Volume 215, number 1,2,3
CHEMICAL PHYSICS LETTERS
26 November 1993
Influence of inhomogeneity and fluctuations of supersaturation on heterogeneous nucleation B.Z. Gorbunov,
E.L. Zapaclinsky
NovosibirskState University,Pirogovastreet2. 630090 Novosibirsk,RussianFederation
K.K. Sabelfeld and M. Ataev ComputingCenter, Lavrentjeva6, 630090 Novosibirsk,RussianFederation Received 16 September 199 1; in final form 16 September 1993
The influence of fluctuations of vapour concentration on the heterogeneous nucleation rate is investigated. Computer calculations based on the Monte Carlo method show this influence to be substantial at realistic values of non-thermodynamic fluctuations. Thermodynamic fluctuations do not affect the nucleation rate in the model under consideration.
1. Introduction Nucleation is an important factor in many physical and chemical processes and is often used for practical purposes. The theory of nucleation rate is based on the law of local equilibrium, the kinetic theory of gases and the thermodynamic theory of fluctuations. The rate of heterogeneous nucleation (J) at constant temperature (T) and pressure depends on the critical value of Gibbs free energy of the new phase embryo formation AG*(S) [ 1,2], J=Bexp[
-AG*(S)/kT]
,
(1)
where k is the Boltzmann constant, S is the supersaturation for nucleation from gases or supercooling for nucleation from liquids. Further, let us consider, more particularly, the nucleation from a mixture of two gases when the concentration of a nucleating component is much lower than that of another gas. This situation is typical for many applications, e.g. for water condensation in the atmosphere. The preexponential factor B is determined by the number of molecular collisions with the embryo surface, monomer concentration and the form of the barrier near the maximum (Zeldovich factor [ 3 ] ) . The factor B actually remains unchanged with varying external conditions and substrate [ 2 1. Elsevier Science Publishers B.V.
Expression ( 1) is derived for nucleation in a uniform medium with time-independent characteristics. However, when the macroscopic system characteristics are the same, the local environment of a microscopic embryo may vary. This is determined by both thermodynamic and non-thermodynamic fluctuations, e.g. due to turbulence and convection as well as due to the depletion of a metastable medium. Hence, the pressure, temperature and supersaturation in the local environment of a growing embryo will depend on time and space. The fluctuations of temperature, pressure and concentration cannot but influence the nucleation rate. Supersaturation has an especially strong effect on the nucleation rate [ 21. Therefore, supersaturation fluctuations should be taken into consideration first of all. There are practically no publications that highlight the fluctuation influence on nucleation rate. Only in refs. [ 4,5 ] do the authors consider the role of the temperature fluctuations of a liquid droplet embryo for homogeneous nucleation. These fluctuations make droplets with different temperatures overcome the different activation barriers to pass to the state of a stable embryo. Supersaturation flue tuations might be expected to influence the nucleation rate to a larger extent than temperature. The aim of the present Letter is to study the influence of su31
Volume 215, number 1,2,3
persaturation fluctuations on the heterogeneous nucleation from vapour.
2. Calculation principles
the coefficients of free energy of interface between substrate and gas, substrate and liquid, liquid and gas, respectively. In spite of the fact that thermodynamic fluctuations seem to be much less important than non-thermodynamic ones for acceleration of the nucleation process, let us nevertheless start by considering thermodynamic fluctuations in a gaseous mixture. Let the embryo grow from only one substance whose concentration is much lower than that of another gas. The fluctuations of the nucleating gas concentration as well as the temperature and pressure fluctuations will contribute to the fluctuations of its supersaturation. However, the main contribution to the supersaturation fluctuations will be made by the fluctuations of the concentration of vapour molecules. Indeed, for a certain volume V of the mixture of ideal gases, the relative mean-square deviations of any thermodynamic parameter characterizing this mixture as a whole (let us denote this parameter by C), obeys the expression m / (0’ = 1/N [ 8 1, where N is the total number of mixture molecules in volume K For the number of molecules of nucleating admixture n in the volume V, we have [ 81
Eq. (1) is obtained for the ensemble of clusters representing the whole range of embryos, i.e. from monomers to embryos of a critical size. For ergodic systems it can be used in two cases. First, when the number of embryos is large enough and represents the whole range, and second, when only one embryo is considered for a long time. In the second case it is necessary that the nucleation time (time of observation) t is shorter than the fluctuation lifetime tr. In the first case the volume of fluctuation must be large enough to include subcritical embryos of different sizes. On the other hand, it is possible to use formula ( 1) for t,, B tf, but there is another requirement - the lifetime of fluctuations must be greater than the time of non-stationary nucleation tns. This characteristic is well-investigated for homogeneous nucleation (see for example refs. [ 6,7] ). A typical value for water nucleation is &x lO-‘j s. For heterogeneous nucleation this time could be longer by one order because of the lower value of supersaturation. Here we study nucleation on aerosol particles moving through media with inhomogeneous and fluctuating supersaturation. In order to use formula (1) we suggest the lifetime of fluctuations around aerosol particles to be tf> 10B5 s. We also suggest that the time of observation r be much greater than tf. In this case fluctuations can be regarded as non-correlated in time. The critical free energy for embryo formation on a flat surface involved in formula ( 1) has the form
(1Ls)2
[21
(s)2
AC*=
16dvf(
3(n&Tln
m1
S)2 ’
(2)
where S=p/p,, p is the vapour pressure, PO0is the pressure of saturated vapour upon the flat surface of liquid at the given temperature, nL is the number of molecules per unit of liquid volume, f(m) = 4 (2 + m) ( 1 -m)2, m is the cosine of a contact angle between the droplet of liquid embryo and a solid surface. The value of m can be calculated from the relation m = ( aov - ccL) /rrLv, where crcv, acL, aLv are 32
26 November 1993
CHEMICAL PHYSICS LETTERS
(an)21 oz=ff*
(3)
Since N>n, the relative fluctuations for n are observed to be higher than for thermodynamic parameters characterizing the mixture as a whole. Thus, considering the mixture pressure and temperature as homogeneous and constant in time, and using expression ( 3 ) , for supersaturation in volume V we obtain kT = svp,
(4)
*
In this case, the probability density for determination of S within the range S to S+ dS [ 81 is w(S) = [2r(A$
] 1,2exp (-
;Zf).
(5)
Expression (5) is derived for thermodynamic fluctuations. Let us suggest the probability density of nonthermodynamic fluctuations to have the same form with dispersion (LLs)z determined by an equation different from (4).
Volume 215, number 1,2,3
CHEMICAL. PHYSICS LETTERS
At constant pressure and temperature, taking into account expression ( 2 ) , formula ( 1) can be reduced to
(6)
J=Bexp(-*).
Using expression (6) one can obtain the probability of formation of a stable embryo with time 7 P(r)=l-exp(-
lJ(t)dt).
(exp(-
!JCOdt)),
, -I/.rr.c LWF?
0.0
1.0
,/
/’
-2.01
,
0928
(8)
where the angular brackets mean averaging over distribution ( 5 ) . The probability of embryo formation on a substrate without taking into account supersaturation fluctuations (Pz( 7) ) is determined from expression (7), where J is substituted by Jr= Bexp[ -Af(m)/ln’S], i.e. (9)
Let us also introduce the functional (10)
The Jenssen inequality gives Pi ( 7) < P3 ( 7). Calculations were performed using the Monte Carlo simulation assuming S fluctuations to be time-uncorrelated. The value of A was calculated with nL=3.3x 1O22 cms3, a,,=756 erg/cm2 (corresponding to water parameters at 273 K [ 21). The values B and 7 were determined to be 1Or2s- ’ and 1O3s, respectively.
3. Results and discussion Figs. 1 and21istthevalueSfOrP,(7),P~(7),P3(7)
I
I
0.936 0.940
0.944 m
1
I
0.910
0.930
0.920
m
r
Logq,xxxx~~~----“~~-
0.0
(AS12=104
- 4.0
_--I
P,(r)=l-exp{-B(exp(-$!$j))r).
I
0.932
/’
-4.0
0.913
P2(7)=l-exp(-Jg7).
lz?=10-6
/’
(7)
Jis time-dependent because S, in general, is also timedependent. The average probability of appearance of a stable embryo in the time interval 7 (Pi (7) ) is determined as the average over all possible realizations of an arbitrary supersaturation, P,(7)=1-
26 November 1993
__-I
0.915
__----I
0917
I
0.919m
Fis. 1. The dependence of the probability of stable embryo formation calculatedfromformulas (8), (9), (10) onthe cosineof acontactangleatS=l.l. (-) i=l; (---) i=2; (X) i=3. calculated from formulas (8), (9) and ( lo), respectively, for various 3, (A,!$)’ in dependence on the cosine of a water-on-substrate contact angle. It is seenthatoverthewholerangeP2(7)~P1(7)gPj(7). Note that the difference in Pi (7) and P2( 7) can be substantial, and P3 (7) is a good estimate for P1 (7). In this case the influence of fluctuation is stronger at lower supersaturations. Thus, for s= 1.1 this influence is already noticeable for m = 10m6 (fig. 1 ), and for s= 1.5 the influence is noticeable only starting from (LLs)zx: lob4 (fig. 2). Stable embryo formation is treated as a fluctuating process. Is there any sense in considering the influence of thermodynamic fluctuations of supersaturation on the rate of nucleation? For homogeneous nucleation it seems not because interaction of different thermodynamic fluctuations is a very rare event. For heterogeneous nucleation on moving aerosol particles with a big surface this consideration can be valid. Let us imagine an aerosol particle with a set of subcritical embryos on its surface that moves 33
Volume 215, number 1,2,3
-1.2
1
CHEMICALPHYSICS LETTERS
J
,
0.700
0.680
I
0.720
0.72 0
0.740
m
is also required that the lifetime of fluctuations must be longer than the time of non-stationary nucleation. As has been mentioned above in our case tn.= 10V5 s. Then tf must be longer than 10s5 s. For thermodynamic fluctuations tfx R2/D, where D is the diffusion coefficient of vapour. For water vapour in air at 281 K and atmospheric pressure D=O.239 cm2/s [ 9 1. Hence, R must be greater than 1.5 X 10s3 cm. For the volume of a sphere of such radius m = 5 x 1O-lo. As can be seen from fig. 1 these fluctuations do not affect the probability of stable embryo formation. Thus, it is possible to conclude that in the range of validity of formula ( 1) thermodynamic fluctuations of the supersaturation have no influence on the probability of stable embryo formation. As for non-thermodynamic fluctuations connected with turbulence, convection, inhomogeneity of heat and mass transfer, etc., it is possible to expect m > 10m6for areas with dimension greater than 1.5 x 10 -3 cm. Thus, even a sufficiently small nonthermodynamic fluctuation can affect the rate of heterogeneous nucleation, which from our point of view should be taken into account in the treatment of experimental data.
0.760rn
Fig. 2. The dependence of the probability of stable embryo formation calculated from formulas (8), (9), ( 10) on the cosine of acontactangleatS=lS. (-) i=l; (---) i=2; (X) i=3.
through a medium with inhomogeneous and fluctuating supersaturation. In this case thermodynamic
fluctuations of this medium can be regarded as external for our aerosol particle. Let us estimate the value of thermodynamic fluctuations of supersaturation. As has been mentioned above, the value of (AS)‘, at a given mean supersaturation, depends on the volume V. In our case the value of V is approximately the volume of aerosol particles. In turn the surface of the aerosol particle must have enough room for the great number of different subcritical embryos. According to ref. [ 21, r,=2aLv/nLkT In S. Hence, with s= 1.1 and the above values for T, nL and crLv,the critical radius r,= 10s6 cm. Thus, to arrange an ensemble of embryos the aerosol particle must have a radius R of at least 1OB5cm. For the validity of our calculations it
34
26 November 1993
Acknowledgement This work is partly supported foundation.
by the Soros
References [ 1] M. Volmer, Kinetikder Phasenbildung (Germany, Dresden and Leipzig, 1939).
[ 2 ] N.H. Fletcher, The physics of rainclouds (Cambridge Univ. Press, Cambridge, 1962). [3] Ya. B. Zeldovich, Zh. Eksp. Teor. Fiz. 12 (1942) 525. [4] C.F. Clement, Atmospheric aerosols and nucleation (Springer, Berlin, 1988). [ 51I.J. Ford and C.F. Clement, J. Phys. A 22 (1989) 4007. [6] A.A. Lushnikov and A.G. Sutugin, Usp. Khim. 25 (1976) 385. [7] F.F. Abraham, J. Chem. Phys. 51 (1969) 1632. [8] L.D. Landau and E.M. Lifshitz, Statisticheskaya Phi&a (USSR, Moscow, 1964). [ 91 CRC handbook of chemistry and physics (CRC Press, Boca Raton, 1987) p. F-45.