#### Abstract

Titanium dioxide (titania) is used in chemical sensors, pigments, and paints and holds promise as an antimicrobial agent. This is due to its photoinduced activity and, in nanostructured form, its high specific surface area. Particle size and surface area result from the interplay of fluid, chemical, and thermal dynamics as well as nucleation, condensation and coagulation. After nucleation, condensation, and coagulation are the dominant phenomena affecting the particle size distribution. Manufacture of nanostructured titania via gas-phase synthesis often occurs under turbulent flow conditions. This study examines the competition between coagulation and condensation in the growth of nanostructured titania. Direct numerical simulation is utilized in simulating the hydrolysis of titanium tetrachloride to produce titania in a turbulent, planar jet. The fluid, chemical, and particle fields are resolved as a function of space and time. As a result, knowledge of titania is available as a function of space, time, and phase (vapor or particle), facilitating the analysis of the particle dynamics by mechanism. Results show that in the proximal region of the jet nucleation and condensation are the dominant mechanisms. However once the jet potential core collapses and turbulent mixing begins, coagulation is the dominant mechanism. The data also shows that the coagulation growth-rate is as much as twice the condensation growth-rate.

#### 1. Introduction

High rate synthesis of nanoparticles from vapor requires operation in the turbulent flow regime [1]. The variety of length and time scales present in turbulent multiphase flows makes them not very amenable to physical observation or analysis. This is especially true in the early stages of the nanoparticle formation and growth processes [2]. Computational fluid dynamics (CFD) has been developed for many years and a variety of numerical techniques have been developed and utilized for studying particle dynamics [3–10]. CFD has enabled engineers to achieve their goals more rapidly and cost effectively [11]. It has also become an effective tool for understanding physicochemical dynamics.

Nakaso et al. [12] modeled the titania nanoparticle growth of both agglomerates and primary particles simultaneously by using spatial zero-dimensional fluid dynamics. Johannessen et al. [13] combined CFD with a mathematical model for the particle dynamics to compare with experimental data from the synthesis of titanium dioxide particles in diffusion flames. Tsantilis et al. [5] used a moving sectional aerosol dynamics model accounting for gas-phase chemical reactions, coagulation, surface growth, and sintering with zero-dimensional fluid dynamics to investigate flame synthesis of titania nanoparticles. Also one study by Moody and Collins [14] considered titania nanoparticle nucleation and growth in a turbulent “box” located near the center of the reactor via three-dimensional DNS coupled with moment method in particle dynamics. Researchers have also considered the effects of turbulence on particle growth. Strakey et al. [15] studied the role of turbulence on the characteristics of powder made by oxidation and they found that the increased turbulence intensity narrowed the size distribution of the product powder indicating that the particle growth may be dominated by reactant mixing rather than by particle-particle collisions. These computations are quite compute intensive as they resolve all of the appropriate length and time scales. Garrick and his group studied the effects of turbulence on particle coagulation intensively and indicated that turbulence has a positive effect on the particle growth [16, 17]. The effects of turbulence on nucleation have also been elucidated [18–20]. To obtain physically accurate data, simulations must be three-dimensional and must be model-free. That is, the results must be obtained without the use of turbulence or subgrid scale models [21]. A review of the literature reveals a lack of the detailed information, especially via high-resolution direct simulation, on the interplay between the different mechanisms affecting nanoparticle growth.

In this work, the hydrolysis of titanium tetrachloride () to produce titanium dioxide () is simulated via direct numerical simulation. The turbulent, reacting, multiphase flow is obtained by solving the Navier-Stokes equations in conjunction with transport equations for all of the relevant chemical species and a nodal approach is used to represent the particle field [22, 23]. With the chemical and particle fields available as a function of space, time, and size, particle nucleation, condensation, and coagulation are illustrated individually. Additionally the two growth mechanisms are elucidated.

#### 2. Methodology

The mass, momentum, and energy equations are solved to obtain the fluid velocity , pressure , density , and the enthalpy . These variables are governed by the following conservation equations:where is the viscous stress tensor for a Newtonian fluid, is the coefficient of thermal conduction, and is the specific heat at constant pressure.

##### 2.1. Chemical Transport

The fluid contains five chemical species, the transport of which is given by the conservation of species equations:where is the mass concentration of species and is the diffusion coefficient of species . The reactants, and water vapor , undergo an irreversible one-step, isothermal chemical reaction at a temperature of 300 K and atmospheric pressure, which produces titanium dioxide and hydrochloric acid :The source term in (2) represents the effects of chemical reaction, the rate of creation or consumption of species The system is closed with the ideal gas equation of state, and , where and are the gas constant and molecular weight of species , respectively. The fluid temperature, , is obtained using the enthalpy via The fifth chemical species, nitrogen (), does not participate in any chemical reactions and serves as the carrier gas.

##### 2.2. Particle Field

The aerosol general dynamic equation (GDE) describes particle dynamics under the influence of various physicochemical phenomena, convection, diffusion, coagulation, surface growth, nucleation, and other internal/external forces. The GDE is utilized in discrete form as a population balance on each cluster or particle size. The methodology uses the nodal/sectional method of to approximate the GDE [23–26]. This approach effectively divides the aerosol population into three classes, monomers, clusters, and particles [16]. The GDE is therefore solved as a set of transport equations, one for each bin , [27]. Monomers of size nm in diameter populate bin while bins and are populated by clusters of molecules. Molecular clusters of size nm and larger are considered “particles” [28]. The general transport equation for the concentration of monomers, clusters, and particles in bin , , is written aswhere is the diffusivity given bywhere is the Boltzmann constant, is the Cunningham correction factor, and is the mean volume particle diameter [29]. The source term represents particle formation and growth processes and is given bywhere accounts for the formation of monomers in bin via chemical reaction [30–33] and is given byThe source term in (6), , represents the effects of nucleation, condensation (via monomer-cluster, monomer-particle and cluster-particle collisions), and Brownian coagulation. In this study, all particle formation and growth processes occur at 300 K. This means that all particles considered are agglomerates, meaning growth via coagulation results in fractal-like aggregate particles, consisting of primary particles (monomers). The collision frequency function, , is well documented in both the free-molecular and continuum regimes [29, 34–37]. Collisions of all particles (monomers, clusters, etc.) are considered and is given bywherewhere is the primary particle volume, is the volume of an agglomerate in the th section, is the particle density, and are the primary particle numbers in bins and , respectively, and is the fractal dimension. The particles generated by the industrial aerosol processes are usually nonspherical and the products are composed of groups of adhering particles ranging from loosely linked particles, sometimes termed agglomerates, to strongly necked particles, called aggregates or hard agglomerates. The fractal dimension is used to represent the shape of the aggregate or particle structure. A statistical concept, the fractal (or Hausdorff) dimension , was introduced to describe the shape of the agglomerate, which is obtained after averaging over many agglomerates with the same number of primary particles. The value of the fractal dimension ranges from to depending on the details of the agglomerate formation process [29]. The bins are organized in such a manner that the volume of particles in two successive bins is doubled; that is, [23, 38].

#### 3. Results

##### 3.1. Flow Configuration

The flow under consideration is a three-dimensional, isothermal, turbulent reacting jet issuing from an orifice of diameter into a coflowing stream. The jet is composed of titanium tetrachloride diluted in nitrogen , while the coflowing stream is composed of water vapor diluted in . The initial velocities are m/s for the jet and for the coflowing stream. The fluid field is characterized by the velocity ratio . The Reynolds number based on velocity of the high-speed stream and the jet diameter is = 3,000. The simulation is performed at 300 K and atmosphere pressure (1atm). To accelerate the development of large-scale structures, random perturbations with a maximum intensity of 5% are added to the cross-stream -velocity. The chemical composition of the jet stream is 0.01% TiCl_{4} and 99.99% N_{2} by mass. The simulation utilizes stoichiometric mixtures and the molar ratio of 1 : 2 for and .

In this work, several assumptions and approximations are utilized. These are stated below for clarity.(1)Ceramic powders such as have low equilibrium vapor pressures implying that single molecules may be considered particles [31]. As is formed, it appears as 0.5 nm diameter free spherical “monomers,” populating bin ; hence nucleation is treated as an instantaneous process.(2)The nanoparticles are small enough to follow the fluid path lines. Additionally, the particle volume fraction is of order . As a result the presence of the particles does not affect the fluid field.(3)Condensation is dominated by the collision rate of monomers with other monomers, clusters, and particles. This means the condensable species in the simulations is the “vapor."(4)The clusters and particles are stable because of the high supersaturation of the monomers. As a result, at these temperatures, there is no evaporation or sublimation of particles back to the gas-phase.(5)The fractal dimension, , is based on the collision time, , and sintering time, . In this work, all processes occur at K. At this temperature, particles do not sinter as [39].(6)A fractal dimension of is used when collisions occur between monomers while is used when monomers and dimers collide. For all other particle interactions (collisions between larger particles) a fractal dimension of is used [39, 40].

##### 3.2. Numerical Specifications

Ten bins are used to discretize the particle field (). The computational domain is of size and is comprised of grid points in the -, -, and -directions, respectively. The governing transport equations representing both the fluid and particle fields are solved using a MacCormack-based finite difference scheme [41, 42]. The scheme is of second order, accurate in time, and of fourth order, accurate in space. The boundary conditions are periodic in the spanwise -direction and zero-derivative in the cross-stream -direction and nonreflecting boundary conditions are used in both inflow and outflow boundaries (-direction) [43]. The simulation is performed up to a nondimensional time of which corresponds to a physical time of 0.48 ms. Both instantaneous and mean or averaged data are presented. We average in the -direction as it is the spatially homogeneous direction in planar jets. Quantities such as contours, isosurfaces, and mean data are useful in making qualitative and quantitative assessments of the nanoparticle growth dynamics as well as the underlying fluid and particle fields.

##### 3.3. Flow Field

The vorticity is the curl of the velocity vector and is an indicator of fluid mixing. The vorticity magnitude is the local rate of rotation. In nonpremixed chemically reacting flows, vorticity has the effect of increasing the interfacial area between the reactants. An isosurface of the instantaneous vorticity magnitude, the level-set, is shown in Figure 1 at time . The image shows that the flow is initially laminar and becomes turbulent as the jet travels downstream. Near , the two boundary layers initially located at merge and the jet spreads across the domain. This is aided by the presence of vortex braids, the tubular structures oriented in the stream-wise -direction, which act to draw the surrounding fluid into contact with the fluid issuing through the nozzle. Further downstream, vortex bending and stretching acts to generate small-scale structures as the flow becomes fully turbulent. It is evident that the jet suddenly spreads/amplifies near Near , the contours reveal a high concentration of intense mixing that persists throughout the latter half of the computational domain. These small-scale structures result in an increased chemical reaction as they serve to bring the reactants into contact. High-resolution DNS facilitates the capturing of the small-scale structures. If turbulence models were used, then the effects of the small-scale interactions on the chemical reaction (and particle formation and growth) would need to be accounted for [18, 44].

##### 3.4. Titanium Dioxide

The chemical conversion of the and to produce is the first step in the particle synthesis process. Instantaneous contours of the mass fraction at the plane and time are shown in Figure 2. The image reveals that is initially formed along the interface of the two streams and subsequently where the reactants are well mixed. The is convected downstream and across the jet. The maximum mass fraction appears far downstream and mass spreads out. Because of the infinite-rate chemistry, at least one of the reactants is consumed immediately upon contact. Turbulent mixing brings “fresh” reactants together (via large-scale transport) to produce (via molecular scale transport). As the is produced, molecular diffusion acts to transport it from -rich to -free regions.

##### 3.5. Particle Concentrations

As the chemical reaction proceeds, more titania is produced. The monomers collide with each other to produce dimers; those dimers collide with monomers (condensation) to produce trimers and collide with each other to produce larger particles (coagulation). An advantage of the nodal approach is the fact that the particle field is obtained as a function of size (in addition to space and time). A detailed view of the nanoparticle field can be obtained by observing the particle number concentrations distributed throughout the domain. Instantaneous contours of the -direction averaged particle number concentrations of monomers, nm and nm particles at , are shown in Figure 3. Figure 3(a) shows that monomers initially appear at the interface of the two streams and the concentration increases significantly near , after collapse of the jet potential core. Figure 3(b) shows that the nm diameter particles begin to appear near , and high concentrations are found between and . Figure 3(c) shows a similar trend for the nm particles.

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**(b)**

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To convey the spatial inhomogeneity of the particle field, a three-dimensional view is presented in Figure 4. The figure shows three isosurfaces colored to show the large concentration of 1 nm, 2 nm, and 3 nm nanoparticles. The presence of 1 nm particles (colored green) throughout the domain reflects the ongoing chemical reaction and nucleation that occurs when large-scale convective mixing brings and into contact. The increase in particle size with downstream distance is evident in the image as the 3 nm diameter particles are only found in the last third of the domain.

##### 3.6. Mean Nanoparticle Size and Geometric Standard Deviation

Particle size distributions (PSDs) are often characterized by the mean diameter and the geometric standard deviation (GSD). Though the nodal approach employed contains the full PSD, conveying that all of the information is not trivial [45, 46], the information conveyed by the first two moments can be quite useful. The mean diameter used here is the volume-equivalent mean particle diameter and is given by , where the mean volume is given byThe GSD represents the width of the PSD and is given bywhere is a number mean diameter given byLarger GSD values indicate that the size distribution is relatively broad while smaller GSD values () indicate a relatively narrow distribution [47]. The GSD for the hydrolysis of in a planar jet is shown on an isosurface of vorticity magnitude at time in Figure 5. The figure reveals that the GSD generally increases as the jet travels downstream. This implies that mixing (due to turbulence) is a significant contribution to particle polydispersity. If nucleation were absent, there would be no monomers, and the coagulating particles would achieve the so-called self-preserving value () [48, 49]. Locations where , in Figure 5, reflect where nucleation, condensation, and coagulation simultaneously occur. That is, “newly” formed particles by nucleation/mixing and old particles generated by coagulation exist.

More insight into the particle field may be obtained by considering the relationship between the mean diameter and the GSD. A scatter plot of versus is shown in Figure 6. The figure shows that at the two ends of the size distribution, nm and nm, the distribution is fairly narrow or unimodal, while the GSD is largest () when the mean diameter is near nm. This relatively large GSD indicates that there are a variety of particle dynamics present in regions of the flow where the mean diameter is nm.

##### 3.7. Particle Growth

The particle growth-rate is an important parameter to consider as, in combination with residence time or reactor size, it is a predictor of particle size. A diameter-based growth-rate (with units of nm/s) is defined based on where is the particle volumetric growth-rate given byA contour plot of the -averaged growth-rate, , is shown in Figure 7. The figure shows that within the first 6 diameters, , growth is confined to roughly nm/s at the interface of the jet and the coflowing stream, where the hydrolysis reaction produces vapor. Near , when the shear layers merge, the particles begin to grow faster, with the rate approaching nm/s. This occurs after the potential core collapses. Further downstream, the growth-rate increases to roughly nm/s and this value is maintained for . (It should be noted that this is the average growth-rate and at other points in the domain the values of may be smaller or larger.)

The particles grow via two mechanisms: condensation (collisions between monomers and particles) and coagulation (collisions between particles). Because the particle data is available as a function of size, the contribution of each mechanism is readily available. Particle growth, by mechanism, is shown in Figure 8. The interactions between monomers and particles, or the condensation growth-rate, is shown in Figure 8(a). (This image looks similar to the monomer number concentration shown in Figure 3(a) because the condensation is represented by the collision between monomers and particles.) The contours show that large condensation growth occurs both in the proximal region of the jet and after collapse of the jet core. This reflects the ongoing hydrolysis of and production of and its deposition on existing particles.

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**(b)**

Particle growth by coagulation is shown in Figure 8(b). The contours of the -averaged growth-rate shows that growth by coagulation begins after collapse of the jet potential core. The growth-rate in the region is as high as nm/s. Farther downstream the -averaged coagulation growth-rate doubles. This region of the flow is dominated by mixing and small-scale turbulence (evident in Figure 1). That region of the flow also contains particles of a variety of sizes, small and large, as reflected by in Figure 5. The small-scale turbulence means dissipation and increased residence times, while the disparate particle sizes mean an efficient collision efficiency. These two combine to increase coagulation.

The spatial relationship between condensation growth and coagulation growth is elucidated by showing the contribution of each at every grid point in the computational domain. A scatter plot of the two growth-rates is shown in Figure 9. The growth-rates are not averaged in the -direction and while a spatial relationship is not directly evident from this figure, one may be reliably inferred along with the previous data. Figure 9 shows that where condensation growth is low, coagulation growth is high. This trend is evident in Figure 8 as well. However, Figure 9 shows that in these regions the coagulation growth-rate can be as much as an order of magnitude greater than the condensation growth-rate.

#### 4. Summary and Conclusions

The growth mechanisms of titania during hydrolysis of titanium tetrachloride in a three-dimensional planar jet are studied via direct numerical simulation. The mass, momentum, enthalpy, and species transport equations are solved in a model-free manner. Titania was produced via the hydrolysis of titanium tetrachloride, modeled via a 1-step infinitely fast chemical mechanism. The particle field was represented using a nodal method and solves for the evolution of the concentration of particles of various sizes in an Eulerian manner. When coupled to the Navier-Stokes solver, the fluid, thermal, chemical, and particle fields are obtained as a function of space and time.

The results show that fluid turbulence/gas mixing plays a very important role in particle growth. The results indicate that the particle formation and growth are greatly affected if not dominated by mixing and chemical reaction. Reactant conversion or titania production is limited by the ability of the flow to bring the and into contact. Evidence for this is the particle formation throughout the domain. Additionally, the results show that where the turbulence is more intense, the particles are larger and the particle size distribution, as characterized by the geometric standard deviation, is wider. Particle growth in the proximal region of the jet is dominated by condensation. After collapse of the jet core, nanoparticle growth due to coagulation increases significantly. Though is produced after core collapse, producing condensible species, particle growth due to coagulation is as much as an order of magnitude greater than that due to condensation. While both condensation and coagulation act to increase polydispersity, coagulation has been shown to increase the width of the particle size distribution faster, or greater, than condensation [50].

These results help to shed light on, and improve our understanding of, the underlying growth dynamics occurring in nanoparticle synthesis processes. The change from condensation-dominated to coagulation-dominated growth is useful in modeling the complete synthesis process, including sintering, and the formation of hard and soft agglomerates. Spanning the size range from single molecules (particle inception) to hundreds of nanometers, as the particles found in many industrial processes and applications, is compute intensive [18]. However, knowing that condensation and molecular growth is relatively minor could facilitate the use of more affordable modeling strategies that do not need to account for every phenomena [6, 51]. Additionally, a highly desired attribute of metal oxide nanoparticles is their specific surface area. The ability to synthesize narrow size distributions which are desired is advantageous in that it removes the processing necessary to separate particles by size. This is advantageous in that it lowers cost [52–55].

One strategy to reduce polydispersity may be to delay the transition to turbulence vis a vis delaying collapse of the jet core. While the particle field is known as a function of size, it should be noted that in this work we are unable to distinguish the various phases of titania, rutile, anatase, or brookite [56, 57]. The phase composition of titania is very much a function of the synthesis process, for example, precursor composition and temperature history, which are greatly simplified in this work. Such capability requires phenomenological models that account for particle dynamics as a function of space, time, and temperature, as well as solid-state diffusion processes, and is quite beyond the scope of this work.

#### Conflict of Interests

The author declares that there is no conflict of interests regarding the publication of this paper.

#### Acknowledgments

Funding for this research was provided by the University of Minnesota. Computational resources were provided by Minnesota Supercomputing Institute.