverage due to repulsive elastic contributions.The commonly present dispersion forces can be

V k Tcdepletion B mF�

COLLOIDAL SYSTEMS


overcome to ensure colloidal stability by adjusting thethickness of the polymer layer. This must simply extendbeyond the range of the dispersion forces, which meansthat by attaching sufficiently long macromolecules to theparticle surface a very robust stabilization can berealized. If such a system is originally prepared in agood solvent, aggregation can, nevertheless, be initiatedby making the solvent a poor one for the polymer byeither adding a suitable co-solvent or changing thetemperature.

Macromolecules that are not irreversibly adsorbedshare substantially similar features as a function ofcoverage, layer thickness (which is proportional to themolecular weight) and solvent quality, although some oftheir features are rather complex. Firstly, they can desorbfrom the gap between approaching particles to avoid anincreasing osmotic pressure originating from theircompression. Secondly, the end of the macromoleculethat is not adsorbed onto the surface of the referenceparticle and reaches out into solution can then adsorbwith this free end on another particle with free surfacepatches while remaining attached to the referenceparticle. This provides an effective bonding betweenparticles and is called bridging aggregation. Theseadditional factors are particularly important for lowcoverage where surface patches remain free of polymer.Nevertheless, given good or theta solvents, a sufficientlayer thickness and high coverage, these reversiblyadsorbed macromolecules also provide sufficientstability for the colloidal particles. For poor solvents andat low coverage, however, attractions dominate theinteractions and aggregation is observed. The interactionpotential for the attached macromolecules isconveniently described as the sum of an osmotic and anelastic contribution.

Combination of interaction forces and additivity of potentials

In most applications a combination of interactionforces between the polymer particles is present, not justone type. All of these forces have to be properlycombined in order to estimate the total interactionpotential. The simplest and also most widely usedapproach is found in the so-called DLVO (Derjaguin,Landau, Verwey, Overbeek) theory, which assumessimple additivity of the interaction potentials, as given by:

The original version of the theory has beendeveloped for lyophobic colloids and incorporates thecontribution from dispersion and electrostatic forces.Later extensions accounted for steric contributionsexpressed in terms of elastic and osmotic contributions.While the additivity assumption seems to be a suitableapproximation when combining steric and dispersioninteractions, the situation is less clear when treatingdispersion and electrostatic forces, both ofelectromagnetic nature, where more detailedconsiderations about the electromagnetic field have to bemade. Nevertheless, expressions of the interparticle

interaction potentials based on the simplified argumentsabove provide a valuable tool for understanding thestability behaviour of colloidal dispersions.

Aggregation kinetics and stability ratio The shape of the interparticle interaction potential

curve controls the aggregation of a colloidaldispersion. In case the potential has a mainly repulsivecharacter, aggregation of the particles is prevented,whereas a strong, short-range attractive potential leadsto aggregation. The bonds between the particles in thecontext of the aggregation process considered here arepermanent and irreversible on the energy scale of theBrownian motion, i.e. the thermal energy, kBT. Astrong, purely attractive potential thus allows any twoparticles that collide due to their Brownian motion toform an aggregate. Accordingly, the correspondingaggregation kinetics between two primary particles isthe fastest one.

In the case where a repulsive contribution of theinterparticle potential is present in addition to theattractive one, this maximum rate constant is reduced bya factor called the Fuchs stability ratio, defined as theratio between the fastest aggregation rate constant andthe observed one. The evaluation of this importantparameter will be discussed later in greater detail. Hereit is sufficient to note that this can be done based on theexpression of the total interaction potential, such asthose discussed in the previous section. The magnitudeof the stability ratio sets the time scale of aggregation fora given particle size and number concentration ofprimary particles. In particular, increasing values of thestability ratio correspond to increasingly kineticallystable colloids.

Equilibrium, phase transition and phase separation The interaction potential also determines processes

like phase transitions. The conditions in a colloidaldispersion that determine whether thermodynamicequilibrium can be reached depend strongly on the timescale of the underlying processes. The aggregationdescribed in the previous section is an example of aprocess far from equilibrium, because the bonds betweenany numbers of primary particles are essentiallypermanent. The spatial configuration that the particlesassume upon aggregating cannot rearrange into athermodynamically more favourable one. It is a ‘frozen’structure.

On the other hand, in the case where attractions areweaker and comparable to thermal energy, bonds can bebroken and the structure can rearrange to achievethermodynamic equilibrium. A simple way to quantifythis is to estimate the time scale for bond breakage bythermal energy as a function of the depth of theattractive interaction potential. This can be done byconsidering the bond breakage as a diffusion problemout of a sink and over an energy barrier, which in thiscase is provided by the minimum of the attractivepotential. By doing so we obtain the followingexpression for the corresponding characteristic time:

τ µ�

6 3π Rk T

Vk TB B

exp −

max

Velasti+ cc osmotic depletionV V+ +

V V Vtotal dispersion electrostatic= + +



where m is the liquid viscosity. Simple estimates showthat for attractive minima of about 1-5 kBT, the break upof bonds is sufficiently fast to allow for equilibriumbehaviour of the colloidal system, whereas minima of10-30 kBT typically guarantee permanent bonds onexperimental time scales.

Colloids can exhibit a wide variety of behaviourswith respect to phase transitions, that need to beaddressed, at least briefly. A characteristic behaviour isdriven by the small attractive forces discussed above.Under suitable experimental conditions, includingcolloid-liquid relative density and solid volume fraction,the dispersion can phase separate into a colloid-rich,denser phase and an aqueous phase with a solid volumefraction much lower than average. In addition, thecolloid-rich phase can exhibit a crystalline structurewhere the particles are arranged in periodic arrays. Thisis in contrast to the behaviour in more diluted stablecolloids, where there is, typically, no structural order onlength scales larger than the primary particle size.

Another typical transition, where short-range repulsiveforces dominate and no attractions are present (hardspheres), is the colloidal glass transition. Upon increasingthe solid volume fraction of repulsive spheres, the systembecomes more and more crowded and the particles hindereach other in their diffusion and relaxation processes. Byeffectively blocking each other, the particles, although notexhibiting any significant spatial order, undergo a phasetransition, being now in a non-equilibrium state. At thisstage, the system behaves macroscopically as a solid,which means that it can resist applied shear forces and doesnot flow. Particles cannot change position and rearrange toaccommodate the stresses. The mechanical response of thecolloidal system is partially elastic.

The two-phase transitions described above, whichare certainly not the only possible ones, play some rolein the applications of polymeric latexes. Phase separationcan in fact limit the shelf-life of paints, while theso-called ‘jamming’ processes, where dense particleconfigurations resist flow as in the colloidal glasstransition, can increase the apparent viscosity of latexesat very high solid volume fractions and render theirprocessing difficult.

Aggregation kinetics and aggregatestructure

Fast aggregation

Kinetics of the aggregation processIn the case where no repulsive barrier exists between

the particles, i.e. Vtotal(r) is a monotonically increasingfunction, the rate of particle aggregation is entirelycontrolled by Brownian motion. We refer to this situationas fast aggregation or Diffusion Limited ClusterAggregation (DLCA). The flow rate, F, of particlesaggregating on a single reference particle is computed inthe following. For this, note that the flow rate, F, ofidentical particles diffusing through a sphere centredaround a given reference particle, is given by:

with N�N0 at r�� (bulk), where N is the particleconcentration, r the radial coordinate centred at the particlecentre and D11 the mutual diffusion coefficient which, sinceboth colliding particles undergo Brownian motion, is equalto twice the self-diffusion coefficient, i.e. D11�2D. Atsteady state, i.e. F�const, the particle concentration profileis given by:

Assuming that upon contact the second particledisappears because of the coagulation event (whichcorresponds to the boundary condition N�0 at r�R11, whereR11�2R is the collision radius and represents the conditionof largest possible gradient), the following expression for therate of particles aggregating on the single particle underconsideration is obtained:

We can now calculate the rate of decrease of the particleconcentration in bulk due to the aggregation process, R0

agg, asfollows:

where the factor 1/2 is needed to count each event only once.In the above expression the aggregation rate constantb11�4pD11R11 is introduced, which plays the same role asthe reaction rate constant in second order kinetics.

If the particles are of equal size, this reduces to:

Using the Stokes-Einstein equation to computeD�kBT/(6phR), where h is the dynamic viscosity of themedium, the size independent rate constant for fastaggregation is obtained:

In the case of two particles of different radius, Ri and Rj,we have instead:

A close examination of this equation reveals that theaggregation rate constant is particle-size-independent forequal-sized particles, but which can increasesubstantially by several orders of magnitude when twoparticles of largely unequal size are considered.Therefore, we can conclude that in these conditions theaggregation between small and large particles is muchmore effective than aggregation between equal-sizedparticles.

The cluster mass distribution During the aggregation process aggregates of

various sizes and masses are formed. In order todescribe this often wide distribution of aggregatemasses, a mass balance is written for each singleaggregate mass that is present in the system. Eachaggregate of mass, k, can be formed through various

ηB

i j i j

k TR R R R= + +−2

3

1( )( −−1)

βij i j i jD D R R= + + =4π( )( )

βη11

8

3=

k TB

β11

16= π DR

R N F DR N Nagg0

0 11 0

2

11 0

21

24

1

2= − = − = −π β

F D R N= 411 11 0

π

N N FD r

= −0

114

1

π

F r D dNdr

= 4 2

11π

COLLOIDAL SYSTEMS


combinations of smaller aggregates, but can itself alsoaggregate with every other aggregate to form largerones. The equation resulting from considering theaggregate formation and further aggregation ofaggregates of mass k is known as the PopulationBalance Equation (PBE), also referred to as theSmoluchowski aggregation equation:

where Nk is the number concentration of aggregates ofmass k and the first term on the right-hand siderepresents all possible collisions leading to the formationof an aggregate of mass, k, while the second is the rate ofdisappearance of the aggregates of mass, k, due toaggregation with aggregates of any mass. The solution ofthe PBE allows the cluster mass distribution to beobtained for several different aggregation regimes, beingsummarized in the particular form of the aggregationrate constant, bij, which is determined by the prevailingaggregation mechanism.

An analytical solution of the PBE above can bederived in the case where the aggregation rate constant isconstant with size, i.e. bij�b11. In particular, bysumming up both sides of the equation the totalaggregate concentration of any mass, that is:

can be computed as follows:

which upon integration with the initial condition N(t�0)�N1i

leads to:

It is useful to define a characteristic time of coagulation,in this case of fast coagulation, tRC, which is the time neededto halve the initial number of particles:

For particles of equal size the previous equation reduces to:

which for a water suspension at room temperature and ata typical particle number density concentration ofN1

i�2�1014 cm�3 predicts a tRC value in the range ofmilliseconds.

The corresponding expression of the cluster massdistribution is given by:

where time has been made dimensionless using the rapidcoagulation time, i.e. t�tRC. It is worth noting that thisexpression turns out to provide a quite reasonableapproximation of the actual cluster mass distributionobtained in conditions of fast aggregation.

Aggregate structureIn order to solve the Smoluchowski equation in the

general case where the aggregation rate constant is sizedependent, in other words:

we need to correlate bij directly to the masses i and j of thetwo colliding clusters. Accordingly, we need to find arelation between the size, Ri, of an aggregate and its mass, i,which would allow the description of how large an aggregateof a certain dimensionless mass, i, is. In other words we needa quantitative description of the aggregate structure.

If we consider aggregates to be made up of solid sphereswith no voids, this relation is simple:

where ρ is the particle density and m0 the mass of theprimary particles. This relation leads to a straight line withslope 1/3, when reported on a log-log plot of Ri versus i. Thisexpression is used in the case of coalescence, where smallerdrops (e.g. of a liquid) aggregate to form a single largerdrop. In this case, by substitution in the above equation, weget the following expression for the coalescence rateconstant:

Using Monte Carlo simulations and light scatteringmeasurements it has been shown that a similar straightline is also obtained for the non-coalescing aggregates ina non-sheared environment if we define as aggregatesize, Ri, the radius of the smallest sphere enveloping theentire cluster. This is the size that we tacitly used abovein calculating the self-diffusion coefficient of thenon-coalescing aggregate using the Stokes-Einsteinequation.

The structure of these aggregates has been found tobe self-similar on several length scales, meaning that thestructural characteristics repeat themselves on larger andalso smaller length scales. This is the definition of self-similarity, called the fractality, or fractal structure of anobject, in this case of the colloidal aggregates.

The difference with respect to coalescence is that inthis case the slope of the straight line is much larger dueto the many voids present in the aggregate structure, anddefined as 1/df , where df is the fractal dimension. Inmany applications df ranges from 1.5 to 3.0. If we nowaccept that the aggregates have a fractal structure, thenwe use the following scaling of their size with mass:

which leads to the following expression for the aggregationrate constant:

which can be used to solve numerically the Smoluchowskiaggregation equation in diffusion limited conditions. Indiffusion limited aggregation conditions the value of the

βηijB

d d

d dk T

i ji j

f f

f f= +

+( )2

3

1 11 1

1 1

/ /

/ /

R iid f≈ 1

βηijBk T

i ji j= +

+( )2

3

1 11 3 1 3

1 3 1 3

/ /

/ /

iR

mi

=

4

3

0

r p 3

βηijBk T

=2

3( + )( + )1 1R R R Ri j i j

− −

N Nt

tk

i

k

k=( )+( ) +1

1

1

1

τ

τ

-

tk TN NRC

Bi i= ≈ ⋅3

4

2 10

1

11

1

η

t N D R NRCi i= =2 1

211

1 11 11 1βπ

1 1 1

21

11N Nti= + β

dNdt

N N N= − = −1

2

1

211

2

11

2

11

2β β β

N Nkk

==

∞

∑1

dNdt

N N N Nkij i j

i j kk ik i

i= −

+ =∑ ∑1

2β β



fractal dimension typically takes rather small values, in therange from 1.6 to 1.9. This means that DLCA clusters arequite open structures.

Slow aggregation

Kinetics of the aggregation processIn the case where a repulsive energy barrier is present,

i.e. the interparticle potential, Vtotal(r) exhibits a maximumvalue, the aggregation process becomes slower than thediffusion limited process described above and its kinetics iscontrolled by the magnitude of the energy barrier that has tobe overcome for aggregation to occur. This is often referredto as slow aggregation or Reaction Limited ClusterAggregation (RLCA). In this case, in computing the particleflow rate, F, entering a sphere of radius, r, centred on agiven particle, we have to consider not only the particleconcentration gradient, i.e. Brownian diffusion, but also theinterparticle potential gradient.

In particular, the potential gradient produces on theparticle a ‘phantom’ force:

This is contrasted by the friction force given by Stokeslaw, fF�Bu, where B�6phRp represents the frictioncoefficient for a sphere of radius, Rp, in a medium withdynamic viscosity, h. The two forces equilibrate each otheronce the particle reaches the relative terminal velocity, ud.This implies a convective flux of particles given by:

which superimposes to the diffusion flux:

where 2D is the mutual diffusion coefficient of the particles.Accordingly, the flow rate of particles aggregating on asingle particle, F, which at steady state conditions isconstant with r, is given by:

where, using the Stokes-Einstein equation, the friction factorcan be represented in terms of the self-diffusion coefficientby B�kBT/D. Using the boundary condition: Vtotal�0, N�N0at r��, this equation can be integrated and leads to:

from which, considering N�0 at r�2Rp, the following canbe computed:

Note that for Vtotal�0 we obtain F�8pDN0(2Rp),which is the same value computed earlier for diffusionlimited aggregation. Considering the definition ofaggregation rate constant, b�FN, we can conclude thatFDLCA/F�bDLCA/b. This parameter, which we introducedearlier in the context of latex stability, is very importantin colloidal science and is usually referred to as theFuchs or stability ratio, W.

A close look at the behaviour of the function(Rp /r)2exp[Vtotal/(kBT )] in the equation for the stabilityratio shows that the value of its integral can be related tothe maximum of the interparticle potential, Vtotal, max, sothat the following approximate expression for W canbeen derived:

where 1/k is the thickness of the diffusive layer as definedearlier in the section on colloidal stability. Using this relationwe can compute the W values corresponding to values ofVtotal, max typical in applications and, from this, following thesame procedure used in the previous section on DLCA, wecan estimate the corresponding half coagulation time, that ist1�2�2N1

t �b11. Considering again the typical value of particlenumber concentration N1

t�2�1014 cm�3, the results reportedin Table 1 are obtained.

These results clearly indicate the meaning of kineticstabilization, since with a modest energy barrier thecoagulation process can be delayed from milliseconds todays or months. Note that the value of W tends to bevery sensitive to small changes in the surface charge orpotential. In particular, in typical conditions of practicalinterest, a ten-fold variation of W is obtained with abouta 15% change in Vtotal,max and a 7% change in the surfacepotential, ψs. This makes it infeasible to compute W frommeasurements of ψs, while it is recommended that it beestimated by directly measuring the kinetics of primaryparticle aggregation, i.e. b11.

Since the ionic strength of the solution stronglyaffects the interparticle interaction potential through theDebye parameter, k�1, a corresponding effect is alsoexpected on the stability ratio. Fig. 2 shows, the values ofthe stability ratio, W, of a negatively charged colloid as afunction of the electrolyte concentration, cE, forelectrolytes characterized by different valences. Tworegions can be identified. In the first, where W�1, it isseen that the stability ratio decreases strongly with theelectrolyte concentration and that its value is alsostrongly affected by the valence of the counterionswhich, in this case, the latex being negatively charged,are cations. Such loss of stability is due to the double-layer compression effect mentioned above, whichimplies that as the ionic strength of the solutionincreases the double layer thickness, i.e. the Debye

WR

Vk Tp

total

B

≈

1

2kexp

,max

FDNVk T

drr

total

BRp

=

∞

∫

80

22

π

exp

FVk T

total

B+⋅ −exp

8 2π DVk T

drr

r

total

B

��

exp

N NVk T

total

B

= −

+

0exp

F r D dNdr

NB

dVdr

total= +

4 22π

J D dNdr

= −2

J u N NB

dVdrd

total= = −

fdV

drdtotal= −

COLLOIDAL SYSTEMS


Table 1. Parameters in aggregation processes

Vtotal,max � 0 kBT 15 kBT 25 kBT

W 1 105 109

t1/2 (s) tRC�10�3 102 106

parameter, k�1, shrinks. The second region, where W�1,corresponds to conditions where the aggregation iscontrolled by diffusion, i.e. DLCA, which is unaffectedby the presence of an electrolyte. The intersection of thetwo straight lines indicates the so called criticalcoagulation concentration (ccc), which is the minimumelectrolyte concentration at which the energy barrier foraggregation vanishes.

Aggregate structureWe have seen that in the case of primary particles under

RLCA conditions the aggregation rate constant is given by:

We now need to extend this to the aggregation of twoclusters of mass i and j, respectively. One way to considerthis problem is to assume that the aggregation of the twoclusters occurs through the interaction between the twoprimary particles which first come into contact on eachaggregate. This has two relevant consequences. Firstly, theaggregate-aggregate interaction can be approximated withthe interaction between the two colliding primary particles,which is described by the stability ratio, W. The interactionswith the other particles can be assumed, as a firstapproximation, to be too short-range to be relevant.Secondly, the rate of aggregation depends upon theprobability of collision between the primary particles on thecluster surface, which is proportional to the probability ofhaving a primary particle on the external surface of a cluster.

Using fractal scaling which describes the aggregatestructure, it is possible, though in a very simplistic way, tocompute for a generic cluster of mass, k, the change in thenumber of primary particles with its size, Rk, as follows:

By multiplying this by the primary particle radius, Rp,the number of primary particles added on the last layer of theaggregate is obtained:

This means that the aggregation rate constant should beaugmented by a factor k(df �1)�df when considering theprobability of collision of the primary particles instead of theprobability of collision of the clusters. With this result a finalexpression for the aggregation rate constant in RLCAconditions can be derived:

where l�(df�1)�df , which for a typical value of df�2 leadsto l�0,5, which has in fact received some experimentalvalidation as discussed below.

Due to the presence of the energy barrier, in RLCAconditions the aggregates do not stick to each other atthe first collision, but they have to probe many possiblecollision configurations before the aggregation event canactually take place. For this, the resulting aggregates aredenser than the corresponding ones formed in DLCAconditions, since they can penetrate deeper into eachother without immediately aggregating. This results infractal aggregates characterized by a lower fractaldimension equal to about 2.1.

Cluster mass distributionFrom the numerical solution of the Smoluchowski

equation, using the aggregation rate constant, bij, derivedabove for RLCA conditions, the corresponding clustermass distribution can be obtained. It has been found thatthese distributions exhibit some general features whichare characteristic of RLCA in stagnant conditions anddiffer significantly from those discussed above in thecase of DLCA. A general form of the cluster massdistribution, which applies for a wide range ofaggregation rate constant values, is given by:

A being a normalization constant determined from theconservation of total mass in the system and is given by:

where G(x) is the gamma function and kc is the timedependent cut-off mass of the distribution which growsexponentially in time, that is kc(t)�p�exp(t/t0), where t0 isa system specific time constant and p is a system specificfitting parameter. The parameter, t, for which a reasonable

A N kc= −( )−0

2 2τ τΓ

N t Ak k kk c( ) exp /= −( )−τ

ββ λ

ijij

Wij= ( )

DLCA

R dkdR

d kpk

fd df f= −( ) /1

dkdR

dR

RR

dR

kk

f

p

k

p

d

f

p

d df

f f=

=

−−( )

1

1 /

ββ

11

11 =

DLCA

W



log

W

4

�1

0

1

2

3

log cE

�1�2 3210

Ba(NO2)2 KNO3La(NO3)3Fig. 2. Typical plots of the stability ratio, W, of a negatively charged colloidas a function of the electrolyteconcentration, cE, for electrolytes with different valences.

value is 1.5, is an empirical power-law exponentdescribing the shape of the cluster mass distribution.

Comparison of slow and fast aggregation Summarizing the discussion above on DLCA and

RLCA processes, we can cast the aggregation rateconstant in the following general form:

Most important in determining the aggregationkinetics is the magnitude of the aggregation rate foraggregates of equal size, bii and how this changes as afunction of the aggregate mass, i. This function isdominated by the parameter, l, as can be seen in Fig. 3,where the aggregation rate constant of equal-sizedclusters is shown as a function of the cluster mass. It isseen that in the case of DLCA, as discussed earlier, thatW�1, l�0 and bii is independent of the cluster size. Onthe other hand, in RLCA conditions W is very large andl�0.5, which leads to a strong mass dependence of bii.This means that although in RLCA conditions the biivalue for primary particles is in general much smallerthan in DLCA conditions, as the aggregation proceedsand the aggregates grow, bii increases rapidly and caneventually reach the upper limit given by the diffusioncontrolled conditions.

Another characteristic of the aggregation rateconstant is that, while in DLCA small-big aggregationsare favoured over small-small and big-big aggregations,in RLCA the aggregation rate always increases with thesize of the involved clusters. This has significantconsequences on the shape of the cluster massdistributions obtained in the two aggregation regimes.

As discussed above, the cluster mass distributionsobtained in DLCA and RLCA conditions can berepresented by the following two approximateexpressions, respectively:

and

Typical cluster mass distributions obtained by thenumerical solution of the Smoluchowski equation, usingthe expression of the relevant aggregation rate constantsderived above, are compared with those computed fromthese approximate expressions in Fig. 4. We see that thedistribution for DLCA is flat when calculated with theapproximate expression, while the correct distributionexhibits a bell shape. However, the approximateexpression provides a good representation of thegrowing front of the distribution. On the other hand, it isseen that the approximate expression for RLCAconditions is rather accurate. Nevertheless, we shouldemphasize that not all aggregation problems can besolved with approximate expressions which must in anycase be treated with caution.

NN k

k k k tkc

c=−

− −

− 0

2

2

ττ

τΓ ( )exp / ( )

N tN t

tk

i k

k( ) =( )+( )

−

+1

1

1

1

τ

τβ β

β η

ij ij ij

B

W B P

k T

=

=

−

con

11

1

118 3

DLCA

DLCA /

WW

B i j i jijd d df f f

=

= +( ) +− −

β β11 11

1 1 11 4

DLCA /

/ / /

11

0 0 5

/

,

d

ij

f

P ij

( )= ( ) = =

λλ λ ( in DLCA e in RLLCA)

COLLOIDAL SYSTEMS


log biiDLCA, l�0

RLCA, l�0.5

log i

W

Fig. 3. Aggregation rate constant, bii, for equal-sized clusters as a function of the cluster mass, i.

B

A

105

100

102 104 106 108100

1010

1015

105

100

102 104 106 108100

1010

1015

num

ber

dens

ity

(1/m

3 )nu

mbe

r de

nsit

y (1

/m3 )

dimensionless aggregate mass

dimensionless aggregate mass

Fig. 4. Cluster mass distributions computed using the Smoluchowski equation (continuous line) and the approximate expressions given in the text, via diffusion (A) and reaction (B). In B, the approximatecalculation is performed assuming t�1.5.

From the cluster mass distributions anotherimportant feature is evident: in DLCA the bell-shapeddistribution does not include very small particles, whichremain instead present even for very long times inRLCA. This is a consequence of the fact that big-smallaggregations are favoured in DLCA but not in RLCA.This result has consequences in applications, where thepresence of significant fractions of fines is not desired.

As mentioned above, also the aggregate structure isdifferent in the two regimes. We have in fact fractalaggregates in both cases, but those produced in RLCAconditions are characterized by larger values of thefractal dimension, df , (2.0-2.2 vs. 1.6-1.9 in DLCA).

There are several ways to measure the fractal dimension,such as sedimentation experiments, light scattering or imageanalysis. Direct images can for example be obtained byTransmission Electron Microscopy (TEM). It is worth notingthat the agreement between TEM images and thosegenerated through Monte Carlo simulations is verysatisfactory and indeed supports the assumption of theaggregates having a fractal structure.

Comparison to experimental dataA direct comparison between the cluster mass

distribution computed through the Smoluchowskiequation and experimental data is not straightforward,because direct experimental measurements of the clustermass distribution are not possible, or are possible withgreat difficulty. Therefore light scattering techniques areoften used, where certain specific averages (or moments)of the distribution can in fact be measured. Usingappropriate light scattering models, these same momentscan also be computed from the predicted cluster massdistribution and the obtained results can be compared tothe experimental data. This allows the validation of theaggregation models presented in this section. In Fig. 5two examples of such a validation are shown in terms ofthe time evolution of the average hydrodynamic radius ofthe clusters: one in DLCA and the other in RLCAconditions. The points represent experimental datameasured by dynamic light scattering, while the curves

have been computed through the relevant Smoluchowskiequation, at various scattering angles.

Aggregation regimes different from classical DLCA and RLCA

Industrially relevant polymer emulsions are oftensignificantly more complex than the colloidal modelsystems discussed in the previous sections. Theiraggregation behaviour often goes beyond the simpleclassification in DLCA and RLCA with their respectivecharacteristics. This complexity originates from the factthat several different surfactants are often usedsimultaneously in order to achieve the desired stabilitycharacteristics, and that several monomers, co-monomers and various additives are used in order totailor certain final polymer properties, such as the glasstransition temperature, the crosslinking degree or thedegree of hydrophobicity. All of this strongly affects therheological and colloidal characteristics of the finallatex, as well as its aggregation behaviour. Both DLCAand RLCA and their characteristics mentioned above canbe changed, so that for example fast aggregation mightshow power-law growth but with a fractal dimensionsignificantly higher than the expected 1.8, or likewiseslow aggregation can exhibit smaller fractal dimensionsthan the expected 2.1 with a variety of power-law growthbehaviours unlike the expected exponential one. Ourcurrent understanding of the richness of the possibleaggregation behaviours is still limited and a completeclassification is not possible. In any case, carefulexamination and evaluation of the aggregation behaviourof latexes is deemed necessary prior to assuming that acertain classical aggregation regime has to prevail for agiven latex.

Colloidal gelation Aggregation processes at solid volume fractions of

more than 1%, which is indeed the case of most practicalapplications, almost ubiquitously result in the formationof a solid phase, usually referred to as a colloidal gel.This is made of a network of the fractal aggregates that



�Rh,

eff�

(nm

)

101

102

103

time (min)10�1 100 102101

�Rh,

eff�

(nm

)

101

102

103

time (h)

A B

0 2 864

Fig. 5. Average hydrodynamic radius as a function of time computed through the Smoluchowski equation (curves) and measured by dynamic light scattering (symbols) at various scattering angles. A, DLCA; B, RLCA.

have grown to such an extent that they occupy the entirevolume of the dispersion. In order to form the network,the aggregates have to interconnect with each other. Thisinterconnection step is quite different from the precedingaggregation step in that the aggregates do not diffuserandomly but directly influence their nearest neighbours.Therefore, they have to move cooperatively in order torearrange their spatial configuration substantially, whichbecomes increasingly unlikely when nearing the gelpoint.

The aggregation step for either DLCA or RLCAconditions is well described by the population balancemodel. In order to identify when the aggregates becomecaged and then aggregation crosses over tointerconnection, a cumulatively occupied volumefraction can be defined as:

where Ri is the radius of the smallest sphere enclosingthe aggregate. This occupied volume fraction quantifiesthe space occupied by the growing clusters which, due totheir highly open structure, grows continuously in time.Once this reaches a value of /(t)�0.5 the volume iscompletely packed with aggregates which can no longermove freely. The corresponding time, ta, is defined as thearrest time of the aggregation step. At this point theaggregation process crosses over to the interconnectionprocess, and the system changes from a viscous liquid(the colloidal dispersion) to a (visco-) elastic solid whichcan bear external stresses.

The gelation kinetics and the properties of the finalgel are strongly affected by the characteristics of theinitial aggregation process which leads to the stopping ofthe aggregation, that is whether it is conducted underDLCA or RLCA conditions.

In DLCA conditions each collision between particlesand clusters results in a new aggregate. During theaggregation process, once /(ta)�0.5 is reached, theaggregates are in close vicinity to each other and start tointerconnect. This interconnection process is very fastfor two reasons. Firstly, the cluster mass distributiongenerated in DLCA is rather monodisperse and thereforeall aggregates experience approximately the sameenvironment and are on average separated by the sameshort distance. Secondly, almost any short distancediffusion of an aggregate or of one of its subunits leadsto the connection of the aggregate to its neighbours dueto the absence of repulsion barriers. This view isconfirmed by experiments performed on polystyrenecolloids, where the gel formation times and the arresttimes measured as a function of solid volume fractionare very similar. Both times are overlapping within theexperimental error and this confirms that theinterconnection step is extremely short compared to theaggregation step.

The picture is different in the case of RLCA wherethe interconnection step is not instantaneous, and theinterconnection time is significantly longer than thearrest time. This is likely due to the low reactivity ofaggregates in RLCA and their reduced diffusivity, whichis hindered by the surrounding aggregates of varioussizes.

Characterization of aggregating colloidal dispersions

Light scatteringScattering techniques, involving radiation sources

like neutrons, x-rays and light, are widely used tocharacterize colloidal systems and soft condensed matterin general. Among these, light scattering is indeed themost convenient, due to the availability of relatively low-cost lasers, detectors and fast correlators controlled bycomputers.

In a typical light scattering experiment, amonochromatic laser is directed onto the centre of asample of dispersion and the light scattered in thehorizontal plane is collected as a function of the angleformed with the direction of the primary beam. Twodifferent experiments can be performed with thisarrangement. In a Static Light Scattering (SLS)experiment, the time-averaged scattered intensity ismeasured as a function of the scattering angle andprovides useful information about the spatial structure ofthe sample under investigation. In the case of aggregatesof colloidal primary particles we can learn how theseparticles are distributed within the clusters and oftengain information about the previously mentioned fractaldimension.

The same holds true for the second technique,Dynamic Light Scattering (DLS), in which the temporalfluctuation of the scattered intensity is analyzed withsuitable correlation schemes. In this analysis thefluctuations at a certain time, t, are correlated to a timeshifted by a certain delay time constant, t*, that is at t�t*.If the signals at these two times are correlated, whichmeans that the spatial configurations of the particles in thesample are similar, a certain nonzero value is obtained forthis delay time interval, t*. If the particles have not movedat all in this time interval, the correlation value is one. Ifthe particles have moved significantly and have erased allcorrelations between their positions at the initial time, t,and at the delayed time, t�t*, a value of zero is obtained.Accordingly, from DLS measurements we can see howfast a particle (or another object of interest) diffuses awayfrom its original position. This can be related to aneffective diffusion coefficient, from which, using theStokes-Einstein equation, we can estimate an effectivehydrodynamic radius.

The interpretation of light scattering data is nottrivial and is typically based on suitable simplificationsof the general theory of light scattering. For example theRayleigh-Debye-Gans (RDG) theory is limited to singlescattering and assumes that the incoming light is greatlyunaffected by the scattering particles. The condition forwhich the RDG theory is applicable is:

where Rp is the radius of a particle, l the wavelength ofthe incident light and np and ns the refractive indices ofthe particle and the solvent, respectively. If theseconditions do not hold, one has the possibility to use themore complex Mie theory.

It is important to note that when applied to apopulation of clusters dispersed in a liquid, both SLS and

41 1

π R nn

p p

sλ− <

φ( )t R Ni ii

= ∑ 4

3

3π

COLLOIDAL SYSTEMS


DLS provide average information, i.e. the signalobtained is an average of all the different shapes andsizes of the particles and clusters present in the sample.This is the reason why the interpretation of SLS datarequires considerable care. In particular, one shouldknow the cluster mass distribution of the aggregates andtheir individual scattering properties so as to reconstructthe average experimentally measured signal through asuitable mathematical model.

RheologyThe second technique, which is most often used in

the study of concentrated colloidal dispersions, is smallamplitude oscillatory shear rheology. This is particularlyuseful for the investigation of the gel formation kinetics.This technique has a wide range of applications andnumerous experimental setups are possible. Whenmonitoring gel formation in colloidal dispersions,(which is a liquid-solid transition), one measures theevolution of the viscoelasticity of the dispersion. Thiscan be quantified, for example by measuring the elasticmodulus, G’, and the loss modulus, G”. Alternatively,the complex conjugate pair of the moduli, the complexviscosities, can be measured. In any case, a typical latexis initially characterized through its dominantly viscousbehaviour, but once sufficient aggregation has occurred,the dispersion develops a measurable elasticity, whichgives rise to the complex viscoelastic, solid-likebehaviour typical of colloidal gels.

Shear and turbulent aggregation The industrial aggregation or coagulation operations,

which are used to separate the polymer from the aqueousphase, are considerably more complex than theaggregation processes in non-sheared or stagnantdispersions that we have discussed in the previoussections. This is because in order to increase theefficiency of the process at the large scales typical ofindustrial applications, a vigorous mixing of the latexwith the appropriate coagulant solution (typically anelectrolyte) is applied, thus introducing into the systemsignificant shear forces. The most widely used set upconsists of a simple stirred tank filled with the coagulantsolution in which a latex dispersion is fed through asuitable pipe. A plume formed by the latex entering thestirred solution occurs; the presence of such a plumedepends on the competition between mixing andaggregation/coagulation processes and will be discussedlater.

The presence of laminar or in most cases turbulentflows induces new processes in colloidal dispersionwith respect to the aggregation described above whichoccur in stagnant conditions. A turbulent flow can bedescribed as a cascade of eddies. In a stirred tank thelargest eddies are generated by the impeller and sohave the size of the tank diameter. These are too largefor the dissipative forces to be significant, andtherefore they accumulate kinetic energy, becomeunstable and break down into smaller eddies to whichthey transfer their energy. This process is repeatedthrough the so-called turbulent eddy cascade until theenergy is transferred to sufficiently small eddies and isdissipated through viscous forces. The length scale of

these eddies is known as the Kolmogorov length scaleof turbulence and is given by:

where v is the kinematic viscosity of the liquid and e therate of energy dissipation.

Above this length scale, turbulent fluid motiondetermines the mixing and transport processes in the dispersion, whereas below this length scale,molecular diffusive and shear motions dominate thetransport within the smallest eddies. In typicalapplications, such as stirred vessels, this length scale isin the range 1-10 mm.

Aggregation kinetics in turbulent conditionsTypically, aggregation processes are operated at salt

concentrations where the colloidal particles arecompletely destabilized, in other words in DLCAconditions. The primary particles resulting fromemulsion polymerization often have diameters smallerthan 200 nm and therefore are hardly affected byturbulent fluid motion. Initially, they collide andaggregate mainly by diffusive motion within theKolmogorov eddies, until the resulting aggregates havegrown to sizes at which they begin to experience theshear stresses. In this initial period, particle collisions aredue either to Brownian motions or to the presence ofspatial fluid velocity gradients which induce differentvelocities in the particle located at different points withinthe eddy. We can ignore aggregation due to settling whenmixing is sufficiently vigorous. Since we are consideringDLCA conditions, each collision leads to an aggregationevent. The overall aggregation rate constant can then becalculated as the sum of two terms: The first, bij

DLCA, dueto Brownian motion is the same derived above instagnant conditions and results in perikineticaggregation. The second, bij

shear, due to shear motionresults in orthokinetic aggregation. This secondcontribution is proportional to the velocity gradient orshear rate, G, prevailing in the Kolmogorov turbulenteddies, whose flow pattern can be described as a laminarextensional flow model which leads to G�(e�v)1/2. Thetwo aggregation mechanisms prevail at different particlesizes. When the particles are small (submicron), thedominant mechanism is Brownian motion. As particlesgrow larger, shear motion dominates. The completeexpression for the aggregation rate constant is given by:

where it is seen that the shear component of theaggregation rate constant is proportional to the shearrate, to the third power of the aggregate collision crosssection and to the efficiency aij to be discussed later.

It is worth noting that this expression is valid wheninertial effects can be neglected (i.e. when particles arenot very large or their density is not very different from

+ ( )1 2941 2

ij. α ε ν ii jd df f1 13

+( )ηB

d

k T

i f=

2

3

11/

++

+( ) +1

1

1 1

ji jd

d d

f

f f

/

/ /

β β βijturbulent

ij ij= + =DLCA shear

λ νεk =

3

1

4



that of the suspending medium), so that the particlesfollow the fluid flow closely.

Besides aggregation at least two more mechanismsaffect the evolution of the aggregate population. First, assoon as the aggregates become comparable in size to theeddy scale and experience fluid stresses, they tend tobreak into smaller units. This process might beenvisioned as a pulling and tearing of the aggregate untilit breaks at its weakest point. Second, the fluid stressescan also restructure the aggregates and by thismechanism the aggregates are typically densified. Open,weaker structures undergo together a destructuringprocess or even recombine to yield higher densityaggregates. It is indeed quite questionable whether theconcept of fractality introduced in stagnant systems canbe applied under these conditions as well.

The Smoluchowski or population balance equationdescribing the evolution of the cluster mass distributionin turbulent systems has to be modified in order toaccount for the breakage events. By assuming somedegree of homogeneity in the coagulator, the followingequation can be written:

where the first two terms on the right hand side representthe usual aggregation processes, with the rate constantsmodified in order to account for the contribution ofshear aggregation as discussed above. The last two termsrepresent the effect of breakage (restructuring is notincluded) and bi

B is the breakage rate constant ofaggregates of mass i, while Lkj is the breakagedistribution function which defines the number offragments with mass k produced by the breakage of acluster with mass j. Summarizing, this means that, in theabove population balance equation the first term is dueto the formation of aggregates of mass i by collision of apair of smaller clusters. The second and third termsdescribe the loss of aggregates of mass i by aggregationand breakage, respectively. Finally, the last term is due tothe formation of aggregates of mass i by breakage oflarger clusters.

Role of colloidal stabilityWhen the electrolyte concentration in the coagulator is

lower than the critical coagulation concentration, thesystem operates in RLCA conditions and the aggregationof primary particles is more complicated and is influencedby two factors. On the one hand, the turbulent fluid motionand the Brownian diffusion induce many collisions amongthe particles; on the other hand, the repulsive component ofthe interparticle interaction potential prevents many ofthese from successfully producing an aggregation event.Therefore, the aggregation efficiency and the totalaggregation rate can be significantly smaller than in DLCAconditions, although its quantification is currently notpossible with simple relations.

Collision efficiency As mentioned above, the shear aggregation rate

constant bijscorr includes the collision efficiency, aij. This

accounts for the hydrodynamic interaction between twoapproaching aggregates, which with their presencemodify the surrounding fluid flow, thus modifying theircollision paths with respect to the case of undisturbedflow. A common collision efficiency model uses atrajectory analysis following the spatial path of porousaggregates considering their relative flow fields andfluid forces. A quantitative description of the structure ofthe porous aggregate is needed for this like the oneproposed by the so-called shell-core model whichenvisions the aggregates as impermeable cores surroundedby completely permeable shells. With these models thevalues of the collision efficiency are computed whichdecrease for larger and/or less permeable aggregates, suchas those shown in Fig. 6. Higher efficiency values areobtained for aggregates of comparable sizes, lower valuesfor unequal size aggregates.

Role of breakageThe breakage of clusters in turbulent flow is caused

by hydrodynamic stresses. The larger an aggregategrows, the more susceptible it is to breakage. Moreover,the aggregates with more open structure can break moreeasily. A typical expression for the breakage kinetic rateconstant is given by:

where e is the energy dissipation rate and eb,i is theminimum energy dissipation rate capable of breakingaggregates of mass i. The larger the aggregate, the smallerthis critical energy value is expected to be, so that thefollowing relation is used to compute it:

where B is an empirical parameter which has often beenfound to change with the shear rate, G, according to apower law. In the relation above, the breakage rateconstant is the product of the first two factors,

εb idB i f

,= 1

β εν

εεi

B b i=

−

4

15

1 2 1 2

π

/ /

,exp

NjB

j≤

βkjj

+ Λkk

k kN+1

∑ − β B

dNdt

N N N Nki j

i j kk i

i= −

+ =∑ ∑1

2β βij ik +

COLLOIDAL SYSTEMS


224

24

29

214

219

i24 29 224214 219

j

0.01

0.01

0.2

0.2

0.4

0.4

0.6

0.60.7

0.7

0.8

Fig. 6. Values of the collision efficiency, aij, as a function of the mass (number of primary particles) of the two collidingaggregates.

representing the maximum breakage rate, and the thirdfactor representing the probability that the aggregate of agiven mass breaks at the given value of e.

Breakage distribution function The number of fragments into which aggregates

break is very difficult to determine and currentlyconstitutes a major unknown in the understanding of theevolution of cluster mass distribution in turbulentconditions. The breakage distribution function Lkj givesthe number of fragments of mass k that are produced bybreakage of an aggregate with mass j. An often usedassumption to compute this function is that eachbreakage event leads to two fragments. These can be ofequal size in the case of symmetric fragmentation or onecan be very small and the other very large in the case oferosion. It is clear that using one or the other of thesetwo models has strong influences on the final shape ofthe cluster mass distribution.

Role of restructuring and mixingAs mentioned above, larger aggregates with size

comparable to the smallest turbulent eddies tend not onlyto break but also to restructure. Flexible and openstructures like the fractals discussed in the section onDLCA and RLCA are in all likelihood not sufficientlystrong to withstand the shear forces and thereforeundergo significant restructuring and reshaping. Thefinal structure of the aggregates is indeed expected to bedenser, but it is difficult to provide a priori quantitativeestimates since it depends in a complex way on theinteraction forces between the colloidal particles andhow dense the ‘network’ structure is within the aggregateat a certain point during the process.

Another aspect that strongly affects aggregationprocesses in turbulent systems is the interactionbetween the mixing and aggregation processes thatwere mentioned above. This is particularly important inthe large scale aggregation units used in industrialapplications, which typically deal with concentratedlatexes, where the details of the mixing processbetween the latex and the coagulant solution play a keyrole. When the concentrated latex enters the coagulantsolution, it generally takes some time before theparticles are completely mixed and diluted into thewhole tank. Due to the high latex concentration,aggregation is often faster than this mixing process andtherefore most of the aggregation process occurs in theconcentrated latex as soon as this enters the tank. This

leads to the formation of a plume consisting of theaggregating latex.

If the concentration in the plume is sufficiently high,the aggregates can actually gel in this volume element,forming a three-dimensional network structure havingthe same shape of the fluid element they are imbeddedin. Because the elastic modulus of such gels at thevolume fractions often found in typical applications isquite high, the gel units can withstand even turbulentshear stresses and retain their shape. Since this is mainlydetermined by the mixing and the fluid flow, veryirregular shapes like crumbled sheets or hollow shellscan be created in this process. In Fig. 7 some of theseaggregates are shown for increasing values of the stirringspeed. At low stirring speed, mixing is slower thanaggregation and the aggregates exhibit a lamellar shape;this agrees with the above discussion that the latexvolume, after being stretched and folded, is frozen inlamellar shape by gelation. As the stirring speedincreases, mixing become faster and this lamellarstructure is gradually destroyed.

These aggregate shapes can have undesired effectson the properties of the final product and therefore haveto be controlled. This example shows how important themixing process between the latex and the coagulant is inorder to obtain good quality materials. In Fig. 8, for fixedstirring conditions, the operating conditions are shownqualitatively in terms of solid volume fraction, /, andprimary particle diameter, dP0, which lead to controlledor non-mixed aggregation processes. This information isclearly crucial in order to obtain final aggregates withdesired structure characteristics.

Industrial coagulation unitsIndustrially, stirred-tank coagulators are mainly used

to perform aggregation processes, because of their easeof operation in continuous, semi-batch or batch mode. InFig. 9 A a stirred-tank reactor is shown with thecorresponding fluid flow field, calculated with anappropriate Computational Fluid Mechanics (CFD)code. It is apparent that the energy dissipation rate isquite unevenly distributed in the unit, reaching verylarge values close to the impeller and much lower onesclose to the top of the tank. It is clear that this leads to acorrespondingly non-uniform distribution of theproduced aggregates. Better performances couldprobably be achieved by separating theaggregation/gelation step from therestructuring/granulating step. This could be realized



200 mm 70 mm40 mm

Fig. 7. Scanning electron microscopy pictures of polymer aggregates produced at increasing stirring speed, S.S

using a high pressure drop capillary device to producethe gel and then feeding it into a stirred vessel for thesecond step of the operation.

In the case of continuous operation, single stirredtanks are not suitable due to the very large residencetime distribution which would lead to the presence of alarge fraction of fines in the final product. For thisreason a series of continuous-stirred tanks are typicallyused in applications. An alternative device whichcombines a narrow residence time distribution with auniform distribution of the energy dissipation rate isthe Couette-Taylor unit. This device consists of twocylinders, of which only the inner one rotates. Thisproduces a series of vortexes which realize the peculiarmixing characteristics mentioned above. A schematicdiagram of this unit is shown in Fig. 9 B with thecorresponding energy dissipation rate distributionobtained through computational fluid dynamics

simulations. This and other devices are undercontinuous development through experimentalinvestigations and computational fluid dynamicsimulations. Based on our understanding of thecomplex interplay between colloidal interactions,aggregation and breakage kinetics, and fluidmechanics, we can develop appropriate units capable ofproducing materials with the desired structuralcharacteristics.

Bibliography

Brunk B.K. et al. (1997) Hydrodynamic pair diffusion in isotropicrandom velocity fields with application to turbulent coagulation,«Physics of Fluids», 9, 2670-2691.

Evans D.F., Wennerström H. (1999) The colloidal domain: wherephysics, chemistry, biology, and technology meet, New York, JohnWiley.

Hunter R.J. (2001) Foundations of colloid science, Oxford, OxfordUniversity Press.

Larson R.G. (1999) The structure and rheology of complex fluids,Oxford, Oxford University Press.

Melis S. et al. (1999) Effect of fluid motion on the aggregation of smallparticles subject to interaction forces, «American Institute ofChemical Engineers Journal», 45, 1383-1393.

Pusey P. (1991) Colloid suspensions, in: Hansen J.P. et al. (edited by)Liquids, freezing and glass transition, Amsterdam, North-Holland,2v.; v.II, 763.

Russel W. et al. (1989) Colloidal dispersions, Cambridge, CambridgeUniversity Press.

Peter Sandküler

Institut für Chemie- und BioingenieurwissenschaftenEidenössische Technische Hochschule-Hönggerberg

Zürich, Switzerland

Massimo Morbidelli

Institut für Chemie- und BioingenieurwissenschaftenEidenössische Technische Hochschule-Hönggerberg

Zürich, Switzerland

COLLOIDAL SYSTEMS


no mixingcontrol

mixingcontrol

10�6

10�4

10�2

100

dP0 (nm)10 100 1,000

/

Fig. 8. Mixing regimes in turbulent conditions as a function of solid volume fraction, /, and primary particle diameter, dP0, for given mixing conditions.

rotatinginner

cylinder

staticouter

cylinderturbulentenergy

dissipationrate

(m2/s3)

turbulentenergy

dissipationrate

(m2/s3)

A B

2.43·10�51.00·10�22.00·10�23.00·10�24.00·10�25.00·10�26.00·10�27.00·10�28.00·10�29.00·10�21.00·10�1

1.00·10�2

5.09·10�1

1.01·100

1.51·100

2.01·100

2.51·100

3.00·100

3.50·100

4.00·100

4.50·100

5.00·100

Fig. 9. Two differentcoagulation units: the stirred tank (A) and the Couette-Taylor unit (B). A comparison is made in terms of the distribution of energydissipation rate(by unit mass of the fluid).

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3.2 Colloidal systems - Treccani, il portale del sapere