Numerical Study of the Nanoparticle Formation Mechanism in a Titania Flame Combustion Synthesis Process

In this study, a monodisperse particle formation (MPF) model was built and a five-zone diagram (FZ diagram) was used to examine the Titania (TiO2) combustion synthesis process. The effects of chemical reactions, Brownian motion, sintering reactions and diffusion were considered, while the polydispersity of aggregates and primary particles were neglected in the MPF model. The effective collision frequency was used to modify the collision frequency influenced by van der Waal interactions. There were precursor-heated, chemical reaction/nucleation, high-temperature, coagulation/coalescence and aggregation zones in FZ diagram. Results of the FZ diagram as well as particle size (dp) were investigated via three parameters in the MPF model: particle number density (N), total aggregate volume per unit mass of gas (V) and total aggregate surface area per unit mass of gas (A). The inlet oxygen/nitrogen ratios (O2/N2) change from 20/80 to 50/50 will enhance the hightemperature zone, which increases the collective particle sizes from 81.4 to 120.9 nm; the increasing Titanium isopropoxide (TTIP) concentrations (XTTIP) will also increase the particle sizes from 85.7 to 99.3 nm, due to the reinforcement in the chemical reaction/nucleation zone. The particle sizes increase rapidly as the height of particle collection becomes higher, which showed an important factor about choosing a flame type to synthesis particles that are small enough.


INTRODUCTION
Flame combustion synthesis technology has been developed many years and the simulation model of the process is taken seriously.Kruis et al. (1993) proposed a simple monodisperse model for describing the particle morphology, size and number density during coagulation and sintering, neglecting the polydispersity of aggregates and primary particles in the synthesis process.With their simplified model, it was possible to evaluate the main sintering mechanism and predict particle size during synthesis.Jonhannessen et al. (2001) combined computational fluid dynamics with mathematical modeling to simulate experimental data from the TiO 2 combustion synthesis process, considering the effects of dilution modification to the monodisperse model of Kruis et al. (1993).Employing a model based on that of Kruis, Wang (2004) investigated three different assumptions in the combustion synthesis process and more precisely predicted particle sizes.Wang found that the average particle size would noticeably decrease when the molar fraction of oxygen was higher than 40%, but that higher inlet TiO 2 concentrations would induce larger particle sizes.More recently, Yu (2008) used the differential value method to simulate TiO 2 particle formation in the diffusion flame showing that, at fixed oxygen flow rates, the TiO 2 particle size would quickly increase along the axial position before slowly decreasing due to dissipation effects.In contrast, if the flow rate of oxygen increased from 2 L/min to 6 L/min, the TiO 2 particle size would decrease from 18.5 nm to 5.8 nm.Chang et al. (2008) investigated the synthesis routes for TiO 2 particles in the spray flame pyrolysis, proposing mechanisms for the generation and growth of TiO 2 particles within high-temperature surroundings.These mechanisms, included oxidation of precursor vapor, nucleation and condensation, collision and coalescence.In most of these simulation studies, the collision efficiency was assumed to be 100%.However, when the Knudsen number, Kn, is between 10 -2 and 10 2 , the van der Waal force between particles cannot be ignored.Feng and Lin (2008) investigated the collision efficiency in coagulation processes and with modifications to the collision frequency.In a further refinement, Widiyastuti (2010) studied polydisperse and non-spherical precursors in the simulation.
Our previous study of the TiO 2 particle synthesis via TTIP oxidation in a premixed methane/air flame was conducted for mass production and getting the particle sizes smaller than the commercially available products (Ma et al., 2010a, b).Previous works studied the influences of flame temperature, precursor concentration, oxygen molar fraction and particle collection height on the composition of the crystalline phases and particle morphology of TiO 2 particles.Nonetheless, investigations of the formation mechanism in the synthesis process were not well characterized.In this study, a monodisperse particle formation (MPF) model with effective collision frequency (β) is developed to investigate the influence of different experimental conditions via three parameters: particle number density (N), total aggregate volume per unit mass of gas (V) and total aggregate surface area per unit mass of gas (A).The five-zone diagram (FZ diagram) provides a distinct way to describe the mechanisms in the TiO 2 combustion synthesis process, which may predict particles' size and analyze the influence of different inlet oxygen-nitrogen ratios (O 2 /N 2 ), mass flow rates (Q), TTIP concentrations (X TTIP ) and height of particle collection.This study can be used to select the proper burner to prevent collecting oversized particles.

Mechanisms of the Combustion Synthesis Process
Fig. 1 shows the mechanisms of the combustion synthesis process.The TTIP precursor is first heated to reach the gas phase and then converted to TiO 2 seed particles.Seed particles collide and are sintered into larger sizes within the high-temperature surroundings.The mechanisms in the combustion synthesis process are discussed below:

Heating Process
The liquid TTIP precursor is injected with methane/air into the flame, absorbing heat from the flame and is vaporizing to become the gas phase TTIP precursor.

Chemical-Reaction Process
In the high-temperature surroundings, the TTIP vapor is thermally decomposed to produce TiO 2 vapor, as described by Okuyama et al. (1990): The TiO 2 molecular production rate per volume of gas can be calculated as:

Nucleation Process
The TiO 2 vapor produced from the chemical reaction may be nucleated to seed particles.The minimum nucleation size is assumed to be 1 nm by Wang et al. (2004).

Collision Process
Collisions between TiO 2 particles induce formation of aggregates.If collisions are the major mechanism in the process, the aggregate will become a cluster.Let n be the particle number.The rate of change of particle number with respect to time can be described as: where β 0 is the collision frequency.The aggregate volume and surface area rates of change can be written as: After Fuchs (1964), the collision frequency can be written as: where Kn D is the particle Knudsen number: λ p is the mean free path of particles: D is the diffusivity of particles, which is given by Stokes-Einstein relation: Eq. ( 9) should be modified with the Cunningham correction factor by Phillips (1975) if the particle size is smaller than 1 μm: The particles' mean velocity i c can be calculated via the gas molecular mean velocity: Practically, particles may not combine together after every collision, thus the collision efficiency should be considered in the model.Considering a perfectly elastic collision but with van der Waals interactions between particles.If the kinetic energy of colliding particles is larger than the combined energy induced by the van der Waals interactions, then they will separate after the collision; otherwise they will coagulate together.The collision efficiency can be written via McQuarrie et al. (1997) as where Ф is the potential energy caused by van der Waals interactions.According to Hamaker's theory: where d p1 and d p2 are the diameters of the two colliding particles and H is the Hamaker constant, which is estimated by Bergström (1997) to be 1.5 × 10 -9 J for TiO 2 .The parameter h is the distance between the two particles.
Considering that there are n particles inside a mesh whose volume is v m , and each particle's diameter is d p .By monodisperse assumption and assuming all particles are uniformly distributed in the mesh, h can be approximately written as: The collision efficiencies under different surrounding temperature are shown in Fig. 2, which indicates that γ plays an important role in the high temperature surroundings, especially for particles whose diameter is less than 15 nm.Finally, the rate of change of particle number with respect to time in MPF model is modified as below: where β is defined as the effective collision frequency.

Sintering Process
A non-spherical aggregate will be sintered to a spherical particle within the high-temperature surroundings.The aggregate surface area rate of change via sintering can be written as: where a m is the completely fused aggregate surface area and τ f is the characteristic sintering time which can be described by Kobata (1991): where w = 2.714 × 10 -10 m, D gb = 0.18 × exp(-2.3× 10 5 /RT), σ = 1.0J/m 2 , and Ω = 1.204 × 10 -5 m 3 /mol.

Monodisperse Particle Formation (MPF) Model
Considering a control volume with TiO 2 particles, the influences of flow field, Brownian motion diffusion, chemical reaction, nucleation, collision, coagulation and sintering reaction are studied.The three governing equations describing particle numbers, aggregate volume, and aggregate surface area are, respectively: The particle number rate of change via diffusion can be written as: For calculating the MPF model by FLUENT's UDS function, Eqs. ( 18)-( 20) are changed into an Eulerian coordinate system, becoming: Finally, the particle size can be described by N, V and A directly: Experimental Setup and Simulation Conditions Fig. 3 is the experimental setup used by our previous study (Yeh et al., 2004).TiO 2 particles were synthesized in premixed flame with TTIP as the precursor.Premixed combustible gases were prepared using methane, oxygen and nitrogen, and mixed with TTIP vapor in the mixing chamber.The three cases investigated in this study are the following: Case I: Different oxygen/nitrogen ratios.
Case II: Different inlet flow rates.
Case III: Different inlet TTIP concentrations.The overall experimental conditions are shown in Table 1.The temperature field and Eqs. ( 22)-( 24) were calculated using FLUENT, with the SIMPLE algorithm.The computational grid size was x grid × y grid = 50 × 20.The temperature field was first calculated to obtain the stable solution, and the equations of the MPF model, Eqs. ( 22)-( 24), were solved under the stable temperature field.Simulation assumptions included the following: 1.The velocity field is not influenced by TiO 2 particles because the concentration of TiO 2 particles is quite low.2. The temperature field is not influenced by TTIP-TiO 2 chemical reaction particles because the concentrati on of TTIP vapor is low.3. The minimum nucleation size is assumed to be 1 nm (Wang, 2004).4. The turbulent diffusion effect is neglected because the particle and aggregate sizes in this study are smaller than 100 nm.

Five-Zone Diagram (FZ Diagram) of Particle Formation
The FZ diagram is classified by the particle number density (N), total aggregate volume per unit mass of gas (V) and total aggregate surface area per unit mass of gas (A).The distribution of reaction zones are shown in Fig. 4.

Precursor-Heated Zone I: (N = V = A = 0)
N, V and A are zero in the precursor-heated zone, indicating that the TiO 2 particles have not yet been generated in the synthesis process.As the surrounding temperature rises slightly, TTIP is converted from the liquid phase to the gas phase in preparation for synthesis.

Chemical Reaction/Nucleation Zone II: (N↑, V↑, A↑)
The magnitude of N, V and A increase rapidly due to the increasing flame temperature in this zone.The increase of V indicates that the reaction converting TTIP to TiO 2 is ongoing, and the increase of N indicates that most of the produced TiO 2 seed particles are separated after nucleation.Additionally, the value of A increases because there are increasing numbers of TiO 2 seed particles in the process, but there's no obvious sintering reaction yet.

High-Temperature Zone III: (N↓, V↑, A↓)
The rising magnitude of V indicates that the TTIP-TiO 2 chemical reaction is ongoing and that new TiO 2 seed particles are still being produced.Within the high-temperature surroundings, particle collisions become more intense and the collided TiO 2 seed particles may aggregate, reducing the value of N. Additionally, the sintering process is also intense within high-temperature surroundings, which reduces the value of A.

Coagulation/Coalescence Zone IV: (N↓, V = const., A↓)
In this zone, most TTIP are used up, and the surrounding temperature is not high enough to drive the TTIP-TiO 2 chemical reaction, which keeps V constant.N and A are decreased due to the effects of coagulation and coalescence.The main reaction in this zone is the sintering process, and TiO 2 particles become spherical due to the strong sintering effect.

Aggregation Zone V: (N↓, V = const., A = const.)
No additional sintering or chemical reactions occur as the surrounding temperature gradually decreases.The value of N is slightly decreasing, indicating that there aggregation is still occurring.

Particle Morphology in Different Zones of FZ Diagram
A comparison of the predicted results with previous experimental TEM results for zone III at the axial distance of 2.5 cm is shown in Fig. 5(a) (Yeh, 2004).The rapid decrease of N indicates that the collected aggregate at this height will include abundant numbers of particle.When the axial distance increases to 4.5 cm, which is in zone IV, the value of V remains constant but those of N and A are decreasing.Therefore, as seen in Fig. 5(b), the aggregate consists of a few spherical particles.At an axial distance of 7.5 cm, corresponding to the zone V, the sintering reaction is terminated, and thus, V and A remain constant.The value of N is still decreasing slightly, indicating that the aggregate will include more spherical particles than at 4.5 cm, as seen in Fig. 5(c).

Effect of the Oxygen/Nitrogen Ratio in FZ Diagram (CASE I)
The temperature distribution is highly related to the inlet oxygen/nitrogen ratio (O 2 /N 2 ), with a shorter flame and higher temperature as O 2 /N 2 increases.Fig. 6 shows the distribution of N with different O 2 /N 2 ratios.In case I d, there are more TTIP precursors converted to TiO 2 particles as the chemical reaction rate is higher in the high-temperature surroundings.Fig. 7 shows the distribution of V with different O 2 /N 2 ratios.V is only related to the chemical reaction/nucleation effects, and thus, the peak value of V is observed in case I d.Values of A and A m with different O 2 /N 2 ratios are shown in Fig. 8, indicating that under the conditions of case I a and case I d, the TiO 2 particles are spherical prior to axial distances of 6.8 cm and 6 cm, respectively.It can be shown in Fig. 9 that in the zone III, the particle size for the 50/50 case increases faster than in the 20/80 case.The final particles sizes are also influenced by different O 2 /N 2 ratios.As seen in Table 2, the particle sizes in case I a and in case I d are 40.9nm and 54.2 nm, respectively, at an axial distance of 2.5 cm, with these sizes increasing to 81.4 nm and 120.9 nm, respectively, at an axial distance of 7.5 cm.

Effect of Flow Rate in FZ Diagram (CASE II)
The flow rate (Q) will stretch the reaction zones in the synthesis process, meaning that the ranges of zones I-V are increased by the larger flow rate.Fig. 10 shows the  particle sizes under different flow rate conditions.Comparing the particle sizes at three different flow rates for axial distances of 2.5 cm and 7.5 cm indicates that no obvious trend in particle sizes as Q increases.This is due to the different zones with different flow rate conditions at a specific axial distance.For example, an axial distance of 7.5 cm corresponds to zone V in case II a, but corresponds to the start of zone IV in case II c.

Effect of TTIP Concentration in FZ Diagram (CASE III)
The trends of results with different TTIP concentrations (X TTIP ) are similar as shown in Fig. 11.The particles' size of different TTIP concentrations at x = 2.5 cm and 7.5 cm indicates that the higher TTIP concentration will increase the TiO 2 particle size, because there are more TiO 2 seed particles produced in zone II, increasing the effective collision frequency in the synthesis process.

CONCLUSIONS
The TiO 2 monodisperse particle formation model (MPF model) has been used to investigate the characteristics of     The Five-Zone diagram constructed by parameters N, V, and A, provides a distinct way to determine the major reaction at the specific height of particle collection.The effect of inlet oxygen/nitrogen ratio, flow rate, TTIP concentration and height of particle collection, respectively, on synthesized TiO 2 particle size are the following: 1.A higher inlet oxygen/nitrogen ratio strengthens the reaction in the coagulation/coalescence zone (zone IV) and increases TiO 2 particle sizes with higher inlet oxygen/nitrogen ratios.2. The inlet flow rate stretches reaction zones.Therefore, at a specific axial distance, there are different zones with different inlet flow rates.3. The inlet TTIP concentration influences the value of N, V and A in the chemical reaction/nucleation zone (zone II).There are more TiO 2 seed particles produced by the chemical reaction with higher inlet TTIP concentrations, which induces larger particle sizes.4. The particle size increases rapidly as the height of particle collection increases.In case I a, the particle size is 22 nm at an axial distance of 1cm; however it becomes 38 nm at 2 cm.
These conclusions may use in choosing a suitable burner and operation conditions to produce the smaller nanoparticles.

Fig. 4 .
Fig. 4. Classified zones of combustion synthesis process in FZ Diagram (Case I b).

Fig. 6 .
Fig. 6.Profiles of particle number density with different inlet oxygen-nitrogen ratios (case I).

Fig. 8 .
Fig. 8. Profiles of aggregate surface area with different inlet oxygen-nitrogen ratios (case I).

Fig. 10 .
Fig. 10.Profiles of particle sizes with different inlet flow rates (case II).

Table 1 .
Experimental conditions of TiO 2 combustion synthesis.