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Contribution to the theory of aerosol formation in an inert gas atmosphere. Pre-nucleation kinetics
JOURNAL OF COLLOID AND INTERFACE SCIENCE 2 2 , 214--220
(1966)
Contribution to the Theory of Aerosol Formation in an Inert Gas Atmosphere. Pre-nucleation Kinetics 1 K E U N G P. L U K E 2
Unified Science Associates, Inc., Pasadena, California Received August 12, 1965 ABSTRACT
The kinetics of aerosol formation in an inert gas atmosphere may be considered as proceeding in the following steps: prenucleation, nucleation, growth, and coagulation. In this paper the prenucleation kinetics or the cooling-down process of a hot vapor in the presence of a cooler inert gas is examined using the model that the cooling mechanism is the elastic collision process between vapor and gas atoms. Computed values of average energy loss per collision, asymptotic time constant for the coolingdown process, and the time required for cooling a vapor from 1800°K. to 900°K. and 450°K., respectively, are presented for sixteen differgnt vapor-gas pairs. Our results are consistent with the conclusion of Kerker and co-workers that the rate of energy transfer from a body of hot NaC1 vapor to a cooler helium gas is extremely rapid. INTRODUCTION
time and certain other aspects of such a cooling process.
The kinetics of aerosol formation in an inert gas atmosphere m a y be considered as proceeding in the following steps: prenucleation, nucleation, growth, and coagulation or agglomeration. There are a number of theories on nucleation (l, 2), growth (1, 2), and coagulation (3-7), but there seems to be lacking an analysis that could be conveniently used to obtain quantitative information about the time-dependent, cooling process of the vapor. This cooling process, defined here as the prenucleation kinetics, is an important step in the production of aerosols because it transforms a vapor from a nearly saturated or saturated state to a supersaturated state, a necessary initial condition for the homogeneous nucleation from the vapor phase. The purpose of this paper is to consider on a theoretical basis the characteristic
STATEMENT OF THE PROBLEM
1 This work was performed under the sponsorship of the Office of Aerospace Research of the United States Air Force under Contract AF19(628)-4168, and administered by the Air Force Cambridge Research Laboratories. 2Address after September 1, 1966: Department of Physics, California State College at Long Beach, Long Beach, California 90804
We consider the cooling process for the time-dependent case where at zero time a very short burst of vapor atoms is suddenly released in an inert gas. The initial energy of the vapor atoms is several times that of the gas atoms. The vapor atoms subsequently collide with the gas atoms and lose energy, i.e., the vapor is cooled b y the inert gas. Our objective is to obtain quantitative informa~ tion on how the cooling process proceeds with time, assuming that during the cooling-down period a negligible amount of nucleation takes place so that only vapor and gas atoms are present. Because of the similarity of this process to the process of neutron thermalization in a moderator, we shall draw upon pertinent methods of analysis and resuits from studies carried out in that field of nuclear reactor physics. ANALYSIS The basic problem in the study of the cooling process of vapor atoms in a carrier gas involves the determination of the vapor
214
AEROSOL FORMATION IN AN INERT GAS ATMOSPHERE atom velocity distribution .fl(r~, vl, t) such that
function
f1(rl, Vl, t) dr1, dr1 represents the nmnber of vapor atoms in the volume element dr~ and in the velocity range dv~ at position rl and velocity vl in the six-dimensional (position-velocity) phase space at time t. Here, the subscript 1 is used to denote the vapor; 0 will be used for the carrier gas. In principle, the development of fl as a function of time can be calculated because it obeys the Boltzmann equation. Owing to the interaction between the vapor and the cartier gas, the velocity distribution function of the carrier gas, f0(r0, v0, t), m a y also change with time. Thus, the time-dependent behavior of the vapor atoms is governed b y the Boltzmann equation (8) for f0 as well as for
215
We shall start b y introducing the first five of the seven assumptions which make up our model for the cooling process. This particular model is chosen because it is simple enough to allow calculation, while still retaining the essential features we wish to study. 1. The carrier gas is infinite in extent, and it has a Maxwellian velocity distribution characterized by a constant temperature To
4(mo~ 3/2 f(vo)dvo = no ~ \2kTo] •v02 exp
(
[3]
2tcTo/dr0,
where no is the number gas atoms per cubic centimeter, m0 is its mass, Vo is its velocity, and k is the Boltzmann constant. 2. The vapor atoms are distributed uniformly in the carrier gas. 3. The collision frequency between vapor fl. atoms is negligible in comparison t o the colOfo %_ v0" Vfo %_ T0"V~ofo = Coo %_ C01; [1] lision frequency between vapor and gas 0t atoms. 4. There are no external forces acting on Vl' Vfl %_ 71" V~fl -- C10 %_ Cn. [2] the vapor atoms. 0I 5. The cooling of the vapor is due only to These two equations are coupled b y the colelastic collisions. lision terms C01 and C10 • Here Coo, Cn, and Assumption 1 eliminates the need for Eq. C01 or C~0 represent, respectively, the mutual [2] entirely. Assumptions 2, 3, and 4 elimicollision between two gas atoms, between nate, respectively, the terms Vr Vfl, Cn, and two vapor atoms, and between a gas and a 71" Vvfl from Eq. [1]. Thus, applying assumpvapor atom. The other terms in Eqs. [1] tions 1 through 4 to Eqs. [1] and [2] leads to and [2] may be interpreted as follows. Of~Or the result expresses the variation of the velocity distribution function as a function of time at ayl(Vl, t) _ c10. [4] any point of the gas. The term - v . Vf repreOt sents the effect of the diffusion due to a density gradient, and - 7" V~f represents the I t has been shown that C10 is (9-11a) effect of applied forces and velocity gradient. The ratio of the external force to the mass of Clo - - v l f l ( vl , t) l~(vl) the atom is represented b y 7, V~ is the gradi[5] ent operator for differentiation with respect + f, I noz~(v; --~ v l ) v l'f l ( v ~ ', t) dye,' to the components of the velocity vectors v rather than with respect to the space cowhere ls(vl) is the mean free path of the ordinates. Equations [1] and [2] represent the most vapor atoms in the carrier gas, and the kernel general formulation of the cooling problem ~s(vl' --+ vl) is the differential cross section we are considering here. Although they can- for a collision in which the velocity of the not be solved to yield f0 and f l , they permit vapor atom is changed from v1' to a unit concise specification of distinction between velocity interval at vl. In order to calculate the kernel, it is necesthe general cooling-down case just considered and the approximate cooling-down cases sary to have complete information concerning the interaction between the vapor atoms which we wish to consider below.
of1%_
216
LUKE
and the gas atoms. For this calculation, we need the following additional assumption. 6. The gas and vapor atoms are considered as rigid elastic spheres with diameters do and d~, and mass m0 and mx, respectively. A collision is defined as the approach of the vapor atom and gas atom so that their centers are ~/~(d0 + dl) apart• Combining assumptions 1 and 6 leads to the following equation for a / v / --+ vx) (11b, 12)
Because of its complexity, the differential cross section as given by Eq. [8] renders Eq. [5] insolvable. However, it is a very useful quantity and can be profitably used in an approximate treatment of the cooling-down problem• For this purpose, we introduce now the seventh and final assumption. 7. The velocity distribution of the vapor is Maxwellian at all times, characterized by a single parameter, the vapor temperature TiP(t), which decreases during the coolingdown process and eventually approaches the temperature To of the gas, i.e., (ml
• L ~ v,~oo(v,)fo(vo)
[6] f l ( vl , t ) = nl ~Tl21t~ , ~
[10]
• y12 Iexp ( _
• P ( v t i --+ Vl ;~)0, COS O~) d v o ,
where v~ is the relative velocity of the colliding vapor and gas atoms. Here P(v~' --* v~ ; Vo, cos a) is the probability that, in a collision of a vapor atom and a gas atom of velocities vl' and v0 forming an angle a, the vapor atom is left with velocity Vl. The term z,0(v,) is given by [7]
Integration of Eq. [6] yields (lib, 12)
We also rewrite Eq. [8] into the following form (9, 10) O'S(
E'1 ---) E l )
!
± Err ( ~ m + ~i'v~) (3vv'~ -- 3~lv~)
where the upper signs refer to vl < v~' and the lower signs to v~ > vl' and [9a]
fl = (mff2kTo)~/2; + (mffmo)lJ2]; _ (mffmo)~/2];
e ~ du.
[11]
• [Erf (~J~ - ~ l )
[9b] [9c] [9d]
(el'/]cTo)l/2;
[12a]
~l = (E1/kTo)~/L [12b] Because of assumption 7, the vapor ternperature Tl'(t) is related to the average energy E~'(t) of a vapor atom through the simple equation E,~' (t) = 3~]~Tl' (t).
[13]
Thus, the vapor temperature during the cooling-down process is completely specified by "El'(t), which can be shown to be governed by the following equation (9, 11c) : d E'l ( t ) dt •
2f(_
(m0 + mi) 2 --~ ~o 8mom~ E1
• { E ~ (.~l - ~J1)
wl ' = [8]
exp [/~2(p"12-- /)12)] [Erf
Erf (x) = ~
=
where
• {E~ ( ~ v l - ~ S )
= l[(mo/ml)l/2
(t y J dvl.
Erf (v~'l + ~'wl)]},
' (m0 + ml) 2 vl ~(vl --+ vl) = (r~o - -w2 4moml Vl
v = l[(mo/ml)l/2
mlvi2 ~G
± Erf (.~l + ~J1) + exp (~(2 _ ~ )
7~
~,o(v,) = ~ (do + dl) 2.
+
~312
f
no nl
[14]
crt(EO(E'~ -- e~)v'ff~(v', , t ) d v l .
Here, d - E ( ( t ) / d t represents the rate of energy
AEROSOL FORMATION IN AN INERT GAS ATMOSPHERE loss per vapor atom at a particular instant of time. ¢t(ElO is the total collision cross section. a t ( E l ) is given by
where c is the integration constant, and a0 is oo
¢t( E '~) -- J0 ~,(E~ ~ E~)dE1
---- ~0
] -~-~
Err (a)
+ ~
\ ~rml /
nozso )
[20] ( ml/mo) (1 + ml/mo) 8/2 "
[15]
Since the average velocity ~10 and the mean free path l~0 of a vapor atom in the vaporgas mixture of temperature To are (14)
exp ( -- a2)
where
[21]
~o = -(8kT1/2 °~ \ ~rm: /
[16]
a = (moEl'/mlkTo) 112.
217
and The term (El' -- E~)' represents the average energy loss per collision.
(E'I
-
El)
1
noz~o(1 + ml/mo) 1/2'
[22]
Eq. [20] may also be written as follows:
fo~ o's(E1,
El)(E"I -- El) gel
~
~o = 4 k
1
~.1/2 a3 + ~ a
Substituting Eqs. [10], [15], and [17] into Eq. [14] yields (9, 10, 11c, 13)
c=
cosh -1
2 ( 1 + ml/mo) ]
1 + (~£/©
(ml/mo) (l + ml/mo) ~/2 (nca~o)
[18]
l J"
[25]
T i ( t ) -- To To
"]l/2k( T ' I - To).
+ To(1 + m l / m o ) J
[
2(1 + ml/mo) 2(1 + ml/mo) ~
]26]
1 + (~(~(~/~o) - 1]
The solution of Eq. [18] is (9, 10) T'l(t) -- To
"exp ( - - a 0 t / 3
To 2(1 + ml/mo) ÷
-
For those cases in which the values of cosh (c + aot/3/~k) are much greater than unity, Eq. [19] becomes
\ 7rm~/
cosh
[24]
is just the average time between successive collisions involving a vapor atom and a gas atom. The integration constant c is determined by the initial condition that at t = O, Tl'(t) is equal to T{(0). Thus, we have
2 ~rl/~ a exp (--a 2) + (2a 2 + 1) Erf a
-----
/lo ~)10
tl0 = =-
[17]
exp (_a2)
dF,'l(t) dt
[23]
where
El) dE1
4\To 2\To = - (1 + ~rto/ml) -~- (1 + m o / m j 2
[
(ml/~o)
(1 + ml/mo) 2 tlo'
~(E'l -~
"L 1
110 =
[19]
1'
k) •
Thus, the final approach of Tl'(t) to To occurs exponentially with a time constant r~ given by
218
LUKE "rl --
3 k 2 oLo
[
~-r
l
12,aj
or
3 (1 + m~/mo) 2 Z~o. 8 (m~/mo)
[27b]
In this section we have attempted to present a logical development of an appropriate equation for describing the cooling process of a body of hot vapor in a cooler carrier gas. We started with the coupled Boltzmann equations of [1] and [2]. Through the use of five simplifying assumptions, the space- and angle-independent Boltzmann equation of [4] was obtained for determining the velocity distribution of the vapor f1@1, t) as a function of time. However, the complexity of the differential cross section z,(s~' --~ vl) as given b y Eq. [8] rendered Eq. [4] insolvable. At that point, the seventh and key assumption of this paper was made by introducing Eq. [10] for the time-dependent velocity distribution of the vapor. Within the framework of that assumption and the preceding ones, we determined the desired expression [19] for describing the temperature T~'(t) of the vapor as a function of time. It is to be noted that Eq. [19] is the exact solution of the differential equation given b y [18]. An approximate solution is given b y Eq. [26], which shows that the vapor temperature T~'(t) approaches the gas temperature To exponentially with a time constant rl given b y Eq. [27a] or [27b]. The average energy lost by a vapor atom in a collision with a gas atom, given b y Eq. [17], is a quantity which plays an important role in determining the value of T1 . APPLICATION The following quantities were computed for sixteen different vapor-gas combinations: 1. Average energy loss per collision in units of the initial vapor atom energy, Eq.
[17]. 2. Time constant of the cooling process in units of tl0, Eq. [27b]. 3. Times for cooling a vapor from an initial temperature of 1800°K. to intermediate temperatures of 900°K. and 450°K., respectively, Eqs. [19], [20], and [25].
Handbook values (15) were-used for the atomic weights ahd atomic diameters. T h e only exception was the use of~the gas:kinetic diameter (16) for the helium atom. The temperature and pressure of the carrier gas was taken to be 300°K., and 1 ram. Hg, respectively, and the initial temperature of the vapor was assumed to be 1800°K. Computational results are presented in Table I. As noted in Table I, one or more groups of investigators have experimented with the vapor-gas combinations (pairs) we are considering here. Although two different techniques were employed to produce the vapors, i.e., evaporation-sublimation [17-19) and exploding wire (20, 21), the use of an inert gas to contain and to cool the hot vapor was common to both. The effectiveness of a carrier gas in its role as an energy absorber for a vapor m a y be assessed by considering the values of (El' - E l i / E l ' , T1/tl0, and t given in Table I for the various vapor-gas combinations. As expected, because of the mass mismatch, the values o f ( E l ' - E l ) / E l ' are the smallest for the Au-He a n d :Bi-He pairs. There seems to be a maximum around the vapor-to-gas mass ratio of 1.47, which is the Ni-A pair. At this mass ratio, the average energy loss per collision is more than 40 % of the initial energy of the vapor atom. I t is interesting to note that while the mass ratios vary by a factor of ~--86 in going from the Mg-A pair (0.608) to the Bi-He pair (52.2), the values of (E~' - EO ~El t stay within the region between 0.04 to 0.41. The fourth column of Table I gives the values of the asymptotic time constant, r~, of the cooling process in units of t~0, the average time between successive collisions involving a vapor atom and a gas atom when both the vapor and the carrier gas are at the temperature To. There is a theoretical minimum at (ml/mo) = 1, and its value is 1.50. I t is seen that the values for 5/Ig-A, A]-A, Ni-A, and Cu-A are quite close to this minimum value and are located on the near and far side of this minimum. For the cases of (ml/mo) > 10 and (ml/mo) < 0.1, the values of r~/Z10 are approximately given b y 0.38 (ml/mo) and 0.38 (m0/m0, respectively. For the cases considered here, the spread in the values of rl/Z10 are not large, i.e.,
AEROSOL FORMATION IN AN INERT GAS ATMOSPHERE
219
TABLE I COMPUTED V A L U E S OF ( E t '
-- El)/El'
"rl/[io , AND t FOI~ VARIOUS M A s s
,
RATIOS
Tlq0) = 1800°K. %'(t) = 900°K. T~'(t) = 450°K. Vapor-gas
Mg-A A1-A Ni A Cu-A Pd-A Ag-A Au-A Bi-A Mg-He A1-He Ni-He Cu-He Ag-He Cd-He Au-He Bi-He
ml/mo
0.608 0.675 1.47 1.59 2.66 2.70 4.92 5.23 6.08 6.74 14.7 15.9 27.0 28.1 49.2 52.2
( E { -- E O / E ; a
0,381 0,391 0,408 0.404 0.353 0.352 0.264 0,254 0.233 0,218 0,122 0,115 0.0731 0.0708 0.0421 0.0397
~'l~he
1.59 1.56 1.56 1.58 1.89 1.90 2.67 2.78 3.09 3.33 6.27 6.72 10.9 11.3 19.2 20.3
t(lO-S sec.) b
0.105 0.120 0,181 0.185 0.248 0.241 0.431 0,344 0.235 0.302 0.729 0.767 1.14 1.11 2.07 1.66
t(lO-a sec.) b
0,319 0.363 0,529 0.538 0.702 0.680 1.18 0.922 0.636 0.812 1.90 1.99 2.92 2.87 5.28 4.22
References
17 17,18,20 17,21 17,20,21 21 17 1717 19 18,19 19 19 19 19 19 19
Computed for E{ = 3/~ kTt'(O) = 3/~ (6/cT0) = 9kT0. b Computed for carrier gas pressure of 1 mm. Hg. 1.56-20.3, even though the operating conditions and physical properties of the various vapor-gas pairs m a y differ considerably. The reason is that several important parameters such as collision cross section, pressure and temperature of the carrier gas, and mass of the vapor atom are hidden in tl0 • Thus, wh.en these parameters are taken into eonsideration in the calculation of 71, one can expect a larger spread in its values. The last two columns of Table I give the time in microseconds for a vapor initially at a temperature of 1800°K. to cool down to 900°K. and 450°K., respectively. I t is interesting to note that it takes a body of gold vapor about 4}{ times as long to cool down in helium as in argon, and it takes gold vapor 16}3 times as long to cool down in helium as magnesimn vapor in argon. Since rl is inversely proportional to the first power of the carrier gas pressure, the cooling time presented here should be multiplied b y the factor 1 ram./760 ram. if a carrier gas pressure of 760 ram. (instead of 1 ram.) is used. If cooling through inelastic collisions is negligible, the values given in Table I for the cooling times should represent the lower limits. For those cases in which aerosols are
produced in an inert gas atmospb.ere by: the evaporation-sublimation technique, this condition is generally satisfied. Local h.eaibing of the carrier gas by the vapor tends to lengthen the time required to cool the vapor, as the vapor then has to transfer its energy to a carrier gas with an effective temperature higher than its temperature before its interaction with. the vapor. I t is reasonable, however, to expect that the amount of heating is less for an open system employing a flowing carrier gas than for a closed one employing a stationary gas. How much less is difficult to assess and is probably strongly dependent on the flow rate of the carrier gas. For example, suppose that the vapor generation rate is constant, it is clear then that a larger gas flow rate means a higher gas to vapor atom density ratio in the vapor-gas mixture, and thus the gas temperature is less likely to be affected by the vapor temperature. Finally, we m a y mention an interesting conclusion made recently by Kerker et al. (22) concerning the cooling rate of NaC1 vapor in a helium gas. In connection with their aerosol experiments, they prepared NaC1 aerosols b y continuously evaporating and condensing NaC1 vapor in a stream of
220
LUKE
flowing helium gas of one atmosphere pressure. T h e a m o r p h o u s n a t u r e of the resultant NaC1 aerosols led t h e m to conclude t h a t the NaC1 v a p o r was quenched very rapidly b y the helium gas. This conclusion, t h o u g h only qualitative, implies t h a t the c o m p u t e d values of the cooling-down time given in T a b l e I are quite reasonable. I t is to be n o t e d t h a t even for a carrier gas pressure of 1 ram. ttg, the cooling-down times are already in the microsecond region. T h e corresponding coolingd o w n time for a carrier gas pressure of one atmosphere is a b o u t a t h o u s a n d times shorter, or in the milli-microseeond region. REFERENCES 1. COURTNEY, W. G., Symp. Combustion 9th, p. 799 (1963). 2. HIRTH, J. P., AND POUND, G. M., "Condensation and Evaporation: Nucleation and Growth Kinetics." Macmillan, New York, 1963. 3. COUnTNEY,W. G., A R S J. 31,751 (1961). 4. Fucks, N. A., "The Mechanics of Aerosols," Chap. 7. Macmillan, New York, 1964. 5. GREEN, H. L., AND LANE, W. R., "Particulate
Clouds : Dusts, Smokes, and Mists," 2nd ed., Chap. 5. E. and F. N. Spon, London, 1964. 6. M£RTYNOV,G. A., ANE BAI~ANOV,S. P., In B. V. Deryagin, ed., "Research in Surface Forces," p. 182. Consultants Bureau, New York, 1963. 7. BROCK,J. R., AND HIDY, G. M., J. Appl. Phys. 36, 1857 (1965). 8. DELCROIX,J. L., "Introduction to the Theory
of Ionized Gases," pp. 35-37. Interscience, New York, 1960. 9. VON DARNEL,G. F., Phys. Rev. 94, 1272 (1954). 10. VON I)ARDEL, G. F., Trans. Roy. Inst. Technol. Stockholm No. 75 (1954). ll(a). A~ALm, E., In S. Fltigge, ed., "Handbuch der Physik," 38, Part II, p. 212. Springer, Berlin, 1959. (b) ibid., pp. 397-399. (c). ibid., p. 501. 12. COHEN, E. R., "Peaceful Uses of Atomic Energy: Proceedings of the International Conference in Geneva, August, 1955," 5, 405. Published by the United Nations. 13. CnAVATn,A. M., Phys. Rev. 36, 248 (1930). 14. MOELWYN-HUGHES, E. A., "Physical Chemistry," p. 79. Cambridge University Press, Cambridge, 1940. 151 GRAY, D. E., ed., "American Institute of Physics Handbook," pp. 7-9 to 7-12. McGraw-Hill, New York, 1957. 16. KEESOM, W. H., "Helium," p. 114. Elsevier, Amsterdam, 1942. 17. KI~OTO, K., KAMIYA,Y., NONOYAMA,M., AND UYEDA, R., Japan. J. Appl. Phys. 2, 702 (1963). 18. GEN, M. S., ZISKIN, M. S., AND PETROV, Yu. I., Dokl. Akad. Nauk SSSR. 127,366 (1959). 19. TILL, P. H., AnD TURKEVICH,J., U.S. Atomic
Energy Commission Report NYO-3435 (1956). 20. KARmRIS, F. G., FISR, B. R., AND I~OYSTER, G. W., JR., In W. G. Chace and H. K. Moore, eds., "Exploding Wires," Vol. 1, pp. 299-311. Plenum Press, New York, 1962. 21. NAsa, C. P., AND DE SIENO, R. P., J. Phys. Chem. 69, 2139 (1965). 22. ESPENSCHEID, W. F., MATIJEVI~, E., AND KERKER, M., J. Phys. Chem. 68, 2831 (1964).
(1966)
Contribution to the Theory of Aerosol Formation in an Inert Gas Atmosphere. Pre-nucleation Kinetics 1 K E U N G P. L U K E 2
Unified Science Associates, Inc., Pasadena, California Received August 12, 1965 ABSTRACT
The kinetics of aerosol formation in an inert gas atmosphere may be considered as proceeding in the following steps: prenucleation, nucleation, growth, and coagulation. In this paper the prenucleation kinetics or the cooling-down process of a hot vapor in the presence of a cooler inert gas is examined using the model that the cooling mechanism is the elastic collision process between vapor and gas atoms. Computed values of average energy loss per collision, asymptotic time constant for the coolingdown process, and the time required for cooling a vapor from 1800°K. to 900°K. and 450°K., respectively, are presented for sixteen differgnt vapor-gas pairs. Our results are consistent with the conclusion of Kerker and co-workers that the rate of energy transfer from a body of hot NaC1 vapor to a cooler helium gas is extremely rapid. INTRODUCTION
time and certain other aspects of such a cooling process.
The kinetics of aerosol formation in an inert gas atmosphere m a y be considered as proceeding in the following steps: prenucleation, nucleation, growth, and coagulation or agglomeration. There are a number of theories on nucleation (l, 2), growth (1, 2), and coagulation (3-7), but there seems to be lacking an analysis that could be conveniently used to obtain quantitative information about the time-dependent, cooling process of the vapor. This cooling process, defined here as the prenucleation kinetics, is an important step in the production of aerosols because it transforms a vapor from a nearly saturated or saturated state to a supersaturated state, a necessary initial condition for the homogeneous nucleation from the vapor phase. The purpose of this paper is to consider on a theoretical basis the characteristic
STATEMENT OF THE PROBLEM
1 This work was performed under the sponsorship of the Office of Aerospace Research of the United States Air Force under Contract AF19(628)-4168, and administered by the Air Force Cambridge Research Laboratories. 2Address after September 1, 1966: Department of Physics, California State College at Long Beach, Long Beach, California 90804
We consider the cooling process for the time-dependent case where at zero time a very short burst of vapor atoms is suddenly released in an inert gas. The initial energy of the vapor atoms is several times that of the gas atoms. The vapor atoms subsequently collide with the gas atoms and lose energy, i.e., the vapor is cooled b y the inert gas. Our objective is to obtain quantitative informa~ tion on how the cooling process proceeds with time, assuming that during the cooling-down period a negligible amount of nucleation takes place so that only vapor and gas atoms are present. Because of the similarity of this process to the process of neutron thermalization in a moderator, we shall draw upon pertinent methods of analysis and resuits from studies carried out in that field of nuclear reactor physics. ANALYSIS The basic problem in the study of the cooling process of vapor atoms in a carrier gas involves the determination of the vapor
214
AEROSOL FORMATION IN AN INERT GAS ATMOSPHERE atom velocity distribution .fl(r~, vl, t) such that
function
f1(rl, Vl, t) dr1, dr1 represents the nmnber of vapor atoms in the volume element dr~ and in the velocity range dv~ at position rl and velocity vl in the six-dimensional (position-velocity) phase space at time t. Here, the subscript 1 is used to denote the vapor; 0 will be used for the carrier gas. In principle, the development of fl as a function of time can be calculated because it obeys the Boltzmann equation. Owing to the interaction between the vapor and the cartier gas, the velocity distribution function of the carrier gas, f0(r0, v0, t), m a y also change with time. Thus, the time-dependent behavior of the vapor atoms is governed b y the Boltzmann equation (8) for f0 as well as for
215
We shall start b y introducing the first five of the seven assumptions which make up our model for the cooling process. This particular model is chosen because it is simple enough to allow calculation, while still retaining the essential features we wish to study. 1. The carrier gas is infinite in extent, and it has a Maxwellian velocity distribution characterized by a constant temperature To
4(mo~ 3/2 f(vo)dvo = no ~ \2kTo] •v02 exp
(
[3]
2tcTo/dr0,
where no is the number gas atoms per cubic centimeter, m0 is its mass, Vo is its velocity, and k is the Boltzmann constant. 2. The vapor atoms are distributed uniformly in the carrier gas. 3. The collision frequency between vapor fl. atoms is negligible in comparison t o the colOfo %_ v0" Vfo %_ T0"V~ofo = Coo %_ C01; [1] lision frequency between vapor and gas 0t atoms. 4. There are no external forces acting on Vl' Vfl %_ 71" V~fl -- C10 %_ Cn. [2] the vapor atoms. 0I 5. The cooling of the vapor is due only to These two equations are coupled b y the colelastic collisions. lision terms C01 and C10 • Here Coo, Cn, and Assumption 1 eliminates the need for Eq. C01 or C~0 represent, respectively, the mutual [2] entirely. Assumptions 2, 3, and 4 elimicollision between two gas atoms, between nate, respectively, the terms Vr Vfl, Cn, and two vapor atoms, and between a gas and a 71" Vvfl from Eq. [1]. Thus, applying assumpvapor atom. The other terms in Eqs. [1] tions 1 through 4 to Eqs. [1] and [2] leads to and [2] may be interpreted as follows. Of~Or the result expresses the variation of the velocity distribution function as a function of time at ayl(Vl, t) _ c10. [4] any point of the gas. The term - v . Vf repreOt sents the effect of the diffusion due to a density gradient, and - 7" V~f represents the I t has been shown that C10 is (9-11a) effect of applied forces and velocity gradient. The ratio of the external force to the mass of Clo - - v l f l ( vl , t) l~(vl) the atom is represented b y 7, V~ is the gradi[5] ent operator for differentiation with respect + f, I noz~(v; --~ v l ) v l'f l ( v ~ ', t) dye,' to the components of the velocity vectors v rather than with respect to the space cowhere ls(vl) is the mean free path of the ordinates. Equations [1] and [2] represent the most vapor atoms in the carrier gas, and the kernel general formulation of the cooling problem ~s(vl' --+ vl) is the differential cross section we are considering here. Although they can- for a collision in which the velocity of the not be solved to yield f0 and f l , they permit vapor atom is changed from v1' to a unit concise specification of distinction between velocity interval at vl. In order to calculate the kernel, it is necesthe general cooling-down case just considered and the approximate cooling-down cases sary to have complete information concerning the interaction between the vapor atoms which we wish to consider below.
of1%_
216
LUKE
and the gas atoms. For this calculation, we need the following additional assumption. 6. The gas and vapor atoms are considered as rigid elastic spheres with diameters do and d~, and mass m0 and mx, respectively. A collision is defined as the approach of the vapor atom and gas atom so that their centers are ~/~(d0 + dl) apart• Combining assumptions 1 and 6 leads to the following equation for a / v / --+ vx) (11b, 12)
Because of its complexity, the differential cross section as given by Eq. [8] renders Eq. [5] insolvable. However, it is a very useful quantity and can be profitably used in an approximate treatment of the cooling-down problem• For this purpose, we introduce now the seventh and final assumption. 7. The velocity distribution of the vapor is Maxwellian at all times, characterized by a single parameter, the vapor temperature TiP(t), which decreases during the coolingdown process and eventually approaches the temperature To of the gas, i.e., (ml
• L ~ v,~oo(v,)fo(vo)
[6] f l ( vl , t ) = nl ~Tl21t~ , ~
[10]
• y12 Iexp ( _
• P ( v t i --+ Vl ;~)0, COS O~) d v o ,
where v~ is the relative velocity of the colliding vapor and gas atoms. Here P(v~' --* v~ ; Vo, cos a) is the probability that, in a collision of a vapor atom and a gas atom of velocities vl' and v0 forming an angle a, the vapor atom is left with velocity Vl. The term z,0(v,) is given by [7]
Integration of Eq. [6] yields (lib, 12)
We also rewrite Eq. [8] into the following form (9, 10) O'S(
E'1 ---) E l )
!
± Err ( ~ m + ~i'v~) (3vv'~ -- 3~lv~)
where the upper signs refer to vl < v~' and the lower signs to v~ > vl' and [9a]
fl = (mff2kTo)~/2; + (mffmo)lJ2]; _ (mffmo)~/2];
e ~ du.
[11]
• [Erf (~J~ - ~ l )
[9b] [9c] [9d]
(el'/]cTo)l/2;
[12a]
~l = (E1/kTo)~/L [12b] Because of assumption 7, the vapor ternperature Tl'(t) is related to the average energy E~'(t) of a vapor atom through the simple equation E,~' (t) = 3~]~Tl' (t).
[13]
Thus, the vapor temperature during the cooling-down process is completely specified by "El'(t), which can be shown to be governed by the following equation (9, 11c) : d E'l ( t ) dt •
2f(_
(m0 + mi) 2 --~ ~o 8mom~ E1
• { E ~ (.~l - ~J1)
wl ' = [8]
exp [/~2(p"12-- /)12)] [Erf
Erf (x) = ~
=
where
• {E~ ( ~ v l - ~ S )
= l[(mo/ml)l/2
(t y J dvl.
Erf (v~'l + ~'wl)]},
' (m0 + ml) 2 vl ~(vl --+ vl) = (r~o - -w2 4moml Vl
v = l[(mo/ml)l/2
mlvi2 ~G
± Erf (.~l + ~J1) + exp (~(2 _ ~ )
7~
~,o(v,) = ~ (do + dl) 2.
+
~312
f
no nl
[14]
crt(EO(E'~ -- e~)v'ff~(v', , t ) d v l .
Here, d - E ( ( t ) / d t represents the rate of energy
AEROSOL FORMATION IN AN INERT GAS ATMOSPHERE loss per vapor atom at a particular instant of time. ¢t(ElO is the total collision cross section. a t ( E l ) is given by
where c is the integration constant, and a0 is oo
¢t( E '~) -- J0 ~,(E~ ~ E~)dE1
---- ~0
] -~-~
Err (a)
+ ~
\ ~rml /
nozso )
[20] ( ml/mo) (1 + ml/mo) 8/2 "
[15]
Since the average velocity ~10 and the mean free path l~0 of a vapor atom in the vaporgas mixture of temperature To are (14)
exp ( -- a2)
where
[21]
~o = -(8kT1/2 °~ \ ~rm: /
[16]
a = (moEl'/mlkTo) 112.
217
and The term (El' -- E~)' represents the average energy loss per collision.
(E'I
-
El)
1
noz~o(1 + ml/mo) 1/2'
[22]
Eq. [20] may also be written as follows:
fo~ o's(E1,
El)(E"I -- El) gel
~
~o = 4 k
1
~.1/2 a3 + ~ a
Substituting Eqs. [10], [15], and [17] into Eq. [14] yields (9, 10, 11c, 13)
c=
cosh -1
2 ( 1 + ml/mo) ]
1 + (~£/©
(ml/mo) (l + ml/mo) ~/2 (nca~o)
[18]
l J"
[25]
T i ( t ) -- To To
"]l/2k( T ' I - To).
+ To(1 + m l / m o ) J
[
2(1 + ml/mo) 2(1 + ml/mo) ~
]26]
1 + (~(~(~/~o) - 1]
The solution of Eq. [18] is (9, 10) T'l(t) -- To
"exp ( - - a 0 t / 3
To 2(1 + ml/mo) ÷
-
For those cases in which the values of cosh (c + aot/3/~k) are much greater than unity, Eq. [19] becomes
\ 7rm~/
cosh
[24]
is just the average time between successive collisions involving a vapor atom and a gas atom. The integration constant c is determined by the initial condition that at t = O, Tl'(t) is equal to T{(0). Thus, we have
2 ~rl/~ a exp (--a 2) + (2a 2 + 1) Erf a
-----
/lo ~)10
tl0 = =-
[17]
exp (_a2)
dF,'l(t) dt
[23]
where
El) dE1
4\To 2\To = - (1 + ~rto/ml) -~- (1 + m o / m j 2
[
(ml/~o)
(1 + ml/mo) 2 tlo'
~(E'l -~
"L 1
110 =
[19]
1'
k) •
Thus, the final approach of Tl'(t) to To occurs exponentially with a time constant r~ given by
218
LUKE "rl --
3 k 2 oLo
[
~-r
l
12,aj
or
3 (1 + m~/mo) 2 Z~o. 8 (m~/mo)
[27b]
In this section we have attempted to present a logical development of an appropriate equation for describing the cooling process of a body of hot vapor in a cooler carrier gas. We started with the coupled Boltzmann equations of [1] and [2]. Through the use of five simplifying assumptions, the space- and angle-independent Boltzmann equation of [4] was obtained for determining the velocity distribution of the vapor f1@1, t) as a function of time. However, the complexity of the differential cross section z,(s~' --~ vl) as given b y Eq. [8] rendered Eq. [4] insolvable. At that point, the seventh and key assumption of this paper was made by introducing Eq. [10] for the time-dependent velocity distribution of the vapor. Within the framework of that assumption and the preceding ones, we determined the desired expression [19] for describing the temperature T~'(t) of the vapor as a function of time. It is to be noted that Eq. [19] is the exact solution of the differential equation given b y [18]. An approximate solution is given b y Eq. [26], which shows that the vapor temperature T~'(t) approaches the gas temperature To exponentially with a time constant rl given b y Eq. [27a] or [27b]. The average energy lost by a vapor atom in a collision with a gas atom, given b y Eq. [17], is a quantity which plays an important role in determining the value of T1 . APPLICATION The following quantities were computed for sixteen different vapor-gas combinations: 1. Average energy loss per collision in units of the initial vapor atom energy, Eq.
[17]. 2. Time constant of the cooling process in units of tl0, Eq. [27b]. 3. Times for cooling a vapor from an initial temperature of 1800°K. to intermediate temperatures of 900°K. and 450°K., respectively, Eqs. [19], [20], and [25].
Handbook values (15) were-used for the atomic weights ahd atomic diameters. T h e only exception was the use of~the gas:kinetic diameter (16) for the helium atom. The temperature and pressure of the carrier gas was taken to be 300°K., and 1 ram. Hg, respectively, and the initial temperature of the vapor was assumed to be 1800°K. Computational results are presented in Table I. As noted in Table I, one or more groups of investigators have experimented with the vapor-gas combinations (pairs) we are considering here. Although two different techniques were employed to produce the vapors, i.e., evaporation-sublimation [17-19) and exploding wire (20, 21), the use of an inert gas to contain and to cool the hot vapor was common to both. The effectiveness of a carrier gas in its role as an energy absorber for a vapor m a y be assessed by considering the values of (El' - E l i / E l ' , T1/tl0, and t given in Table I for the various vapor-gas combinations. As expected, because of the mass mismatch, the values o f ( E l ' - E l ) / E l ' are the smallest for the Au-He a n d :Bi-He pairs. There seems to be a maximum around the vapor-to-gas mass ratio of 1.47, which is the Ni-A pair. At this mass ratio, the average energy loss per collision is more than 40 % of the initial energy of the vapor atom. I t is interesting to note that while the mass ratios vary by a factor of ~--86 in going from the Mg-A pair (0.608) to the Bi-He pair (52.2), the values of (E~' - EO ~El t stay within the region between 0.04 to 0.41. The fourth column of Table I gives the values of the asymptotic time constant, r~, of the cooling process in units of t~0, the average time between successive collisions involving a vapor atom and a gas atom when both the vapor and the carrier gas are at the temperature To. There is a theoretical minimum at (ml/mo) = 1, and its value is 1.50. I t is seen that the values for 5/Ig-A, A]-A, Ni-A, and Cu-A are quite close to this minimum value and are located on the near and far side of this minimum. For the cases of (ml/mo) > 10 and (ml/mo) < 0.1, the values of r~/Z10 are approximately given b y 0.38 (ml/mo) and 0.38 (m0/m0, respectively. For the cases considered here, the spread in the values of rl/Z10 are not large, i.e.,
AEROSOL FORMATION IN AN INERT GAS ATMOSPHERE
219
TABLE I COMPUTED V A L U E S OF ( E t '
-- El)/El'
"rl/[io , AND t FOI~ VARIOUS M A s s
,
RATIOS
Tlq0) = 1800°K. %'(t) = 900°K. T~'(t) = 450°K. Vapor-gas
Mg-A A1-A Ni A Cu-A Pd-A Ag-A Au-A Bi-A Mg-He A1-He Ni-He Cu-He Ag-He Cd-He Au-He Bi-He
ml/mo
0.608 0.675 1.47 1.59 2.66 2.70 4.92 5.23 6.08 6.74 14.7 15.9 27.0 28.1 49.2 52.2
( E { -- E O / E ; a
0,381 0,391 0,408 0.404 0.353 0.352 0.264 0,254 0.233 0,218 0,122 0,115 0.0731 0.0708 0.0421 0.0397
~'l~he
1.59 1.56 1.56 1.58 1.89 1.90 2.67 2.78 3.09 3.33 6.27 6.72 10.9 11.3 19.2 20.3
t(lO-S sec.) b
0.105 0.120 0,181 0.185 0.248 0.241 0.431 0,344 0.235 0.302 0.729 0.767 1.14 1.11 2.07 1.66
t(lO-a sec.) b
0,319 0.363 0,529 0.538 0.702 0.680 1.18 0.922 0.636 0.812 1.90 1.99 2.92 2.87 5.28 4.22
References
17 17,18,20 17,21 17,20,21 21 17 1717 19 18,19 19 19 19 19 19 19
Computed for E{ = 3/~ kTt'(O) = 3/~ (6/cT0) = 9kT0. b Computed for carrier gas pressure of 1 mm. Hg. 1.56-20.3, even though the operating conditions and physical properties of the various vapor-gas pairs m a y differ considerably. The reason is that several important parameters such as collision cross section, pressure and temperature of the carrier gas, and mass of the vapor atom are hidden in tl0 • Thus, wh.en these parameters are taken into eonsideration in the calculation of 71, one can expect a larger spread in its values. The last two columns of Table I give the time in microseconds for a vapor initially at a temperature of 1800°K. to cool down to 900°K. and 450°K., respectively. I t is interesting to note that it takes a body of gold vapor about 4}{ times as long to cool down in helium as in argon, and it takes gold vapor 16}3 times as long to cool down in helium as magnesimn vapor in argon. Since rl is inversely proportional to the first power of the carrier gas pressure, the cooling time presented here should be multiplied b y the factor 1 ram./760 ram. if a carrier gas pressure of 760 ram. (instead of 1 ram.) is used. If cooling through inelastic collisions is negligible, the values given in Table I for the cooling times should represent the lower limits. For those cases in which aerosols are
produced in an inert gas atmospb.ere by: the evaporation-sublimation technique, this condition is generally satisfied. Local h.eaibing of the carrier gas by the vapor tends to lengthen the time required to cool the vapor, as the vapor then has to transfer its energy to a carrier gas with an effective temperature higher than its temperature before its interaction with. the vapor. I t is reasonable, however, to expect that the amount of heating is less for an open system employing a flowing carrier gas than for a closed one employing a stationary gas. How much less is difficult to assess and is probably strongly dependent on the flow rate of the carrier gas. For example, suppose that the vapor generation rate is constant, it is clear then that a larger gas flow rate means a higher gas to vapor atom density ratio in the vapor-gas mixture, and thus the gas temperature is less likely to be affected by the vapor temperature. Finally, we m a y mention an interesting conclusion made recently by Kerker et al. (22) concerning the cooling rate of NaC1 vapor in a helium gas. In connection with their aerosol experiments, they prepared NaC1 aerosols b y continuously evaporating and condensing NaC1 vapor in a stream of
220
LUKE
flowing helium gas of one atmosphere pressure. T h e a m o r p h o u s n a t u r e of the resultant NaC1 aerosols led t h e m to conclude t h a t the NaC1 v a p o r was quenched very rapidly b y the helium gas. This conclusion, t h o u g h only qualitative, implies t h a t the c o m p u t e d values of the cooling-down time given in T a b l e I are quite reasonable. I t is to be n o t e d t h a t even for a carrier gas pressure of 1 ram. ttg, the cooling-down times are already in the microsecond region. T h e corresponding coolingd o w n time for a carrier gas pressure of one atmosphere is a b o u t a t h o u s a n d times shorter, or in the milli-microseeond region. REFERENCES 1. COURTNEY, W. G., Symp. Combustion 9th, p. 799 (1963). 2. HIRTH, J. P., AND POUND, G. M., "Condensation and Evaporation: Nucleation and Growth Kinetics." Macmillan, New York, 1963. 3. COUnTNEY,W. G., A R S J. 31,751 (1961). 4. Fucks, N. A., "The Mechanics of Aerosols," Chap. 7. Macmillan, New York, 1964. 5. GREEN, H. L., AND LANE, W. R., "Particulate
Clouds : Dusts, Smokes, and Mists," 2nd ed., Chap. 5. E. and F. N. Spon, London, 1964. 6. M£RTYNOV,G. A., ANE BAI~ANOV,S. P., In B. V. Deryagin, ed., "Research in Surface Forces," p. 182. Consultants Bureau, New York, 1963. 7. BROCK,J. R., AND HIDY, G. M., J. Appl. Phys. 36, 1857 (1965). 8. DELCROIX,J. L., "Introduction to the Theory
of Ionized Gases," pp. 35-37. Interscience, New York, 1960. 9. VON DARNEL,G. F., Phys. Rev. 94, 1272 (1954). 10. VON I)ARDEL, G. F., Trans. Roy. Inst. Technol. Stockholm No. 75 (1954). ll(a). A~ALm, E., In S. Fltigge, ed., "Handbuch der Physik," 38, Part II, p. 212. Springer, Berlin, 1959. (b) ibid., pp. 397-399. (c). ibid., p. 501. 12. COHEN, E. R., "Peaceful Uses of Atomic Energy: Proceedings of the International Conference in Geneva, August, 1955," 5, 405. Published by the United Nations. 13. CnAVATn,A. M., Phys. Rev. 36, 248 (1930). 14. MOELWYN-HUGHES, E. A., "Physical Chemistry," p. 79. Cambridge University Press, Cambridge, 1940. 151 GRAY, D. E., ed., "American Institute of Physics Handbook," pp. 7-9 to 7-12. McGraw-Hill, New York, 1957. 16. KEESOM, W. H., "Helium," p. 114. Elsevier, Amsterdam, 1942. 17. KI~OTO, K., KAMIYA,Y., NONOYAMA,M., AND UYEDA, R., Japan. J. Appl. Phys. 2, 702 (1963). 18. GEN, M. S., ZISKIN, M. S., AND PETROV, Yu. I., Dokl. Akad. Nauk SSSR. 127,366 (1959). 19. TILL, P. H., AnD TURKEVICH,J., U.S. Atomic
Energy Commission Report NYO-3435 (1956). 20. KARmRIS, F. G., FISR, B. R., AND I~OYSTER, G. W., JR., In W. G. Chace and H. K. Moore, eds., "Exploding Wires," Vol. 1, pp. 299-311. Plenum Press, New York, 1962. 21. NAsa, C. P., AND DE SIENO, R. P., J. Phys. Chem. 69, 2139 (1965). 22. ESPENSCHEID, W. F., MATIJEVI~, E., AND KERKER, M., J. Phys. Chem. 68, 2831 (1964).