Universal Approximation Formulas of Curvature Correction and Equilibrium Sizes of Hygroscopic Particles

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  • 1 Department of Atmospheric Science, Colorado State University, Fort Collins, Colorado
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Abstract

Theoretical analysis shows that when water activity is larger than its threshold value and the dry radius of a particle is larger than 0.005 µm, the deviation of curvature correction from unity can be accurately represented by the product of two terms, with one term strongly depending upon water activity and the other depending upon dry radius. Moreover, experimental data show that the water-activity-dependent term can be approximated by linear and one-third power functions of water activity. According to the approximation made to curvature correction, water activity is solved as analytical functions of relative humidity (RH). The analytically solved water activity is then used to compute particle equilibrium sizes using a known (observed) relationship between water activity and water uptake by unit mass of dry material. The accuracy of equilibrium sizes calculated with this method is checked with seven typical classes of aerosols. Results show that when RH ≤ 99.99%, the equilibrium radius computed with this method is accurate to within 3% (6%) if the dry radius of a particle is larger (smaller) than 0.02 µm and that when RH > 99.99%, equilibrium sizes can be estimated with an accuracy higher than 84%. Analytical approximation formulas are also derived for calculating critical equilibrium radii and critical supersaturation. The maximum relative errors of these formulas range from 3% to 15%.

Abstract

Theoretical analysis shows that when water activity is larger than its threshold value and the dry radius of a particle is larger than 0.005 µm, the deviation of curvature correction from unity can be accurately represented by the product of two terms, with one term strongly depending upon water activity and the other depending upon dry radius. Moreover, experimental data show that the water-activity-dependent term can be approximated by linear and one-third power functions of water activity. According to the approximation made to curvature correction, water activity is solved as analytical functions of relative humidity (RH). The analytically solved water activity is then used to compute particle equilibrium sizes using a known (observed) relationship between water activity and water uptake by unit mass of dry material. The accuracy of equilibrium sizes calculated with this method is checked with seven typical classes of aerosols. Results show that when RH ≤ 99.99%, the equilibrium radius computed with this method is accurate to within 3% (6%) if the dry radius of a particle is larger (smaller) than 0.02 µm and that when RH > 99.99%, equilibrium sizes can be estimated with an accuracy higher than 84%. Analytical approximation formulas are also derived for calculating critical equilibrium radii and critical supersaturation. The maximum relative errors of these formulas range from 3% to 15%.

DECEMBER 1996 YANG AND COTTON 2261Universal Approximation Formulas of Curvature Correction and Equilibrium Sizes of Hygroscopic Particles SHUOWEN YANG AND WILLIAM R. COTTONDepartment of Atmospheric Science, Colorado State University, Fort Collins, Colorado(Manuscript received 1 November 1995, in final form 30 May 1996)ABSTRACT Theoretical analysis shows that when water activity is larger than its threshold value and the dry radius of aparticle is larger than 0.005/~m, the deviation of curvature correction from unity can be accurately representedby the product of two terms, with one term strongly depending upon water activity and the other dependingupon dry radius. Moreover, experimental data show that the water-activity-dependent term can be approximatedby linear and one-third power functions of water activity. According to the approximation made to curvaturecorrection, water activity is solved as analytical functions of relative humidity (RH). The analytically solvedwater activity is then used to compute particle equilibrium sizes using a known (observed) relationship betweenwater activity and water uptake by unit mass of dry material. The accuracy of equilibrium sizes calculated withthis method is checked with seven typical classes of aerosols. Results show that when RH ~ 99.99%, theequilibrium radius computed with this method is accurate to within 3% (6%) if the dry radius of a particle islarger (smaller) than 0.02 ,ttm and that when RH ) 99.99%, equilibrium sizes can be estimated with an accuracyhigher than 84%. Analytical approximation formulas are also derived for calculating critical equilibrium radiiand critical supersaturation. The maximum relative errors of these formulas range from 3% to 15%.1. Introduction Equilibrium sizes of hygroscopic particles have beenextensively studied by a number of investigators (e.g.,Orr et al. 1958; Fletcher 1962; Mason 1971; Junge andMclaren 1971; Winkler 1973; Fitzgerald 1975; H/inel1976; Pruppacher and Klett 1978; H~inel and Lehman1981; Tang 1980; d'Almeida et al. 1991; Tang andMunkelwitz 1994). The K6hler equation (Pmppacherand Klett 1978) associates the equilibrium radius of aparticle with its dry radius, chemical composition, andambient relative humidity (RH). Because the KOhlerequation is a nonlinear equation, iterative methodswere used to obtain an accurate solution of equilibriumradius. Computation of equilibrium radius by iterativemethods is very time consuming when a large numberof such computations are involved, as, for example, incertain studies associated with fog formation, cloud nucleation processes, and remote sensing applications. To facilitate the ~omputation of the equilibrium sizeof a hydrophilic particle, Kasten (1969) proposed asimple radius-RH formula for quantitative predictionof visibility, but the formula can only be applied to thecases in which RH ~< 95% and dry radii are larger than Corresponding author address: Dr. Shuowen Yang, Dept. of Atmospheric Science, Colorado State University, Fort Collins, CO80523.E-mail: yang @ tofu.atmos.colostate.edu0.1 pm. Fitzgerald (1975) developed approximationformulas that express the equilibrium radius as an explicit function of relative humidity and dry radius.These formulas are accurate to within 5% for RH between 81% and 99.5% if a particle has a dry radius inthe range of 0.05-3.0 tzm and is composed greater than60%, by mass, of any one of the common electrolytes.H~nel (1976) also proposed an explicit formula [Eq.(6.2) in his paper] to calculate the equilibrium radiusof a particle. For maritime and urban aerosols, the approximation formula is accurate to within 3% if RH isin the range of 70%-99% and the dry radius is greaterthan 0.04 tzm. Obviously, the applicable ranges of theabove-mentioned methods do not cover the entireranges of RH and particle sizes of atmospheric interest(e.g., Feingold and Grand 1994). The purpose of this paper is to present approximation formulas for computing equilibrium sizes ofhygroscopic particles in wide ranges of RH and dryradii. Specifically, this paper demonstrates how toapproximate curvature correction in the KOhler equation as a function of water activity. The paper is organized as follows. In section 2, the H~inel (1976)model is briefly described. Based on this model, insection 3 a universal approximation form of curvature correction is derived theoretically and verifiedwith observed data. According to the universal properties of curvature correction, approximation formulas for water activity are proposed in section 4 andtested with observed data. In section 5 we investigatec 1996 American Meteorological Society2262 JOURNAL OF APPLIED METEOROLOGY VOLUME 35the critical phenomenon, and a summary and conclusions are presented in section 6.2. The H~inel model The Kthler equation (see, e.g., Pruppacher and Klett1978) is the basis for calculating the equilibrium radiusof a hygroscopic particle. However, it is not valid whena particle is not fully covered by a liquid layer, whichhappens when water activity (denoted as aw) is smallerthan a threshold: water activity (denoted as aw,). In order to compute the equilibrium radius of a wet particlein the entire range of water activity from 0 to 1, theH~inel (1976) model is used as a framework in the present paper. The H~inel model associates the equilibrium radiusof a wet particle with relative humidity (denoted as f)by the following equations: r=ro I + P--9--W P~ (1)f = awt, (2)aw = exp(- -), (3) 1, 0 ~ aw ~ awdt,, (4a) 6= --, awt/tt < aw < awt, (4b) awt exp , aw,~<aw~< 1, (4c) be cr = awo- a(T- To) + W ' (5)where r0 is the dry radius of the particle; pw and po are,respectively, the densities of water and dry material ofthe particle; W is the water uptake of a unit mass of drymaterial; r/is the exponential mass increase coefficient;6 is the curvature correction; 6, is the threshold valueof 6; Rw is the specific gas constant of water ( =:4.6151X 106 erg g-t K-I); rr is the surface tension of theparticle; a~0 (=75.6 dyn cm-~) is the surface tensionof pure water at standard temperature To; a is a constant(=0.153 dyn cm-~); b is a constant depending on thechemical properties of water-soluble substance; and eis the mass fraction of soluble material in the dry particle. The c. ombination of (2) and (4c) is the. Kthlerequation. Equations (1)- (5) are not a closed system; an additional equation (or relationship) connecting the unknown variables (r, W, aw, rl, 6, and a) is needed forthe determination of r- As shown by H~inel (1976) andH~inel and Lehman ( 1981 ), measurements can be madeto give a relationship between aw and W or between awand ~7 through tables or analytical formulas. Therefore,if one can obtain aw as a function of f from ( 1 ) - (5),then from the known aw- W relationship one can obtainW and calculate the equilibrium radius according to( 1 ). This paper assumes that the aw-Wrelationship (orequivalently, the aw-r~ relationship) is known and focuses on how to solve aw as a function off. For the trivial case f ~< aw,6,, one can obtain ananalytical solution of aw as a function off. But for thecase f >~ aw,6,, to our knowledge aw cannot be solvedas an analytical function off, due to the nonlinearityof ( 1 ) - (3), (4c), and (5). However, from (2) onecan find that if curvature correction 6 is a simple function of water activity, for example, 6 is a linear or quadratic function of aw, then from.(2) aw can be solved asan analytical function off. To examine whether 6 canbe approximated by simple functions of aw, curvaturecorrection is investigated in the next section.3. Universal approximation form of curvature correction In this section, a universal approximation form ofcurvature correction is derived and verified with sevenrepresentative aerosol samples measured by H~inel andLehman (1981). Curvature correction is approximatedas a simple function of water activity.a. Universal approximation form of curvature correction In the following, curvature correction is studied forthe case aw >~ aw,. Substituting r in (4c) with ( 1 ) yields 6 = (e~)t/~-, (6)where 20 ( 1 po )-~/3 ~ - pw~wr ~ + --Pw W . (7)In (7), R is only a function of W that uniquely dependson water activity (aw), and therefore R solely dependson water activity, not on the dry radius. Because R decreases as W (or a~) increases, from (4) and (6) onecan deduce that the threshold curvature correction of aparticle is also the maximum curvature correction ofthe particle. In H~inel (1976), threshold water activityis assumed to be the value of aw when W = W, = pw/P0, and therefore from (1) and (4) one has rt = 2~/3ro, (8) 2o 6,=exp(- ~ (9) ~,pwRwrr,? 'where r, and 6, represent, respectively, the thresholdradius and the threshold curvature correction. For particles of meteorological interest, curvaturecorrection usually varies between 1 and 1.3. It is thedeviation of curvature correction from unity that makesDECEMBER 1996 YANG AND COTTON 2263TABLE 1. The characterization and physicochemical properties of aerosol samples measured by H~nel and Lehman (1981). Po b Main water-soluble substancesSample Type of aerosol (g cm-3) (dyn c~n-~) e (composition in mass fraction)1 Desert, normal condition 2.59 17.3 0.312 Desert, sand storm 2.72 15.6 0.393 Industrial 1.87 13.2 0.384 Maritime 1.93 27.8 0.896 Background** 3.32 20.5 0.608 Background** 1.92 13.1 0.4010 Background** 1.93 24.6 0.560.55 Na2SO4 + 0.45 NaNO30.59 Na2SO4 + 0.41 NH4NO30.76 NaNO3 + 0.24 H2SO4SS*0.43 NH4NO3 + 0.57 SS0.57 (NH4)2SO4 + 0.23 NH4NO3 + 0.20 H2SO40.19 NH4NO3 + 0.81 SS* SS--Sea salt.** Also combined with particles from surrounding industrial zones and local sources.the difference for different RH and particle size. Todistinctly describe the deviation of 5 from unity, wedefine a scaled deviation of curvature correction fromunity, N0, as 6-1 N0 - (10) St- 1Expanding (10) into Taylor series of R/ro and R,/roand neglecting terms higher than third order give 1+E7- +~ro/where Rt is the value of R at threshold water activity,Rt = R(awO. Note that R ~< R~, and therefore if Rdro~ 1, then ( 11 ) can be further approximated as R No m -- (12) Rt'Since R only depends on aw and R, exclusively on aw,,No in (12) is only a function of aw and independent ofdry radius. The meaning of (12) is that for any particlewith dry radius far larger than Rt, the scaled deviationof curvature correction from unity No follows the sameuniversal form (12) that depends only on water activity. To check the accuracy of (12), it is sufficient toexamine the difference (denoted as DV) between ( 11 )and (12). Subtracting (12) from (ll) gives R DV - (13) Rt 2ro + Rt'It can be shown that the maximum absolute value ofDV (denoted as DVmax) occurs at R = Rmax = 0.5Rtand N0 m 0.5. Therefore, using 5~ = exp(Rt/ro), onecan obtain lmSt DVm= - (14) 8 + 4 ln6, 'Equation (14) indicates that the maximum differencebetween (11) and (12) is proportional to lnSt. From(4c), (5), and (14), one can estimate that for r0= 0.005/~m, (5, m 1.3 and DVma~ = 0.029; and that forro = 0.1/~m, 8t ~ 1.01 and DVmax = 0.0012. Note thatat R = R .... N0 ~ 0.5. Therefore, at R = Rm~, therelative error in No estimated by (12) is about 6% forro = 0.005/2m and 0.2% for ro = 0.1 /~m. Thus, onecan conclude that the universal approximation formula(12) is very accurate for ro ~> 0.005/~m.b. N~-aw relationship from experimental data To see whether the above theoretical analysis is correct and how curvature correction changes with wateractivity, seven typical tropospheric aerosol samplesmeasured by H~nel and Lehman (1981) were used toexamine the N,-a~ relationship. The characterizationand physicochemical properties of aerosol samples arelisted in Table 1. As can be seen in Table 1, the aerosolsamples are composed of a diversity of water-solublesubstances, so these samples provide a wide scope forone to investigate the characteristics of the No-a~ relationship. In the original data of Hfinel and Lehman (1981),the water uptake W of a unit mass of dry aerosol islisted separately for RH increasing and RH decreasingcases to account for the RH hysteresis effect (e.g., Orret al. 1958; H'~nel 1976). In this paper, this hysteresiseffect is not considered. For a given water activity, Wused in this paper is the average of W for the cases ofincreasing and decreasing relative humidity. Figure 1 shows how No changes with aw for particleshaving different sizes in sample 10. In Fig. 1, Noreaches a maximum value of 1 at threshold water activity, in the section where a~ < awt the curves for different particles separate remarkably, and while in thesection where aw >~ awt, the difference in No of differentparticles is very small. The maximum difference between N~ for ro = 0.005/~m and that for r0 = 0.1 t~mis about 0.035, and appears in the region of No ~ 0.5.This is consistent with previous analysis. Figure 2 displays the No-aw relationship for particlesin sample 3. It can be seen in Fig. 2 that the curves fordifferent particles change with a~ in a way much likethat in Fig. 1, although the chemical composition andorigins of the two samples are very different.2264 JOURNAL OF APPLIED METEOROLOGY VOLUME351.2.8.4.2I ~ I i I.6 .7 .8 .9 1 .0 FIG. 1. Scaled deviation of curvature correction from unity (N~)versus water activity (a,~) for particles having different dry radii (ro)in sample 10. Solid line--r0 = 0.1/~m, dash-dotted line--ro = 0.01/~m, dotted line--r0 = 0.005/~m. Apart from the above-mentioned features of the N,aw relationship, there is another striking feature', sharedby the curves in Figs. 1 and 2. In Fig. 1, for a,, <~ a~~< 0.75, N~ of different particles decreases with aw inan approximately linear way, and in the range 0.75~< a,, ~< a,~,~ = 0.96, it also decreases quasi-linearly withaw but with a different slope. Also, in Fig. 2, N~ decreases approximately linearly with aw when aw is inthe range aw, ~< aw ~< 0.96. The linear relationship between N~ and aw makes it very easy for one to get anapproximate solution of a,~ as a function of relative humidity by using (2). The No-aw relationship was also investigated forother samples. Results indicate that the universal properties of No-aw curves shown in Figs. 1 and 2 still existregardless of the different chemical compositions andorigins that other sample, s have.c. Approximation of curvature correction Based on the findings in the previous subsections, N0is approximated with two pieces of the linear functionof aw when aw, ~ a,~ ~ awm = 0.96. Given the characteristics of No-aw curves in Figs. 1 and 2 and also forthe sake of simplicity, the common point (denoted asa,~j) of the two pieces of the linear function is definedas the aw at which the true N0 has maximum deviationfrom the linear function 1 + (N~(0.96) - 1)(aw- aw,)(0.96 - awt)-~. Since N0 has little dependenceon r0, the common point for any ro is assumed to bethe same. For the case aw ~> 0.96, from Figs. 1 and 2 one cansee that N, decreases very rapidly with aw when aw islarger than about 0.96 and approaches zero when aw -~1. This indicates that N0 can only be fit with certainfractional power functions of ( 1 - aw). In the following, No's properties are examined for the case aw -~ 1.When aw approaches 1, the exponential mass increasecoefficient in (3) approaches a constant value r/o. Assuming r/is a constant, from (1), (3), and (4c) andusing lnaw -~ aw.- 1, one can obtain 6 = 1 + c~(1 - aw)1/3, (15)N6 - c~ (1 - aw)1/3, (16) 6t- 1where 2[-'w--a(T-T-)] (p2w-~o~o)m= (17)Equation (16) indicates that N, is a power function of(I - aw) to the one-third when aw--' I. Equation (16)was tested against the seven samples listed in Table 1.1.2.8.4.2i I I I.6 .7 .8 .9FiG. 2. As in Fig. 1 but for sample 3.DECEMBER 1996 YANG AND COTTON 2265It was found that (16) can underestimate N, by as muchas about 10% at aw -- 0.96. This error is mainly causedby the variation of ~7 with a~~. H~nel and Lehman( 1981 ) pointed out that % could be 17% to 55% largerthan the ~7 at aw = awm = 0.96, and therefore using ~= % in (17) leads to the underestimation of N,. To ensure that 6 is a continuous function of aw, Cl isestimated directly from the 6 at aw = 0.96--that is, c~ = 0.04-~/318(0.96) - 1]. (18)Substituting c~ in (16) with (18) gives _ --~.,_1'1 aw\1/3 N~ = N~(0.96)~0-~) . (19).Equation (19) was also tested against the seven samples. In Fig. 3, N~ calculated with (19) agrees very wellwith that computed with (4c) for a~ E (0.96, 1 ). Moreover, N~ calculated with (19) matches very well withthat obtained with (4c) even when a~ is as small as 0.8.Tests on the six other samples also demonstrate that N~computed with (19) can compare very well with thatobtained from (4c) when 0.96 ~< aw ~< 1. In addition,results show that when aw < 0.96, for all seven samplesN~ computed with (19) is still very close to (within0.04) that obtained from (4c) if W > 2W,.4. Approximation formulas of water activity as functions of RH In this section, (2) is solved with the N~-aw (or 6aw) relationship obtained in the previous section. Approximation formulas for water activity are derived asfunctions of relative humidity. Accuracy of equilibriumsizes calculated with water activity given by these formulas is examined. In this paper, the performance of the approximationformulas was investigated only for the stable subcriticalcase aw ~ aw~, where a~ is the critical water activity[df/da~l~=~w~ = 0]. In the case where aw ~< a~, awmonotonically increases with f. For a given f, thismonotonic relationship between aw and f determineswhat approximation formula of 6 should be used tosolve (2).a. Water activity as a function of RH For the case aw ~ (a~,, 0.96), t5 and N6 can be approximated by two piecewise linear functions of aw.Assuming~ = al + blaw, (20)from (2) and (20) one can obtain 1 a~ = ~ I-a1 +-- (a~2 + 4b~f)~/2]. (21)For the two roots in (21), one can easily determine thatonly one root has physical meaning..8.7.6.5.3.2.1.0 .80 .84 .88 ,g2 .96 1.00 Fro. 3. Scaled deviation of curvature correction from unity (No)versus water activity (aw) for a particle with dry radius of 0.1 /~m insample 10. Solid line is calculated with Eq. (19), and dotted line isthe tree value calculated with Eqs. (4c) and (10). For the case where aw~ (0.96, awc), from (2) and(15) [note that cl is defined by (18)] one can obtain(f _ aw)3 3 3 = cla~(1 - a~). (22)Equation (22) is a fourth-order algebraic equation. Although it has analytical solutions, the roots are complicated functions off. Since a~ is in the narrow rangefrom 0.96 to 1, the term a3w on the rhs of (22) is approximated with 0.983 = 0.9412. Thus, (22) is reducedto a cubic equation. This cubic equation may have tworeal roots of a,~ if aw~ > 0.96, with one of the roots lessthan a~ and the other greater than aw~.. It is worth mentioning that formulas (21) and (22)use only three ~5 values at the three interpolation pointsof a~ (i.e., a}~, a~, and a~,~ = 0.96). Because (21) and(22) are in simple forms, the computation cost to calculate the equilibrium radius of a particle is very lowwhen compared with iterative methods. When the approximation formulas are applied to a group of particlesof different sizes or to a particle in a RH varying environment, (21) and (22) can further lower the computation cost when compared with iterative methodsbecause R in (6) is independent of ro and-RH, andtherefore needs to be calculated only once at the threeinterpolation points of aw.2266 JOURNAL OF APPLIED METEOROLOGY VOLUME35b. Accuracy and error analysis In order to check the accuracy of (21) and (22),highly accurate solutions of ( 1 ) - (5) are needed as reference values. To this end, an iterative method wasemployed to solve ( 1 ) - (5). The convergence criterionof the iterative method is that (2) is accurate within 1.0x i0-8 at the last iteration step. Hereafter, solutions of(1)- (5) obtained with the iterative method are referred to as tree values (or solutions). To quantitatively examine the accuracy of (21 ) and(22), the relative error of equilibrium radius (denotedas ER) computed with water activity given by (21 ) and(22) is defined as follows: ER - --, (23)where ? represents the true value of equilibrium radiusand r is the equilibrium radius calculated with wateractivity given by (21) or (22). The relative error of equilibrium radius obtainedfrom water activity given by (21) and (22) was examined for RH ranging from the threshold relative humidity f, = aw,6, to the critical relative humidity fc of aparticle. Listed in Table 2 is the maximum ER in theRH rangeft < f ~< 99.99%. From Table 2 one can findthat the maximum ER is usually around 1%-3% whenthe dry radius is larger than 0.02/~m and is below 5.5%when ro < 0.02/~m. These results demonstrate that theapproximation method developed in this paper can accurately estimate particle equilibrium sizes when relative humidity does not exceed 99.99%. Results (not shown) also demonstrate that when relative humidity is larger than 99.99% and is close to thecritical relative humidity, the relative error of equilibrium radius computed with the proposed method canbe as large as 30% if ro > 0.01 ~m. The sensitivity ofequilibrium sizes to water activity can be estimated asfollows. Assuming that r] does not change with W, thenfrom (1) and (3) one has Ar 1 Pw lnaw 1 (24) ~- = ~ ~7 aw lnaw 'where Aaw stands for the error in aw and Ar/r is therelative error of r. Equation (24) shows that the relativeerror of equilibrium radius becomes more and moresensitive to the error in water activity as water activityincreases--that is, as RH increases. When a~ --~ 1, Ar/r is extremely sensitive to the error in a~. Fortunately,the approximate N,-a~ relationship (19) is accurate towithin 1.5% when a~ > 0.96. Results also show that overall, the relative error ofequilibrium radius decreases with the increase of dryradius for a fixed aw if a~ is not very close to 1. Thisis because the error in a,~ obtained from (21) and (22)decreases with the increase of r0. This conclusion canbe verified through examining (2), (10), and (124) (de T^U[E 2. The maximum relative errors (%) of equilibrium radiuscomputed with water activity from Eqs. (21) and (22) in the relativehumidity range f, ~< RH ~< 99.99% for different aerosol samples anddifferent ranges of dry radius r0. Sample 1 2 3 4 6 8 10r0 < 0.02/~m 1.0 1.5 1.0 5.5 2.2 1.0 2.80.02/~m ~< r0 ~< 10/zm 1.2 3.3 1.0 1.1 2.0 1.5 1.7tails are omitted here). When a,~ is very close to 1,although from (2) and (10) one has Aaw -~ 0, the termAaw/lnaw in (24) may not be small, so certain errorsmay remain in the equilibrium radius computed withthe proposed method. This kind of error occurs mostdistinctly in the vicinity of the critical point. It is discussed thoroughly in the next section.5. Problems associated with the critical point In this section, critical equilibrium radius and supersaturation are studied with the proposed 6-a~ relationship. Error analysis is performed to equilibrium sizescomputed with aw given by (22) when 100% <' RH~< f,. Modified formulas are developed to reduce errorsin the critical supersaturation and equilibrium radius.a. Critical equilibrium radius and supersaturation When ambient relative humidity is larger than 100%,particles with critical humidity less than the ambient relative humidity will be activated. To investigate ther-f relationship near the critical point, investigators(e.g., Fletcher 1962; Mason 1971; Pmppacher and Klett1978) developed some methods to approximate the Kthler equation when aw ~ 1. As pointed out by H'finel(1976), for some representative salts such as NaC1 and(NI-I4)2SO4, previous approximation methods give equilibrium sizes remarkably larger than true values. For most hygroscopic particles, the critical water activity is larger than 0.96, and hence one can use (15)to find the critical water activity. Using (2) and (15)[note that Cl is defined by ( 18)] and assuming that df/da~ = 0, one can obtainwhere4 y2 _ c~ = 0, (25)5 c~y3 + '~ y = (1 - aw)1/3. (26)When c~ < 1.0, a very precise solution of (25) is itssecond-order approximated root, 1 )~~Yc = Y~ + ~ ( I + 4c~y~ x [-c2+ [c22- 16(1+4c~yt)y~]m}, (27)DECEMBER 1996 YANG AND COTTONwhere .1 E-01y~ = , (28) 4 c~ + 2y]. (29) .1E-02 c2 ~-~From (2), (15), (26), and (27), one can obtain thecritical supersaturation (denoted as &) Sc~-fc- 1 = -yc3 + ct(1 -Y~)Yc (30) rt~ .1--~3and critical water activity awc = 1 - yc3. (31)As y- is a unique function of c], so in (30) and (31) .1E-04the critical supersaturation and water activity areunique functions of c~. If a dry radius is large enough,from (18) one has Cl -o 0. Thus, (30) and (31) reduceto 2 c~/2 S~ ~ ~-~ for c~ -~ 0 (32)andfor c~ ~0. (33) In Fig. 4, the critical supersaturation calculated with(30) compares well with its true value, especially whena dry radius is small. The relative error of S~ calculatedwith (30) is about 5% when ro = 0.04 /zm, is about10% when ro = 0.13 ~m, and approaches a maximumvalue of about 15% at ro = 10 /.tm. The increase ofrelative error of Sc with r0 is caused by the inaccuracyof c~ when a,, -~ 1. Results show that S~ computed with(32) differs by less than 1% from that computed with(30) when r0 > 0.05 ktm, so (32) can be used to estimate the error in &. From (32), one can deduce thatthe relative error of Sc is 1.5 times as large as the relative error of c~. For sample 8, the c~ determined by(18) is about 12% larger than the ct determined by(17), which corresponds to the case of infinite dilution(aw = i ). Therefore, (30) and (32) overestimate S~ byabout 18% at a~ 2-- 1. From (24), (32), and (33), one can prove that therelative error of critical equilibrium radius is about onethird of the relative error of S-. For all samples listedin Table 1, results show that if ro >~ 0.01 /~m, S~ estimated by (32) is accurate to within 15% and the criticalequilibrium radius (rc) computed with water activitygiven by (33) is accurate to within 5%.b. Errors and improvements around the critical point As mentioned earlier, ER is very sensitive to the error in a~ when aw -~ 1 [see (24)]. The inaccuracy ofwater activity computed with (22) is partially caused2267 - 1 E-05 .01 .10 1 .00 10.00 ro (~rn) F~G. 4. Critical supersaturation & versus dry radius r0 for sample8. Solid line is computed with Eq. (30), and dotted line is the truevalue.by the assumption that ct is independent ofa~. To moreaccurately estimate c~, based on (17), we modify (18)into c](a~) = c~(0.96) [r/(0'9--~6) ~~/3- (34) L r/(a~)Since r/(a~) is unknown before a~ is determined, a onestep iteration is used to estimate ~l(aw). Computationsshow that the error in the aw calculated with (22) and(34) is about half of that in the aw computed with (22)and a constant value of c~ [see (18)]. As a result, the equilibrium radii computed with awgiven by (22) and (34) have a higher accuracy. For allsamples in Table 1, the relative error (ER) of equilibrium radius computed with (22) and (34) is less than2% when r0 ~> 0.02/.tm and RH ~< 100%. For the case100% < RH ~< f~. and ro ~< 10/.tm, results show thatthe maximum ER calculated with (22) and (34) rangesfrom 9% to 16% for the seven samples in Table 1,which is about half of that calculated with (22) and aconstant value of c~. Similarly, by taking c~ as a function of aw, the accuracy of critical equilibrium radius and supersaturation computed with the approximation method can beincreased. Using (2), (15), and (34), one can modify(25) into 0 ln~ ~ ~ y2 c~ c-234 ~ na ]1--~ly + ---3 =0' (35)2268 JOURNAL OF APPLIED METEOROLOGY VOLUME35Equation (35) differs from (25) only by the coefficientof the first term. Again, a one-step iteration is used toestimate c~ and 0 lnr//0 lnaw, which is approximatedwith finite difference. Computation shows that for 0.01/~m ~< r0 ~< 10/~m andfc ~< 102%, the critical equilibrium radius and critical supersaturation obtained from(35) are accurate within 3.3% and 5.2%, respectively.For sample 8, the maximum relative error of S~ computed with (35) is about one-third of the maximumrelative error of Sc obtained from (25). As mentioned in section 3c, the exponential massincrease coefficient (r/) is 17%-55% larger at aw = 1than at aw = 0.96. The above results indicate that if ~7changes more than 17% from a~ = 0.96 to a~ = 1,considering the change of c~ with aw can greatly increase the accuracy of estimated critical supersaturationand equilibrium sizes.6. Summary and conclusions To study the relationship between curvature correction (6) and water activity (aw), a scaled deviation of curvature correction from unity, N,, is defined as N, = (6 - 1)(6, - 1)-~, where 6t is thethreshold curvature correction at the threshold wateractivity (aw,) (H~inel 1976). It was shown theoretically that N~ can be very accurately represented bya universal function of water activity if the dry radius of a particle is larger than 0.005 /~m and aw> aw,. Furthermore, through studying seven representative aerosol samples measured by H~inel andLehman (1981), it was found that N, can ~be approximated by two piecewise linear functions of awwhen aw, < a~ < 0.96 and by a 1/3 power fi~nctionof ( 1 - a~) when aw >~ 0.96. Using these simple approximation formulas of Neand the K6hler equation, water activity is solved asan analytical function of relative humidity, and theequilibrium size of a wet particle is analyticallycomputed from a given relationship between wateractivity and water uptake by unit mass of dry material. Moreover, the approximation formulas of N,are used to derive analytical formulas for calculating critical supersaturation S~ and critical equilibrium radius r~.T^m~E 3. Relative errors (%) of equilibrium radiusin different ranges of RH and dry radius r0.fi ~< RH ~< 99.99%99.99% Equations for computing a~ ro ~< 0.02 ~m 0.02 < r0(18), (21), and (22) ~<5.5 ~<3.3 ~<30(21), (22), and (34) <~5.5 ~<2.0 ~<16 TABLE 4. Relative errors (%) of critical equilibrium radius r~ andcritical supersaturation Sc when dry radius is less than 10 tzm and S-<~ 2%.Equations forcomputing aw~ Relative error of rc Relative error of S- (18) and (25) ~<5.0 ~<15 (34) and (35) ~<3.3 <~5.2 The accuracy of equilibrium radius r, rc, and Scobtained from the above-mentioned method waschecked with seven aerosol samples measured byH~inel and Lehman ( 1981 ). The relative errors of r,S~, and rc, along with the equations used to calculatewater activity and its ranges of application, are listedin Tables 3 and 4. For formulas listed in Tables.3 and4, (21) was derived from the linear relationship between N~ and aw, while (22), (25), and (35) werederived from the one-third power relationship between N~ and a~. In (18), the parameter c~ was defined as a constant, whereas in (34) c~ was treated asa function of aw. Formulas in Table 3 are applicableto the case where RH is greater than the thresholdrelative humidity (ft = aw,6,). For the case where RH< fi, one can use the H~inel model [ see (1)- (5)] toobtain an analytical relationship between aw and RH.In addition, results (not shown) also demonstrate thatthe hysteresis effect of RH on equilibrium radiusdoes not affect the accuracy of the approximationformulas developed in this paper. Acknowledgments. The authors are indebted to Graham Feingold, Bjorn Stevens, Bob Walko, Scot Rafkin,Michael Meyers, Robert Czys, and the two anonymousreviewers for their critical comments and helpful suggestions. Brenda Thompson and Abby Hodges provided technical assistance with word processing. Thisresearch is supported by National Oceanic and Atmospheric Administration Contract NA37RJ0202.REFERENCESd'Almeida~ G. A., P. Koepke, and E. P. Shettle, 1991: Atmospheric Aerosols: Global Climatology and Radiative Characteristics. A. Deepak Publishing, 561 pp.Feingold, G., and C. J. 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Mclaren, 1971: Relationship of cloud nuclei spectra to aerosol size distribution and composition. J. Atmos. Sci., 28, 382-390.Kasten, F., 1969: Visibility forecast in the phase of pre-condensation. Tellus, 21, 631-635.Mason, B. J., 1971: The Physics of Clouds. 2d ed. Clarendon Press, 671 pp.Orr, C., F. K. Hurd, and W. J. Corbett, 1958: Aerosol size and relative humidity. J. Colloid Sci., 13, 472-482.Pmppacher, H. R., and J. D. Klett, 1978: Microphysics of Clouds and Precipitation. D. Reidel, 714 pp.Tang, I. N., 1980: Deliquescence properties and particle size change of hygroscopic aerosols. Generation of Aerosols, K. Willeke, Ed., Ann Arbor Science Publishers, 153-167.--., and H. R. Munkelwitz, 1994: Aerosol phase transformation and growth in the atmosphere. J. Appl. Meteor., 33, 791 796.Winkler, P., 1973: The growth of atmospheric aerosol particles as a function of the relative humidity II. An improved concept of mixed nuclei. Aerosol Sci., 4, 373-387.

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