Atmos. Chem. Phys., 3, 251 257, 23 Atmosheric Chemistry an Physics A moel for article formation an growth in the atmoshere with molecular resolution in size K. E. J. Lehtinen an M. Kulmala Helsinki University, Det. Physical Sciences, P.O. Box 64, 14 Univ. of Helsinki, Finlan Receive: 23 August 22 Publishe in Atmos. Chem. Phys. Discuss.: 28 October 22 Revise: 23 January 23 Accete: 3 February 23 Publishe: 21 February 23 Abstract. The formation an growth of atmosheric aerosol articles is consiere using an exact iscrete metho with molecular resolution in size sace. The metho is immune to numerical iffusion roblems that are a nuisance for tyical simulation methos using a sectional reresentation for the article size istribution. For conensational growth, a slight moification is roose for the Fuchs-Sutugin exression, which imroves the reiction of the growth rate of nanosize articles by as much as a factor of two. The resente metho is alie to article formation in a Finnish Boreal forest an is shown to cature the essential features of the ynamics quite nicely. Furthermore, it is shown that the growth of the articles is roughly linear, which means that the amount of conensable vaour is constant (of the orer 13 1/m 3 ). 1 Introuction The formation an growth of atmosheric aerosols has recently receive growing exerimental an theoretical interest ue to climate an health relate effects of fine articles (Charlson an Wigley, 1994; Dockery an Poe, 1994). The increase aerosol concentrations are largely ue to seconary article rouction, i.e. homogeneous nucleation an subsequent growth from vaours. Instrument techniques for measuring freshly forme article concentrations have been recently eveloe, an articles with a iameter of about 3 nm can be etecte. These small articles have been foun in the free trooshere (Clarke, 1992; Schröer an Ström, 1997; Raes et al., 1997), in the marine bounary layer (Covert et al., 1992; Hoel et al., 1994; O Dow et al., 1999), in the vicinity of evaorating clous (Hegg et al., 1991), in Arctic areas (Wieensohler et al., 1996; Pirjola et al., 1998), Corresonence to: K. E. J. Lehtinen (kari.lehtinen@helsinki.fi) in urban areas an in stack lumes (Kerminen an Wexler, 1996), in continental bounary layer (Birmilli an Wieensohler, 2) an recently also in boreal forests (Mäkelä et al., 1997; Kulmala et al., 1998, 21a). The freshly forme aerosols become climatically imortant only if they are able to grow to sizes of 5 nm an larger. Particles in this size range can act as clou conensation nuclei, an therefore they may contribute to the so-calle inirect aerosol cooling effect of the climate. Furthermore, if the articles grow to sizes above nm, they become to scatter light very efficiently, an have thereby a irect cooling effect on the climate. Whether or not the new articles ever reach these sizes is to a large extent ecie while they are still smaller than nm (Kulmala et al., 2, 21b). Due to their Brownian movement, articles with iameters of a few nm coagulate very efficiently with larger articles, which imlies that the freshly nucleate articles have to grow fairly raily (within a few hours) ast the nm limit or they will be lost in the collision rocesses (Kulmala et al., 2). Such growth is ossible only if there is a suersaturate vaour resent at concentrations above 7 molecules er cubic centimeter of air (Kulmala et al., 21b). In orer to etermine the climatic imortance of atmosheric nucleation events, we nee to know what are the vaours causing the article growth, what are the chemical mechanisms controlling their formation, an what if any is the anthroogenic influence on their concentrations. At the moment, we only know that the growth of articles after continental nucleation events is with high robability cause by organic vaours (O Dow et al., 22). The longest time sequence of observations is from Hyytiälä (61 51 N 24 17 E) in Finlan, where measurements with a ifferential mobility article sizer (DMPS, measures the aerosol number concentration from 3 6 nm ry iameter), begun in January 1996. U to May 22 there were aroun 3 ays with nucleation. The article formation is usually observe in the late morning as the aearance of 3 nm articles at the lower size range of c Euroean Geosciences Union 23
252 K. E. J. Lehtinen an M. Kulmala: Particle formation an growth in the atmoshere the DMPS (Mäkelä at al., 1997, Kulmala et al., 21a). In this aer we consier the conensational growth of freshly nucleate articles. The methoology inclues three novel techniques: (1) an imrove moel in free molecular regime conensation, base on a simle moification to the well known Fuchs-Sutugin exression (Fuchs an Sutugin, 1971) (2) a iscrete metho for solving the general ynamic equation (GDE, Frielaner, 2) for the article size istribution, that is free of numerical iscretization errors an (3) analysis of the time evolution of oint wise values of the measure article size istribution to valiate the assumtion of a constant concentration of conensable vaour, resulting in linear growth in terms of article iameter. 2 Theory 2.1 Free-molecular conensation The traitional way to escribe free-molecular conensation (Seinfel an Panis, 1998) is to assume vaour molecules as oint masses an use the so-calle Fuchs-Sutugin exression (Fuchs an Sutugin, 1971) for the article growth rates. However, this aroach inuces serious errors when the articles are very small (say, uner nm). Then, it is not ossible anymore to ignore molecular imensions in comarison with article size. Also, the article iffusion coefficient, which tyically is also assume negligible, has to be accounte for. Our aroach is to use the following equation for the collision rate of molecules with articles β 1i : β 1i = 2π( 1 + i )(D 1 + D i ) Kn + 1.377Kn + 1 + 3α 4 (Kn2 + Kn), (1) which is comrise of the continuum regime conensation rate multilie by the semi-emirical Fuchs-Sutugin interolation function (Fuchs an Sutugin, 1971) (the last term in the equation). It has been obtaine by fitting Sahni s (1966) numerical results for the solution of Boltzmann s equation, an extens the valiity of Eq. (1) to cover all article sizes. In Eq. (1), 1 an i are the iameters an D 1 an D i the iffusion coefficients of the conensing molecule an article in class i, resectively. The factor α is the mass accommoation coefficient. Traitionally, 1 an D i are left out of Eq. (2) (Seinfel an Panis, 1998) since the molecular iameter is tyically negligible comare with article iameter an article iffusion coefficient negligible comare with molecular iffusion coefficient. However, if one is intereste in the really initial stages of article growth, i.e. when their iameters are of orer 1 nm, such aroximations are obviously not ossible. Atmos. Chem. Phys., 3, 251 257, 23 In this formulation, to obtain correct asymtotical behaviour in the small article limit, the Knusen number Kn shoul be efine as Kn = 2λ ( 1 + i ) in which the mean free ath of the conensation rocess is λ = 3(D 1 + D i ) c 2 1 + c2 i (2). (3) Here c 1 an c i are the thermal sees of the molecule an article, resectively. The iffusion coefficients are calculate using simle kinetic theory. Again, the ifference to the stanar way is that now the molecular imensions as well as article iffusion are not neglecte. This moification of conensation theory aroaches free-molecular coagulation theory in the limit of free-molecular articles. 2.2 Discrete solution of the size istribution Next, we turn to the issue of solving the article size istribution in etail. The aerosol general ynamic equation (Frielaner, 2) is tyically solve using either a sectional metho with fixe sections or with moving sections. The iea behin the sectional metho is to aroximate the article size istribution with a histogram. Then, the various aerosol ynamical rocesses (nucleation, conensation, coagulation, eosition, transort) are treate by moelling the concentration an/or location of each section. Using fixe sections is naturally ieal for nucleation an coagulation rocesses. However, for conensation/evaoration this metho has a serious rawback, numerical iffusion (Zhang et al., 1999) It can be circumvente by using moving sections - then, however, we might encounter a situation in which there are no sections in a size range of imortance (for examle, the size of nucleating articles). In this aer, we resent an aroach that has neither of the abovementione rawbacks: the size sections are fixe but the sacing is one moleculeby-molecule. Then, the actual hysics of the roblem are mimicke in etail, an rovie that the comutational buren is not insuerable, the metho shoul, in rincile, be free of any iscretization errors. The ynamics of nucleation moe article size istribution is simulate using a iscrete coagulation scheme with aroriate sources: N 1 t N k t N k t = I 1 k I k N 1 i=k β 1i N i = I k β 1k N 1 N k N k k k i=k β k,in k N i = β 1,k 1 N 1 N k 1 + 1 β i,k i N i N k i 2 i=k (4a) (4b)
K. E. J. Lehtinen an M. Kulmala: Particle formation an growth in the atmoshere 253 β 1k N 1 N k N k β ik N i i=k (k > k ) (4c) using the Fuchs exression (Fuchs an Sutugin, 1971) for the collision frequency functions β ij. In our aroach, we select class 1 to reresent molecules. I 1 is a source rate for the molecules. Class k reresents the smallest stable articles, with I k as their nucleation rate. The article ynamics, in aition to the nucleation source, is riven by coagulation an conensation/evaoration. In this metho, conensation is treate as collisions of classes i with class 1, an is thus free of any numerical iffusion roblems. The conensation rates β 1i are calculate from the moifie Fuchs-Sutugin exressions as exlaine before. In equations 4, the evaoration of articles is not taken into account. However, ossible surface ressures of the vaours can be straightforwarly introuce into the equations, resulting essentially in the birth an eath formalism by Goorich (1964). iameter 3. 2.5 2. 1.5 1. 5 15 2 25 3 35 time [s] amu, correct amu, stanar 5amu, correct 5amu, stanar 2amu, correct 2amu, stanar 6 3 Results An imortant ractical system to investigate the imrove growth exression is the growth of articles from nucleation size (say 1 nm in iameter) to 3 nm. The reason for this is that available article size istribution measurement instrumentation tyically has a cut-off at 3 nm. This means that when trying to obtain the article nucleation rate from these measurements, one has to back-calculate own to 1 nm (Kulmala et al., 21b; Kerminen an Kulmala, 22). As soon as the articles have been forme, they grow by conensation to larger sizes, but at the same time their concentration is iminishe because of collisions with other articles (coagulation). Thus, to get the correct nucleation rate from exerimental ata, it is essential to know the conensational growth rate in etail. In the atmoshere, in many cases, it is not known what is the vaour that is conensing onto the freshly nucleate articles, thus causing their growth to observable sizes. The arameters affecting the growth rate are mainly the vaour molecular mass an iameter. In our samle calculations, we thus use several ifferent values for the molecular mass, thus not secifying any etaile secies. In Fig. 1, the growth of a article of ensity kg/m 3 is simulate. The conense material is also assume to be of the same ensity, with molecular mass of 5, an 2 amu. The vaour concentration is assume constant (4 13 1/m 3 ). The stanar Fuchs-Sutugin results are shown with otte lines an the results obtaine using the correcte moel with soli lines. As seen in Fig. 1, it is obvious that article growth is faster with a vaour with larger molecular size. Also, there is a significant ifference in growth rates, when using the correcte exressions comare with the stanar Fuchs-Sutugin moel, esecially at the initial stages of growth when the article is still very small. Figure 1a shows the evolution of the article growth time [s] 4 3 2 correct stanar 5 15 2 25 mass of conensing molecule [amu] Fig. 1. Conensational growth of freshly nucleate articles from 1 nm to 3 nm in iameter, assuming a constant vaour concentration of 4 13 1/m 3 : (a) Three ifferent vaour molecular masses (5 amu, amu, 2 amu) are use for temoral growth, (b) growth time as a function of molecular mass of conensing molecule. The label correct refers to the metho resente in this aer, stanar to the unmoifie Fuchs-Sutugin exression. iameter with time for various values of the molecular mass an Fig. 1b the growth time as a function of vaour molecular mass. In growth time from 1 nm to 3 nm, errors more than % can be obtaine, if the molecular imensions an the article iffusion coefficient are neglecte. It must be note that each simulation is base on assuming constant vaour concentration uring growth, an that the same value is use for each run using ifferent molecular masses. Also, since the ientity of the vaour(s) causing the growth an thus also their saturation vaour ressures are unknown, a zero surface ressure on the articles was assume. Atmos. Chem. Phys., 3, 251 257, 23
254 K. E. J. Lehtinen an M. Kulmala: Particle formation an growth in the atmoshere N/(log ) - contours measure result 1 8 12 16 2 iscrete moel result 1 8 12 16 2 tim e of ay sectional moel result 1 tim e of ay 8 12 16 2 tim e of ay Fig. 2. The measure vs. the moelle article size istribution in 2 May 1998 in Hyytiälä, Finlan. The contour-lot color inicates the article size istribution height in 1/cm 3. To emonstrate the usefulness of the resente metho, we show two simulation results of nucleation events from Hyytiälä measurement station in Finlan, in which clear bursts of article formation, accomanie with subsequent growth, were observe above a boreal forest. One of the events (2 May 1998) was simulate reviously by Kulmala et al. (2), but using a sectional metho with a geometrical sacing of size classes. The qualitative agreement between the exeriment an the simulation was satisfactory. However, the use sectional metho suffers from numerical iffusion an the size istribution narrowing effect, often resent in article growth riven by vaour conensation, is estroye. The new result, using the iscrete metho is shown in Fig. 2, together with the exerimental result an the result using the sectional metho by Kulmala et al. (2). When comare with the sectional moel result there is one very noticeable ifference. Now, the maximum values for the istribution ensity function are not locate at 3 nm (bottom of figure) as is the case for the simulation with numerical iffusion. Instea, now the istribution is first broaer, but graually evolves to a more narrow but higher istribution. The nucleation rate for the simulation was chosen to be ientical with the simulation by Kulmala et al. (2), which means that it was a result of ternary water sulfuric aci ammonia nucleation. The backgroun article concentration was selecte to be the measure concentration before the event at 7: LT. The henomenon can be seen even more clearly, when the exerimental results (for 2 May 1998) are resente as in Fig. 3, in which the oint wise values of the size istribution are resente for four ifferent article iameters (6.82 nm, 9.353 nm, 12.3 nm an 15.83 nm), as a function of time. The values correson to ifferent classes of the articlesizing instrument. In all curves the region left of the eak corresons to early stages in which the nucleation burst has not even starte or growth has not roagate to the size of interest yet. Then the concentration starts increasing an reaches its maximum when the actual maximum of the nucleation moe size istribution is at the corresoning location. The region right to the eak corresons to time when the eak has alreay asse the corresoning size. There are, however some articles resent, which most robably come from mixing of air. The moel assumes a homogenous air arcel, in contrast to the exerimental set-u in which the measuring location is fixe but the air aroun it moves. The clearly observable nucleation eaks in Fig. 3 were fitte by normal istributions (soli lines), from which the eak locations an heights were then extracte. From the ata resente Fig. 3, it is ossible to analyse the exerimental growth rocess further. In Fig. 4a the location of the size istribution eak is shown as a function of time, an in Fig. 4b the height of the eak is shown as a function of corresoning article iameter. From Eq. (1) it is straightforwar to euce that the growth rate of articles in the kinetic regime is almost ineenent of size, resulting in linear growth (see, e.g. Seinfel an Panis, 1998). Atmos. Chem. Phys., 3, 251 257, 23
K. E. J. Lehtinen an M. Kulmala: Particle formation an growth in the atmoshere 255 6.82nm 9.353nm 2 2 N/log( ) [1/cm3] 1 1 4 6 8 12 14 16 18 2 22 4 6 8 12 14 16 18 2 22 12.3nm 15.63nm N/log( ) [1/cm3] 2 1 2 1 4 6 8 12 14 16 18 2 22 4 6 8 12 14 16 18 2 22 Fig. 3. Evolution of oint wise values (at 6.82 nm, 9.353 nm, 12.3 nm an 15.63 nm) of the article size istribution in 2 May 1998 in Hyytiälä, Finlan. The ots reresent the measure ata, the lines least-squares-fitte Gaussian eaks. Base on Fig. 4a, growth inee is linear, suorting theory as well as the assumtion that the amount of vaour contributing to the growth rocess is roughly constant. In Fig. 4b, the eak height value is base on the tyical logarithmic N/(log ) form of the size istribution. Thus, since the growth rate in iameter sace is constant, an N (log ) = N, (5) the eak height shoul increase linearly with iameter (of location). However, this simle chain of thought isregars the fact that the eak height is also lowere because of collisions with backgroun articles. It is, though, evient in Fig. 4b: the eak height increases almost linearly with iameter. The highest chosen iameter in the analysis, 15.63 nm, corresons to the size for which there is a clear visible nucleation-originate eak assing through. Above this size, the backgroun articles isturb the analysis. The exerimentally observe growth rate (6.6 nm/s), for assume vaour molecules of atomic mass units, is obtaine by using a constant value of 4 13 1/m 3 for the vaour concentration. The same analysis works even better for a secon moelle event ay 19 May 1999, as shown in Figs. 5a an b. In this case the backgroun article concentration is lower, resulting in a smaller coagulation sink for the forme articles. Thus there is clear linear growth to much higher sizes than in 2 May 1998. Atmos. Chem. Phys., 3, 251 257, 23
256 a) K. E. J. Lehtinen an M. Kulmala: Particle formation an growth in the atmoshere a) 19 May 1999 eak location 15 eak location 15 6.6 nm/h 6.6 nm/h eak location 25 2 15 eak location vs. time 5.5 nm/h 5 11 12 5 5 [h] 11 12 [h] b) 11 12 13 14 15 16 [h] 3 b) eak height vs. size 3 2 3 eak height [1/cm 3 ] eak height [1/cm 3 ] 2 2 2 1 1 eak height [1/cm 3 ] 2 2 1 5 15 5 15 2 25 3 35 eak location Fig. 4. The eak location 5 as a function of time (a) an 15eak height as a function of eak location (b) for the nucleation event of 2 May 1998 in Hyytiälä, Finlan. Fig. 5. The eak location as a function of time (a) an eak height as a function of eak location (b) for the nucleation event of 19 May 1999 in Hyytiälä, Finlan. 4 Conclusions In this aer, a new moel for escribing nucleation moe article growth has been resente. The main new features of the work are a new formulation of the Fuchs-Sutugin exressions for conensational article growth an the use of a iscrete, molecule-by-molecule, simulation metho to solve the general ynamic equation of aerosols. The moification of the Fuchs-Sutugin exression consists of taking into account the molecular imensions in calculating collision cross sections as well as the article iffusion coefficient when calculating relative movement of the article-molecule-system. There is, in rincile, nothing novel in this. It is qualitatively how article-article col- lision frequency functions have been calculate for a long time. However, for conensation it is very tyical to use the Fuchs-Sutugin exressions instea of collision theory. It is shown that these effects are far from negligible. For instance, when calculating article growth from 1 nm to 3 nm, the effect can be more than a factor of two! For many formationgrowth occurrences in the nature, the molecules resonsible for conensational growth can be comlex organic or other rather large molecules. Then it is increasingly imortant to take the above-mentione effects into account, as shown in the samle calculations of this aer. The iscrete metho use in this aer has the avantage of being very easy to rogram an being free of artificial numerical iffusion roblems, that are so often resent in sec- Atmos. Chem. Phys., 3, 251 257, 23
K. E. J. Lehtinen an M. Kulmala: Particle formation an growth in the atmoshere 257 tional moels use in article size istribution stuies. The reason is that, in rincile, all hysically ossible article sizes are chosen to be size classes. Of course the metho is comutationally heavy, but as shown in this aer, still feasible for covering the entire nucleation moe ynamics. A comarison with a real natural nucleation-growth event over a boreal forest in Finlan, the metho seems to cature the key qualitative behaviour of such a system. In a conensation ominate growth rocess, the numerical iffusion associate with the fixe sectional metho makes the istribution artificially wier. However, as shown by the the etaile analysis of the exerimentally obtaine size istributions in a Finnish forest, the nucleation moe becomes narrower but higher as it roagates through size sace. This effect is nicely cature by the iscrete metho. Furthermore, the metho can be use to valiate more aroximate but faster methos, in which nucleation, conensation an coagulation are simultaneously resent. Analytical solutions for such systems are non-existent, thus a metho free of iscretisation errors is certainly of value for such valiation stuies. References Birmilli, W. an Wieensohler, A.: New article formation in the continental bounary layer: Meteorological an gas hase arameter influence, Geohys. Res. Lett., 27, 3325, 2. Charlson, R. J. an Wigley, T. M. L.: Sulhate aerosol an climatic change, Scientific American, 27, 48 57, 1994. Clarke, A. D.: Atmosheric nuclei in the remote free trooshere, J. Atmos. Chem., 14, 479 488, 1992. Covert, D. S., Kaustin, V. N., Quinn, P. K., an Bates, T. S.: New article formation in the marine bounary layer, J. Geohys. Res., 97, 2 581 2 587, 1992. Dockery, D. W. an Poe, C. A.: Acute resiratory effects of articulate air ollution, Annu. Rev. Public. Health, 15, 7 132, 1994. Frielaner, S. 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M.: Observations of aerosols in the free trooshere an marine bounary layer of the subtroical Northeast Atlantic: Discussion of rocesses etermining their size istribution, J. Geohys. Res., 2, 21 315 21 328, 1997. Schröer, F. an Ström, J.: Aircraft measurements of submicrometer aerosol articles (> 7 nm) in the milatitue free trooshere an trooause region, Atmos. Res., 44, 333 356, 1997. Sahni, D. C.: The effect of a black shere on the flux istribution of an infinite moerator, J. Nucl. Energy 2, 915 92, 1966. Seinfel, J. H. an Panis, S. N.: Atmosheric Chemistry an Physics, Wiley, New York, 1998. Wieensohler, A., Covert, D. S., Swietlicki, E., Aalto, P., Heintzenberg, J., an Leck, C.: Occurrence of an ultrafine article moe less than 2 nm in iameter in the marine bounary layer uring Arctic summer an autumn, Tellus, 48B, 213 222, 1996. Zhang, Y., Seigneur, C., Seinfel, J. H., Jacobson, M. Z., an Binkowski, F.: Simulation of aerosol ynamics: a comarative review of algorithms use in air quality moels, Aerosol Sci. Tech. 31, 487 514, 1999. Atmos. Chem. Phys., 3, 251 257, 23