Chemical Nucleation Theory for Various Humidities and Pollutants BY C. S. KIANGAND D. STAUFFER Physics Department Clark College Atlanta Georgia 30314 U.S.A. Received 28th November 1972 The Flood-Neumann-Doring-Reiss-Doyle theory gives a strong dependence of the nucleation rate on the humidity (for r.h. < 100 %) and shows that under atmospheric condition H2SO4 but not HN03 SO2 or NH3 contributes directly to the aerosol nucleation without pre-existing nuclei. Typically lo9 H2S04 molecules per cm3 are enough to form droplets of aqueous sulphuric acid solutions. One possible mechanism for the formation of atmospheric aerosols is the formation of small droplets consisting of a liquid mixture of water and a pollutant e.g. H2S04. Even around pre-existing nuclei pure water can form droplets that grow to infinite size only if the relative humidity of the atmosphere is greater than 100% ; and a pure pollutant like H,S04 can form large droplets only if its partial gas pressure is greater than the equilibrium vapour pressure over liquid H2S04 i.e.only if its “activity”is greater than one. But droplets consisting of an aqueous solution of the pollutant can grow even for activities smaller than one and humidities smaller than 100% since the partial pressures of both components over a liquid mixture can be much smaller than over the pure materials. If a small mixture droplet is formed out of the gas phase with at least two compon- ents (e.g. H2S04 and H20) then we call this process “chemical nucleation” to distinguish it from other nucleation processes.This paper applies the theory of chemical nucleation to various materials as a function of the relative atmospheric humidity. Our calculations are based on the work of Flood,’ Neumann and Doring,2 re is^,^ and Doyle.4 Our main results and the connection with experiments were reported earlier. REVIEW OF CHEMICAL NUCLEATION THEORY The nucleation process is determined mainly by the free energy AG = A(E-TS-pN) necessary to form a droplet. We write for the formation energy of a droplet consisting of nA water and nB pollutant molecules AG = PA)^ +(AB -P&B +S(~A, ~z&J(x) (1) where the p are the chemical potentials of the two materials (A = H20 B = pollutant e.g. H2S04)if gas and liquid are in equilibrium over a flat mixture surface.The p are the actual chemical potentials in the supersaturated atmosphere ; S is the surface area of the droplet and depends on nA and nB ;y is the concentration-dependent surface tension of the liquid mixture; x = nelh 4-4 (2) is the mol fraction of the pollutant in the droplet. We assume the volume of a droplet 26 C. S. KIANG AND D. STAUFFER to be given by VA??A+ VBnB = 47rr3/3,where Y is the droplet radius and VAand V are the volumes per molecule of pure water and pure pollutant. Thus S = 4nr2. Large mixture droplets can be formed from a gas only if both chemical potentials pAand pBare greater than their values pcAand pcB on the coexistence curve ; that means supersaturation with respect to both vapours is required.In this case the first two terms of the right-hand side of (1) are negative ; the last term is positive and dominates for small droplets. Thus due to the surface tension the droplets have to overcome an energy maximum before they can grow further. For a binary mixture the free energy AG depends on nA and nBand thus can be represented by a surface (“mountain”) in three dimensions. In order to grow the droplets have to overcome the lowest free energy maximum (saddle point). In the growth process most droplets approach this saddle point along the deepest valley leading to this mountain pass.2 This saddle point condition yields two equations = (aG/anA), = O and (aG/dt~B),~0. In the evaluation of these two derivatives the changes of the pc with the mol fraction x (eqn (2)) cancel out because of the Gibbs-Duhem-Margule equation,2-n~d~cA+nBd~cB = 0 (3) (or (1 -x)dpcA/dx = -x dpcB/dx).Then the saddle point condition gives for the composition x* and the droplet radius Y* at the saddle point A& = pA-p,*A = (2yvA/r*)/(l-a*x*) > 0 (44 Apg = ,uB-,u~,*B = (2yvB/r*)/(l +a*(l -x*)) > 0 (4b) or equivalently Ap;/A& = (VA/VB)(1+a*( 1-x*))/(1 -a*x*) (44 r* = 2yvA/((1 -a*x*)Ap;) (44 with = 1.5(1- X 4-VB/VA)(dy/dX)/y. (44 (The star * denotes values at the saddle point.) On insertion of (4) into (l) the a*-corrections cancel out AG* = (47~/3)y*r*~ (5) The equilibrium number of droplets of critical size r* and critical composition x* is proportional to exp (-AG*/RT) ; the factor of proportionality is assumed by Reiss and Doyle to be the total number NA+ NB of gaseous water and pollutant molecules.Under usual atmospheric conditions there exists much less pollutant than water NA 9NB. Droplets at the saddle point grow by incorporation of single molecules such that the composition x remains roughly unchanged (at least for the examples used in this paper). Therefore the growth rate at the saddle point is determined by the pollutant concentrations NB whereas the proportionality factor for the droplet numbers is given by NA for NA$ NB. (Most of the water molecules impinging on the critical size droplets will evaporate again ; only if additional H2S04 molecules have been incorporated into the droplet then also more water molecules can remain with the droplet and keep its composition near x*.) Thus the rate at which new molecules are incorporated into the droplets is given roughly by the product of surface area 47cP2 and pollutant impinging rate PB = NB kT/(2nm~ kT)+ CHEMICAL NUCLEATION THEORY where ing is the mass of a pollutant molecule.The nucleation rate J is the rate at which droplets grow over the saddle point (per cm3 per s) and thus can be approxi- mated as J = 47Cr"2&NA exp (-AG*/kT). (6) The Ap appearing in (4) can easily be evaluated since in the atmosphere the gas densities are very low and therefore application of the ideal gas law gives where PA and Ps are the actual partial pressures of the water and the pollutant whereas PcoA(x) are the equilibrium partial pressures over a large solution and PoDB(x) with composition x (mol fraction).Historically this theory of "chemical nucleation " was developped by Flood who applied it to water+alcohol mixtures. Neumann and Doring introduced the saddle point picture and took into account the possible enrichment of one phase near the droplet surface. (This effect is neglected by us; for the HzO+H2SOo nucleation we found its influence on J to be negligible if calculated as in the theory of ref. (2)). In ref. (1) and ref. (2) the pre-exponential factor for the nucleation rate was simply chosen to be the same as that of pure water which would be too high an estimate for H20+H2SO4nucleation. Reiss derived in detail the pre-exponential factor but used only special limiting cases in the evaluation of the exponential term (e.g.dilute solutions in the liquid droplet). In ref. (1)-(3) the a*-corrections in (4) were neglected. They were introduced by Doyle who applied this theory to H20+H2S0 mixtures at 50 % relative humidity. Bricard et al.' recently evaluated J also for 30 % and 70 % relative humidity. In the next section Doyle's formulae are applied to different materials and humidities in order to give a more complete picture than known to us before. NUMERICAL RESULTS Our figures show the main results from a numerical solution of (4c) and (7) (iteration on an IBM 1130 computer) and the application of (4d) and (6) for the nucleation rate J. The H,0+H2S04 nucleation rate is evaluated at 25"C the H,O +HN03 nucleation at 20°C ; data is taken from ref.(6). Fig. 1 shows some cuts through a free energy surface. Fig. 2 gives some nucleation rates J for chemical nucleation of H,O +H2S04 and H,O +HNOJ mixtures. The " activity " in the figures is simply the actual pollutant partial pressure in units of the pressure of a pure pollutant Torr for H2S04from ref. (4) ; 45 Torr for HNO from ref. (6)). Thus the activity is for the pollutant what the relative humidity is for water. For atmospheric applications it seems to be useful to know the characteristic time z during which the atmospheric content of gaseous pollutant is reduced appreci- ably due to chemical nucleation rates. Fig. 3 gives this time as estimated from z = IVB/x*J-NB/J. Fig. 4 shows the activities necessary to achieve a fixed J = one droplet per cm3/s or a fixed z = one second or ofie month.Critical composition x* and number of mole-cules in a critical size droplet are given by fig. 5. All these results describe homogeneous chemical nucleation without preexisting nuclei. Nucleation of a mixture can also occur as heterogeneous chemical nucleation on pre-existing nuclei * like flat surfaces insoluble particles soluble particles or ions. But also for such heterogeneous nucleation the actual partial pressures of C. S. KIANG AND D. STAUPFER FIG. 1.-Droplet formation energy AG as a function of droplet radius r (HzO+HzS04 at 50 % relative humidity ; activity = 0.005). The numbers on the curves give the mol fraction s of HzS04 in the droplet ; the star indicates the saddle point.r.h. FIG.2.-Nucleation rate J as a function of relative humidity r.h. for H20+ HzS04 (left) and HzO+ HNOJ (right). The numbers on the curves give the activity (= partial pollutant gas pressure in units of Torr for HzSO4 (25‘C) and of 45 Torr for HNOJ (20’0). CHEMICAL NUCLEATION THEORY 1 O6 I 8 -ul 10' I r.h. FIG.3.-Characteristic decay time T-NB/J (cp. fig. 2) as a function of relative humidity r.h. ; the numbers on the curves give the activity. The insert shows the gaseous pollutant concentration NB(t)/NB(t = 0) as a function of time t/T (solid line) and the tangent to this curve at t = 0 (dashed line). Initial activity 0.005 r.h. = SO% 7-lh. x c, .-> .d Y 0 -10 lo3-20% 40% 100 r.h.FIG.4.-Activities and pollutant concentrations N~/crn~ for fixed J or fixed T. The dashed-dotted line is the " zero supersaturation " curve for H20+ HNOJ (lower limit for heterogeneous chemical nucleation). Heterogeneous nucleation on ions or wetted particles with 8 8,radius is indicated by full (H20+ HNO,) and open (H20+ H,SO,) circles. (J = 1 droplet per second and ion or particle for HN03 10-3/sfor H2S04.) C. S. KIANG AND D. STAUFFEK water and pollutant cannot be smaller than those over a flat surface of a liquid mixture in equilibrium with its vapours. This condition (actual partial pressures = equilibrium partial pressures) determines the “ zero supersaturation curve ” of fig. 4. 8C 5c * $ 2c r i 30°/0 60% 900/0 r.h.Fic. 5.-Number of molecules IZ* (left scale) composition x* (centre scale) and radius Y* (right scale) of ” critical ” saddle point droplets for fixed J (left) or fixed activity (right ; as given by the numbers on the curves). Roughly Y* is the minimum radius of wetted particles which can produce heterogeneous chemical nucleation for the given activity and humidity. Nucleation whether heterogeneous or homogeneous does not occur for activities smaller than the activity on the zero supersaturation curve. (For H20+ H2S04 activity = at 20 % relative humidity activity = 10-9-5at 80 % relative humidity.) Heterogeneous chemical nucleation processes may be important for activities between the one on the zero supersaturation curve and the higher ones shown in fig.4for homo- geneous chemical nucleation. Using the same models as in ref. (10) we find for nucleation on ions or on “ dust ” particles of 8 A radius the activities indicated by the circles in fig. 4. (For H20+H2S04 the pre-exponential factor for Jion/Nion is smaller than 1 s-I ; thus Jion/Nion cannot be greater than I s-I for the activities for which nucleation theory is valid(AG* % kT). The same holds for “dust” nucleation.) DISCUSSION This theory shares with others the disadvantage of applying macroscopic concepts to small droplets. Recent indirect evidence indicates that this is a much better approximation than usually assumed ; the results of ref. (1 1) leave little space for a correction factor like lo1’ for the droplet concentrations.But it may be necessary to employ a “ microscopic surface tension ” for the free energy of the droplet. For pure water near O’C this microscopic surface tension was found in ref. (12) to agree with the measured bulk surface tension. However the activities of H2S04or HN03 CHEMICAL NUCLEATION THEORY predicted in our theory might well be wrong by one order of magnitude. (The "zero supersaturation curve " of fig. 4 should be more accurate since no surface tension enters there.) Hydration effects could produce additional complications. For the H20 +H2S04system an additional source of error is the vapour pressure over pure H2S04 taken as Torr (from ref. (4)). If this pressure is actually larger by an order of magnitude then our activities remain roughly unchanged but the partial pressures or concentrations of H2S04 would be increased by a factor 10.s-l to lo3 ~m-~ The nucleation rate is increased from 1 ~m-~ s-' if the volume of a droplet is calculated l4 from the measured density of H20 +H2S04 mixtures and not simply from VAnA+ VBnB = 4zr3/3. Some other corrections were found to be rather unimportant. In the work of Reiss and Doyle the impinging rate is not simply PB but PA/3~(l +tan2 ~)/(PB+PA tan2 4). We found tan 4 = x*/(1 -x*) to be a sufficient approximation ; then this s-I. correction increases J by about one order of magnitude near J = 1 ~m-~ In ref. (12) we used corrections to the classical nucleation theory for the number of droplets with a given size (pure substances) ; instead of number of droplets = N1exp (-AG/ kT),we used number of droplets = qo exp (-AG/kT),where qo near 0°Cfor water is of the order of the particle density at the liquid-gas critical point ~m-~ instead of ATl = monomer concentration = 1017*5 ~m-~).But AG in ref.(12) also contained a logarithmic term giving an additional factor (number of molecules in a for the number of droplets. Thus the three effects discussed in this paragraph roughly cancel each other out of the final results if we may apply our previous experience l2 for "pure " nucleation also to chemical nucleation. Reiss and Doyle also introduced two "Zeldovich " factors for the saddle point ; but the nucleation rate contains the quotient of them which is near 1 (see ref.(4)). For nucleation on ions the product of two other Zeldovich factors for the energy minimum appears in the droplet numbers; it is of order lo-' and was taken into account in fig. 4. We conclude that our results may be inaccurate but should give the correct trends for the dependence on humidities activities ions etc. This could be con- firmed by experiment.l* s Our fig. 4 shows that for different materials the activities are roughly the same; only the transformation from activity to partial pressure or concentration differed by many orders of magnitude for H2S04and HN03. Thus under usual atmospheric conditions on earth where impurities are measured in parts per million or less the high concentration of HN03 necessary for chemical nucleation cannot be reached.Similarly SO2 C2H50H,or NH3 cannot form directly liquid aqueous solution droplets in the atmosphere because of their high pressure whereas H,S04 forms very easily a mixture droplet. The typical partial pressure of the pollutant vapour over a pollutant +water mixture thus indicates directly whether this pollutant can form droplets with water by (homogeneous) chemical nucleation. On the other hand the vapour pressure of NaCl is so small at room temperature Torr from an extrapolation of the results of ref. (13)) that the atmosphere should practically not contain gaseous neutral NaCl molecules. Then again chemical nucleation does not occur. Only for intermediate vapour pressures e.g. near the Torr of H2S04,is this chemical nucleation process of direct importance in the atmosphere.Indirectly both SO2 and NaCl can contribute to aerosol formation by chemical nucleation. As discussed in ref. (4) (15) concentrations of SO2 in the parts per million range are sufficient to produce H20+H2S04aerosols via photoxidation of SO2 to H2S04. And solid NaCl particles in theatmosphere can serve as soluble condensation nuclei for heterogeneous nucleation of pure water ** (for relative C. S. KIANG AND D. STAUFFER humidities > 100 %) or solution droplets (humidities < 100 %). For relative humid- ities less than 100 % and activities less than 1 our chemical nucleation theory describes the only mechanism for the formation of large aerosols from the gas phase. For relative humidities greater than 100 % this chemical nucleation mechanism can be regarded as the initial stage of other aerosol formation processes (e.g.heterogeneous water nucleation on soluble particles 8* '). We hope to discuss this second nucleation process in a later paper.I5 We thank Prof. V. A. Mohnen for drawing our attention to this problem and for encouraging discussions and the National Center for Atmospheric Research Boulder for its hospitality. This investigation is partially supported by NSF Grant GA-33422 Atmospheric Science section and by NIH Grant RR 8006 from the General Research Support Branch Division of General Resources National Institute of Health. ' H. Flood Z. phys. Chem. A 1934 170,286. K. Neumann and W. Doring Z. phys. Chem. A 1940 186,203. H.Ress J. Chem. Phys. 1950 18 840. G. J. Doyle J. Chem. Phys. 1961 35 795. C. S. Kiang D. Stauffer V. A. Mohnen J. Bricard and D. Vigla submitted to Atmospheric Environment. W. R. Forsythe and W. F. Giauque J. Atner. Chem. SOC. 1941,64,48 ; W. F. Giauque E. W. Hornung J. E. Kunzler and T. R. Rubin ibid. 1960,82,62 ; Int. Critical Tables vol. IV (E. W. Washburn et al. ed.) (McGraw Hill Book Company New York and London 1928) Landolt- Bornstein Zahlenwerte und Funktionen vol. I12a (K. Schafer and E. Lax ed.) (Springer Verlag Berlin 1960). Private communication of J. Bricard September 1972. 'B. J. Mason The Physics of Clouds (Ciarendon Press Oxford 1957). K. G. Vohra and P. V. N. Nair J. Arm. Sci. 1971 28 280. Clark College Research Group Phys. Rev. B 1972 6 2780.lo C. S. Kiang Phys. Rev. Letters 1970,24,47 ; K. Binder et a/.,J. Stat. Phys. 1972 6,49 ; and I' Phys. Rev. B 1972,6,2777 ; C. Carlier and H. L. Frisch J. Chem. Phys. 1972 to be published. l2 Clark College Research Group J. Atm. Sci. 1971 28 1222. l3 B. H. Zimm and J. E. Mayer J. Chem. Phys. 1944 12 362. l4 We thank Mr. L. Roland for help on this calculation. 15 D. Stauffer V. A. Mohnen C. S. Kiang to be submitted to J. Aerosol Sc. Notes added in proof 1 The calculation of T in fig. 3 overestimates the time after which the gaseous pollutant is con-sumed since it neglects condensation on already nucleated droplet^.'^ 2 At the third Chemist-Meteorologist Workshop (Ft. Lauderdale Florida Jan. 1973) the name "heteromolecular nucleation " was proposed for what we called here "chemical nucleation." S7-2