Modelling particle formation and growth in a …...Modelling Particle Growth in Reactors 149 mass of gas, ni. We define a dimensionless i-mer density r~i as ni ~ nil nO (5) where no

Plasma Chemistry and Plasma Processing, Vol. 8, No. 2, 1988

Model l ing Particle Formation and Growth in a P lasma

Synthesis Reactor

Steven L. Girshick, 1 Chia-Pin Chiu, I and Peter H. McMurry 1

Received August 14, 1987; revised December 7, 1987

A modelforparticle nucleation andgrowth in a thermalplasma reactor is discussed. A nondimensionaIform of the aerosol general dynamic equation is derived under a set of simplifying assumptions which are appropriate to plasma powder synthesis, and the resulting set of equations is solved numerically. The results are converted to dimensional form for the case of iron powder, for which experimental data are available, and for silicon carbide. Calculated particle sizes increase significantly with increasing reactant concentrations and with decreasing cooling rate, although the influence of cooling rate is mainly a residence time effect.

KEY WORDS: Thermal plasmas; particle nucleation and growth; modelling.

1. I N T R O D U C T I O N

Powders synthesized in thermal plasma reactors have been produced in a variety of submicron sizes. Mean diameters are most commonly quoted in the range 10-100 nm, where "mean diameter" is usually taken as a number determined by visual inspection of electron micrographs. On the other hand, silicon carbide and ti tanium diboride powders have been plasma-produced with mass-mean diameters of 0.3-0.6/~m. (1) In any case, at least as far as is evidenced by the open literature, the relation between particle size and process operating parameters has been almost completely empirical. This paper describes a preliminary model for simulating the particle nucleation and growth process in a thermal plasma reactor.

There are three routes which are available for conversion from the gas phase to a condensed phase:

(1) homogeneous nucleation;

Department of Mechanical Engineering, University of Minnesota, Minneapolis, Minnesota 55455.

145

0272-4324/88/0600-0145506.00/0 �9 t988 Plenum Publishing Corporalion

146 Girshick, Chiu, and McMurry

(2) condensation to an existing particle; (3) heterogeneous chemistry at a particle surface.

The occurrence of homogeneous nucleation is favored by the combina- tion of fast high-temperature chemistry and rapid cooling which typically exists in a plasma reactor. Following particle inception, growth may in general occur by either of (2) or (3), as well as by coagulation.

In some plasma syntheses it might be reasonable to assume that the chemical generation of condensible vapor was infinitely fast, or in any case was completed before the onset of homogeneous nucleation. In other cases either finite-rate chemical kinetics or the equilibrium thermochemistry itself might allow the continuing generation of monomer while particles were already growing.

The importance of surface-induced chemistry as a particle-growth mechanism depends on the particular chemical system. MacKinnon, Hamblyn, and Reuben, (2) in a study of boron and of boron carbide synthesis from the reaction of BCI3 with methane and hydrogen in an argon plasma, concluded that the dominant growth mechanism for boron particles was chemical reaction of BC1 vapor at the particle surface. Similarly for boron carbide, they claimed the dominant mechanism to be chemical reaction of BC1 on precipitated carbon nuclei, with diffusion of boron into the carbon structure.

In an attempt to examine the dominant physical (as opposed to chemical) mechanisms, Yoshida and Akashi (3) injected pure iron powder into an RF argon plasma, allowing it to vaporize and then recondense. They varied the powder feed rate and the distance from the coil region to the powder collection plate. They found a reasonable agreement between their observed mean diameters ( - 1 0 n m ) and sizes calculated by a single Brownian collision-coalescence model for particle growth (4) in which at a given instant all particles are assumed to have the same size.

We consider a model for particle growth in a plasma reactor that is more realistic than that used in Ref. 3, in that particle size distributions are calculated, and the gas density is allowed to vary. The computer code AEROSOL, developed by Gelbard and Seinfeld, ~5) has been used to simulate particle growth in a plasma reactor, under the formulation discussed in the following section. Recently Okuyama et al. used the Gelbard and Seinfeld code to predict size distributions of titania, silica, and alumina powders produced by thermal decomposition of metal alkoxide vapors, and compared these calculations to experimental data. (6~ However, although measured temperatures'down the length of the furnace varied by a factor of about 2, their calculations did not account for variable gas density. In a study of fine particle formation in pulverized coal combustion, Flagan

Modelling Particle Growth in Reactors 147

and Friedlander have formulated the aerosol GDE SO as to account for variable gas density. (7) Their calculations, however, were made before the development of a numerical solution, and their analysis proceeded from the assumption of a self-preserving size distribution.

2. M O D E L

The dominant consideration in plasma synthesis is that the flow cools rapidly from a very high temperature, - 10 4 K. Thus we expect that chemical reactions that generate the condensible vapor are fast, the saturation vapor pressure drops rapidly, and hence the supersaturation ratio mounts quickly over a narrow temperature range until it can no longer be sustained and homogeneous nucleation occurs. Because saturation ratios are large, evaporation rates are small in comparison with condensation, and can be neglected. Particle growth occurs through collisions or through further chemical reaction. We will here assume that all the relevant chemistry has occurred upstream of the homogeneous nucleation event.

In the situation where both evaporation and chemical effects are absent, and neglecting migration of particles into or out of an elemental mass moving with the flow, the evolution of the particle size distribution function is a purely collisional process.

We make the following assumptions:

(1) The initial monomer concentration is given. Particle growth is initiated at a temperature To such that this monomer concentration corresponds to a specified value of the supersaturation ratio, e.g., 10. (The choice of this value has only a minor effect on the results.)

(2) A spherical particle of size k is formed by the collision of an i-met with a j-mer, where i + j = k.

(3) Particles are assumed to be spherical and electrically neutral. The latter assumption might not seem warranted for plasma synthesis, since by definition a plasma has a significant population of free charges, and it is well known that particles injected into a thermal plasma become negatively charged quite quickly. However, virtually no materials would nucleate until gas temperatures were at least below 3000 K. At 104K and atmospheric pressure, 2% of argon atoms are ionized; at 3000 K, the ionized fraction equals 2 • 10 -1~ assuming local thermodynamic equilibrium in both cases. In cases of very fast cooling, finite electron recombination rates may cause the actual electron concentration at 3000 K to be higher than the equilibrium value, or species may be present with lower ionization potentials than argon. It is probably finite-rate molecular chemistry at temperatures in the range 3000-2000 K, rather than the fact that the upstream gas is significantly ionized, which could cause the assumption of electrical neutrality to be


questioned. In any case the extent of particle charging is not likely to be nearly as great as one might suppose for a "plasma" reactor.

Van der Waals forces are neglected. The flow field is assumed to vary only in the axial direction. An elemental mass following the flow cools at a constant rate

(4) (5) (6)

s(K/s). (7) (8)

Effects on collision rates of gas nonuniformities are neglected. Particle Knudsen numbers are sufficiently large that Brownian

collision rates are accurately represented by the expression from kinetic theory. That is, the collision frequency function /3~j (m3/s) for Brownian collisions between i-mers and j-mers is given by

-(3vl~'/6F6k----r(1+1111/2(i,/~+j,/3) 2 (1) /3g- \ 4"rr / L P~ \ i j / _l

where vl is the monomer volume, k is Boltzmann's constant, T is the gas temperature, and pp is the particle density.

(9) All collisions are assumed to be effective. (10) Particle growth terminates at a specified "crystallization tem-

perature" To. The physical rationale for this is that beyond this point the primary particle size is fixed although these primary particles may continue to agglomerate. In powder synthesis it is the primary particle size that is of most importance for the consolidated material properties.

(11) Losses to walls are neglected.

The form of Eq. (1) allows us to define a dimensionless collision frequency Yo

1 T/J~ fll-'-~--4(2) 1/---'--'~ \ i j~

With "initial" referring to the time at which the supersaturation ratio equals a specified value, let No denote the initial monomer concentration per unit volume, while 130 is the initial value of /311. We then define a dimensionless time, r, and a dimensionless cooling rate, S, as

~" =-- No/3ot (3)

where t is dimensional time, and

s S --= - - (4)

No/3o To Allowing for the fact that the gas density can vary in the flow direction,

the aerosol GDE is written in terms of particle number densities per unit


mass of gas, ni. We define a dimensionless i-mer density r~i as

ni ~ n i l nO (5)

where no is the initial value of n~. Under the stated assumptions and definitions, the time evolution of

monomer concentration in an elemental mass of gas following the flow can be expressed in dimensionless form by

Dril 1

D, , / l - S ; Z ,j (6)

j=l

while the time evolution of k-mers, k > 1, is given by

Dz - ~ Y~ ~ YkinJ i+j=k j=l

The solution to Eqs. (6) and (7) yields the evolution of the dimensionless particle size distribution {~i} versus dimensionless time z, with the dimensionless cooling rate S as a parameter. The factor (1 -S . r ) -~/2 can be incorporated into a modified dimensionless collision rate c0,

~'0 (8) co - , / i - s~,

in which case Eqs. (6) and (7) are identical in form to the discrete form of the dimensional equations solved in the computer code AEROSOL described in Ref. 5. This code involves the solution of a coupled set of these equations for particle sizes k less than or equal to some specified value; we have used k = 5 for computat ional economy. For larger sizes a continuous formulation is used in which the summations are replaced by integrals. To ensure that the results were not too sensitive to the value chosen for k, we ran a typical case using several different values. Varying k by a factor of 3 produced only about a 2% difference in mean number diameter and a virtually identical size distribution. The discrete formulation is retained for small values of i, however, as its use facilitates handling the initial distribution consisting of 100% monomer.

The initial conditions are

1, i = 1 t~,(0)= 0, i > 1 (9)

3. RESULTS AND D I S C U S S I O N

In their study of iron powder formation in an RF argon plasma, Yoshida and Akashi presented data for two different iron injection rates as well as

150 Girshiek, Chiu, and McMurry

for different powder collection locations. A number of factors limit the validity of a direct comparison between our model and these data. However, this does represent a clear case in which chemical effects are absent, and no other data for this case are available. Therefore we have solved for the particle size distributions predicted by our model using properties of iron (i.e., molecular volume and mass density). The value of initial temperature To depends on the iron vapor mole fraction. Assuming a total pressure of 1 atm., the temperature To at which the supersaturation ratio has a value of 10 ranges in our calculations from 2095 K when the iron mole fraction equals 1.0%, to 1845 K when it equals 0.05%.

Solutions to Eqs. (6), (7), and (9), for various values of S and assuming To = 2095 K, are shown in Fig. 1. The calculations lie close to the straight line in the figure, which is given by

(D) = (0.70)~.o.4o (10) DI

where D1 is the monomer diameter. However, the final mean d i ame te r

t _

Z

f 10 2

"~ 1

1 6

A 0.43 n 4.3

�9 30

10 3

100 , , ' '

10 2 10 3 10 4 10 5 10 6 10 7

D i m e n s i o n l e s s t ime ,

Fig. 1. Calculated normalized number mean diameter versus dimensionless time. Results for different values of S terminate at different locations, as indicated.

depends on 5;, as each curve terminates at a value of r given by

T o - t o (1~) r a , ~ - STo

40

The form of Eq. (10) is similar to that which can be derived from the self-preserving size distribution (SPSD) obtained by Lai et al. ~8) for the case of free-molecule regime coagulation in a constant-density gas. In separate work C9~ we show that the SPSD of Ref. 8 applies as well to the variable- density case; the cooling rate enters in the inversion of the similarity transformation to dimensional variables.

In converting from nondimensional to dimensional variables, we have assumed that particles stop growing by Brownian coalescence at a temperature T~-- 1000 K; that sets the total residence time available for a given cooling rate. Yoshida and Akashi estimated their cooling rate to be 30,000 K/s. Figure 2 shows our calculations for final mean diameter as a

0 " "

0 . 0

E

30

E . ~

t_. 20

E 10 X•/D BX

x C A~t

C a l cu l a t i ons

x Da ta o f Y o s h i d a and A k a s h i

I ~ I --T T

o.2 0.4 0.6 o.s 1.0 1.2

I r on v a p o r mole f r ac t ion (%)


Fig. 2. Calculated final number mean diameter versus iron mole fraction for the case s = 30 K/ms . Data from Yoshida and Akashi(3): Case A, L = 12; B, 14; C, 16; D, 20; L = distance (cm) from last induction coil to powder collection plate.

152 Girshick, Chiu, and MeMurry

function of the mole fraction of iron in the plasma, for a cooling rate of 30,000 K/s. The data from Ref. 3 are shown for comparison. Agreement is close at the lower iron concentration (cases C and D), while the measured sizes are smaller than the calculations indicate at the higher concentration (cases A and B). A possible explanation for this is as follows. The distance from the last induction coil to the powder collection plate was smaller in cases A and B (12 and 14 cm, respectively) than in cases C and D (16 and 20 cm). It is possible that the shorter time-of-flight to the collection plate in cases A and B may have terminated the particle growth at a residence time shorter than that assumed in the calculations. It should also be noted that there are questions regarding the reliability of this method for obtaining a representative sample of particles.

Figure 3 shows a typical result for the calculated evolution of the particle size distribution, in this case with a 1% iron vapor mole fraction and a 10,000 K/s cooling rate, which corresponds to a total residence time in the particle growth window of approximately 100 ms.

~ 3 N

" ~ 2

t--- 13ms

36 ms

XFo= 1% { s = 10 K/ms ]

0 50 100 150

Pa r t i c l e D i a m e t e r , n m

Fig. 3. Calculated evolution of the particle size distribution of iron powder, with a 1.0% iron vapor concentration and a 10 K/ms cooling rate.


The calculated effect on the final mean diameter of varying the cooling rate and the vapor concentration is shown in Fig. 4. The main effect of cooling rate is primarily caused by the differences in available residence time for particle growth. The strong shift toward larger particles with increasing vapor concentration is a result of the increase in collision rates, as is evident from Eqs. (2) and (3). There is also a smaller effect present, which is that higher concentrations cause nucleation to occur sooner and thus increase the residence time available for growth at a given cooling rate. For example, the particle growth residence time for a 1% iron mole fraction is 30% greater than that for a 0.05% mole fraction, all else being equal.

At typical flow rates for an RF plasma reactoL the gas velocity in the region of particle growth would be ~ 1 m/s. Then, a 1-meter-long growth chamber with insulated walls could provide a residence time within the temperature window for coagulation of - 1 s. For a 1% vapor concentration, our results indicate that mean iron particle sizes at the end of such a chamber would be about 140 nm. The same analysis for production of a molecular powder would yield a larger particle size by virtue of the fact that the monomer volume would be larger. At a diameter of 140 nm and a temperature of 1000 K, the Knudsen number (Kn-= l/rp, where l is the mean

?

~ 150

loo

~ 50 or mole fraction (%) =

~ 1.00

0 1 , -f T

0 20 40 60 80

C o o l i n g r a t e ( K / m s )

Fig. 4. Calculated effect of iron vapor concentration and of cooling rate on final number mean particle diameter.


free path for collisions between argon atoms and rp is particle radius) would have a value of about 9. As Eq. (1) is generally restricted to K n > 10, the collision frequency function could then be more accurately expressed by using the Fuchs interpolation formula (~~ if Knudsen numbers this small are expected. In this case, however, yu becomes a complicated function of gas properties as well as of Kni and Knj, as compared to the simple form of Eq. (2). In principle this does not pose a problem, although the non- dimensionality of the procedure is compromised, in the sense that the temperature at each time step must be known.

Our analysis suggests two obvious routes to controlling particle size: through the reactant concentration and through the cooling rate (equivalently, residence time) in the temperature region for particle growth. Thus reactant stoichiometries, plasma compositions, or injection methods that favor higher yields are also conducive to larger particle sizes, as are flow geometries or reactor configurations that allow for longer residence times, assuming that losses to walls are suppressed.

While our dimensional calculations have been presented only for the case of iron powder, it is evident that this model could be extended to other powders as long as the assumptions remained valid. The nondimensional solution is not material-dependent. In particular, for cases involving chemical synthesis, this approach would remain valid as long as the relevant chemical reactions were completed before the nucleation event. Then a conversion of the nondimensional results to dimensional form would be relatively straightforward. A number of required inputs would depend on the specific material being produced. These would include the molecular weight, the mass density, the vapor pressure curve, and an effective crystallization temperature. Some materials produced by plasma synthesis, e.g., silicon carbide, are believed to form solid-phase nuclei directly (or via a short-lived intermediate) from a reaction involving gas-phase precursors.(~l~ In such cases it would be more appropriate to assume an effective nucleation temperature based on chemical equilibrium considerations or, when available, experimental data. SiC possesses a peritectic point at 3100 + 40 K, (12) so one might assume a value for To of roughly 3000 K.

For the case of refractory materials such as SiC, for given values of vapor concentration and cooling rate, several factors would cause the results to shift toward larger particle sizes than the results presented for iron would suggest. These factors include larger monomer volume, longer residence times available for growth, and higher coagulation rates.

The first two effects are obvious. When (D)/D~ is converted to (D), a molecular powder is likely to have a larger value of D~, hence of (D). The diameter of atomic iron is 0.28 nm, while that for SiC it is 0.345 rim. Likewise, regarding residence time, if we assume that growth proceeds in both cases


until the gas temperature drops to 1000 K, then SiC (To ~ 3000 K) has twice as much available growth time as iron ( To ~ 2000 K). However, the assumption that growth ceases at 1000 K is in each case arbitrary. The issue here is whether at a certain point particles of a given phase cease to coagulate to form larger primary particles, but, instead, upon colliding, attach to form agglomerates, e.g., chains of separately identifiable particles. This point is undoubtedly material specific, and it may depend on the cooling rate.

The effect on coagulation rates is seen from Eq. (1). Evidently, 13o- (D1To/pp) 1/2. Comparing SiC to iron, the value of/3o is greater for SiC by a factor of 2.2. The effects of /30 and To on the conversion from the nondimensional solution to dimensional quantities can be seen by rewriting Eq. (3), using the perfect gas law:

X/3oP r = - - ~ - o t (12)

where X is the mole fraction of condensible monomer and P is the total gas pressure. The ratio (/3o/To) for SiC is about 1.4 times that for iron.

Taking all these factors into account, we conclude that the mean particle sizes would have been greater by a factor of about 2.0 had we based the calculations on SiC rather than on Fe, as shown in Fig. 5. The maximum diameter indicated for SiC is about 0.29/~m, whereas earlier we noted that plasma-produced SiC has been reported with a mean diameter of about 0.3-0.6/zm. (1) However, based on the presentation of Ref. 1, it appears that the effective vapor mole fraction of SiC in that work was about 2%. It is also true, of course, that factors such as particle growth by heterogeneous chemical reactions may occur in SiC synthesis whereas they would be absent in iron formation. Additionally, iron particle growth is a clear liquid- coalescence type of coagulation, whereas SiC is not believed to have a liquid phase, so that coalescence in this case required in-flight sintering. This presumably would alter the growth rate of primary particles as opposed to aggregates.

For cases where the relevant chemistry would not be expected to be completed before the onset of nucleation, the chemical generation (or depletion) of monomer could be included as a source term in Eq. (2). A nondimensional ~I~E formulation that included chemical monomer generation was derived by one of us (McMurry) in a study of photochemical aerosol formation from SO2. (13~ The time variation of this source term could be obtained either from equilibrium thermochemistry or from appropriate reaction rate constants if these are available.

Although random Brownian motion is often assumed to be the dominant collision mechanism for ultrafine particles, conditions in a thermal plasma reactor could cause other factors to substantially increase collision

156 Girshiek, Chiu, and McMurry

3t~.~

~ 0 -

r 2 ~ "

o l

e-

150 r $ L

e- 5O o ~

Vapor mole fraction = 1.00% ]

SiC

0 t , , ,

0 20 40 60 80

Cooling rate, s (K/ms)

Fig. 5. Compar i son of final number mean diameter for iron and SiC, assuming vapor mole fraction in each case equals 1.00%.

rates. In particular, electric charging effects, London-van der Waals forces, thermophoresis, and diffusion across concentration gradients might all be suspected of playing a significant role. An accurate assessment of these effects would require a far more detailed study of plasma powder synthesis than has thus far been attempted.

ACKNOWLEDGMENTS

One of the authors (Girshick) acknowledges support for this work in the form of a grant-in-aid from the Graduate School, University of Min- nesota.

REFERENCES

1, T. N. Meyer, A. J. Becker, J. R. Edd, F. N. Smith, and J. Lui, "Plasma synthesis of ceramic powders," Proc. 8th International Symposium on Plasma Chemistry, Tokyo, August 31-September 4, 1987, Vol. 4, pp. 2006-2011.


2, I. M. MacKinnon, S. M. L. Hamblyn, and B. G. Reuben, "Mechanism of formation of boron and boron carbide by reduction of boron trichloride in an r.f. plasma," Proc. 3rd International Symposium on Plasma Chemistry, Limoges, France, 1977.

3. T. Yoshida, and K. Akashi, "Preparation of ultrafine iron particles using an r.f. plasma, ' ' Trans. Jpn. Inst. Met. 22, 371-378 (1981).

4. G.D. Ulrich, "Theory of particle formation and growth in oxide synthesis flames," Combust. Sci. Teehnol. 4, 47-57 (1971).

5. F. Gelbard, and J. H. Seinfeld, "Numerical solution of the dynamic equation for particulate systems," J. Comput. Phys. 28, 357-375 (1978).

6. K. Okuyama, Y. Kousaka, N. Tohge, S, Yamamoto, J. J. Wu, R. C. Flagan, and J. H. Seinfeld, "Production of ultrafine metal oxide aerosol particles by thermal decomposition of metal alkoxide vapors," AIChEJ . 32, 2010-2019 (1986).

7. R. C. Flagan, and S. K. Friedlander, "Particle formation in pulverized coal combustion--a review, "Symposium on Aerosol Science and Technology, 82nd National Meeting of the American Institute of Chemical Engineers, Atlantic City, New Jersey, August 29-September 1, 1976.

8. F. S. Lai, S. K. Friedlander, J. Pich, and G. M. Hidy, "The self-preserving particle size distribution for Brownian coagulation in the free-molecule regime," J. Colloid Interface Sci. 39, 395-405 (1972).

9. Work in progress. 10. N. A. Fuchs, The Mechanics o f Aerosols, Pergamon, Oxford (1964). 11. P. Kong, T. T. Huang, and E. Pfender, "Synthesis of ultrafine silicon carbide powders in

thermal arc plasmas," IEEE Trans. Plasma Sei. PS-14, 357-369 (1986). 12. R. I. Scace, and G. A. Slack, "Solubility of carbon in silicon and germanium," J. Chem.

Phys. 30, 1551-1555 (1959). 13. P. H. McMurry, "Photochemical aerosol formation from SO2: a theoretical analysis of

smog chamber data," J. Colloid Interface Sci. 78, 513-527 (1980).

Documents

Modelling particle formation and growth in a …...Modelling Particle Growth in Reactors 149 mass of gas, ni. We define a dimensionless i-mer density r~i as ni ~ nil nO (5) where no