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Determination of Agglomeration Kinetics in Nanoparticle Dispersions Herbert Loria* and Pedro Pereira-Almao Department of Chemical and Petroleum Engineering, Schulich School of Engineering, University of Calgary, Calgary, AB T2N 1N4, Canada
Carlos E. Scott Alberta Ingenuity Centre for In Situ Energy, University of Calgary, Calgary, AB T2N 1N4, Canada ABSTRACT: The direct application of nanoparticles as nonsupported adsorbents and catalysts is of high interest since they offer high surface areas with reduced mass transfer limitations. However, the natural tendency of these materials to aggregate, even faster when at high temperatures, makes the agglomeration process an important phenomenon to be studied, understood and, eventually controlled. A method to obtain the kinetics of nanoparticle agglomeration processes is presented in this paper. This analysis was based on the change of particle diameter during aggregation. The kinetic expression was validated with a series of experiments where the growth of Fe2O3 nanoparticles immersed in base oil was followed at different times, temperatures, and particle concentrations. Results revealed the nature of the particle agglomeration process in the ranges of the experimental conditions; they indicated that physical adhesion, more than chemical binding, is the determining mechanism for agglomeration of Fe2O3 nanoparticles immersed in base oil.
1. INTRODUCTION Agglomeration of particulate solids is a mass-conserving, but number-reducing, process that shifts the particle distribution toward larger sizes.1 This phenomenon has an important impact in particle transport as larger particles tend to settle more rapidly under gravity but diffuse more slowly. Therefore, an accurate modeling of agglomeration is essential for understanding and predicting particle transport. Agglomeration literature provides models based on basically three main processes: Brownian motion, gravitational agglomeration, and turbulent agglomeration. Brownian motion, first studied by Robert Brown in the 19th century, refers to the continuous random movement (or diffusion) of particles suspended in a fluid. Brownian agglomeration occurs when, as a result of their random motion, particles collide and stick together.2 Brownian agglomeration is important for submicrometer particles and is probably the best understood of the agglomeration mechanisms, it has been treated by several authors and extensive reviews have been presented.25 Gravitational agglomeration occurs as a result of the size dependence of the terminal velocity of the particles. The slowly settling (generally smaller) particles are captured by the more rapidly settling (generally larger) particles. Thus, the problem is reduced to the determination of the critical velocity of the particles. This mechanism is important for supermicrometer particles and its theory has been well established.69 Saffman and Turner10 subdivide turbulent agglomeration into two processes: turbulent shear agglomeration and turbulent inertial agglomeration. Turbulent shear can cause particles following flow pathlines to collide with one another. This occurs because particles on different streamlines are traveling at different speeds. Turbulent shear agglomeration is a result of this effect. Turbulent inertial agglomeration results when particle trajectories depart from r 2011 American Chemical Society
flow streamlines and such departures cause collisions. Turbulence agglomeration is still beset with difficulties and consequently is still a not well understood agglomeration process.11 The model from Saffman and Turner10 provides reasonable order of magnitude estimates for this process.1 Although particle agglomeration has been investigated extensively in the past, agglomeration kinetics, especially the activation energy necessary to start this process, has received much less attention.12,13 The most relevant literature regarding agglomeration kinetics comes from the works on shear-induced agglomeration of particles performed by Chimmili et al.13 and Agarwal et al.14 and the study on kinetics rearrangement of aerosol agglomerates presented by Weber and Friedlander.15 Agglomeration reduces the particle surface area for condensation and/or chemical reaction. Consequently, this is an important phenomenon to be studied, understood, and controlled in the direct application of nanoparticles as nonsupported adsorbents and catalysts. The high surface area, with reduced mass transfer limitations, offered by nanoparticles can be affected by the natural tendency of these very active materials to aggregate, even faster when they are exposed to high reaction temperatures. In the present work, the main objective is to experimentally examine the agglomeration of a model system with a view toward developing a novel fundamental understanding of the kinetics of particle size enlargement, which departs from the classical mechanisms of agglomeration (Brownian, gravitational, and turbulent). This paper also aims to reveal the nature of the agglomeration process (being either a physical or chemical process). Received: January 19, 2011 Accepted: June 2, 2011 Revised: May 28, 2011 Published: June 02, 2011 8529
dx.doi.org/10.1021/ie200135r | Ind. Eng. Chem. Res. 2011, 50, 8529–8535
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The kinetic expression developed in this study is based on the fact that the very act of dispersing a solid powder in a liquid leads to particleparticle collisions that can result in coalescence if a binding mechanism develops. Commonly encountered binding mechanisms include molecular and capillary forces, mechanical interlocking, and liquid bridges.12 The kinetic modeling presented in this work takes into account that the number of particles decreases proportionally with the particle size when particleparticle collisions result in agglomerate formation.13 It is important to notice that agglomerate size does not increase indefinitely; as the aggregates become larger, equilibrium is reached between the binding forces and the forces of separation, this phenomenon is also acknowledged in the present paper. In a previous work,16 a convectivedispersive model that predicts the necessary conditions to maintain particle suspension in a viscous fluid enclosed in a horizontal cylindrical geometry was mathematically developed and experimentally validated. This model allows finding the value of the critical diameter for which particles will remain suspended in a specific system. In this model, particle agglomeration was not taken into account; however, the agglomeration kinetic expression developed in this paper can be added to the previously developed model in order to better predict the conditions for which particles will remain suspended in the viscous medium or will sediment in order to recover and reuse them in a recirculation system. The experimental part of this work was developed by preparing dispersions of Fe2O3 nanoparticles in base oil with different particle concentrations and measuring their particle diameter at different times and temperatures. These experimental data was used to optimize the parameters for the proposed kinetic expression. Simulation results from the previously mentioned convectivedispersive model16 were employed to obtain the experimental conditions for the present study. These conditions guarantee that particles remain suspended in the fluid medium of interest. Before discussing more experimental details, the theoretical analysis of the nanoparticle agglomeration process is presented in the following section.
where Dmax is the maximum diameter that particles can reach when a minimum number of particles remain in the dispersion (analogous to the minimum Gibbs free energy) and P is a proportionality constant that takes into account the shape factor of the particles. Glasgow and Luecke17 presented a mechanistic expression that describes the change in particle number as a result of collisional forces. Thompson et al.18 proposed that in a dispersed system of particles with small diameter and low concentration, the agglomeration of particles can be treated like a simple chemical reaction and employed a power law expression for this purpose. These two independent works yielded a similar expression to represent the variation of the number of particles with respect to time. The form of this expression is
2. THEORY In this part of the work, an analogy is made between the agglomeration process present in an aerosol and the one happening in a system of solid particles dispersed in a liquid. Aerosol agglomerates are often formed by collision between primary particles which can range from few to several tens of nanometers. Weber and Friedlander15 proposed an analysis of the aerosol agglomeration process where the excess Gibbs free energy drives the agglomerates to restructure toward the condensed state. They established that the change in coordination number (the number of the nearest particles around a central one) is proportional to the Gibbs free energy change. An arrangement with minimum Gibbs free energy for active particles leads to a close-packed structure (agglomerate with a large coordination number); on the other hand when the Gibbs free energy is maximum, the number of bonds between particles is zero. Taking this idea into account, it can be assumed that the number of spherical solid particles dispersed in a liquid system (N) (analogous to the Gibbs free energy) is proportional to the change of average particle diameter (D) (equivalent to the coordination number) at any given time (t). This can be expressed as:
where D0 is the particle diameter at the beginning of the process (t = 0). Rearranging eq 6 by dividing by D0, the following expression is obtained:
N ¼ PðDmax DÞ
ð1Þ
dN ¼ kN n dt
ð2Þ
where k is the agglomeration rate coefficient, t is time and, n is the reaction order. Deriving eq 1: dN ¼ PdðDÞ
ð3Þ
The substitution of eqs 1 and 3 in eq 2 provides: P
dðDÞ ¼ k½PðDmax DÞn dt
ð4Þ
Based on the work by Thompson et al.,18 it can be assumed that n = 1; then, dðDÞ ¼ kðDmax DÞ dt
ð5Þ
Considering that at t = 0, D = D0, eq 5 can be solved: D ¼ Dmax expðktÞðDmax D0 Þ
d ¼ deq expðktÞðdeq 1Þ
ð6Þ
ð7Þ
where d = D/D0 and deq = Dmax/ D0. Equation 7 represents the behavior of the particle diameter as a function of time for n = 1. The agglomeration rate coefficient (k) is a function of temperature; by employing the appropriate experimental data, the values of k and deq can be estimated. The calculation of the activation energy is necessary to find the nature of the particle agglomeration process. By making use of the experimental measurements of particle diameters at different temperatures along with the Arrhenius equation, the pre-exponential factor and activation energy for the particle agglomeration of a specific system of study can be found. The well-known Arrhenius equation is described as k ¼ AexpðEa=RTÞ
ð8Þ
where A is the pre-exponential factor, Ea is the activation energy, R is the gas constant (8.314 J/mol K) and, T is the temperature in 8530
dx.doi.org/10.1021/ie200135r |Ind. Eng. Chem. Res. 2011, 50, 8529–8535
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Table 1. Particle Sizes Obtained by DLS for Each Sample at Different Particle Concentrations (CP), Temperatures (T), and Times (t), D0 = 193 nm d = D/D0
t, s
CP, ppm
CP, ppm
CP, ppm
100
300
500
T, °C
T, °C
T, °C
150
200
250
300
350
150
200
250
300
350
150
200
250
300
350
0
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
1.00
600 900
1.14 1.33
1.67 1.95
2.84 3.02
3.33 3.78
4.27 4.95
1.41 1.52
1.87 2.12
2.81 3.47
3.80 4.29
4.05 4.91
1.73 1.87
1.87 2.33
2.95 3.82
3.72 4.63
4.62 5.31
1200
1.35
2.19
3.40
4.36
5.38
1.65
2.50
3.71
4.62
5.35
2.03
2.91
3.87
5.05
5.41
1500
1.52
2.40
3.82
4.63
5.64
1.88
2.53
3.95
5.04
5.68
2.36
3.18
4.39
5.09
5.52
1800
1.61
2.65
4.07
4.83
5.81
2.00
2.87
4.13
5.30
5.82
2.55
3.38
4.60
5.23
5.67
2700
1.84
3.02
4.54
5.36
6.04
2.19
3.17
4.66
5.39
5.88
2.70
3.63
4.71
5.34
6.10
3600
2.02
3.14
4.83
5.46
6.10
2.41
3.46
4.88
5.51
6.17
2.86
3.72
4.91
5.49
6.19
5400
2.36
3.48
4.98
5.54
6.14
2.69
3.78
5.03
5.61
6.16
2.89
3.80
5.07
5.71
6.25
7200 10800
2.66 2.76
3.84 3.99
5.03 5.10
5.60 5.66
6.13 6.15
2.92 2.93
3.91 3.93
5.10 5.11
5.69 5.70
6.20 6.19
2.98 3.01
3.91 3.98
5.12 5.12
5.72 5.73
6.27 6.27
14400
2.79
4.03
5.10
5.66
6.16
2.94
3.97
5.11
5.69
6.19
2.99
3.97
5.12
5.72
6.27
K. The substitution of the Arrhenius expression in eq 7 yields a function for the calculation of the particle diameter at any given time and temperature: d ¼ deq exp½AtexpðEa=RTÞðdeq 1Þ
ð9Þ
3. EXPERIMENTAL SECTION The variation in particle diameter of Fe2O3 nanoparticles dispersed in base oil with respect to time at different temperatures and concentrations was followed to validate and calculate the different parameters from the previously developed kinetic expression. For this, iron(III) oxide (Fe2O3) (NanoAmor laboratories) nanoparticles were dispersed in 100 mL of base oil (Imperial Oil MCT 10) at particle concentrations of 100, 300, and 500 ppm (g/m3). The translucent base oil was selected as the dispersant phase to enable particle diameter determination via Dynamic Light Scattering (DLS). The nanoparticles were dispersed in a sonication bath at 40 °C for 1 h. After this time, 2-mL Pyrex bottles were filled up to 1.5 mL with the dispersed nanoparticles solution with a specific concentration and sealed with metal caps. The bottles were placed in a furnace (Barnstead Thermolyne 62700) under an inert nitrogen atmosphere. The furnace had been previously programmed with a ramp of 10 °C/min and heated until it reached the desired temperature. Five different temperatures were studied: 150, 200, 250, 300, and 350 °C for each temperature, the bottles were retired from the furnace, one by one, at different times in order to perform a DLS analysis to obtain the diameter of the nanoparticles suspended in the solution at that particular time and temperature. The temperature values were deliberately chosen because this work is part of the fundamental research of a multidisciplinary project focused on the ultradispersed catalytic upgrading of heavy oils at low temperature conditions;19 in addition, according to simulation results from the previously developed convectivedispersive model,16 particle sedimentation in the fluid medium is avoided at these experimental conditions.
Once the samples were retired from the furnace, they were cooled until they reached room temperature (25 °C) before being measured via DLS; the DLS equipment that was employed for these measurements cannot handle temperatures above 25 °C.20 Furthermore, this cooling effect serves as a form of quenching for the agglomeration process, stopping any further particle agglomeration once the samples are cooled. Five different DLS measurements were performed for each sample, and the average of these measurements is reported in Table 1 as the average particle diameter. Particle diameter standard deviations varied from 0.63 to 0.96 nm. Samples were taken at different positions inside the vial to ensure the absence of particle sedimentation. Due to the submicrometer nature of the Fe2O3 particles, their particle diameters were determined using a DLS technique with a Malvern Instruments Zetasizer Nano analyzer. The DLS instrument was equipped with a 4.0 mW HeNe laser (633 nm) and operated at an angle of 173° and a temperature of 25 °C to obtain the average particle diameter of the suspended particles. In organic media an ion charge layer (i.e., stern layer) is not present reducing the interaction of the particles with the solvent media, thus not affecting the direct measurement by the DLS technique.21 Particle concentration is also an important parameter in DLS analysis. High particle concentrations produce an increment in particle diameters due to the effects of the agglomeration. These particle diameters can be increased to degrees beyond the measurement limitations of the instrument (>10 μm). High particle concentrations also lead to high sample opacities and multiple scattering which also prevent the instrument from providing meaningful measurements.20 For these reasons, reduced particle concentrations (100500 ppm) were used in these experiments.
4. RESULTS AND DISCUSSION Using the least-squares method, parameters k and deq were optimized from eq 9 employing particle diameters obtained at different times by the DLS analysis for a specific temperature (T) 8531
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Table 2. Optimized Parameters k and deq and AAE% at Different Temperatures and Particle Concentrations
T, °C deq
CP, ppm
CP, ppm
CP, ppm
100
300
500
-1
k, s
AAE% deq
-1
k, s
AAE% deq
k, s-1
AAE%
150 2.97 0.000206
3.4
2.98 0.000361
1.6
2.99 0.000713
1.8
200 3.94 0.000432 250 5.06 0.000801
4.3 1.7
3.94 0.000535 5.06 0.000918
1.6 1.6
3.95 0.000790 5.07 0.001143
3.3 1.9
300 5.60 0.001069
1.2
5.62 0.001406
1.4
5.63 0.001552
2.1
350 6.14 0.001529
0.5
6.16 0.001638
0.8
6.16 0.001806
2.4
Figure 2. Experimental (markers) and model calculated (solid lines) particle growth with respect to time and temperature of a dispersion with a particle concentration of 300 ppm.
Figure 1. Experimental (markers) and model calculated (solid lines) particle growth with respect to time and temperature of a dispersion with a particle concentration of 100 ppm.
and particle concentration (CP). Table 2 shows the different values of the optimized parameters k and deq, and the average percent absolute error (AAE%) corresponding to each studied case. AAE% is the average of all the absolute values of the differences between the experimental and estimated values divided by the experimental ones and multiplied by 100. The comparison between experimental (markers) and model calculated (solid lines) particle diameters at different times for specific temperatures is shown in Figures 13, with each figure corresponding to a different particle concentration. The model calculated results were obtained by employing eq 9 and the data in Table 2. Figures 13 also represent the consequence of changing the suspending liquid viscosity due to a change in temperature while keeping the same particle concentration. As it can be observed in Figures 13 and Table 2, both k and deq increase with increasing the temperature which is equivalent to decreasing the viscosity, the former due to an increase of particleparticle collisions resulting from an increase of particle velocity at lower viscosities or higher temperatures, and the latter because of the ease with which the suspending liquid is squeezed out from between colliding particles.22 The effect of changing the concentration of the dispersed particles while keeping the same temperature, and, therefore, the suspending liquid viscosity is shown in Figures 48. The model calculated results from eq 9 (solid lines) again fit the experimental data (markers) quite well. As it can be seen Figures 48 and Table 2, the value of deq is independent of the particle concentration because this value
Figure 3. Experimental (markers) and model calculated (solid lines) particle growth with respect to time and temperature of a dispersion with a particle concentration of 500 ppm.
Figure 4. Experimental (markers) and model calculated (solid lines) particle growth with respect to time and particle concentration of a dispersion with a temperature of 150 °C.
depends only on a balance between colliding forces and agglomerate strength and not on the amount of suspended particles.13 k is only dependent on temperature; however, there are some 8532
dx.doi.org/10.1021/ie200135r |Ind. Eng. Chem. Res. 2011, 50, 8529–8535
Industrial & Engineering Chemistry Research
Figure 5. Experimental (markers) and model calculated (solid lines) particle growth with respect to time and particle concentration of a dispersion with a temperature of 200 °C.
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Figure 8. Experimental (markers) and model calculated (solid lines) particle growth with respect to time and particle concentration of a dispersion with a temperature of 350 °C.
Table 3. Viscosities of the Solutions Containing Fe2O3 Nanoparticles Immersed in Base Oil in the Range of 150 to 350 °C
Figure 6. Experimental (markers) and model calculated (solid lines) particle growth with respect to time and particle concentration of a dispersion with a temperature of 250 °C.
Figure 7. Experimental (markers) and model calculated (solid lines) particle growth with respect to time and particle concentration of a dispersion with a temperature of 300 °C.
changes observed in this parameter with respect to particle concentration, especially at low temperatures (250 °C), k becomes independent of particle concentration because the number of particleparticle collisions raise due to an increment in particle velocity as a result of a reduction on the viscosity of the solution. Table 3 shows the behavior of the solution viscosity with respect to the temperature; it can be observed that at temperatures >250 °C a more pronounced decrease in viscosity is presented. These viscosities were determined using a Brookfield viscometer model DV-IIþ Pro. Based on the previous comments, it can be affirmed that the effect of the viscosity change with the temperature is already included in the agglomeration rate coefficient (k) and therefore, this effect will be included in the activation energy values of this process. The outcome of this study suggests the occurrence of two different particle agglomeration mechanisms: one at low temperatures, where an increase in particle concentration is the major cause of the agglomeration, and another one at high temperatures, where agglomeration is mainly due to particle collisions generated by an increment in the velocity of the particles. These two apparent agglomeration mechanisms could indicate that the assumption of a first order agglomeration kinetics is not quite appropriate at lower temperatures; since the agglomeration kinetics order should take care of the particle concentration dependence instead of the agglomeration rate coefficient. The assumption of a first order was an approximation based on the work by Thompson et al.18 and as it is indicated in that work the agglomeration kinetics order could also be of order zero. 8533
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particles agglomerated in base oil required activation energies in the order of 36 56 kJ/mol.
Figure 9. Arrhenius plots for the calculation of the activation energy at different particle concentrations.
Table 4. Arrhenius Equation Parameters for Different Initial Particle Concentrations CP, ppm
Ea, kJ/mol
ln A
R2
100
21.9
2.07
0.9939
300
16.7
3.11
0.9795
500
10.8
4.22
0.9526
The activation energy (Ea) for this process was calculated using the linearized Arrhenius expression: ln k ¼ ln A
Ea RT
ð10Þ
Figure 9 presents Arrhenius plots for each particle concentration which were constructed using the data from Table 2. The activation energies (Ea) and frequency factors (A) from these experiments as well as the linear regression coefficients (R2) for eq 10 are presented in Table 4. As it can be seen in Table 4, the values of the activation energies have a slight variation with respect to the particle concentration in the dispersion; this effect is due to the apparent dependence of k with respect to the particle concentration in the dispersion, the explanation of which was presented in previous paragraphs. Results in Figure 9 show that there is a tendency to obtain a single Arrhenius plot for temperatures above 350 °C; thus, if experimental results at temperatures higher than 350 °C were available, a single value of activation energy, independent of particle concentration, could have been obtained. The process of nanoparticle agglomeration may occur via reconstruction of large structural domains. For such a case, the process would be of a chemical nature and it should be expected that, with the formation of multiple bonds leading to larger stable solid particles, a highly exothermic reaction occurs with a high activation energy value.18 However, the obtained activation energies of such low requirement (
