Monitoring particle aggregation processes

Advances in Colloid and Interface Science 147–148 (2009) 109–123

Contents lists available at ScienceDirect

Advances in Colloid and Interface Science

j ourna l homepage: www.e lsev ie r.com/ locate /c is

Monitoring particle aggregation processes

John GregoryDepartment of Civil, Environmental and Geomatic Engineering. University College London, Gower Street, London WC1E 6BT, UK

E-mail address: [email protected].

0001-8686/$ – see front matter © 2008 Elsevier B.V. Aldoi:10.1016/j.cis.2008.09.003

a b s t r a c t
a r t i c l e i n f o
Available online 18 September 2008

Keywords:

A wide range of test methtechniques for measuring

AggregationFlocculationFractal dimensionLight scatteringMonitoring

monitoring and control of practical coagulation/flocculation processes. Most emphasis is on optical methods,including light transmission (turbidity) and light scattering measurements and the fundamentals of thesephenomena are briefly introduced. It is shown that in some cases, absolute aggregation rates can be derived.However, evenwhen only relative rates can be obtained, these can still be very useful, for instance in definingoptimum flocculation conditions. Some of the methods available for investigating properties of aggregates(flocs), such as size, strength and fractal dimension are also discussed, along with some related propertiessuch as sedimentation rate and filterability of flocculated suspensions.

ods for monitoring particle aggregation processes is reviewed. These includeaggregation rates in fundamental studies and those which are useful in the

© 2008 Elsevier B.V. All rights reserved.

Contents

1. Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1102. Aggregation kinetics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 110

2.1. Perikinetic aggregation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1112.2. Orthokinetic aggregation. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1112.3. Fractal aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

3. Light scattering by aggregates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.1. Light scattering and turbidity. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1123.2. Scattering by fractal aggregates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

4. Particle counting and sizing. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.1. Microscopic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1144.2. Sensing zone techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.2.1. Electrozone methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1154.2.2. Optical methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 115

4.3. Focused beam reflectance measurement (FBRM) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165. Light scattering methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

5.1. Turbidity methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.1.1. Aggregation rate measurement . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1165.1.2. Turbidity-wavelength spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1175.1.3. Turbidity fluctuations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 117

5.2. Small-angle light scattering (SALS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.2.1. Aggregation rates by SALS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.2.2. Measurement of aggregate size . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1205.2.3. Fractal dimensions of aggregates. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121

5.3. Dynamic light scattering . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1216. Other techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

6.1. Electro-optical effects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.2. Ultrasonic methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.3. Sedimentation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1226.4. Filterability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 123

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110 J. Gregory / Advances in Colloid and Interface Science 147–148 (2009) 109–123

1. Introduction

Aggregation (coagulation/flocculation) processes are in wide-spread use in many industries, including papermaking, mineralprocessing, and water and wastewater treatment, among others. Inmost cases it is very useful to have methods of monitoring theaggregation process. This may be aimed at optimizing the dosage ofadditives, measuring the rate of aggregation or assessing the proper-ties of the formed aggregates, such as size, density and strength. Aswell as being of great practical value, these methods are also essentialin fundamental studies of particle aggregation.

A schematic view of a typical aggregation process is shown in Fig.1,which also shows some possible test methods. The initial suspensionis assumed to be colloidally stable (usually by virtue of particle chargein the case of aqueous suspensions). In order to promote aggregationthe particles need to be destabilized and this can sometimes beachieved simply by a change in pH of the suspension. However inmostpractical cases certain additives (coagulants or flocculants) areneeded. These may be simple inorganic salts, hydrolyzing metalcoagulants or polymeric flocculants.

At this stage it may be possible to assess the degree of particledestabilization by measuring the particle charge, by one of severalelectrokinetic techniques, such as particle electrophoresis or stream-ing current. The Streaming Current Detector (SCD) is quite often usedto optimize coagulant dosages inwater treatment [1]. Such techniquesare only applicable when charge neutralization is the main destabi-lization mechanism.

When particles are adequately destabilized they will aggregate oncollision with other particles and collisions can occur through twoimportant mechanisms:

• Brownian diffusion (perikinetic aggregation)• Fluid motion (orthokinetic aggregation)

Aggregation takes place at a rate that depends on the collisionfrequency and the collision efficiency factor (or its inverse, the stabilityratio), which is governed by the degree of particle destabilizationachieved. Aggregation rates can be measured in a number of ways.

When aggregates (or flocs) are formed, several of their propertiescan be measured, including size, density, fractal dimension, strength,settling rate and filterability. After settling, the degree of particleseparation can be assessed by measuring the residual turbidity of thesupernatant. This is one of the most common test methods in practice(as in the standard Jar Test), since the aim of many industrialflocculation processes is to achieve a high degree of solid–liquidseparation.

Fig. 1. Schematic aggregation process, with possible test methods.

In this review, we shall focus mainly on methods for measuringrates of aggregation and aggregate properties. Many of the availablemethods are based on light scattering. For these reasons, the next twosections deal with aggregation kinetics and light scattering byaggregates.

2. Aggregation kinetics

Aggregation depends on binary collisions of particles and so isa second-order rate process. (Three-body collisions are usuallyignored — they only become significant at very high particleconcentrations). Most theoretical treatments of aggregation kineticsare based on the early work of Smoluchowski [2] and there have beenmany more recent treatments (e.g. [3]). The number of collisions, Jijoccurring between particles of type i and j in unit time and unitvolume is given by:

Jij=kijninj ð1Þ

where ni and nj are the number concentrations of i and j particles andkij is a second-order rate coefficient, which depends on a number offactors such as particle size and transport mechanism.

In the following, it is convenient to assume that the suspensionconsists initially of monodisperse primary particles and that the labelsi, j, etc. refer to the numbers of primary particles in aggregates. Thusan i-particle is an aggregate of i primary particles. If it is assumed thatall collisions lead to aggregation, then it is possible to write anexpression for the rate of change of k-fold particles, originallyproposed by Smoluchowski:

dnk

dt=12

Xi=k−1i+jYk

i=1

kijninj−nk

X∞k=1

kikni ð2Þ

The first term on the right hand side represents the rate offormation of k-fold aggregates by collision of any pair of aggregates, iand j, such that i+ j=k. Carrying out the summation by this methodwould mean counting each collision twice and hence the factor 1/2 isincluded. The second term accounts for the loss of k-fold aggregates bycollisions with any other aggregates. The terms kij and kik are theappropriate rate coefficients. It is important to note that Eq. (2) is forirreversible aggregation, since no allowance is made for break-up ofaggregates. Also, it has been assumed that every collision is effective informing an aggregate. If the particles are not fully-destabilized, thenonly a fraction of collisions are successful. This fraction is the collisionefficiency, which depends on colloidal interactions between particlesand hydrodynamic effects. This aspect will not be considered furtherhere.

In the very early stages of aggregation, when most of the particlesare still single, only the second term on the r.h.s. of Eq. (2) needs to beconsidered. In that case, the rate of loss of primary particles is simply:

dn1

dt

� �tY0

=−k11n21 ð3Þ

Each collision leads to the loss of two primary particles and theformation of one doublet, so that there is a net loss of one particle.Thus the rate of decrease in the total number of particles in the earlystages could be written:

dnT

dt

� �tY0

=−k112

n21=−kan

21 ð4Þ

This can be regarded as the aggregation rate and ka, the aggrega-tion rate coefficient, is just half the collision rate coefficient for primaryparticles.

The rate coefficients depend greatly on the collision mechanismand there are two cases of practical importance.

Fig. 2. Relative decrease in total particle concentration for perikinetic (Eq. (8)) andorthokinetic (Eq. (14)) aggregation. Conditions: aqueous suspension at 25 °C,n0=1015 m−3, d1=1 µm, G=6 s−1.

111J. Gregory / Advances in Colloid and Interface Science 147–148 (2009) 109–123

2.1. Perikinetic aggregation

All suspended particles undergo Brownian motion to some extentand this can cause particle collisions and aggregation. The Smolu-chowski result for the perikinetic collision rate coefficient for unequalspheres is:

kij=2kBT3η

ai+aj� �2aiaj

ð5Þ

where kB is Boltzmann's constant, T the absolute temperature, ηthe viscosity of the fluid and ai, aj are the radii of the collidingparticles.

When the particles are nearly equal in size, the size term in Eq. (4)is approximately constant, with a value close to 4, so that:

kij≈8kBT3η

ð6Þ

(For aqueous suspensions at 25 °C, the coefficient is about 1.23×10−17 m3s−1.)

By assuming a constant collision rate coefficient, Smoluchowskishowed that the previous expression for the early stages ofaggregation, Eq. (4), would apply throughout the aggregation process.Thus:

dnT

dt=−kan2

T ð7Þ

where the aggregation rate coefficient, ka=4kBT /3η.Eq. (7) can be integrated to give an expression for the total number

of particles at time t:

nT=n0

1+kan0t=

n0

1+t=tað8Þ

where n0 is the initial concentration of (primary) particles and ta is theaggregation time, in which the number of particles is reduced to onehalf of the initial value (ta=1/kan0). For aqueous colloids at 25 °C,ta=1.63×1017/n0.

The ratio n0/nT can be regarded as the mean aggregation number,k̄̄ , i.e. the average number of particles per aggregate.

k=1+kan0t=1+t=ta ð9Þ

The above expressions are based on the assumption of sphericalparticles and aggregates. For hard particles, spherical aggregates areunlikely, but, for the perikinetic case, this assumption does not lead tolarge errors.

Another effect is hydrodynamic interaction between particles [4],which can typically reduce the aggregation rate by a factor of about 2.As aggregates grow, they generally form rather open fractal structures(see below), for which hydrodynamic interactions may not be sosignificant.

2.2. Orthokinetic aggregation

The simplest case to consider is that of spherical particles in auniform, laminar shear field. Collisions of particles are brought aboutbecause of their different velocities at different layers of the shearfield. For a shear rate (velocity gradient) G, Smoluchowski showedthat the collision rate coefficient for unequal particles is:

kij=43G ai+aj� �3 ð10Þ

In contrast to the perikinetic case, particle size has a huge influenceon the collision rate, which is of great practical importance. If Eq. (10)is restricted to the case of equal primary particles, radius a1, then

combination with Eq. (4) gives the following expression for the initialrate of aggregation:

dnT

dt=−

163

n2TGa

31 ð11Þ

Because of the great dependence of collision rate on particle size,this expression only applies to the very early stages of aggregation,where most of the particles are still single. Nevertheless, a simpletransformation is possible since the volume fraction, ϕ, of particles inthe suspension is:

�=4πa31nT=3 ð12Þ

This allows Eq. (11) to be written in a pseudo first-order form:

dnT

dt=−

4G�nT

πð13Þ

If the volume fraction of particles is assumed to remain constantduring aggregation then Eq. (13) can be integrated to give:

nT

n0= exp

−4G�tπ

� �ð14Þ

The assumption of constant volume fraction is questionable forfractal aggregates, since their effective volume grows with increasingaggregate size. This means that Eq. (13) will likely give an under-estimate of the actual aggregation rate. However, hydrodynamicinteractions become more significant for larger particles [5] and canreduce the capture efficiency of aggregates [6]. This makes quantita-tive modeling of the orthokinetic aggregation process quite difficult.

The most important implication of Eq. (14) is that the total particleconcentration should decrease exponentially with time and hence themean aggregation number should show exponential growth. This is inmarked contrast to the perikinetic case, where aggregates shouldgrow in a linear manner with time. This is illustrated in Fig. 2, wherethe decrease in particle number with time is shown for both cases. Thecalculations are for an aqueous suspension containing 1 µm diameterparticles at a concentration of 1015 m−3 and a temperature of 25 °C.This gives a characteristic perikinetic aggregation time ta=163 s, and avolume fraction ϕ=5.24×10−4. For the orthokinetic case, the shearrate is chosen as the very low value of G=6 s−1, which give about thesame initial rate of aggregation (τ=173 s). Note that these calculationsare for the individual collision mechanisms acting separately. Ofcourse, in reality, Brownian diffusion cannot be ‘switched off’ and thetwo effects can be treated as approximately additive.

Fig. 3. Measurement of transmitted and scattered light.


From Fig. 2 it is clear that, for the first few minutes, aggregationoccurs at about the same rate for both cases. However, at longer times,the orthokinetic rate becomes relatively much faster. These results arebased on rather unrealistic assumptions, especially for the orthoki-netic case. Nevertheless they are sufficient to show the enormouslydifferent rates at long aggregation times and to explain whyperikinetic aggregation alone is rarely sufficient to form very largeaggregates.

Because orthokinetic aggregation occurs by some form of fluidshear (e.g. in a stirred vessel), aggregate strength and breakage alsohave to be considered. In the original Smoluchowski treatmentaggregation was assumed to be irreversible, with no breakage. Inpractice, aggregates are often found to reach a limiting size undergiven shear conditions and this is generally assumed to be a result of abalance between aggregate growth and breakage [7]. However, adiminishing collision efficiencywith increasing aggregate size can alsolimit aggregate growth [6]. Although a quantitative description ofaggregate breakage is difficult [8], a simple empirical expression forlimiting aggregate size, dmax, in terms of the applied shear rate, G, isoften used:

dmax=KG−γ ð15Þwhere K and γ are empirical constants. The exponent γ depends onthe mode of aggregate breakage, but values in the region of γ=0.5 areoften found.

2.3. Fractal aggregates

In most cases, aggregates are known to be fractal, scale invariantobjects [9], which have a rather open structure. For aggregates themass fractal dimension is an important parameter. This relates themass of an aggregate with a convenient linear dimension, such asdiameter or radius of gyration. Generally, the massM and length L arerelated by:

M~LD ð16Þwhere D is the mass fractal dimension, which, in principle can varybetween 1 and 3 for objects in 3-dimensional space. Typicalaggregates have values of D between about 1.7 and 2.5, dependingon several factors. Generally, diffusion-limited (perikinetic) aggre-gates have fractal dimensions at the lower end of this range, whereasorthokinetic aggregation gives somewhat higher values (i.e. morecompact aggregates).

For an aggregate of identical primary particles, the mass is directlyproportional to the aggregation number. If an aggregate contains iprimary particles, radius a1, then the following expression applies:

i=B Rg=a1� �D ð17Þ

where Rg is the radius of gyration of the aggregate and B is a constantof order unity. Wang and Sorensen [10] quote a value of B=1.3±0.1,from several studies of fractal aggregates.

The fractal nature of aggregates has very important practicalimplications. For instance, aggregate density decreases with aggregatesize according to:

ρE~d−y ð18Þ

where ρE is the effective (or buoyant) density of an aggregate, withdiameter d. This depends of the densities of the particles and fluid andon the solid volume fraction within the aggregate, ϕS:

ρE=ρA−ρL=�S ρS−ρLð Þ ð19Þwhere ρA, ρL and ρS are the densities of the aggregate, liquid andparticles, respectively.

It is easy to show that the exponent y in Eq. (18) is given by:

y=3−D ð20Þ

For solid, non-fractal objects (D=3), the density is independent ofsize, but for low fractal dimensions there can be a marked decrease indensity as aggregates grow larger. Large aggregates can have very lowdensity and his has significant consequences for solid–liquid separa-tion [11]. Also, by measuring aggregate density over a range ofaggregate size, it should be possible to derive the fractal dimension[12].

The fractal nature of aggregates has a great effect on their lightscattering properties, as discussed in the next section.

3. Light scattering by aggregates

3.1. Light scattering and turbidity

For solid particles of regular shape, such as spheres, light scatteringis a verywell studied phenomenon, with an extensive literature [13]. Abasic light scattering set-up is shown schematically in Fig. 3. Here, alight beam illuminates a suspension of particles in a suitable cell andthe scattered light intensity can be measured by a detector placed atany desired angle to the incident beam. By placing a detector directlyin line with the incident beam it is also possible to measure thetransmitted light intensity and hence the turbidity of the suspension.Both of these approaches have been used to derive information onaggregate properties.

A general theory of light scattering has been available since theearly 20th Century. This is usually attributed to Mie, although, asKerker [14] has pointed out, there were many other importantcontributions to the theory. Without going into detail, the intensity ofscattered light depends on the following parameters:

• Particle size, e.g. radius, a• Light wavelength, λ• Particle refractive index relative to the suspension medium, m• Scattering angle, θ

Very often, particle size and light wavelength are combined in asingle dimensionless parameter, α=2πa /λ.

In general light may be absorbed as well as scattered by particles, inwhich case the refractive index takes complex values. For simplicity,we shall assume that absorption is negligible, so thatm is always real.In that case scattering occurs with no net loss of energy from the lightbeam (elastic scattering).

Fig. 4 shows scattered light intensity against scattering angle forspherical particles with a range of α values from 1–50 (correspondingto particle diameters in the region of about 0.2–10 µm for light in the

Fig. 5. Scattering coefficient vs. size parameter for different values of relative refractiveindex, m (values shown on curves).

Fig. 6. Specific turbidity vs. particle diameter for different m values.

Fig. 4. Relative scattered light intensity from spherical particles as a function ofscattering angle, for various values of the size parameter α (values shown on curves).


visible wavelength range). The relative refractive index was assumedto be m=1.20, which is about the value for polystyrene latex particlesin water. The results shown are for unpolarized light and thecomputations were carried out using Mieplot v3418 (www.philipla-ven.com). For smaller particles (αb1), light scattering behaviorapproaches that predicted by Rayleigh theory, where the scatteringintensity is proportional to the 6th power of particle size and has littledependence on scattering angle.

There are two major effects of increasing particle size:

• Avery large increase in scattered light intensity (overmany orders ofmagnitude)

• An increase in the proportion of light scattered at low angles

Both of these effects are very important in monitoring aggregation.By integrating the scattered light at all angles around a particle, it is

possible to calculate the scattering cross-section, C. The total lightenergy scattered by a particle is effectively that within an area of theincident light beam, C. This is related to the geometric cross-section ofthe particle, through the scattering coefficient, Q. Thus, for a sphericalparticle:

C=Qπa2 ð21Þ

The scattering coefficient varies with the size parameter α, asshown in Fig. 5, for different values of refractive index. Low m valuesare included, since these may be relevant to aggregates (see below).Typically, Q rises from very low values and passes through a series ofmaxima and minima, before eventually reaching a constant valueQ=2. However, for low m, the first maximum is not reached until theparticle size is quite large.

When light transmitted through a suspension is measured (asshown in Fig. 3) the turbidity can be defined in terms of the pathlength, L, the particle number concentration, n, and the scatteringcross-section of the particles. (A monodisperse suspension isassumed). If the incident light intensity is I0, then the transmittedlight intensity is given by:

I=I0 exp −τLð Þ ð22Þ

where τ is the turbidity, given by:

τ=nC ð23Þ

It is convenient to introduce the specific turbidity, τ/ϕ, where ϕ isthe concentration of particles expressed as a volume fraction. Since,

for a monodisperse suspension, the volume fraction is just (4/3)nπa3,the specific turbidity can be derived from Eqs. (21) and (23) as:

τ�=3Q4a

ð24Þ

For very large particles, where Q becomes almost constant, thisshows that the specific turbidity should be inversely proportional toparticle size. However, for smaller particles the turbidity shows morecomplex behavior. The curves in Fig. 6 show specific turbidity as afunction of particle diameter, for the same refractive index values as inFig. 5. In this case, since the actual size is plotted, we need to specifythe light wavelength and a value of 650 nm is chosen. (This is the valuein the suspension. For aqueous suspensions this would correspond toa vacuum wavelength of about 865 nm.) For very small particles, theturbidity is low, but it increases markedly as particle size increases.This effect is most pronounced with higher m values, where the firstturbidity maximum is quite sharp. For lower refractive index theturbidity rises less steeply and the maxima occur at progressivelylarger particle sizes. Particle sizing by light scattering, includingFraunhofer diffraction, can be significantly affected by the assumedrefractive index of particles [15].

From the curves in Fig. 6 it is clear that a single turbiditymeasurement, for particles of known refractive index, would not givean unambiguous value of particle size. However, measurements at two

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Fig. 8. Log I(q) vs. log q, for aggregates of particles, based on the RGD approximation (seetext).


or more wavelengths can, in principle, be used to derive particle size[14].

A significant problemwith turbidity methods, especially for largerparticles and aggregates, is that, in any practical method, someforward-scattered light must reach the detector (see Fig. 7), thusincreasing the apparent transmitted light intensity and decreasing theapparent scattering coefficient. This effect has been know for a longtime [16] and can have major effects on the apparent turbidity ofaggregating suspensions [17]. Typical acceptance angles in commer-cial spectrophotometers are in the region of a few degrees, but withcareful design it is possible to reduce this to close to zero [18]. Morerecently, this problem has been discussed in some detail [19].

3.2. Scattering by fractal aggregates

The problem of light scattering by aggregates is much moredifficult than for solid spheres, although some simplifications may bepossible in certain cases. The subject has been comprehensivelyreviewed by Sorensen [20].

The simplest approach is to assume that the primary particles in anaggregate are small enough to behave as Rayleigh scatterers and to usethe Rayleigh-Gans-Debye (RGD) approximation (see e.g. Schmidt [21]).In this case the scattering by an aggregate of size L depends on thequantity qL, where q is the magnitude of the scattering vector, given,for a scattering angle θ and a wavelength λ, by:

q=4πλ

sinθ2

� �ð25Þ

When the RGD approximation applies, the same equations can beused for a given value of q, irrespective of the wavelength of theradiation. So, in principle, the method applies equally to scattering ofvisible light, X-rays or neutrons. Since thewavelength of visible light isaround 1000 times that of X-rays or neutrons, visible light scatteringat large angles corresponds to small-angle scattering of X-rays orneutrons. An important physical implication of the scattering vector isthat the length 1/q defines the scale probed by the scatteringmeasurement. When 1/q is much smaller than the primary particlesin an aggregate, the aggregate structure plays no part and thescattering from a k-fold aggregate would be just the same as that fromk isolated particles. Thus at large scattering angles, aggregation of asuspension would give no change in scattered light intensity.Conversely, for small q values, where 1/q is much larger than theaggregate size, the scattered intensity varies as k2, so that thescattering from an aggregating suspension increases with the averageaggregation number. For this reason, small-angle light scattering(SALS) can be a useful method for determining aggregate size, even forrelatively large primary particles [22]. At intermediate q values it can

Fig. 7. Showing that light transmission is affected by forward-scattered light. Lightscattered by particles at angles less than θ will reach the detector and thus increase theapparent transmitted light intensity. (The detector size is exaggerated tomake the effectclearer. In practice, the effect can be reduced by making the detector smaller andlocating it further from the cell). After Latimer [17].

be shown that the scattered light intensity depends on the fractaldimension of the aggregates, according to:

I qð Þ~q−D ð26Þ

So, in principle, the fractal dimension for aggregates can be deriveddirectly from the slope of a plot of log I(q) vs. log q, as shown in Fig. 8.It must be remembered that Eq. (26) is based on the RGDapproximation (small primary particles of fairly low refractiveindex), although light scattering methods have been used to deriveinformation on the fractal properties of aggregates of larger particles[23]. In such cases, it is better to refer to the slope of the linear regionin Fig. 8 as a “scattering exponent” or “scaling exponent” [24].

We now review some methods for monitoring aggregationprocesses.

4. Particle counting and sizing

The oldest method of following particle aggregation is by countingthe number of particles (and aggregates) at intervals during theprocess. This may be done by direct microscopic observation or by anautomated counting method. The latter involves passing the suspen-sion through some kind of sensing zonewhere individual particles canbe counted (and sized in most cases). Where sizing is possible, a sizedistribution can be derived without any a priori assumptions aboutthe form of the distribution.

4.1. Microscopic methods

Many early measurements of aggregation rates and stability ratiosof colloidal dispersions were made by direct observation of particlesby optical microscopes (e.g. Freundlich [25]). Suchmethods have beenwidely used over many decades, for instance in the study of particleaggregation in natural waters [26]. Samples of the suspension arewithdrawn at intervals, diluted if necessary, and microscopicallyexamined. The number of particles in a given volume can then bedetermined simply by counting. Aggregation of large latex particles innarrow tubes has been directly observed by optical microscopy [27].Ordinary optical microscopes cannot easily resolve particles less thanabout 1 µm in size, although the ultramicroscope, using dark fieldillumination, allows much smaller particles to be detected (down toaround 5 nm diameter for gold sols [28]).

It is not generally possible to determine the size of aggregates byoptical microscopy, unless they grow quite large and the method isnormally used to give just the total number of particles. Nevertheless,this enables the aggregation rate coefficient, ka, to be determined,using Eq. (8) and the average aggregation number from Eq. (9). Whenaggregates grow quite large (greater than about 15 µm) it is possible to


apply image analysis to digitized optical micrographs. This methodhas been used to derive information on the size shape and fractalproperties of flocs produced in orthokinetic aggregation [29–31]. Thedependence of perceived aggregate size from image analysis on pixelresolution has recently been discussed [32].

Transmission electron microscopy provides much higher resolu-tion than the light microscope, but the sample preparation, includinga drying stage, makes it rather impractical for kinetic measurements.Aggregate structure can be examined by electron microscopy,provided that changes do not occur during sample preparation. Thefreeze-fracture technique has proved useful in this respect [33].

4.2. Sensing zone techniques

Automated particle counting can be achieved by allowing particlesand aggregates to pass singly through a zone in which their presencecan be detected by a suitable sensor. Particles passing through thiszone give a series of pulses, which can be counted. Such a device isshown schematically in Fig. 9. If the sensor response depends onparticle size, then the pulse height is size-dependent and can be usedto discriminate between particles of different size. These methodsdepend on there being only one particle in the sensing zone at a time.Otherwise two or more particles would give just one pulse and appearas one larger particle. This is known as the coincidence effect and canbe avoided by adequate dilution of the suspension or by making thesensing zone sufficiently small. Since the pulse count per unit time ismeasured, the flow rate must be known and kept constant.

There are two commonly used sensing methods in particlecounting:

• Electrical (or Electrozone)• Optical (light scattering)

These have their own advantages and disadvantages.

4.2.1. Electrozone methodsElectrical sensing zone (Electrozone) counters are based on a

principle developed by Coulter in the 1940s and commercialized asthe Coulter Counter [34]. Particles in an electrolyte solution passthrough an orifice and cause a momentary change in electricalresistance and hence a voltage pulse if the current is kept constant.Electrodes are located on either side of the orifice and a particlepassing through displaces a volume of electrolyte equal to the particle

Fig. 9. Schematic illustration of particle counting and sizing by the sensing zonemethod. The presence of a particle in the sensing zone is detected by a suitable probeand detector system (usually either electrical or optical).

volume. Most particles have effectively infinite resistance and so thevoltage pulse is proportional to the particle volume. It is capable ofquite high speed counting (5000 or more particles per second) andcan resolve particles only slightly different in size. A unique feature ofthe electrozone method, in comparison with other (especially lightscattering) techniques, is that it is virtually independent of the shapeor composition of the particles.

The Coulter technique has been used very widely for particle sizeanalysis and is the subject of several reviews (e.g. Lines [35]). Byaround 1990 there were already many thousands of references to usesof the technique in a large range of applications.

Use of the electrozone method for particle aggregation studies hasbeen more limited. Among the first to use this technique wereMatthews and Rhodes [36] and there have been several laterexamples, including flocculation of bacterial suspensions [37] andlatex [38–40]. In principle, the aggregate size reported by the Coultertechnique should be just that corresponding to the volume of theprimary particles within the aggregate and not the included fluid(since this would be the supporting electrolyte, of high conductivity).Thus the apparent Coulter size should be less than, say, the aggregatesize seen by optical microscopy [37]. This could be a means ofdetermining the effective aggregate density, and hence fractaldimension [41,42].

A possible difficulty with the electrozone technique is the break-upof aggregates as they pass through the orifice, where the shear ratecan be very high. However, the fragments should have the same totalvolume as the original aggregate, so that breakage within the orificemight still produce only one pulse, corresponding to the aggregatevolume. Breakage occurring in the elongational flow field justupstream of the orifice could give several smaller pulses.

Another problem is the need for dilution of the suspensionwith anelectrolyte solution (typically around 2% NaCl), which could causecolloid destabilization. Because of the rather low particle concentra-tion in the diluted suspension (usually less than 106 particles/mL toavoid coincidence effects) the aggregation rate would be very slowand in the short time needed for a measurement it should not be asignificant effect. However, dilution of a weakly-aggregated suspen-sion may cause some disaggregation.

The electrozone technique cannot be used over a wide range ofaggregate size, which should be between about 2% and 40% of theorifice diameter. A practical lower size limit is about 1 µm, which is aserious limitation for colloids. For large aggregates, orifice blockagebecomes a major problem. It is possible to use different orifice sizes tocover a wider size range, but this is rather inconvenient.

4.2.2. Optical methodsIn this case, particles are made to pass singly through a focused

light beam (usually a laser beam) and either the transmitted orscattered light intensity is monitored. The transmitted light method(light blockage) is not as sensitive as light scattering, although it formsthe basis of several commercial particle counters. Particle countingand sizing by scattered light measurement is often referred to as flowultramicroscopy.

For aggregation studies small-angle light scattering (SALS) is mostappropriate, although some older instruments used scattering ataround 90° to the incident beam [43]. At very small scattering angles(low q values) the scattered light intensity varies as the square of theaggregate volume (or aggregation number, k), at least under con-ditions where the RGD approximation is acceptable (see Section 3.2).There have been several single particle optical counting and sizinginstruments developed and the subject has been reviewed byLichtenfeld et al. [44]. Features include feedback control of theincident light intensity to enable a wide range of aggregate size to bemeasured without exceeding the dynamic range of the photomulti-plier [45]. Also, hydrodynamic focusing can be used to reduce the sizeof the sample volume and hence minimize the coincidence effect [46].


Most applications of these techniques have been with mono-disperse suspensions, such as polystyrene latex [45,47] and silica [48]undergoing perikinetic aggregation. In such cases it is possible todiscriminate between aggregates up to about 7-fold. These measure-ments allow experimental tests of Smoluchowski kinetics, at least inthe early stages of aggregation [48] and can give useful information onaggregation by polymers [47].

Hydrodynamic aspects can be quite important, either because oforthokinetic aggregation or the break-up of aggregates as a result offlow [49]. In a commercial light blockage particle counter (Hiac), it hasbeen shown that aggregate breakage can be a significant problem [50].

4.3. Focused beam reflectance measurement (FBRM)

This is a more recent method of counting and sizing particles andaggregates and is available in commercial versions (Gallai, Lasentec). Inthe FBRM technique, a tightly-focused laser beam is projected from aprobe into the suspension through awindow, to forma small spot closeto thewindow. The spot is caused to rotate at high speed (about 2m/s).Particles close to the window will be scanned by the spot and light isreflected back to the probe. The reflected pulse is of a certain duration,which depends on the scanning speed and on the appropriate chordlength (see Fig. 10). Thousands of chords can be measured per second,giving a chord length distribution from below 1 µm tomore than 1mm[51]. Themethodworks well withmuch higher suspension concentra-tions (up to 20%) and much larger flocs than are possible withelectrozone or optical counters. In most cases, no dilution is needed,which is a great advantage. The suspension must be agitated in someway, so that the scanned sample is renewed frequently. This meansthat themethod is only suitable for studies of orthokinetic aggregation.Another benefit of the FBRM method is that aggregating suspensionscan be monitored in situ (e.g. in a stirred vessel).

The relationship between chord length and particle size is notstraightforward. This problem has been examined in some detail byHeath et al. [52], who found that an empirical method was moresuccessful than a theoretical approach.

The FBRM technique has been used to follow aggregation andaggregate breakage in latex suspensions in Couette flow [53]. Thisstudy clearly showed the expected dependence of maximumaggregate size on shear rate (see Eq. (15)) and gave useful informationon the effect of particle concentration. There have also been morepractical applications, such as in mineral processing [51] andpapermaking [54].

5. Light scattering methods

Previous sections have dealt with optical techniques for thecounting and sizing of individual particles. We now turn to light

Fig. 10. The FBRM principle. A rapidly moving light spot gives reflected light fromparticles or aggregates in its path. The lengths of the light pulses (below) areproportional to the chord lengths, which may differ considerably, even for particles ofsimilar size.

scattering methods where there may be many particles or aggregatesin the light beam and information on average properties is derived.Such methods are widely used, but cannot give such detailedinformation on particle or aggregate size distribution as single particlecounting.

Available techniques involve measurement of transmitted light(turbidity), or of light scattered at one or more angles to the incidentbeam (static light scattering). Another method is dynamic lightscattering, which derives information from the diffusion of particles.

5.1. Turbidity methods

5.1.1. Aggregation rate measurementFor particles smaller than the light wavelength, turbidity increases

with particle size (see Fig. 6) and also increases as small particlesaggregate. This has often been used as a semi-empirical method offollowing aggregation, for instance to determine relative coagulationrates and stability ratios [55]. It is usually assumed that the initial rateof increase in turbidity is proportional to the initial aggregation rate.For perikinetic aggregation, the stability ratio, W, is defined as theratio of the rate of aggregation of the fully-destabilized suspension (i.e.the diffusion-controlled case) to that of the partially-destabilizedsuspension (where there is some repulsion between the particles andnot every collision is effective). This is just the same as the ratio of theappropriate rate coefficients, ka, in Eq. (7). If we assume that the rate ofaggregation is proportional to the corresponding rate of change ofturbidity, then it follows that the stability ratio is given by:

W=ka;max

ka=

dτ=dtð Þ0;max

dτ=dtð Þ0ð27Þ

where the numerators are for a fully-destabilized sol and thedenominators are for some other experimental condition. Both ratesare those at the onset of aggregation.

This method is still used to give relative aggregation rates [56].In many cases relative rates are quite adequate to give valuable

information on mechanisms of particle destabilization with variousadditives and under different conditions. However, there have beenmany attempts to derive absolute rates of aggregation from turbiditymeasurements. Even for spherical, monodisperse primary particles,we need information on scattering cross-sections of aggregates. Theturbidity of an aggregating suspension, by analogy with Eq. (13),should be given by:

τ=n1C1+n2C2+n3C3+:::: ð28Þ

where n1, C1, etc. are the number concentrations and scattering cross-sections of singlets, doublets and triplets, etc. after a certain period ofaggregation.

The concentrations of the various aggregates can be easily derivedfrom Smoluchowski theory, but the scattering coefficients are muchmore difficult to calculate. The simplest case is when the primaryparticles are much smaller than the light wavelength and Rayleightheory applies. The scattering cross-section is then proportional to thesquare of the particle volume, so that C2=4C1. By considering only theinitial stage of aggregation where only singlets and doublets arepresent and assuming that the doublets also act as Rayleigh scatterers,it can be shown that:

1τ0

dτdt

� �0=2ta=2kan0 ð29Þ

where τ0 is the initial turbidity, ta is the aggregation time (Eq. (8)), kais the aggregation rate coefficient and n0 is the initial concentration ofprimary particles.

According to this simple analysis, the turbidity of a sol shouldinitially increase linearlywith time and the aggregation rate coefficient

Fig. 12. Plot of n=d(log τ) /d(log λ) volume fraction of suspension for a weakly-flocculating system. The change of slope indicates the onset of flocculation.


could be derived from the initial slope. It has been implicitly assumedthat the doublets are spherical particles, so that coalescence occurs onaggregation. This is an unrealistic assumption for hard particles, wherereal doublets would be in the form of dumbbells. A more realisticexpression is [57]:

1τ0

dτdt

� �0=2ka C2=2C1−1ð Þn0 ð30Þ

If C2=4C1, as in the ‘coalescence’ assumption, then Eq. (30) isequivalent to Eq. (29). Otherwise the scattering cross-section of ‘real’doublets needs to be considered. The usual approach has been to useRGD theory, which requires the particle size and the relative refractiveindex to be quite small. For doublets of equal spheres, application ofRGD theory is straightforward, since there is no doubt about theshape. For higher aggregates, this is not the case. Lichtenbelt et al. [57]computed the optical correction factor (the bracketed termon the r.h.s.of Eq. (30)) for ‘real’ doublets over a range of values of the sizeparameter α (=2πa /λ) from RGD theory. Their results are plotted inFig. 11, along with computations fromMie theory for polystyrene latexparticles, assuming that the doublet coalesces to form a sphere of thesame total volume. It is clear that for α values greater than about 0.3the RGD results give a much smaller factor. The coalescence assump-tion overestimates the rate of turbidity increase and hence would giveaggregation rate coefficients that were too low.

An improved method of calculating the scattering cross-section ofdoublets is the T-matrix procedure [58], which is not subject to thelimitations of the RGD approach. Results from this method, forpolystyrene latex are also shown in Fig. 11 and it appears that there isreasonable agreement with the RGD results for fairly small α values.For larger particles the agreement is less good and for α values greaterthan about 6, the optical factor becomes negative, indicating that theturbidity would decrease on aggregation. It is experimentallyobserved that latex particles of around 1 µm diameter or larger,with visible light, do showa decreased turbidity as aggregation occurs.It is doubtful whether a simple turbidity approach would be suitablefor aggregation rate measurements in such cases.

5.1.2. Turbidity-wavelength spectraA simple qualitative way of assessing the state of aggregation of

dilute suspensions was described in 1973 by Vincent and co-workers[59] and has been frequently used since then [60]. It is especiallyuseful in cases of weak flocculation, which may be influenced by thesuspension concentration or temperature.

Fig. 11. The ‘optical factor’ from Eq. (30) calculated from: Mie theory assumingcoalescence of doublets [57], RGD theory for ‘real’ doublets [57] and from the T-matrixprocedure [58].

The basis of the technique is that the specific turbidity of asuspension depends on the light wavelength. This follows fromEq. (24) since the scattering coefficient Q is wavelength dependent(except for particles much larger than the light wavelength).Generally, the relationship can be written:

τ=�=kλ−n ð31Þwhere k and n are constants for a given particle size.

For very small particles, where the Rayleigh approximation holds,n=4, but for larger particles it has lower values. By plotting log τagainst log λ for a given sol, a value of n can be derived from the slope.If this is done over a range of conditions in which the state ofaggregation changes, then a sudden change in n may be observed, asshown schematically in Fig. 12. In this case n is plotted against the solconcentration and there is a clear break at a certain concentration.This is characteristic of equilibrium flocculation of a sterically-stabilized suspension, where there is a shallow minimum in theinteraction energy curve. Reversible flocculation occurs at somecritical particle concentration, which is marked by the break in the nvs. ϕ plot. In other cases a sudden change of n may occur at a certaintemperature [60]. A decrease in n indicates an increase in particle sizeand hence aggregation.

5.1.3. Turbidity fluctuationsIf a small volume of a colloidal suspension is observed micro-

scopically, then Brownian diffusion causes particles tomove in and out

Fig. 13. Observed counts of gold particles in a small volume [61] compared with valuespredicted from the Poisson distribution.


of the observed volume, so that the total number of particles viewedvaries in a random manner. It can be shown that these variationsfollow the Poisson distribution:

P nð Þ= exp −�ð Þ�nn!

ð31Þ

where ν is the mean number of particles in the observed volume andP(n) is the probability of finding n particles in the volume.

Svedberg [61] observed a gold sol by ultramicroscope and countedthe number of particles in a defined volume at intervals over severalminutes. In 518 observations he counted a total of 801 particles, so thatthemeanwas about 1.55. He counted the number of observationswith0,1, 2, 3 etc particles and the results are shown in Fig. 13, togetherwiththe corresponding counts expected from the Poisson distribution. It isclear that the agreement is excellent. A very important consequence ofthe Poisson distribution is that the variance is equal to themean and sothe standard deviation is the square root of themean. From Svedberg'sdata the variance is 1.53, very close to the actual mean (1.55).

Exactly the same distribution would be found if particles werecounted in a series of equal small volumes withdrawn from asuspension. In fact, non-Brownian particles would also show thesame distribution, since, however well mixed, a suspension alwaysshows non-uniformity on a microscopic scale and Poisson statisticsapply in this case also. Instead of taking a sequence of samples wecould simply cause a suspension to flow through a suitable device inwhich particle number variation could be monitored. This is the basisof the turbidity fluctuation technique.

The method was developed in the 1980s by Gregory and Nelson[62] and theoretical aspects have also been discussed [63,64].Essentially, a suspension flows through an optical cell (or simply atransparent plastic tube) and is illuminated by a narrow light beam.The transmitted light intensity is measured by a photodiode andshows random variations about a mean value (see Fig. 14). The meantransmitted light intensity can be used to calculate the turbidity of thesuspension, from Eq. (22). The fluctuations arise from randomvariations in number (and size) of particles in the light beam. Thephotodiode output would typically show a mean (DC) value of severalvolts and the fluctuations (AC) might be of the order of only a fewmV.The normal procedure is to derive the root mean square (RMS) value ofthe voltage fluctuations and to divide this by the mean (DC) value, to

Fig. 14. Schematic illustration of the

give a Ratio value R. Unlike single particle counting methods, theturbidity fluctuation technique is valid when there are large numbersof particles in the light beam and works well up to quite highconcentrations. In practice, suspension concentration is limited byturbidity — there must be a measurable transmitted light intensity.The lower concentration limit is determined by random electronicnoise in the equipment, which means that measurable R values musttypically be greater than about 3×10−5. For larger particles, this maycorrespond, on average, to less than one particle in the light beam, butPoisson statistics still apply in this case.

It can be shown that, for a monodisperse suspension, the Ratio isgiven by:

R=VRMS

VDC=

ffiffiffiffiffinLA

rC ð32Þ

where n is the average number of particles per unit volume, L is theoptical path length, A the effective cross-sectional area of the lightbeam and C is the scattering cross-section of the particles.

The dependence of R on the square root of particle concentration isa consequence of the Poisson distribution for the number fluctuations.The RMS value of the fluctuating signal is analogous to the standarddeviation, which varies as the square root of concentration. Since theturbidity depends directly on the particle concentration, it is possibleto combine Eqs. (23) and (32) to give an expression for the particleconcentration:

n=τR

� �2LA

ð33Þ

The scattering cross-section does not appear in Eq. (32), so that theparticle number concentration could be derived without any knowl-edge of the optical properties of the particles. However, this onlyapplies to a monodisperse suspension. For a heterodisperse suspen-sion, the equivalent expression to Eq. (32) is:

R=LA

� �1=2 XniC2

i

� �1=2ð34Þ

where ni and Ci are the concentration and scattering cross-section ofparticles of type i and the sum is taken over all particle types.

‘turbidity fluctuation’ method.

Fig. 15. Typical experimental set-up for flocculation monitoring by the turbidity fluctuation method.


An important consequence of Eq. (34) is that R is heavily influencedby larger particles (with higher scattering cross-sections). It is notpossible to derive a simple expression for particle concentration inthis case, but it can be shown [63] that, even for rather broad particlesize distributions, the total number concentration derived fromEq. (33) does not differ from the true value by more than a factor ofabout 2. This suggests that, for an aggregating suspension a similarapproach could give an indication of the mean aggregation number(inversely proportional to the total particle number concentration —

see Section 2.1).Experimentally, it is found that R always increases as aggregation

occurs and this provides a very sensitive monitoring method. Acommercial version of this technique (Photometric DispersionAnalyzer, PDA 2000, Rank Brothers, UK) has been available for sometime and the method has been widely used to monitor flocculationprocesses in a variety of applications. These include papermakingsystems [65], asphaltene aggregation [66], flocculation of bacteria[67], water [68] and wastewater [69] treatment and floc strengthinvestigations [70].

For laboratory flocculation studies, a set-up like that in Fig. 15 maybe used. Suspension from a stirred vessel is circulated continuouslythrough the monitor and the outputs (usually DC and RMS values) are

Fig. 16. Change of Flocculation Index with time for (a) ferric sulfate alone and (b) ferric sulfatdetails.

recorded. In such applications, the Ratio value, R, is often called theFlocculation Index (FI). For a given system, this value is stronglycorrelated with the aggregate (floc) size, although, because the FIvalue depends on the optical properties of flocs, as well as their size,absolute values of floc size cannot usually be derived. The methodshould be regarded as a semi-empirical technique, which can givevery useful comparative information on floc growth and breakage,showing the effects of different additives, mixing conditions and otherfactors.

As an example of the kind of information that can be derived fromthe turbidity fluctuation technique, some recent unpublished resultsare given in Fig. 16. This shows flocculation of clay suspensions byferric sulfate (a common water treatment coagulant). Fig. 16a showsthe effect of adding ferric sulfate (2 mg/L Fe) alone to filtered Londontap water. The pH and alkalinity of this water are such thatprecipitation of hydrous ferric oxide (ferrihydrite) occurs; initially asvery small (a few nm) crystallites, which then aggregate to form fractalstructures [71]. These would have a very low density and so a loweffective refractive index. It can be shown that such aggregates have alow scattering coefficient and hence a very low specific turbidity (seeFig. 6). For this reason they are not easily visible and the FlocculationIndex does not show a large increase.

e with different concentrations of kaolin (values in mg/L shown on curves). See text for


The ferric sulfate was added after 120 s of stirring at 100 rpm. At900 s the stirring rate was increased to 400 rpm for 10 s and thenreduced to 100 rpm. There is a significant lag time (2–3 min) beforethe FI value begins to rise. During this period aggregates are growingbut are too small to be detected by the instrument. The FI reaches alimiting value, corresponding to a maximum floc size. The increasedstirring speed at 900 s causes an immediate and rapid reduction in FI,as a result of floc breakage, due to the greatly increased effective shearrate. On returning to the lower stirring speed, some re-growth of flocsoccurs but not to the previous maximum FI value. This irreversibilityof floc breakage with hydrolyzing coagulants is well-known [72], buthas not been adequately explained.

Fig. 16b shows the results of similar trials, but in the presence ofdifferent concentrations (0–10 mg/L) of clay (SPS kaolin, Imerys, UK).The result for 0 mg/L kaolin is the same curve as in Fig. 16a, but with adifferent scale. It is clear that increasing kaolin concentrations giveprogressively larger FI values, but the pattern of the results isremarkably similar. A simple re-scaling of the FI values causes all ofthe curves to very nearly collapse onto a single curve. The kinetics ofthe process (lag time and time to reach maximum FI) are very similarin all cases. Also floc breakage and recovery show the same relativebehavior, independent of clay concentration.

It seems likely from these results that the different FI values inFig. 16b are not due to different floc sizes, but are a consequence ofdifferent scattering cross-sections of flocs which have about the samesize. Flocs formed with ferric sulfate under these conditions are aresult of process known as “sweep flocculation” in water treatment[73]. The clay particles are effectively enmeshed in the growinghydrous oxide floc. The implication of the results in Fig. 16b is thatferric flocs grow to a certain size, under given shear conditions, whichis not greatly affected by the included clay particles. However, kaolinparticles aremuchmore optically dense (higher scattering coefficient)than the amorphous hydroxide precipitate and so their presencecauses the effective scattering cross-section of the flocs to becomesignificantly larger. (This suggestion is supported by the work of Stoneet al. [74], who concluded that, for aggregates of alumina particlesformed with ferric chloride, the scattering pattern was determinedmainly by the alumina particles rather than the amorphous Fe(OH)3precipitate.) It also appears that the breakage and reformation of theflocs is not greatly affected by the enmeshed clay particles.

It is clear from Eq. (34) that ratio (FI) values from the turbidityfluctuation technique depend directly on the light scattering cross-sections of flocs, which may be much smaller than the projected area,especially for low-density flocs. This is the main reason why absolutefloc sizes are difficult to derive. It should be clear that this difficultywould apply to any light transmission technique, where quite largeaggregates are monitored. Another problem, mentioned earlier(Section 3.1), is that caused by forward-scattered light reaching thedetector. This becomes more important for larger aggregates, wheremost of the scattering may be at quite low forward angles.

5.2. Small-angle light scattering (SALS)

Many investigations of aggregation involve measurement of lightscattered at quite low forward angles to the incident beam. There areseveral possible aims of such studies, including determination of:

• Aggregation rates• Aggregate size distribution• Fractal dimensions of aggregates

Some of the relevant fundamental principles have already beendiscussed and only fairly brief accounts will be given here.

5.2.1. Aggregation rates by SALSAs mentioned earlier (Section 4.2.2) light scattered at sufficiently

small forward angles (low q values) is proportional to the square of

the particle volume, provided that the RGD approximation is valid.This effect is exploited in single particle optical counters, but alsoapplies in the case of conventional light scattering from suspensions.Ofoli and Prieve [22] showed experimentally that the RGD approachwas good for latex particles up to about 1 µm diameter for light from aHe–Ne laser and a scattering angle of 2°. With laser illumination it iseasily possible to measure scattered light at such low angles. Underthese conditions, for monodisperse suspensions, it is possible toderive absolute aggregation rates, at least in the early stages, whereaggregate size distribution does not need to be considered. It can beshown [75] that the scattered light intensity for a given q value, I(q,t)varies with time initially as:

1I q;0ð Þ

dI q; tð Þdt

� �tY0

=2kan0sinqd0qd0

ð35Þ

where I(q,0) is the scattered light intensity for the initial, unaggre-gated suspension, ka is the perikinetic aggregation rate coefficient, Eq.(7), n0 the initial particle concentration and d0 the diameter of theprimary particles.

For low scattering angles, such that qd0bb1, Eq. (35) becomes:

1I0

dIdt

� �tY0

=2kan0 ð36Þ

(Note the similarity to Eq. (29) for the initial rate of turbidity increasefor very small particles. The relative rates of increase are the same inboth cases. However, Eq. (29) only applies to Rayleigh scatterers,whereas Eq. (36) is acceptable for considerably larger particles for lowscattering angles.)

This approach has often been used to determine absolute aggre-gation rate coefficients, mostly for perikinetic aggregation (e.g. [76]).Similar methods have been used study particle aggregation underturbulent conditions [77,78]. Lu et al. [77] suggested an empiricalmethod of extending the SALS method to coarser particles.

5.2.2. Measurement of aggregate sizeMeasurements of aggregate size are mainly carried out with

commercial instruments such as the Malvern Mastersizer (e.g. [79]).These typically have an array of detectors so that light scattered atdifferent angles can be monitored simultaneously. From the angulardistribution of scattered light intensity particle size distributions arederived. For particles larger than around 15 µm, it is possible to useFraunhofer diffraction theory, in which scattering is treated as aproblem in geometric optics. A large spherical particle in a light beamcan be treated as a circular disc (or hole) with the same diameter. Atthe edge of the disc, light is diffracted, giving a characteristic pattern oflight and dark bands far from the particle. These bands correspond tomaxima and minima in the intensity of diffracted light and theirpositions depend only on the particle size and the light wavelength,not on the refractive index of the particles. For large particles and forvisible light, these bands occur at quite lowangles. Nevertheless, usinga laser beam and high-quality optics, including a Fourier transformlens and an array of concentric detectors, it is possible to derivedetailed information [80].

For smaller particles, Fraunhofer diffraction is not an adequatedescription of light scattering and the full Mie theory has to be used.Thismeans that optical properties of the particles, especially refractiveindex, need to be known. Modern instruments incorporate softwarefor carrying out Mie computations, but input of correct refractiveindex data is necessary, otherwise the results may be misleading [81].This makes analysis of suspensions with different types of particlesvery difficult.

Particle sizing equipment is intended for use with solid particles,rather than aggregates, although such instruments arewidely used foraggregate size determination. Rasteiro et al. [79] have recently


measured floc formation, breakage and re-growth using a MalvernMastersizer. They used precipitated calcium carbonate and twocationic polyacrylamides as flocculants and compared floc breakageunder the influence of sonication and hydrodynamic shear. Theysuggested that the floc size derived by SALS is similar to that observedby optical microscopy. Soos et al. [24] also used the Mastersizer toinvestigate the effect of shear rate on aggregate size under turbulentconditions.

However, in other cases, especially for low-density aggregates,there is some doubt about the applicability of Fraunhofer theory [23].An approximate way of predicting optical properties of aggregates[82] is to calculate an effective aggregate refractive index,ma, from therefractive index of the primary particles, m, and their volume fractionwithin the aggregate, ϕS (see Eq. (19)). The Maxwell–Garnett effectivemedium theory [20] leads to the following result:

m2a−1

m2a+2

=�Sm2−1m2+2

ð37Þ

Assuming spherical aggregates, the light scattering properties canthen be calculated fromMie theory. Although this approach is far fromexact, it can give a useful idea of the likely effect of aggregate densityon scattering properties.

For fractal aggregates, especially when the primary particles arevery small, the effective density and hence ϕS can be very low (lessthan 1%) giving correspondingly low effective refractive index. (ma lessthan 1.01), evenwhen the primary particles have a highm value. This,in turn, can give low values of the scattering coefficient (see Fig. 5) andlow values of the effective scattering cross-section. (This is a likelyexplanation of results like those in Fig. 16). It can be shown that forparticles of low refractive index, Fraunhofer diffraction differssignificantly from Mie scattering, even for quite large particles.Fig. 17 shows computed scattering intensities for spheres of 100 µmdiameter and refractive index m=1.01, over a range of angles from0–5°. The light wavelength is 650 nm. Both Mie and diffraction resultsare shown and it is clear that the first fewmaxima andminima occur atsignificantly different angles. Similar computations for m=1.10 andgreater show the first 4 maxima and minima are at very nearly equalangles for both cases.

If these computations are any guide to the behavior of aggregateswith low average refractive index, then it is unlikely that Fraunhoferdiffraction would be a reliable assumption in the derivation ofaggregate size.

5.2.3. Fractal dimensions of aggregatesThe basic ideas concerning light scattering by fractal aggregates

have already been discussed in Section 3.2, as well as a possiblemethod of deriving the fractal dimension. A detailed review of

Fig. 17. Relative scattered light intensity at low forward angles for spheres of 100 µmdiameter and relative refractive index 1.01. Computations based on Mie theory andFraunhofer diffraction.

methods for the measurement of mass fractal dimensions has beengiven by Bushell et al. [83], with emphasis on light scatteringmethods.

Many experimental studies, for a wide range of aggregates, haveshown log I(q) vs. log q plots of the form shown in Fig. 8. There hasbeen a lot of discussion of the range of validity of the RGD theory forfractal aggregates based on theoretical [84] and experimental [10]studies. Generally, it seems that RGD theory can be used without toomuch error for primary particles which are not strictly within the RGDcriteria. For instance,Wang and Sorensen found that optical propertiesof aggregates formed from aerosols of TiO2 nanoparticles were not toodifferent from RGD predictions, despite the high refractive index ofTiO2. This is probably due to a fortuitous canceling of errors in the RGDapproximation [84]. When the RGD assumption is acceptable, theslope of plots like those in Fig. x should give a reasonable estimate ofthe aggregate's fractal dimension. Where comparisons have beenmade, there is fairly good agreement between values obtained in thisway with those derived by other methods and with theoreticalpredictions [85].

Nevertheless there are many practical cases where the primaryparticles are much too large to behave as Rayleigh scatterers andfractal dimensions obtained from the power-law slopes are notreliable [74]. In these cases it is better to refer to the slope as a“scattering exponent”. Liao et al. [86] showed that aggregates of finecoal particles gave scattering exponents somewhat higher than fractaldimensions derived by othermethods, but the trendswere in the rightdirection. Since the coal particles had amean diameter of about 12 µm,the agreement was surprisingly good.

5.3. Dynamic light scattering

The basis of this method is that light scattered from a movingparticle has a slightly different frequency from the incident light (as inthe well-known Döppler effect). In colloidal dispersions, randomBrownian motion of particles causes scattered light to vary randomlyin frequency and interference between light scattered from differentparticles causes random fluctuations in intensity measured by astationary detector. A characteristic speckle pattern is visible when acolloidal dispersion is illuminated by a laser beam. Analysis of thiseffect involves autocorrelation of the scattered light intensity,monitored as a train of pulses from a photomultiplier tube. From theautocorrelation function it is possible to derive the diffusioncoefficient of the particles and hence the effective particle size. Thenature of the technique is such that it may be called photon correlationspectroscopy (PCS) or quasi-elastic light scattering (QELS) as well asdynamic light scattering. Several commercial instruments are avail-able and these are routinely used for the sizing of particles (andpolymer molecules) in the submicron range. Because the effect isdependent on particle diffusion, a practical upper limit of particle sizeis about 3 µm.

Although the restriction to small particles limits the application ofdynamic light scattering in aggregation studies, it has been used todetermine absolute aggregation rates of colloids [87–89]. Generally,the PCS data are used to derive hydrodynamic radii of the growingaggregates and to relate these to aggregate size. Because this is not astraightforward procedure for higher fractal aggregates, it is easier torestrict attention to the earliest stages of aggregation, where onlysinglets and doublets are present. This enables absolute aggregationrate coefficients to be derived and this method has been used to studythe aggregation of positively charged latex particles with anionicpolyelectrolytes [89]. Midmore [90] has discussed methods forderiving moments of aggregate size distributions from the auto-correlation function.

A novel instrument combining both static and dynamic light scat-tering measurements has been described [91]. This allows simulta-neous measurements of light scattered at nine equally-spaced angles,but by changing the direction of the incident laser beam, scattering at


up to 198 angles, between about 4° and 144° can bemeasured. Holthoffet al. [92] have shown that this technique can be used to deriveabsolute coagulation rate coefficients for monodisperse colloids, atleast in the early stages of the aggregation process. The combination ofstatic and dynamic scattering allows the rate coefficients to be deter-mined without explicit knowledge of the light scattering properties ofthe aggregates.

6. Other techniques

In this section, some alternative methods for assessing the state ofaggregation of a dispersion will be briefly discussed.

6.1. Electro-optical effects

Charged particles which have some degree of anisotropy may beorientated in an electric field. Because light scattering by anisotropicparticles usually depends on their alignment in the light beam,changes in particle orientation can give measurable effects onscattered or transmitted light. This is the electro-optic effect [93](sometimes called “electric birefringence” [94]). The effect can bequantified by a parameter αE, which is defined as the relative changein light scattering intensity (measured under given conditions) as aresult of applying an electric field:

αE=IE−II

ð38Þ

where IE and I are the intensities in the presence and absence of thefield.

Rather than deriving information from absolute values of αE, amore common approach is to observe the change in this parameterafter the field is switched on or off. For instance, when the field isswitched off particles adopt random orientations by rotary Brownianmotion and the rate of this relaxation process is highly dependent onparticle size. The rotary diffusion coefficient, DR of a spherical particleis inversely dependent on the cube of particle size and can be derivedfrom the initial slope of the αE vs. time curve after the field is switchedoff. The rate of decay decreases greatly as particle size is increased andthis could, in principle, be used as a measure of aggregation.

There are complications in using the electro-optic effect inaggregation studies, the most important of which is that aggregationnot only gives an increase in size, but may also cause a significantchange in anisotropy. In some cases, particles that are initially highlyanisotropic may become less so on aggregation. An importantexample is kaolinite, which has plate-like particles and shows astrong electro-optic effect. Aggregates of these particles are morespherically symmetric and give much lower αE than the originalplatelets. This effect has been used to monitor the flocculation of clayparticles at quite high concentrations [94]. It has also been used tomonitor the dispersion (deflocculation) of kaolin aggregates by theaddition of sodium polyacrylate [95].

Aggregation of various oxide particles has been studied by electro-optic methods, to give hydrodynamic radii of aggregates and theirfractal properties [96–98].

Although these methods can give useful information on aggregat-ing suspensions, they do not seem to have been adopted byresearchers other than the original investigators.

6.2. Ultrasonic methods

Ultrasound techniques are quite commonly used to characterizecolloidal dispersions [99]. The attenuation of ultrasonic waves as afunction of frequency depends on the concentration, size and densityof suspended particles, among other properties. The method issuitable for concentrated emulsions and suspensions, where opticaltechniques are not applicable because of opacity.

McClements [100] found that the scattering of ultrasound by a 20%oil-in-water emulsion decreased greatly when droplets flocculatedand used this method for examining the effect of surfactantconcentration on emulsion flocculation. The same technique hasbeen used to study depletion flocculation and shear-induced flocdisruption in a 10% corn oil emulsion [101] and flocculation of latexparticles with adsorbed layers [102].

6.3. Sedimentation

Sedimentation of particles depends essentially on their size anddensity and on the viscosity of the fluid. The same is true for particleaggregates, but there are serious complications. For instance,aggregates are usually far from spherical shape and they may haveappreciable permeability. The first of these is usually dealt with byincluding an empirical shape factor [103], but the effect of perme-ability is more difficult to deal with. The density of fractal aggregates isknown to decreasewith increasing size, according to Eqs. (18) and (20)and this effect has a major effect on settling rate.

Sedimentation rates of flocculated suspensions have long beenused to assess the performance of flocculants. For quite concentratedsuspensions, as encountered in mineral processing, it is possible toobserve the interface between the clear supernatant and the settlingparticles. Simple visual observation of the settling rate is a commonand very useful empirical test. Gravimetric and optical (photosedi-mentation) methods are also used [104]. However, it is difficult toderive quantitative information from such measurements for con-centrated suspensions [105,106].

In more fundamental studies, the sedimentation of individualaggregates is observed directly, so that both the size and settling ratemay be determined. With these parameters an effective floc densitymay be derived, from Stokes law, although permeability is acomplicating factor [107]. In the design of settling columns for suchmeasurements, the mode of introducing single flocs and the need toavoid the effects of convection currents have to be considered. A goodaccount of these points has been given by Nobbs et al. [108], who alsogave detailed technical information on the design of a suitable set-up.

Experimental results are usually plotted as log (sedimentationvelocity) vs. log (aggregate size) and the points always show a gooddeal of scatter about a linear regression line. Nevertheless, it is possibleto derive fractal dimensions from the slope, which are usually in theexpected range. Theoretical aspects and a discussion of the effects ofaggregate permeability have recently been given by Tian et al. [12].

As well as sedimentation rate, it is also possible to measure thefinal sediment volume. For flocculated suspensions, this can be muchhigher than for the sediment formed from the original suspension,which can be quite tightly packed. This approach is mostly used in anempirical manner [109], but it is possible to interpret the results interms of floc density and fractal dimension [110]. For lower density,(lower fractal dimension) aggregates the sediment volume becomesgreater. This method has recently been used in a study of depletionflocculation of silica dispersions, to derive information on aggregatemorphology [56].

6.4. Filterability

The flow of liquid through a packed bed of particles depends greatlyon the size of the particles. When a suspension of particles is filtered(e.g. through amembranefilterwithpores smaller than theparticle size)a filter cake is built up which gives an increasing resistance to filtration.For small particles, the permeability is lowand the filtration ratemay bevery slow. It is commonly observed that aggregation of particles cangreatly increase permeability and hence filtration rates. This is whyflocculation is often used with mineral wastes such as ‘red mud’ fromalumina processing and clays produced as a result of phosphate mining.Flocculation of these slurries, usually with high molecular weight


polymers, gives greatly increased permeability and hence higherdewatering rates, which is of enormous practical benefit.

Although some quantitative analysis is possible [111], filterabilitymethods are often used in an empirical manner to evaluate theperformance polymeric flocculants. In early work of La Mer [112], arefiltration rate techniquewas used, inwhich a flocculated suspensionwas filtered through a suitable medium and the filtrate was thenpassed oncemore through the filter cake under constant pressure. Thetime for a given volume to flow was measured, giving the refiltrationrate. This method can give an indication of optimum flocculationconditions, although unflocculated fine particles can have a dispro-portionate effect on the refiltration rate [113].

A convenientmeasureof thefilterabilityof concentrated suspensionsis the capillary suction time (CST), developed in the 1960s by Gale andBaskerville [114]. A commercial version of the CST device is available(Triton Electronics, UK) and is now widely used. Essentially, the CSTdevice measures the time taken for a certain volume of liquid to bedrawn from the suspension by a standard filter paper. The time isdetermined automatically as the advancing liquid front passes betweenelectrical contacts. The CST method is mainly used for empirical studiesof the dewatering of slurries or sludges, particularly in assessingoptimum flocculation conditions. Interpretation of CST data has beendiscussed [115] and the method is still the subject of research [116].

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