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Nonstationary homogeneous nucleation in the atmosphere — a numerical solution
Atmospheric Research, 25 (1990) 417-430
417
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
Nonstationary homogeneous nucleation in the atmosphere - a numerical solution Nikolay Miloshev and Georgi Miloshev Institute of Geophysics, Bulgarian Academy of Sciences, Acad. G. BonchevStr., Block 3, 1113 Sofia (Bulgaria) Bulgarian Hydrometeorological Sevice, Bulgarian Academy of Sciences, Sofia (Bulgaria) (Received March 6, 1989; accepted after revision September 6, 1989 )
ABSTRACT Miloshev, N. and Miloshev, G., ! 990. Nonstationary homogeneous nucleation in the atmosphere - a numerical solution. Atmos. Res., 25:417-430. The present study deals with the problem of nonsteadiness of the formation process of a new phase in the atmosphere. Using a numerical method, the solution of the main kinetic equation of phase formation is found in the cases ofnonstationary homogeneous condensation, deposition and freezing. The results obtained are compared with those derived by the analytical approach, and the advantages of the numerical solution are stressed. The differences between the time characteristics of the nonstationary phase formation process, delay time rn, transient time tn, effective time lag 0n and induction time or time lag z are pointed out. The period of nonsteadiness is shown as being essential in the process of formation of a new phase in the atmosphere, which has to be thoroughly studied. It is emphasised that the numerical approach is also applicable to over-critical sizes, i.e. it is applicable in order to obtain the description of the process of condensational growth leading to the formation of visible clusters under experimental conditions. This provides a possibility for immediate comparison between theory and experiment. RESUME Le pr6sent travail de recherche traite le probl~me de la non-stationnarit6 du processus de formation d'une phase nouvelle dans l'atmosphbre. On trouve par voie num6rique la solution de 1'6quation cin6tique principale pour la formation de la phase en ce qui concerne les cas de condensation, de sublimation et de congdation homog~nes non-stationnaires. Les r6sultats obtenus sont compar6s avec les r6sultats 6tablis d'apr~s la m6thode de r6solution analytique, et on montre les avantages de la r6solution num6rique. On explique la signification de la formation non-stationnaire de la phase, et on indique les diff6rences entre ses caract6ristiques temporelles: temps de retard rn, temps de transition tn, temps de retard effectif 0, et temps d'induction ou temps retardateur r. On montre que la pdriode de la non-stationnarit6 est une p6riode significative dans le processus de formation de la nouvelle phase dans l'atmosphbre, laquelle doit &re 6tudi6e d'une mani~re approfondie. On souligne que la m6thode de r6solution utilis6e est applicable 6galement aux dimensions surcritiques, c'est-h-dire qu'on peut aussi l'employer pour obtenir une description ad6quate du processus d'accroissement de la condensation jusqu'aux complexes visibles, ce qui permet de faire une confrontation imm6diate entre la th6orie et l'exp6rience.
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© 1990 Elsevier Science Publishers B.V.
4 18
N MILOSHEV AND G. MtLOSHEV
INTRODUCTION
The process of formation of a new phase is a nonstationary, discrete process of joining/detachment of monomers to/from already established clusters. The kinetics of this process has been the object of numerous investigations in the framework of the theory of phase transitions. Concerning the cases evolving in the atmosphere, all authors make two essential restricting assumptions: ( l ) the process is continuous; and (2) the process is a steady-state one. The formulae for the stationary rate of phase formation are thus obtained (Becker and Doting, 1935; Fletcher, 1962; Krastanov and Miloshev, 1980; Pruppacher and Klett, 1980). Expressions for the nonstationary rate of phase formation are derived in physical chemistry by a number of analytical approaches assuming a continuous process. The complexity of the assumptions involved in these approaches reveals, in numerous cases, that the analytical descriptions of nonstationary kinetics is only approximative. The solution proposed by Kashchiev ( 1969 ), is accepted as the most precise solution approved after such an approach. During the recent years numerical solutions to the basic nonstationary kinetic equation of phase formation were also found (Courtney, 1962; Abraham, 1969; Kelton et al., 1983; Volterra and Cooper, 1985; Kozisek, 1988 ). They account for the discrete character of the process as well. The considerably smaller number of restricting assumptions in these solutions determines their advantages over the analytical ones, namely: ( 1 ) more precise results both for the nonstationary and the stationary rates of phase formation; (2) illustration of the process in time not only for complexes of size close to the critical one but also for various sizes; and ( 3 ) consideration from the same view point of the formation process of the nuclei and their further condensational growth to experimentally observable sizes, enabling the adequate connections between theory and experiment. These advantages led us to study the nonstationary kinetics of phase formation in the atmosphere using numerical methods. The present paper deals with the cases of homogeneous nonstationary phase formation in the atmosphere - condensation, deposition and freezing. FORMULATION OF THE PROBLEM
Szilard's model for "bimolecular reactions" between complexes of molecules (clusters) and monomers can be given as follows: .ftz- I
Z,,_ j + Zl ~
6,
Z~
1~
Z,, +ZI ~ Z , , +
i
l~+ l
According to this model, the basic kinetic equation of phase formation is
NONSTATIONARY HOMOGENEOUS NUCLEATION IN THE ATMOSPHERE
419
given by Zeldovich ( 1942 ), and further developed by Kashchiev ( 1969 ) for the cases of supersaturation variable in time. The general form of this equation is as follows: dZ,(t) --=f._,(t)Z._,(t)-[f~(t)+f~(t)]Z.(t)+f~+t(t)Z,,+~(t) dt
(1)
for each n = e [ U, V] or:
dZ,(t) --=In_l(t)--I.(t) dt
(2)
I.(t)=f.(t) Z.(t)-f.+~(t) Z . + l ( t )
(3)
where Z. (t) is the number of clusters, consisting of n molecules ( monomers ) at the moment t; f. (t) and f~ (t) are the respective frequencies of the monomer joining to, or detaching from, the n-sized complex at moment t; I. (t) is the rate of formation of n-sized clusters at moment t; U and Vare respectively the lower and upper bounds of n. The conditions imposed in our considerations are as follows. (a) There are no clusters existing in the system before.
Zl(t=O)=N~
(4)
(total number of molecules in the system under examination )
Z.(t=O)=On>2 (b) Each growing cluster reaching size Vis taken out of the system. Z, ( t ) = 0 n>_ V
(4')
Eqs. 2 and 3 enable us to define the following two specific states. ( 1 ) Steady state: Z, does not vary with time, Z, = const. = Z~, Therefore, I , in eq. 2 has the same value for each n. This is the stationary rate of phase formation Is which, in the general case, differs from zero. (2) Equilibrium state: Is= 0. Then from eq. 3:
fnN. = f . + , Nn+ l
(5)
where N. is the equilibrium value of Z. (t). In the nonstationary state, after the new phase nuclei are formed, their size distribution approaches with time to steady state distribution Z~,, and the phase formation rate grows from 0 to Is. In order to present the problem in its integrality, a brief review of the previous investigations related to the kinetics of the phase formation has to be made.
420
N. M I L O S H E ' v A N D O. M I L O S H E V
REVIEW OF THE PREVIOUS INVESTIGATIONS
Using eqs. 3, 4 and 5, the following solution for the stationary rate of phase formation is obtained:
L,=
(6) \i=1
,
After some simplifications:
Is=ZN*f,*
(7)
Similarly for the stationary number of complexes with relevant size n: t--I
1
Z~;=LN, ~ ~N i=nJi "
(8)
l
where Z is the Zeldovich factor; Nn is the equilibrium number of complexes with relevant size n; n* is the size of the critical cluster. Based on the classical theory of phase formation, the work to generate a cluster of the new phase in a homogeneous initial phase is given by relation:
AGn=-n AG'+Ancr
(9)
where A G' is the difference in the Gibbs's free energy per molecule between the new phase and the initial phase; An is the cluster surface; a is the surface tension. For the cases of condensation and deposition under examination, i.e. phase formation from a gaseous initial phase:
A G'=KTIn S
(10)
where S is the supersaturation in the initial phase. In the case of freezing, i.e. the initial phase is liquid water:
A G ' - Sc(T-T'n) N,,
(11)
where S/is the molar entropy of fusion; Tm is the melting temperature; T the temperature; N~ is Avogadro's number. To simplify our examinations we assume spheric complexes of the new phase. Then: A,, = (36zt)1/2/)I/2/,/1/2
(12)
where v is the molecular volume. As already mentioned, a number of approaches have been made to obtain an analytical solution of eq. 1. The main assumption is that Zn (t) and Fn (t) are continuous functions of n. As stated by Kelton et al. ( 1983 ) and Volterra
NONSTATIONARY HOMOGENEOUS NUCLEATION IN THE ATMOSPHERE
421
and Cooper ( 1985 ), the best analytical solution for the phase formation rate is the solution proposed by Kashchiev (1969) in the following general form:
I*t=Is 1+
(-1yexp
-
(13)
where r = 4/n3f..Z z is the co-called induction time. NUMERICAL RESOLUTION
Despite the simplicity of the initial system of eq. 1, the accuracy of the obtained results is greatly affected by the adequate choice of a suitable numerical m e t h o d which can describe the process correctly. This holds especially for those cases, similar to ours, when the system ( 1 ) consists of a great n u m b e r of equations. For our purposes we adopted initially the algorithm given by Kelton et al. (1983 ), with an error of approximation O(dt), where 8t is the time-step. Taken into account the obvious necessity of minimizing the computer time required, we worked out an algorithm after the RoungeKutta method with an error 0 (cSt3 ). The principal advantage of this approach over that proposed is the possibility to use a longer time-step keeping the same accuracy. Let Zn (t) be the number of clusters with size n at m o m e n t t. Then:
Zn,,+a, =Z~,, + ~ [ - (Z.., +Z*,) (f. +L) + (Z._ ,., +Z*_l.t)f~_, +(Z.+l,,+Z*+,.,)~+l]
(14)
where:
Z*,,=Z..t+St[-Z..t(f.+f~)+Z._~,tL_l + Z.+ l.,f.+ l ]
(15)
To satisfy the physical character of the process, the number of clusters joining or detaching more than one molecule for one time step 8t must be insignificant. The probability a cluster of size n to form a cluster of size n + 1 is 8t.f~. Hence the upper bound of the time step is tmax= 1/f~ The algorithm works with a variable time step and starts from c~t~ 10-10 8tmax. When getting near to the steady state, it is possible to enlarge the time step, keeping it, however, below C~/max. The program is over when the modification of Z*, for one time interval is less than 10-8. Another problem to be solved in the numerical realization is the correct selection of initial and boundary conditions. As in Abraham ( 1969 ) and Kelton et al. (1983), the lower bound selected is U= 10, since for very small clusters the classical theory for the work of phase formation is inaccurate, because the macrophysical characteristics used in it are undefined for very small sizes. V= 3* is taken for the upper bound. Our numerical experiments proved this value to be reasonable in order to obtain results accurate enough
422
N. MILOSHEV AND G. MILOSHEV
for the critical cluster. As an initial condition, we assume that the clusters of 10 molecules are in equilibrium concentration, retained during the entire process. Although artificial, this condition is accurate enough for the cases in the atmosphere where the number of molecules in the initial phase, involved in the formation of critical clusters of the new phase, is found to be insignificant. The problem thus formulated is to solve eq. 1 using eqs. 14-15 under the conditions: Zlo.o =N~0=NI exp ( - A I 0 / K T ) Zn.o =0forn>_-10 Zio.t =Nlo Z~., =0forV>~3*
(16)
The determination of the frequencies of joining and detachment of monomers is important for the numerical solution. In the present treatments we will confine to supersaturation steady in time, i.e. to time-independent frequencies. Their general form is:
f,,
=A,,WF~ (zlg,,)
f,+,
=AnWF2(dgn)
(17)
where W is the flux density; dg,, =AG,,+t-AG,, is the difference between the works of formation of clusters, composed of n + 1 molecules and n molecules. For gaseous initial phase:
SPo W - (2nmKT) I/2
(18)
where S is the supersaturation; Po is the saturation pressure over flat water surface; m is the molecular mass. As for the liquid initial phase W= N~exp
( - ~-~)
(19)
where K,h,R are respectively the constants of Boltzman, Planck and the universal gas constant; Nc is the number of water monomers in contact with the unit area of the new phase; AF is the activation energy for diffusion through the phase boundary which is approximated with sufficient accuracy by the activation energy for self-diffusion. An empirical expression of its temperature dependence is given by Pruppacher and Klett (1980) as follows: ZIF= 5.5 exp ( - 1.33.10 - 2 T+2.74.10
-4 T2+
1.085-
10 - 6 T 3 )
(20)
As it is seen from eq. 5: f,+,-
N~ =exp
-
This ratio has to be taken into account in eq. 20 using F~ (Agn) and/72 (zig,,). For this purpose different approaches are proposed in the literature.
NONSTATIONARYHOMOGENEOUSNUCLEATIONIN THE ATMOSPHERE
423
Abraham (1969) and Volterra and Cooper ( 1985 ) assume this exponent to be involved in the expression forf~+ 1 . Kozisek ( 1988 ) assumes that for n < n* the exponent is included in the expression forfn, while for n > n* it is included in the expression for f~+ ~. Similarly to the procedure adopted by Kelton et al. ( 1983 ), we shared this exponent between the two frequencies so that: F1 = exp
(_Agn~ 2KT] and/72 = (Ag_~__T)
(21 )
Nevertheless we agree with the opinion stated by Volterra and Cooper (1985) that only a detailed treatment on molecular level could shed more light on the exact form of these expressions. RESULTS AND DISCUSSIONS
Steady-statenucleation One of the criteria for the applicability of our numerical solution is the accuracy of determination of the steady-state rate of the phase formation (I NvM). As we have already shown, the most accurate value of the steadystate rate is given by direct s u m m a t i o n (eq. 6): I suM. For the residual error we have: l/SUM I N U M I
• ~
- =0.000035 to0.0003
(22)
for the three cases under consideration (condensation, deposition and freezing) Analogously, for the steady-state distribution of clusters with critical size we derive: 1 7 S U M __ 7 N U M [
•
~s--6-~ '
=0.00005 to 0.0006
(23)
r/~,S
It is seen that the numerical values coincide perfectly with those calculated according to the formulae 6 and 8. As expected the numerical solution is of much higher accuracy than the analytical solution suggested in eq. 7. Actually: IISUM-- IAN = 0.01 ~ to 0.0207
(24)
where I AN is the value calculated according to eq. 7. Fig. 1 shows the curves for steady state Z~, and equilibrium N. number of clusters with size close to the critical n * = 2 7 in the case of freezing, at T = 213.15 K. Our results (circles) follow quite precisely the trend of the curve.
424
N. MILOSHEV AND G. MILOSHEV
/~]0
J
6L..
2
I O't 20
2t
22
23
2~
25
26
27
28
2.q
30
3¢
32
Fig. 1. E q u i l i b r i u m (N.=N~ exp ( - G . / k T ) ) and steady-state ( Z s ) n u m b e r o f clusters as a function o f the cluster size in the case o f freezing at T = 215.15 K a n d critical cluster size N * = 27.
Nonstationary nucleation Fig. 2 shows the time-dependence of the number of clusters composed of n,t, for clusters with different sizes at T = 233.15 K and supersaturation S = 11 for condensation (continuous curves ); S = 17 for deposition ( dashed curves ); melting temperature T = 273.15 K for freezing (dotted curves). The sizes of the critical complexes are: n * = 38 for condensation; n * = 32 for deposition; and n* = 126 for freezing. The time of appearance of complexes of a given size .................... ~0
cm"3, Zn, ~
I0 °
~'
...... i:: ..............~60
-t
0
_
//
i
ii
'
I
/
Eo9~ (s) -6
-5
-~
-5
-2
I
Fig. 2. Time-dependence of the number of clusters, composed ofn molecules ( Z . , t ) f o r clusters w i l h different sizes. Condensation: continuous lines; deposition: dashed lines; freezing: d o t t e d lines•
NONSTATIONARY
HOMOGENEOUS
NUCLEATION
425
IN THE ATMOSPHERE
and their growth in n u m b e r is seen to depend mostly on the size. The bigger the complexes, the longer the period of time necessary for their appearance and growing in number. Here, it is possible to formulate two time characteristics of the nonstationary process of phase formation. ( 1 ) Time of delay zn, given by Abraham (1969) as delay time. This is the time elapsing after the beginning of the process, for which Zn,t-< 10- 3o (corn_ plexes/cm3). As indicated in Fig. 2, for clusters with critical size n*, Zcond.= 3.5" 10 -5 S; Zdep.= 1.2" 10 -5 S; and Zfreezing= 4 . 0 " 10 -2 S. (2) Time of nonsteadiness t,,, pointed out as transient time by Courtney ( 1962 ). Here this is the time from the beginning of the process until the moment when Z,,,t varies in one time interval by less than l0 -s. The m o m e n t when Z,,.t=95%(SS.) is considered as transient time by Courtney. In the studied cases, for critical clusters tcond.= 1.2608" 10-2 S; tdep. = 5.8167" 10--3 S; and /freezing= 12.669 S. Fig. 3 shows the variations with time of the rate of phase formation I,,t for complexes of different size, the other parameters being the same as in Fig. 2. During the period of nonsteadiness this rate as well as Z,,,t depends on the size for which it is calculated, in contrast to the steady-state case where the rate is the same for each size. Similarly to the variation of Z,,t, once appearing, I,,,t is characterized by a rapid increase with time to its steady-state value. Another specific feature noted by many authors is that for complexes with subcritical sizes, I,,t passes through a m a x i m u m which is higher than the steadystate value. The smaller the cluster size, the higher this maximum. For critical and larger clusters, I,,t increases smoothly to the steady-state value, in the case 4o
...........~i:..':.:'.'..:.....
c m - 3 5 -I
/oIo
/
/
/
/ ./ ' //
.
~
i0~
/
/
/ t2~;" ....... ." ~6¢
.:
_
_
f
/-
/
/ // /
/
/1
/
""
'
:I -
-
// I i
i
/
:/
/
,/~i'/80 I I 60/ /
/ 180 -2
0
4'
Fig. 3. Time-dependence of the rate of phase formation I,,,t for clusters with different sizes. Condensation: continuous lines; deposition: dashed lines; freezing: dotted lines.
426
N. MILOSHEVAND G. MILOSHEV ~0
.~-r~ d Z n , t
,°1
20
.." . ~"" ......:.:~%~,, f: .. .:
';
40 o
/ i 0 I~
/
/
/
/
lllll/f
la~t
I
¢6:_ 6
I ,
-5
-z¢
-3
-2
~¢
0
!
Fig. 4. Time dependence of the total derivative dZ.,t/dt for clusters with different sizes. Condensation: continuous lines; deposition: dashed lines; freezing: dotted lines.
0,g I~ o,7 o,6 o,5 0,0
O,2
o,t
% t
)
5
i
~
"
Fig. 5. Variation of I , , f f I s as a function of t/r. M = o u r numerical solution: K=the analytical solution by Kashchiev ( 1969 ).
of sub-critical sizes, the probability for detachment is higher than that for joining and "accumulation" of the rate values towards smaller sizes is produced. For this reason the smaller the complexes, the higher their rate peak. Such an accumulation could not be observed with the critical size, since the probabilities of joining and detachment are equal and subsequently the rate does not reach a maximum. The trend of the curves for dZ, Jdt (Fig. 4) is similar to that of the curves
NONSTATIONARYHOMOGENEOUSNUCLEATIONIN THE ATMOSPHERE
427
TABLE I Variation of l,,t/Is from the analytical solution of Kashchiev (1969) (IK~sh./ls), and from our numerical solution (IMit./I,) depending on the change of the ratio t/z (t is time time from the beginning of the process; r is induction time) Process
time
lv,~h.
IMiL
1,
L
Freezing
1.3414 2.5684 3.7954 5.0224 6.2455
0.4864 0.8464 0.9551 0.9868 0.9961
0.2909 0.8147 0.9530 0.9878 0.9994
Condensation
0.5838 1.0069 1.5064 3.2429 5.838
0.0678 0.3047 0.561 0.9198 0.994
0.0009 0.0642 0.3367 0.9254 0.996
Deposition
0.631 0.8464 1.206 2.02 5.50
8.92- 10 -2 0.2088 0.417 0.736 0.991
3.44-10 -3 0.0308 0.185 0.674 0.998
c,.rrJl ;( n',t I00 90
i f f "¢2
80 io 6
70
I I t I I I
60 50 ~o 30
I
20
I
** i0 3
1I
~o
J. -/+
-3
-2
-~
, .// o
tog ~ :O ;
Fig. 6. Variation with time of the concentration of clusters with critical size X,,.,t. Condensation: continuous lines; deposition: dashed lines; freezing: dotted lines.
428
N. MILOSHEV A N D G. MILOSHEV
for In.,. The peculiarity here is that dZ,,t/dt does not reach the steady-state value and decreases with time, tending to zero. This trend results from eq. 2 where at I~,,~I~, dZ,,,/dt--,O. The comparison between our results and the analytical solution of Kashchiev ( 1969 eq. 13 ) about the rate of phase formation at the critical size n* is shown in Fig. 5. The analytical solution is seen to be close to the numerical results for t> v where r is the induction time in eq. 13. The respective exact values are presented in Table I. Another specific feature of the process of nonstationary phase tbrmation is the concentration of clusters of a given size as a function of time X,,,,. This is given by the time integral of/,., in the form:
X,,,t '-" i I,,,dt
(25)
0
In Fig. 6 the changes in X*, of critical clusters are given for the cases of condensation (continuous curves) deposition (dashed curves) and freezing (dotted curves). It is worth noting that this characteristic of the process is of great importance since it can be measured experimentally. As it is seen, a period of time should elapse after the beginning of the process until the appearance in the volume of a considerable number of complexes with a given size. Here it is important to formulate a third time characteristic of the process of nonstationary phase formation. (3) Time of appearance 0n, defined by Kelton et al. (1983), as effective time lag. The respective mathematical expression is:
tJ
In our study 0,*con0.= 1.7318" 10-3 S; 0n*dep. = 7.2547- 10-a s; 0~*rre,-zing= 2.6521 S. It is interesting to compare the time of appearance 0n, thus obtained and that derived by the analytical solution the so-called induction time or time lag r. As indicated in Kelton et al. ( 1983 ), by applying the analytical solution of Kashchiev ( 1969 ), the relation between them for the critical size is: 7~2T
0~.- 6
(27)
In our considerations rco.a. = 8.86" 10- 4 s; ~'dep.= 3.90" 10-- 4S; rfreezmg= 1.399 S; or (0,-/r)~ona.= 1.95; (0,*/r)aep.= 1.86; (0,*/V)freezin~= 1.89, these values being fairly close to fez~6 = 1.645. The respective differences are due to the
NONSTATIONARY HOMOGENEOUS NUCLEATION IN THE ATMOSPHERE
429
On..lO~ (s)
/ 20
~0
60
/
80
Fig. 7. Variation of the effective time lag 0, with growing cluster size.
fact that the treatments of Kashchiev are related to a critical region close to n* for which lAG,-AGn* [ < KT. In this manner the time for sub-critical cluster formation is excluded. For that reason the ratio is always higher then
n2/6. Naturally X,., as well as 0,, are functions of the size for which they are calculated. The size dependence of 0, in the case of condensation at T = 261.15 K and S = 7 is illustrated in Fig. 7. On is seen to increase with size ,. CONCLUSION
The examining of the nucleation process in the atmosphere as a nonstationary one is of great importance. Firstly the numerical solution proposed here is appliable to over-critical sizes, i.e. it is useful for the description of the condensational growth process, leading to the formation of visible clusters under experimental conditions. The latter provides a possibility for immediate comparison between theory and experiment. On the other hand the numerical solution, used here, gives the most exact results because of the minim u m of restrictive assumptions. Homogeneous nucleation in the atmosphere, although it is a case of idealization, has it significance when it refers to weather modification with cooled reagents. At last the results exposed here provide a basis for developing the nonstationary nucleation theory for the more real, heterogeneous cases observed in the atmosphere. REFERENCES Abraham, F.F., 1969. Multistate kinetics in nonsteady-state nucleation. A numberical solution. J. Chem. Phys., 51: 1632-1638. Becker, R. and Doring, W., 1935. Kinetische Behandlung der Keimbildung in 0bersattigen D~impfen. Ann. Phys., 24: 719-731.
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N. MILOSHEV AND G. MILOSHEV
Courtney, W.G., 1962. See Kelton et al., 1983. Fletcher, N.H., 1962. The Physics of Rain Clouds. Cambrige University Press, Cambridge, Ch. 3, pp. 37-63. Kashchiev, D.B., 1969. Solution of the non-steady State problem in nucleation kinetics. Surface Sci., 14: 209-220. Kelton, K.F., Greer, A.L. and Thompson, G.V., 1983. Transient Nucleation in Condensed Systems. J. Chem. Phys., 79: 6261-6276. Kozisek, Z., 1988. Two step nucleation in glass-forming systems. Cryst. Res. Technol. In Press. Krastanov, L. and Miloshev. G., 1980. Theoretical Bases of Phase Transitions of Water in the Atmosphere. Meteorological Service of the Hungarian People's Republic, Budapest, CH. 3, pp. 51-65. Pruppacher, H. and Klett, J., 1980. Microphysics of Clouds and Precipitation. D. Reidel Publishing Company, Dordrecht, Ch. 7, pp. 162-182. Volterra,. and Cooper, A., 1985. Numerical calculation of induction times for crystallization of glasses. J. Non-Cryst. Solids, 74: 85-95. Zeldovich, J.B., 1942. K. Teorii Obrasovania Novoi Fazi. Kavitacia. Zh. Eksp. Teor. Fiz., 12: 525-537.
417
Elsevier Science Publishers B.V., Amsterdam - - Printed in The Netherlands
Nonstationary homogeneous nucleation in the atmosphere - a numerical solution Nikolay Miloshev and Georgi Miloshev Institute of Geophysics, Bulgarian Academy of Sciences, Acad. G. BonchevStr., Block 3, 1113 Sofia (Bulgaria) Bulgarian Hydrometeorological Sevice, Bulgarian Academy of Sciences, Sofia (Bulgaria) (Received March 6, 1989; accepted after revision September 6, 1989 )
ABSTRACT Miloshev, N. and Miloshev, G., ! 990. Nonstationary homogeneous nucleation in the atmosphere - a numerical solution. Atmos. Res., 25:417-430. The present study deals with the problem of nonsteadiness of the formation process of a new phase in the atmosphere. Using a numerical method, the solution of the main kinetic equation of phase formation is found in the cases ofnonstationary homogeneous condensation, deposition and freezing. The results obtained are compared with those derived by the analytical approach, and the advantages of the numerical solution are stressed. The differences between the time characteristics of the nonstationary phase formation process, delay time rn, transient time tn, effective time lag 0n and induction time or time lag z are pointed out. The period of nonsteadiness is shown as being essential in the process of formation of a new phase in the atmosphere, which has to be thoroughly studied. It is emphasised that the numerical approach is also applicable to over-critical sizes, i.e. it is applicable in order to obtain the description of the process of condensational growth leading to the formation of visible clusters under experimental conditions. This provides a possibility for immediate comparison between theory and experiment. RESUME Le pr6sent travail de recherche traite le probl~me de la non-stationnarit6 du processus de formation d'une phase nouvelle dans l'atmosphbre. On trouve par voie num6rique la solution de 1'6quation cin6tique principale pour la formation de la phase en ce qui concerne les cas de condensation, de sublimation et de congdation homog~nes non-stationnaires. Les r6sultats obtenus sont compar6s avec les r6sultats 6tablis d'apr~s la m6thode de r6solution analytique, et on montre les avantages de la r6solution num6rique. On explique la signification de la formation non-stationnaire de la phase, et on indique les diff6rences entre ses caract6ristiques temporelles: temps de retard rn, temps de transition tn, temps de retard effectif 0, et temps d'induction ou temps retardateur r. On montre que la pdriode de la non-stationnarit6 est une p6riode significative dans le processus de formation de la nouvelle phase dans l'atmosphbre, laquelle doit &re 6tudi6e d'une mani~re approfondie. On souligne que la m6thode de r6solution utilis6e est applicable 6galement aux dimensions surcritiques, c'est-h-dire qu'on peut aussi l'employer pour obtenir une description ad6quate du processus d'accroissement de la condensation jusqu'aux complexes visibles, ce qui permet de faire une confrontation imm6diate entre la th6orie et l'exp6rience.
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4 18
N MILOSHEV AND G. MtLOSHEV
INTRODUCTION
The process of formation of a new phase is a nonstationary, discrete process of joining/detachment of monomers to/from already established clusters. The kinetics of this process has been the object of numerous investigations in the framework of the theory of phase transitions. Concerning the cases evolving in the atmosphere, all authors make two essential restricting assumptions: ( l ) the process is continuous; and (2) the process is a steady-state one. The formulae for the stationary rate of phase formation are thus obtained (Becker and Doting, 1935; Fletcher, 1962; Krastanov and Miloshev, 1980; Pruppacher and Klett, 1980). Expressions for the nonstationary rate of phase formation are derived in physical chemistry by a number of analytical approaches assuming a continuous process. The complexity of the assumptions involved in these approaches reveals, in numerous cases, that the analytical descriptions of nonstationary kinetics is only approximative. The solution proposed by Kashchiev ( 1969 ), is accepted as the most precise solution approved after such an approach. During the recent years numerical solutions to the basic nonstationary kinetic equation of phase formation were also found (Courtney, 1962; Abraham, 1969; Kelton et al., 1983; Volterra and Cooper, 1985; Kozisek, 1988 ). They account for the discrete character of the process as well. The considerably smaller number of restricting assumptions in these solutions determines their advantages over the analytical ones, namely: ( 1 ) more precise results both for the nonstationary and the stationary rates of phase formation; (2) illustration of the process in time not only for complexes of size close to the critical one but also for various sizes; and ( 3 ) consideration from the same view point of the formation process of the nuclei and their further condensational growth to experimentally observable sizes, enabling the adequate connections between theory and experiment. These advantages led us to study the nonstationary kinetics of phase formation in the atmosphere using numerical methods. The present paper deals with the cases of homogeneous nonstationary phase formation in the atmosphere - condensation, deposition and freezing. FORMULATION OF THE PROBLEM
Szilard's model for "bimolecular reactions" between complexes of molecules (clusters) and monomers can be given as follows: .ftz- I
Z,,_ j + Zl ~
6,
Z~
1~
Z,, +ZI ~ Z , , +
i
l~+ l
According to this model, the basic kinetic equation of phase formation is
NONSTATIONARY HOMOGENEOUS NUCLEATION IN THE ATMOSPHERE
419
given by Zeldovich ( 1942 ), and further developed by Kashchiev ( 1969 ) for the cases of supersaturation variable in time. The general form of this equation is as follows: dZ,(t) --=f._,(t)Z._,(t)-[f~(t)+f~(t)]Z.(t)+f~+t(t)Z,,+~(t) dt
(1)
for each n = e [ U, V] or:
dZ,(t) --=In_l(t)--I.(t) dt
(2)
I.(t)=f.(t) Z.(t)-f.+~(t) Z . + l ( t )
(3)
where Z. (t) is the number of clusters, consisting of n molecules ( monomers ) at the moment t; f. (t) and f~ (t) are the respective frequencies of the monomer joining to, or detaching from, the n-sized complex at moment t; I. (t) is the rate of formation of n-sized clusters at moment t; U and Vare respectively the lower and upper bounds of n. The conditions imposed in our considerations are as follows. (a) There are no clusters existing in the system before.
Zl(t=O)=N~
(4)
(total number of molecules in the system under examination )
Z.(t=O)=On>2 (b) Each growing cluster reaching size Vis taken out of the system. Z, ( t ) = 0 n>_ V
(4')
Eqs. 2 and 3 enable us to define the following two specific states. ( 1 ) Steady state: Z, does not vary with time, Z, = const. = Z~, Therefore, I , in eq. 2 has the same value for each n. This is the stationary rate of phase formation Is which, in the general case, differs from zero. (2) Equilibrium state: Is= 0. Then from eq. 3:
fnN. = f . + , Nn+ l
(5)
where N. is the equilibrium value of Z. (t). In the nonstationary state, after the new phase nuclei are formed, their size distribution approaches with time to steady state distribution Z~,, and the phase formation rate grows from 0 to Is. In order to present the problem in its integrality, a brief review of the previous investigations related to the kinetics of the phase formation has to be made.
420
N. M I L O S H E ' v A N D O. M I L O S H E V
REVIEW OF THE PREVIOUS INVESTIGATIONS
Using eqs. 3, 4 and 5, the following solution for the stationary rate of phase formation is obtained:
L,=
(6) \i=1
,
After some simplifications:
Is=ZN*f,*
(7)
Similarly for the stationary number of complexes with relevant size n: t--I
1
Z~;=LN, ~ ~N i=nJi "
(8)
l
where Z is the Zeldovich factor; Nn is the equilibrium number of complexes with relevant size n; n* is the size of the critical cluster. Based on the classical theory of phase formation, the work to generate a cluster of the new phase in a homogeneous initial phase is given by relation:
AGn=-n AG'+Ancr
(9)
where A G' is the difference in the Gibbs's free energy per molecule between the new phase and the initial phase; An is the cluster surface; a is the surface tension. For the cases of condensation and deposition under examination, i.e. phase formation from a gaseous initial phase:
A G'=KTIn S
(10)
where S is the supersaturation in the initial phase. In the case of freezing, i.e. the initial phase is liquid water:
A G ' - Sc(T-T'n) N,,
(11)
where S/is the molar entropy of fusion; Tm is the melting temperature; T the temperature; N~ is Avogadro's number. To simplify our examinations we assume spheric complexes of the new phase. Then: A,, = (36zt)1/2/)I/2/,/1/2
(12)
where v is the molecular volume. As already mentioned, a number of approaches have been made to obtain an analytical solution of eq. 1. The main assumption is that Zn (t) and Fn (t) are continuous functions of n. As stated by Kelton et al. ( 1983 ) and Volterra
NONSTATIONARY HOMOGENEOUS NUCLEATION IN THE ATMOSPHERE
421
and Cooper ( 1985 ), the best analytical solution for the phase formation rate is the solution proposed by Kashchiev (1969) in the following general form:
I*t=Is 1+
(-1yexp
-
(13)
where r = 4/n3f..Z z is the co-called induction time. NUMERICAL RESOLUTION
Despite the simplicity of the initial system of eq. 1, the accuracy of the obtained results is greatly affected by the adequate choice of a suitable numerical m e t h o d which can describe the process correctly. This holds especially for those cases, similar to ours, when the system ( 1 ) consists of a great n u m b e r of equations. For our purposes we adopted initially the algorithm given by Kelton et al. (1983 ), with an error of approximation O(dt), where 8t is the time-step. Taken into account the obvious necessity of minimizing the computer time required, we worked out an algorithm after the RoungeKutta method with an error 0 (cSt3 ). The principal advantage of this approach over that proposed is the possibility to use a longer time-step keeping the same accuracy. Let Zn (t) be the number of clusters with size n at m o m e n t t. Then:
Zn,,+a, =Z~,, + ~ [ - (Z.., +Z*,) (f. +L) + (Z._ ,., +Z*_l.t)f~_, +(Z.+l,,+Z*+,.,)~+l]
(14)
where:
Z*,,=Z..t+St[-Z..t(f.+f~)+Z._~,tL_l + Z.+ l.,f.+ l ]
(15)
To satisfy the physical character of the process, the number of clusters joining or detaching more than one molecule for one time step 8t must be insignificant. The probability a cluster of size n to form a cluster of size n + 1 is 8t.f~. Hence the upper bound of the time step is tmax= 1/f~ The algorithm works with a variable time step and starts from c~t~ 10-10 8tmax. When getting near to the steady state, it is possible to enlarge the time step, keeping it, however, below C~/max. The program is over when the modification of Z*, for one time interval is less than 10-8. Another problem to be solved in the numerical realization is the correct selection of initial and boundary conditions. As in Abraham ( 1969 ) and Kelton et al. (1983), the lower bound selected is U= 10, since for very small clusters the classical theory for the work of phase formation is inaccurate, because the macrophysical characteristics used in it are undefined for very small sizes. V= 3* is taken for the upper bound. Our numerical experiments proved this value to be reasonable in order to obtain results accurate enough
422
N. MILOSHEV AND G. MILOSHEV
for the critical cluster. As an initial condition, we assume that the clusters of 10 molecules are in equilibrium concentration, retained during the entire process. Although artificial, this condition is accurate enough for the cases in the atmosphere where the number of molecules in the initial phase, involved in the formation of critical clusters of the new phase, is found to be insignificant. The problem thus formulated is to solve eq. 1 using eqs. 14-15 under the conditions: Zlo.o =N~0=NI exp ( - A I 0 / K T ) Zn.o =0forn>_-10 Zio.t =Nlo Z~., =0forV>~3*
(16)
The determination of the frequencies of joining and detachment of monomers is important for the numerical solution. In the present treatments we will confine to supersaturation steady in time, i.e. to time-independent frequencies. Their general form is:
f,,
=A,,WF~ (zlg,,)
f,+,
=AnWF2(dgn)
(17)
where W is the flux density; dg,, =AG,,+t-AG,, is the difference between the works of formation of clusters, composed of n + 1 molecules and n molecules. For gaseous initial phase:
SPo W - (2nmKT) I/2
(18)
where S is the supersaturation; Po is the saturation pressure over flat water surface; m is the molecular mass. As for the liquid initial phase W= N~exp
( - ~-~)
(19)
where K,h,R are respectively the constants of Boltzman, Planck and the universal gas constant; Nc is the number of water monomers in contact with the unit area of the new phase; AF is the activation energy for diffusion through the phase boundary which is approximated with sufficient accuracy by the activation energy for self-diffusion. An empirical expression of its temperature dependence is given by Pruppacher and Klett (1980) as follows: ZIF= 5.5 exp ( - 1.33.10 - 2 T+2.74.10
-4 T2+
1.085-
10 - 6 T 3 )
(20)
As it is seen from eq. 5: f,+,-
N~ =exp
-
This ratio has to be taken into account in eq. 20 using F~ (Agn) and/72 (zig,,). For this purpose different approaches are proposed in the literature.
NONSTATIONARYHOMOGENEOUSNUCLEATIONIN THE ATMOSPHERE
423
Abraham (1969) and Volterra and Cooper ( 1985 ) assume this exponent to be involved in the expression forf~+ 1 . Kozisek ( 1988 ) assumes that for n < n* the exponent is included in the expression forfn, while for n > n* it is included in the expression for f~+ ~. Similarly to the procedure adopted by Kelton et al. ( 1983 ), we shared this exponent between the two frequencies so that: F1 = exp
(_Agn~ 2KT] and/72 = (Ag_~__T)
(21 )
Nevertheless we agree with the opinion stated by Volterra and Cooper (1985) that only a detailed treatment on molecular level could shed more light on the exact form of these expressions. RESULTS AND DISCUSSIONS
Steady-statenucleation One of the criteria for the applicability of our numerical solution is the accuracy of determination of the steady-state rate of the phase formation (I NvM). As we have already shown, the most accurate value of the steadystate rate is given by direct s u m m a t i o n (eq. 6): I suM. For the residual error we have: l/SUM I N U M I
• ~
- =0.000035 to0.0003
(22)
for the three cases under consideration (condensation, deposition and freezing) Analogously, for the steady-state distribution of clusters with critical size we derive: 1 7 S U M __ 7 N U M [
•
~s--6-~ '
=0.00005 to 0.0006
(23)
r/~,S
It is seen that the numerical values coincide perfectly with those calculated according to the formulae 6 and 8. As expected the numerical solution is of much higher accuracy than the analytical solution suggested in eq. 7. Actually: IISUM-- IAN = 0.01 ~ to 0.0207
(24)
where I AN is the value calculated according to eq. 7. Fig. 1 shows the curves for steady state Z~, and equilibrium N. number of clusters with size close to the critical n * = 2 7 in the case of freezing, at T = 213.15 K. Our results (circles) follow quite precisely the trend of the curve.
424
N. MILOSHEV AND G. MILOSHEV
/~]0
J
6L..
2
I O't 20
2t
22
23
2~
25
26
27
28
2.q
30
3¢
32
Fig. 1. E q u i l i b r i u m (N.=N~ exp ( - G . / k T ) ) and steady-state ( Z s ) n u m b e r o f clusters as a function o f the cluster size in the case o f freezing at T = 215.15 K a n d critical cluster size N * = 27.
Nonstationary nucleation Fig. 2 shows the time-dependence of the number of clusters composed of n,t, for clusters with different sizes at T = 233.15 K and supersaturation S = 11 for condensation (continuous curves ); S = 17 for deposition ( dashed curves ); melting temperature T = 273.15 K for freezing (dotted curves). The sizes of the critical complexes are: n * = 38 for condensation; n * = 32 for deposition; and n* = 126 for freezing. The time of appearance of complexes of a given size .................... ~0
cm"3, Zn, ~
I0 °
~'
...... i:: ..............~60
-t
0
_
//
i
ii
'
I
/
Eo9~ (s) -6
-5
-~
-5
-2
I
Fig. 2. Time-dependence of the number of clusters, composed ofn molecules ( Z . , t ) f o r clusters w i l h different sizes. Condensation: continuous lines; deposition: dashed lines; freezing: d o t t e d lines•
NONSTATIONARY
HOMOGENEOUS
NUCLEATION
425
IN THE ATMOSPHERE
and their growth in n u m b e r is seen to depend mostly on the size. The bigger the complexes, the longer the period of time necessary for their appearance and growing in number. Here, it is possible to formulate two time characteristics of the nonstationary process of phase formation. ( 1 ) Time of delay zn, given by Abraham (1969) as delay time. This is the time elapsing after the beginning of the process, for which Zn,t-< 10- 3o (corn_ plexes/cm3). As indicated in Fig. 2, for clusters with critical size n*, Zcond.= 3.5" 10 -5 S; Zdep.= 1.2" 10 -5 S; and Zfreezing= 4 . 0 " 10 -2 S. (2) Time of nonsteadiness t,,, pointed out as transient time by Courtney ( 1962 ). Here this is the time from the beginning of the process until the moment when Z,,,t varies in one time interval by less than l0 -s. The m o m e n t when Z,,.t=95%(SS.) is considered as transient time by Courtney. In the studied cases, for critical clusters tcond.= 1.2608" 10-2 S; tdep. = 5.8167" 10--3 S; and /freezing= 12.669 S. Fig. 3 shows the variations with time of the rate of phase formation I,,t for complexes of different size, the other parameters being the same as in Fig. 2. During the period of nonsteadiness this rate as well as Z,,,t depends on the size for which it is calculated, in contrast to the steady-state case where the rate is the same for each size. Similarly to the variation of Z,,t, once appearing, I,,,t is characterized by a rapid increase with time to its steady-state value. Another specific feature noted by many authors is that for complexes with subcritical sizes, I,,t passes through a m a x i m u m which is higher than the steadystate value. The smaller the cluster size, the higher this maximum. For critical and larger clusters, I,,t increases smoothly to the steady-state value, in the case 4o
...........~i:..':.:'.'..:.....
c m - 3 5 -I
/oIo
/
/
/
/ ./ ' //
.
~
i0~
/
/
/ t2~;" ....... ." ~6¢
.:
_
_
f
/-
/
/ // /
/
/1
/
""
'
:I -
-
// I i
i
/
:/
/
,/~i'/80 I I 60/ /
/ 180 -2
0
4'
Fig. 3. Time-dependence of the rate of phase formation I,,,t for clusters with different sizes. Condensation: continuous lines; deposition: dashed lines; freezing: dotted lines.
426
N. MILOSHEVAND G. MILOSHEV ~0
.~-r~ d Z n , t
,°1
20
.." . ~"" ......:.:~%~,, f: .. .:
';
40 o
/ i 0 I~
/
/
/
/
lllll/f
la~t
I
¢6:_ 6
I ,
-5
-z¢
-3
-2
~¢
0
!
Fig. 4. Time dependence of the total derivative dZ.,t/dt for clusters with different sizes. Condensation: continuous lines; deposition: dashed lines; freezing: dotted lines.
0,g I~ o,7 o,6 o,5 0,0
O,2
o,t
% t
)
5
i
~
"
Fig. 5. Variation of I , , f f I s as a function of t/r. M = o u r numerical solution: K=the analytical solution by Kashchiev ( 1969 ).
of sub-critical sizes, the probability for detachment is higher than that for joining and "accumulation" of the rate values towards smaller sizes is produced. For this reason the smaller the complexes, the higher their rate peak. Such an accumulation could not be observed with the critical size, since the probabilities of joining and detachment are equal and subsequently the rate does not reach a maximum. The trend of the curves for dZ, Jdt (Fig. 4) is similar to that of the curves
NONSTATIONARYHOMOGENEOUSNUCLEATIONIN THE ATMOSPHERE
427
TABLE I Variation of l,,t/Is from the analytical solution of Kashchiev (1969) (IK~sh./ls), and from our numerical solution (IMit./I,) depending on the change of the ratio t/z (t is time time from the beginning of the process; r is induction time) Process
time
lv,~h.
IMiL
1,
L
Freezing
1.3414 2.5684 3.7954 5.0224 6.2455
0.4864 0.8464 0.9551 0.9868 0.9961
0.2909 0.8147 0.9530 0.9878 0.9994
Condensation
0.5838 1.0069 1.5064 3.2429 5.838
0.0678 0.3047 0.561 0.9198 0.994
0.0009 0.0642 0.3367 0.9254 0.996
Deposition
0.631 0.8464 1.206 2.02 5.50
8.92- 10 -2 0.2088 0.417 0.736 0.991
3.44-10 -3 0.0308 0.185 0.674 0.998
c,.rrJl ;( n',t I00 90
i f f "¢2
80 io 6
70
I I t I I I
60 50 ~o 30
I
20
I
** i0 3
1I
~o
J. -/+
-3
-2
-~
, .// o
tog ~ :O ;
Fig. 6. Variation with time of the concentration of clusters with critical size X,,.,t. Condensation: continuous lines; deposition: dashed lines; freezing: dotted lines.
428
N. MILOSHEV A N D G. MILOSHEV
for In.,. The peculiarity here is that dZ,,t/dt does not reach the steady-state value and decreases with time, tending to zero. This trend results from eq. 2 where at I~,,~I~, dZ,,,/dt--,O. The comparison between our results and the analytical solution of Kashchiev ( 1969 eq. 13 ) about the rate of phase formation at the critical size n* is shown in Fig. 5. The analytical solution is seen to be close to the numerical results for t> v where r is the induction time in eq. 13. The respective exact values are presented in Table I. Another specific feature of the process of nonstationary phase tbrmation is the concentration of clusters of a given size as a function of time X,,,,. This is given by the time integral of/,., in the form:
X,,,t '-" i I,,,dt
(25)
0
In Fig. 6 the changes in X*, of critical clusters are given for the cases of condensation (continuous curves) deposition (dashed curves) and freezing (dotted curves). It is worth noting that this characteristic of the process is of great importance since it can be measured experimentally. As it is seen, a period of time should elapse after the beginning of the process until the appearance in the volume of a considerable number of complexes with a given size. Here it is important to formulate a third time characteristic of the process of nonstationary phase formation. (3) Time of appearance 0n, defined by Kelton et al. (1983), as effective time lag. The respective mathematical expression is:
tJ
In our study 0,*con0.= 1.7318" 10-3 S; 0n*dep. = 7.2547- 10-a s; 0~*rre,-zing= 2.6521 S. It is interesting to compare the time of appearance 0n, thus obtained and that derived by the analytical solution the so-called induction time or time lag r. As indicated in Kelton et al. ( 1983 ), by applying the analytical solution of Kashchiev ( 1969 ), the relation between them for the critical size is: 7~2T
0~.- 6
(27)
In our considerations rco.a. = 8.86" 10- 4 s; ~'dep.= 3.90" 10-- 4S; rfreezmg= 1.399 S; or (0,-/r)~ona.= 1.95; (0,*/r)aep.= 1.86; (0,*/V)freezin~= 1.89, these values being fairly close to fez~6 = 1.645. The respective differences are due to the
NONSTATIONARY HOMOGENEOUS NUCLEATION IN THE ATMOSPHERE
429
On..lO~ (s)
/ 20
~0
60
/
80
Fig. 7. Variation of the effective time lag 0, with growing cluster size.
fact that the treatments of Kashchiev are related to a critical region close to n* for which lAG,-AGn* [ < KT. In this manner the time for sub-critical cluster formation is excluded. For that reason the ratio is always higher then
n2/6. Naturally X,., as well as 0,, are functions of the size for which they are calculated. The size dependence of 0, in the case of condensation at T = 261.15 K and S = 7 is illustrated in Fig. 7. On is seen to increase with size ,. CONCLUSION
The examining of the nucleation process in the atmosphere as a nonstationary one is of great importance. Firstly the numerical solution proposed here is appliable to over-critical sizes, i.e. it is useful for the description of the condensational growth process, leading to the formation of visible clusters under experimental conditions. The latter provides a possibility for immediate comparison between theory and experiment. On the other hand the numerical solution, used here, gives the most exact results because of the minim u m of restrictive assumptions. Homogeneous nucleation in the atmosphere, although it is a case of idealization, has it significance when it refers to weather modification with cooled reagents. At last the results exposed here provide a basis for developing the nonstationary nucleation theory for the more real, heterogeneous cases observed in the atmosphere. REFERENCES Abraham, F.F., 1969. Multistate kinetics in nonsteady-state nucleation. A numberical solution. J. Chem. Phys., 51: 1632-1638. Becker, R. and Doring, W., 1935. Kinetische Behandlung der Keimbildung in 0bersattigen D~impfen. Ann. Phys., 24: 719-731.
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N. MILOSHEV AND G. MILOSHEV
Courtney, W.G., 1962. See Kelton et al., 1983. Fletcher, N.H., 1962. The Physics of Rain Clouds. Cambrige University Press, Cambridge, Ch. 3, pp. 37-63. Kashchiev, D.B., 1969. Solution of the non-steady State problem in nucleation kinetics. Surface Sci., 14: 209-220. Kelton, K.F., Greer, A.L. and Thompson, G.V., 1983. Transient Nucleation in Condensed Systems. J. Chem. Phys., 79: 6261-6276. Kozisek, Z., 1988. Two step nucleation in glass-forming systems. Cryst. Res. Technol. In Press. Krastanov, L. and Miloshev. G., 1980. Theoretical Bases of Phase Transitions of Water in the Atmosphere. Meteorological Service of the Hungarian People's Republic, Budapest, CH. 3, pp. 51-65. Pruppacher, H. and Klett, J., 1980. Microphysics of Clouds and Precipitation. D. Reidel Publishing Company, Dordrecht, Ch. 7, pp. 162-182. Volterra,. and Cooper, A., 1985. Numerical calculation of induction times for crystallization of glasses. J. Non-Cryst. Solids, 74: 85-95. Zeldovich, J.B., 1942. K. Teorii Obrasovania Novoi Fazi. Kavitacia. Zh. Eksp. Teor. Fiz., 12: 525-537.