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Nucleation burst in coagulating aerosol
J Aenmol Sci. Vol. 3 I, Suppl. I. pp. $568-$569, 2000
Pergamon www.elsevier.com/locate/jaerosci S e s s i o n 6 D - Particle f o r m a t i o n a n d c o a g u l a t i o n NUCLEATION BURST IN COAGULATING
AEROSOL
A.A. Lushnikov 1 and M. Kulmala 2
1Karpov Institute of Physical Chemistry, 10, Vorontsovo pole, 103064 Moscow, Russia.
2Department of Physicsi University of Helsinki, P.O. Box 9, FIN-00014, Helsinki, Finland.
Keywords: CONDENSATION, COAGULATION, NUCLEATION, NANOPARTICLES.
Formation of disperse phase by spontaneous nucleation plays an extremely important role in numerous atmospheric and technological processes. Very diverse manifestations of this process like formation of aerosol in random atmospheric conditions or well managed technological processes of nanomaterial production via aerosol state prevent for describing this process uniquely: general models taking into account everything are too hard even for very powerful modern computers. So in attempting to treat gas-to-particle formation the models sacrificing some details are in order. This communication reports on a simple model of the particle formation-growth process which takes into account nucleation, condensational growth and coagulation. The particles are assumed to form in free molecular regime which defines the size dependence of the rates of particle growth due to coagulation and condensation. We consider the free molecular regime not only because of its practical importance. The simple and specific dependence of the condensational efficiency on the particle mass (c~(9) = c~92/a, 9 being the number of molecules in the particle) allows for restricting the whole consideration by three moments of the particle mass distribution which meet the set of four first-order differential equations together with the vapour concentration. Some complications appear if the process involves other moments. In this case the assumption on lognormal shape of the particle mass distribution function saves the simplicity of the scheme. We assume next that there is a spacially uniform source of condensable vapour of productivity I, and only small amount of condensable vapour is spent to nucleation. Of course, newly born particles grow after the nucleation burst condensing the nonvolatile vapour onto their surfaces, and change their total number concentration by coagulation. Although we consider here the barrierless nucleation (the nucleation rate J is proportional to the squared vapour concentration C) the results can be easily extended to arbitrary dependence of the nucleation rate on supersaturation. This is absolutely clear because the time of the nucleation burst is much shorter other characteristic time scales. On the other hand, the barrierless nucleation is very often met in the processes of formation of nanomaterials and functionally is rather simple to operate with. So we assume Y = A C 2. Our model uses the moment method that suits very well for considering the particle formation-growth process in free moleclar regime. Lushnikov & Kulmala, 1998, had discussed the application of this method to the growth processes in free molecular regime and found that three moments of particle size distribution and the concentration of condensable
S568
Abstracts of the 2000 European Aerosol Conference
$569
vapour can be described in term of universal functions, with all details of the process being hidden in the scales of the time and concentraton axes. This work extends this approach by including the coagulation process into consideration. We show that there are two different scales of time the shorter of which defines the dynamics of the nucleation-condensation stage while the longer one scales coagulation aging. [t is found that each the stage is described by the set of four universal functions that meet four (different. for each stage) first-order differential equations the right-hand sides of which contain coagulation integrals. These integrals are evaluated and expressed in terms of the parameters of the lognornml particle mass distribution. In contrast to commonly accepted approaches the latters include: particle number concentration, and two moments of the order of 1/3 and 2/3. This step allows one to formulate the close set of equations for these three values and the vapour concenration. This set contains the smallness parameter p = A/(~ which is not treatable by a straightforward application of the perturbation theory. However, two rather nontrivial rescalings allow for separating the nucleation-condensation and coagulation- condensation stages of the particle formation-growth process and formulating two closed sets of equations not containing the smallness parameter at all. It is shown that the time for condensation-nucleation stage is longer than the characteristic condensation time 1 / v / ~ by the factor p-Vs. The particle number concentration contains the smallness parameter to the power 5/8: 6o ~ #5/Sv/~. The coagulation stage is longer than the condensation stage by /t -3/16. The asymptotic analysis shows that the moments and particle number concentration are power flmctions of time: 61/a ,x t -a/5, 62/3 o< t 1/5 and 6o o( t -7/5. These values of exponents correspond to the predictions of the selfpreservation theory for source enhanced coagulation in free molecular regime. Numerical analysis confirms these power laws and gives the values of tile constants before the powers. Above asymptotic dependencies correspond to the constant width of the lognormal function (s = 0.69 for source enhanced growth process).
ACKNOWLEDGEMENT One of us (A.L.) thanks grant #521-98 of ISTC for partial financial support of this research.
REFERENCES
Lushnikov, A.A., and Kulmala, M., (1998) Nucleation Controlled Formation and Gwwth of Disperse Particles, Phys. Rev. Lett., 81, 5165 - 5168.
Pergamon www.elsevier.com/locate/jaerosci S e s s i o n 6 D - Particle f o r m a t i o n a n d c o a g u l a t i o n NUCLEATION BURST IN COAGULATING
AEROSOL
A.A. Lushnikov 1 and M. Kulmala 2
1Karpov Institute of Physical Chemistry, 10, Vorontsovo pole, 103064 Moscow, Russia.
2Department of Physicsi University of Helsinki, P.O. Box 9, FIN-00014, Helsinki, Finland.
Keywords: CONDENSATION, COAGULATION, NUCLEATION, NANOPARTICLES.
Formation of disperse phase by spontaneous nucleation plays an extremely important role in numerous atmospheric and technological processes. Very diverse manifestations of this process like formation of aerosol in random atmospheric conditions or well managed technological processes of nanomaterial production via aerosol state prevent for describing this process uniquely: general models taking into account everything are too hard even for very powerful modern computers. So in attempting to treat gas-to-particle formation the models sacrificing some details are in order. This communication reports on a simple model of the particle formation-growth process which takes into account nucleation, condensational growth and coagulation. The particles are assumed to form in free molecular regime which defines the size dependence of the rates of particle growth due to coagulation and condensation. We consider the free molecular regime not only because of its practical importance. The simple and specific dependence of the condensational efficiency on the particle mass (c~(9) = c~92/a, 9 being the number of molecules in the particle) allows for restricting the whole consideration by three moments of the particle mass distribution which meet the set of four first-order differential equations together with the vapour concentration. Some complications appear if the process involves other moments. In this case the assumption on lognormal shape of the particle mass distribution function saves the simplicity of the scheme. We assume next that there is a spacially uniform source of condensable vapour of productivity I, and only small amount of condensable vapour is spent to nucleation. Of course, newly born particles grow after the nucleation burst condensing the nonvolatile vapour onto their surfaces, and change their total number concentration by coagulation. Although we consider here the barrierless nucleation (the nucleation rate J is proportional to the squared vapour concentration C) the results can be easily extended to arbitrary dependence of the nucleation rate on supersaturation. This is absolutely clear because the time of the nucleation burst is much shorter other characteristic time scales. On the other hand, the barrierless nucleation is very often met in the processes of formation of nanomaterials and functionally is rather simple to operate with. So we assume Y = A C 2. Our model uses the moment method that suits very well for considering the particle formation-growth process in free moleclar regime. Lushnikov & Kulmala, 1998, had discussed the application of this method to the growth processes in free molecular regime and found that three moments of particle size distribution and the concentration of condensable
S568
Abstracts of the 2000 European Aerosol Conference
$569
vapour can be described in term of universal functions, with all details of the process being hidden in the scales of the time and concentraton axes. This work extends this approach by including the coagulation process into consideration. We show that there are two different scales of time the shorter of which defines the dynamics of the nucleation-condensation stage while the longer one scales coagulation aging. [t is found that each the stage is described by the set of four universal functions that meet four (different. for each stage) first-order differential equations the right-hand sides of which contain coagulation integrals. These integrals are evaluated and expressed in terms of the parameters of the lognornml particle mass distribution. In contrast to commonly accepted approaches the latters include: particle number concentration, and two moments of the order of 1/3 and 2/3. This step allows one to formulate the close set of equations for these three values and the vapour concenration. This set contains the smallness parameter p = A/(~ which is not treatable by a straightforward application of the perturbation theory. However, two rather nontrivial rescalings allow for separating the nucleation-condensation and coagulation- condensation stages of the particle formation-growth process and formulating two closed sets of equations not containing the smallness parameter at all. It is shown that the time for condensation-nucleation stage is longer than the characteristic condensation time 1 / v / ~ by the factor p-Vs. The particle number concentration contains the smallness parameter to the power 5/8: 6o ~ #5/Sv/~. The coagulation stage is longer than the condensation stage by /t -3/16. The asymptotic analysis shows that the moments and particle number concentration are power flmctions of time: 61/a ,x t -a/5, 62/3 o< t 1/5 and 6o o( t -7/5. These values of exponents correspond to the predictions of the selfpreservation theory for source enhanced coagulation in free molecular regime. Numerical analysis confirms these power laws and gives the values of tile constants before the powers. Above asymptotic dependencies correspond to the constant width of the lognormal function (s = 0.69 for source enhanced growth process).
ACKNOWLEDGEMENT One of us (A.L.) thanks grant #521-98 of ISTC for partial financial support of this research.
REFERENCES
Lushnikov, A.A., and Kulmala, M., (1998) Nucleation Controlled Formation and Gwwth of Disperse Particles, Phys. Rev. Lett., 81, 5165 - 5168.