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Advances in Colloid and Interface Science j o u r n a l h o m e p a g e : w w w. e l s e v i e r. c o m / l o c a t e / c i s

Monitoring particle aggregation processes John Gregory Department of Civil, Environmental and Geomatic Engineering. University College London, Gower Street, London WC1E 6BT, UK

a r t i c l e

i n f o

Available online 18 September 2008 Keywords: Aggregation Flocculation Fractal dimension Light scattering Monitoring

a b s t r a c t A wide range of test methods for monitoring particle aggregation processes is reviewed. These include techniques for measuring aggregation rates in fundamental studies and those which are useful in the 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 these phenomena are brieﬂy introduced. It is shown that in some cases, absolute aggregation rates can be derived. However, even when only relative rates can be obtained, these can still be very useful, for instance in deﬁning optimum 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 properties such as sedimentation rate and ﬁlterability of ﬂocculated suspensions. © 2008 Elsevier B.V. All rights reserved.

Contents 1. 2.

Introduction . . . . . . . . . . . . . . . . . . . . . Aggregation kinetics . . . . . . . . . . . . . . . . . 2.1. Perikinetic aggregation. . . . . . . . . . . . . 2.2. Orthokinetic aggregation. . . . . . . . . . . . 2.3. Fractal aggregates . . . . . . . . . . . . . . . 3. Light scattering by aggregates. . . . . . . . . . . . . 3.1. Light scattering and turbidity. . . . . . . . . . 3.2. Scattering by fractal aggregates . . . . . . . . 4. Particle counting and sizing. . . . . . . . . . . . . . 4.1. Microscopic methods . . . . . . . . . . . . . 4.2. Sensing zone techniques . . . . . . . . . . . . 4.2.1. Electrozone methods . . . . . . . . . 4.2.2. Optical methods . . . . . . . . . . . 4.3. Focused beam reﬂectance measurement (FBRM) 5. Light scattering methods . . . . . . . . . . . . . . . 5.1. Turbidity methods . . . . . . . . . . . . . . . 5.1.1. Aggregation rate measurement . . . . . 5.1.2. Turbidity-wavelength spectra . . . . . 5.1.3. Turbidity ﬂuctuations . . . . . . . . . 5.2. Small-angle light scattering (SALS) . . . . . . . 5.2.1. Aggregation rates by SALS . . . . . . . 5.2.2. Measurement of aggregate size . . . . 5.2.3. Fractal dimensions of aggregates. . . . 5.3. Dynamic light scattering . . . . . . . . . . . . 6. Other techniques . . . . . . . . . . . . . . . . . . . 6.1. Electro-optical effects . . . . . . . . . . . . . 6.2. Ultrasonic methods . . . . . . . . . . . . . . 6.3. Sedimentation . . . . . . . . . . . . . . . . 6.4. Filterability . . . . . . . . . . . . . . . . . . References . . . . . . . . . . . . . . . . . . . . . . . .

E-mail address: [email protected] 0001-8686/$ – see front matter © 2008 Elsevier B.V. All rights reserved. doi:10.1016/j.cis.2008.09.003

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1. Introduction Aggregation (coagulation/ﬂocculation) processes are in widespread use in many industries, including papermaking, mineral processing, and water and wastewater treatment, among others. In most cases it is very useful to have methods of monitoring the aggregation process. This may be aimed at optimizing the dosage of additives, measuring the rate of aggregation or assessing the properties of the formed aggregates, such as size, density and strength. As well as being of great practical value, these methods are also essential in 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 suspension is assumed to be colloidally stable (usually by virtue of particle charge in the case of aqueous suspensions). In order to promote aggregation the particles need to be destabilized and this can sometimes be achieved simply by a change in pH of the suspension. However in most practical cases certain additives (coagulants or ﬂocculants) are needed. These may be simple inorganic salts, hydrolyzing metal coagulants or polymeric ﬂocculants. At this stage it may be possible to assess the degree of particle destabilization by measuring the particle charge, by one of several electrokinetic techniques, such as particle electrophoresis or streaming current. The Streaming Current Detector (SCD) is quite often used to optimize coagulant dosages in water treatment [1]. Such techniques are only applicable when charge neutralization is the main destabilization mechanism. When particles are adequately destabilized they will aggregate on collision with other particles and collisions can occur through two important mechanisms: • Brownian diffusion (perikinetic aggregation) • Fluid motion (orthokinetic aggregation) Aggregation takes place at a rate that depends on the collision frequency and the collision efﬁciency factor (or its inverse, the stability ratio), which is governed by the degree of particle destabilization achieved. Aggregation rates can be measured in a number of ways. When aggregates (or ﬂocs) are formed, several of their properties can be measured, including size, density, fractal dimension, strength, settling rate and ﬁlterability. After settling, the degree of particle separation can be assessed by measuring the residual turbidity of the supernatant. This is one of the most common test methods in practice (as in the standard Jar Test), since the aim of many industrial ﬂocculation processes is to achieve a high degree of solid–liquid separation.

In this review, we shall focus mainly on methods for measuring rates of aggregation and aggregate properties. Many of the available methods are based on light scattering. For these reasons, the next two sections deal with aggregation kinetics and light scattering by aggregates. 2. Aggregation kinetics Aggregation depends on binary collisions of particles and so is a second-order rate process. (Three-body collisions are usually ignored — they only become signiﬁcant at very high particle concentrations). Most theoretical treatments of aggregation kinetics are based on the early work of Smoluchowski [2] and there have been many more recent treatments (e.g. [3]). The number of collisions, Jij occurring between particles of type i and j in unit time and unit volume is given by: Jij =kij ni nj

ð1Þ

where ni and nj are the number concentrations of i and j particles and kij is a second-order rate coefﬁcient, which depends on a number of factors such as particle size and transport mechanism. In the following, it is convenient to assume that the suspension consists initially of monodisperse primary particles and that the labels i, j, etc. refer to the numbers of primary particles in aggregates. Thus an i-particle is an aggregate of i primary particles. If it is assumed that all collisions lead to aggregation, then it is possible to write an expression for the rate of change of k-fold particles, originally proposed by Smoluchowski: ∞ X X dnk 1 i=k−1 = kij ni nj −nk kik ni dt 2 i+jYk k=1

ð2Þ

i=1

The ﬁrst term on the right hand side represents the rate of formation of k-fold aggregates by collision of any pair of aggregates, i and j, such that i + j = k. Carrying out the summation by this method would mean counting each collision twice and hence the factor 1/2 is included. The second term accounts for the loss of k-fold aggregates by collisions with any other aggregates. The terms kij and kik are the appropriate rate coefﬁcients. It is important to note that Eq. (2) is for irreversible aggregation, since no allowance is made for break-up of aggregates. Also, it has been assumed that every collision is effective in forming an aggregate. If the particles are not fully-destabilized, then only a fraction of collisions are successful. This fraction is the collision efﬁciency, which depends on colloidal interactions between particles and hydrodynamic effects. This aspect will not be considered further here. In the very early stages of aggregation, when most of the particles are still single, only the second term on the r.h.s. of Eq. (2) needs to be considered. In that case, the rate of loss of primary particles is simply: dn1 =−k11 n21 dt tY0

ð3Þ

Each collision leads to the loss of two primary particles and the formation 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 early stages could be written: dnT k11 2 n =−ka n21 =− dt tY0 2 1

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

ð4Þ

This can be regarded as the aggregation rate and ka, the aggregation rate coefﬁcient, is just half the collision rate coefﬁcient for primary particles. The rate coefﬁcients depend greatly on the collision mechanism and there are two cases of practical importance.

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

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

2.1. Perikinetic aggregation All suspended particles undergo Brownian motion to some extent and this can cause particle collisions and aggregation. The Smoluchowski result for the perikinetic collision rate coefﬁcient for unequal spheres is:

kij =

2kB T ai +aj 3η ai aj

2 ð5Þ

where kB is Boltzmann's constant, T the absolute temperature, η the viscosity of the ﬂuid and ai, aj are the radii of the colliding particles. 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: 8kB T kij ≈ 3η

ð6Þ

(For aqueous suspensions at 25 °C, the coefﬁcient is about 1.23 × 10− 17 m3s− 1.) By assuming a constant collision rate coefﬁcient, Smoluchowski showed that the previous expression for the early stages of aggregation, Eq. (4), would apply throughout the aggregation process. Thus: dnT =−ka n2T dt

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 n0 = 1+ka n0 t 1+t=ta

ð8Þ

where n0 is the initial concentration of (primary) particles and ta is the aggregation time, in which the number of particles is reduced to one half 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, ¯¯, i.e. the average number of particles per aggregate. k k=1+ka n0 t=1+t=ta

111

ð9Þ

The above expressions are based on the assumption of spherical particles and aggregates. For hard particles, spherical aggregates are unlikely, but, for the perikinetic case, this assumption does not lead to large 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 so signiﬁcant.

dnT 16 =− n2T Ga31 3 dt

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 simple transformation is possible since the volume fraction, ϕ, of particles in the suspension is: =4πa31 nT =3

ð12Þ

This allows Eq. (11) to be written in a pseudo ﬁrst-order form: dnT 4GnT =− dt π

ð13Þ

If the volume fraction of particles is assumed to remain constant during aggregation then Eq. (13) can be integrated to give: nT −4Gt = exp π n0

ð14Þ

The assumption of constant volume fraction is questionable for fractal aggregates, since their effective volume grows with increasing aggregate size. This means that Eq. (13) will likely give an underestimate of the actual aggregation rate. However, hydrodynamic interactions become more signiﬁcant for larger particles [5] and can reduce the capture efﬁciency of aggregates [6]. This makes quantitative modeling of the orthokinetic aggregation process quite difﬁcult. The most important implication of Eq. (14) is that the total particle concentration should decrease exponentially with time and hence the mean aggregation number should show exponential growth. This is in marked contrast to the perikinetic case, where aggregates should grow in a linear manner with time. This is illustrated in Fig. 2, where the decrease in particle number with time is shown for both cases. The calculations are for an aqueous suspension containing 1 µm diameter particles at a concentration of 1015 m− 3 and a temperature of 25 °C. This gives a characteristic perikinetic aggregation time ta = 163 s, and a volume fraction ϕ = 5.24 × 10− 4. For the orthokinetic case, the shear rate is chosen as the very low value of G = 6 s− 1, which give about the same initial rate of aggregation (τ = 173 s). Note that these calculations are for the individual collision mechanisms acting separately. Of course, in reality, Brownian diffusion cannot be ‘switched off’ and the two effects can be treated as approximately additive.

2.2. Orthokinetic aggregation The simplest case to consider is that of spherical particles in a uniform, laminar shear ﬁeld. Collisions of particles are brought about because of their different velocities at different layers of the shear ﬁeld. For a shear rate (velocity gradient) G, Smoluchowski showed that the collision rate coefﬁcient for unequal particles is: 3 4 kij = G ai +aj 3

ð10Þ

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

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

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From Fig. 2 it is clear that, for the ﬁrst few minutes, aggregation occurs at about the same rate for both cases. However, at longer times, the orthokinetic rate becomes relatively much faster. These results are based on rather unrealistic assumptions, especially for the orthokinetic case. Nevertheless they are sufﬁcient to show the enormously different rates at long aggregation times and to explain why perikinetic aggregation alone is rarely sufﬁcient to form very large aggregates. Because orthokinetic aggregation occurs by some form of ﬂuid shear (e.g. in a stirred vessel), aggregate strength and breakage also have to be considered. In the original Smoluchowski treatment aggregation was assumed to be irreversible, with no breakage. In practice, aggregates are often found to reach a limiting size under given shear conditions and this is generally assumed to be a result of a balance between aggregate growth and breakage [7]. However, a diminishing collision efﬁciency with increasing aggregate size can also limit aggregate growth [6]. Although a quantitative description of aggregate breakage is difﬁcult [8], a simple empirical expression for limiting aggregate size, dmax, in terms of the applied shear rate, G, is often used: dmax =KG−γ

ð15Þ

where K and γ are empirical constants. The exponent γ depends on the mode of aggregate breakage, but values in the region of γ = 0.5 are often found. 2.3. Fractal aggregates In most cases, aggregates are known to be fractal, scale invariant objects [9], which have a rather open structure. For aggregates the mass fractal dimension is an important parameter. This relates the mass of an aggregate with a convenient linear dimension, such as diameter or radius of gyration. Generally, the mass M and length L are related by: M~LD

ð16Þ

where D is the mass fractal dimension, which, in principle can vary between 1 and 3 for objects in 3-dimensional space. Typical aggregates have values of D between about 1.7 and 2.5, depending on several factors. Generally, diffusion-limited (perikinetic) aggregates have fractal dimensions at the lower end of this range, whereas orthokinetic aggregation gives somewhat higher values (i.e. more compact aggregates). For an aggregate of identical primary particles, the mass is directly proportional to the aggregation number. If an aggregate contains i primary particles, radius a1, then the following expression applies: D i=B Rg =a1

It is easy to show that the exponent y in Eq. (18) is given by: ð20Þ

y=3−D

For solid, non-fractal objects (D = 3), the density is independent of size, but for low fractal dimensions there can be a marked decrease in density as aggregates grow larger. Large aggregates can have very low density and his has signiﬁcant consequences for solid–liquid separation [11]. Also, by measuring aggregate density over a range of aggregate size, it should be possible to derive the fractal dimension [12]. The fractal nature of aggregates has a great effect on their light scattering 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 scattering is a very well studied phenomenon, with an extensive literature [13]. A basic light scattering set-up is shown schematically in Fig. 3. Here, a light beam illuminates a suspension of particles in a suitable cell and the scattered light intensity can be measured by a detector placed at any desired angle to the incident beam. By placing a detector directly in line with the incident beam it is also possible to measure the transmitted light intensity and hence the turbidity of the suspension. Both of these approaches have been used to derive information on aggregate properties. A general theory of light scattering has been available since the early 20th Century. This is usually attributed to Mie, although, as Kerker [14] has pointed out, there were many other important contributions to the theory. Without going into detail, the intensity of scattered 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 a single dimensionless parameter, α = 2πa / λ. In general light may be absorbed as well as scattered by particles, in which case the refractive index takes complex values. For simplicity, we shall assume that absorption is negligible, so that m is always real. In that case scattering occurs with no net loss of energy from the light beam (elastic scattering). Fig. 4 shows scattered light intensity against scattering angle for spherical particles with a range of α values from 1–50 (corresponding to particle diameters in the region of about 0.2–10 µm for light in the

ð17Þ

where Rg is the radius of gyration of the aggregate and B is a constant of 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 practical implications. For instance, aggregate density decreases with aggregate size according to: ρE ~d−y

ð18Þ

where ρE is the effective (or buoyant) density of an aggregate, with diameter d. This depends of the densities of the particles and ﬂuid and on the solid volume fraction within the aggregate, ϕS: ρE =ρA −ρL =S ðρS −ρL Þ

ð19Þ

where ρA, ρL and ρS are the densities of the aggregate, liquid and particles, respectively.

Fig. 3. Measurement of transmitted and scattered light.

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

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

visible wavelength range). The relative refractive index was assumed to be m = 1.20, which is about the value for polystyrene latex particles in water. The results shown are for unpolarized light and the computations were carried out using Mieplot v3418 (www.philiplaven.com). For smaller particles (α b 1), light scattering behavior approaches that predicted by Rayleigh theory, where the scattering intensity is proportional to the 6th power of particle size and has little dependence on scattering angle. There are two major effects of increasing particle size: • A very large increase in scattered light intensity (over many orders of magnitude) • 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 light energy scattered by a particle is effectively that within an area of the incident light beam, C. This is related to the geometric cross-section of the particle, through the scattering coefﬁcient, Q. Thus, for a spherical particle: C=Qπa2

ð21Þ

The scattering coefﬁcient varies with the size parameter α, as shown in Fig. 5, for different values of refractive index. Low m values are included, since these may be relevant to aggregates (see below). Typically, Q rises from very low values and passes through a series of maxima and minima, before eventually reaching a constant value Q = 2. However, for low m, the ﬁrst maximum is not reached until the particle size is quite large. When light transmitted through a suspension is measured (as shown in Fig. 3) the turbidity can be deﬁned in terms of the path length, L, the particle number concentration, n, and the scattering cross-section of the particles. (A monodisperse suspension is assumed). If the incident light intensity is I0, then the transmitted light intensity is given by: I=I0 expð−τLÞ

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

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 3Q = 4a

ð24Þ

For very large particles, where Q becomes almost constant, this shows that the speciﬁc turbidity should be inversely proportional to particle size. However, for smaller particles the turbidity shows more complex behavior. The curves in Fig. 6 show speciﬁc turbidity as a function of particle diameter, for the same refractive index values as in Fig. 5. In this case, since the actual size is plotted, we need to specify the light wavelength and a value of 650 nm is chosen. (This is the value in the suspension. For aqueous suspensions this would correspond to a vacuum wavelength of about 865 nm.) For very small particles, the turbidity is low, but it increases markedly as particle size increases. This effect is most pronounced with higher m values, where the ﬁrst turbidity maximum is quite sharp. For lower refractive index the turbidity rises less steeply and the maxima occur at progressively larger particle sizes. Particle sizing by light scattering, including Fraunhofer diffraction, can be signiﬁcantly affected by the assumed refractive index of particles [15]. From the curves in Fig. 6 it is clear that a single turbidity measurement, for particles of known refractive index, would not give an unambiguous value of particle size. However, measurements at two

ð22Þ

where τ is the turbidity, given by: τ=nC

ð23Þ

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

113

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

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or more wavelengths can, in principle, be used to derive particle size [14]. A signiﬁcant problem with turbidity methods, especially for larger particles and aggregates, is that, in any practical method, some forward-scattered light must reach the detector (see Fig. 7), thus increasing the apparent transmitted light intensity and decreasing the apparent scattering coefﬁcient. This effect has been know for a long time [16] and can have major effects on the apparent turbidity of aggregating suspensions [17]. Typical acceptance angles in commercial spectrophotometers are in the region of a few degrees, but with careful design it is possible to reduce this to close to zero [18]. More recently, this problem has been discussed in some detail [19]. 3.2. Scattering by fractal aggregates The problem of light scattering by aggregates is much more difﬁcult than for solid spheres, although some simpliﬁcations may be possible in certain cases. The subject has been comprehensively reviewed by Sorensen [20]. The simplest approach is to assume that the primary particles in an aggregate are small enough to behave as Rayleigh scatterers and to use the Rayleigh-Gans-Debye (RGD) approximation (see e.g. Schmidt [21]). In this case the scattering by an aggregate of size L depends on the quantity qL, where q is the magnitude of the scattering vector, given, for a scattering angle θ and a wavelength λ, by: q=

4π θ sin λ 2

ð25Þ

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

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

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

be shown that the scattered light intensity depends on the fractal dimension of the aggregates, according to: IðqÞ~q−D

ð26Þ

So, in principle, the fractal dimension for aggregates can be derived directly 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 RGD approximation (small primary particles of fairly low refractive index), although light scattering methods have been used to derive information on the fractal properties of aggregates of larger particles [23]. In such cases, it is better to refer to the slope of the linear region in Fig. 8 as a “scattering exponent” or “scaling exponent” [24]. We now review some methods for monitoring aggregation processes. 4. Particle counting and sizing The oldest method of following particle aggregation is by counting the number of particles (and aggregates) at intervals during the process. This may be done by direct microscopic observation or by an automated counting method. The latter involves passing the suspension through some kind of sensing zone where individual particles can be counted (and sized in most cases). Where sizing is possible, a size distribution can be derived without any a priori assumptions about the form of the distribution. 4.1. Microscopic methods Many early measurements of aggregation rates and stability ratios of colloidal dispersions were made by direct observation of particles by optical microscopes (e.g. Freundlich [25]). Such methods have been widely used over many decades, for instance in the study of particle aggregation in natural waters [26]. Samples of the suspension are withdrawn at intervals, diluted if necessary, and microscopically examined. The number of particles in a given volume can then be determined simply by counting. Aggregation of large latex particles in narrow tubes has been directly observed by optical microscopy [27]. Ordinary optical microscopes cannot easily resolve particles less than about 1 µm in size, although the ultramicroscope, using dark ﬁeld illumination, allows much smaller particles to be detected (down to around 5 nm diameter for gold sols [28]). It is not generally possible to determine the size of aggregates by optical microscopy, unless they grow quite large and the method is normally 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). When aggregates grow quite large (greater than about 15 µm) it is possible to

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apply image analysis to digitized optical micrographs. This method has been used to derive information on the size shape and fractal properties of ﬂocs produced in orthokinetic aggregation [29–31]. The dependence of perceived aggregate size from image analysis on pixel resolution has recently been discussed [32]. Transmission electron microscopy provides much higher resolution than the light microscope, but the sample preparation, including a 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. The freeze-fracture technique has proved useful in this respect [33]. 4.2. Sensing zone techniques Automated particle counting can be achieved by allowing particles and aggregates to pass singly through a zone in which their presence can be detected by a suitable sensor. Particles passing through this zone give a series of pulses, which can be counted. Such a device is shown schematically in Fig. 9. If the sensor response depends on particle size, then the pulse height is size-dependent and can be used to discriminate between particles of different size. These methods depend on there being only one particle in the sensing zone at a time. Otherwise two or more particles would give just one pulse and appear as one larger particle. This is known as the coincidence effect and can be avoided by adequate dilution of the suspension or by making the sensing zone sufﬁciently small. Since the pulse count per unit time is measured, the ﬂow rate must be known and kept constant. There are two commonly used sensing methods in particle counting: • Electrical (or Electrozone) • Optical (light scattering) These have their own advantages and disadvantages. 4.2.1. Electrozone methods Electrical sensing zone (Electrozone) counters are based on a principle developed by Coulter in the 1940s and commercialized as the Coulter Counter [34]. Particles in an electrolyte solution pass through an oriﬁce and cause a momentary change in electrical resistance and hence a voltage pulse if the current is kept constant. Electrodes are located on either side of the oriﬁce and a particle passing through displaces a volume of electrolyte equal to the particle

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

115

volume. Most particles have effectively inﬁnite resistance and so the voltage pulse is proportional to the particle volume. It is capable of quite high speed counting (5000 or more particles per second) and can resolve particles only slightly different in size. A unique feature of the electrozone method, in comparison with other (especially light scattering) techniques, is that it is virtually independent of the shape or composition of the particles. The Coulter technique has been used very widely for particle size analysis and is the subject of several reviews (e.g. Lines [35]). By around 1990 there were already many thousands of references to uses of the technique in a large range of applications. Use of the electrozone method for particle aggregation studies has been more limited. Among the ﬁrst to use this technique were Matthews and Rhodes [36] and there have been several later examples, including ﬂocculation of bacterial suspensions [37] and latex [38–40]. In principle, the aggregate size reported by the Coulter technique should be just that corresponding to the volume of the primary 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 aggregate size seen by optical microscopy [37]. This could be a means of determining the effective aggregate density, and hence fractal dimension [41,42]. A possible difﬁculty with the electrozone technique is the break-up of aggregates as they pass through the oriﬁce, where the shear rate can be very high. However, the fragments should have the same total volume as the original aggregate, so that breakage within the oriﬁce might still produce only one pulse, corresponding to the aggregate volume. Breakage occurring in the elongational ﬂow ﬁeld just upstream of the oriﬁce could give several smaller pulses. Another problem is the need for dilution of the suspension with an electrolyte solution (typically around 2% NaCl), which could cause colloid destabilization. Because of the rather low particle concentration in the diluted suspension (usually less than 106 particles/mL to avoid coincidence effects) the aggregation rate would be very slow and in the short time needed for a measurement it should not be a signiﬁcant effect. However, dilution of a weakly-aggregated suspension may cause some disaggregation. The electrozone technique cannot be used over a wide range of aggregate size, which should be between about 2% and 40% of the oriﬁce diameter. A practical lower size limit is about 1 µm, which is a serious limitation for colloids. For large aggregates, oriﬁce blockage becomes a major problem. It is possible to use different oriﬁce sizes to cover a wider size range, but this is rather inconvenient. 4.2.2. Optical methods In this case, particles are made to pass singly through a focused light beam (usually a laser beam) and either the transmitted or scattered light intensity is monitored. The transmitted light method (light blockage) is not as sensitive as light scattering, although it forms the basis of several commercial particle counters. Particle counting and sizing by scattered light measurement is often referred to as ﬂow ultramicroscopy. For aggregation studies small-angle light scattering (SALS) is most appropriate, although some older instruments used scattering at around 90° to the incident beam [43]. At very small scattering angles (low q values) the scattered light intensity varies as the square of the aggregate volume (or aggregation number, k), at least under conditions where the RGD approximation is acceptable (see Section 3.2). There have been several single particle optical counting and sizing instruments developed and the subject has been reviewed by Lichtenfeld et al. [44]. Features include feedback control of the incident light intensity to enable a wide range of aggregate size to be measured without exceeding the dynamic range of the photomultiplier [45]. Also, hydrodynamic focusing can be used to reduce the size of the sample volume and hence minimize the coincidence effect [46].

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Most applications of these techniques have been with monodisperse suspensions, such as polystyrene latex [45,47] and silica [48] undergoing perikinetic aggregation. In such cases it is possible to discriminate between aggregates up to about 7-fold. These measurements allow experimental tests of Smoluchowski kinetics, at least in the early stages of aggregation [48] and can give useful information on aggregation by polymers [47]. Hydrodynamic aspects can be quite important, either because of orthokinetic aggregation or the break-up of aggregates as a result of ﬂow [49]. In a commercial light blockage particle counter (Hiac), it has been shown that aggregate breakage can be a signiﬁcant problem [50].

scattering methods where there may be many particles or aggregates in the light beam and information on average properties is derived. Such methods are widely used, but cannot give such detailed information on particle or aggregate size distribution as single particle counting. Available techniques involve measurement of transmitted light (turbidity), or of light scattered at one or more angles to the incident beam (static light scattering). Another method is dynamic light scattering, which derives information from the diffusion of particles.

4.3. Focused beam reﬂectance measurement (FBRM)

5.1.1. Aggregation rate measurement For particles smaller than the light wavelength, turbidity increases with particle size (see Fig. 6) and also increases as small particles aggregate. This has often been used as a semi-empirical method of following aggregation, for instance to determine relative coagulation rates and stability ratios [55]. It is usually assumed that the initial rate of increase in turbidity is proportional to the initial aggregation rate. For perikinetic aggregation, the stability ratio, W, is deﬁned as the ratio of the rate of aggregation of the fully-destabilized suspension (i.e. the diffusion-controlled case) to that of the partially-destabilized suspension (where there is some repulsion between the particles and not every collision is effective). This is just the same as the ratio of the appropriate rate coefﬁcients, ka, in Eq. (7). If we assume that the rate of aggregation is proportional to the corresponding rate of change of turbidity, then it follows that the stability ratio is given by:

This is a more recent method of counting and sizing particles and aggregates and is available in commercial versions (Gallai, Lasentec). In the FBRM technique, a tightly-focused laser beam is projected from a probe into the suspension through a window, to form a small spot close to the window. The spot is caused to rotate at high speed (about 2 m/s). Particles close to the window will be scanned by the spot and light is reﬂected back to the probe. The reﬂected pulse is of a certain duration, which depends on the scanning speed and on the appropriate chord length (see Fig. 10). Thousands of chords can be measured per second, giving a chord length distribution from below 1 µm to more than 1 mm [51]. The method works well with much higher suspension concentrations (up to 20%) and much larger ﬂocs than are possible with electrozone or optical counters. In most cases, no dilution is needed, which is a great advantage. The suspension must be agitated in some way, so that the scanned sample is renewed frequently. This means that the method is only suitable for studies of orthokinetic aggregation. Another beneﬁt of the FBRM method is that aggregating suspensions can be monitored in situ (e.g. in a stirred vessel). The relationship between chord length and particle size is not straightforward. This problem has been examined in some detail by Heath et al. [52], who found that an empirical method was more successful than a theoretical approach. The FBRM technique has been used to follow aggregation and aggregate breakage in latex suspensions in Couette ﬂow [53]. This study clearly showed the expected dependence of maximum aggregate size on shear rate (see Eq. (15)) and gave useful information on the effect of particle concentration. There have also been more practical applications, such as in mineral processing [51] and papermaking [54]. 5. Light scattering methods Previous sections have dealt with optical techniques for the counting and sizing of individual particles. We now turn to light

5.1. Turbidity methods

W=

ka;max ðdτ=dt Þ0;max = ka ðdτ=dt Þ0

where the numerators are for a fully-destabilized sol and the denominators are for some other experimental condition. Both rates are 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 various additives and under different conditions. However, there have been many attempts to derive absolute rates of aggregation from turbidity measurements. Even for spherical, monodisperse primary particles, we need information on scattering cross-sections of aggregates. The turbidity of an aggregating suspension, by analogy with Eq. (13), should be given by: τ=n1 C1 +n2 C2 +n3 C3 +::::

ð28Þ

where n1, C1, etc. are the number concentrations and scattering crosssections of singlets, doublets and triplets, etc. after a certain period of aggregation. The concentrations of the various aggregates can be easily derived from Smoluchowski theory, but the scattering coefﬁcients are much more difﬁcult to calculate. The simplest case is when the primary particles are much smaller than the light wavelength and Rayleigh theory applies. The scattering cross-section is then proportional to the square of the particle volume, so that C2 = 4C1. By considering only the initial stage of aggregation where only singlets and doublets are present and assuming that the doublets also act as Rayleigh scatterers, it can be shown that: 1 dτ 2 = =2ka n0 τ0 dt 0 ta

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

ð27Þ

ð29Þ

where τ0 is the initial turbidity, ta is the aggregation time (Eq. (8)), ka is the aggregation rate coefﬁcient and n0 is the initial concentration of primary particles. According to this simple analysis, the turbidity of a sol should initially increase linearly with time and the aggregation rate coefﬁcient

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could be derived from the initial slope. It has been implicitly assumed that the doublets are spherical particles, so that coalescence occurs on aggregation. This is an unrealistic assumption for hard particles, where real doublets would be in the form of dumbbells. A more realistic expression is [57]: 1 dτ =2ka ðC2 =2C1 −1Þn0 τ0 dt 0

ð30Þ

If C2 = 4C1, as in the ‘coalescence’ assumption, then Eq. (30) is equivalent to Eq. (29). Otherwise the scattering cross-section of ‘real’ doublets needs to be considered. The usual approach has been to use RGD theory, which requires the particle size and the relative refractive index to be quite small. For doublets of equal spheres, application of RGD theory is straightforward, since there is no doubt about the shape. For higher aggregates, this is not the case. Lichtenbelt et al. [57] computed the optical correction factor (the bracketed term on the r.h.s. of Eq. (30)) for ‘real’ doublets over a range of values of the size parameter α (=2πa / λ) from RGD theory. Their results are plotted in Fig. 11, along with computations from Mie theory for polystyrene latex particles, assuming that the doublet coalesces to form a sphere of the same total volume. It is clear that for α values greater than about 0.3 the RGD results give a much smaller factor. The coalescence assumption overestimates the rate of turbidity increase and hence would give aggregation rate coefﬁcients that were too low. An improved method of calculating the scattering cross-section of doublets is the T-matrix procedure [58], which is not subject to the limitations of the RGD approach. Results from this method, for polystyrene latex are also shown in Fig. 11 and it appears that there is reasonable agreement with the RGD results for fairly small α values. For larger particles the agreement is less good and for α values greater than about 6, the optical factor becomes negative, indicating that the turbidity would decrease on aggregation. It is experimentally observed that latex particles of around 1 µm diameter or larger, with visible light, do show a decreased turbidity as aggregation occurs. It is doubtful whether a simple turbidity approach would be suitable for aggregation rate measurements in such cases. 5.1.2. Turbidity-wavelength spectra A 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 especially useful in cases of weak ﬂocculation, which may be inﬂuenced by the suspension concentration or temperature.

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

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

The basis of the technique is that the speciﬁc turbidity of a suspension depends on the light wavelength. This follows from Eq. (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

ð31Þ

where 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 τ against 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 of aggregation changes, then a sudden change in n may be observed, as shown schematically in Fig. 12. In this case n is plotted against the sol concentration and there is a clear break at a certain concentration. This is characteristic of equilibrium ﬂocculation of a stericallystabilized suspension, where there is a shallow minimum in the interaction energy curve. Reversible ﬂocculation occurs at some critical particle concentration, which is marked by the break in the n vs. ϕ plot. In other cases a sudden change of n may occur at a certain temperature [60]. A decrease in n indicates an increase in particle size and hence aggregation. 5.1.3. Turbidity ﬂuctuations If a small volume of a colloidal suspension is observed microscopically, then Brownian diffusion causes particles to move in and out

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

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of the observed volume, so that the total number of particles viewed varies in a random manner. It can be shown that these variations follow the Poisson distribution: P ðnÞ=

expð− Þ n n!

ð31Þ

where ν is the mean number of particles in the observed volume and P(n) is the probability of ﬁnding n particles in the volume. Svedberg [61] observed a gold sol by ultramicroscope and counted the number of particles in a deﬁned volume at intervals over several minutes. In 518 observations he counted a total of 801 particles, so that the mean was about 1.55. He counted the number of observations with 0, 1, 2, 3 etc particles and the results are shown in Fig. 13, together with the corresponding counts expected from the Poisson distribution. It is clear that the agreement is excellent. A very important consequence of the Poisson distribution is that the variance is equal to the mean and so the standard deviation is the square root of the mean. From Svedberg's data the variance is 1.53, very close to the actual mean (1.55). Exactly the same distribution would be found if particles were counted in a series of equal small volumes withdrawn from a suspension. In fact, non-Brownian particles would also show the same distribution, since, however well mixed, a suspension always shows non-uniformity on a microscopic scale and Poisson statistics apply in this case also. Instead of taking a sequence of samples we could simply cause a suspension to ﬂow through a suitable device in which particle number variation could be monitored. This is the basis of 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 a transparent plastic tube) and is illuminated by a narrow light beam. The transmitted light intensity is measured by a photodiode and shows random variations about a mean value (see Fig. 14). The mean transmitted light intensity can be used to calculate the turbidity of the suspension, from Eq. (22). The ﬂuctuations arise from random variations in number (and size) of particles in the light beam. The photodiode output would typically show a mean (DC) value of several volts and the ﬂuctuations (AC) might be of the order of only a few mV. The normal procedure is to derive the root mean square (RMS) value of the voltage ﬂuctuations and to divide this by the mean (DC) value, to

give a Ratio value R. Unlike single particle counting methods, the turbidity ﬂuctuation technique is valid when there are large numbers of particles in the light beam and works well up to quite high concentrations. In practice, suspension concentration is limited by turbidity — there must be a measurable transmitted light intensity. The lower concentration limit is determined by random electronic noise in the equipment, which means that measurable R values must typically be greater than about 3 × 10− 5. For larger particles, this may correspond, on average, to less than one particle in the light beam, but Poisson statistics still apply in this case. It can be shown that, for a monodisperse suspension, the Ratio is given by: R=

rﬃﬃﬃﬃﬃ VRMS nL = C A VDC

ð32Þ

where n is the average number of particles per unit volume, L is the optical path length, A the effective cross-sectional area of the light beam and C is the scattering cross-section of the particles. The dependence of R on the square root of particle concentration is a consequence of the Poisson distribution for the number ﬂuctuations. The RMS value of the ﬂuctuating signal is analogous to the standard deviation, which varies as the square root of concentration. Since the turbidity depends directly on the particle concentration, it is possible to combine Eqs. (23) and (32) to give an expression for the particle concentration: τ 2 L n= R A

ð33Þ

The scattering cross-section does not appear in Eq. (32), so that the particle number concentration could be derived without any knowledge of the optical properties of the particles. However, this only applies to a monodisperse suspension. For a heterodisperse suspension, the equivalent expression to Eq. (32) is: 1=2 X 1=2 L R= ni Ci2 A

ð34Þ

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

Fig. 14. Schematic illustration of the ‘turbidity ﬂuctuation’ method.

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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 inﬂuenced by larger particles (with higher scattering cross-sections). It is not possible to derive a simple expression for particle concentration in this case, but it can be shown [63] that, even for rather broad particle size distributions, the total number concentration derived from Eq. (33) does not differ from the true value by more than a factor of about 2. This suggests that, for an aggregating suspension a similar approach 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. A commercial version of this technique (Photometric Dispersion Analyzer, PDA 2000, Rank Brothers, UK) has been available for some time and the method has been widely used to monitor ﬂocculation processes in a variety of applications. These include papermaking systems [65], asphaltene aggregation [66], ﬂocculation of bacteria [67], water [68] and wastewater [69] treatment and ﬂoc strength investigations [70]. For laboratory ﬂocculation studies, a set-up like that in Fig. 15 may be used. Suspension from a stirred vessel is circulated continuously through the monitor and the outputs (usually DC and RMS values) are

recorded. In such applications, the Ratio value, R, is often called the Flocculation Index (FI). For a given system, this value is strongly correlated with the aggregate (ﬂoc) size, although, because the FI value depends on the optical properties of ﬂocs, as well as their size, absolute values of ﬂoc size cannot usually be derived. The method should be regarded as a semi-empirical technique, which can give very useful comparative information on ﬂoc growth and breakage, showing the effects of different additives, mixing conditions and other factors. As an example of the kind of information that can be derived from the turbidity ﬂuctuation technique, some recent unpublished results are given in Fig. 16. This shows ﬂocculation of clay suspensions by ferric sulfate (a common water treatment coagulant). Fig. 16a shows the effect of adding ferric sulfate (2 mg/L Fe) alone to ﬁltered London tap water. The pH and alkalinity of this water are such that precipitation of hydrous ferric oxide (ferrihydrite) occurs; initially as very small (a few nm) crystallites, which then aggregate to form fractal structures [71]. These would have a very low density and so a low effective refractive index. It can be shown that such aggregates have a low scattering coefﬁcient and hence a very low speciﬁc turbidity (see Fig. 6). For this reason they are not easily visible and the Flocculation Index does not show a large increase.

Fig. 16. Change of Flocculation Index with time for (a) ferric sulfate alone and (b) ferric sulfate with different concentrations of kaolin (values in mg/L shown on curves). See text for details.

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The ferric sulfate was added after 120 s of stirring at 100 rpm. At 900 s the stirring rate was increased to 400 rpm for 10 s and then reduced to 100 rpm. There is a signiﬁcant lag time (2–3 min) before the FI value begins to rise. During this period aggregates are growing but are too small to be detected by the instrument. The FI reaches a limiting value, corresponding to a maximum ﬂoc size. The increased stirring speed at 900 s causes an immediate and rapid reduction in FI, as a result of ﬂoc breakage, due to the greatly increased effective shear rate. On returning to the lower stirring speed, some re-growth of ﬂocs occurs but not to the previous maximum FI value. This irreversibility of ﬂoc breakage with hydrolyzing coagulants is well-known [72], but has not been adequately explained. Fig. 16b shows the results of similar trials, but in the presence of different 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 a different scale. It is clear that increasing kaolin concentrations give progressively larger FI values, but the pattern of the results is remarkably similar. A simple re-scaling of the FI values causes all of the curves to very nearly collapse onto a single curve. The kinetics of the process (lag time and time to reach maximum FI) are very similar in all cases. Also ﬂoc breakage and recovery show the same relative behavior, independent of clay concentration. It seems likely from these results that the different FI values in Fig. 16b are not due to different ﬂoc sizes, but are a consequence of different scattering cross-sections of ﬂocs which have about the same size. Flocs formed with ferric sulfate under these conditions are a result of process known as “sweep ﬂocculation” in water treatment [73]. The clay particles are effectively enmeshed in the growing hydrous oxide ﬂoc. The implication of the results in Fig. 16b is that ferric ﬂocs grow to a certain size, under given shear conditions, which is not greatly affected by the included clay particles. However, kaolin particles are much more optically dense (higher scattering coefﬁcient) than the amorphous hydroxide precipitate and so their presence causes the effective scattering cross-section of the ﬂocs to become signiﬁcantly larger. (This suggestion is supported by the work of Stone et al. [74], who concluded that, for aggregates of alumina particles formed with ferric chloride, the scattering pattern was determined mainly by the alumina particles rather than the amorphous Fe(OH)3 precipitate.) It also appears that the breakage and reformation of the ﬂocs is not greatly affected by the enmeshed clay particles. It is clear from Eq. (34) that ratio (FI) values from the turbidity ﬂuctuation technique depend directly on the light scattering crosssections of ﬂocs, which may be much smaller than the projected area, especially for low-density ﬂocs. This is the main reason why absolute ﬂoc sizes are difﬁcult to derive. It should be clear that this difﬁculty would apply to any light transmission technique, where quite large aggregates are monitored. Another problem, mentioned earlier (Section 3.1), is that caused by forward-scattered light reaching the detector. This becomes more important for larger aggregates, where most of the scattering may be at quite low forward angles. 5.2. Small-angle light scattering (SALS) Many investigations of aggregation involve measurement of light scattered at quite low forward angles to the incident beam. There are several 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 been discussed and only fairly brief accounts will be given here. 5.2.1. Aggregation rates by SALS As 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 also applies in the case of conventional light scattering from suspensions. Ofoli and Prieve [22] showed experimentally that the RGD approach was good for latex particles up to about 1 µm diameter for light from a He–Ne laser and a scattering angle of 2°. With laser illumination it is easily possible to measure scattered light at such low angles. Under these conditions, for monodisperse suspensions, it is possible to derive absolute aggregation rates, at least in the early stages, where aggregate size distribution does not need to be considered. It can be shown [75] that the scattered light intensity for a given q value, I(q,t) varies with time initially as:

1 dI ðq; t Þ sinqd0 =2ka n0 Iðq; 0Þ dt qd0 tY0

ð35Þ

where I(q,0) is the scattered light intensity for the initial, unaggregated suspension, ka is the perikinetic aggregation rate coefﬁcient, Eq. (7), n0 the initial particle concentration and d0 the diameter of the primary particles. For low scattering angles, such that qd0 bb1, Eq. (35) becomes:

1 dI I0 dt

=2ka n0

ð36Þ

tY0

(Note the similarity to Eq. (29) for the initial rate of turbidity increase for very small particles. The relative rates of increase are the same in both cases. However, Eq. (29) only applies to Rayleigh scatterers, whereas Eq. (36) is acceptable for considerably larger particles for low scattering angles.) This approach has often been used to determine absolute aggregation rate coefﬁcients, mostly for perikinetic aggregation (e.g. [76]). Similar methods have been used study particle aggregation under turbulent conditions [77,78]. Lu et al. [77] suggested an empirical method of extending the SALS method to coarser particles. 5.2.2. Measurement of aggregate size Measurements 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 at different angles can be monitored simultaneously. From the angular distribution of scattered light intensity particle size distributions are derived. For particles larger than around 15 µm, it is possible to use Fraunhofer diffraction theory, in which scattering is treated as a problem in geometric optics. A large spherical particle in a light beam can be treated as a circular disc (or hole) with the same diameter. At the edge of the disc, light is diffracted, giving a characteristic pattern of light and dark bands far from the particle. These bands correspond to maxima and minima in the intensity of diffracted light and their positions depend only on the particle size and the light wavelength, not on the refractive index of the particles. For large particles and for visible light, these bands occur at quite low angles. Nevertheless, using a laser beam and high-quality optics, including a Fourier transform lens and an array of concentric detectors, it is possible to derive detailed information [80]. For smaller particles, Fraunhofer diffraction is not an adequate description of light scattering and the full Mie theory has to be used. This means that optical properties of the particles, especially refractive index, need to be known. Modern instruments incorporate software for carrying out Mie computations, but input of correct refractive index data is necessary, otherwise the results may be misleading [81]. This makes analysis of suspensions with different types of particles very difﬁcult. Particle sizing equipment is intended for use with solid particles, rather than aggregates, although such instruments are widely used for aggregate size determination. Rasteiro et al. [79] have recently

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measured ﬂoc formation, breakage and re-growth using a Malvern Mastersizer. They used precipitated calcium carbonate and two cationic polyacrylamides as ﬂocculants and compared ﬂoc breakage under the inﬂuence of sonication and hydrodynamic shear. They suggested that the ﬂoc size derived by SALS is similar to that observed by optical microscopy. Soos et al. [24] also used the Mastersizer to investigate the effect of shear rate on aggregate size under turbulent conditions. 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 the refractive index of the primary particles, m, and their volume fraction within the aggregate, ϕS (see Eq. (19)). The Maxwell–Garnett effective medium theory [20] leads to the following result: m2a −1 m2 −1 =S 2 2 ma +2 m +2

ð37Þ

Assuming spherical aggregates, the light scattering properties can then be calculated from Mie theory. Although this approach is far from exact, it can give a useful idea of the likely effect of aggregate density on scattering properties. For fractal aggregates, especially when the primary particles are very small, the effective density and hence ϕS can be very low (less than 1%) giving correspondingly low effective refractive index. (ma less than 1.01), even when the primary particles have a high m value. This, in turn, can give low values of the scattering coefﬁcient (see Fig. 5) and low values of the effective scattering cross-section. (This is a likely explanation of results like those in Fig. 16). It can be shown that for particles of low refractive index, Fraunhofer diffraction differs signiﬁcantly from Mie scattering, even for quite large particles. Fig. 17 shows computed scattering intensities for spheres of 100 µm diameter and refractive index m = 1.01, over a range of angles from 0–5°. The light wavelength is 650 nm. Both Mie and diffraction results are shown and it is clear that the ﬁrst few maxima and minima occur at signiﬁcantly different angles. Similar computations for m = 1.10 and greater show the ﬁrst 4 maxima and minima are at very nearly equal angles for both cases. If these computations are any guide to the behavior of aggregates with low average refractive index, then it is unlikely that Fraunhofer diffraction would be a reliable assumption in the derivation of aggregate size. 5.2.3. Fractal dimensions of aggregates The basic ideas concerning light scattering by fractal aggregates have already been discussed in Section 3.2, as well as a possible method of deriving the fractal dimension. A detailed review of

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

121

methods for the measurement of mass fractal dimensions has been given by Bushell et al. [83], with emphasis on light scattering methods. Many experimental studies, for a wide range of aggregates, have shown log I(q) vs. log q plots of the form shown in Fig. 8. There has been a lot of discussion of the range of validity of the RGD theory for fractal aggregates based on theoretical [84] and experimental [10] studies. Generally, it seems that RGD theory can be used without too much error for primary particles which are not strictly within the RGD criteria. For instance, Wang and Sorensen found that optical properties of aggregates formed from aerosols of TiO2 nanoparticles were not too different from RGD predictions, despite the high refractive index of TiO2. This is probably due to a fortuitous canceling of errors in the RGD approximation [84]. When the RGD assumption is acceptable, the slope of plots like those in Fig. x should give a reasonable estimate of the aggregate's fractal dimension. Where comparisons have been made, there is fairly good agreement between values obtained in this way with those derived by other methods and with theoretical predictions [85]. Nevertheless there are many practical cases where the primary particles are much too large to behave as Rayleigh scatterers and fractal dimensions obtained from the power-law slopes are not reliable [74]. In these cases it is better to refer to the slope as a “scattering exponent”. Liao et al. [86] showed that aggregates of ﬁne coal particles gave scattering exponents somewhat higher than fractal dimensions derived by other methods, but the trends were in the right direction. Since the coal particles had a mean 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 moving particle has a slightly different frequency from the incident light (as in the well-known Döppler effect). In colloidal dispersions, random Brownian motion of particles causes scattered light to vary randomly in frequency and interference between light scattered from different particles causes random ﬂuctuations in intensity measured by a stationary detector. A characteristic speckle pattern is visible when a colloidal dispersion is illuminated by a laser beam. Analysis of this effect involves autocorrelation of the scattered light intensity, monitored as a train of pulses from a photomultiplier tube. From the autocorrelation function it is possible to derive the diffusion coefﬁcient of the particles and hence the effective particle size. The nature of the technique is such that it may be called photon correlation spectroscopy (PCS) or quasi-elastic light scattering (QELS) as well as dynamic light scattering. Several commercial instruments are available and these are routinely used for the sizing of particles (and polymer molecules) in the submicron range. Because the effect is dependent on particle diffusion, a practical upper limit of particle size is about 3 µm. Although the restriction to small particles limits the application of dynamic light scattering in aggregation studies, it has been used to determine absolute aggregation rates of colloids [87–89]. Generally, the PCS data are used to derive hydrodynamic radii of the growing aggregates and to relate these to aggregate size. Because this is not a straightforward procedure for higher fractal aggregates, it is easier to restrict attention to the earliest stages of aggregation, where only singlets and doublets are present. This enables absolute aggregation rate coefﬁcients to be derived and this method has been used to study the aggregation of positively charged latex particles with anionic polyelectrolytes [89]. Midmore [90] has discussed methods for deriving moments of aggregate size distributions from the autocorrelation function. A novel instrument combining both static and dynamic light scattering measurements has been described [91]. This allows simultaneous measurements of light scattered at nine equally-spaced angles, but by changing the direction of the incident laser beam, scattering at

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up to 198 angles, between about 4° and 144° can be measured. Holthoff et al. [92] have shown that this technique can be used to derive absolute coagulation rate coefﬁcients for monodisperse colloids, at least in the early stages of the aggregation process. The combination of static and dynamic scattering allows the rate coefﬁcients to be determined without explicit knowledge of the light scattering properties of the aggregates.

McClements [100] found that the scattering of ultrasound by a 20% oil-in-water emulsion decreased greatly when droplets ﬂocculated and used this method for examining the effect of surfactant concentration on emulsion ﬂocculation. The same technique has been used to study depletion ﬂocculation and shear-induced ﬂoc disruption in a 10% corn oil emulsion [101] and ﬂocculation of latex particles with adsorbed layers [102].

6. Other techniques

6.3. Sedimentation

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

Sedimentation of particles depends essentially on their size and density and on the viscosity of the ﬂuid. The same is true for particle aggregates, but there are serious complications. For instance, aggregates are usually far from spherical shape and they may have appreciable permeability. The ﬁrst of these is usually dealt with by including an empirical shape factor [103], but the effect of permeability is more difﬁcult to deal with. The density of fractal aggregates is known to decrease with 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 been used to assess the performance of ﬂocculants. For quite concentrated suspensions, as encountered in mineral processing, it is possible to observe the interface between the clear supernatant and the settling particles. Simple visual observation of the settling rate is a common and very useful empirical test. Gravimetric and optical (photosedimentation) methods are also used [104]. However, it is difﬁcult to derive quantitative information from such measurements for concentrated suspensions [105,106]. In more fundamental studies, the sedimentation of individual aggregates is observed directly, so that both the size and settling rate may be determined. With these parameters an effective ﬂoc density may be derived, from Stokes law, although permeability is a complicating factor [107]. In the design of settling columns for such measurements, the mode of introducing single ﬂocs and the need to avoid the effects of convection currents have to be considered. A good account of these points has been given by Nobbs et al. [108], who also gave detailed technical information on the design of a suitable set-up. Experimental results are usually plotted as log (sedimentation velocity) vs. log (aggregate size) and the points always show a good deal of scatter about a linear regression line. Nevertheless, it is possible to derive fractal dimensions from the slope, which are usually in the expected range. Theoretical aspects and a discussion of the effects of aggregate permeability have recently been given by Tian et al. [12]. As well as sedimentation rate, it is also possible to measure the ﬁnal sediment volume. For ﬂocculated suspensions, this can be much higher than for the sediment formed from the original suspension, which can be quite tightly packed. This approach is mostly used in an empirical manner [109], but it is possible to interpret the results in terms of ﬂoc density and fractal dimension [110]. For lower density, (lower fractal dimension) aggregates the sediment volume becomes greater. This method has recently been used in a study of depletion ﬂocculation of silica dispersions, to derive information on aggregate morphology [56].

6.1. Electro-optical effects Charged particles which have some degree of anisotropy may be orientated in an electric ﬁeld. Because light scattering by anisotropic particles usually depends on their alignment in the light beam, changes in particle orientation can give measurable effects on scattered or transmitted light. This is the electro-optic effect [93] (sometimes called “electric birefringence” [94]). The effect can be quantiﬁed by a parameter αE, which is deﬁned as the relative change in light scattering intensity (measured under given conditions) as a result of applying an electric ﬁeld: αE =

IE −I I

ð38Þ

where IE and I are the intensities in the presence and absence of the ﬁeld. Rather than deriving information from absolute values of αE, a more common approach is to observe the change in this parameter after the ﬁeld is switched on or off. For instance, when the ﬁeld is switched off particles adopt random orientations by rotary Brownian motion and the rate of this relaxation process is highly dependent on particle size. The rotary diffusion coefﬁcient, DR of a spherical particle is inversely dependent on the cube of particle size and can be derived from the initial slope of the αE vs. time curve after the ﬁeld is switched off. The rate of decay decreases greatly as particle size is increased and this could, in principle, be used as a measure of aggregation. There are complications in using the electro-optic effect in aggregation studies, the most important of which is that aggregation not only gives an increase in size, but may also cause a signiﬁcant change in anisotropy. In some cases, particles that are initially highly anisotropic may become less so on aggregation. An important example is kaolinite, which has plate-like particles and shows a strong electro-optic effect. Aggregates of these particles are more spherically symmetric and give much lower αE than the original platelets. This effect has been used to monitor the ﬂocculation of clay particles at quite high concentrations [94]. It has also been used to monitor the dispersion (deﬂocculation) of kaolin aggregates by the addition of sodium polyacrylate [95]. Aggregation of various oxide particles has been studied by electrooptic methods, to give hydrodynamic radii of aggregates and their fractal properties [96–98]. Although these methods can give useful information on aggregating suspensions, they do not seem to have been adopted by researchers other than the original investigators. 6.2. Ultrasonic methods Ultrasound techniques are quite commonly used to characterize colloidal dispersions [99]. The attenuation of ultrasonic waves as a function of frequency depends on the concentration, size and density of suspended particles, among other properties. The method is suitable for concentrated emulsions and suspensions, where optical techniques are not applicable because of opacity.

6.4. Filterability The ﬂow of liquid through a packed bed of particles depends greatly on the size of the particles. When a suspension of particles is ﬁltered (e.g. through a membrane ﬁlter with pores smaller than the particle size) a ﬁlter cake is built up which gives an increasing resistance to ﬁltration. For small particles, the permeability is low and the ﬁltration rate may be very slow. It is commonly observed that aggregation of particles can greatly increase permeability and hence ﬁltration rates. This is why ﬂocculation is often used with mineral wastes such as ‘red mud’ from alumina processing and clays produced as a result of phosphate mining. Flocculation of these slurries, usually with high molecular weight

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polymers, gives greatly increased permeability and hence higher dewatering rates, which is of enormous practical beneﬁt. Although some quantitative analysis is possible [111], ﬁlterability methods are often used in an empirical manner to evaluate the performance polymeric ﬂocculants. In early work of La Mer [112], a reﬁltration rate technique was used, in which a ﬂocculated suspension was ﬁltered through a suitable medium and the ﬁltrate was then passed once more through the ﬁlter cake under constant pressure. The time for a given volume to ﬂow was measured, giving the reﬁltration rate. This method can give an indication of optimum ﬂocculation conditions, although unﬂocculated ﬁne particles can have a disproportionate effect on the reﬁltration rate [113]. A convenient measure of the ﬁlterability of concentrated suspensions is the capillary suction time (CST), developed in the 1960s by Gale and Baskerville [114]. A commercial version of the CST device is available (Triton Electronics, UK) and is now widely used. Essentially, the CST device measures the time taken for a certain volume of liquid to be drawn from the suspension by a standard ﬁlter paper. The time is determined automatically as the advancing liquid front passes between electrical contacts. The CST method is mainly used for empirical studies of the dewatering of slurries or sludges, particularly in assessing optimum ﬂocculation conditions. Interpretation of CST data has been discussed [115] and the method is still the subject of research [116]. References [1] Dentel SK. Crit Rev Environ Control 1991;21:41. [2] Smoluchowski M. Z Phys Chem 1917;92:129. [3] Elimelech M, Gregory J, Jia X, Williams RA. Particle Deposition and Aggregation. Measurement, modelling and simulation; 1995. [4] Spielman LA. J Colloid Interface Sci 1970;33:562. [5] van de Ven TGM, Mason SG. Colloid Polym Sci 1977;255:468. [6] Brakalov LB. Chem Eng Sci 1987;42:2373. [7] Spicer PT, Pratsinis SE. AIChE J 1996;42:1612. [8] Jarvis P, Jefferson B, Gregory J, Parsons SA. Water Res 2005;39:3121. [9] Meakin P. Adv Colloid Interface Sci 1988;28:249. [10] Wang GM, Sorensen CM. Appl Opt 2002;41:4645. [11] Gregory J. Filtr Sep 1998;35:367. [12] Tian WJ, Nakayama T, Huang JP, Yu KW. EPL 2007;78. [13] Bohren CF, Huffman DR. Absorption and Scattering of Light by Small Particles. New York: Wiley; 1998. [14] Kerker M. The Scattering of Light and other Electromagnetic Radiation. New York: Academic Press; 1969. [15] Zhang HJ, Xu GD. Powder Technol 1992;70:189. [16] Rose HE. J Appl Chem 1952;2:80. [17] Latimer P. Appl Opt 1985;24:3231. [18] Bryant FD, Seiber BA, Latimer P. Arch Biochem Biophys 1969;135:97. [19] Wind L, Szymanski WW. Meas Sci Technol 2002;13:270. [20] Sorensen CM. Aerosol Sci Tech 2001;35:648. [21] Schmidt PW. In: Avnir D, editor. The Fractal Approach to Heterogeneous Chemistry. Chichester: John Wiley; 1989. p. 67. [22] Ofoli RY, Prieve DC. Langmuir 1997;13:4837. [23] Bushell G. Chem Eng J 2005;111:145. [24] Soos M, Moussa AS, Ehrl L, Sefcik J, Wu H, Morbidelli M. J Colloid Interface Sci 2008;319:577. [25] Freundlich H. Colloid and Capillary Chemistry. London: Methuen; 1926. [26] Findlay AD, Thompson DW, Tipping E. Colloids Surf A Physicochem Eng Asp 1996;118:97. [27] Reynolds PA, Goodwin JW. Colloids Surf 1987;23:273. [28] Mysels KJ. Introduction to Colloid Chemistry. New York: Interscience; 1959. [29] Spicer PT, Pratsinis SE. Water Res 1996;30:1049. [30] Chakraborti RK, Gardner KH, Kaur J, Atkinson JF. J Water Supply Res Technol AQUA 2007;56:1. [31] Coufort C, Dumas C, Bouyer D, Line A. Chem Eng Process 2008;47:287. [32] Chakraborti RK, Atkinson JF. J Water Supply Res Technol AQUA 2006;55:439. [33] Donaldson CC, Mcmahon J, Stewart RF, Sutton D. Colloids Surf 1986;18:373. [34] W.H. Coulter, US Patent 2,656,508, 1953. [35] Lines RW. In: Stanley-Wood NG, Lines RW, editors. Particle Size Analysis. Cambridge: Royal Society of Chemistry; 1992. p. 350. [36] Matthews BA, Rhodes CT. J Colloid Interface Sci 1970;32:339. [37] Treweek GP, Morgan JJ. Environ Sci Technol 1977;11:707. [38] Casson LW, Lawler DF. J - Am Water Works Assoc 1990;82:54. [39] Pefferkorn E, Pichot C, Varoqui R. J Phys 1988;49:983. [40] Le Berre F, Chauveteau G, Pefferkorn E. J Colloid Interface Sci 1998;199:1. [41] Jackson GA, Logan BE, Alldredge AL, Dam HG. Deep-Sea Res Part II-Top Stud Oceanogr 1995;42:139. [42] Sterling MC, Bonner JS, Ernest ANS, Page CA, Autenrieth RL. Mar Poll Bull 2004;48:533.

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