JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 105, NO. D3, PAGES 3917-3928, FEBRUARY 16, 2000

Uptake on fractal particles 2. Applications

Slimane Bekki, 1 Christine David, 1 Kathy Law, 2 Dwight M. Smith, 3 Daniel Coelho, 4 Jean-Francois Thovert, and Pierre M. Adler 4

Abstract. This first part of the paper is devoted to the validation of the theoretical framework developed in the companion paper [Coelho et al., this issue] against laboratory data. Although there are limited data suitable for a full-scale validation of the model, model-calculated surface to mass ratios of soot aggregates are found to be consistent with laboratory measurements. Then we use the framework to estimate errors which can be generated in the derivation of fractal aggregate properties from measurements of equivalent sizes. It is shown that the derivations of the aggregate mass and the surface area enhancement factor can be in error by at least an order of magnitude. The calculations are substantially improved when the fractal character of soot is accounted for. Since the gyration radius is a key parameter of the model, useful relationships are provided for converting aggregate equivalent sizes (mass transfer equivalent radius, hydro dynamic radius) into gyration radius. Finally, uptake of chemical species on atmospheric soot is investigated for the lower stratosphere and the boundary layer. The uptake is found to be mostly reaction limited, justifying the assumption of proportionality between the soot surface area and the uptake rate. However, the uptake occurs in the transition regime for the relatively compact aggregates found in urban areas.

1. Introduction

Carbonaceous aerosols, also called soot or smoke, rep- resent a substantial fraction of the ambient particulate matter. They are generally produced by incomplete combustion of fuel with the main sources being fossil fuel and biomass burning [Penner et al., 1993]. They are composed of elemental carbon (black carbon) and of a variety of organic compounds. The respective ratio of elemental to organic carbon depends on the condi- tions under which combustion takes place and the type of fuel [$cinfcld and Pandis, 1998]. These factors along with the age determine the structure and morphology of carbonaceous aerosols [Reid and Hobbs, 1998; Martins et al., 1998]. These aerosols strongly absorb the in- earn{ha .qalnr rndint. inn and t, horohv cont, rib•t,e directlv

x Service d'A4ronomie du Centre National de la Recherche Sci-

entifique, Universit6 P. et M. Curie, Paris. aGentre For Atmospheric Science, Department of Chemistry,

University of Cambridge, England. 3Department of Chemistry and Biochemistry, University of

Denver, Denver, Colorado. 4Institut de Physique du Globe de Paris, Paris. 5 Laboratoire des Ph6nomsnes de Transport dans les M6langes,

Puturoscope, France.

Copyright 2000 by the American Geophysical Union.

Paper number 1999JD900816. 0148-0227 / 00 / 1999J D900816509.00

to climate forcing [Intergovernmental Panel on climate Change, 1996]. They also have the potential to influence the climate indirectly via the cloud cover. Indeed, they may act as cloud condensation nuclei and are also in- volved in the formation of contrails, which can develop into large scale cirrus [Karcher et al., 1996; Minnis et al., 1998; Strom and Ohlsson, 1998]. More recently, there have been a renewed interest in their heteroge- neous chemical reactivity. Reactive uptake of nitrogen species and ozone on carbonaceous aerosols have been proposed to explain discrepancies between observations and model calculations [$myth et al., 1996; Jacob et al., 1996; Hauglustaine et al., 1996; Lary et al., 1997; Bekki, 1997; Aumont et al., 1999; Lary et al., 1999].

A study of heterogeneous processes associated with carbonaceous aerosols and, in pargiculm', up[ake of wa- ter and chemically active species is difficult from the theoretical point of view because of the complex ge- ometry of these aerosols. Carbonaceous aerosols are often aggregates or clusters of small rather spherical monomers, which are also called elementary particles. The size, shape, and morphology of the atmospheric soot aggregates vary widely, but the small monomers are a consistent feature [Seinfeld and Pandis, 1998]. The structure of these aggregates have been described as self-similar or fractal because they satisfy a power law relation between the aggregate mass or the number of primary particles and its radius of gyration [Forrest and Witten, 1979; Meakin, 1991].

3917

3918 BEKKI ET AL.: UPTAKE ON FRACTAL PARTICLES, 2

While general analytical solutions to the equations describing properties and dynamics of isolated spheri- cal aerosols are well known [$einfeld and Pandis, 1998], there are no general analytical solutions for fractal ag- gregates. The few solutions to transport properties equations of fractal aggregates are generally derived from numerical simulations. This lack of solid frame-

work might be one of the reasons why derivations of soot aggregate properties can differ so considerably. For •ex- ample, the surface area of soot aggregates is sometimes calculated as the sum of the surfaces of the monomers

composing the aggregates, whereas, in other studies, soot aggregates are assumed to be spheres. Certain simplifications such as the widely used assumption of spherical geometry may not be justified in some cases.

One of the aims of the present paper is to use the model developed by Coelho et al. [this issue] and esti- mate the errors generated by some of the simplifications made in the derivation of soot properties from equiva- lent size measurements. The first section presents a preliminary validation of the framework against labo- ratory data. Although there is limited amount of data suitable for a full-scale validation of the model, model- calculated soot surface to mass ratios are found to be

consistent with laboratory measurements. In the sec- ond section, different approaches for calculating aggre- gate properties from equivalent sizes measurements are evaluated. It is shown that derivations of the aggre- gate mass and the surface area enhancement factor can be in error by at least an order of magnitude. The calculation is substantially improved by accounting for the fractal character of soot. The gyration radius is a key parameter of the model. The third section pro- vides useful relationships for converting equivalent sizes (mass transfer equivalent radius, hydrodynamic radius) into gyration radius. The strong link between the mass transfer equivalent radius and the hydrodynamic radius is discussed. In the fourth section, we consider uptake processes on atmospheric soot for two regions of the at- mosphere where uptake on soot is potentially important for the overall chemistry, the lower stratosphere and the boundary layer. Sensitivity of the results to various morphological parameters is investigated. The uptake is found to be reaction limited for most cases, justify- ing the assumption of proportionality between the soot surface area and the uptake rate. However, the uptake occurs in the transition regime for relatively large and compact aggregates (i.e., aggregates composed a large number of monomers). They represent a very large frac- tion of soot in urban areas. The last section is devoted

to some concluding remarks.

2. Surface to Mass Ratio: Model

Predictions Versus Laboratory Data

Let us consider a carbonaceous aggregate composed of monomers, which are assumed to be cubes of size a for numerical reasons [Coelho et al., this issue]. The

aggregate is said to be fractal when, for example, the number N (or mass) of primary particles composing it and its gyration radius RG are linked by a power law relationship [Forrest and Witten, 1979; Mandelbrot, 1982; A. Schmidt-Oft, 1988; Meakin, 1991],

- ½ a

where ( and Dr are the compacity and the fractal di- mension of the aggregate, respectively.

The mass of a fractal aggregate can be obtained by multiplying by the mass of a monomer by the number of primary particles N composing the aggregate,

Mfractal -- a 3 p • ( RG )Dr (2) -- a

where p is the density of the carbon material. In the same manner, the aggregate surface area

can be obtained by multiplying the average available monomer surface area by the number of monomers,

Sfractal- F6a 2 • , (3)

where F is the fraction of free surface area (surface in contact with the air assuming that no closed air cavities are present within the aggregate) out of the total possi- ble surface area if the monomers were not covering each other's surfaces. This total possible surface area can be thought of as the total free surface area of the isolated monomers, before the formation of the aggregate.

Combining (2) and (3)leads to the following expres- sion for the aggregate surface to mass ratio, also called specific surface area,

6F (•/M)fractal - . (4)

pa

Apart from F, all the terms in (4) can be measured directly in laboratory. Surface to mass ratios are tra- ditionally measured with the Brunauer, Emmett and Teller method of nitrogen adsorption. The carbon den- sity p is well known (-• 2 g/cm3). Estimating the size of the monomers is less trivial. For numerical reasons, the monomers are supposed to be cubic in the theoretical framework, whereas micropictures reveal that real soot aggregates are composed of rather spherical monomers. Nonetheless, this difference should not be critical be- cause of the equivalence between diameter of spheres and size of cubes concerning some properties. If the cubic monomers composing the modeled aggregates are replaced by spheres of a diameter equal to the size of the cubic monomers, the radius of gyration remains un- changed. And, more importantly, the surface to mass ratio of a cube of size a equals the surface to mass ra- tio of a sphere with a diameter equal to the size of the cube: (S/M)sphere -- 6/(pa). Therefore, at first or- der, diameters of spherical monomers and sizes of cubic monomers will be considered to be equivalent in the present work. In order to compare F estimated from

BEKKI ET AL.: UPTAKE ON FRACTAL PARTICLES, 2 3919

laboratory measurements to F predicted by the model, (4) can be rewritten as

F -- 0.17pa ($/M)fractal ß (5)

Recall that F is equal to the ratio of the aggregate surface area to the sum of the surface areas of the iso-

lated monomers. Conversely, the difference (1- F) can be seen as the fraction of aggregate surface area lost from attachments between monomers. If the surface of

attachment was negligible, F would tend towards unity. Some values of F estimated from laboratory data are listed in Table 1.

No obvious explanations can be found for the F value equal to 2 in Table 1; it is twice the maximum theoret- ical value. It might be the result of the composition of the Degussa FW-2 carbon used by Rogaski et al. [1997]; this material is posttreated with NOs by the manufac- turer and may bear little resemblance to the others in its surface structure and chemistry. Discarding this value, F ranges from 0.003 to 0.7 in laboratory experiments. In the numerical simulations, between a third and half of the total possible surface area is lost when aggregates are formed (0.67 > F > 0.53). These values appear lie in the upper range of the experimental estimates listed in Table 1. It should be kept in mind that the compar- ison does not strictly refer to the same structural type of soot. The modeled soot corresponds to isolated ag- gregates suspended in air whereas laboratory data are likely to refer to some sort of powder, possibly obtained by grinding soot [$orensen et al., 1998].

Soot surface areas have been calculated previously as the sum of the surface areas of the toohomers [Blake and Karo, 1995], assuming implicitly that the surface of at- tachment between monomers was nil (F = 1). Modeled and experimental values listed in Table i suggest that this is not the case. This finding seems to be supported by micropictures of soot aggregates [Akhter et al., 1985;

Schmidt-Oft, 1988; Colbeck et al., 1990; Blake and Karo, 1995; Pueschel, 1996] showing that monomers appear to cover each other's surfaces and that the amount of

surface lost from attachments is not negligible. The as- sumption of unity for F leads to an overestimation of the surface area, possibly even in the case of elongated chain-like aggregates, where each monomer is linked to relatively few monomers.

As expected, the model predicts that F decreases when the compacity of the aggregate increases. Inter- estingly, like the number N of monomers or the aggre- gate mass, the aggregate surface area Sfractal was found to be proportional to R• DF with a proportionality coef- ficient independent of the mode of construction of the aggregates [Coelho et al., this issue]

$fr•c•l - 14-4 a2 (_•)Dr which, combined with (3), yields to

(6)

F.(• ,-- 2.4. (7)

As indicated by (1), the compacity ( can be deter- mined from a full characterization of the soot mor-

phology. Cai et al. [1995] measured the radius of the monomers, gyration radius, and number of monomers of soot aggregates in order to derive the fractal dimen- sion and the structural coefficient k0 which is related

to the compacity by (• ,.• 2 Dr . ko [Coelho et al., this issue]). Unfortunately, no corresponding measurements of surface to mass ratios were provided, preventing an estimation of F and hence a direct test of the value

of the proportionality coefficient in (7). If (7) is sup- posed to be correct, F can easily be derived from the compacity. For example, in the case of soot with known values of structural coefficient (k0 • 1.2 - 1.3) and frac- tal dimension (DF • 1.8) [Cai et al., 1995; $orensen

Table 1. Fraction F of Free Surface Area Estimated From Laboratory Soot Data (See Text)

F S/M, m 2/g a, nm

n2 190" 5 •' 0.5 300 c 5 •' 0.7 395 d 5 •'

0.05-0.003 60 e 40-70

2 460 ! 13 !

• Kamm et aI. [1998]. b Elementary particles produced by carbon spark discharges [$chwyn et al., 1988,

Helsper et al., 1993]. • Atomann et al. [1998]. d HeIsper et aI. [1993]. • Soot generated during the combustion of n-hexane, diesel fuel and aircraft JPL48 fuel

according to the procedure described in Smith and Chugbrai [1996]. Radii of hexane soot toohomers, also called toohomers, were found to average 40-70 nm.

f Commercial soot whose morphological characteristics are reported in a technical doc- ument referenced by Rogaski et al. [1997]. This soot is treated with NO• by the manu- facturer (Degussa FW-2).

3920 BEKKI ET AL.: UPTAKE ON FRACTAL PARTICLES, 2

and Feke, 1996; $orensen and Roberts, 1997; Oh and $orensen, 1997], F=0.57-0.53, which is in agreement with our modeled values and the experimental values listed in Table 1.

3. Derivation of Aggregate Properties From Equivalent Sizes Measurements

Dynamics of spherical aerosols have been extensively studied, and the complete solutions to the equations describing these processes are usually well defined [$e- infeld and Pandis, 1998]. For convenience, dynamics of nonspherical aerosols such as fractal aggregates tend to be defined by comparison to properties of spherical aerosols. Typically, the length scale of a single non- spherical aerosol with respect to a particular property may be defined by determining the size of a sphere which behaves like the nonspherical aerosol under the same conditions [Kasper, 1982]. For example, the hy- drodynamic radius of an aggregate moving within a fluid is defined as the radius of the sphere which expe- riences the same drag as the agglomerate when moving in the same fluid. This approach can be extended to most properties. Size characterizations in atmospheric applications mostly consist of measuring an equivalent size integrated over the aerosol size distribution or di- rectly a size distribution expressed in terms of an equiv- alent size provided by a particular instrument. The var- ious measurement techniques describe a wide range of particle properties such as inertia, optical properties, diffusion, mass transfer, or electrical mobility [Kasper, 1982; Kandlikar and Ramachandran, 1999]. An equiva- lent size specific to a particular property is generally not interchangeable with another property equivalent size. Nonetheless, when two aerosol properties are controlled by similar processes, their equivalent sizes might be ex- pected to be very close [Meakin et al., 1989; Schmidt-Oft et al., 1990; Rogak et al., 1991]. For isolated spherical aerosols, comparing equivalent sizes of different proper- ties/dynamics is usually straighforward because general analytical solutions to the equations describing these properties are often well known [$einfeld and Pandis, 1998]. On the other hand, there are no general analyt- ical solutions for fractal aggregates. For instance, the few solutions to transport properties equations of frac- tal aggregates are often derived from numerical simu- lations and are not widely known. It might be one of the reasons why properties of fractal aggregates are still derived using simplifications such as the assumption of spherical geometry. The following section is devoted to estimating the errors introduced by some of theses simplifications.

3.1. Mass

Soot mass loadings in the upper troposphere/lower stratosphere have been determined from soot collected with wire impactors and characterized by scanning elec- tron microscope (SEM)images [Pueschel et al., 1992,

1998; Blake and Karo, 1995]. Volume equivalent radii of individual aggregates were approximated by Rmean, the half of the mean between the maximum and min-

imum dimensions of each soot aggregate, as measured on SEM images. Soot loadings were then calculated by multiplying aggregates volumes by the carbon density and the concentrations of soot aggregates. Blake and Karo [1995] mentioned that, due to the fractal geom- etry, their estimates could be in error by as much as an order of magnitude. The framework developed by Coelho et al. [this issue] can be used to estimate the errors.

The perimeter radius Rp is defined as half of the longest dimension of the aggregate and is is related to the gyration radius by [Oh and $orensen, 1997; $orensen et al., 1998]

Rp _ (DF + 2) 1/2 (8) For sufficiently long linear chain-like aggregates, D•

tends towards unity, and, consequently, Rp = 1.7 since the longest dimension of such aggregates is much larger than the minimum dimension, Rm•a• • Rp/2 = 0.87 R6. For closely packed spherical aggregates, D• tends towards 3 and, hence, Rp = 1.3 R6. Since the ma•mum and minimum dimensions for this type of ag- gregates are very similar, R•, • Rp = 1.3

The results for the two extremes suggest that, for typical aggregates, Rm•a• may be approfimated by for our estimation purposes. Using this approfimation, it is straightforward to show that the aggregate mass calculated by Blake and Karo [1995] and Pueschel et al. [1992, 1998] is given by

M • 4•p RG3. (9) 3

where p is the carbon density (2 g/cmS); it was set to a slightly lower value (1.5 g/cm s) by Pueschel et al. [1998] in an attempt to account for the loose structure of the aggregate.

It follows that the ratio of the mass of a fractal ag- gregate given by (2) to the mass as expressed by (9) is

Mfractal!M • 4.19 . (10) Since ( ranges from 4.48 to 3.66 in our simulations

[Coelho et al., this issue], (10) may be further appro•- mated by

Mf•t•l/M • . (11)

The mass ratio is displayed as a function of R6/a and D• in Figure 1. Even accounting for a factor 2 uncertainty generated by the various appro•mations, Figure I clearly shows that neglecting the kact• char- acter in this type of calculation may lead to an overes-

BEKKI ET AL.. UPTAKE ON FRACTAL PARTICLES, 2 3921

'"• 10- 13:

6-

I I I

I

30

i i I

1.5 2.0 2.5

D. Figure 1. Ratio Mfractal/M as a function of D F and RG/a.

timation of aggregate masses by up to orders of magni- tude depending on the fractal dimension and the size of the monomers with respect to the aggregate size. In many cases, the use of (2) combined with reason- able estimates for the morphological parameters a and DF should lead to a more accurate estimation of the aggregate mass. For soot aggregates found in the up- per troposphere/lower stratosphere (UT/LS), the aver- age radius is about 100 nm, whereas the typical size of the monomers composing the aggregates is of the or- der of 20 nm [Blake and Kato, 1995]. Therefore, for typical UT/LS morphologies (RG=100 nm, a=20 nm and D•- 1.5- 2.0 from Table 2), the masses of in- dividual aggregates derived from (9) could have been overestimated by a factor 5-10. On the other hand, there was another approximation by Blake and Kato [1995] mass derivation which led to an underestimation of soot loadings. As stressed by the authors, the col- lection efficiency on wire impact ors was assumed to be 100% whereas, in fact, the collection efficiency drops rapidly with the aggregate size. There is also an ad- ditional process which may have been overlooked and could have contributed to a reduced collection efficiency. Depending on factors such as the velocity, dryness, size and structure of the aggregates, solid aggregates may bounce or break up and escape when impacting the wire [$trawa et al., 1999]. The effective sticking coefficient of soot on wire is unknown. It is important to work out reliable estimates of soot loadings in the upper tro- posphere/lower stratosphere because aircraft appear to be the major source of soot in the lower stratosphere [Blake and Kato, 1995; Bekki, 1997]. Estimates of soot loadings are used to validate model-calculated soot dis- tributions in aircraft assessment studies [Bekki, 1997; Rahmes et al., 1998; Danilin et al., 1998].

3.2. Surface Area Enhancement

The surface area enhancement factor, which we call E•, has been used in some studies to derive soot sur-

face areas from mass measurements [Blake and Karo, 1995; Hauglustaine et al., 1996; Bekki, 1997]. This fac- tor can be seen as a measure of the enhancement in

- the surface area of an aerosol when its geometry de- viates from the spherical one; it is necessarily an en-

_

hancement because the spherical geometry corresponds to the minimum surface area for a given mass. In the

-- present context, E• can be calculated as the ratio of the _

- surface area of a fractal aggregate to the surface area of _

- an equivalent sphere of the same mass. Since the mass of a fractal aggregate is given by (2), the radius R of

- a sphere with the same mass as the aggregate can be expressed by

R - a • -- . (12) This radius is also called the bulk radius, the radius

after melting of the aggregate. Combining (6) and (12) yields to the following expression for the enhancement factor

E• = 4•rR 2 -- (2/3 -- . (13) Since ( varies within 10% of its mean value 4.07 in

the numerical simulations [Coelho et al., this issue], (13) can be approximated by

E•-,• 1.2 -- . (14)

E• is plotted as a function of R•/a and D• in Fig- ure 2. For soot aggregates typical of the upper tropo- sphere/lower stratosphere (R•=100 nm and a=20 nm, giving R•/a=5), the factor of enhancement due to the fractal geometry is 2.7 - 3.5 for D• = 1.5 - 2.0. This is about an order of magnitude smaller than the enhance- ment factor of 30 derived previously [Blake and Kato, 1995] and used in modeling studies [Blake and Kato, 1995; Hauglustaine et al., 1996; Bekki, 1997]. S.G.

s

"'• lO

6

I I I

1.5 2.0 2.5

D F Figure 2. Fractal surface enhancement factor E• as a function of D• and R•/a.

3922 BEKKI ET AL.: UPTAKE ON FRACTAL PARTICLES, 2

Jennings at al. (Carbonaceous aerosol fractal surface area: Implications for heterogeneous chemistry, submit- ted to Geophysical Research Letters, 1999) (hereinafter referred to as (Jennings at al., submitted manuscript, 1999)) reached a similar conclusion from semianalyti- cal calculations. They found that previously used en- hancement factors have been largely overestimated. In the extreme case of closely packed spherical aggregates of radius R (Dr-• 3 and RG-• v•.R), (14) gives E• -• 0.93 R/a. This value can be compared to the value of E• calculated assuming that the total volume of the closely packed spherical aggregate is composed of spherical monomers with a diameter a [Blake and Karo, 1995]; taking into account the voids in the aggregate with a packing factor of 0.7 (Jennings at al., submitted manuscript, 1999), this approach gives E• ,• 1.4R/a, which is 50% higher than the value calculated from (14). This difference is not surprising because (14) is implic- itly derived from results of numerical experiments on aggregates with a fractal dimension between 1.6 and 2.6. (14) should not be applied to closely packed spher- ical aggregates. This 50% difference may be seen as a conservative estimate of the error on E• calculated from

(14).

3.3. Gyration Radius

The gyration radius is a key parameter in our flame- work. Its knowledge is required for the derivation of many aggregate properties. Instruments do not provide a direct measure of it but rather a measure of a prop- erty equivalent radius. These equivalent radii have to be converted into a gyration radius before being used in the flamework. For example, (8) indicates how to con- vert the longest dimension of a fractal aggregate into the gyration radius. The following section provides expres- sions for converting other equivalent radii, namely, the mass transfer equivalent radius and the hydrodynamic radius, into gyration radii.

We first consider the mass transfer equivalent radius. The epiphaniometer is an instrument designed to pro- vide a measure of the mass transfer equivalent radius.

-1

i ) (15) Jsphere -- 4• D C• R 1 + PeDa ' where

PeDa = D '

where D, C• and K• are the gas phase diffusion co- efficient of the sticking/reacting species, the concentra- tion of this species far from the sphere, and the surface reaction rate constant (which may be called the up- take velocity), respectively. PeDa is called the Peclet- Dahmkohler number and compares the chemical rate to the diffusion rate. For clarity, we will call it the reac- tion/diffusion ratio. The uptake velocity K• is related to the reaction probability 7 (or sticking probability in the case of a nonreactive uptake) by

K•= 7.v 4 '

where v is the molecular velocity of the species. Coelho et al. [this issue] calculated numerically the

mass flux of a chemical species onto ffactal aggregates instead of spheres. The numerical results were used to express the uptake flux J•t• as a function of the gas-phase diffusion coe•cient, uptake velocity and mor- phological parameters in a single formula,

( 1) Jfracta• - 4• D C• 0.45 Dr RG 1 + PeDa

where

-1

(16)

D DF

Without considering mass transfer, the question of defining a characteristic length scale for a fractal aggre- gate arises in the calculation of the Knudsen number Kn [Schmidt-Ott et al., 1990]. The value of this number with respect to unity provides indications about which

It works by exposing aerosols to gaseous radionuclide equations, continuum mechanics or kinetic theory of atoms and then, after trapping, monitors the decay of gases, can be applied. In the present context, the mass the radionuclide atoms activity, which is proportional to transfer equivalent radius RMT is the radius of a sphere the number of atoms which have stuck to the aerosols whose uptake flux equals the uptake flux of the consid- [Gaggeler et al., 1989; Harrison et al., 1999]. The stick- ered fractal aggregate (Jsphere=Jfractal). RMT can be ing coefficient of the radionuclide atoms to the surface is determined by comparing the expressions of Zsphere and expected to be unity [Rogak et al., 1991]. In this case, Jf•ct• in the diffusion limited regime (PeDa>>l) and the limiting process can be determined by comparing reaction limited regime (PeDa<<l). In the diffusion lim- the mean free path of the diffusing radionuclide atom ired regime, and the size of the aggregate.

Let us recall the expression for the mass flux of a diffusing species which sticks or/and reacts at the sur- face of an isolated sphere immersed in an infinite fluid [Coelho et al., this issue]; this will be used for deter- mining the mass transfer equivalent radius of fractal aggregates,

RM•, = Rc 0.45 DF ,

whereas in the reaction limited regime,

RMT -- R• 1.07 (R•/a) (Dr-2)/2

(17)

(18)

As expected, in the reaction limited regime regime,

BEKKI ET AL.: UPTAKE ON FRACTAL PARTICLES, 2 3923

RMT equals the surface area equivalent radius, also called the surface area mean radius (see (6) which pro- vides the expression for the aggregate surface area). Equations (17) and (18) show how the gyration radius and the mass transfer equivalent radius can be inter- changed.

The hydrodynamic radius is a commonly measured equivalent radius. It is measured typically by cen- trifuges and inertial impactors [Kasper, 1982]. Accord- ing to the equations describing the motion of a parti- cle in a fluid, the relevant dimensionless numbers are the Reynolds number, which compares inertial to vis- cous forces, and the Knudsen number which compares the mean free path of air molecules to the particle size. For most aerosols, the Reynolds number appears to be much smaller than unity in the lower atmosphere [Se- infeld and Pandis, 1998]. Therefore the Navier-Stokes equations can be simplified to the Stokes equations where only the Knudsen number is relevant. Interest- ingiy, the hydrodynamical properties of the fractal ag- gregates considered by Uoelho et al. [this issue] have been previously investigated by solving numerically the three-dimensional Stokes equations [ Uoelho et al., 1997]; only the continuum regime was considered (i.e., Kn<<l, mean free path of air molecules much smaller than the aggregate size). For an isolated fractal aggregate im- mersed in an infinite fluid, which is the relevant con- figuration for atmospheric applications, a simple rela- tionship between the hydro dynamic radius R• and the gyration radius was derived from the numerical results

(0.3DF + 0.4) (19)

This expression was found to be consistent with pre- vious calculations [Meakin et al., 1985; Rogak and Fla- gan, 1990; Veerapaneni and Wiesner, 1996]. Equation (19) is also valid for dense spheres (DF -- 3).

Theoretical and experimental studies have suggested that the hydro dynamic radius and the mass transfer equivalent radius should be very close. Equations (17) and (19) can be combined to derive the ratio of these two equivalent radii and check whether our expressions lead to the same conclusion in the diffusion limited

regime (which necessarily implies a continuum regime),

0.45 De

RMT/R• -- 0.3D• + 0.4 ' (20) For our range offractal dimension (D•=1.5-2.6 [Coelho

et al., 1997, this issue]), RMT/R•t varies from 0.8 to 1. Therefore, within 20%, the mass transfer equivalent radius and the hydrodynamic radius can be considered to be interchangeable in the continuum regime, which is in agreement with previous studies [Meakin et al., 1989; $chmidt-Ott et al., 1990; Rogak et al., 1991]. According to (20), it is only in the case of very elongated chain-like

aggregates that the mass transfer equivalent radius and the hydrodynamic radius differ very significantly.

Hydrodynamic properties of the aggregates consid- ered in the numerical experiments [Coelho et al., 1997, this issue] have not been investigated in the free molecu- lar regime. It is likely that the mass transfer equivalent radius and the hydrodynamic radius also exhibit some equivalence in this regime. In the free molecular regime, which corresponds to the reaction limited regime for sufficiently high reaction or sticking probabilities, the mean free path of the gaseous species is much larger than the size of the aggregates. In these conditions, the mass and momentum transfer processes are controlled primarily by the rate of molecules impinging the sur- face of the aggregate [Meakin et al., 1989; Rogak et al., 1991]. For mass transfer the relevant molecules are the species reacting or sticking to the aggregate surface. For momentum transfer the relevant molecules are all the

molecules, which are essentially the air molecules in the atmosphere. Since both processes appear analogous, the mass transfer and hydro dynamic radii may expect to be strongly correlated or even somewhat equivalent. This has been supported by some theoretical and nu- merical experiments [Meakin et al., 1989; Schmidt-Oft et al., 1990; Rogak et al., 1991]. More laboratory data are required in order to test whether this equivalence holds over all environmental conditions and determine

the possible influence of the aggregate morphology. It is important to notice that measurements of mass transfer radii and of hydrodynamic radii of atmospheric aggre- gate populations are not always comparable. Measure- ments of the mass transfer radius are commonly per- formed with an epiphaniometer which integrates over the entire size distribution, whereas hydrodynamic ra- dius measurements usually provide size distributions [Schmidt-Oft et al., 1990].

4. Uptake on Atmospheric Soot

We now consider uptake processes on atmospheric soot. Coelho et al. [this issue] show in the compan- ion paper that neglecting the fractal structure in the calculation of uptake rates on single isolated aggregates rnav lead t,o flawed romllt,,q_ Here t. ho n.irn i.q ka ielon•.ifv

the key morphological parameters in this type of calcu- lation for atmospheric situations of interest. Equation (16) is only strictly valid for monodispersed size distri- butions of aggregates. In order to be applied to realistic atmospheric soot, it has to be generalized to aggregate populations with a certain width in the size distribu- tion.

The uptake flax Jt of a species onto a population of fractal aggregates can be expressed by

at -- ,]'fra½t;al N(RG)dRo, (21)

where N(Ro) is the size distribution.

3924 BEKKI ET AL.: UPTAKE ON FRACTAL PARTICLES, 2

Table 2. Soot Morphological Parameters

RG, nm • • a, nm DF

Lower troposphere b 50 - 1000 1.3- 1.9 5 - 30 1.2 - 2.1 Lower stratosphere c 15- 90 1.4- 1.6 few- 20 1.5- 2.0 a

Laboratory' 10- <1000 ? 2.5- 50 1.1 - 2.9

•Sizes refer to different types of measurements and, hence, to different equivalent sizes without conversion to gyration radii.

b Heintzenberg[1982], Meszaros and Meszaros [1989], Katrinak et al. [1993; $einfeld and Pandis [1998 and references therein], Reid and Hobbs [1998], Reid et al. [1998], Jennings et al. (submitted manuscript, 1999).

c Aircraft-generated soot are included, Pueschel et al. [1992], Sheridan et al. [1994], Blake and Idato [1995], Pueschel [1996], Hagen et al. [1996], Pueschel et al. [1998], Petzold and $chroder [1998].

a Values represent the most likely range because no measurements of Dr have been performed on stratospheric soot.

e Predominantly soot produced in spark discharges or during combustion processes, Akhter et al. [1985], $chmidt-Ott [1988], Colbeck et al. [1990], Katrinak et al. [1993], Helspev et al. [1993], Nyeki and Colbeck [1995], Ramachandran and Reist [1995], Kalberer et al. [1996], Colbeck et al. [1997], Weingavtnev et al. [1997], Kamm et al. [1998], Skillas et al. [1998].

If the size distribution is lognormal, (21) can be inte- grated analytically in the two asymptotic regimes. If the lognormal distribution is not too wide, (21) can also be integrated analytically in the transition regime, yielding an approximate expression for the uptake flux [Coelho et al., this issue],

Jr(approx.) -- 5.65 D C• Dr RG,mod No .

( ) ( 1 ) -x exp ln2 (ø') 1 + 2 PeDa

with

K• RG,mod 2.55 D Dr

(22)

RG,mod) (DF - 2) a

where RG,moa, No and a are the mode radius, the to- tal concentration, and the geometric standard deviation (also called the width) of the distribution of soot aggre- gates.

Expression (22) provides exact results in the diffu- sion limited regime and reaction limited regime. The time constant for the uptake of a gaseous species on a population of fractal aggregate is simply the reciprocal of the total flux multiplied by the species concentration far from the aggregate.

Values of soot morphological parameters are given in Table 2. Some ranges are only meant to be indicative because comprehensive morphological measurements of atmospheric soot are scarce. In order to assess the role of soot morphology in uptake processes, we investigate the dependency of the uptake on morphological param- eters for two regions of the atmosphere where uptake on soot is potentially important for the overall chemistry.

4.1. Lower Stratosphere

Figure 3 illustrates the dependence of the uptake time scale •- on the fractal dimension for soot distributions

typical of the lower stratosphere and a relatively reac- tive species. This species is HNO3; its reaction proba- bility 7 is of the same order as NO2 [Tabor et al., 1994; Rogaski et al., 1997]. Following the Lary et al. study

500

100

5o

lO

Figure 3. DF

Uptake time constant •- as a func- tion of Dr for different soot distributions (N0=0.625 particles/cm3; RG,,od=100 nm; (r=1.2, 1.5 and 1.8; a=10, 20, and 40 nm) at T=220 K and pressure=50 mbar. The chemical species is HNO3 and the reaction probability 7 is set to 0.028 [Hauglustaine et al., 1996; Lary et al., 1997]. The heavy line corresponds to a typ- ical background soot distribution in the lower strato- sphere (rr=l.5; a=20 nm) [Pueschel et al., 1992; Blake and Karo, 1995]; the No value has been chosen to give a soot mass concentration of I ng/m 3 at D i•=1.8 (solid circle).

BEKKI ET AL.' UPTAKE ON FRACTAL PARTICLES, 2 3925

a-40 nrn •x --": ....

1

- I ' I ' I

- 1,5 2.0 2.5

D F Figure 4. Same as Figure 3 but for the reac- tion/diffusion ratio PeDa.

Soot loading (/zg/m - 10 • ' • .

•" 0.3

o. O. oeo

, 2 _ • .5 2.0 2.5

- Figure •. Soo• loading as a function of D• and R•,•od (N0-10 p•id•/•m•; •=1.•; •=• •m [S•i•]d• Pa•di•, 1998; Jennings e• al., submitted manuscript, 1999]) a• T=298 K and pressure=900 mbar.

[1997], 7 has been set to 0.028. For a given gyration radius and size of monomers, the uptake rate can in- crease by about an order of magnitude from a chain- like aggregate (Dr= 1.1) to a highly compact aggregate (Dr=2.7). The dependence of •- on Dr decreases when the size of the monomers increases. For Dr close to 2, •- is independent of the size of the monomers. The dependence of •- on the width of the distribution is weak except for high values of Dr. Figure 4 shows the reaction/diffusion ratio PeDa as a function of the fractal dimension for the same conditions as Figure 3. PeDa has a strong dependence on the fractal dimen- sion, varying by up to an order of magnitude. PeDa is always smaller than 0.025 for the conditions explored here. This indicates that uptake can be assumed to be reaction limited for most species of interest in the lower stratosphere, justifying the assumption of proportional- ity between the uptake rate and the soot surface area. Much faster uptake rates are expected in aircraft plumes where concentrations of soot particles are few orders of magnitude higher than in the background lower strato- sphere [Schumann et al., 1996; Petzold and $chroder, 1998].

4.2. Boundary Layer

A wide range of soot sizes (see Table 2) and load- ings are found in the boundary layer. Figures 5, 6, and 7 show the soot mass loading, the uptake time constant •- and the reaction/diffusion ratio PeDa as a function of Dr and RG,mo,• for typical tropospheric soot and a relatively reactive species (3,=0.028). The bottom parts of the figures correspond to soot loadings typical of free tropospheric/remote air ,whereas the top right-hand part of the figure is more characteristic of soot loadings in urban areas. The uptake time constant •- varies by up to 3 orders of magnitude between these

two extremes. Note that time constants calculated from

(22) do not differ by more than about 10% from those obtained by integrating numerically (21) for the condi- tions explored here. It is only in the case of wide size distribution that differences are substantial.

For similar soot loadings, uptake rates can differ by up to an order of magnitude depending on the soot morphological characteristics. This shows that a mass or size characterization of soot is not sufficient for the

calculation of uptake rates. The uptake is found to be mostly reaction limited, even for species reacting strongly on soot (7 > 10-2) (see Figure 7). The only exception is for large relatively compact soot aggregates where the uptake occurs in the transition regime and al- most in the diffusion limited regime. This type of aggre- gate is found in large numbers in polluted areas where soot mass distributions are bimodal with one of the

T (day)

n' 0.1 •.00

-4 -' •O. oo 0.06• , "' •'u ------..__.__•1 ' , _ 1.5 2.0 2.5

D. Figure 6. Same as Figure 5 but for the uptake constant •; 7=0.028.

3926 BEKKI ET AL.: UPTAKE ON FRACTAL PARTICLES, 2

PeDa

. O. Zo o. -_ 'øo_ f_

1.5 2.0 2.5

Fig•e 7, Same as Figure 6 but for the reac- tion/diffusion ratio PeDa.

peaks around 0.5-1 pm [Seinfeld and Pandis, 1998, and references therein]. Tropospheric uptake rates are some- times calculated assuming that uptake is reaction lim- ited (proportionality between uptake rate and surface area) [Hauglustaine et al., 1996; Aumont et al., 1999]. This assumption has to be used with some caution when considering uptake of strongly reactive species on urban soot [Aumont et al., 1999]. For moderately or weakly reactive species (7 • 10-3), the uptake is reaction lim- ited.

5. Concluding Remarks

Overall, the results demonstrate that accounting for the fractal morphology of carbonaceous aggregates can substantially improve calculations of their properties or dynamics. Errors in deriving soot surface areas or up- take rates may prevent quantitative analysis of the soot impact on atmospheric chemistry. In particular, accu- rate calculations are required when attempting to re- solve discrepancies between photochemical model cal- culations and atmospheric observations.

There are some limitations to the use of the frame-

work developed in Coelho et al. [this issue] and the present paper. First, morphological characterisations of carbonaceous aggregates are often required for the cal- culations of properties/dynamics. Unfortunately, mor- phological measurements are scarce. They are also not always reliable because collecting and handling atmo- spheric soot may result in breaking up or restructur- ing of the aggregates, thereby altering the initial mor- phology. Second, the uptake on the soot surface is as- sumed to be a pseudo-first-order process in the frame- work. Surface uptake can be a complex multistep pro- cess [Chughtai et al., 1994, 1996; Smith and Chughtai, 1996, 1997]. The uptake cannot always be reduced to a pseudo-first-order process, depending on the environ-

mental conditions. Third, carbonaceous aggregates can contain a very large organic fraction which is mixed with an insoluble fraction composed predominantly of elemental carbon. The organic fraction is also strongly reactive [Gouw and Lovejoy, 1998]. Since this type of aggregate looks more like "blobby" particles rather than fractal aggregates [Sheridan et al., 1994], it might be preferable to treat them as spherical aerosols. Finally, the framework is based on results from numerical sim- ulations. More detailed laboratory data are needed for validating some of the predicted relationships such the one between the mass transfer equivalent radius and the hydrodynamic radius or the scaling behavior of the surface area.

It is worth noticing that the morphology of carbona- ceous aerosols evolves with time. They tend to be branched, chain-like, and fluffy near the source regions but become more compacted with age [Reid and Hobbs, 1998; Martins et al., 1998] and reaction [Sergides et al., 1987]. There is some evidence from laboratory experi- ments [Colbeck et al., 1990; Ramachandran and Reist, 1995] that carbon aggregates subjected to some form of cloud processing tend to collapse to a more regular and compact structure. One of the reasons for the re- structuring is that adsorption of gaseous species such as water, nitrogen dioxides, or ozone modifies the na- ture of the surface functional groups [Smith and Chugh- tai, 1996, 1997], altering the bounds that the monomers can establish between them. This aggregate restructur- ing modifies the surface area available for heterogeneous reactions. Adsorption and exchange of molecules at the surface also lead to partial or complete saturation of the chemically active sites. It is at the origin of the time dependence of reaction and sticking probabilities found in laboratory studies.

Acknowledgments. A.R. Chughtai and J. C. Wilson axe thanked for helpful discussions. The support of the French Programme National de la Chimie Atmosph•rique (PNCA) is gratefully acknowledged.

References

Akhter, M.S., A.R. Chughtai, and D.M. Smith, The struc- ture of hexane soot, 1. Spectroscopic studies, Appl. Spec- trosc., 39, 143-153, 1985.

Ammann, M., M. Kalberer, D.T. Jost, L. Tobler, E. Rosslet, D. Piguet, H.W. Gaggeler, and U. Baltensperger, Hetero- geneous production of nitrous acid on soot in polluted air masses, Nature, 395, 157-160,1998.

Aumont, B., S. Madronich, M. Ammann, M. Kalberer, U. Baltensperger, D. Hauglustaine, and F. Brocheton, On the NO2 plus soot reaction in the atmosphere J. Geophys. Res., 104, 1729-1736, 1999.

Bekki, S., On the possible role of aircraft generated soot in middle latitude ozone depletion, J. Geophys. Res., 10œ, 10751-10758, 1997.

Blake, D.F., and K. Karo, Latitudinal distribution of black carbon soot in the upper troposphere and lower strato- sphere, J. Geophys. Res., 100, 7195-7202, 1995.

Cai, J., N. Lu, and C.M. Sorensen, Analysis of fractal cluster

BEKKI ET AL.: UPTAKE ON FRACTAL PARTICLES, 2 3927

morphology parameters: Structural coefficient and den- sity autocorrelation function cutoff, J. Colloid Interface Sci., 171, 470-473, 1995.

Chughtai, A.R., S.A. Gordon, and D.M. Smith, Kinetics of the hexane soot reaction with NO2/N204 at low concen- tration, Carbon, 3œ, 405-416, 1994.

Chugbrai, A.R., M.E. Brooks, and D.M. Smith, Hydration of black carbon, J. Geophys. Res., 101, 19,505-19,514, 1996.

Coelho, D., J.F. Thovert, R. Thouy, and P.M. Adler, Hydro- dynamic drag and electrophoresis of suspensions of fractal aggregates, Fractals, 5, 507-522, 1997.

Coelho, D., S. Bekki, J.F. Thovert, and P.M. Adler, Uptake on fractal particles, 1, Theoretical framework, J. Geophys. Res., this issue.

Colbeck, I., L. Appleby, E.J. Hardman, and R.M. Harrison, The optical-properties and morphology of cloud-processed carbonaceous smoke, J. Aerosol Sci., 21, 527-538, 1990.

Colbeck, I., B. Atkinson, and Y. Jobar, The morphology and optical properties of soot produced by different fuels, J. Aerosol Sci., 28, 715-723, 1997.

Danilin, M.Y., et al., Aviation fuel tracer simulation: Model intercomparison and implications, Geophys. Res. Left., 25, 3947-3950, 1998.

Forrest, S.R., and T.A. Witten, Long-range correlations in smoke-particule aggregates, J. Phys. A Math Gen., 12, 109-117, 1979.

Gaggeler, H.W., U. Baltensperger, M. Emmenegger, D.T. Jost, A. Schmidt-Oft, P. Haller, and M. Hofmann, The epiphaniometer, a new device for continuous aerosol mon- itoring, J. Aerosol Sci., 20, 557-564, 1989.

Gouw, J.A., and E.R. Lovejoy, Reactive uptake of ozone by liquid organic compounds, Geophys. Res. Left., 25, 931- 934, 1998.

Hagen, D.E., P.D. Whitefield, and H. Schlager, Particulate- emissions in the exhaust plume from commercial jet air- craft under cruise conditions, J. Geophys. Res., 101, 19551-19557, 1996.

Harrison, R.M., M. Jones, and G. Collins, Measurements of the physical properties of particles in the urban atmo- sphere, Atmos. Environ., 33, 309-321, 1999.

Hauglustaine, D.A., B.A. Ridley, S. Solomon, P.G. Hess, and S. Madronich, HNOa/NOx ratio in the remote tro- posphere during MLOPEX 2: Evidence for nitric acid re- duction on carbonaceous aerosols?, Geophys. Res. Left., 23, 2609-2612, 1996.

Heintzenberg, J., Size-segregated measurements of particu- late elemental carbon and aerosol light-absorption at re- mote Arctic locations, Atmos. Environ., 16, 2461-2469, 1982.

Helsper, C., W. Molter, F. Loftier, C. Wadenpohl, S. Kauf- mann, and G. Wenninger, Investigations of a new aerosol generator for the production of carbon aggregate parti- cles, Atmos. Environ., 27, 1271-1275, 1993.

Intergovernmental Panel on Climate Change, Climate Change 1995: The Science of Climate Change, edited by J.T. Houghton et al., Cambridge Univ. Press, New York, 1996.

Jacob, D.J., et al., Origin of ozone and NO x in the tropical troposphere: A photochemical analysis of aircraft obser- vations over the South Atlantic basin, J. Geophys. Res., 101, 24,235-24,250, 1996.

Kalberer, M., et al., Heterogeneous chemical-processing of (NO2)-N-13 by monodisperse carbon aerosols at very-low concentrations, J. Phys. Chem., 100, 15487-15493, 1996.

Karore, S., et al., Heterogeneous interaction of ozone, NO2, N2Os with soot aerosols, paper presented at CACGP/IGAC 1998 Symposium, CACGP/IGAC, Seat- tle, Wash., 19-25 August 1998.

Kandlikar, M., and G. Ramachandran, Inverse methods for

analyzing aerosol spectrometer measurements: A critical review, J. Aerosol Sci., 30, 413-437, 1999.

Karcher, B., T. Peter, U.M. Biermann, and U. Schumann, The initial composition of jet condensation trails, J. At- mos. Sci., 53, 3066-3083, 1996.

Kasper, G., Dynamics and measurements of smokes, I, Size characterization of nonspherical particles, Aerosol Sci. Technol., 1, 187-199, 1982.

Katrinak, K.A., P. Rez, P.R. Perkes, and P.R. Buseck, Frac- tal geometry of carbonaceous aggregates from an urban aerosol, Environ. Sci. Technol., œ7, 539-547, 1993.

Lary, D.J., A.M. Lee, R. Toumi, M.J. Newchurch, M. Pirre, and J.B. Renard, Carbon aerosols and atmospheric pho- tochemistry, J. Geophys. Res., 10œ, 3671-3682, 1997.

Lary, D.J., D.E. Shallcross, and R. Toumi, Carbonaceous aerosols and their potential role in atmospheric chemistry, J. Geophys. Res., 10•, 15,929-15,940, 1999.

Mandelbrot, B.B., The Fraetat Geometry of Nature, W.H. Freeman, New York, 1982.

Martins, J.V., P.V. Hobbs, R.E. Weiss, and P. Ar- taxo, Sphericity and morphology of smoke particles from biomass burning in Brazil, J. Geophys. Res., 103, 32,051- 32,057, 1998.

Me&in, P., Fractal aggregates in geophysics, Rev. Geophys., œ9, 317-354, 1991.

Meakin, P., Z.Y. Chen, and J.M. Deutch, The translational friction coefficient and time-dependent cluster size dis- tribution of 3 dimensional cluster cluster aggregation, J. Chem. Phys., 82, 3786-3789, 1985.

Meakin, P., B. Donn, and G.W. Mulholland. Collisions be- tween point masses and fractal aggregates, Langmuir, 5, 510-518, 1989.

Meszaros, A., and E. Meszaros, Sulfate formation on ele- mental carbon particles, Aerosol $ci. Technol., 10, 337- 342, 1989.

Minnis, P., D.F. Young, D.P. Garber, L. Nguyen, W.L. Smith Jr., and R. Palikonda, Transformation of contrails into cirrus during SUCCESS, Geophys. Res. Left., 25, 1157-1160, 1998.

Nyeki, S., and I. Colbeck, Fractal dimension analysis of single, in-situ, restructured carbonaceous aggregates, Aerosol Sci. Technol., 23, 109-120, 1995.

Oh, C., and C.M. Sorensen, The effect of overlap between toohomers on the determination of fractal cluster mor-

phology, J. Colloid Interface Sci., 193, 17-25, 1997. Penner, J.E., H. Eddieman, and T. Novakov, Towards the

development of a global inventory for black carbon emis- sions, Atmos. Environ., 27, 1277-1295, 1993.

Petzold, A., and F.P. Schroder, Jet engine exhaust aerosol characterization, Aerosol Sci. Technot., 28, 62-76, 1998.

Pueschel, R.F., Stratospheric aerosols: Formation, proper- ties, effects, J. Aerosol Sci., 27, 383-402, 1996.

Pueschel, R.F., D.F. Blake. K.G. Suetsinger, A.D.A. Hansen, S. Verma, and K. Karo, Black carbon (soot) aerosol in the lower stratosphere and upper troposphere, Geophys. Res. Left., 19, 1659-1662, 1992.

Pueschel, R.F., S. Verma, G.V. Ferry, S.D. Howard, S. Vay, S.A. Kinne, J. Goodman, and A.W. Strawa, Sulfuric acid and soot particle formation in aircraft exhaust, Geophys. Res. Lett., 25, 1685-1688, 1998.

Rahmes, T.F., A.H. Omar, and D.J. Wuebbles, Atmospheric distributions of soot, particles by current and future air- craft fleets and resulting radiative forcing on climate, J. Geophys. Res., 103, 31,657-31,667, 1998.

Ramachandran, G., and P.C. Reist, Characterization of morphological changes in agglomerates subject to conden- sation and evaporation using multiple fractal dimensions, Aerosol $ci. Technol., 23, 431-442, 1995.

Reid, J.S., and P.V. Hobbs, Physical and optical properties

3928 BEKKI ET AL.: UPTAKE ON FRACTAL PARTICLES, 2

of young smoke from individual biomass fires in Brazil, J. Geophys. Res., 103, 32,013-32,030, 1998.

Reid, J.S., P.V. Hobbs, R.J. Ferek, D.R. Blake, J.V. Mar- tins, M.R.Dunlap, and C. Liousse, Physical, chemical, and optical properties of regional hazes dominated by smoke in Brazil, J. Geophys. Res., 103, 32,059-32,080, 1998.

Rogak, S.N., and R.C. Flagan, Stokes drag on self-similar clusters of spheres, J. Colloid Interface $ci., 13•, 206-218, 1990.

Rogak, S.N., U. Baltensperger, and R.C. Flagan, Measure- ment of mass-transfer to agglomerate aerosols, Aerosol $ci. Technol., 1•, 447-458, 1991.

Rogaski, C.A., D.M. Golden, and L.R. Williams, Reactive uptake and hydration experiments on amorphous carbon treated with NO2, SO2, 03, HNO3, and H•SO4, Geophys. Res. Lett., œ•, 381-384, 1997.

Schmidt-Oft, A., New approaches to in situ characterization of ultrafine agglomerates, J. Aerosol $ci., 19, 553-563, 1988.

Schmidt-Oft, A., U. Baltensperger, H.W. Gaggeler, and D.T. Jost, Scaling behavior of physical parameters de- scribing agglomerates J. Aerosol $ci., œ1, 711-717, 1990.

Schumann, U., J. Strom, R. Busen, R. Baumann, K. Gierens, M. Krautstrunk, F.P. Schroder, and J. Stingl, In situ observations of particles in jet aircraft exhausts and contrails for different sulfur-containing fuels, J. Geophys. Res., 101, 6853-6869, 1996.

Schwyn, S., E. Carwin, and A. Schmidt-Oft, Aerosol gen- eration in spark discharges, J. Aerosol $ci., 19, 639-642, 1988.

Seinfeld, J.H., and S.N. Pandis, Atmospheric Chemistry and Physics, John Wiley, New York, 1998.

Sergides, C.A., J.A. Jassim, A.R. Chugbrai, and D.M. Smith, The structure of hexane soot, III, Ozonation stud- ies, Appl. $pectrosc., •1, 482-492, 1987.

Sheridan, P.J., C.A. Brock, and J.C. Wilson, Aerosol par- ticles in the upper troposphere and lower stratosphere: Elemental composition and morphology of individual par- ticles in northern midlatitudes, Geophys. Res. Lett., œ1, 2587-2590, 1994.

Skillas, G., S. Kunzel, H. Burrschef, U. Baltensperger, and K. Siegmann, High fractal-like dimension of diesel soot agglomerates, J. Aerosol $ci., œ9, 411-419, 1998.

Smith, D.M., and A.R. Chugbrai, Reaction kinetics of ozone at low concentrations with n-hexane soot, J. Geophys. Res., 101, 19,607-19,620, 1996.

Smith, D.M., and A.R. Chugbrai, Photochemical effects in the heterogeneous reaction of soot with ozone at low con- centrations, J. Atmos. Chem., œ6, 77-91, 1997.

Smyth, S.B., et al., Factors influencing the upper free tro- pospheric distribution of reactive nitrogen over the South Atlantic during the TRACE A experiment, J. Geophys. Res., 101, 24,165-24,186, 1996.

Sorensen, C.M., and G.D. Feke, The morphology of macro- scopic soot, Aerosol $ci. Technol., œ5, 328-337, 1996.

Sorensen, C.M., and G. Roberts, The prefactor of fractal aggregates, J. Colloid Interface $ci., 186, 447-452, 1997.

Sorensen, C.M., C. Oh, P.W. Schmidt, and T.P. Rieker, Scaling description of the structure factor of fractal soot composites, Phys. Rev. E, 56, 4666-4672, 1998.

Strawa, A.W., et al., Carbonaceous aerosol (soot) measured in the lower stratosphere during POLARIS and its role in stratospheric photochemistry, J. Geophys. Res., 10•, 26,753-26,766, 1999.

Strom, J., and S. Ohlsson, In situ measurements of enhanced crystal number densities in cirrus clouds caused by aircraft exhaust, J. Geophys. Res., 103, 11,355-11,361, 1998.

Tabor, K., L. Gutzwiller, and M.J. Rossi, The heterogeneous interaction of NO•. with amorphous carbon at ambient temperatures, J. Phys. Chem., 98, 6172-6186, 1994.

Veerapaneni, S., and M.R. Wiesner, Hydrodynamics of frac- tal aggregates with radially varying permeability, J. Col- loid Interface $ci., 177, 45-57, 1996.

Weingartner, E., H. Burtscher, and U. Baltensperger, Hy- groscopic properties of carbon and diesel soot particles, Atmos. Environ., 31, 2311-232.7, 1997.

P.M. Adler and D. Coelho, Institut de Physique du Globe de Paris, Tour 24, 4 Place Jussieu, F-75252 Paris, France. (e-mail: adler@n32.rhea.cnusc.fr)

S. Bekki (corresponding author) and C. David, Service d'A•ronomie du Centre National de la Recherche Scien- tifique, Universit• P. et M. Curie, Paris, Boite 102, Couloir 15-14, 5 •'• etage, 4 place Jussieu, 75230 Paris Cedex 05, France. (e-mail: slimane@aero.jussieu.fr)

K.S. Law, Centre For Atmospheric Science, Depart- ment of Chemistry, University of Cambridge, Lens- field Road, Cambridge, CB2 1EW, England. (e-mail: kathy@ atm.ch.cam.ac. uk)

D.M. Smith, Department of Chemistry and Biochemistry, University of Denver, Denver, CO 80208. (e-mail: dwi- smith@du.edu)

J.-F. Thorerr, Laboratoire des Ph•nom[nes de Trans- port dans les M•langes (LPTM), SP2MI, BP 179, 86960- Fnturoscope Cedex, France. (e-mail: jft@io.cnusc.fr)

(Received April 7, 1999; revised July 27, 1999; accepted August 5, 1999.)