STUDY ON REQUIRED ENERGY TO DEAGGLOMERATE PIGMENTARY TITANIUM DIOXIDE IN WATER

Transcription

1 STUDY ON REQUIRED ENERGY TO DEAGGLOMERATE PIGMENTARY TITANIUM DIOXIDE IN WATER INTRODUCTION Ricardo Tadeu Abrahao, Victor Postal, Robert Guardani DuPont do Brasil S.A. and University of São Paulo, Chemical Engineering Department. Titanium dioxide (TiO 2 ) is the most important white pigment used in several industries because of its ability to conciliate light scattering and white color. (1, 2, 3, 4) The focus of this study was to understand how the properties of TiO 2 influences the interaction with a liquid medium and the amount of energy required to break agglomerates. In order to achieve an adequate performance, coatings based on TiO 2 pigments must consist of individual particles with a minimum distance from particle to particle (5). This condition is influenced by the characteristics of the liquid suspension, which is used in the preparation of coatings. Suspended particles must be adequately dispersed, and the suspension must be stable. It is a common practice among pigment grade TiO2 manufacturers to enhance the easiness of dispersion characteristics. However, it is not clear what is meant with this terminology. Users in the market for coatings seems to accept and desire this feature, but, usually, there is not a clear justification of this claim. In addition, recent studies investigating this property, does not directly correlate with the energy required for dispersion and the particle properties, although it elucidated important insights on particle dispersions (6-9). Therefore, the motivation of this work was the lack of a link among the properties of the TiO2 pigmentation particles, the dispergent liquid characteristics and the energy required to promote the full incorporation and dispersion of these particles into the dispergent. This study contributes to the current knowledge of techniques and it established correlations between the properties of different pigment grade titanium dioxide particles and their behavior in relation to the dispersion process, which is the main step in the production of liquid pigment suspensions. The dispersion process of solid particles in water or any liquid medium can be divided into the following three main stages (10): Wetting, which refers to the displacement of gases or other species (such as water) that are adsorbed at the surface of the pigment particles, followed by a subsequent attachment of the wetting vehicle to the particle surface. Deagglomeration, which refers to the separation of the particle clusters into isolated primary particles or aggregates, followed by the movement of the wetted particles into the body of the liquid vehicle in order to achieve a permanent particle separation. Stabilization, which is the desired state where the deagglomerated particles remain separated and distant from each other. This state is achieved if the particle-to-particle forces are balanced to make this separation thermodynamically favorable. The energy required to deagglomerate powders is influenced by the wetting efficiency(11). The maximum stress resulting from fluid oscillation is represented by Equation 1 (12). The equation also considers the bonding Van der Waals forces between primary particles inside the agglomerates as well as the electrostatic repulsion forces. 4 F τ r 0 = πl 0 ( x, L) 2 Equation 1 Whereτ 0 is the maximum stress,f 0 is the particle interaction force in the agglomerates and L is the agglomerate size. Equation 1 was deduced by Baldyga et al in 2008 and it takes into account parameters of the particle, the agglomerates and the fluid motion and cavitation parameters. Figure 3 illustrates these parameters for spherical agglomerate.

2 2 Figure 1 Schematic indication of the parameters in Equation 1 (12). Erosion occurs in the periphery of agglomerates, where small fragments are gradually sheared off from agglomerate surfaces. The intensity of this process is highly influenced by the particle wetting, liquid viscosity and agglomerate pore radius, the latter being influenced by the unitary particle-packing factor. The particle packing and shape factor also rule the forces that bond the particles inside the agglomerate, and this determines the number of contact points and the number of bonds connecting each other. When liquid penetrates the pores of the agglomerates, repulsion and attractive forces take place, and when the erosion forces (generated by cavitation) and the repulsion forces are larger than the attractive forces deagglomeration is successful (12). As shown in figure 2, wetting is the key step in deagglomeration, TiO 2 particles treated with organic hydrophilic surfactants need a lower torque during dispersion in a Cowles disperser. This indicates that the pigment properties influence the wetting stage (13). While wetting occurs, the cohesiveness of the particles increases because of the forces involved in liquid bonding. The liquid may be present as either a mobile phase or an immobile adsorbed layer. The successive forms that a mobile liquid can take are described as pendular, funicular, and capillary states (Figure 2) (14). Figure 2 - Successive states of mobile liquid in a powder agglomerate. (A) pendular, (B) funicular, and (C) capillary (14) In the pendular state, the void space in the agglomerate is partially filled with liquid, which forms bridges between adjacent particles. In the funicular state, adjacent pendular rings coalesce to form a continuous liquid network interspersed with pockets of air. In this region, the strength of the agglomerate results from both pendular liquid rings and capillary suction pressure. In the capillary state, liquid fills the entire void space in such a way that there is a curvature of the liquid surface at the outer layer of the agglomerate. This results in capillary suction pressure. The maximum liquid bonding force is achieved between the capillary and funicular states. The next stage is the slurry form. In order for the deagglomeration process to occur, the capillary state should be achieved once the energy required to break the agglomerate decreases further. The general dependence of the strength of the agglomerate on the saturation level is shown in Figure 4. Figure 3 Agglomerate strength due to the mobile liquid versus saturation level. (A) Pendular, (B) funicular and (C) Capillary (14)

3 3 Saturation is calculated according to Equation 2, where ε is the porosity of the agglomerates in percentage, mo is the moisture content, ρ s is the density of the solid and ρ l is the liquid density (14) Equation 2 Critical saturation, where the liquid rings begin contact, is calculated according to Equation 3, where k is the coordination number, ε is agglomerate s porosity, V p is pendular volume (Zwickel volumens). The latter is a function of, ,, Refer to, in Figure 4. Equation 3 Figure 4 Graphical presentation of liquid bridge between 2 spherical particles (14) The tensile strength in the capillary tensile strength given by Equation 4, where S is the saturation and p is the capillary pressure.! Equation 4 The penetration of liquid into particle packages is comparable to the penetration of liquid into porous material or into small tubes. Such phenomena are described as capillary flow (15). The flow through a cylindrical capillary tube can be described by Equation 5, where V is the capillary volume in cm³, t is the time in seconds (s), p is the capillary pressure in (N/cm²), r is the capillary radius (cm), η is the dispergent medium viscosity and L is the tube length. "! #$% 8'( ) Equation 5 Since V/t is equal to the product of the average liquid velocity u and the tube cross-sectional area πr², Equation 5 can be rearranged as Equation 6.! * + 8'( $, - Equation 6 The faster the liquid penetrates into the agglomerate, the faster the agglomerate passes through the three saturation level phases, achieving a smaller cohesive force point. Applying equation 5 to the critical saturation (S*) results in equation 8 that describe the maximum tensile strength of the agglomerate. / ! Equation 7

4 4 All theories regarding light scattering, wetting and other phenomena related to particle or even particle fluid interactions have been developed to describe the behavior of spherical smooth-surface particles. However, as shown in Figure 5, TiO2 particles are neither round nor smooth Figure 5 Scanning electron microscopy of TiO2 particles (16) The relationship between a rough and a smooth surface can be described by the ratio of particle surface areas as expressed in Equation 8: Equation 8 Where Ai is the actual contour area of the particle surface and Ae is the projected area. In general for liquid surfaces i is equal to 1, while for solids i is larger than 1. Since TiO2 particles are not spherical, it is important to specify which particle dimension is adopted as the size parameter. It is common to adopt x as the diameter of a hypothetical sphere with equivalent properties as the one that is measured in each specific technique. For instance, when laser diffraction is used as a technique to obtain the particle size distribution of dispersed TiO2 particles, it is convenient to adopt x as the diameter of a sphere with equivalent scattering properties as the particle. EXPERIMENTAL METHODS AND MATERIALS The samples were selected according to the following criteria: different size ranges, whilst keeping an approximately constant specific surface area; and similar size ranges with different surface properties. The characteristics of the samples are listed in Table 1. The mean particle sizes D(3,2) and D(4,3) correspond with the ratios of the moments (Erro! Fonte de referência não encontrada.) M3/M2 and M4/M3, and represent the Sauter and volumetric mean diameter, respectively. Samples A, A and A originate from the same material and were obtained by applying different levels of micronization, thus resulting in different particle sizes. The surface area, based on the laser diffraction, is a relation between the particle surface area obtained by the moments calculation and the mass, which is calculated using the volume-based moment (M3) and multiplying it by the particle density. Table 1 Particle properties Property * Mean particle size D(3,2) by LD µm * Mean particle size D(4,3) by LD µm Capillary volume w/ n-hexane cm³ Surface area.m²/g (by B.E.T.) * Surface area by LD m²/g i(equation 8) Surface tension**(mn/m) A * Laser diffraction ** Calculated based on capillary rise methodology A A B C

5 5 The surface area calculated using the BET methodology, whose model can be represented by Equation 9. (16) 89 Equation 9 /67 1 9:1 1 89; Where V is the adsorbed gas volume, V mon is the adsorbed gas volume considering a monolayer of adsorbed molecules, z is the ratio between the equilibrium and saturation pressures and c is the BET constant expressed as Equation 10.(18) 8 exp AB - Equation 10 WhereE 1 is the first layer heat of adsorption, E L is the liquefaction heat of the adsorbate, R is the ideal gas constant, and T is the absolute temperature. The linearization of Equation 9 generates Equation 11 allowing the determination of the adsorbed gas volume as a function of the applied pressure Equation 11 C 8 /67 8 /67 By plotting the left side of Equation 11 as a function of z, both V mon and c can be obtained. The equipment used to measure the BET surface area was a NOVA 1200e Surface Area & Pore Sizer Analyzer (Quantachrome Instruments). This equipment provides information regarding surface area as well as the constant c, which relates to the particle surface free energy. The wettability of the samples was based on the Washburn capillary rise method. According to the literature that describes this methodology (19-21), it utilizes the capillary constant (C w ), liquid density and viscosity (ρ and η), time (t), liquid surface tension (γ) and the liquid-solid surface tension interaction (cosθ) to determine the mass adsorbed by the powder (m),as described by Equation 12. Equation 12 Where θ is the contact angle between the liquid and solid. For the same TiO 2 mass (5g), assuming a capillary constant (C w ), and adjusting the liquid density and viscosity (ρ and η) to each liquid in use, the equilibrium between solid and liquid is achieved for long periods of time. The mass of the adsorbed liquid, m, is proportional to the wettability of the agglomerates. N-hexane is used as a reference liquid in this method due to its low surface energy. Since n-hexane is a low surface tension liquid, it completely wets the solid. Therefore, the volume of n-hexane adsorbed is adopted as the maximum volume that can be adsorbed by the powder, for the volume of pores formed by such particle packing. Although, in this study, the reference liquid is non-polar and the other liquids (water and water-plus-surfactant) are polar or mainly polar, hexane was adopted as a reference liquid in order to enable the estimation of the packing properties and the Cw value. Similar approaches were adopted in other published studies (e.g., 22) where the reference liquid (toluene or n-hexane) is non-polar and the liquid under study was water. The equipment to evaluate the Washburn wettability was a K100 (KRÜSS) system, which provides information on capillarity and solid-liquid affinity. Since this methodology is highly dependent on the powder packing, the sample preparation procedure was carried out so as to ensure that the mass of powder (5g) was adequately tapered and compressed by the piston, in order to meet the assumption of full compaction. The intention was to simulate a worst case scenario of highly compacted agglomerates where all particles touch each other and the void space is smallest. Highly compact agglomerates are frequently found as a result of inefficient dispersion processes.

6 2 In this study 3 different liquids were used, n-hexane, deionized (D.I.) water and a third liquid was the D.I. water with its surface tension altered using a commercially available hydrophobically modified sodium salt of a carboxylate polyelectrolyte. This type of dispersant is widely used in the coatings industry. Its ph varies from 9.5 to 10.5 and has 25% solids (w/w). The liquid surface tension was measured using the Du Noüy ring method. The method involves slowly lifting a ring made of platinum from the surface of a liquid. The force required to raise the ring from the liquid surface is measured and then related to the liquid surface tension (23). The equipment used is a LAUDA model TD1, which is composed of an electronic unit where calibration and surface tension readings are performed and a mechanical unit where the sample and the ring is placed. Before using the equipment, the calibration is performed using a standard weight of 500 mg. A Malvern Mastersizer X laser diffractometer with a sampling unit that is equipped with an agitation and sonication unit was used to monitor the particle size distribution of the samples in the dispersion experiments. The equipment is based on the angular distribution of forward scattered light, and it operates with a collimated laser beam at 640 nm and is able to provide the particle size distribution in the range between0.1 to 600 µm. Although different optical models can be adopted to estimate the particle sizes, the present study adopted the Mie model. To generate the shear force that provides energy to break the agglomerates into aggregates and individual particles, a high-energy ultrasound generator was used (UP100H Ultrasonic Processor, Hielscher).The output of this equipment, as specified by the supplier, is 100 W at 30 khz, which can be adjusted for better control of the energy applied to the suspension, down to 20% of its full power. RESULTS AND DISCUSSIONS Table 3 shows the results of the experimental data of the Washburn methodology, the adsorbed volume, the adsorption speed in grams per seconds and the adsorption speed in milliliters per seconds. The sample mass was kept constant at 5g in all Washburn analyses. Table 2 Washburn results Liquid surface tension (mn/m) TiO2 grade Adsorbed Volume (ml) Adsorption Speed (g/s) Adsorption Speed (ml/s) A X X10-1 A' X X10-1 A" X X10-2 B X X10-2 C X X10-2 A X X10-4 A' X X10-4 A" X X10-3 B X X10-4 C X X10-4 A X X10-4 A' X X10-4 A" X X10-3 B X X10-3 C X X10-4 With the data on Table 3 it is possible to calculate the contact angle between the liquid and solid (θ). By using this contact angle, the surface tension of the liquid-solid interface can be estimated by Equation 14 (8). The values of the contact angle obtained by means of these calculations are shown in Table 4. D D E 8 cose Equation 13

7 3 The calculated values of the solid surface tension are shown in Table 4. Table 3 Calculated contact angle between the samples and liquids with different surface tension. Grade Shape Solid surface Liquid surface Contact angle Cw factor (i) tension (mn/m) tension (mn/m) Radian Degrees A A' A" B C It is interesting to note that sample A presented a significantly different surface tension than samples A and A, although the three samples have same surface treatment, the sample A also presents a significantly higher shape factor and the particle size distribution of this sample is also significantly different from the other two, as shown in table 4. Equation 4 and 5 are used to estimate the meniscus radius, the capillary radius and the capillary pressure. The results are shown in Table 4. Table 4 Estimated values of the meniscus radius, capillary radius and capillary pressure of the particle - liquid interaction, based on Equations 2 and 4. TiO2 Grade Liquid surface tension (mn/m) Capillary radius (µm) Capillary Pressure (mn/m²) Meniscus radius (µm) A A A B C Figure 6 shows the adsorbed liquid volume for all samples and for liquids with different surface tension. The adsorbed liquid volume is smaller for bigger particles and for liquids with larger surface tension. The relation to particle size is understandable, since bigger particles have larger bulk density, and thus the packed bed has less void space per unit mass. This can be observed in the results for n-hexane (surface tension equal to 18 mn/m). Particles A and A adsorb higher liquid volume than the bigger particles A, C and D. The lower adsorbed volume for liquids with higher surface tension, as shown in Figure 6, can be further explained by the meniscus radius formed by the interaction of the liquid and solid. For samples A, A and A, for example, which have the same surface treatment, it is possible to observe that even though a reduction in adsorbed volume occurred for particle A, it adsorbed more liquid than samples A and A, for the three liquids tested.

8 4 Figure 6 Adsorbed liquid volume by all samples for liquids with different surface tension. Based on these results, the ratio between the meniscus/particle radii ratio and the adsorbed volume of liquid was verified for samples A, A, and A, by means of the plot shown in Figure 7. Data could be correlated by means of a power function with coefficient of determination of Thus, a clear correlation between the particle properties considered in this study and the ability for liquid adsorption, for liquids with different surface tension. Figure 8 shows the result of extending the correlation to all particles, i.e., including particles with different surface treatment. Although the quality of the correlation has decreased, with a coefficient of determination of 0.75, the observed tendency remains clear. Figure 7 Liquid adsorbed volume as a function of the meniscus to particle radii ratio for grades A. A and A. Figure 8 Liquid volume adsorption as a function of meniscus and particle radii ratio for particles A. A. A. B and C. Based on the experimental results herein, it is possible to observe that the ratio between liquid and solid surface tensions affects the wettability of the solid by the liquid; the data of this study were used to

9 5 generate a response surface for the wetting speed of the solid by the liquids as a function of the following variables: Liquid to solid surface tension ratio (D J D ); Meniscus to particle radii ratio ($ / $ ); Shape factor, i. The objective was to generate an empirical correlation that enables to evaluate the quality of the correlation and the relative importance of these properties. The empirical correlation considered the individual linear and quadratic effect of each variable, and interactions of pairs of the variables. The empirical correlation is presented in Equation 14 K C $ / $ D J D C :$ / $ D J D ; :$ / $ 2;C :D J D 2; :$ / $ D J D 2; Equation 14 Where WS is the wetting speed (in ml/s). The adjusted coefficient of determination for the model based on such factors is 95.46%, meaning that the model is representative of the wettability process. Response surfaces describing the wettability according to the empirical model are shown in Figure 17. The plots show the calculated liquid adsorption speed as a function of the liquid to solid surface tension ratio (D J D ), and meniscus to particle radii ratio ($ / $ ), for different values of the shape factor, i. The minimum value of wetting speed is obtained for combinations of large meniscus/particle radii ratio, and large solid-liquid surface tension ratio. The maximum wetting speed is achieved mainly when the meniscus radius is much smaller than the particle radius. The results are presented in Figure 9 in terms of the percentual change in wettability (wetting speed) as a function of the values of each input variable between minimum (0) and maximum (100) of the range of each variable, by keeping the other variables at the mean value (50). The model used in this sensitive analysis has an r² adjustment of %. Figure 9 Sensitivity analysis for wetting speed.

10 6 As shown in Figure 9, the most impacting factor in wettability speed is the ratio between the meniscus and particle radii. The particle wetting speed is inversely proportional to this factor and means that if the meniscus is excessively large the wettability speed is greatly reduced. If the particle radius is small enough this ratio is enhanced and, as consequence, the wetting speed is also reduced. Thus, the meniscus particle radii ratio may represent an important limitation to the wetting speed. It was also found that the wetting speed is inversely proportional to the liquid and solid surface tension ratio, but the impact is much smaller when compared to the meniscus and particle radii ratio. However, as shown in Equatio 4, there is interdependency between the meniscus radius and liquid surface tension, since the meniscus is a function of the liquid surface tension and the capillary pressure. The wetting speed is directly proportional to the shape factor, since for larger values of the shape factor, or smaller capillary radius, the capillary pressure is increased, as shown in Figure 10. An increased capillary pressure causes faster penetration of the liquid into the agglomerate. However, this factor has less effect than the ratio between the meniscus and particle radii. Figure 10 Capillary pressure as a function of the shape factor (i) The deagglomeration energy was obtained for each sample when it was submitted to dispersion in different liquids. Figure 11 shows a compilation of the energy that was required to deagglomerate the samples until a stable particle size distribution could be attained in the liquids with different values of the surface tension, for particles with different shape factors. The results show that the liquid surface tension plays an important role in determining the required energy to achieve minimum particle size. It seems as if there is a critical value for the liquid surface tension, above which the dispersion process requires a significantly larger energy and the particle shape factor affects the process. Figure 11 shows that a drop in required energy when changing the liquid from pure water to water plus dispersant additive is significant. 500 Required energy deagglomerate (kj) Shape factor Liquid surface tension (mn/m) Figure 11 Scatter plot of energy required in the dispersion for liquids with different surface tension and particles with different shape factor (i). The graph shown in Figure 12 was compiled with the data that were collected from the dispersion experiments under the effect of ultrasound. The x-axis represents the liquid surface tension, and the y axis is the energy applied to reduce the mean particle size, D(4,3), to its minimum observed value.

11 7 Figure 12 Ultrasound energy required to achieve minimum mean particle size, D(4,3), for the samples used in this study. Figure 12 shows that particles presenting different surface properties behave differently. Since particles A, A, and A had the same surface treatment, they were expected to behave in the same way. However, this was true for particles A and A, only. The difference observed for sample A may be due to its different particle size distribution. When the distribution of particle size is spread over a wide size range, then the particle-to-particle interaction is much larger than for monodisperse particles, since the void space is smaller. The particle size distribution curves of all samples, used in this study, are shown in Figure 13. Sample A shows a significantly different particle size distribution curve compared to the other samples. Figure 13 Particle size distribution of samples A, A, A, B, and C. The relationship between energy required for dispersion and the properties of the liquids and particles can be seen in Figure 14, which shows the same information presented in Figure 11, but in the form of a loglog plot. The important information shown in Figure 14 is that the required energy for dispersion can be correlated logarithmically to the liquid surface tension. Figure 14 Required energy as a function of the liquid surface tension for all samples.

12 8 In Figure 16 the data are plotted in a similar log-log scale, but specific correlations are shown for each value of the particle shape factor (i). In this case, the fitting became significantly better. The plots show that the slope of the fitted straight lines varied with the particle shape factor. Based on this, the parameters of the equations shown in Figure 16, i.e. slope and exponent, are plotted as a function of the particle shape factor, I, in Figures 17 and 18. Figure 15 Required surface tension as a function of liquid surface tension and the particle shape factor i (4.5; 8.7 and 7.4). Based on these correlations, an empirical expression was adjusted to the data. For the particles in this study, the required energy (RE) as a function of the liquid surface tension and particle shape factor were correlated according to Equation 15: lna? 2² 0,56 0,0058 D 2 0,0525 D 5,74C0,3478 D 13,47 Equation 15 Equation 15 can be applied to predict the energy that is required to achieve the minimum mean particle size for the particle samples used in this study. A comparison between predicted and observed values of the required energy is shown in figure 19. As shown in the figure, the agreement between observed and predicted values was good. Figure 16 Comparison between predicted values of required energy for dispersion of the TiO 2 particles (log scales)

13 9 Figure 17 shows the correlation between the required deagglomeration energy and the relative agglomerate maximum tensile strength calculated according to Equation 8. The maximum tensile strength seems to correspond with the potential required energy to deagglomerate the samples for the three values of the shape factor, 4.5, 7.4 and 8,7, used in this study. It is possible to see from Equation 8 that the particle shape has an important role in the required energy or the energy required to reduce the mean particle size to its minimum value. / Equation 16.! If a shape factor value of 4.5 is taken as reference, then the sample with a shape factor equal to 8.7 requires more energy to deagglomerate than the reference and the sample with a shape factor equal to 7.4 requires less energy than the reference to achieve the minimum particle size. This behavior can be explained when the origin of the particles are taken into account. In this study, particles with shape factor of 8.7 are the same particles with shape factor 4.5 but with no micronization, meaning that the particles are physically attached to one another. This physical attachment changes the shape of the particles but not the surface roughness, meaning that the average number of particle-to-particle interaction per particle increases. In this particular case, most particles of the sample consist of aggregates and not of single unitary primary particles (that have no, or too little physical bounding to each other). Figure 17 - Correlation between calculated relative agglomerates tensile strength and required energy to deagglomerate the powder. The results of the experiments on wetting and dispersion are summarized in a qualitative way as a diagram (shown in Figure 22). The left column shows the behavior of individual particles: as the particles become less spherical, they tend to attain a dendritic structure and their sphericity decreases, whilst the shape factor, i, increases. This tendency is associated with a reduction of interparticle contact, resulting in lower values of the required energy in dispersion operations. This behavior is different for the agglomerates of irregular particles. As the particles become less spherical, sphericity decreases, and the shape factor increases. However, as illustrated in Figure 21, agglomerates of irregular particles tend to exhibit larger inter-particle contact areas, and this, results in an increased resistance to deagglomerate. Thus, the required energy for achieving a minimum mean particle size in dispersion operations is increased.

14 10 Figure 22 Qualitative sketch to summarize the effect the particle shape on the required energy to achieve minimum mean particle size. CONCLUSIONS The wetting speed of different pigment grade titanium dioxide samples has been investigated experimentally by means of wetting tests carried out with liquids of different surface tension. By organizing the experimental results in terms of pertinent properties of the particles and liquids, the wetting behavior of the particles has been correlated in a clear and consistent way. Based on the results of this study, the main factors that influence the wetting process are described as follows. The first factor is the ratio between the particle size and the meniscus formed by the liquid when this flows into the porous agglomerates. If the meniscus is small enough to allow the liquid to permeate throughout the pores formed by the particles, then wetting is favored. If the particle packing forms pores that are smaller than the meniscus formed by the liquid, then the wetting process will be stopped. The second factor is the ratio between the liquid and the solid surface tensions. This effect is predicted in the literature, and indicates that, the higher the surface tension of the liquid compared to the surface tension of the solid the lower will be the wetting speed. This ratio also rules the meniscus radius formation and surface interaction phenomena. Therefore, the ratio between surface tensions is an important factor ruling the wetting process. A third factor influencing the wetting speed is the shape factor of the particles. It can be concluded from this study that the shape of the particles is important in determining the required energy in dispersion because it defines the number of interactions among the particles. The larger the number of interactions, the more difficult it is to deagglomerate the particles. In this study, the shape factor of particles of sample A (A, A and A ) was larger than the others, which resulted the increased required energy. It is also possible to conclude that a correlation between particle and liquid properties and the required energy to disperse the particles was adjusted to the experimental results. The correlation (Equation 15) can predict the energy required to achieve maximum deagglomeration of pigmentary TiO 2 powders for the samples used in this study. The energy required for pigment dispersion is described as a function of the ratio between the liquid and the particle surface tensions and the particle shape factor. The energy required decreases as this ratio decreases. This ratio is driven by changes in the liquid surface tension because the solid surface tension does not change during the process. Therefore, this ratio is an important factor for the dispersion process of pigmentary TiO 2 particles.

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