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High Concentration Agglomerate Dynamics at High Temperatures M. C. Heine and S. E. Pratsinis* Particle Technology Laboratory, Institute of Process Engineering, Department of Mechanical and Process Engineering, ETH Zurich, 8092 Zu¨rich, Switzerland ReceiVed July 12, 2006. In Final Form: September 8, 2006 The dynamics of agglomerate aerosols are investigated at high solids concentrations that are typical in industrial scale manufacture of fine particles (precursor mole fraction larger than 10 mol %). In particular, formation and growth of fumed silica at such concentrations by chemical reaction, coagulation, and sintering is simulated at nonisothermal conditions and compared to limited experimental data and commercial product specifications. Using recent chemical kinetics for silica formation by SiCl4 hydrolysis and neglecting aerosol polydispersity, the evolution of the diameter of primary particles (specific surface area, SSA), hard- and soft-agglomerates, along with agglomerate effective volume fraction (volume occupied by agglomerate) is investigated. Classic Smoluchowski theory is fundamentally limited for description of soft-agglomerate Brownian coagulation at high solids concentrations. In fact, these high concentrations affect little the primary particle diameter (or SSA) but dominate the soft-agglomerate diameter, structure, and volume fraction, leading to gelation consistent with experimental data. This indicates that restructuring and fragmentation should affect product particle characteristics during high-temperature synthesis of nanostructured particles at high concentrations in aerosol flow reactors.
1. Introduction Industrial aerosol reactors are widely used for the synthesis of carbon black, fumed silica, alumina, nickel, pigmentary titania, and optical fibers. In these reactors, product primary and agglomerate particle characteristics are determined by reactant concentration and particle residence time at high temperatures.1 Population balance models such as monodisperse2 and one-3 or two-dimensional sectional4 have been developed, revealing the details of particle growth accounting for chemical reaction, particle formation, coagulation, and sintering. By interfacing such models with fluid dynamics in the reactors, the design of the latter has been better understood, facilitating their construction and operation.5,6 The primary particle size distribution always affects product properties (e.g., pigments, catalysts, etc.); however, morphology, degree of agglomeration, surface chemistry, and phase composition can be equally important for final product performance.1 Agglomerates of primary particles can be distinguished by the strength of forces that hold them together. These can be either weak physical van der Waals forces (soft-agglomerates) or stronger chemical (sinter) bonds (hard-agglomerates).7 Suspended particle properties and the suspendability of particles such as titania and zinc oxide (e.g., paints or sunscreens) or silica (e.g., rheological additive in paints and printing inks) are strongly influenced by agglomerate size, strength, and concentration among other variables.8 Suspended agglomerates immobilize part of * Corresponding author. Tel.: +41-44-632-3180. Fax: +41-44-632-1595. E-mail:
[email protected]. (1) Pratsinis, S. E. Prog. Energy Combust. Sci. 1998, 24, 197-219. (2) Kruis, F. E.; Kusters, K. A.; Pratsinis, S. E.; Scarlett, B. Aerosol Sci. Technol. 1993, 19, 514-526. (3) Tsantilis, S.; Pratsinis, S. E. AIChE J. 2000, 46, 407-415. (4) Xiong, Y.; Pratsinis, S. E. J. Aerosol Sci. 1993, 24, 283-300. (5) Mu¨hlenweg, H.; Gutsch, A.; Schild, A.; Pratsinis, S. E. Chem. Eng. Sci. 2002, 57, 2305-2322. (6) Johannessen, T.; Pratsinis, S. E.; Livbjerg, H. Chem. Eng. Sci. 2000, 55, 177-191. (7) Tsantilis, S.; Pratsinis, S. E. Langmuir 2004, 20, 5933-5939. (8) Ullmann. Ullmann’s Encyclopedia of Industrial Chemistry; Wiley-VCH: New York, 2005.
the matrix liquid, leading to an increased viscosity, while breakage of soft-agglomerate bonds results in shear thinning or thixotropic behavior.9 Industrial aerosol reactors are usually operated at the maximum possible initial precursor concentrations (close to stoichiometry) to achieve high reactor volume yields.10 For nanosized product powder, this results in large fractal-like or filamentary agglomerates that can be much larger than the constitutive primary particles. The corresponding effective particle volume fraction that includes gas entrapped in these agglomerates11 increases with increasing number of primary particles per agglomerate. As a result, there is less “free” volume per agglomerate preventing the development of the so-called steady-state concentration profile away from the receiving particle surface during coagulation, which is the basic assumption in Smoluchowski’s classic theory for Brownian coagulation (eqs 4-10).12 Additionally, soft-agglomerates might be restructured or broken by shear during their cooling as they flow through industrial reactors as in agitation of particle suspensions in liquids.13 Evidence for fragmentation in aerosols was reported during soot formation where breakage of soot agglomerates could explain better experimentally observed deviations form the self-preserving particle size distribution.14 Here, the transition from dilute to concentrated aerosol dynamics is investigated during formation and growth of SiO2 particles by gas-phase hydrolysis of SiCl4 at high concentrations (close to stoichiometry) that are typical10 in manufacture of fumed silica with specific surface areas between 90 and 380 m2/g (e.g., Cabot, Degussa, Wacker). This is the third largest industrial aerosol-made product by value (after carbon black and pigmentary titania)15 and quite relevant also for manufacture of fumed alumina (9) Oles, V. J. Colloid Interface Sci. 1992, 154, 351-358. (10) Hannebauer, B.; Menzel, F. Z. Anorg. Allg. Chem. 2003, 629, 14851490. (11) Matsoukas, T.; Friedlander, S. K. J. Colloid Interface Sci. 1991, 146, 495-506. (12) Smoluchowski, M. Z. Phys. Chem. 1917, 92, 129-168. (13) Spicer, P. T.; Pratsinis, S. E.; Raper, J.; Amal, R.; Bushell, G.; Meesters, G. H. M. Powder Technol. 1998, 97, 26-34. (14) Harris, S. J.; Maricq, M. M. J. Aerosol Sci. 2002, 33, 935-942.
10.1021/la062022q CCC: $33.50 © 2006 American Chemical Society Published on Web 10/20/2006
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and other metal oxides. Spherical as well as hard- and softagglomerate diameters are identified, and the degree of agglomeration is reported as a function of primary particle size, maximum flame temperature, and average cooling rate. The effect of high particle concentration on coagulation is investigated, showing that the basic assumption of classic Smoluchowski theory for coagulation no longer holds at these conditions. Subsequently, the likelihood of soft-agglomerate restructuring and fragmentation is explored.
2. Theory 2.1. Chemical Kinetics. Silica particles are produced by hydrolysis of SiCl4 in an oxygen/hydrogen flame.16 A preheated SiCl4/H2/air mixture reacts by fast exothermic oxidation of H2. The resulting water steam then reacts stepwise with SiCl4 following the pathway of:10 k1
SiCl4 + H2O 98 SiCl3OH + HCl
k3
SiCl2O + H2O 98 SiClOOH + HCl k4
SiClOOH 98 SiO2 + HCl with
( ) -∆Ei RT
; ∆E2 ) 209 kJ/mol
m3 ; ∆E3 ) 70 kJ/mol K3 ) 2.4 × 109 mol‚s 1 K4 ) 3.7 × 1013 ; ∆E4 ) 205 kJ/mol s The rate constant ki(T) of reaction i is expressed by the Arrhenius pre-exponential factor Ki and activation energy ∆Ei. The change in species concentration j is then:
dC dt
4
|j )
emax
C dQ
(νi,jki∏Ck) ∑ Q dt i)1 k)e
(5)
(6)
1 A dQ dA NA dC˙ SiO2 ) a - (A - NRs) dt QFg dt m τs Q dt
(7)
V dQ dV NA dC˙ SiO2 ) V dt QFg dt m Q dt
(8)
and
(3)
131
s
)
dN NA dC˙ SiO2 1 2 N dQ ) - βN Fg dt QFg dt 2 Q dt
and
K2 ) 2.8 × 10
∏
Gas properties are calculated continuously by averaging all gaseous species j ) {SiCl4, H2O, HCl, N2, O2, SiCl3OH, SiCl2O, SiClOOH} except solid SiO2 particles according to their molar fractions. 2.2. Particle Dynamics. Silica particles are formed by the above reaction sequence followed by coagulation and sintering. Neglecting the spread of the size distribution, surface growth,17,18 and the variation of the fractal dimension,2,19 the evolution of primary, dp, and collision, dc, particle diameters as well as that of the total particle number, N (1/kggas), surface area, A (m2/ kggas), and volume concentration, V (m3/kggas), is calculated by following the rate of change in N:
(2)
m3 K1 ) 8.2 × 108 ; ∆E1 ) 128 kJ/mol mol‚s
e
where NA is Avogadro’s number and β (m3/s) is the collision frequency function for monodisperse coagulation calculated by the so-called Fuchs interpolation from the free-molecular to the continuum regime replacing the particle diameter with the collision diameter for fractal-like particles.2,20 Terms on the righthand side of eq 6 from left to right describe the formation of new particles by chemical reaction, the decrease in particle number concentration by particle collision and growth, and finally the influence of the overall gas flow rate, Q, on N. The rates of change of the total particle area and volume concentrations are:
SiCl3OH 98 SiCl2O + HCl
ki ) Ki exp
∑
(1)
k2
(
max C 4 ˙k dC˙ νi,jki |j ) Q dt i)1 k)e1 Q
where Rs (m2) is the surface area of a completely fused (spherical) particle of volume V/N; while am and Vm are the area and volume of a monomer (SiO2 molecule), respectively. Similar to eq 6, the last term in eqs 7 and 8 describes the effect of changing reactor gas volume flow on particle dynamics. These terms can be avoided as for eq 4 by balancing the rate of change of N˙ ) NQ, A˙ ) AQ, and V˙ ) VQ instead of N, A, and V. The effect of agglomerate structure in β is incorporated by the collision particle diameter dc:2
dc ) (4)
1
where νi,j is the stoichiometry of species j in reaction i with the educts e1 to emax and Cj is the concentration (mol/m3) of species j. The last term in eq 4 accounts for changes in Cj by a changing reactor gas volume flow rate, Q (m3/s), for example, by changes in gas density at nonisothermal conditions or by nonstoichiometric reactions (e.g., eq 1, 2nd reaction). The molar flow rate of species j, C˙ j ) CjQ (mol/s), is independent of gas density or stoichiometry and only changes by chemical reaction, so balancing C˙ j leads to: (15) Wegner, K.; Pratsinis, S. E. Chem. Eng. Sci. 2003, 58, 4581-4589. (16) Bohnet, M. Ullmann’s Encyclopedia of Industrial Chemistry, 6th ed.; Wiley-VCH: Weinheim, 2003.
( )(
)
6V A3 A 36πNV2
1/Df
) dp(np)1/Df
(9)
where np is the average number of primaries per agglomerate, and Df is the mass fractal dimension of 2.4 or 1.8. The latter value is common for particles generated by cluster-cluster agglomeration, while the former is for monomer-cluster agglomeration.21 These Df values are attained very quickly given the high number particle concentrations here.3 Once 10 particles have been clustered together, they have reached the Df dictated (17) Talukdar, S. S.; Swihart, M. T. J. Aerosol Sci. 2004, 35, 889-908. (18) Tsantilis, S.; Pratsinis, S. E. J. Aerosol Sci. 2004, 35, 405-420. (19) Tsantilis, S.; Briesen, H.; Pratsinis, S. E. Aerosol Sci. Technol. 2001, 34, 237-246. (20) Seinfeld, J. H. Atmospheric Chemistry and Physics of Air Pollution; John Wiley and Sons: New York, 1986. (21) Schaefer, D. W.; Hurd, A. J. Aerosol Sci. Technol. 1990, 12, 876-890.
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Heine and Pratsinis
by their agglomeration mechanism.22 It should be noted that for high soot agglomerate concentrations (near gelation) even a Df ) 2.6 has been reported.23 The characteristic time for viscous flow sintering of silica is:19
τs ) 6.5 × 10-13dp exp
(
(
))
dp,min 8.3 × 104 1T dp
(10)
where dp,min ) 1 nm, and τs f 0 (full coalescence) for dp e dp,min. 2.3. State of Agglomeration. At early stages of particle formation and growth at high temperatures, spherical particles form as they fully coalesce upon collision. Then, agglomerate collision diameter and primary particle diameter are indistinguishable (dc ) dp), while the characteristic time for coagulation is much larger than that for sintering (τcoag . τs). As particles grow and reactor temperature decreases, sintering slows down and τs increases faster than τc.24,25 Particles then do not coalesce completely upon collision, so dc grows faster than dp (τcoag ≈ τs). Strong sintering necks form between colliding particles, leading to so-called hard-agglomerates (aggregates). When sintering stops and dp ceases to grow, collisions between particles do no lead to hard (sinter) bonds, ending the formation of hard-agglomerates and starting the formation of soft-agglomerates so the total aerosol area concentration is constant.7 The transition from spheres to hard-agglomerates is defined when dc is 1% larger than dp.7 The transition from hard- to soft-agglomerates is defined when dp becomes 99% of its final constant value, dpF (when dpF/dp ) 1.01). The corresponding dc then becomes the hard-agglomerate collision diameter, dcH.7 The effective agglomerate volume fraction, φc, is calculated from the agglomerate collision diameter by accounting for the entrapped gas:26
π φc ) NFgdc3 6
(11)
For fully coalesced particles, φc is identical to the solid volume fraction, φsol, while φc increases for increasing degree of agglomeration. 2.4. Effect of Turbulent Shear. Turbulence can affect particle dynamics in aerosol reactors where the collision diameter is comparable to Kolmogorov microscales of turbulence.27 The shear-induced coagulation rate of fractal-like monodisperse particles is defined as:28
4 βSh ) dc3G 3
(12)
where G is the effective shear rate determined from the Kolmogorov scale of turbulence.29 The total coagulation rate is then the sum of βBC and βSh.30 Shear might also lead to fragmentation once agglomerates reach a critical size.31 2.5. Process Conditions and Numerical Integration. Aerosol synthesis at typical industrial conditions proceeds at high (22) Mitchell, P.; Frenklach, M. Phys. ReV. E 2003, 67, 061407. (23) Sorensen, C. M.; Kim, W.; Fry, D.; Shi, D.; Chakrabarti, A. Langmuir 2003, 19, 7560-7563. (24) Lehtinen, K. E. J.; Windeler, R. S.; Friedlander, S. J. Aerosol Sci. 1996, 27, 883-896. (25) Windeler, R. S.; Lehtinen, K. E. J.; Friedlander, S. K. Aerosol Sci. Technol. 1997, 27, 191-205. (26) Kusters, K. A.; Wijers, J. G.; Thoenes, D. Chem. Eng. Sci. 1997, 52, 107-121. (27) Xiong, Y.; Pratsinis, S. E. J. Aerosol Sci. 1991, 22, 637-655. (28) Saffman, P. G.; Turner, J. S. J. Fluid Mech. 1956, 1, 16-30. (29) Pope, S. B. Turbulent Flows; Cambridge University Press: Cambridge, 2000. (30) Swift, D. L.; Friedlander, S. K. J. Colloid Sci. 1964, 19, 621-647. (31) Spicer, P. T.; Pratsinis, S. E. AIChE J. 1996, 42, 1612-1620.
Figure 1. SiCl4 conversion (solid lines) and temperature profile (dash-double-dot lines) for F130 (Tmax ) 2000 K and CR ) 40 000 K/s; thin lines) and F380 (Tmax ) 1740 K and CR ) 50 000 K/s; bold lines), respectively, as a function of particle residence time.
production rates.27 For example, for a Reynolds number of 2 × 105 and a pipe diameter of 0.25 m, the turbulent intensity can be estimated as f 0.5 ) 6%, where f is the friction factor calculated from the Ka´rma´n-Nikuradse equation.32 For a kinetic viscosity of 2 × 10-4 m2/s, the Kolmogorov shear rate is then G ) 4300 1/s. In fumed SiO2 manufacture, the highest possible precursor (SiCl4) concentrations are employed, for 5 and 10 vol % of excess hydrogen and oxygen, respectively, are used as initial conditions.10 Hence, the resulting initial gas composition of SiCl4/H2/O2/N2 ) 1.0/2.1/1.1/4.3 corresponds to a precursor SiCl4 fraction of 11.8 mol %. Detailed simulations of the H2/O2 reaction and the SiCl4 combustion showed that the H2/O2 reaction is about 2 orders of magnitude faster and is almost finished before the SiCl4 oxidation essentially starts.10 Here, for simplicity, the H2/ O2 combustion is not simulated and the initial stoichiometry is adapted assuming complete conversion of H2 and O2 to H2O. This leads to a modified initial molar ratio of SiCl4/H2O/O2/N2 ) 1.0/2.1/0.05/4.3 that is used at t ) 0. No particles then are present, and N, A, and V are initialized with very small but finite values (N0 ) 1, A0 ) am‚N0, and V0 ) Vm‚N0) to avoid division by zero without affecting the solution. The resulting set of ordinary differential equations is solved by the implicit Euler method.33
3. Results and Discussion 3.1. SiCl4 Hydrolysis and Fumed SiO2 Formation. Reaction rate simulations were validated by reproducing Figures 8 and 10 of Hannebauer and Menzel,10 while the particle growth model was validated by comparing its predictions with Figure 3 of Kruis et al.2 and Figure 2 of Tsantilis and Pratsinis.7 The reactor temperature profile is typically adjusted by the inlet temperature of the gases, the initial gas stoichiometry, and controlled cooling of the reactor walls. Two distinct flame temperature profiles (dash-double-dot lines) and the corresponding precursor conversion (solid lines) are shown in Figure 1, corresponding to the synthesis of powders with 130 and 380 m2/g specific surface area (SSA). Flames are called Fxyz where xyz is the SSA of the product. The gas inlet temperature is T0 ) 1000 K,10 which is low enough to avoid SiCl4 decomposition outside the reactor. The initial temperature increase in the reactor (32) Deen, W. M. Analysis of Transport Phenomena; Oxford University Press: New York, 1998. (33) Ehrig, R.; Nowak, U. Konrad-Zuse-Zentrum fu¨r Informationstechnik, Berlin (ZIB), 2000.
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Figure 2. Evolution of the gaseous mole fractions during hydrolysis of SiCl4 (solid line) and H2O (dashed line) to HCl (dash-double-dot line) for F380 along with the intermediate products SiCl3OH (dashdot line), SiCl2O (short dashed line), and SiClOOH (dotted line).
results from the H2/O2 combustion and is estimated by a linear temperature profile from T0 to Tmax within t1 ) 20 µs for the given conditions.10 The maximum reactor temperature, Tmax, and cooling rate, CR, are the main parameters determining product specific surface area (SSA) and hard-agglomerate diameter, dcH. Vice versa, knowing the product properties allows estimating Tmax and CR. The dcH typically ranges between 200 and 300 nm for untreated CAB-O-SIL34 measured for dry powders. For suspended AEROSIL (AERODISP), dcH ranges from 160 to 300 nm.35 The flame temperature profiles for F130 (thin line) and F380 (bold line) first increase linearly from T0 to Tmax ) 2000 and 1740 K, respectively (Figure 1). This increase is followed by a linear decrease at a constant cooling rate of CR ) 40 000 and 50 000 K/s, respectively. Once the reactor temperature rises above 1500 K during initial heat-up, SiCl4 hydrolysis sets in leading to 99% precursor conversion after 8 × 10-5 and 2 × 10-4 s for F130 and F380, respectively. For the synthesis of F380 silica with the temperature profile of Figure 1 and eqs 1-4, Figure 2 shows the detailed evolution of SiCl4 (solid line), H2O (dashed line), HCl (dash-double-dot line), along with the intermediate products SiCl3OH (dash-dot line), SiCl2O (short dashed line), and SiClOOH (dotted line). The intermediate products quickly form, leading to rather low concentrations adding up to a maximum of 1.3 mol % at about 1.2 × 10-5 s but disappear at 5 × 10-5 s. Hydrolysis of the initially 11.8 mol % SiCl4 leads to 10.6 mol % (18.3 wt %) SiO2 molecules after complete SiCl4 conversion (eq 1). Accepting that silica monomers (molecules) are formed in the absence of a nucleation barrier,36 they then grow by coagulation and sintering to form silica nanoparticles, leaving little, if any, SiO2 in the gas phase. So even at high SiCl4 loadings the effect of SiO2 on the gas-phase composition is negligible at all times for both F380 and F130 powders. This allows one to model silica monomers as particles only, resulting in a more robust and computationally efficient set of equations, decoupling essentially particle dynamics from chemical kinetics. 3.2. Fumed SiO2 Particle Dynamics. Figure 3 shows the evolution of dp (solid line) and dc (broken line) for (a) F380 and (b) F130 powders along with the effective volume fraction (dashdouble-dot line) of the agglomerates (eq 11). Once particle (34) Cabot. Product data sheet: Cabosil EH-5, H-5, LM-150, M-5; Cabot Corp., 2002; w1.cabot-corp.com/index.jsp. (35) Degussa. Product data sheet: Aerodisp W 1226, 1714, 1824, 1836; Aerosil, 2004; www.aerosil.com. (36) Ulrich, G. D. Combust. Sci. Technol. 1971, 4, 47-57.
Figure 3. Evolution of the primary (solid line) and agglomerate (broken line) particle size and the effective aggregate volume fraction, φc (dash-double-dot line), for (a) F380 and (b) F130 resulting in a primary particle size of dp ) 7.2 nm (SSA ) 380 m2/g) and 21.0 nm (SSA ) 130 m2/g), respectively. Symbols indicate the end of full coalescence (4) and the hard-agglomerate (O) diameter and effective agglomerate volume fraction (b) at the onset of softagglomerate formation.
formation sets in at about 10-5 s for F380 (Figure 1, bold line), molecular clusters grow initially by fully coalescing upon collision. As a result, dp and dc are identical up to 5 × 10-5 s (Figure 3a, 4) so particles are nonagglomerated. As particles grow further and temperature drops (Figure 1), their sintering rate slows down so particles coalesce only partially. As a result, hard-agglomerates form, until sintering effectively stops at t ) 0.0024 s when primary particles reach their final size of 7.2 nm (380 m2/g), while the corresponding hard-agglomerate collision diameter (O) is dcH ) 249 nm.7 Beyond this time, particle collisions and coagulation continue, however, without any sintering. As a result, during the synthesis of F380 powders, after t ) 0.0024 s, soft-agglomerates are formed that are held together by physical rather than chemical forces. The effective volume fraction φc (dash-double-dot line) in the pure coalescence period (t e 5 × 10-5 s) where no agglomerates are formed equals the solid volume loading that is between 0.0022% and 0.013% (not shown in Figure 3) at T ) 1740 and 300 K, respectively, after full precursor conversion. This is too low for the particles to affect the overall fluid flow by volume even though gas properties can be affected by the high solid SiO2 fraction of 18.3 wt % (solid-to-gas ratio 1:4.5). However, φc increases when agglomerates are formed (t > 0.06 ms) as they occupy far more volume than their equivalent solid sphere mass.26,37 In fact, during the onset of soft-agglomerate formation, it is already 0.17% (b) and steadily increasing with residence time. (37) Flesch, J. C.; Spicer, P. T.; Pratsinis, S. E. AIChE J. 1999, 45, 11141124.
10242 Langmuir, Vol. 22, No. 24, 2006
Figure 4. Degree of agglomeration (h ) dcH/dp) for silica particles in the process parameter space of the cooling rate, CR, and maximum temperature, Tmax, for an initial precursor mole fraction of φ0(SiCl4) ) 11.8%. Contours (yellow lines) map the final primary particle diameter. The specific conditions for F380 (Tmax ) 1740 K and CR ) 50 000 K/s), F200 (Tmax ) 1910 K and CR ) 48 000 K/s), and F130 (Tmax ) 2000 K and CR ) 40 000 K/s) are marked (O).
Higher flame temperatures in F130 (Figure 1) lead to enhanced particle sintering, resulting in longer full particle coalescence7 periods up to 8 × 10-5 s (Figure 3b, 4) than for F380 powders (Figure 3a). For t > 8 × 10-5 s, faster sintering in F130 leads to larger primary particles of 21 nm than the 7.2 nm for F380 but slightly smaller agglomerates. As a result, F130 agglomerates consist of fewer primary particles, leading to a slower increase in φc (eqs 9, 11) than that of F380 powders. For F130, the φc at the onset of the soft-agglomerate period is 0.04% (0.0042 s, b). Within t ) 1 s, the φc reaches 43% and 5.1% for F380 and F130, respectively. This indicates that for the industrially employed high particle loadings and high specific surface area flame-made powders the effective volume fraction of particles can become significant. However, even for the low SSA F130 powders, φc is bigger than 5% for t ) 1 s and bigger than 20% for t ) 10 s. For the high SSA F380, the φc becomes unrealistically high (>60%) within 2 s (Figure 3a), even exceeding 100% if agglomerates could coagulate without restructuring or fragmentation! Clearly, for these conditions classic coagulation theory no longer holds. In other words, dp and dcH are reasonably well predicted by classic Smoluchowski coagulation and sintering as they have attained their final values at rather early process times when φc was quite low (φc < 0.2%). In stark contrast, the soft-agglomerate dc may not be predicted at all by classic Smoluchowski theory. This is important as current models for coagulation and sintering cast in computation fluid dynamics simulations may predict well commonly measured particle properties such as specific surface area even at the high particle concentrations that are encountered in industrial processes.1 They may, however, fail quite miserably when used to predict the degree of agglomeration. This may not be easy to detect as measurements of agglomerate state are not standardized, yet its effects are felt in a number of end processes such as chemical-mechanical polishing, paints, or nanocomposites. 3.3. Hard-Agglomerate Formation Maps. The above results show that the effective volume fraction of hard- and softagglomerates depends on particle concentration and hightemperature residence time (Tmax and CR). Particle concentration is defined by the initial SiCl4 mole fraction, φ0, and the transient gas density. Economics dictate that φ0 is chosen as high as possible10 to achieve maximum yield for a given reactor size and
Heine and Pratsinis
Figure 5. Effective hard-agglomerate volume fraction, φcH, for silica particles in the process parameter space of cooling rate and maximum temperate. Contours show dpF (yellow lines) and the line of 99.5% precursor conversion (broken bold yellow line) for an initial precursor mole fraction of φ0(SiCl4) ) 11.8%.
filter area. The influence of the two remaining parameters, Tmax and CR, on dcH and dc is discussed below. Figure 4 illustrates the degree of hard-agglomeration, h ) dcH/dp, as a function of Tmax and CR for initial mole fraction of SiCl4 φ0 ) 11.8. Contours (yellow lines) show the corresponding product dp, which increases for increasing Tmax and decreasing CR. The highest h occurs at low Tmax and low CR, as a low Tmax leads to the formation of high concentrations of small primary particles, while low CR results in prolonged particle residence times in medium temperatures, leading to extended particle necking and to the formation of large hard-agglomerates as particles only partially sinter upon collision.7 Contour lines of dp (Figure 4) show that different operation conditions can be chosen to form powders with selected dp. For dp ) 10 nm and φ0 ) 11.8, these are bound between CR ) 2600-106 K/s and Tmax ) 1700-2150 K, leading to powders with h ) 3.4-82 (highly agglomerated particles) following the color label in Figure 4. In general, high flame temperatures and fast cooling result in hardly agglomerated silicas (h ) 1, mostly blue), while lower Tmax and CR will result in large hard-agglomerates (h . 1, mostly red).7,25 The F380 (h ) 35, dp ) 7.2 nm), F200 (h ) 20, dp ) 13.6 nm), and F130 (h ) 12, dp ) 21.0 nm) (Figure 4, O) are specific process conditions where dp is similar and the hard-agglomerate (aggregate) diameter, dcH, comparable to commercial fumed silica powders. Even though the SSA of F380 is almost 3 times higher than that of F130, the corresponding hard-agglomerate diameters of dc ) 249 and 246 nm, respectively, are almost identical for chosen process conditions of similar CR. This is consistent with the powder specifications of CAB-O-SIL, stating hard-agglomerate (aggregate) diameters between 200 and 300 nm for all untreated silica powders with SSAs from 130 to 380 m2/g. Similar results were found for silica produced in lab-scale vaporfed flames where ultra-small-angle X-ray scattering showed that the average hard-agglomerate (aggregate) diameter only varies within 20% when increasing the primary particle diameter from 12 to 37 nm by increasing the production rate from 125 to 700 g/h.38 As was shown in Figure 3, the Tmax and CR of Figure 1 affect dramatically the agglomerate volume fractions. Figure 5 explores the effect of Tmax and CR on the effective volume fraction, φc, (38) Kammler, H. K.; Beaucage, G.; Mueller, R.; Pratsinis, S. E. Langmuir 2004, 20, 1915-1921.
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Figure 6. Effective soft-agglomerate volume fraction, φcS, for silica at (a) t ) 0.1, (b) 1, (c) 10, and (d) 100 s in the process parameter space of cooling rate and maximum temperate. Contours (yellow lines) show the primary particle diameter at the corresponding time and the line of 99.5% precursor conversion (broken green lines).
of hard-agglomerates, φcH (b in Figure 3), for the conditions of Figure 4. Precursor conversion is always above 99.5% except in its top left corner (yellow broken line) where Tmax is too low and CR is too steep for complete reaction. Contours of dp (yellow lines) are essentially identical to those in Figure 4. The corresponding residence time tH at the end of hard-agglomerate formation depends on the specific process conditions and is not constant in Figure 5 but increases for decreasing CRs as expected from Figure 3. At low Tmax and low CR, φcH is largest as h is highest (Figure 4). However, even for Tmax ) 1700 K and CR ) 100 K/s, the φcH is below 0.85% so that the influence of hard-agglomerates on the overall particle and fluid dynamics is rather minor.39 3.4. Soft-Agglomerate Formation Maps. Figure 6 shows the evolution of soft-agglomerate volume fraction, φcS (eqs 9, 11), in color-coded form (φcS ) 0.1% blue, 1% light blue, 10% green, and 100% red) for t equal to (a) 0.1, (b) 1, (c) 10, and (d) 100 s along with the corresponding product dp as contours (yellow lines) at the conditions of Figure 4. At t ) 0.1 s, dp still grows at high Tmax and low CRs as reactor temperatures are still high enough for particle growth by sintering. For CR > 8000 K/s, however, the final dp is already reached at t ) 0.1 s so softagglomerate bonding takes place as the yellow isopleths do not change at longer residence times in this region (Figure 6b-d). At t ≈ 1 s (Figure 6b), however, all primary particles reached their final size so sintering has ceased in the entire Tmax-CR parameter space, forming agglomerates by coagulation that are (39) Quemada, D. Eur. Phys. J.: Appl. Phys 1998, 1, 119-127.
no longer held by chemical (sinter) bonding. As a result, regions other than the dark blue represent soft-agglomerates possessing an effective volume fraction larger than that of particles with equivalent mass. Effective volume fractions, φcS, are largest when products contain small primary particles (top left corner, high CR-low Tmax). For those, the total number of primary particles per volume is highest, resulting in large fractal agglomerates with large φcH (Figure 5). In contrast, large primary particles of low concentrations are formed at high Tmax and low CR (bottom right corner at all four t values), so the resulting agglomerates are fewer and smaller resulting in small φcS. Increasing residence times at low temperature (Figure 1) increase the size of these agglomerates and the associated φcS so the blue region in Figure 6 decreases with increasing t. This results in contours for the effective volume fraction that are almost mirroring those of primary particle size. As time increases from t ) 1 to 10 and 100 s (Figure 6c,d), soft-agglomerates grow continuously, increasing the overall φcS. Starting at t ) 1 s, all process conditions with CR > 2000 K/s and T e 1950 K result in effective volume fractions above 1%, so particle dynamics and flow characteristics will affect each other. For CR > 5000 K/s and Tmax < 1800 K, φcS is even above 10% at t ) 1 s. For larger residence times, the φcS even becomes larger than unity (e.g., t ) 10 s, CR > 25 000 K/s and Tmax < 1800 K). This indicates that agglomerate growth may no longer proceed by mere coagulation, but some restructuring and even break-up could be taking place.
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Figure 7. Evolution of the collision particle diameter, dc, and the effective volume fraction, φc, in F380 for Brownian coagulation (solid lines) and the superposition of Brownian and shear-induced coagulation (dash-double-dot lines) for Df ) 1.8 (bold) and 2.4 (thin) at G ) 5000 1/s. The dcH at the end of the hard-agglomerate region is marked (O).
Heine and Pratsinis
coagulation. At the onset of soft-agglomerate formation (t ) 0.0024 s), denser agglomerates (Df ) 2.4) have already smaller dcH (114 nm) and φcH (0.01%) values than the more open ones (Df ) 1.8) with dcH ) 249 nm and φcH ) 0.17%. During softagglomerate formation, once dc rises above 300 nm at t ) 0.003 s for Df ) 1.8, shear-induced coagulation sets in, leading to exponentially fast particle growth. At t ) 0.028 s, φcH reaches and exceeds 100%, indicating gelation as known from liquid suspensions (Figure 7). For Df ) 2.4, the onset of shear-induced coagulation is shifted to higher residence times (t ) 0.03 s) as agglomerate diameters are smaller. Again, exponentially fast shear-induced growth can be observed; however, gelation does not set in until t ) 0.4 s (Figure 7). Clearly, such results for flowing particle suspensions call for fragmentation of these soft-agglomerates (t > 2.6 ms). Recently, it was shown that soft-agglomerated fumed silica particles can be broken, for example, by high-pressure dispersion.41 Similarly, shear forces in the turbulent industrial reactors will probably overcome the weakest physical (van der Waals) bonds of softagglomerates. Typically the fragmentation probability increases dramatically during soft-agglomerate growth37 until coagulation and fragmentation dynamics balance, essentially stopping softagglomerate growth. This shows that at high residence times, dc and φc are determined by flow properties such as fluid viscosity and Kolmogorov shear rate and particle properties that determine the attractive forces between adjacent primary particles. Vice versa, φc of soft-agglomerates can become so significant that particles will influence the overall fluid flow and heat transfer from the suspension. Depending on the onset of high concentration particle dynamics, the latter will delay particle cooling and may lead to larger primary particles and hard-agglomerates.
4. Conclusions
As a result, particle formation at high concentrations and long residence will be determined by particle concentration-dependent coagulation,40 restructuring, and fragmentation that will mostly affect soft-agglomerate size and structure (Figure 6), while hardagglomerate structure and primary particle size may be affected to a lesser extent (Figures 4 and 5). 3.5. Consequences of High Concentrations: Restructuring and Fragmentation. High concentration particle dynamics will increase the rate of coagulation given by the classic Smoluchowski theory. Depending on when these dynamics set in, they may affect dp, dcH, and h but most dramatically the soft-agglomerate collision diameter. As a result, it is reasonable to expect setting in of other phenomena such as restructuring and fragmentation. Here, an estimate of restructuring on the evolution of softagglomerate diameter and effective volume fraction is explored. Figure 7 shows the increased growth of agglomerate size (top) and effective volume fraction (bottom) when accounting for superimposed Brownian and shear-induced coagulation for G ) 5000 1/s (dash-double-dot lines) as compared to pure Brownian coagulation (solid lines). To explain the effect of restructuring, two limiting cases of Df ) 1.8 (bold lines) and Df ) 2.4 (thin lines) are shown for F380 powders. Below 5 × 10-5 s, nothing matters as spherical particles are formed (Figure 3a), so φc ) φsol. For larger t, denser agglomerates with smaller collision diameters are formed for Df ) 2.4 than for 1.8, resulting in slower agglomerate particle growth and lower φc. Prior to the onset of soft-agglomeration (Figure 3a: t < 0.0024 s), particle growth is not affected by shear (Figure 7), so regardless of Df, both dcH and φcH are accurately predicted by pure Brownian
The synthesis of nanostructured SiO2 by SiCl4 hydrolysis was investigated at high concentrations (∼12% SiCl4 by volume) that are typically encountered in the manufacture of fumed silica. Accounting for detailed reaction kinetics, coagulation and sintering, neglecting particle polydispersity, and evolution of fractal structures, conditions for synthesis of hard- and softagglomerates were identified and related to the effective particle concentration in the reactor. In particular, for high specific surface area powders, a transition from dilute to concentrated particle dynamics takes place. The influence of maximum reactor temperature and cooling rate on primary particle, hard- and soft-agglomerate sizes, and volume fractions was investigated and found consistent with industrial product specifications and limited experimental data. The degree of hard-agglomeration and primary particle diameter seems to be barely affected by high concentration particle dynamics. The effective volume fraction of the hard-agglomerates is below 0.9% in the entire parameter space at the employed conditions. In contrast, the effective volume fraction of softagglomerates quickly rises above 1% or even 10% for most process conditions. This indicates that classic Smoluchowski theory may not accurately describe their coagulation and most importantly there should be a significant interaction between fluid and particle dynamics during their transport, cooling, or collection in industrial aerosol reactors. At high particle concentrations, this enhanced Smoluchowski Brownian and/or shear-induced coagulation could lead to rapid soft-agglomerate
(40) Trzeciak, T. M.; Podgorski, A.; Marijnissen, J. C. M. Inz. Chem. Procesowa 2004, 25, 1741-1746.
(41) Wengeler, R.; Teleki, A.; Vetter, M.; Pratsinis, S. E.; Nirschl, H. Langmuir 2006, 22, 4928-4935.
High Concentration Agglomerate Dynamics
growth and eventual gelation. Restructuring and shear-induced fragmentation, however, should slow and stop soft-agglomerate growth in industrial flow reactors. Acknowledgment. Financial support by the Swiss National Science Foundation (SNF), 200020-107947/1, is gratefully acknowledged. Nomenclature a A A˙ C C˙ CR d Df e ∆E f G h i j k K N N˙ NA Q R t
particle surface area [m2] total particle surface area concentration [m2 kg-1] particle surface area flow rate [m2 s-1] gas-phase concentration [mol m-3] molar flow rate [mol s-1] cooling rate [K s-1] particle diameter [m] mass fractal dimension [-] educt [-] activation energy [J mol-1] friction factor [-] shear rate [s-1] degree of agglomeration [-] index for chemical reaction [-] index for gas component [-] reaction rate constant [m3 mol-1 s-1 or s-1] Arrhenius pre-exponential factor [m3 mol-1 s-1 or s-1] particle number concentration [kg-1] particle number flow rate [s-1] Avogadro’s number [mol-1] gas flow rate [m3 kg-1] ideal gas constant [J mol-1 K-1] time [s]
Langmuir, Vol. 22, No. 24, 2006 10245 T v V V˙
temperature [K] particle volume [m3] total particle volume concentration [m3 kg-1] particle volume flow rate [m3 s-1]
Greek Symbols R β φ ν F τ
surface area of completely fused (spherical) particle [m2] particle collision frequency [m3 s-1] volume fraction [-] stoichiometry in chemical reaction [-] density [kg m-3] characteristic time scale [s]
Indices BC c coag F g H 0 m max min N p s S Sh sol
Brownian continuum regime collision coagulation final gas hard initial (t ) 0) monomer (molecule) maximum minimum nonagglomerated (spherical) primary sintering soft shear solid LA062022Q
