Multidimensional Nature of Fluidized Nanoparticle Agglomerates

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Multidimensional Nature of Fluidized Nanoparticle Agglomerates Lilian de Martín,*,† Wim G. Bouwman,‡ and J. Ruud van Ommen† †

Department of Chemical Engineering and ‡Department of Radiation, Science and Technology, Delft University of Technology, Delft, The Netherlands S Supporting Information *

ABSTRACT: We show that fluidized nanoparticle agglomerates are hierarchical fractal structures with three fractal dimensions: one characterizing sintered aggregates formed during nanoparticle synthesis, one that is also found in stored agglomerates and represents unbroken agglomerates, and one describing the large agglomerates broken during fluidization. This has been possible by using spin-echo small-angle neutron scatteringa relatively novel technique that, for the first time, allowed to characterize in situ the structure of fluidized nanoparticle agglomerates from 21 nm to ∼20 μm. The results show that serial agglomeration mechanisms in the gas phase can generate nanoparticle clusters with different fractal dimensions, contradicting the common approach that considers fluidized nanoparticle agglomerates as single fractals, in analogy to the agglomerates formed by micron-sized particles. This work has important implications for the fluidization field but also has a wider impact. Current studies deal with the formation and properties of clusters where the building blocks are particles and the structure can be characterized by only one fractal dimension. However, fluidized nanoparticle agglomerates are low-dimensional clusters formed by higher-dimensional clusters that are formed by low-dimensional clusters. This multifractality demands a new type of multiscale model able to capture the interplay between different scales.

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During fluidization, drag and collision forces start playing a role and the agglomerates that were formed during storage break in smaller pieces that recombine continuously to rebuild new agglomerates, exchanging aggregates during the process.13,14 These dynamics are evident from the experiments conducted by Hakim et al.,15 who fluidized colored batches of nanopowders and obtained agglomerates with a mixture of the different colors. Yao et al.16 classified fluidized nanoparticle agglomerates as aggregates (1−100 nm), simple agglomerates (1−100 μm), and complex agglomerates (200−400 μm) based on microscopy images. This apparent hierarchy made some authors wonder if fluidized nanoparticle agglomerates might have different fractal dimensions.13 However, this question has remained unanswered due to the fragile nature of the agglomerates and the lack of proper in situ techniques. As a consequence, most of the authors simply assume fluidized nanoparticle agglomerates are single fractals and estimate the fractal dimension from their density. A quantitative description of the agglomerate structure from nanoparticle to agglomerate scale for fluidized agglomerates is still missing. This knowledge has obvious implications in the fluidization of nanopowders but also would give fundamental insights about the behavior of nanopowders in the gas phase.

INTRODUCTION

Nanoparticles are extensively used in diverse applications, such as solar cells, batteries, catalysts, and cosmetics, because of the unique properties of the nanoscale. Often, the bare surfaces of nanoparticles cannot be used directly as synthesized for applications but must be coated to give them specific properties or functional groups. Gas−solid fluidization has been successfully employed to coat nanoparticles with atomic layer deposition1−3 and is a promising method to process large amounts of nanopowder due to the high production capacity of fluidized beds, compared to aerosol-based reactors. The type of nanopowder mostly used in fluidization studies due to its availability is in flame produced nanoparticles. Nanoparticles sinter in the flame reactor to form fractal aggregates that are very strong due to the covalent nature of the interparticle bonds.4 These aggregates have been extensively studied by small-angle neutron scattering (SANS), small-angle X-ray scattering (SAXS), and light scattering and are characterized by a low fractal dimension ∼1.8−2.15−8 and a size of approximately 100−1000 nm.7−9 If sintered aggregates come in contact, for instance during storage, they will further link to form larger and soft agglomerates by means of van der Waals force, hydrogen bonds,10 and adsorbed water.11 Opposite to aggregates, soft agglomerates are weak, and their structure and size are affected by the external forces typically present during storage, transport, and processing.12 © 2014 American Chemical Society

Received: July 28, 2014 Revised: September 16, 2014 Published: October 2, 2014 12696

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In this work, we elucidate the structure of fluidized TiO2 nanoparticle agglomerates from nanoparticle (21 nm) to agglomerate scale (20 μm) by using in situ spin-echo smallangle neutron scattering (SESANS). By combining the results obtained from SESANS and previous information reported from video analysis,17 we show for the first time that fluidized nanoparticle agglomerates have three fractal dimensions: one characterizing sintered aggregates formed during nanoparticle synthesis, a second one that is also found in stored agglomerates and represents unbroken agglomerates, and a third one describing the large agglomerates broken during fluidization. This work reveals the structural peculiarities of fluidized nanoparticle agglomerates as a consequence of the different environments experienced by the powder. It also highlights the importance of developing new models able to link the different agglomerate scales.

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Therefore, SESANS measurement represents an average structure of the agglomerates. SESANS has two features very suitable for the study of nanoparticle agglomerates; it covers from nanoparticle scale (20 nm) to agglomerate scale (20 μm) and has the information encoded in real space, instead of reciprocal. The latter allows a more direct comparison between the experimental data and the autocorrelation function of the sample, defined as

ρp (kg/m3)

ρb (kg/m3)

ρt (kg/m3)

stored stored fluidized fluidized

SiO2 R812 TiO2 P25 TiO2 P25 TiO2 P25

7 21 21 21

2200 4000 4000 4000

50 ± 10 130 ± 10 N/A N/A

60 130 N/A N/A

(2)

where Σt is the average number of scattering events of a neutron while going through the sample and G(z) is the Abel projection of γ(r) (eq 3).20

G(z) =

2 Γ

∫z

∞

γ(r )r r 2 − z2

dr (3)

where

Γ=2

∫0

∞

γ(r ) dr

(4)

The number of scattering events Σt can be calculated from Σt = λ2t(Δρ0)2ϕ(1 − ϕ)Γ, where λ is the wavelength of the neutron beam (λ = 0.209 ± 0.001 nm), t is the thickness of the sample, ϕ is the total voidage, and Δρ0 is the contrast. Δρ0 can be easily calculated from the scattering length density of the atoms and atomic density, which is equal to the particle density.23 For stagnant powder ϕ = 1 − ρb/ρp, whereas for fluidized powder ϕ was determined from the voidage of stagnant powder and transmission measurements. The scattering variables are summarize in Table 2.

Table 1. Characteristics of the Nanopowders Investigated, Manufactured by Evonik Industriesa dp (nm)

(1)

P(z) = exp(Σt (G(z) − 1))

Four sets of experiments were carried out: two of them with stagnant (stored) SiO2 R812 and TiO2 P25 and another two fluidizing TiO2 P25 under identical conditions. Stored powder that has not been fluidized was gently deposited in a cuvette for analysis. TiO2 nanoparticles were fluidized in a 4.5 × 4.5 cm square column made of quartz and equipped with a porous plate distributor. This column is the same that was used in our earlier work.17 Fluidization is conducted under atmospheric pressure by flowing nitrogen gas through the bed at a superficial velocity about 0.10 m/s. The height of the static bed was approximately 4 cm. The properties of the nanopowers are shown in Table 1.

powder

∫ ρ(r′)ρ(r′ + r) dr

where ρ(r) is the scattering length density distribution and V is the volume. γ(r) gives the probability of observing the same density when looking at (r′ + r) away from a point r′ in the density distribution.20 The outcome of SESANS is a curve with polarization of the neutron beam P(z) as a function of the spin-echo lengthor length scalez. γ(r) and P(z) are related by

EXPERIMENTAL SECTION

state

1 V

γ(r) =

Table 2. Scattering Variables of the Nanopowdersa

The tamped density ρt is obtained from ref 18. The bulk density of the stagnant bed ρb was determined experimentally. Experiments for fluidized TiO2 were duplicated.

state

powder

t (mm)

Δρ0 (μm−2)

stored stored fluidized fluidized

SiO2 R812 TiO2 P25 TiO2 P25 TiO2 P25

15 40 40 40

347 246 246 246

a

Nanoparticle agglomerates were analyzed in situ by spin-echo smallangle neutron scattering (SESANS) at the Reactor Institute Delft (Delft, The Netherlands).19−21 SESANS measures the polarization of a neutron beam that passes through two vertical parallelogram-shaped magnetic fields with opposite field directions. Neutrons traveling through these magnetic fields experience a precession with exactly opposite precession angles in the first and second field. The magnitude of the angle depends on the time spent by the neutrons in the field. Without scattering, the precession angle in the first field is reversed (spin echo) in the second field, recovering the initial polarization. However, if the beam scatters after passing through the first field, the precession angle is not echoed in the second field because the time spent in both fields is different, depending on the scattering angle. This is the way SESANS encodes the scattering angles. For a detailed description of SESANS measurements, see Rekveldt et al.22 Nanoparticle agglomerates have been fluidized in some of the experiments conducted in this work. These agglomerates move, break, and recombine continuously while taking the SESANS measurements. Although fluidized agglomerates are dynamical, their average structure reaches a stationary state after some minutes. The scattering process operates on a much smaller time scale than the dynamics of the agglomerates, so we effectively get a snapshot of their structure. These snap shots are accumulated over a longer period, in the order of hours.

a

Σt/Γ (μm−1)

ϕ 0.978 0.968 0.984 0.987

± ± ± ±

0.005 0.003 0.005 0.005

1.7 3.3 1.6 1.4

± ± ± ±

0.4 0.3 0.5 0.3

Experiments for fluidized TiO2 were duplicated.

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MODELING Autocorrelation Function of Stored Nanoparticle Agglomerates. It is well known that nanoparticles sinter in the flame reactor and form aggregates that are fractals,5−8 having therefore an autocorrelation function γ(r) ∝ (r/rp)(Df,1−3),24 where Df,1 is the fractal dimension and rp is the particle size. Aggregates are collected in a filter where they further link to form larger and soft agglomerates. During storage, these agglomerates experience repeated fragmentation and reagglomeration due to random forces, such as shocks and vibrations. Schwager et al.25 were the first authors to simulate these dynamics, finding in 2D simulations that fragmentation and reagglomeration lead to fractal clusters with a robust fractal dimension of 1.7 up to the fragmentation scale. A fractal structure was also found in compressed cohesive powder using 12697

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SESANS.26 Furthermore, fractality is common in gelation processes, which share many features with aggregation of granular solids.27 If these observations are aplicable to our powder, and sintered aggregates form larger fractal agglomerates of radius rg,2, the autocorrelation function at scales larger than the size of the aggregates is γ(r) ∝ (r/a)(Df,2−3), where Df,2 is the fractal dimension of the large agglomerates, and a = f(rp,rg,1,Df,1,Df,2), where rg,1 is the gyration radius of the aggregates. Different agglomeration mechanisms during nanoparticle synthesis and storage will form agglomerates with different Df,1 and Df,2. Note the similarities between this bidimensional autocorrelation function and that observed by Sorensen et al.28 in what they call superaggregates. These authors report the formation in flame reactors of 10 μm soot agglomerates with Df,2 = 2.6, obtained after reagglomeration of smaller aggregates with Df,1 = 1.8. They propose aggregate percolation or restructuring as possible mechanisms. This bifractality can be translated into the relevant SESAN variables with the following piecewise autocorrelation function: γ1(r ) = (r /rp + 1)Df,1− 3

Figure 1. Sketch of the change from γ1(r) to γ2(r). Black segments represent power-law regions, red segment from 0.3 to 0.7 μm represents transition between hierarchical levels, and dashed line represents eq 5.

done considering that stored fluidized agglomerates are 200 μm two-dimensional fractals. Autocorrelation Function of Fluidized Nanoparticle Agglomerates. During fluidization, the agglomerates that were formed during storage fragment and reagglomerate continuously with mechanisms that are characteristic of the fluidized state. Nevertheless, these dynamics do not affect all scales. Hakim et al.15 fluidized colored batches of nanopowders and obtained agglomerates with a mixture of the different colors. However, the microscope images also show areas of several microns uniformly colored, revealing that the smallest agglomerates, were not broken during fluidization. A similar result was reported by Nakamura and Watano34 working in a rotatory fluidized bed, finding that mixing occurs at scales of several tens of microns. Valverde and Castellanos35 found that the degree of powder compaction prior to fluidization influences the fluidization dynamics. This verifies that the powder has memory and some scales are preserved during fluidization. If all scales up to aggregate level were broken, the powder would not retain any information about storage. We define here a f ragmentation scale (rc,2). Agglomerates larger than the fragmentation scale have a low relative strength (low ratio between bond and hydrodynamic forces36) and can be broken during conventional fluidization. Agglomerates smaller than the fragmentation scale have a large relative strength (large ratio between bond and hydrodynamic forces36) and are not broken during conventional fluidization. Note that the fragmentation scale comes from a balance between attractive and separating forces and therefore depends on the environment. The autocorrelation function proposed for fluidized agglomerates has therefore three scales: (1) aggregate scale (r < rc,1), (2) intermediate scale (rc,1 ≤ r < rc,2), and (3) large scales r ≥ rc,2. If the structure of the aggregate and intermediate scale preserves the fractal scaling during fluidization, γ1(r) ∝ (r/rp)(Df,1−3) and γ2(r) ∝ (r/a1)(Df,2−3), respectively, as discussed for stored agglomerates. Large agglomerate scales have been recently characterized by video analysis, finding a power law behavior between agglomerate density and size in the range 40−400 μm.37 This reveals that large scales also have fractal scaling and therefore γ3(r) ∝ (r/a2)(Df,3−3). According to what has been discussed, the autocorrelation function for fluidized agglomerates can be expressed as the following piecewise function:

for r ≤ rc,1

γ2(r ) = (r /a + 1)Df,2 − 3 h(r , ξ2) for r > rc,1

(5)

where a is obtained equating γ1(r) = γ2(r) at r = rc,1. The parameter rc,1 represents the length scale at which the correlation changes from γ1(r) to γ2(r). The +1 in eq 5 does not appear in the autocorrelation of a fractal but is a modification introduced by Andersson et al.20 to have correlation γ(0) = 1. This modification is necessary to avoid divergence at r = 0 and does not affect the estimated fractal dimensions. h(r,ξ2) is a cutoff function that accounts for the finite size of the agglomerates and forces γ2(r) to quickly decay at length scales larger than ξ2.29,30 Thus, ξi represents the length scale at which the correlation starts to change due to approaching the aggregate/agglomerate boundaries. Here h(r, ξ2) = e−(r/ξ2),31 where ξi 2 =

2rg, i 2 Df, i(Df, i + 1)

(6)

Equation 5 considers a sharp change between dimensions, as reported by Sorensen et al.28 However, in reality rc,1 might be somewhere between the length scale at which γ1(r) starts to vanish due to aggregate boundaries, this is ξ1, and the length scale at which γ2(r) is fully developed, assumed 4rg,1 (Figure 1) (see Supporting Information). Accordingly, ξi ≤ rc,i < 4rg,i and rc, i /4 < rg, i ≤ rc, i

Df, i(1 + Df, i) 2

(7)

The average size of stored agglomerates (2rg,2) is unknown but in the order of a few hundreds of microns.32 Actually, it is common to find agglomerates larger than 400 μm that must be removed before fluidization. Moreover, in dense gas−solid suspensions, where the agglomerates suffer stronger breaking forces than in the stagnant situation, the average agglomerate size ranges from 70 to 700 μm.33 A sensitivity analysis of the influence of agglomerate size on the estimated fractal dimensions (see Supporting Information) reveals that rg,2 plays a minor role in the results if rg,2 > 100 μm, something that can be safely assumed. Thus, subsequent analysis will be 12698

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γ1(r ) = (r /rp + 1)Df,1− 3

for r ≤ rc,1

γ2(r ) = (r /a1 + 1)Df,2 − 3

for rc,1 < r ≤ rc,2

γ3(r ) = (r /a 2 + 1)Df,3 − 3 h(r , ξ3) for r > rc,2

Stored Aerosil SiO2 R812 consists of 25−180 nm aggregates with Df,1 ∼ 2.1 that link to form agglomerates with Df,2 ∼ 2.72. Df,1 is in excellent agreement with reported values 2.15−2.2 for Aerosil silica.38,39 We have not found reported values for SiO2 R812 aggregate size; however, Aerosil SiO2 200 forms 150 nm aggregates,38,39 in agreement with our findings. Apart from the reasons discussed in the Modeling section, the good agreement of Df,1 and rg,1 with reported values further supports the choice of a bidimensional autocorrelation function γ(r), instead of any other three-parameter model. On top of this, Df,2 = 2.7 is able to approximate the tamped density of stored nanopowder. Note how polarization is estimated (eqs 2 and 3). The polarization at scale z is calculated by integrating γ(r) from z to ∞. An error in the model would greatly affect the polarization at small scales and the parameters obtained. Both powders have a surprisingly similar structure, revealing that they went through similar agglomeration mechanisms, not only in the flame reactor but also during reagglomeration of primary aggregates. This supports the observations of Schwager et al.,25 who found in 2D simulations that fragmentation and recombination of agglomerates form fractals with a robust fractal dimension. The origin of Df,2 ∼ 2.7 is unclear as nobody has proposed a mechanism for fragmentation, recombination, and compression of agglomerates in 3D. Sorensen et al.28 propose percolation and shear-induced restructuring as mechanisms to explain the fractal dimension 2.6 found in the superaggregates formed by smaller aggregates. Percolation forms clusters with fractal dimension ∼2.55, and compression raises the fractal dimension of the resultant agglomerates because it increases the overlap of the building blocks.40 More research is needed to clarify if fragmentation and recombination in 3D generates universal agglomerates, as suggested by Schwager et al.25 and supported by our results, or the fractal dimension 2.7 is the result of a complex interplay between different mechanisms. Fluidized Agglomerates. The procedure to characterize fluidized TiO2 nanoparticle agglomerates is identical to that discussed for stored agglomerates. Experimental polarizations are fitted by using eq 8, which contains six parameters: Df,1, Df,2, Df,3, rc,1, rc,2, and ξ3; too many to be left as fitting parameters. However, Df,1 and rc,1 characterize strong, sintered aggregates that will not be broken during fluidization; hence, these two parameters can be taken from the results for stored agglomerates. Df,3 represents the fractal dimension of the very large agglomerate scales (>40 μm) and was reported in our earlier work17 (Df,3 = 2.24 ± 0.02) using the same powder and similar experimental conditions. Note that we are combining information obtained from very different techniques: video and SESANS analysis. Furthermore, the expected values for ξ3 and Df,3 are outside the experimental sensitivity window of SESANS. ξ3 is calculated from eq 6 and rg,3 = 100 μm, as discussed. Although Df,2 and rc,2 are the only unknown terms, Df,1 will be left as fitting parameter as well. The reason is that the results are extremely sensitive to Df,1 and rc,1, and the error in these parameters is rather large (Table 3). Thus, we prefer to leave Df,1 as fitting parameter and check that the new value is consistent with the value previously obtained for stored agglomerates. Results are equivalent leaving rc,1 as fitting parameter. Results are shown in Figure 3 and Table 4. As expected, the fractal dimension Df,1 is consistent with the value obtained for stored agglomerates, confirming that sintered aggregates do not change in structure. According to SESANS,

(8)

a1 and a2 make the function continuous. An average size of 199 μm for fluidized TiO2 agglomerates was found by Martı ́n et al.37 from visual analysis under similar experimental conditions. Therefore, we assume rg,3 = 100 μm for subsequent analysis, where rg,3 is the gyration radius of fluidized agglomerates.

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RESULTS AND DISCUSSION Stored Agglomerates. Experimental polarizations for stored nanoparticle agglomerates have been correlated with the model described by eq 5 and the procedure explained through eqs 3 and 4. ξ2 was obtained from eq 6, given rg,2 = 100 μm. The fractal dimensions Df,1 and Df,2 and the critical length rc,1 are the values that minimize the sum of squares between experimental and calculated polarizations. In our experience, up to three structural parameters can be analyzed in this way, more than that could give results with no physical meaning. The results are shown in Figure 2 and Table 3.

Figure 2. Dots: experimental polarizations for stored SiO2 and TiO2; error bars are the standard deviation of the polarization for the neutrons detected. Red lines: polarizations obtained from bidimensional γ(r) (eq 5). Dashed lines: polarization obtained from monodimensional γ(r).

Table 3. Fractal Dimensions and Critical Length for Stored Agglomerates (Figure 2); Confidence Intervals Are 2σ TiO2 SiO2

Df,1

rc,1 (μm)

Df,2

1.8 ± 0.2 2.1 ± 0.3

0.07 ± 0.02 0.05 ± 0.04

2.67 ± 0.02 2.72 ± 0.03

It is clear that a monodimensional fractal autocorrelation function γ(r) = (r/rp + 1)Df−3h(r,ξ) (dashed lines) is not able to describe the experimental polarizations in a reasonable way, as expected. This reveals the unique structural properties of nanoparticle agglomerates compared to the monodimensional agglomerates formed by micron-sized particles. In view of the results, stored TiO2 P25 consists of aggregates with a size 2rg,1 between 35 and 220 nm (eq 7) and Df,1 ∼ 1.8 that link to form agglomerates with Df,2 ∼ 2.67. This size interval is in agreement with the 200 nm reported for in-flame produced TiO2 P25.9,38 We have not found values in the literature for the fractal dimension of TiO2 P25 aggregates; however, Df,1 is in good agreement with the dimensions typically found for in-flame produced nanoparticle aggregates of other metal oxides, typical of the cluster−cluster agglomeration mechanism. 12699

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limited to just that field: it demonstrates the existence of multifractal nanoparticle agglomerates in the gas phase as a consequence of different serial agglomeration mechanisms. This phenomenon poses numerous questions of a very fundamental nature, the main one being: how are the properties of a multifractal cluster (i.e., strength, conductivity, permeability, etc.) determined by the interplay between the different scales? There is an extensive amount of literature dealing with agglomeration of particles and the properties of the resultant fractal clusters. However, little work has been done on modeling the agglomeration of preexisting clusters, which would generate multifractal structures. The structural peculiarities of fluidized nanoparticle agglomerates demands new multiscale approaches that will also be useful in fields other than fluidization. On top of this, this work suggests that it is possible to form clusters with controlled multifractal structures using serial agglomeration mechanisms in the gas phase. These clusters can be used, for instance, as a template of porous materials.

Figure 3. Dots: experimental polarizations for duplicated experiments with fluidized TiO2; error bars are the standard deviation of the polarization for the neutrons detected. Red lines: polarization obtained from bidimensional γ(r) (eq 8).

these hard aggregates form secondary agglomerates with a size 5−45 μm (eq 7) and a fractal dimension about 2.7, which build the final large agglomerates observed in our earlier work17 (Figure 4). Df,2 is nearly the same as for stored agglomerates, suggesting that 5−45 μm agglomerates were not broken during fluidization. The existence of a fragmentation scale explains the history dependency of fluidized nanopowder.35 Moreover, the order of magnitude of the fragmentation scale found in this work is in agreement with previous observations. Nam et al.41 obtained agglomerates of average size ∼35 μm after preparing a nanopowder sample for countering. Nakamura and Watano34 reports that mixing occurs at scales of several tens of microns. The extremely simple model proposed in this work gathers all the information available in the literaturetheoretical and practicalfor the structure of fluidized agglomerates from 20 nm to 400 μm, which is remarkable. Opposite to a tridimensional γ(r), a bidimensional γ(r) for fluidized nanoparticle agglomerates fails to explain the experimental observations (see Supporting Information). These results contradict the common approach that considers fluidized nanoparticle agglomerates as single fractals, in analogy to the agglomerates formed by micron-sized particles. For the sake of illustration, let us summarize in Figure 5 the combined results from this work and de Martı ́n et al.17 Stored and fluidized agglomerates share sintered aggregates and also a common structure for scales smaller than the fragmentation scale rc,2. On the contrary, agglomerates larger than rc,2 are broken and recombined with a mechanism that produces lowdimensional agglomerates. Under our experimental conditions rc,2 ∼ 10 μm. If an assistance method, such as vibration or microjets, decreased the fragmentation scale adding energy to the system and broke the strong agglomerates with Df,2 = 2.7 formed during storage, the new formed agglomerates would have a significantly lower density, as fluidization forms agglomerates with Df,3 = 2.0−2.2.17 Agglomerates with lower density would lead to larger bed expansions, as observed.13 In this work, we have shown that fluidized nanoparticle agglomerates are hierarchical structures with three fractal dimensions. This observation has important implications for the fluidization of nanopowders, but its significance is not

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CONCLUSIONS Combining neutron scattering and image analysis, we have shown that fluidized nanoparticle agglomerates have three fractal dimensions as a result of the three different agglomeration mechanisms experienced by the powder during synthesis, storage, and fluidization. These fractal dimensions correspond to the following: • Aggregate scale (r < rc,1). It spans a few hundreds of nanometers and represents the structure of the sintered aggregates formed in the flame reactor. This scale is common for stored and fluidized agglomerates formed by in-flame produced nanoparticles and is characterized by a fractal dimension about two, in agreement with reported values. • Unbreakable scales (rc,1 ≤ r < rc,2). The clusters at this scale are formed during agglomerate storage and are not broken during fluidization. These scales are smaller than the fragmentation scale rc,2, which is defined as the scale where bonding and separating forces are equal. Under our experimental conditions rc,2 ∼ 10 μm. Unbreakable scales are characterized by a large fractal dimension of ∼2.7, but the formation mechanism is unknown. These small scales are the responsible for the history dependency of the fluidized nanopowder and might be the key to understanding how to enhance nanopowder fluidization by assistance methods. • Breakable scales (r ≥ rc,2). The clusters at this scale are large enough to have a low relative strength and to be broken during fluidization. Scales larger than the fragmentation scale are characterized by a low fractal dimension of about two. This work shows how to characterize in situ fluidized nanopowder agglomerates from particle to agglomerate scale integrating the information obtained from two techniques of a very different nature. The insights provided here will help to understand, model, and control the behavior of nanopowders in dense gas suspensions and other hierarchical granular systems.

Table 4. Fractal Dimensions and Critical Length for Fluidized Agglomerates (Figure 3); Confidence Intervals Are 2σ

TiO2 TiO2

Df,1

rc,1 (μm) (known)

Df,2

rc,2 (μm)

Df,3 (known)

1.9 ± 0.4 1.9 ± 0.3

0.07 ± 0.02 0.07 ± 0.02

2.67 ± 0.04 2.71 ± 0.02

10 ± 2 9.9 ± 0.8

2.24 ± 0.02 2.24 ± 0.02

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Figure 4. Sketch of the agglomerate structure from nanoparticle synthesis to fluidization. 2rg,1 = 35−200 nm, Df,1 = 1.8−2.1, 2rg,2 ∼ 200 μm, Df,2 = 2.7, 2r*g,2 = 5−45 μm, 2rg,3 ∼ 200 μm, and Df,3 ∼ 2.2.

Seventh Framework Programme (FP/2007-2013)/ERC Grant, agreement no. 279632.

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Figure 5. Conceptual variation of the agglomerate density as a function of length scale for fluidized (blue line) and stored (red line) TiO2 P25. Black dots are experimental data for fluidized agglomerates.17,37 The red dot is the tamped density of TiO2 P25 (Table 1). Stored agglomerates, represented by the large blue agglomerate and the red dot, break in smaller pieces that recombine to form fluidized agglomerates, represented by the yellow agglomerate and the black dots.

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ASSOCIATED CONTENT

S Supporting Information *

Testing the multidimensional autocorrelation functions with a simulated agglomerate; influence of agglomerate size on the estimated fractal dimensions; polarization of fluidized nanoparticle agglomerates predicted with a bidimensional autocorrelation function. This material is available free of charge via the Internet at http://pubs.acs.org.

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AUTHOR INFORMATION

Corresponding Author

*E-mail [email protected]; Ph +(44) 20 7679 0663; Fax +(44) 20 7383 2348 (L.d.M.). Present Address

L.d.M.: Department of Chemical Engineering, University College London, UK. Notes

The authors declare no competing financial interest.

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ACKNOWLEDGMENTS We acknowledge Chris Duif, Maarten Weeber, and Jörgen Konings for their assistance in the experimental work, and Michael Nigra for commenting on the manuscript. The research leading to these results has received funding from the European Research Council under the European Union’s 12701

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