1854 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57
Predicting Particle Critical Supersaturation from Hygroscopic Growth Measurements in the Humidi®ed TDMA. Part I: Theory and Sensitivity Studies
FRED J. BRECHTEL* AND SONIA M. KREIDENWEIS Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado
(Manuscript received 21 October 1998, in ®nal form 22 July 1999)
ABSTRACT A method is described to estimate the critical supersaturation of quasi-monodisperse, dry particles using measurements of hygroscopic growth at relative humidities below 100%. KoÈhler theory is used to derive two chemical composition±dependent parameters, with appropriate accounting for solution effects through a simpli®ed model of the osmotic coef®cient. The two unknown chemical parameters are determined by ®tting the KoÈhler model to data obtained from humidi®ed tandem differential mobility analyzer (HTDMA) measurements, and used to calculate the critical supersaturation for a given dry particle size. In this work the theory and methodology are presented, and sensitivity studies are performed, with respect to assumptions made and uncertainties in key input parameters to the KoÈhler model. Results show that for particle diameters of 40 and 100 nm, the average error between critical supersaturations derived using the proposed method and theoretical values is Ϫ7.5% (1 ϭ 10%, n ϭ 16). This error is similar to experimental uncertainties in critical supersaturations determined from laboratory studies on particles of known chemical composition (Ϫ0.6%, 1 ϭ 11%, n ϭ 16).
1. Introduction values of Scrit. Knowledge of how the critical supersat- uration varies among similar-sized particles and across A recent report by the Intergovernmental Panel on the size spectrum can reveal the size ranges of particles Climate Change (IPCC) indicates that current estimates that would form cloud droplets in different cloud-form- of the uncertainty in predictions of the indirect effect of aerosols on climate are at least twice as large as the ing regimes (Fitzgerald et al. 1982; Hudson and Da 1996). Conventional CCN spectrometers typically mea- total direct effect of CO2, and of opposite sign (IPCC 1995). The so-called indirect effect is the ability of par- sure the integral number of particles of all sizes that are ticulate matter to alter climate by acting as cloud con- activated at a given supersaturation. densation nuclei (CCN) and thereby altering cloud ra- In Part I of this work, a technique is proposed to diative properties. In order to better understand the roles directly relate particle size and chemical composition to played by various sources of ambient particles in de- its critical supersaturation by performing a mathematical termining the CCN population, experimental techniques regression of a modi®ed form of the KoÈhler equation are needed to link particle size and chemical compo- to hygroscopic growth measurements at relative humid- sition to CCN activity. ities (RHs) below 100%. A modi®ed KoÈhler theory is A key parameter in the relationship among particle required in this application in order to constrain the size, chemical composition, and CCN activity is the solutions to realistic values and to minimize the number
critical supersaturation, Scrit, the water supersaturation of chemical composition±dependent unknowns that necessary to activate a cloud condensation nucleus and must be determined by the mathematical regression. Nu- form a cloud droplet. At any given size, particles having merical sensitivity studies are performed to test the as- different chemical compositions can exhibit different sumptions made in the modi®ed theory and demonstrate the accuracy of predictions of the critical supersaturation for particle diameters of 40 and 100 nm. The 40±100- nm size range is important relative to CCN activity since * Current af®liation: Environmental Chemistry Division, Brook- haven National Laboratory, Upton, New York. a large fraction of the CCN number concentration at Scrit values around 0.5% occurs in this size range (Covert et al. 1998; Hallberg et al. 1994). We note that the Corresponding author address: Sonia M. Kreidenweis, Dept. of current version of the modi®ed KoÈhler theory does not Atmospheric Science, Colorado State University, Fort Collins, CO 80523. explicitly account for the effects of undissolved mate- E-mail: [email protected] rial, soluble gases, and surface-active compounds on Scrit
᭧ 2000 American Meteorological Society
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TABLE 1. Summary of previous HTDMA studies in both clean and in¯uenced maritime (CM and IM) and continental conditions (CC and IC). ``All'' designates all air mass conditions, ``I'' ϭ internally mixed, ``E'' ϭ externally mixed chemical composition and ``Ð'' implies not observed. A less hygroscopic particle subpopulation, if observed, is designated as ``less hygro,'' while a more hygroscopic population is designated as ``more hygro.'' Simultaneous chemical composition measurements designated by ``Y'' in ``chem. meas.'' column. The size ranges of particle samples depend on the individual study, but the 30±200-nm size range is representative of the sizes sampled.
Ddrop/Dp,sol Ddrop/Dp,sol Location Study Less More Inferred Chem. Study (cond.) RH (%) hygro hygro mixture meas. a IC 80 Ð 1.2±1.8 I N b IC 90 1.0 1.12±1.49 E Y c IC 90 1.07 1.38±1.44 E Y d CC, IC 85 1.15 1.36±1.51 E, I Y e CC, IC 90 1.05 1.44±1.48 E Y f IC Ð Ð Ð E N g IC 85 1.1 1.44 E Y h IC, CC 85 1.02±1.11 1.34±1.37 E, I Y i CC, IC 85 1.15 1.43 E, I Y j IC 79 1.02 1.3 E, I N k CM, IM 50, 85 1.1 1.4 E, I Y l All 90 1.1 1.5 E, I Y m IC 90 1.0 1.4 E Y n Lab 80±97 1.0 0.94 I Y
a: Sekigawa (1983); insoluble fraction varies with Dp.
b: McMurry and Stolzenburg (1989); Los Angeles (SCAQS); larger Dp more hygroscopic (1.33ϫ). c: Covert and Heintzenberg (1984); carbon in less hygroscopic fraction. d: Zhang et al. (1993); Grand Canyon winter 1990 (NGS and SCAQS); insoluble inclusions; hygroscopicity was a function of [ions]/ [carbon]. e: Covert and Heintzenberg (1993); sulfur and carbon in more hygroscopic fraction. f: Juozaitis et al. (1993); high soot; observed hydrophobic size distribution, 60% hydrophobic fraction at 0.02 m, 15% at 0.05 m, 4% at 0.45 m.
g: Svenningsson et al. (1992); potential large organic in¯uence; hygroscopic growth not f(Dp), nonhygroscopic and hygroscopic fractions equal. h: Svenningsson et al. (1994); 50% of CN concentration nonhygroscopic. i: Pitchford and McMurry (1994); NGS and SCAQS; 85% of CN volume in nonhygroscopic fraction nonsoluble.
j: Liu et al. (1978); Minneapolis, MN; focused on H2SO4. k: Berg et al. (1998); ACE-1 study; Discoverer. l: Covert et al. (1998); ACE-1 study; Cape Grim. m: McMurry et al. (1996); less hygro ϭ C chains, more hygro ϭ liquid drops with S and some C. n: Weingartner et al. (1997); combustion particles; particles shrink with increasing RH up to 95% RH.
(Laaksonen et al. 1998; Schulmann et al. 1996). In Part ticle counter (CPC) is used to count the particles exiting II (Brechtel and Kreidenweis 2000) results from labo- the second DMA. By observing the electrical mobility ratory studies performed to test the method are pre- where the CPC measures the maximum number of par- sented. ticles, the size of the wetted particles can be determined, and the water content can be derived from the difference between the wet and dry sizes. By controlling the hu- 2. Previous work midities before and within the second DMA at several The hygroscopicity of submicrometer particles can be different values, the growth as a function of relative investigated as a function of size using the humidi®ed humidity may be determined for each input, dry mono- tandem differential mobility analyzer technique disperse particle population. (HTDMA; Rader and McMurry 1986). In the HTDMA, There have been several previous studies of particle dry (RH Ͻ 10%), quasi-monodisperse particles are se- hygroscopic growth using the HTDMA technique, but lected from a polydisperse size distribution using a dif- only a few of these also involved a determination of ferential mobility analyzer (DMA). The DMA selects Scrit. A summary of several HTDMA studies is provided particles based upon their electrical mobility, which is in Table 1. In their HTDMA measurements on ambient directly proportional to the number of elementary elec- particles in the polluted Po Valley of Italy, Svenningsson trical charges carried by the particle and inversely pro- et al. (1992) determined that the chemical composition portional to its size. The monodisperse particles are ex- of individual particles was just as important as size in posed to a controlled humidity environment before pass- determining whether they are CCN. Other HTDMA ing through a second DMA. If the particles take up studies (Covert and Heitzenberg 1993; McMurry and water, they will grow in size and exhibit a lower elec- Stolzenburg 1989) demonstrated that the chemical com- trical mobility in the second DMA. A condensation par- position of ambient particles was heterogeneous with
Unauthenticated | Downloaded 09/30/21 01:19 AM UTC 1856 JOURNAL OF THE ATMOSPHERIC SCIENCES VOLUME 57 respect to particle size. Zhang et al. (1993) and Pitchford 4 RH ϭ 100a expdrop l , (1) and McMurry (1994) inferred that the ambient particle w RTD []drop chemical composition was heterogeneous over a narrow size range, based on observations of more than one hy- where RH is in percent, drop is the droplet surface ten- groscopic mode at a single particle size during both sion, l is the partial molar volume of water in solution, anthropogenically in¯uenced and more remote conti- R is the universal gas constant, T is the droplet tem- nental conditions. Covert et al. (1998) found that even perature, Ddrop is the droplet diameter, and aw is the water in remote marine regions, the CCN activity could best activity of the solution. Here drop, l, and aw are func- be determined from hygroscopic growth measurements tions of temperature and solution composition. The ex- during periods characterized by relatively simple par- ponential term on the right-hand side of Eq. (1) is gen- ticle chemical compositions. erally referred to as the Kelvin, or curvature, term. For aqueous solutions of ionic compounds, aw may There have been only a few studies where Scrit was derived using HTDMA measurements. Svenningson et be written (Robinson and Stokes 1959) al. (1992) and Covert et al. (1998) used simultaneous composition measurements from impactor samples to ϪMw⌽m aw ϭ exp , (2) determine the chemical composition of the hygroscopic []1000 fraction of ambient particulate matter. Weingartner et al. where Mw is the molecular weight of pure water, is (1997) performed HTDMA measurements on combus- the total number of ions produced by the dissociation tion particles to predict their critical supersaturation. No of one molecule of solute, ⌽ is the practical osmotic simultaneous CCN measurements were conducted, al- coef®cient of the solution, and m is the solution molality. though derived values of Scrit compared favorably with For an in®nitely dilute solution, aw ϭ 1. The term ln(aw) historical measurements of the CCN activity of com- is often referred to as the solute or Raoult term, where bustion particles. the often used van't Hoff factor has been replaced by Recently, organics have been recognized as playing the more exact form represented by the product ⌽. The a role toward determining particle hygroscopicity (Sax- common assumption that the solution is dilute (⌽ϭ1; ena et al. 1995; Saxena and Hildemann 1996). Cruz and Pruppacher and Klett 1997), appropriate for cloud drop- Pandis (1997) measured the critical supersaturations lets, is not valid under most HTDMA measurement con- necessary to activate monodisperse particles composed ditions. Our calculations show that even at the critical of organic acids, compounds that have been found in size, the assumption that ⌽ϭ1 corresponds to errors ambient particles (Rogge et al. 1993). Cruz and Pandis in predicted values of Scrit of 20% for a dry 10-nm- (1997) demonstrated that KoÈhler theory could be used diameter particle composed of ammonium sulfate and to predict Scrit for less soluble organics within the ex- 5% for a 300-nm dry diameter and the same composition perimental uncertainties of their method. (Brechtel 1998). Therefore, we retain the variation of In this work, a methodology is proposed to extend the osmotic coef®cient with solution composition. estimates of hygroscopic growth to RHs above 100% by incorporating a thermodynamic model of water ac- a. Reference KoÈhler model tivity with correct limiting behavior. The approach en- ables Scrit to be determined from a series of hygroscopic In this section we develop the full, or reference, KoÈh- growth measurements made at relative humidities less ler model in terms of parameters to be determined from than 100%. The applicability of the method to particles HTDMA studies. Then we develop the modi®ed form composed of single and mixed solutes is examined. of the KoÈhler equation used in the HTDMA data re- gression procedure and show the sensitivities of the pre-
dicted RH and Scrit values to the various assumptions. 3. Theoretical approach The droplet surface tension is corrected for temper- ature and solute effects using the relationship proposed Droplets formed on soluble particles at relative hu- by HaÈnel (1976) midities between 80% and 92%, a typical range for HTDMA studies, experience molalities between about drop ϭ o(To) ϩ a(To Ϫ T) ϩ bm, (3) 1 and 6 molal. At these relatively high concentrations, where o(To) is the surface tension of pure water at To inter-ion interactions modify the vapor pressure low- ϭ 273.15 K (0.0756 N mϪ1), a is a coef®cient to account ering over the droplet surface as described by Raoult's for the temperature dependence of the droplet surface law. The KoÈhler equation, which describes the equilib- tension [a ϭ 1.53 ϫ 10Ϫ4 N(mK)Ϫ1], b is a compo- rium water vapor pressure over a droplet formed on a sition-dependent factor listed in Table 2, and m is the soluble nucleus, must include an appropriate model for solution molality. Values of drop as a function of solute the change in water activity with solution molality. mass fraction are shown in Fig. 1 for most of the solutes The KoÈhler equation for a droplet containing water- investigated in this work. soluble material may be written (Seinfeld 1986) The molality m is de®ned as the number of moles of
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TABLE 2. Values of s, Ms, 0, surface tension `b' parameter, deliquescence RH, and crystallization RH for solute chemical compositions examined in this work. Values for mixed composition solutes assume an equimolar mixture of the two salts.
bc
a a N molw d d Ratio s Ms ϫ 10Ϫ3 RHdel RHcrys Ϫ3 Ϫ1b Solute Cation/Anion kg m g mol 0 mkgsol % % NaCl 1:1 2165 58.44 0.1082 1.64 75.3 45±48
(NH4)2SO4 2:1 1769 132.1 0.04763 2.17 80 37±40
NH4HSO4 1:1 1780 115.1 0.04494 2.3 40 0.05±22
H2SO4 2:1 1841 98.01 Ϫ0.0933 0.67
NH4NO3 1:1 1725 80.04 Ϫ0.0154 2.2 62
(NH4)3H(SO4)2 4:2 1830 247.0 0.0463 2.2 69 35±44
(NH4)2SO4±NaCl 3:2 1874 95.30 0.078 1.91
(NH4)2SO4±NH4NO3 3:2 1752 106.1 0.016 2.185
NaCl±NH4NO3 2:2 1887 69.24 0.0464 1.6 e e NaCl±Na2SO4 3:2 2506 100.2 0.048 2.05 84 53
a Weast (1988). b Pitzer and Mayorga (1973). c Chen (1994). d Tang and Munkelwitz (1994). e Tang (1997). solute per kilogram of water and may be written for the 1973; Pitzer 1973; Kim et al. 1993; Clegg and Pitzer droplet solution as 1992). Guidance in choosing a parameterization for ⌽ comes from the model of Pitzer and Mayorga (1973), 1000m 1000x m ϭϭss, (4) which was capable of reproducing the variation of ⌽ Mmsw M s(1 Ϫ x s) with molality for solutions containing a single solute for over two hundred 1:1, 1:2, and 2:1 solutes. They where ms and mw are the masses of solute and water in proposed the following functional form for ⌽ for elec- the droplet, respectively; Ms is the molecular weight of trolyte solutions: the solute; and xs is the mass fraction of solute in so- lution, de®ned by ͙I 3 m D ⌽ϭ1 Ϫ |zz12| A s sp,sol 1 ϩ b I xs ϭϭ, (5) []pit͙ 3 mswϩ m dropD drop 2 2()3/2 where and are the densities of the solute and ϩ m 12[ ϩ  eϪ␣͙I ] ϩ mC2 12 . (7) s drop 01 solution, respectively; and Dp,sol and Ddrop are the dry particle and droplet size, respectively. Equation (5) as- In the above equation, ␣ and b are 2 and 1.2 at 298 sumes that the dry particle is completely dissolved in pit K, respectively, for all of the solutes examined; z1 and solution. z are the respective charges on the ions in electronic The partial molar volume of water in the droplet so- 2 units; 1 and 2 are the number of molecules of ions of lution may be written (Atkins 1994) species 1 and 2, respectively, produced by the disso-
Mxddrop ciation of one molecule of solute; is 1 ϩ 2; and I ws 1 l ϭ 1 ϩ , (6) is the ionic strength of the solution, m z2, where m dx 2 ⌺ i i i drop[] drop s is the molality of species i and the sum is carried out where ddrop/dxs is the change in solution density with over the total number of ionic species present. We note mass fraction of solute and re¯ects the fact that inter- that Eq. (7) is explicitly written for solutes that disso- molecular forces in solution change with changing sol- ciate into two ions in solution. The relation for ⌽ for ute mass fraction. The drop and ddrop/dxs may be cal- solutions with more ions requires additional terms to culated for droplet solutions containing the solutes stud- account for the additional ion-pair interactions in so- ied here from polynomials of as functions of x drop s lution. The coef®cients  0, 1, and C depend on the from Tang and Munkelwitz (1994), Tang et al. (1981), chemical composition of the solute and are tabulated by and Tang (1997). Values of l as a function of solute Pitzer and Mayorga (1973) and Kim et al. (1993). The mass fraction are shown in Fig. 1. Debye±HuÈckel coef®cient for the osmotic function A The most dif®cult part of predicting Scrit using results is calculated as a function of temperature (T) for tem- from HTDMA studies is dealing with ⌽. No purely peratures between 10Њ and 50ЊC using the following theoretical model of the osmotic coef®cient currently polynomial ®t to data from Pitzer (1979) exists. Only semiempirical models based on theoretical Ϫ4 considerations are available for ⌽ (Pitzer and Mayorga A ϭ 6.97714 ϫ 10 T ϩ 0.3741. (8)
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FIG. 1. Partial molar volume of water and solution surface tension values vs solute mass fraction for solute chemical compositions examined in this work. The mass fraction ranges correspond to HTDMA relative humidities between 80% and 95%. The solid horizontal lines represent the respective values for pure water.
The value of A is 0.392 for water at 25ЊC. Equation ionic strength of the solution also approaches zero, and (7) will be referred to as the ``full Pitzer'' model of ⌽. ⌽ approaches unity, the thermodynamic limit. The terms
In the full Pitzer model of ⌽, the term involving A involving m model the interaction forces between ions models the long-range forces through which ions in so- and are important in more concentrated solutions. lution interact and determines the value of ⌽ when the Substituting the above relations for drop, l, and m solution becomes very dilute. As m approaches zero the into the KoÈhler equation Eq. (1) and Eq. (2) we ®nd
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TABLE 3. Values of critical diameters (nm), supersaturations (%), (1988). Equation (9) will be referred to as the reference and osmotic coef®cients at the critical supersaturation for chemical KoÈhler model. compositions studied in this work. Results are reported for indicated dry particle sizes and assumed temperature of 29.5ЊC.
Dp,sol (nm) b. Evaluation of mixture properties 40 70 100 200 Equation (9) has been written for a single solute. In Sodium Chloride NaCl order to study the hygroscopic growth of particles con-
Dcrit 321 748 1285 3668 taining two salts, expressions are required to calculate Scrit 0.475 0.203 0.119 0.042 the chemical composition-dependent parameters for in- ⌽ 0.937 0.951 0.960 0.974 crit ternally mixed solutes. Designating the mole fractions Amm. Bisulf. NH HSO 4 4 of solute ``a'' and solute ``b'' as f a and f b, respectively, Dcrit 210 484 829 2364 the molecular weight of the mixed solute particle is Scrit 0.730 0.314 0.184 0.064 ⌽ 0.925 0.942 0.952 0.968 crit Ms ϭ f aMs,a ϩ f bMs,b, (10)
Amm. Sulf. (NH4)2SO4 D 210 504 879 2575 where Ms,a is the molecular weight of solute a and Ms,b crit is the molecular weight of solute b. The mass fractions Scrit 0.705 0.295 0.170 0.058
⌽crit 0.778 0.832 0.863 0.911 of the two solutes can be determined from
Sulfuric Acid H2SO4 fMa s,a Dcrit 239 590 1039 1818 xs,a ϭ , and similarly (11) Scrit 0.603 0.249 0.143 0.012 Ms
⌽crit 0.752 0.826 0.863 0.840 Amm. Nitrate NH NO fMb s,b 4 3 xs,b ϭ . (12) Dcrit 255 598 1029 2944 Ms
Scrit 0.534 0.228 0.133 0.046
⌽crit 0.922 0.943 0.954 0.971 Using the calculated mass fractions, the solute density s may be determined from Sodium Chloride±Sodium Sulf. NaCl±Na2SO4 D 278 685 1185 3407 crit 1 S 0.447 0.190 0.110 0.038 crit s ϭ , (13) ⌽ 0.858 0.895 0.914 0.943 xx crit s,a ϩ s,b Sodium Chloride±Amm. Sulf. NaCl±(NH4)2SO4 s,a s,b
Dcrit 166 421 730 2109
Scrit 0.709 0.304 0.176 0.061 where s,a is the density of pure solute a and s,b is the ⌽crit 0.823 0.869 0.891 0.928 density of pure solute b. Furthermore, the mole fraction
Sodium Chloride±Amm. Nit. NaCl±NH4NO3 weighted number of ions produced when one molecule
Dcrit 187 467 804 2288 of the mixed solute dissolves in solution may be cal- Scrit 0.644 0.278 0.162 0.057 culated from ⌽crit 0.916 0.939 0.950 0.968 ϭ fa(1,a ϩ 2,a) ϩ fb(1,b ϩ 2,b), (14) Amm. Sulf.±Amm. Nit. (NH4)2SO4±NH4NO3 D 147 380 662 1915 crit where and are the number of anions and cations, S 0.780 0.334 0.193 0.066 1,a 2,a crit respectively, resulting from the dissociation of one mol- ⌽crit 0.799 0.855 0.881 0.922 ecule of solute a and 1,b and 2,b are the respective values for solute b. Assuming volume additivity, the droplet solution density may be calculated using the expression from Tang (1997), 1000x s 4 0 ϩ bMw 1 Mss(1 Ϫ x ) d []xs drop drop ϭ , (15) RH ϭ 100 exp 1 ϩ xxs,a s,b RTD dx drop drop[] drop s ϩ drop,a drop,b ϪMx⌽ where and are the corresponding solution ϫ expws , (9) drop,a drop,b M (1 Ϫ x ) densities of the binary solutions evaluated at the total []ss mass fraction of solute in solution (xs). For internally where the temperature dependence of the surface tension mixed solutes, the term involving the derivative of drop has been absorbed into the o term. We have applied with respect to the solute mass fraction in Eq. (9) is Eq. (9) to a variety of compounds selected as illustrative evaluated by differentiating Eq. (15) with respect to xs, examples, as shown in Table 3. Parameters for Pitzer's where the dependence of drop on xs is through drop,a and model of ⌽ were taken from Pitzer and Mayorga (1973). drop,b in Eq. (15). The surface tension of the mixed salt All other property data were obtained from Weast solution is determined from (Chen 1994)
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FIG. 2. (a) Comparison of RHs predicted from results of Tang and Munkelwitz (1994) (dotted lines) and reference KoÈhler model (solid lines) for particles composed of single solutes.
ϭ o ϩ bama ϩ bbmb, (16) expressions were used in Eq. (9) to calculate the equi- librium RH over droplets consisting of ternary solutions where b and b are listed in Table 2, and m and m are a b a b of water and the mixed solutes listed in Table 2. As the molalities of solutes a and b in the ternary solution, described below, the results were successfully validated respectively. Following Pruppacher and Klett (1997), against data reported by Tang (1997). we use a molality weighting approach to evaluate the osmotic coef®cient of the solution containing two sol- utes, 4. Reference model calculations m ( ϩ )⌽ϩm ( ϩ )⌽ Correct values of RH in equilibrium with the vapor ⌽ϭa 1,a 2,a s,a b 1,b 2,b s,b , (17) s,mix m ( ϩ ) ϩ m ( ϩ ) pressure over a solution having a given mass fraction a 1,a 2,a b 1,b 2,b of solute are required to compare subsequent values of where ⌽s,a is the osmotic coef®cient for a binary solution RH predicted from modi®ed versions of the KoÈhler of solute a and water, and ⌽s,b is the osmotic coef®cient equation. For calculations involving particles containing for a binary solution of solute b and water. The above single solutes, the mass fraction, xs, of solute in a drop
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FIG.2.(Continued) (b). Comparison of RHs predicted from ZSR±Tang relation (dotted lines) and reference KoÈhler model (solid lines) for internally mixed, equimolar solutes.
of diameter Ddrop formed on a dry particle of size Dp,sol Tang and Munkelwitz (1994), and the corresponding RH was found by iteratively solving calculated using Eq. (1). The values of aw from the polynomial ®ts are generally regarded as more accurate (x )x DD33Ϫ ϭ 0, (18) drop s s drop s p,sol than values from other techniques (Kim et al. 1993). where drop(xs) designates the polynomial ®t of solution Values of Dcrit,Scrit, and ⌽ at Scrit (⌽crit) calculated using density to solute mass fraction. The molality corre- the reference KoÈhler model are shown in Table 3 for sponding to this mass fraction was used to compute drop various dry particle sizes and compositions. Values of [Eq. (3)] and ⌽ [Eq. (7)]. The equilibrium RH for the RH calculated using the reference KoÈhler model (solid particular Dp,sol and Ddrop was then determined using the lines) are compared with values determined using the reference KoÈhler model [Eq. (9)] and is referred to in results of Tang and Munkelwitz (1994) (dotted lines) in this work as the ``reference value'' of RH. To examine Fig. 2a for binary solutions in order to validate the the accuracy of the expression used to compute ⌽, the results from the reference model. Also shown is the same value of xs used in the reference KoÈhler model percent error between the values of RH predicted by the was also used in the polynomial ®ts for aw compiled by two methods. The same droplet diameter was used in
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FIG. 3. Errors (%) in xs, ⌽, aw, natural log of Kelvin term, and RH due to assumptions of volume
additivity, drop ϭ w, and ⌽ϭ⌽f , for different solute chemical compositions studied. Circles designate
NaCl, squares designate NH4HSO4, diamonds designate (NH4)2SO4, and triangles designate NH4NO3. A dry particle size of 40 nm was assumed for all calculations. Solid horizontal lines designate 0% error. the two different methods for calculating RH, therefore, still between 0% and 2.5%. Uncertainties in the values the percent error in RH is presented in Fig. 3. Results of aw and RH determined from the reference model are in Fig. 2a demonstrate agreement between the two dif- estimated between Ϯ1% and Ϯ2.5%, based on errors ferent methods for determining RH within 0.5% for in Pitzer's model of ⌽, and errors in the polynomial ®ts
NaCl and (NH4) 2SO4 over the range of RHs experienced to solution density data. during HTDMA studies. The agreement for NH4HSO4 The RH values predicted by the reference KoÈhler and NH4NO3 is not as good as for the other solutes, but model for internally mixed solutes were compared with
Unauthenticated | Downloaded 09/30/21 01:19 AM UTC 15 JUNE 2000 BRECHTEL AND KREIDENWEIS 1863 results from the ZSR relation (Stokes and Robinson dicted RHs is not as good as for the single solutes ex- 1966). In the ZSR relation, the assumption is made that amined. For the last three mixtures examined, the per- the total water uptake by an internally mixed compo- cent error in RH is less than 3.3% for the RH range sition particle is equal to the sum of the water uptake 80%±92%. Better agreement is found at a higher RH, by the individual solutes. The total mass fraction of corresponding to more dilute solutions where the as- solute in the ternary solution (xs) was determined for sumption that the osmotic coef®cient of the ternary so- the assumed mass fractions of solutes a and b in the dry lution can be determined by the molality weighting ap- particle and for a given drop diameter by iteratively proach is more valid. For the one case where the ex- solving Eq. (18), and using Eq. (15) to compute drop. tended version of the ZSR relation was used (NaCl± First, the mass fractions of solutes a and b in the dry Na2SO4±H2O), the agreement is worse than that found particle were ®xed. To calculate s, Eq. (13) was used. for the other ternary solutions, but is within 2% for RH's The total mass of solute could then be determined for greater than 89%. This demonstrates that interactions the assumed dry particle size and the calculated value between solutes in solution can be important for ternary of s assuming the particle was spherical in shape. Next, solutions. Parameters for the extended ZSR relation are for an assumed value of Ddrop, an initial guess for the not available for any of the other ternary solutions ex- water content was proposed, from which xs was cal- amined. culated. The value of drop in Eq. (15) corresponding to xs could then be determined. The Newton±Raphson technique was used to iteratively solve for the value of 5. Development of the modi®ed KoÈhler model xs in Eq. (18) so that the difference between the two In this section we describe the simplifying assump- terms on the left side was less than 10Ϫ6. The value of tions made to the reference KoÈhler model and our meth- xs determined from the iteration was then used to cal- odology for testing their validity. The assumptions in- culate the solution molality, m, and the solution density. clude volume additivity of solute and solvent within the Using the polynomials of molality as a function of water drop; substitution of the surface tension of pure water activity from Tang and Munkelwitz (1994) in the ZSR for the solution surface tension; and the use of a sim- relation, the water activity solving the ZSR relation was pli®ed, parameterized form of the osmotic coef®cient determined using the following form in the Raoult term. The sensitivity of predicted values of RH to the various assumptions is examined for par- fm fm abϩϭ1, (19) ticles composed of single solutes. The validity of the mma,0 b,0 assumptions is checked over the range of RHs of HTDMA studies (80%±92%) to ensure that the ®nal where m and m are the molalities of the binary so- a,0 b,0 modi®ed KoÈhler model can accurately describe the hy- lutions of solutes a and b calculated using Tang's poly- groscopic growth of particles over this range. nomials as a function of aw for the given water activity. Equation (19) was solved iteratively using a procedure similar to that outlined above, and the value of aw cor- a. Assumption of volume additivity responding to x determined. The RH corresponding to s Assuming the volumes of solute and solvent may be the value of aw was then calculated by applying the Kelvin correction as shown in Eq. (1). We note that the added to determine the solution total volume is equiv- form of the ZSR relation given by Eq. (19) does not alent to assuming the partial molar volume of water in account for interactions between the two solutes in so- solution is equal to the molar volume of pure water. If lution. According to Tang (1997), the assumption that volume additivity is assumed, the mass fraction may be the solutes in ternary solutions do not interact can lead determined as follows: 3 to errors in predicted values of aw using the ZSR relation sp,D sol between 1% and 6%. Tang (1997) presents results for xs ϭ , (20) (D33Ϫ D ) ϩ D 3 an extended ZSR relation that better ®ts experimental w drop p,sol sp,sol data of aw as a function of m for a variety of mixed where w is the density of pure water. The partial molar solutes in ternary solutions where the two solutes have volume of water for a variety of solution compositions one common ion. is shown in Fig. 1. The difference between the partial Values of RH determined from the ZSR relation using molar volume of water and the molar volume of pure Tang's density and water activity polynomials are com- water increases with increasing mass fraction. The larg- pared with those from the reference model for ternary est difference is found for (NH4) 2SO4 and amounts to solutions in Fig. 2b. In the ®gure, the ZSR results are Ϫ4% for xs ϭ 40%. shown as dotted lines and the reference model results We tested the sensitivity of values of xs, ⌽, aw, the are shown as solid lines. The percent difference between natural log of the Kelvin term, and equilibrium RH pre- the RH values calculated using the two different tech- dicted by Eq. (9) using the Pitzer model for ⌽ to the niques is shown in the right column. The agreement assumption of volume additivity. Calculations show that between the ZSR-predicted and reference model±pre- differences in xs when volume additivity is assumed lead
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to maximum errors in ⌽ of about 1%, with similar errors (21), were compared with those from Eq. (7). Calcu-
propagating to aw. The Kelvin term depends on xs lations were performed for RH values between 80% and through drop and exhibits relatively larger error than slightly over 100% for all of the compositions inves- aw. However, for RHs below 95% the Kelvin term has tigated in this work as well as for H2SO4. For H2SO4, a relatively smaller effect on the predicted equilibrium the two neglected terms in Pitzer's model are important, RH than the Raoult term, so the in¯uence of the error and poor agreement was found between the full and in the Kelvin term on RH is very small. The errors in simpli®ed versions of ⌽ at RHs below 99%. Therefore, equilibrium RHs due to the assumption of volume ad- the proposed simpli®ed version of ⌽ cannot be applied
ditivity are between 0.6% and Ϫ0.9%, which is smaller to particles composed of H2SO4. For NaCl, NH4HSO4, than uncertainties in the reference model RH values. (NH4) 2SO4,NH4NO3, and the internally mixed solutes The differences between the reference RHs and RHs examined here, Eq. (21) reproduced the values of ⌽ predicted by the modi®ed KoÈhler model assuming vol- from the full Pitzer model within 10% for all compounds
ume additivity were found to decrease with increasing except NH4HSO4 and internally mixed NaCl±Na2SO4. RH, consistent with the fact that as the solution becomes The worst agreement was for NH4HSO4, where ⌽ was more dilute, the assumption that the volumes may be overpredicted by as much as 19% at an RH of 80%. added becomes more valid. Based on the results of this For the binary solutions examined, the simpli®ed model analysis, the assumption of volume additivity is made for ⌽ tended to overpredict ⌽ for RHs between 80% in this work, therefore, the molar volume of pure water and 92%. Overpredictions of ⌽ will result in under-
is used in place of l. predictions of Scrit since the larger values of ⌽ will ef- fectively increase the solute effect, resulting in a lower b. Assumption of ϭ predicted equilibrium RH for a given solution compo- drop w sition. The maximum errors in ⌽ corresponded to small- As shown in Fig. 1, for the solute mass fraction range er maximum errors, within Ϫ2.5%, in predicted values encountered in HTDMA studies, the difference between of aw and RH. The maximum errors in aw and RH were and can be as high as 28% for NH NO . This drop w 4 3 found for NH4HSO4 and (NH4) 2SO4, and were due to difference is similar to the maximum difference noted the relatively poorer approximation made in using Eq. by Schulmann et al. (1996) between the surface tensions (21) to determine ⌽. At 92% RH, the maximum errors of various organic acid aqueous solutions and pure wa- in aw and RH were Ϯ0.5%. ter. Assuming drop ϭ w only affects the Kelvin term Assuming volume additivity in Eq. (4), and an using and equilibrium RH. Although the errors in the Kelvin relations for the masses of solute (ms) and water (mw) term are similar to the differences between drop and w in the droplet, the molality, m, may be written in Fig. 1, the associated error in RH is small because 3 the Kelvin term is relatively smaller than the Raoult 1000sp,D sol m ϭ , (22) term. As the solution becomes more dilute, the as- 33 Msw (Ddrop Ϫ Dp,sol) sumption that drop ϭ w becomes more valid. The max- imum error in RH incurred by assuming drop ϭ w where w is the density of pure water. We can de®ne a (Ϫ1.5%) is less than the maximum uncertainty in the coef®cient c as reference model RH values, therefore we assume that 3 , equivalent to setting b 0 in the reference 1000Dp,sol drop ϭ w ϭ c ϭ , (23) 33 KoÈhler model. w(Ddrop Ϫ Dp,sol) which can be calculated for each HTDMA (D , RH) c. Parameterization of ⌽ drop data pair for a known Dp,sol. The solution molality may The full model for ⌽ [Eq. (7)] is too complex to be be rewritten in terms of c, easily applied to determining S from HTDMA study crit c results. Pitzer and Mayorga (1973) suggest that for most m ϭ s , (24) solutes and solution ionic strengths below 6 molal, the Ms term involving m 2 is very small and may be neglected; therefore, we do so. Furthermore, our investigations and the ionic strength of a solution with a solute dis- show that for solution ionic strengths encountered in sociating into two ions may be written our studies, the 1 term is very small compared to the 11 c  term; therefore, we choose to set  ϭ 0. With the I ϭ mz2 ϭ [z22 ϩ z ]s , (25) 0 1 ii 11 22 above simpli®cations, the original expression for ⌽, Eq. 22 Ms (7), becomes since the molality of each of the ionic species in solution (m ) is just the solution molality (m) multiplied by the ͙I 212 0 i ⌽ϭ1 Ϫ |zz12| A ϩ m . (21) number of ions from the dissociation of one molecule []1 ϩ bpit͙I of the solute (i). Substituting the above expressions for Values of ⌽ predicted by the truncated expression, Eq. m and I into Eq. (21) we ®nd
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TABLE 4. Values of Y ϭ s /Ms, 0, Ya,Yb,Yc,Yf, and 0, f for solute chemical compositions examined in this work. The units of all Y Ϫ3 Ϫ3 parameters are 10 m and 0 is dimensionless. Values of Yf and 0, f were determined by ®tting the product Yf ⌽f to calculated values of the product Y ⌽.
Solute Y 0 Ya Yb Yc Yf 0,f NaCl 74.09 0.1082 74.1 74.1 18.52 77.4 0.018
(NH4)2SO4 40.16 0.04763 321.3 80.3 8.93 29.1 0.008
NH4HSO4 30.93 0.04494 30.93 30.93 7.73 33.3 0.006
H2SO4 56.35 Ϫ0.0933 450.8 112.7 12.5 2.0 21.8
NH4NO3 43.10 Ϫ0.0154 43.1 43.1 10.78 44.1 Ϫ0.004
NaCl±(NH4)2SO4 49.2 0.078 197.7 77.2 13.7 43.1 0.013
(NH4)2SO4±NH4NO3 41.3 0.016 182.2 61.7 9.9 36.3 Ϫ0.002
NaCl±NH4NO3 54.5 0.046 58.6 58.6 14.6 57.6 0.006
NaCl±Na2SO4 62.5 0.064 263.5 93.7 15.6 45.9 0.03
1 c s [z22 ϩ z ] s Y ϭ . (31) 11 22 Ms 2 Ms Ί ⌽ϭ1 Ϫ |zz12| A Values of Y are listed in Table 4. The various Y differ 1 c i 1 ϩ b [z22 ϩ z ] s from Y by a constant that depends on the ratio of cations pit2 1 1 2 2 M Ί s to anions for a given solute. Although a single de®nition of the combination of i, zi, and s/Ms cannot be sub- 212 0 sc ϩ . (26) stituted into Eq. (30) for ⌽ for the different Yi,we Ms substitute a single unknown Yf for each of the Yi and assume that the retained functional dependence on /M Equations (25) and (26) are explicitly written for a solute s s of the resulting parameterization is suf®cient to accu- that dissociates into two ions in solution. rately model the variation of ⌽ with RH for HTDMA In an effort to further simplify the form for ⌽ given conditions and to predict Scrit. This assumption is tested by Eq. (21), we de®ne different parameters, Yi, and substitute them into Eq. (26). In the numerator of the below in our numerical simulations of the derivation of S . In the modi®ed KoÈhler equation, the product ⌽ Y c Debye±HuÈckel term, a parameter Y can be de®ned for crit f f a ϭ m⌽ will be substituted for the product of molality a solute molecule that dissociates into two ions, f and ⌽. With all of the aforementioned assumptions, the 22 2s parameterized form for ⌽ becomes Ya ϭ |zz12|[ 11z ϩ 22z ] , (27) Ms 1/2 1/2 Ac Yf while in the denominator a parameter Y can be de®ned ⌽ϭf 1 Ϫϩ2c0,ffY , (32) b ͙2 ϩ bc1/2 Y 1/2 as pit f
where ⌽f denotes the proposed simpli®ed form for ⌽, 22s Yb ϭ [11z ϩ 22z ] . (28) and Yf and  0, f are the only unknowns and depend only Ms on the dry particle chemical composition. Note that in
In the  0 term, a parameter Yc can be de®ned as the above equation for ⌽f , the parameter  0, f no longer has the same value as Pitzer's  0 coef®cient if the single 12 s de®nition of Yf is used. Although bpit, another adjustable Yc ϭ . (29) Ms parameter in Pitzer's model, would also be different in ⌽ , we assume the same value used by Pitzer. The theoretically correct values of Y , Y , and Y are f a b c To test the validity of assuming Y ϭ Y ϭ Y ϭ Y shown in Table 4. Substituting the above de®nitions for f a b c in Eq. (32), we ®t the product Y ⌽ to data of Y⌽ as a each of the Y into Eq. (26), we ®nd Eq. (30), equivalent f f i function of x and determined best-®t values of Y and to Eq. (21): s f  0, f . The product Yf ⌽f was ®t since it is this product A͙Yca that appears in the relation for aw. All terms are retained ⌽ϭ1 Ϫϩ2Ycc 0. (30) in the osmotic coef®cient expression used to calculate ͙2 ϩ bpit͙Ycb the product Y⌽. The Dp,sol, Ddrop pairs were chosen for
The above de®nitions of Ya, Yb, and Yc have the com- a given solute chemical composition and ⌽ calculated mon factor s/Ms and prefactors involving i and zi that as a function of molality using the full Pitzer model. depend upon the chemical composition of the solute. The ®ts were performed for molality ranges expected Furthermore, the common factor also appears in the during HTDMA conditions, between approximately 1
coef®cient of ⌽ in the reference KoÈhler model [Eq. (9)] and 6 molal. Best-®t values of Yf and  0, f are shown
through xs and Ms. We de®ne the following unknown in Table 4 for each solute. Except for H2SO4, Yf tends variable to be close to Y, suggesting that some information about
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the chemical composition of the particle nuclei can be Dp,sol, Ddrop, RH, and T are measured during HTDMA inferred from the ®t of Yf ⌽f to Y⌽ data. studies; w, w, and A are a function of T only; and c Differences were examined between values of aw and is known if Dp,sol, w, and Ddrop are known. This leaves RH predicted using the full model for Y⌽ and those Yf and  0, f as the two chemical composition±dependent calculated using the best-®t parameters in Yf ⌽f . The unknowns that remain to be determined. By ®tting Eq. best-®t values of Yf and  0, f shown in Table 4 were (33) to (Ddrop, RH) data pairs determined from HTDMA used in Eq. (32) to calculate ⌽f , and the product of c, studies performed on the same monodisperse dry par- Yf , and ⌽f was then used in place of the product of m ticle size at different RHs, the two unknowns, Yf and and ⌽ to calculate aw [Eq. (2)]. This value of aw was  0, f may be determined. Once known, the values of Yf substituted into Eq. (1) to examine how errors in aw and  0, f may be used in Eq. (33) to ®nd the maximum propagated to errors in RH. The maximum errors in aw equilibrium RH, which is Scrit. and RH were between ϩ1% and Ϫ2.5%, similar to the Another, simpler KoÈhler model was proposed by uncertainties in the reference model values for these Weingartner et al. (1997) to ®t HTDMA study results parameters. Based on these results, we accept the pro- in order to derive Scrit. It is of interest to compare the posed parameterization of ⌽f , Eq. (32), for use in the results from their model with those from the model pro- modi®ed KoÈhler model. posed in this work. Their model took the form
␣ ϪN d. Combined assumptions of ϭ , ϭ , RH ϭ 100 exp expi , (35) l w drop w DD33Ϫ D and ⌽ϭ⌽ [][drop drop p,sol ] f where was taken equal to 2.155 nm, was set equal In Fig. 3 we explore errors in the various parameters ␣  to 5.712 10Ϫ2 nm3, and the droplet and particle sizes when all of the aforementioned assumptions are made ϫ were in nanometers (Weingartner et al. 1997). The num- simultaneously. Each chemical composition is repre- ber of dissociated molecules in the water layer of the sented by a different symbol in the ®gure. Results are droplet, N , was the one unknown in the model. By reported as percent error relative to the values from the i comparison of Eqs. (35) and (33), N is de®ned as reference model for the same dry and wet particle sizes. i The errors in a and RH are between 0% and 1% and w N ϵ YN D3 , (36) Ϯ0.5%, respectively, for all compounds except i av 6 p,sol NH4HSO4. The errors for NH4HSO4 are relatively larg- er due to the relatively poorer approximation made for where Nav is Avogadro's number and Y is de®ned by this compound, as discussed above. Based on the results Eq. (31). Implicit in the form of Eq. (36) is the as- shown in Fig. 3, except for NH4HSO4, the modi®ed sumption that ⌽ is constant and does not vary with RH, KoÈhler model is shown to correctly predict RH and aw whereas the variation of ⌽ with RH is built into the within reference model uncertainties (Ϯ3%) and ex- modi®ed KoÈhler model proposed in this work [Eq. (33)]. perimental uncertainties in RH (Ϯ1%) during HTDMA Below, we compare theoretical values of Scrit predicted studies. by the reference KoÈhler model to values of Scrit derived by ®tting Eqs. (33) and (35) to (Ddrop, RH) data pairs representing hypothetical HTDMA study results. e. Modi®ed form of KoÈhler equation Substituting the above relation for ⌽ [Eq. (32)] into f 6. Estimating Scrit using simulated HTDMA data Eq. (9) and incorporating the simpli®cations discussed in the previous section, the KoÈhler equation for a droplet For the numerical sensitivity studies, a dry particle containing a single, completely dissociated solute may size and composition were chosen. The reference KoÈhler be written model was used to calculate the theoretical equilibrium RH for assumed droplet sizes. The (Ddrop, RH) data pairs and the assumed dry particle size were used in a non- 4wwM RH ϭ 100 exp linear, least squares ®t routine to derive values of Y and RTD f []w drop  0, f in Eq. (33) and Ni in Eq. (35). Two input data pairs 1/2 3/2 ϪMc Ac Yf were insuf®cient to capture the curvature of the KoÈher w 2 ϫ exp Yf Ϫϩ2c0,ffY , curve. Ten pairs were used for each of the sensitivity 1000 2 bc1/2 Y 1/2 []͙ ϩ pit f studies described below. Trial and error sensitivity stud- (33) ies on the minimum number of input data pairs required to maintain acceptable error in derived values of S where c is de®ned as crit from Eq. (33) showed that 3 input data pairs equally 3 1000Dp,sol spaced between 80% and 92% RH produced the same c ϵ . (34) error in S as 10 data pairs over the same RH range. (D33Ϫ D ) crit w drop p,sol Four different cases were investigated. First, we de-
In the above equations, Mw, R, and bpit are constants; termined Yf and  0, f from a ®t of Eq. (33) with perfect
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input values for Dp,sol, Ddrop, and RH. Second, we ver- The relatively larger value for  0, f is due to the lower i®ed that the results from the ®t routine were not sen- sensitivity of the goodness of ®t to the value of this sitive to the initial guess for Yf and  0, f . Third, we parameter compared to Yf . Tests of the sensitivity of determined the sensitivities of the derived values for Yf the ®t routine to assumed initial conditions showed it and  0, f to assumed Ϯ0.5% random errors in RH, Ddrop, to be insensitive, therefore, we report results from these and Dp,sol. Fourth, we examined the sensitivity of de- cases for NaCl only. For the cases with Dp,sol ϭ 100 rived values of the unknowns to Ϯ1% random error in nm, there were no signi®cant differences between re- RH, dry particle size, and droplet size. The random sults where Ϯ0.5% and Ϯ1% random errors were ap- errors in the three input parameters were applied si- plied; therefore, only results for cases with Ϯ1% errors multaneously, and the values chosen were guided by are shown. expected experimental error. As discussed in Part II, the Values of the critical droplet diameter (Dcrit) are observed, average experimental uncertainty in RH dur- shown in Tables 5 and 6 for the single and mixed solutes ing laboratory HTDMA studies was 0.7% Ϯ0.2%, there- examined. The results demonstrate that the ®t routine fore, the 0.5%±1% range of applied random RH error derived values of Yf and  0, f return values of Dcrit that is relevant for investigating the in¯uence of RH uncer- agree with theoretical values within Ϫ10% and 12% tainties on ®t routine results. The model of Weingartner when used in the modi®ed KoÈhler model. Also shown et al. (1997) and that proposed in this work were run in the tables is the 2 convergence value of the ®t in simultaneously only for the fourth case, and dry particle units of RH. In general, the 2 values are smaller than sizes of 40 and 100 nm were examined. For studies on 0.5%; however, for NH4NO3 the values for the case with internally mixed particles, only the cases described perfect input data are larger. This is most likely due to above with perfect input and with assumed Ϯ1% ran- the poorer approximations made by assuming volume dom uncertainties applied to all input data were inves- additivity (xs, Fig. 3) and drop ϭ w (Fig. 1) for tigated. These uncertainties were chosen to re¯ect those NH4NO3 compared to other solutes. expected in experimental data. Rader and McMurry The percent errors in derived values of Scrit are listed (1986) have studied the sizing errors in the TDMA. in Table 5 for the ®ve different single solute compo- They determined that the error in the measured midpoint sitions examined and in Table 6 for the mixed solutes diameter in the second DMA was 0.11% Ϯ 0.06% for examined. For comparison, values of Scrit from the ref- particle diameters less than 200 nm. For ¯ow uncer- erence KoÈhler model are listed in Table 3 for the various tainties in the HTDMA measurement system, the esti- dry particle sizes and chemical compositions examined. mated experimental uncertainty in particle sizing is be- Results of the ®t routine using the modi®ed KoÈhler mod- tween 0.2% and 1%. el of this work and applied error have been plotted as A single case study of the effect that a single set of solid dots in Fig. 4, with the solid line designating the applied random errors has on values of Yf ,  0, f , and Scrit 1:1 line. The results from the model of Weingartner et from the ®t procedure would be less representative of al. (1997) with applied error are shown as squares in the average in¯uence of experimental uncertainties than the ®gure and are discussed below. The plus symbols averaging the results from several cases of assumed ran- designate results where no errors were applied to the dom errors. Therefore, 100 cases were run on the same input data. Results from the model of this work and an input Dp,sol, Ddrop, and RH data for each assumed range assumed 40-nm-diameter particle have been labeled in of random errors in order to obtain a statistically sig- Fig. 4 with the corresponding chemical composition for ni®cant estimate of the in¯uence of uncertainties on the particles composed of single solutes. Results from the derived values of Yf and  0, f . model with internally mixed solutes are not labeled in Results from our numerical simulations are shown Fig. 4. As can be seen from the ®gure, the agreement in Table 5 for dry diameters of 40 and 100 nm. The between theoretical and predicted values of Scrit is very values of Yf and  0, f in Table 5 for the cases of perfect good. Also, the results for the cases where no errors input, are close to, but not exactly the same as values were applied are similar to those with error. Therefore, in Table 4. The reason for this is that the values in the assumed uncertainties in the input data do not ad-
Table 4 were determined by ®tting the product of Yf ⌽f versely affect the ability of the ®t routine and modi®ed to values of Y⌽ as described above, whereas the values KoÈhler model to predict accurate values of Scrit. The in Table 5 were determined by ®tting the modi®ed errors in predicted values of Scrit are between Ϫ20% and KoÈhler model to (Ddrop, RH) data pairs calculated using ϩ15%, similar to the uncertainties in experimental es- the reference model. The values of Yf and  0, f shown timates of Scrit determined from CCN studies on known in the table for cases where errors were applied are particles (Ϫ0.6%, 1 ϭ 11%, n ϭ 16; Brechtel and obtained by averaging the results from the 100 different Kreidenweis 2000). For comparison, the supersatura- cases of applied error. Ratios of the standard deviation tions in a typical stratocumulus cloud can vary between computed from the 100 different cases to the average 0.1% and 1% (Rogers and Yau 1989). Although cloud values of Yf ,  0, f , and Scrit were less than 0.2, 1.3, and supersaturations cannot be measured directly, Yum et 0.16, respectively. The small standard deviations for al. (1998) have estimated supersaturations and standard
Yf and Scrit indicate that the ®t routine is quite stable. deviations of supersaturations for stratocumulus clouds
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TABLE 5. Sensitivity of ®t routine results as a function of particle composition, initial guess and assumed Dp,sol, RH and Ddrop errors for 40- and 100-nm dry particle diameters and indicated compositions. Ratios of the standard deviation computed from the 100 different cases of applied error to the average values of Yf, 0, f , and Scrit were always less than 0.2, 1.3, and 0.16, respectively. Input RHs were calculated using the reference model. The simpli®ed KoÈhler model was run with initial conditions equal to ®ve times the perfect initial conditions for the cases including assumed uncertainties.
Dp,sol Scrit Error Dcrit Error 2 Case (nm) Yf 0,f (%) (%) (nm) (%) Sodium Chloride Perfect input 40 81.653 0.01488 0.43 Ϫ10 315 Ϫ1 5.22 ϫ 10Ϫ1 5 ϫ ICs 40 81.653 0.01488 0.43 Ϫ10 315 Ϫ1 5.22 ϫ 10Ϫ1 0.5% error 40 80.944 0.01532 0.43 Ϫ10 316 Ϫ1 3.67 ϫ 10Ϫ2 1% error 40 78.504 0.01754 0.43 Ϫ9 313 Ϫ2 3.90 ϫ 10Ϫ1 Perfect input 100 80.098 0.01598 0.11 Ϫ9 1265 Ϫ1 7.14 ϫ 10Ϫ1 1% error 100 78.349 0.01838 0.11 Ϫ8 1251 Ϫ2 3.63 ϫ 10Ϫ1 Ammonium Sulfate Perfect input 40 28.723 0.01023 0.65 Ϫ8 208 0 2.84 ϫ 10Ϫ1 0.5% error 40 28.549 0.01059 0.65 Ϫ7 208 0 4.02 ϫ 10Ϫ2 1% error 40 28.367 0.01086 0.65 Ϫ7 206 Ϫ1 4.88 ϫ 10Ϫ1 Perfect input 100 28.295 0.01093 0.16 Ϫ5 839 Ϫ4 2.66 ϫ 10Ϫ1 1% error 100 26.889 0.01374 0.17 Ϫ2 816 Ϫ7 3.62 ϫ 10Ϫ1 Ammonium Bisulfate Perfect input 40 35.801 0.00064 0.58 Ϫ20 232 10 7.48 ϫ 10Ϫ3 0.5% error 40 35.653 0.00068 0.58 Ϫ20 231 10 3.70 ϫ 10Ϫ2 1% error 40 34.939 0.00099 0.59 Ϫ19 229 9 3.75 ϫ 10Ϫ1 Perfect input 100 34.784 0.00158 0.15 Ϫ20 929 12 1.09 ϫ 10Ϫ3 1% error 100 32.987 0.00307 0.15 Ϫ19 918 10 3.84 ϫ 10Ϫ1 Sulfuric Acid Perfect input 40 37.340 0.05529 0.56 Ϫ7 243 1 8.14 ϫ 10Ϫ1 0.5% error 40 36.843 0.05864 0.56 Ϫ6 241 0 9.18 ϫ 10Ϫ2 1% error 40 33.323 0.08634 0.59 Ϫ1 232 Ϫ2 3.82 ϫ 10Ϫ1 Perfect input 100 35.986 0.06038 0.14 0 951 Ϫ8 1.46 1% error 100 34.050 0.08118 0.15 1 934 Ϫ10 3.80 ϫ 10Ϫ1 Ammonium Nitrate Perfect input 40 44.386 Ϫ0.00418 0.52 Ϫ2 258 1 4.25 0.5% error 40 44.201 Ϫ0.00419 0.52 Ϫ2 258 1 1.91 ϫ 10Ϫ2 1% error 40 42.899 Ϫ0.00415 0.53 0 252 0 3.31 ϫ 10Ϫ1 Perfect input 100 44.073 Ϫ0.00388 0.13 Ϫ2 1047 1 1.44 1% error 100 42.812 Ϫ0.00377 0.13 Ϫ2 1045 1 3.36 ϫ 10Ϫ1
from measurements of CCN spectra, cloud droplet spec- H2SO4, especially given the poor agreement between tra, and other parameters. The results of Yum et al. values of ⌽ from the full Pitzer model and values cal- (1998) showed that predicted standard deviations in su- culated using the simpli®ed model of ⌽. The values of persaturations were between 40% and 100% of the pre- Yf and  0, f in Table 5 are signi®cantly different from dicted supersaturations. The reported standard devia- the values of Y and  0 and the values of Yf and  0, f in tions may be representative of the variability in super- Table 4. Even though the parameterization for ⌽ does saturations in clouds, and we note that the modi®ed not accurately reproduce the actual behavior of ⌽ for
KoÈhler model of this work can predict values of Scrit H2SO4, for this compound ®t parameters could be de- with lower uncertainty. rived that allowed the modi®ed KoÈhler model to ac-
In Fig. 4, the Scrit values from the modi®ed KoÈhler curately reproduce the droplet growth with increasing model of this work for particles composed of single RH as well as values of Scrit. The results in Fig. 4 dem- solutes are typically underpredicted by the ®t routine. onstrate that for a single dry particle size, the various This is caused by the systematic underprediction of the compositions examined correspond to about a factor of
Kelvin term due to the assumption that the surface ten- 2 change in Scrit. Therefore, particle chemical compo- sion of the solution is equal to that for pure water and sition plays an important role in determining critical by the overprediction of ⌽ by the assumed simpli®ed supersaturation. form, ⌽f . The average agreement between predicted and The largest errors in predicted values of Scrit are found theoretical values of Scrit for all compounds listed in for NH4HSO4 and internally mixed NaCl±Na2SO4. This Tables 5 and 6 is Ϫ7.5% (1 ϭ 10%, n ϭ 9). A sur- is not surprising for at least three reasons. First, it is prising result in Table 5 is the relatively good agreement possible that the modi®ed KoÈhler model is less capable between predicted and theoretical values of Scrit for of capturing the hygroscopic behavior of certain particle
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TABLE 6. Sensitivity of ®t routine results as a function of particle composition, and assumed Dp,sol, RH, and Ddrop errors for 40- and 100- nm dry particle diameters and indicated equimolar mixtures of solutes. Ratios of the standard deviation computed from the 100 different cases of applied error to the average values of Yf, 0, f , and Scrit were always less than 0.2, 1.3, and 0.16, respectively. The input RHs used in the ®t routine were calculated using the reference model. The simpli®ed KoÈhler model was run with initial conditions equal to the perfect initial conditions for the cases including assumed uncertainties.
Dp,sol Scrit Error Dcrit Error 2 Case (nm) Yf 0,f (%) (%) (nm) (%) Sodium Chloride±Ammonium Sulfate Perfect input 40 43.140 0.01317 0.74 4 183 10 0.589 With 1% error 40 40.835 0.01727 0.76 7 177 6 0.376 Perfect input 100 42.096 0.01456 0.19 5 728 0 0.691 With 1% error 100 40.336 0.01719 0.19 6 718 Ϫ1 0.337 Ammonium Sulfate±Ammonium Nitrate Perfect input 40 36.247 Ϫ0.00233 0.81 3 166 12 0.148 With 1% error 40 34.850 Ϫ0.00169 0.83 6 162 10 0.301 Perfect input 100 35.611 Ϫ0.00183 0.20 4 670 1 0.119 With 1% error 100 33.536 Ϫ0.00043 0.21 6 654 Ϫ1 0.279 Sodium Chloride±Ammonium Nitrate Perfect input 40 57.578 0.00625 0.64 0 210 12 0.107 With 1% error 40 55.991 0.00734 0.64 0 210 12 0.336 Perfect input 100 56.508 0.00717 0.16 Ϫ1 848 5 0.171 With 1% error 100 54.910 0.00823 0.16 Ϫ1 845 5 0.302 Sodium Chloride±Sodium Sulfate Perfect input 40 45.869 0.03092 0.71 13 190 0 1.52 With 1% error 40 45.040 0.03302 0.72 15 188 Ϫ1 0.379 Perfect input 100 46.140 0.02974 0.18 13 766 Ϫ8 1.01 With 1% error 100 44.288 0.03403 0.18 15 758 Ϫ9 0.432
chemical compositions compared to others. Second, the
full Pitzer model of ⌽ is less accurate for NH4HSO4 (Kim et al. 1993). Third, (other than H2SO4)NH4HSO4 and internally mixed NaCl±Na2SO4 compositions ex- hibited the poorest agreement between values of ⌽ from the full Pitzer model and values calculated using the simpli®ed model of ⌽. The underprediction of ⌽ by the
simpler model for internally mixed NaCl±Na2SO4 re- sults in an overprediction of Scrit.
Comparison of results from different KoÈhler models The same numerical sensitivity studies conducted with the modi®ed KoÈhler model proposed here were performed on the model proposed by Weingartner et al. (1997). Results from their model are shown in Fig. 4 as squares and demonstrate reasonable agreement with theory. For 40-nm particles, the average agreement be-
tween predicted values of Scrit and theoretical values was Ϫ2.6% (1 ϭ 16%, n ϭ 9) for the model of Weingartner et al. and Ϫ1.1% (1 ϭ 10%, n ϭ 9) for the model of this work. Results for 100-nm particles were similar. The model of Weingartner et al. performs most poorly for solution droplets formed on particles composed of
NH4NO3,H2SO4, and internally mixed (NH4) 2SO4± FIG. 4. Values of Scrit from proposed KoÈhler model and ®t routine vs theoretical values of Scrit for 40- and 10-nm-diameter particles and NH4NO3. Errors in Scrit were between 18% and 26% for all chemical compositions examined in this work. Solid dots represent these compounds. The osmotic coef®cient varies more results from KoÈhler model of this work with applied error and squares strongly with RH for NH NO and H SO , and exhibits represent results from model of Weingartner et al. (1997) with applied 4 3 2 4 error. Solid line is 1:1 line. Plus symbols represent results with no a value far from unity (ഠ0.6) for NH4NO3,H2SO4, and applied errors to the input data. Results for 40-nm internally mixed (NH4) 2SO4±NH4NO3 over the RH range typical of solutes are not labeled. HTDMA experiments. The other compounds studied
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here exhibit osmotic coef®cient values that are closer posed of NaCl, (NH 4 ) 2SO 4 ,NH4HSO 4 ,H2SO 4 , to unity and do not vary as much as those for solutions NH4NO3, and for the interally mixed compositions ex- containing NH4NO3 and (NH4)2SO4±NH4NO3. There- amined. Comparisons between values of Scrit predicted fore, the simpler model of Weingartner et al. is a rea- by the KoÈhler model proposed in this work and the sonable approach for solutions with osmotic coef®cients simpler model proposed by Weingartner et al. (1997) that do not vary signi®cantly over the range of RHs for indicated that both models produced results that agreed
HTDMA studies and that are not signi®cantly different well with theoretical values of Scrit, except for com- from the value of ⌽ for solution droplets at their critical pounds where the osmotic coef®cient varied signi®- size (at Scrit). This model would predict less accurate cantly from its value at Scrit. In those cases, the model values of Scrit for droplet solutions of unknown chemical proposed in this work predicted Scrit more accurately. composition where the above conditions may not exist. Numerical sensitivity studies showed that values of Scrit Although the model proposed in this work also has lim- can be predicted within Ϫ7.5% Ϯ 10% using the pro- itations, the inclusion of an additional adjustable pa- posed technique, similar to observed experimental un- rameter expands its range of applicability. certainties in Scrit determined from laboratory CCN stud- ies discussed in Part II (Brechtel and Kreidenweis 2000). 7. Conclusions Acknowledgments. This work was sponsored by the A technique for using hygroscopic growth measure- Environmental Protection Agency under STAR Grad- ments at RHs below 100% to derive the critical super- uate Fellowship U-914726-01-0. The comments of J. saturation required to activate a particle to a cloud drop- Hudson and an anonymous reviewer are greatly appre- let has been described. A modi®ed KoÈhler model is ciated. derived, including a parameterization for the solution osmotic coef®cient that approaches the correct ther- modynamic limit as the solution becomes more dilute. REFERENCES The method involves ®tting the proposed KoÈhler model to (Ddrop, RH) data pairs obtained from HTDMA studies Atkins, P. W., 1994: Physical Chemistry. Oxford University Press, on a single dry particle size. Chemical composition± 1031 pp. Berg, O. H., E. Swietlicki, and R. 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