Research ArticleGrowth Mechanisms of Nanostructured Titania inTurbulent Reacting Flows
Sean C Garrick
Department of Mechanical Engineering University of Minnesota 111 Church Street SE Minneapolis MN 55455-0111 USA
Correspondence should be addressed to Sean C Garrick sgarrickumnedu
Received 5 June 2015 Accepted 22 July 2015
Academic Editor Lin-Hua Xu
Copyright copy 2015 Sean C Garrick This is an open access article distributed under the Creative Commons Attribution Licensewhich permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited
Titanium dioxide (titania) is used in chemical sensors pigments and paints and holds promise as an antimicrobial agent Thisis due to its photoinduced activity and in nanostructured form its high specific surface area Particle size and surface arearesult from the interplay of fluid chemical and thermal dynamics as well as nucleation condensation and coagulation Afternucleation condensation and coagulation are the dominant phenomena affecting the particle size distribution Manufacture ofnanostructured titania via gas-phase synthesis often occurs under turbulent flow conditions This study examines the competitionbetween coagulation and condensation in the growth of nanostructured titaniaDirect numerical simulation is utilized in simulatingthe hydrolysis of titanium tetrachloride to produce titania in a turbulent planar jet The fluid chemical and particle fields areresolved as a function of space and time As a result knowledge of titania is available as a function of space time and phase (vaporor particle) facilitating the analysis of the particle dynamics by mechanism Results show that in the proximal region of the jetnucleation and condensation are the dominant mechanisms However once the jet potential core collapses and turbulent mixingbegins coagulation is the dominant mechanism The data also shows that the coagulation growth-rate is as much as twice thecondensation growth-rate
1 Introduction
High rate synthesis of nanoparticles from vapor requiresoperation in the turbulent flow regime [1] The variety oflength and time scales present in turbulent multiphase flowsmakes them not very amenable to physical observation oranalysis This is especially true in the early stages of thenanoparticle formation and growth processes [2] Com-putational fluid dynamics (CFD) has been developed formany years and a variety of numerical techniques have beendeveloped and utilized for studying particle dynamics [3ndash10] CFD has enabled engineers to achieve their goals morerapidly and cost effectively [11] It has also become an effectivetool for understanding physicochemical dynamics
Nakaso et al [12]modeled the titania nanoparticle growthof both agglomerates andprimary particles simultaneously byusing spatial zero-dimensional fluid dynamics Johannessenet al [13] combined CFD with a mathematical model forthe particle dynamics to compare with experimental datafrom the synthesis of titanium dioxide particles in diffusion
flames Tsantilis et al [5] used a moving sectional aerosoldynamics model accounting for gas-phase chemical reac-tions coagulation surface growth and sintering with zero-dimensional fluid dynamics to investigate flame synthesis oftitania nanoparticles Also one study by Moody and Collins[14] considered titania nanoparticle nucleation and growthin a turbulent ldquoboxrdquo located near the center of the reactorvia three-dimensional DNS coupled with moment methodin particle dynamics Researchers have also considered theeffects of turbulence on particle growth Strakey et al [15]studied the role of turbulence on the characteristics of TiO
2
powder made by TiCl4oxidation and they found that the
increased turbulence intensity narrowed the size distributionof the product powder indicating that the particle growthmay be dominated by reactantmixing rather than by particle-particle collisions These computations are quite computeintensive as they resolve all of the appropriate length andtime scales Garrick and his group studied the effects ofturbulence on particle coagulation intensively and indicatedthat turbulence has a positive effect on the particle growth
Hindawi Publishing CorporationJournal of NanotechnologyVolume 2015 Article ID 642014 10 pageshttpdxdoiorg1011552015642014
2 Journal of Nanotechnology
[16 17] The effects of turbulence on nucleation have alsobeen elucidated [18ndash20] To obtain physically accurate datasimulations must be three-dimensional and must be model-free That is the results must be obtained without the use ofturbulence or subgrid scale models [21] A review of the liter-ature reveals a lack of the detailed information especially viahigh-resolution direct simulation on the interplay betweenthe different mechanisms affecting nanoparticle growth
In this work the hydrolysis of titanium tetrachloride(TiCl4) to produce titanium dioxide (TiO
2) is simulated
via direct numerical simulation The turbulent reactingmultiphase flow is obtained by solving the Navier-Stokesequations in conjunction with transport equations for all ofthe relevant chemical species and a nodal approach is usedto represent the particle field [22 23] With the chemicaland particle fields available as a function of space timeand size particle nucleation condensation and coagulationare illustrated individually Additionally the two growthmechanisms are elucidated
2 Methodology
The mass momentum and energy equations are solved toobtain the fluid velocity 119906
119894(119909 119905) pressure 119901(119909 119905) density
120588(119909 119905) and the enthalpy ℎ(119909 119905) These variables are governedby the following conservation equations
120597120588
120597119905
+
120597120588119906119895
120597119909119895
= 0
120597120588119906119894
120597119905
+
120597120588119906119894119906119895
120597119909119895
= minus
120597119901
120597119909119894
+
120597120591119894119895
120597119909119895
120597120588ℎ
120597119905
+
120597120588119906119895ℎ
120597119909119895
=
120597
120597119909119895
(
119896
119862119901
120597ℎ
120597119909119895
)
(1)
where 120591119894119895is the viscous stress tensor for a Newtonian fluid 119896
is the coefficient of thermal conduction and119862119901is the specific
heat at constant pressure
21 Chemical Transport The fluid contains five chemicalspecies the transport of which is given by the conservationof species equations
120597120588119884119898
120597119905
+
120597120588119906119895119884119898
120597119909119895
=
120597
120597119909119895
(120588D119898
120597119884119898
120597119909119895
) + 119898 (2)
where 119884119898is the mass concentration of species 119898 and D
119898
is the diffusion coefficient of species 119898 The reactants TiCl4
and water vapor (H2O) undergo an irreversible one-step
isothermal chemical reaction at a temperature of 300Kand atmospheric pressure which produces titanium dioxide(TiO2) and hydrochloric acid (HCl)
TiCl4+ 2H2O119896119891
997888997888rarr TiO2+ 4HCl (3)
The source term 119898in (2) represents the effects of chemical
reaction the rate of creation or consumption of species 119898
The system is closed with the ideal gas equation of state 119901 =120588119877119879 and 119877 = sum119877
119898119884119898119872119882119898 where 119877
119898and119872119882
119898are the
gas constant and molecular weight of species119898 respectivelyThe fluid temperature 119879 is obtained using the enthalpy via119889ℎ = 119862
119901119889119879The fifth chemical species nitrogen (N
2) does
not participate in any chemical reactions and serves as thecarrier gas
22 Particle Field The aerosol general dynamic equation(GDE) describes particle dynamics under the influence ofvarious physicochemical phenomena convection diffusioncoagulation surface growth nucleation and other inter-nalexternal forces The GDE is utilized in discrete formas a population balance on each cluster or particle sizeThe methodology uses the nodalsectional method of toapproximate the GDE [23ndash26] This approach effectivelydivides the aerosol population into three classes monomersclusters and particles [16] The GDE is therefore solvedas a set of 119873
119904transport equations one for each bin 119876
119896
119896 = 1 2 119873119904[27] Monomers of size 05 nm in diameter
populate bin 1 while bins 2 and 3 are populated by clustersof molecules Molecular clusters of size 1 nm and larger areconsidered ldquoparticlesrdquo [28] The general transport equationfor the concentration of monomers clusters and particles inbin 119896 119876
119896 is written as
120597120588119876119896
120597119905
+
120597120588119906119895119876119896
120597119909119895
=
120597
120597119909119895
(120588119863119876
120597119876119896
120597119909119895
) + 119876
119896 (4)
where119863119876is the diffusivity given by
119863119876= 119896119887119879
119862119888
3120587120583119889119901
(5)
where 119896119887is the Boltzmann constant 119862
119888is the Cunningham
correction factor and 119889119901is the mean volume particle diam-
eter [29] The source term 119876119896represents particle formation
where 119869 accounts for the formation of monomers in bin 119896 = 1via chemical reaction [30ndash33] and 120594
119894119895119896is given by
120594119894119895119896=
V119896+1minus (V119894+ V119895)
V119896+1minus V119896
if V119896le V119894+ V119895lt V119896+1
(V119894+ V119895) minus V119896minus1
V119896minus V119896minus1
if V119896minus1le V119894+ V119895lt V119896
0 otherwise
(7)
Journal of Nanotechnology 3
The source term in (6) 120596119876119896 represents the effects of
nucleation condensation (via monomer-cluster monomer-particle and cluster-particle collisions) and Brownian coag-ulation In this study all particle formation and growth pro-cesses occur at 300KThismeans that all particles considered
are agglomerates meaning growth via coagulation results infractal-like aggregate particles consisting of primary parti-cles (monomers) The collision frequency function 120573
119894119895 is
well documented in both the free-molecular and continuumregimes [29 34ndash37] Collisions of all particles (monomersclusters etc) are considered and 120573
119894119895is given by
120573119894119895=
1198861(1198941119863119891
+ 1198951119863119891)
119863119891(
1
119894
+
1
119895
)
12
119863119891lt 2 free-molecular regime
119886 (1198941119863119891
+ 1198951119863119891)
2
(
1
119894
+
1
119895
)
12
119863119891ge 2 free-molecular regime
2119896119887119879
3120583
(
1
V1119863119891119894
+
1
V1119863119891119895
)(V1119863119891119894
+ V1119863119891119895) in continuum regime
(8)
where
119886 = (
3V119900
4120587
)
16
(
6119896119887119879
120588119901
)
12
1198861=
2119863119891119886
489
(9)
where V119900is the primary particle volume V
119894is the volume of
an agglomerate in the 119894th section 120588119901is the particle density
119894 and 119895 are the primary particle numbers in bins 119894 and 119895respectively and 119863
119891is the fractal dimension The particles
generated by the industrial aerosol processes are usuallynonspherical and the products are composed of groups ofadhering particles ranging from loosely linked particlessometimes termed agglomerates to strongly necked particlescalled aggregates or hard agglomeratesThe fractal dimensionis used to represent the shape of the aggregate or particlestructure A statistical concept the fractal (or Hausdorff)dimension 119863
119891 was introduced to describe the shape of the
agglomerate which is obtained after averaging over manyagglomerates with the same number of primary particlesThevalue of the fractal dimension ranges from 119863
119891= 1 to 119863
119891=
3 depending on the details of the agglomerate formationprocess [29] The bins are organized in such a manner thatthe volume of particles in two successive bins is doubled thatis V119896= 2 times V
119896minus1[23 38]
3 Results
31 Flow Configuration The flow under consideration is athree-dimensional isothermal turbulent reacting jet issuingfrom an orifice of diameter 119863 into a coflowing stream Thejet is composed of titanium tetrachloride (TiCl
4) diluted in
nitrogen (N2) while the coflowing stream is composed of
water vapor (H2O) diluted in N
2 The initial velocities are
119880119900= 100ms for the jet and119880
infinfor the coflowing streamThe
fluid field is characterized by the velocity ratio 119880infin119880119900= 02
The Reynolds number based on velocity of the high-speedstream and the jet diameter is Re = 119880
119900119863] = 3000 The
simulation is performed at 300K and atmosphere pressure(1 atm) To accelerate the development of large-scale struc-tures random perturbations with a maximum intensity of5 are added to the cross-stream V-velocity The chemicalcomposition of the jet stream is 001 TiCl
4and 9999 N
2
by mass The simulation utilizes stoichiometric mixtures andthe molar ratio of 1 2 for TiCl
4and H
2O
In this work several assumptions and approximations areutilized These are stated below for clarity
(1) Ceramic powders such as TiO2have low equilibrium
vapor pressures implying that single molecules maybe considered particles [31] As TiO
2is formed
it appears as 05 nm diameter free spherical TiO2
ldquomonomersrdquo populating bin 119896 = 1 hence nucleationis treated as an instantaneous process
(2) The nanoparticles are small enough to follow the fluidpath lines Additionally the particle volume fractionis of order 10minus7 As a result the presence of theparticles does not affect the fluid field
(3) Condensation is dominated by the collision rateof monomers with other monomers clusters andparticles This means the condensable species in thesimulations is the TiO
2ldquovaporrdquo
(4) The clusters and particles are stable because of thehigh supersaturation of the monomers As a resultat these temperatures there is no evaporation orsublimation of particles back to the gas-phase
(5) The fractal dimension 119863119891 is based on the collision
time 120591119888 and sintering time 120591
119891 In this work all
processes occur at 119879 = 300K At this temperatureTiO2particles do not sinter as 120591
119888≪ 120591119891[39]
(6) A fractal dimension of119863119891= 3 is used when collisions
occur between monomers while119863119891= 2 is used when
monomers and dimers collide For all other particleinteractions (collisions between larger particles) afractal dimension of119863
119891= 18 is used [39 40]
4 Journal of Nanotechnology
32 Numerical Specifications Ten bins are used to discretizethe particle field (119873
119904= 10) The computational domain is of
size 20119863 times 15119863 times 4119863 and is comprised of 500 times 375 times 100grid points in the 119909- 119910- and 119911-directions respectively Thegoverning transport equations representing both the fluidand particle fields are solved using a MacCormack-basedfinite difference scheme [41 42] The scheme is of secondorder accurate in time and of fourth order accurate inspace The boundary conditions are periodic in the span-wise 119911-direction and zero-derivative in the cross-stream 119910-direction and nonreflecting boundary conditions are usedin both inflow and outflow boundaries (119909-direction) [43]The simulation is performed up to a nondimensional time of119905⋆= 119880119900119905119863 = 100 which corresponds to a physical time of
048ms Both instantaneous and mean or averaged data arepresented We average in the 119911-direction as it is the spatiallyhomogeneous direction in planar jets Quantities such ascontours isosurfaces and mean data are useful in makingqualitative and quantitative assessments of the nanoparticlegrowth dynamics as well as the underlying fluid and particlefields
33 Flow Field The vorticity is the curl of the velocity vectorand is an indicator of fluidmixingThe vorticity magnitude isthe local rate of rotation In nonpremixed chemically reactingflows vorticity has the effect of increasing the interfacial areabetween the reactants An isosurface of the instantaneousvorticity magnitude the |120596| = 3 level-set is shown inFigure 1 at time 119905⋆ = 100 The image shows that the flowis initially laminar and becomes turbulent as the jet travelsdownstream Near 119909119863 = 6 the two boundary layers initiallylocated at 119910119863 = plusmn05 merge and the jet spreads across thedomain This is aided by the presence of vortex braids thetubular structures oriented in the stream-wise 119909-directionwhich act to draw the surrounding fluid into contact with thefluid issuing through the nozzle Further downstream vortexbending and stretching acts to generate small-scale structuresas the flow becomes fully turbulent It is evident that the jetsuddenly spreadsamplifies near 119909119863 = 8 Near 119909119863 = 10the contours reveal a high concentration of intense mixingthat persists throughout the latter half of the computationaldomain These small-scale structures result in an increasedchemical reaction as they serve to bring the reactants intocontact High-resolution DNS facilitates the capturing of thesmall-scale structures If turbulence models were used thenthe effects of the small-scale interactions on the chemicalreaction (and particle formation and growth) would need tobe accounted for [18 44]
34 Titanium Dioxide The chemical conversion of the TiCl4
and H2O to produce TiO
2is the first step in the particle
synthesis process Instantaneous contours of the TiO2mass
fraction at the 119911 = 0 plane and time 119905⋆ = 100 are shown inFigure 2The image reveals that TiO
2is initially formed along
the interface of the two streams and subsequently where thereactants are well mixed The TiO
2is convected downstream
and across the jetThemaximum TiO2mass fraction appears
far downstream and TiO2mass spreads out Because of
Figure 1 An instantaneous isosurface of the vorticity magnitude|120596| = 3 at time 119905⋆ = 100
0 5e minus 4
Figure 2 Instantaneous contour of the TiO2mass fraction at 119911 = 0
plane at 119905⋆ = 100
the infinite-rate chemistry at least one of the reactantsis consumed immediately upon contact Turbulent mixingbrings ldquofreshrdquo reactants together (via large-scale transport) toproduce TiO
2(via molecular scale transport) As the TiO
2is
produced molecular diffusion acts to transport it fromTiO2-
rich to TiO2-free regions
35 Particle Concentrations As the chemical reaction pro-ceeds more titania is produced The monomers collide witheach other to produce dimers those dimers collide withmonomers (condensation) to produce trimers and collidewith each other to produce larger particles (coagulation) Anadvantage of the nodal approach is the fact that the particlefield is obtained as a function of size (in addition to spaceand time) A detailed view of the TiO
2nanoparticle field can
be obtained by observing the particle number concentrationsdistributed throughout the domain Instantaneous contours
Journal of Nanotechnology 5
0 14e20
(a)64e180
(b)
0 75e17
(c)
Figure 3 Instantaneous contours of the 119911-direction averaged particle number concentrations (a) monomers (b) 1 nm (c) 2 nm
of the 119911-direction averaged particle number concentrationsof monomers 1 nm and 2 nm particles at 119905⋆ = 100are shown in Figure 3 Figure 3(a) shows that monomersinitially appear at the interface of the two streams and theconcentration increases significantly near 119909119863 = 10 aftercollapse of the jet potential core Figure 3(b) shows that the1 nm diameter particles begin to appear near 119909119863 = 5 andhigh concentrations are found between 119909119863 = 10 and 119909119863 =20 Figure 3(c) shows a similar trend for the 2 nm particles
To convey the spatial inhomogeneity of the particlefield a three-dimensional view is presented in Figure 4 Thefigure shows three isosurfaces colored to show the largeconcentration of 1 nm 2 nm and 3 nm nanoparticles Thepresence of 1 nm particles (colored green) throughout thedomain reflects the ongoing chemical reaction andnucleationthat occurs when large-scale convective mixing brings TiCl
4
and H2O into contact The increase in particle size with
downstream distance is evident in the image as the 3 nmdiameter particles are only found in the last third of thedomain
36 Mean Nanoparticle Size and Geometric Standard Devi-ation Particle size distributions (PSDs) are often charac-terized by the mean diameter and the geometric standard
deviation (GSD) Though the nodal approach employedcontains the full PSD conveying that all of the informationis not trivial [45 46] the information conveyed by the firsttwo moments can be quite useful The mean diameter usedhere is the volume-equivalent mean particle diameter and isgiven by 119889
119901= (6V119901120587)13 where the mean volume is given by
V119901=
sum119873119904
119894=1119876119894V119894
sum119873119904
119894=1119876119894
(10)
The GSD represents the width of the PSD and is given by
[log (120590119892)]
2
=
sum119873119904
119896=1119876119896(log (119889
119901119896) minus log ( 119889
119901119896))
2
sum119873119904
119896=1119876119896
(11)
where 119889119901119896is a number mean diameter given by
log ( 119889119901119896) =
sum119873119904
119896=1119876119896log (119889
119901119896)
sum119873119904
119896=1119876119896
(12)
Larger GSD values indicate that the size distribution isrelatively broad while smaller GSD values (120590
119892= 1) indi-
cate a relatively narrow distribution [47] The GSD for the
6 Journal of Nanotechnology
3nm
3nm
2nm
2nm
1nm
1nm
Figure 4 Instantaneous isosurfaces of 1 nm 2 nm and 3 nmparticles at 119905⋆ = 100
2
19
18
17
16
15
14
13
12
11
1
Figure 5 An instantaneous isosurface of vorticity colored by thegeometric standard deviation 120590
119892
hydrolysis of TiCl4in a planar jet is shown on an isosurface of
vorticity magnitude at time 119905⋆ = 100 in Figure 5 The figurereveals that the GSD generally increases as the jet travelsdownstream This implies that mixing (due to turbulence)is a significant contribution to particle polydispersity Ifnucleation were absent there would be no monomers andthe coagulating particles would achieve the so-called self-preserving value (120590
119892= 15) [48 49] Locations where 120590
119892gt
15 in Figure 5 reflect where nucleation condensation andcoagulation simultaneously occur That is ldquonewlyrdquo formedparticles by nucleationmixing and old particles generated bycoagulation exist
More insight into the particle field may be obtained byconsidering the relationship between the mean diameter andthe GSD A scatter plot of 120590
119892versus 119889
119901is shown in Figure 6
1 2 3 41
12
14
16
18
2
120590
dp (nm)
Figure 6 Scatter plot of the geometric standard deviation120590119892 versus
the mean particle diameter 119889119901
The figure shows that at the two ends of the size distribution119889119901= 1 nm and 119889
119901= 4 nm the distribution is fairly narrow
or unimodal while the GSD is largest (18 lt 120590119892lt 2) when
the mean diameter is near 119889119901= 32 nm This relatively large
GSD indicates that there are a variety of particle dynamicspresent in regions of the flow where the mean diameter is119889119901= 32 nm
37 Particle Growth Theparticle growth-rate is an importantparameter to consider as in combination with residence timeor reactor size it is a predictor of particle size A diameter-based growth-rate Ωnm (with units of nms) is defined basedon
Ωnm = (6Ω
120587
)
13
times 109 (13)
whereΩ is the particle volumetric growth-rate given by
Ω =
sum119873119904
119896(V119896)2
120596119896
sum119873119904
119896119876119896V119896
minus
sum119873119904
119896(V119896)2
119876119896
(sum119873119904
119896119876119896V119896)
2
119873119904
sum
119896
V119896120596119896 (14)
A contour plot of the 119911-averaged growth-rate ⟨Ωnm⟩119911 isshown in Figure 7 The figure shows that within the first 6diameters 0 le 119909119863 le 6 growth is confined to roughly⟨Ωnm⟩119911 = 10 nms at the interface of the jet and thecoflowing stream where the hydrolysis reaction producesTiO2vapor Near 119909119863 = 10 when the shear layers merge
the particles begin to grow faster with the rate approaching⟨Ωnm⟩119911 = 25 nms This occurs after the potential corecollapses Further downstream the growth-rate increases toroughly ⟨Ωnm⟩119911 = 50 nms and this value is maintained for119909119863 gt 14 (It should be noted that this is the average growth-rate and at other points in the domain the values ofΩnm maybe smaller or larger)
The particles grow via two mechanisms condensation(collisions between monomers and particles) and
Journal of Nanotechnology 7
0 53
Figure 7 Instantaneous contours of the 119911-direction averaged particle growth-rate ⟨Ωnm⟩119911 at 119905⋆= 100
0 28(a)
0 33(b)
Figure 8 Instantaneous contours of the nanoparticle growth-rate decomposed by mechanism 119911-direction averaged particle growth-rate (a)by condensation (b) by coagulation
coagulation (collisions between particles) Becausethe particle data is available as a function of size thecontribution of each mechanism is readily availableParticle growth by mechanism is shown in Figure 8The interactions between monomers and particles orthe condensation growth-rate is shown in Figure 8(a)(This image looks similar to the monomer numberconcentration shown in Figure 3(a) because thecondensation is represented by the collision betweenmonomers and particles) The contours show that largecondensation growth occurs both in the proximal regionof the jet and after collapse of the jet core This reflects theongoing hydrolysis of TiCl
4and production of TiO
2and its
deposition on existing particlesParticle growth by coagulation is shown in Figure 8(b)
The contours of the 119911-averaged growth-rate shows thatgrowth by coagulation begins after collapse of the jet potentialcore The growth-rate in the region 4 lt 119909119863 lt 8 is ashigh as Ωnm = 16 nms Farther downstream the 119911-averaged
coagulation growth-rate doubles This region of the flow isdominated by mixing and small-scale turbulence (evident inFigure 1) That region of the flow also contains particles of avariety of sizes small and large as reflected by 120590
119892in Figure 5
The small-scale turbulence means dissipation and increasedresidence times while the disparate particle sizes mean anefficient collision efficiency These two combine to increasecoagulation
The spatial relationship between condensation growthand coagulation growth is elucidated by showing the con-tribution of each at every grid point in the computationaldomain A scatter plot of the two growth-rates is shown inFigure 9The growth-rates are not averaged in the 119911-directionand while a spatial relationship is not directly evident fromthis figure one may be reliably inferred along with the pre-vious data Figure 9 shows that where condensation growthis low coagulation growth is high This trend is evidentin Figure 8 as well However Figure 9 shows that in theseregions the coagulation growth-rate can be as much as an
8 Journal of Nanotechnology
0 10 20 30 40 500
10
20
30
40
50
Ωnm
coagulation(nms)
Ωnmcondensation (nms)
Figure 9 Scatter plot of the particle growth-rate by condensationversus the particle growth-rate by coagulation
order of magnitude greater than the condensation growth-rate
4 Summary and Conclusions
The growth mechanisms of titania during hydrolysis oftitanium tetrachloride in a three-dimensional planar jet arestudied via direct numerical simulation The mass momen-tum enthalpy and species transport equations are solved in amodel-free manner Titania was produced via the hydrolysisof titanium tetrachloride modeled via a 1-step infinitelyfast chemical mechanism The particle field was representedusing a nodal method and solves for the evolution of theconcentration of particles of various sizes in an Eulerianmanner When coupled to the Navier-Stokes solver thefluid thermal chemical and particle fields are obtained as afunction of space and time
The results show that fluid turbulencegas mixing plays avery important role in particle growth The results indicatethat the particle formation and growth are greatly affectedif not dominated by mixing and chemical reaction Reactantconversion or titania production is limited by the ability ofthe flow to bring the TiCl
4and H
2O into contact Evidence
for this is the particle formation throughout the domainAdditionally the results show that where the turbulenceis more intense the particles are larger and the particlesize distribution as characterized by the geometric standarddeviation is wider Particle growth in the proximal regionof the jet is dominated by condensation After collapseof the jet core nanoparticle growth due to coagulationincreases significantly Though TiO
2is produced after core
collapse producing condensible species particle growth dueto coagulation is as much as an order of magnitude greaterthan that due to condensation While both condensationand coagulation act to increase polydispersity coagulationhas been shown to increase the width of the particle sizedistribution faster or greater than condensation [50]
These results help to shed light on and improve ourunderstanding of the underlying growth dynamics occur-ring in nanoparticle synthesis processes The change fromcondensation-dominated to coagulation-dominated growthis useful in modeling the complete synthesis process includ-ing sintering and the formation of hard and soft agglomer-ates Spanning the size range from single molecules (particleinception) to hundreds of nanometers as the particles foundin many industrial processes and applications is computeintensive [18] However knowing that condensation andmolecular growth is relatively minor could facilitate the useof more affordable modeling strategies that do not needto account for every phenomena [6 51] Additionally ahighly desired attribute of metal oxide nanoparticles is theirspecific surface area The ability to synthesize narrow sizedistributions which are desired is advantageous in that itremoves the processing necessary to separate particles by sizeThis is advantageous in that it lowers cost [52ndash55]
One strategy to reduce polydispersity may be to delay thetransition to turbulence vis a vis delaying collapse of the jetcore While the particle field is known as a function of size itshould be noted that in this work we are unable to distinguishthe various phases of titania rutile anatase or brookite [5657]The phase composition of titania is very much a functionof the synthesis process for example precursor compositionand temperature history which are greatly simplified in thiswork Such capability requires phenomenologicalmodels thataccount for particle dynamics as a function of space time andtemperature as well as solid-state diffusion processes and isquite beyond the scope of this work
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Funding for this research was provided by the Universityof Minnesota Computational resources were provided byMinnesota Supercomputing Institute
References
[1] S E Pratsinis and S Vemury ldquoParticle formation in gases areviewrdquo Powder Technology vol 88 no 3 pp 267ndash273 1996
[2] G Beaucage H Kammler P Mueller et al ldquoProbing thedynamics of nanoparticle growth in a flame using synchrotronradiationrdquo Nature Materials vol 3 no 6 pp 370ndash373 2004
[3] S Panda and S E Pratsinis ldquoModeling the synthesis ofaluminum particles by evaporation-condensation in an aerosolflow reactorrdquo Nanostructured Materials vol 5 no 7-8 pp 755ndash767 1995
[4] S Yuu K Ikeda and T Umekage ldquoFlow-field prediction andexperimental verification of low Reynolds number gas-particleturbulent jetsrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 109 pp 13ndash27 1996
[5] S Tsantilis H K Kammler and S E Pratsinis ldquoPopulationbalance modeling of flame synthesis of titania nanoparticlesrdquo
Journal of Nanotechnology 9
Chemical Engineering Science vol 57 no 12 pp 2139ndash21562002
[6] N Settumba and S C Garrick ldquoDirect numerical simulationof nanoparticle coagulation in a temporal mixing layer via amoment methodrdquo Journal of Aerosol Science vol 34 no 2 pp149ndash167 2003
[7] D L Marchisio and R O Fox ldquoSolution of population balanceequations using the direct quadrature mehtod of momentsrdquoJournal of Aerosol Science vol 36 pp 43ndash73 2005
[8] F Aristizabal R J Munz and D Berk ldquoModeling of the pro-duction of ultra fine Aluminium particles in rapid quenchingturbulent flowrdquo Journal of Aerosol Science vol 37 no 2 pp 162ndash186 2006
[9] S Rigopoulos ldquoPDF method for population balance in turbu-lent reactive flowrdquo Chemical Engineering Science vol 62 no 23pp 6865ndash6878 2007
[10] K Zhou A Attili A Alshaarawi and F Bisetti ldquoSimulationof aerosol nucleation and growth in a turbulent mixing layerrdquoPhysics of Fluids vol 26 no 6 Article ID 065106 2014
[11] S A Orszag and I Staroselsky ldquoCFD progress and problemsrdquoComputer Physics Communications vol 127 no 1 pp 165ndash1712000
[12] K Nakaso T Fujimoto T Seto M Shimada K Okuyamaand M M Lunden ldquoSize distribution change of titania nano-particle agglomerates generated by gas phase reaction agglom-eration and sinteringrdquo Aerosol Science and Technology vol 35no 5 pp 929ndash947 2001
[13] T Johannessen S E Pratsinis andH Livbjerg ldquoComputationalanalysis of coagulation and coalescence in the flame synthesis oftitania particlesrdquoPowder Technology vol 118 no 3 pp 242ndash2502001
[14] E G Moody and L R Collins ldquoEffect of mixing on thenucleation and growth of titania particlesrdquo Aerosol Science andTechnology vol 37 no 5 pp 403ndash424 2003
[15] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[16] S Das and S C Garrick ldquoThe effects of turbulence onnanoparticle growth in turbulent reacting jetsrdquo Physics of Fluidsvol 22 no 10 Article ID 103303 2010
[17] S C Garrick ldquoEffects of turbulent fluctuations on nanoparticlecoagulation in shear flowsrdquo Aerosol Science and Technology vol45 no 10 pp 1272ndash1285 2011
[18] A J Fager J Liu and S C Garrick ldquoHybrid simulations ofmetal particle nucleation a priori and a posteriori analyses ofthe effects of unresolved scalar interactions on nanoparticlenucleationrdquo Physics of Fluids vol 24 no 7 Article ID 0751102012
[19] N J Murfield and S C Garrick ldquoLarge eddy simulation anddirect numerical simulation of homogeneous nucleation inturbulent wakesrdquo Journal of Aerosol Science vol 60 pp 21ndash332013
[20] N JMurfield and S CGarrick ldquoThe effects of unresolved scalarfluctuations during homogeneous nucleationrdquo Aerosol Scienceand Technology vol 47 no 7 pp 806ndash817 2013
[21] P Givi ldquoModel free simulations of turbulent reactive flowsrdquoProgress in Energy and Combustion Science vol 15 no 1 pp 1ndash107 1989
[22] K E J Lehtinen and M R Zachariah ldquoSelf-preserving theoryfor the volume distribution of particles undergoing browniancoagulationrdquo Journal of Colloid and Interface Science vol 242no 2 pp 314ndash318 2001
[23] S C Garrick K E J Lehtinen andM R Zachariah ldquoNanopar-ticle coagulation via a Navier-Stokesnodal methodology evo-lution of the particle fieldrdquo Journal of Aerosol Science vol 37 no5 pp 555ndash576 2006
[24] F Gelbard and J H Seinfeld ldquoSimulation of multicomponentaerosol dynamicsrdquo Journal of Colloid And Interface Science vol78 no 2 pp 485ndash501 1980
[25] P Biswas C Y Wu M R Zachariah and B McMillin ldquoChar-acterization of iron oxide-silica nanocomposites in flamespart II comparison of discrete-sectional model predictions toexperimental datardquo Journal of Materials Research vol 12 no 3pp 714ndash723 1997
[26] K E J Lehtinen and M R Zachariah ldquoEnergy accumulationin nanoparticle collision and coalescence processesrdquo Journal ofAerosol Science vol 33 no 2 pp 357ndash368 2002
[27] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[28] J Loeffler S Das and S C Garrick ldquoLarge eddy simulationof titanium dioxide nanoparticle formation and growth inturbulent jetsrdquoAerosol Science and Technology vol 45 no 5 pp616ndash628 2011
[29] K S Friedlander Smoke Dust and Haze Fundamentals ofAerosol Dynamics Oxford University Press New York NYUSA 2000
[30] M Frenklach and S J Harris ldquoAerosol dynamics modelingusing the method of momentsrdquo Journal of Colloid and InterfaceScience vol 118 no 1 pp 252ndash261 1987
[31] S E Pratsinis ldquoParticle production by gas-to-particle conver-sion in turbulent flowsrdquo Journal of Aerosol Science vol 20 no8 pp 1461ndash1464 1989
[32] J D Landgrebe and S E Pratsinis ldquoA discrete-sectional modelfor particulate production by gas-phase chemical reaction andaerosol coagulation in the free-molecular regimerdquo Journal ofColloid and Interface Science vol 139 no 1 pp 63ndash86 1990
[33] GWang and S C Garrick ldquoModeling and simulation of titaniasynthesis in two-dimensional methane-air flamesrdquo Journal ofNanoparticle Research vol 7 no 6 pp 621ndash632 2005
[34] G W Mulholland R J Samson R D Mountain and M HErnst ldquoCluster size distribution for free molecular agglomera-tionrdquo Energy amp Fuels vol 2 no 4 pp 481ndash486 1988
[35] J Cai N Lu and C M Sorensen ldquoAnalysis of fractal clus-ter morphology parameters structural coefficient and densityautocorrelation function cutoffrdquo Journal of ColloidAnd InterfaceScience vol 171 no 2 pp 470ndash473 1995
[36] R Jullien and PMeakin ldquoSimplemodels for the restructuring ofthree-dimensional ballistic aggregatesrdquo Journal of Colloid AndInterface Science vol 127 no 1 pp 265ndash272 1989
[37] S N Rogak and R C Flagan ldquoCoagulation of aerosol agglom-erates in the transition regimerdquo Journal of Colloid and InterfaceScience vol 151 no 1 pp 203ndash224 1992
[38] S Modem S C Garrick M R Zachariah and K E J LehtinenldquoDirect numerical simulation of nanoparticle coagulation in atemporal mixing layerrdquo in Proceedings of the 29th Symposium(International) on Combustion pp 1071ndash1077 The CombustionInstitute Pittsburgh Pa USA 2002
[39] S C Garrick and G Wang ldquoModeling and simulation of tita-nium dioxide nanoparticle synthesis with finite-rate sinteringin planar jetsrdquo Journal of Nanoparticle Research vol 13 no 3pp 973ndash984 2011
10 Journal of Nanotechnology
[40] M C Heine and S E Pratsinis ldquoPolydispersity of primaryparticles in agglomerates made by coagulation and sinteringrdquoJournal of Aerosol Science vol 38 no 1 pp 17ndash38 2007
[41] R W MacCormack ldquoThe effect of viscosity in hypervelocityimpact cateringrdquo AIAA Paper 69-354 1969
[42] M H Carpenter ldquoA high-order compact numerical algorithmfor supersonic flowsrdquo in Twelfth International Conference onNumerical Methods in Fluid Dynamics K W Morton Ed vol371 of Lecture Notes in Physics pp 254ndash258 Springer BerlinGermany 1990
[43] D H Rudy and J C Strikwerda ldquoBoundary conditions forsubsonic compressible navier-stokes calculationsrdquo Computersand Fluids vol 9 no 3 pp 327ndash338 1981
[44] P Givi ldquoFiltered density function for subgrid scale modeling ofturbulent combustionrdquo AIAA Journal vol 44 no 1 pp 16ndash232006
[45] S Modem and S C Garrick ldquoNanoparticle coagulation in atemporal mixing layer mean and size-selected imagesrdquo Journalof Visualization vol 6 no 3 pp 293ndash302 2003
[46] D L Wright S Yu P S Kasibhatla et al ldquoRetrieval of aerosolproperties from moments of the particle size distribution forkernels involving the step function cloud droplet activationrdquoJournal of Aerosol Science vol 33 no 2 pp 319ndash337 2002
[47] H K Kammler R Jossen PWMorrison Jr S E Pratsinis andG Beaucage ldquoThe effect of external electric fields during flamesynthesis of titaniardquo Powder Technology vol 135-136 pp 310ndash320 2003
[48] W C Hinds Aerosol Technology Properties Behavior andMeasurement of Air-Borne Particles John Wiley amp Sons NewYork NY USA 2nd edition 1999
[49] N Settumba and S C Garrick ldquoA comparison of diffusivetransport in a moment method for nanoparticle coagulationrdquoJournal of Aerosol Science vol 35 no 1 pp 93ndash101 2004
[50] S E Pratsinis ldquoSimultaneous nucleation condensation andcoagulation in aerosol reactorsrdquo Journal of Colloid And InterfaceScience vol 124 no 2 pp 416ndash427 1988
[51] RMcGraw ldquoDescription of aerosol dynamics by the quadraturemethod of momentsrdquo Aerosol Science and Technology vol 27no 2 pp 255ndash265 1997
[52] J Bai Y-H Xu and J-P Wang ldquoCubic and spherical high-moment FeCo nanoparticles with narrow size distributionrdquoIEEE Transactions on Magnetics vol 43 no 7 pp 3340ndash33422007
[53] A Khorsand Zak R Razali W H Abd Majid and MDarroudi ldquoSynthesis and characterization of a narrow sizedistribution of zinc oxide nanoparticlesrdquo International Journalof Nanomedicine vol 6 no 1 pp 1399ndash1403 2011
[54] M Asemi and M Ghanaatshoar ldquoPreparation of CuCrO2
nanoparticles with narrow size distribution by sol-gel methodrdquoJournal of Sol-Gel Science and Technology vol 70 no 3 pp 416ndash421 2014
[55] A M Ahadi O Polonskyi U Schurmann T Strunskus and FFaupel ldquoStable production of TiOx nanoparticles with narrowsize distribution by reactive pulsed dc magnetron sputteringrdquoJournal of Physics D Applied Physics vol 48 no 3 Article ID035501 2015
[56] W W So S B Park K J Kim C J Shin and S J Moon ldquoThecrystalline phase stability of titania particles prepared at roomtemperature by the sol-gelmethodrdquo Journal ofMaterials Sciencevol 36 no 17 pp 4299ndash4305 2001
[57] A Teleki R Wengeler L Wengeler H Nirschl and S EPratsinis ldquoDistinguishing between aggregates and agglomeratesof flame-made TiO
[16 17] The effects of turbulence on nucleation have alsobeen elucidated [18ndash20] To obtain physically accurate datasimulations must be three-dimensional and must be model-free That is the results must be obtained without the use ofturbulence or subgrid scale models [21] A review of the liter-ature reveals a lack of the detailed information especially viahigh-resolution direct simulation on the interplay betweenthe different mechanisms affecting nanoparticle growth
In this work the hydrolysis of titanium tetrachloride(TiCl4) to produce titanium dioxide (TiO
2) is simulated
via direct numerical simulation The turbulent reactingmultiphase flow is obtained by solving the Navier-Stokesequations in conjunction with transport equations for all ofthe relevant chemical species and a nodal approach is usedto represent the particle field [22 23] With the chemicaland particle fields available as a function of space timeand size particle nucleation condensation and coagulationare illustrated individually Additionally the two growthmechanisms are elucidated
2 Methodology
The mass momentum and energy equations are solved toobtain the fluid velocity 119906
119894(119909 119905) pressure 119901(119909 119905) density
120588(119909 119905) and the enthalpy ℎ(119909 119905) These variables are governedby the following conservation equations
120597120588
120597119905
+
120597120588119906119895
120597119909119895
= 0
120597120588119906119894
120597119905
+
120597120588119906119894119906119895
120597119909119895
= minus
120597119901
120597119909119894
+
120597120591119894119895
120597119909119895
120597120588ℎ
120597119905
+
120597120588119906119895ℎ
120597119909119895
=
120597
120597119909119895
(
119896
119862119901
120597ℎ
120597119909119895
)
(1)
where 120591119894119895is the viscous stress tensor for a Newtonian fluid 119896
is the coefficient of thermal conduction and119862119901is the specific
heat at constant pressure
21 Chemical Transport The fluid contains five chemicalspecies the transport of which is given by the conservationof species equations
120597120588119884119898
120597119905
+
120597120588119906119895119884119898
120597119909119895
=
120597
120597119909119895
(120588D119898
120597119884119898
120597119909119895
) + 119898 (2)
where 119884119898is the mass concentration of species 119898 and D
119898
is the diffusion coefficient of species 119898 The reactants TiCl4
and water vapor (H2O) undergo an irreversible one-step
isothermal chemical reaction at a temperature of 300Kand atmospheric pressure which produces titanium dioxide(TiO2) and hydrochloric acid (HCl)
TiCl4+ 2H2O119896119891
997888997888rarr TiO2+ 4HCl (3)
The source term 119898in (2) represents the effects of chemical
reaction the rate of creation or consumption of species 119898
The system is closed with the ideal gas equation of state 119901 =120588119877119879 and 119877 = sum119877
119898119884119898119872119882119898 where 119877
119898and119872119882
119898are the
gas constant and molecular weight of species119898 respectivelyThe fluid temperature 119879 is obtained using the enthalpy via119889ℎ = 119862
119901119889119879The fifth chemical species nitrogen (N
2) does
not participate in any chemical reactions and serves as thecarrier gas
22 Particle Field The aerosol general dynamic equation(GDE) describes particle dynamics under the influence ofvarious physicochemical phenomena convection diffusioncoagulation surface growth nucleation and other inter-nalexternal forces The GDE is utilized in discrete formas a population balance on each cluster or particle sizeThe methodology uses the nodalsectional method of toapproximate the GDE [23ndash26] This approach effectivelydivides the aerosol population into three classes monomersclusters and particles [16] The GDE is therefore solvedas a set of 119873
119904transport equations one for each bin 119876
119896
119896 = 1 2 119873119904[27] Monomers of size 05 nm in diameter
populate bin 1 while bins 2 and 3 are populated by clustersof molecules Molecular clusters of size 1 nm and larger areconsidered ldquoparticlesrdquo [28] The general transport equationfor the concentration of monomers clusters and particles inbin 119896 119876
119896 is written as
120597120588119876119896
120597119905
+
120597120588119906119895119876119896
120597119909119895
=
120597
120597119909119895
(120588119863119876
120597119876119896
120597119909119895
) + 119876
119896 (4)
where119863119876is the diffusivity given by
119863119876= 119896119887119879
119862119888
3120587120583119889119901
(5)
where 119896119887is the Boltzmann constant 119862
119888is the Cunningham
correction factor and 119889119901is the mean volume particle diam-
eter [29] The source term 119876119896represents particle formation
where 119869 accounts for the formation of monomers in bin 119896 = 1via chemical reaction [30ndash33] and 120594
119894119895119896is given by
120594119894119895119896=
V119896+1minus (V119894+ V119895)
V119896+1minus V119896
if V119896le V119894+ V119895lt V119896+1
(V119894+ V119895) minus V119896minus1
V119896minus V119896minus1
if V119896minus1le V119894+ V119895lt V119896
0 otherwise
(7)
Journal of Nanotechnology 3
The source term in (6) 120596119876119896 represents the effects of
nucleation condensation (via monomer-cluster monomer-particle and cluster-particle collisions) and Brownian coag-ulation In this study all particle formation and growth pro-cesses occur at 300KThismeans that all particles considered
are agglomerates meaning growth via coagulation results infractal-like aggregate particles consisting of primary parti-cles (monomers) The collision frequency function 120573
119894119895 is
well documented in both the free-molecular and continuumregimes [29 34ndash37] Collisions of all particles (monomersclusters etc) are considered and 120573
119894119895is given by
120573119894119895=
1198861(1198941119863119891
+ 1198951119863119891)
119863119891(
1
119894
+
1
119895
)
12
119863119891lt 2 free-molecular regime
119886 (1198941119863119891
+ 1198951119863119891)
2
(
1
119894
+
1
119895
)
12
119863119891ge 2 free-molecular regime
2119896119887119879
3120583
(
1
V1119863119891119894
+
1
V1119863119891119895
)(V1119863119891119894
+ V1119863119891119895) in continuum regime
(8)
where
119886 = (
3V119900
4120587
)
16
(
6119896119887119879
120588119901
)
12
1198861=
2119863119891119886
489
(9)
where V119900is the primary particle volume V
119894is the volume of
an agglomerate in the 119894th section 120588119901is the particle density
119894 and 119895 are the primary particle numbers in bins 119894 and 119895respectively and 119863
119891is the fractal dimension The particles
generated by the industrial aerosol processes are usuallynonspherical and the products are composed of groups ofadhering particles ranging from loosely linked particlessometimes termed agglomerates to strongly necked particlescalled aggregates or hard agglomeratesThe fractal dimensionis used to represent the shape of the aggregate or particlestructure A statistical concept the fractal (or Hausdorff)dimension 119863
119891 was introduced to describe the shape of the
agglomerate which is obtained after averaging over manyagglomerates with the same number of primary particlesThevalue of the fractal dimension ranges from 119863
119891= 1 to 119863
119891=
3 depending on the details of the agglomerate formationprocess [29] The bins are organized in such a manner thatthe volume of particles in two successive bins is doubled thatis V119896= 2 times V
119896minus1[23 38]
3 Results
31 Flow Configuration The flow under consideration is athree-dimensional isothermal turbulent reacting jet issuingfrom an orifice of diameter 119863 into a coflowing stream Thejet is composed of titanium tetrachloride (TiCl
4) diluted in
nitrogen (N2) while the coflowing stream is composed of
water vapor (H2O) diluted in N
2 The initial velocities are
119880119900= 100ms for the jet and119880
infinfor the coflowing streamThe
fluid field is characterized by the velocity ratio 119880infin119880119900= 02
The Reynolds number based on velocity of the high-speedstream and the jet diameter is Re = 119880
119900119863] = 3000 The
simulation is performed at 300K and atmosphere pressure(1 atm) To accelerate the development of large-scale struc-tures random perturbations with a maximum intensity of5 are added to the cross-stream V-velocity The chemicalcomposition of the jet stream is 001 TiCl
4and 9999 N
2
by mass The simulation utilizes stoichiometric mixtures andthe molar ratio of 1 2 for TiCl
4and H
2O
In this work several assumptions and approximations areutilized These are stated below for clarity
(1) Ceramic powders such as TiO2have low equilibrium
vapor pressures implying that single molecules maybe considered particles [31] As TiO
2is formed
it appears as 05 nm diameter free spherical TiO2
ldquomonomersrdquo populating bin 119896 = 1 hence nucleationis treated as an instantaneous process
(2) The nanoparticles are small enough to follow the fluidpath lines Additionally the particle volume fractionis of order 10minus7 As a result the presence of theparticles does not affect the fluid field
(3) Condensation is dominated by the collision rateof monomers with other monomers clusters andparticles This means the condensable species in thesimulations is the TiO
2ldquovaporrdquo
(4) The clusters and particles are stable because of thehigh supersaturation of the monomers As a resultat these temperatures there is no evaporation orsublimation of particles back to the gas-phase
(5) The fractal dimension 119863119891 is based on the collision
time 120591119888 and sintering time 120591
119891 In this work all
processes occur at 119879 = 300K At this temperatureTiO2particles do not sinter as 120591
119888≪ 120591119891[39]
(6) A fractal dimension of119863119891= 3 is used when collisions
occur between monomers while119863119891= 2 is used when
monomers and dimers collide For all other particleinteractions (collisions between larger particles) afractal dimension of119863
119891= 18 is used [39 40]
4 Journal of Nanotechnology
32 Numerical Specifications Ten bins are used to discretizethe particle field (119873
119904= 10) The computational domain is of
size 20119863 times 15119863 times 4119863 and is comprised of 500 times 375 times 100grid points in the 119909- 119910- and 119911-directions respectively Thegoverning transport equations representing both the fluidand particle fields are solved using a MacCormack-basedfinite difference scheme [41 42] The scheme is of secondorder accurate in time and of fourth order accurate inspace The boundary conditions are periodic in the span-wise 119911-direction and zero-derivative in the cross-stream 119910-direction and nonreflecting boundary conditions are usedin both inflow and outflow boundaries (119909-direction) [43]The simulation is performed up to a nondimensional time of119905⋆= 119880119900119905119863 = 100 which corresponds to a physical time of
048ms Both instantaneous and mean or averaged data arepresented We average in the 119911-direction as it is the spatiallyhomogeneous direction in planar jets Quantities such ascontours isosurfaces and mean data are useful in makingqualitative and quantitative assessments of the nanoparticlegrowth dynamics as well as the underlying fluid and particlefields
33 Flow Field The vorticity is the curl of the velocity vectorand is an indicator of fluidmixingThe vorticity magnitude isthe local rate of rotation In nonpremixed chemically reactingflows vorticity has the effect of increasing the interfacial areabetween the reactants An isosurface of the instantaneousvorticity magnitude the |120596| = 3 level-set is shown inFigure 1 at time 119905⋆ = 100 The image shows that the flowis initially laminar and becomes turbulent as the jet travelsdownstream Near 119909119863 = 6 the two boundary layers initiallylocated at 119910119863 = plusmn05 merge and the jet spreads across thedomain This is aided by the presence of vortex braids thetubular structures oriented in the stream-wise 119909-directionwhich act to draw the surrounding fluid into contact with thefluid issuing through the nozzle Further downstream vortexbending and stretching acts to generate small-scale structuresas the flow becomes fully turbulent It is evident that the jetsuddenly spreadsamplifies near 119909119863 = 8 Near 119909119863 = 10the contours reveal a high concentration of intense mixingthat persists throughout the latter half of the computationaldomain These small-scale structures result in an increasedchemical reaction as they serve to bring the reactants intocontact High-resolution DNS facilitates the capturing of thesmall-scale structures If turbulence models were used thenthe effects of the small-scale interactions on the chemicalreaction (and particle formation and growth) would need tobe accounted for [18 44]
34 Titanium Dioxide The chemical conversion of the TiCl4
and H2O to produce TiO
2is the first step in the particle
synthesis process Instantaneous contours of the TiO2mass
fraction at the 119911 = 0 plane and time 119905⋆ = 100 are shown inFigure 2The image reveals that TiO
2is initially formed along
the interface of the two streams and subsequently where thereactants are well mixed The TiO
2is convected downstream
and across the jetThemaximum TiO2mass fraction appears
far downstream and TiO2mass spreads out Because of
Figure 1 An instantaneous isosurface of the vorticity magnitude|120596| = 3 at time 119905⋆ = 100
0 5e minus 4
Figure 2 Instantaneous contour of the TiO2mass fraction at 119911 = 0
plane at 119905⋆ = 100
the infinite-rate chemistry at least one of the reactantsis consumed immediately upon contact Turbulent mixingbrings ldquofreshrdquo reactants together (via large-scale transport) toproduce TiO
2(via molecular scale transport) As the TiO
2is
produced molecular diffusion acts to transport it fromTiO2-
rich to TiO2-free regions
35 Particle Concentrations As the chemical reaction pro-ceeds more titania is produced The monomers collide witheach other to produce dimers those dimers collide withmonomers (condensation) to produce trimers and collidewith each other to produce larger particles (coagulation) Anadvantage of the nodal approach is the fact that the particlefield is obtained as a function of size (in addition to spaceand time) A detailed view of the TiO
2nanoparticle field can
be obtained by observing the particle number concentrationsdistributed throughout the domain Instantaneous contours
Journal of Nanotechnology 5
0 14e20
(a)64e180
(b)
0 75e17
(c)
Figure 3 Instantaneous contours of the 119911-direction averaged particle number concentrations (a) monomers (b) 1 nm (c) 2 nm
of the 119911-direction averaged particle number concentrationsof monomers 1 nm and 2 nm particles at 119905⋆ = 100are shown in Figure 3 Figure 3(a) shows that monomersinitially appear at the interface of the two streams and theconcentration increases significantly near 119909119863 = 10 aftercollapse of the jet potential core Figure 3(b) shows that the1 nm diameter particles begin to appear near 119909119863 = 5 andhigh concentrations are found between 119909119863 = 10 and 119909119863 =20 Figure 3(c) shows a similar trend for the 2 nm particles
To convey the spatial inhomogeneity of the particlefield a three-dimensional view is presented in Figure 4 Thefigure shows three isosurfaces colored to show the largeconcentration of 1 nm 2 nm and 3 nm nanoparticles Thepresence of 1 nm particles (colored green) throughout thedomain reflects the ongoing chemical reaction andnucleationthat occurs when large-scale convective mixing brings TiCl
4
and H2O into contact The increase in particle size with
downstream distance is evident in the image as the 3 nmdiameter particles are only found in the last third of thedomain
36 Mean Nanoparticle Size and Geometric Standard Devi-ation Particle size distributions (PSDs) are often charac-terized by the mean diameter and the geometric standard
deviation (GSD) Though the nodal approach employedcontains the full PSD conveying that all of the informationis not trivial [45 46] the information conveyed by the firsttwo moments can be quite useful The mean diameter usedhere is the volume-equivalent mean particle diameter and isgiven by 119889
119901= (6V119901120587)13 where the mean volume is given by
V119901=
sum119873119904
119894=1119876119894V119894
sum119873119904
119894=1119876119894
(10)
The GSD represents the width of the PSD and is given by
[log (120590119892)]
2
=
sum119873119904
119896=1119876119896(log (119889
119901119896) minus log ( 119889
119901119896))
2
sum119873119904
119896=1119876119896
(11)
where 119889119901119896is a number mean diameter given by
log ( 119889119901119896) =
sum119873119904
119896=1119876119896log (119889
119901119896)
sum119873119904
119896=1119876119896
(12)
Larger GSD values indicate that the size distribution isrelatively broad while smaller GSD values (120590
119892= 1) indi-
cate a relatively narrow distribution [47] The GSD for the
6 Journal of Nanotechnology
3nm
3nm
2nm
2nm
1nm
1nm
Figure 4 Instantaneous isosurfaces of 1 nm 2 nm and 3 nmparticles at 119905⋆ = 100
2
19
18
17
16
15
14
13
12
11
1
Figure 5 An instantaneous isosurface of vorticity colored by thegeometric standard deviation 120590
119892
hydrolysis of TiCl4in a planar jet is shown on an isosurface of
vorticity magnitude at time 119905⋆ = 100 in Figure 5 The figurereveals that the GSD generally increases as the jet travelsdownstream This implies that mixing (due to turbulence)is a significant contribution to particle polydispersity Ifnucleation were absent there would be no monomers andthe coagulating particles would achieve the so-called self-preserving value (120590
119892= 15) [48 49] Locations where 120590
119892gt
15 in Figure 5 reflect where nucleation condensation andcoagulation simultaneously occur That is ldquonewlyrdquo formedparticles by nucleationmixing and old particles generated bycoagulation exist
More insight into the particle field may be obtained byconsidering the relationship between the mean diameter andthe GSD A scatter plot of 120590
119892versus 119889
119901is shown in Figure 6
1 2 3 41
12
14
16
18
2
120590
dp (nm)
Figure 6 Scatter plot of the geometric standard deviation120590119892 versus
the mean particle diameter 119889119901
The figure shows that at the two ends of the size distribution119889119901= 1 nm and 119889
119901= 4 nm the distribution is fairly narrow
or unimodal while the GSD is largest (18 lt 120590119892lt 2) when
the mean diameter is near 119889119901= 32 nm This relatively large
GSD indicates that there are a variety of particle dynamicspresent in regions of the flow where the mean diameter is119889119901= 32 nm
37 Particle Growth Theparticle growth-rate is an importantparameter to consider as in combination with residence timeor reactor size it is a predictor of particle size A diameter-based growth-rate Ωnm (with units of nms) is defined basedon
Ωnm = (6Ω
120587
)
13
times 109 (13)
whereΩ is the particle volumetric growth-rate given by
Ω =
sum119873119904
119896(V119896)2
120596119896
sum119873119904
119896119876119896V119896
minus
sum119873119904
119896(V119896)2
119876119896
(sum119873119904
119896119876119896V119896)
2
119873119904
sum
119896
V119896120596119896 (14)
A contour plot of the 119911-averaged growth-rate ⟨Ωnm⟩119911 isshown in Figure 7 The figure shows that within the first 6diameters 0 le 119909119863 le 6 growth is confined to roughly⟨Ωnm⟩119911 = 10 nms at the interface of the jet and thecoflowing stream where the hydrolysis reaction producesTiO2vapor Near 119909119863 = 10 when the shear layers merge
the particles begin to grow faster with the rate approaching⟨Ωnm⟩119911 = 25 nms This occurs after the potential corecollapses Further downstream the growth-rate increases toroughly ⟨Ωnm⟩119911 = 50 nms and this value is maintained for119909119863 gt 14 (It should be noted that this is the average growth-rate and at other points in the domain the values ofΩnm maybe smaller or larger)
The particles grow via two mechanisms condensation(collisions between monomers and particles) and
Journal of Nanotechnology 7
0 53
Figure 7 Instantaneous contours of the 119911-direction averaged particle growth-rate ⟨Ωnm⟩119911 at 119905⋆= 100
0 28(a)
0 33(b)
Figure 8 Instantaneous contours of the nanoparticle growth-rate decomposed by mechanism 119911-direction averaged particle growth-rate (a)by condensation (b) by coagulation
coagulation (collisions between particles) Becausethe particle data is available as a function of size thecontribution of each mechanism is readily availableParticle growth by mechanism is shown in Figure 8The interactions between monomers and particles orthe condensation growth-rate is shown in Figure 8(a)(This image looks similar to the monomer numberconcentration shown in Figure 3(a) because thecondensation is represented by the collision betweenmonomers and particles) The contours show that largecondensation growth occurs both in the proximal regionof the jet and after collapse of the jet core This reflects theongoing hydrolysis of TiCl
4and production of TiO
2and its
deposition on existing particlesParticle growth by coagulation is shown in Figure 8(b)
The contours of the 119911-averaged growth-rate shows thatgrowth by coagulation begins after collapse of the jet potentialcore The growth-rate in the region 4 lt 119909119863 lt 8 is ashigh as Ωnm = 16 nms Farther downstream the 119911-averaged
coagulation growth-rate doubles This region of the flow isdominated by mixing and small-scale turbulence (evident inFigure 1) That region of the flow also contains particles of avariety of sizes small and large as reflected by 120590
119892in Figure 5
The small-scale turbulence means dissipation and increasedresidence times while the disparate particle sizes mean anefficient collision efficiency These two combine to increasecoagulation
The spatial relationship between condensation growthand coagulation growth is elucidated by showing the con-tribution of each at every grid point in the computationaldomain A scatter plot of the two growth-rates is shown inFigure 9The growth-rates are not averaged in the 119911-directionand while a spatial relationship is not directly evident fromthis figure one may be reliably inferred along with the pre-vious data Figure 9 shows that where condensation growthis low coagulation growth is high This trend is evidentin Figure 8 as well However Figure 9 shows that in theseregions the coagulation growth-rate can be as much as an
8 Journal of Nanotechnology
0 10 20 30 40 500
10
20
30
40
50
Ωnm
coagulation(nms)
Ωnmcondensation (nms)
Figure 9 Scatter plot of the particle growth-rate by condensationversus the particle growth-rate by coagulation
order of magnitude greater than the condensation growth-rate
4 Summary and Conclusions
The growth mechanisms of titania during hydrolysis oftitanium tetrachloride in a three-dimensional planar jet arestudied via direct numerical simulation The mass momen-tum enthalpy and species transport equations are solved in amodel-free manner Titania was produced via the hydrolysisof titanium tetrachloride modeled via a 1-step infinitelyfast chemical mechanism The particle field was representedusing a nodal method and solves for the evolution of theconcentration of particles of various sizes in an Eulerianmanner When coupled to the Navier-Stokes solver thefluid thermal chemical and particle fields are obtained as afunction of space and time
The results show that fluid turbulencegas mixing plays avery important role in particle growth The results indicatethat the particle formation and growth are greatly affectedif not dominated by mixing and chemical reaction Reactantconversion or titania production is limited by the ability ofthe flow to bring the TiCl
4and H
2O into contact Evidence
for this is the particle formation throughout the domainAdditionally the results show that where the turbulenceis more intense the particles are larger and the particlesize distribution as characterized by the geometric standarddeviation is wider Particle growth in the proximal regionof the jet is dominated by condensation After collapseof the jet core nanoparticle growth due to coagulationincreases significantly Though TiO
2is produced after core
collapse producing condensible species particle growth dueto coagulation is as much as an order of magnitude greaterthan that due to condensation While both condensationand coagulation act to increase polydispersity coagulationhas been shown to increase the width of the particle sizedistribution faster or greater than condensation [50]
These results help to shed light on and improve ourunderstanding of the underlying growth dynamics occur-ring in nanoparticle synthesis processes The change fromcondensation-dominated to coagulation-dominated growthis useful in modeling the complete synthesis process includ-ing sintering and the formation of hard and soft agglomer-ates Spanning the size range from single molecules (particleinception) to hundreds of nanometers as the particles foundin many industrial processes and applications is computeintensive [18] However knowing that condensation andmolecular growth is relatively minor could facilitate the useof more affordable modeling strategies that do not needto account for every phenomena [6 51] Additionally ahighly desired attribute of metal oxide nanoparticles is theirspecific surface area The ability to synthesize narrow sizedistributions which are desired is advantageous in that itremoves the processing necessary to separate particles by sizeThis is advantageous in that it lowers cost [52ndash55]
One strategy to reduce polydispersity may be to delay thetransition to turbulence vis a vis delaying collapse of the jetcore While the particle field is known as a function of size itshould be noted that in this work we are unable to distinguishthe various phases of titania rutile anatase or brookite [5657]The phase composition of titania is very much a functionof the synthesis process for example precursor compositionand temperature history which are greatly simplified in thiswork Such capability requires phenomenologicalmodels thataccount for particle dynamics as a function of space time andtemperature as well as solid-state diffusion processes and isquite beyond the scope of this work
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Funding for this research was provided by the Universityof Minnesota Computational resources were provided byMinnesota Supercomputing Institute
References
[1] S E Pratsinis and S Vemury ldquoParticle formation in gases areviewrdquo Powder Technology vol 88 no 3 pp 267ndash273 1996
[2] G Beaucage H Kammler P Mueller et al ldquoProbing thedynamics of nanoparticle growth in a flame using synchrotronradiationrdquo Nature Materials vol 3 no 6 pp 370ndash373 2004
[3] S Panda and S E Pratsinis ldquoModeling the synthesis ofaluminum particles by evaporation-condensation in an aerosolflow reactorrdquo Nanostructured Materials vol 5 no 7-8 pp 755ndash767 1995
[4] S Yuu K Ikeda and T Umekage ldquoFlow-field prediction andexperimental verification of low Reynolds number gas-particleturbulent jetsrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 109 pp 13ndash27 1996
[5] S Tsantilis H K Kammler and S E Pratsinis ldquoPopulationbalance modeling of flame synthesis of titania nanoparticlesrdquo
Journal of Nanotechnology 9
Chemical Engineering Science vol 57 no 12 pp 2139ndash21562002
[6] N Settumba and S C Garrick ldquoDirect numerical simulationof nanoparticle coagulation in a temporal mixing layer via amoment methodrdquo Journal of Aerosol Science vol 34 no 2 pp149ndash167 2003
[7] D L Marchisio and R O Fox ldquoSolution of population balanceequations using the direct quadrature mehtod of momentsrdquoJournal of Aerosol Science vol 36 pp 43ndash73 2005
[8] F Aristizabal R J Munz and D Berk ldquoModeling of the pro-duction of ultra fine Aluminium particles in rapid quenchingturbulent flowrdquo Journal of Aerosol Science vol 37 no 2 pp 162ndash186 2006
[9] S Rigopoulos ldquoPDF method for population balance in turbu-lent reactive flowrdquo Chemical Engineering Science vol 62 no 23pp 6865ndash6878 2007
[10] K Zhou A Attili A Alshaarawi and F Bisetti ldquoSimulationof aerosol nucleation and growth in a turbulent mixing layerrdquoPhysics of Fluids vol 26 no 6 Article ID 065106 2014
[11] S A Orszag and I Staroselsky ldquoCFD progress and problemsrdquoComputer Physics Communications vol 127 no 1 pp 165ndash1712000
[12] K Nakaso T Fujimoto T Seto M Shimada K Okuyamaand M M Lunden ldquoSize distribution change of titania nano-particle agglomerates generated by gas phase reaction agglom-eration and sinteringrdquo Aerosol Science and Technology vol 35no 5 pp 929ndash947 2001
[13] T Johannessen S E Pratsinis andH Livbjerg ldquoComputationalanalysis of coagulation and coalescence in the flame synthesis oftitania particlesrdquoPowder Technology vol 118 no 3 pp 242ndash2502001
[14] E G Moody and L R Collins ldquoEffect of mixing on thenucleation and growth of titania particlesrdquo Aerosol Science andTechnology vol 37 no 5 pp 403ndash424 2003
[15] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[16] S Das and S C Garrick ldquoThe effects of turbulence onnanoparticle growth in turbulent reacting jetsrdquo Physics of Fluidsvol 22 no 10 Article ID 103303 2010
[17] S C Garrick ldquoEffects of turbulent fluctuations on nanoparticlecoagulation in shear flowsrdquo Aerosol Science and Technology vol45 no 10 pp 1272ndash1285 2011
[18] A J Fager J Liu and S C Garrick ldquoHybrid simulations ofmetal particle nucleation a priori and a posteriori analyses ofthe effects of unresolved scalar interactions on nanoparticlenucleationrdquo Physics of Fluids vol 24 no 7 Article ID 0751102012
[19] N J Murfield and S C Garrick ldquoLarge eddy simulation anddirect numerical simulation of homogeneous nucleation inturbulent wakesrdquo Journal of Aerosol Science vol 60 pp 21ndash332013
[20] N JMurfield and S CGarrick ldquoThe effects of unresolved scalarfluctuations during homogeneous nucleationrdquo Aerosol Scienceand Technology vol 47 no 7 pp 806ndash817 2013
[21] P Givi ldquoModel free simulations of turbulent reactive flowsrdquoProgress in Energy and Combustion Science vol 15 no 1 pp 1ndash107 1989
[22] K E J Lehtinen and M R Zachariah ldquoSelf-preserving theoryfor the volume distribution of particles undergoing browniancoagulationrdquo Journal of Colloid and Interface Science vol 242no 2 pp 314ndash318 2001
[23] S C Garrick K E J Lehtinen andM R Zachariah ldquoNanopar-ticle coagulation via a Navier-Stokesnodal methodology evo-lution of the particle fieldrdquo Journal of Aerosol Science vol 37 no5 pp 555ndash576 2006
[24] F Gelbard and J H Seinfeld ldquoSimulation of multicomponentaerosol dynamicsrdquo Journal of Colloid And Interface Science vol78 no 2 pp 485ndash501 1980
[25] P Biswas C Y Wu M R Zachariah and B McMillin ldquoChar-acterization of iron oxide-silica nanocomposites in flamespart II comparison of discrete-sectional model predictions toexperimental datardquo Journal of Materials Research vol 12 no 3pp 714ndash723 1997
[26] K E J Lehtinen and M R Zachariah ldquoEnergy accumulationin nanoparticle collision and coalescence processesrdquo Journal ofAerosol Science vol 33 no 2 pp 357ndash368 2002
[27] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[28] J Loeffler S Das and S C Garrick ldquoLarge eddy simulationof titanium dioxide nanoparticle formation and growth inturbulent jetsrdquoAerosol Science and Technology vol 45 no 5 pp616ndash628 2011
[29] K S Friedlander Smoke Dust and Haze Fundamentals ofAerosol Dynamics Oxford University Press New York NYUSA 2000
[30] M Frenklach and S J Harris ldquoAerosol dynamics modelingusing the method of momentsrdquo Journal of Colloid and InterfaceScience vol 118 no 1 pp 252ndash261 1987
[31] S E Pratsinis ldquoParticle production by gas-to-particle conver-sion in turbulent flowsrdquo Journal of Aerosol Science vol 20 no8 pp 1461ndash1464 1989
[32] J D Landgrebe and S E Pratsinis ldquoA discrete-sectional modelfor particulate production by gas-phase chemical reaction andaerosol coagulation in the free-molecular regimerdquo Journal ofColloid and Interface Science vol 139 no 1 pp 63ndash86 1990
[33] GWang and S C Garrick ldquoModeling and simulation of titaniasynthesis in two-dimensional methane-air flamesrdquo Journal ofNanoparticle Research vol 7 no 6 pp 621ndash632 2005
[34] G W Mulholland R J Samson R D Mountain and M HErnst ldquoCluster size distribution for free molecular agglomera-tionrdquo Energy amp Fuels vol 2 no 4 pp 481ndash486 1988
[35] J Cai N Lu and C M Sorensen ldquoAnalysis of fractal clus-ter morphology parameters structural coefficient and densityautocorrelation function cutoffrdquo Journal of ColloidAnd InterfaceScience vol 171 no 2 pp 470ndash473 1995
[36] R Jullien and PMeakin ldquoSimplemodels for the restructuring ofthree-dimensional ballistic aggregatesrdquo Journal of Colloid AndInterface Science vol 127 no 1 pp 265ndash272 1989
[37] S N Rogak and R C Flagan ldquoCoagulation of aerosol agglom-erates in the transition regimerdquo Journal of Colloid and InterfaceScience vol 151 no 1 pp 203ndash224 1992
[38] S Modem S C Garrick M R Zachariah and K E J LehtinenldquoDirect numerical simulation of nanoparticle coagulation in atemporal mixing layerrdquo in Proceedings of the 29th Symposium(International) on Combustion pp 1071ndash1077 The CombustionInstitute Pittsburgh Pa USA 2002
[39] S C Garrick and G Wang ldquoModeling and simulation of tita-nium dioxide nanoparticle synthesis with finite-rate sinteringin planar jetsrdquo Journal of Nanoparticle Research vol 13 no 3pp 973ndash984 2011
10 Journal of Nanotechnology
[40] M C Heine and S E Pratsinis ldquoPolydispersity of primaryparticles in agglomerates made by coagulation and sinteringrdquoJournal of Aerosol Science vol 38 no 1 pp 17ndash38 2007
[41] R W MacCormack ldquoThe effect of viscosity in hypervelocityimpact cateringrdquo AIAA Paper 69-354 1969
[42] M H Carpenter ldquoA high-order compact numerical algorithmfor supersonic flowsrdquo in Twelfth International Conference onNumerical Methods in Fluid Dynamics K W Morton Ed vol371 of Lecture Notes in Physics pp 254ndash258 Springer BerlinGermany 1990
[43] D H Rudy and J C Strikwerda ldquoBoundary conditions forsubsonic compressible navier-stokes calculationsrdquo Computersand Fluids vol 9 no 3 pp 327ndash338 1981
[44] P Givi ldquoFiltered density function for subgrid scale modeling ofturbulent combustionrdquo AIAA Journal vol 44 no 1 pp 16ndash232006
[45] S Modem and S C Garrick ldquoNanoparticle coagulation in atemporal mixing layer mean and size-selected imagesrdquo Journalof Visualization vol 6 no 3 pp 293ndash302 2003
[46] D L Wright S Yu P S Kasibhatla et al ldquoRetrieval of aerosolproperties from moments of the particle size distribution forkernels involving the step function cloud droplet activationrdquoJournal of Aerosol Science vol 33 no 2 pp 319ndash337 2002
[47] H K Kammler R Jossen PWMorrison Jr S E Pratsinis andG Beaucage ldquoThe effect of external electric fields during flamesynthesis of titaniardquo Powder Technology vol 135-136 pp 310ndash320 2003
[48] W C Hinds Aerosol Technology Properties Behavior andMeasurement of Air-Borne Particles John Wiley amp Sons NewYork NY USA 2nd edition 1999
[49] N Settumba and S C Garrick ldquoA comparison of diffusivetransport in a moment method for nanoparticle coagulationrdquoJournal of Aerosol Science vol 35 no 1 pp 93ndash101 2004
[50] S E Pratsinis ldquoSimultaneous nucleation condensation andcoagulation in aerosol reactorsrdquo Journal of Colloid And InterfaceScience vol 124 no 2 pp 416ndash427 1988
[51] RMcGraw ldquoDescription of aerosol dynamics by the quadraturemethod of momentsrdquo Aerosol Science and Technology vol 27no 2 pp 255ndash265 1997
[52] J Bai Y-H Xu and J-P Wang ldquoCubic and spherical high-moment FeCo nanoparticles with narrow size distributionrdquoIEEE Transactions on Magnetics vol 43 no 7 pp 3340ndash33422007
[53] A Khorsand Zak R Razali W H Abd Majid and MDarroudi ldquoSynthesis and characterization of a narrow sizedistribution of zinc oxide nanoparticlesrdquo International Journalof Nanomedicine vol 6 no 1 pp 1399ndash1403 2011
[54] M Asemi and M Ghanaatshoar ldquoPreparation of CuCrO2
nanoparticles with narrow size distribution by sol-gel methodrdquoJournal of Sol-Gel Science and Technology vol 70 no 3 pp 416ndash421 2014
[55] A M Ahadi O Polonskyi U Schurmann T Strunskus and FFaupel ldquoStable production of TiOx nanoparticles with narrowsize distribution by reactive pulsed dc magnetron sputteringrdquoJournal of Physics D Applied Physics vol 48 no 3 Article ID035501 2015
[56] W W So S B Park K J Kim C J Shin and S J Moon ldquoThecrystalline phase stability of titania particles prepared at roomtemperature by the sol-gelmethodrdquo Journal ofMaterials Sciencevol 36 no 17 pp 4299ndash4305 2001
[57] A Teleki R Wengeler L Wengeler H Nirschl and S EPratsinis ldquoDistinguishing between aggregates and agglomeratesof flame-made TiO
The source term in (6) 120596119876119896 represents the effects of
nucleation condensation (via monomer-cluster monomer-particle and cluster-particle collisions) and Brownian coag-ulation In this study all particle formation and growth pro-cesses occur at 300KThismeans that all particles considered
are agglomerates meaning growth via coagulation results infractal-like aggregate particles consisting of primary parti-cles (monomers) The collision frequency function 120573
119894119895 is
well documented in both the free-molecular and continuumregimes [29 34ndash37] Collisions of all particles (monomersclusters etc) are considered and 120573
119894119895is given by
120573119894119895=
1198861(1198941119863119891
+ 1198951119863119891)
119863119891(
1
119894
+
1
119895
)
12
119863119891lt 2 free-molecular regime
119886 (1198941119863119891
+ 1198951119863119891)
2
(
1
119894
+
1
119895
)
12
119863119891ge 2 free-molecular regime
2119896119887119879
3120583
(
1
V1119863119891119894
+
1
V1119863119891119895
)(V1119863119891119894
+ V1119863119891119895) in continuum regime
(8)
where
119886 = (
3V119900
4120587
)
16
(
6119896119887119879
120588119901
)
12
1198861=
2119863119891119886
489
(9)
where V119900is the primary particle volume V
119894is the volume of
an agglomerate in the 119894th section 120588119901is the particle density
119894 and 119895 are the primary particle numbers in bins 119894 and 119895respectively and 119863
119891is the fractal dimension The particles
generated by the industrial aerosol processes are usuallynonspherical and the products are composed of groups ofadhering particles ranging from loosely linked particlessometimes termed agglomerates to strongly necked particlescalled aggregates or hard agglomeratesThe fractal dimensionis used to represent the shape of the aggregate or particlestructure A statistical concept the fractal (or Hausdorff)dimension 119863
119891 was introduced to describe the shape of the
agglomerate which is obtained after averaging over manyagglomerates with the same number of primary particlesThevalue of the fractal dimension ranges from 119863
119891= 1 to 119863
119891=
3 depending on the details of the agglomerate formationprocess [29] The bins are organized in such a manner thatthe volume of particles in two successive bins is doubled thatis V119896= 2 times V
119896minus1[23 38]
3 Results
31 Flow Configuration The flow under consideration is athree-dimensional isothermal turbulent reacting jet issuingfrom an orifice of diameter 119863 into a coflowing stream Thejet is composed of titanium tetrachloride (TiCl
4) diluted in
nitrogen (N2) while the coflowing stream is composed of
water vapor (H2O) diluted in N
2 The initial velocities are
119880119900= 100ms for the jet and119880
infinfor the coflowing streamThe
fluid field is characterized by the velocity ratio 119880infin119880119900= 02
The Reynolds number based on velocity of the high-speedstream and the jet diameter is Re = 119880
119900119863] = 3000 The
simulation is performed at 300K and atmosphere pressure(1 atm) To accelerate the development of large-scale struc-tures random perturbations with a maximum intensity of5 are added to the cross-stream V-velocity The chemicalcomposition of the jet stream is 001 TiCl
4and 9999 N
2
by mass The simulation utilizes stoichiometric mixtures andthe molar ratio of 1 2 for TiCl
4and H
2O
In this work several assumptions and approximations areutilized These are stated below for clarity
(1) Ceramic powders such as TiO2have low equilibrium
vapor pressures implying that single molecules maybe considered particles [31] As TiO
2is formed
it appears as 05 nm diameter free spherical TiO2
ldquomonomersrdquo populating bin 119896 = 1 hence nucleationis treated as an instantaneous process
(2) The nanoparticles are small enough to follow the fluidpath lines Additionally the particle volume fractionis of order 10minus7 As a result the presence of theparticles does not affect the fluid field
(3) Condensation is dominated by the collision rateof monomers with other monomers clusters andparticles This means the condensable species in thesimulations is the TiO
2ldquovaporrdquo
(4) The clusters and particles are stable because of thehigh supersaturation of the monomers As a resultat these temperatures there is no evaporation orsublimation of particles back to the gas-phase
(5) The fractal dimension 119863119891 is based on the collision
time 120591119888 and sintering time 120591
119891 In this work all
processes occur at 119879 = 300K At this temperatureTiO2particles do not sinter as 120591
119888≪ 120591119891[39]
(6) A fractal dimension of119863119891= 3 is used when collisions
occur between monomers while119863119891= 2 is used when
monomers and dimers collide For all other particleinteractions (collisions between larger particles) afractal dimension of119863
119891= 18 is used [39 40]
4 Journal of Nanotechnology
32 Numerical Specifications Ten bins are used to discretizethe particle field (119873
119904= 10) The computational domain is of
size 20119863 times 15119863 times 4119863 and is comprised of 500 times 375 times 100grid points in the 119909- 119910- and 119911-directions respectively Thegoverning transport equations representing both the fluidand particle fields are solved using a MacCormack-basedfinite difference scheme [41 42] The scheme is of secondorder accurate in time and of fourth order accurate inspace The boundary conditions are periodic in the span-wise 119911-direction and zero-derivative in the cross-stream 119910-direction and nonreflecting boundary conditions are usedin both inflow and outflow boundaries (119909-direction) [43]The simulation is performed up to a nondimensional time of119905⋆= 119880119900119905119863 = 100 which corresponds to a physical time of
048ms Both instantaneous and mean or averaged data arepresented We average in the 119911-direction as it is the spatiallyhomogeneous direction in planar jets Quantities such ascontours isosurfaces and mean data are useful in makingqualitative and quantitative assessments of the nanoparticlegrowth dynamics as well as the underlying fluid and particlefields
33 Flow Field The vorticity is the curl of the velocity vectorand is an indicator of fluidmixingThe vorticity magnitude isthe local rate of rotation In nonpremixed chemically reactingflows vorticity has the effect of increasing the interfacial areabetween the reactants An isosurface of the instantaneousvorticity magnitude the |120596| = 3 level-set is shown inFigure 1 at time 119905⋆ = 100 The image shows that the flowis initially laminar and becomes turbulent as the jet travelsdownstream Near 119909119863 = 6 the two boundary layers initiallylocated at 119910119863 = plusmn05 merge and the jet spreads across thedomain This is aided by the presence of vortex braids thetubular structures oriented in the stream-wise 119909-directionwhich act to draw the surrounding fluid into contact with thefluid issuing through the nozzle Further downstream vortexbending and stretching acts to generate small-scale structuresas the flow becomes fully turbulent It is evident that the jetsuddenly spreadsamplifies near 119909119863 = 8 Near 119909119863 = 10the contours reveal a high concentration of intense mixingthat persists throughout the latter half of the computationaldomain These small-scale structures result in an increasedchemical reaction as they serve to bring the reactants intocontact High-resolution DNS facilitates the capturing of thesmall-scale structures If turbulence models were used thenthe effects of the small-scale interactions on the chemicalreaction (and particle formation and growth) would need tobe accounted for [18 44]
34 Titanium Dioxide The chemical conversion of the TiCl4
and H2O to produce TiO
2is the first step in the particle
synthesis process Instantaneous contours of the TiO2mass
fraction at the 119911 = 0 plane and time 119905⋆ = 100 are shown inFigure 2The image reveals that TiO
2is initially formed along
the interface of the two streams and subsequently where thereactants are well mixed The TiO
2is convected downstream
and across the jetThemaximum TiO2mass fraction appears
far downstream and TiO2mass spreads out Because of
Figure 1 An instantaneous isosurface of the vorticity magnitude|120596| = 3 at time 119905⋆ = 100
0 5e minus 4
Figure 2 Instantaneous contour of the TiO2mass fraction at 119911 = 0
plane at 119905⋆ = 100
the infinite-rate chemistry at least one of the reactantsis consumed immediately upon contact Turbulent mixingbrings ldquofreshrdquo reactants together (via large-scale transport) toproduce TiO
2(via molecular scale transport) As the TiO
2is
produced molecular diffusion acts to transport it fromTiO2-
rich to TiO2-free regions
35 Particle Concentrations As the chemical reaction pro-ceeds more titania is produced The monomers collide witheach other to produce dimers those dimers collide withmonomers (condensation) to produce trimers and collidewith each other to produce larger particles (coagulation) Anadvantage of the nodal approach is the fact that the particlefield is obtained as a function of size (in addition to spaceand time) A detailed view of the TiO
2nanoparticle field can
be obtained by observing the particle number concentrationsdistributed throughout the domain Instantaneous contours
Journal of Nanotechnology 5
0 14e20
(a)64e180
(b)
0 75e17
(c)
Figure 3 Instantaneous contours of the 119911-direction averaged particle number concentrations (a) monomers (b) 1 nm (c) 2 nm
of the 119911-direction averaged particle number concentrationsof monomers 1 nm and 2 nm particles at 119905⋆ = 100are shown in Figure 3 Figure 3(a) shows that monomersinitially appear at the interface of the two streams and theconcentration increases significantly near 119909119863 = 10 aftercollapse of the jet potential core Figure 3(b) shows that the1 nm diameter particles begin to appear near 119909119863 = 5 andhigh concentrations are found between 119909119863 = 10 and 119909119863 =20 Figure 3(c) shows a similar trend for the 2 nm particles
To convey the spatial inhomogeneity of the particlefield a three-dimensional view is presented in Figure 4 Thefigure shows three isosurfaces colored to show the largeconcentration of 1 nm 2 nm and 3 nm nanoparticles Thepresence of 1 nm particles (colored green) throughout thedomain reflects the ongoing chemical reaction andnucleationthat occurs when large-scale convective mixing brings TiCl
4
and H2O into contact The increase in particle size with
downstream distance is evident in the image as the 3 nmdiameter particles are only found in the last third of thedomain
36 Mean Nanoparticle Size and Geometric Standard Devi-ation Particle size distributions (PSDs) are often charac-terized by the mean diameter and the geometric standard
deviation (GSD) Though the nodal approach employedcontains the full PSD conveying that all of the informationis not trivial [45 46] the information conveyed by the firsttwo moments can be quite useful The mean diameter usedhere is the volume-equivalent mean particle diameter and isgiven by 119889
119901= (6V119901120587)13 where the mean volume is given by
V119901=
sum119873119904
119894=1119876119894V119894
sum119873119904
119894=1119876119894
(10)
The GSD represents the width of the PSD and is given by
[log (120590119892)]
2
=
sum119873119904
119896=1119876119896(log (119889
119901119896) minus log ( 119889
119901119896))
2
sum119873119904
119896=1119876119896
(11)
where 119889119901119896is a number mean diameter given by
log ( 119889119901119896) =
sum119873119904
119896=1119876119896log (119889
119901119896)
sum119873119904
119896=1119876119896
(12)
Larger GSD values indicate that the size distribution isrelatively broad while smaller GSD values (120590
119892= 1) indi-
cate a relatively narrow distribution [47] The GSD for the
6 Journal of Nanotechnology
3nm
3nm
2nm
2nm
1nm
1nm
Figure 4 Instantaneous isosurfaces of 1 nm 2 nm and 3 nmparticles at 119905⋆ = 100
2
19
18
17
16
15
14
13
12
11
1
Figure 5 An instantaneous isosurface of vorticity colored by thegeometric standard deviation 120590
119892
hydrolysis of TiCl4in a planar jet is shown on an isosurface of
vorticity magnitude at time 119905⋆ = 100 in Figure 5 The figurereveals that the GSD generally increases as the jet travelsdownstream This implies that mixing (due to turbulence)is a significant contribution to particle polydispersity Ifnucleation were absent there would be no monomers andthe coagulating particles would achieve the so-called self-preserving value (120590
119892= 15) [48 49] Locations where 120590
119892gt
15 in Figure 5 reflect where nucleation condensation andcoagulation simultaneously occur That is ldquonewlyrdquo formedparticles by nucleationmixing and old particles generated bycoagulation exist
More insight into the particle field may be obtained byconsidering the relationship between the mean diameter andthe GSD A scatter plot of 120590
119892versus 119889
119901is shown in Figure 6
1 2 3 41
12
14
16
18
2
120590
dp (nm)
Figure 6 Scatter plot of the geometric standard deviation120590119892 versus
the mean particle diameter 119889119901
The figure shows that at the two ends of the size distribution119889119901= 1 nm and 119889
119901= 4 nm the distribution is fairly narrow
or unimodal while the GSD is largest (18 lt 120590119892lt 2) when
the mean diameter is near 119889119901= 32 nm This relatively large
GSD indicates that there are a variety of particle dynamicspresent in regions of the flow where the mean diameter is119889119901= 32 nm
37 Particle Growth Theparticle growth-rate is an importantparameter to consider as in combination with residence timeor reactor size it is a predictor of particle size A diameter-based growth-rate Ωnm (with units of nms) is defined basedon
Ωnm = (6Ω
120587
)
13
times 109 (13)
whereΩ is the particle volumetric growth-rate given by
Ω =
sum119873119904
119896(V119896)2
120596119896
sum119873119904
119896119876119896V119896
minus
sum119873119904
119896(V119896)2
119876119896
(sum119873119904
119896119876119896V119896)
2
119873119904
sum
119896
V119896120596119896 (14)
A contour plot of the 119911-averaged growth-rate ⟨Ωnm⟩119911 isshown in Figure 7 The figure shows that within the first 6diameters 0 le 119909119863 le 6 growth is confined to roughly⟨Ωnm⟩119911 = 10 nms at the interface of the jet and thecoflowing stream where the hydrolysis reaction producesTiO2vapor Near 119909119863 = 10 when the shear layers merge
the particles begin to grow faster with the rate approaching⟨Ωnm⟩119911 = 25 nms This occurs after the potential corecollapses Further downstream the growth-rate increases toroughly ⟨Ωnm⟩119911 = 50 nms and this value is maintained for119909119863 gt 14 (It should be noted that this is the average growth-rate and at other points in the domain the values ofΩnm maybe smaller or larger)
The particles grow via two mechanisms condensation(collisions between monomers and particles) and
Journal of Nanotechnology 7
0 53
Figure 7 Instantaneous contours of the 119911-direction averaged particle growth-rate ⟨Ωnm⟩119911 at 119905⋆= 100
0 28(a)
0 33(b)
Figure 8 Instantaneous contours of the nanoparticle growth-rate decomposed by mechanism 119911-direction averaged particle growth-rate (a)by condensation (b) by coagulation
coagulation (collisions between particles) Becausethe particle data is available as a function of size thecontribution of each mechanism is readily availableParticle growth by mechanism is shown in Figure 8The interactions between monomers and particles orthe condensation growth-rate is shown in Figure 8(a)(This image looks similar to the monomer numberconcentration shown in Figure 3(a) because thecondensation is represented by the collision betweenmonomers and particles) The contours show that largecondensation growth occurs both in the proximal regionof the jet and after collapse of the jet core This reflects theongoing hydrolysis of TiCl
4and production of TiO
2and its
deposition on existing particlesParticle growth by coagulation is shown in Figure 8(b)
The contours of the 119911-averaged growth-rate shows thatgrowth by coagulation begins after collapse of the jet potentialcore The growth-rate in the region 4 lt 119909119863 lt 8 is ashigh as Ωnm = 16 nms Farther downstream the 119911-averaged
coagulation growth-rate doubles This region of the flow isdominated by mixing and small-scale turbulence (evident inFigure 1) That region of the flow also contains particles of avariety of sizes small and large as reflected by 120590
119892in Figure 5
The small-scale turbulence means dissipation and increasedresidence times while the disparate particle sizes mean anefficient collision efficiency These two combine to increasecoagulation
The spatial relationship between condensation growthand coagulation growth is elucidated by showing the con-tribution of each at every grid point in the computationaldomain A scatter plot of the two growth-rates is shown inFigure 9The growth-rates are not averaged in the 119911-directionand while a spatial relationship is not directly evident fromthis figure one may be reliably inferred along with the pre-vious data Figure 9 shows that where condensation growthis low coagulation growth is high This trend is evidentin Figure 8 as well However Figure 9 shows that in theseregions the coagulation growth-rate can be as much as an
8 Journal of Nanotechnology
0 10 20 30 40 500
10
20
30
40
50
Ωnm
coagulation(nms)
Ωnmcondensation (nms)
Figure 9 Scatter plot of the particle growth-rate by condensationversus the particle growth-rate by coagulation
order of magnitude greater than the condensation growth-rate
4 Summary and Conclusions
The growth mechanisms of titania during hydrolysis oftitanium tetrachloride in a three-dimensional planar jet arestudied via direct numerical simulation The mass momen-tum enthalpy and species transport equations are solved in amodel-free manner Titania was produced via the hydrolysisof titanium tetrachloride modeled via a 1-step infinitelyfast chemical mechanism The particle field was representedusing a nodal method and solves for the evolution of theconcentration of particles of various sizes in an Eulerianmanner When coupled to the Navier-Stokes solver thefluid thermal chemical and particle fields are obtained as afunction of space and time
The results show that fluid turbulencegas mixing plays avery important role in particle growth The results indicatethat the particle formation and growth are greatly affectedif not dominated by mixing and chemical reaction Reactantconversion or titania production is limited by the ability ofthe flow to bring the TiCl
4and H
2O into contact Evidence
for this is the particle formation throughout the domainAdditionally the results show that where the turbulenceis more intense the particles are larger and the particlesize distribution as characterized by the geometric standarddeviation is wider Particle growth in the proximal regionof the jet is dominated by condensation After collapseof the jet core nanoparticle growth due to coagulationincreases significantly Though TiO
2is produced after core
collapse producing condensible species particle growth dueto coagulation is as much as an order of magnitude greaterthan that due to condensation While both condensationand coagulation act to increase polydispersity coagulationhas been shown to increase the width of the particle sizedistribution faster or greater than condensation [50]
These results help to shed light on and improve ourunderstanding of the underlying growth dynamics occur-ring in nanoparticle synthesis processes The change fromcondensation-dominated to coagulation-dominated growthis useful in modeling the complete synthesis process includ-ing sintering and the formation of hard and soft agglomer-ates Spanning the size range from single molecules (particleinception) to hundreds of nanometers as the particles foundin many industrial processes and applications is computeintensive [18] However knowing that condensation andmolecular growth is relatively minor could facilitate the useof more affordable modeling strategies that do not needto account for every phenomena [6 51] Additionally ahighly desired attribute of metal oxide nanoparticles is theirspecific surface area The ability to synthesize narrow sizedistributions which are desired is advantageous in that itremoves the processing necessary to separate particles by sizeThis is advantageous in that it lowers cost [52ndash55]
One strategy to reduce polydispersity may be to delay thetransition to turbulence vis a vis delaying collapse of the jetcore While the particle field is known as a function of size itshould be noted that in this work we are unable to distinguishthe various phases of titania rutile anatase or brookite [5657]The phase composition of titania is very much a functionof the synthesis process for example precursor compositionand temperature history which are greatly simplified in thiswork Such capability requires phenomenologicalmodels thataccount for particle dynamics as a function of space time andtemperature as well as solid-state diffusion processes and isquite beyond the scope of this work
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Funding for this research was provided by the Universityof Minnesota Computational resources were provided byMinnesota Supercomputing Institute
References
[1] S E Pratsinis and S Vemury ldquoParticle formation in gases areviewrdquo Powder Technology vol 88 no 3 pp 267ndash273 1996
[2] G Beaucage H Kammler P Mueller et al ldquoProbing thedynamics of nanoparticle growth in a flame using synchrotronradiationrdquo Nature Materials vol 3 no 6 pp 370ndash373 2004
[3] S Panda and S E Pratsinis ldquoModeling the synthesis ofaluminum particles by evaporation-condensation in an aerosolflow reactorrdquo Nanostructured Materials vol 5 no 7-8 pp 755ndash767 1995
[4] S Yuu K Ikeda and T Umekage ldquoFlow-field prediction andexperimental verification of low Reynolds number gas-particleturbulent jetsrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 109 pp 13ndash27 1996
[5] S Tsantilis H K Kammler and S E Pratsinis ldquoPopulationbalance modeling of flame synthesis of titania nanoparticlesrdquo
Journal of Nanotechnology 9
Chemical Engineering Science vol 57 no 12 pp 2139ndash21562002
[6] N Settumba and S C Garrick ldquoDirect numerical simulationof nanoparticle coagulation in a temporal mixing layer via amoment methodrdquo Journal of Aerosol Science vol 34 no 2 pp149ndash167 2003
[7] D L Marchisio and R O Fox ldquoSolution of population balanceequations using the direct quadrature mehtod of momentsrdquoJournal of Aerosol Science vol 36 pp 43ndash73 2005
[8] F Aristizabal R J Munz and D Berk ldquoModeling of the pro-duction of ultra fine Aluminium particles in rapid quenchingturbulent flowrdquo Journal of Aerosol Science vol 37 no 2 pp 162ndash186 2006
[9] S Rigopoulos ldquoPDF method for population balance in turbu-lent reactive flowrdquo Chemical Engineering Science vol 62 no 23pp 6865ndash6878 2007
[10] K Zhou A Attili A Alshaarawi and F Bisetti ldquoSimulationof aerosol nucleation and growth in a turbulent mixing layerrdquoPhysics of Fluids vol 26 no 6 Article ID 065106 2014
[11] S A Orszag and I Staroselsky ldquoCFD progress and problemsrdquoComputer Physics Communications vol 127 no 1 pp 165ndash1712000
[12] K Nakaso T Fujimoto T Seto M Shimada K Okuyamaand M M Lunden ldquoSize distribution change of titania nano-particle agglomerates generated by gas phase reaction agglom-eration and sinteringrdquo Aerosol Science and Technology vol 35no 5 pp 929ndash947 2001
[13] T Johannessen S E Pratsinis andH Livbjerg ldquoComputationalanalysis of coagulation and coalescence in the flame synthesis oftitania particlesrdquoPowder Technology vol 118 no 3 pp 242ndash2502001
[14] E G Moody and L R Collins ldquoEffect of mixing on thenucleation and growth of titania particlesrdquo Aerosol Science andTechnology vol 37 no 5 pp 403ndash424 2003
[15] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[16] S Das and S C Garrick ldquoThe effects of turbulence onnanoparticle growth in turbulent reacting jetsrdquo Physics of Fluidsvol 22 no 10 Article ID 103303 2010
[17] S C Garrick ldquoEffects of turbulent fluctuations on nanoparticlecoagulation in shear flowsrdquo Aerosol Science and Technology vol45 no 10 pp 1272ndash1285 2011
[18] A J Fager J Liu and S C Garrick ldquoHybrid simulations ofmetal particle nucleation a priori and a posteriori analyses ofthe effects of unresolved scalar interactions on nanoparticlenucleationrdquo Physics of Fluids vol 24 no 7 Article ID 0751102012
[19] N J Murfield and S C Garrick ldquoLarge eddy simulation anddirect numerical simulation of homogeneous nucleation inturbulent wakesrdquo Journal of Aerosol Science vol 60 pp 21ndash332013
[20] N JMurfield and S CGarrick ldquoThe effects of unresolved scalarfluctuations during homogeneous nucleationrdquo Aerosol Scienceand Technology vol 47 no 7 pp 806ndash817 2013
[21] P Givi ldquoModel free simulations of turbulent reactive flowsrdquoProgress in Energy and Combustion Science vol 15 no 1 pp 1ndash107 1989
[22] K E J Lehtinen and M R Zachariah ldquoSelf-preserving theoryfor the volume distribution of particles undergoing browniancoagulationrdquo Journal of Colloid and Interface Science vol 242no 2 pp 314ndash318 2001
[23] S C Garrick K E J Lehtinen andM R Zachariah ldquoNanopar-ticle coagulation via a Navier-Stokesnodal methodology evo-lution of the particle fieldrdquo Journal of Aerosol Science vol 37 no5 pp 555ndash576 2006
[24] F Gelbard and J H Seinfeld ldquoSimulation of multicomponentaerosol dynamicsrdquo Journal of Colloid And Interface Science vol78 no 2 pp 485ndash501 1980
[25] P Biswas C Y Wu M R Zachariah and B McMillin ldquoChar-acterization of iron oxide-silica nanocomposites in flamespart II comparison of discrete-sectional model predictions toexperimental datardquo Journal of Materials Research vol 12 no 3pp 714ndash723 1997
[26] K E J Lehtinen and M R Zachariah ldquoEnergy accumulationin nanoparticle collision and coalescence processesrdquo Journal ofAerosol Science vol 33 no 2 pp 357ndash368 2002
[27] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[28] J Loeffler S Das and S C Garrick ldquoLarge eddy simulationof titanium dioxide nanoparticle formation and growth inturbulent jetsrdquoAerosol Science and Technology vol 45 no 5 pp616ndash628 2011
[29] K S Friedlander Smoke Dust and Haze Fundamentals ofAerosol Dynamics Oxford University Press New York NYUSA 2000
[30] M Frenklach and S J Harris ldquoAerosol dynamics modelingusing the method of momentsrdquo Journal of Colloid and InterfaceScience vol 118 no 1 pp 252ndash261 1987
[31] S E Pratsinis ldquoParticle production by gas-to-particle conver-sion in turbulent flowsrdquo Journal of Aerosol Science vol 20 no8 pp 1461ndash1464 1989
[32] J D Landgrebe and S E Pratsinis ldquoA discrete-sectional modelfor particulate production by gas-phase chemical reaction andaerosol coagulation in the free-molecular regimerdquo Journal ofColloid and Interface Science vol 139 no 1 pp 63ndash86 1990
[33] GWang and S C Garrick ldquoModeling and simulation of titaniasynthesis in two-dimensional methane-air flamesrdquo Journal ofNanoparticle Research vol 7 no 6 pp 621ndash632 2005
[34] G W Mulholland R J Samson R D Mountain and M HErnst ldquoCluster size distribution for free molecular agglomera-tionrdquo Energy amp Fuels vol 2 no 4 pp 481ndash486 1988
[35] J Cai N Lu and C M Sorensen ldquoAnalysis of fractal clus-ter morphology parameters structural coefficient and densityautocorrelation function cutoffrdquo Journal of ColloidAnd InterfaceScience vol 171 no 2 pp 470ndash473 1995
[36] R Jullien and PMeakin ldquoSimplemodels for the restructuring ofthree-dimensional ballistic aggregatesrdquo Journal of Colloid AndInterface Science vol 127 no 1 pp 265ndash272 1989
[37] S N Rogak and R C Flagan ldquoCoagulation of aerosol agglom-erates in the transition regimerdquo Journal of Colloid and InterfaceScience vol 151 no 1 pp 203ndash224 1992
[38] S Modem S C Garrick M R Zachariah and K E J LehtinenldquoDirect numerical simulation of nanoparticle coagulation in atemporal mixing layerrdquo in Proceedings of the 29th Symposium(International) on Combustion pp 1071ndash1077 The CombustionInstitute Pittsburgh Pa USA 2002
[39] S C Garrick and G Wang ldquoModeling and simulation of tita-nium dioxide nanoparticle synthesis with finite-rate sinteringin planar jetsrdquo Journal of Nanoparticle Research vol 13 no 3pp 973ndash984 2011
10 Journal of Nanotechnology
[40] M C Heine and S E Pratsinis ldquoPolydispersity of primaryparticles in agglomerates made by coagulation and sinteringrdquoJournal of Aerosol Science vol 38 no 1 pp 17ndash38 2007
[41] R W MacCormack ldquoThe effect of viscosity in hypervelocityimpact cateringrdquo AIAA Paper 69-354 1969
[42] M H Carpenter ldquoA high-order compact numerical algorithmfor supersonic flowsrdquo in Twelfth International Conference onNumerical Methods in Fluid Dynamics K W Morton Ed vol371 of Lecture Notes in Physics pp 254ndash258 Springer BerlinGermany 1990
[43] D H Rudy and J C Strikwerda ldquoBoundary conditions forsubsonic compressible navier-stokes calculationsrdquo Computersand Fluids vol 9 no 3 pp 327ndash338 1981
[44] P Givi ldquoFiltered density function for subgrid scale modeling ofturbulent combustionrdquo AIAA Journal vol 44 no 1 pp 16ndash232006
[45] S Modem and S C Garrick ldquoNanoparticle coagulation in atemporal mixing layer mean and size-selected imagesrdquo Journalof Visualization vol 6 no 3 pp 293ndash302 2003
[46] D L Wright S Yu P S Kasibhatla et al ldquoRetrieval of aerosolproperties from moments of the particle size distribution forkernels involving the step function cloud droplet activationrdquoJournal of Aerosol Science vol 33 no 2 pp 319ndash337 2002
[47] H K Kammler R Jossen PWMorrison Jr S E Pratsinis andG Beaucage ldquoThe effect of external electric fields during flamesynthesis of titaniardquo Powder Technology vol 135-136 pp 310ndash320 2003
[48] W C Hinds Aerosol Technology Properties Behavior andMeasurement of Air-Borne Particles John Wiley amp Sons NewYork NY USA 2nd edition 1999
[49] N Settumba and S C Garrick ldquoA comparison of diffusivetransport in a moment method for nanoparticle coagulationrdquoJournal of Aerosol Science vol 35 no 1 pp 93ndash101 2004
[50] S E Pratsinis ldquoSimultaneous nucleation condensation andcoagulation in aerosol reactorsrdquo Journal of Colloid And InterfaceScience vol 124 no 2 pp 416ndash427 1988
[51] RMcGraw ldquoDescription of aerosol dynamics by the quadraturemethod of momentsrdquo Aerosol Science and Technology vol 27no 2 pp 255ndash265 1997
[52] J Bai Y-H Xu and J-P Wang ldquoCubic and spherical high-moment FeCo nanoparticles with narrow size distributionrdquoIEEE Transactions on Magnetics vol 43 no 7 pp 3340ndash33422007
[53] A Khorsand Zak R Razali W H Abd Majid and MDarroudi ldquoSynthesis and characterization of a narrow sizedistribution of zinc oxide nanoparticlesrdquo International Journalof Nanomedicine vol 6 no 1 pp 1399ndash1403 2011
[54] M Asemi and M Ghanaatshoar ldquoPreparation of CuCrO2
nanoparticles with narrow size distribution by sol-gel methodrdquoJournal of Sol-Gel Science and Technology vol 70 no 3 pp 416ndash421 2014
[55] A M Ahadi O Polonskyi U Schurmann T Strunskus and FFaupel ldquoStable production of TiOx nanoparticles with narrowsize distribution by reactive pulsed dc magnetron sputteringrdquoJournal of Physics D Applied Physics vol 48 no 3 Article ID035501 2015
[56] W W So S B Park K J Kim C J Shin and S J Moon ldquoThecrystalline phase stability of titania particles prepared at roomtemperature by the sol-gelmethodrdquo Journal ofMaterials Sciencevol 36 no 17 pp 4299ndash4305 2001
[57] A Teleki R Wengeler L Wengeler H Nirschl and S EPratsinis ldquoDistinguishing between aggregates and agglomeratesof flame-made TiO
32 Numerical Specifications Ten bins are used to discretizethe particle field (119873
119904= 10) The computational domain is of
size 20119863 times 15119863 times 4119863 and is comprised of 500 times 375 times 100grid points in the 119909- 119910- and 119911-directions respectively Thegoverning transport equations representing both the fluidand particle fields are solved using a MacCormack-basedfinite difference scheme [41 42] The scheme is of secondorder accurate in time and of fourth order accurate inspace The boundary conditions are periodic in the span-wise 119911-direction and zero-derivative in the cross-stream 119910-direction and nonreflecting boundary conditions are usedin both inflow and outflow boundaries (119909-direction) [43]The simulation is performed up to a nondimensional time of119905⋆= 119880119900119905119863 = 100 which corresponds to a physical time of
048ms Both instantaneous and mean or averaged data arepresented We average in the 119911-direction as it is the spatiallyhomogeneous direction in planar jets Quantities such ascontours isosurfaces and mean data are useful in makingqualitative and quantitative assessments of the nanoparticlegrowth dynamics as well as the underlying fluid and particlefields
33 Flow Field The vorticity is the curl of the velocity vectorand is an indicator of fluidmixingThe vorticity magnitude isthe local rate of rotation In nonpremixed chemically reactingflows vorticity has the effect of increasing the interfacial areabetween the reactants An isosurface of the instantaneousvorticity magnitude the |120596| = 3 level-set is shown inFigure 1 at time 119905⋆ = 100 The image shows that the flowis initially laminar and becomes turbulent as the jet travelsdownstream Near 119909119863 = 6 the two boundary layers initiallylocated at 119910119863 = plusmn05 merge and the jet spreads across thedomain This is aided by the presence of vortex braids thetubular structures oriented in the stream-wise 119909-directionwhich act to draw the surrounding fluid into contact with thefluid issuing through the nozzle Further downstream vortexbending and stretching acts to generate small-scale structuresas the flow becomes fully turbulent It is evident that the jetsuddenly spreadsamplifies near 119909119863 = 8 Near 119909119863 = 10the contours reveal a high concentration of intense mixingthat persists throughout the latter half of the computationaldomain These small-scale structures result in an increasedchemical reaction as they serve to bring the reactants intocontact High-resolution DNS facilitates the capturing of thesmall-scale structures If turbulence models were used thenthe effects of the small-scale interactions on the chemicalreaction (and particle formation and growth) would need tobe accounted for [18 44]
34 Titanium Dioxide The chemical conversion of the TiCl4
and H2O to produce TiO
2is the first step in the particle
synthesis process Instantaneous contours of the TiO2mass
fraction at the 119911 = 0 plane and time 119905⋆ = 100 are shown inFigure 2The image reveals that TiO
2is initially formed along
the interface of the two streams and subsequently where thereactants are well mixed The TiO
2is convected downstream
and across the jetThemaximum TiO2mass fraction appears
far downstream and TiO2mass spreads out Because of
Figure 1 An instantaneous isosurface of the vorticity magnitude|120596| = 3 at time 119905⋆ = 100
0 5e minus 4
Figure 2 Instantaneous contour of the TiO2mass fraction at 119911 = 0
plane at 119905⋆ = 100
the infinite-rate chemistry at least one of the reactantsis consumed immediately upon contact Turbulent mixingbrings ldquofreshrdquo reactants together (via large-scale transport) toproduce TiO
2(via molecular scale transport) As the TiO
2is
produced molecular diffusion acts to transport it fromTiO2-
rich to TiO2-free regions
35 Particle Concentrations As the chemical reaction pro-ceeds more titania is produced The monomers collide witheach other to produce dimers those dimers collide withmonomers (condensation) to produce trimers and collidewith each other to produce larger particles (coagulation) Anadvantage of the nodal approach is the fact that the particlefield is obtained as a function of size (in addition to spaceand time) A detailed view of the TiO
2nanoparticle field can
be obtained by observing the particle number concentrationsdistributed throughout the domain Instantaneous contours
Journal of Nanotechnology 5
0 14e20
(a)64e180
(b)
0 75e17
(c)
Figure 3 Instantaneous contours of the 119911-direction averaged particle number concentrations (a) monomers (b) 1 nm (c) 2 nm
of the 119911-direction averaged particle number concentrationsof monomers 1 nm and 2 nm particles at 119905⋆ = 100are shown in Figure 3 Figure 3(a) shows that monomersinitially appear at the interface of the two streams and theconcentration increases significantly near 119909119863 = 10 aftercollapse of the jet potential core Figure 3(b) shows that the1 nm diameter particles begin to appear near 119909119863 = 5 andhigh concentrations are found between 119909119863 = 10 and 119909119863 =20 Figure 3(c) shows a similar trend for the 2 nm particles
To convey the spatial inhomogeneity of the particlefield a three-dimensional view is presented in Figure 4 Thefigure shows three isosurfaces colored to show the largeconcentration of 1 nm 2 nm and 3 nm nanoparticles Thepresence of 1 nm particles (colored green) throughout thedomain reflects the ongoing chemical reaction andnucleationthat occurs when large-scale convective mixing brings TiCl
4
and H2O into contact The increase in particle size with
downstream distance is evident in the image as the 3 nmdiameter particles are only found in the last third of thedomain
36 Mean Nanoparticle Size and Geometric Standard Devi-ation Particle size distributions (PSDs) are often charac-terized by the mean diameter and the geometric standard
deviation (GSD) Though the nodal approach employedcontains the full PSD conveying that all of the informationis not trivial [45 46] the information conveyed by the firsttwo moments can be quite useful The mean diameter usedhere is the volume-equivalent mean particle diameter and isgiven by 119889
119901= (6V119901120587)13 where the mean volume is given by
V119901=
sum119873119904
119894=1119876119894V119894
sum119873119904
119894=1119876119894
(10)
The GSD represents the width of the PSD and is given by
[log (120590119892)]
2
=
sum119873119904
119896=1119876119896(log (119889
119901119896) minus log ( 119889
119901119896))
2
sum119873119904
119896=1119876119896
(11)
where 119889119901119896is a number mean diameter given by
log ( 119889119901119896) =
sum119873119904
119896=1119876119896log (119889
119901119896)
sum119873119904
119896=1119876119896
(12)
Larger GSD values indicate that the size distribution isrelatively broad while smaller GSD values (120590
119892= 1) indi-
cate a relatively narrow distribution [47] The GSD for the
6 Journal of Nanotechnology
3nm
3nm
2nm
2nm
1nm
1nm
Figure 4 Instantaneous isosurfaces of 1 nm 2 nm and 3 nmparticles at 119905⋆ = 100
2
19
18
17
16
15
14
13
12
11
1
Figure 5 An instantaneous isosurface of vorticity colored by thegeometric standard deviation 120590
119892
hydrolysis of TiCl4in a planar jet is shown on an isosurface of
vorticity magnitude at time 119905⋆ = 100 in Figure 5 The figurereveals that the GSD generally increases as the jet travelsdownstream This implies that mixing (due to turbulence)is a significant contribution to particle polydispersity Ifnucleation were absent there would be no monomers andthe coagulating particles would achieve the so-called self-preserving value (120590
119892= 15) [48 49] Locations where 120590
119892gt
15 in Figure 5 reflect where nucleation condensation andcoagulation simultaneously occur That is ldquonewlyrdquo formedparticles by nucleationmixing and old particles generated bycoagulation exist
More insight into the particle field may be obtained byconsidering the relationship between the mean diameter andthe GSD A scatter plot of 120590
119892versus 119889
119901is shown in Figure 6
1 2 3 41
12
14
16
18
2
120590
dp (nm)
Figure 6 Scatter plot of the geometric standard deviation120590119892 versus
the mean particle diameter 119889119901
The figure shows that at the two ends of the size distribution119889119901= 1 nm and 119889
119901= 4 nm the distribution is fairly narrow
or unimodal while the GSD is largest (18 lt 120590119892lt 2) when
the mean diameter is near 119889119901= 32 nm This relatively large
GSD indicates that there are a variety of particle dynamicspresent in regions of the flow where the mean diameter is119889119901= 32 nm
37 Particle Growth Theparticle growth-rate is an importantparameter to consider as in combination with residence timeor reactor size it is a predictor of particle size A diameter-based growth-rate Ωnm (with units of nms) is defined basedon
Ωnm = (6Ω
120587
)
13
times 109 (13)
whereΩ is the particle volumetric growth-rate given by
Ω =
sum119873119904
119896(V119896)2
120596119896
sum119873119904
119896119876119896V119896
minus
sum119873119904
119896(V119896)2
119876119896
(sum119873119904
119896119876119896V119896)
2
119873119904
sum
119896
V119896120596119896 (14)
A contour plot of the 119911-averaged growth-rate ⟨Ωnm⟩119911 isshown in Figure 7 The figure shows that within the first 6diameters 0 le 119909119863 le 6 growth is confined to roughly⟨Ωnm⟩119911 = 10 nms at the interface of the jet and thecoflowing stream where the hydrolysis reaction producesTiO2vapor Near 119909119863 = 10 when the shear layers merge
the particles begin to grow faster with the rate approaching⟨Ωnm⟩119911 = 25 nms This occurs after the potential corecollapses Further downstream the growth-rate increases toroughly ⟨Ωnm⟩119911 = 50 nms and this value is maintained for119909119863 gt 14 (It should be noted that this is the average growth-rate and at other points in the domain the values ofΩnm maybe smaller or larger)
The particles grow via two mechanisms condensation(collisions between monomers and particles) and
Journal of Nanotechnology 7
0 53
Figure 7 Instantaneous contours of the 119911-direction averaged particle growth-rate ⟨Ωnm⟩119911 at 119905⋆= 100
0 28(a)
0 33(b)
Figure 8 Instantaneous contours of the nanoparticle growth-rate decomposed by mechanism 119911-direction averaged particle growth-rate (a)by condensation (b) by coagulation
coagulation (collisions between particles) Becausethe particle data is available as a function of size thecontribution of each mechanism is readily availableParticle growth by mechanism is shown in Figure 8The interactions between monomers and particles orthe condensation growth-rate is shown in Figure 8(a)(This image looks similar to the monomer numberconcentration shown in Figure 3(a) because thecondensation is represented by the collision betweenmonomers and particles) The contours show that largecondensation growth occurs both in the proximal regionof the jet and after collapse of the jet core This reflects theongoing hydrolysis of TiCl
4and production of TiO
2and its
deposition on existing particlesParticle growth by coagulation is shown in Figure 8(b)
The contours of the 119911-averaged growth-rate shows thatgrowth by coagulation begins after collapse of the jet potentialcore The growth-rate in the region 4 lt 119909119863 lt 8 is ashigh as Ωnm = 16 nms Farther downstream the 119911-averaged
coagulation growth-rate doubles This region of the flow isdominated by mixing and small-scale turbulence (evident inFigure 1) That region of the flow also contains particles of avariety of sizes small and large as reflected by 120590
119892in Figure 5
The small-scale turbulence means dissipation and increasedresidence times while the disparate particle sizes mean anefficient collision efficiency These two combine to increasecoagulation
The spatial relationship between condensation growthand coagulation growth is elucidated by showing the con-tribution of each at every grid point in the computationaldomain A scatter plot of the two growth-rates is shown inFigure 9The growth-rates are not averaged in the 119911-directionand while a spatial relationship is not directly evident fromthis figure one may be reliably inferred along with the pre-vious data Figure 9 shows that where condensation growthis low coagulation growth is high This trend is evidentin Figure 8 as well However Figure 9 shows that in theseregions the coagulation growth-rate can be as much as an
8 Journal of Nanotechnology
0 10 20 30 40 500
10
20
30
40
50
Ωnm
coagulation(nms)
Ωnmcondensation (nms)
Figure 9 Scatter plot of the particle growth-rate by condensationversus the particle growth-rate by coagulation
order of magnitude greater than the condensation growth-rate
4 Summary and Conclusions
The growth mechanisms of titania during hydrolysis oftitanium tetrachloride in a three-dimensional planar jet arestudied via direct numerical simulation The mass momen-tum enthalpy and species transport equations are solved in amodel-free manner Titania was produced via the hydrolysisof titanium tetrachloride modeled via a 1-step infinitelyfast chemical mechanism The particle field was representedusing a nodal method and solves for the evolution of theconcentration of particles of various sizes in an Eulerianmanner When coupled to the Navier-Stokes solver thefluid thermal chemical and particle fields are obtained as afunction of space and time
The results show that fluid turbulencegas mixing plays avery important role in particle growth The results indicatethat the particle formation and growth are greatly affectedif not dominated by mixing and chemical reaction Reactantconversion or titania production is limited by the ability ofthe flow to bring the TiCl
4and H
2O into contact Evidence
for this is the particle formation throughout the domainAdditionally the results show that where the turbulenceis more intense the particles are larger and the particlesize distribution as characterized by the geometric standarddeviation is wider Particle growth in the proximal regionof the jet is dominated by condensation After collapseof the jet core nanoparticle growth due to coagulationincreases significantly Though TiO
2is produced after core
collapse producing condensible species particle growth dueto coagulation is as much as an order of magnitude greaterthan that due to condensation While both condensationand coagulation act to increase polydispersity coagulationhas been shown to increase the width of the particle sizedistribution faster or greater than condensation [50]
These results help to shed light on and improve ourunderstanding of the underlying growth dynamics occur-ring in nanoparticle synthesis processes The change fromcondensation-dominated to coagulation-dominated growthis useful in modeling the complete synthesis process includ-ing sintering and the formation of hard and soft agglomer-ates Spanning the size range from single molecules (particleinception) to hundreds of nanometers as the particles foundin many industrial processes and applications is computeintensive [18] However knowing that condensation andmolecular growth is relatively minor could facilitate the useof more affordable modeling strategies that do not needto account for every phenomena [6 51] Additionally ahighly desired attribute of metal oxide nanoparticles is theirspecific surface area The ability to synthesize narrow sizedistributions which are desired is advantageous in that itremoves the processing necessary to separate particles by sizeThis is advantageous in that it lowers cost [52ndash55]
One strategy to reduce polydispersity may be to delay thetransition to turbulence vis a vis delaying collapse of the jetcore While the particle field is known as a function of size itshould be noted that in this work we are unable to distinguishthe various phases of titania rutile anatase or brookite [5657]The phase composition of titania is very much a functionof the synthesis process for example precursor compositionand temperature history which are greatly simplified in thiswork Such capability requires phenomenologicalmodels thataccount for particle dynamics as a function of space time andtemperature as well as solid-state diffusion processes and isquite beyond the scope of this work
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Funding for this research was provided by the Universityof Minnesota Computational resources were provided byMinnesota Supercomputing Institute
References
[1] S E Pratsinis and S Vemury ldquoParticle formation in gases areviewrdquo Powder Technology vol 88 no 3 pp 267ndash273 1996
[2] G Beaucage H Kammler P Mueller et al ldquoProbing thedynamics of nanoparticle growth in a flame using synchrotronradiationrdquo Nature Materials vol 3 no 6 pp 370ndash373 2004
[3] S Panda and S E Pratsinis ldquoModeling the synthesis ofaluminum particles by evaporation-condensation in an aerosolflow reactorrdquo Nanostructured Materials vol 5 no 7-8 pp 755ndash767 1995
[4] S Yuu K Ikeda and T Umekage ldquoFlow-field prediction andexperimental verification of low Reynolds number gas-particleturbulent jetsrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 109 pp 13ndash27 1996
[5] S Tsantilis H K Kammler and S E Pratsinis ldquoPopulationbalance modeling of flame synthesis of titania nanoparticlesrdquo
Journal of Nanotechnology 9
Chemical Engineering Science vol 57 no 12 pp 2139ndash21562002
[6] N Settumba and S C Garrick ldquoDirect numerical simulationof nanoparticle coagulation in a temporal mixing layer via amoment methodrdquo Journal of Aerosol Science vol 34 no 2 pp149ndash167 2003
[7] D L Marchisio and R O Fox ldquoSolution of population balanceequations using the direct quadrature mehtod of momentsrdquoJournal of Aerosol Science vol 36 pp 43ndash73 2005
[8] F Aristizabal R J Munz and D Berk ldquoModeling of the pro-duction of ultra fine Aluminium particles in rapid quenchingturbulent flowrdquo Journal of Aerosol Science vol 37 no 2 pp 162ndash186 2006
[9] S Rigopoulos ldquoPDF method for population balance in turbu-lent reactive flowrdquo Chemical Engineering Science vol 62 no 23pp 6865ndash6878 2007
[10] K Zhou A Attili A Alshaarawi and F Bisetti ldquoSimulationof aerosol nucleation and growth in a turbulent mixing layerrdquoPhysics of Fluids vol 26 no 6 Article ID 065106 2014
[11] S A Orszag and I Staroselsky ldquoCFD progress and problemsrdquoComputer Physics Communications vol 127 no 1 pp 165ndash1712000
[12] K Nakaso T Fujimoto T Seto M Shimada K Okuyamaand M M Lunden ldquoSize distribution change of titania nano-particle agglomerates generated by gas phase reaction agglom-eration and sinteringrdquo Aerosol Science and Technology vol 35no 5 pp 929ndash947 2001
[13] T Johannessen S E Pratsinis andH Livbjerg ldquoComputationalanalysis of coagulation and coalescence in the flame synthesis oftitania particlesrdquoPowder Technology vol 118 no 3 pp 242ndash2502001
[14] E G Moody and L R Collins ldquoEffect of mixing on thenucleation and growth of titania particlesrdquo Aerosol Science andTechnology vol 37 no 5 pp 403ndash424 2003
[15] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[16] S Das and S C Garrick ldquoThe effects of turbulence onnanoparticle growth in turbulent reacting jetsrdquo Physics of Fluidsvol 22 no 10 Article ID 103303 2010
[17] S C Garrick ldquoEffects of turbulent fluctuations on nanoparticlecoagulation in shear flowsrdquo Aerosol Science and Technology vol45 no 10 pp 1272ndash1285 2011
[18] A J Fager J Liu and S C Garrick ldquoHybrid simulations ofmetal particle nucleation a priori and a posteriori analyses ofthe effects of unresolved scalar interactions on nanoparticlenucleationrdquo Physics of Fluids vol 24 no 7 Article ID 0751102012
[19] N J Murfield and S C Garrick ldquoLarge eddy simulation anddirect numerical simulation of homogeneous nucleation inturbulent wakesrdquo Journal of Aerosol Science vol 60 pp 21ndash332013
[20] N JMurfield and S CGarrick ldquoThe effects of unresolved scalarfluctuations during homogeneous nucleationrdquo Aerosol Scienceand Technology vol 47 no 7 pp 806ndash817 2013
[21] P Givi ldquoModel free simulations of turbulent reactive flowsrdquoProgress in Energy and Combustion Science vol 15 no 1 pp 1ndash107 1989
[22] K E J Lehtinen and M R Zachariah ldquoSelf-preserving theoryfor the volume distribution of particles undergoing browniancoagulationrdquo Journal of Colloid and Interface Science vol 242no 2 pp 314ndash318 2001
[23] S C Garrick K E J Lehtinen andM R Zachariah ldquoNanopar-ticle coagulation via a Navier-Stokesnodal methodology evo-lution of the particle fieldrdquo Journal of Aerosol Science vol 37 no5 pp 555ndash576 2006
[24] F Gelbard and J H Seinfeld ldquoSimulation of multicomponentaerosol dynamicsrdquo Journal of Colloid And Interface Science vol78 no 2 pp 485ndash501 1980
[25] P Biswas C Y Wu M R Zachariah and B McMillin ldquoChar-acterization of iron oxide-silica nanocomposites in flamespart II comparison of discrete-sectional model predictions toexperimental datardquo Journal of Materials Research vol 12 no 3pp 714ndash723 1997
[26] K E J Lehtinen and M R Zachariah ldquoEnergy accumulationin nanoparticle collision and coalescence processesrdquo Journal ofAerosol Science vol 33 no 2 pp 357ndash368 2002
[27] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[28] J Loeffler S Das and S C Garrick ldquoLarge eddy simulationof titanium dioxide nanoparticle formation and growth inturbulent jetsrdquoAerosol Science and Technology vol 45 no 5 pp616ndash628 2011
[29] K S Friedlander Smoke Dust and Haze Fundamentals ofAerosol Dynamics Oxford University Press New York NYUSA 2000
[30] M Frenklach and S J Harris ldquoAerosol dynamics modelingusing the method of momentsrdquo Journal of Colloid and InterfaceScience vol 118 no 1 pp 252ndash261 1987
[31] S E Pratsinis ldquoParticle production by gas-to-particle conver-sion in turbulent flowsrdquo Journal of Aerosol Science vol 20 no8 pp 1461ndash1464 1989
[32] J D Landgrebe and S E Pratsinis ldquoA discrete-sectional modelfor particulate production by gas-phase chemical reaction andaerosol coagulation in the free-molecular regimerdquo Journal ofColloid and Interface Science vol 139 no 1 pp 63ndash86 1990
[33] GWang and S C Garrick ldquoModeling and simulation of titaniasynthesis in two-dimensional methane-air flamesrdquo Journal ofNanoparticle Research vol 7 no 6 pp 621ndash632 2005
[34] G W Mulholland R J Samson R D Mountain and M HErnst ldquoCluster size distribution for free molecular agglomera-tionrdquo Energy amp Fuels vol 2 no 4 pp 481ndash486 1988
[35] J Cai N Lu and C M Sorensen ldquoAnalysis of fractal clus-ter morphology parameters structural coefficient and densityautocorrelation function cutoffrdquo Journal of ColloidAnd InterfaceScience vol 171 no 2 pp 470ndash473 1995
[36] R Jullien and PMeakin ldquoSimplemodels for the restructuring ofthree-dimensional ballistic aggregatesrdquo Journal of Colloid AndInterface Science vol 127 no 1 pp 265ndash272 1989
[37] S N Rogak and R C Flagan ldquoCoagulation of aerosol agglom-erates in the transition regimerdquo Journal of Colloid and InterfaceScience vol 151 no 1 pp 203ndash224 1992
[38] S Modem S C Garrick M R Zachariah and K E J LehtinenldquoDirect numerical simulation of nanoparticle coagulation in atemporal mixing layerrdquo in Proceedings of the 29th Symposium(International) on Combustion pp 1071ndash1077 The CombustionInstitute Pittsburgh Pa USA 2002
[39] S C Garrick and G Wang ldquoModeling and simulation of tita-nium dioxide nanoparticle synthesis with finite-rate sinteringin planar jetsrdquo Journal of Nanoparticle Research vol 13 no 3pp 973ndash984 2011
10 Journal of Nanotechnology
[40] M C Heine and S E Pratsinis ldquoPolydispersity of primaryparticles in agglomerates made by coagulation and sinteringrdquoJournal of Aerosol Science vol 38 no 1 pp 17ndash38 2007
[41] R W MacCormack ldquoThe effect of viscosity in hypervelocityimpact cateringrdquo AIAA Paper 69-354 1969
[42] M H Carpenter ldquoA high-order compact numerical algorithmfor supersonic flowsrdquo in Twelfth International Conference onNumerical Methods in Fluid Dynamics K W Morton Ed vol371 of Lecture Notes in Physics pp 254ndash258 Springer BerlinGermany 1990
[43] D H Rudy and J C Strikwerda ldquoBoundary conditions forsubsonic compressible navier-stokes calculationsrdquo Computersand Fluids vol 9 no 3 pp 327ndash338 1981
[44] P Givi ldquoFiltered density function for subgrid scale modeling ofturbulent combustionrdquo AIAA Journal vol 44 no 1 pp 16ndash232006
[45] S Modem and S C Garrick ldquoNanoparticle coagulation in atemporal mixing layer mean and size-selected imagesrdquo Journalof Visualization vol 6 no 3 pp 293ndash302 2003
[46] D L Wright S Yu P S Kasibhatla et al ldquoRetrieval of aerosolproperties from moments of the particle size distribution forkernels involving the step function cloud droplet activationrdquoJournal of Aerosol Science vol 33 no 2 pp 319ndash337 2002
[47] H K Kammler R Jossen PWMorrison Jr S E Pratsinis andG Beaucage ldquoThe effect of external electric fields during flamesynthesis of titaniardquo Powder Technology vol 135-136 pp 310ndash320 2003
[48] W C Hinds Aerosol Technology Properties Behavior andMeasurement of Air-Borne Particles John Wiley amp Sons NewYork NY USA 2nd edition 1999
[49] N Settumba and S C Garrick ldquoA comparison of diffusivetransport in a moment method for nanoparticle coagulationrdquoJournal of Aerosol Science vol 35 no 1 pp 93ndash101 2004
[50] S E Pratsinis ldquoSimultaneous nucleation condensation andcoagulation in aerosol reactorsrdquo Journal of Colloid And InterfaceScience vol 124 no 2 pp 416ndash427 1988
[51] RMcGraw ldquoDescription of aerosol dynamics by the quadraturemethod of momentsrdquo Aerosol Science and Technology vol 27no 2 pp 255ndash265 1997
[52] J Bai Y-H Xu and J-P Wang ldquoCubic and spherical high-moment FeCo nanoparticles with narrow size distributionrdquoIEEE Transactions on Magnetics vol 43 no 7 pp 3340ndash33422007
[53] A Khorsand Zak R Razali W H Abd Majid and MDarroudi ldquoSynthesis and characterization of a narrow sizedistribution of zinc oxide nanoparticlesrdquo International Journalof Nanomedicine vol 6 no 1 pp 1399ndash1403 2011
[54] M Asemi and M Ghanaatshoar ldquoPreparation of CuCrO2
nanoparticles with narrow size distribution by sol-gel methodrdquoJournal of Sol-Gel Science and Technology vol 70 no 3 pp 416ndash421 2014
[55] A M Ahadi O Polonskyi U Schurmann T Strunskus and FFaupel ldquoStable production of TiOx nanoparticles with narrowsize distribution by reactive pulsed dc magnetron sputteringrdquoJournal of Physics D Applied Physics vol 48 no 3 Article ID035501 2015
[56] W W So S B Park K J Kim C J Shin and S J Moon ldquoThecrystalline phase stability of titania particles prepared at roomtemperature by the sol-gelmethodrdquo Journal ofMaterials Sciencevol 36 no 17 pp 4299ndash4305 2001
[57] A Teleki R Wengeler L Wengeler H Nirschl and S EPratsinis ldquoDistinguishing between aggregates and agglomeratesof flame-made TiO
Figure 3 Instantaneous contours of the 119911-direction averaged particle number concentrations (a) monomers (b) 1 nm (c) 2 nm
of the 119911-direction averaged particle number concentrationsof monomers 1 nm and 2 nm particles at 119905⋆ = 100are shown in Figure 3 Figure 3(a) shows that monomersinitially appear at the interface of the two streams and theconcentration increases significantly near 119909119863 = 10 aftercollapse of the jet potential core Figure 3(b) shows that the1 nm diameter particles begin to appear near 119909119863 = 5 andhigh concentrations are found between 119909119863 = 10 and 119909119863 =20 Figure 3(c) shows a similar trend for the 2 nm particles
To convey the spatial inhomogeneity of the particlefield a three-dimensional view is presented in Figure 4 Thefigure shows three isosurfaces colored to show the largeconcentration of 1 nm 2 nm and 3 nm nanoparticles Thepresence of 1 nm particles (colored green) throughout thedomain reflects the ongoing chemical reaction andnucleationthat occurs when large-scale convective mixing brings TiCl
4
and H2O into contact The increase in particle size with
downstream distance is evident in the image as the 3 nmdiameter particles are only found in the last third of thedomain
36 Mean Nanoparticle Size and Geometric Standard Devi-ation Particle size distributions (PSDs) are often charac-terized by the mean diameter and the geometric standard
deviation (GSD) Though the nodal approach employedcontains the full PSD conveying that all of the informationis not trivial [45 46] the information conveyed by the firsttwo moments can be quite useful The mean diameter usedhere is the volume-equivalent mean particle diameter and isgiven by 119889
119901= (6V119901120587)13 where the mean volume is given by
V119901=
sum119873119904
119894=1119876119894V119894
sum119873119904
119894=1119876119894
(10)
The GSD represents the width of the PSD and is given by
[log (120590119892)]
2
=
sum119873119904
119896=1119876119896(log (119889
119901119896) minus log ( 119889
119901119896))
2
sum119873119904
119896=1119876119896
(11)
where 119889119901119896is a number mean diameter given by
log ( 119889119901119896) =
sum119873119904
119896=1119876119896log (119889
119901119896)
sum119873119904
119896=1119876119896
(12)
Larger GSD values indicate that the size distribution isrelatively broad while smaller GSD values (120590
119892= 1) indi-
cate a relatively narrow distribution [47] The GSD for the
6 Journal of Nanotechnology
3nm
3nm
2nm
2nm
1nm
1nm
Figure 4 Instantaneous isosurfaces of 1 nm 2 nm and 3 nmparticles at 119905⋆ = 100
2
19
18
17
16
15
14
13
12
11
1
Figure 5 An instantaneous isosurface of vorticity colored by thegeometric standard deviation 120590
119892
hydrolysis of TiCl4in a planar jet is shown on an isosurface of
vorticity magnitude at time 119905⋆ = 100 in Figure 5 The figurereveals that the GSD generally increases as the jet travelsdownstream This implies that mixing (due to turbulence)is a significant contribution to particle polydispersity Ifnucleation were absent there would be no monomers andthe coagulating particles would achieve the so-called self-preserving value (120590
119892= 15) [48 49] Locations where 120590
119892gt
15 in Figure 5 reflect where nucleation condensation andcoagulation simultaneously occur That is ldquonewlyrdquo formedparticles by nucleationmixing and old particles generated bycoagulation exist
More insight into the particle field may be obtained byconsidering the relationship between the mean diameter andthe GSD A scatter plot of 120590
119892versus 119889
119901is shown in Figure 6
1 2 3 41
12
14
16
18
2
120590
dp (nm)
Figure 6 Scatter plot of the geometric standard deviation120590119892 versus
the mean particle diameter 119889119901
The figure shows that at the two ends of the size distribution119889119901= 1 nm and 119889
119901= 4 nm the distribution is fairly narrow
or unimodal while the GSD is largest (18 lt 120590119892lt 2) when
the mean diameter is near 119889119901= 32 nm This relatively large
GSD indicates that there are a variety of particle dynamicspresent in regions of the flow where the mean diameter is119889119901= 32 nm
37 Particle Growth Theparticle growth-rate is an importantparameter to consider as in combination with residence timeor reactor size it is a predictor of particle size A diameter-based growth-rate Ωnm (with units of nms) is defined basedon
Ωnm = (6Ω
120587
)
13
times 109 (13)
whereΩ is the particle volumetric growth-rate given by
Ω =
sum119873119904
119896(V119896)2
120596119896
sum119873119904
119896119876119896V119896
minus
sum119873119904
119896(V119896)2
119876119896
(sum119873119904
119896119876119896V119896)
2
119873119904
sum
119896
V119896120596119896 (14)
A contour plot of the 119911-averaged growth-rate ⟨Ωnm⟩119911 isshown in Figure 7 The figure shows that within the first 6diameters 0 le 119909119863 le 6 growth is confined to roughly⟨Ωnm⟩119911 = 10 nms at the interface of the jet and thecoflowing stream where the hydrolysis reaction producesTiO2vapor Near 119909119863 = 10 when the shear layers merge
the particles begin to grow faster with the rate approaching⟨Ωnm⟩119911 = 25 nms This occurs after the potential corecollapses Further downstream the growth-rate increases toroughly ⟨Ωnm⟩119911 = 50 nms and this value is maintained for119909119863 gt 14 (It should be noted that this is the average growth-rate and at other points in the domain the values ofΩnm maybe smaller or larger)
The particles grow via two mechanisms condensation(collisions between monomers and particles) and
Journal of Nanotechnology 7
0 53
Figure 7 Instantaneous contours of the 119911-direction averaged particle growth-rate ⟨Ωnm⟩119911 at 119905⋆= 100
0 28(a)
0 33(b)
Figure 8 Instantaneous contours of the nanoparticle growth-rate decomposed by mechanism 119911-direction averaged particle growth-rate (a)by condensation (b) by coagulation
coagulation (collisions between particles) Becausethe particle data is available as a function of size thecontribution of each mechanism is readily availableParticle growth by mechanism is shown in Figure 8The interactions between monomers and particles orthe condensation growth-rate is shown in Figure 8(a)(This image looks similar to the monomer numberconcentration shown in Figure 3(a) because thecondensation is represented by the collision betweenmonomers and particles) The contours show that largecondensation growth occurs both in the proximal regionof the jet and after collapse of the jet core This reflects theongoing hydrolysis of TiCl
4and production of TiO
2and its
deposition on existing particlesParticle growth by coagulation is shown in Figure 8(b)
The contours of the 119911-averaged growth-rate shows thatgrowth by coagulation begins after collapse of the jet potentialcore The growth-rate in the region 4 lt 119909119863 lt 8 is ashigh as Ωnm = 16 nms Farther downstream the 119911-averaged
coagulation growth-rate doubles This region of the flow isdominated by mixing and small-scale turbulence (evident inFigure 1) That region of the flow also contains particles of avariety of sizes small and large as reflected by 120590
119892in Figure 5
The small-scale turbulence means dissipation and increasedresidence times while the disparate particle sizes mean anefficient collision efficiency These two combine to increasecoagulation
The spatial relationship between condensation growthand coagulation growth is elucidated by showing the con-tribution of each at every grid point in the computationaldomain A scatter plot of the two growth-rates is shown inFigure 9The growth-rates are not averaged in the 119911-directionand while a spatial relationship is not directly evident fromthis figure one may be reliably inferred along with the pre-vious data Figure 9 shows that where condensation growthis low coagulation growth is high This trend is evidentin Figure 8 as well However Figure 9 shows that in theseregions the coagulation growth-rate can be as much as an
8 Journal of Nanotechnology
0 10 20 30 40 500
10
20
30
40
50
Ωnm
coagulation(nms)
Ωnmcondensation (nms)
Figure 9 Scatter plot of the particle growth-rate by condensationversus the particle growth-rate by coagulation
order of magnitude greater than the condensation growth-rate
4 Summary and Conclusions
The growth mechanisms of titania during hydrolysis oftitanium tetrachloride in a three-dimensional planar jet arestudied via direct numerical simulation The mass momen-tum enthalpy and species transport equations are solved in amodel-free manner Titania was produced via the hydrolysisof titanium tetrachloride modeled via a 1-step infinitelyfast chemical mechanism The particle field was representedusing a nodal method and solves for the evolution of theconcentration of particles of various sizes in an Eulerianmanner When coupled to the Navier-Stokes solver thefluid thermal chemical and particle fields are obtained as afunction of space and time
The results show that fluid turbulencegas mixing plays avery important role in particle growth The results indicatethat the particle formation and growth are greatly affectedif not dominated by mixing and chemical reaction Reactantconversion or titania production is limited by the ability ofthe flow to bring the TiCl
4and H
2O into contact Evidence
for this is the particle formation throughout the domainAdditionally the results show that where the turbulenceis more intense the particles are larger and the particlesize distribution as characterized by the geometric standarddeviation is wider Particle growth in the proximal regionof the jet is dominated by condensation After collapseof the jet core nanoparticle growth due to coagulationincreases significantly Though TiO
2is produced after core
collapse producing condensible species particle growth dueto coagulation is as much as an order of magnitude greaterthan that due to condensation While both condensationand coagulation act to increase polydispersity coagulationhas been shown to increase the width of the particle sizedistribution faster or greater than condensation [50]
These results help to shed light on and improve ourunderstanding of the underlying growth dynamics occur-ring in nanoparticle synthesis processes The change fromcondensation-dominated to coagulation-dominated growthis useful in modeling the complete synthesis process includ-ing sintering and the formation of hard and soft agglomer-ates Spanning the size range from single molecules (particleinception) to hundreds of nanometers as the particles foundin many industrial processes and applications is computeintensive [18] However knowing that condensation andmolecular growth is relatively minor could facilitate the useof more affordable modeling strategies that do not needto account for every phenomena [6 51] Additionally ahighly desired attribute of metal oxide nanoparticles is theirspecific surface area The ability to synthesize narrow sizedistributions which are desired is advantageous in that itremoves the processing necessary to separate particles by sizeThis is advantageous in that it lowers cost [52ndash55]
One strategy to reduce polydispersity may be to delay thetransition to turbulence vis a vis delaying collapse of the jetcore While the particle field is known as a function of size itshould be noted that in this work we are unable to distinguishthe various phases of titania rutile anatase or brookite [5657]The phase composition of titania is very much a functionof the synthesis process for example precursor compositionand temperature history which are greatly simplified in thiswork Such capability requires phenomenologicalmodels thataccount for particle dynamics as a function of space time andtemperature as well as solid-state diffusion processes and isquite beyond the scope of this work
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Funding for this research was provided by the Universityof Minnesota Computational resources were provided byMinnesota Supercomputing Institute
References
[1] S E Pratsinis and S Vemury ldquoParticle formation in gases areviewrdquo Powder Technology vol 88 no 3 pp 267ndash273 1996
[2] G Beaucage H Kammler P Mueller et al ldquoProbing thedynamics of nanoparticle growth in a flame using synchrotronradiationrdquo Nature Materials vol 3 no 6 pp 370ndash373 2004
[3] S Panda and S E Pratsinis ldquoModeling the synthesis ofaluminum particles by evaporation-condensation in an aerosolflow reactorrdquo Nanostructured Materials vol 5 no 7-8 pp 755ndash767 1995
[4] S Yuu K Ikeda and T Umekage ldquoFlow-field prediction andexperimental verification of low Reynolds number gas-particleturbulent jetsrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 109 pp 13ndash27 1996
[5] S Tsantilis H K Kammler and S E Pratsinis ldquoPopulationbalance modeling of flame synthesis of titania nanoparticlesrdquo
Journal of Nanotechnology 9
Chemical Engineering Science vol 57 no 12 pp 2139ndash21562002
[6] N Settumba and S C Garrick ldquoDirect numerical simulationof nanoparticle coagulation in a temporal mixing layer via amoment methodrdquo Journal of Aerosol Science vol 34 no 2 pp149ndash167 2003
[7] D L Marchisio and R O Fox ldquoSolution of population balanceequations using the direct quadrature mehtod of momentsrdquoJournal of Aerosol Science vol 36 pp 43ndash73 2005
[8] F Aristizabal R J Munz and D Berk ldquoModeling of the pro-duction of ultra fine Aluminium particles in rapid quenchingturbulent flowrdquo Journal of Aerosol Science vol 37 no 2 pp 162ndash186 2006
[9] S Rigopoulos ldquoPDF method for population balance in turbu-lent reactive flowrdquo Chemical Engineering Science vol 62 no 23pp 6865ndash6878 2007
[10] K Zhou A Attili A Alshaarawi and F Bisetti ldquoSimulationof aerosol nucleation and growth in a turbulent mixing layerrdquoPhysics of Fluids vol 26 no 6 Article ID 065106 2014
[11] S A Orszag and I Staroselsky ldquoCFD progress and problemsrdquoComputer Physics Communications vol 127 no 1 pp 165ndash1712000
[12] K Nakaso T Fujimoto T Seto M Shimada K Okuyamaand M M Lunden ldquoSize distribution change of titania nano-particle agglomerates generated by gas phase reaction agglom-eration and sinteringrdquo Aerosol Science and Technology vol 35no 5 pp 929ndash947 2001
[13] T Johannessen S E Pratsinis andH Livbjerg ldquoComputationalanalysis of coagulation and coalescence in the flame synthesis oftitania particlesrdquoPowder Technology vol 118 no 3 pp 242ndash2502001
[14] E G Moody and L R Collins ldquoEffect of mixing on thenucleation and growth of titania particlesrdquo Aerosol Science andTechnology vol 37 no 5 pp 403ndash424 2003
[15] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[16] S Das and S C Garrick ldquoThe effects of turbulence onnanoparticle growth in turbulent reacting jetsrdquo Physics of Fluidsvol 22 no 10 Article ID 103303 2010
[17] S C Garrick ldquoEffects of turbulent fluctuations on nanoparticlecoagulation in shear flowsrdquo Aerosol Science and Technology vol45 no 10 pp 1272ndash1285 2011
[18] A J Fager J Liu and S C Garrick ldquoHybrid simulations ofmetal particle nucleation a priori and a posteriori analyses ofthe effects of unresolved scalar interactions on nanoparticlenucleationrdquo Physics of Fluids vol 24 no 7 Article ID 0751102012
[19] N J Murfield and S C Garrick ldquoLarge eddy simulation anddirect numerical simulation of homogeneous nucleation inturbulent wakesrdquo Journal of Aerosol Science vol 60 pp 21ndash332013
[20] N JMurfield and S CGarrick ldquoThe effects of unresolved scalarfluctuations during homogeneous nucleationrdquo Aerosol Scienceand Technology vol 47 no 7 pp 806ndash817 2013
[21] P Givi ldquoModel free simulations of turbulent reactive flowsrdquoProgress in Energy and Combustion Science vol 15 no 1 pp 1ndash107 1989
[22] K E J Lehtinen and M R Zachariah ldquoSelf-preserving theoryfor the volume distribution of particles undergoing browniancoagulationrdquo Journal of Colloid and Interface Science vol 242no 2 pp 314ndash318 2001
[23] S C Garrick K E J Lehtinen andM R Zachariah ldquoNanopar-ticle coagulation via a Navier-Stokesnodal methodology evo-lution of the particle fieldrdquo Journal of Aerosol Science vol 37 no5 pp 555ndash576 2006
[24] F Gelbard and J H Seinfeld ldquoSimulation of multicomponentaerosol dynamicsrdquo Journal of Colloid And Interface Science vol78 no 2 pp 485ndash501 1980
[25] P Biswas C Y Wu M R Zachariah and B McMillin ldquoChar-acterization of iron oxide-silica nanocomposites in flamespart II comparison of discrete-sectional model predictions toexperimental datardquo Journal of Materials Research vol 12 no 3pp 714ndash723 1997
[26] K E J Lehtinen and M R Zachariah ldquoEnergy accumulationin nanoparticle collision and coalescence processesrdquo Journal ofAerosol Science vol 33 no 2 pp 357ndash368 2002
[27] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[28] J Loeffler S Das and S C Garrick ldquoLarge eddy simulationof titanium dioxide nanoparticle formation and growth inturbulent jetsrdquoAerosol Science and Technology vol 45 no 5 pp616ndash628 2011
[29] K S Friedlander Smoke Dust and Haze Fundamentals ofAerosol Dynamics Oxford University Press New York NYUSA 2000
[30] M Frenklach and S J Harris ldquoAerosol dynamics modelingusing the method of momentsrdquo Journal of Colloid and InterfaceScience vol 118 no 1 pp 252ndash261 1987
[31] S E Pratsinis ldquoParticle production by gas-to-particle conver-sion in turbulent flowsrdquo Journal of Aerosol Science vol 20 no8 pp 1461ndash1464 1989
[32] J D Landgrebe and S E Pratsinis ldquoA discrete-sectional modelfor particulate production by gas-phase chemical reaction andaerosol coagulation in the free-molecular regimerdquo Journal ofColloid and Interface Science vol 139 no 1 pp 63ndash86 1990
[33] GWang and S C Garrick ldquoModeling and simulation of titaniasynthesis in two-dimensional methane-air flamesrdquo Journal ofNanoparticle Research vol 7 no 6 pp 621ndash632 2005
[34] G W Mulholland R J Samson R D Mountain and M HErnst ldquoCluster size distribution for free molecular agglomera-tionrdquo Energy amp Fuels vol 2 no 4 pp 481ndash486 1988
[35] J Cai N Lu and C M Sorensen ldquoAnalysis of fractal clus-ter morphology parameters structural coefficient and densityautocorrelation function cutoffrdquo Journal of ColloidAnd InterfaceScience vol 171 no 2 pp 470ndash473 1995
[36] R Jullien and PMeakin ldquoSimplemodels for the restructuring ofthree-dimensional ballistic aggregatesrdquo Journal of Colloid AndInterface Science vol 127 no 1 pp 265ndash272 1989
[37] S N Rogak and R C Flagan ldquoCoagulation of aerosol agglom-erates in the transition regimerdquo Journal of Colloid and InterfaceScience vol 151 no 1 pp 203ndash224 1992
[38] S Modem S C Garrick M R Zachariah and K E J LehtinenldquoDirect numerical simulation of nanoparticle coagulation in atemporal mixing layerrdquo in Proceedings of the 29th Symposium(International) on Combustion pp 1071ndash1077 The CombustionInstitute Pittsburgh Pa USA 2002
[39] S C Garrick and G Wang ldquoModeling and simulation of tita-nium dioxide nanoparticle synthesis with finite-rate sinteringin planar jetsrdquo Journal of Nanoparticle Research vol 13 no 3pp 973ndash984 2011
10 Journal of Nanotechnology
[40] M C Heine and S E Pratsinis ldquoPolydispersity of primaryparticles in agglomerates made by coagulation and sinteringrdquoJournal of Aerosol Science vol 38 no 1 pp 17ndash38 2007
[41] R W MacCormack ldquoThe effect of viscosity in hypervelocityimpact cateringrdquo AIAA Paper 69-354 1969
[42] M H Carpenter ldquoA high-order compact numerical algorithmfor supersonic flowsrdquo in Twelfth International Conference onNumerical Methods in Fluid Dynamics K W Morton Ed vol371 of Lecture Notes in Physics pp 254ndash258 Springer BerlinGermany 1990
[43] D H Rudy and J C Strikwerda ldquoBoundary conditions forsubsonic compressible navier-stokes calculationsrdquo Computersand Fluids vol 9 no 3 pp 327ndash338 1981
[44] P Givi ldquoFiltered density function for subgrid scale modeling ofturbulent combustionrdquo AIAA Journal vol 44 no 1 pp 16ndash232006
[45] S Modem and S C Garrick ldquoNanoparticle coagulation in atemporal mixing layer mean and size-selected imagesrdquo Journalof Visualization vol 6 no 3 pp 293ndash302 2003
[46] D L Wright S Yu P S Kasibhatla et al ldquoRetrieval of aerosolproperties from moments of the particle size distribution forkernels involving the step function cloud droplet activationrdquoJournal of Aerosol Science vol 33 no 2 pp 319ndash337 2002
[47] H K Kammler R Jossen PWMorrison Jr S E Pratsinis andG Beaucage ldquoThe effect of external electric fields during flamesynthesis of titaniardquo Powder Technology vol 135-136 pp 310ndash320 2003
[48] W C Hinds Aerosol Technology Properties Behavior andMeasurement of Air-Borne Particles John Wiley amp Sons NewYork NY USA 2nd edition 1999
[49] N Settumba and S C Garrick ldquoA comparison of diffusivetransport in a moment method for nanoparticle coagulationrdquoJournal of Aerosol Science vol 35 no 1 pp 93ndash101 2004
[50] S E Pratsinis ldquoSimultaneous nucleation condensation andcoagulation in aerosol reactorsrdquo Journal of Colloid And InterfaceScience vol 124 no 2 pp 416ndash427 1988
[51] RMcGraw ldquoDescription of aerosol dynamics by the quadraturemethod of momentsrdquo Aerosol Science and Technology vol 27no 2 pp 255ndash265 1997
[52] J Bai Y-H Xu and J-P Wang ldquoCubic and spherical high-moment FeCo nanoparticles with narrow size distributionrdquoIEEE Transactions on Magnetics vol 43 no 7 pp 3340ndash33422007
[53] A Khorsand Zak R Razali W H Abd Majid and MDarroudi ldquoSynthesis and characterization of a narrow sizedistribution of zinc oxide nanoparticlesrdquo International Journalof Nanomedicine vol 6 no 1 pp 1399ndash1403 2011
[54] M Asemi and M Ghanaatshoar ldquoPreparation of CuCrO2
nanoparticles with narrow size distribution by sol-gel methodrdquoJournal of Sol-Gel Science and Technology vol 70 no 3 pp 416ndash421 2014
[55] A M Ahadi O Polonskyi U Schurmann T Strunskus and FFaupel ldquoStable production of TiOx nanoparticles with narrowsize distribution by reactive pulsed dc magnetron sputteringrdquoJournal of Physics D Applied Physics vol 48 no 3 Article ID035501 2015
[56] W W So S B Park K J Kim C J Shin and S J Moon ldquoThecrystalline phase stability of titania particles prepared at roomtemperature by the sol-gelmethodrdquo Journal ofMaterials Sciencevol 36 no 17 pp 4299ndash4305 2001
[57] A Teleki R Wengeler L Wengeler H Nirschl and S EPratsinis ldquoDistinguishing between aggregates and agglomeratesof flame-made TiO
Figure 4 Instantaneous isosurfaces of 1 nm 2 nm and 3 nmparticles at 119905⋆ = 100
2
19
18
17
16
15
14
13
12
11
1
Figure 5 An instantaneous isosurface of vorticity colored by thegeometric standard deviation 120590
119892
hydrolysis of TiCl4in a planar jet is shown on an isosurface of
vorticity magnitude at time 119905⋆ = 100 in Figure 5 The figurereveals that the GSD generally increases as the jet travelsdownstream This implies that mixing (due to turbulence)is a significant contribution to particle polydispersity Ifnucleation were absent there would be no monomers andthe coagulating particles would achieve the so-called self-preserving value (120590
119892= 15) [48 49] Locations where 120590
119892gt
15 in Figure 5 reflect where nucleation condensation andcoagulation simultaneously occur That is ldquonewlyrdquo formedparticles by nucleationmixing and old particles generated bycoagulation exist
More insight into the particle field may be obtained byconsidering the relationship between the mean diameter andthe GSD A scatter plot of 120590
119892versus 119889
119901is shown in Figure 6
1 2 3 41
12
14
16
18
2
120590
dp (nm)
Figure 6 Scatter plot of the geometric standard deviation120590119892 versus
the mean particle diameter 119889119901
The figure shows that at the two ends of the size distribution119889119901= 1 nm and 119889
119901= 4 nm the distribution is fairly narrow
or unimodal while the GSD is largest (18 lt 120590119892lt 2) when
the mean diameter is near 119889119901= 32 nm This relatively large
GSD indicates that there are a variety of particle dynamicspresent in regions of the flow where the mean diameter is119889119901= 32 nm
37 Particle Growth Theparticle growth-rate is an importantparameter to consider as in combination with residence timeor reactor size it is a predictor of particle size A diameter-based growth-rate Ωnm (with units of nms) is defined basedon
Ωnm = (6Ω
120587
)
13
times 109 (13)
whereΩ is the particle volumetric growth-rate given by
Ω =
sum119873119904
119896(V119896)2
120596119896
sum119873119904
119896119876119896V119896
minus
sum119873119904
119896(V119896)2
119876119896
(sum119873119904
119896119876119896V119896)
2
119873119904
sum
119896
V119896120596119896 (14)
A contour plot of the 119911-averaged growth-rate ⟨Ωnm⟩119911 isshown in Figure 7 The figure shows that within the first 6diameters 0 le 119909119863 le 6 growth is confined to roughly⟨Ωnm⟩119911 = 10 nms at the interface of the jet and thecoflowing stream where the hydrolysis reaction producesTiO2vapor Near 119909119863 = 10 when the shear layers merge
the particles begin to grow faster with the rate approaching⟨Ωnm⟩119911 = 25 nms This occurs after the potential corecollapses Further downstream the growth-rate increases toroughly ⟨Ωnm⟩119911 = 50 nms and this value is maintained for119909119863 gt 14 (It should be noted that this is the average growth-rate and at other points in the domain the values ofΩnm maybe smaller or larger)
The particles grow via two mechanisms condensation(collisions between monomers and particles) and
Journal of Nanotechnology 7
0 53
Figure 7 Instantaneous contours of the 119911-direction averaged particle growth-rate ⟨Ωnm⟩119911 at 119905⋆= 100
0 28(a)
0 33(b)
Figure 8 Instantaneous contours of the nanoparticle growth-rate decomposed by mechanism 119911-direction averaged particle growth-rate (a)by condensation (b) by coagulation
coagulation (collisions between particles) Becausethe particle data is available as a function of size thecontribution of each mechanism is readily availableParticle growth by mechanism is shown in Figure 8The interactions between monomers and particles orthe condensation growth-rate is shown in Figure 8(a)(This image looks similar to the monomer numberconcentration shown in Figure 3(a) because thecondensation is represented by the collision betweenmonomers and particles) The contours show that largecondensation growth occurs both in the proximal regionof the jet and after collapse of the jet core This reflects theongoing hydrolysis of TiCl
4and production of TiO
2and its
deposition on existing particlesParticle growth by coagulation is shown in Figure 8(b)
The contours of the 119911-averaged growth-rate shows thatgrowth by coagulation begins after collapse of the jet potentialcore The growth-rate in the region 4 lt 119909119863 lt 8 is ashigh as Ωnm = 16 nms Farther downstream the 119911-averaged
coagulation growth-rate doubles This region of the flow isdominated by mixing and small-scale turbulence (evident inFigure 1) That region of the flow also contains particles of avariety of sizes small and large as reflected by 120590
119892in Figure 5
The small-scale turbulence means dissipation and increasedresidence times while the disparate particle sizes mean anefficient collision efficiency These two combine to increasecoagulation
The spatial relationship between condensation growthand coagulation growth is elucidated by showing the con-tribution of each at every grid point in the computationaldomain A scatter plot of the two growth-rates is shown inFigure 9The growth-rates are not averaged in the 119911-directionand while a spatial relationship is not directly evident fromthis figure one may be reliably inferred along with the pre-vious data Figure 9 shows that where condensation growthis low coagulation growth is high This trend is evidentin Figure 8 as well However Figure 9 shows that in theseregions the coagulation growth-rate can be as much as an
8 Journal of Nanotechnology
0 10 20 30 40 500
10
20
30
40
50
Ωnm
coagulation(nms)
Ωnmcondensation (nms)
Figure 9 Scatter plot of the particle growth-rate by condensationversus the particle growth-rate by coagulation
order of magnitude greater than the condensation growth-rate
4 Summary and Conclusions
The growth mechanisms of titania during hydrolysis oftitanium tetrachloride in a three-dimensional planar jet arestudied via direct numerical simulation The mass momen-tum enthalpy and species transport equations are solved in amodel-free manner Titania was produced via the hydrolysisof titanium tetrachloride modeled via a 1-step infinitelyfast chemical mechanism The particle field was representedusing a nodal method and solves for the evolution of theconcentration of particles of various sizes in an Eulerianmanner When coupled to the Navier-Stokes solver thefluid thermal chemical and particle fields are obtained as afunction of space and time
The results show that fluid turbulencegas mixing plays avery important role in particle growth The results indicatethat the particle formation and growth are greatly affectedif not dominated by mixing and chemical reaction Reactantconversion or titania production is limited by the ability ofthe flow to bring the TiCl
4and H
2O into contact Evidence
for this is the particle formation throughout the domainAdditionally the results show that where the turbulenceis more intense the particles are larger and the particlesize distribution as characterized by the geometric standarddeviation is wider Particle growth in the proximal regionof the jet is dominated by condensation After collapseof the jet core nanoparticle growth due to coagulationincreases significantly Though TiO
2is produced after core
collapse producing condensible species particle growth dueto coagulation is as much as an order of magnitude greaterthan that due to condensation While both condensationand coagulation act to increase polydispersity coagulationhas been shown to increase the width of the particle sizedistribution faster or greater than condensation [50]
These results help to shed light on and improve ourunderstanding of the underlying growth dynamics occur-ring in nanoparticle synthesis processes The change fromcondensation-dominated to coagulation-dominated growthis useful in modeling the complete synthesis process includ-ing sintering and the formation of hard and soft agglomer-ates Spanning the size range from single molecules (particleinception) to hundreds of nanometers as the particles foundin many industrial processes and applications is computeintensive [18] However knowing that condensation andmolecular growth is relatively minor could facilitate the useof more affordable modeling strategies that do not needto account for every phenomena [6 51] Additionally ahighly desired attribute of metal oxide nanoparticles is theirspecific surface area The ability to synthesize narrow sizedistributions which are desired is advantageous in that itremoves the processing necessary to separate particles by sizeThis is advantageous in that it lowers cost [52ndash55]
One strategy to reduce polydispersity may be to delay thetransition to turbulence vis a vis delaying collapse of the jetcore While the particle field is known as a function of size itshould be noted that in this work we are unable to distinguishthe various phases of titania rutile anatase or brookite [5657]The phase composition of titania is very much a functionof the synthesis process for example precursor compositionand temperature history which are greatly simplified in thiswork Such capability requires phenomenologicalmodels thataccount for particle dynamics as a function of space time andtemperature as well as solid-state diffusion processes and isquite beyond the scope of this work
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Funding for this research was provided by the Universityof Minnesota Computational resources were provided byMinnesota Supercomputing Institute
References
[1] S E Pratsinis and S Vemury ldquoParticle formation in gases areviewrdquo Powder Technology vol 88 no 3 pp 267ndash273 1996
[2] G Beaucage H Kammler P Mueller et al ldquoProbing thedynamics of nanoparticle growth in a flame using synchrotronradiationrdquo Nature Materials vol 3 no 6 pp 370ndash373 2004
[3] S Panda and S E Pratsinis ldquoModeling the synthesis ofaluminum particles by evaporation-condensation in an aerosolflow reactorrdquo Nanostructured Materials vol 5 no 7-8 pp 755ndash767 1995
[4] S Yuu K Ikeda and T Umekage ldquoFlow-field prediction andexperimental verification of low Reynolds number gas-particleturbulent jetsrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 109 pp 13ndash27 1996
[5] S Tsantilis H K Kammler and S E Pratsinis ldquoPopulationbalance modeling of flame synthesis of titania nanoparticlesrdquo
Journal of Nanotechnology 9
Chemical Engineering Science vol 57 no 12 pp 2139ndash21562002
[6] N Settumba and S C Garrick ldquoDirect numerical simulationof nanoparticle coagulation in a temporal mixing layer via amoment methodrdquo Journal of Aerosol Science vol 34 no 2 pp149ndash167 2003
[7] D L Marchisio and R O Fox ldquoSolution of population balanceequations using the direct quadrature mehtod of momentsrdquoJournal of Aerosol Science vol 36 pp 43ndash73 2005
[8] F Aristizabal R J Munz and D Berk ldquoModeling of the pro-duction of ultra fine Aluminium particles in rapid quenchingturbulent flowrdquo Journal of Aerosol Science vol 37 no 2 pp 162ndash186 2006
[9] S Rigopoulos ldquoPDF method for population balance in turbu-lent reactive flowrdquo Chemical Engineering Science vol 62 no 23pp 6865ndash6878 2007
[10] K Zhou A Attili A Alshaarawi and F Bisetti ldquoSimulationof aerosol nucleation and growth in a turbulent mixing layerrdquoPhysics of Fluids vol 26 no 6 Article ID 065106 2014
[11] S A Orszag and I Staroselsky ldquoCFD progress and problemsrdquoComputer Physics Communications vol 127 no 1 pp 165ndash1712000
[12] K Nakaso T Fujimoto T Seto M Shimada K Okuyamaand M M Lunden ldquoSize distribution change of titania nano-particle agglomerates generated by gas phase reaction agglom-eration and sinteringrdquo Aerosol Science and Technology vol 35no 5 pp 929ndash947 2001
[13] T Johannessen S E Pratsinis andH Livbjerg ldquoComputationalanalysis of coagulation and coalescence in the flame synthesis oftitania particlesrdquoPowder Technology vol 118 no 3 pp 242ndash2502001
[14] E G Moody and L R Collins ldquoEffect of mixing on thenucleation and growth of titania particlesrdquo Aerosol Science andTechnology vol 37 no 5 pp 403ndash424 2003
[15] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[16] S Das and S C Garrick ldquoThe effects of turbulence onnanoparticle growth in turbulent reacting jetsrdquo Physics of Fluidsvol 22 no 10 Article ID 103303 2010
[17] S C Garrick ldquoEffects of turbulent fluctuations on nanoparticlecoagulation in shear flowsrdquo Aerosol Science and Technology vol45 no 10 pp 1272ndash1285 2011
[18] A J Fager J Liu and S C Garrick ldquoHybrid simulations ofmetal particle nucleation a priori and a posteriori analyses ofthe effects of unresolved scalar interactions on nanoparticlenucleationrdquo Physics of Fluids vol 24 no 7 Article ID 0751102012
[19] N J Murfield and S C Garrick ldquoLarge eddy simulation anddirect numerical simulation of homogeneous nucleation inturbulent wakesrdquo Journal of Aerosol Science vol 60 pp 21ndash332013
[20] N JMurfield and S CGarrick ldquoThe effects of unresolved scalarfluctuations during homogeneous nucleationrdquo Aerosol Scienceand Technology vol 47 no 7 pp 806ndash817 2013
[21] P Givi ldquoModel free simulations of turbulent reactive flowsrdquoProgress in Energy and Combustion Science vol 15 no 1 pp 1ndash107 1989
[22] K E J Lehtinen and M R Zachariah ldquoSelf-preserving theoryfor the volume distribution of particles undergoing browniancoagulationrdquo Journal of Colloid and Interface Science vol 242no 2 pp 314ndash318 2001
[23] S C Garrick K E J Lehtinen andM R Zachariah ldquoNanopar-ticle coagulation via a Navier-Stokesnodal methodology evo-lution of the particle fieldrdquo Journal of Aerosol Science vol 37 no5 pp 555ndash576 2006
[24] F Gelbard and J H Seinfeld ldquoSimulation of multicomponentaerosol dynamicsrdquo Journal of Colloid And Interface Science vol78 no 2 pp 485ndash501 1980
[25] P Biswas C Y Wu M R Zachariah and B McMillin ldquoChar-acterization of iron oxide-silica nanocomposites in flamespart II comparison of discrete-sectional model predictions toexperimental datardquo Journal of Materials Research vol 12 no 3pp 714ndash723 1997
[26] K E J Lehtinen and M R Zachariah ldquoEnergy accumulationin nanoparticle collision and coalescence processesrdquo Journal ofAerosol Science vol 33 no 2 pp 357ndash368 2002
[27] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[28] J Loeffler S Das and S C Garrick ldquoLarge eddy simulationof titanium dioxide nanoparticle formation and growth inturbulent jetsrdquoAerosol Science and Technology vol 45 no 5 pp616ndash628 2011
[29] K S Friedlander Smoke Dust and Haze Fundamentals ofAerosol Dynamics Oxford University Press New York NYUSA 2000
[30] M Frenklach and S J Harris ldquoAerosol dynamics modelingusing the method of momentsrdquo Journal of Colloid and InterfaceScience vol 118 no 1 pp 252ndash261 1987
[31] S E Pratsinis ldquoParticle production by gas-to-particle conver-sion in turbulent flowsrdquo Journal of Aerosol Science vol 20 no8 pp 1461ndash1464 1989
[32] J D Landgrebe and S E Pratsinis ldquoA discrete-sectional modelfor particulate production by gas-phase chemical reaction andaerosol coagulation in the free-molecular regimerdquo Journal ofColloid and Interface Science vol 139 no 1 pp 63ndash86 1990
[33] GWang and S C Garrick ldquoModeling and simulation of titaniasynthesis in two-dimensional methane-air flamesrdquo Journal ofNanoparticle Research vol 7 no 6 pp 621ndash632 2005
[34] G W Mulholland R J Samson R D Mountain and M HErnst ldquoCluster size distribution for free molecular agglomera-tionrdquo Energy amp Fuels vol 2 no 4 pp 481ndash486 1988
[35] J Cai N Lu and C M Sorensen ldquoAnalysis of fractal clus-ter morphology parameters structural coefficient and densityautocorrelation function cutoffrdquo Journal of ColloidAnd InterfaceScience vol 171 no 2 pp 470ndash473 1995
[36] R Jullien and PMeakin ldquoSimplemodels for the restructuring ofthree-dimensional ballistic aggregatesrdquo Journal of Colloid AndInterface Science vol 127 no 1 pp 265ndash272 1989
[37] S N Rogak and R C Flagan ldquoCoagulation of aerosol agglom-erates in the transition regimerdquo Journal of Colloid and InterfaceScience vol 151 no 1 pp 203ndash224 1992
[38] S Modem S C Garrick M R Zachariah and K E J LehtinenldquoDirect numerical simulation of nanoparticle coagulation in atemporal mixing layerrdquo in Proceedings of the 29th Symposium(International) on Combustion pp 1071ndash1077 The CombustionInstitute Pittsburgh Pa USA 2002
[39] S C Garrick and G Wang ldquoModeling and simulation of tita-nium dioxide nanoparticle synthesis with finite-rate sinteringin planar jetsrdquo Journal of Nanoparticle Research vol 13 no 3pp 973ndash984 2011
10 Journal of Nanotechnology
[40] M C Heine and S E Pratsinis ldquoPolydispersity of primaryparticles in agglomerates made by coagulation and sinteringrdquoJournal of Aerosol Science vol 38 no 1 pp 17ndash38 2007
[41] R W MacCormack ldquoThe effect of viscosity in hypervelocityimpact cateringrdquo AIAA Paper 69-354 1969
[42] M H Carpenter ldquoA high-order compact numerical algorithmfor supersonic flowsrdquo in Twelfth International Conference onNumerical Methods in Fluid Dynamics K W Morton Ed vol371 of Lecture Notes in Physics pp 254ndash258 Springer BerlinGermany 1990
[43] D H Rudy and J C Strikwerda ldquoBoundary conditions forsubsonic compressible navier-stokes calculationsrdquo Computersand Fluids vol 9 no 3 pp 327ndash338 1981
[44] P Givi ldquoFiltered density function for subgrid scale modeling ofturbulent combustionrdquo AIAA Journal vol 44 no 1 pp 16ndash232006
[45] S Modem and S C Garrick ldquoNanoparticle coagulation in atemporal mixing layer mean and size-selected imagesrdquo Journalof Visualization vol 6 no 3 pp 293ndash302 2003
[46] D L Wright S Yu P S Kasibhatla et al ldquoRetrieval of aerosolproperties from moments of the particle size distribution forkernels involving the step function cloud droplet activationrdquoJournal of Aerosol Science vol 33 no 2 pp 319ndash337 2002
[47] H K Kammler R Jossen PWMorrison Jr S E Pratsinis andG Beaucage ldquoThe effect of external electric fields during flamesynthesis of titaniardquo Powder Technology vol 135-136 pp 310ndash320 2003
[48] W C Hinds Aerosol Technology Properties Behavior andMeasurement of Air-Borne Particles John Wiley amp Sons NewYork NY USA 2nd edition 1999
[49] N Settumba and S C Garrick ldquoA comparison of diffusivetransport in a moment method for nanoparticle coagulationrdquoJournal of Aerosol Science vol 35 no 1 pp 93ndash101 2004
[50] S E Pratsinis ldquoSimultaneous nucleation condensation andcoagulation in aerosol reactorsrdquo Journal of Colloid And InterfaceScience vol 124 no 2 pp 416ndash427 1988
[51] RMcGraw ldquoDescription of aerosol dynamics by the quadraturemethod of momentsrdquo Aerosol Science and Technology vol 27no 2 pp 255ndash265 1997
[52] J Bai Y-H Xu and J-P Wang ldquoCubic and spherical high-moment FeCo nanoparticles with narrow size distributionrdquoIEEE Transactions on Magnetics vol 43 no 7 pp 3340ndash33422007
[53] A Khorsand Zak R Razali W H Abd Majid and MDarroudi ldquoSynthesis and characterization of a narrow sizedistribution of zinc oxide nanoparticlesrdquo International Journalof Nanomedicine vol 6 no 1 pp 1399ndash1403 2011
[54] M Asemi and M Ghanaatshoar ldquoPreparation of CuCrO2
nanoparticles with narrow size distribution by sol-gel methodrdquoJournal of Sol-Gel Science and Technology vol 70 no 3 pp 416ndash421 2014
[55] A M Ahadi O Polonskyi U Schurmann T Strunskus and FFaupel ldquoStable production of TiOx nanoparticles with narrowsize distribution by reactive pulsed dc magnetron sputteringrdquoJournal of Physics D Applied Physics vol 48 no 3 Article ID035501 2015
[56] W W So S B Park K J Kim C J Shin and S J Moon ldquoThecrystalline phase stability of titania particles prepared at roomtemperature by the sol-gelmethodrdquo Journal ofMaterials Sciencevol 36 no 17 pp 4299ndash4305 2001
[57] A Teleki R Wengeler L Wengeler H Nirschl and S EPratsinis ldquoDistinguishing between aggregates and agglomeratesof flame-made TiO
Figure 7 Instantaneous contours of the 119911-direction averaged particle growth-rate ⟨Ωnm⟩119911 at 119905⋆= 100
0 28(a)
0 33(b)
Figure 8 Instantaneous contours of the nanoparticle growth-rate decomposed by mechanism 119911-direction averaged particle growth-rate (a)by condensation (b) by coagulation
coagulation (collisions between particles) Becausethe particle data is available as a function of size thecontribution of each mechanism is readily availableParticle growth by mechanism is shown in Figure 8The interactions between monomers and particles orthe condensation growth-rate is shown in Figure 8(a)(This image looks similar to the monomer numberconcentration shown in Figure 3(a) because thecondensation is represented by the collision betweenmonomers and particles) The contours show that largecondensation growth occurs both in the proximal regionof the jet and after collapse of the jet core This reflects theongoing hydrolysis of TiCl
4and production of TiO
2and its
deposition on existing particlesParticle growth by coagulation is shown in Figure 8(b)
The contours of the 119911-averaged growth-rate shows thatgrowth by coagulation begins after collapse of the jet potentialcore The growth-rate in the region 4 lt 119909119863 lt 8 is ashigh as Ωnm = 16 nms Farther downstream the 119911-averaged
coagulation growth-rate doubles This region of the flow isdominated by mixing and small-scale turbulence (evident inFigure 1) That region of the flow also contains particles of avariety of sizes small and large as reflected by 120590
119892in Figure 5
The small-scale turbulence means dissipation and increasedresidence times while the disparate particle sizes mean anefficient collision efficiency These two combine to increasecoagulation
The spatial relationship between condensation growthand coagulation growth is elucidated by showing the con-tribution of each at every grid point in the computationaldomain A scatter plot of the two growth-rates is shown inFigure 9The growth-rates are not averaged in the 119911-directionand while a spatial relationship is not directly evident fromthis figure one may be reliably inferred along with the pre-vious data Figure 9 shows that where condensation growthis low coagulation growth is high This trend is evidentin Figure 8 as well However Figure 9 shows that in theseregions the coagulation growth-rate can be as much as an
8 Journal of Nanotechnology
0 10 20 30 40 500
10
20
30
40
50
Ωnm
coagulation(nms)
Ωnmcondensation (nms)
Figure 9 Scatter plot of the particle growth-rate by condensationversus the particle growth-rate by coagulation
order of magnitude greater than the condensation growth-rate
4 Summary and Conclusions
The growth mechanisms of titania during hydrolysis oftitanium tetrachloride in a three-dimensional planar jet arestudied via direct numerical simulation The mass momen-tum enthalpy and species transport equations are solved in amodel-free manner Titania was produced via the hydrolysisof titanium tetrachloride modeled via a 1-step infinitelyfast chemical mechanism The particle field was representedusing a nodal method and solves for the evolution of theconcentration of particles of various sizes in an Eulerianmanner When coupled to the Navier-Stokes solver thefluid thermal chemical and particle fields are obtained as afunction of space and time
The results show that fluid turbulencegas mixing plays avery important role in particle growth The results indicatethat the particle formation and growth are greatly affectedif not dominated by mixing and chemical reaction Reactantconversion or titania production is limited by the ability ofthe flow to bring the TiCl
4and H
2O into contact Evidence
for this is the particle formation throughout the domainAdditionally the results show that where the turbulenceis more intense the particles are larger and the particlesize distribution as characterized by the geometric standarddeviation is wider Particle growth in the proximal regionof the jet is dominated by condensation After collapseof the jet core nanoparticle growth due to coagulationincreases significantly Though TiO
2is produced after core
collapse producing condensible species particle growth dueto coagulation is as much as an order of magnitude greaterthan that due to condensation While both condensationand coagulation act to increase polydispersity coagulationhas been shown to increase the width of the particle sizedistribution faster or greater than condensation [50]
These results help to shed light on and improve ourunderstanding of the underlying growth dynamics occur-ring in nanoparticle synthesis processes The change fromcondensation-dominated to coagulation-dominated growthis useful in modeling the complete synthesis process includ-ing sintering and the formation of hard and soft agglomer-ates Spanning the size range from single molecules (particleinception) to hundreds of nanometers as the particles foundin many industrial processes and applications is computeintensive [18] However knowing that condensation andmolecular growth is relatively minor could facilitate the useof more affordable modeling strategies that do not needto account for every phenomena [6 51] Additionally ahighly desired attribute of metal oxide nanoparticles is theirspecific surface area The ability to synthesize narrow sizedistributions which are desired is advantageous in that itremoves the processing necessary to separate particles by sizeThis is advantageous in that it lowers cost [52ndash55]
One strategy to reduce polydispersity may be to delay thetransition to turbulence vis a vis delaying collapse of the jetcore While the particle field is known as a function of size itshould be noted that in this work we are unable to distinguishthe various phases of titania rutile anatase or brookite [5657]The phase composition of titania is very much a functionof the synthesis process for example precursor compositionand temperature history which are greatly simplified in thiswork Such capability requires phenomenologicalmodels thataccount for particle dynamics as a function of space time andtemperature as well as solid-state diffusion processes and isquite beyond the scope of this work
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Funding for this research was provided by the Universityof Minnesota Computational resources were provided byMinnesota Supercomputing Institute
References
[1] S E Pratsinis and S Vemury ldquoParticle formation in gases areviewrdquo Powder Technology vol 88 no 3 pp 267ndash273 1996
[2] G Beaucage H Kammler P Mueller et al ldquoProbing thedynamics of nanoparticle growth in a flame using synchrotronradiationrdquo Nature Materials vol 3 no 6 pp 370ndash373 2004
[3] S Panda and S E Pratsinis ldquoModeling the synthesis ofaluminum particles by evaporation-condensation in an aerosolflow reactorrdquo Nanostructured Materials vol 5 no 7-8 pp 755ndash767 1995
[4] S Yuu K Ikeda and T Umekage ldquoFlow-field prediction andexperimental verification of low Reynolds number gas-particleturbulent jetsrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 109 pp 13ndash27 1996
[5] S Tsantilis H K Kammler and S E Pratsinis ldquoPopulationbalance modeling of flame synthesis of titania nanoparticlesrdquo
Journal of Nanotechnology 9
Chemical Engineering Science vol 57 no 12 pp 2139ndash21562002
[6] N Settumba and S C Garrick ldquoDirect numerical simulationof nanoparticle coagulation in a temporal mixing layer via amoment methodrdquo Journal of Aerosol Science vol 34 no 2 pp149ndash167 2003
[7] D L Marchisio and R O Fox ldquoSolution of population balanceequations using the direct quadrature mehtod of momentsrdquoJournal of Aerosol Science vol 36 pp 43ndash73 2005
[8] F Aristizabal R J Munz and D Berk ldquoModeling of the pro-duction of ultra fine Aluminium particles in rapid quenchingturbulent flowrdquo Journal of Aerosol Science vol 37 no 2 pp 162ndash186 2006
[9] S Rigopoulos ldquoPDF method for population balance in turbu-lent reactive flowrdquo Chemical Engineering Science vol 62 no 23pp 6865ndash6878 2007
[10] K Zhou A Attili A Alshaarawi and F Bisetti ldquoSimulationof aerosol nucleation and growth in a turbulent mixing layerrdquoPhysics of Fluids vol 26 no 6 Article ID 065106 2014
[11] S A Orszag and I Staroselsky ldquoCFD progress and problemsrdquoComputer Physics Communications vol 127 no 1 pp 165ndash1712000
[12] K Nakaso T Fujimoto T Seto M Shimada K Okuyamaand M M Lunden ldquoSize distribution change of titania nano-particle agglomerates generated by gas phase reaction agglom-eration and sinteringrdquo Aerosol Science and Technology vol 35no 5 pp 929ndash947 2001
[13] T Johannessen S E Pratsinis andH Livbjerg ldquoComputationalanalysis of coagulation and coalescence in the flame synthesis oftitania particlesrdquoPowder Technology vol 118 no 3 pp 242ndash2502001
[14] E G Moody and L R Collins ldquoEffect of mixing on thenucleation and growth of titania particlesrdquo Aerosol Science andTechnology vol 37 no 5 pp 403ndash424 2003
[15] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[16] S Das and S C Garrick ldquoThe effects of turbulence onnanoparticle growth in turbulent reacting jetsrdquo Physics of Fluidsvol 22 no 10 Article ID 103303 2010
[17] S C Garrick ldquoEffects of turbulent fluctuations on nanoparticlecoagulation in shear flowsrdquo Aerosol Science and Technology vol45 no 10 pp 1272ndash1285 2011
[18] A J Fager J Liu and S C Garrick ldquoHybrid simulations ofmetal particle nucleation a priori and a posteriori analyses ofthe effects of unresolved scalar interactions on nanoparticlenucleationrdquo Physics of Fluids vol 24 no 7 Article ID 0751102012
[19] N J Murfield and S C Garrick ldquoLarge eddy simulation anddirect numerical simulation of homogeneous nucleation inturbulent wakesrdquo Journal of Aerosol Science vol 60 pp 21ndash332013
[20] N JMurfield and S CGarrick ldquoThe effects of unresolved scalarfluctuations during homogeneous nucleationrdquo Aerosol Scienceand Technology vol 47 no 7 pp 806ndash817 2013
[21] P Givi ldquoModel free simulations of turbulent reactive flowsrdquoProgress in Energy and Combustion Science vol 15 no 1 pp 1ndash107 1989
[22] K E J Lehtinen and M R Zachariah ldquoSelf-preserving theoryfor the volume distribution of particles undergoing browniancoagulationrdquo Journal of Colloid and Interface Science vol 242no 2 pp 314ndash318 2001
[23] S C Garrick K E J Lehtinen andM R Zachariah ldquoNanopar-ticle coagulation via a Navier-Stokesnodal methodology evo-lution of the particle fieldrdquo Journal of Aerosol Science vol 37 no5 pp 555ndash576 2006
[24] F Gelbard and J H Seinfeld ldquoSimulation of multicomponentaerosol dynamicsrdquo Journal of Colloid And Interface Science vol78 no 2 pp 485ndash501 1980
[25] P Biswas C Y Wu M R Zachariah and B McMillin ldquoChar-acterization of iron oxide-silica nanocomposites in flamespart II comparison of discrete-sectional model predictions toexperimental datardquo Journal of Materials Research vol 12 no 3pp 714ndash723 1997
[26] K E J Lehtinen and M R Zachariah ldquoEnergy accumulationin nanoparticle collision and coalescence processesrdquo Journal ofAerosol Science vol 33 no 2 pp 357ndash368 2002
[27] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[28] J Loeffler S Das and S C Garrick ldquoLarge eddy simulationof titanium dioxide nanoparticle formation and growth inturbulent jetsrdquoAerosol Science and Technology vol 45 no 5 pp616ndash628 2011
[29] K S Friedlander Smoke Dust and Haze Fundamentals ofAerosol Dynamics Oxford University Press New York NYUSA 2000
[30] M Frenklach and S J Harris ldquoAerosol dynamics modelingusing the method of momentsrdquo Journal of Colloid and InterfaceScience vol 118 no 1 pp 252ndash261 1987
[31] S E Pratsinis ldquoParticle production by gas-to-particle conver-sion in turbulent flowsrdquo Journal of Aerosol Science vol 20 no8 pp 1461ndash1464 1989
[32] J D Landgrebe and S E Pratsinis ldquoA discrete-sectional modelfor particulate production by gas-phase chemical reaction andaerosol coagulation in the free-molecular regimerdquo Journal ofColloid and Interface Science vol 139 no 1 pp 63ndash86 1990
[33] GWang and S C Garrick ldquoModeling and simulation of titaniasynthesis in two-dimensional methane-air flamesrdquo Journal ofNanoparticle Research vol 7 no 6 pp 621ndash632 2005
[34] G W Mulholland R J Samson R D Mountain and M HErnst ldquoCluster size distribution for free molecular agglomera-tionrdquo Energy amp Fuels vol 2 no 4 pp 481ndash486 1988
[35] J Cai N Lu and C M Sorensen ldquoAnalysis of fractal clus-ter morphology parameters structural coefficient and densityautocorrelation function cutoffrdquo Journal of ColloidAnd InterfaceScience vol 171 no 2 pp 470ndash473 1995
[36] R Jullien and PMeakin ldquoSimplemodels for the restructuring ofthree-dimensional ballistic aggregatesrdquo Journal of Colloid AndInterface Science vol 127 no 1 pp 265ndash272 1989
[37] S N Rogak and R C Flagan ldquoCoagulation of aerosol agglom-erates in the transition regimerdquo Journal of Colloid and InterfaceScience vol 151 no 1 pp 203ndash224 1992
[38] S Modem S C Garrick M R Zachariah and K E J LehtinenldquoDirect numerical simulation of nanoparticle coagulation in atemporal mixing layerrdquo in Proceedings of the 29th Symposium(International) on Combustion pp 1071ndash1077 The CombustionInstitute Pittsburgh Pa USA 2002
[39] S C Garrick and G Wang ldquoModeling and simulation of tita-nium dioxide nanoparticle synthesis with finite-rate sinteringin planar jetsrdquo Journal of Nanoparticle Research vol 13 no 3pp 973ndash984 2011
10 Journal of Nanotechnology
[40] M C Heine and S E Pratsinis ldquoPolydispersity of primaryparticles in agglomerates made by coagulation and sinteringrdquoJournal of Aerosol Science vol 38 no 1 pp 17ndash38 2007
[41] R W MacCormack ldquoThe effect of viscosity in hypervelocityimpact cateringrdquo AIAA Paper 69-354 1969
[42] M H Carpenter ldquoA high-order compact numerical algorithmfor supersonic flowsrdquo in Twelfth International Conference onNumerical Methods in Fluid Dynamics K W Morton Ed vol371 of Lecture Notes in Physics pp 254ndash258 Springer BerlinGermany 1990
[43] D H Rudy and J C Strikwerda ldquoBoundary conditions forsubsonic compressible navier-stokes calculationsrdquo Computersand Fluids vol 9 no 3 pp 327ndash338 1981
[44] P Givi ldquoFiltered density function for subgrid scale modeling ofturbulent combustionrdquo AIAA Journal vol 44 no 1 pp 16ndash232006
[45] S Modem and S C Garrick ldquoNanoparticle coagulation in atemporal mixing layer mean and size-selected imagesrdquo Journalof Visualization vol 6 no 3 pp 293ndash302 2003
[46] D L Wright S Yu P S Kasibhatla et al ldquoRetrieval of aerosolproperties from moments of the particle size distribution forkernels involving the step function cloud droplet activationrdquoJournal of Aerosol Science vol 33 no 2 pp 319ndash337 2002
[47] H K Kammler R Jossen PWMorrison Jr S E Pratsinis andG Beaucage ldquoThe effect of external electric fields during flamesynthesis of titaniardquo Powder Technology vol 135-136 pp 310ndash320 2003
[48] W C Hinds Aerosol Technology Properties Behavior andMeasurement of Air-Borne Particles John Wiley amp Sons NewYork NY USA 2nd edition 1999
[49] N Settumba and S C Garrick ldquoA comparison of diffusivetransport in a moment method for nanoparticle coagulationrdquoJournal of Aerosol Science vol 35 no 1 pp 93ndash101 2004
[50] S E Pratsinis ldquoSimultaneous nucleation condensation andcoagulation in aerosol reactorsrdquo Journal of Colloid And InterfaceScience vol 124 no 2 pp 416ndash427 1988
[51] RMcGraw ldquoDescription of aerosol dynamics by the quadraturemethod of momentsrdquo Aerosol Science and Technology vol 27no 2 pp 255ndash265 1997
[52] J Bai Y-H Xu and J-P Wang ldquoCubic and spherical high-moment FeCo nanoparticles with narrow size distributionrdquoIEEE Transactions on Magnetics vol 43 no 7 pp 3340ndash33422007
[53] A Khorsand Zak R Razali W H Abd Majid and MDarroudi ldquoSynthesis and characterization of a narrow sizedistribution of zinc oxide nanoparticlesrdquo International Journalof Nanomedicine vol 6 no 1 pp 1399ndash1403 2011
[54] M Asemi and M Ghanaatshoar ldquoPreparation of CuCrO2
nanoparticles with narrow size distribution by sol-gel methodrdquoJournal of Sol-Gel Science and Technology vol 70 no 3 pp 416ndash421 2014
[55] A M Ahadi O Polonskyi U Schurmann T Strunskus and FFaupel ldquoStable production of TiOx nanoparticles with narrowsize distribution by reactive pulsed dc magnetron sputteringrdquoJournal of Physics D Applied Physics vol 48 no 3 Article ID035501 2015
[56] W W So S B Park K J Kim C J Shin and S J Moon ldquoThecrystalline phase stability of titania particles prepared at roomtemperature by the sol-gelmethodrdquo Journal ofMaterials Sciencevol 36 no 17 pp 4299ndash4305 2001
[57] A Teleki R Wengeler L Wengeler H Nirschl and S EPratsinis ldquoDistinguishing between aggregates and agglomeratesof flame-made TiO
Figure 9 Scatter plot of the particle growth-rate by condensationversus the particle growth-rate by coagulation
order of magnitude greater than the condensation growth-rate
4 Summary and Conclusions
The growth mechanisms of titania during hydrolysis oftitanium tetrachloride in a three-dimensional planar jet arestudied via direct numerical simulation The mass momen-tum enthalpy and species transport equations are solved in amodel-free manner Titania was produced via the hydrolysisof titanium tetrachloride modeled via a 1-step infinitelyfast chemical mechanism The particle field was representedusing a nodal method and solves for the evolution of theconcentration of particles of various sizes in an Eulerianmanner When coupled to the Navier-Stokes solver thefluid thermal chemical and particle fields are obtained as afunction of space and time
The results show that fluid turbulencegas mixing plays avery important role in particle growth The results indicatethat the particle formation and growth are greatly affectedif not dominated by mixing and chemical reaction Reactantconversion or titania production is limited by the ability ofthe flow to bring the TiCl
4and H
2O into contact Evidence
for this is the particle formation throughout the domainAdditionally the results show that where the turbulenceis more intense the particles are larger and the particlesize distribution as characterized by the geometric standarddeviation is wider Particle growth in the proximal regionof the jet is dominated by condensation After collapseof the jet core nanoparticle growth due to coagulationincreases significantly Though TiO
2is produced after core
collapse producing condensible species particle growth dueto coagulation is as much as an order of magnitude greaterthan that due to condensation While both condensationand coagulation act to increase polydispersity coagulationhas been shown to increase the width of the particle sizedistribution faster or greater than condensation [50]
These results help to shed light on and improve ourunderstanding of the underlying growth dynamics occur-ring in nanoparticle synthesis processes The change fromcondensation-dominated to coagulation-dominated growthis useful in modeling the complete synthesis process includ-ing sintering and the formation of hard and soft agglomer-ates Spanning the size range from single molecules (particleinception) to hundreds of nanometers as the particles foundin many industrial processes and applications is computeintensive [18] However knowing that condensation andmolecular growth is relatively minor could facilitate the useof more affordable modeling strategies that do not needto account for every phenomena [6 51] Additionally ahighly desired attribute of metal oxide nanoparticles is theirspecific surface area The ability to synthesize narrow sizedistributions which are desired is advantageous in that itremoves the processing necessary to separate particles by sizeThis is advantageous in that it lowers cost [52ndash55]
One strategy to reduce polydispersity may be to delay thetransition to turbulence vis a vis delaying collapse of the jetcore While the particle field is known as a function of size itshould be noted that in this work we are unable to distinguishthe various phases of titania rutile anatase or brookite [5657]The phase composition of titania is very much a functionof the synthesis process for example precursor compositionand temperature history which are greatly simplified in thiswork Such capability requires phenomenologicalmodels thataccount for particle dynamics as a function of space time andtemperature as well as solid-state diffusion processes and isquite beyond the scope of this work
Conflict of Interests
The author declares that there is no conflict of interestsregarding the publication of this paper
Acknowledgments
Funding for this research was provided by the Universityof Minnesota Computational resources were provided byMinnesota Supercomputing Institute
References
[1] S E Pratsinis and S Vemury ldquoParticle formation in gases areviewrdquo Powder Technology vol 88 no 3 pp 267ndash273 1996
[2] G Beaucage H Kammler P Mueller et al ldquoProbing thedynamics of nanoparticle growth in a flame using synchrotronradiationrdquo Nature Materials vol 3 no 6 pp 370ndash373 2004
[3] S Panda and S E Pratsinis ldquoModeling the synthesis ofaluminum particles by evaporation-condensation in an aerosolflow reactorrdquo Nanostructured Materials vol 5 no 7-8 pp 755ndash767 1995
[4] S Yuu K Ikeda and T Umekage ldquoFlow-field prediction andexperimental verification of low Reynolds number gas-particleturbulent jetsrdquo Colloids and Surfaces A Physicochemical andEngineering Aspects vol 109 pp 13ndash27 1996
[5] S Tsantilis H K Kammler and S E Pratsinis ldquoPopulationbalance modeling of flame synthesis of titania nanoparticlesrdquo
Journal of Nanotechnology 9
Chemical Engineering Science vol 57 no 12 pp 2139ndash21562002
[6] N Settumba and S C Garrick ldquoDirect numerical simulationof nanoparticle coagulation in a temporal mixing layer via amoment methodrdquo Journal of Aerosol Science vol 34 no 2 pp149ndash167 2003
[7] D L Marchisio and R O Fox ldquoSolution of population balanceequations using the direct quadrature mehtod of momentsrdquoJournal of Aerosol Science vol 36 pp 43ndash73 2005
[8] F Aristizabal R J Munz and D Berk ldquoModeling of the pro-duction of ultra fine Aluminium particles in rapid quenchingturbulent flowrdquo Journal of Aerosol Science vol 37 no 2 pp 162ndash186 2006
[9] S Rigopoulos ldquoPDF method for population balance in turbu-lent reactive flowrdquo Chemical Engineering Science vol 62 no 23pp 6865ndash6878 2007
[10] K Zhou A Attili A Alshaarawi and F Bisetti ldquoSimulationof aerosol nucleation and growth in a turbulent mixing layerrdquoPhysics of Fluids vol 26 no 6 Article ID 065106 2014
[11] S A Orszag and I Staroselsky ldquoCFD progress and problemsrdquoComputer Physics Communications vol 127 no 1 pp 165ndash1712000
[12] K Nakaso T Fujimoto T Seto M Shimada K Okuyamaand M M Lunden ldquoSize distribution change of titania nano-particle agglomerates generated by gas phase reaction agglom-eration and sinteringrdquo Aerosol Science and Technology vol 35no 5 pp 929ndash947 2001
[13] T Johannessen S E Pratsinis andH Livbjerg ldquoComputationalanalysis of coagulation and coalescence in the flame synthesis oftitania particlesrdquoPowder Technology vol 118 no 3 pp 242ndash2502001
[14] E G Moody and L R Collins ldquoEffect of mixing on thenucleation and growth of titania particlesrdquo Aerosol Science andTechnology vol 37 no 5 pp 403ndash424 2003
[15] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[16] S Das and S C Garrick ldquoThe effects of turbulence onnanoparticle growth in turbulent reacting jetsrdquo Physics of Fluidsvol 22 no 10 Article ID 103303 2010
[17] S C Garrick ldquoEffects of turbulent fluctuations on nanoparticlecoagulation in shear flowsrdquo Aerosol Science and Technology vol45 no 10 pp 1272ndash1285 2011
[18] A J Fager J Liu and S C Garrick ldquoHybrid simulations ofmetal particle nucleation a priori and a posteriori analyses ofthe effects of unresolved scalar interactions on nanoparticlenucleationrdquo Physics of Fluids vol 24 no 7 Article ID 0751102012
[19] N J Murfield and S C Garrick ldquoLarge eddy simulation anddirect numerical simulation of homogeneous nucleation inturbulent wakesrdquo Journal of Aerosol Science vol 60 pp 21ndash332013
[20] N JMurfield and S CGarrick ldquoThe effects of unresolved scalarfluctuations during homogeneous nucleationrdquo Aerosol Scienceand Technology vol 47 no 7 pp 806ndash817 2013
[21] P Givi ldquoModel free simulations of turbulent reactive flowsrdquoProgress in Energy and Combustion Science vol 15 no 1 pp 1ndash107 1989
[22] K E J Lehtinen and M R Zachariah ldquoSelf-preserving theoryfor the volume distribution of particles undergoing browniancoagulationrdquo Journal of Colloid and Interface Science vol 242no 2 pp 314ndash318 2001
[23] S C Garrick K E J Lehtinen andM R Zachariah ldquoNanopar-ticle coagulation via a Navier-Stokesnodal methodology evo-lution of the particle fieldrdquo Journal of Aerosol Science vol 37 no5 pp 555ndash576 2006
[24] F Gelbard and J H Seinfeld ldquoSimulation of multicomponentaerosol dynamicsrdquo Journal of Colloid And Interface Science vol78 no 2 pp 485ndash501 1980
[25] P Biswas C Y Wu M R Zachariah and B McMillin ldquoChar-acterization of iron oxide-silica nanocomposites in flamespart II comparison of discrete-sectional model predictions toexperimental datardquo Journal of Materials Research vol 12 no 3pp 714ndash723 1997
[26] K E J Lehtinen and M R Zachariah ldquoEnergy accumulationin nanoparticle collision and coalescence processesrdquo Journal ofAerosol Science vol 33 no 2 pp 357ndash368 2002
[27] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[28] J Loeffler S Das and S C Garrick ldquoLarge eddy simulationof titanium dioxide nanoparticle formation and growth inturbulent jetsrdquoAerosol Science and Technology vol 45 no 5 pp616ndash628 2011
[29] K S Friedlander Smoke Dust and Haze Fundamentals ofAerosol Dynamics Oxford University Press New York NYUSA 2000
[30] M Frenklach and S J Harris ldquoAerosol dynamics modelingusing the method of momentsrdquo Journal of Colloid and InterfaceScience vol 118 no 1 pp 252ndash261 1987
[31] S E Pratsinis ldquoParticle production by gas-to-particle conver-sion in turbulent flowsrdquo Journal of Aerosol Science vol 20 no8 pp 1461ndash1464 1989
[32] J D Landgrebe and S E Pratsinis ldquoA discrete-sectional modelfor particulate production by gas-phase chemical reaction andaerosol coagulation in the free-molecular regimerdquo Journal ofColloid and Interface Science vol 139 no 1 pp 63ndash86 1990
[33] GWang and S C Garrick ldquoModeling and simulation of titaniasynthesis in two-dimensional methane-air flamesrdquo Journal ofNanoparticle Research vol 7 no 6 pp 621ndash632 2005
[34] G W Mulholland R J Samson R D Mountain and M HErnst ldquoCluster size distribution for free molecular agglomera-tionrdquo Energy amp Fuels vol 2 no 4 pp 481ndash486 1988
[35] J Cai N Lu and C M Sorensen ldquoAnalysis of fractal clus-ter morphology parameters structural coefficient and densityautocorrelation function cutoffrdquo Journal of ColloidAnd InterfaceScience vol 171 no 2 pp 470ndash473 1995
[36] R Jullien and PMeakin ldquoSimplemodels for the restructuring ofthree-dimensional ballistic aggregatesrdquo Journal of Colloid AndInterface Science vol 127 no 1 pp 265ndash272 1989
[37] S N Rogak and R C Flagan ldquoCoagulation of aerosol agglom-erates in the transition regimerdquo Journal of Colloid and InterfaceScience vol 151 no 1 pp 203ndash224 1992
[38] S Modem S C Garrick M R Zachariah and K E J LehtinenldquoDirect numerical simulation of nanoparticle coagulation in atemporal mixing layerrdquo in Proceedings of the 29th Symposium(International) on Combustion pp 1071ndash1077 The CombustionInstitute Pittsburgh Pa USA 2002
[39] S C Garrick and G Wang ldquoModeling and simulation of tita-nium dioxide nanoparticle synthesis with finite-rate sinteringin planar jetsrdquo Journal of Nanoparticle Research vol 13 no 3pp 973ndash984 2011
10 Journal of Nanotechnology
[40] M C Heine and S E Pratsinis ldquoPolydispersity of primaryparticles in agglomerates made by coagulation and sinteringrdquoJournal of Aerosol Science vol 38 no 1 pp 17ndash38 2007
[41] R W MacCormack ldquoThe effect of viscosity in hypervelocityimpact cateringrdquo AIAA Paper 69-354 1969
[42] M H Carpenter ldquoA high-order compact numerical algorithmfor supersonic flowsrdquo in Twelfth International Conference onNumerical Methods in Fluid Dynamics K W Morton Ed vol371 of Lecture Notes in Physics pp 254ndash258 Springer BerlinGermany 1990
[43] D H Rudy and J C Strikwerda ldquoBoundary conditions forsubsonic compressible navier-stokes calculationsrdquo Computersand Fluids vol 9 no 3 pp 327ndash338 1981
[44] P Givi ldquoFiltered density function for subgrid scale modeling ofturbulent combustionrdquo AIAA Journal vol 44 no 1 pp 16ndash232006
[45] S Modem and S C Garrick ldquoNanoparticle coagulation in atemporal mixing layer mean and size-selected imagesrdquo Journalof Visualization vol 6 no 3 pp 293ndash302 2003
[46] D L Wright S Yu P S Kasibhatla et al ldquoRetrieval of aerosolproperties from moments of the particle size distribution forkernels involving the step function cloud droplet activationrdquoJournal of Aerosol Science vol 33 no 2 pp 319ndash337 2002
[47] H K Kammler R Jossen PWMorrison Jr S E Pratsinis andG Beaucage ldquoThe effect of external electric fields during flamesynthesis of titaniardquo Powder Technology vol 135-136 pp 310ndash320 2003
[48] W C Hinds Aerosol Technology Properties Behavior andMeasurement of Air-Borne Particles John Wiley amp Sons NewYork NY USA 2nd edition 1999
[49] N Settumba and S C Garrick ldquoA comparison of diffusivetransport in a moment method for nanoparticle coagulationrdquoJournal of Aerosol Science vol 35 no 1 pp 93ndash101 2004
[50] S E Pratsinis ldquoSimultaneous nucleation condensation andcoagulation in aerosol reactorsrdquo Journal of Colloid And InterfaceScience vol 124 no 2 pp 416ndash427 1988
[51] RMcGraw ldquoDescription of aerosol dynamics by the quadraturemethod of momentsrdquo Aerosol Science and Technology vol 27no 2 pp 255ndash265 1997
[52] J Bai Y-H Xu and J-P Wang ldquoCubic and spherical high-moment FeCo nanoparticles with narrow size distributionrdquoIEEE Transactions on Magnetics vol 43 no 7 pp 3340ndash33422007
[53] A Khorsand Zak R Razali W H Abd Majid and MDarroudi ldquoSynthesis and characterization of a narrow sizedistribution of zinc oxide nanoparticlesrdquo International Journalof Nanomedicine vol 6 no 1 pp 1399ndash1403 2011
[54] M Asemi and M Ghanaatshoar ldquoPreparation of CuCrO2
nanoparticles with narrow size distribution by sol-gel methodrdquoJournal of Sol-Gel Science and Technology vol 70 no 3 pp 416ndash421 2014
[55] A M Ahadi O Polonskyi U Schurmann T Strunskus and FFaupel ldquoStable production of TiOx nanoparticles with narrowsize distribution by reactive pulsed dc magnetron sputteringrdquoJournal of Physics D Applied Physics vol 48 no 3 Article ID035501 2015
[56] W W So S B Park K J Kim C J Shin and S J Moon ldquoThecrystalline phase stability of titania particles prepared at roomtemperature by the sol-gelmethodrdquo Journal ofMaterials Sciencevol 36 no 17 pp 4299ndash4305 2001
[57] A Teleki R Wengeler L Wengeler H Nirschl and S EPratsinis ldquoDistinguishing between aggregates and agglomeratesof flame-made TiO
Chemical Engineering Science vol 57 no 12 pp 2139ndash21562002
[6] N Settumba and S C Garrick ldquoDirect numerical simulationof nanoparticle coagulation in a temporal mixing layer via amoment methodrdquo Journal of Aerosol Science vol 34 no 2 pp149ndash167 2003
[7] D L Marchisio and R O Fox ldquoSolution of population balanceequations using the direct quadrature mehtod of momentsrdquoJournal of Aerosol Science vol 36 pp 43ndash73 2005
[8] F Aristizabal R J Munz and D Berk ldquoModeling of the pro-duction of ultra fine Aluminium particles in rapid quenchingturbulent flowrdquo Journal of Aerosol Science vol 37 no 2 pp 162ndash186 2006
[9] S Rigopoulos ldquoPDF method for population balance in turbu-lent reactive flowrdquo Chemical Engineering Science vol 62 no 23pp 6865ndash6878 2007
[10] K Zhou A Attili A Alshaarawi and F Bisetti ldquoSimulationof aerosol nucleation and growth in a turbulent mixing layerrdquoPhysics of Fluids vol 26 no 6 Article ID 065106 2014
[11] S A Orszag and I Staroselsky ldquoCFD progress and problemsrdquoComputer Physics Communications vol 127 no 1 pp 165ndash1712000
[12] K Nakaso T Fujimoto T Seto M Shimada K Okuyamaand M M Lunden ldquoSize distribution change of titania nano-particle agglomerates generated by gas phase reaction agglom-eration and sinteringrdquo Aerosol Science and Technology vol 35no 5 pp 929ndash947 2001
[13] T Johannessen S E Pratsinis andH Livbjerg ldquoComputationalanalysis of coagulation and coalescence in the flame synthesis oftitania particlesrdquoPowder Technology vol 118 no 3 pp 242ndash2502001
[14] E G Moody and L R Collins ldquoEffect of mixing on thenucleation and growth of titania particlesrdquo Aerosol Science andTechnology vol 37 no 5 pp 403ndash424 2003
[15] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[16] S Das and S C Garrick ldquoThe effects of turbulence onnanoparticle growth in turbulent reacting jetsrdquo Physics of Fluidsvol 22 no 10 Article ID 103303 2010
[17] S C Garrick ldquoEffects of turbulent fluctuations on nanoparticlecoagulation in shear flowsrdquo Aerosol Science and Technology vol45 no 10 pp 1272ndash1285 2011
[18] A J Fager J Liu and S C Garrick ldquoHybrid simulations ofmetal particle nucleation a priori and a posteriori analyses ofthe effects of unresolved scalar interactions on nanoparticlenucleationrdquo Physics of Fluids vol 24 no 7 Article ID 0751102012
[19] N J Murfield and S C Garrick ldquoLarge eddy simulation anddirect numerical simulation of homogeneous nucleation inturbulent wakesrdquo Journal of Aerosol Science vol 60 pp 21ndash332013
[20] N JMurfield and S CGarrick ldquoThe effects of unresolved scalarfluctuations during homogeneous nucleationrdquo Aerosol Scienceand Technology vol 47 no 7 pp 806ndash817 2013
[21] P Givi ldquoModel free simulations of turbulent reactive flowsrdquoProgress in Energy and Combustion Science vol 15 no 1 pp 1ndash107 1989
[22] K E J Lehtinen and M R Zachariah ldquoSelf-preserving theoryfor the volume distribution of particles undergoing browniancoagulationrdquo Journal of Colloid and Interface Science vol 242no 2 pp 314ndash318 2001
[23] S C Garrick K E J Lehtinen andM R Zachariah ldquoNanopar-ticle coagulation via a Navier-Stokesnodal methodology evo-lution of the particle fieldrdquo Journal of Aerosol Science vol 37 no5 pp 555ndash576 2006
[24] F Gelbard and J H Seinfeld ldquoSimulation of multicomponentaerosol dynamicsrdquo Journal of Colloid And Interface Science vol78 no 2 pp 485ndash501 1980
[25] P Biswas C Y Wu M R Zachariah and B McMillin ldquoChar-acterization of iron oxide-silica nanocomposites in flamespart II comparison of discrete-sectional model predictions toexperimental datardquo Journal of Materials Research vol 12 no 3pp 714ndash723 1997
[26] K E J Lehtinen and M R Zachariah ldquoEnergy accumulationin nanoparticle collision and coalescence processesrdquo Journal ofAerosol Science vol 33 no 2 pp 357ndash368 2002
[27] GWang and S C Garrick ldquoModeling and simulation of titaniaformation and growth in temporal mixing layersrdquo Journal ofAerosol Science vol 37 no 4 pp 431ndash451 2006
[28] J Loeffler S Das and S C Garrick ldquoLarge eddy simulationof titanium dioxide nanoparticle formation and growth inturbulent jetsrdquoAerosol Science and Technology vol 45 no 5 pp616ndash628 2011
[29] K S Friedlander Smoke Dust and Haze Fundamentals ofAerosol Dynamics Oxford University Press New York NYUSA 2000
[30] M Frenklach and S J Harris ldquoAerosol dynamics modelingusing the method of momentsrdquo Journal of Colloid and InterfaceScience vol 118 no 1 pp 252ndash261 1987
[31] S E Pratsinis ldquoParticle production by gas-to-particle conver-sion in turbulent flowsrdquo Journal of Aerosol Science vol 20 no8 pp 1461ndash1464 1989
[32] J D Landgrebe and S E Pratsinis ldquoA discrete-sectional modelfor particulate production by gas-phase chemical reaction andaerosol coagulation in the free-molecular regimerdquo Journal ofColloid and Interface Science vol 139 no 1 pp 63ndash86 1990
[33] GWang and S C Garrick ldquoModeling and simulation of titaniasynthesis in two-dimensional methane-air flamesrdquo Journal ofNanoparticle Research vol 7 no 6 pp 621ndash632 2005
[34] G W Mulholland R J Samson R D Mountain and M HErnst ldquoCluster size distribution for free molecular agglomera-tionrdquo Energy amp Fuels vol 2 no 4 pp 481ndash486 1988
[35] J Cai N Lu and C M Sorensen ldquoAnalysis of fractal clus-ter morphology parameters structural coefficient and densityautocorrelation function cutoffrdquo Journal of ColloidAnd InterfaceScience vol 171 no 2 pp 470ndash473 1995
[36] R Jullien and PMeakin ldquoSimplemodels for the restructuring ofthree-dimensional ballistic aggregatesrdquo Journal of Colloid AndInterface Science vol 127 no 1 pp 265ndash272 1989
[37] S N Rogak and R C Flagan ldquoCoagulation of aerosol agglom-erates in the transition regimerdquo Journal of Colloid and InterfaceScience vol 151 no 1 pp 203ndash224 1992
[38] S Modem S C Garrick M R Zachariah and K E J LehtinenldquoDirect numerical simulation of nanoparticle coagulation in atemporal mixing layerrdquo in Proceedings of the 29th Symposium(International) on Combustion pp 1071ndash1077 The CombustionInstitute Pittsburgh Pa USA 2002
[39] S C Garrick and G Wang ldquoModeling and simulation of tita-nium dioxide nanoparticle synthesis with finite-rate sinteringin planar jetsrdquo Journal of Nanoparticle Research vol 13 no 3pp 973ndash984 2011
10 Journal of Nanotechnology
[40] M C Heine and S E Pratsinis ldquoPolydispersity of primaryparticles in agglomerates made by coagulation and sinteringrdquoJournal of Aerosol Science vol 38 no 1 pp 17ndash38 2007
[41] R W MacCormack ldquoThe effect of viscosity in hypervelocityimpact cateringrdquo AIAA Paper 69-354 1969
[42] M H Carpenter ldquoA high-order compact numerical algorithmfor supersonic flowsrdquo in Twelfth International Conference onNumerical Methods in Fluid Dynamics K W Morton Ed vol371 of Lecture Notes in Physics pp 254ndash258 Springer BerlinGermany 1990
[43] D H Rudy and J C Strikwerda ldquoBoundary conditions forsubsonic compressible navier-stokes calculationsrdquo Computersand Fluids vol 9 no 3 pp 327ndash338 1981
[44] P Givi ldquoFiltered density function for subgrid scale modeling ofturbulent combustionrdquo AIAA Journal vol 44 no 1 pp 16ndash232006
[45] S Modem and S C Garrick ldquoNanoparticle coagulation in atemporal mixing layer mean and size-selected imagesrdquo Journalof Visualization vol 6 no 3 pp 293ndash302 2003
[46] D L Wright S Yu P S Kasibhatla et al ldquoRetrieval of aerosolproperties from moments of the particle size distribution forkernels involving the step function cloud droplet activationrdquoJournal of Aerosol Science vol 33 no 2 pp 319ndash337 2002
[47] H K Kammler R Jossen PWMorrison Jr S E Pratsinis andG Beaucage ldquoThe effect of external electric fields during flamesynthesis of titaniardquo Powder Technology vol 135-136 pp 310ndash320 2003
[48] W C Hinds Aerosol Technology Properties Behavior andMeasurement of Air-Borne Particles John Wiley amp Sons NewYork NY USA 2nd edition 1999
[49] N Settumba and S C Garrick ldquoA comparison of diffusivetransport in a moment method for nanoparticle coagulationrdquoJournal of Aerosol Science vol 35 no 1 pp 93ndash101 2004
[50] S E Pratsinis ldquoSimultaneous nucleation condensation andcoagulation in aerosol reactorsrdquo Journal of Colloid And InterfaceScience vol 124 no 2 pp 416ndash427 1988
[51] RMcGraw ldquoDescription of aerosol dynamics by the quadraturemethod of momentsrdquo Aerosol Science and Technology vol 27no 2 pp 255ndash265 1997
[52] J Bai Y-H Xu and J-P Wang ldquoCubic and spherical high-moment FeCo nanoparticles with narrow size distributionrdquoIEEE Transactions on Magnetics vol 43 no 7 pp 3340ndash33422007
[53] A Khorsand Zak R Razali W H Abd Majid and MDarroudi ldquoSynthesis and characterization of a narrow sizedistribution of zinc oxide nanoparticlesrdquo International Journalof Nanomedicine vol 6 no 1 pp 1399ndash1403 2011
[54] M Asemi and M Ghanaatshoar ldquoPreparation of CuCrO2
nanoparticles with narrow size distribution by sol-gel methodrdquoJournal of Sol-Gel Science and Technology vol 70 no 3 pp 416ndash421 2014
[55] A M Ahadi O Polonskyi U Schurmann T Strunskus and FFaupel ldquoStable production of TiOx nanoparticles with narrowsize distribution by reactive pulsed dc magnetron sputteringrdquoJournal of Physics D Applied Physics vol 48 no 3 Article ID035501 2015
[56] W W So S B Park K J Kim C J Shin and S J Moon ldquoThecrystalline phase stability of titania particles prepared at roomtemperature by the sol-gelmethodrdquo Journal ofMaterials Sciencevol 36 no 17 pp 4299ndash4305 2001
[57] A Teleki R Wengeler L Wengeler H Nirschl and S EPratsinis ldquoDistinguishing between aggregates and agglomeratesof flame-made TiO
[40] M C Heine and S E Pratsinis ldquoPolydispersity of primaryparticles in agglomerates made by coagulation and sinteringrdquoJournal of Aerosol Science vol 38 no 1 pp 17ndash38 2007
[41] R W MacCormack ldquoThe effect of viscosity in hypervelocityimpact cateringrdquo AIAA Paper 69-354 1969
[42] M H Carpenter ldquoA high-order compact numerical algorithmfor supersonic flowsrdquo in Twelfth International Conference onNumerical Methods in Fluid Dynamics K W Morton Ed vol371 of Lecture Notes in Physics pp 254ndash258 Springer BerlinGermany 1990
[43] D H Rudy and J C Strikwerda ldquoBoundary conditions forsubsonic compressible navier-stokes calculationsrdquo Computersand Fluids vol 9 no 3 pp 327ndash338 1981
[44] P Givi ldquoFiltered density function for subgrid scale modeling ofturbulent combustionrdquo AIAA Journal vol 44 no 1 pp 16ndash232006
[45] S Modem and S C Garrick ldquoNanoparticle coagulation in atemporal mixing layer mean and size-selected imagesrdquo Journalof Visualization vol 6 no 3 pp 293ndash302 2003
[46] D L Wright S Yu P S Kasibhatla et al ldquoRetrieval of aerosolproperties from moments of the particle size distribution forkernels involving the step function cloud droplet activationrdquoJournal of Aerosol Science vol 33 no 2 pp 319ndash337 2002
[47] H K Kammler R Jossen PWMorrison Jr S E Pratsinis andG Beaucage ldquoThe effect of external electric fields during flamesynthesis of titaniardquo Powder Technology vol 135-136 pp 310ndash320 2003
[48] W C Hinds Aerosol Technology Properties Behavior andMeasurement of Air-Borne Particles John Wiley amp Sons NewYork NY USA 2nd edition 1999
[49] N Settumba and S C Garrick ldquoA comparison of diffusivetransport in a moment method for nanoparticle coagulationrdquoJournal of Aerosol Science vol 35 no 1 pp 93ndash101 2004
[50] S E Pratsinis ldquoSimultaneous nucleation condensation andcoagulation in aerosol reactorsrdquo Journal of Colloid And InterfaceScience vol 124 no 2 pp 416ndash427 1988
[51] RMcGraw ldquoDescription of aerosol dynamics by the quadraturemethod of momentsrdquo Aerosol Science and Technology vol 27no 2 pp 255ndash265 1997
[52] J Bai Y-H Xu and J-P Wang ldquoCubic and spherical high-moment FeCo nanoparticles with narrow size distributionrdquoIEEE Transactions on Magnetics vol 43 no 7 pp 3340ndash33422007
[53] A Khorsand Zak R Razali W H Abd Majid and MDarroudi ldquoSynthesis and characterization of a narrow sizedistribution of zinc oxide nanoparticlesrdquo International Journalof Nanomedicine vol 6 no 1 pp 1399ndash1403 2011
[54] M Asemi and M Ghanaatshoar ldquoPreparation of CuCrO2
nanoparticles with narrow size distribution by sol-gel methodrdquoJournal of Sol-Gel Science and Technology vol 70 no 3 pp 416ndash421 2014
[55] A M Ahadi O Polonskyi U Schurmann T Strunskus and FFaupel ldquoStable production of TiOx nanoparticles with narrowsize distribution by reactive pulsed dc magnetron sputteringrdquoJournal of Physics D Applied Physics vol 48 no 3 Article ID035501 2015
[56] W W So S B Park K J Kim C J Shin and S J Moon ldquoThecrystalline phase stability of titania particles prepared at roomtemperature by the sol-gelmethodrdquo Journal ofMaterials Sciencevol 36 no 17 pp 4299ndash4305 2001
[57] A Teleki R Wengeler L Wengeler H Nirschl and S EPratsinis ldquoDistinguishing between aggregates and agglomeratesof flame-made TiO
