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Shear-Induced Aggregation of Anatase Dispersions Investigated by Oscillation and Low Shear Rate Viscometry
Journal of Colloid and Interface Science 242, 82–89 (2001) doi:10.1006/jcis.2001.7764, available online at http://www.idealibrary.com on
Shear-Induced Aggregation of Anatase Dispersions Investigated by Oscillation and Low Shear Rate Viscometry Jan Gustafsson,1 Erik Nordenswan, and Jarl B. Rosenholm ˚ Akademi University, FIN-20500 Abo, ˚ Department of Physical Chemistry, Abo Finland Received December 28, 2000; accepted June 8, 2001; published online August 22, 2001
sion near the IEP, where the particles or flocs attract each other. Obtaining the interaction potential between particles from the ζ potential is therefore of direct relevance to the colloidal stability and the flow properties of a dispersion. Measurements and correlation of the ζ potential (or surface charge) and rheology have also been presented by others for kaolinite (4), zirconia (5), silica (6), alumina (6, 7), and titania (6, 8) dispersions. Different measuring techniques and approaches were used, but correlations obtained were similar, with the highest resistance to shear found at or close to the IEP. In the previous papers by Kosmulski et al. (1, 2), rheology was measured in a traditional way, that is, the shear stress (τ ) was measured as a function of shear rate (γ˙ ) at stepwise increasing and decreasing shear rate. They discovered a local maximum in the shear stress at around 0.5–1 s−1 on the increasing shear rate curve but not on the decreasing shear rate curve. This phenomenon is known as stress overshoot. In contrast to using the Bingham model, using this stress overshoot as a measure of the flow properties is a direct method, that is, no extrapolations are needed to obtain the yield stress. It is noteworthy that the experimental results, measured at low or medium shear rate, gave a poor fit to the Bingham model. The Bingham yield stress value is obtained by extrapolating the linear part of the shear stress at the high shear rate end of the rheogram to zero shear rate. Moreover, results obtained at high shear rate would not give any information about the kinetics of the aggregation. A similar stress overshoot has also been found for suspensions of clay by Pignon et al. (9), for silica particles by Kawaguchi et al. (10), and in our laboratory for calcium carbonate (11) and alumina (unpublished results). The presence of stress overshoot was believed to originate from the balance between aggregation and breakup of a network structure in the dispersion. Since many colloidal materials aggregate during shearing, the build up of these time-dependent shear-induced structures has been investigated, for example, for micelles (12, 13), silica particles (14), and latex (14, 15). Different patterns in the rheological behavior have been distinguished for the various measured samples. Hu et al. (12) found a plateau value in the viscosity as a function of time for micellar solution. Lee et al. (16) observed a shear thickening behavior (increasing viscosity) at shear rates of the order of 100 s−1 for silica suspensions. Myska and Stern (13)
The evolution of shear-induced network structures of aggregates in concentrated anatase dispersions was studied in a Couette cell by using long sequence times, low constant shear rates, and singlefrequency oscillation at various shear stresses. The ζ potential was used as a measure of the interaction forces between the particles. Dispersions at the vicinity of the isoelectric point were observed to aggregate during the early stages of shearing in a Couette cell. The aggregates formed a network of new weak interaction bonds. Further shearing caused a breakup of the network structure. The rheological response to this process was a peak in the shear stress (the structural yield stress) as a function of cumulative shear strain at sufficiently low constant shear rate. The observed structural yield stress increased with decreasing shear rate and was an indication of the induction of a network structure. The behavior of unstable flocculated dispersions at low shear rates was noticed to be strongly dependent of the shear rate history. Single-frequency oscillation measurements were performed and repeated after short intervals of monotonic shearing. The elastic modulus G 0 increased rapidly (exceeding G 00 ) after a certain length of shearing, indicating the growth of a structure possessing weak gel-like properties. ° C 2001
Academic Press
Key Words: shear-induced aggregation; anatase; concentrated dispersion; yield stress; oscillation; isoelectric point.
INTRODUCTION
Kosmulski et al. (1, 2) have in recent papers observed a maximum in the shear stress at low shear rates in a shear rate sweep test with TiO2 dispersions. The maximum shear stress was designated “yield stress” and was found to be ζ potential dependent. At the vicinity of the isoelectric point (IEP) the yield stress experienced a maximum. The surface is neutrally charged and van der Waals attraction dominates the particle interaction at the IEP. According to the DLVO theory (3) the total interparticle potential energy between two particles is the sum of van der Waals attraction (which flocculates the dispersion) and the electrical double-layer repulsion (which stabilizes the dispersion). The resistance against flow is therefore stronger for a disper1 To whom correspondence should be addressed. Fax: 358-2-215 4706. E-mail: [email protected].
0021-9797/01 $35.00
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and Liu and Pine (17), on the other hand, found a maximum in the viscosity at around 10 s−1 for micellar solution suspensions. Cabane et al. (18) noticed that above a critical shear rate a shearinduced gelation occurs, leading to an increase in viscosity for silica dispersions by adsorbing macromolecules and resulting in an unsaturated (flocculated) state. The aggregation of latex particles in Couette shear cells has recently been widely studied by several authors (19–22) who characterized the structures of aggregates by a fractal dimension. Hansen et al. (22) concluded that aggregation of monodisperse systems proceeds through homogenous aggregation, with large clusters sticking to large clusters, while for bidisperse systems heterogeneous aggregation was more dominant. When measuring the kinetics of shear-induced aggregation for flocculated dispersions the measurement procedure may be of great importance and the results obtained can depend on the parameters used. It is also important to find under which conditions network structures develop when studying sedimentation (23), since they enhance the sedimentation rate and leave a particlefree supernatant. Network structures have stronger resistance against centrifugation and pressure filtration and may also be interesting for colloidal processing of ceramic bodies because they are rigid at low shear rates. However, these structures show poorer consolidation behavior than stabilized dispersions (24), since they create loose porous bodies. Therefore it is also of importance from a practical point of view to obtain knowledge of structures of aggregates, as they may negatively affect the quality in practical applications, for examples, in the coating of ceramic honeycombs (25). Most of the studies on shear-induced structures have been performed with cone–plate or Couette systems. The Couette system was chosen for this work since this geometry shows the interesting rheological behavior of narrow two-dimensional shear zones. The aim of this work is to relate shear-induced aggregation to the ζ potential. Of particular interest is also to gain some understanding at which shear rates the aggregation and breakup occurs in the shear cell. The interpretation of a rheogram may be misleading if aggregation is not considered. The low shear rate rheology may also serve as a link between rheology and sedimentation by gravity, since in both cases small forces are applied on the network structure. Parallel investigations of sedimentation have been made and will be discussed in a forthcoming paper (26). Despite the extensive research on dispersions, few investigations on aggregation and breakup of network structures in “electrostatically” flocculated particle dispersions have combined rheology with ζ potential. EXPERIMENTAL
Materials The titanium dioxide (anatase from Aldrich) was of 99.9% purity. The chosen anatase has been thoroughly investigated (1, 27). The powder has been characterized by ESCA (not
shown), which detected phosphate added in the manufacturing stage. No attempt was made to remove the phosphates. The particles were quite spherical and rather monodisperse around 0.3 µm. Hydrochloric acid (analytical concentrate) and sodium hydroxide pellets (both from J. T. Baker) were used in aqueous solutions of 1 mol dm−3 to adjust the pH. The water used was distilled and ion exchanged (Milli-Q) and had a conductivity of 0.054 µS cm−1 . Rheological Measurements A Bohlin VOR (Bohlin Reologi) rheometer was used with a C14 (Couette) cup–piston system with a gap of 0.5 mm. Two different torsion bars were used; a 0.0126 mNm (12.4 g × cm) torsion bar was used in the steady shear rate experiments and a 0.00427 mNm (4.19 g × cm) torsion bar was used in the combined oscillation/steady shear rate experiments. The dispersions were prepared separately in airtight polycarbonate bottles, and stirred for 1 h prior to the experiment. The solids content was 38 wt% (φ = 14 vol%). Since some of the rheological experiments took several hours to complete, the pH was followed during the experiment as a function of time in a separate bottle to ensure that no leakage of ions from the particle surface took place, which would have changed the ζ potential. The pH was observed to remain constant within a range of ±0.1 units. Separate sedimentation tests showed that very slow or negligible sedimentation occurred at this solids content due to hindered settling. In steady shear rate measurements the outer cylinder was rotated with known shear rates and the stress of the sample which acts on the inner cylinder results in a displacement (twist) of the torsion bar with known stiffness connected to the inner cylinder. The shear stress, τ , at an applied shear rate, γ˙ , was measured from this displacement. The viscosity, η, can then be calculated from the shear stress by dividing by the applied shear rate: η=
τ . γ˙
[1]
Normally, a viscometry measurement with a Bohlin VOR was carried out according to the following procedure: At a chosen shear rate the sample has time to equilibrate itself during a defined delay time, after which the recording takes place during a defined measuring time. After this procedure the shear rate is increased or decreased to the next shear rate and the procedure is repeated. The (delay + measuring) time will in this paper be referred to as “sequence time.” Wall slip may occur during the measurements, but this was not corrected for. When the particles have a low affinity to the wall of the measuring cell a thin layer of particle-free fluid may develop, which causes the slip. The magnitude of the slip is dependent on the thickness of the slip layer and results in a discontinuous drop in viscosity, especially at low stress (28). Wall slip would therefore be expected to increase toward low shear rates. The shear stress was measured at 25◦ C as a function of increasing shear rate at both normal and extended sequence times.
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The maximum observed shear stress (the peak value) will be referred to as “yield stress.” The shear stress was also measured as a function of time at constant shear rates ranging from 0.0073 s−1 to 1.16 s−1 . The maximum observed shear stress for the constant shear rates will be referred to as “structural yield stress,” to distinguish the two methods. Single-frequency oscillation measurements were performed after preset times of the continuous shearing to study the process of structure changes at corresponding shear stresses. The oscillation measurements were performed according to the following procedure: After shearing at a constant shear rate for a preset time, the shearing was stopped and the oscillation was performed, after which the shearing was continued in the original direction. The frequency used was 0.5 Hz and the amplitude was 0.3 mrad (1.5%). With these parameters the oscillation movements were so small and fast that they did not disturb the sample, which was verified by parallel tests without oscillation. In oscillation (dynamic) measurements a sinusoidal strain with known amplitude, γ 0 , was applied and the resulting sinusoidal stress amplitude, τ 0 , was measured. The phase angle shift, δ, is zero for a perfectly elastic Hookean solid and π/2 radians for a perfect viscous Newtonian liquid. For a viscoelastic material the phase angle shift is in between these two extremes. The complex modulus, G ∗ , is the ratio between the stress and strain amplitude: G ∗ = τ 0 /γ 0 .
[2]
The elastic (or storage) modulus G 0 and the viscous (or loss) modulus G 00 can be determined from the complex modulus and the phase angle shift: G 0 = G ∗ cos δ, 00
∗
G = G sin δ.
RESULTS AND DISCUSSION
ζ Potential and Particle Size Since the phosphate impurities observed with ESCA are chelated to the surface and therefore determine the ζ potential, the isoelectric point was found at very low pH values, as expected (see Fig. 1). Thus, the ζ potential was highly negative at neutral pH and the dispersion was electrostatically stable at low salt concentrations. The ζ potential at native condition (pH = 6.7) was −60.6 mV and the IEP was found at pH = 1.7. The particle size determined by the Acoustosizer at the pH where the dispersion was stable, pH = 3.5–6.8, was quite monodisperse around 0.35–0.4 µm, which was in accordance with previous results (27, 30). The ζ potential at the constant pH values used in the rheology measurements are presented in Table 1. Dependency of the Yield Stress on the Sequence Time The shear stress was found more appropriate for presenting the results than (shear) viscosity because viscosity for concentrated particle dispersions showed both a strong shear rate dependency (pseudoplastic behavior) and a shear rate history dependency (thixotropic behavior). Using viscosity would therefore in itself give a multiplied effect when comparing the rheological data with different shear rates, since the viscosity is obtained by dividing the shear stress by the shear rate according to Eq. [1]. The apparent (shear) viscosity observed actually changed by 4 orders of magnitude for the same dispersion with increasing shear rate. In traditional viscometry the obtained shear stress versus shear rate curve is normally expected to be dependent only of the shear rate. Figure 2a shows, however, that the yield stress increased with increased sequence time. At the same time the position of the peak moved toward lower shear rates. This rheological behavior proves clearly that there exists a shear rate history dependency. At longer sequence times the structure of aggregates had a longer time to build up a strong network,
[3] [4]
ζ Potential Measurements The ζ potential and the particle size were determined at 25◦ C using an Acoustosizer (MATEC Instruments). In the experiments 10 wt% powder (φ = 2.8 vol%) was dispersed in aqueous solution and was treated in an ultrasonic bath for 5 min to break up aggregates in the dispersion. Afterwards the dispersion was stirred for 1 h. The ζ potential was measured under stirring from native state (neutral pH) to pH = 1.5 and back by automatic titration with 1.0 M HCl/NaOH. The ζ potential calculated by the Acoustosizer from the acid titration curve was determined by the equation developed by O’Brien et al. (29). No attempts were made to subtract the background signal from the data received, since the electrolyte concentration was rather low.
FIG. 1. The ζ potential as a function of pH.
SHEAR-INDUCED AGGREGATION OF ANATASE DISPERSIONS
TABLE 1 The (O’Brien) ζ Potential at the Constant pH Values Used in Rheology Obtained from Fig. 1 pH
ζ /mV
2.2 2.7 3.7 4.5 5.1
−13.6 −22.4 −35.8 −47.9 −51.9
because the network building took place at low shear rates. This will be discussed further below. In Fig. 2a the “10 s” sequence time was chosen to represent the sequence time normally used. It should be noted that the yield stress only occurred on the increasing shear rate curve, an example of which is shown in Fig. 2b. A high shear rate increased the breakup rate and the aggregates in the dispersion were broken down to smaller units. This resulted in the hysteresis (the difference in shear stress at increasing and decreasing shear rate) observed in Fig. 2b. Hu and Matthys (31) found that surfactant solutions need a certain rest time in order to reconstruct the original state of the solution. A similar behavior can probably be expected for anatase. Instead, a fresh undisturbed sample was used for every measuring sequence. It should also be pointed out that the shear rate at which the yield stress occurred was only strongly dependent on the sequence time and not on the ζ potential (not shown). For the solids content examined, the strong networks were only found for dispersions that were close to the isoelectric point. Further away from the isoelectric point the dispersion had a rheological behavior closer to a Newtonian fluid, since the electrostatic forces were strong enough to stabilize the dispersion.
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induced by shearing. A similar phenomenon with a maximum shear stress at constant shear rate measurements (γ˙ = 100 s−1 ) was also observed by Hu and Matthys (31) in their rheological study of surfactant solutions. The maximum or overshoot in stress increased with decreasing rest time after a constant preshearing. They concluded that the overshoot indicated a build up of a shear-induced structure, which after a rest time returns to almost the original state. The shear stress in continued shearing decreased after reaching its maximal value (see Fig. 3a), due to breakup and rearrangement of the aggregate structure. It can also be seen from Fig. 3a that there was a difference in the slopes of the descending part of the curves for the constant shear rates. The structures possessed different strengths due to dissimilar shear rate histories. The curve was steeper at lower shear rates, which indicated
The Low Constant Shear Rate The shear stress was recorded as a function of time for various constant shear rates. To compare the constant shear rates ranging over three decades of magnitude we chose to plot the shear stress as a function of the cumulative shear strain. The cumulative shear strain, γcum , is the length (in radians) the shearing has proceeded and is defined as Zt γcum =
γ˙ dt,
[5]
0
where γ˙ is the shear rate and t is the elapsed time. Figure 3a shows the shear stress (the left y-axis) obtained from four of the constant shear rates at a constant ζ potential of −22.4 mV. It can clearly be seen that for the three lowest constant shear rates the structural yield stress occurs at almost the same strain, around 20–40 rad, regardless of the applied shear rate. The structural yield stress was, however, higher at lower shear rates. The value of the structural yield stress is a measure of the strength of the aggregate network, which has been
FIG. 2. (a) Shear stress as a function of the increasing shear rate at different sequence times. (b) Shear stress as a function of the increasing and decreasing shear rate showing the stress overshoot and hysteresis.
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that the breakup rate (as a function of the length of shear) was higher at lower shear rates, owing to the more rigid structure of the network induced by the low shear rate. This was believed to reflect the higher rigidity and consequently more brittle structures. Brownian motion would probably compete with shearing at a shear rate of less than 10 s−1 according to Serra et al. (20). Since the shear rates used in our measurements were even lower, we needed a value that would predict if the process was orthokinetic (induced by shear) or perikinetic (quiescent). We tried two methods of calculating the ratio between Brownian motion and shearing, one using the P´eclet number and the other using the relative particle displacement ratio. The P´eclet number, Pe, is described as the ratio of the viscous force experienced by a particle to the Brownian force (32) and is defined as (22) Pe =
3π ηa 3 γ˙ , kT
[6]
where η is the viscosity of the dispersion, a is the particle radius, γ˙ is the shear rate, and kT is the thermal energy. Since the shear stress, τ , is equal to the shear viscosity multiplied by the shear rate obtained from Eq. [1], the P´eclet number is linearly correlated to the shear stress for an experiment with constant shear rate. This would for our measurements give P´eclet numbers from approximately 100 up to almost 1000; these data are plotted in Fig. 3a as the right y-axis. This indicated that the influence of Brownian motion on the flocculation kinetics was negligible, but then again the P´eclet number may not be the right choice for these strongly flocculated dispersions with high solids content. For the second method, we theoretically calculated the relative particle displacement ratio, which is the ratio of the length a particle has moved during a specific time induced by the Brownian motion to that induced by the constant shearing. This was performed as follows: Einstein’s equation gives the mean Brownian displacement, x¯ , of a particle from its original position along a given axis after a time t and is defined as (33) x¯ = (2Dt)1/2 ,
[7]
where D is the diffusion coefficient. The diffusion coefficient D is defined as (33) D=
FIG. 3. (a) Shear stress (left axis) and the P´eclet number (right axis) as a function of cumulative shear strain for constant shear rates. (b) Same as Fig. 3a but with x-axis lengthened showing the slow declination of the shear stress at high constant shear rate. (c) Shear stress as a function of time for the constant shear rates in Fig. 3a.
kT , 6π ηa
[8]
where kT is the thermal energy, η is the viscosity of the solution, and a is the particle radius. Next we calculate the relative displacement ratio of particles in a shear field. After calculating the mean distance, dm , between the centers of two particles from the solids content, the distance, ds , a particle has moved during a time t due to shearing with a constant shear rate γ˙ can
SHEAR-INDUCED AGGREGATION OF ANATASE DISPERSIONS
TABLE 2 The Relative Particle Displacement Ratio and the P´eclet Number for the Various Constant Shear Rates Constant shear rate (s−1 )
Relative particle displacement ratio
P´eclet number for a particle in an infinitely diluted suspension
0.0073 0.0116 0.0461 0.0921 0.116 0.292 0.731 1.16
0.11 0.18 0.72 1.44 1.81 4.55 11.40 18.09
7.99E-05 1.27E-04 5.05E-04 1.01E-03 1.27E-03 3.20E-03 8.01E-03 1.27E-02
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aggregation is reduced and the aggregation proceeds mainly through particle attachment to growing clusters. To get a more complete picture of the breakup Fig. 3c shows the shear stress plotted as a function of time for the four constant shear rates shown in Fig. 3a. From Fig. 3c it can be deduced that the rate of the breakup was slower at the lowest shear rates and becomes faster at higher shear rates. Relationship between Structural Yield Stress and Constant Shear Rate The structural yield stress is plotted in Fig. 4a as a function of constant shear rate for the ζ potential values used. The value for the critical shear rate, 1.16 s−1 , is not a real structural yield stress value but the plateau value of the shear stress at 20–40 rad of shear strain (see Fig. 3a), used only as a comparison with
be obtained from ds = dm γ˙ t.
[9]
Dividing x¯ from Eq. 7 by ds we can obtain the relative particle displacement ratio between Brownian motion and shearing. These values are collected for the constant shear rates studied in Table 2. Also collected in Table 2 are the theoretical P´eclet numbers calculated for particles in an infinitely diluted suspension at the constant shear rates used. The very low values for the P´eclet number shows that the flocculation would only be induced by Brownian motion at diluted conditions. This was not, however, observed for our concentrated dispersions. Using the relative particle displacement ratio indicates that the Brownian motion was more dominant at the three lowest shear rates, but this was expected to be true only at the absolute beginning of the experiment. The method using the relative particle displacement ratio gives only theoretical values, which are valid for unflocculated particle dispersions (lattice model). During the evolution of the aggregation process the role of the Brownian motion was reduced, since the amount of free particles diminished drastically. It seems that neither the P´eclet number nor the relative particle displacement ratio showed an exact measure of the influence of Brownian motion on the kinetics of aggregation. Brownian motion may still influence the results, leading to scatter in the measurements, as discussed below. At a critical shear rate and above it (γ˙ = 1.16 s−1 in Fig. 3a) no structural yield stress was observed as a clear peak value. Instead, a plateau was found. At high shear rates the breakup of the network required more shearing than at low rates. The shear stress decreased to a shear stress value of 1.4 Pa after shearing 750 rad, which is indicated in Fig. 3b. A strong structure of aggregates would at these conditions never be subjected to sufficient cumulative shearing to be built up before it was broken down. The growth and breakup of the network clearly follow different shear rate dependencies. Axford (34 ) stated that the probability of bond formation during cluster collision decreases at higher shear rates. Despite the increase in overall collision frequency and rate of formation of large clusters, homogenous
FIG. 4. (a) Structural yield stress as a function of constant shear rate for the ζ potentials used. (b) Structural yield stress as a function of ζ potential for the constant shear rates.
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GUSTAFSSON, NORDENSWAN, AND ROSENHOLM
the real structural yield stress values. The structural yield stress increased with decreasing constant shear rate, but at the two lowest shear rates the inhomogeneities of the samples are more pronounced, especially for unstable dispersions with low ζ potential. The scatter was therefore large for these dispersions, due to network instabilities at high stresses or competing flocculation induced by Brownian motion. The uncertain values are therefore indicated with dotted lines in Fig. 4a. A fairly reliable trend can still be obtained. A maximum structural yield stress of about 70 Pa was found for dispersions close to the IEP. This upper limit is a measure of the finite strength of the structure of aggregates. The limit is due to the weakness of the attractive interaction and the changing geometrical orientation of the bonds between the particles. For the “−51.9 mV” experiment, the structural yield stress was negligible for all but the two lowest shear rates. At these conditions the repulsion forces were already sufficiently high to prevent the aggregate structure from being built up or from having any significant strength. In Fig. 4b the yield stress from Fig. 4a is plotted against the ζ potential for the constant shear rates. The structural yield stress increased successively with decreasing ζ potential. The trends for the two lowest constant shear rates (0.0073 s−1 and 0.0116 s−1 ) are indicated with dotted lines due to scatter in the results. Relationship between Yield Stress and ζ Potential at Different Sequence Times In Fig. 5 the yield stress is shown as a function of ζ potential obtained by measurements with various sequence times (at increasing shear rate). Because of the use of increasing shear rate the yield stress values were lower than with the constant shear rate experiments (see Fig. 4b). When the shear rate was increased stepwise the structure was weaker than when using a single and hence lower shear rate. The shear rate history was different from
FIG. 6. Shear stress (filled squares) as a function of cumulative strain by shearing at 0.0461 s−1 . G0 (filled circles) and G00 (open circles) at 0.5 Hz measured at the corresponding cumulative shear strain and shear stress as indicated in the figure.
that of the constant shear rate experiments. The contribution of the first (lowest) shear rate value of the shear rate sweep (seen from Fig. 2a) was of minor significance, since the low shear rates contributed only with very small shear strain increments. The yield stress was more effected by extended sequence time at ζ potential values close to zero than further away from it. However, the two longest sequence times showed a decline in the yield stress before reaching the IEP. This was not expected, since the maximum yield stress was found for low electrolyte concentrations approximately at the IEP (1). The yield stress at different sequence times may not have the same ζ potential dependency though, owing to slightly different mechanisms in the aggregation. Two different solids contents were required for the ζ potential and rheology measurements, a low one for ζ potential and a high one for rheology. Hence, the interparticular distances at which they were measured were not the same. The Influence of G 0 and G 00 on Monotonic Shearing
FIG. 5. Yield stress as a function of ζ potential for the sequence times used.
Figure 6 shows a combined shear and oscillation measurement. The experiment was carried out according to the following procedure: A constant shear rate of 0.0461 s−1 was applied and the shear stress was measured. The shearing was stopped and an oscillation measurement with 0.5 Hz was performed, after which the shearing was continued. Before the first and after every shearing step an oscillation measurement was performed. The oscillation was adjusted so that it did not modify the state of the shear-induced aggregation. This was compared and verified by shearing with the same shear rate without oscillation. The program was set to auto-zero itself before every new shearing, which is the reason for the low starting value for the otherwise high stress shearing. The viscous (loss) modulus G00 followed the development of the shear stress with quite good accuracy. We believe that the viscous component was mainly a measure of the energy consumed to break up the aggregates. The higher shear
SHEAR-INDUCED AGGREGATION OF ANATASE DISPERSIONS
stress indicated that the number of interaction bonds between the aggregates increased and therefore the number of bonds that were broken down also increased. In the initial phase of the shearing the elastic modulus G0 increased negligibly but showed a rapid increase after a certain number of interaction bonds developed. This increase of G0 and the fact that G0 exceeded G00 indicated that the structure was growing more rigid and becoming gel-like during shearing. Faers and Luckham ( 35) have illustrated the contribution to G0 of the three-dimensional floc network of polystyrene latex particles. The more flocculated network gave an increased G0 . This was in good agreement with our experience. The good correlation between the oscillation and constant shear measurements would also support the prediction that the stress overshoot phenomenon was caused by a structure buildup. Furthermore it is argued that wall slip, although presumably to some extent present in the measurements, would not change the shape of the shear stress curve, only the magnitude of stress. As the test in Fig. 6 proceeded G0 decreased faster than both 00 G and the shear stress. This indicated that the dispersion lost its elasticity, when the breakup rate of the interaction bonds exceeded the growth rate. This took place at the observed peak value of the shear stress (i.e., at the structural yield stress). CONCLUSIONS
The investigated flocculated anatase dispersions showed at moderately low shear rates not only a shear rate dependency but also clearly a shear rate history dependency. Both the growth and breakup of the aggregates are dependent on the shear rate but the dependencies are different. At low shear rate the growth was favored over breakup, which leads to stronger aggregate structures. At higher shear rates the strong network structure did not have time to build up. The observed structural yield stress increased with decreasing constant shear rates. The yield stress increased with longer sequence time and also toward lower ζ potential (pH closer to the IEP). The (structural) yield stress was an indication of the strength of the shear-induced aggregate network at the specific applied conditions (shear rate, sequence time, and ζ potential). The elastic properties of the dispersion were studied by oscillation measurements. The increase of the elastic modulus G0 to values greater than the viscous modulus G00 indicated that the network structure became gel-like. This elasticity occurred when the shearing had proceeded long enough to aggregate a sufficient amount of interaction bonds. The dispersion lost its elasticity when the breakup rate of the bonds exceeded the growth rate. The viscous modulus G00 followed the development of the shear stress and was believed to correspond to the energy consumed to break down the bonds. The traditional presentation of shear rate dependency can be misleading as the results strongly depend on the measurement procedure and the shear rate history. The methods presented here can also be used for studies of flocculation kinetics as these too are sensitive to shear rate.
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ACKNOWLEDGMENTS The Technology Development Centre of Finland (TEKES) and The Ministry of Education (Finland) (Graduate School of Materials Research) are acknowledged for financial support.
REFERENCES 1. Kosmulski, M., Gustafsson, J., and Rosenholm, J. B., J. Colloid Interface Sci. 209, 200 (1999), doi:10.1006/jcis.1998.5884; Erratum, J. Colloid Interface Sci. 233, 367 (2001), doi:10.1006/jcis.2000.7332. 2. Kosmulski, M., Gustafsson, J., and Rosenholm, J. B., Colloid Polym. Sci. 277, 550 (1999). 3. Verwey, E. J. W., and Overbeek, J. Th. G., “Theory of the Stability of Lyophobic Colloids.” Elsevier, Amsterdam, 1948. 4. Hunter, R. J., and Nicol, S. K., J. Colloid Interface Sci. 28, 250 (1968). 5. Leong, Y. K., Scales, P. J., Healy, T. W., Boger, D. V., and Buscall, R., J. Chem. Soc., Faraday Trans. 89, 2473 (1993). 6. Tomb´acz, E., Szekeres, M., Kert´esz, I., and Turi, L., Prog. Colloid Polym. Sci. 98, 160 (1995). 7. Chang, J. C., Lange, F. F., and Pearson, D. S., J. Am. Ceram. Soc. 77, 19 (1994). 8. Morris, G. E., Skinner, W. A., Self, P. G., and Smart, R. St. C., Colloids Surf. A 155, 27 (1999). 9. Pignon, F., Magnin, A., and Piau, J.-M., J. Rheol. 42, 1349 (1998). 10. Kawaguchi, M., Mizutani, A., Matsushita, Y., and Kato, T., Langmuir 13, 6339 (1997). 11. Nystr¨om, R., Linden, M., and Rosenholm, J. B., J. Colloid Interface Sci., in press (2001). 12. Hu, Y. T., Boltenhagen, P., and Pine, D. J., J. Rheol. 42, 1185 (1998). 13. Myska, J., and Stern, P., Colloid Polym. Sci. 276, 816 (1998). 14. Buscall, R., Mills, P. D., Goodwin, J. W., and Lawson, D. W., J. Chem. Soc., Faraday Trans. 84, 4249 (1988). 15. Sonntag, R. C., and Russel, W. B., J. Colloid Interface Sci. 116, 485 (1987). 16. Lee, J.-D., So, J.-H., and Yang, S.-M., J. Rheol. 43, 1117 (1999). 17. Liu, C.-H., and Pine, D. J., Phys. Rev. Lett. 77, 2121 (1996). 18. Cabane, B., Wong, K., Lindner, P., and Lafuma, F., J. Rheol. 41, 531 (1997). 19. Oles, V., J. Colloid Interface Sci. 154, 351 (1992). 20. Serra, T., and Casamitjana, X., J. Colloid Interface Sci. 206, 505 (1998). 21. Le Berre, F., Chauveteau, G., and Pfefferkorn, E., J. Colloid Interface Sci. 199, 13 (1998). 22. Hansen, P. H. F., Malmsten, M., Bergenst˚ahl, B., and Bergstr¨om, L., J. Colloid Interface Sci. 220, 269 (1999), doi:10.1006/jcis.1999.6531. 23. Buscall, R., and McGowan, I. J., Faraday Discuss. Chem. Soc. 76, 277 (1983). 24. Bergstr¨om, L., Schilling, C. H., and Aksay, A., J. Am. Ceram. Soc. 75, 1 (1992). 25. Agrafiotis, C., Tsetsekou, A., and Leon, I., J. Am. Ceram. Soc. 83, 1033 (2000). 26. Gustafsson, J., Nordenswan, E., and Rosenholm, J. B., J. Eur. Ceram. Soc., submitted (2001). 27. Kosmulski, M., and Rosenholm, J. B., J. Phys. Chem. 100, 11681 (1996). 28. Russel, W. B., and Grant, M. C., Colloids Surf. A 161, 271 (2000). 29. O’Brien, R. W., Cannon, D. W., and Rowlands, W. N., J. Colloid Interface Sci. 173, 406 (1995). 30. Kosmulski, M., Durand-Vidal, S., Gustafsson, J., and Rosenholm, J. B., Colloids Surf. A, 157, 245 (1999). 31. Hu, Y., and Matthys, E. F., J. Rheol. 41, 151 (1997). 32. Barnes, H. A., Hutton, J. F., and Walters, K., “An Introduction to Rheology,” p. 126. Elsevier Science, Amsterdam, 1989 . 33. Shaw, D. J., “Introduction to Colloid and Surface Chemistry,” pp. 22–23. Butterworths, London, 1980. 34. Axford, S. D. T., J. Chem. Soc., Faraday Trans. 92, 1007 (1996). 35. Faers, M. A., and Luckham, P. F., Langmuir 13, 2922 (1997).
Shear-Induced Aggregation of Anatase Dispersions Investigated by Oscillation and Low Shear Rate Viscometry Jan Gustafsson,1 Erik Nordenswan, and Jarl B. Rosenholm ˚ Akademi University, FIN-20500 Abo, ˚ Department of Physical Chemistry, Abo Finland Received December 28, 2000; accepted June 8, 2001; published online August 22, 2001
sion near the IEP, where the particles or flocs attract each other. Obtaining the interaction potential between particles from the ζ potential is therefore of direct relevance to the colloidal stability and the flow properties of a dispersion. Measurements and correlation of the ζ potential (or surface charge) and rheology have also been presented by others for kaolinite (4), zirconia (5), silica (6), alumina (6, 7), and titania (6, 8) dispersions. Different measuring techniques and approaches were used, but correlations obtained were similar, with the highest resistance to shear found at or close to the IEP. In the previous papers by Kosmulski et al. (1, 2), rheology was measured in a traditional way, that is, the shear stress (τ ) was measured as a function of shear rate (γ˙ ) at stepwise increasing and decreasing shear rate. They discovered a local maximum in the shear stress at around 0.5–1 s−1 on the increasing shear rate curve but not on the decreasing shear rate curve. This phenomenon is known as stress overshoot. In contrast to using the Bingham model, using this stress overshoot as a measure of the flow properties is a direct method, that is, no extrapolations are needed to obtain the yield stress. It is noteworthy that the experimental results, measured at low or medium shear rate, gave a poor fit to the Bingham model. The Bingham yield stress value is obtained by extrapolating the linear part of the shear stress at the high shear rate end of the rheogram to zero shear rate. Moreover, results obtained at high shear rate would not give any information about the kinetics of the aggregation. A similar stress overshoot has also been found for suspensions of clay by Pignon et al. (9), for silica particles by Kawaguchi et al. (10), and in our laboratory for calcium carbonate (11) and alumina (unpublished results). The presence of stress overshoot was believed to originate from the balance between aggregation and breakup of a network structure in the dispersion. Since many colloidal materials aggregate during shearing, the build up of these time-dependent shear-induced structures has been investigated, for example, for micelles (12, 13), silica particles (14), and latex (14, 15). Different patterns in the rheological behavior have been distinguished for the various measured samples. Hu et al. (12) found a plateau value in the viscosity as a function of time for micellar solution. Lee et al. (16) observed a shear thickening behavior (increasing viscosity) at shear rates of the order of 100 s−1 for silica suspensions. Myska and Stern (13)
The evolution of shear-induced network structures of aggregates in concentrated anatase dispersions was studied in a Couette cell by using long sequence times, low constant shear rates, and singlefrequency oscillation at various shear stresses. The ζ potential was used as a measure of the interaction forces between the particles. Dispersions at the vicinity of the isoelectric point were observed to aggregate during the early stages of shearing in a Couette cell. The aggregates formed a network of new weak interaction bonds. Further shearing caused a breakup of the network structure. The rheological response to this process was a peak in the shear stress (the structural yield stress) as a function of cumulative shear strain at sufficiently low constant shear rate. The observed structural yield stress increased with decreasing shear rate and was an indication of the induction of a network structure. The behavior of unstable flocculated dispersions at low shear rates was noticed to be strongly dependent of the shear rate history. Single-frequency oscillation measurements were performed and repeated after short intervals of monotonic shearing. The elastic modulus G 0 increased rapidly (exceeding G 00 ) after a certain length of shearing, indicating the growth of a structure possessing weak gel-like properties. ° C 2001
Academic Press
Key Words: shear-induced aggregation; anatase; concentrated dispersion; yield stress; oscillation; isoelectric point.
INTRODUCTION
Kosmulski et al. (1, 2) have in recent papers observed a maximum in the shear stress at low shear rates in a shear rate sweep test with TiO2 dispersions. The maximum shear stress was designated “yield stress” and was found to be ζ potential dependent. At the vicinity of the isoelectric point (IEP) the yield stress experienced a maximum. The surface is neutrally charged and van der Waals attraction dominates the particle interaction at the IEP. According to the DLVO theory (3) the total interparticle potential energy between two particles is the sum of van der Waals attraction (which flocculates the dispersion) and the electrical double-layer repulsion (which stabilizes the dispersion). The resistance against flow is therefore stronger for a disper1 To whom correspondence should be addressed. Fax: 358-2-215 4706. E-mail: [email protected].
0021-9797/01 $35.00
C 2001 by Academic Press Copyright ° All rights of reproduction in any form reserved.
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SHEAR-INDUCED AGGREGATION OF ANATASE DISPERSIONS
and Liu and Pine (17), on the other hand, found a maximum in the viscosity at around 10 s−1 for micellar solution suspensions. Cabane et al. (18) noticed that above a critical shear rate a shearinduced gelation occurs, leading to an increase in viscosity for silica dispersions by adsorbing macromolecules and resulting in an unsaturated (flocculated) state. The aggregation of latex particles in Couette shear cells has recently been widely studied by several authors (19–22) who characterized the structures of aggregates by a fractal dimension. Hansen et al. (22) concluded that aggregation of monodisperse systems proceeds through homogenous aggregation, with large clusters sticking to large clusters, while for bidisperse systems heterogeneous aggregation was more dominant. When measuring the kinetics of shear-induced aggregation for flocculated dispersions the measurement procedure may be of great importance and the results obtained can depend on the parameters used. It is also important to find under which conditions network structures develop when studying sedimentation (23), since they enhance the sedimentation rate and leave a particlefree supernatant. Network structures have stronger resistance against centrifugation and pressure filtration and may also be interesting for colloidal processing of ceramic bodies because they are rigid at low shear rates. However, these structures show poorer consolidation behavior than stabilized dispersions (24), since they create loose porous bodies. Therefore it is also of importance from a practical point of view to obtain knowledge of structures of aggregates, as they may negatively affect the quality in practical applications, for examples, in the coating of ceramic honeycombs (25). Most of the studies on shear-induced structures have been performed with cone–plate or Couette systems. The Couette system was chosen for this work since this geometry shows the interesting rheological behavior of narrow two-dimensional shear zones. The aim of this work is to relate shear-induced aggregation to the ζ potential. Of particular interest is also to gain some understanding at which shear rates the aggregation and breakup occurs in the shear cell. The interpretation of a rheogram may be misleading if aggregation is not considered. The low shear rate rheology may also serve as a link between rheology and sedimentation by gravity, since in both cases small forces are applied on the network structure. Parallel investigations of sedimentation have been made and will be discussed in a forthcoming paper (26). Despite the extensive research on dispersions, few investigations on aggregation and breakup of network structures in “electrostatically” flocculated particle dispersions have combined rheology with ζ potential. EXPERIMENTAL
Materials The titanium dioxide (anatase from Aldrich) was of 99.9% purity. The chosen anatase has been thoroughly investigated (1, 27). The powder has been characterized by ESCA (not
shown), which detected phosphate added in the manufacturing stage. No attempt was made to remove the phosphates. The particles were quite spherical and rather monodisperse around 0.3 µm. Hydrochloric acid (analytical concentrate) and sodium hydroxide pellets (both from J. T. Baker) were used in aqueous solutions of 1 mol dm−3 to adjust the pH. The water used was distilled and ion exchanged (Milli-Q) and had a conductivity of 0.054 µS cm−1 . Rheological Measurements A Bohlin VOR (Bohlin Reologi) rheometer was used with a C14 (Couette) cup–piston system with a gap of 0.5 mm. Two different torsion bars were used; a 0.0126 mNm (12.4 g × cm) torsion bar was used in the steady shear rate experiments and a 0.00427 mNm (4.19 g × cm) torsion bar was used in the combined oscillation/steady shear rate experiments. The dispersions were prepared separately in airtight polycarbonate bottles, and stirred for 1 h prior to the experiment. The solids content was 38 wt% (φ = 14 vol%). Since some of the rheological experiments took several hours to complete, the pH was followed during the experiment as a function of time in a separate bottle to ensure that no leakage of ions from the particle surface took place, which would have changed the ζ potential. The pH was observed to remain constant within a range of ±0.1 units. Separate sedimentation tests showed that very slow or negligible sedimentation occurred at this solids content due to hindered settling. In steady shear rate measurements the outer cylinder was rotated with known shear rates and the stress of the sample which acts on the inner cylinder results in a displacement (twist) of the torsion bar with known stiffness connected to the inner cylinder. The shear stress, τ , at an applied shear rate, γ˙ , was measured from this displacement. The viscosity, η, can then be calculated from the shear stress by dividing by the applied shear rate: η=
τ . γ˙
[1]
Normally, a viscometry measurement with a Bohlin VOR was carried out according to the following procedure: At a chosen shear rate the sample has time to equilibrate itself during a defined delay time, after which the recording takes place during a defined measuring time. After this procedure the shear rate is increased or decreased to the next shear rate and the procedure is repeated. The (delay + measuring) time will in this paper be referred to as “sequence time.” Wall slip may occur during the measurements, but this was not corrected for. When the particles have a low affinity to the wall of the measuring cell a thin layer of particle-free fluid may develop, which causes the slip. The magnitude of the slip is dependent on the thickness of the slip layer and results in a discontinuous drop in viscosity, especially at low stress (28). Wall slip would therefore be expected to increase toward low shear rates. The shear stress was measured at 25◦ C as a function of increasing shear rate at both normal and extended sequence times.
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The maximum observed shear stress (the peak value) will be referred to as “yield stress.” The shear stress was also measured as a function of time at constant shear rates ranging from 0.0073 s−1 to 1.16 s−1 . The maximum observed shear stress for the constant shear rates will be referred to as “structural yield stress,” to distinguish the two methods. Single-frequency oscillation measurements were performed after preset times of the continuous shearing to study the process of structure changes at corresponding shear stresses. The oscillation measurements were performed according to the following procedure: After shearing at a constant shear rate for a preset time, the shearing was stopped and the oscillation was performed, after which the shearing was continued in the original direction. The frequency used was 0.5 Hz and the amplitude was 0.3 mrad (1.5%). With these parameters the oscillation movements were so small and fast that they did not disturb the sample, which was verified by parallel tests without oscillation. In oscillation (dynamic) measurements a sinusoidal strain with known amplitude, γ 0 , was applied and the resulting sinusoidal stress amplitude, τ 0 , was measured. The phase angle shift, δ, is zero for a perfectly elastic Hookean solid and π/2 radians for a perfect viscous Newtonian liquid. For a viscoelastic material the phase angle shift is in between these two extremes. The complex modulus, G ∗ , is the ratio between the stress and strain amplitude: G ∗ = τ 0 /γ 0 .
[2]
The elastic (or storage) modulus G 0 and the viscous (or loss) modulus G 00 can be determined from the complex modulus and the phase angle shift: G 0 = G ∗ cos δ, 00
∗
G = G sin δ.
RESULTS AND DISCUSSION
ζ Potential and Particle Size Since the phosphate impurities observed with ESCA are chelated to the surface and therefore determine the ζ potential, the isoelectric point was found at very low pH values, as expected (see Fig. 1). Thus, the ζ potential was highly negative at neutral pH and the dispersion was electrostatically stable at low salt concentrations. The ζ potential at native condition (pH = 6.7) was −60.6 mV and the IEP was found at pH = 1.7. The particle size determined by the Acoustosizer at the pH where the dispersion was stable, pH = 3.5–6.8, was quite monodisperse around 0.35–0.4 µm, which was in accordance with previous results (27, 30). The ζ potential at the constant pH values used in the rheology measurements are presented in Table 1. Dependency of the Yield Stress on the Sequence Time The shear stress was found more appropriate for presenting the results than (shear) viscosity because viscosity for concentrated particle dispersions showed both a strong shear rate dependency (pseudoplastic behavior) and a shear rate history dependency (thixotropic behavior). Using viscosity would therefore in itself give a multiplied effect when comparing the rheological data with different shear rates, since the viscosity is obtained by dividing the shear stress by the shear rate according to Eq. [1]. The apparent (shear) viscosity observed actually changed by 4 orders of magnitude for the same dispersion with increasing shear rate. In traditional viscometry the obtained shear stress versus shear rate curve is normally expected to be dependent only of the shear rate. Figure 2a shows, however, that the yield stress increased with increased sequence time. At the same time the position of the peak moved toward lower shear rates. This rheological behavior proves clearly that there exists a shear rate history dependency. At longer sequence times the structure of aggregates had a longer time to build up a strong network,
[3] [4]
ζ Potential Measurements The ζ potential and the particle size were determined at 25◦ C using an Acoustosizer (MATEC Instruments). In the experiments 10 wt% powder (φ = 2.8 vol%) was dispersed in aqueous solution and was treated in an ultrasonic bath for 5 min to break up aggregates in the dispersion. Afterwards the dispersion was stirred for 1 h. The ζ potential was measured under stirring from native state (neutral pH) to pH = 1.5 and back by automatic titration with 1.0 M HCl/NaOH. The ζ potential calculated by the Acoustosizer from the acid titration curve was determined by the equation developed by O’Brien et al. (29). No attempts were made to subtract the background signal from the data received, since the electrolyte concentration was rather low.
FIG. 1. The ζ potential as a function of pH.
SHEAR-INDUCED AGGREGATION OF ANATASE DISPERSIONS
TABLE 1 The (O’Brien) ζ Potential at the Constant pH Values Used in Rheology Obtained from Fig. 1 pH
ζ /mV
2.2 2.7 3.7 4.5 5.1
−13.6 −22.4 −35.8 −47.9 −51.9
because the network building took place at low shear rates. This will be discussed further below. In Fig. 2a the “10 s” sequence time was chosen to represent the sequence time normally used. It should be noted that the yield stress only occurred on the increasing shear rate curve, an example of which is shown in Fig. 2b. A high shear rate increased the breakup rate and the aggregates in the dispersion were broken down to smaller units. This resulted in the hysteresis (the difference in shear stress at increasing and decreasing shear rate) observed in Fig. 2b. Hu and Matthys (31) found that surfactant solutions need a certain rest time in order to reconstruct the original state of the solution. A similar behavior can probably be expected for anatase. Instead, a fresh undisturbed sample was used for every measuring sequence. It should also be pointed out that the shear rate at which the yield stress occurred was only strongly dependent on the sequence time and not on the ζ potential (not shown). For the solids content examined, the strong networks were only found for dispersions that were close to the isoelectric point. Further away from the isoelectric point the dispersion had a rheological behavior closer to a Newtonian fluid, since the electrostatic forces were strong enough to stabilize the dispersion.
85
induced by shearing. A similar phenomenon with a maximum shear stress at constant shear rate measurements (γ˙ = 100 s−1 ) was also observed by Hu and Matthys (31) in their rheological study of surfactant solutions. The maximum or overshoot in stress increased with decreasing rest time after a constant preshearing. They concluded that the overshoot indicated a build up of a shear-induced structure, which after a rest time returns to almost the original state. The shear stress in continued shearing decreased after reaching its maximal value (see Fig. 3a), due to breakup and rearrangement of the aggregate structure. It can also be seen from Fig. 3a that there was a difference in the slopes of the descending part of the curves for the constant shear rates. The structures possessed different strengths due to dissimilar shear rate histories. The curve was steeper at lower shear rates, which indicated
The Low Constant Shear Rate The shear stress was recorded as a function of time for various constant shear rates. To compare the constant shear rates ranging over three decades of magnitude we chose to plot the shear stress as a function of the cumulative shear strain. The cumulative shear strain, γcum , is the length (in radians) the shearing has proceeded and is defined as Zt γcum =
γ˙ dt,
[5]
0
where γ˙ is the shear rate and t is the elapsed time. Figure 3a shows the shear stress (the left y-axis) obtained from four of the constant shear rates at a constant ζ potential of −22.4 mV. It can clearly be seen that for the three lowest constant shear rates the structural yield stress occurs at almost the same strain, around 20–40 rad, regardless of the applied shear rate. The structural yield stress was, however, higher at lower shear rates. The value of the structural yield stress is a measure of the strength of the aggregate network, which has been
FIG. 2. (a) Shear stress as a function of the increasing shear rate at different sequence times. (b) Shear stress as a function of the increasing and decreasing shear rate showing the stress overshoot and hysteresis.
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GUSTAFSSON, NORDENSWAN, AND ROSENHOLM
that the breakup rate (as a function of the length of shear) was higher at lower shear rates, owing to the more rigid structure of the network induced by the low shear rate. This was believed to reflect the higher rigidity and consequently more brittle structures. Brownian motion would probably compete with shearing at a shear rate of less than 10 s−1 according to Serra et al. (20). Since the shear rates used in our measurements were even lower, we needed a value that would predict if the process was orthokinetic (induced by shear) or perikinetic (quiescent). We tried two methods of calculating the ratio between Brownian motion and shearing, one using the P´eclet number and the other using the relative particle displacement ratio. The P´eclet number, Pe, is described as the ratio of the viscous force experienced by a particle to the Brownian force (32) and is defined as (22) Pe =
3π ηa 3 γ˙ , kT
[6]
where η is the viscosity of the dispersion, a is the particle radius, γ˙ is the shear rate, and kT is the thermal energy. Since the shear stress, τ , is equal to the shear viscosity multiplied by the shear rate obtained from Eq. [1], the P´eclet number is linearly correlated to the shear stress for an experiment with constant shear rate. This would for our measurements give P´eclet numbers from approximately 100 up to almost 1000; these data are plotted in Fig. 3a as the right y-axis. This indicated that the influence of Brownian motion on the flocculation kinetics was negligible, but then again the P´eclet number may not be the right choice for these strongly flocculated dispersions with high solids content. For the second method, we theoretically calculated the relative particle displacement ratio, which is the ratio of the length a particle has moved during a specific time induced by the Brownian motion to that induced by the constant shearing. This was performed as follows: Einstein’s equation gives the mean Brownian displacement, x¯ , of a particle from its original position along a given axis after a time t and is defined as (33) x¯ = (2Dt)1/2 ,
[7]
where D is the diffusion coefficient. The diffusion coefficient D is defined as (33) D=
FIG. 3. (a) Shear stress (left axis) and the P´eclet number (right axis) as a function of cumulative shear strain for constant shear rates. (b) Same as Fig. 3a but with x-axis lengthened showing the slow declination of the shear stress at high constant shear rate. (c) Shear stress as a function of time for the constant shear rates in Fig. 3a.
kT , 6π ηa
[8]
where kT is the thermal energy, η is the viscosity of the solution, and a is the particle radius. Next we calculate the relative displacement ratio of particles in a shear field. After calculating the mean distance, dm , between the centers of two particles from the solids content, the distance, ds , a particle has moved during a time t due to shearing with a constant shear rate γ˙ can
SHEAR-INDUCED AGGREGATION OF ANATASE DISPERSIONS
TABLE 2 The Relative Particle Displacement Ratio and the P´eclet Number for the Various Constant Shear Rates Constant shear rate (s−1 )
Relative particle displacement ratio
P´eclet number for a particle in an infinitely diluted suspension
0.0073 0.0116 0.0461 0.0921 0.116 0.292 0.731 1.16
0.11 0.18 0.72 1.44 1.81 4.55 11.40 18.09
7.99E-05 1.27E-04 5.05E-04 1.01E-03 1.27E-03 3.20E-03 8.01E-03 1.27E-02
87
aggregation is reduced and the aggregation proceeds mainly through particle attachment to growing clusters. To get a more complete picture of the breakup Fig. 3c shows the shear stress plotted as a function of time for the four constant shear rates shown in Fig. 3a. From Fig. 3c it can be deduced that the rate of the breakup was slower at the lowest shear rates and becomes faster at higher shear rates. Relationship between Structural Yield Stress and Constant Shear Rate The structural yield stress is plotted in Fig. 4a as a function of constant shear rate for the ζ potential values used. The value for the critical shear rate, 1.16 s−1 , is not a real structural yield stress value but the plateau value of the shear stress at 20–40 rad of shear strain (see Fig. 3a), used only as a comparison with
be obtained from ds = dm γ˙ t.
[9]
Dividing x¯ from Eq. 7 by ds we can obtain the relative particle displacement ratio between Brownian motion and shearing. These values are collected for the constant shear rates studied in Table 2. Also collected in Table 2 are the theoretical P´eclet numbers calculated for particles in an infinitely diluted suspension at the constant shear rates used. The very low values for the P´eclet number shows that the flocculation would only be induced by Brownian motion at diluted conditions. This was not, however, observed for our concentrated dispersions. Using the relative particle displacement ratio indicates that the Brownian motion was more dominant at the three lowest shear rates, but this was expected to be true only at the absolute beginning of the experiment. The method using the relative particle displacement ratio gives only theoretical values, which are valid for unflocculated particle dispersions (lattice model). During the evolution of the aggregation process the role of the Brownian motion was reduced, since the amount of free particles diminished drastically. It seems that neither the P´eclet number nor the relative particle displacement ratio showed an exact measure of the influence of Brownian motion on the kinetics of aggregation. Brownian motion may still influence the results, leading to scatter in the measurements, as discussed below. At a critical shear rate and above it (γ˙ = 1.16 s−1 in Fig. 3a) no structural yield stress was observed as a clear peak value. Instead, a plateau was found. At high shear rates the breakup of the network required more shearing than at low rates. The shear stress decreased to a shear stress value of 1.4 Pa after shearing 750 rad, which is indicated in Fig. 3b. A strong structure of aggregates would at these conditions never be subjected to sufficient cumulative shearing to be built up before it was broken down. The growth and breakup of the network clearly follow different shear rate dependencies. Axford (34 ) stated that the probability of bond formation during cluster collision decreases at higher shear rates. Despite the increase in overall collision frequency and rate of formation of large clusters, homogenous
FIG. 4. (a) Structural yield stress as a function of constant shear rate for the ζ potentials used. (b) Structural yield stress as a function of ζ potential for the constant shear rates.
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GUSTAFSSON, NORDENSWAN, AND ROSENHOLM
the real structural yield stress values. The structural yield stress increased with decreasing constant shear rate, but at the two lowest shear rates the inhomogeneities of the samples are more pronounced, especially for unstable dispersions with low ζ potential. The scatter was therefore large for these dispersions, due to network instabilities at high stresses or competing flocculation induced by Brownian motion. The uncertain values are therefore indicated with dotted lines in Fig. 4a. A fairly reliable trend can still be obtained. A maximum structural yield stress of about 70 Pa was found for dispersions close to the IEP. This upper limit is a measure of the finite strength of the structure of aggregates. The limit is due to the weakness of the attractive interaction and the changing geometrical orientation of the bonds between the particles. For the “−51.9 mV” experiment, the structural yield stress was negligible for all but the two lowest shear rates. At these conditions the repulsion forces were already sufficiently high to prevent the aggregate structure from being built up or from having any significant strength. In Fig. 4b the yield stress from Fig. 4a is plotted against the ζ potential for the constant shear rates. The structural yield stress increased successively with decreasing ζ potential. The trends for the two lowest constant shear rates (0.0073 s−1 and 0.0116 s−1 ) are indicated with dotted lines due to scatter in the results. Relationship between Yield Stress and ζ Potential at Different Sequence Times In Fig. 5 the yield stress is shown as a function of ζ potential obtained by measurements with various sequence times (at increasing shear rate). Because of the use of increasing shear rate the yield stress values were lower than with the constant shear rate experiments (see Fig. 4b). When the shear rate was increased stepwise the structure was weaker than when using a single and hence lower shear rate. The shear rate history was different from
FIG. 6. Shear stress (filled squares) as a function of cumulative strain by shearing at 0.0461 s−1 . G0 (filled circles) and G00 (open circles) at 0.5 Hz measured at the corresponding cumulative shear strain and shear stress as indicated in the figure.
that of the constant shear rate experiments. The contribution of the first (lowest) shear rate value of the shear rate sweep (seen from Fig. 2a) was of minor significance, since the low shear rates contributed only with very small shear strain increments. The yield stress was more effected by extended sequence time at ζ potential values close to zero than further away from it. However, the two longest sequence times showed a decline in the yield stress before reaching the IEP. This was not expected, since the maximum yield stress was found for low electrolyte concentrations approximately at the IEP (1). The yield stress at different sequence times may not have the same ζ potential dependency though, owing to slightly different mechanisms in the aggregation. Two different solids contents were required for the ζ potential and rheology measurements, a low one for ζ potential and a high one for rheology. Hence, the interparticular distances at which they were measured were not the same. The Influence of G 0 and G 00 on Monotonic Shearing
FIG. 5. Yield stress as a function of ζ potential for the sequence times used.
Figure 6 shows a combined shear and oscillation measurement. The experiment was carried out according to the following procedure: A constant shear rate of 0.0461 s−1 was applied and the shear stress was measured. The shearing was stopped and an oscillation measurement with 0.5 Hz was performed, after which the shearing was continued. Before the first and after every shearing step an oscillation measurement was performed. The oscillation was adjusted so that it did not modify the state of the shear-induced aggregation. This was compared and verified by shearing with the same shear rate without oscillation. The program was set to auto-zero itself before every new shearing, which is the reason for the low starting value for the otherwise high stress shearing. The viscous (loss) modulus G00 followed the development of the shear stress with quite good accuracy. We believe that the viscous component was mainly a measure of the energy consumed to break up the aggregates. The higher shear
SHEAR-INDUCED AGGREGATION OF ANATASE DISPERSIONS
stress indicated that the number of interaction bonds between the aggregates increased and therefore the number of bonds that were broken down also increased. In the initial phase of the shearing the elastic modulus G0 increased negligibly but showed a rapid increase after a certain number of interaction bonds developed. This increase of G0 and the fact that G0 exceeded G00 indicated that the structure was growing more rigid and becoming gel-like during shearing. Faers and Luckham ( 35) have illustrated the contribution to G0 of the three-dimensional floc network of polystyrene latex particles. The more flocculated network gave an increased G0 . This was in good agreement with our experience. The good correlation between the oscillation and constant shear measurements would also support the prediction that the stress overshoot phenomenon was caused by a structure buildup. Furthermore it is argued that wall slip, although presumably to some extent present in the measurements, would not change the shape of the shear stress curve, only the magnitude of stress. As the test in Fig. 6 proceeded G0 decreased faster than both 00 G and the shear stress. This indicated that the dispersion lost its elasticity, when the breakup rate of the interaction bonds exceeded the growth rate. This took place at the observed peak value of the shear stress (i.e., at the structural yield stress). CONCLUSIONS
The investigated flocculated anatase dispersions showed at moderately low shear rates not only a shear rate dependency but also clearly a shear rate history dependency. Both the growth and breakup of the aggregates are dependent on the shear rate but the dependencies are different. At low shear rate the growth was favored over breakup, which leads to stronger aggregate structures. At higher shear rates the strong network structure did not have time to build up. The observed structural yield stress increased with decreasing constant shear rates. The yield stress increased with longer sequence time and also toward lower ζ potential (pH closer to the IEP). The (structural) yield stress was an indication of the strength of the shear-induced aggregate network at the specific applied conditions (shear rate, sequence time, and ζ potential). The elastic properties of the dispersion were studied by oscillation measurements. The increase of the elastic modulus G0 to values greater than the viscous modulus G00 indicated that the network structure became gel-like. This elasticity occurred when the shearing had proceeded long enough to aggregate a sufficient amount of interaction bonds. The dispersion lost its elasticity when the breakup rate of the bonds exceeded the growth rate. The viscous modulus G00 followed the development of the shear stress and was believed to correspond to the energy consumed to break down the bonds. The traditional presentation of shear rate dependency can be misleading as the results strongly depend on the measurement procedure and the shear rate history. The methods presented here can also be used for studies of flocculation kinetics as these too are sensitive to shear rate.
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ACKNOWLEDGMENTS The Technology Development Centre of Finland (TEKES) and The Ministry of Education (Finland) (Graduate School of Materials Research) are acknowledged for financial support.
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