Secondary Particle Formation during the Nonaqueous Synthesis of

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Secondary particle formation during the nonaqueous synthesis of metal oxide nanocrystals Pierre Stolzenburg, Benjamin Hämisch, Sebastian Richter, Klaus Huber, and Georg Garnweitner Langmuir, Just Accepted Manuscript • DOI: 10.1021/acs.langmuir.8b00020 • Publication Date (Web): 01 Oct 2018 Downloaded from http://pubs.acs.org on October 4, 2018

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Secondary particle formation during the nonaqueous synthesis of metal oxide nanocrystals Pierre Stolzenburg,a Benjamin Hämisch,b Sebastian Richter,a Klaus Huber,b Georg Garnweitner∗a a

Institute for Particle Technology and Laboratory for Emerging Nanometrology, Technische Universität Braunschweig, Volkmaroder Str. 5, 38104 Braunschweig, Germany

b

Physical Chemistry, Universität Paderborn, Warburger Str. 100, 33098 Paderborn, Germany *Corresponding author

Abstract This study aims to elucidate the aggregation and agglomeration behavior of TiO2 and ZrO2 nanoparticles during the nonaqueous synthesis. We found that zirconia nanoparticles immediately form spherical-like aggregates after nucleation with a homogeneous size of 200 nm, which can be related to the metastable state of the nuclei and the reduction of surface free energy. These aggregates further agglomerate, following a diffusion-limited colloid agglomeration mechanism that is additionally supported by the high fractal dimension of the resulting agglomerates. In contrast, TiO2 nanoparticles randomly orient and follow a reaction limited colloid agglomeration mechanism that leads to a dense network of particles throughout the entire reaction volume. We performed in situ laser light transmission measurements and showed that particle formation starts earlier than previously reported. A complex population balance equation model was developed that is able to simulate particle aggregation as well as agglomeration which eventually allowed us to distinguish between both phenomena. Hence, we were able to investigate the respective agglomeration kinetics with great agreement to our experimental data. 1.

Introduction

The nonaqueous sol-gel-method1-6 has become a versatile route to tailor highly crystalline metal oxide nanoparticles for various applications.7-17 Undoubtedly, one needs to understand the mechanisms that lead to particle formation in order to develop rational approaches for customizing final nanoparticle 1 ACS Paragon Plus Environment

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properties through the synthesis conditions. In recent works, tremendous progress has been made towards unravelling the formation phenomena that lead to binary,18-22 ternary23-24 and doped10,

25-26

metal oxide nanoparticles. However, the mechanisms of secondary particle formation during the nonaqueous synthesis have not been studied in detail yet, although it has a major influence on subsequent post processing steps like dispersing,27 functionalization28-29 or surface property tailoring3031

. In contrast, the secondary particle formation via agglomeration of nanoparticles is studied in great

detail in various fields such as cancer research32 or environmental science, where numerous works examine the aggregation behavior of nanoparticles in natural waters to perform risk assessment of a possible contamination33-34 or to remove nanoparticles in drinking water.35 All these works are based on the fundamental knowledge of colloid aggregation and agglomeration which Smoluchowski first approached with a mathematical model theory.36 He proposed a particle size-dependent model that describes the interparticle collision rate driven by Brownian motion, called the diffusion limited colloid agglomeration model (DLCA). Particle agglomeration induced by Brownian motion is denoted perikinetic agglomeration whereas agglomeration caused by hydrodynamic forces is referred to as orthokinetic agglomeration.37 Another perikinetic agglomeration model is the reaction limited colloid agglomeration (RLCA) model, where agglomeration is limited by the binding reaction between particles. This limitation is described by the attachment efficiency α which corresponds to the probability of interparticle adherence after collision.38 One can also express this limitation by the Fuchs stability ratio W which refers to the colloidal stability (the inverse of the attachment efficiency, W=1/α).39 In other words, the attachment efficiency and Fuchs ratio represent the deviation between DLCA and RLCA which stems from the colloid integration reaction or stabilizing mechanisms that hinder the particle-particle adherence. Both factors can be derived from the DLVO theory which is still the dominant theory to describe nanoparticle agglomeration in aqueous media.40 The agglomeration behavior of titania nanoparticles from stabilized dispersions synthesized by other solgel processes has been addressed by various authors,41-42 showing that titania agglomeration kinetics highly depend on pH, temperature and ionic strength. For instance, Shih et al. investigated the stability of TiO2 nanoparticle suspensions and showed that at a certain salt concentration, the so-called critical coagulation concentration, the system destabilizes and nanoparticles start to agglomerate.43 Padovini et 2 ACS Paragon Plus Environment

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al. investigated a hydrothermal zirconia nanoparticle synthesis and found that ZrO2 nanoparticles directly form amorphous microspheres before the primary particles crystallize within these structures.44 To the best of our knowledge, no specific studies on the agglomeration kinetics of ZrO2 or similar metal oxide nanoparticles during a nonaqueous or solvothermal synthesis have been carried out so far.

Figure 1. Particle formation mechanisms during the nonaqueous synthesis with an emphasis on the aggregation and agglomeration processes

This study investigates the secondary particle formation processes during the nonaqueous syntheses of zirconia and titania nanoparticles and attempts to understand the mechanisms that drive aggregation as well as agglomeration (Figure 1). We found that densely packed aggregates composed of primary nanoparticles are formed initially, which then further agglomerate to so-called superstructures. The development of the aggregates and the agglomerates was investigated by following their structure and size over the course of the synthesis. Hence, we derived a model for the aggregation kinetics which was used to simulate the entire particle formation process by the method of population balance equations (PBE) with great accordance to the experimentally obtained data. Moreover, we elucidated similarities between both model systems and derived conclusions that are valid for the nonaqueous synthesis in general. A thorough knowledge about these phenomena may not only serve as the groundwork for further research, e.g. on colloidal stability or a possible sol-gel transition, it is a prerequisite for fully understanding the synthesis and allowing its rational adjustment to enhance subsequent processing steps such as stabilization, dispersing and even drying. 3 ACS Paragon Plus Environment

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2.

Materials and methods

2.1

Experimental

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The experiments were conducted in an agitated 250 ml steel autoclave under solvothermal conditions at temperatures ranging from 175 to 250 °C. The reaction vessel was equipped with a light transmission measurement system (wavelength of 635 nm), which allowed us to follow the particle formation process in situ by means of turbidity measurements. A detailed description of the reactor system can be found elsewhere.45 For the experiments, the liquid precursors zirconium n-propoxide (Zr(OnPr)4, 70 wt-% in 1-propanol, Sigma Aldrich) and titanium isopropoxide (Ti(OiPr)4, 97%, Sigma Aldrich) were mixed with the solvent benzyl alcohol (BnOH, 97 wt-%, Merck) according to initial precursor concentrations [Zr]0 and [Ti]0 varying from 180 mmol L-1 to 540 mmol L-1. Afterwards, the mixture was heated to reaction temperature, whereby standard syntheses were performed with an initial precursor concentration of 180 mmol L-1 at 250 °C for the zirconia system and at 200 °C for the titania system. Samples of the reaction mixture were taken through a special outlet valve and further analyses were carried out to follow the characteristics of the solid as well as the liquid phase over the course of the synthesis. 2.2

Characterization

Dynamic light scattering (DLS) using a Zetasizer Nano ZS (Malvern Instruments) was used to investigate the aggregation and agglomeration process. The samples were cooled down and diluted with benzyl alcohol to assure free particle diffusion when determining a volume size distribution at an angle of 173 ° (backscattering detection). Crystallite sizes and phase composition were determined by powder X-ray diffraction (Cu Kα radiation; Empyrean Cu LEF HR goniometer; Empyrean series 2, PANalytical, PIXcel-3D detector; Si wafer; 20–90°, step size 0.05°) and subsequent Rietveld refinement. Scanning electron microscopy (SEM, LEO 1550, Zeiss) and transmission electron microscopy (TEM, JEM-2100F-UHR, JEOL) was used to investigate the secondary particle structures and primary particle sizes. We further processed the micrographs with the open-source software ImageJ to calculate the fractal dimension DF of the agglomerates from the two-dimensional properties of the projected area and the Feret diameter. The fractal dimension DF of the aggregates was 4 ACS Paragon Plus Environment

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determined via static light scattering using an ALV 5000E CGS from ALV-Laser GmbH after dilution of the reaction mixture with the solvent benzyl alcohol. We analyzed the liquid phase of the reaction mixture, determining the normalized metal oxide concentration [MO]i to track the progress of the overall chemical reaction. The solid and liquid (supernatant) phases of a sample were separated by centrifugation and 5 mL supernatant were heated up to 1000 °C. Dynamic light scattering analyses as well as small-angle X-ray scattering of the supernatant revealed only low scattering signals and showed no reasonable evidence of nanoparticles in the system. Thus, a possible fraction of stable nanoparticles in the supernatant was neglected. Gravimetric analysis of the resulting metal oxide enabled us to calculate the metal ion concentration in solution [M]i – the concentration of all soluble metal species – which was used to determine the solid metal oxide expressed as a concentration [MO]i in the reaction volume according to

= − .

2.3

(1)

Modelling

The experimental concentration data was fitted with Matlab (MathWorks, R2017a) to the differential model equations for the chemical reaction kinetics. Subsequently, the particle formation and agglomeration process were simulated by solving the integrodifferential form of the population balance for both model systems with the software Parsival© (Wulkow CIT GmbH, Version 7.6a) applying the Garlekin h-p-method. 3.

Results and discussion

3.1

Nanoparticle aggregation and agglomeration

Both aggregation and agglomeration were investigated over time via dynamic light scattering as this technique yielded the most reproducible results. Figure 2 shows the development of the particle size over the course of a standard ZrO2 and TiO2 nanoparticle synthesis. Interestingly, only aggregates above a size of 200 – 300 nm were detected, which is due to a rapid aggregation step that we have mentioned in our previous report to occur instantly after primary particle nucleation.45 Unfortunately, we were not able to follow this aggregation process in more detail as the characteristic aggregation time was too short to be resolved by our sampling procedure. In a similar fashion, Rizzuti et al. investigated a hydrothermal zirconia nanoparticle synthesis and observed very similar spherical 5 ACS Paragon Plus Environment

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aggregate structures that are composed of 8 nm particles.46 Likewise in our study, these aggregate structures can be regarded as discrete units that further agglomerate to superstructures. Our data shows that the aggregates dominate at the beginning, but then agglomerate to bigger superstructures over the course of the synthesis. The development in size of these structures follows an exponential trend (Figure 2) as agglomeration is promoted by the constant flux of new particles (aggregates) that are formed by the chemical reaction. The increasing amount of particles raises the probability of interparticle collisions and consequently accelerates the agglomerate growth rate, leading to an apparent exponential trend. Additionally, the particles are partly sterically stabilized through the surface-bound organic ligands,47-49 whereby this effect might be surpassed when the particle density becomes too high and interparticle spacing decreases.50-51 This becomes apparent for the TiO2 synthesis and may explain the abrupt change in the agglomeration rate after 2 hours (Figure 2) which is induced by a sudden further step in the particle formation reaction that increases vastly the amount of particles within a short time span. Zimmermann et al. thoroughly investigated the reaction mechanism during the nonaqueous synthesis of TiO2 nanoparticles and showed that a spontaneous water release autocatalyzes particle formation and thereby causes the sudden burst of particles.52 Hence, we will discuss the kinetics of this reaction in combination with our agglomeration model in section 3.2. Agglomerate growth seems to stop at a size above 2000 nm which is however due to the detection limit of the DLS device. Furthermore, particles with larger sizes undergo sedimentation caused by the gravimetric force, which thus prevents larger particles from being detected. In addition, one can observe larger standard deviations when particles approach the micron size regime due to the broadening of the particle size distribution as a result of the agglomeration process. To follow the agglomeration process at the micron scale, we applied laser diffraction, which however only revealed polydisperse particle size distributions over a broad size range that vary randomly over time (data shown in supplementary information Figure S1). These findings might be explained by the effect of shear forces from the agitator, which leads to the breakage of agglomerates. Simultaneous growth via agglomeration and breakage eventually results in fluctuations characterized by the destruction and rearrangement of fractal agglomerates. Schlomach et al. described this phenomenon for a precipitation process of silica nanoparticles and stated that such behavior likely occurs when the system has formed 6 ACS Paragon Plus Environment

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a network of particles throughout the reaction volume.53 However, constant agitation of the reaction system is necessary to realize homogenous process conditions as well as to prevent the sedimentation of particles to the bottom of the reactor.

Figure 2. Agglomeration during a zirconia and titania synthesis under standard conditions measured with dynamic light scattering

Electron microscopy for the zirconia system (Figure 3) revealed that the primary zirconia nanoparticles form spherical-like aggregates with an average size of 263 nm (σ = 62 nm, supplementary information Figure S2), which is in good accordance with the results from dynamic light scattering (Figure 2). The aggregates are formed with a homogenous size throughout the entire synthesis as a result of an isotropic aggregation mechanism. Such a behavior is frequently observed for amorphous and metastable particles below 10 nm as these particles tend to reduce their free surface energy by forming a spherical shape.54 This leads to isotropic aggregation and inevitably to sphericallike aggregates that do not grow in size but only in number. Figure 3a shows that the aggregate structure consists of small zirconia nanoparticles with a size in the low-nanometer regime. To determine the primary particle size more accurately, we stabilized the zirconia nanoparticles after the 7 ACS Paragon Plus Environment

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experiment and performed TEM investigation to obtain a better resolution of the primary nanoparticles (supplementary information section, Figure S3), revealing a primary particle size of 3 nm that corresponds well to the crystallite size of 3.2 nm as determined from the corresponding X-ray diffraction pattern of the sample (XRD, Figure 4). The diffractogram also reveals that the primary particles consist of the tetragonal zirconia polymorph, which we have shown in our previous works to be the metastable form and occur before the particles transform into the monoclinic phase.20 Therefore, the zirconia primary particles might self-arrange instantaneously after nucleating in the tetragonal phase, exhibiting also an isotropic aggregation mechanism, to compact spherical-like structures with a fractal surface and then further agglomerate to fractal superstructures.

Figure 3. TEM micrographs showing (a) primary zirconia crystallites within an aggregate structure and (b) a fractal superstructure (agglomerate) composed of spherical-like aggregates; SEM micrograph (c) showing a large network of agglomerates

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Figure 4. XRD pattern obtained from a sample in the beginning of the agglomeration process showing that the particles are mostly tetragonal with a crystallite size of 3.2 nm determined from the characteristic reflection for the (001) lattice plane

In contrast, the TiO2 nanoparticles are formed with a cubic crystal habit and defined facets (Figure 5a) in the anatase phase. Although anatase is regarded as the metastable form for bulk titania, it is the stable polymorph for small titania nanoparticles as obtained by the nonaqueous method.55 The kinetics of aggregation mechanisms of titania nanoparticles during sol-gel synthesis has been investigated by Penn who observed that primary particles align along their crystal planes and form highly ordered aggregates.56 Penn also stated that surface-bound ligands play a crucial role as they can passivate crystal faces and prevent particles from aligning, which leads to randomly oriented aggregates. The cubic shape of the nanoparticles and the high specific surface energy of the crystal facets favor oriented aggregation, but the chemisorbed organic ligands hinder particles from aligning their crystal planes during aggregation, 3,57 resulting in randomly oriented structures as illustrated in Figure 5b. We determined the amount of chemisorbed ligands on the nanoparticle surface (supplementary information, Figure S5) via thermogravimetric analysis and found that the amount of ligands decreases over the course of the synthesis from 32 to 8 wt.-%, which is in good accordance with the findings by Penn and the observed structures. The amount of chemisorbed ligands on the surface of zirconia nanoparticles is significantly lower than on the titania nanoparticles, which might explain the differences between the observed more dense aggregate structures and also indicates the influence of ligands involved in the aggregation process of metal oxide nanoparticles. Moreover, the TiO2 9 ACS Paragon Plus Environment

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aggregates form superstructures (Figure 5c), whereby one can assume that the agglomeration of aggregates is superimposed by primary particle aggregation, which is typical for Smoluchowski-type aggregation kinetics.36 Again, the primary particle size was determined from TEM images of stabilized nanoparticles (supporting information section, Figure S6), resulting in an average size of about 7 nm that corresponds well to the crystallite size determined from XRD data (supporting information section, Figure S7). In a recent study on the formation and agglomeration of titania nanoparticles by hydrolysis of various titanium alkoxide precursors in benzyl alcohol, Cadman et al. showed that the presence of water promotes the aggregation as well as agglomeration behavior of the formed primary nanoparticles.57 Cadman concluded that −OH groups present on the particle surface after the addition of water eventually causes a reaction-limited agglomeration of the formed nanoparticles and the formation of fractal structures. Therefore, the formation of randomly oriented structures and the observed acceleration of the agglomeration kinetics in our study might be influenced by traces of water being present in the utilized benzyl alcohol solvent or formed during the synthesis. Nevertheless, we did not find strongly different results for a synthesis performed in benzyl alcohol with higher purity, and benzyl alcohol is a hydrophobic solvent with limited water miscibility. Zeta potential measurements yielded insufficient results (data not shown) to characterize the charge of the particle surface for both systems. This indicates that electrostatic mechanisms of colloidal stabilization only exert a minor impact in the studied systems. Both model systems show aggregate structures in the beginning of the synthesis which further agglomerate to larger superstructures and finally form a dense network of particles throughout the entire reaction volume (illustration in supplementary information section Figure S4). This is commonly referred to as the gel point and marks the end of the aggregation and agglomeration processes on the submicron scale. The gel point can be determined by detecting an increase in viscosity (data not shown), which we do not discuss in this study as we focus solely on the particle aggregation and agglomeration mechanisms.


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Figure 5. TEM micrographs (a) showing primary TiO2 particles with a cubic shape; (b) showing a randomly formed aggregate composed of primary particles; (c) demonstrating a fractal agglomerate structure which is predominant in the system at later synthesis stages

In order to elucidate the structure of the aggregates, we used static light scattering to determine the fractal dimension DF by applying the method of Sorensen.58 Thus, the scattering intensity S was plotted over the scattering vector q, which allows the determination of DF according to the relation ∝

(2)

with

=

4π

λ

sin 2

θ

(3)

in the scattering angle range of 30° < θ < 150°. Figure 6 shows the results for both model systems for samples obtained from the reaction solution after 90 min for the ZrO2 synthesis and after 135 min for the TiO2 system. The data shows a fractal dimension DF of 2.97 for the zirconia system, which corresponds most likely to the very dense and randomly packed nanoparticle that assemble to spherical-like structures as observed by TEM. Similar results were detected over the course of the entire synthesis, indicating that static light scattering detected the structure of the aggregates but not of the agglomerates. One has to take into account that the reaction mixture had to be diluted before the measurement which might have caused deagglomeration. This substantiates the assumption that the aggregates form metastable bonds within the agglomerates. These bonds tend to break upon dilution or when exposed to shear forces. The fractal dimension of the TiO2 aggregates was determined to 2.74 which is close to DF of the zirconia aggregates and indicates that the aggregates are randomly shaped as well but are less densely packed as found in the transmission electron microscopy images (cf. Figure 5). In comparison to the zirconia system, the fractal dimension of the TiO2 aggregates might 11 ACS Paragon Plus Environment

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also be valid for the agglomerated superstructures, for primary particles as well as aggregates can be regarded as entities that integrate into larger particle clusters. Such a particle-cluster aggregation mechanism leads to fractal dimensions between 2.5 and 3.0 and is described as typical for ceramic materials.59

Figure 6. Results from static light scattering indicating compact zirconia aggregates with a course surface structure (DF 2.97) and densely packed but randomly shaped structure for the titania aggregates (DF 2.74)

As the fractal dimension of the ZrO2 agglomerates could not be detected by static light scattering, we used an image analysis approach as introduced by Tang.60 Thereby, we processed transmission electron micrographs with an automated routine, using the open source software ImageJ, to determine the projection area A and maximum length L (Feret diameter) of a large number of agglomerates. This enabled us to derive the average fractal dimension DF from the slope of a Log-Log plot of the following relation:

∝

(4)

A DF value of 1.74 resulted from this determination. Strictly, this method only provides the fractal dimension of the two-dimensional projection, but simulations and experimental studies of Meakin 12 ACS Paragon Plus Environment

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have shown that two-dimensional and three-dimensional fractal dimensions are equal if DF is less than 2.61 A fractal dimension less than 2 indicates diffusion-limited colloid agglomeration (DLCA) which is conclusive given the branched structure of the agglomerates and the assumption that the aggregates easily form loose bonds with each other.62 Yet, the densely packed structure of the aggregates for zirconia and titania nanoparticles as well as for the titania superstructures rather indicate a reaction limited colloid aggregation (RLCA) mechanism.63 For better illustration, a schematic overview of the secondary particle formation processes for both model systems is shown in Figure 8. We address the question whether the aggregation mechanism corresponds rather to the DLCA- or the RLCA model by population balance equation modelling in section 3.2.

Figure 7. Agglomerate projection area A as a function of the maximum length (Feret diameter) L of ZrO2 agglomerates obtained from electron microscopy and image analysis; the inset illustrates the projection area and maximum length of an agglomerate as processed with image analysis


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Figure 8. Schematic comparison of the aggregation and agglomeration phenomena during the synthesis of ZrO2 and TiO2 nanoparticles, illustrating that the agglomeration process of zirconia nanoparticles is determined by diffusion-limited colloid agglomeration kinetics, while for titania nanoparticle reaction-limited colloid agglomeration kinetics are dominant.

3.2

Population balance equation modelling

Investigating aggregation and agglomeration processes during a nanoparticle synthesis is demanding as the particle number concentration is determined by two events. First, the number of particles is constantly decreasing due to smaller particles aggregating to fewer bigger particles. Secondly, the chemical reaction yields a constant flux of formed primary particles that increases the number concentration and provides further monomer species that facilitate primary particle growth. To approach this problem, we use the population balance equations method (PBE) in combination with chemical reaction kinetics to model the agglomeration mechanisms and validate the simulation with our experimental results. For the sake of simplicity, we assume that primary nanoparticles are monodisperse and neglect growth processes. Thus, the kinetics of the following overall reactions Zr OnPr4 4 BnOH "####$ ZrO2 4 ROR

R:Bn/nPr

(5)

and

Ti OiPr4 4 BnOH "####$ TiO2 4 ROR,

R:Bn/iPr

(6)

identified previously as the main reaction mechanisms,20, 52 are the basis for the mass balances which eventually can be used to calculate the quantity of primary particles entering the system. The kinetics 14 ACS Paragon Plus Environment

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for the zirconia nanoparticle synthesis as a function of the process conditions are obtained from our previous process simulation model,45 whereas we determined the autocatalytic reaction kinetics for the titania nanoparticle synthesis in this study. In combination with the mass balance and the number of primary particles, we can use the aggregate sizes to calculate the actual number concentration of particles in the system. Thereby, one can distinguish the changes in particle number concentration due to the synthesis from the changes due to agglomeration. The general dynamic population balance equation can be written according to Friedlander64 as follows

*+ ,, - *+ ,, - *, + ,, - − . / = 0 **, *-

(7)

*+ ,, - − .1223. /1223. − .1224. = 0 *-

(8)

which can in this study be simplified to Equation 8.

The change in the number of particles is represented by the differential term of n(x,t), the birth terms B describe the formation of new particles through agglomeration and aggregation and the death term D represents the loss of smaller particles agglomerating to bigger particles. As the aggregation term Baggr. is determined by the flux of nucleating nanoparticles that instantly aggregate, it serves in this model as the nucleation term. Baggr. is expressed by a power law function (Equation 9) that yields the temporary number of formed aggregates with a size x0 and is driven by the chemical reactions that lead to nucleation and indirectly to aggregation. Hence, this aggregation term comprises the metal oxide monomer concentration [MO]m that represents the amount of metal oxide species in solution and is calculated from the differential equations that reflect the chemical reaction kinetics and therefore couple the mass- with population balances.45 In addition, kaggr. is the aggregation rate constant and

φ(x0) a normal distribution function with x0 corresponding to the formed particle size of the respective model system.

.5667. = 81224. :9 φ ,

(9)

As mentioned above, the first step corresponding to the aggregation of the zirconia nanoparticles is too fast to be resolved by our measuring techniques and directly leads to spherical aggregates. Hence, with our population balance equation model we focus on the formation of the agglomerate 15 ACS Paragon Plus Environment

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superstructures during synthesis and attempt to investigate the underlying agglomeration mechanisms by solving the respective model equations. The zirconia agglomerates each consist of a number of spherical aggregates with an average diameter x0 (Figure 9). The apparent diameter of a fractal agglomerate can be described by the collision diameter xC which accounts for the influence of the agglomerate structure on the collision behavior and depends on the fractal dimension DF as well as on the number of aggregates per agglomerate.65-66. Since the collision diameter is a measure for the apparent and effective diameter of the agglomerates in solution, we assume that it is also in good accordance with the hydrodynamic diameter measured in the experiment. Therefore, the experimentally obtained agglomerate sizes from DLS are compared to the calculated values for the collision diameter from our simulations. Furthermore, we introduce the volume equivalent diameter xL which corresponds to the diameter of a sphere having the same volume as the sum of all aggregates within the agglomerate (as illustrated in Figure 9). This is necessary as our simulation tool Parsival© solves the mass balances along the PBE and thus requires a volume equivalent diameter for mass preservation within the calculations. For the population balance equations model, the volume equivalent diameter xL is used as the size (property) coordinate, whereby our model takes into account the collision diameter with Equation 10 to represent the agglomeration behavior of the fractal agglomerates. The relation between the collision diameter xC and the volume equivalent diameter xL describes the geometric deviation from a fractal agglomerate to a spherical agglomerate by the fractal dimension DF as follows 67 @

,=> AB ,; = < > ? , , ,

(10)

whereby the term xL3/x03 represents the number of particles x0 within an agglomerate.


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Figure 9. Illustration of an aggregate with the size x0, the respective particle as a volume equivalent sphere with the diameter xL and the real structure of an agglomerate described by the collision diameter xC

Thus, the agglomeration terms can be formulated as follows:65 @ I

@ @ 1 FGH JGH KLMN .1223. ,, - = D E ,=> − ,=R> I , , R = , -+ ,=> − ,=R> I , -+,=R , -S,= ′ 2 GH I

I

OPQ

and

U

/1223. ,, - = D E ,= , , R = , -+ ,=R , -S,=R ,

(11)

(12)

where β denotes the so called agglomeration kernel which describes the collision rate of two particles. One of the most common agglomeration rate models is the diffusion limited colloid aggregation (DLCA) model by Smoluchowski which mathematically describes the size dependent interparticle collision rate for colloids exposed to Brownian motion.36 The DLCA model is valid for submicron particles (< 1 µm) and hypothesizes that the particles directly adhere after collision. Thus, the respective model Equation 13 for βDLCA is a function of the viscosity η of the solvent, the temperature T as well the sizes of two colliding particles x1 and x2. β=;V =

28: W ,Y ,Z Z 3η ,Y ,Z

(13)

To take into account the fractal structure of the agglomerates, the collision diameter xC in the form of Equation 10 is used to rearrange Equation 13 to β∗ =

I I 28: W 1 1 AB AB \ I I ] < ,=,Y ,=,Z ?. 3η AB AB ,=,Y ,=,Z

(14)

We solved the population balance model over the course of the synthesis (Figure 10) and found that the size-dependent DLCA model does not properly describe the agglomeration process during the 17 ACS Paragon Plus Environment

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ZrO2 synthesis. One might have to take into account that the flux of new particles and the larger collision diameters of the agglomerates lead to an increase of the total agglomerate volume and a higher probability of interparticle collisions. We attribute this by assuming that the effective volume fraction of the agglomerate superstructures ϕeff is proportional to the agglomeration rate in Equation 15. β^_` = β∗ ϕ_aa = β∗

b_aa bcdc13

(15)

The total volume Vtotal is the reaction volume, whereas the effective volume Veff amounts to the sum of all agglomerate collision volumes. A collision volume corresponds to the rotational volume of an agglomerate rotating around its center and can be described by the volume of a sphere with the collision diameter xC. Thus the volume fraction accounts to ϕ_aa = + ,; S,; ?

1 bcdc13

(16)

and is rearranged in Equation 16 with respect to xL to ϕ_aa = gD

U

Gf,KLM 6

π

ABhI AB

,= ? + ,= S,= i

1 . bcdc13

(17)

Figure 10 shows the development of the simulated agglomerate sizes for various model parameters and the experimental agglomerate sizes over the course of a standard ZrO2 synthesis. Our model best agrees with the experimental data if one uses a fractal dimension of 1.7 which is in good accordance with the value for the agglomerates obtained by image analysis. The simulation using a fractal dimension of 2.3 showed no agreement with our experimental data and therefore illustrates the strong influence of the fractal dimension on the agglomeration model. Our simulation data (Figure 11) shows that the number of particles, i.e. particle entities, first increases in the beginning due to the rapid formation of aggregates and then drastically decreases due to the enhanced agglomeration process. The number of particles passes a maximum before agglomeration accelerates exponentially, which might indicate that a vast number of aggregates are stable in solution until a critical concentration is exceeded. As discussed in section 3.1, high concentrations decrease the interparticle spacing as well as increase collision probability which both promote agglomeration.


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Figure 10. Development of the simulated agglomerates sizes in comparison to the experimental data of a standard ZrO2 synthesis

Figure 11. Development of the aggregate size obtained from experiment and simulation as well as the development of the number of particle entities in the entire system over the course of a standard ZrO2 synthesis


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In order to simulate the agglomeration behavior of the TiO2 particles during synthesis, we had to model the chemical reaction kinetics. As the particle formation is autocatalyzed by the formation of water,52 we chose autocatalytic reaction kinetics to model this synthesis.68 Then we determined the respective reaction rates by fitting the set of differential equations of the involved chemical reactions to our experimental data. Figure 12 illustrates that the measured molar concentration of the formed TiO2 is in good accordance with the kinetic model. In contrast to Zimmermann et al.,52 we found that titania nanoparticles are already present in the solution at a small concentration before the spontaneous water release (accompanied by an increase in pressure), which is inevitably due to the autocatalytic reaction mechanism.

Figure 12. Kinetic model for autocatalytic reactions describing the evolution of the solid TiO2 concentration and showing the nucleation of titania nanoparticles before the spontaneous water release (as seen by the increase in pressure)

Likewise, we used the chemical reaction kinetics for the TiO2 system to calculate the metal oxide monomer concentration [MO]m and thereby described the flux of aggregates with a size x0 in the aggregation birth term Baggr. (Equation 9). We found that a size-dependent reaction limited colloid 20 ACS Paragon Plus Environment

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agglomeration mechanism (RLCA) best describes the titania agglomeration process. The RLCA model takes into account the colloidal stability or Fuchs stability ratio W which is formulated as follows: j=

β=;V . βk=;V

(18)

In this respect, the agglomeration kernel for reaction limited aggregation denotes to βk=;V =

1 28: W ,Y ,Z Z . j 3η ,Y ,Z

(19)

Figure 13 illustrates the development of the particle size obtained from experiment and simulation as well as the laser light transmission through the reaction medium over the course of a standard TiO2 synthesis. The experimental data corresponds well to the respective simulation data when an RLCA agglomeration model is used. We calculated a stability ratio W of 0.4 from our simulation results which indicates that particles agglomerate slightly faster than under a diffusion limited colloid aggregation regime, which is somewhat surprising as we expected a value greater than 1. The DLCA model respects solely the agglomeration and aggregation of colloids from a stable suspension and does not take into account the formation of new primary particles by chemical reactions. In contrast, our population balance model incorporates the nucleation of new particles through the aggregation birth term and thus factors the agglomeration kinetics. This results in a higher value of the RLCA kernel which consequently explains the unexpected result of a smaller stability ratio W. Interestingly, in this case we did not incorporate the fractal dimension DF in the calculation which indicates that the experimentally determined value of 2.7 (section 3.1) might be valid as it is close to a standard value of 3 for densely and randomly packed structures. We were able to precisely detect the onset of particle formation and agglomeration via in situ laser light transmission measurements (Figure 13) and found that agglomerates are only detected by DLS above a size of 250 nm. Similar measurements were performed by Soloviev et al. for determining the aggregation kinetics during an aqueous sol-gel process based on titanium tetraisopropoxide.42 They found that an RLCA mechanism dominates the formation of aggregates shortly after nucleation and moreover, the decrease of the light transmission was reported to refer to the kinetics of particle formation. This is conclusive with our results, as we state that the particle formation process starts prior to the spontaneous water release which was followed by the decrease in light transmission. 21 ACS Paragon Plus Environment

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Figure 13. Particle sizes obtained from experiment and simulation during a standard TiO2 nanoparticle synthesis and corresponding laser light transmission showing the onset of first particle formation

4.

Conclusion

Our investigations show that the secondary particle formation processes differ between the zirconia and titania systems. Although both systems immediately form stable aggregate structures in the beginning of the synthesis, the mechanisms that drive further agglomeration highly depend on the characteristics of the nanoparticles themselves and lead to distinct superstructures. Zirconia nuclei insprimartantly form spherical aggregates with a homogenous size of 200 nm, due to the tendency of the metastable tetragonal polymorph to reduce surface free energy. The zirconia aggregates further agglomerate to fractal superstructures, following a diffusion limited colloid agglomeration mechanism. This mechanism is promoted through the increasing volume fraction of the fractal agglomerates and thereby is highly dependent on the fractal dimension DF. We propose a new agglomeration model that incorporates the DLCA mechanism as well as the effective volume fraction of the agglomerates and is


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thereby able to simulate the agglomeration process using a fractal dimension that is in great accordance to the values observed experimentally by static light scattering and TEM. Titania nanoparticles nucleate in the anatase polymorph and subsequently form randomly oriented aggregates. Although the defined crystal facets of the primary nanoparticles should result in highly oriented structures, we propose that the reaction medium or side products of the synthesis bind to the particle surface and form an organic ligand shell that plays a crucial role and hinder crystal alignment, leading to randomly ordered aggregates. The aggregates furthermore agglomerate, interestingly showing no structural change with respect to the fractal dimensions, according to a reaction limited colloid agglomeration mechanism which we were able simulate with the population balance equation model. Our results provide an insight into the particle formation mechanisms after nucleation and explain the pathway to the resulting aggregate and agglomerate structures during the nonaqueous synthesis. The synthesis was performed in the absence of stabilizing agents, as it is common for the nonaqueous method, to realize an experimentally simple system and achieve products of high purity. Hence, these findings may be the groundwork for in situ agglomeration studies and simplify further investigations on primary particle stabilization as well as post synthesis processing such as dispersing and functionalization. Acknowledgments This study was financed by the Deutsche Forschungsgemeinschaft (grant GA 1492/3-3). The authors would like to thank Bilal Temel for TEM and XRD measurements, and the Laboratory of Nano and Quanum Engineering (LNQE, Leibniz University Hannover) for the TEM instrument. Moreover, the authors are grateful to Ramazan Aydemir and Kaite Qin for great experimental assistance.

Keywords Metal oxide nanoparticles, nonaqueous sol-gel method, aggregation, agglomeration, simulation, population balance equations


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Secondary Particle Formation during the Nonaqueous Synthesis of

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