homogeneous nucleation of particles from the vapor … · plasma chemistry and plasma processing,...

Plasma Chemistry and Plasma Processing, "coL 9, No. 3, 1989

Homogeneous Nucleat ion of Particles from the Vapor

Phase in Thermal P lasma Synthesis I

S. L. Girshick 2 and C.-P. Chiu 2

Received September 21, 1988, revised December 21, 1988

Particle nucleation and growth are simulated for iron vapor in a thermal plasma reactor with an assumed one-dimensional flow field and decoupled chemistry and aerosol dynamics. Including both evaporation and coagulation terms in the set of cluster-balance rate equations, a sharply defined homogeneous nucleation event is calculated. Following nueleation the vapor phase is rapidly depleted by condensation, and thereafter particle growth occurs purely by Browman coagulation. The size and number of nucleated particles are found to be affected strongly by the cooling rate and by the initial monomer concentration. An explanation is presented in terms of the response time of the aerosol to changing thermodynamic conditions.

KEY WORDS: Plasma synthesis; homogeneous nucleation; ultrafine powders.

1. INTRODUCTION

The dynamics of particle formation in a thermal plasma reactor is an important problem in plasma synthesis of ultrafine powders of advanced materials. A number of factors can be present which make analysis of this problem extremely complicated, even aside from the aerosol dynamics, for example, nonuniform temperature and velocity profiles, imperfect mixing of reactants, and finite-rate chemical kinetics. However, considerable insight can be gained if we focus on the aerosol dynamics by considering an idealized one-dimensional flow reactor in which the relevant gas-phase chemistry is completed upstream of particle nucleation.

This work appears in abbreviated form in the proceedings of the International Symposium on' Combustion and Plasma Synthesis of High Temperature Materials, San Francisco, Oct. 24-26, 1988, to be published as Combustion and Plasma Synthesis of High Temperature Materials, Z. A. Munir and J. B. Holt (eds.), VCH, New York (in press).

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

355

0272-4324/89/0900-0355506.00/0 �9 I989 Plenum Publishing Corporation

356 Girshick and Chi.

This situation is shown schematically in Fig. 1, The plasma itself, whether it is generated by DC or RF methods, is at a temperature of about 10,000 K. We assume that reactants, if injected in powder or droplet form, are completely vaporized. Molecular gases are largely dissociated in the hottest regions, but as the gas cools chemical reactions generate one or more condensable species. For simplicity we oss_umr that there is only one condensable species; molecules of that species are referred to as monomers. In the absence of preexisting particles, particle nucleation would not be expected to occur until the gas becomes significantly undercooled with respect to the condensable species.

In previou~ work Girshick et aL (~) modeled the scenario just described using a two-region model. Initially a fixed monomer concentration exists but the gas is so hot that the saturation ratio is not high enough to overcome the energy barrier due to the surface tension of very small droplets. In this region it is assumed that no particles can form, because evaporation rates far exceed condensation rates. But upon sufficient cooling the supersaturation reaches such a high value that evaporation can be neglected, and particle nucleation and coagulation can both be described as irreversible agglomeration processes. Obviously these two regions must be bridged by a transition region in which both evaporation and coagulation are significant, but under conditions of very fast cooling, such as may occur in a plasma reactor, it was assumed in Ref. 1 that the transition regime would be extremely brief and could be neglected. An analogous "collision-controlled nucleation" model was developed by McMurry and Friedlander (2) and similar approaches have been adopted by McMurry ~3~ in a study of atmo- spheric aerosol formation, by Wu et aL ~4~ to simulate the production of silicon particles by pyrolysis of silane, and by Bolsaitis et al, (5~ in a study of metal oxide particle nucleation in a supersaturated laminar jet.

One practical reason for the neglect of evaporation in the earlier work was the lack of efficient numerical schemes for solving the aerosol dynamics equations with simultaneous coagulation and evaporation. Recently efficient techniques have been developed which we have found to be effective for a

Centerllne Temperature 10,000 K 5000 - 2000 2000 - 1000

Fig. 1. One-dimensional schematic of plasma synthesis.

Homogeneous Nucleation of Particles in Plasma Synthesis 357

certain locus of conditions regarding monomer concentrations and cooling rates. Fortunately these conditions are of considerable relevance to the conditions of a plasma reactor. Thus the present study discusses calculations in which evaporation and coagulation terms are simultaneously included. To the authors' knowledge the only previous case in which the full set of cluster-balance equations was solved with simultaneous coagulation and evaporation was the recent work of Rao and McMurry (6) in which particle nucleation and growth were calculated for a constant-temperature system with a constant rate of monomer generation.

We find that including evaporation terms in the cluster-balance equations makes a decided difference in the particle size evolution. Whereas the calculations in which evaporation were neglected produced continuously "smooth" growth curves, the inclusion of evaporation produces a sharply defined nucleation event in which particle sizes jump so rapidly that mean size versus time is almost a step function; thereafter growth follows the slower curve predicted by coagulation theory.

These new calculations indicate a strong dependence of the particle size immediately following the nucleation burst on the cooling rate at the point of nucleation. In addition to what is already known about the importance of reactant concentrations and total residence time, this suggests effective practical strategies for controlling particle size in thermal plasma synthesis.

2. THEORY

Our concern is the evolution of the particle size distribution function n(v, t), where v is the particle volume, as the gas cools from an initially high-temperature homogeneous state. Let nk represent the concentration of particles (m -3) of size k (i.e., containing k monomers, or a "k-mer").

Monomers, clusters, and particles collide due to random Brownian motion. For most cases of interest to plasma synthesis particle Knudsen numbers are sufficiently large that the collision-frequency function from gas kinetic theory may be used: particles collide just as if they were large molecules in a dilute gas. Then the collision-frequency function flu (m3/sec) for collisions between an i-mer and a j-mer is given by

/3v# fii,=\-4-~ j l | ~ 7 ~7+~) (i'/3+j*/3) = (1)

Here v~ is the monomer volume, k is Boltzmann:s constant, T is the absolute temperature, and pp is the particle mass density. This expression assumes spherical, electrically neutral particles and neglects London-van der Waals forces. Approaches such as shape factors or fractal theory appear promising

358 Girshiek and Chiu

for modeling coagulation of nonspherical particles; our main interest here is in ultrafine particles during and shortly following homogeneous nucleation, for which sphericity seems a reasonably good model for collision cross sections. Electrical neutrality might seem an unwarranted assumption for a plasma reactor. Yet particles would rarely be expected to nucleate until temperatures dropped to about 2000 K or lower, at which temperatures equilibrium electron number densities are negligible. If preexisting particles were present, however (we assume they are not), they would carry charge from the plasma. The neglect of dispersion forces probably causes collision frequencies to be somewhat underestimated. Graham and Homer (7> esti- mated for a lead aerosol at 940 K that dispersion forces increased coagulation rates by a factor of about two. However, the magnitude of this effect decreases with increasing temperature. (8)

The monomer population balance for a constant-temperature system can be written as

dnl--n 1 ~ ~xjnj-t-~ (1-~-~2j)Ejl'I j (2) dt j=~ :=2

The first term on the right-hand side represents loss of monomers due to collisions, with an assumed mass accommodation coefficient of unity. The second terms accounts for production of monomers due to evaporation. The Kronecker delta function 62j accounts for the production of two monomers when a dimer dissociates. The evaporation coefficient Ej is given by (9)

fa, r ( j - I)2/3]} Ej = [~ljl'ls e x p ~ - ~ tY'=/3 _ (3)

where ns is the equilibrium monomer concentration based on the saturation vapor pressure at temperature T, A~ is the monomer surface area, and cr is surface tension.

The analogous population balance equations for particles of dimer size or larger can be written

dnk_l ~ flijninj--nk ~ ~jknj.-kEk+lnk+l--Eknk, k>-2 (4) dt 2 ~+/=k j=l

A plasma reactor is far from being a constant-temperature, constant- density system. To ensure that the flow satisfies mass continuity it is more convenient to cast the particle size distribution function in terms of particles per unit mass of gas rather than per unit volume. We assume for simplicity that the gas is at constant pressure and cools at a constant rate b(K/sec). By writing Eqs. (2) and (4) for a flow, defining suitable dimensionless

Homogeneous NucLeation of Particles in Plasma Synthesis 359

quantities, and invoking the ideal gas law, we obtain the following form of the population balance equations:

Dnl Dr

cx~ 1 nlJ --~I l~ljl~j..[..~-f_B3 " ~ (]..~2j)~3j~j j=2

and

(5)

t l Dr ~ i+j=k j=l

+ V~-- B~" [Ek+l nk+l-- Ekt~k], k--->2 (6)

The left-hand sides are written as substantial derivatives to emphasize that the equations are written for an elemental mass of gas following the flow. The nondimensional quantities, which are defined under Nomen- clature, are the size distribution function r~k, time r, cooling rate B, collision- frequency function/ti j , and evaporation coefficient/~k. In practice the upper limits of the coagulation summations are set to finite values which depend on the largest expected particle size and on the numerical scheme employed.

In Ref. 1 initial conditions were chosen based on starting the calculations at a value of the supersaturation for which the neglect of evaporation seemed reasonably valid; a value of 10 was arbitrarily chosen. In the present case there is no need to make such an arbitrary assumption. Indeed, this is an important advantage to being able to include evaporation terms. The value of the saturation ratio at which homogeneous nucleation occurs is not known a priori, but it obviously cannot be less than unity. Therefore we have begun our calculations at time zero corresponding to a point at which the saturation ratio equals unity. With the nondimensional particle size distribution normalized by the monomer concentration at that point, the initial conditions for Eqs. (5) and (6) are

1, i = 1 (7) ~i(O)= O, i>-2

This choice of initial conditions is an approximation since we would expect a subcritical cluster size distribution to exist already at time zero. We justify this choice by observing that while a distribution of subcritical clusters develops rapidly in the calculations, a relatively long time elapses before the occurrence of nucleation. The initial value of the temperature To is that value for which the saturation ratio equals unity, based on the given monomer concentration and on the vapor pressure curve for the substance in question.

As in Ref. 1 the present calculations are based on elemental iron, for which necessary property inputs are known. The surface tension of iron is

360 Girshick and Chiu

known to vary somewhat with temperature, (a~ but for simplicity we have assumed a constant value of 1.7 N/m. Implicit in this are the assumptions that the surface energy of microscopic clusters can be represented by macroscopic measurements and that the iron nucleates as liquid droplets rather than as solid particles, even though our calculated nucleation temperatures are somewhat undercooled from the viewpoint of solidification. The latter assumption is consistent with the discussion by Akashi and Yoshida, (11) who produced iron powder in an RF induction plasma by evaporating iron feed powder and letting it nucleate and recondense. The iron vapor pressure curve is taken from the compilation by Hultgren et aL (~2)

3. NUMERICAL METHOD

In principle Eqs. (5) and (6) can be written for particles of every discrete size over the entire expected range of sizes, yielding a set of coupled ordinary differential equations to be solved at each time step. However, a particle with a diameter of 0.1/xm contains o v e r 10 7 monomers, so this approach is obviously impractical. In Ref. 1 a "discrete-continuous" formu- lation was used, in which separate equations are written for the smallest sizes but larger sizes are represented by a continuous size distribution, the coagulation summations being replaced by integrals. In the present work we have used the numerically more efficient "discrete-sectional" technique, in which larger sizes are represented by separate discrete sections. The section widths (in particle volume space) are logarithmically scaled so as to give highest resolution for the smallest sizes. Within a given section the size distribution function is assumed to have a uniform value. The sectional approach itself was originally developed by Gelbard et aL (13) A discrete- sectional code was recently developed by Rao and McMurry (6) and, indepen- dently, by Wu et aL (4)

In the present work we have modified the code written by Rao and McMurry. Depending on input parameters, the total set of equations con- sisted of 10-15 equations for the discrete regime plus 25-30 equations to represent the sections. The given vapor concentration sets the value of To. At each time step the temperature is updated according to the given cooling rate, and all temperature-dependent terms are recalculated.

A major cause of numerical difficulties is the stiffness of the set of equations, especially in the region where evaporation and coagulation rates are both important. We used the stiff equation code EPISODE. (~4) We found that the primary determinant of computer time was the value of the nondimensional cooling rate B; lower values of B resulted in longer calculation times. Total calculation time ranged from only a few minutes on a Macintosh II microcomputer to about 200 CPU seconds on a Cray-2 supercomputer.


RESULTS AND DISCUSSION

In order to illuminate the physical processes which drive particle nucleation and growth, this section focuses on a single base case, followed by cases in which either the cooling rate or the monomer concentration was varied.

4.1. Base Case

For the base case we consider an iron vapor partial pressure of 5 x 10 -4 atm and a cooling rate of 30 K/msec. At this concentration the saturation ratio equals unity when the temperature equals 2018 K, so the calculation is begun at that point. Figure 2 shows the calculated value of the saturation ratio as the gas cools, where the saturation ratio S at each point is defined by S = na/n~. The thick line in the figure, labeled "nucleation suppressed," indicates how the supersaturation would increase if particles larger than monomers were not allowed to form. The calculated saturation ratio follows this line closely until the gas cools to about 1600 K, corresponding to a supersaturation of about 280. At that point the calculated saturation

~

10 4

10 3

10 2

10 1

i0 0 ~ ,

I000 1200

~,,.nucleation suppressed

1400 1600 t800 2000

Temperature (K)

Fig. 2. Calculated saturation ratio for the base case: Pve = 5 x 1 0 - 4 atm, b = 30 K/msec. The "nucleat ion suppressed" curve indicates the saturation ratio if the vapor were constrained to remain entirely as monomers.

362 G i r s h i c k and Chiu

ratio peaks and then drops rapidly; by 1400 K the calculated monomer concentration has relaxed to its equilibrium value.

To understand this behavior it is instructive to consider the critical size for a stable nude,us, which is the size for which the condensation rate balances the evaporation rate: smaller particles tend to evaporate and larger ones to grow. In terms of particle diameter the critical nucleus size can be written (8)

d* = 4o-vl (8) kT In S

Figure 3 shows the calculated variation in the critical cluster size k* (number of monomers comprising the critical size nucleus). In early times the critical cluster size is very large, and only monomers and smaller clusters are present, so no stable particles can form and the supersaturation mounts. But the rapid rise in the supersaturation eventually causes k* to drop to a very small value, in this case equal to 12. When the critical size becomes this small, there finally are clusters present whose size is equal to or larger than k*, because the prenucleation size distribution does not consist entirely of monomers but includes a distribution of subcritical clusters. Thus the cooling itself, rather than particle growth, causes a previously subcritical cluster to become stable, and the highly supersaturated vapor has nucleation sites available where previously there were none.

41. ~e

t~

t..

.o t . ,

10 5

10 4

10 3

10 2

10 1

1000 I

1200 I I i I

1400 1600 1800 2000

Temperature (K)

2200

Fig. 3. The critical size for a stable nucleus, calculated for the base case.


120000

1OO0O3

8O000

A ,.~ 60000" V

40000"

20000"

0 ~000 1200 1400 1600 asoo 20oo

T e m p e r a t u r e (K)

Fig. 4. The evotutiofi &the ~ean particle size, calculated for the base case. A "particle" here is defined as any entity of dimer sl~e or laNer, and (k) is the mean number of monomers per particle.

10 -3

10"4 t 11! 1567 K

I f "

10"6 ]

10-7 ,1 l 10 100 1000

Dimensionless diameter, ~r

Fig, 5~ Size distributions calculated at various temperatures for the base case.


Most of the monomer population condenses to these particles in a remarkably short time, producing a "nucleation burst." This result is seen clearly in Fig. 4. In a very short temperature span the mean particle size (k) jumps from essentially unit up to about 60,000. Thereafter growth follows the almost linear trend produced by coagulation, (~5) for which ( k ) ~ t6/5~

The detailed result of these calculations is the evolution of the particle size distribution, as shown in Fig. 5, in which the distributions in the neighborhood of the nucleation burst are emphasized. At T = 1647 K the distribution consists mostly of monomers, with a monotonically decreasing distribution of subcritical clusters. At 1620 K, less than ! msec later, the distribution has changed dramatically, showing the sudden presence of stable particles. By 1567 K the distribution has the mature appearance of an aerosol which is growing by pure coagulation. At a considerably lower temperature, 1090 K, the distribution has shifted by coagulation toward larger particles with a correspondingly smaller total number density.

4.2. Effect of Cooling Rate

The time response of the aerosol to the rise in supersaturation and resulting decrease in critical cluster size is governed by collision rates. For the situation considered we may write a characteristic response time for the aerosol dynamics as

1 ~'c ~" (9)

/'/0/30

where no is the initial value of the monomer concentration and/3o is the initial value of/3H. Strictly speaking rc should be adjusted to account for the T 1/2 dependence /311. Neglecting this relatively small modification, it is evident that the aerosol response time varies inversely as the monomer concentration.

Therefore if the cooling rate is increased for a given monomer concentration, the result is that the supersaturation shoots to a higher value before the nucleation burst occurs, as shown in Fig. 6, in which the monomer concentration for both cooling rates shown is the same as in the base case. Shooting to a higher supersaturation causes a decrease in the critical nucleus size before the onset of nucleation occurs. Recalling that the cluster size distribution for early times is monotonically decreasing, the result is that a larger number of mostly smaller stable nuclei are made available for condensation. Therefore we would expect a higher cooling rate to produce more and smaller particles. The calculations clearly support this conclusion, as seen in Fig. 7. For example, doubling the cooling rate from 10 to 20 K/msec decreases the mass-mean diameter immediately following the

H o m o g e n e o u s Nucleat ion o f Particles in Plasma Synthesis 365

10 3

o * a m

10 2

.o

t_

101

10 0 ,

looo

cooling rate = ~ 40 K/ms

10 K/ms

___d 1200 I400 1600 I800 2000

Temperature (K) Fig.6. Effect of cooling rate onthe critical supersaturation for nucleation. Monomer concentra-

tion is the same as in the base case.

30

10'

O' 1000

cooling rate = 10 K/ms

20

1200 1400 1600 1800 2000

Temperature (K)

Fig. 7. Effect of cooling rate on the evolution of the mass-mean diameter. Monomer concentration is the same as in the base case.


nucleation event from about 23 nm to about 14.5 nm and produces about four times as many particles.

It is worth reemphasizing that we have assumed a constant cooling rate. In fact the cooling rate in a plasma reactor would typically be expected to decrease in the flow direction: it can be of the order of 10 6 K/sec in the plasma core and less than 10 3 K / s e c at the reactor exit. Since the cooling rate effect on nucleation appears to be so strong, we conclude that the local cooling rate at the location of the nucleation burst is an important determinant in the final size. In contrast, after the nucleation burst the total residence time, i.e., the average cooling rate, is of importance for coagulation. (However, local cooling rates can affect phase transformations within the particles.)

4.3. Effect o f Monomer Concentration

The discussion accompanying Eq. (9) suggests that increasing the monomer concentration should have an effect opposite to increasing the cooling rate. For a given cooling rate, increasing no decreases ~c. Thus the aerosol can respond more quickly to the decrease in critical cluster size, as is evident from the calculations in Fig. 8, in which the saturation ratio in the base case is compared to a case with the same cooling rate but twice the monomer concentration. Changing the monomer concentration alters the prenucleation saturation ratio, but in addition, the higher-concentration case peaks at a lower supersaturation (190) than does the base ease (280). Therefore increasing the monomer concentration results in the nucleation of larger particles, as seen in Fig. 9.

5. CONCLUSIONS

We have considered an idealized view of a plasma reactor in which the flow field is one-dimensional, the relevant chemistry is completed before homogeneous nucleation occurs, and there is only one condensing species. Obviously particle size distributions are broadened by nonuniform temperature and velocity profiles and by particle deposition to walls. The assumption regarding the decoupling of monomer generation and the aerosol dynamics may not always be valid. In principle, however, Eq. (2) can account for monomer generation by including it as a source term if the appropriate reaction rate is known.

The present calculations are more realistic than those reported in Ref. 1, in that the latter neglected evaporation from small clusters. It was found that by including evaporation terms the homogeneous nucleation event could be resolved and predicted and that it is a major determinant of the ultimate powder size distribution. Homogeneous nucleation in a plasma


10 3

~

i,,.

L

10 2

10 1

lo ~ 4-- 1000

Pc, = 5 x lO-4ama

10 -3 arm

1200 1400 1600 !800 2000

Tempera tu r e (K)

Fig. 8. Effect of monomer concentration on the supersaturation history. Cooling rate is the same as in the base case,

25

E 20

t _

E 15

10

5'

O- I "r

1000 1200 1400 1600

Pr, = lO-Satm

1800 2000

Tempera tu r e (K)

Fig. 9. Effect of monomer concentration on the evolution of the mass-mean diameter. Cooling rate is the same as in the base case.


reactor can be expected to occur as a rapid event in which a large number of particles are suddenly formed by condensation to stable clusters and particles. The size and number of nucleated particles can be significantly affected by two factors, the cooling rate at the location of nucleation and the monomer concentration. Lower local cooling rates and higher concentrations both favor the nucleation of larger (and fewer) particles. The nucleation burst is followed by steady growth by coagulation.

ACKNOWLEDGMENTS

The authors thank their colleague Peter H. McMurry for his valuable comments and suggestions regarding this work.

This work was supported by the National Science Foundation under Grant CBT-8805934, by the University of Minnesota Graduate School, and by the Minnesota Supercomputer Institute.

NOMENCLATURE

Roman Letters

A 1

b d* E i,j,k k rli

l,'l s

no

0

S t T D1

Monomer surface area Cooling rate (K/sec) Critical diameter for stable nucleus Evaporation coefficient (s -~) Number of monomers comprising a particle Boltzmann's constant Concentration (m -3) of particles of size i Equilibrium monomer concentration Initial monomer concentration Subscript denoting conditions at initial point where saturation ratio equals unity Vapor saturation ratio Time Temperature Monomer volume

Greek Letters

/3 ~o 8

Pp (7"

7"c

Collision frequency function (m3/sec) Value of/311 at To Kronecker delta function Particle mass density Surface tension (N/m) Characteristic time for aerosol dynamics


Nondimensional Terms

b B -

nol3o To j_=a_

dl

=- n~Blj exp 2 / 3 ( j _ 1)2/3

~___ni r/o

~o T ~ Hot~oI

Cooling rate

Particle diameter

Evaporation coefficient

Size distribution function

Collision frequency function

Time

REFERENCES

1. S. L. Girshick, C.-P. Chiu and P. H. McMurry, Plasma Chem. Plasma Process. 8, 145 (1988). 2. P. H. McMurry and S. K. Friedlander, J. Colloid Interface Sci. 78, 513 (1979). 3. P. H. McMurry, J. Colloid Interface Sci. 78, 513 (1980). 4. J. J. Wu, H. V. Nguyen and R. C. Flagan, Langmuir 3, 266-271 (1987). 5. P. P. Bolsaitis, J. F. McCarthy, G. Mohiuddin and J. F. Elliott, Aerosol Sci. Teeh. 6, 225

(1987). 6. N. P. Rao and P. H. McMurry, Aerosol Sci. Tech. (in press). 7. S. C. Graham and J. B. Homer, Faraday Syrup. 7, 85 (1973). 8. S. K. Friedlander, Smoke, Dust and Haze; Fundamentals of Aerosol Behavior, Wiley, New

York (1977). 9. J. Frankel, Kinetic Theory of Liquids, Dover, New York (t955).

10. P. Sahoo and T. Debroy, Met. Trans. B 18B, 597 (1987). 11. T. Yosbida and K. Akashi, Trans. Jap. Inst. Metals 22, 371 (1981). 12. R. Hultgren, et aL, Selected Values of Thermodynamic Properties of the Elements, American

Society for Metals, Metal Parks, Ohio (1973). 13. F. Gelbard, Y. Tambour and J. H. Seinfeld, J. Colloid Interface Sci. 76, 541 (1980). 14. A. C. Hindmarsh and G. D. Byrne, EPISODE: An effective package for the integration

of systems of ordinary differential equations, Lawrence Livermore Laboratory Report UCID-30112, Rev. 1.

15. E S. Lai, S. K. Friedlander, J. Pich and G. M. Hidy, J. Colloid Interface Sci. 39, 395 (1972).

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