Turbulent Aggregation of Titania in Water

Journal of Colloid and Interface Science 229, 511–525 (2000) doi:10.1006/jcis.2000.6994, available online at http://www.idealibrary.com on

Turbulent Aggregation of Titania in Water Christoph Tontrup, Fr´ed´eric Gruy, and Michel Cournil1 Ecole des Mines de Saint-Etienne, 158 Cours Fauriel, 42023 Saint Etienne Cedex 2, France Received January 4, 2000; accepted May 30, 2000

This paper deals with the application of two in situ particle sizing methods to the study of aggregation of titanium dioxide in turbulent aqueous solutions. Turbidity measurements are used to characterize aggregation of diluted suspensions (solid volume fraction less than 10−4 ). Analysis of backscattered light can be applied to highly concentrated suspensions. Because in situ measurements are possible, the aggregation process can be followed from its beginning to the final steady state which is always observed. The influences of stirring rate and solid volume fraction on the aggregation dynamics are presented. They are interpreted in the framework of a model of aggregation which takes into account the morphology and the related optical properties of the aggregates and the physical and hydrodynamic interactions between aggregates. The complex relationship between aggregate restructuring and aggregation on one hand, and fragmentation on the other hand, is considered. °C 2000 Academic Press Key Words: aggregation; fractal; turbulence; breakage; turbidimetry; light backscattering.

INTRODUCTION

Aggregation occurs in many biological, chemical, and physical processes. It often concerns suspension of small particles in a liquid. The dynamics of aggregation mainly depends on the hydrodynamic conditions and on the particle size. In still media, aggregation of submicron particles is due to Brownian encounters whereas larger particles undergo sedimentation with different settling velocities and therefore collide and may aggregate. However, in many practical situations, it is necessary to put the solid–liquid suspension in motion to homogenize or to convey it. In this case, whatever the nature of the flow, the role of the local shear flow in collisions becomes predominant. A particularly frequent application concerns the behavior of slurries in a stirred tank. Most of the experimental studies are relative to aggregation in diluted suspensions, with solid volume fraction φ currently smaller than 10−5 . This limit is too low to be representative of standard industrial suspensions. The lack of studies related to realistic systems is due to the scarcity of investigation tools for characterizing concentrated or at least less diluted suspensions. Generally speaking, in situ analysis methods present many advantages over sampling methods. For instance, they are particularly effective for investigating rapid 1

To whom correspondence should be addressed.

phenomena, the characteristic time of which is comparable to the sampling time. This is precisely the case in aggregation of concentrated suspensions, because the aggregation characteristic time is inversely proportional to the solid volume fraction and may be very short. In the past 10 years, we have developed several optical devices to measure in situ the particle size distribution (PSD) or the moments of particulate systems. For diluted suspensions (φ < 10−5 ), an optical sensor based on spectral turbidimetry principles can be used to determine the PSD of suspensions composed of submicron and micron particles (1). For less diluted and concentrated suspensions, an optical device based on backscattered light analysis can be applied to the determination of the mean particle size (2). The aim of this work is to study particle aggregation kinetics in suspension over the widest range possible of volume fractions in solid. We chose titania as the experimental material because of the importance of its aggregation in pigment production and painting handling. This paper is organized as follows: after a review of the theoretical background connected with aggregation and fragmentation, we present the experimental data available on titania aggregation; next, the different steps of the model, each corresponding to a theoretical aspect, are discussed and adapted to the present case. Last, theoretical predictions are compared to the experimental results and a global interpretation is proposed. THEORETICAL BACKGROUND

Turbulence In this work, we adopt the classical description of turbulence as presented in (3, 4). At any instant and in any place of the turbulent flow, eddies are supposed to appear and to take part in the energy dissipation. Energy is supplied to the flow by an external source, such as mechanical stirring. Energy is transferred from the largest eddies to the smallest eddies in which it is dissipated through viscous interactions. The size of these smallest eddies is the Kolmogorov microscale η; from dimensionless considerations η is expressed as a function of the kinematic viscosity ν and the energy dissipation rate per unit mass εm in the following form: µ 3 ¶1/4 ν η= . [1] εm

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The velocity gradient γ˙ in each eddy is proportional to (εm /ν)1/2 . The resulting shear stress is τS = µl γ˙ .

[2]

In this expression, µl denotes the dynamic viscosity. Aggregation Aggregation is the consequence of a collision between particles. The mechanism which brings particles into close proximity results from the hydrodynamics of the suspension. An aggregate is characterized by its number i of primary particles (supposed identical). Aggregation between i-mer and j-mer may be represented by the following quasi-chemical equation: i-mer + j-mer → (i + j)-mer. The corresponding reaction rate can be written as d Ni+ j = k i j Ni N j , dt

[3]

where ki j is the kinetic constant or kernel. ki j is the product of two terms ki0j and αi j : ki j = ki0j αi j .

[4]

The collision frequency function ki0j is dependent on the origin of the encounters between particles: Brownian motion, differential settling velocity, and shear flow. The collision efficiency αi j depends on the different interactions between particles. London–Van der Waals attractive interactions. These forces have their origin in interactions between instantaneous induced dipoles. For two spheres of radii ai and a j separated by a distance H , an attractive interaction potential VA can be calculated (5). In fact, a more correct expression is obtained by taking into account a “retardation” effect (6) due to propagation of electromagnetic waves in a dielectric medium. Schenkel and Kitchener (7) have proposed an empirical correlation to calculate the attractive potential as a function of the geometrical parameters (ai , a j , H ) and of A, the Hamaker constant of the particle in its medium. Double layer repulsive interactions. A solid particle located in an electrolyte generally takes a surface charge and is surrounded by a layer of dissolved ions. Several models exist to describe this double layer (5, 8, 9). The zeta potential is the only characteristic parameter of this double layer which is experimentally available; it is currently assimilated to the surface potential 9o in the case of diluted liquid media. When two particles come into contact, the interactions of their respective double layers result in a repulsive potential VR . In this paper, only experiments for which the double layer interactions were zero or negligible, are presented. For this reason, we do not give further details on this subject.

Hydrodynamic interactions. When particles come close together they do not continue their approach along rectilinear paths because hydrodynamic and colloidal forces modify their trajectories. The hydrodynamic interaction arises mainly from the draining off of fluid between the particles and tends to hinder collisions between them. (a) Collision efficiency for spherical particles. Calculation of collision efficiency should take into account the contribution of the three types of interactions. Spielman (10) has proposed analytical expression for αi j when collisions are due to Brownian motion. More recent studies (11–14) have extended Spielman’s work to shear flow. The model of Van de Ven and Mason (11) includes attractive, repulsive, and hydrodynamic interactions in the case of equally sized particles, whereas the model of Higashitani et al. (13, 14) can be applied to any pair of spherical particles, but in the case of attractive and hydrodynamic interactions only. The case of aggregation in turbulent flows has been little investigated; two tracks have been explored so far. For particles smaller than the Kolmogorov microscale, aggregation is assumed to take place in the smallest eddies in local shear flow conditions. This makes possible the use of the previous models (13–17). Brunk et al. (18) have achieved simulations based on Langevin’s equation describing collision between two equally sized particles in a fluctuating shear field. Previous hypotheses on collision conditions are confirmed and αi j is expressed by the same expression as that used in Van de Ven et al. (11). Recently, two of us (19) proposed to use the Spielman equation (10), replacing the Brownian diffusion coefficient Di by the Levich (20) turbulent diffusion coefficient Dturb ; they obtained an analytical expression for αi j in the most general case of interactions and radii values, in a turbulent flow. (b) Collision efficiency for noncompact aggregates. The morphology of the aggregates depends on both the physicochemical and hydrodynamic conditions of their formation and their intrinsic mechanical properties. However, the aggregation dynamics also depend on the morphology of the colliding particles. Recent experiments have shown that aggregates have a fractal structure (21–26). An aggregate containing i primary particles of radius a1 is characterized by the fractal dimension Df , the outer radius ai , and the hydrodynamic radius aHi ; as the structure of the aggregates is nonuniform, their volume density φa (r ) depends on the distance r from the center of mass of the aggregate; the average volume density is denoted φ¯ a . These different characteristics are linked by the following relations (22, 27), µ ¶1/Df i ai = a1 [5] S µ ¶Df −3 S r [6] φa (r ) = Df 3 a1 µ ¶ Df −3 ai ¯ , [7] φa = S a1 where S is a structure factor.

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The authors differ in the way they express the ratio between the hydrodynamic radius and the outer radius; this ratio is assumed to be constant by (21, 23, 28), whereas, according to (22), it depends on the aggregate volume density. However, for fractal dimensions close to 2, the two models are in agreement. From computer simulations, Gmachowski (27) has found a relationship between S and Df which can be represented by the following correlation: S ≈ 0.42Df − 0.22.

[8]

Sorensen et al. (29, 30) also studied the function S(Df ). Contrary to Gmachowski’s results, they found that S was a decreasing function of Df . However, for fractal dimension values corresponding to shear flow aggregation (Df ≈ 2.3) both models give the same S values, i.e., about 0.74. Several authors have shown that only large aggregates (i > i lim ) have fractal structure. For instance, Kyriakidis et al. (23) achieved simulations with a given aggregation mechanism. They observed that the fractal dimension of large aggregates was equal to 1.86 whereas they proved that the small aggregates were denser than the large ones. A transition value i lim was found equal to 16. Adachi et al. (31) also found that aggregates with more than 50 primary particles have fractal structure. However, small aggregates are more porous than large ones. Careful experiments of Takayasu and Galenbeck (32) showed that even small aggregates (i < 5) have a fractal-like structure. In this work we follow the procedure proposed by Kusters et al. (22, 23) (shell-core approach) to calculate the collision efficiency αi j between two porous aggregates (i ≥ j > 1); it consists of the following steps: (i) determination of the permeability κi of the bigger particle (33); (ii) determination of the Debye shielding ratio of this parti1/2 cle: ξl = ai /κi ; (iii) determination of the collision efficiency αi,Kj as a function of ξl (Kusters model); (iv) determination of the collision efficiency αi,Hj (Higashitani model) for compact aggregates (13, 14); and (v) comparison between αi,Kj and αi,Hj and choice of the αi, j value (22). Fragmentation of Aggregates In the aggregation processes, a maximum aggregate size is almost always observed (17, 33, 34). This can be due to several factors: breakage, or collision efficiency becoming zero beyond a critical size or settling which could remove the largest aggregates from the system. The third factor requires particle sizes much larger than the maximum size reached in our experiments. We shall not consider this mechanism in the following. We shall successively examine the two first factors. (i) Breakage. The occurrence of breakage depends on the balance between the disaggregation effects due to the action of the fluid and the overall cohesion of the aggregate due to the

interactions between primary particles. The hydrodynamic effects are of different nature because the aggregate is larger or smaller than the Kolmogorov microscale. Only the latter case is compatible with the experimental conditions of this study. It corresponds to a shear stress originating from the local velocity gradient and acting on the aggregate. The authors do not agree on the way to express the competition between the disaggregation and the cohesion effects. According to the models, the relevant parameter is either the ratio E c /E t , or the ratio σ/τs ; E c and E t are respectively the aggregate cohesion energy and the turbulent energy acting on this aggregate; σ is the mean mechanical strength of the aggregate and τs is the mean shear stress (21, 33, 35, 36). The breakage rate or the fragmentation kernel kf is proportional respectively to γ˙ e−Ec /Et and to γ˙ e−σ/τs . Another discussion is relative to the size of the fragments produced by breakage. Two cases are currently envisaged: —erosion of single or small groups of particles from the aggregate surface (21); —production of fragments of comparable sizes (33, 35). In all cases, the breakage rate depends on the hydrodynamic conditions of the flow, via εm and µl , and on the characteristics of the aggregates: outer radius, fractal dimension, primary particle radius, and cohesion force between two primary particles. This force F can be expressed in the form, F=

Aa1 . 12H02

[9]

H0 is the separation between two primary particles in the aggregates. Several authors (37–40) propose another kind of expression for the fragmentation kernel: 0

kf α γ˙ b i 1/Df ,

[10]

with b0 ≈ 1.6 Serra et al. (37–39) have observed that kf is dependent on φ for φ > 10−5 ; this proves that fragmentation would be due to collision between aggregates and not only between aggregate and eddies. The competition between aggregation and fragmentation leads to a steady PSD. The corresponding mean particle size aL depends on shear rate, according to aL 0 α γ˙ −c . a1

[11]

Exponent c0 is close to 1. Its reported value varies according to the authors: 0.56 in (41), 0.7 in (37), 1 in (39), and 0.4 in (42). (ii) Zero collision efficiency. This approach was especially developed by Brakalov (43). The collision efficiency between two equally sized spherical particles decreases with the particle size. The decrease is sharper as the particles (aggregates) are more porous. Otherwise, the aggregate, which results from two

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smaller aggregates, can be too loose to survive. Brakalov shows that there exists a limit value for the aggregate size. However, the assumption of an additional short-range interaction force is necessary to interpret the experimental results. Dynamics of Aggregation–Fragmentation The variation of the volume density Ni of aggregates Ai versus time t is given by the population balance equation in its discrete form (44), — for i > 1:

Light Scattering by Suspensions

X 1 X d Ni = k j,i− j N j Ni− j − ki,k Ni Nk dt 2 j=1,i−1 k=1,∞ −

X 1 X f f k j,i− j Ni + ki,k−i Nk 2 j=1,i−1 k=i+1,∞

[12]

— for i = 1: X f X d N1 k1,k N1 Nk + k1,k−1 Nk =− dt k=1,∞ k=2,∞

[13]

where ki j is the aggregation kernel (i-mer + j-mer → (i + j)f mer) and k j,i− j the breakage kernel (i-mer → j-mer + (i − j)mer). As particle (aggregate) size is smaller than the Kolmogorov microscale, Brownian kernels and turbulent kernels should both be considered. Adachi et al. (45) propose to express ki j as the sum of two contributions, ki j = (ki j )Br + (ki j )turb ,

[14]

in which (ki j )Br is the Brownian kernel given by Smoluchowski and possibly corrected by the collision efficiency αi j (equal to 0.5 in the experimental work of Adachi et al. (45) and Kyriakidis et al. (23)); the turbulent kernel (ki j )turb is derived from one of the turbulent models presented above. For instance, in the case of particles smaller than the Kolmogorov microscale, the following expression is currently used: (ki j )turb

µ ¶ µ ¶ 4 3π 1/2 ε¯ m 1/2 = δ(ai + a j )3 αi j . 3 10 ν

[15]

This relation comes from the Saffmann–Turner approach (15); εm is the mean energy dissipation rate, and δ is a correction coefficient which is introduced to take into account different deviations from this ideal model. Many expressions are found in the literature for the mean value of εm , for instance, ε¯m =

Np ω3 Ds5 , V

given for instance by (46) (Np is the power number, Ds the stirrer diameter, ω the rotation rate of the stirrer, and V the volume of the suspension). This type of expression should be used with precautions because, except for highly turbulent media, εm is not uniform in a stirred vessel. For instance, in a recent review paper, Kresta (47) considers different stirred tank models. A reactor consists of one, two, or three zones, each one characterized by its volume and a turbulent dissipation rate value. The consequence of the turbulence heterogeneity can be taken into account via δ in relation [15] (for instance, δ = 0.7 in (33)).

[16]

Diluted suspensions. In the case of diluted suspensions, aggregation can be studied using an in situ turbidimetric sensor (1). This device allows us to record continuously a signal directly dependent on the particle size distribution (PSD) in the reactor and so avoids us two drawbacks: —sampling from the suspension, which could damage the aggregates and anyway introduces systematic counting errors, in particular concerning the finest particles which are difficult to collect; —a by-pass through the measurement cell of a commercial particle sizer, with again the risk of damaging the aggregates by pumping and anyway uncontrolled hydrodynamical disturbances. Turbidity τ of a monodisperse suspension of spherical particles (radius a) is given by the Mie theory (48, 49): τ (λ) = l˜−1 = N Csca (m, 2πa/λ).

[17]

N and Csca are respectively the volume density of particles and their scattering cross section. λ and m are respectively the light wavelength and the ratio of particle refractive index to surrounding medium refractive index. τ is also the inverse mean free path length l˜−1 of a photon in the suspension. The turbidity of a polydisperse suspension contains the contribution of each size class i of particles (1 ≤ i ≤ n): τ (λ) =

X i=1,n

l˜i−1 =

X

Ni Csca i .

[18]

i=1,n

The reverse problem, i.e., the determination of the PSD from the measured turbidity spectrum, is acknowledged to be an illconditioned mathematical problem which can be solved by the use of constrained least-squares methods (50, 51). In (1) we discussed and defined the potential and limits of such a method for particle sizing. For instance, a low volume fraction φ of solid is required—generally φ < 10−4 for micron particles. Concentrated suspensions. In a concentrated suspension, multiple light scattering takes place. The transmitted light intensity recorded by the turbidimetric sensor becomes too small to be exploited. In this case it is better to analyze the light which is backscattered by the suspension. Recently (2), we perfected

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an experimental set-up to measure in situ the ratio of backscattered light intensity Ib to incident light intensity I0 . We found that, for a monodisperse suspension, the normalized backscattered intensity Ib /I0 is a function of the inverse transport mean free path l˜∗−1 = N Csca (1 − µ): Ib = f (l˜∗−1 ) = f (N Csca (1 − µ)). I0

—in the case of more concentrated suspensions the calculated inverse transport mean free path is compared to the experimental value deduced from backscattering measurements. Of course in both cases, the use of in situ sensors will enable us to compare predicted and measured data throughout the aggregation process.

[19]

The asymmetry factor µ is the mean value of the cosine of the light scattering angle. polydisperse suspension, l˜∗−1 should be replaced by PFor a ∗−1 ˜l where l˜i∗−1 is the contribution of the ith class of i=1,n i particles, thus: ! Ã Ã ! X X Ib ∗−1 l˜i = f = f Ni Csca i (1 − µi ) . [20] I0 i=1,n i=1,n P The shape of the curve Ib /I0 = f ( i=1,n l˜i∗−1 ) only depends on the optical probe geometry. The particle size measurement ranges of turbidimetry and backscattered light analysis are complementary (2). Turbidimetry is applicable for 0.1 ≤ N Csca L ≤ 3 (i.e., 0.1 ≤ N Csca ≤ 3 cm−1 for an optical path length L commonly equal to 1 cm), whereas the backscattering sensor can be used for l˜∗−1 ≥ 5 cm−1 . Optical properties of aggregates. Aggregates in a stirred tank are generally small and slightly porous. Each aggregate is characterized by its diameter and its mean inner volume fraction φa . An easier way to determine the optical properties of an aggregate is to calculate its effective refractive index m a (52, 53). For our purpose, the equation derived by Maxwell-Garnett has been proved to be suitable for titania aggregates (54): ¡ 2 ¢ ¢ ¡ 2 ma − 1 m −1 ¢ = φa ¡ ¢. ¡ [21] m 2a + 2 m2 + 2

EXPERIMENTAL PART

Experimental Setup, Material, and Procedure Experimental setup. The reactor used for this study of aggregation is a stirred tank, the diameter D 0 of which is equal to 150 mm (Fig. 1). This reactor is equipped with four baffles of width b. The liquid depth H in the vessel is equal to diameter D 0 . The bottom part of the tank is rounded. Agitation is ensured by a propeller of diameter 50 mm pumping downward. Temperature is kept constant at 25.00◦ C ± 0.01◦ C by a double-wall jacket. The liquid volume is always equal to 2.5 l. The turbidity probe and the backscattering sensor used in this work were respectively described by Crawley (1) and Tontrup (2). They were located at the third of the vessel radius midway between two baffles and mounted at z height above the bottom of the tank with z/H = 0.3. When necessary, off-line PSD determinations were carried out by means of laser diffraction particle sizers (Coulter LS 130 and Malvern Mastersizer S). Microphotographs of materials were obtained from a scanning electronic microscope (Zeiss Gemini) and a transmission electronic microscope (Phillips CM 200). Zeta potential was measured by acoustophoresis (Dispersion Technology DT 1200) for high solid volume fraction (φ > 0.01)

m and m a are the relative refractive indices respectively for titania primary particles and titania aggregates. Knowing the diameter and the effective refractive index of an aggregate, the Mie theory (48, 49) allows us to calculate the scattering cross section Csca and asymmetry factor µ at a given wavelength. In the following the time evolution of a suspension due to aggregation will be studied. In order to interpret the experimental results and to determine the concerned phenomena, we will adopt the following approach: —calculation of the optical properties (in fact Csca i ) of the agglomerates possibly involved in the process; —prediction of the density (Ni (t)) of these agglomerates from numerical solution of the population balance equation; —in the case of diluted suspensions, turbidity or equivalently inverse mean free path length are calculated and then compared to the respective experimental values;

FIG. 1. Schematic representation of the aggregation reactor (H = D 0 ; dag = 0.33D 0 ; b = 0.1D 0 ; c = 0.33D 0 ; f = 0.02D 0 ).

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TABLE 1 Surface Layer Composition of RL60 Titania Metallic oxide

Mass fraction

SiO2 Al2 O3 P2 O5

0.0024 0.0322 0.0026

and by microelectrophoresis (Sephy, Zetaphorometer) for low solid volume fraction. Material. The aggregation experiments were performed on rutile titanium dioxide. This material is commonly used as white painting pigment. We purchased it from Millenium Inorganic Chemicals (RL60 Titania). Due to its use as a pigment for painting, the titania grain surface is covered by a thin layer of several metallic oxides. Its composition—determined by X fluorescence—is given in Table 1. Alumina is the main component of the surface layer. The mean layer thickness is equal to 3 nm. Figure 2 represents a TEM microphotograph of RL60 Titania which appears to be composed of polyhedral particles. The PSD obtained from laser diffraction is shown in Fig. 3. The main size characteristics are summarized in Table 2. Figure 4 represents the zeta potential variation against pH for different values of the ionic strength I at high volume fraction in solid (φ = 0.01). The measurements have been obtained by acoustophoresis. Ionic strength is fixed at a constant value by initial addition of a potassium nitrate solution. The zero value

FIG. 3. Size distribution of initial titania samples: q3 , volume density distribution (bold line); q0 , number density distribution.

of zeta potential (PZC) is reached for pH 8.6. Zeta potential measurements have been performed at low solid volume fraction using microelectrophoresis; they confirm the PZC value. The effect of ionic strength on zeta potential is consistent with other researchers’ experimental works (59). The PZC of rutile (55) is commonly found to be equal to 6. The value obtained for RL60 Titania should be compared with data known for α alumina, for which the PZC value is in the range 8–9.5 (55). This result is quite consistent with the surface layer composition. Determination of the suspension optical properties requires the values of the refractive index of RL60 Titania. A calculation, inspired by Kerker (49), shows that the thin alumina coating has no influence on the optical properties of titania grains. Therefore, we will consider that RL60 Titania has the same optical properties as pure titania. Experimental procedures. (a) Diluted suspension. For the sake of simplifying the modeling, we chose to study diluted suspension aggregation under experimental conditions for which the repulsive forces between particles are very weak. A preliminary study has allowed us to determine the optimal conditions (pH = 6 and I = 0.15 M) for our study (63). Each experiment consists of the different steps: (i) The reactor is filled with 2.15 l of water, and then starting thermal control and stirring. (ii) The blank turbidity is measured (without solid). (iii) Titania powder is dispersed for 5 min with ultrasonics in a small vessel containing 0.2 l of potassium hydroxide solution (pH 10). This suspension is then poured into the reactor. TABLE 2 RL60 Titania PSD Characteristics

FIG. 2. MIC RL 60 titanium dioxide particles (MET microphotographs).

Average volume diameter (µm)

Average number diameter (µm)

Standard deviation (µm)

0.44

0.35

0.27

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from the reactor and put into the laser particle sizer measurement cell. These measurements are only carried out for suspensions with the highest solid volume fractions. RESULTS

Diluted Suspensions

FIG. 4. Titania zeta potential as a function of pH at different ionic strength values I (measurements by acoustophoresis).

(iv) The pH is fixed at 6 by addition of a few drops of nitric acid solution. Next, 0.15 l of potassium nitrate solution (2.5 M) is rapidly added into the reactor for suspension destabilization by ionic strength effect. At the end of this operation the pH is still equal to 6. Depending on the added amount of titania, the solid volume fraction is in the range 2 × 10−6 –2 × 10−5 . The ionic strength is equal to 0.15 M. (b) Concentrated suspension. At higher φ, aggregation without repulsive forces is too rapid to be studied. By lowering the pH value, it is possible to slow the aggregation process, the study of which becomes feasible under these conditions (pH = 4 and I = 10−1 M). Each experiment consists of the different steps: (i) The reactor is filled with 2.24 l of acidified water (pH 4); thermal control and stirring are started. (ii) Titania powder is dispersed for 5 min with ultrasonics in a small vessel containing 0.2 l of water (pH 3.5). This operation modifies the pH value which is again brought to 3.5. Ultrasonics and pH fitting have to be repeated until dispersion is completed. This suspension is then poured into the reactor. (iii) Ionic strength and pH are respectively fixed at 10−2 M and 4 by adding potassium nitrate and nitric acid solutions. (iv) The suspension is destabilized by adding rapidly 60 ml of potassium nitrate solution (3.75 M) (pH 4). Depending on the added amount of titania, the solid volume fraction is in the range 5 × 10−5 –5 × 10−3 . For higher φ, aggregation characteristic time is smaller than the time required to mix potassium nitrate solution and suspension. Therefore, aggregation cannot be studied any more using this method. In the case of aggregation in concentrated suspensions, a steady state seems to be reached after a short time. In addition to the in situ size determinations by backscattered light analysis, off-line PSD characterizations of this steady state were performed; they involve a special procedure: a solution composed of dispersing agent (Darvan C) and potassium hydroxide is added to the suspension while the stirrer speed is decreased to 100 rpm. At this moment, a suspension sample is withdrawn

Figure 5 presents the turbidity (or the inverse mean free path length) time evolution for a typical experiment. Aggregation starts at zero time. Turbidity decreases with time and reaches a plateau corresponding to a presumed steady state. Remark: The variation of turbidity with particle size is generally complex due to the complex variation of scattering cross section Csca . At increasing particle diameter, turbidity of a given volume of identical titania spherical particles first sharply increases and reaches a maximum for a diameter equal to 0.35 µm. Afterward, turbidity decreases versus the particle diameter. Consequently, for an aggregating system of titania particles, turbidity initially increases only if the initial particle diameter is lower than 0.35 µm; however, it decreases in the opposite case. In the investigated system, the initial particle diameter ranges roughly between 0.3 and 0.5 µm. This means that turbidity should decrease throughout the aggregation process as observed in our experiments. Influence of the stirring rate. Figures 6a–6d represent turbidity time evolution for different stirring rate and solid volume fraction values. Aggregation in still suspensions (ω = 0 rpm) was studied for shorter periods, because after 15 min, sedimentation takes place. Generally speaking, the behavior of titania diluted suspensions is quite similar to the behavior we observed for alumina diluted suspensions in a previous study (56). We mean that the same qualitative dependence of the initial and large time characteristics of the aggregating suspension against the stirring rate is observed here. As expected, the higher the stirring rate, the

FIG. 5. Typical turbidity time evolution due to the aggregation process. Experimental conditions: pH = 6.0; T = 25◦ C; 1084 rpm; φ = 8 × 10−6 ; I = 0.15 mol.l−1 .

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FIG. 6. Influence of stirring rate on turbidity time evolution during titania aggregation. Common experimental conditions: pH = 6.0; T = 25◦ C; I = 0.15 mol.l−1 . (a) φ = 2 × 10−6 ; (b) φ = 8 × 10−6 ; (c) φ = 1.6 × 10−5 ; (d) φ = 3.2 × 10−5 .

higher the initial aggregation rate (initial slope of the turbidity– time curve). The final turbidity value can be easily linked to the granular state of the system. For a given weight of large particles indeed, turbidity is a decreasing function of mean radius or equivalently an increasing function of the number density (48, 49). Thus, from Figs. 6, it appears that increasing the stirring rate results in reducing the aggregate size and increasing the aggregate number at the steady state (large time experiments). Influence of solid volume fraction. The higher the solid volume fraction, the smaller the time needed for the plateau to be reached. Whatever the stirring rate, the time–turbidity curve initial slope is proportional to φ 1.8 , instead of φ 2 which is the expected law for collisions between two particles. Long-term turbidity is proportional to φ. Hence, steady state PSD does not depend on φ.

backscattering is observed (Fig. 8). For small (resp. high) φ, l˜∗−1 (0) is smaller (resp. higher) than l˜∗−1 max (see Fig. 8). After a sharp initial variation, the normalized backscattered intensity reaches a plateau (limit value). Influence of the stirring rate. Figures 9a–9d show the time evolution of normalized backscattered intensity for different

Concentrated Suspensions Figure 7 shows the time evolution of the normalized backscattered intensity for two solid volume fraction values. The normalized backscattered intensity decreases (resp. increases) for the smaller (resp. higher) φ value. These different behaviors are due to the location of the initial inverse transport mean free path l˜∗−1 (0) compared to the value l˜∗−1 max for which the maximum of

FIG. 7. Normalized backscattered intensity time evolution due to the aggregation process for two values of the solid volume fraction. Experimental conditions: pH = 4.0; T = 25◦ C; 1084 rpm; I = 0.10 mol.l−1 .

TURBULENT AGGREGATION OF TITANIA

FIG. 8. Normalized backscattered intensity versus inverse transport mean free path l˜∗−1 (titania suspensions in water; φ = 3.13 × 10−4 corresponds to l˜∗−1 = 2.5 × 10−3 µm−1 ; φ = 5 × 10−3 corresponds to l˜∗−1 = 9.0 × 10−2 µm−1 ).

stirring rate and solid volume fraction values. As expected, the higher the stirring rate, the smaller the aggregate limit (t → ∞) size. The dependence of the limit backscattered intensity on the stirring rate is stronger than for small solids volume fraction (Figs. 6a–6c). Then, it seems that the aggregates’ cohesion strength decreases with the solid volume fraction.

519

FIG. 10. Influence of the stirring rate on the aggregate normalized maximum size (laser diffraction measurements).

In Figs. 10 and 11 are shown the limit mean size aL (normalized by al ) and the corresponding limit standard deviation σL (normalized by aL ) against stirring rate. These data were obtained by means of laser diffraction sizing. aL is proved to be proportional to ω−0.33 ; however, it does not depend on φ. As the stirring rate increases, the aggregate PSD nearly keeps the same shape.

FIG. 9. Influence of stirring rate on normalized backscattered intensity time evolution during titania aggregation. Common experimental conditions: pH = 4.0; T = 25◦ C; I = 0.10 mol.l−1 ; wavelength = 632.8 nm. (a) φ = 7.81 × 10−5 ; (b) φ = 3.13 × 10−4 ; (c) φ = 1.25 × 10−3 ; (d) φ = 5 × 10−3 .

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FIG. 11. Influence of the stirring rate on the aggregate size normalized standard deviation (laser diffraction measurements).

Influence of solid volume fraction φ. The higher φ, the smaller the time needed for the plateau to be reached. MODELS

Models of Aggregates Whatever the experiments, the titania aggregates formed in the system are small. From their maximum size (Fig. 10), it appears that they contain less than 64 primary particles. We will use the standard relations (5, 8) to express the relationship between their size and the number of monomers of which they are comprised. Under the present operating conditions (pH, ionic strength) only long-range attractive forces between particles should be considered. Therefore, the total interparticle interaction potential should not present an energy barrier. However, observations by optical microscopy and particle sizing by laser diffraction indicate that the aggregates formed at pH 4 are smaller than those formed at pH 6. This can be explained by the presence of a short-range interparticle repulsive force at pH 4. This has been already observed by Ducker et al. (57) in a similar system (alumina, pH 3, I = 0.1 M; we remind the reader that the surface of our titania grains is coated with alumina). At higher pH or smaller ionic strength, such repulsive forces disappear. So far this behavior is not well understood. If we take into account this short-range repulsive force, the total interaction potential shows a secondary minimum for an interparticular distance equal to 2 nm (63). In fact, short-range repulsive forces only play a minor part before collision of particles; however, they have a significant role in aggregate cohesion strength. Due to these repulsive forces, the aggregates which are formed at pH 4 are loose and small.

FIG. 12. Comparison between experimental and predicted inverse mean free path l˜−1 for different aggregate fractal dimension values Df . Experimental conditions: pH = 6.0; T = 25◦ C; 610 rpm; φ = 8 × 10−6 ; I = 0.15 mol.l−1 ; no fragmentation considered in the simulations.

been taken into account using the same value of the correction coefficient as that used by this author. This appeared to us to be a reasonable assumption, considering the analogies between our two reactors. In these calculations, the power number Np has been taken as equal to 0.36, a value commonly assumed for a marine-type mixing propeller (58). The global aggregation kernel, i.e., including the turbulent and Brownian contributions, has been determined following the Adachi relation [14]. As said before, no repulsive interaction has been considered. In our experimental system the Hamaker constant is equal to 5.35 × 10−20 J in water; from this value, the Brownian collision efficiency αi j has been found close to 0.5. Preliminary simulations show that modeling based on fragmentation kernels (corresponding to surface erosion or breakage mechanisms) does not give good agreement between experimental and simulated time–turbidity curves (63). Experimentally, turbidity decreases smoothly with time whereas the simulated time–turbidity curves present a strong bend just before the steady

Aggregation Kernels The turbulent kernels have been calculated from relation [15] using the method proposed by Kusters and recalled above. The nonuniformity of the energy dissipation rate in the reactor has

FIG. 13. Comparison between experimental and predicted inverse mean free path l˜−1 : importance of the assumption of existence of a maximum aggregation size. Experimental conditions: pH = 6.0; T = 25◦ C; 610 rpm; φ = 8 × 10−6 ; I = 0.15 mol.l−1 ; fractal dimension, 2.4.

TURBULENT AGGREGATION OF TITANIA

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FIG. 14. Comparison between experimental and simulated l˜−1 (t) curves for various stirring rates during aggregation at low solid volume fraction. Experimental conditions: pH = 6.0; T = 25◦ C; 610 rpm; φ = 2 × 10−6 ; I = 0.15 mol.l−1 ; wavelength, 652.1 nm. (a) No stirring, Df = 1.8; (b) 343 rpm, Df = 2.4; (c) 610 rpm, Df = 2.4; (d) 1084 rpm, Df = 2.4.

state: from this point on, competition between aggregation and fragmentation suddenly appears. In fact, the usual fragmentation kernels are suitable for describing the breakage behavior of aggregates which have “forgotten” the conditions of their formation, i.e., after restructuring (59–62). When an aggregate is newly formed by the collision of two smaller aggregates and has no time for restructuring, the local flow shear can disrupt it into its original two components. This behavior is particularly expected for the large aggregates which are produced during the aggregation process. One may consider that the collision efficiency is equal to 0 when the resulting aggregate size is larger than a limit value aL . Under these conditions, simulations predict that aggregation is a smooth process as observed in the experiments. Limit size and shear rate are related by Eq. [11].

The size domain is divided into adjacent sections numbered by index i. With the ith section is associated a volume Vi that is the average volume of the aggregates contained in this section: Vi = (bi + bi−1 )/2, where bi is the upper boundary volume of section i; the volume of section i is obtained from the volume

Numerical Methods The time evolution of the particle size distribution is simulated by solving the population balance equations [12], [13]. The necessary discretization over the wide size range investigated, that is to say 0.2–5 µm, may require excessive computation time. To avoid this difficulty, we have adapted a discretization method proposed by (40) and known as particularly efficient and rapid in the aggregation–fragmentation problems.

FIG. 15. Comparison between the experimental and two simulated l˜−1 (t) curves for two values of the aggregate fractal dimension. Experimental conditions: pH = 6.0; T = 25◦ C; 1084 rpm; φ = 1.6 × 10−5 ; I = 0.15 mol.l−1 ; wavelength, 652.1 nm.

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FIG. 16. Comparison between experimental and simulated inverse transport mean free path l˜∗−1 (t) for short aggregation time in more concentrated suspensions. Experimental conditions: pH = 4.0; T = 25◦ C; φ = 1.25 × 10−3 ; I = 0.10 mol.l−1 ; wavelength, 652.1 nm; aggregate fractal dimension, Df = 2.4. (a) 175 rpm; (b) 343 rpm; (c) 610 rpm; (d) 1084 rpm; (e) 1500 rpm.

of the previous section i − 1 by the relation Vi = 2Vi−1 (V1 corresponding to the volume of a primary particle). For high solid volume fractions and stirring rates, the aggregates are small. Then, a more accurate size discretization is necessary: in this case, section i contains only identical aggregates with i primary particles. The so-obtained ordinary differential equations system has been solved using the explicit Euler method (which was revealed to be as efficient as the Runge–Kutta algorithm, currently used in this sort of problem).

For each calculated PSD, the corresponding suspension optical properties can be determined using Eqs. [18] and [20]. The validation of the different models and the estimation of the unknown parameters were carried out by comparison between experimental and simulated X (t) at a given wavelength: X = l˜−1 at 652 nm for diluted suspension X = l˜∗−1 at 632.8 nm for concentrated suspension.

TURBULENT AGGREGATION OF TITANIA

COMPARISON OF EXPERIMENTS AND SIMULATIONS—DISCUSSION

Diluted Suspensions In Fig. 12 the experimental and simulated inverse mean free paths l˜−1 are compared for different Df values. The best agreement is obtained for Df = 2.4, which is the fractal dimension expected for turbulent aggregation (56). The simulations shown in Fig. 12 do not include the fragmentation process (aL → ∞). The effect of a maximum aggregate size can be seen in Fig. 13. A good agreement is obtained if we take aL = 165γ˙ −0.6 . al

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Remark: The oscillations of turbidity observed in Fig. 16a are not explained so far. Their reality is not certain. However their existence, only at low stirring rate, could be linked to an hydrodynamic phenomenon. Figures 17a–17c correspond to long time behaviors. Other experimental and simulated curves obtained for φ = 7.81 × 10−5 and 3.14 × 10−4 can be found in (63). In all the simulations,

[22]

The exponent value in this power law is in the usual range, 0.5– 0.8 (37, 41, 62). Figures 14a–14d represent experimental and simulated l˜−1 (t) curves for various stirring rates and for φ = 2 × 10−6 . Experiments and simulations have been performed for other solid volume fraction values (φ = 8 × 10−6 , 1.6 × 10−5 , 3.2 × 10−5 ); they can be found in (63). In all cases, the same law for limit aggregate size [22] is used. Fractal dimension value is fixed to 2.4 for turbulent aggregation and 1.8 for aggregation in a still medium (23). Agreement is good for the smallest solid volume fraction values (2 × 10−6 and 8 × 10−6 ). However, for high φ(1.6 × 10−5 , 3.2 × 10−5 ) and ω (610, 1084 rpm) values, the simulated aggregation rate in the early stage of the process is higher than the experimental value. However, experimental and simulated final states are in good agreement. This discrepancy can be explained as follows: As restructuring occurs throughout the aggregation process, we have to consider a sort of competition between aggregation and restructuring. For small φ, the aggregation rate is smaller than the restructuring rate; thus, the aggregates have time for restructuring and rapidly tend to a more compact structure, the porosity of which does not change with time. As φ increases, aggregates appear faster and have no time to undergo restructuring before the next collision. Thus, their porosity will slowly decrease with time. Figure 15 shows the experimental curve and two simulated curves obtained for Df equal respectively to 2.2 and 2.4. As expected, Df = 2.2 (resp. 2.4) allows the best fit for small (resp. large) times. This confirms the assumption of change of the fractal dimension due to the slow restructuring of the aggregates. The fractal dimension ranges between 2.2 and 2.4. This is in agreement with other works (33, 56). Concentrated Suspensions Figures 16a–16e represent experimental and simulated l˜∗−1 (t) for various stirring rates and for φ equal to 1.25 × 10−3 for short time.

FIG. 17. Comparison between experimental and simulated inverse transport mean free path l˜∗−1 (t) for long aggregation time in more concentrated suspensions. Experimental conditions: pH = 4.0; T = 25◦ C; φ = 1.25 × 10−3 ; I = 0.10 mol.l−1 ; wavelength, 652.1 nm; aggregate fractal dimension, Df = 2.4. (a) 175 rpm and 1084 rpm; (b) 343 rpm and 1500 rpm; (c) 610 rpm.

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aL (γ˙ ) is expressed by the equation aL = 9.16γ˙ −0.22 . al

[23]

This law has been derived from the size measurements obtained by laser diffraction. As short-range repulsive force decreases the aggregation rate a little, the Brownian and turbulent kernels have been multiplied by a factor equal to 0.6. Fractal dimension is fixed to 2.4. Simulations correctly describe experimental results at short and long times. Nevertheless, we observe some discrepancy at intermediate times, for which the experimental aggregation rate is smaller as expected. We can explain this behavior as follows: at pH 4 the occurrence of short-range repulsive force makes certain aggregates not durable in the presence of shear flow. Thus, collision efficiency becomes smaller as expected for large aggregates. CONCLUSION

Titania aggregation has been investigated in a wide range of solid volume fractions. This study has been achieved by means of in situ optical characterization methods. Two subranges of solid volume fractions were examined. The first one (2 × 10−6 –2 × 10−5 ) corresponds to a diluted suspension for which spectral turbidimetry is particularly suitable. Aggregation was carried out under physicochemical conditions corresponding to only attractive forces between particles. Experimental results can be explained in the framework of Kusters’s aggregation modeling. Aggregates were considered as fractal-like with a fractal dimension equal to 2.4. Modeling does not involve a fragmentation kernel, but a zero collision efficiency, when the resulting aggregate size is larger than a maximum value aL , which is given by the relation aL = 165γ˙ −0.6 . al Rapid restructuring always occurs after collision and aggregate formation. However, as the solid volume fraction increases, the aggregation rate increases too and the aggregate restructuring is slower. Then, aggregates are more porous and their fractal dimension tends to 2.2. The second solid volume fraction range (5 × 10−5 –5 × 10−3 ) corresponds to more concentrated suspensions for which light backscattering measurements are suitable. Aggregation was carried out under physicochemical conditions corresponding to long-range attractive forces and short-range repulsive forces. Kusters’s modeling can once again be used with the same fractal dimension as previously. Doing so, the experimental behaviors corresponding to the early and final stages of the process can be understood. The maximum aggregate size is now given by the relation aL = 9.16γ˙ −0.22 . al

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