Formation of binary ion clusters from polar vapours: effect of the dipole-charge interaction

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1 Atmos. Chem. Phys., 4, , 24 SRef-ID: /acp/ Atmospheric Chemistry and Physics Formation of inary ion clusters from polar vapours: effect of the dipole-charge interaction A. B. Nadykto and F. Yu Atmospheric Sciences Research Center, State University of New York at Alany, 25 Fuller Road, Alany, NY 223, USA Received: July 23 Pulished in Atmos. Chem. Phys. Discuss.: 2 Octoer 23 Revised: 9 Feruary 24 Accepted: 5 Feruary 24 Pulished: 27 Feruary 24 Astract. Formation of inary cluster ions from polar vapours is considered. The effect of vapour polarity on the size and composition of the critical clusters is investigated theoretically and a corrected version of classical Kelvin- Thomson theory of inary ion-induced nucleation is derived. The model predictions of the derived theory are compared to the results given y classical inary homogeneous nucleation theory and ion-induced nucleation theory. The calculations are performed in wide range of the amient conditions for a system composed of sulfuric acid and water vapour. It is shown that dipole-charge interaction significantly decreases the size of the critical clusters, especially under the atmospheric conditions when the size of critical clusters is predicted to e small. Introduction Formation of ultrafine aerosols has received increasing attention in the last few decades due to its importance for atmospheric physics and chemistry, chemical technology and health research. The possile role of air ions in aerosol formation, which was intensively studied during seventies and earlier eighties e.g. Mohnen, 97; Castleman et al., 978; Arnold, 98; Hamill et al., 982, has received renewed attention in recent years e.g. Yu and Turco, 2; Carslaw et al., 22; Eichkorn et al., 22; Yu, 22, 23. Classical theory of ion-induced nucleation e.g. Hamill et al., 982; Raes et al., 986; Laakso et al., 22 treats the cluster formation using capillary approximation and it accounts for the charge effect on the pressure in the condensed phase only. The classical model is derived assuming a flat monomer concentration profile in the vicinity of the nucleating cluster, which is a good approximation for non-polar vapours. How- Correspondence to: A. B. Nadykto alexn@asrc.cestm.alany.edu ever, the interaction of polar monomers with the electrical field of the charged particle leads to the enhancement in the monomer concentration near the particle surface Korshunov, 98; Nadykto et al., 23. Recent studies Nadykto et al., 23 showed that the interaction of polar vapour molecules with the electrical field of charged particles may e important for the formation of small ion clusters. Nadykto et al. 23 considered the ion-induced formation of singlecomponent particle and they concluded that the contriution of the dipole-charge attraction potential to the size of the ion clusters is significant, when polar vapours are involved in the nucleation process. In addition to the reduction of evaporation, the dipole-charge interaction enhances the condensation rate through enlargement of the effective collision cross section Nadykto and Yu, 23. There is a clear difference etween the effect due to the dipole-charge interaction in the gas phase and the Thomson effect. The Thomson effect relates to the properties of the condensed phase while the dipole-charge interaction modifies the chemical potential of the condensing monomers in the electrical field of the charged particle/cluster. The purpose of this paper is to study the effect of dipolecharge interaction on the formation of inary cluster ions. We will derive the generalized Kelvin-Thomson equation accounting for vapour polarity, calculate the critical size of inary sulfuric acid-water ion clusters and compare our model predictions with the results of the earlier theories. 2 Model In the classical inary theory e.g. Hamill et al., 982; Raes et al., 986; Laakso et al., 22 in the prevailing temperature and vapor pressures of two condensing components a and, the critical size of inary cluster ion is determined European Geosciences Union 24

2 386 A. B. Nadykto and F. Yu: Formation of inary ion clusters from polar vapours from the Kelvin-Thomson equation e.g. Laakso et al., 22, S = V 4σ kt D p ] qe 2 ε g ε r 2π 2, ε D 4 p where S =S X a S X, V = X v a + X v, πd 3 p 6 ρ = n a m a + n m. Here S is the generalized saturation ratio, S a and S are the saturation ratios for component a and respectively, V is the average molecular volume, v a and v are the partial molecular volumes for components a and respectively, n a and n are the numer of molecules in the particle for components a and respectively, m a and m are the molecule mass of components a and respectively, q is the numer of the charges in the cluster, e is the elementary charge, σ is the surface tension, ε r is the relative permittivity of particle, ε g is the relative permittivity of the condensale vapour, ε is the vacuum permittivity, k is the Boltzman constant, ρ is the particle/cluster density, X a and X are molar fractions for components a and respectively, and D p is the diameter of the cluster. The composition of the charged particle is decided y the following equation v a = v, 2 µ a µ where µ i =µ il µ ig is the chemical potential change from gas phase µ ig to the condensed /liquid phase µ il of component i i=a,. In a inary system, change in the Gis free energy can e expressed y the following equation G = µ a n a + µ n + σ A + q2 e 2 o 8πε ε g ε r r r 3, where r is the cluster radius, A is the cluster surface area, and r is the radius of the core ion. In the case of non-polar vapours we get, assuming flat vapour profile in the vicinity of the cluster nucleating, the following conventional equation G = µ al µ a n a + µ L µ n + σ A + q2 eo 2 8πε ε g ε r r, 4 r The change in the Gis free energy due to the phase transition relates to the difference in the chemical potentials of the gas phase molecules located near the interphase oundary over the particle surface and molecules in the condensed phase. Since the isothermal chemical potential of the vapour is a function of the vapour pressure only, the correction to the chemical potential is derived through the calculation of the vapour pressure of over the charged particle surface Korshunov, 98; Nadykto et al., 23. The electrical field of the cluster/particle attracts polar monomers and, thus, their concentration in the vicinity of the nucleating particle and the vapour pressure over the particle surface rises. This may modify the chemical potential of the vapour molecules and change in the Gis free energy significantly. In the electrical field of the charged cluster/particle, the difference etween chemical potentials of the polar molecules in the condensed phase and in the gas phase is given y e.g. Nadykto et al., 23 µ il µ ig = kt Aig A il ] +. 5 where A il is activity of component i in the condensed phase, A ig is activity of component i in the gas phase,p ig is the vapour pressure over the particle surface, is the amient vapour pressure. Term relates to the change in the monomer concentration near the particle surface and it is descried y the following equation Nadykto et al., 23 + sinh = α i qe 2 32π 2 ε 2r4 kt qe l i 4πε kt r 2 qe l i 4πε kt r 2 = C i r, l, T. 6 where sinhz=expz exp z]/2, l i is the dipole moment of component i and α i is the polarizaility of component i and C i r, l, T is the correction to the condensing vapour pressure due to the dipole-charge interaction. Now we insert the expression for the change in chemical potentials 5 6 into Eq. 3, and get the analytical expression for the change in the Gis free energy G = kt n a + kt C a n a + C n ] + σ Ar + q2 eo 2 ] 8πε ε g ε r r ] r Ag A L n ] Applying the Gis-Duhem identity Renninger et al., 98 to function G at constant temperature and pressure, d G =, 8 we otain, after differentiation with rearranging the terms, the following set kt + C a + m a ρ 2σ r q2 e 2 o 32π 2 ε ε g ε r Ag kt + C A L 2σ + m q2 e 2 o ρ r 32π 2 ε ε g ε r 7 ] ] ] r 4 =, 9 ] ] ] r 4 =, Atmos. Chem. Phys., 4, , 24

3 A. B. Nadykto and F. Yu: Formation of inary ion clusters from polar vapours 387 Numers of molecules in a critical cluster, T=273.5 K, RH=.85 Numers of molecules in critical cluster. T=273.5 K, RH= INN Eqs Fig.. Numers of molecules in critical cluster as a function of correspond to the sulfuric acid concentration of 7, squares Figure. Numers of molecules in critical cluster as a function of temperature, relative humidity to thatand of sulfuric 8 and acid diamonds concentration. to that Triangles of correspond 9 cm 3. to T=273.5 the sulfuric K, acid concentration RH=.85. of 7, squares to that of 8 and diamonds to that of 9 cm -3. T=273.5 K, RH=.85. Now we multiply sides of Eqs. 8 and 9 y m a and m respectively and sum the equations otained to get + C a Ag = ν a, A L + C ν In prevailing temperature and saturation ratios of the components a and, solution to the set of Eqs., or 9, gives us the numers of molecules of components a and in the cluster and the corresponding the cluster size. In the case of non-polar vapours l a, l set reduces to Eqs. 2 predicting the composition and size of the critical cluster in the classical theory. The derived model can e considered as a generalization of the classical inary theory ecause it not only accounts for all the mechanisms involved in classical inary ut also include the effect of the dipole -charge interaction neglected in the classical inary theory. Since the present model is derived, as well as the classical theory, assuming the ulk surface tension, density and dielectric constants of the condensed matter, it might e generally limited when the critical cluster is composed of n molecules. The quality of the measurements of the ulk surface tension applied in the nucleation models is another important issue. It is well known that the values of the surface tension given y different methods such as maximum ule pressure, capillary rise and Wilhelmi plate or other contact methods often deviates y several dynes. Since the thermodynamics of the cluster formation depends strongly on the surface tension of the sustance nucleating, in the case of disagreement etween theoretical predictions and experimental data it is difficult to figure out whether the capillary approximation is imperfect or the ulk surface tension is measured inaccurately Fig. 2. Numers of molecules in critical cluster as a function of correspond to the sulfuric acid concentration of 7, squares to that of 8 and diamonds to that of 9 cm 3. T=273.5 K, RH=.95. Figure 2. Numers of molecules in critical cluster as a function of temperature, relative concentration of 7, squares to that of 8 and diamonds to that of 9 cm -3. T=273.5 K, RH=.95. Although the present theory is not focused on the sign effect, it possesses some potential for explaining the sign preference ecause oth the enhancement factor for the condensation/nucleation rates Nadykto and Yu, 23 and correction to the chemical potential of the condensale vapour molecules due to the dipole-charge interaction strongly depend on the stretch of electrical field and the mean cluster density, which may e different for positive and negative ions due to different geometry and charge distriution. In order to study the sigh effect quantitatively, the detailed information aout structure and properties of the cluster ions have to e otained. 3 Results and Discussion Calculations were performed using Eqs. for inary sulfuric acid-water vapour mixture. Binary clusters are singly charged. Values of input parameters have een adopted from CRC Handook of Chemistry and Physics 22, Kulmala et al. 998 and Myhre et al Figures 4 show the comparisons of the cluster sizes as functions of amient temperature T, relative humidity RH, and the concentration of sulfuric acid vapour calculated from Eqs. 2 q= and q=, inary homogeneous nucleation, and Eqs. this study. As may e seen from Figs. 4, dipole-charge interaction significantly influences the formation of small cluster ions, reducing the numer of molecules in the critical cluster and, consequently, decreasing the critical size. Difference etween results given y the considered theories rises as the cluster size decreases. For small clusters, the difference in the numers of molecules in the critical cluster may e as Atmos. Chem. Phys., 4, , 24

4 388 A. B. Nadykto and F. Yu: Formation of inary ion clusters from polar vapours Numers of molecules in critical cluster. T=283.5K, RH=.85 Size of the critical cluster. T=273.5 K, RH= Fig. 3. Numers of molecules in critical cluster as a function of correspond to the sulfuric acid concentration of 7, squares to that of 8 and diamonds to that of 9 cm 3. T=283.5 K, RH=.85. Figure 3. Numers of molecules in critical cluster as a function of temperature, relative concentration of 7, squares to that of 8 and diamonds to that of 9 cm -3. T=283.5 K, RH= Numers of molecules in critical cluster. T=283.5 K, RH=.95 ig as more than 2 times. The deviation etween theory and present theory rises when the relative humidity and sulfuric acid concentration are growing. The contriution of the Thomson effect is smaller than that of the dipole-charge interaction that is essential for the nucleation from highly polar vapours. As may e seen from Figs. 4, the classical Kelvin-Thomson equation significantly overestimates the Fig. 4. Numers of molecules in critical cluster as a function of correspond to the sulfuric acid concentration of 7, squares to that of 8 and diamonds to that of 9 cm 3. T=283.5 K, Figure 4. Numers of molecules in critical cluster as a function of temperature, relative concentration RH=.95. of 7, squares to that of 8 and diamonds to that of 9 cm -3. T=283.5 K, RH=.95. numer of the molecules in the critical cluster compared to Cluster diameter, nm E+7.E+8.E+9 Sulfuric acid concentration cm-3 Figure 5. Size of the critical cluster given y different models. Triangles correspond to, squares Fig. to 5. classical Size of the and diamonds critical present cluster the given results of ythe different present study. models. T=273.5 Triangles K, correspond to, squares to classical and RH=.95. diamonds present the results of the present study. T=273.5 K, RH=.95. results predicted y the present theory in all the cases studied here. Since oth the sulfuric acid l=2.72 Deyes and water l=.85 Deyes are highly polar, such a ig effect of the dipole charge-interaction is not surprising. We would like to emphasize that the domination of the effect related to the dipole-charge interaction over the Kelvin-Thomson effect is not a must. This effect is essential for polar gases only. To illustrate the consequences of Eqs. in terms of the cluster size and thermodynamics of the cluster formation, we calculated the critical cluster sizes Fig. 5. As seen from Fig. 5, the deviation etween the classical theory and the present study is 5 percent. The complete theory of the nucleation rates requires corrections to oth the Gis free energy and forward condensation rate to e accounted for simultaneously. The contriution of the dipole-charge interaction to the growth kinetics may e significant Nadykto and Yu, 23 and, thus, the derivation of the nucleation rates is not as straightforward as it could e expected. The work on the model of the nucleation rates is in progress and we plan to pulish it elsewhere. 4 Summary In this paper we developed the model of ion-induced nucleation of two-component polar vapours. It has een shown that the formation of small ion clusters is influenced y the vapour polarity and the dipole-charge interaction decreases the size of critical clusters formed. It has een demonstrated that the actual size of small inary ion clusters may deviate significantly from the size predicted y classical Kelvin- Thomson theory, when the highly polar vapours are nucleating. The derived model can e considered as a generalized reformulation of the classical theory extended to the nucleation in polar vapours. Based on the results otained we suggest that the dipole moment of the condensing monomers is likely to e a new parameter controlling the inary ioninduced nucleation. Atmos. Chem. Phys., 4, , 24

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