Evaluation of Vapor Pressure Estimation Methods for Use in Simulating the Dynamic of Atmospheric Organic Aerosols

The modified Mackay (mM), the Grain-Watson (GW), Myrdal and Yalkovsky (MY), Lee and Kesler (LK), and Ambrose-Walton (AW) methods for estimating vapor pressures (Pvap) are tested against experimental data for a set of volatile organic compounds (VOC). Pvap required to determine gas-particle partitioning of such organic compounds is used as a parameter for simulating the dynamic of atmospheric aerosols. Here, we use the structure-property relationships of VOC to estimate Pvap. The accuracy of each of the aforementionedmethods is also assessed for each class of compounds (hydrocarbons, monofunctionalized, difunctionalized, and triandmore functionalized volatile organic species). It is found that the bestmethod for eachVOCdepends on its functionality.


Introduction
Atmospheric aerosols (AA) have a strong influence on the earth's energy balance [1] and a great importance in the understanding of climate change and human health (respiratory and cardiac diseases, cancer).They are complex mixtures of inorganic and organic compounds, with composition varying over the size range from a few nanometers to several micrometers.Given this complexity and the desire to control AA concentration, models that accurately describe the important processes that affect size distribution are crucial.Therefore, the representation of particle size distribution is of interest in aerosol dynamics modeling.However, in spite of the impressive advances in the recent years, our knowledge of AA and physical and chemical processes in which they participate is still very limited, compared to the gas phase [2].Several models have been developed that include a very thorough treatment of AA processes such as in Adams and Seinfeld [3], Gons et al. [4], and Whitby and McMurry [5].Indeed, the evolution of size distribution of AA is made by a mathematical formulation of processes called the general dynamic equation (GDE).It is well known that the first step in developing a numerical aerosol model is to assemble expressions for the relevant physical processes.The second step is to approximate the particle size distribution with a mathematical size distribution function.Thus, the time evolution of the particle size distribution of aerosols undergoing coagulation, deposition, nucleation, and condensation/evaporation phenomena is finally governed by GDE [1].This latter phenomenon is characterized by the mass flux   for volatile species  between gas phase and particle which is computed using the following expression [6]: FS describes the noncontinuous effects [7].When   ≥ 0 (  ≤ 0), there is condensation (evaporation).   is assumed to be at local thermodynamic equilibrium with the particle composition [8] and can be obtained from the vapour pressure ( vap  ) of each volatile compound  with average molar mass of the atmospheric aerosol, mole fraction, and activity coefficient, through the following equation: where  vap  is given by the Clausius-Clapeyron law: Parameters in (1)-( 3) are named in Table 1.In (3),  vap is equal to 156 kJ/mol as stated by Derby et al. [8].Consequently, the vapour pressures of all aerosol compounds are needed to calculate the mass flux of condensing and evaporating compounds.The knowledge of  vap  for organic compound  at the atmospheric temperature  is required whenever phase equilibrium between gas phase and particle is of interest.Often, most of the compounds able to condense have experimental vapor pressures unavailable, and because of that, their estimation becomes necessary.To solve this problem of estimation, many methods have been developed.For example, in Tong et al. [9], a method based on atomic simulation is applied only for compounds bearing acid moieties.Quantum-mechanical calculations are making steady effort in vapor pressure prediction (Banerjee et al. [10]; Diedenhofen et al. [11]).Furthermore, current models describing gas-particle partitioning use semiempirical methods for vapor pressure estimation based on molecular structure, often in the form of a group contribution approach.Therefore, these methods require in most cases molecular structures (e.g., boiling point   , critical temperature   , and critical pressure   ), which usually have themselves to be estimated.For example, The MY method [12] was used by Griffin et al. [13] and Pun et al. [14] for modeling the formation of secondary organic aerosol.Jenkin [15], in a gas particle partitioning model, used the modified form of the Mackay method [16].Some methods (Pankow and Asher [17], Capouet and Müller [18]) assume a linear logarithmic dependence (ln  vap  ) on several functional groups, but this consideration fails when multiple hydrogen bonding groups are present.Furthermore, more investigations are needed to clarify which method can give values closer to the experimental data.Camredon and Aumont [19], Compernolle et al. [20], and Barley and McFiggans [21] have made an assessment of different vapor pressure estimation methods with experimental data for compounds of relatively higher volatility.For this latter reason, large differences in the estimated vapor pressure have been reported.
In this paper, our focus will be on the (i) evaluation of a number of vapor pressure estimation methods against experimental data using all volatile organic compounds present in our database and (ii) assessment of the accuracy of each of these methods on the base of each class of compounds.
The rest of the paper is organized as follows.In Section 2, we describe experimental data and methods.The results are presented in Section 3. Finally, we summarize our finding in Section 4.

Data and Vapor Pressure Estimation Methods
2.1.Experimental Data.The molecules selected in this study have been identified during in situ campaigns [22] and during chamber experiments [23].In fact, they are hydrocarbons, monofunctionalized and multifunctionalized species, and bearing alcohol, aldehyde, ketone, carboxylic acid, ester, ether, and alkyl nitrate functions.The experimental vapor pressures are taken from NIST chemistry  [12], Asher et al. [24], Lide [25], Yams [26], and Boulik et al. [27], and they have a range from 10 −8 atm to 1 atm.Molecular properties (boiling point, critical temperature, and critical pressure) are also taken from the NIST chemistry website.All vapor pressure estimation methods used in this study take into account these properties.In most cases, these properties have also to be estimated.

Estimation of Molecular Properties
2.2.1.Boiling Temperature   .Using the boiling temperature of Joback [28] and its extension, the group contribution technique denoted by  Job  is written as where   is the contribution of group ( International Journal of Geophysics 3 New group contributions have been added by Camredon and Aumont [19] for hydroperoxide, alkyl nitrate, and peroxyacyl nitrate moieties: where  is the molar mass,  is the number of atoms in the molecule, and   is the critical pressure contribution of group .The new group contributions described in the previous subsection are taken into account here.

The Myrdal and Yalkowsky (MY)
Method.The MY method [12] starts from the extended form of the Clausius-Clapeyron equation obtained by using Euler's cyclic relation.
Here, the expression of  vap is given by where Δ  is the vaporization entropy at the boiling temperature,   is the gas-liquid heat capacity, and  is the gas constant.Δ  used in this method is an empirical expression given by Myrdal et al. [31]: In ( 9), the parameters  and HBN which characterize the molecular structure represent the torsional bond (see Vidal [32]) and the hydrogen bonding number (see [19, Section 3.1.2]),respectively.Δ  is a linear dependence of  [32]: Thus, in the MY method, vapor pressure is estimated by the relatively simplified formula ln  vap = − (21, 2 + 0, 3 + 177HBN) (   −   ) + (10, 8 + 0, 25) ln    . (11) 2.3.2.The Modified Mackay (mM) Method.Often called simplified expression of Baum [33], this method is also based on the extended form of the Clausius-Clapeyron equation.
The simplifying assumption here is to consider the ratio Δ  /Δ  to be constant [34]: In ( 12), the vaporization entropy, Δ  , takes into account the van der Waals interactions and is based upon the Trouton's rule.For its calculation, Lyman [35] has suggested the following expression: where   is a structural factor of Fishtine [36] which corrects many polar interactions.It has different values as follows: (i)   = 1 for nonpolar and monopolar compounds; (ii)   = 1.04 for compounds with a weak bipolar character; (iii)   = 1.1 for primary amines; (iv)   = 1.3 for aliphatic alcohols.
Finally, the mM method is reduced to the following equation:

The Grain-Watson (GW) Method.
The GW method is based on the following equation [37]: where  = 0.4133−0.2575 and   is inversely proportional to   (  = /  ).Δ has the same form like that used in the mM method.

The Lee and Kesler (LK)
Method.Like the methods described previously, the LK method required critical temperature, critical pressure, and boiling temperature.Here, the vapor pressure is estimated on the base of Pitzer expansion [38]: where  vap  is reduced vapor pressure,   is reduced temperature, and  is the Pitzer's acentric factor which accounts for the nonsphericity of molecules: with  =   /  .In ( 16) and ( 17),  0 and  1 are the Pitzer's functions which are polynomials in   .Lee and Kesler have suggested the following equations [39]: ( 2.3.5.The Ambrose-Walton (AW) Method.AW method [40] is also based on the Pitzer expansion.They have reported their analytical expressions of Pitzer's functions in the form of a Wagner type of vapor pressure equation: In (19),  = 1 −   .

Molecular Properties.
We present in this section the results obtained for the Joback and Lydersen techniques described previously.The accuracy of each of the five vapor pressure estimation methods used in this study is assessed taking into account the reliability of pure substance property estimates.The reliability of the two techniques presented in Section 2.2 is therefore crucial.
In Figure 1, where the results of Joback technique are displayed,  Job  is plotted against experimental values for a set of 253 volatile organic compounds.The scatter tends to be larger for boiling temperature higher than 500 K.The correlation coefficient ( 2 = 0.97) shows that estimated values match very well experimental   .The root mean square error (RMSE) and the mean absolute error (MAE) are, respectively, 17.60 K and 12.65 K.This MAE agrees with 12.9 K and 12.1 K calculated in Reid et al. [39] and Camredon and Aumont [19] for a set of 252 and 438 volatile organic compounds, respectively.Hence, these results show that  Job Joback technique shows a negative bias for   higher than 700 K.This technique gives for the overall compounds an RMSE of 24.98 K.This value is higher than the 19.81 K provided by the Lydersen technique.The mean bias error (MBE) for  Job  and  Lyd  is −4.9 and −1.9, respectively.These results and Figures 2(a) and 2(b) show clearly that Joback technique underpredicts critical temperature, mostly for compounds which can be condensed onto particle phase, with high boiling temperature.The experimental group contributions provided by Lydersen are therefore more accurate than those provided by Joback.Thus, the Lydersen technique is more reliable than the Joback technique to estimate   .
Figure 3 shows  Job  and  Lyd  versus experimental values for a set of 117 compounds.The RMSE is 4.5 atm and 2.6 atm for Joback and Lydersen, respectively.According to Figures 3(a) and 3(b),   estimated by Lydersen matches fairly better ( 2 = 0.95) experimental data than   estimated by Joback ( 2 = 0.85).Joback technique considerably overpredicts critical pressure with MBE of 1,95 K higher than 0,39 K obtained with the Lydersen technique.In fact, besides group contribution, Lydersen technique takes into account molecular weight.Therefore, the Lydersen technique is retained to the critical pressure estimation in this paper.This is in agreement with Poling et al. [38] who found that the Lydersen technique is one of the best techniques for estimating critical properties.For the five vapor pressure estimation methods described previously, we will use estimated   ,   , and   because there is in general a lack of experimental data.Some of the five methods described in Section 2.3 need   ,   , and   , while others need only   and   .This last pure substance property is estimated by the Joback technique, and the two critical properties are estimated by the Lydersen technique.

Evaluation of Vapor Pressure Estimation Methods.
The accuracy of each method is assessed in terms of the mean absolute error (MAE), the main bias error (MBE), and the root mean square error (RMSE) (Table 3).The MBE measures the average difference between the estimated and experimental values, while the MAE measures the average magnitude of the error.The RMSE also measures the error magnitude, but gives some greater weight to the larger errors.( 6 International Journal of Geophysics In (20),  vap est, and  vap exp, are the estimated and experimental values of VOC , respectively, and  is the total number of VOC.We have also used linear correlation coefficient which measures the degree of correspondence between the estimated and experimental distributions.
The logarithms of vapor pressures estimated at  = 298 K for different methods are compared in the scatter plots shown in Figure 4 for a set of 262 VOC.Corresponding MAE, MBE, and RMSE are given in Table 1.In this figure, it is clear that all the five methods give similar scatter for vapor pressures higher than 10 −5 atm.The species concerned are hydrocarbons (Figure 5).For the set of 28 tri-and more functionalized species used in this study, vapor pressures are lower than 10 −2 atm (Figure 8).
Figure 4(d) shows that MY method is well correlated with experimental values.For this method, we have one of the best correlation coefficients  2 = 0.968.This method shows no systematic bias for vapor pressure lower than 10 −5 atm, while it is not the case for other methods.The MBE found here is 0.027.This value is one of the lowest ones of the total VOC (see Table 1).Thus, the MY method does not show any systematic bias.This method also provides the smallest values of MAE and RMSE (0.265 and 0.381, resp.).These results are in agreement with those found by Camredon and Aumont [19] using Ambrose technique to estimate critical properties.It is also found that this method provides the smallest values of these errors for a set of 74 hydrocarbons (see Table 2).Those are compounds with vapor pressure higher than 10 −2 atm.This result is not the same for mono-and difunctionalized species whose errors are some of the largest ones.The vapor pressures higher than 10 −4 atm are fairly well estimated.Using a set of 45 multifunctional compounds, Barley and McFiggans [21] found that MY method tends to overpredict vapor pressure of lower volatility compounds.Furthermore, it is important to note that MY method provides one of the poor results for difunctionalized VOC (see Table 2).Thus, for a set of 32 difunctionalized VOC, estimation values fit the experimental ones with a coefficient  2 = 0.957 (Figure 7) which is the smallest value obtained for this class of species.Vapor pressures are overpredicted with a bias of 0.24, while the RMSE = 0.54 is of the same order of magnitude as those obtained by other methods.In contrast, Figure 8 shows that MY method has the best correlation for tri-and more functionalized species and is therefore the best method to estimate vapor pressure for this class of species.The systematic errors reported in Table 2 allow us to conclude that assumption.Indeed, the peculiarity of the MY method is that it takes into account the molecular structure.
Figure 4(c) displays the results for the mM method.This method gives one of the lowest scatterings with a coefficient  2 = 0.957 and agrees with other methods for the highest vapor pressures.It shows a positive bias (MBE = 0.026) for the total set of 262 VOC.As the GW method, the mM method tends to overpredict vapor pressures lower than 10 −6 atm.Furthermore, these methods describe vaporisation entropy by taking into account van der Waals interactions.The root mean square error (RMSE = 0.442) is close to those provided by GW and AW methods.The mM method is then less appropriate than the four others for all classes of VOC. Figure 5(c) shows that the predicted values match the experimental values with a coefficient  2 = 0.96 for 32 difunctionalized species.For this class of species, estimates are provided with a positive bias (MBE = 0.208) and an RMSE of 0.512.These are the best values obtained from all the five methods (see Table 2) for difunctionalized species.Thus, mM method is more accurate than others to estimate vapor pressure for difunctionalized species, but does not provide International Journal of Geophysics good results for monofunctionalized (Figure 3) and tri-and more functionalized species (Figure 5).It can be seen in Figure 7 that estimated values provided by GW method are strongly correlated with experimental values ( 2 = 0.960) for difunctionalized species.This method tends to overpredict vapor pressure for this class of species (MBE = 0.288), but does not show any bias for other classes of species (Table 2).Figure 8 shows that we have very acceptable results for tri-and more functionalized species with RMSE and MAE equal to 0.549 and 0.440, respectively.
Except for tri-and more functionalized species (see Figure 8), it is clear from Figures 4 to 7 that the LK method gives accurate values, based upon best correlation coefficient values.This method is the second best one of the five methods, but it has the greatest systematic bias (RMSE = 0.490, MAE = 0.32) for the total set of VOC and for difunctionalized species (RMSE = 0.70, MAE = 0.56).The MAE of hydrocarbons and monofunctionalized species are 0.142 and 0.36, respectively.For these species, Figures 5 and  6 give the best correlations.The AW and LK methods are both based on Antoine's equation.According to all figures plotted, it is clear that these two methods give very similar results.The peculiarity of AW is that, for the monofunctionalized compounds,

Conclusion
We have evaluated in this study five vapor pressure estimation methods useful for simulating the dynamics of atmospheric organic aerosols.These are the Myrdal and Yalkovsky (MY), the Lee and Kesler (LK), the Grain-Watson (GW), the modified Mackay (mM), and the Ambrose-Walton (AW) methods.Some of them are based on the Antoine equation, while others are based on the extended form of the Clausius-Clapeyron equation.But all of them take into account boiling temperature   and (or) critical temperature   .Therefore, Joback technique has been used to estimate   , while the Lydersen technique was found to be better for   estimation.When using Joback to provide the   values, LK, AW, and MY are the best three methods for all classes of species.Moreover, for a set of 262 volatile organic compounds and as illustrated in the scatter plots and errors computed, the MY method which appears to be the best one fails for difunctionalized species.For these latter species, the mM method provides good results, according to the correlation coefficient  2 = 0.960 and the least errors reported in Table 2. GW method is the least reliable, which provides the lowest results for all VOC and also for each class of species.Predictions made with the AW method for monofunctionalized species are more reliable than those made with the other four methods employing the Joback technique to provide the   .For vapor pressure higher than 10 −2 atm, all the five methods give similar results.
This work highlights that the choice of a method to predict vapor pressure of volatile organic compounds depends on the number of functional groups existing in the species.
Methods.As said earlier, many methods for vapor pressure estimation have been developed and are based on the Antoine or on the extended form of the Clausius-Clapeyron equation.Let us present in what follows each of the five methods used.

Figure 1 :
Figure 1: Estimated boiling point for a set of 253 species versus experimental values for the Joback technique.The black line is the 1 : 1 diagonal.

Figure 2 :Figure 3 :
Figure 2: Estimated critical temperature for a set of 138 species versus experimental values for the (a) Joback technique and (b) Lydersen technique.The black line is the 1 : 1 diagonal.

Figure 4 :
Figure 4: Logarithm of estimated vapor pressure of all VOCs used in this study versus experimental values for the (a) GW, (b) LK, (c) mM, (d) MY, and (e) AW methods.The black line is the 1 : 1 diagonal.

Figure 5 :
Figure 5: Logarithm of estimated vapor pressure for a set of 74 hydrocarbons versus experimental values for the (a) GW, (b) LK, (c) mM, (d) MY, and (e) AW methods.The black line is the 1 : 1 diagonal.

Figure 6 :
Figure 6: Logarithm of estimated vapor pressure for a set of 128 monofunctionalized species versus experimental values for the (a) GW, (b) LK, (c) mM, (d) MY, and (e) AW methods.The black line is the 1 : 1 diagonal.

Figure 8 :
Figure 8: Logarithm of estimated vapor pressure for a set of 28 tri-and more functionalized species versus experimental values for the (a) GW, (b) LK, (c) mM, (d) MY, and (e) AW methods.The black line is the 1 : 1 diagonal.

)
2.2.3.Critical Pressure   .The two techniques listed in the previous section have been used to estimate critical pressure   .Here, it is also assumed that   is the sum of group contributions.Therefore, denoting by  Job

Table 2 :
MAE, MBE, and RMSE of vapor pressure computed based on various methods.

Table 3 :
MAE, MBE, and RMSE of vapor pressure computed based on various methods for each class of compounds.