Prediction of Gas/particle Partitioning of Polybrominated Diphenyl Ethers (pbdes) in Global Air: a Theoretical Study

Gas/particle (G/P) partitioning of semi-volatile organic compounds (SVOCs) is an important process that primarily governs their atmospheric fate, long-range atmospheric transport, and their routes of entering the human body. All previous studies on this issue are hypothetically based on equilibrium conditions, the results of which do not predict results from monitoring studies well in most cases. In this study, a steady-state model instead of an equilibrium-state model for the investigation of the G/P partitioning behavior of polybrominated diphenyl ethers (PB-DEs) was established, and an equation for calculating the partition coefficients under steady state (K PS) of PBDEs (log K PS = log K PE + logα) was developed in which an equilibrium term (log K PE = log K OA + logf OM −11.91 where f OM is organic matter content of the particles) and a non-equilibrium term (log α, caused by dry and wet depositions of particles), both being functions of log K OA (octanol–air partition coefficient), are included. It was found that the equilibrium is a special case of steady state when the non-equilibrium term equals zero. A criterion to classify the equilibrium and non-equilibrium status of PBDEs was also established using two threshold values of log K OA , log K OA1 , and log K OA2 , which divide the range of log K OA into three domains: equilibrium, non-equilibrium, and maximum partition domain. Accordingly, two threshold values of temperature t, t TH1 when log K OA = log K OA1 and t TH2 when log K OA = log K OA2 , were identified, which divide the range of temperature also into the same three domains for each PBDE congener. We predicted the existence of the maximum partition domain (the values of log K PS reach a maximum constant of −1.53) that every PBDE congener can reach when log K OA ≥ log K OA2 , or t ≤ t TH2. The novel equation developed in this study was applied to predict the G/P partition coefficients of PBDEs for our Chinese persistent organic pollutants (POPs) Soil and Air Monitoring Program, Phase 2 (China-SAMP-II) program and other monitoring programs worldwide, including in Asia, Europe, North America, and the Arctic, and the results matched well with all the monitoring data, except those obtained at e-waste sites due to the unpredictable PBDE emissions at these sites. This study provided evidence that the newly developed steady-state-based equation is superior …

Abstract.Gas/particle (G/P) partitioning of semi-volatile organic compounds (SVOCs) is an important process that primarily governs their atmospheric fate, long-range atmospheric transport, and their routes of entering the human body.All previous studies on this issue are hypothetically based on equilibrium conditions, the results of which do not predict results from monitoring studies well in most cases.In this study, a steady-state model instead of an equilibrium-state model for the investigation of the G/P partitioning behavior of polybrominated diphenyl ethers (PB-DEs) was established, and an equation for calculating the partition coefficients under steady state (K PS ) of PBDEs (log K PS = log K PE + logα) was developed in which an equilibrium term (log K PE = log K OA + logf OM −11.91 where f OM is organic matter content of the particles) and a nonequilibrium term (log α, caused by dry and wet depositions of particles), both being functions of log K OA (octanol-air partition coefficient), are included.It was found that the equilibrium is a special case of steady state when the nonequilibrium term equals zero.A criterion to classify the equilibrium and non-equilibrium status of PBDEs was also established using two threshold values of log K OA , log K OA1 , and log K OA2 , which divide the range of log K OA into three domains: equilibrium, non-equilibrium, and maximum partition domain.Accordingly, two threshold values of temperature t, t TH1 when log K OA = log K OA1 and t TH2 when log K OA = log K OA2 , were identified, which divide the range of temperature also into the same three domains for each PBDE congener.We predicted the existence of the maximum partition domain (the values of log K PS reach a maximum constant of −1.53) that every PBDE congener can reach when log K OA ≥ log K OA2 , or t ≤ t TH2 .The novel equation developed in this study was applied to predict the G/P partition coefficients of PBDEs for our Chinese persistent organic pollutants (POPs) Soil and Air Monitoring Program, Phase 2 (China-SAMP-II) program and other monitoring programs worldwide, including in Asia, Europe, North America, and the Arctic, and the results matched well with all the monitoring data, except those obtained at e-waste sites due to the unpredictable PBDE emissions at these sites.This study provided evidence that the newly developed steady-state-based equation is superior to the equilibrium-state-based equation that has been used in describing the G/P partitioning behavior over decades.We suggest that the investigation on G/P partitioning behavior for PBDEs should be based on steady state, not equilibrium state, and equilibrium is just a special case of steady state when non-equilibrium factors can be ignored.We also believe that our new equation provides a useful tool for environmental scientists in both monitoring and modeling research on G/P partitioning of PBDEs and can be extended to predict G/P partitioning behavior for other SVOCs as well.

Introduction
Atmospheric transport is a major mechanism of moving semi-volatile organic compounds (SVOCs), including persistent organic pollutants (POPs), from source regions to other Published by Copernicus Publications on behalf of the European Geosciences Union.
Y.-F.Li et al.: Prediction of gas/particle partitioning of PBDEs in global air remote places including the Arctic and Antarctic, where these chemicals have never been produced and used (Barrie et al., 1992;Macdonald et al., 2000;Li et al., 1998;Li and Bidleman, 2003;Eckhardt and Manø, 2007).The gas/particle (also called aerosol) (G/P) partitioning of SVOCs is a very important process that primarily governs their atmospheric fate (Lohmann et al., 2000), since wet and dry depositions and other processes act differently on gaseous and particulate SVOCs, thus affecting the efficiency and scope of their long-range atmospheric transport and fate (Bidleman, 1988).In addition, SVOCs are an important class of indoor pollutants that are of great concern to human health (Weschler and Nazaroff, 2008).The gaseous and particulate SVOCs have different routes of entering the human body, and therefore the G/P partitioning of SVOCs has a significant influence on human exposure (Weschler, 2003).
The G/P partitioning behavior of SVOCs, K P , is commonly defined as (Yamasaki et al., 1982, Pankow 1991, Pankow and Bidleman 1991) where C G and C P are concentrations of SVOCs in gas and particle phases (both in pg m −3 of air), respectively, and TSP is the concentration of total suspended particles in air (µg m −3 ).Thus, K P is measured in units of m 3 µg −1 .In this study, the term of partition quotient instead of partition coefficient is used for K P because partition coefficient is used for equilibrium conditions only, and Eq. ( 1) was not defined under equilibrium conditions.The value of K P , calculated by Eq. ( 1) using the monitoring data TSP, C P , and C G , is denoted as K PM (subscript "M" in K PM means measurement).It has been shown that there is a linear relationship between log K PM and log K OA (K OA is the octanol-air partition coefficient) (Finizio et al., 1997, Harner and Bidleman, 1998, Pankow, 1998) and between logK PM and log P L (P L is sub-cooled liquid vapor pressure) (Pankow, 1987;Bidleman and Foreman, 1987;Pankow andBidleman, 1991, 1992).The log K OA -based model given by log has been widely used in describing the partitioning behavior of SVOCs, where slope m O and intercept b O are fitting constants obtained using regression of logK PM (from Eq. 1) against log K OA .The subscript "R" in K PR indicates regression.Unfortunately, Eq. ( 2) is not very useful to environmental modelers since this equation can only be used when the monitoring data of SVOCs concentrations in both gas and particle phases are known; however, modelers need the equations to predict environmental behavior, including concentrations in air with both gas and particle phases, based on physicochemical properties of SVOCs, their emissions, and climate and meteorological conditions.
Under the conditions of the equilibrium (m O = 1), the dominant absorption processes between gas and particle phases, and equivalence of octanol to the sorbing organic matter in particle, Harner and Bidleman (1998) derived the following equation: where f OM is organic matter content of the particle.The subscript "E" in K PE indicates equilibrium.In comparison to Eq. ( 2), Eq. ( 3) has the advantage of predicting K PE from the values of K OA and f OM without the need for real monitoring data.However, as discussed in previous publications (Finizio et al., 1997;Cetin and Odabasi, 2007;Tian et al., 2011;Yang et al., 2013;Li and Jia, 2014), Eq. ( 3) cannot describe accurately the relationship between gas-and particle-phase polybrominated diphenyl ethers (PBDEs).It is evident that Eq. ( 3) can be applied only in a few cases, such as for lessbrominated PBDE congeners (such as  or at high temperatures, and becomes inaccurate in most cases, especially for highly brominated PBDE congeners, such as or at low temperatures (Yang et al., 2013;Li and Jia, 2014).This has been blamed on the artifacts and non-equilibrium factors (Finizio et al., 1997;Cetin and Odabasi, 2007;Tian et al., 2011;Su et al., 2006).
Based on a large data set of more than 700 pairs of air samples in both gas and particle phases with a wide ambient temperature range of 60 • C from −22 to +38 • C obtained from our Chinese POPs Soil and Air Monitoring Program, Phase 2 (China-SAMP-II), we investigated the G/P partitioning behavior of PBDEs in Chinese air (Yang et al., 2013;Li and Jia, 2014).We derived for the first time empirical equations to predict the values of slopes and intercepts for both K OAbased and P L -based models as functions of temperature, and thus the predicted partition quotient (K P ), without assuming an equilibrium status and free of artifacts (Li and Jia, 2014).The slope m O and the intercept b O were given as functions of temperature (in The temperature t in Eqs. ( 4) and ( 5) is usually a mean value of temperature for a series of sampling events, such as annual or monthly mean temperature at each sampling site.After the values of m O and b O are calculated using Eqs.( 4) and ( 5), we can use Eq. ( 2) to calculate the values of log K P .Since this method can be used to predict the values of log K P , we use log K PP (the second "P" in the subscript K PP indicates prediction) instead of K PR in Eq. ( 2), and rewrite them as where m O (t) and b O (t) are given by Eqs. ( 4) and ( 5), respectively.
It is noteworthy that log K PP in Eq. ( 6) depends on two parameters, temperature t and K OA (which is also a function of temperature); K OA is given by the empirical equation (Harner and Shoeib, 2002): where t (in • C) is the temperature for each sampling event, and the parameters A and B are given in Table S1 in the Supplement for several PBDE congeners.It should be borne in mind that temperature t in Eqs. ( 4) and ( 5) can also be used as the temperature for each sampling event, and thus using Eq. ( 7), we can express log K PP in Eq. ( 6) as a function of a single independent variable of log K OA as (Li and Jia, 2014) log These two Eqs.(6 and 8) have been successfully applied to predict the values of K P for PBDEs as functions of log K OA in the air of China and other countries in the north temperate zone and also at an Arctic site in East Greenland Sea (Li and Jia, 2014), and our results matched the monitoring data well at background, rural, urban, and suburban sites, but not at e-waste sites due to the unpredictable PBDE emissions at these sites.The results indicated that our new equations have a better performance than Eq. ( 3) in describing G/P partitioning behavior of PBDEs in air as functions of log K OA .
We also found for the first time that the G/P partitioning of PBDE congeners can reach a maximum value if the ambient temperature is low enough.A criterion to classify the equilibrium and non-equilibrium status for PBDEs was also established using log K OA (Li and Jia, 2014).These equations, however, suffer from two drawbacks.First, these equations are derived at the temperature range of −22 to +38 • C, and thus cannot be used at temperatures beyond this range; secondly, these equations were obtained empirically, and do not have a strong theoretical foundation.Therefore, in this paper, we study the G/P partitioning behavior of PBDEs in global air in a theoretical way.The objectives of this study are to establish a partitioning model between gaseous and particulate phases for PBDEs, which can reveal the real partitioning mechanism of PBDEs between these two phases, and to predict the partition quotients defined in Eq. ( 1) accurately for PBDEs in air, thus achieving the capability to address a series of G/P partitioning issues of these chemicals, such as those presented previously (Yang et al., 2013;Li and Jia, 2014).

Equilibrium state and steady state
To develop a new model in simulating G/P partitioning behavior, we need to understand the equilibrium state and steady state for SVOCs in environment.The steady state is a state in which no change occurs with time, or all time derivatives are equal to zero."Equilibrium implies that phases have concentrations such that they experience no tendency for net transfer of mass" (Mackay 2001).These two terms have been frequently mistaken with each other.In the author's book (2001), Mackay gave examples to explain the difference between these two states, indicating a chemical is in equilibrium between two media (phases) as long as its fugacities in the two media (phases) are equal no mater the system is steady or unsteady.
We also noticed that although equilibrium is actually an ideal event since such a system cannot exist in a real environment, this state has been successfully applied in some cases.Good examples are to treat the air-water exchange for the gaseous pesticide α-hexachlorocyclohexane (α-HCH) (Jantunen and Bidleman, 1996Bidleman, , 1997;;Li et al., 2004) and airsoil exchange for gaseous polychlorinated biphenyls (PCBs) (Li et al., 2010).In these two examples, the factors other than the diffusion due to random molecular movement were negligible, and the systems can be treated as equilibrium; thus, the net flux of gaseous α-HCH between air and water, and gaseous PCBs between air and soil are zero.We realized, however, that the exchange of SVOCs between the gas and particle phases is different since the advection processes, such as dry and wet depositions, caused by an external force (gravity) on the particles, cannot be ignored in studying the partitioning behavior of SVOCs between these two phases.Therefore, we suggest that the steady state, not equilibrium state, should be applied here.

Basic equation
An equation describing G/P partitioning under steady state of PBDEs is where N G-P is flux of PBDEs from gas phase to particle phase, N P-G is flux of PBDEs from particle phase to gas phase, and N P-S the net flux of particle-bound PBDEs between air and earth surface, such as water body or surface soil.For the sake of simplicity, we only consider dry and wet depositions in N P-S , which is given by (Mackay 2001) where f P is fugacity of particle in air, given by Eq. (S1) in the Supplement with subscript "I" being "P", and D D is D value of dry deposition of particle-phase PBDEs described by (Mackay 2001) where U D is dry deposition velocity (a typical value being 10 m h −1 ), A is the area between air and earth (water or soil), Y.-F.Li et al.: Prediction of gas/particle partitioning of PBDEs in global air and Z P is Z value of aerosol, given by Eq. ( S3) in the Supplement, and ν P is the volume fraction of aerosol, given by where TSP is the concentration of total suspended particles in air (µg m −3 ) and ρ is the density of the particle (kg m −3 ).D W is D value of wet deposition given by (Mackay, 2001) where U R is rain rate, a typical value being 0.5 (m yr −1 ).Q is a scavenging ratio representing the volume of air efficiently scavenged by rain of its particle content, per unit volume of rain.A typical value for Q of 200 000 may be used.Substituting the above two equations in Eq. ( 10) leads to

Gas/particle exchange of PBDEs
One of the most important issues for investigating the G/P partitioning behavior is the exchange of PBDE between air and particle.We treat each particle as a sphere with a mean diameter of d, a volume of v = π d 3 /6, surface area a = π d 2 , and a mass m = ρv, where ρ is the density of the particle.
The number of particles in 1 m 3 of air is n = TSP/m.In air with volume of Ah (h is the height of atmosphere), the total area of the particles is Assuming ρ = 1.5 × 10 6 g m −3 , d = 1.0 × 10 −7 m (Rissler et al., 2006) h = 1.0 × 10 3 m, the above equation becomes To be simplified, we treat the particles as a film, called the particle film, with a thickness of d, a surface area of A P , as shown in Fig. S1 in the Supplement.The ratio between A P and A is which is a linear function of TSP.
In order to study the movement of SVOCs between air and particle, we adapted the method used for the air-soil interface introduced by Mackay (2001).For diffusion, the tworesistance approach is used, and the overall D values is given by where D E is air boundary layer D value, D A and D H are diffusion D values of chemical in air and water sub-phases in particle film, respectively.D E is deduced as the product of the surface area of the particle film, A P (m 2 ), a mass transfer coefficient k V (m h −1 ), and the Z value of air Z G , given by Here, where B a is the chemical's molecular diffusivity in air (0.018 m 2 h −1 was assumed), and l a is an air boundary layer thickness (0.00475 m was assumed) (Mackay 2001).
In comparison to the soil surface in air-soil exchange, the particle film that we suggested in our model keeps moving within the atmosphere and thus has much more chances to intersect with the chemical in gas phase.Therefore, the mass transfer coefficient will be larger than that given by Eq. ( 20), and accordingly, a parameter C is introduced in Eq. ( 20), leading to Thus, the term CB a is the chemical's molecular diffusivity for the particle film in air, and its value will be determined later.
Since most of the SVOCs (including PBDEs) are associated with the organic matter of the particles, again for the sake of simplicity, the two terms, D A and D H , in Eq. ( 18) are neglected, which becomes The flux of PBDEs from gas phase to particle phase, N G-P , becomes and the flux from particle phase to gas phase, N P-G , is If the term N P-S is dropped from Eq. ( 9), i.e., the net flux of particle-bound PBDEs between air and surface is neglected, we will have From Eqs. ( 23) and ( 24), the fugacities of a chemical in gas phase (f G ) and in particle phase (f P ) are equal, and thus the steady state becomes equilibrium.Therefore, it is concluded that equilibrium is just a special case of steady state when N P-S is ignored.

G/P partition coefficient under steady state
We use Eqs.( 10), ( 23), and (24) into Eq.( 9), leading to By using Eqs.( S6) and (S7) in the Supplement, the above equation leads to where C P (pg m −3 of particle) and C G (pg m −3 of air) are concentrations of SVOCs in particle and gas phases, respectively, and K PG is dimensionless G/P partition coefficient under equilibrium (= Z P /Z G ). Setting parameter α as and the above equation becomes By using Eqs.( S10) and (S11), C p and K PG are replaced by C p and K PE , respectively, and Eq. ( 28) becomes Defining a G/P partition coefficient under steady state, where C G and C P are concentrations of PBDEs in gas and particle phases (both in pg m −3 of air), respectively, at steady state, and the subscript "S" in K PS indicates steady state.
Although Eq. ( 30) seems the same as Eq. ( 1) (for K P ) and Eq.(S8) (for K PE ), they are different since Eq. ( 30) is defined under steady state, Eq. ( S8) is under equilibrium, and Eq. ( 1) was defined at neither steady nor equilibrium state.Thus, Eq. ( 29) becomes In the above equation, log K PE is designated the equilibrium term, given by Eq. ( 3), and log α is the non-equilibrium term.Therefore, we have two predicted partition coefficients: partition coefficientK PS under steady state when the system is at steady state, and partition coefficientK PE under equilibrium when the system is at equilibrium.Equation (31) indicates that the equilibrium is just a special case of the steady state when log α = 0.

log α as functions of log K OA and temperature
Figure S2 in the Supplement depicts variations of log α as functions of log K OA and temperature t with C = 5.As shown in Fig. S2A, the function of log α versus log K OA is a curve shared by all PBDE congeners, showing that when log K OA <∼ 10.4, log α = 0, the state is equilibrium.In contrast with the function of log α versus log K OA , the functions of log α versus t are different for different PBDE congeners, as shown in Fig. S2B   ( In this case, Eq. (33b) becomes which is equilibrium state. ( In this case, Eq. (33b) becomes If we assume that f OM = 0.1 and C = 5, then we have the first threshold value from Eqs. (33b) and ( 34), which is very close to the threshold value of log K OA = 11.5 suggested by Li and Jia (2014).The physical meaning of log K OA1 is, at this threshold, the data of log K PS deviates from log K PE by an amount of log 2. ( In this case, Eq. (33b) becomes thus log K PS in Eq. ( 31) reaches its maximum value, when G C. This is very close to the maximum value of −1.5 suggested by Li and Jia (2014).
The maximum value of log K PS is clearly shown in Fig. 1 (the thin blue horizontal line), and we define the second threshold value as As shown in Fig. 1, the first threshold value of log K OA divides the whole range of log K OA into equilibrium (EQ) and non-equilibrium (NE) domains.The second threshold indicates the start of the maximum partition (MP)domain, in which the values of log K PS reach a maximum value of log K PSM , which is independent of the values of f OM and K OA .In brief, as shown in Fig. 1, the curve of log K PS , originally coinciding with the straight line of log K PE , increases along with increase of log K OA , and separates visibly (by a mount of log2) from the straight line of log K PE at the first threshold value of log K OA1 , entering the NE domain from the EQ domain.After the second threshold value of log K OA2 , the curve of log K PS enters the MP domain and becomes a horizontal straight line of log The values of log K OA depend on each PBDE congener and the ambient temperature, as discussed previously (Harner and Shoeib, 2002).Accordingly, we defined two threshold values for temperature, the threshold temperatures t TH1 and t TH2 which are the temperatures when log K OA of PBDE congeners equals the threshold values log K OA1 and log K OA2 , respectively.As presented in Fig. 2, while the threshold values of log K OA1 and log K OA2 are constants for all congeners, the threshold values for t TH1 and t TH2 are different for different PBDE congeners.These two threshold temperature values divided the temperature space also into the same three domains; the EQ domain when t > t TH1 , the NE domain when t ≤ t TH1 , and the MP domain when t ≤ t TH2 .Taking BDE-47 as an example, with its t TH1 = +11 • C and t TH2 = −6 • C, BDE-47 is in the EQ domain when t > +11 • C; in the NE domain when t ≤ +11 • C; and in the MP domain at t ≤ −6 • C.

Particle-phase fraction
Another important parameter, the fraction of chemical in the particle phase, φ (= C P /(C G + C P )), can be calculated from K P as where the subscript "PX" can be one of "PS", "PE", and "PR".Thus, the maximum value of particle-phase fraction can be obtained from Eqs. ( 39) and (41) as which indicates that while the maximum partition coefficient log K PSM is a constant for all PBDE congeners, its corresponding maximum value of particle-phase fraction is not, but it depends on TSP.The variation of φ PSM as a function of TSP is depicted in Fig. S3 in the Supplement.
3 Application of the equations

G/P partitioning of PBDEs in Chinese air from China-SAMP-II
In the previous section, we derived Eq. ( 31) to predict the values of partition coefficient under steady state K PS .In this subsection, we used these equations to predict K P for air samples collected at 15 sites across China under our PBDE monitoring program, China-SAMP-II (Yang et al., 2013, Li andJia, 2014), and the results is later compared with the predicted values of partition coefficient under equilibrium state K PE , obtained using Eq. ( 3); and the values of partition quotient, K PR , obtained using Eq. ( 2) with the help of K PM , calculated from Eq. ( 1) using the monitoring data C P and C G .Among the three modeled values (K PS , K PE , and K PR ), K PR values are the ones closest to the values of K PM since log K PR values are obtained directly from log K PM by least squares regression against log K OA , and the accuracy of the equations of K PS and K PE depends on how close their results are to those given by K PR .Figures S4 and S5 in the Supplement depict the variations of log K PS , log K PE , and log K PR as functions of log K OA for the 15 sampling sites and 10 PBDE congeners, respectively, both showing the curve of log K PS is closer to the regression line of logK PR than log K PE .It is worthwhile to point out that for the best match between the results of log K PS and log K PR for PBDEs, C = 5 was used in Eq. ( 5) to calculate log K PS in air at all the sampling sites with an exception of the site of Waliguan, where C = 50 was used.The reason why a much higher value of C was used at this site will be explained later.Figure S5 also shows that, from the light PBDE congeners to the heavy ones, the ranges of log K OA for each PBDE congeners (at temperature range of −22 • C-+38 • C) move from left to right, from values smaller than log K OA1 for BDE-17 to larger than log K OA1 for BDE-183, or the states that these congeners reside in change from EQ domain to the NE domain, and finally reach the MP domain.
The 10 regression lines (log K PR ) for the 10 PBDE congeners shown in Fig. S5 are all presented in Fig. S6 in the Supplement along with the curves of log K PE and log K PS , indicating evidently that these 10 lines of log K PR change their slopes m O along the curve of log K PS , not the straight line of log K PE , and accordingly, the curve of log K PS matches the monitored G/P partitioning data very well for all the 10 PBDE congeners in Chinese air.
We understand that modelers are most interested in K P values as a function of temperature for each PBDE congener.Figure S7 in the Supplement presents variations of log K PS , log K PE , and log K PR as functions of temperature for the 10 PBDE congeners, indicating that the curve of log K PS matches the line of log K PR for each PBDE congener dramatically well, particularly for the highly brominated congeners.It is interesting to note that the two threshold temperatures, t TH1 and t TH2 , designated by two vertical dashed lines, increase from the less brominated to highly brominated PBDEs.For example, the value of t TH1 of BDE-17 equals −16.5 • C, meaning that this compound is in the EQ domain in the most ambient temperature range of ≥ −16.5 • C, while for BDE-183, t TH1 = 36.5 • C and t TH2 = 15 • C, meaning that this compound is in the EQ domain only when t > 36.5 • C, in the NE domain when t ≤ 36.5 • C, and in the MP domain when t ≤15.0 • C. We also calculated the modeled values of log K PS for five typical PBDE congeners as functions of temperature from −50 to +50 • C, and the results are given in Fig. S8 in the Supplement, showing that, along with a decrease of temperature, the values of log K PS for PBDE congeners increase to a maximum partition value; the more highly brominated the congener is, the higher is the value of its first threshold temperature (t TH1 , data are not shown), second threshold temperature (t TH2 ), and thus higher temperatures at which the congener reaches the NE and MP domains.
As shown in Fig. 1, the partitioning behavior of PBDEs depends on ambient temperature of sampling events.There are three squares presented in Fig. 1 indicating the three regions with different temperature ranges, 0 to +50 • C (orange), −30 to +30 • C (green), and −50 to 0 • C (blue).Here, we take the two sampling sites from China-SAMP-II (Yang et al., 2013;Li and Jia, 2014).One is Harbin in the northeast of China, with a sampling temperature range of −22 to +28 • C, within the green square, and the other is Guangzhou in the south of China, with a range of +8 to +38 • C, within the orange square, as examples to show how the threshold values can be used in study the G/P partitioning behavior of PBDEs.
We determined the ranges of log K OA for the 10 PBDEs at the site of Harbin (vertical bars) based on the ambient temperature range of −22 to +28 • C at the site, and the results are presented in Fig. 3.The two threshold values of log K OA , log K OA1 and log K OA2 (the horizontal light blue dashed lines), divide the space of log K OA (the left axis) into three domains, the EQ, the NE, and the MP domains, and accordingly, the 10 PBDEs in Harbin air can be segregated into three groups; BDE-17 and 28 (3-Br homologue) as equilibrium EQ group, BDE-47 and 66 (4-Br homologue) as semi-equilibrium SE group, and others (> 4-Br homologues) as non-equilibrium NE group.The dominant portions of log K OA for the EQ group (purple lines) are under the line of log K OA1 , i.e., these congeners are mainly in the EQ do- main, while the dominant or whole portions of the NE group (blue lines) are above the line of log K OA1 , indicating that these congeners are in the NE and the MP domains.The SE group (green lines) is in both the EQ and the NE domains.It is noteworthy that the major portions of log K OA for the PBDE congeners in the NE group were in the MP domain.These three domains can also be identified in the temperature space.In Fig. 3, the two threshold temperatures, t TH1 (the red diamonds) and t TH2 (the red square), are also shown (the right axis), which is similar to Fig. 2. In the real ambient temperature range, formed by the two red dashed lines (−22 • C and +28 • C) at the Harbin site, the major temperature portions of PBDEs in the EQ group were in the EQ domain (t<t TH1 ), those of the NE group in the NE and MP domains (t ≥ t TH1 ), and those of the SE group in both the EQ and NE domains.
Figure 4 presents the log K P -log K OA graph for the 10 PB-DEs in Harbin, which is almost identical to the one contained in the green square of Fig. 1.The ranges of log K OA for the three groups and their corresponding log K P -log K OA diagram are also shown.For example, the log K P -log K OA diagram for the EQ group (3-Br homologue), bound by the two burgundy dashed lines, is mainly in the EQ domain, with a small portion in the NE domain; the log K P -log K OA diagram for the SE group (4-Br homologue), contained by two green dashed lines, is mainly in the NE domain, with a small por- tion in the MP domain; and the log K P -log K OA diagram for the NE group (>4-Br homologue), formed by the two blue dashed lines, is mainly in the NE and MP domains.
Similar analysis was carried out for the 10 PBDEs at the site of Guangzhou at an ambient temperature range of +8 to +38 • C at the site (Yang et al., 2013), and the results are presented in Figs.S9 and S10 in the Supplement.The 10 PBDEs at Guangzhou air can also be segregated into three groups, BDE-17, -28, and -47 as EQ group, BDE-66, 99, and 100 as SE group, and the others, BDE-85, -99, -100, and -183 as NE group, which are quite different from those for Harbin, caused by the different ambient temperature ranges at the two sites.
We concluded that the PBDEs in Chinese air at 15 sampling sites across China were in the steady state instead of equilibrium state, by realizing that, for less-brominated PBDE congeners, BDE-17 and -28, this steady state can be treated as equilibrium state since their non-equilibrium term (logα) can be ignored in comparison to the equilibrium term (logK P E ) in the temperature range of −22 • C to +38 • C (see Fig. S2B).

G/P partitioning of PBDEs in air from other sources
There are only a few data available in the literature that we can compare to our prediction data.We predicted the partitioning behavior of gaseous-and particle-bound PBDEs in the atmosphere at an e-waste site and a rural site in southern China during 2007-2008 using the information given by Tian et al. (2011).We calculated the values of log K PS , log K PE , and log K PR as functions of log K OA , and the results are presented in Fig. S11.It is noticeable that our predicted results are obviously better than those obtained by the equilibrium model at the rural site, but not at the e-waste site, where the data from equilibrium model matched the monitoring data better than those predicted using our equation.This seemed unexpected but could possibly be explained by the fact that the emissions of PB-DEs from the e-waste site compensated the flux of PBDEs due to dry and wet deposition, leading a situation that seemed to be at equilibrium.We cannot, however, accept the point of view that the PBDEs in air at the rural area cannot reach equilibrium, but those in air at the e-waste sites can.
The G/P partitioning behavior was studied for seven PB-DEs  at four sites (one suburban, two urban, and one industrial) in Izmir, Turkey, in summer and winter in 2004-2005 with a temperature range of 1.8-22.4• C (Cetin and Odabasi, 2007).We calculated the particle-phase PBDE fractions φ PS and φ PE using Eq. ( 41) and compared them with the monitoring data, and the results are depicted in Fig. S12.It was noted by the authors that their monitoring data were much lower than the predicted values by the equilibrium equation (φ PE ) (Cetin and Odabasi, 2007), but it is obvious that their results matched our predicted data (φ PS ) very well, among which the best agreement was observed for BDE-209, the most highly brominated congener of PBDEs.
We calculated the G/P partition coefficients for PBDEs in atmosphere of Kyoto, Japan, which were measured in August 2000, and January and September 2001 (Hayakawa et al., 2004), and the variations of log K PE and log K PS as functions of log K OA are presented in Fig. S13, indicating obviously that the values of log K PS were in a better agreement with the monitoring data than log K PE .
Air samples were collected from one urban, one remote, and two rural sites near the Great Lakes in 1997-1999 as part of the Integrated Atmospheric Deposition Network (IADN), among which, those taken when the ambient atmospheric temperatures were 20 ± 3 • C were analyzed for the G/P partitioning behavior of PBDEs, and the log-log relationship of K P and their sub-cooled liquid vapor pressures (P L ) for BDE-47, -99, -100, -153, and -154 were calculated (Strandberg et al., 2001).By using these data, we calculated both log K P and ϕ P as functions of log K OA for the same five PBDE congeners, using the values of f OM = 0.2 and TSP = 25 µg m −3 suggested by Harner and Shoeib (2002), which are presented in Fig. S14, along with the predicted results under equilibrium state and steady state.Again the results indicated that the prediction by our new equation is more accurate than those by the equilibrium equation.

G/P partitioning of PBDEs in the Arctic air
As discussed in the previous sections, each PBDE congener will reach the maximum partition domain when log K OA ≥ log K OA2 or t ≤ t TH2 .The value of t TH2 of BDE-183 is 15 • C, meaning that BDE-183 in air will be in MP domain when t<15 • C. The value of t TH2 for BDE-209 should be higher (log K OA = 14.98 for BDE-209 was estimated at 25 • C by Cetin and Odabasi (2007) in comparison to log K OA = 11.97 for BDE-183 at the same temperature).Accordingly, BDE-209 in Arctic air should be in the MP domain, with a constant of log K PSM (−1.53) and the corresponding φ PSM (0.23 if TSP = 10 µg m −3 is assumed).This prediction was remarkably in agreement with monitoring data for BDE-209 measured in Arctic air at Alert, Canada from 2007 to 2009 with a temperature range between 10 and −50 • C (NCP 2013), lower than the value of t TH2 for BDE-209 (see Fig. 5).The comparisons between the predicted values and the monitoring data at Alert for other values of TSP (5 and 2 µg m −3 ) given in Fig. S15 3), which means that the values of K PE are from more than 5 orders at 10 • C to 10 orders at −50 • C of magnitude higher than the monitoring data, a huge error that cannot be tolerated.The corresponding values of φ PE are 1, indicating that BDE-209 are all in particle phase in the Arctic air predicted by the equilibrium equation, which was not the case for the monitoring data.In other words, the maximum value 0.23 of φ PSI indicates that, from our prediction, more than half BDE-209 (∼ 0.77) is in gas phase in Arctic air, which was confirmed by the monitoring data shown in Fig. 5.
We also studied the G/P partition of the 10 PBDEs  in the Arctic atmosphere over the East Greenland Sea in August and September 2009 with a temperature range between −0.5 • C and +6.5 • C (Möller et al., 2011).We calculated the values of log K PS , log K PE , and log K PR as functions of log K OA , and the results are shown in Fig. S16.Once again, the equation of log K PS had a better performance than the equation of log K PE , especially for those congeners in the NE domain with log K OA ≥ log K OA1 .

G/P partitioning of PBDEs in global air
The section At a Glance in the Supplement presents G/P partition coefficients of PBDEs (log K PS and log K PE ) as functions of log K OA at ambient temperature ranging from −50 to +50 • C, which can be applied at any sites worldwide (the top middle panel, similar to Fig. 1).The three squares in the panel designate the log K P -log K OA graphs with three different temperature ranges: 0 to +50, −30 to +30, and −50 to 0 • C, representing the tropical and subtropical climate zones, warm temperate climate zone, and boreal and tundra climate zones, respectively.Monitoring data (log K PM ), their regression data (log K PR ), and the predicted results log K PS and log K PE in the three different temperature zones are presented in the figure: the site Guangzhou, China, within the subtropical climate zone, shown in the top-left panel; the site Harbin, China, within the warm temperate climate zone, shown in the bottom panel; and the site Alert, Canada, within tundra climate zone, shown in the top-right panel, all introduced in the previous sections.The data at these three sampling sites all indicated that the curve of our new equation (log K PS ) is superior to the equilibrium equation (log K PE ) in G/P prediction of partitioning behavior for PBDEs in global air, at the sites in warm temperate, boreal, and tundra climate zones.Harner and Bidleman (1998) developed Eq.(3), which was able to predict, for the first time, the partition coefficients of selected SVOCs in air under the conditions of equilibrium between gas and particle phases.Four years later, Harner and Shoeib (2002) used this equation to predict the partitioning behavior of 11 PBDE congeners at 25 • C and 0 • C, the results of which, along with the results from our new equation under steady state, are given in Fig. S17 in the Supplement.As shown in the figure, the equilibrium Eq. (3) predicted that, at 0 • C, the particle fraction of PBDE congeners can reach as high as ∼ 1, which means that PBDE congeners can be completely sorbed to the particles.According to our new Eq.( 31) under steady state, however, the maximum particle fraction of PBDE congeners was about 0.42 when log K OA ≥ log K OA2 , which was less than half of the highest values predicted by Eq. (3).In other words, we predict that the maximum particle fraction of PBDE congeners in air cannot be more than 50 % under steady state if TSP < 30 µg m −3 (See Fig. S3 in the Supplement).In order to support their prediction results, Harner and Shoeib (2002) used the monitoring data of gaseous and particle-bound PBDEs in Great Lakes air at 20 ± 3 • C (Strandberg et al., 2001).However, as demonstrated in Fig. S14, the prediction by our new equation is much accurate than those by the equilibrium equation.This suggests that PBDEs in the Great Lakes atmosphere were in the steady state, not in the equilibrium state.

Equilibrium state vs. steady state
In brief, we cannot treat the gas and particle phases as a closed system for studying G/P partitioning behavior of PB-DEs since the third compartment, the surface of the earth, has to be considered.If the non-equilibrium term, log α, in Eq. ( 31) cannot be ignored, then the fugacities of PBDEs in gas and particle phases are not equal, indicating that the system is not at equilibrium but at steady state.For some lessbrominated PBDEs (such as BDE-17 and -28) at certain temperature, the values of log α is small enough in comparison to the value of log K PE in Eq. ( 31), which is considered as a small perturbation; the system can be considered as equilibrium.

The maximum partition coefficient
In our previous study (Li and Jia, 2014), we predicted for the first time by an empirical approach the existence of a maximum partition coefficient that every PBDE congener can reach, and was wrongly termed as "saturation state".This prediction was confirmed in this study by a theoretical approach.As shown in Fig. 1, the logarithm of the maximum partition coefficient log K PSM is equal to −1.53 (or K PSM = 0.03) when log K OA ≥ log K OA2 (12.5 for all PBDE congeners), or equally when the ambient temperature is smaller than or equal to t TH2 , which is from −34.5 • C for BDE-17 to 15 • C for BDE-183, and cannot increase linearly along with increase of log K OA as predicted by the straight line of log K PE .The difference of prediction data between these two equations can be very great.For example, as shown in Fig. 1, the difference can reach as high as ∼ 5.5 orders of magnitude when log K OA = 17.Obviously, the state in the MP domain is a steady state, but not an equilibrium state since the fugacities of PBDEs in gas and particle phases are not equal.
The best example is the case for BDE-209 in the Canadian Arctic site Alert predicted by our new steady equation discussed previously (Fig. 5).In fact, this is true for any PBDE congener, not only for BDE-209.As shown in Fig. 1, the blue square with a temperature range of −50 to 0 could most likely be the situation for the Arctic atmosphere.Fig. 2 shows that, for the seven PBDE congeners 153,, t TH2 > 0 • C. Thus, we predict that as the G/P partitioning behavior is considered, these seven PBDE congeners do not behave differently in the Arctic air, and are all have the same partition coefficient, log K PSM = −1.53.Unfortunately, there are no data for the PBDE congeners other than BDE-209 available for confirmation of this prediction in the Arctic air.

Comparison to the empirical equations
The two empirical Eqs.(6 and 8) have been successfully applied to predict the values of K P for PBDEs in the air of China and other countries in the north temperate zone and also at an Arctic site in East Greenland Sea (Li and Jia, 2014).The Eq. ( 31) for log K PS derived in this study is superior to these empirical equations log K PP and log K PP (K OA ) in two ways.First, steady-state Eq. ( 31) was derived theoretically, and secondly, this steady-state equation can be used at any ambient temperature, from the Equator to polar regions, while the empirical equations can only be used at a temperature range of −22 to +38 • C. A comparison between steady-state Eq. ( 31) for log K PS and empirical equations log K PP given by Eq. ( 6) and log K PP (K OA ) given by Eq. ( 8) in Harbin air at a temperature range of −22 to +28 • C is presented in Fig. 6, and the equilibrium equation log K PE , given by Eq. (3), is also included for comparison.It is evident from Fig. 6 that the straight line of log K PP deviates apparently from the straight line log K PE at log K OA = log K OA1 and increases linearly with log K OA .Different lines of log K PP (K OA ) for different PBDE congeners are able to predict the G/P partitioning behavior of PBDE more accurately than the straight line log K PP .It is interesting to note that the different lines of log K PP (K OA ) for different PBDE congeners change their trends along with the single line of steady-state equation log K PS , which is the best equation that can be used to predict the G/P partitioning behavior of all PBDE congeners and at all ranges of ambient temperature.

The limitation of applications
In order to derive Eq. ( 31), several assumptions were made, which include, among others, that the G/P partitioning reached steady state and that the annual rainfall was 0.5 m yr −1 , f OM = 0.1, C = 5.This equation, however, has been successfully applied in all situations that we discussed in this study, in which some of the assumptions were not satisfied.We should be aware, however, that the situations in which some abnormal conditions exist, such as heavy wind, heavy rains, or sampling sites close to e-waste or PBDE manufactures, are best treated separately.For example, as described in the previous section, the constant C should be changed from 5 to 50 in Eq. ( 5) for Site Waliguan.The reason why a much higher value of C was used at this site is possibly due to the high wind speed there.At Waliguan, the annual average wind speed reaches 4.6 m s −1 and northwest wind speeds of > 10 m s −1 occur quite often in the winter and spring seasons (http://gaw.empa.ch/gawsis/reports.asp?StationID=12), much higher than the other sites.There may be an analytic equation to determine the relationship between the parameter C and the wind speed (and possibly other factors too), but this equation is not available at present and planned for future studies.The case for the Chinese e-waste site is also worth mentioning.Our equation cannot be used at the e-waste sites and most likely at the PBDE manufacturers as well since the emissions of PBDEs at these sites could be too large and also variable with time so that the steady state cannot be reached or maintained.
It should be borne in mind that the steady state discussed here is still an idealized scenario since only dry and wet depositions were discussed in the study; other factors, such as humidity and artifacts, will also play roles to a certain extent to affect the G/P partitioning.As anticipated, results obtained from this study do not perfectly fit monitoring data.However, this study revealed the major internal factors governing the gas/particle partitioning processes for PBDEs, and explained how these processes can be more correctly treated as being in steady state rather than in equilibrium state.At the least, the steady-state model, not the equilibrium-state model, should be applied to analyze the gas/particle relationship of SVOCs, such as PBDEs.Further studies on other SVOCs, like PCBs and PAHs, are forthcoming.
The Supplement related to this article is available online at doi:10.5194/acp-15-1669-2015-supplement.

Figure 1 .
Figure 1.Variation of log K PE and log K PS as functions of log K OA with a temperature range of −50 to +50 • C. Two threshold values of log K OA (log K OA1 and log K OA2 ) are also shown, which divide the space of log K OA into three domains: the equilibrium (EQ), the non-equilibrium (NE), and the maximum partition (MP) domains.The three squares designate the log K P -log K OA graphs with three different temperature ranges: 0 to +50 • C, −30 to +30 • C, and −50 to 0 • C, representing the tropical and subtropical climate zones, warm temperate climate zone, and boreal and tundra climate zones, respectively.
et al.: Prediction of gas/particle partitioning of PBDEs in global air geners/homologues at an environmental temperature range of −50 to +50 • C, as shown in Fig. 1.The straight line (thick blue) for log K PE and the curve (red) for log K PS can be used for all PBDE congeners/homologues as long as their ranges of log K OA are known.Of course, it should be mentioned that the different PBDE congeners have different ranges of log K OA at the same temperature span, and thus are represented by different portions of the curves in the Fig. 1, accordingly showing different G/P partitioning behaviors.There are three cases for G in Eq. (33b).

Figure 2 .
Figure2.The first and second threshold temperatures, t TH1 and t TH2 for 10 PBDE congeners, which divide the temperature space into the same three domains (EQ, NE, and MP).

Figure 3 .
Figure 3.The range of log K OA (the left axis) and the threshold temperatures (the right axis) for 10 PBDE congeners in Harbin air at a temperature range of −22 to +28 • C. The ranges of log K OA for the 10 PBDE congeners are given by the vertical bars.The two horizontal light blue dashed lines give the two threshold values of log K OA1 and log K OA2, and the red diamonds and red squares present, respectively, the two corresponding threshold temperatures, t TH1 and t TH2 .The former divides the space of log K OA (the left axis) and the latter divides the temperature space (the right axis) into three domains: the EQ domain, the NE domain, and the MP domain.Thus, the PBDE congeners (homologues) in Harbin air can be segregated into three groups; BDE-17 and -28 (3-Br homologue) as equilibrium EQ group, BDE-47 and -66 (4-Br homologue) as semi-equilibrium SE group, and others (> 4-Br homologues) as nonequilibrium NE group.

Figure 4 .
Figure 4.The log K P -log K OA diagram for the 10 PBDE congeners in Harbin air at a temperature range of −22 to +28 • C. The EQ group includes BDE-17 and -28, the SE group contains BDE-47 and -66, and the rest belong to NE group.The range of log K OA for each group and their corresponding log K P -log K OA diagram are also shown.The log K P -log K OA diagram for the EQ group, indicated by two burgundy dashed lines, is mainly in the EQ domain, with a small portion in NE domain; the log K P -log K OA diagram for the SE group, contained by two green dashed lines, is mainly in the NE domain, with a small portion in MP domain; and the log K Plog K OA diagram for the NE group, formed by the two blue dashed lines, is mainly in the NE and MP domains.

Figure 5 .
Figure 5.The temporal trends of concentrations of BDE-209 in the Arctic air in gas and particle phases (blue line) and in particle phase (green line) at Alert, Canada from 2007 to 2009 (NCP 2013).The purple triangles and red diamonds are the values of ϕ and log K P of BDE-209, respectively, calculated using the concentration data, and match well the values of φ PSM (=0.23) and log K PSM (−1.53), respectively (TSP = 10 µg m −3 was assumed).
also showed great consistency.It should be stressed that the figure shows values of log K PE of BDE-209 are from 3.06 at 10 • C to 8.36 at −50 • C calculated by Eq. (

Threshold values of log K OA and temperature
Since all the equations to calculate the parameter K PE and K PS link only the PBDE parameter log K OA , it may be advantageous to explore partitioning behavior according to log K OA , rather than individual PBDE congeners or homolog groups.Under this consideration, we drew log K PSlog K OA and log K PE -log K OA graphs for all PBDE conwww.atmos-chem-phys.net/15/1669/2015/Atmos.Chem.Phys., 15, 1669-1681, 2015

Li et al.: Prediction of gas/particle partitioning of PBDEs in global air 1679
Variation of log K PS (the thick red line, given by Eq. 31), log K PE (the thick dark green line, given by Eq. 3), and log K PP (the thick pink line, given by Eq. 6) as functions of log K OA .The functions of log K PP (K OA ) (the thin lines, given by Eq. 8) vs. log K OA for 10 PBDE congeners in Harbin air at a temperature range of −22 to +28 • C are also included.