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15 O 01 Nucleation theory in non-uniform media. The case of fast nucleation
J. Aerosol Sci.. Vol. 24, Suppl. I, pp. $99--S100. 1993 Printed in Great Britain.
0021-8502/93 $6.00 +0.00 Pergamon Press Ltd
15 0 O1
Nucleation theory in non-uniform media. The case of fast
nucleation Gorbunov B. Institute of Chemical Kinetics and Combustion, 630090 Novosibirsk, Russia
KEYWORDS non-uniform media, nucleation rate, supersaturation, fluctuation The available nucleation theories can be used to calculate the rate of stable embryo formation in ideal uniform conditions. For this case, it is assumed that in every place of metastable medium, the embryo grows under the same conditions. This assumption is an idealization of real systems. There are number of reasons responsible for local non-uniformity in real systems: convection, thermodynamic fluctuations, turbulence, boundary conditions (Gorbunov et al., 1992). An example of this is the formation of atmospheric aerosols, particularly nucleation in stratosphere, and many technological processes. There are two limiting cases: when the nucleation time (r~) is much shorter than the lifetime of local non-uniformity (r:): r~ << r:, and opposite case is: r~ >> r:. The present contribution reports the first case. Consider homogeneous nucleation of either liquid or solid from vapour. In uniform metastable system, the steady-state nucleation rate (Io) is determined by the following relation (Abraham, 1974):
Io = ZC.Oexp \
kT J
(1)
where Z is the Zeldowich factor, C. is the rate at which molecules (atoms) condense on an critical embryo, q~ depends on monomer concentration, rotational and translational partition functions of embryo, k is Boltzmann constant, T is temperature. The magnitude of free energy of formation of a critical embryo (AF2) is determined by the following equation:
AFt, = -i*krlnSo + Xi*~
(2)
where i* is the number of monomers in critical embryo, So is the supersaturation, X is the coefficient depending on the interface free energy between embryo and metastable medium (Abraham, 1974). In the non-uniform system, the local supersaturation (S) depends on coordinates and S ¢ So. Let the spectrum of local supersaturation be defined by function f(S) so that S = So, (S - So) 2 << So2 and
fo ~ f ( S ) d S = 1.
$99
(3)
SIO0
B GORSUNOV For vn << r f, when relatiou (1) carl be used~ we derive the following expression for nuclea¢,~(,n(m
non-uniform metastable mediun~:
: :.
i+
f(s)
\ kWZnSo )
\
,% /
dS
(4)
Thus, relation (4) can be used to calculate nucleation rate in non-uniform media when the de~iation from mean values of supersaturation is not large. The rate of non-uniformity can be estimated from the relation for a two state system in which f(S) = 6 ( S - So ± AS), where 6 is delta function: di(x) = 1 with x --- 0 and 6(x) = 0 for x # 0. In this case we arrive at:
I
( AF:
Io = 1 + 2 \ k f l n S o ~o
(5)
Relation (5) was also obtained providing the smalness of deviations A S << So. In some cases, f(S) can correspond the normal distribution law: f(S) = ( ~ v ~ ) - t exp ( (-S - S °~g) 2 )
(6)
For instance, the thermodynamic fluctuations of supersaturation or non-uniformities in turbulent fluxes are distributed according to the normal taw [1]. In this ease, instead of (4) we obtain: I l+2( A F : ~__)2 :-~ = \ kTlnSo So "
(7)
Note that relations (1- 7) hold not only for homogeneous nucleation from gas. They can also be used to describe other first-order phase transitions. The same method was used to obtain the relations that take into account the influence of local non-uniformity of metastable medinm on the rate of heterogeneous nucleation. Using relations (1- 7) one can calculate nucleation rate in real system in which the local characteristics of the medium in the vicinity of growing embryo differ from the mean values. The range of the influence of non-uniformity on nucleation rate can be estimated from relations (5) or (7) using ice or water nucleations in atmosphere as an example. It is shown that I >> io if AS or cr >> 10 -:~ + i0 -'t Such a range is possible in turbulent fluxes and convection and is observed in atmosphere. So the non-uniformity in metastable media should be taken into account in real systems.
REFERENSES GorbunovB., Sabelfeld K.,Zapadinsky E., Kleitz A.,Laali A.(1992) "Fluctuations in nucleation problems". In: Nucleation and Atmospheric aerosols, Ed. N.Fukuta, P.E.Wagner, A.Deepak Publishing, 95-97. Abraham F. (1974), Homogeneous nucleation theory. Acad.Press, N.-Y., London.
0021-8502/93 $6.00 +0.00 Pergamon Press Ltd
15 0 O1
Nucleation theory in non-uniform media. The case of fast
nucleation Gorbunov B. Institute of Chemical Kinetics and Combustion, 630090 Novosibirsk, Russia
KEYWORDS non-uniform media, nucleation rate, supersaturation, fluctuation The available nucleation theories can be used to calculate the rate of stable embryo formation in ideal uniform conditions. For this case, it is assumed that in every place of metastable medium, the embryo grows under the same conditions. This assumption is an idealization of real systems. There are number of reasons responsible for local non-uniformity in real systems: convection, thermodynamic fluctuations, turbulence, boundary conditions (Gorbunov et al., 1992). An example of this is the formation of atmospheric aerosols, particularly nucleation in stratosphere, and many technological processes. There are two limiting cases: when the nucleation time (r~) is much shorter than the lifetime of local non-uniformity (r:): r~ << r:, and opposite case is: r~ >> r:. The present contribution reports the first case. Consider homogeneous nucleation of either liquid or solid from vapour. In uniform metastable system, the steady-state nucleation rate (Io) is determined by the following relation (Abraham, 1974):
Io = ZC.Oexp \
kT J
(1)
where Z is the Zeldowich factor, C. is the rate at which molecules (atoms) condense on an critical embryo, q~ depends on monomer concentration, rotational and translational partition functions of embryo, k is Boltzmann constant, T is temperature. The magnitude of free energy of formation of a critical embryo (AF2) is determined by the following equation:
AFt, = -i*krlnSo + Xi*~
(2)
where i* is the number of monomers in critical embryo, So is the supersaturation, X is the coefficient depending on the interface free energy between embryo and metastable medium (Abraham, 1974). In the non-uniform system, the local supersaturation (S) depends on coordinates and S ¢ So. Let the spectrum of local supersaturation be defined by function f(S) so that S = So, (S - So) 2 << So2 and
fo ~ f ( S ) d S = 1.
$99
(3)
SIO0
B GORSUNOV For vn << r f, when relatiou (1) carl be used~ we derive the following expression for nuclea¢,~(,n(m
non-uniform metastable mediun~:
: :.
i+
f(s)
\ kWZnSo )
\
,% /
dS
(4)
Thus, relation (4) can be used to calculate nucleation rate in non-uniform media when the de~iation from mean values of supersaturation is not large. The rate of non-uniformity can be estimated from the relation for a two state system in which f(S) = 6 ( S - So ± AS), where 6 is delta function: di(x) = 1 with x --- 0 and 6(x) = 0 for x # 0. In this case we arrive at:
I
( AF:
Io = 1 + 2 \ k f l n S o ~o
(5)
Relation (5) was also obtained providing the smalness of deviations A S << So. In some cases, f(S) can correspond the normal distribution law: f(S) = ( ~ v ~ ) - t exp ( (-S - S °~g) 2 )
(6)
For instance, the thermodynamic fluctuations of supersaturation or non-uniformities in turbulent fluxes are distributed according to the normal taw [1]. In this ease, instead of (4) we obtain: I l+2( A F : ~__)2 :-~ = \ kTlnSo So "
(7)
Note that relations (1- 7) hold not only for homogeneous nucleation from gas. They can also be used to describe other first-order phase transitions. The same method was used to obtain the relations that take into account the influence of local non-uniformity of metastable medinm on the rate of heterogeneous nucleation. Using relations (1- 7) one can calculate nucleation rate in real system in which the local characteristics of the medium in the vicinity of growing embryo differ from the mean values. The range of the influence of non-uniformity on nucleation rate can be estimated from relations (5) or (7) using ice or water nucleations in atmosphere as an example. It is shown that I >> io if AS or cr >> 10 -:~ + i0 -'t Such a range is possible in turbulent fluxes and convection and is observed in atmosphere. So the non-uniformity in metastable media should be taken into account in real systems.
REFERENSES GorbunovB., Sabelfeld K.,Zapadinsky E., Kleitz A.,Laali A.(1992) "Fluctuations in nucleation problems". In: Nucleation and Atmospheric aerosols, Ed. N.Fukuta, P.E.Wagner, A.Deepak Publishing, 95-97. Abraham F. (1974), Homogeneous nucleation theory. Acad.Press, N.-Y., London.