JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 107, NO. D19, 4366, doi:10.1029/2001JD002044, 2002
Analysis of aerosol interactions with numerical techniques for solving coagulation, nucleation, condensation, dissolution, and reversible chemistry among multiple size distributions Mark Z. Jacobson Department of Civil and Environmental Engineering, Stanford University, Stanford, California, USA Received 28 December 2001; revised 11 March 2002; accepted 13 March 2002; published 2 October 2002.
[1] This paper describes new and modified numerical techniques for solving the size- and time-dependent aerosol processes of nucleation, coagulation, condensation, dissolution, and reversible chemistry among multiple aerosol particle size distributions and the gas phase and over the entire relative humidity (RH) range. The techniques treat particle number, mole, and volume concentrations, solution and nonsolution densities, and refractive index consistently. Some findings include the following: (1) Coagulation internally mixes particles of different original composition over the entire size distribution. (2) Coagulation internally mixes a greater fraction of larger than smaller particles. (3) Coagulation internally mixes larger particles with more other distributions than it does smaller particles. (4) Coagulation among multiple distributions produces the same summed size distribution as coagulation of a single distribution when the sum of initial distributions is the same in both cases. (5) In a competition for vapor between homogeneous nucleation and condensation, the relative importance of condensation increases with an increasing number of background particles. (6) In the absence of a continuous source of new particles, coagulation, condensation, dissolution, hydration, and chemical reaction internally mix most particles within half a day under moderately polluted conditions. (7) Condensation increases the fractional coating of small more than large particles. (8) The real refractive index of a particle containing electrolytes is higher at low RHs than at high RHs. (9) The difference between total particle and solution real refractive indices increases with decreasing RH. (10) A solution real refractive index generally increases with decreasing particle size. INDEX TERMS: 0305 Atmospheric Composition and Structure: Aerosols and particles (0345, 4801); 0345 Atmospheric Composition and Structure: Pollution—urban and regional (0305); 0360 Atmospheric Composition and Structure: Transmission and scattering of radiation; 0365 Atmospheric Composition and Structure: Troposphere—composition and chemistry; 1610 Global Change: Atmopshere (0315, 0325); KEYWORDS: aerosols, coagulation, condensation, nucleation, equilibrium, internal mixtures
Citation: Jacobson, M. Z., Analysis of aerosol interactions with numerical techniques for solving coagulation, nucleation, condensation, dissolution, and reversible chemistry among multiple size distributions, J. Geophys. Res., 107(D19), 4366, doi:10.1029/2001JD002044, 2002.
1. Introduction 2000; Bott, 2000; Jeong and Choi, 2001; Sandu, 2002]. Many studies have also examined solutions to condensa- [2] Many studies to date have described numerical tech- tional growth equations [e.g., Mordy, 1959; Middleton and niques for solving the differential coagulation equation Brock, 1976; Gelbard and Seinfeld, 1980; Varoglu and [e.g., Lushnikov, 1975; Turco et al., 1979; Suck and Brock, Finn, 1980; Smolarkiewicz, 1983; Friedlander, 1983; Tsang 1979; Gelbard and Seinfeld, 1980; Tsang and Brock, 1982; and Brock, 1986; Whitby, 1985; Brock et al., 1986; Tsang Seigneur, 1982; Friedlander, 1983; Warren and Seinfeld, and Korgaonkar, 1987; Toon et al., 1988; Rao and 1985; Tzivion et al., 1987; Toon et al., 1988; Strom et al., McMurry, 1989; Tsang and Huang, 1990; Gelbard, 1990; 1992; Kim and Seinfeld, 1992; Jacobson et al., 1994; Pilinis, 1990; Kim and Seinfeld, 1990, 1992; Jacobson and Kostoglou and Karabelas, 1994; Jacobson and Turco, Turco, 1995; Lister et al., 1995; Binkowski and Shankar, 1995; Binkowski and Shankar, 1995; Kumar and Ramk- 1995; Jacobson, 1997c; Kleeman et al., 1997; Gelbard et rishna, 1996; Fassi-Fihri et al., 1997; Russell and Seinfeld, al., 1998; Chock and Winkler, 2000; Nguyen and Dabdub, 1998; Trautmann and Wanner, 1999; Fernandez-Diaz et al., 2001; Sandu and Borden, 2001]. A third group of studies has examined solutions to equations for nonequilibrium Copyright 2002 by the American Geophysical Union. dissolutional growth plus equilibrium reversible chemistry 0148-0227/02/2001JD002044$09.00 of electrolytes [Jacobson et al., 1996; Meng and Seinfeld,
AAC 2 - 1 AAC 2 - 2 JACOBSON: ANALYSIS OF AEROSOL INTERACTIONS
1996; Jacobson, 1997a, 1997b, 1997c; Meng et al., 1998; imaginary refractive from compositions in multiple distri- Sun and Wexler, 1998a, 1998b; Song and Carmichael, butions are also given. 1999; von Salzen and Schlu¨nzen, 1999; Capaldo et al., 2000; Pilinis et al., 2000]. 2. Particle Size Distributions [3] In most studies to date, one aerosol size distribution has been considered. In some studies, more than one [6] The numerical treatments here assume that particles distribution has been considered [e.g., Toon et al., 1988; can be described by any number of size distributions, each Jacobson et al., 1994; Fassi-Fihri et al., 1997; Kleeman et with any number of size bins and any number of compo- al., 1997; Russell and Seinfeld, 1998; Jacobson, 2001]. One nents within each size bin. Each size bin i in each size reason to treat multiple distributions and interactions among distribution N is characterized by a number concentration, them is that real particles exist in a variety of mixing states nNi (number of particles per cubic centimeter of air). Each [e.g., Andreae et al., 1986; Levin et al., 1996; Murphy et al., component q in each size bin of each distribution is 1998; Po´sfai et al., 1999; Silva et al., 2000; Guazzotti et al., characterized by a mole concentration, cq,Ni (moles per 2001; Okada and Hitzenberger, 2001; Naoe and Okada, cubic centimeter of air). 2001]. Particles near their sources are generally externally [7] For many processes, such as coagulation, single- mixed, whereas those far from their sources are often particle volume information is needed. The calculation of partially or completely internally mixed. When only one single-particle volume takes into account the facts that size distribution is treated in a model, all material is particles contain a solution and nonsolution component considered internally mixed at the emission source. Such and solution density varies as a function of electrolyte a representation can distort predicted chemical composi- concentration. Single-particle volumes (cubic centimeters tions [e.g., Kleeman et al., 1997]. When multiple distri- per particle) are calculated as butions without interactions among them are treated, all distributions are treated as externally mixed. This is vNi uNi ¼ ð1Þ unrealistic away from source regions. When multiple nNi distributions with interactions among them are treated, some distributions can be treated as externally mixed where vNi is the volume concentration (cubic centimeters of whereas others can be treated as partially or completely particle per cubic centimeter of air) of the total particle and internally mixed. The radiative effects of externally and is calculated as internally mixed particles differ. For example, the modeled global direct radiative forcing of black carbon (BC) may XNNS XNNS be about a factor of two higher when BC is treated as an mq internally mixed core surrounded by a shell compared vNi ¼ vs;Ni þ vq;Ni ¼ vs;Ni þ cq;Ni ð2Þ rq with when it is treated as externally mixed [Jacobson, q¼1 q¼1 2000]. Simulations of the global scale mixing state and resulting direct forcing of BC (which used the numerical where vs,Ni is the volume concentration of the solution, NNS techniques and size distributions treated here) suggest that is the number of nonsolution components, vq,Ni = mqcq,Ni/rq both may fall between those of an external and internal is the volume concentration of each nonsolution component, mixture but closer to those of an internal mixture [Jacob- mq is the molecular weight (grams per mole) of component son, 2001]. q, and rq is the mass density (grams per cubic centimeter) of [4] In this paper, techniques for solving (1) coagulation, component q. (2) nucleation/condensational growth simultaneously, and [8] The volume concentration of a solution is calculated as (3) coupled nonequilibrium dissolutional growth/equili- brium reversible chemistry of electrolytes, all between the ms;Ni vs;Ni ¼ ð3Þ gas phase and multiple aerosol size distributions, are given. rs;Ni All techniques conserve moles or volume (including between the gas and aerosol phases), all techniques except where the reversible chemistry technique are noniterative, and all techniques are unconditionally stable (where a stable sol- XNS ution is defined as one in which the difference between the ms;Ni ¼ mvcw;Ni þ mqcq;Ni ð4Þ numerical and exact solution is bounded regardless of the q¼1 time step or integration time [e.g., Celia and Gray, 1992, p. 216]). The techniques make use of a consistent treatment of is the mass concentration (grams per cubic centimeter of air) particle number concentration, mole concentration, volume of the solution (mv is the molecular weight of water and concentration, solution density, nonsolution density, and cw,Ni is the mole concentration of liquid water) and refractive index. [5] In the following sections, the numerical techniques 1 r ¼ ð5Þ are described. The techniques are then analyzed in box s;Ni N PE c model simulations to explore physical processes, chemical q;Ni ra;q;Ni processes, and interactions affecting aerosol particle evolu- q¼1 tion. In one case, results are compared when a different method of treating the aerosol size distribution is consid- is the mass density (grams per cubic centimeter of solution) ered. Equations for the calculation of particle real and of the solution [e.g., Tang, 1997]. In equations (4) and (5), JACOBSON: ANALYSIS OF AEROSOL INTERACTIONS AAC 2 - 3
Table 1a. Components in and Sources of New Particles in Each Size Distribution Distribution Name Symbol Potential Components Initialized? Sources of New Particles Sea spray A All in (b) except BC, OM, and soil Yes Ema Soil B All except BC, OM, and species with Na, Ca, Mg, K Yes Em Sulfate D All except BC, OM, soil, and species with Na, Ca, Mg, K Yes Nuc BC with 0–5% shell E1 All except OM, soil, and species with Na, Ca, Mg, K Yes Em BC with 5–20% shell E2 All except OM, soil, and species with Na, Ca, Mg, K – Gr from E1 BC with >20% shell E3 All except OM, soil, and species with Na, Ca, Mg, K – Gr from E1 and E2 OM F All except BC, soil, and species with Na, Ca, Mg, K Yes Em Sea spray–soil AB All except BC and OM – Co Sea spray–sulfate AD All except BC, OM, soil – Co Sea spray–BC AE All except OM and soil – Co Sea spray–OM AF All except BC and soil – Co Soil–sulfate BD All except BC, OM, and species with Na, Ca, Mg, K – Co Soil–BC BE All except OM, and species with Na, Ca, Mg, K – Co Soil–OM BF All except BC and species with Na, Ca, Mg, K – Co Sulfate–BC DE All except OM, soil, and species with Na, Ca, Mg, K – Co Sulfate–OM DF All except BC, soil, and species with Na, Ca, Mg, K – Co BC–OM EF All except soil, and species with Na, Ca, Mg, K – Co Internal mixture MX All – Co a Em = emissions, Nuc = homogeneous nucleation, Co = coagulation, Gr = condensational and/or dissolutional growth. All distributions are affected by self-coagulation loss to larger sizes and heterocoagulation loss to other distributions (except the MX distribution, which has no heterocoagulation loss). In addition, particles in all distributions can grow in size by condensation and dissolution, reversible equilibrium chemistry, and equilibrium hydration. The difference between distributions A and AD is that the only source of new A particles is emissions, whereas that of new AD particles is coagulation of A with D, A with AD, or D with AD particles. The differences among distributions E1, E2, E3, and DE are that the only source of new E1 particles is emissions, the only sources of E2 and E3 particles are growth from E1 and E2, respectively, and the only source of new DE particles is coagulation of E1, E2, or E3 with D or DE and coagulation of D with DE. Each size distribution contains multiple size bins.
NS is the number of electrolytes in the solution. In equation during growth, all particles in the bin are moved to distribu- (5), tion E2, and so on. This allows distribution E1 to always contain relatively externally mixed BC. All distributions mqcq;Ni (including E1, E2, and E3) are affected by self-coagulation cq;Ni ¼ ð6Þ ms;Ni loss to larger sizes and heterocoagulation loss to other distributions (except for the MX distribution, which has no is the mass fraction of component q in solution and ra, q, Ni heterocoagulation loss). Particles in all distributions can is the mass density of electrolyte q as if it were alone in grow by condensation, dissolution, and equilibrium hydra- solution at the same weight percent as the current weight tion, and can be affected by chemistry. percent of total solute. Density polynomials for several [11] Each size bin in each size distribution is character- electrolytes are available in the work of Tang [1997]. ized by a high- and low-edge volume, which equal a [9] The numerical methods described here assume that constant, Vrat, multiplied by the high- and low-edge volume, each distribution can be affected by nucleation, emissions, respectively, of the next-smallest bin. One way to initialize coagulation, condensation, dissolution, chemical equili- the high- and low-edge volume of a bin is given by brium, transport, and removal (although emissions, trans- Jacobson [1999a, equations (14.6) and (14.7)]. Other ways port, and removal are not simulated here). Table 1a shows a are also possible. All particles in the bin, though, have a set of 18 size distributions used here for all the simulations single characteristic volume (uNi) between the low- and discussed. Table 1b lists the components treated in each high-edge volumes of the bin. This single volume can distribution of Table 1a, except as specified in Table 1a. For change as the particles grow, evaporate, coagulate, etc. If each distribution, 60 size bins, ranging from 0.001 to 120 mm in radius, are used. In sum, this study considers 32,400 concentrations in a single spatial grid cell (18 size distribu- tions  an average of 30 species per distribution  60 size Table 1b. Aerosol Components in Each Distribution of Table 1a, bins per species per distribution). Except as Specified in That Table [10] Table 1a identifies the sources of new particles in each Species Formula Species Formula Species Formula 2À distribution. For four distributions (A, B, E1, and F), the only H2O(aq) CO3 KHSO4(s) source of new particles is emissions. For one distribution (D), H2CO3(aq) NH4NO3(s) K2SO4(s) H2SO4(aq) NH4Cl(s) K2CO3(s) the only source of new particles is homogenous nucleation. + H NH4HSO4(s) Ca(NO3)2(s) For 12 distributions (AB, AD, AE, AF, BD, BE, BF, DE, DF, + NH4 (NH4)2SO4(s) CaCl2(s) EF, and MX), the only new-particle source is coagulation + Na (NH4)3H(SO4)2(s) CaSO4(s) 2+ between simpler distributions. For two distributions (E2 and Mg (NH4)2CO3(s) CaCO3(s) 2+ E3), the only new-particle source is condensation onto E1 or Ca NaNO3(s) Mg(NO3)2(s) + K NaCl(s) MgCl2(s) E2, respectively. Distributions E1, E2, and E3 contain black À HSO4 NaHSO4(s) MgSO4(s) carbon (BC) with different volume fractional coatings (0– 2À SO4 Na2SO4(s) MgCO3(s) À 5%, 5–20%, and >20%, respectively). BC is emitted into NO3 Na2CO3(s) Soil À (and initialized in) distribution E1. When the fractional Cl KNO3(s) BC À coating of E1 particles in a size bin increases above 5% HCO3 KCl(s) OM AAC 2 - 4 JACOBSON: ANALYSIS OF AEROSOL INTERACTIONS the volume increases above the high-edge volume of the where h is the time step size, t À h indicates a value at the bin, all particles in the bin are moved to and averaged with beginning of the time step, NT is the total number of size particles in the next-largest bin. This size bin structure is the distributions (18 in Tables 1a and 1b), NB is the number of moving-center structure [Jacobson, 1997a, 1999a, pp. 420– size bins in each distribution, bNi,Mj,tÀh is the coagulation 4201]. Results from it have been compared with other size rate coefficient (cubic centimeters per particle per second) structures by Jacobson [1997a] and Zhang et al. [1999] and between a particle in size bin i of distribution N and a are compared further here with those from another structure, particle in bin j of distribution M, and fIi,Mj,Nk,tÀh is the the full-moving structure. fraction of the summed single-particle volume VIi,Mj,tÀh = uIi,tÀh + uMj,tÀh partitioned into bin k of distribution N at the 3. Coagulation beginning of the time step. This formulation allows the average single-particle volume (u) of a particle in a size bin [12] The coagulation scheme described here is an exten- to change each time step, which is necessary for the sion of an earlier volume-conserving, noniterative, uncondi- moving-center size structure, in which the single-particle tionally stable coagulation scheme in which multiple size volume in a bin can vary between the low- and high-edge distributions were considered [Jacobson et al., 1994]. When volume in the bin. For a size bin structure in which single- external mixtures coagulated in the original scheme, they particle volume does not change in time (e.g., a stationary were sent into a single internally mixed distribution. For this structure), the volume fractions are calculated only once, at work, the scheme was modified to treat coagulation into and the beginning of the simulation. In either case, volume among any number of externally mixed, binary, ternary, etc. fractions are calculated as combinations of distributions and any number of components fIi;Mj;Nk;tÀh ¼ in each distribution (maintaining volume conservation). In 8 > uNkþ1;tÀhÀVIi;Mj;tÀh uNk;tÀh the work of Jacobson et al. [1994], results from the scheme > uNk;t h VIi;Mj;t h < uNk 1;t h k < NB < uNkþ1;tÀhÀuNk;tÀh VIi;Mj;tÀh À À þ À were compared with analytical and numerical solutions for 1 À fIi;Mj;NkÀ1;tÀh uNk;tÀh
Cq;t ¼ stable because the difference between the numerical and exact solution is bounded between 0 and Ctot, regardless of hi PNT PNB hS0 k 0 q;Ni;tÀh q;Ni;tÀh the time step or integration time. Cq;tÀh þ h ð1 À DNiÞðkq;Ni;tÀhSq;Ni;tÀhCq;s;Ni;tÀhÞþDNicq;Ni;tÀh 1 À exp H 0 N 1i 1 q;Ni;tÀh ¼ ¼ nonohi [24] When HNO3 or HCl grows onto a size bin and a PNT PNB 0 0 Hq;Ni;tÀh hSq;Ni;tÀh kq;Ni;tÀh 1 þ h ð1 À DNiÞkq;Ni;tÀh þ DNi 0 1 À exp 0 solid exists in the bin, condensational growth terms are Sq;Ni;tÀh Hq;Ni;tÀh N¼1 i¼1 needed in equations (13)–(17). Such terms require a satu- ration vapor mole concentration. For HNO3, the SVMC is ð16Þ calculated as AAC 2 - 8 JACOBSON: ANALYSIS OF AEROSOL INTERACTIONS
cHNO3;s;Ni ¼ min 2þ À 8 9 MgðNO3Þ2ðsÞ¼Mg þ 2NO3 ð27Þ > c þ c À 2 > >c À 2 H ; Ni NO ; Nig þ > > NO ; Ni 1 1000 3 Ni;H =NOÀ > > 3 ¼ 3 > > 0 * > The corresponding equilibrium coefficient expressions for > H NOÀ R T mvcw; Ni Keq;HNO3 > >3 > these reactions are > 2 > > 1 Ks; NH4NO3 > > ; > > * > 2 2 > R T CNH3 > > > 1000 þ mol > c þ > c ; c À ; 2 þ À ¼ K > 2 > NH4 Ni NO3 Nig Ni;NH =NO eq;NH4NO3 2 > H ; Nig þ À > mvcw;Ni 4 3 kg > 1 Ni;H =NO3 Keq;NH4NO3 > > > > * K c þ 2 > > R T eq;HNO3 NH ; Nig þ À > ð28Þ > 4 Ni;NH =NO3 > > 4 > > c þ 2 > > H ; Nig þ > > 1 Ni;H =NOÀ K > < 3 eq;NaNO3 ; = * c þ 2 R T Keq;HNO3 Na ; Nig þ 2 > Ni;Na =NOÀ > 1000 mol2 > 3 > þ 2 > þ > c ;Ni c À 2 þ À ¼ Keq;NaNO3 ð29Þ > c 2 > Na NO3 ;Nig Ni;Na =NO3 2 > H ; Nig þ > mvcw;Ni kg > 1 Ni;H =NOÀ Keq;KNO > > 3 3 ; > > c þ > > * Keq;HNO 2 > > R T 3 K ; Nig þ À > > Ni;K =NO3 > > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > c þ 2 u > > H ; Nig þ > 2 > 1 Ni;H =NOÀ u 1000 Keq;Ca NO > 2 > 3 t ð 3Þ2 > 1000 þ mol > ; > ; À 2 > * c 2 3 > cK Ni cNO ;NigNi;Kþ=NOÀ ¼ Keq;KNO3 2 ð30Þ > R T Keq;HNO3 mvcw; Ni Ca ; Nig > 3 3 > ; 2þ = À > mvcw;Ni kg > Ni Ca NO3 > > vffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi > > c þ 2 u > > H ; Nig þ À u > > 1 Ni;H =NO 1000 Keq;MgðNO3Þ > > 3 t 2 ; > :> * K m c ; Ni c 2þ 3 ;> R T eq;HNO3 v w Mg ; Nig 2þ À 3 3 Ni;Mg =NO3 1000 2 mol c 2þ ; c À 2þ 3 ¼ K Þ Ca Ni NO ;NigNi;Ca =NO3 eq;CaðNO3 2 3 ð18Þ mvcw;Ni 3 kg ð31Þ which states that the SVMC is the minimum SVMC arising from several processes. All terms in equation (18) are evaluated at the beginning of a growth time step. The use of 3 3 1000 2 mol 2þ 3 equation (18) allows more nitrate mass to transfer to the cMg ;Ni c NOÀ;NigNi;Mg2þ=NO ¼ Keq;MgðNO3 Þ2 m c 3 3 kg3 aerosol phase when a solid is present than the APD scheme v w;Ni originally allowed when only solubility was considered. ð32Þ The first process in equation (18) is dissolution by the reaction, respectively. Coupling these equations with equation (20) þ À and solving for CHNO ,s,Ni gives the SVMC terms in HNO ðgÞ¼H þ NO ð19Þ 3 3 3 equation (18). The SVMCs for HCl are calculated in a similar way. which has the equilibrium coefficient expression, [25] When homogeneous nucleation is coupled with growth, such as in the case of binary homogeneous nucle- 2 þ c À 1000 cH ; Ni NO3 ; Ni ation/condensation of S(VI), the nucleation rate is first mvcw;Ni C R*T converted to a mass transfer rate between the gas and HNO3;s;Ni 2 2 mol aerosol and added to the growth mass transfer rate in the Á gNi;Hþ=NOÀ ¼ Keq;HNO3 ð20Þ 3 kg2 À atm first size bin. For example, if the binary homogeneous
nucleation rate of S(VI)–water particles is RS(VI) – H2O (new particles per cubic centimeter per second), and if this Solving equation (20) for CHNO ,s,Ni gives the SVMC term 3 quantity exceeds zero (and if the initial partial pressure of in equation (18). The second process is the solid–gas 0 reaction S(VI), CS(VI),tÀh, exceeds S S(VI),N1,tÀhCS(VI),s,N1,tÀh),
RS(VI) – H2O is converted to a gas-phase mass transfer rate for nucleation (sÀ1) with NH4NO3ðsÞ¼NH3ðgÞþHNO3ðgÞð21Þ
which has an equilibrium coefficient expression, rSðVIÞuN1FSðVIÞ kSðVIÞ; nuc; N1;tÀh ¼ RSðVIÞÀH2O 2 ÀÁ m * 2 SðVIÞ R T cNH3;NiCHNO3 ¼ Ks;NH4 NO3 atm ð22Þ ! 1 Á 0 ð33Þ All remaining processes are solid–ion reactions coupled CSðVIÞ;tÀh À S CSðVIÞ;s;N1;tÀh + À SðVIÞ;N1;tÀh with the gas–ion reaction HNO3(g) = H +NO3 . The five solid–ion reactions are
þ À À3 NH4NO3ðsÞ¼NH4 þ NO3 ð23Þ where rS(VI) is the density of pure S(VI) (g cm ), uN1 is the þ À single-particle volume of a nucleated particle (cubic NaNO3ðsÞ¼Na þ NO3 ð24Þ centimeters per particle), FS(VI) is the volume fraction of þ À KNO3ðsÞ¼K þ NO3 ð25Þ S(VI) in an S(VI)–H2O nucleated mixture, and mS(VI) is the molecular weight (grams per mole) of S(VI). The denomi- 2þ À CaðNO3Þ2ðsÞ¼Ca þ 2NO3 ð26Þ nator on the right side of equation (33) is the maximum gas JACOBSON: ANALYSIS OF AEROSOL INTERACTIONS AAC 2 - 9 0 1 0 1 available for nucleation at the beginning of the time step (mol , XNNS XNNS À3 B C B C cm ). The mass transfer rate in equation (33) is then added kshell;Ni;l ¼ @vs;Niks;Ni;l þ vq;Nikq;lA @vs;Ni þ vq;NiA to the growth mass transfer rate (kS(VI),grow,N1,tÀh) for the size q¼1 q¼1 bin in which nucleated particles are placed (assumed to be bin q6¼BC q6¼BC 1 in this equation), to give a total mass transfer rate to this bin ð36Þ of kS(VI),N1,tÀh = kS(VI),grow,N1,tÀh + kS(VI),nuc,N1,tÀh. [26] After equations (13)–(17) are solved with the revised where ns,Ni,l and nq,l are the overall solution and individual growth plus nucleation transfer rate, the new number con- component nonsolution real refractive indices, respectively, centration of S(VI)–H2O particles due to homogeneous and ks,Ni,l and kq,l are overall solution and individual nucleation is calculated from the incremental change in component nonsolution imaginary refractive indices. The mole concentration of S(VI) with solution real refractive index is sffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi nN1;t ¼ nN1;tÀh þ MAX vs;Ni þ 2Rs;Ni;lcs;Ni "#ns;Ni;l ¼ ð37Þ ÀÁ v ; À R ; ; c ; mSðVIÞ kSðVIÞ;nuc;N1;tÀh s Ni s Ni l s Ni cSðVIÞ;N1;t À cSðVIÞ;N1;tÀh ; 0 rSðVIÞuN1FSðVIÞ kSðVIÞ;N1;tÀh derived from equation (1) of Stelson [1990], where ð34Þ
XNS Once new particles are added, water, with a volume fraction Rs;Ni;lcs;Ni ¼ RH2O;lcH2O; Ni þ Rq;lcq;Ni ð38Þ of 1 À FS(VI), is added to ensure the total volume added to q¼1 the size distribution is consistent with the number of 3 À1 particles added to the distribution. This water content and is the product of the molar refraction (Rs,Ni,l,cm mol ) À3 the single-particle volume are adjusted subsequently when and mole concentration (cs,Ni, mol cm ) of the solution, 3 À1 equilibrium hydration equations are solved. and Rq,l is the partial molar refraction (cm mol )of [27] The advantage of coupling nucleation with growth individual solution component q (either ion or electrolyte). is that rates for both are fast, and if these processes are Partial molar refractions are given for several ions and operator split, one process will be favored extensively over aqueous electrolytes in the works of Stelson [1990] and the other. Solving both processes together allows for a Tang [1997] at 550 nm. When data are not available, such as more realistic competition among size bins for the limited at solar- and thermal-IR wavelengths and for certain species, amount of sulfuric acid gas available for nucleation and partial molar refractions for aqueous electrolytes are condensation. estimated by combining all ions in solution into hypothe- 28 [ ] The nucleation rate, RS(VI)ÀH2O, is calculated with tical aqueous electrolytes, assuming the refractive index of classical theory in the simulations discussed below but the aqueous electrolyte equals that of the corresponding could also be calculated with a parameterization. Zhang et solid electrolyte, and applying the equation, al. [1999] compare several parameterizations of nucleation ! rates. 2 mq nq;n À 1 Rq;l ¼ 2 ð39Þ rq nq;n þ 2 5. Aerosol Optical Properties
[29] Aerosol optical depths, single-scattering albedos, where nq,l is the real refractive index, mq is the molecular À1 À3 and asymmetry parameters are determined from particle weight (g mol ), and rq is the density (g cm ) of the solid scattering, absorption, and forward-scattering efficiencies electrolyte. The assumption that the partial molar refraction from Mie theory calculations under the assumption that of an aqueous electrolyte is similar to that of the when a particle contains BC, BC is core material and all corresponding solid electrolyte is substantiated by the work other material is shell material. Shell real and imaginary of Stelson [1990, Figure 1]. refractive indices for a given particle size and wavelength [31] The solution imaginary refractive index is estimated are obtained by first calculating solution and nonsolution here with refractive indices of all non-BC substances in the shell, then volume averaging the solution and nonsolution refractive As;Ni;lcs;Ni ks;Ni;l ¼ ð40Þ indices to obtain average shell refractive indices. These vs;Ni refractive indices are then used in a core-shell Mie theory calculation in which BC is core [e.g., Toon and Ackerman, where 1981].
[30] The volume-averaged shell real and imaginary XNS refractive indices for a given wavelength l and particle size As;Ni;lcs;Ni ¼ AH2O;lcH2O;Ni þ Aq;lcq;Ni ð41Þ i in distribution N are calculated as q¼1 0 1 0 1 , 3 À1 is the product of the molar absorption (As,Ni,l,cm mol ) B XNNS C B XNNS C @ A @ A and the mole concentration of the solution, and Aq,l is the nshell;Ni;l ¼ vs;Nins;Ni;l þ vq;Ninq;l vs;i þ vq;Ni 3 À1 q¼1 q¼1 partial molar absorption (cm mol ) of individual solution q6¼BC q6¼BC component (either ion or electrolyte) q. Since partial molar ð35Þ absorption data of ions and aqueous electrolytes are AAC 2 - 10 JACOBSON: ANALYSIS OF AEROSOL INTERACTIONS
106 A (Spray) Initial B (Soil) Initial 4 D (Sulf) Initial
p 10
D E1 (BC<5% shell) Initial
10 F (OM) Initial 102
) / d log 0 -3 10
10-2 Number concentration
dN (No. cm of each distribution 10-4
10-6 0.01 0.1 1 10 100 Particle diameter (µm)
Figure 1. Initial number concentrations of five externally mixed size distributions used for all simulations. unavailable, such quantities are estimated by combining all and mass concentrations, summed over each distribution, ions in solution into hypothetical electrolytes, assuming the were as follows: A, sea spray: 502 cmÀ3, 54.6 mgmÀ3 (84% partial molar absorptions of aqueous electrolytes equal those of which was water); B, soil: 6.1 cmÀ3; 0.72 mgmÀ3;C, of solid electrolytes, then applying the equation, sulfate: 1192 cmÀ3;11.2mgmÀ3 (82% of which was water); E1, BC < 5% shell: 90 cmÀ3, 0.25 mgmÀ3; F, primary À3 À3 mq organic matter (OM): 337 cm , 0.75 mgm . These initial Aq;l ¼ kq;l ð42Þ rq conditions represent sea spray particles interacting with low to moderate loadings of soil, sulfate, BC, and OM at a high where kq,l is the imaginary refractive index of solid RH. The initial number and mass concentrations, summed electrolyte q. Equations (40)–(42) result in a imaginary over all distributions, were 2127 cmÀ3 and 67.5 mgmÀ3 index mixing rule for aerosol solutions that is analogous to a (82% of which was water), respectively. volume-averaged mixing rule, except that here, the imaginary index is scaled by the ratio of the solution 6.1. Coagulation Alone density to the weighted sum of the densities of the [33] In the first simulation, coagulation alone was simu- individual electrolytes and water in solution. Sources of lated for a 12 hour period among the five externally mixed wavelength-dependent imaginary refractive indices for distributions A, B, D, E1, and F to produce larger particles in some species treated here include the following: liquid the same distributions, all possible binary mixtures of dis- water [Hale and Querry, 1973], organic compounds and tributions (e.g., AB, AD, AE, ...), and higher mixtures (MX). black carbon [Krekov, 1993], organic compounds and Distributions E2 and E3 were not needed in this simulation, ammonium nitrate in the UV [Jacobson, 1999c], sodium since no growth onto E1 particles was considered. chloride and ammonium sulfate [Toon et al., 1976], calcium [34] Figure 2a compares the initial and final particle sulfate and potassium chloride [Query, 1987], and sulfuric number concentrations, summed over all distributions, result- acid [Palmer and Williams, 1975]. ing from the simulation. The figure appears to indicate, as past coagulation simulations of one size distribution have 6. 0-D Simulations of the Aerosol/Radiative indicated [e.g., Jacobson, 1997a, Figures 2 and 3], that coagulation among multiple distributions has little effect on Algorithms the total number concentration of particles larger than 0.2 mm [32] Here, the processes described above are analyzed in in diameter over a 12 hour period. Figure 2b clarifies this 0-D case studies. In all cases, except for one, the simula- misconception. It shows that coagulation among multiple tions were run with the moving-center size structure. In all distributions internally mixes particles of different original cases, the distributions and species considered were those in compositions, affecting the composition of the entire size Tables 1a and 1b. In all cases, five distributions (A, B, D, distribution without affecting the number concentration of E1, and F) were initialized. Except where specified, the RH large particles summed over all distributions. was 90% and the temperature was T = 285 K. Initial [35] The results shown in Figure 2b suggest that in an air distributions are shown in Figure 1. The initial number mass containing a moderate loading of particulates, coagu-
Figure 2. (opposite) Coagulation alone. Results of simulation in which the five externally mixed distributions from Figure 1 were coagulated over 12 hours to produce 16 externally and internally mixed distributions. (a) The initial and final number concentrations summed over all distributions. (b) The final number concentrations by distribution. (c–e) The initial and final À 2À mass concentrations of (c) S(VI), (d) BC, and (e) soil in each distribution. S(VI) = H2SO4 (aq) + HSO4 +SO4 . 4 10
p 10 10 A (Spray) 12 hr AF (Spray-OM) 12 hr
D Initial 2 10 10 12 hr B (Soil) 12 hr BD (Soil-sulf) 12 hr 108 D (Sulf) 12 hr BE (Soil-BC) 12 hr 100 E1 (BC<5% shell) 12 hr BF (Soil-OM) 12 hr
) / d log F (OM) 12 hr DE (Sulf-BC) 12 hr
p 6
-3 10 -2 10 D AB (Spray-soil) 12 hr DF (Sulf-OM) 12 hr
10 AD (Spray-sulf) 12 hr EF (BC-OM) 12 hr (a) Number concentration 4 10-4 summed over all distributions 10 AE (Spray-BC) 12 hr MX 12 hr ) / d log dN (No. cm -6 2 (b) Number concentration 10 -3 10 0.01 0.1 1 10 100 of each distribution Particle diameter (µm) 100 dN (No. cm 10-2 INTERACTIONS AEROSOL OF ANALYSIS JACOBSON:
10-4
10-6 0.01 0.1 1 10 100 Particle diameter (µm) 105 105 A (Spray) Initial AD (Spray-BC) 12 hr E1 (BC<5% shell) Initial DE (Sulf-BC) 12 hr
D (Sulf) Initial AF (Spray-OM) 12 hr p p 3 3 10 E1 (BC<5% shell) 12 hr EF (BC-OM) 12 hr A (Spray) 12 hr BD (Soil-sulf) 12 hr D D 10
10 AE (Spray-BC) 12 hr MX 12 hr
10 D (Sulf) 12 hr DE (Sulf-BC) 12 hr AB (Spray-soil) 12 hr DF (Sulf-OM) 12 hr 101 BE (Soil-BC) 12 hr 101 AD (Spray-sulf) 12 hr MX 12 hr
) / d log -1 ) / d log
-3 10 (d) BC by -3 -1 (c) S(VI) by distribution
10 g m
g m distribution -3 µ
µ 10
-3 dM ( dM ( 10 10-5
-7 10-5 10 0.01 0.1 1 10 100 0.01 0.1 1 10 100 µ Particle diameter (µm) Particle diameter ( m)
105 B (Soil) Initial BE (Soil-BC) 12 hr
p 103 B (Soil) 12 hr BF (Soil-OM) 12 hr D AB (Spray-soil) 12 hr MX 12 hr 10 101 BD (Soil-sulf) 12 hr
) / d log 10-1 -3 A 2 AAC (e) Soil by
g m -3
µ 10 distribution
dM ( 10-5
-7
10 - 0.01 0.1 1 10 100 Particle diameter (µm) 11 AAC 2 - 12 JACOBSON: ANALYSIS OF AEROSOL INTERACTIONS lation alone can internally mix most particles to some degree particles cmÀ3). Doubling the initial number of particles within 12 hours. In the simulation, coagulation internally decreased the number of nucleated particles by more than mixed almost all the largest particles and a smaller percent of one half. Thus, as the number concentration of existing the smaller particles, which may explain a result of Okada particles increased, the more condensation of S(VI) was and Hitzenberger [2001], who found that the number fraction favored over homogeneous nucleation. Although most new of mixed particles in Vienna increased with increasing sulfuric acid water particles formed at a diameter near 0.002 particle radius and with increasing particle abundance. mm, condensation onto these particles caused them to grow [36] Not only did coagulation cause a greater fraction of to around 0.008 mm, as illustrated in Figures 3a and 3b. larger particles than smaller particles to mix internally, but it [41] Figure 3c shows that growth of S(VI) caused most also caused mixtures of larger particles to obtain more small BC particles to obtain a shell >20% by volume and components than mixtures of smaller particles. Since the most midsize BC particles to obtain a shell 5–20% by combination of even the smallest with the largest particle is volume after 12 hours. Only larger BC particles avoided considered a mixture, though, an internally mixed large significant growth of S(VI). This result suggests that con- particle can often contain a trivial amount of a second and/ densation increases the fractional coating of small particles or third component. more than large particles. Condensation similarly affected [37] Of the total reduction in particle number concentration primarily small OM particles (Figure 3a), soil particles during the 12 hour simulation, an estimated 97.283% was due (Figures 3a and 3d), and sea spray particles (Figure 3a). to Brownian motion, 2.716% was due to Brownian convec- tive diffusion enhancement, 0.000984% was due to turbulent 6.3. Nucleation, Condensation, and Coagulation inertial motion plus turbulent shear, and 0.000089% was due [42] Figure 4 shows results when coagulation was added to gravitational collection. Aside from Brownian motion, to the simulation of Figure 3, which accounted for homo- convective diffusion enhancement had the greatest effect on geneous nucleation and condensation of S(VI). Figure 4a coagulating small particles with large particles. shows that coagulation reduced the distinct nucleation- [38] Figures 2c–2e show the fate of S(VI), BC, and soil mode number concentration peak, obtained when coagula- mass concentration by distribution resulting from the sim- tion was ignored, by a factor of 40 over the 12 hour period, 5 À3 ulation. Because S(VI) was initially in two distributions (A from 1.2 Â 10 particles cm in Figure 3 to 3000 particles À3 and D) whereas BC and soil were initially in one, coagu- cm in Figure 4. At the same time, coagulation reduced the lation spread S(VI) among more distributions than it spread number concentration of particles >0.3 mm in diameter by À3 the other components. Figure 2d shows that, although BC only 12%, from 86 to 76 particles cm . did not exist in large particles initially, coagulation of BC [43] The large reduction in sulfate particle number con- with large particles from other distributions put BC into centration when coagulation was included versus when it such particles. Most of the largest particles that BC entered was not suggests that heterocoagulation during S(VI) nucle- contained components from at least two other distributions ation might have caused significant internal mixing of to form mixed (MX) particles. Figure 2e shows that most sulfate with other distributions. This can be verified by large soil particles, which existed initially, coagulated with comparing Figure 4 with Figure 2 (the coagulation-only (most likely) small particles from at least two other distri- case). Whereas coagulation alone permitted a large number butions to form mixed particles. of BC and soil particles to remain externally mixed (Figures 2c and 2d), coagulation of newly nucleated S(VI) particles 6.2. Nucleation and Condensation Alone internally mixed almost all BC and soil particles (Figures 4c and 4d), producing sulfate–BC particles (DE), soil–sulfate [39] Figure 3 shows results when condensation of 1 mg mÀ3 of gas-phase S(VI) (as sulfuric acid) was grown particles (BD), and higher mixtures (MX). This internal simultaneously onto the five particle size distributions mixing was due primarily to coagulation rather than growth initialized in Figure 1. Homogeneous nucleation of S(VI) because the only source of BD, DE, and MX particles is was solved simultaneously in distribution D. All nucleated coagulation. Growth of S(VI) onto primary BC particles particles were assumed to enter a bin the size of their critical (E1) produces E2 and E3 particles. E1, E2, and E3 particles radius and to grow in competition with other particles. all coagulate with S(VI) (distribution D) to form DE Coagulation was ignored in this simulation. The RH was particles. Since the concentrations of E1, E2, and E3 90% and the temperature was 285 K. particles were negligible at the end of this simulation, internal mixing of BC must have been due primarily to [40] Figures 3a and 3b show that nucleation and growth modified the number and mass concentrations of the sulfate coagulation, although some of it was due to growth that distribution (D) the most. Including nucleation in this case created E2 and E3 particles, which subsequently coagulated. allowed the production of 1.2 Â 105 new particles per cubic centimeter during the 12 hour simulation. In the absence of 6.4. Nucleation, Condensation, and competitive growth, homogeneous nucleation of 1 mgmÀ3 Dissolution/Chemistry 8 À3 of S(VI) would have produced over 1 Â 10 particles cm [44] Figure 5 shows results when homogeneous nuclea- at the critical radius, so solving growth simultaneously with tion and condensation of S(VI), dissolution of HCl, HNO3, nucleation and conserving mass between the gas and aerosol and NH3, internal particle chemical equilibrium and equili- phases curtailed the runaway particle production that would brium hydration were solved between the gas phase and all have been predicted if an explicit calculation were per- particle size distributions. The initial concentrations of gas- formed. Homogeneous nucleation was not swamped out by phase S(VI), HCl, HNO3, and NH3 were 1, 0, 0.8, and 0.3 growth during the simulation because the initial number mgmÀ3, respectively. The RH was 90%, and T = 285 K. concentration of all particles together was not large (2127 Simulations with each the moving-center and full-moving JACOBSON: ANALYSIS OF AEROSOL INTERACTIONS AAC 2 - 13
108 (a) Number concentration A (Spray) 12 hr of each distribution B (Soil) 12 hr 6 D (Sulf) 12 hr p 10 E1 (BC<5% shell) 12 hr D E2 (BC 5-20% shell) 12 hr 10 4 10 E3 (BC>20% shell) 12 hr F (OM) 12 hr 2
) / d log 10 -3 100
10-2 dN (No. cm
10-4
10-6 0.01 0.1 1 10 100 Particle diameter (µm)
105 A (Spray) Initial E1 (BC>5% shell) Initial
p D (Sulf) Initial E2 (BC 5-20% shell) 12 hr 103 D A (Spray) 12 hr E3 (BC>20% shell) 12 hr 10 B (Soil) 12 hr F (OM) 12 hr 101 D (Sulf) 12 hr
) / d log 10-1 -3 (b) S(VI) by distribution g m -3
µ 10
dM ( 10-5
10-7 0.01 0.1 1 10 100 Particle diameter (µm)
103 103 E1 (BC<5% shell) Initial A (Soil) Initial p
E1 (BC<5% shell) 12 hr p 1 1 A (Soil) 12 hr D 10 E2 (BC 5-20% shell) 12 hr D 10 10 E3 (BC>20% shell) 12 hr 10 10-1 10-1 ) / d log ) / d log
-3 (c) BC by -3 10-3 distribution 10-3 g m
g m (d) Soil by µ µ distribution -5 -5 dM ( 10 dM ( 10
10-7 10-7 0.01 0.1 1 10 100 0.01 0.1 1 10 100 Particle diameter (µm) Particle diameter (µm)
Figure 3. Nucleation and condensation alone. (a) Number concentrations of seven size distributions after 12 hours of nucleation and condensation of S(VI) onto the five distributions shown in Figure 1. The two extra distributions, E2 and E3, resulted by growing S(VI) onto distribution E1, as described in the text. Condensation was solved among all seven distributions and the gas phase. Nucleation was solved simultaneously, but only between the gas phase and distribution D. The initial gas-phase concentration of S(VI) was 1 mgmÀ3, and the RH was 90%. (b–d) The initial and final mass concentrations of (b) S(VI), (c) BC, and (d) soil in each distribution. size structures were run. Coagulation was not solved in this the newly nucleated S(VI) and other particles to grow to case. larger sizes than when these processes were ignored. Dis- [45] Figures 5a.i and 5b show that, in comparison with solutional growth and hydration also caused BC particles to Figure 3, accounting for additional growth processes caused obtain larger shell fractions and grow to slightly larger sizes AAC 2 - 14 JACOBSON: ANALYSIS OF AEROSOL INTERACTIONS
108 A (Spray) 12 hr AD (Spray-sulf) 12 hr B (Soil) 12 hr BD (Soil-sulf) 12 hr 106 D (Sulf) 12 hr DE (Sulf-BC) 12 hr E3 (BC>20% shell) 12 hr DF (Sulf-OM) 12 hr p F (OM) 12 hr MX 12 hr D 4
10 10 (a) Number concentration 102 of each distribution ) / d log -3 100
10-2 dN (No. cm 10-4
10-6 0.01 0.1 1 10 100 Particle diameter (µm)
105 A (spray) Initial BD (Soil-sulf) 12 hr
p 3 D (Sulf) Initial DE (Sulf-BC) 12 hr 10 A (Spray) 12 hr DF (Sulf-OM) 12 hr D
10 D (Sulf) 12 hr MX 12 hr 101 AD (Spray-sulf) 12 hr
) / d log 10-1 -3
g m -3
µ 10
dM ( 10-5 (b) S(VI) by distribution 10-7 0.01 0.1 1 10 100 Particle diameter (µm)
103 102 E1 (BC<5% shell) Initial B (Soil) Initial
p E3 (BC>20% shell) 12 hr
p BD (Soil-sulf) 12 hr
D DE (Sulf-BC) 12 hr 1
D 10 MX 12 hr
10 0 MX 12 hr
10 10 (c) BC by distribution -1 10-2 10 ) / d log ) / d log -3 -3 -4 -3 g m 10 10 g m µ µ (d) Soil by -6 -5 dM ( distribution
10 dM ( 10
-8 10 10-7 0.01 0.1 1 10 100 0.01 0.1 1 10 100 Particle diameter (µm) Particle diameter (µm)
Figure 4. Nucleation, coagulation, and condensation. Same as Figure 3, but when coagulation was treated as well (thus, more distributions were affected).
(Figure 5c versus Figure 3c). Soil similarly grew to larger the solution pH and hydrating more water. Figures 3b and sizes (Figure 5d versus Figure 3d). 5b show that S(VI) condensed on the BC, OM, and soil [46] Figure 5e shows the fate of liquid water by distribu- distributions. Figures 5f–5h show that HCl, NH3,and tion. Water uptake was due solely to equilibrium hydration, HNO3, respectively, dissolved in the resulting S(VI)–H2O thus the presence of water in a distribution indicated the solution. Since no HCl gas existed initially, all the chloride distribution contained hygroscopic material by the end of in distributions aside from the sea spray distribution (A) the simulation. Initially, BC, OM, and soil distributions did must have evaporated from that distribution to the gas phase not contain hygroscopic material. These distributions then dissolved in the other distributions. Indeed, Figure 5f obtained such material only after S(VI) condensed on them, shows that chloride was lost primarily from small particles simultaneously hydrating liquid water. Once liquid water in the sea spray distribution (A). This resulted from acid- was present, HCl, NH3, and HNO3 could dissolve, changing ification of the small sea spray particles by sulfate and 1011 1011 A (Spray) 12 hr (a.ii) Number concentration A (Spray) 12 hr 9 (a.i) Number concentration 9
p B (Soil) 12 hr B (Soil) 12 hr 10 of each distribution p 10 of each distribution D (Sulf) 12 hr D (Sulf) 12 hr D
D (Full moving) 7 (Moving center) 7
10 E1 (BC<5% shell) 12 hr E1 (BC<5% shell) 12 hr 10 10 10 E2 (BC 5-20% shell) 12 hr E2 (BC 5-20% shell) 12 hr 105 E3 (BC>20% shell) 12 hr 105 E3 (BC>20% shell) 12 hr F (OM) 12 hr F (OM) 12 hr ) / d log 3 ) / d log 3
-3 10
-3 10 101 101 10-1 10-1 INTERACTIONS AEROSOL OF ANALYSIS JACOBSON: dN (No. cm 10-3 dN (No. cm 10-3 10-5 10-5 0.01 0.1 1 10 100 0.01 0.1 1 10 100 Particle diameter (µm) Particle diameter (µm)
3 105 103 10 A (Spray) Initial E1 (BC<5% shell) 12 hr E1 (BC<5% shell) Initial B (Soil) Initial
D (Sulf) Initial E2 (BC 5-20% shell) 12 hr E1 (BC<5% shell) 12 hr p p p 3 B (Soil) 12 hr 10 A (Spray) 12 hr E3 (BC>20% shell) 12 hr 1 1 E2 (BC 5-20% shell) 12 hr D D D 10 10
B (Soil) 12 hr F (OM) 12 hr 10 10 10 E3 (BC>20% shell) 12 hr D (Sulf) 12 hr 101 10-1 10-1 ) / d log ) / d log ) / d log -1 (c) BC by -3
10 -3 -3 (b) S(VI) by -3 distribution -3 10 10 (d) Soil by g m g m -3 distribution g m µ µ 10 µ distribution -5 -5 dM ( dM ( 10 dM ( 10 10-5
10-7 10-7 10-7 0.01 0.1 1 10 100 0.01 0.1 1 10 100 0.01 0.1 1 10 100 Particle diameter (µm) Particle diameter (µm) Particle diameter (µm)
Figure 5. Nucleation, condensation, and dissolution/equilibrium. (a.i) Number concentrations of seven distributions after 12 hours of nucleation and condensation of S(VI), dissolution of HCl, HNO3,NH3, internal particle equilibrium, and À3 equilibrium hydration. The initial gas concentrations of S(VI), HCl, HNO3, and NH3 were 1, 0, 0.8, and 0.3 mgm , respectively, and the RH was 90%. Condensation was solved among all distributions and the gas phase. Nucleation was
solved simultaneously with condensation over distribution D. The moving-center size structure was used. (a.ii) Same as 2 AAC (a.i), except the full-moving size structure was used (all nucleated material was placed in the smallest current bin in that À + À case). (b–h) Initial and final mass concentration of (b) S(VI), (c) BC, (d) soil, (e) H2O(l), (f ) Cl , (g) NH4 , and (h) NO3 in each distribution, all obtained when the moving-center structure was used. - 15 A 2 AAC - 6JCBO:AAYI FARSLINTERACTIONS AEROSOL OF ANALYSIS JACOBSON: 16
3 107 10 A (spay) Initial E1 (BC<5% shell) 12 hr A (Spray) Initial E2 (BC 5-20% shell) 12 hr A (Spray) 12 hr E3 (BC>20% shell) 12 hr
5 D (Sulf) Initial E2 (BC 5-20% shell) 12 hr p p 10 A (Spray) 12 hr E3 (BC>20% shell) 12 hr 1 B (Soil) 12 hr F (OM) 12 hr D D 10 D (Sulf) 12 hr
B (Soil) 12 hr F (OM) 12 hr 10 10 103 D (Sulf) 12 hr - 10-1 (f) Cl by 101 distribution ) / d log ) / d log -3 -3 -1 (e) Water by 10 10-3
distribution g m g m µ µ 10-3 -5
dM ( 10 dM ( 10-5 -7 10-7 10 0.01 0.1 1 10 100 0.01 0.1 1 10 100 Particle diameter (µm) Particle diameter (µm)
105 105 A (Spray) 12 hr A (Spray) 12 hr B (Soil) 12 hr B (Soil) 12 hr p p 103 D (Sulf) 12 hr 103 D (Sulf) 12 hr D D E2 (BC 5-20% shell) 12 hr E2 (BC 5-20% shell) 12 hr 10 10 1 E3 (BC>20% shell) 12 hr 1 E3 (BC>20% shell) 12 hr 10 F (OM) 12 hr 10 F (OM) 12 hr (g) NH + 4 -1 ) / d log ) / d log -1 -
10 -3 10 (h) NO -3 3 g m g m -3 -3 µ
µ 10 10 dM ( dM ( 10-5 10-5
10-7 10-7 0.01 0.1 1 10 100 0.01 0.1 1 10 100 Particle diameter (µm) Particle diameter (µm)
Figure 5. (continued) JACOBSON: ANALYSIS OF AEROSOL INTERACTIONS AAC 2 - 17 nitrate, as shown in Figures 5b and 5h, respectively. The tion and dissolution. If continuous emissions were treated growth of sulfate and nitrate onto the smallest sea spray simultaneously with the processes described here, distribu- particles simultaneously increased the size of these particles tion E1 (BC < 5% shell) would always be present to some (Figure 5a.i) so that the smallest sea spray particles not only degree, as shown by Jacobson [2001]. Figure 6d similarly grew, but they also lost their chloride. shows that only a small amount of soil remained uncoagu- [47] Figure 5g shows that ammonium entered both small lated. Figures 6e–6i show the fate of water, soluble species, and large sea spray particles but not particles between and the solid gypsum [CaSO4 Á 2H2O(s)]. Gypsum, which around 0.15 and 1.5 mm. Ammonium entered the small can form when the RH is less than 97%, tied up sulfate in particles to balance sulfate and other anions in excess of the sea spray, sea spray–sulfate, and mixed distributions, Na+,Ca2+,Mg2+, and K+. It entered the large particles to reducing the water contents of those distributions. balance the nitrate and other anions in excess of these cations. Sulfate and nitrate both entered 0.15–1.5 mm 6.6. Solid Formation During Diurnal Variation of RH particles, joining chloride there, but these anions were [51] Figure 7 shows time series concentrations, summed balanced there by the cations, preventing the dissolution over all sizes, from an additional simulation in which of ammonia. condensation and nucleation of S(VI), dissolution of HCl, [48] Figure 5a.ii shows the final number concentration HNO3, and NH3, equilibrium chemistry of all components, distribution for the simulation shown in Figure 5a.i, but and equilibrium hydration were solved simultaneously over when the full-moving size structure was used. The compar- the five initial distributions shown in Figure 1 (except that ison between the full-moving and moving-center structures nucleation was solved for only in the sulfate distribution, was done because the full-moving structure allows zero D). Coagulation was ignored. The initial gas concentrations diffusion during growth. However, a problem with the full- of S(VI), HCl, HNO3, and NH3 were 1, 0, 0.8, and 0.3 mg moving structure, illustrated here, is that, once growth mÀ3, respectively. During the 24 hour simulation, the RH increases the size of the smallest size bin, no real bin exists varied sinusoidally from 95 to 5 back to 95%. The temper- for nucleation to occur within. Since the simulation here ature was held at 285 K. When the RH was decreasing, a included nucleation, all nucleated particles were placed in solid was not allowed to form unless it already existed or the smallest full-moving bin, which was much larger than until the RH decreased below the CRH of the solid. Thus, a the critical radius for nucleation (except at the beginning metastable state was assumed upon a decrease in RH. When of the simulation). In the moving-center structure, a small the RH was increasing, a solid was allowed to form (but did bin always exists for nucleation. Thus, the apparent differ- not have to form) if the RH was less than the DRH of the ence in results in the nucleation mode between the two solid. simulations is due substantially to the inability to treat [52] The only solid that could potentially be present at an nucleation physically in the full-moving structure. The rest initial RH of 95% was gypsum [CaSO4 Á 2H2O(s)], which of the size distribution in both structures are nearly identi- has a DRH of 97%. Figure 7a shows that it formed initially cal, indicating that the moving-center structure was non- in the sea spray distribution and was present to some degree diffusive yet also treated nucleation physically. at all RHs. Figures 7a and 7b show that the water content decreased with decreasing RH and increased with increas- 6.5. Nucleation, Condensation, Coagulation, and ing RH, as expected. Water content was higher for a longer Dissolution/Chemistry period during a decrease in RH than for an increase in RH, [49] Figure 6 shows results when coagulation was added particularly in the sea spray distribution, since a metastable to the moving-center simulation at 90% RH shown in state was assumed during a decrease in RH but not an Figure 5. Thus, the simulation included nucleation, con- increase. Assuming a metastable reduced solid formation in densation, dissolution, equilibrium chemistry, equilibrium comparison with not assuming one, increasing liquid water hydration, and coagulation. A comparison of Figure 6a with content. The most abundant solids formed in the sea spray Figure 5a suggests that coagulation reduced the number of distribution were NaCl(s), gypsum, and NaNO3(s). In the nucleated S(VI) particles, as it did when dissolution was sulfate distribution (which did not contain Na+,Mg2+,Ca2+, ignored. This simulation also reaffirms that coagulation or K+ because coagulation was ignored), the primary solids + internally mixes a greater fraction of larger particles than were NH4HSO4(s) and (NH4)2SO4(s). Much more NH4 smaller particles and leaves only a residual of externally entered the sulfate distribution than the sea spray distribu- mixed small particles unaffected when new sources of those tion because the sea spray distribution had more cations À particles are not considered. After 12 hours, most initial there initially. More NO3 entered the sea spray distribution externally mixed particles from the simulation were inter- than the sulfate distribution for the opposite reason. nally mixed with one or more distributions. The sulfate Although no HCl existed in the gas phase and no ClÀ distribution, which had a new-particle source during the existed in the sulfate distribution initially, some ClÀ was simulation, remained more externally mixed than did other transferred from the sea spray distribution to the gas phase distributions, which did not have new-particle sources. then to the sulfate distribution, as shown in Figure 7b. Figure 6b shows that although a good portion of sulfate remained externally mixed, a large fraction became inter- 6.7. Refractive Indices and Solution Densities nally mixed during the simulation. [53] Figures 8a and 8b show solution and total particle [50] Figure 6c shows that only a small amount of BC real refractive indices, weight percent solute in solution, remained uncoagulated during the simulation. That BC, solution pH, and solution density for the sea spray distri- though, was still internally mixed (with greater than 20% bution (A) after a simulation in which condensation of coating) due to S(VI) condensation and subsequent hydra- S(VI), dissolution of HCl, HNO3,NH3, internal particle A 2 AAC - 1014 105 INTERACTIONS AEROSOL OF ANALYSIS JACOBSON: 18 A (Spray) 12 hr AF (Spray-OM) 12 hr A (Spray) Initial AD (Spray-sulf) 12 hr 12 B (Soil) 12 hr BD (Soil-sulf) 12 hr D (Sulf) Initial BD (Soil-sulf) 12 hr 10 p 3 D (Sulf) 12 hr BE (Soil-BC) 12 hr 10 A (Spray) 12 hr DE (Sulf-BC) 12 hr 10 D
p D (Sulf) 12 hr DF (Sulf-OM) 12 hr 10 E3 (BC>20% shell) 12 hr BF (Soil-OM) 12 hr 10
D F (OM) 12 hr DE (Sulf-BC) 12 hr 1 E3 (BC>20% shell) 12 hr MX 12 hr 8 10 10 10 AB (Spray-soil) 12 hr DF (Sulf-OM) 12 hr F (OM) 12 hr AD (Spray-sulf) 12 hr EF (BC-OM) 12 hr 6 ) / d log 10-1 10 AE (Spray-BC) 12 hr MX 12 hr -3
) / d log 4 g m -3 10 -3
(a) Number concentration µ 10 2 10 of each distribution (b) S(VI) by dM ( 10-5 distribution 100
dN (No. cm 10-2 10-7 0.01 0.1 1 10 100 10-4 Particle diameter (µm) 10-6 0.01 0.1 1 10 100 Particle diameter (µm)
3 10 103 109 E1 (BC<5% shell) Initial B (Soil) Initial A (Spray) Initial F (OM) 12 hr
p E3 (BC>20% shell) 12 hr B (Soil) 12 hr 7 D (Sulf) Initial AD (Spray-sulf) 12 hr p 1 p 10 D 10 DE (Sulf-BC) 12 hr 1 BD (Soil-sulf) 12 hr A (Spray) 12 hr BD (Soil-sulf) 12 hr D
D 10 10 MX 12 hr 5 SOIL DE (Sulf-BC) 12 hr MX 12 hr 10 10 10 D (Sulf) 12 hr DF (Sulf-OM) 12 hr 10-1 -1 (d) Soil by 103 E3 (BC>20% shell) 12 hr MX 12 hr (c) BC by 10 distribution ) / d log
) / d log 1 -3 distribution ) / d log 10 -3
-3 (e) Water by 10-3 -3 g m 10 -1 distribution
g m 10 µ g m µ µ -3 -5 -5 10 dM ( 10 10 dM ( dM ( 10-5 -7 10 10-7 10-7 0.01 0.1 1 10 100 0.01 0.1 1 10 100 0.01 0.1 1 10 100 Particle diameter (µm) Particle diameter (µm) Particle diameter (µm)
Figure 6. Nucleation, coagulation, condensation, and dissolution/equilibrium. Same as Figure 5, but when coagulation was treated as well. The moving-center structure was used to obtain these results. 105 105 A (Spray) Initial BD (Soil-sulf) 12 hr A (Spray) 12 hr BD (Soil-sulf) 12 hr A (Spray) 12 hr DE (Sulf-BC) 12 hr p
3 p D (Sulf) 12 hr DE (Sulf-BC) 12 hr 10 D (Sulf) 12 hr DF (Soil-OM) 12 hr 103 D
D E3 (BC>20% shell) 12 hr DF (Sulf-OM) 12 hr F (OM) 12 hr MX 12 hr 10 10 F (OM) 12 hr MX 12 hr 1 AD (Spray-sulf) 12 hr 10 101 AD (Spray-sulf) 12 hr
- INTERACTIONS AEROSOL OF ANALYSIS JACOBSON: -1 (f) Cl by ) / d log -1 10 ) / d log
-3 distribution 10 -3 (g) NH + by 4 g m -3 g m -3
µ 10 distribution µ 10
dM ( -5 10 dM ( 10-5
10-7 10-7 0.01 0.1 1 10 100 0.01 0.1 1 10 100 Particle diameter (µm) Particle diameter (µm)
105 103 A (Spray) 12 hr A (Spray) Initial D (Sulf) 12 hr p A (Spray) 12 hr p 3 10 AD (Spray-sulf) 12 hr 1 D AD (Spray-sulf) 12 hr D BD (Soil-sulf) 12 hr 10 10 MX 12 hr 10 1 DE (Sulf-BC) 12 hr 10 DF (Sulf-OM) 12 hr MX 12 hr 10-1
-1 ) / d log ) / d log
10 -3 -3 - 10-3 (h) NO g m
g m -3 3 µ
µ 10 (i) Gypsum -5 dM ( dM ( 10-5 10
10-7 10-7 0.01 0.1 1 10 100 0.01 0.1 1 10 100 Particle diameter (µm) Particle diameter (µm) Figure 6. (continued) A 2 AAC - 19 AAC 2 - 20 JACOBSON: ANALYSIS OF AEROSOL INTERACTIONS
105 RH + - KNO (s) Na Cl 3 H O(l) + (NH ) SO (s) KHSO (s) 2 K 4 2 4 4 ) S(VI)(aq) Ca2+ NaNO (s) K SO (s) -3 3 2 4 + 2+ NaCl(s) CaSO -2H O(s) 3 4 2 g m 10 H Mg µ + - Na2SO (s) MgSO (s) NH NO 4 4 4 3 (a) Seaspray distribution (A)
101
10-1
10-3 Concentration, summed over all sizes (
10-5 0 4 8 12162024 Time from start (hours) ) -3 RH H+ Cl-
g m 5
µ 10 H O(l) + NH HSO (s) 2 NH 4 4 4 3 S(VI)(aq) - (NH ) SO (s) 10 NO 4 2 4 3 (b) Sulfate distribution (D) 101
10-1
10-3
10-5 0 4 8 12 16 20 24
Concentration, summed over all sizes ( Time from start (hours)
Figure 7. Time variation of concentrations, summed over all size bins, of ions, liquids, and solids in the (a) sea spray distribution and (b) sulfate distribution during a 24 hour simulation in which condensation of S(VI), nucleation of S(VI), dissolution of HCl, HNO3, and NH3, equilibrium chemistry of all components, and equilibrium hydration were solved simultaneously over the five initial distributions shown in Figure 1. Coagulation was ignored. The initial gas concentrations of S(VI), HCl, HNO3, and À3 NH3 were 1, 0, 0.8, and 0.3 mgm , respectively. During the simulation, the RH (also shown) dropped from 95% to 5% then rose again to 95%. equilibrium, and equilibrium hydration were treated at 90% water has a lower density and refractive index than do most and 40% RH, respectively. Coagulation was ignored. Fig- electrolytes dissolved in solution, solution density and ures 8c and 8d show the concentrations of species in the sea refractive index also peaked at about 0.1 mm. spray distribution in the two cases, respectively, at the time [55] Figure 8b shows that the increase in weight percent of the refractive index calculations. The difference between solute with decreasing particle size was greater at 40% RH the solution and total particle refractive indices for the sea than at 90% RH, causing the solution density to increase spray distribution is the refractive indices due to solids, with decreasing particle size more at 40% RH than at 90% since they are not in solution. RH. At all particle sizes, solids comprised a greater percent [54] Figure 8a shows that the difference between solution of particles at 40% RH than at 90% RH. At 40% RH, the and total refractive indices at 90% RH was small. This was solid fraction increased with decreasing particle size. Due to because the solution volume dominated the total particle the high solid fraction at 40% RH, the total particle real volume at 90% RH. The only nonsolution component of the refractive index at all particle sizes was about 7% higher particles in that case was gypsum. The weight percent solute than was the solution real refractive index, and both were in the distribution peaked at about 0.1 mm diameter. Because higher than the total or solution refractive indices at 90% JACOBSON: ANALYSIS OF AEROSOL INTERACTIONS AAC 2 - 21 Density (g/cm3); Refractive index, n
Density (g/cm3); Refractive index, n 100 100 1.4 1.8 Total particle n Weight % solute in solution Solution n 1.35 1.7
1.3 (a) 90% RH (b) 40% RH 10 Weight % solute 1.6 in solution Total particle n 1.25 10 1.5 1.2 Solution pH Solution n Weight % solute, pH Weight % solute; pH 1.4 1 Solution 1.15 Solution density Solution pH density 1.1 1 1.3 0.01 0.1 1 10 100 0.01 0.1 1 10 100 Particle diameter (µm) Particle diameter (µm)
106 SVI(aq) Na+ K+
p 4 + + H O(l) 10 NH H 2 4 D
10 2+ - CaSO -2H O(s) 102 Ca Cl 4 2 Mg2+ NO - 3 0 ) / d log 10 -3
g m 10-2 µ (c) 90% RH -4 dM ( 10
10-6 0.01 0.1 1 10 100 Particle diameter (µm)
108 + 3- (NH )2SO (s) Na NO 4 4 6 p NH + SVI(aq) CaSO -2H O(s) 10 4 4 2 D 2+ H O(l) K SO (s) 10 104 Ca 2 2 4 2+ Na SO (s) KNO (s) Mg 2 4 3 2 NaCl(s) 10 K+ ) / d log
-3 - NaNO (s) 100 Cl 3 g m µ 10-2 (d) 40% RH dM ( 10-4
10-6 0.01 0.1 1 10 100 Particle diameter (µm)
Figure 8. Weight percent solution, pH, solution density, total particle refractive index, and solution refractive index for the sea spray distribution (A) after a simulation in which condensation of S(VI), dissolution of HCl, HNO3,NH3, internal particle equilibrium, and equilibrium hydration were treated and when the RH was (a) 90% and (b) 40%. Mass concentrations corresponding to Figures 7a and 7b are given in Figures 7c and 7d, respectively.
RH. Solid electrolytes enhanced the real refractive index [e.g., Tang et al., 1997], because particle cross sections compared with when the electrolytes dissolve in solution increase with increasing RHs. because when electrolytes dissolved, they hydrated water, and the real refractive index of water is less than that of most solid electrolytes. 7. Conclusions [56] Although real refractive indices of particles with [57] This paper discussed numerical techniques for solving solutions decrease with increasing RH, scattering extinction coagulation, simultaneous nucleation and condensation, and coefficients of such particles increase with increasing RHs nonequilibrium dissolution/equilibrium reversible chemistry AAC 2 - 22 JACOBSON: ANALYSIS OF AEROSOL INTERACTIONS over multiple size distributions. All schemes conserve moles Celia, M. A., and W. G. Gray, Numerical Methods for Differential Equa- tions, Prentice-Hall, Old Tappan, N. J., 1992. 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