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A Theoretical Study on Gas-Phase Coating of Aerosol Particles
JOURNAL OF COLLOID AND INTERFACE SCIENCE ARTICLE NO.
185, 26–38 (1997)
CS964558
A Theoretical Study on Gas-Phase Coating of Aerosol Particles SANJEEV JAIN, GEORGE P. FOTOU,
AND
TOIVO T. KODAS 1
Center for Micro-Engineered Ceramics, Department of Chemical and Nuclear Engineering, University of New Mexico, Albuquerque, New Mexico 87131 Received February 1, 1996; accepted August 19, 1996
oxides and coating of silica fibers with ultrafine silica particles in the gas phase to increase their surface area (4, 5). An important class of applications of coated particles is the manufacture of titania pigments. Large quantities of titania are produced by gas-phase reaction and are used as a pigment in paper, plastics, paints, and other materials (4). For many applications, coating of TiO2 with various metal oxides is required to reduce photoreactivity and therefore improve the durability as well as the dispersion properties of titania pigments (6, 7). In industrial practice, titania is often produced by the chloride route (gas-phase reaction of TiCl4 and O2 ) and coated by the liquid-phase precipitation of SiO2 , Al2O3 , or Al2O3 /SiO2 mixtures from silicon and/ or aluminum hydroxide solutions followed by drying, calcining, and milling (8). The availability of a gas-phase process for coating would dramatically simplify coated pigment particle production and would have a significant impact on the economics of TiO2 pigment production. Silica-coated titania particles have been synthesized in laboratory flame reactors (9, 10). However, common flame reactor designs involve premixed reactants and do not allow sequential TiO2 formation followed by coating. In a recent experimental study, titanium dioxide particles generated by the prior reaction of titanium tetrachloride with oxygen were coated in situ with alumina, silica, and mixtures of the two in a tubular furnace aerosol reactor (11). The coating thickness and uniformity depended on the coating additive loading and the reactor temperature. This study identified the major qualitative features of the process but no theoretical description was developed. Modeling of this and other related gas-phase coating processes is vital in understanding the complex physicochemical phenomena that take place. Very few theoretical studies on the coating of aerosol particles have been reported so far (5, 12). Gas-phase coating of fibers has been modeled using a simple monodisperse model accounting for coagulation and sintering of the coating material (5). This model assumed that the particle size distribution of the host as well as the coating particles remained monodisperse throughout the reactor length. Scavenging of a fine mode aerosol (coating
In situ coating of aerosol particles by gas-phase and surface reaction in a flow reactor is modeled accounting for scavenging (capture of small particles by large particles) and simultaneous surface reaction along with the finite sintering rate of the scavenged particles. A log-normal size distribution is assumed for the host and coating particles to describe coagulation and a monodisperse size distribution is used for the coating particles to describe sintering. As an example, coating of titania particles with silica in a continuous flow hot-wall reactor was modeled. High temperatures, low reactant concentrations, and large host particle surface areas favored smoother coatings in the parameter range: temperature 1700–1800 K, host particle number concentration 1 1 10 5 – 1 1 10 7 #/cm3 , average host particle size 1 mm, inlet coating reactant concentration (SiCl4 ) 2 1 10 07 –2 1 10 010 mol/cm3 , and various surface reaction rates. The fraction of silica deposited on the TiO2 particles decreased by more than seven times with a hundredfold increase in SiCl4 inlet concentration because of the resulted increase in the average SiO2 particle size under the assumed coating conditions. Increasing the TiO2 particle number concentration resulted in higher scavenging efficiency of SiO2 . In the TiO2 /SiO2 system it is likely that surface reaction as well as scavenging play important roles in the coating process. The results agree qualitatively with experimental observations of TiO2 particles coated in situ with silica. q 1997 Academic Press Key Words: log-normal; scavenging; surface reaction; coatings; titania; silica.
INTRODUCTION
Several gas-phase techniques for coating particulate and fibrous materials have been developed over the past years for diverse applications. Silica fibers have been coated with titania films by chemical vapor deposition (CVD) to improve tensile strength for lightguide applications (1, 2). Silicon nitride and aluminum oxide particles have been coated with titanium nitride and titanium dioxide in the gas phase using fluidized bed reactors (3). Other examples include coating of nuclear fuel pellets with protective layers of metal 1
To whom correspondence should be addressed.
26
0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.
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GAS-PHASE COATING OF AEROSOL PARTICLES
FIG. 1. Schematic illustration of in situ coating of TiO2 with SiO2 .
particles) by a coarse mode aerosol (host particles) has also been modeled assuming different host particle size distributions (12). The latter model was developed to describe a purely physical process, and did not account for sintering or chemical reaction. Furthermore, the effect of surface reaction was not considered in these studies. In the present work, a model has been developed and used to describe in situ coating of particles in aerosol flow reactors to gain insights into the physicochemical phenomena taking place in this process and as a guide for design and scale-up of such a process. The model was applied for the specific case of in situ coating of titania with silica by gas-phase reaction of silicon tetrachloride. The effects of process variables such as inlet reactant concentration, temperature, residence time, and host particle size distribution on the thickness and quality (rough or smooth) of the coatings were studied. PROCESS DESCRIPTION
The in situ gas-phase coating process modeled here involves the generation of host particles by gas-phase chemical reaction of a precursor in a flow reactor which collide and sinter (coalesce) to form larger particles. Once the reaction is complete and sufficient residence time is provided, the particles attain a size distribution which can be adequately represented by a log-normal function (13). Figure 1 depicts this process for the special case of coating titania particles with silica. The model makes no assumption about the types of reactants used to produce the particles or to coat them. In the case of titania, metal halide precursors are most desirable and the coating precursors are likely to be compounds such as SiCl4 , AlCl3 , or ZrCl4 . These reactants are introduced in the vapor phase downstream into the same flow reactor and may react on the surfaces of the host particles to form a coating. Simultaneously, the precursors may react in the gas phase to form particles of the coating material. These particles collide and coalesce to form larger particles.
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The particles of the coating material are simultaneously captured (scavenging) by the larger host particles. Therefore, the coating of the host particles may take place by surface reaction as well as by scavenging of the smaller particles (SiO2 , Al2O3 , or ZrO2 ). The present model assumes a log-normal size distribution for both the host and coating particles, accounts for coating by surface reaction and scavenging, and takes into account the finite sintering rate of the coating material in the gas phase and on the host particles to describe the coating thickness and roughness. The model is used here to describe coating of titania particles with silica but it is generic and can be used for other systems also. THEORY
In the present and the following sections, the host particles are termed as A particles (titania) and particles of the coating material as B particles (silica). The assumed process conditions used in the model simulations are shown in Table 1. The following assumptions were made: 1. Formation and growth of A particles is complete and A particles with a log-normal size distribution enter the coating TABLE 1 Process Conditions Assumed in the Simulations A particle number concentration Average A particle size Geometric standard deviation of A particle size distribution Initial specific surface area of A particles Inlet reactant concentration Residence time Density of B particles Monomer size of B particles
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1 1 106 (#/cm3) 1 ( mm) 1.5 0.93 (m2/g) 2 1 1009 (mol/cm3) 0.6 (s) 2.2 (g/cm3) ˚) 4.4 (A
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JAIN, FOTOU, AND KODAS
zone. The size distribution remains constant throughout the coating zone. The assumption of the log-normal size distribution can be relaxed as shown later. 2. The B particles are much smaller than A particles and lie in the free-molecular regime whereas A particles lie in the continuum regime. This is in agreement with experimental observations (11). 3. Collisions between A particles are negligible in the coating zone because the characteristic collision time ( Ç10 s for particle number concentration 10 8 /cm3 ) is greater than the residence time ( õ1 s). 4. CVD of the coating reactant on the walls and other wall losses are negligible. 5. The flow in the reactor is one-dimensional plug flow under isothermal conditions. 6. The gain in surface area of A or B particles by surface reaction is negligible. 7. Simultaneous CVD on sintering particles has a negligible effect on the sintering rate.
Size distribution of B particles. The equations presented here are similar to those developed by Friedlander et al. ( 12 ) to describe a purely scavenging process except that the equations presented below account for surface reaction as well as the finite sintering rate. The change in the zeroth moment ( or total B particle number concentration ) of the B particle size distribution with time is given by
dM0 dNB Å dt dt Å
£g Å
M 21 , M 30 / 2 M 12 / 2
1
S
[3]
Silica is formed in the gas phase by the reaction SiCl4 (g) / O2 (g) r 2Cl2 (g) / SiO2 (s),
[4]
where the rate of the gas-phase reaction is given by (15)
D
402kJ C, RT
[5]
where the rate is in mol/(cm3 )(s), T is in Kelvin, and C is expressed as mol/cm3 .
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dNB dt
coagulation
25 2 ln s 8
exp
5 2 1 2 ln s / exp ln s 8 8
SD
6 Å 3p p
kfmr Å
D
S
scavenging
S D
,
[6a]
r
01 / 3
kTm 1 1 2p rg (1 / pa/8)
[6b]
[2]
9 2 2 k ln s . 2
rSiO2 Å 1.7 1 10 14 exp 0
/
and ,
and the kth moment is given by (14) Mk Å M0£ gk exp
reaction
dNB dt
where the coagulation coefficient for collisions between B and A particles, kscav , is (12)
kscav
S D
gas-phase
S D
F S D S D S DG
[1]
1 M0 M2 ln 9 M 21
/
/ 2 exp
with the geometric standard deviation as (14) ln 2s Å
dNB dt
Å kgCA£ 0 kscav M1 / 3A M02 / 3 0 b0kfmr M 20 £ 1g / 6
Log-Normal Model In this section, the equations describing the gas-phase reaction rate of the coating reactant and the particle size distribution of the coating material are presented. For a lognormal size distribution function, the geometric mean volume £g is given by (14)
S D
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S DS D 3 4p
1/6
6kT rB
1/2
.
[6c]
The calculation of kscav was based on Epstein’s equation ( 16 ) following Friedlander et al.’s approach ( 12 ) . The first term on the right hand side describes the generation of particles by gas-phase reaction, the second term describes the loss due to scavenging by A particles, and the third term describes the change in the particle number concentration due to collisions between the B particles. The total particle number concentration is not affected by surface reaction. Equation [ 6b ] can also be applied for turbulent conditions which are very common in industrial aerosol reactors because of the small average B particle size ( 17 ) . The change in the total volume of B particles or the first moment with time is given by
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GAS-PHASE COATING OF AEROSOL PARTICLES
dM1 dVB Å dt dt Å
S D dVB dt
/
gas-phase reaction
S D dVB dt
/ scavenging
Å kgCA££ * 0 kscav M1 / 3A M1 / 3 /
S D dVB dt
surface reaction
a£ *CA£ AB , [7] (2pmc /kT ) 1 / 2
where a is the number of collisions which result in reaction which is normally much less than unity. The parameter a is given by an Arrhenius-type expression (18):
F G
a Å a0 exp 0
E RT
.
dM2 Å kgCA££ * 2 0 kscav M1 / 3A M4 / 3 dt
1
dAB dt
F S D S D S DG S D 3 2 exp ln s . 2
/ gas-phase reaction
S D dAB dt
/ scavenging
Å kgCA£a* 0 kscav M1 / 3A M02 / 3aB 0
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and t1 is the characteristic sintering time for silica particles in the gas phase and is given by (21) t1 Å 6.5 1 10 015 dB exp
aB Å
/
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83000 T
,
[10b]
AB NB
[11a]
and dB (the primary particle diameter) is given by (22) dB Å
dAA Å dt
6VB . AB
[11b]
dAA dt
/
scavenging
S D
[8]
sintering
(AB 0 ABF ) , t1
S D
dAA dt
Å kscav M1 / 3A M02 / 3aB 0
S D dAB dt
S D
where dB in Eq. [10b] is in cm. Using the monodisperse assumption for sintering, aB (the area of one B particle) is given by
25 2 5 2 exp ln s / 2 exp ln s 8 8
The values of the constants b0 and b2 in Eqs. [6a] and [8] were adopted from (13). The change in the total surface area of B particles ( per unit volume of aerosol ) with time is approximated by ( 20 )
S D
[10a]
Surface area and coating thickness of A particles. The change in the total surface area of A particles with time is approximated by (20)
aCA££ *aBVB / 2b2kfmr M 21 £ 1g / 6 (2pmc /kT ) 1 / 2
1 2 / exp ln s 8
dAB Å dt
ABF Å (36p ) 1 / 3 M2 / 3
[7a]
The surface reaction term in Eq. [7] and for the following equations is valid for the case of fast gas-phase diffusion of the reactant to the A particles as has been discussed in detail elsewhere (18, 19). The first term on the right hand side of Eq. [7] describes the generation of volume due to gas-phase chemical reaction, the second term describes the loss in volume due to scavenging by A particles, and the third term describes the gain in volume due to surface reaction on the B particles. The change of the second moment with time is given by
/2
where the first term on the right hand side describes the generation of area by gas-phase reaction, the second term describes the loss of area due to scavenging, and the third term describes the loss in area due to sintering. The area generation due to surface reaction is assumed negligible for thin coatings. In Eq. [9], ABF is the total area of B particles which are completely fused and is given by
(AA 0 A0 ) , t
[12a]
where A0 is the initial surface area of A particles. For thin coatings the increase in the total surface area of A particles will be negligible on complete sintering and the final area can be approximated as A0 . In the above equation the first term on the right hand side describes the gain in area by scavenging and the second term describes the loss in area due to sintering of the scavenged silica particles. In Eq. [12a] t is the characteristic sintering time of silica particles on the titania surface. The specific surface area of A particles is defined as Sp Å
[9]
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sintering
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AA , WA0
[12b]
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JAIN, FOTOU, AND KODAS
where WA0 is the total initial mass of A particles (per unit volume of aerosol). The increase in the mass of A particles is assumed to be negligible for thin coatings. The specific surface area is used as a measure of the extent of sintering. A higher specific surface area implies that the scavenged particles have not completely sintered, whereas a specific surface area close to the initial specific surface area implies almost full coalescence. The characteristic sintering time for silica particles on the surface of titania particles is calculated using Eq. [10b] where dB is calculated as dB Å
6VBA , ABA
[12c]
where VBA Å VA 0 V0 and ABA Å AA 0 A0 are the volume and surface area, respectively, of silica particles on the titania particles. The increase in the average coating thickness with time is calculated from the total volume gain of A particles: d g dV/dt (dV/dt)scavenging / (dV/dt) surface reaction Å Å 1/3 dt A0 (36p ) M2 / 3A
Monodisperse Model
kscav M1 / 3A M1 / 3 aCA££ *AA Å /q . [13] 1/3 (36p ) M2 / 3A 2pmc /kT (36p ) 1 / 3 M2 / 3A Equation [13] describes the evolution of coating thickness assuming complete coalescence of the scavenged B particles. In practice, due to incomplete sintering the coating will be porous and the thickness will be greater than that predicted by Eq. [13]. The consumption rate of the coating reactant is given by dC Å dt
S D dC dt
/
gas-phase reaction
Å 0kgC 0
S D dC dt
surface reaction
aC(AA / AB ) q
2pmc /kT
,
[14]
where the first term on the right hand side describes consumption by gas-phase reaction and the second term describes consumption by surface reaction on the surfaces of both A and B particles. Because of the lack of surface reaction kinetic data ( a ), surface reaction kinetics were assumed and varied over a range of values to examine the role of surface reaction in the coating process as shown later. Equations [6] – [9], [12a], [13], and [14] along with the auxiliary equations were solved simultaneously using the IMSL subroutine DIVPAG which is based on the backward Gear method (23).
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FIG. 2. Comparison of the present model with that of Friedlander et al. (12).
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Monodisperse models have been used in the past to describe various aerosol processes (5, 12, 22). Their major advantage compared to the more detailed and complex models is that although they are more simplistic in nature they often retain the qualitative features of the physicochemical phenomena that take place in the process under study. Therefore, the log-normal model was compared with a monodisperse model in terms of coating thickness and specific surface area of titania (A) particles. By assuming a monodisperse B particle size distribution ( s Å 1), the equations presented in the previous section can be significantly simplified. Furthermore, the number of equations needed to describe the process decreases by one because a monodisperse size distribution can be completely characterized by NB , AB , and VB whereas the log-normal distribution is characterized by an additional parameter s, the geometric standard deviation (14). In the monodisperse model the kth moment can still be calculated from Eq. [3] by putting s Å 1 and £g is VB /NB . RESULTS AND DISCUSSION
Validation In the absence of surface reaction the results were compared to the simulation results by Friedlander et al. (12) for a purely scavenging process. This comparison is shown in Fig. 2 where the normalized volume of B particles is plotted
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GAS-PHASE COATING OF AEROSOL PARTICLES
as a function of time normalized with respect to the characteristic diffusion time td (12). The monodisperse model too was compared to the work by Friedlander et al. (12) and good agreement was obtained. In the absence of surface reaction and A particles, the log-normal model was compared with the model developed by Xiong and Pratsinis (17) assuming instantaneous sintering for the same simulation conditions. The average particle size and geometric standard deviation of the B particle size distribution were in good agreement ( õ1% deviation). The model was further validated by comparing the results of the surface reaction neglecting the gas-phase reaction for monodisperse A particles and constant reactant concentration with the analytical solution (growth law) for the surface reaction (16). The maximum deviation of the coating thickness was õ1% for different residence times. In addition, a mass balance was maintained in the reactor for all simulations. Simulations in the following sections were all done under the conditions shown in Table 1 and in the absence of the surface reaction unless stated otherwise. Coating Modes Before the discussion of the results of the simulation, the various coating regimes are discussed qualitatively. In complex aerosol processes it is often useful to use characteristic times to evaluate the extent to which the different particle growth mechanisms take place (12). In the coating process discussed here there exist four different characteristic times that can describe the process: gas-phase reaction time ( trxn ), scavenging time ( tscav ), sintering time ( t ), and residence time ( tres ). Another characteristic time which is not considered here is that for the surface reaction. Based on these characteristic times, the coating process can be broadly classified into two modes: instantaneous reaction mode and constant rate reaction mode as shown in Fig. 3. In the instantaneous reaction mode ( trxn ! tres ), the precursor is consumed rapidly and monomer generation by gasphase reaction is limited to the early stages in the coating zone. The significant processes occurring downstream are scavenging and sintering. The coating particles can be completely scavenged by the host particles if tscav õ tres . If the sintering rate is high enough and the particles are given sufficient time the coating particles on the surfaces of the host particles will sinter completely forming smooth coatings as shown in Fig. 3. Otherwise, if t @ tscav rough coatings will be obtained. In the constant rate reaction mode ( trxn § tres ) particle generation takes place throughout the coating zone. If the sintering rate of the coating material is much higher than the scavenging rate ( t ! tscav ) then the process will be collision-limited while if the collision rate is higher than the sintering rate ( t @ tscav ) the process will be sintering-limited
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(Fig. 3). In the case of a collision-limited process, the coatings will be smooth and the host particles will have a low specific surface area because sintering is instantaneous (compared to the collision rate) and the coating thickness will be limited by the frequency of collisions between the coating and the host particles. On the other hand, if the sintering rate is low compared to the collision rate the coating will be uneven and the host particles will have a high specific surface area as a result of incomplete sintering of the coating particles on the surface of the host particles. From the qualitative picture of the possible coating modes it is apparent that it is desirable to operate in the collisionlimited mode for both instantaneous and constant rate reaction for smoother coatings, e.g., coatings on pigmentary titania (6, 7). On the other hand, it is desirable to operate in the sintering-limited mode for the generation of host particles with high surface area, e.g., catalytic or chromatographic supports (4, 5). Effect of Temperature Simulations were carried out at 1700 and 1800 K under the conditions summarized in Table 1 using both the monodisperse and the log-normal models. These are typical reactor temperatures encountered in the industrial production of TiO2 as well as the temperatures used in the experimental study of titania coatings with metal oxides (11). The results are expressed in terms of coating thickness and specific surface area of the titania particles. The latter is used here as a measure of the roughness of the coating. At 1700 K (Fig. 4) good agreement between the log-normal and monodisperse models was obtained and the final coating thickness predicted by the two models was Ç10.5 nm. The specific surface area was Ç45 m2 /g. The specific surface area (Fig. 4) initially increased because the scavenging rate of silica particles was greater than the sintering rate due to the high silica particle number concentration. At longer residence times ( ú5 1 10 02 s) the sintering rate of silica particles on the titania particle surfaces was greater than the scavenging rate and therefore the specific surface area of titania particles decreased. The final specific surface area of Ç45 m2 /g for the SiO2coated TiO2 corresponds to an average silica particle size of Ç1 nm (specific surface area of a spherical particle Ç2500 m2 /g) as calculated from the following equation): Sp Å
SpB 1 WB . WA
[15]
These unrealistically small silica particle sizes arise from the assumed sintering kinetics (Eq. [10b]) which were derived for micrometer-size silica spheres. The model results suggest that extrapolation of these kinetics to nanosize silica
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JAIN, FOTOU, AND KODAS
FIG. 3. Coating modes with respect to reaction rate and characteristic times for scavenging and sintering.
particles underpredicts the sintering rate resulting in smaller particle sizes. This problem has been dealt with before using accessibility factors to obtain realistic values for the specific surface area of the host particles (5). Simulations were also conducted using the Einstein diffusion equation (16) corrected with the Cunningham factor in the calculation of kscav and were compared with the more accurate Epstein equation. The coating thicknesses and spe-
FIG. 4. Evolution of coating thickness and specific surface area of TiO2 at 1700 K. Monodisperse versus log-normal model.
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cific surface areas of the host particles were in this case approximately 30% higher than those predicted using Epstein’s equation under the process conditions considered in this study (Table 1). The overall qualitative behavior of the coating process can be better understood by considering the characteristic sintering and scavenging times of the process which are plotted in Fig. 5 as a function of residence time for 1700 K. It is worthwhile to notice that the abrupt increase in the characteristic scavenging time that occurs roughly at 5 1 10 02 s corresponds to the peak of the specific surface area in Fig. 4. This suggests that for tres ú 5 1 10 02 s sintering takes place much faster than scavenging. At the point of intersection of the two curves in Fig. 5 ( tres Ç 5 1 10 02 s), the characteristic times for scavenging and sintering are equal. At shorter residence times, the characteristic time for scavenging is shorter than the characteristic time for sintering and therefore the process is limited by sintering. Even at 1800 K the two models were in close agreement with respect to the coating thickness and specific surface area of titania particles (Fig. 6). At this temperature, SiCl4 is consumed faster and over a shorter length of the coating zone. Therefore, scavenging and coagulation between silica particles take place to a greater extent compared to 1700 K and reduce the silica particle number concentration and geometric standard deviation of the silica particle size distribution. The good agreement between the log-normal and monodisperse models is attributed to the continuous scav-
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33
FIG. 5. Characteristic times for sintering and scavenging as a function of residence time at 1700 K.
FIG. 7. Effect of initial titania particle number concentration on the coating thickness at 1800 K.
enging of the fine mode (coating particles) by the coarse mode (host particles) and the relatively fast B particle generation rate by gas-phase reaction. Under those conditions, the particles in the fine mode do not have sufficient time to grow appreciably and change substantially the spread of their size distribution. Both the monodisperse and the log-normal models predicted a higher final coating thickness at lower temperatures (10.5 nm at 1700 K versus 9 nm at 1800 K). At lower temperatures, silica particles are continuously generated (constant reaction mode) and their number concentration is
high while their size is small throughout the coating zone. This translates to a higher scavenging rate of silica particles and consequently, higher specific surface areas and thicker coatings for the titania particles. In contrast, at T ú 1800 K the enhanced reaction rate results in a higher concentration of silica monomers which leads to increased collision rate between silica particles (which results in increased scavenging rate at the early stages) thereby increasing the average size of silica particles and decreasing their particle number concentration. The fewer and larger silica particles are scavenged less effectively leading to thinner coatings. Because of the strong dependence of the characteristic sintering time on temperature (Eq. [10b]), the sintering rate of silica at 1800 K is much higher compared to 1700 K and the coatings are smooth. Effect of A Particle Size Distribution and Inlet Reactant Concentration
FIG. 6. Evolution of coating thickness and specific surface area of TiO2 at 1800 K. Monodisperse versus log-normal model.
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It is important to note that in the equations describing the coating process, the only property of A particles that affects the scavenging rate of B particles by the A particles is the M1 / 3A moment. Simulations were performed to assess the effect of the one-third moment of titania particles on the coating thickness by varying their number concentration and average particle size. Results of these simulations for constant titania particle diameter of 1 mm and two initial titania particle number concentrations are shown in Fig. 7 for 1800 K. The coating thickness decreased with increased total initial number concentration of titania particles. A higher particle number concentration of titania particles (or higher surface area) re-
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JAIN, FOTOU, AND KODAS
sulted in more effective scavenging of SiO2 particles. The overall increase of the extent of scavenging of SiO2 particles by the TiO2 particles was offset by the distribution of the total silica volume over a larger surface area. Therefore, an increase in the number concentration of titania particles (assuming all other variables constant) led to a thinner coating. Conversely, the coating thickness increased with decreased initial titania particle number concentration. A similar effect was observed when the initial titania particle number concentration was held constant and the average particle size was varied. The effect of inlet reactant (SiCl4 ) concentration on the coating characteristics was also examined (Fig. 8). Increasing the initial reactant concentration resulted in an increase of the coating thickness. This is attributed to the increase in the generation rate of silica particles by the gas-phase reaction leading to a higher silica particle number concentration which resulted in an increased scavenging rate. Figure 8 shows that a hundredfold increase in the inlet reactant concentration to 2 1 10 07 mol/cm3 led to a twentyfold increase in the coating thickness. Moreover, decreasing the inlet concentration tenfold resulted in a decrease of the coating thickness by a factor of 10. Although Eq. [13] indicates that the coating thickness is directly proportional to SiCl4 concentration, the latter is coupled with NB in a nonlinear manner which also affects AA . The net effect is a nonlinear dependence of coating thickness on SiCl4 concentration. Role of Surface Reaction The effect of the SiCl4 surface reaction rate on the coating characteristics of TiO2 was also studied. This phenomenon
FIG. 8. Effect of initial coating reactant (SiCl4 ) concentration on the coating thickness at 1800 K.
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FIG. 9. Effect of surface reaction on the coating thickness of titania particles at 1800 K.
is usually neglected in modeling studies of gas-phase particle formation although it can be an important particle growth process as will be shown later. Unfortunately, no kinetic data were found on the surface reaction of silicon tetrachloride and oxygen for the conditions studied here. For this reason, it was assumed that the surface reaction kinetics of SiCl4 are similar to those of TiCl4 because both compounds are chlorides of the elements of group IV in the periodic table. The ratio of the rate constants of the surface reaction and the gas-phase reaction for silicon tetrachloride were assumed to be the same as that for the titanium tetrachloride– oxygen system (24). Moreover, the difference in the gasphase and surface reaction activation energies for silicon tetrachloride was assumed to be the same as that for titanium tetrachloride (24). Therefore, the activation energy for surface reaction was assumed to be E Å 335 kJ/mol and the pre-exponential constant in Eq. [7a], a0 Å 5 1 10 8 s 01 . With surface reaction, the predicted coating thickness is lower compared to the predicted thickness in the absence of surface reaction as can be observed by comparing Figs. 6 and 9. When the surface reaction takes place both on the titania and on the silica particles, less mass of SiCl4 is consumed to produce SiO2 particles resulting in lower number concentrations of SiO2 particles. Furthermore, silica particles grow now not only by coagulation and sintering but also by surface reaction. The surface reaction results in increased growth rates of the SiO2 particles. The lower mobility of these fewer and larger silica particles results in a lower scavenging rate and eventually thinner coatings. Since the exact surface reaction kinetics for the SiCl4 /O2 system is not known, two extreme cases were also studied
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FIG. 10. Scavenging efficiency as a function of inlet coating reactant concentration at 1800 K in the presence and absence of surface reaction.
for the sake of comparison. When the activation energy for the surface reaction was increased up to E Å 419 kJ/mol (virtually no surface reaction) the trend in the evolution of the coating thickness in Fig. 9 matched that obtained in the absence of surface reaction shown in Fig. 6. On the other hand, when E Å 251 kJ/mol the surface reaction dominated over all other particle growth mechanisms. In this case, SiCl4 is rapidly consumed by the surface reaction and the coating thickness reaches its final value very early.
35
decrease in the number concentration decreased the scavenging rate and consequently the coating efficiency. The coating efficiency would approach unity when the SiCl4 concentration is close to zero. Increasing the titania particle concentration from 10 6 to 10 7 #/cm3 resulted in a substantial increase in coating efficiency for the same SiCl4 starting concentration range because of the increase in the total area of host particles. For instance, at C0 Å 10 09 mol/cm3 , the coating efficiency increased from 90 to 100% (virtually complete scavenging of silica particles) when the surface reaction was neglected. This increase in coating efficiency was at the expense of coating thickness which under the same conditions decreased from 18 to 1 nm (Fig. 7). The above results indicate that for industrial applications where the host number concentration is normally very high ( ú10 8 #/cm3 ), the window with respect to loading of coating additive is in principle broader. Figure 10 shows that the coating efficiency for NA Å 10 7 #/cm3 remains almost 100% even when the SiCl4 initial concentration is increased up to 10 07 mol/cm3 . Therefore, higher coating reactant loadings can be used without sacrificing the coating efficiency. The surface reaction can significantly affect the coating efficiency as shown in Fig. 10. When the surface reaction was considered in the simulations, the coating efficiency was in all cases lower compared to that obtained in the absence of the surface reaction. As discussed earlier, the surface reaction results in further increase of the average silica particle size which leads to faster depletion of these particles. Because of the lower mobility of those fewer and larger silica particles, lower coating efficiencies are obtained when surface reaction is considered as a coating mechanism.
Efficiency of Coating Process As mentioned in the process description (Fig. 1), in the most general case not all of the particles formed by the gasphase reaction are scavenged. Some of them may be collected as separate particles at the end of the coating zone. For economic and scale-up considerations it is important to know what fraction of the total mass of the coating additive (SiO2 in this case) is utilized to coat the host particles. The latter provides a measure of the efficiency of the gas-phase coating process. Figure 10 shows the fraction of the total inlet mass of silica that ends up in the coating for the assumed residence time (0.6 s) as a function of the initial SiCl4 concentration. This fraction was calculated for two different titania particle concentrations (10 6 and 10 7 #/cm3 ) in the presence and absence of surface reaction. In all cases, the coating efficiency decreased with increased SiCl4 concentration. Higher SiCl4 concentrations resulted in formation of a larger number of SiO2 monomers and therefore a higher collision rate between the silica particles leading to larger SiO2 particles. The increase in size and the simultaneous
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Comparison with Experimental Observations Figures 11a and 11b are transmission electron microscope (TEM) pictures of silica-coated titania (11), obtained under conditions similar to the simulation conditions listed in Table 1 at 1573 and 1773 K, respectively. They show an average coating thickness of about 40 nm at 1573 K and about 8 nm at 1773 K. The average thickness predicted by the two models was Ç10.5 nm at 1700 K. At 1800 K, the thickness calculated was Ç9.5 nm. These are the thicknesses calculated assuming no surface reaction. When the log-normal model with assumed surface reaction kinetics was used, it predicted a thickness of Ç8 nm at 1800 K (Fig. 9). The predicted coating thickness was lower than the observed thickness at lower temperatures and in the absence of surface reaction because it was calculated assuming complete sintering. The coating experiments showed that sintering at 1573 K was not complete because distinct particles of silica could be seen on the surface of titania particles (Fig. 11a). This suggests that the sintering law used in this study (Eq. [10b]) does not success-
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JAIN, FOTOU, AND KODAS
FIG. 11. TEM picture of SiO2-coated TiO2 particles at (a) 1573 K, SiO2 /TiO2 : 4 wt% and (b) 1773 K, SiO2 /TiO2 : 6 wt%.
fully describe the sintering behavior of ultrafine silica particles at low temperatures resulting in unrealistically high specific surface areas for titania particles. In contrast, the average thicknesses calculated with and without the assumed surface reaction kinetics were in close agreement with the observed thickness at 1800 K. Without knowledge of the exact surface reaction kinetics it is not possible to conclude that coating of titania particles with silica is influenced by surface reaction. However, from the above comparison it seems possible that the surface reaction can play a significant role in the gas-phase coating of aerosol particles. CONCLUSIONS
A model was developed to simulate gas-phase coating of aerosol particles. Collision and sintering are the physical processes that control coating characteristics such as thickness and smoothness. Although the proposed model was used in this study for thin coatings by assuming that the A particle size distribution remains constant, it can be extended to thick coatings by accounting for a change in the area of A particles with time.
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Smoother coatings were obtained at high temperatures ( ú1800 K ) , low inlet coating reactant concentration, and large host particle surface area. The coating thickness increases with an increase in the inlet reactant concentration and a decrease in the total surface area of host particles. The coating efficiency expressed as the fraction of the total mass of silica that is used to coat the host particles depended strongly on the inlet SiCl4 concentration. A hundredfold increase in SiCl4 concentration resulted in a decrease of coating efficiency by 5 – 8 times under the conditions examined in this study. The surface reaction can be an important mechanism in the coating process; therefore knowledge of the surface reaction kinetics for each particular chemical system is vital in fully assessing its effect. Coating of titania particles in the gas phase would be a great advantage over the currently used wet process, especially when the titania particles are made by the chloride process (gas phase). Gas-phase coating would not only be more economical but would also simplify the production process of coated titania particles.
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37
FIG. 11 —Continued
APPENDIX: NOMENCLATURE
A0 aB
accommodation coefficient (5.3 1 10 04 ) area of B monomer (cm2 ) total surface area of A particles (cm2 /cm3 ) total surface area of B particles (cm2 /cm3 ) total surface area of B particles on titania surfaces (cm2 ) total surface area of B particles at complete coalescence (cm2 /cm3 ) initial total surface area of A particles (cm2 /cm3 ) surface area of a single B particle (cm2 )
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a a* AA AB ABA ABF
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Av b0 b2 C dB E k kfmr kg kscav m mc Mk
Avogadro’s number (6.02 1 10 23 mol 01 ) 0.666 / 0.992s 2 0 0.22s 3 (constant (13)) 0.39 / 0.5s 0 0.214s 2 / 0.29s 3 (constant (13)) coating reactant concentration (mol/cm3 ) diameter of a primary B particle (cm) activation energy for surface reaction (kJ/mol) Boltzmann constant (g cm2 /s 2 K) defined in Eq. [7a] (cm5 / 2 /s) gas phase reaction rate constant (s 01 ) defined in Eq. [6b] (cm02 s 01 ) mass of a gas molecule (g) mass of a coating reactant molecule (g) kth moment of B particle size distribution (cm3k03 )
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38 Mk A NB n( £ ) R rSiO2 Sp SpB t T VB VBA £A £* £g WA WB
JAIN, FOTOU, AND KODAS
kth moment of A particle size distribution (cm3k03 ) total B particle number concentration (cm03 ) size distribution of B particles (cm06 ) universal gas constant (82.3 cm3 atm/mol K) rate of generation of SiO2 (mol/cm3 s) specific surface area of A particles (cm2 /g) specific surface area of B particles (cm2 /g) time (s) gas and particle temperature (K) total B particle volume (cm3 /cm3 ) total B particle volume on titania surface (cm3 ) volume of a single A particle (cm3 ) volume of a monomer of B particle (cm3 ) geometric mean volume of B particles (cm3 ) total mass of A particles (g/cm3 ) total mass of B particles (g/cm3 )
Greek
a0 a bBB
trxn tscav tres
pre-exponential coefficient used in Eq. [7a] number of effective collisions coagulation coefficient for collisions between B particles (cm3 /s) coagulation coefficient for coagulation between A and B particles (cm3 /s) coating thickness (cm) density of B particles (g/cm3 ) density of the gas (g/cm3 ) geometric standard deviation of B particle size distribution characteristic sintering time of silica particles on titania surface (s) characteristic sintering time of silica particles in gasphase (s) characteristic diffusion time of silica particles to titania particles (s) (12) characteristic time for gas-phase reaction (s) characteristic scavenging time (s) residence time (s)
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bAB g rB rg s t t1 td
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ACKNOWLEDGMENTS The authors acknowledge stimulating discussions and financial support from Kemira Pigments, Inc. Dr. T. T. Kodas acknowledges support from the Office Naval Research (ONR).
REFERENCES 1. Ehrburger, P., Musielak, A., and Lahaye, J., Am. Ceram. Soc. Bull. 59, 844 (1980). 2. Nath, D. K., Am. Ceram. Soc. Bull. 65, 752 (1986). 3. Morooka, S., Kobata, A., and Kusakabe, K., AIChE Symp. Series 281, 32 (1991). 4. Powell, C. F., Oxley, J. H., and Blocher, J. M., ‘‘Vapor Deposition.’’ Wiley, New York, 1966. 5. Fotou, G. P., Pratsinis, S. E., and Baron, P. A., Chem. Eng. Sci. 49, 1651 (1994). 6. Santacessaria, E., Carra, S., Pace, R. C., and Scotti, C., Ind. Eng. Chem. Prod. Res. Dev. 21, 496 (1982). 7. Santacessaria, E., Carra, S., Pace, R. C., and Scotti, C., Ind. Eng. Chem. Prod. Res. Dev. 21, 501 (1982). 8. Clark, H. B., in ‘‘Pigments Part 1’’ (R. M. Myers and J. S. Lond, Eds.). Dekker, New York, 1975. 9. Hung, C. H., and Katz, J. L., J. Mater. Res. 7, 1861 (1992). 10. Hung, C. H., and Katz, Miguel, P. F., and J. L., J. Mater. Res. 7, 1870 (1992). 11. Powell, Q., ‘‘Gas-Phase Production and Coating of Titania in Tubular Hot-Wall Reactors,’’ Masters Thesis, University of New Mexico, 1995. 12. Friedlander, S. K., Koch, W., and Main, H. H., J. Aerosol Sci. 22, 1 (1991). 13. Pratsinis, S. E., J. Colloid Interface Sci. 124, 416 (1988). 14. Lee, K. W., Chen, H., and Gieseke, J. A., Aerosol Sci. Technol. 3, 53 (1984). 15. Powers, D. R., J. Am. Ceram. Soc. 61, 295 (1978). 16. Friedlander, S. K., ‘‘Smoke, Dust and Haze.’’ Wiley, New York, 1977. 17. Xiong, Y., and Pratsinis, S. E., J. Aerosol Sci. 22, 637 (1991). 18. Kodas, T. T., Baurm, T. H., and Comita, P. B., J. Appl. Phys. 62, 281 (1987). 19. Kodas, T. T., and Hampden-Smith, M., ‘‘The Chemistry of Metal CVD,’’ p. 479. VCH, Weinheim, 1994. 20. Koch, W., and Friedlander, S. K., J. Colloid Interface Sci. 140, 419 (1990). 21. Kingery, W. D., Bowen, H. K., and Uhlmann, D. R., ‘‘Introduction to Ceramics.’’ Wiley, New York, 1976. 22. Kruis, F. E., Kusters, K. A., Pratsinis, S. E., and Scarlett, B., Aerosol Sci. Technol. 19, 514 (1993). 23. IMSL ‘‘User’s Manual,’’ Math/Library, version 1.1, Houston, 1989. 24. Jain, S., Kodas, T. T., Wu, M., and Preston, P., J. Aerosol Sci., in press, 1996.
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CS964558
A Theoretical Study on Gas-Phase Coating of Aerosol Particles SANJEEV JAIN, GEORGE P. FOTOU,
AND
TOIVO T. KODAS 1
Center for Micro-Engineered Ceramics, Department of Chemical and Nuclear Engineering, University of New Mexico, Albuquerque, New Mexico 87131 Received February 1, 1996; accepted August 19, 1996
oxides and coating of silica fibers with ultrafine silica particles in the gas phase to increase their surface area (4, 5). An important class of applications of coated particles is the manufacture of titania pigments. Large quantities of titania are produced by gas-phase reaction and are used as a pigment in paper, plastics, paints, and other materials (4). For many applications, coating of TiO2 with various metal oxides is required to reduce photoreactivity and therefore improve the durability as well as the dispersion properties of titania pigments (6, 7). In industrial practice, titania is often produced by the chloride route (gas-phase reaction of TiCl4 and O2 ) and coated by the liquid-phase precipitation of SiO2 , Al2O3 , or Al2O3 /SiO2 mixtures from silicon and/ or aluminum hydroxide solutions followed by drying, calcining, and milling (8). The availability of a gas-phase process for coating would dramatically simplify coated pigment particle production and would have a significant impact on the economics of TiO2 pigment production. Silica-coated titania particles have been synthesized in laboratory flame reactors (9, 10). However, common flame reactor designs involve premixed reactants and do not allow sequential TiO2 formation followed by coating. In a recent experimental study, titanium dioxide particles generated by the prior reaction of titanium tetrachloride with oxygen were coated in situ with alumina, silica, and mixtures of the two in a tubular furnace aerosol reactor (11). The coating thickness and uniformity depended on the coating additive loading and the reactor temperature. This study identified the major qualitative features of the process but no theoretical description was developed. Modeling of this and other related gas-phase coating processes is vital in understanding the complex physicochemical phenomena that take place. Very few theoretical studies on the coating of aerosol particles have been reported so far (5, 12). Gas-phase coating of fibers has been modeled using a simple monodisperse model accounting for coagulation and sintering of the coating material (5). This model assumed that the particle size distribution of the host as well as the coating particles remained monodisperse throughout the reactor length. Scavenging of a fine mode aerosol (coating
In situ coating of aerosol particles by gas-phase and surface reaction in a flow reactor is modeled accounting for scavenging (capture of small particles by large particles) and simultaneous surface reaction along with the finite sintering rate of the scavenged particles. A log-normal size distribution is assumed for the host and coating particles to describe coagulation and a monodisperse size distribution is used for the coating particles to describe sintering. As an example, coating of titania particles with silica in a continuous flow hot-wall reactor was modeled. High temperatures, low reactant concentrations, and large host particle surface areas favored smoother coatings in the parameter range: temperature 1700–1800 K, host particle number concentration 1 1 10 5 – 1 1 10 7 #/cm3 , average host particle size 1 mm, inlet coating reactant concentration (SiCl4 ) 2 1 10 07 –2 1 10 010 mol/cm3 , and various surface reaction rates. The fraction of silica deposited on the TiO2 particles decreased by more than seven times with a hundredfold increase in SiCl4 inlet concentration because of the resulted increase in the average SiO2 particle size under the assumed coating conditions. Increasing the TiO2 particle number concentration resulted in higher scavenging efficiency of SiO2 . In the TiO2 /SiO2 system it is likely that surface reaction as well as scavenging play important roles in the coating process. The results agree qualitatively with experimental observations of TiO2 particles coated in situ with silica. q 1997 Academic Press Key Words: log-normal; scavenging; surface reaction; coatings; titania; silica.
INTRODUCTION
Several gas-phase techniques for coating particulate and fibrous materials have been developed over the past years for diverse applications. Silica fibers have been coated with titania films by chemical vapor deposition (CVD) to improve tensile strength for lightguide applications (1, 2). Silicon nitride and aluminum oxide particles have been coated with titanium nitride and titanium dioxide in the gas phase using fluidized bed reactors (3). Other examples include coating of nuclear fuel pellets with protective layers of metal 1
To whom correspondence should be addressed.
26
0021-9797/97 $25.00 Copyright q 1997 by Academic Press All rights of reproduction in any form reserved.
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GAS-PHASE COATING OF AEROSOL PARTICLES
FIG. 1. Schematic illustration of in situ coating of TiO2 with SiO2 .
particles) by a coarse mode aerosol (host particles) has also been modeled assuming different host particle size distributions (12). The latter model was developed to describe a purely physical process, and did not account for sintering or chemical reaction. Furthermore, the effect of surface reaction was not considered in these studies. In the present work, a model has been developed and used to describe in situ coating of particles in aerosol flow reactors to gain insights into the physicochemical phenomena taking place in this process and as a guide for design and scale-up of such a process. The model was applied for the specific case of in situ coating of titania with silica by gas-phase reaction of silicon tetrachloride. The effects of process variables such as inlet reactant concentration, temperature, residence time, and host particle size distribution on the thickness and quality (rough or smooth) of the coatings were studied. PROCESS DESCRIPTION
The in situ gas-phase coating process modeled here involves the generation of host particles by gas-phase chemical reaction of a precursor in a flow reactor which collide and sinter (coalesce) to form larger particles. Once the reaction is complete and sufficient residence time is provided, the particles attain a size distribution which can be adequately represented by a log-normal function (13). Figure 1 depicts this process for the special case of coating titania particles with silica. The model makes no assumption about the types of reactants used to produce the particles or to coat them. In the case of titania, metal halide precursors are most desirable and the coating precursors are likely to be compounds such as SiCl4 , AlCl3 , or ZrCl4 . These reactants are introduced in the vapor phase downstream into the same flow reactor and may react on the surfaces of the host particles to form a coating. Simultaneously, the precursors may react in the gas phase to form particles of the coating material. These particles collide and coalesce to form larger particles.
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The particles of the coating material are simultaneously captured (scavenging) by the larger host particles. Therefore, the coating of the host particles may take place by surface reaction as well as by scavenging of the smaller particles (SiO2 , Al2O3 , or ZrO2 ). The present model assumes a log-normal size distribution for both the host and coating particles, accounts for coating by surface reaction and scavenging, and takes into account the finite sintering rate of the coating material in the gas phase and on the host particles to describe the coating thickness and roughness. The model is used here to describe coating of titania particles with silica but it is generic and can be used for other systems also. THEORY
In the present and the following sections, the host particles are termed as A particles (titania) and particles of the coating material as B particles (silica). The assumed process conditions used in the model simulations are shown in Table 1. The following assumptions were made: 1. Formation and growth of A particles is complete and A particles with a log-normal size distribution enter the coating TABLE 1 Process Conditions Assumed in the Simulations A particle number concentration Average A particle size Geometric standard deviation of A particle size distribution Initial specific surface area of A particles Inlet reactant concentration Residence time Density of B particles Monomer size of B particles
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1 1 106 (#/cm3) 1 ( mm) 1.5 0.93 (m2/g) 2 1 1009 (mol/cm3) 0.6 (s) 2.2 (g/cm3) ˚) 4.4 (A
28
JAIN, FOTOU, AND KODAS
zone. The size distribution remains constant throughout the coating zone. The assumption of the log-normal size distribution can be relaxed as shown later. 2. The B particles are much smaller than A particles and lie in the free-molecular regime whereas A particles lie in the continuum regime. This is in agreement with experimental observations (11). 3. Collisions between A particles are negligible in the coating zone because the characteristic collision time ( Ç10 s for particle number concentration 10 8 /cm3 ) is greater than the residence time ( õ1 s). 4. CVD of the coating reactant on the walls and other wall losses are negligible. 5. The flow in the reactor is one-dimensional plug flow under isothermal conditions. 6. The gain in surface area of A or B particles by surface reaction is negligible. 7. Simultaneous CVD on sintering particles has a negligible effect on the sintering rate.
Size distribution of B particles. The equations presented here are similar to those developed by Friedlander et al. ( 12 ) to describe a purely scavenging process except that the equations presented below account for surface reaction as well as the finite sintering rate. The change in the zeroth moment ( or total B particle number concentration ) of the B particle size distribution with time is given by
dM0 dNB Å dt dt Å
£g Å
M 21 , M 30 / 2 M 12 / 2
1
S
[3]
Silica is formed in the gas phase by the reaction SiCl4 (g) / O2 (g) r 2Cl2 (g) / SiO2 (s),
[4]
where the rate of the gas-phase reaction is given by (15)
D
402kJ C, RT
[5]
where the rate is in mol/(cm3 )(s), T is in Kelvin, and C is expressed as mol/cm3 .
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dNB dt
coagulation
25 2 ln s 8
exp
5 2 1 2 ln s / exp ln s 8 8
SD
6 Å 3p p
kfmr Å
D
S
scavenging
S D
,
[6a]
r
01 / 3
kTm 1 1 2p rg (1 / pa/8)
[6b]
[2]
9 2 2 k ln s . 2
rSiO2 Å 1.7 1 10 14 exp 0
/
and ,
and the kth moment is given by (14) Mk Å M0£ gk exp
reaction
dNB dt
where the coagulation coefficient for collisions between B and A particles, kscav , is (12)
kscav
S D
gas-phase
S D
F S D S D S DG
[1]
1 M0 M2 ln 9 M 21
/
/ 2 exp
with the geometric standard deviation as (14) ln 2s Å
dNB dt
Å kgCA£ 0 kscav M1 / 3A M02 / 3 0 b0kfmr M 20 £ 1g / 6
Log-Normal Model In this section, the equations describing the gas-phase reaction rate of the coating reactant and the particle size distribution of the coating material are presented. For a lognormal size distribution function, the geometric mean volume £g is given by (14)
S D
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S DS D 3 4p
1/6
6kT rB
1/2
.
[6c]
The calculation of kscav was based on Epstein’s equation ( 16 ) following Friedlander et al.’s approach ( 12 ) . The first term on the right hand side describes the generation of particles by gas-phase reaction, the second term describes the loss due to scavenging by A particles, and the third term describes the change in the particle number concentration due to collisions between the B particles. The total particle number concentration is not affected by surface reaction. Equation [ 6b ] can also be applied for turbulent conditions which are very common in industrial aerosol reactors because of the small average B particle size ( 17 ) . The change in the total volume of B particles or the first moment with time is given by
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GAS-PHASE COATING OF AEROSOL PARTICLES
dM1 dVB Å dt dt Å
S D dVB dt
/
gas-phase reaction
S D dVB dt
/ scavenging
Å kgCA££ * 0 kscav M1 / 3A M1 / 3 /
S D dVB dt
surface reaction
a£ *CA£ AB , [7] (2pmc /kT ) 1 / 2
where a is the number of collisions which result in reaction which is normally much less than unity. The parameter a is given by an Arrhenius-type expression (18):
F G
a Å a0 exp 0
E RT
.
dM2 Å kgCA££ * 2 0 kscav M1 / 3A M4 / 3 dt
1
dAB dt
F S D S D S DG S D 3 2 exp ln s . 2
/ gas-phase reaction
S D dAB dt
/ scavenging
Å kgCA£a* 0 kscav M1 / 3A M02 / 3aB 0
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and t1 is the characteristic sintering time for silica particles in the gas phase and is given by (21) t1 Å 6.5 1 10 015 dB exp
aB Å
/
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83000 T
,
[10b]
AB NB
[11a]
and dB (the primary particle diameter) is given by (22) dB Å
dAA Å dt
6VB . AB
[11b]
dAA dt
/
scavenging
S D
[8]
sintering
(AB 0 ABF ) , t1
S D
dAA dt
Å kscav M1 / 3A M02 / 3aB 0
S D dAB dt
S D
where dB in Eq. [10b] is in cm. Using the monodisperse assumption for sintering, aB (the area of one B particle) is given by
25 2 5 2 exp ln s / 2 exp ln s 8 8
The values of the constants b0 and b2 in Eqs. [6a] and [8] were adopted from (13). The change in the total surface area of B particles ( per unit volume of aerosol ) with time is approximated by ( 20 )
S D
[10a]
Surface area and coating thickness of A particles. The change in the total surface area of A particles with time is approximated by (20)
aCA££ *aBVB / 2b2kfmr M 21 £ 1g / 6 (2pmc /kT ) 1 / 2
1 2 / exp ln s 8
dAB Å dt
ABF Å (36p ) 1 / 3 M2 / 3
[7a]
The surface reaction term in Eq. [7] and for the following equations is valid for the case of fast gas-phase diffusion of the reactant to the A particles as has been discussed in detail elsewhere (18, 19). The first term on the right hand side of Eq. [7] describes the generation of volume due to gas-phase chemical reaction, the second term describes the loss in volume due to scavenging by A particles, and the third term describes the gain in volume due to surface reaction on the B particles. The change of the second moment with time is given by
/2
where the first term on the right hand side describes the generation of area by gas-phase reaction, the second term describes the loss of area due to scavenging, and the third term describes the loss in area due to sintering. The area generation due to surface reaction is assumed negligible for thin coatings. In Eq. [9], ABF is the total area of B particles which are completely fused and is given by
(AA 0 A0 ) , t
[12a]
where A0 is the initial surface area of A particles. For thin coatings the increase in the total surface area of A particles will be negligible on complete sintering and the final area can be approximated as A0 . In the above equation the first term on the right hand side describes the gain in area by scavenging and the second term describes the loss in area due to sintering of the scavenged silica particles. In Eq. [12a] t is the characteristic sintering time of silica particles on the titania surface. The specific surface area of A particles is defined as Sp Å
[9]
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sintering
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AA , WA0
[12b]
30
JAIN, FOTOU, AND KODAS
where WA0 is the total initial mass of A particles (per unit volume of aerosol). The increase in the mass of A particles is assumed to be negligible for thin coatings. The specific surface area is used as a measure of the extent of sintering. A higher specific surface area implies that the scavenged particles have not completely sintered, whereas a specific surface area close to the initial specific surface area implies almost full coalescence. The characteristic sintering time for silica particles on the surface of titania particles is calculated using Eq. [10b] where dB is calculated as dB Å
6VBA , ABA
[12c]
where VBA Å VA 0 V0 and ABA Å AA 0 A0 are the volume and surface area, respectively, of silica particles on the titania particles. The increase in the average coating thickness with time is calculated from the total volume gain of A particles: d g dV/dt (dV/dt)scavenging / (dV/dt) surface reaction Å Å 1/3 dt A0 (36p ) M2 / 3A
Monodisperse Model
kscav M1 / 3A M1 / 3 aCA££ *AA Å /q . [13] 1/3 (36p ) M2 / 3A 2pmc /kT (36p ) 1 / 3 M2 / 3A Equation [13] describes the evolution of coating thickness assuming complete coalescence of the scavenged B particles. In practice, due to incomplete sintering the coating will be porous and the thickness will be greater than that predicted by Eq. [13]. The consumption rate of the coating reactant is given by dC Å dt
S D dC dt
/
gas-phase reaction
Å 0kgC 0
S D dC dt
surface reaction
aC(AA / AB ) q
2pmc /kT
,
[14]
where the first term on the right hand side describes consumption by gas-phase reaction and the second term describes consumption by surface reaction on the surfaces of both A and B particles. Because of the lack of surface reaction kinetic data ( a ), surface reaction kinetics were assumed and varied over a range of values to examine the role of surface reaction in the coating process as shown later. Equations [6] – [9], [12a], [13], and [14] along with the auxiliary equations were solved simultaneously using the IMSL subroutine DIVPAG which is based on the backward Gear method (23).
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FIG. 2. Comparison of the present model with that of Friedlander et al. (12).
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Monodisperse models have been used in the past to describe various aerosol processes (5, 12, 22). Their major advantage compared to the more detailed and complex models is that although they are more simplistic in nature they often retain the qualitative features of the physicochemical phenomena that take place in the process under study. Therefore, the log-normal model was compared with a monodisperse model in terms of coating thickness and specific surface area of titania (A) particles. By assuming a monodisperse B particle size distribution ( s Å 1), the equations presented in the previous section can be significantly simplified. Furthermore, the number of equations needed to describe the process decreases by one because a monodisperse size distribution can be completely characterized by NB , AB , and VB whereas the log-normal distribution is characterized by an additional parameter s, the geometric standard deviation (14). In the monodisperse model the kth moment can still be calculated from Eq. [3] by putting s Å 1 and £g is VB /NB . RESULTS AND DISCUSSION
Validation In the absence of surface reaction the results were compared to the simulation results by Friedlander et al. (12) for a purely scavenging process. This comparison is shown in Fig. 2 where the normalized volume of B particles is plotted
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GAS-PHASE COATING OF AEROSOL PARTICLES
as a function of time normalized with respect to the characteristic diffusion time td (12). The monodisperse model too was compared to the work by Friedlander et al. (12) and good agreement was obtained. In the absence of surface reaction and A particles, the log-normal model was compared with the model developed by Xiong and Pratsinis (17) assuming instantaneous sintering for the same simulation conditions. The average particle size and geometric standard deviation of the B particle size distribution were in good agreement ( õ1% deviation). The model was further validated by comparing the results of the surface reaction neglecting the gas-phase reaction for monodisperse A particles and constant reactant concentration with the analytical solution (growth law) for the surface reaction (16). The maximum deviation of the coating thickness was õ1% for different residence times. In addition, a mass balance was maintained in the reactor for all simulations. Simulations in the following sections were all done under the conditions shown in Table 1 and in the absence of the surface reaction unless stated otherwise. Coating Modes Before the discussion of the results of the simulation, the various coating regimes are discussed qualitatively. In complex aerosol processes it is often useful to use characteristic times to evaluate the extent to which the different particle growth mechanisms take place (12). In the coating process discussed here there exist four different characteristic times that can describe the process: gas-phase reaction time ( trxn ), scavenging time ( tscav ), sintering time ( t ), and residence time ( tres ). Another characteristic time which is not considered here is that for the surface reaction. Based on these characteristic times, the coating process can be broadly classified into two modes: instantaneous reaction mode and constant rate reaction mode as shown in Fig. 3. In the instantaneous reaction mode ( trxn ! tres ), the precursor is consumed rapidly and monomer generation by gasphase reaction is limited to the early stages in the coating zone. The significant processes occurring downstream are scavenging and sintering. The coating particles can be completely scavenged by the host particles if tscav õ tres . If the sintering rate is high enough and the particles are given sufficient time the coating particles on the surfaces of the host particles will sinter completely forming smooth coatings as shown in Fig. 3. Otherwise, if t @ tscav rough coatings will be obtained. In the constant rate reaction mode ( trxn § tres ) particle generation takes place throughout the coating zone. If the sintering rate of the coating material is much higher than the scavenging rate ( t ! tscav ) then the process will be collision-limited while if the collision rate is higher than the sintering rate ( t @ tscav ) the process will be sintering-limited
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(Fig. 3). In the case of a collision-limited process, the coatings will be smooth and the host particles will have a low specific surface area because sintering is instantaneous (compared to the collision rate) and the coating thickness will be limited by the frequency of collisions between the coating and the host particles. On the other hand, if the sintering rate is low compared to the collision rate the coating will be uneven and the host particles will have a high specific surface area as a result of incomplete sintering of the coating particles on the surface of the host particles. From the qualitative picture of the possible coating modes it is apparent that it is desirable to operate in the collisionlimited mode for both instantaneous and constant rate reaction for smoother coatings, e.g., coatings on pigmentary titania (6, 7). On the other hand, it is desirable to operate in the sintering-limited mode for the generation of host particles with high surface area, e.g., catalytic or chromatographic supports (4, 5). Effect of Temperature Simulations were carried out at 1700 and 1800 K under the conditions summarized in Table 1 using both the monodisperse and the log-normal models. These are typical reactor temperatures encountered in the industrial production of TiO2 as well as the temperatures used in the experimental study of titania coatings with metal oxides (11). The results are expressed in terms of coating thickness and specific surface area of the titania particles. The latter is used here as a measure of the roughness of the coating. At 1700 K (Fig. 4) good agreement between the log-normal and monodisperse models was obtained and the final coating thickness predicted by the two models was Ç10.5 nm. The specific surface area was Ç45 m2 /g. The specific surface area (Fig. 4) initially increased because the scavenging rate of silica particles was greater than the sintering rate due to the high silica particle number concentration. At longer residence times ( ú5 1 10 02 s) the sintering rate of silica particles on the titania particle surfaces was greater than the scavenging rate and therefore the specific surface area of titania particles decreased. The final specific surface area of Ç45 m2 /g for the SiO2coated TiO2 corresponds to an average silica particle size of Ç1 nm (specific surface area of a spherical particle Ç2500 m2 /g) as calculated from the following equation): Sp Å
SpB 1 WB . WA
[15]
These unrealistically small silica particle sizes arise from the assumed sintering kinetics (Eq. [10b]) which were derived for micrometer-size silica spheres. The model results suggest that extrapolation of these kinetics to nanosize silica
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FIG. 3. Coating modes with respect to reaction rate and characteristic times for scavenging and sintering.
particles underpredicts the sintering rate resulting in smaller particle sizes. This problem has been dealt with before using accessibility factors to obtain realistic values for the specific surface area of the host particles (5). Simulations were also conducted using the Einstein diffusion equation (16) corrected with the Cunningham factor in the calculation of kscav and were compared with the more accurate Epstein equation. The coating thicknesses and spe-
FIG. 4. Evolution of coating thickness and specific surface area of TiO2 at 1700 K. Monodisperse versus log-normal model.
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cific surface areas of the host particles were in this case approximately 30% higher than those predicted using Epstein’s equation under the process conditions considered in this study (Table 1). The overall qualitative behavior of the coating process can be better understood by considering the characteristic sintering and scavenging times of the process which are plotted in Fig. 5 as a function of residence time for 1700 K. It is worthwhile to notice that the abrupt increase in the characteristic scavenging time that occurs roughly at 5 1 10 02 s corresponds to the peak of the specific surface area in Fig. 4. This suggests that for tres ú 5 1 10 02 s sintering takes place much faster than scavenging. At the point of intersection of the two curves in Fig. 5 ( tres Ç 5 1 10 02 s), the characteristic times for scavenging and sintering are equal. At shorter residence times, the characteristic time for scavenging is shorter than the characteristic time for sintering and therefore the process is limited by sintering. Even at 1800 K the two models were in close agreement with respect to the coating thickness and specific surface area of titania particles (Fig. 6). At this temperature, SiCl4 is consumed faster and over a shorter length of the coating zone. Therefore, scavenging and coagulation between silica particles take place to a greater extent compared to 1700 K and reduce the silica particle number concentration and geometric standard deviation of the silica particle size distribution. The good agreement between the log-normal and monodisperse models is attributed to the continuous scav-
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33
FIG. 5. Characteristic times for sintering and scavenging as a function of residence time at 1700 K.
FIG. 7. Effect of initial titania particle number concentration on the coating thickness at 1800 K.
enging of the fine mode (coating particles) by the coarse mode (host particles) and the relatively fast B particle generation rate by gas-phase reaction. Under those conditions, the particles in the fine mode do not have sufficient time to grow appreciably and change substantially the spread of their size distribution. Both the monodisperse and the log-normal models predicted a higher final coating thickness at lower temperatures (10.5 nm at 1700 K versus 9 nm at 1800 K). At lower temperatures, silica particles are continuously generated (constant reaction mode) and their number concentration is
high while their size is small throughout the coating zone. This translates to a higher scavenging rate of silica particles and consequently, higher specific surface areas and thicker coatings for the titania particles. In contrast, at T ú 1800 K the enhanced reaction rate results in a higher concentration of silica monomers which leads to increased collision rate between silica particles (which results in increased scavenging rate at the early stages) thereby increasing the average size of silica particles and decreasing their particle number concentration. The fewer and larger silica particles are scavenged less effectively leading to thinner coatings. Because of the strong dependence of the characteristic sintering time on temperature (Eq. [10b]), the sintering rate of silica at 1800 K is much higher compared to 1700 K and the coatings are smooth. Effect of A Particle Size Distribution and Inlet Reactant Concentration
FIG. 6. Evolution of coating thickness and specific surface area of TiO2 at 1800 K. Monodisperse versus log-normal model.
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It is important to note that in the equations describing the coating process, the only property of A particles that affects the scavenging rate of B particles by the A particles is the M1 / 3A moment. Simulations were performed to assess the effect of the one-third moment of titania particles on the coating thickness by varying their number concentration and average particle size. Results of these simulations for constant titania particle diameter of 1 mm and two initial titania particle number concentrations are shown in Fig. 7 for 1800 K. The coating thickness decreased with increased total initial number concentration of titania particles. A higher particle number concentration of titania particles (or higher surface area) re-
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JAIN, FOTOU, AND KODAS
sulted in more effective scavenging of SiO2 particles. The overall increase of the extent of scavenging of SiO2 particles by the TiO2 particles was offset by the distribution of the total silica volume over a larger surface area. Therefore, an increase in the number concentration of titania particles (assuming all other variables constant) led to a thinner coating. Conversely, the coating thickness increased with decreased initial titania particle number concentration. A similar effect was observed when the initial titania particle number concentration was held constant and the average particle size was varied. The effect of inlet reactant (SiCl4 ) concentration on the coating characteristics was also examined (Fig. 8). Increasing the initial reactant concentration resulted in an increase of the coating thickness. This is attributed to the increase in the generation rate of silica particles by the gas-phase reaction leading to a higher silica particle number concentration which resulted in an increased scavenging rate. Figure 8 shows that a hundredfold increase in the inlet reactant concentration to 2 1 10 07 mol/cm3 led to a twentyfold increase in the coating thickness. Moreover, decreasing the inlet concentration tenfold resulted in a decrease of the coating thickness by a factor of 10. Although Eq. [13] indicates that the coating thickness is directly proportional to SiCl4 concentration, the latter is coupled with NB in a nonlinear manner which also affects AA . The net effect is a nonlinear dependence of coating thickness on SiCl4 concentration. Role of Surface Reaction The effect of the SiCl4 surface reaction rate on the coating characteristics of TiO2 was also studied. This phenomenon
FIG. 8. Effect of initial coating reactant (SiCl4 ) concentration on the coating thickness at 1800 K.
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FIG. 9. Effect of surface reaction on the coating thickness of titania particles at 1800 K.
is usually neglected in modeling studies of gas-phase particle formation although it can be an important particle growth process as will be shown later. Unfortunately, no kinetic data were found on the surface reaction of silicon tetrachloride and oxygen for the conditions studied here. For this reason, it was assumed that the surface reaction kinetics of SiCl4 are similar to those of TiCl4 because both compounds are chlorides of the elements of group IV in the periodic table. The ratio of the rate constants of the surface reaction and the gas-phase reaction for silicon tetrachloride were assumed to be the same as that for the titanium tetrachloride– oxygen system (24). Moreover, the difference in the gasphase and surface reaction activation energies for silicon tetrachloride was assumed to be the same as that for titanium tetrachloride (24). Therefore, the activation energy for surface reaction was assumed to be E Å 335 kJ/mol and the pre-exponential constant in Eq. [7a], a0 Å 5 1 10 8 s 01 . With surface reaction, the predicted coating thickness is lower compared to the predicted thickness in the absence of surface reaction as can be observed by comparing Figs. 6 and 9. When the surface reaction takes place both on the titania and on the silica particles, less mass of SiCl4 is consumed to produce SiO2 particles resulting in lower number concentrations of SiO2 particles. Furthermore, silica particles grow now not only by coagulation and sintering but also by surface reaction. The surface reaction results in increased growth rates of the SiO2 particles. The lower mobility of these fewer and larger silica particles results in a lower scavenging rate and eventually thinner coatings. Since the exact surface reaction kinetics for the SiCl4 /O2 system is not known, two extreme cases were also studied
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FIG. 10. Scavenging efficiency as a function of inlet coating reactant concentration at 1800 K in the presence and absence of surface reaction.
for the sake of comparison. When the activation energy for the surface reaction was increased up to E Å 419 kJ/mol (virtually no surface reaction) the trend in the evolution of the coating thickness in Fig. 9 matched that obtained in the absence of surface reaction shown in Fig. 6. On the other hand, when E Å 251 kJ/mol the surface reaction dominated over all other particle growth mechanisms. In this case, SiCl4 is rapidly consumed by the surface reaction and the coating thickness reaches its final value very early.
35
decrease in the number concentration decreased the scavenging rate and consequently the coating efficiency. The coating efficiency would approach unity when the SiCl4 concentration is close to zero. Increasing the titania particle concentration from 10 6 to 10 7 #/cm3 resulted in a substantial increase in coating efficiency for the same SiCl4 starting concentration range because of the increase in the total area of host particles. For instance, at C0 Å 10 09 mol/cm3 , the coating efficiency increased from 90 to 100% (virtually complete scavenging of silica particles) when the surface reaction was neglected. This increase in coating efficiency was at the expense of coating thickness which under the same conditions decreased from 18 to 1 nm (Fig. 7). The above results indicate that for industrial applications where the host number concentration is normally very high ( ú10 8 #/cm3 ), the window with respect to loading of coating additive is in principle broader. Figure 10 shows that the coating efficiency for NA Å 10 7 #/cm3 remains almost 100% even when the SiCl4 initial concentration is increased up to 10 07 mol/cm3 . Therefore, higher coating reactant loadings can be used without sacrificing the coating efficiency. The surface reaction can significantly affect the coating efficiency as shown in Fig. 10. When the surface reaction was considered in the simulations, the coating efficiency was in all cases lower compared to that obtained in the absence of the surface reaction. As discussed earlier, the surface reaction results in further increase of the average silica particle size which leads to faster depletion of these particles. Because of the lower mobility of those fewer and larger silica particles, lower coating efficiencies are obtained when surface reaction is considered as a coating mechanism.
Efficiency of Coating Process As mentioned in the process description (Fig. 1), in the most general case not all of the particles formed by the gasphase reaction are scavenged. Some of them may be collected as separate particles at the end of the coating zone. For economic and scale-up considerations it is important to know what fraction of the total mass of the coating additive (SiO2 in this case) is utilized to coat the host particles. The latter provides a measure of the efficiency of the gas-phase coating process. Figure 10 shows the fraction of the total inlet mass of silica that ends up in the coating for the assumed residence time (0.6 s) as a function of the initial SiCl4 concentration. This fraction was calculated for two different titania particle concentrations (10 6 and 10 7 #/cm3 ) in the presence and absence of surface reaction. In all cases, the coating efficiency decreased with increased SiCl4 concentration. Higher SiCl4 concentrations resulted in formation of a larger number of SiO2 monomers and therefore a higher collision rate between the silica particles leading to larger SiO2 particles. The increase in size and the simultaneous
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Comparison with Experimental Observations Figures 11a and 11b are transmission electron microscope (TEM) pictures of silica-coated titania (11), obtained under conditions similar to the simulation conditions listed in Table 1 at 1573 and 1773 K, respectively. They show an average coating thickness of about 40 nm at 1573 K and about 8 nm at 1773 K. The average thickness predicted by the two models was Ç10.5 nm at 1700 K. At 1800 K, the thickness calculated was Ç9.5 nm. These are the thicknesses calculated assuming no surface reaction. When the log-normal model with assumed surface reaction kinetics was used, it predicted a thickness of Ç8 nm at 1800 K (Fig. 9). The predicted coating thickness was lower than the observed thickness at lower temperatures and in the absence of surface reaction because it was calculated assuming complete sintering. The coating experiments showed that sintering at 1573 K was not complete because distinct particles of silica could be seen on the surface of titania particles (Fig. 11a). This suggests that the sintering law used in this study (Eq. [10b]) does not success-
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FIG. 11. TEM picture of SiO2-coated TiO2 particles at (a) 1573 K, SiO2 /TiO2 : 4 wt% and (b) 1773 K, SiO2 /TiO2 : 6 wt%.
fully describe the sintering behavior of ultrafine silica particles at low temperatures resulting in unrealistically high specific surface areas for titania particles. In contrast, the average thicknesses calculated with and without the assumed surface reaction kinetics were in close agreement with the observed thickness at 1800 K. Without knowledge of the exact surface reaction kinetics it is not possible to conclude that coating of titania particles with silica is influenced by surface reaction. However, from the above comparison it seems possible that the surface reaction can play a significant role in the gas-phase coating of aerosol particles. CONCLUSIONS
A model was developed to simulate gas-phase coating of aerosol particles. Collision and sintering are the physical processes that control coating characteristics such as thickness and smoothness. Although the proposed model was used in this study for thin coatings by assuming that the A particle size distribution remains constant, it can be extended to thick coatings by accounting for a change in the area of A particles with time.
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Smoother coatings were obtained at high temperatures ( ú1800 K ) , low inlet coating reactant concentration, and large host particle surface area. The coating thickness increases with an increase in the inlet reactant concentration and a decrease in the total surface area of host particles. The coating efficiency expressed as the fraction of the total mass of silica that is used to coat the host particles depended strongly on the inlet SiCl4 concentration. A hundredfold increase in SiCl4 concentration resulted in a decrease of coating efficiency by 5 – 8 times under the conditions examined in this study. The surface reaction can be an important mechanism in the coating process; therefore knowledge of the surface reaction kinetics for each particular chemical system is vital in fully assessing its effect. Coating of titania particles in the gas phase would be a great advantage over the currently used wet process, especially when the titania particles are made by the chloride process (gas phase). Gas-phase coating would not only be more economical but would also simplify the production process of coated titania particles.
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FIG. 11 —Continued
APPENDIX: NOMENCLATURE
A0 aB
accommodation coefficient (5.3 1 10 04 ) area of B monomer (cm2 ) total surface area of A particles (cm2 /cm3 ) total surface area of B particles (cm2 /cm3 ) total surface area of B particles on titania surfaces (cm2 ) total surface area of B particles at complete coalescence (cm2 /cm3 ) initial total surface area of A particles (cm2 /cm3 ) surface area of a single B particle (cm2 )
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a a* AA AB ABA ABF
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Av b0 b2 C dB E k kfmr kg kscav m mc Mk
Avogadro’s number (6.02 1 10 23 mol 01 ) 0.666 / 0.992s 2 0 0.22s 3 (constant (13)) 0.39 / 0.5s 0 0.214s 2 / 0.29s 3 (constant (13)) coating reactant concentration (mol/cm3 ) diameter of a primary B particle (cm) activation energy for surface reaction (kJ/mol) Boltzmann constant (g cm2 /s 2 K) defined in Eq. [7a] (cm5 / 2 /s) gas phase reaction rate constant (s 01 ) defined in Eq. [6b] (cm02 s 01 ) mass of a gas molecule (g) mass of a coating reactant molecule (g) kth moment of B particle size distribution (cm3k03 )
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JAIN, FOTOU, AND KODAS
kth moment of A particle size distribution (cm3k03 ) total B particle number concentration (cm03 ) size distribution of B particles (cm06 ) universal gas constant (82.3 cm3 atm/mol K) rate of generation of SiO2 (mol/cm3 s) specific surface area of A particles (cm2 /g) specific surface area of B particles (cm2 /g) time (s) gas and particle temperature (K) total B particle volume (cm3 /cm3 ) total B particle volume on titania surface (cm3 ) volume of a single A particle (cm3 ) volume of a monomer of B particle (cm3 ) geometric mean volume of B particles (cm3 ) total mass of A particles (g/cm3 ) total mass of B particles (g/cm3 )
Greek
a0 a bBB
trxn tscav tres
pre-exponential coefficient used in Eq. [7a] number of effective collisions coagulation coefficient for collisions between B particles (cm3 /s) coagulation coefficient for coagulation between A and B particles (cm3 /s) coating thickness (cm) density of B particles (g/cm3 ) density of the gas (g/cm3 ) geometric standard deviation of B particle size distribution characteristic sintering time of silica particles on titania surface (s) characteristic sintering time of silica particles in gasphase (s) characteristic diffusion time of silica particles to titania particles (s) (12) characteristic time for gas-phase reaction (s) characteristic scavenging time (s) residence time (s)
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ACKNOWLEDGMENTS The authors acknowledge stimulating discussions and financial support from Kemira Pigments, Inc. Dr. T. T. Kodas acknowledges support from the Office Naval Research (ONR).
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