particles containing ammonium, sulfate, and nitrate

A parameterization was developed for the hetero- geneous reaction probability ( ) of N2O5 as a function of temperature, relative humidity (RH), particle composition, and phase state, for use in advanced air quality models. The reaction probabilities on aqueous NH4HSO4, (NH4)2SO4, and NH4NO3 were modeled statistically using data and un- certainty values compiled from seven different laboratory studies. A separate regression model was fit to laboratory data for dry NH4HSO4 and (NH4)2SO4 particles, yielding lower values than the corresponding aqueous parameter- izations. The regression equations reproduced 80% of the laboratory data within a factor of two and 63% within a factor of 1.5. A fixed value was selected for on ice- containing particles based on a review of the literature. The combined parameterization was applied under atmospheric conditions representative of the eastern United States using 3-dimensional fields of temperature, RH, sulfate, nitrate, and ammonium. The resulting spatial distributions of were contrasted with three other parameterizations that have been applied in air quality models in the past and with atmo- spheric observational determinations of . Our equations lay the foundation for future research that will parameterize the suppression of when inorganic ammoniated particles are mixed or coated with organic material. Our analyses draw attention to a major uncertainty in the available laboratory data at high RH and highlight a critical need for future labo- ratory measurements of at low temperature and high RH to improve model simulations of N2O5 hydrolysis during win- tertime conditions.


Introduction
Heterogeneous reactions of N 2 O 5 have a substantial influence on gaseous and particulate pollutant concentrations Correspondence to: J. M. Davis (davisj@ncsu.edu)(Dentener and Crutzen, 1993;Riemer et al., 2003;Aldener et al., 2006).To accurately simulate nighttime nitrogen chemistry and the resulting impacts on ozone, particulate nitrate, and nitrogen oxides, air quality models must contain a reliable parameterization of the heterogeneous reaction probability (γ ), which is defined in a relative frequency sense as the fraction of collisions between gaseous N 2 O 5 molecules and particle surfaces that lead to the production of HNO 3 .
Several recent laboratory studies demonstrate that γ varies substantially with temperature (T ), relative humidity (RH), and particle composition (Mentel et al., 1999;Kane et al., 2001;Hallquist et al., 2003).Mechanistic explanations for these variations have been proposed (Mozurkewich and Calvert, 1988;Wahner et al., 1998;Hallquist et al., 2003), but the mechanisms are semi-quantitative at best and not yet suitable for inclusion in air quality models.In the absence of a quantitative heterogeneous reaction mechanism, some investigators have developed statistical parameterizations for γ based on laboratory data and incorporated those into air quality models.In a pioneering modeling study, Dentener and Crutzen (1993) selected 0.1 as a representative value of γ based on the available data at that time.Riemer et al. (2003) parameterized γ as a function of the particulate sulfate and nitrate content using laboratory data of Mentel et al. (1999).Most recently, Evans and Jacob (2005) modeled γ on sulfate particles as a function of RH and T using the laboratory data of Kane et al. (2001) and Hallquist et al. (2003), respectively.
In the present study, we build upon these past efforts to develop a parameterization of γ as a function of T , RH, particle composition, and phase state, for use in advanced air quality models.Our work has several advantages over previous studies: (1) our parameterization uses all published laboratory measurements of γ on ammoniated sulfate and nitrate particles; (2) rigorous statistical methods are employed (e.g., significance test of each independent variable, weighting of each data point by the measurement uncertainty, defining extrapolation limits) leading to a simple formula that captures the most important features in the laboratory data; and (3) Published by Copernicus Publications on behalf of the European Geosciences Union.e Measurements were collected between 293 and 297 K. Midpoint of this range, 295 K, is used in our analysis.f BAD06 does not state the temperature at which their experiments were conducted.The 293 K value is obtained from the experimental description by Badger et al. (2006b).
phase changes are considered explicitly in our parameterization such that γ on aqueous particles exceeds that on solid particles.

Laboratory data
All the data used for our examination of γ with respect to T , RH, and particle composition come from laboratory work documented in the following publications: Mozurkewich and Calvert (1988); Hu and Abbatt (1997); Folkers (2001); Kane et al. (2001); Folkers et al. (2003); Hallquist et al. (2003); Badger et al. (2006a).For brevity, these documents will be referred to hereafter as MOZ88, HU97, FOL01, KAN01, FOL03, HAL03, and BAD06, respectively.Each publication contains a description of the procedures that were followed to determine γ from analytical measurements of the reactants and/or products of the heterogeneous reaction.In some cases, the laboratory procedures differ considerably from one another (e.g., FOL03 vs. HAL03); however, accounting for the possible impacts of procedural differences on the reported values of γ is beyond the scope of this work.Therefore, we used published values of γ directly rather than attempting to compute them from raw laboratory measurements.
To our knowledge, the seven publications listed above contain all of the published laboratory measurements of γ on ammoniated sulfate and nitrate particles at conditions relevant to the troposphere.We focus on these particle compositions because they constitute a substantial portion of the aerosol surface area in continental air masses and because laboratory measurements of γ on these surfaces are relatively abundant.Measurements of γ on organic acids, soot, and dust have been parameterized adequately in a recent modeling study (Evans and Jacob, 2005), so those particle compositions are not reconsidered here.The impact of organic coatings on γ is an active area of research (Folkers et al., 2003;Thornton and Abbatt, 2005;Anttila et al., 2006;McNeill et al., 2006;Cosman et al., 2008;Cosman and Bertram, 2008) and should be incorporated into future parameterizations.

Aqueous particles
Our review of the literature identified 67 published values of γ on aqueous particles, including 35 data points for NH 4 HSO 4 , 28 for (NH 4 ) 2 SO 4 , and 4 for NH 4 NO 3 (see Table 1).Included in Table 1 is a HAL03 data point on (NH 4 ) 2 SO 4 particles at 20% RH and 298 K.At those conditions, (NH 4 ) 2 SO 4 is expected to be dry (Martin et al., 2003) but HAL03 reports a measurable increase in aerosol volume in these experiments relative to their own dry experiments at 20% RH.Therefore, this datum is retained for statistical modeling of the aqueous particle data.
Figure 1a to c summarize all of the laboratory measurements used to develop our statistical model for aqueous particles, and illustrate the dependence of γ on RH, T , and particle composition.For all three compositions studied here, a prominent feature is that γ increases with RH although the reliability of this trend at high RH is unclear (see discussion in Sect.5.2 and Appendix A for further details).For a fixed RH and   Badger et al. (2006b).g These data were not used in our analysis due to the missing standard error and uncertain γ value.
A successful statistical model must capture these features of the laboratory data.

Dry particles
Our literature review also identified 24 published values of γ on dry particles, including 11 data points for NH 4 HSO 4 and 13 for (NH 4 ) 2 SO 4 (see Table 2).Measurements of γ on dry NH 4 NO 3 were not found.All of the MOZ88 data points were removed from our analysis based on the description in MOZ88 of an artificial enhancement of γ at RH<40% due to small amounts of NH 3 evaporating from the NH 4 HSO 4 particle surfaces.Theoretically, none of the NH 4 HSO 4 particles analyzed in KAN01 should be dry because their experiments began with wet particles and pure NH 4 HSO 4 does not crystallize at 295 K (Martin et al., 2003).However, we include three KAN01 data points in Table 2 because of their extremely low γ values coupled with the knowledge that supersaturated aqueous NH 4 HSO 4 particles are difficult to sustain below 22% RH (Tang and Munkelwitz, 1994).The (NH 4 ) 2 SO 4 particles analyzed in KAN01 below 40% RH and in BAD06 below 34% RH were reported as dry, so they are included in Table 2. Similarly, six of the (NH 4 ) 2 SO 4 data points from HAL03 are reported on dry particles.All of the data reported by HU97, FOL01, and FOL03 were obtained on aqueous particles, so those publications are not listed in Table 2.An analysis of the 15 valid laboratory values of γ on dry particles indicates that they are generally lower than γ on aqueous particles.The very limited data reveal an apparent influence of RH and T on γ , which can be seen in Fig. 1d.

Statistical models for aqueous particles
The primary objective of this study is to derive a robust statistical parameterization for γ on aqueous particles using the data summarized in Fig. 1.Prior to our regression analyses, the laboratory data were transformed using the logit function: This transformation ensures that all model estimates of γ lie between zero and one, which is consistent with the relativefrequency definition of γ .One regression equation was estimated from the union of NH 4 HSO 4 and (NH 4 ) 2 SO 4 data for which observations are most abundant.A model based on the NH 4 NO 3 data was fit separately since those measurements were reported in a separate study from all the ammoniated sulfate data and include only four unique combinations of T and RH values (see FOL01 data in Table 1).
On NH 4 HSO 4 and (NH 4 ) 2 SO 4 particles, a dependence of γ on temperature is difficult to observe in the measurements.The only cross sections of data where temperature was varied over a large range are at 50% RH (see Fig. 1a and b).Along those cross sections, the γ values do not change monotonically with temperature.Based on the work by Hallquist et al. (2003), Evans and Jacob (2005), and our own data analyses, a temperature threshold, j , was applied within our statistical model.The threshold temperature variable is defined as: where j is an integer whose value is determined below.(2) The regression coefficients, β, represent scalar quantities and the model errors, ε i , were assumed to be normally and independently distributed.Higher order polynomial terms were considered for RH, but not T j , based on our inspection of plots of λ versus RH and T .Note that during the regression analysis, Eqs. ( 1) and (2) were solved simultaneously using the combination of NH 4 HSO 4 and (NH 4 ) 2 SO 4 data.The extra coefficientsβ 20 , β 21 , β 22 , β 23 , β 24 , and β 25 -in Eq. ( 2) were included to test whether the linear relationship between λ, RH, and T j , differs significantly for different particle compositions.
Coefficients of the linear model were estimated using weighted least squares (Kutner et al., 2005).Each observation was assigned a weight, w, that is a function of the standard error values, σ , reported with each laboratory measurement of γ : These weights were derived based on a Taylor approximation for the estimate of the variance of λ (Casella and Berger, 2002, p. 241).
Stepwise model selection for regression analysis was used to determine which of the variables in Eqs. ( 1) and ( 2) are statistically significant.At each stage of the model selection process, the Akaike Information Criterion (AIC) was used to decide whether linear terms, cross product terms, and higher order terms should be included or excluded.This process was repeated for each integer value of j between 275 K and 295 K as well as for a model with no temperature threshold (i.e.j =0 K).The AIC value was used to determine the most parsimonious statistical model that best fits the data with a minimum number of covariates (Akaike, 1973).This procedure yielded a model with five statistically significant terms and a threshold temperature of j =291 K.The individual equations for NH 4 HSO 4 and (NH 4 ) 2 SO 4 differ in the intercept term and the slope term for threshold temperature.
The best-fit values and standard errors of each coefficient are given in Table 3.In each equation, RH is expressed as a percentage.Several diagnostics were used to assess the adequacy of the regression function.Plots of the residuals from the linear model indicate the error terms are reasonably normally distributed.Correlation between the covariates is very low, indicating the effects of multicollinearity on the estimated regression parameters should not be a problem.While the above model satisfies several statistical criteria, there is a lingering concern about this parameterization because the relationship between γ and RH lacks consistency from one laboratory to another.We discuss this source of uncertainty in Sect.5.2 and provide a second possible parameterization for γ on aqueous NH 4 HSO 4 and (NH 4 ) 2 SO 4 particles in Appendix A.
For aqueous NH 4 NO 3 particles, all data were collected in a narrow temperature range so it was not possible to identify a temperature trend.The model for NH 4 NO 3 was selected to match the form of Eq. ( 1), and is given by the following equation: The estimated coefficients for this model are also given in Table 3.The estimated values of λ 1 , λ 2 , and λ 3 , are back transformed to compute the reaction probability values for the three particle compositions.
The parameterized functions for γ are plotted on a log scale versus RH in the upper panel of Fig. 2 for a selected set of temperatures.The lower panel of Fig. 2 shows contour plots of the estimated γ values spanning the ranges of T and RH encountered in the lower troposphere.Each laboratory measurement is shown as a discrete point in the lower panel.Laboratory data are not available over the entire range of tropospheric conditions, so caution must be exercised when extrapolating parameterizations derived from the available data.For example, extrapolation of the regression equations to the regime of low T and high RH yields γ values which exceed all of the lab measurements (e.g., γ 1 >0.15).To prevent erroneous extrapolations when these functions are used at different atmospheric conditions, such as the application in Sect.4.3, the estimated γ values were constrained to be no greater than the maximum observed γ : These upper limits are depicted by horizontal gray lines in Fig. 2a-c, and the lightly-shaded irregular pentagons in the lower-right portions of Fig. 2d-f.Extrapolation of Eqs.(4-7) must also be limited by thermodynamic considerations.At RH<32.8%, (NH 4 ) 2 SO 4 crystallizes to form a solid phase (Martin et al., 2003) so γ 2 is expected to decline abruptly with decreasing RH across this phase boundary.The regions of Fig. 2b and e   Multiple R-Squared: 0.8991, Adjusted R-squared: 0.8823 to illustrate this limit.In contrast, aqueous droplets containing pure NH 4 HSO 4 or pure NH 4 NO 3 do not crystallize (Martin et al., 2003) so our equations for γ 1 and γ 3 may be applicable at very low RH.Under the conditions of very high RH and low T encountered occasionally in the lower troposphere, particle-bound water will freeze to form ice (Martin, 2000).The point at which this phase change occurs is determined by the ratio of the saturation vapor pressure over ice (pw ice ) to that over water (pw water ).This portion of the phase diagram is shaded in the lower panel of Fig. 2, using the Goff-Gratch equations to compute pw ice and pw water (Goff and Gratch, 1946).As RH increases across this phase boundary, we expect γ to be far less than the upper limits imposed by Eq. ( 8).

Statistical model for dry particles
When the ambient RH is below the deliquescence relative humidity (DRH), ammoniated sulfate and nitrate can exist as dry particles that are devoid of liquid water.For dry particles, a separate parameterization of γ is needed.The statistical methodology outlined in Sect.3.1 was repeated using the 15 dry particle data points selected from Table 2 except, this time, higher order polynomial terms of RH were not consid-ered due to the very limited sample size.The variable selection procedure yielded a model with three statistically significant terms and a threshold temperature of 293 K. Analogous to Eq. ( 2), extra coefficients were included to test whether the linear relationship between λ, RH, and T j , differs for different particle compositions, but none of those coefficients were deemed significant.The model for dry particles (denoted with subscript "d" for dry) is given by The best-fit values and standard errors of each coefficient are given in Table 3.The estimated values of λ d can be back transformed using Eq. ( 7).As with the aqueous parameterizations, estimated γ values were constrained to be no greater than the maximum observed γ : Recall that no laboratory measurements of γ on dry NH 4 NO 3 were found, so Eqs.(9-10) are intended primarily for dry ammoniated sulfate particles.

Model for internally-mixed particles
In urban-and regional-scale air quality models, inorganic fine particles are often assumed to exist as an internal mixture of SO 2− 4 , NO − 3 , NH + 4 , and H 2 O (Wexler et al., 1994;Binkowski and Roselle, 2003).To date, laboratory measurements of γ on such mixtures have not been reported.As a provisional measure, it is useful to express γ as a molar-weighted average of the single-component parameterizations provided above.Assuming all of the NO − 3 is present as NH 4 NO 3 and the leftover NH + 4 is distributed between (NH 4 ) 2 SO 4 and NH 4 HSO 4 , we computed mole fractions where γ * i is defined in Eq. ( 8).The corresponding expression for mixed dry particles is In the absence of laboratory measurements on dry NH 4 NO 3 particles, the min function is used in Eq. ( 13) to reflect our expectation that γ on dry NH 4 NO 3 particles is similar to γ on dry ammoniated sulfate particles, but should not exceed γ on aqueous NH 4 NO 3 particles.
No laboratory measurements of γ on particles containing a mixture of ice and ammoniated sulfate or nitrate were found, but γ on pure ice has been studied extensively for stratospheric applications (e.g., Leu, 1988;Hanson and Ravishankara, 1991).A representative γ value of 0.02 was selected from those data (IUPAC, 2006).This fixed value was applied to all ice-containing particles in the present study.

Mechanistic evaluation
Our statistical parameterizations capture all prominent features of the laboratory data and are qualitatively consistent with the ionic hydrolysis mechanism proposed by Mozurkewich and Calvert (1988) and later extended by Wahner et al. (1998) and Stewart et al. (2004): First, the aqueous parameterizations of γ increase with RH at each temperature for all three particle compositions (see Fig. 2a-c).This is consistent with the KAN01 and FOL01 data (see Fig. 1a-c), which indicate that aerosol water content controls γ via the dissolution step R2a.However, laboratory data on several other water-soluble particle types indicate no discernable change in γ above 50% RH.This issue is explored in Sect.5.2 and Appendix A. Second, the parameterization of γ on NH 4 HSO 4 exceeds that on (NH 4 ) 2 SO 4 (i.e., γ 1 >γ 2 ) for T ≤298 K due to the negative value of our β 20 coefficient (see Eq. 5 and Table 3).This result is consistent with the KAN01 and HAL03 data (compare Fig. 1a  and b).Kane et al. (2001) speculate that the rise in γ may be due to the small amount of H + in NH 4 HSO 4 solutions, which could catalyze Reaction (R2a) (Robinson et al., 1997).
Third, the parameterizations of γ 1 and γ 2 decrease with increasing temperature for T >291 K.This is consistent with the MOZ88 and HAL03 data (see vertical transects at 45%, 50%, 55%, 66%, and 76% RH in Fig. 1a, and at 50% RH in Fig. 1b).Mozurkewich and Calvert (1988) speculate that this negative temperature dependence may be due to the N 2 O 5 evaporation rate (R1b) increasing faster with temperature than the N 2 O 5 dissolution rate (R2a).
Fourth, the parameterization of γ on NH 4 NO 3 is lower than that on (NH 4 ) 2 SO 4 (i.e., γ 3 <γ 2 ).This can be seen most clearly by comparing Fig. 2c with the 293 and 298 K isotherms in Fig. 2b because all of the NH 4 NO 3 measurements were collected in that temperature range.This "nitrate effect" is described in detail by Mentel et al. (1999), and is attributed to an enhanced rate of N 2 O 5 recombination (R2b) in aqueous particles composed of NO − 3 .Note that the NO − 3 concentration leading to recombination is dominated by the substrate composition rather than the small amount which comes from N 2 O 5 dissolution.Comparing our parameterizations of γ 2 and γ 3 over the ranges of T and RH where NH 4 NO 3 lab data are available (i.e., 293<T <298 K and 50<RH< 80%), we see that the nitrate effect ranges from a factor of 1.3 to 5.2.This rather broad range is due to the larger RH dependence in Eq. ( 6) versus (5) (compare β 31 with β 11 in Table 3).More importantly, the nitrate effect deduced from our analyses is much less pronounced than the factor of ten applied in model simulations by Riemer et al. (2003).Our result is probably more appropriate for continental air masses, because it is derived from a comparison between (NH 4 ) 2 SO 4 and NH 4 NO 3 particles rather than a comparison between Na 2 SO 4 and NaNO 3 particles.Fifth, our parameterization of γ on aqueous ammoniated sulfate is 1.6 to 7.5 times greater than that on dry particles within the RH range where those components can exist in either a solid or aqueous phase due to hysteresis (i.e., γ 1 >γ d and γ 2 >γ d for CRH <RH<DRH).This result is consistent with the HAL03 data, in which γ on wet (NH 4 ) 2 SO 4 was found to be 1.2-13 times greater than that on dry (NH 4 ) 2 SO 4 (compare 5 paired values in Tables 1 and 2).The difference in γ on dry versus wet particles can be attributed to the scarcity of water in the dry particles, which inhibits dissolution.The statistical significance of the RH term in Eq. ( 9) puzzled us initially, because we expected the heterogeneous uptake of N 2 O 5 on dry particles to be independent of RH.However, it has been shown that the heterogeneous uptake of HNO 3 on solid salts can increase with RH due to increases in surface-adsorbed water (Goodman et al., 2000;Ghosal and Hemminger, 2004).It is possible that the reactivity of N 2 O 5 on solid ammoniated sulfate particles is influenced by a similar process.An alternative explanation is that homogeneous gas-phase reactions between N 2 O 5 and water vapor were taking place during the laboratory experiments, leading to an apparent enhancement of γ d (Davis et al., 2008a).

Statistical evaluation
Figure 3 illustrates a comparison between the parameterized γ values and the underlying laboratory data used for statistical modeling.Out of the 35 aqueous NH 4 HSO 4 data points, 22 are fit within a factor of 1.5 and 27 within a factor of two (see Fig. 3a).Seven of the remaining eight points are overestimated by Eq. ( 4).Four of these are KAN01 data points with RH between 29 and 37%, which is in the metastable regime where the NH 4 HSO 4 phase is very sensitive to trace impurities.Figure 6 in KAN01 shows the same four points falling below their best-fit line, while one data point in this RH range has a much higher γ value.It is possible that crystallization occurred during a subset of their experiments leading to the four low γ measurements.Figure 3a also reveals that Eq. ( 4) overestimates two HAL03 data points by a factor of four.Those values were measured at 268 and 273 K, but are a factor of three lower than the HAL03 observations at the surrounding temperatures of 263 and 288 K. To fit all four of these γ values would require a third order polynomial of temperature, but such a term would have no physical basis.Instead, Eq. ( 4) yields a fixed value of 0.052 for all 4 data points because they are all below the threshold temperature of 291 K.
Out of the 28 aqueous (NH 4 ) 2 SO 4 data points, 15 are fit within a factor of 1.5 and 22 within a factor of two (see Fig. 3b).All four data points that Eq. ( 5) overestimates by more than a factor of two are from KAN01 and in the metastable RH regime (42, 45, 48, and 62% RH). Figure 5 in KAN01 shows the same four points falling below their best-fit line, while three data points in this RH range have much higher γ values.As noted by Hallquist et al. (2003) and analogous to our line of reasoning above, we speculate that crystallization occurred during a subset of the KAN01 (NH 4 ) 2 SO 4 experiments leading to the four low γ measurements.Figure 3b also reveals two HU97 data points that we underestimate by a factor of three or more.A number of authors (e.g., Kane et al., 2001;Hallquist et al., 2003) have commented that the HU97 data are unlike those reported in any other study, both in their magnitude and RH dependence, but no physical explanation could be found to justify the exclusion of these data from our statistical model.Given the large differences compared to other laboratory data, we do

BAD06 HAL03 KAN01
q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q q 0.00

NH4HSO4 (aqueous) NH4HSO4 (dry) (NH4)2SO4 (aqueous) (NH4)2SO4 (dry) NH4NO3
(aqueous) q q q q q q q q q q q q q q q q q q q q qq q q q q q q q q q q q q q q q q 0.00 not expect our parameterization to fit the HU97 data well and this expectation is verified in Fig. 3b. Figure 3c shows excellent agreement between the fitted and observed γ values on aqueous NH 4 NO 3 particles, with Eq. ( 6) capturing all four data points within ±7%. Figure 3d reveals good agreement between Eq. ( 9) and the dry particle data.Out of the 15 data points, 11 are fit within a factor of 1.5 and 13 within a factor of two.One of the outlying data points has an extremely large measurement uncertainty (see HAL03 datum at 308 K in Table 2) so our weighted regression model is not expected to capture this point.Equation ( 9) reproduces the other outlying point -from KAN01 at 8% RH -within two times the measurement uncertainty.
In Fig. 3e, we summarize a comparison between all of the fitted and observed γ values used to develop our regression equations.Among the 82 data points reported in seven different laboratory studies, our parameterizations reproduce 63% within a factor of 1.5 and 80% within a factor of two.For comparison, we contrasted this performance with the T -and RH-dependent parameterization reported by Evans and Jacob (2005) for γ on (NH 4 ) 2 SO 4 particles (referred to hereafter as EJ05).In the process, we discovered a typographical error in Table 1 of that paper.The form of their equation should read γ =α×10 −β but the negative sign was omitted in the publication (Mathew Evans, personal communication).Figure 3f summarizes the EJ05 performance after correction of the typographical error.When compared against all 40 (NH 4 ) 2 SO 4 data points (dry and wet), EJ05 underestimates half of them by 15% or more whereas our parameterization exhibits a smaller median relative bias (+10%).The EJ05 parameterization is not intended for NH 4 HSO 4 particles and an erroneous application of EJ05 to such acidic particle surfaces would underestimate most of the NH 4 HSO 4 data points by nearly a factor of 2 (see circles in Fig. 3f).Similarly, EJ05 should not be applied to NH 4 NO 3 particles because it was not designed to capture the nitrate effect on γ (see plus symbols in Fig. 3f).

Atmospheric implications
To illustrate the atmospheric relevance of this work, we used the parameterizations developed above to compute γ at conditions encountered over the eastern United States.The inputs for this calculation (i.e., 3-dimensional fields of  (Appel et al., 2008), so the calculation does not account for any feedback on γ that would result from changes to particle composition if our new parameterization was implemented in an air quality model.In the CMAQ model run, transitions in the aerosol phase state were not simulated.To utilize all of the parameterizations developed in this study, we assume here that the modeled particles are wet above their CRH and dry below their CRH.This is a reasonable assumption because several field campaigns have demonstrated that tropospheric particles often retain water well below their DRH (Rood et al., 1989;Santarpia et al., 2004;Khlystov et al., 2005).The CRH for an internally mixed particle composed of SO 2− 4 , NO − 3 , and NH + 4 was calculated using the equation for complete crystallization by Martin et al. (2003).Although that equation was only validated at 293 K, we applied it at all T because there currently exists very little data to justify a temperaturedependent CRH (see discussion by Schlenker et al., 2004).To compute γ over the full range of conditions encountered in the atmosphere, γ ambient , we applied Eqs.(12-14) as follows: where IRH is the relative humidity of ice formation, which varies with temperature (see discussion in Sect.3.1).
Values of γ ambient were computed for each hour of January, February, July, and August 2001, using T , RH, SO 2− 4 , NO − 3 , and NH + 4 in each 12-km grid cell.For comparison, we also computed hourly spatial fields of γ on NH 4 HSO 4 (i.e., γ * 1 ), (NH 4 ) 2 SO 4 (i.e., γ * 2 when RH ≥32.8% and γ * d otherwise), and NH 4 NO 3 (i.e., γ * 3 ), each with the IRH constraint imposed, as well as three other parameterizations of γ that have been applied in air quality models in the past.For illustrative purposes, the hourly γ values were averaged over two-month periods representative of the winter (January-February 2001) and summer (July-August 2001).When averaging hourly values, we included only the nighttime hours of 04:00-09:00 GMT because N 2 O 5 concentrations are negligible in the daylight (Brown et al., 2003a) so values of γ at those times have no practical importance.Though vertical distributions of γ were computed, we focused our attention on the model layer that is 75-150 m above the surface because the CMAQ results indicate that the product of N 2 O 5 and aerosol surface area is highest in this layer during both the summer and winter periods.Recall that the N 2 O 5 hydrolysis rate is proportional to this product (Riemer et al., 2003).Vertical profiles of NO 3 and N 2 O 5 measured by Stutz et al. (2004) and Brown et al. (2007aBrown et al. ( , 2007b)), respectively, support our decision to focus on γ ∼100 m above the surface.
Figure 4 displays results of the above calculations for the January-February 2001 period.Across the eastern US, values of γ ambient range from 0.02 over the Midwest to 0.05 over Florida (see Fig. 4d).Low values over the Midwest result from high NO − 3 concentrations, typical of this region during winter, combined with below-freezing temperatures and high RH that occasionally exceeds the IRH.The high γ ambient values over Florida result from low NO − 3 concentrations coupled with the absence of ice-containing particles.Comparing Fig. 4d with e, it can be seen that our γ ambient values fall below the fixed value of 0.1 used in the early work by Dentener and Crutzen (1993) which is now recognized as an upper estimate of γ (Evans and Jacob, 2005).Comparing Fig. 4d with f, we see that the γ parameterization by Riemer et al. (2003) is consistently lower than γ ambient .The Riemer parameterization has an upper bound of 0.02, based on measurements of γ on Na 2 SO 4 and NaHSO 4 particles (Mentel et al., 1999).As noted above, our parameterization is more appropriate for continental air masses because it is derived from measurements on ammoniated particles rather than sodiumcontaining particles.Moreover, the Riemer parameterization does not depend on T and RH due to insufficient laboratory data at the time of that study.
The comparison between γ ambient and EJ05 is the most intriguing (compare Fig. 4d and g) because EJ05 is the current parameterization of choice in many regional-and globalchemistry models such as CMAQv4.6 and GEOS-CHEM.Across the northern half of the domain, EJ05 often exceeds 0.10 whereas γ ambient is between 0.02 and 0.06.At lower latitudes, EJ05 exceeds γ ambient by 0.01 to 0.02.Recognizing that γ ambient is suppressed by the nitrate effect whereas EJ05 is not, a more equitable comparison is between EJ05 and our parameterization of γ on (NH 4 ) 2 SO 4 (i.e., Fig. 4g versus b).Surprisingly, this comparison also revealed rather large differences.The main reason for differences in the northern latitudes is that our parameterization was bounded by the maximum laboratory value of 0.053 (see Fig. 2b and e) whereas EJ05 yields very large values when extrapolated to low T and high RH conditions.In addition, our values of γ in the north and along the Appalachian Mountains are suppressed by ice formation (see upper-left corner of Fig. 4b for example) whereas the effects of that phase transition are not considered in the EJ05 parameterization.In fact, the very highest values of γ (∼0.13) are obtained when EJ05 is extrapolated to the regime where ice formation is favored.In the southern half of the domain, EJ05 exceeds γ * 2 by 0.02 to 0.03.These locations are characterized by average nighttime conditions of 80% RH and 280 K, where EJ05 yields a value of 0.08 and our parameterization reaches its upper-limit value of 0.053.
In the summer, γ ambient ranges from 0.01 over Texas and Oklahoma to 0.07 over the Appalachian Mountains.The lowest values correspond to the portion of our domain where the nighttime average temperature was highest (302 K), RH was lowest (60%), and the particle compositions were dominated by (NH 4 ) 2 SO 4 (x 2 =0.9).Over the Appalachian Mountains, high values of γ ambient resulted from the particle compositions being dominated by NH 4 HSO 4 (x 1 =0.8) along with T <290 K and RH>80%.Across the entire domain, γ ambient and EJ05 were in very good agreement except in locations where x 1 =0.8.Under those acidic conditions, γ ambient exceeded EJ05 by nearly 0.02 as would be expected from a comparison of Fig. 3e and f.

Comparisons with atmospheric data
Historically, attempts to estimate γ from atmospheric observations were obstructed by the inability to measure ambient N 2 O 5 concentrations.This obstacle was overcome by Brown et al. (2001), leading to ambient measurements of N 2 O 5 in a handful of intensive field campaigns (Brown et al., 2003a;Brown et al., 2004;Wood et al., 2005;Brown et al., 2006;Matsumoto et al., 2006;Ayers and Simpson, 2006;Nakayama et al., 2008).Brown et al. (2003b) also developed an innovative method for estimating γ from simultaneous, high-resolution measurements of N 2 O 5 , NO 3 , NO 2 , O 3 , and aerosol surface area.Applications of this technique have been reported in two field campaigns to date, yielding ambient estimates of γ =0.03±0.02 in the marine boundary layer off the coast of New England during summer 2002 (Aldener et al., 2006) and γ =0.017±0.004over Ohio and western Pennsylvania during summer 2004 (Brown et al., 2006).Using the CMAQ results which simulated July -August 2001, our parameterization yields slightly larger values: γ ambient =0.05 off the New England coast, and 0.03-0.05over Ohio and western Pennsylvania.However, as mentioned in Sect.2, our parameterization neglects the influence of organic coatings on γ ambient .Laboratory and field evidence suggests that the organic influence is very important (Folkers et al., 2003;Brown et al., 2006) and additional research in that area should help close the gap between the observationbased and statistically-parameterized estimates of γ during the summer.
Several modeling studies concur that gaseous and particulate pollutant concentrations are far more sensitive to γ during the winter than during summer (Dentener and Crutzen, 1993;Evans and Jacob, 2005;Gilliland et al., 2006).However, N 2 O 5 has seldom been measured in wintertime field campaigns (Wood et al., 2005;Ayers and Simpson, 2006;Nakayama et al., 2008) and none of those campaigns collected the full suite of measurements required to estimate γ .Moreover, it is not clear that the observation-based estimation method mentioned above will be successful during winter months because the underlying steady state approximation is not applicable at cold temperatures (Brown et al., 2003b).Thus, our ability to evaluate the statistical parameterizations of γ against atmospheric data is hindered during winter months which, unfortunately, correspond to the time www.atmos-chem-phys.net/8/5295/2008/Atmos.Chem.Phys., 8, 5295-5311, 2008 period when the heterogeneous reaction probability of N 2 O 5 has maximum importance.

Uncertainty in γ values at high RH
A prominent feature in our parameterization of γ on aqueous ammoniated sulfate particles is the increase with RH across the entire range of ambient conditions.While mechanistic support for this trend exists at low RH, its validity at high RH is uncertain (Anonymous Referee #1, 2007).Having surveyed all ammoniated sulfate data in the published literature, only KAN01 show a uniform increase in γ above 50% RH while the remaining publications contain very few data points at high RH and their trends are mixed (see Fig. 1a  and b).Given this dearth of data coupled with the sizeable influence of γ on atmospheric concentrations during winter, all laboratory measurements on water-soluble particles were examined to seek independent confirmation of the trend reported by KAN01.Measurements on NaNO 3 (Wahner et al., 1998;HAL03) and NH 4 NO 3 (FOL01) show a pronounced increase in γ as RH exceeds 50%, but this increase may be due in part to the dilution of NO − 3 which decreases the rate of N 2 O 5 recombination via reaction R2b.In contrast, laboratory measurements of γ on NaCl (Behnke et al., 1997;Stewart et al., 2004;McNeill et al., 2006), sea-salt (Stewart et al., 2004;Thornton and Abbatt, 2005), and malonic-acid particles (Thornton et al., 2003) show no discernable effect of RH above 50%.Likewise, field-based determinations of γ indicate no clear dependence on RH in this range (Brown, 2008).Based on this survey, we conclude that the trend reported by KAN01 at high RH has not been confirmed in other studies.
An alternative parameterization is described in Appendix A, in which we repeated the procedure described in Sect.3.1 without using the KAN01 data.Until more data are collected at high RH, we recommend the alternative parameterization for use in advanced air quality models such as CMAQ, GEOS-CHEM, and WRF-Chem (Davis et al., 2008c).A negative effect of discarding the KAN01 measurements is a 40% loss of available data for use in the development of the statistical model.

Critical gaps in the laboratory data
Our statistical analysis of the laboratory data and our assessment of the atmospheric implications have helped to identify the most important areas where more laboratory measurements of γ are needed.First, laboratory studies at low T and high RH conditions are of utmost importance.This need is motivated by the substantial differences between Eqs. (4-5) and (A1-A2), and between Fig. 4b and g, which arise from different extrapolations of the available laboratory data to typical winter conditions.A set of laboratory measurements at ∼260 K with RH increasing from 50-95% would be particularly illuminating as it would help fill in the data gap mentioned above and test our hypothesis that γ decreases abruptly at the phase boundary between liquid water and ice.A transect of laboratory measurements at ∼85% RH with T decreasing from 295 to 260 K would also be valuable to double-check the validity of the temperature threshold in Eqs.(4-5) and (A1-A2).It is important to emphasize that no laboratory measurements of γ on aqueous (NH 4 ) 2 SO 4 particles are available below 288 K (see Fig. 2e) and that considerable variations exist in the available data at high RH (see Sect. 5.2) and on pure liquid water (Davis et al., 2008b).Additional data at these conditions are essential for atmospheric modeling during winter periods, especially given the fact that there are no ambient data to evaluate our statistical parameterizations during winter months.
Second, there is a need for more laboratory data on aqueous NH 4 NO 3 particles.The available measurements adequately characterize the regime of 293-298 K and 50-80% RH (see Fig. 2f).In the atmosphere, NH 4 NO 3 concentrations are highest in the winter because of low SO 2− 4 concentrations and because NH 4 NO 3 formation is most favorable at low T and high RH (Stelson and Seinfeld, 1982).Therefore, future experiments on NH 4 NO 3 particles should be conducted at conditions representative of winter.Third, a few laboratory experiments ought to be conducted on internally-mixed particles consisting of aqueous SO 2− 4 , NO − 3 , and NH + 4 .Such experiments would help evaluate and refine the simple mixing rule given in Eq. ( 12).Fourth, a few measurements are needed on particles containing a mixture of ice and ammoniated sulfate or nitrate.As noted in Sect.3.3, we opted to use a γ value for pure ice in the present study (see Eq. 14) due to the absence of such measurements.
Although there are far fewer measurements of γ on dry particle surfaces than on wetted surfaces, we believe that additional experiments on dry particles will not be very valuable.In our analyses of the 3-dimensional CMAQ fields, the ambient RH fell below the internally-mixed CRH in only 0.02 and 0.7% of the grid cells during summer and winter, respectively.However, this conclusion may require reevaluation if future studies elucidate a strong temperature dependence in the CRH of internally mixed particles.

Conclusions
We have developed a parameterization for the heterogeneous reaction probability of N 2 O 5 as a function of RH, T , particle composition, and phase state, for use in advanced air quality models.We used all published measurements of γ on ammoniated sulfate and nitrate particles, which were compiled from seven different laboratory studies.The final equations are relatively simple, non-linear functions of RH and T that were selected in an objective and statistically rigorous manner.Our parameterization reproduced 80% of the laboratory data within a factor of two and 63% within a factor of 1.5.To our knowledge, this is the first parameterization in which phase changes are considered explicitly such that γ on aqueous particles exceeds that on solid particles.It is also the first parameterization to capture the difference in γ on NH 4 HSO 4 particle surfaces relative to (NH 4 ) 2 SO 4 .Another important result comes from our reevaluation of the nitrate effect using laboratory data on ammoniated particles.That effect, which was previously considered to decrease γ by a factor of ten (Mentel et al., 1999;Riemer et al., 2003), is now believed to be a factor of 1.3 to 5.2.Though we account for various effects of inorganic constituents, our parameterization may provide only an upper limit for γ on ambient particles.Additional research is necessary to parameterize the suppression of γ when inorganic particles are mixed or coated with organic material.
Our parameterization was used to compute γ under winter and summer conditions representative of the eastern United States.The resulting spatial distributions of γ were contrasted with three other parameterizations that have been applied in air quality models in the past.Our ambient estimates of γ fall between the upper value proposed by Dentener and Crutzen (1993) and the rather low values used by Riemer et al. (2003).Under winter conditions, large differences were found between our parameterization and that of Evans and Jacob (2005).Our parameterization yields summertime values that are slightly larger than the ambient-observational estimates of Aldener et al. (2006) and Brown et al. (2006).These comparisons helped us identify critical gaps in the laboratory data that will be most valuable for future refinements of the γ parameterization.

Appendix A Alternative parameterization on aqueous sulfate particles
Following the discussion in Sect.5.2, an alternative parameterization on aqueous sulfate particles is estimated by repeating the procedure described in Sect.3.1 without the KAN01 data.In this analysis, a new variable is used to allow for the possibility that no effect on γ is discernible at high RH.The threshold relative humidity variable is defined as: where k is an integer between 0 and 100.
The variable selection procedure yielded a model with a threshold temperature of j =291 K and a threshold RH of k=46%.The following equations may be viewed as a potential substitute for Eqs.(4-5), with best fit values and standard errors of each coefficient given in Table A1.Multiple R-squared: 0.7541, Adjusted R-squared: 0.7223 46%, which is in fair agreement with the studies on NaCl, sea-salt, and malonic-acid particles, as well as the HAL03 data on ammoniated sulfate particles (see Sect. 5.2).Below the RH threshold, the alternative parameterization has a much stronger dependence on RH than the original equations (compare β 11 in Tables 3 and A1).For NH 4 HSO 4 particles, the dependence on T j in the alternative parameterization is similar to Eq. ( 4), but the dependence of λ 2 on T j is found to be negligible (i.e., β 12 ≈ −β 22 ) and is therefore omitted from Eq. (A2).
Although the alternative parameterization may exhibit a defensible trend at high RH, evaluation against the underlying laboratory data reveals some shortcomings.For example, Eq. (A2) yields a fixed, temperature-independent value of γ 2 =0.025 whenever RH exceeds 46% (see Fig. A1b).In contrast, the 14 laboratory measurements of γ on aqueous (NH 4 ) 2 SO 4 particles above 46% RH range widely from 0.004 to 0.053 (see Table 1 without KAN01 data).Similarly, Eq. (A1) yields a fixed γ 1 value of 0.065 for RH>46% and T <291 K whereas the laboratory measurements taken under these conditions range from 0.0129 to 0.08585 (compare Fig. A1a to Fig. 1a without the KAN01 data).These results suggest that much of the variability in the measurements of γ above 46% RH may be due to factors other than T and RH.Overall, we find that omission of the KAN01 data points from our analysis greatly reduces the structure of the dataset.
Computations of γ 1 , γ 2 , and γ ambient using the alternative parameterization are illustrated in Fig. A2a, A2b, and  A2d.Under the wintertime conditions described in Sect.4.3, γ ambient values are approximately 0.01 to 0.015 less than those shown in Fig. 4d along the Gulf coast, Appalachian Mountains, and Great Lakes regions.The reductions in γ ambient are smaller (0.001 to 0.007) across the remainder of the eastern and central US.In the future, numerical modeling studies can be conducted using both the original and alternative parameterizations to gauge the impact of uncertainties at high RH on predicted pollutant concentrations.
In Appendix A, we provide an alternative parameterization for γ 1 and γ 2 that is based on a statistical analysis of all data points except for the KAN01 measurements.In this supplement, plots are presented of γ 1 , γ 2 , and γ ambient , based on that alternative parameterization.

Fig. 2 .
Fig. 2. Parameterization of γ as a function of RH and T for (a and d) aqueous NH 4 HSO 4 , (b and e) aqueous (NH 4 ) 2 SO 4 , and (c and f) aqueous NH4NO3.In (d-f), contour lines show the parameterized value of γ and discrete points show the combinations of T and RH at which laboratory measurements were collected.Shaded regions are described in the text.

Fig. 3 .
Fig. 3. Comparison between the parameterized and observed γ values using (a-e) our regression equations and (f) the Evans and Jacob (2005) parameterization for sulfate particles.Dashed and dotted lines bound the data points which are fit within factors of 2 and 1.5, respectively.Solid lines depict 1:1 agreement.

Fig. 4 .
Fig. 4. Average nighttime values of γ over the eastern United States during January-February 2001, comparing (a-d) our parameterizations with (e-g) those used in previous model applications.In (a, c, and g), γ is a function of RH and T .In (b), γ is a function of RH, T , and phase state.In (e), γ is fixed at 0.1.In (f), γ is a function of SO 4 2− and NO 3 − .In (d), γ is a function of RH, T , SO 4 2− , NH 4 + , NO 3 − , and phase state.All γ values greater than 0.1 are plotted in black.

Fig. S2 .
Fig. S2.Reconstruction of Fig. 4 using the alternative parameterization given in Appendix A.

Fig. A2 .
Fig. A2.Reconstruction of Fig. 4 using the alternative parameterization given in Appendix A. Only (a, b, and d) differ from Fig. 4.

Table 1 .
Continued.Published γ values from MOZ88 were multiplied by 0.85 in this table based on correction factors for the plug flow approximation described byFried et al. (1994).bAverage of separate standard errors reported above and below the γ value.Species 1 = NH 4 HSO 4 , Species 2 = (NH 4 ) 2 SO 4 , and Species 3 = NH 4 NO 3 .
c Published values from MOZ88 are 2 times standard error.Those values are divided by 2 in this table for consistency with other papers which report 1 standard error.d T , γ is typically highest on NH 4 HSO 4 , lowest on NH 4 NO 3 , and exhibits intermediate values on (NH 4 ) 2 SO 4 .
a Not used in our statistical analysis based on the discussion in MOZ88 (see text for details).b Published γ values from MOZ88 were multiplied by 0.